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ESSAYS ON APPLIED ECONOMICS
By
Mingli Zheng
A thesis submitted in conformity with the requirements
For the degree of
DOCTOR OF PHILOSOPHY
Department of Economics
University of Toronto
©Copyright of Mingli Zheng (2002)
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ABSTRACT
ESSAYS ON APPLIED ECONOMICS
By
Mingli Zheng
DOCTOR OF PHILOSOPHY
Department of Economics
University of Toronto
This dissertation consists of three essays on applied economics.
In the first essay, I provide an empirical assessment of competing auction theory.
Specifically, it uses an extensive new data set containing detailed information about
bids placed on eBay computer CPU auctions to explore bidding strategies in the
presence of competing auctions. The evidence indicates that a significant proportion
of bidders bid across several competing auctions at the same time and that bidders
tend to submit bids on auctions with the lowest standing bid. We also find that
winning bidders who bid across competing auctions pay lower prices than winning
bidders who do not cross-bid. These findings jointly amount to the first evidence
lending empirical support to competing auction theory.
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In the second essay, I consider the convergence properties of behavior under
a comparative negligence rule (CN) and under a rule of negligence with contributory
negligence (NCN), assuming bilateral care with three care levels. Using an
evolutionary model, we show that CN reduces the proportion of the population
using low care more rapidly than does NCN. However NCN increases the
proportion of the population using high (efficient) care more rapidly than does CN.
As a result, the mean care level increases more rapidly and the mean social cost falls
more rapidly under CN than under NCN.
In the third essay, I provide a model for rational legal decision-making by
considering the consistency of social decision-making on resource allocations. In the
model, legal decision-making is not based on individual utilities. The purpose of law
is described by a subjective social welfare function, with a social value judgment
and an attitude towards distributional inequality as its parameters. Thus a legal
decision-making always involves a social value judgment. Social efficiency can be
different from the maximization of social wealth. In tort law, the determination of
the efficient precaution level and the award of compensation depend on the social
value judgment, and the compensation scheme implicitly transfers wealth from the
less socially valued party to the more socially valued party. The model also
provides another explanation for the award of punitive damage.
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ACKNOLEDGEMENTS
I am deeply indebted to my supervisor Michael Peters, committee members Don
Dewees and Robert McMillan. Their generous help, stimulating suggestions and
constant encouragements helped me in all the time of research for and writing of this
thesis. My gratitude to them cannot be expressed by words.
I could not have survived the PHD study without the understanding and the support
by my wife Fang Xiao. I would like to give her special thanks.
I also thank Mohr Siebeck for authorizing me to include the paper published in JITE
in my dissertation.
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Table of Contents
Abstract……………………………………………………………………………….iiList of Tables……………………………………………………….………………..viList of Figures……………………………………...………………………………..vii
Chapters
1. Bidding Behavior with Competing Auctions: Evidence from ebayAbstract………………………………………………………………………11. Introduction………………………………………………………………..32. Theory of Competing Auctions………………………………………..…..83. Mechanism in eBay and Summary of Data………………………………124. Bidding Behavior with Competing Auctions in ebay…………………….205. Does Cross Bidders Pay Lower Price?…………………………………...286. Conclusion………………………………………………………………..30References…………………………………………………………………..31
2. Liability Rules and Evolutionary DynamicsAbstract……………………………………………………………………..451. Introduction………………………………………………………………462. Liability Rules and Nash Equilibrium……………………………………493.Evolutionary Dynamics…………………………………………………...524. Conclusion………………………………………………………………..70References…………………………………………………………………...72
3. Rational Legal Decision-Making, Value Judgment and the Efficient Precaution in Tort Law
Abstract…………………………………………………………………….741. Introduction…………………………………………………………… 752. Rationality, Subjective Social Welfare Function and Value Judgment
……………………………………………………………………...813. The Determination of Efficient Precaution Level and the Damage
Compensation in Tort Law……………………………………...……..944. Summary……………………………………………………..…….…..105References………………………………………………………..……….106
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List of Tables
Table Page
1.1 Sample statistics for CPU auctions ………………………………………34
1.2 Sample statistics for samples of competing auctions……………………..35
1.3 Statistics of Cross Bidding in the Whole Process…………………………38
1.3B Percentage of Cross Bidders in All bidders………………………………..39
1.4 Statistics of Cross Bidders for the Last Day……………………………….40
1.5 Result on Bidding on the Auction with the Lowest Standing Bid…………41
1.6 Result on Bidding on the Auction with the Lowest Standing Bid
(For groups with cross bidders only)………………………………42
1.7 Result on Bidding on Group with Zero Bid Auctions……………………...43
1.8 Average Number of Bids Submitted in a Group by a Bidder……………....43
1.9 Price Paid by Cross Bidders and Non-Cross Bidders………………………44
2.1 Payoff Under CN………………………………………………………...…51
2.2 Simulation Results………………………………………………………….65
3.1 Sensitivity of the Award of the Punitive Damage to the Estimation
Bias of …………………………………………………………104
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List of Figures
Figure Page
1.1 A Bidding History Page From ebay………………………………….33
1.2 Histogram of Bidders’ Feedback…………………………………….36
1.3 Histogram of Bids Submission Time for Daily Sample……………..37
2.1 Proportion of Individuals Taking High care…………………………66
2.2 Proportion of Individuals Taking Medium Care……………………..66
2.3 The Composition of the Population in the Simulation……………….67
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Chapter 1
Bidding Behavior with Competing Auctions:
Evidence from ebay
Abstract
The existing auctions literature treats on-line auctions as running independently of one
another with each bidder choosing to participate in only one auction. This characterization is
less than perfect: in on-line auctions, many substitutable goods are auctioned concurrently,
and bidders can bid in several auctions at the same time. Some recent theoretical research by
Peters and Severinov (2001) is more relevant to the study of bidding behavior in on-line
auctions, showing that bidders can gain from the existence of competing auctions.
Specifically, a strategy in which bidders bid on the auction with the lowest standing (or
prevailing) bid is a Bayesian Nash equilibrium. In the light of this work, the current paper
provides the first empirical assessment of competing auction theory. Specifically, it uses an
extensive new data set containing detailed information about bids placed on eBay computer
CPU auctions to explore bidding strategies in the presence of competing auctions. The
evidence indicates that a significant proportion of bidders bid across several competing
auctions at the same time and that bidders tend to submit bids on auctions with the lowest
standing bid. We find that for homogeneous items, as the difference in ending time across
competing auctions becomes smaller, so more bidders bid across competing auctions and bid
on the auction with the lowest standing bid. We also find that winning bidders who bid
across competing auctions pay lower prices than winning bidders who do not cross-bid.
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These findings jointly amount to the first evidence lending empirical support to competing
auction theory.
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1. Introduction
There is a tremendous amount of interest in auctions as a means of selling items,
both for vendors and theorists. The appeal of auction is understandable: as shown in
the mechanism design literature (e.g., Myerson (1981)), when a seller wants to sell
an object to several buyers, an auction is the best way to do it.
In standard auction theory, the typical assumption is that there is a single seller and
several bidders. The seller acts as a monopoly and gets all the information rent from
bidders. In practice, sellers often do not have monopoly power, but rather have to
compete against other sellers. For example, in on-line auctions, many sellers sell
their goods at the same time and some of the items are almost indistinguishable.
Thus buyers can choose among many auctions and decide whether to buy from one
among many sellers.
A few papers in the literature consider the case in which sellers compete
against each other (see for instance McAfee (1993), and Peters and Severinov
(1997)). But these papers assume that bidders can only choose to buy from one
seller, and the only equilibrium involves buyers randomizing over available sellers.
In this case, it is natural that some auctions have many bidders while other auctions
have few or no bidders, and consequently that some profitable trades may not be
realized.
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For on-line auctions, such as those on eBay, bidders are not constrained to
participate only one auction. eBay has evolved to act as a clearinghouse for a large
number of homogeneous goods. At any time, there are many similar items on sale,
the bidding cost for on-line auctions is very low, and bidders can easily monitor
several auctions at the same time. Thus it is possible for bidders to bid across several
competing auctions simultaneously.
The central question addressed in this study is: Are bidders responsive to the
existence of competing auctions? If they can choose among competing auctions,
how do bidders bid? Suppose that bidders bid across several competing auctions at
the same time, even if they only need one item, as they search for the best deal.
Doing so exposes them to the risk that they may win more than one item. But this
consequence can be avoided if bidders use a specific strategy: always bid on an
auction with the lowest ‘standing’ (or prevailing) bid, and bid with the minimum
increment; if the bidder becomes the highest bidder in one auction, pause bidding
until other bidders outbid him/her.
This strategy ensures that a bidder never wins more than one auction.
Another advantage of this strategy is that bidders are never trapped in very
competitive auctions. For example, suppose there are two competing auctions and
four bidders, sellers’ valuations are all 0, and bidders’ valuations are 10,10, 7, and 6,
respectively. If all bidders choose one auction and bid their true valuation, those two
bidders with valuation 10 may end up bidding on the same auction; whoever wins
the auction has to pay 10. If bidders bid across competing auctions and bid with the
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minimum increment, these two high valuation bidders will always end up winning
two different auctions and paying a much lower price.
Most existing work on on-line auctions treats them as many independently
running auctions and allows bidders to bid on only one auction. The emphasis in
prior work has typically been on studying the strategic behavior of market players.
For instance, Roth and Ockenfels (2000) explain the phenomenon of late bidding in
eBay by the existence of a fixed auction ending time. Bidders bid late because very
late bids have a positive probability of not being successfully submitted, and this
provides a way for bidders to implicitly collude and avoid bidding wars. Bajari and
Hortacsu (2000) study costly entry for bidders and the choice of reserve price on the
part of sellers. Both models assume that bidders only bid on one auction.
A recent paper by Peters and Severinov (2001) studies the market equilibrium
with competing auctions similar to those in eBay. If there is no bidding cost and no
fixed ending time for auctions, the paper proves that the strategy in which bidders
always submit a bid on an auction with the lowest standing bid and bid with the
minimum increment is actually a (weak) perfect Bayesian equilibrium. Contrary to
standard second-price auctions, in this environment bidding once and bidding one’s
true valuation is no longer equilibrium. Intuitively, if bidders bid their true valuation
and bid only once, they may be trapped in very competitive auctions and not have
opportunity to switch to other less competitive auctions. Consequently, the final
price of one auction is affected by the existence of other auctions. Further, prices
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tend to be uniform for competing auctions, and in addition, the price is the same as
the price under a double auction.
The strategy needs two assumptions: that there is no bidding cost and no
fixed ending time for auctions. In eBay, these assumptions are not perfectly met.
There is a bidding cost, though it is very low; and all auctions have a fixed ending
time. For identical or very similar items, auctions with almost the same ending time
compete against each other, while auctions with different ending times do not
perfectly compete with each other. Obviously, those auctions ending early cannot
compete with those ending later once they are finished, especially when many bids
are clustered at the very late period, as documented by Roth and Ockenfels (2000)
and Bajari and Hortacsu (2000).
Despite this discrepancy, eBay provides a valuable opportunity to see how far
actual bidding behavior in the presence of competing auctions corresponds to the
strategy prescribed in the theory. We have assembled data on competing auctions
for CPU’s taking place in one month, the period of September 20 to October 19.
Each group of competing auctions consists of auctions with the same description, the
same starting price and delivery method, and with a similar ending time. We classify
auctions into three categories: auctions ending in the same day, ending within the
same hour, and auctions ending within the same minute. Doing so allows us to
identify the effect of increasing the degree of substitutability on the behavior of
auction participants.
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Our results provide convincing evidence that bidders bid across competing
auctions (or “cross-bid”), and they tend to bid on the auction with the lowest
standing bid, as the theory would predict. Further, this tendency becomes stronger as
auctions become closer substitutes. Thus we find that for competing auctions that
end within the same minute, bidders are more likely to cross-bid and bid on auctions
with the lowest standing bid than is the case for auctions that end further apart. We
also find that, on average, bidders revise their bids more often when the auctions
they bid on end closer together; bidders also revise their bids more often when they
cross-bid. To assess the potential gains from following the strategy outlined in the
theory, we compare the winning price for winning bidders who bid across competing
auctions and for winning bidders who do not, finding that bidders who bid across
competing auctions pay lower prices on average than those bidders who do not. In
total, these results provide the first compelling evidence in support of competing
auction theory.
The paper is organized as follows: We first briefly summarize a theory with
competing auctions. Then in section 3, we describe the data. Section 4 reports
results of bidding behavior in eBay and in section 5, we compare the winning price
for bidders who bid across competing auctions with those for bidders who do not.
Section 6 concludes.
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2. Theory of competing auctions
There are few papers on auctions with many sellers and many bidders. When many
bidders with independent valuations simultaneously choose among many sellers, the
only equilibrium (as noted above) has buyers randomizing over available sellers (see
McAfee (1993) and Peters and Severinov (1997)). When there are many competing
auctioneers, if bidders cannot bid across auctions, independently run auctions lead to
inefficient trades, in the sense that the sum of all agents’ welfare is not maximized:
some very low valuation sellers do not successfully sell their goods while some high
valuation buyers cannot buy a good, because of the mismatch between buyers and
sellers.
In a double auction, the outcome is much more efficient. There, potential
buyers and sellers of a single good move simultaneously, with buyers submitting
bids and sellers submitting asking prices. An auctioneer then chooses a price that
clears the market: all sellers who ask less than sell, all buyers who bid more than
buy, and the total number of units supplied at price equals the number demanded.
In Wilson’s (1985) research, buyers and sellers’ valuations are drawn independently.
When the number of buyers and the number of sellers are large enough, a double
auction yields an efficient allocation; the sum of all agents’ welfare is maximized,
and all sellers with low valuations and all buyers with high valuations successfully
make trades.
The assumption that bidders have to choose one and only one auction
simultaneously is critical for the efficiency outcome in independent auctions with
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many buyers and many sellers. For on-line auctions, this assumption is unlikely to
hold. Peters and Severinov (2001) ask the question whether independently organized
auctions in a centralized exchange such as eBay can overcome the inefficiency of
random matching. In their model, there are many sellers and many bidders. Each
seller has a single good for sale and all goods for sale are identical. Each bidder only
needs one good, so that winning more than one auction is undesirable – the
additional good provides no additional utility. Auctions follow a similar pattern to
eBay: the standing bid is the second highest bid and the highest bid is never revealed.
However, bidders are not required to confine their attention to only one auction.
Under the assumption that bidding is costless and there is no fixed ending time for
auctions, the paper shows that competing auctions can overcome the inefficiency of
random matching. The paper gives a symmetric strategy for bidders and proves that
this strategy is a perfect Bayesian equilibrium. In equilibrium, all trades occur at the
same price and the price is the same as that in a double auction.
The main results of Peters and Severinov (2001) are repeated here:
Lemma: The symmetric equilibrium is defined as follows:
(a) if the buyer is the current high bidder at any auction, or if the buyer’s valuation is
less than or equal to the lowest standing bid, the buyer should pass;
(b) otherwise, if there is a unique lowest standing bid, the buyer should submit a bid
with the seller offering the lowest standing bid. The bid should be equal to the
smallest valuation that exceeds this lowest standing bid;
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(c) otherwise, let be the set of sellers who have the lowest standing bid. Let
be the subset of sellers in whose current high bidder submitted his bid
while the standing bid was strictly below its current level; let be the set of
sellers in who have not received a bid. If is not empty, the buyer should
bid with equal probability with every sellers in . If is empty but is
non-empty, the buyer should bid with equal probability with every seller in .
Otherwise, if both subsets are empty, the buyer should bid with equal probability
with every seller in .
Suppose there are sellers and bidders. Let be the lowest
valuation of all traders. Their theorem states:
Theorem: The outcome in which all buyers use the strategy is a (weak) perfect
Bayesian equilibrium. Buyers trade if and only if their valuation is above .
Sellers whose reserve prices are lower than trade for sure. All trades occur at the
price .
Bidders’ strategies can be summarized as follows: if a bidder is currently the
highest bidder in any auction, he pauses until he is outbid. Otherwise, he always bids
on an auction with the lowest standing bid and bids with the minimum increment.
The essential idea is that with cross-bidding, bidders bid up prices with each seller as
slowly as possible. They try to pick sellers where they can become the highest
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bidders by bidding the minimum increment. In this way, high valuation buyers are
never trapped into bidding a high price by having another high valuation buyer
accidentally bid against them. Cross bidding ensures that any mismatch between
buyers and sellers can be solved by giving bidders the opportunity to bid on other
auctions with low standing bids. As a consequence, all trade occurs at the same
price. Most importantly, the outcome of trading is the same as in a double auction.
We are interested to know whether bidders behave as the theory predicts.
Economists usually seek clean theoretical results, which lie in the territory of a
complete market and rational individuals. In case there is incomplete information,
players are required to make probabilistic calculations and forecast future outcomes.
Each player is assumed to be fully aware of the strategic value of his own private
information and to know the structure of other players’ strategies. Such analysis
seeks to characterize the Bayesian Nash equilibrium of the incomplete information
game defined by the trading institution and the trading environment. Yet research in
psychology and behavioral science is replete with examples in which individuals do
not, or perhaps even cannot, perceive circumstance objectively. This raises serious
questions about whether markets can achieve efficiency in practice.
There are more and more experiments in economics. Most results seem to
support that the efficient outcome of the market can be achieved. One import result
gaining support is the Hayek Hypothesis: markets economize on information in the
sense that strict privacy together with the public information (about prices) in the
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market are sufficient to produce efficient competitive equilibrium outcomes.1 How
and why this is true, there is no general answer. The information flow and the
contracting rule in the market may be more important than some traditional structural
characteristics in determining the market outcome. However, whether players use
strategies in Bayesian Nash equilibrium is dubious. In research by Forsythe et al.
(1992), a market worked extremely well, yet traders in the market exhibited
substantial amount of judgment bias.
3. Mechanism in eBay and Summary of Data
eBay provides a rich resource for the empirical study of auctions, and since there
exists a large quantity of similar auctions at any given time, eBay also provides an
excellent resource for the study of competing auctions.
eBay is a list e-commerce site. It provides a central market for buyers and
sellers to meet each other by way of auctions. The income eBay earns comes from a
fee charged to sellers, which varies from a fixed fee per listing, or a small proportion
of the final sale price. Buyers on eBay do not pay anything to participate.
Sellers choose an auction type in which they sell their good.2 They set a
starting bid, a bid increment, and the duration of the auction. Sellers also provide a
detailed description of the item, which usually also includes the method of delivery
1 This may refer to an efficient market allocation and a competitive price, or simply refer to an efficient market allocation with price different to the competitive price.2 There are 6 other types of auctions in eBay: reserve price auctions, Buy It Now, NEW eBay Store Buy It Now listings, Private Auctions, Dutch Auctions, Restricted-Access Auctions, see http://pages.ebay.com/help/sellerguide/selling-type.html.
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and method of payment. Sellers have the option to set a secret reserve price. If they
do so, during the process of the auction, eBay will indicate whether the reserve is
met or not. At the end of the auction, if the reserve is not met, sellers have the right
not to sell the item with the final price.
eBay use the mechanism of a second price auction. At any time, eBay shows
the current standing bid of the auction, which is the current second highest bid (if
there is no bid or there is only one bid, the current bid is the starting bid). When a
bidder submits a bid, he has access to the level of present standing bid, the identity of
the seller (with the seller’s “feedback,” discussed below), the starting and ending
time and the description of the item. He also has access to information as to how
many bids have already been submitted, the identity of the bidders, and the time of
the bids. However, the exact amount of each bid is not revealed until the end of the
auction. The final price is the second highest bid plus the bid increment.
At the end of an auction, eBay does not intervene in the actual transaction
between the seller and the winner of the auction; the seller and the winner contact
each other themselves to complete the transaction. Since bidders cannot inspect the
good directly, sellers have incentives to provide false information; winners may
regret the high price they have to pay and may not contact the seller to finish the
purchase. To promote the faithful implementation of the transaction, eBay uses a
feedback system. After each transaction (whether successful or not), the seller and
the bidder can send feedback about the other party to eBay, marked as positive,
neutral and negative, with values of +1, 0, -1 respectively, plus brief comments. A
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trader is given a feedback number, which is the sum of value of each feedback. The
feedback information is public, and always associated with the trader, though eBay
cannot prevent the traders from changing identity. Indeed, traders with negative
feedbacks have a strong incentive to change their identities in subsequent trades,
while traders with high feedbacks and good comments own an asset of good
reputation. There is some evidence that the reputation associated with the feedbacks
of sellers have an effect on the final price of auctions (see Houser and Wooders
(2001)).
When an auction has ended, eBay provides detailed information about the bid
history.3 Figure 1 is an example of an auction history extracted from eBay’s website.
The first half page shows the basic information about the auction. It is a three-day
auction, started at Oct-31-01 22:51:27 PST and ended at Nov-03-01 22:51:27 PST.
The seller has feedback with value 11. The starting bid set by the seller is $10 and
the increment is set to $0.50. The auction received 10 bids from 4 different bidders.
There was a shipping cost of $5 and optional shipping insurance $5.
The second half page shows the detailed bidding history. The bid history is
sorted by the amount of each bid. The auction received its first bid of $17.50 by
planetorb around 23 hours after the start of the auction. Then 4 hours before the end
of the auction, bidder raheem112 started to bid. He could observe that there was
already a bidder, but would not know the exact amount of bid. Since there was only
one bidder at that time, the standing bid was still $10. Bidder raheem112 first bid
3 In 2000, eBay provided only the highest bid of a bidder and his last bidding time.
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$11, and found that the standing bid increased to $11 and he was not the highest
bidder. Then he increased his bid subsequently in 2 minutes to $13, $14, $15, until at
last he became the highest bidder with bid $20. The standing bid became $17.50. In
the last hour of the auction, the bidder iteachcomputers submitted a bid $20,
increasing the standing bid to $20. Bidder raheem increased his bid to $21, followed
by another bidder who bid $23.99. Bidder planetorb finally won the auction with an
unknown bid. Since the minimum increment is $0.50, the final price was $24.49,
which is the second highest bid $23.99 plus the bid increment $0.50. There was no
bid retraction or cancellation for this auction. eBay keeps the information of
completed auctions public for one month.
Competing auctions in our sample should satisfy the following conditions: 1.
they should be reasonably homogeneous in quality (including warranty) and have
similar delivery method and shipping cost, 2. they should end at approximately the
same time. As documented by Roth and Ockenfels (2000) and Bajari and Hortacsu
(2000), bids on eBay tend to be clustered towards the end of the auction.
We use eBay CPU auctions data from a one-month period - September 20 to
October 19. These are drawn for the category of “Computer, Component, CPUs” in
eBay, which includes subcategories “AMD”, “Cyrix”, “Intel” and “Others”, and each
subcategory includes further subcategories. At the time of writing in November
2001, there are 800 to 900 new CPU auctions every day. We choose those auctions
with only one CPU for sale, and only those auctions with the method of standard
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auction.4 Most of the CPUs are second hand. Table 1 reports the basic statistics of the
sample.
In the sample, there are 7910 auctions involving a single CPU for sale. Not
all items were sold. Among all auctions, 1452 of them did not receive any bids. 899
of them, which is more than 10% of the auctions, have a secret reserve price. In all
899 auctions with secret reserve price, 515 auctions had the reserve price met.
The CPUs in the sample are very different. The mean final price is $60.60,
with standard deviation of $82.34. The number of bids received is 6.66 per auction,
with standard deviation of 6.91. The maximum number of bids in the sample is 49.
Also the starting bids of these auctions are very different, with a mean of $29.25 and
standard deviation of $61.60.
Though the conditions of properly working CPUs in the same specific
category (such as Pentium III 800 retail box) are largely the same, they may be very
different in many respects: some are new and never opened, others are used for
several months or years; some are still under warranty, others aren’t; some are with
both box and complete manual, others without. And the method of the delivery, the
shipping cost and the method of payment can differ.
To overcome this complication in obtaining the competing auctions, we use a
sample consisting of groups of auctions with the same product description, the same
4 eBay has introduced the ‘Buy it Now’ features for auction. Sellers can set a fixed Buy It Now price so that if any buyer wants of buy at that price, he can buy it immediately without go through the auction price. Bidders can bid on auction with Buy It Now feature before the item is sold. Since the Buy It Now method and the standard auction are very different, we exclude all CPUs sold by Buy It Now from the sample. (We will study the buy it now feature in a separate paper).
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delivery method and the same shipping cost. This is done by choosing the auctions
sold by the same sellers.5 Some big second CPU sellers sell many items with the
same description, the same delivery method and the same shipping cost. Without
considering the different ending time, these auctions are completely
indistinguishable to buyers.
Such auctions with identical items started and ended at different time.
Auctions with almost the same ending time compete directly against each other, and
auctions with large differences in ending time compete less directly. We get three
different samples for competing auctions. Each observation in the samples is a group
of auctions, which consists of 2 or more competing auctions. In the first sample (the
daily sample), each group of competing auctions consists of homogenous auctions
ending on the same day. In the second sample (the hourly sample), each observation
is a group of competing auctions consisting of homogenous auctions with the
difference of ending time being less than 1 hour. In the third sample (the minute
sample), each observation is a group of competing auctions consisting of
homogenous auctions with the difference of ending time being less than 1 minute.
Auctions appearing in the minute sample must appear in the hourly sample
and auctions appear in the hourly sample must appear in the daily sample. The
minute sample consists of groups of auctions that compete directly against each
5 This method of selection may not get all possible competing auctions. To get exhausted set of competing auctions, we have to include all similar items, including those sold by different sellers. However, such selection does not affect the validity of our result.
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other, and the hourly and daily data consists of groups of auctions that compete
against each other less directly. Table 2 is a sample description of the three samples.
In the daily sample, there are 550 groups of competing auctions, consisting of
1247 different auctions. Among all auctions, 305 (24%) of them do not receive any
bids and 106 auctions have secret reserve price (40 of them with the reserve price not
met). The hourly sample has a relatively smaller size, with 321 groups of competing
auctions, consisting of 748 different auctions. Among these auctions, 196 (26%) of
them do not receive any bids and 66 of them have secret reserve price (30 of them
with reserve price not met). The minute sample is less than half of the size of the
hourly data, with 139 groups of competing auctions, consisting of 346 different
auctions. Among them 115 auctions (33% of the sample) do not receive any bids and
24 have secret reserve price (19 of them with reserve price not met).
We find that in each sample, there are bidders who are winners for more than
one auction in a group of competing auctions. In the daily sample, there are 24
groups of competing auctions with bidders winning more than one auction,
representing 4% of the total groups. In the hourly sample and minute sample, the
number of the groups with a bidder winning more than one auctions is 18 and 10,
representing 6% and 7% of the total groups.
These bidders may happen to need more than one item for themselves, or
they can be professional dealers. Buyers with multiple demands can bid across more
than one auction at the same time. If we consider all bidders with single unit demand,
the existence of buyers with multiple demands will exaggerate the result that bidders
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bid across auctions. However, the proportion of such bidders is low, and the number
of group in which such bidders win more than one auctions is low. In the following,
we will address this problem in more detail and distinguish true cross bidders from
the multiple demand bidders. Our results are not affected significantly by the
existence of those multiple demand buyers.
One may be concerned that bidders, especially the novice, may not fully
understand the mechanism in eBay auction. They might not use optimized strategies.
We use the feedback as an indicator of traders’ experience in eBay6. eBay also uses
heavily the number of feedback in daily trade. For example, to use the Buy It Now
feature, sellers must have a feedback greater than 10 or be ID verified7. Figure 2 is
the distribution of bidders’ number of feedbacks. In the daily sample, there are 2286
bidders and 21 bidders with negative feedback (1 bidder with feedback –4, 2 bidders
with feedback –3, 8 bidders with feedback –2, the rest 10 with feedback –1). 60% of
the bidders have feedback greater than 8. Most of the bidders have history of
transactions in eBay. We can be confident that the observed behavior is not very
different from the optimal one for most individuals.
4. Bidding behavior with competing auctions in eBay
Since we have very detailed bidding history for each auction, we can use it to
study how bidders bid in face of competing auctions.
6 The feedback number in eBay is the sum of positive and negative feedbacks.7 See http://pages.ebay.com/services/buyandsell/buyitnow-seller.html.
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The most interesting result we observe is that bidders, even with single unit
demand, bid across competing auctions.
In table 3 we report the statistics of the bidders and cross bidders in each
sample. For each group of competing auctions, we use a Java program to get all
different bidders and all bidders that bid on more than one auction.
We only consider the group of competing auctions with positive bid for at
least one auction. In the daily sample, there are 458 such groups of competing
auctions. On average, there are 2.28 auctions in each group and the maximum
number of auctions in a group is 9. On average, there are 6.56 bidders for each
group and 1.53 bidders bid across auctions in the group. On average, 23% of them
bid across competing auctions in the group they bid.
In the hourly sample, there are 258 groups of competing auctions with
positive bids for at least one auction. The number of different bidders per group is
almost the same as that in the daily sample, and slightly few numbers of bidders who
bid across auctions in a given group. 21% of the bidders bid across auctions in the
group they bid.
In the minute sample, there are only 101 groups of competing auctions with
positive bids for at least one auction. On average, there are fewer bidders in each
group, with an average of 4.868. And there are 1.48 bidders who bid across auctions
in a given group. The proportion of bidders who cross bid is higher than the other
8 It may be a little puzzle to see that there are fewer bidders on average for the minute sample. This is because that in the minute sample, a higher proportion of auctions have high starting bid and receive only one or zero bids.
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two samples, but it is not statistically significant. (The t-statistics for hourly sample
and minute sample is 1.33 and the t-statistics for daily sample and minute sample is
1.14).
The data shows that a significant proportion of bidders bid across auctions in
a competing group. For auctions with all most the same ending time, significantly
higher proportion of bidders bid across competing auctions. The proportion of
bidders who bid across competing auctions is not very different for daily sample and
hourly sample (with a t-statistics of 0.54).
For competing auctions ending at all most the same time, the strategy for
bidders to bid across auction is more effective in avoiding in trapping in highly
competitive auctions. However, if there is a significant time difference in ending
time, bidders cannot effectively coordinate bidding across auctions.
As documented at Roth and Ockenfels (2000) and Bajari and Hortacsu
(2000), bids are clustered at the ending period of the auction. This is also true for our
competing auction data. We have information about all the bids and the time the bids
are submitted. Figure 3 shows the bid submission time for all auctions in the daily
sample.
Bids are clustered at the ending period. Almost all auctions received their last
bid in the last several hours. In addition, 28% of the samples receive their last bid
within the last 60 seconds.
In Roth and Ockenfels (2000) the reason for delaying bids is that eBay
auction has a fixed ending time. If the bids are submitted at the last minute, there is
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some probability that bids may not be submitted successfully. This can explain the
very last minute bidding, but cannot explain late bidding. Roth and Ockenfels (2000)
also find that even in auctions like Amazon auctions that have no fixed ending time,
there are significant late biddings. Simply bidding late does not have effect that bids
might not be submitted successfully. Bajari and Hortacsu (2000) argue that in a
common value environment, bidders refrain from bidding earlier to avoid revealing
their private information.9
Therefore, it is sometimes argued that early biddings are not serious, and only
late biddings are serious. We look at the data of the last day of each auction, with all
different bidders in the last day and the different bidders who bid across competing
auctions. We use the sample consisting of the groups with at least one auction
receiving positive bids in the last day. We find that for all bidders bidding in the last
day of the auctions, significantly more bidders bid across competing auctions when
the difference in ending time is less than 1 minute. (See Table 4.) For the minute
sample, on average, 27% of the bidders in the last day bid across competing auctions,
compared to the percentage of 0.20 in the hourly sample and 0.17 in the sample. For
the hypothesis of having the same percentage of bidders who bid across competing
auctions, the t-test for the minute data and the hourly data is 1.65 and the t-test for
9 The existence of competing auction can provide an explanation for late bidding. With the existence of competing auctions, bidders bid across competing auctions to avoid being trapped in an auction if other high valuation bidder happens to bid against them. If there is no bidding cost, bidders should bid the minimum increment each time and bid again and again. With bidding cost, it is too costly to bid very often. Bidders can bid with a significant jump and bid less frequently. However, this increases the risk of being trapped in a very competitive auction. Another alternative for bidders is to bid only at the late period, and bid with small increment. In this way, they do not need to bid too frequently, at the same time they can effectively cross bid and avoid being trapped in too competitive auctions.
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the minute data and daily data is 2.52, which is significant at the 0.01 level. And
though there is slightly higher percentage of bidders who bid across competing
auctions for hourly data than for daily data, the result is not statistically significant,
with t-test 1.23.
We may wonder if the cross bidding observed here is only because of the
existence of bidders who want more than one item at the same time. There is not a
simple way to check directly if a bidder wants more than one item. We use the
following methodology: bidders who need more than one item is very likely to be the
highest bidders of more than one competing auction. In the first case, we define true
cross bidders as all those bidders who bid across competing auctions and who are
never the highest bidder of more than one auction at any time (type I). In the second
case, we consider true cross bidders as all those bidders who bid across competing
auctions and who are not the highest bidders of more than one competing auction in
the last day (type II).
The first criterion for true cross bidder is stricter than the second criterion.
The second criterion has already excluded all bidders who win more than one
competing auctions. Some bidders excluded by the first criterion may still be true
cross bidders, since at the early stage of the auctions, bidders may feel safe to be
high bidders of more than auction even if they only need one item.
Table 3B reports the result of such considerations. When we exclude those
possible bidders with more than one demand, we still observe high percentage of
bidders who bid across competing auctions. For the minute data, 28 bidders are high
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bidders for more than one competing auction in some point of time of auction
process and 21 are high bidder for more than one competing auction in some pint of
the last day of the auctions. The bidders who truly bid across auctions are 25% in the
first criterion and 26% in the second criterion. Similarly, for the hourly data, we
observe that among all bidders, 21% of them bid across competing auctions. Apart
from those who need more than one item, the percentage of true cross bidder is 18%
(first criterion) and 19% (second criterion). For the daily data, we observe 23% of
bidders who bid across competing auctions. The percentage of true cross bidders is
21% (under the first criterion) and 22% (under the second criterion).
Another feature of bidding with competing auctions is that bidders tend to bid
on the auction with the lowest standing bid. This strategy not only guarantee that a
bidder never wins two auctions, it also let bidders to avoid being trapped in very
competitive auction. (This also makes the price of the auction become uniform.) We
report the result on whether bidders bid on the auction with the lowest standing bid
in table 6. 10
When all auctions in a competing auction group are ended except the last
one, bids submitted on this last auction thereafter are considered as bidding on
auction with the lowest standing bid. This way of calculation tend to increase the
number of bids submitted on the auction with the lowest standing bid for group of
auctions with big difference in ending time, such as in the daily and hourly sample.
10 It is not trivial to see if a bid is submitted on the auction with the lowest standing bid. We use some data structure such as TreeSet in Java. Interested reader may have a look at the appendix of Zheng (2001).
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In the daily sample, the average number of bids a group of competing
auctions received is 13.29 and the maximum number of bids a group received is 79.
The average number of bids submitted on auctions with the lowest standing bid is
8.95 and the maximum number of such bids is 37. For the hourly sample, the average
number of bid in a competing group and the average number of bids on auction with
the lowest standing bid is roughly the same. The proportion of the bids submitted on
the auction with the lowest standing bid is also roughly the same, representing 79%
for daily sample and 77% for hourly sample (the t-test for the proportion for daily
and hourly sample is 1.10).
For the minute sample, the average number of bids a group receives is
relatively less, with a value of 10.89, and the maximum number of bids that a group
receives is 7.86. The proportion of the bids submitted on the auction of the lowest
standing bid is significant higher, with an average of 87%. The t-test for the minute
data and the hourly data is 4.49 and the t-test for the minute data and the daily data is
4.33.
Bidder who bid across competing auctions are more likely to bid on the one
with the lowest standing bid. In table 6, we report the result about the proportion of
bids submitted by bidders who bid across competing auctions.
In all three samples, there are groups in which some auctions do not receive
any bids while others receive positive number of bids. It happens usually that if there
is any auction does not receive bids, then other competing auctions either receive no
bids or receive only one bid. For same items, there is no reason that bidders bid on
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auctions with positive bids and not bid on auctions with zero bids. It is interesting to
see how often it happens that in a group of competing auctions, some auctions
receive no bids while other auctions receive more than one bid. Table 7 reports the
result for three samples.
In the daily sample, there are 74 groups of competing auctions with positive
bids and with zero bid auctions. 30% of them with auctions receiving more than 1
bid. In the minute sample, there are 20 groups of competing auctions with positive
bids and with zero bid auctions. However, only 15% of them have auctions receiving
more than 1 bid.
It is obvious from the above analysis that bidders are not following the
strategy as in independent second price auction: bid once and bid the true valuation.
Bidders are not following the advice form eBay to submit the true valuation and let
the proxy to bid for them. The proxy bid mechanism in eBay is believed to save
bidders from revising their bid. eBay’s help page about proxy bid explains the proxy
bid as the following:
A proxy bid and a maximum bid are the same thing. To place a
proxy bid, just enter the maximum amount you are willing to pay. eBay
will then automatically bid up to your maximum amount for you.
(http://pages.ebay.com/help/basics/e_item1.html)
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In table 8, we report the result of the average number of bids a bidder
submitted in each group. We only consider the sample consisting of groups with
bidders who bid across competing auctions. With competing auctions, bidders do not
bid on one auction, but rather on a group of competing auctions. We calculate the
bids each bidder submitted on each competing group. There is a tendency that
bidders bid more frequently with competing auctions with almost the same time. In
the minute sample, on average, each bidder submit 2.24 bids on a group of
competing auctions (On average, there are 2.6 auctions in each group. Each bidder
bid less than 1 bid on each auction of a group). However, for bidders bidding cross
auctions, the average number of bids they submitted on a group is significantly
higher. For the minute sample, each bidder who bid across auctions submits 3.97
bids on a given group of competing auctions.
Bid revising can be a consequence of the existence of competing auctions. If
there is no bidding cost, bidders should bid with the minimum increment and bid
many times. If bidders bid true valuation and bid only once, they may be trapped in
very competitive auctions and do not have opportunity to switch to other less
competitive auctions. Even if with bidding cost, bidders still revise their bids very
often if the cost is not too high compared to the risk of bidding with another high
bidding bidder. (They may not always bid the minimum increment, which is too
costly).
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5. Does cross bidders pay lower price?
Some winners of auctions bid across competing auctions, while others do not.
Bidding across competing auctions make bidders have more choice in bidding and in
trying to find auction with low price. The theory predicts that bidders who bid across
competing auctions can win auction with lower price.
To test this result, we divide each group of competing auctions into two
subgroup: a group consisting of auctions in which winners bid across competing
auctions and with single unit demand, another group consisting of the rest of
auctions. Here we define a bidder who only needs one item as in the sense that he
was never the highest bidder for more than one auction in any time of the auction
process (criterion 1). Then we compare the average final price in these two sub-
groups by calculating the ratio of price paid by cross bidder and the price paid by
non-cross bidder. In some group all winners bid across competing auctions while in
other groups neither of the winner bid across competing auctions. These groups are
excluded from the analysis. We keep the groups in which there are winners who bid
across competing auctions and winners who do not bid across auctions. Table 9
reports the result.
In the minute data, there are only 17 groups of competing auctions with both
winners who bid across competing auctions and those winners who do not. The ratio
of the price paid by cross bidder and by non-cross bidder is 0.89, with a standard
deviation of 0.19. We consider the test ratio<1. For minute data: we have
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. At level 0.01, the winners who bid across competing auctions pay a
price statistically lower than those winners who do not. On average, the winners who
bid across competing auctions pay a price that is only 90% of the price paid by those
winners who do not.
This is also true for the daily data. There are 103 groups of competing
auctions with winners who bid across competing auctions and winners who do not.
For the test ratio<1, we have , and at level 0.01, the winners
who bid across competing auctions pay a price statistically lower than the winners
who do not.
However, for the hourly sample, the price paid by the cross bidding winner
and non-cross bidding winner is not significantly different. For the test ratio<1,
we have .
Therefore, it is strictly benefit for bidders to bid across competing auctions
even when some other bidders do not.
6. Conclusion
This paper has provided the first related pieces of evidence in the literature that
together lend strong support to competing auctions theory. First, in the presence of
competing auctions, bidders use a very different strategy than they would with
independent auctions. Bidders tend to bid across competing auctions and bid on the
auction with the lowest standing bid. We further demonstrate that when homogenous
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auctions end at almost the same time, more bidders are likely to bid across
competing auctions than when auctions end at different times. We also find that
winners who bid across competing auctions pay a lower price than those winners
who do not bid across competing auctions. Therefore, since there are many
homogenous auctions competing against each other in on-line auctions, it is
inappropriate to consider on-line auctions as independent, ignoring the existence of
other competing auctions.
In this paper, we do not consider whether the auction prices are uniform with
the existence of competing auctions. According to the theory, competing auctions
tend to make prices more uniform. We will study such price effect of the existence of
competing auction on eBay in a separate paper. In the current paper, we also take the
behavior of sellers as exogenous. It will be interesting to see what the equilibrium
behavior of sellers is given the existence of competing auctions. This will be
especially useful for those sellers who want to sell many identical items one by one
in a short time.
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Reference
Bajari, Patrick and Ali Hortacsu. (2000) “Winner’s curse, Reserve Prices and
Endogenous Entry: Empirical Insights from eBay auctions”, Working paper,
Stanford University.
Hooser, Daniel and Daniel Wooders. (2001) Reputation in Auctions: Theory and
Evidence from eBay, working paper, University of Arizona.
Lucking-Reiley, David. (1999) “Auctions on the Internet: What’s being auctioned,
and How?” Working paper, Vanderbilt University.
Myerson, R. B. (1981) Optimal Auction Design. Mathematical of Operations
Research, 6, 58-73.
McAfee, P. (1993) “Mechanism Design by Competing Sellers,” Econometrica 61
(1993) no.6, 1281-1312.
Peters, Michael and Sergei Severinov. (1997) “Competition among sellers who offer
auctions instead of prices,” Journal of Economic Theory, 75, 141-197.
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Peters, Michael and Sergei Severinov. (2001) “Internet Auctions with Many
Traders,” Working paper, University of Toronto.
Forsythe, Robert et al. (1992) “Anatomy of an Experimental Political Stock Market,”
American Economic Review, vol. 5, 1142-1161.
Roth, Alvin and Axel Ockenfels. (2000) “Last Minute Bidding and the Rules for
Ending Second Price Auctions: Theory and Evidence from a Natural Experiment in
eBay,” Tech. Report, Harvard University
Wilson, R. (1985) “Incentive efficiency of double auctions,” Econometrica 53, 1101-
1117.
Zheng, Mingli (2001) Real time simulations of eBay auctions with competing
auctions, working paper, University of Toronto.
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Figure 1.1
A bidding History Page from ebay
EBay Bid History forIntel Pentium II Xeon 450MHz 512k - Used (Item # 1292121814)
Currently $24.49 First bid $10.00
Quantity 1 # of bids 10
Time left Auction has ended. Started Oct-31-01 22:51:27 PSTEnds Nov-03-01 22:51:27 PSTSeller (Rating) nhahmad (11)
View page with email addresses (Accessible by Seller only) Learn more.
Bidding History (Highest bids first) User ID Bid Amount Date of Bid planetorb (9) $24.49 Nov-03-01 22:30:37 PSTibgeek (4) $23.99 Nov-03-01 22:18:01 PSTraheem112 (0) $21.00 Nov-03-01 22:03:35 PSTiteachcomputers (1) $20.00 Nov-03-01 22:06:49 PSTraheem112 (0) $20.00 Nov-03-01 18:33:27 PSTplanetorb (9) $17.50 Nov-01-01 21:11:05 PSTraheem112 (0) $15.00 Nov-03-01 18:33:16 PSTraheem112 (0) $14.00 Nov-03-01 18:32:50 PSTraheem112 (0) $13.00 Nov-03-01 18:32:39 PSTraheem112 (0) $11.00 Nov-03-01 18:31:52 PSTRemember that earlier bids of the same amount take precedence. Bid Retraction and Cancellation History There are no bid retractions or cancellations.
Table 1.1
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Sample Statistics For CPU Auctions (Number of auction=7910)
Variable Mean Std Dev Minimum Maximum
Number of bids 6.66 6.91 0 49
Last bid (unit: $) 60.60 82.34 0.01 981
Starting Bid (unit: $) 29.25 61.60 0.01 899
1452 auction receive no bids899 auctions have secret reserve price515 auctions have met reserve prices
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Table 1.2
Sample Statistics for Samples of Competing Auctions
Daily sample Hourly sample Minute sample
Number of auctions 1247 748 346
Number of groups 550 321 139
Number of auction with reserve met 66 36 5
Number of auctions with reserve not met 40 30 19
Number of auctions not receiving bids 305 196 115
Number of bidders winning more than one item in a
group 24 18 10
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Figure 1.2
Histogram of Bidders’ Feedback
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Figure 1.3
Histogram of Bids Submission Time for Daily Sample
0
200
400
600
800
1000
1200
1400
1600
1800
Fraction of auction duration
Freq
uenc
y
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Table 1.3
Statistics of Cross Bidding in the Whole Process
Groups Auctions per group Cross bidder per group Bidder per group ProportionMean Std Min Max Mean Std Min Max Mean Std Max Min Mean
Daily 458 2.28 0.87 2 9 1.53 1.78 0 10 6.56 5.14 32 1 0.23Hourly 258 2.36 0.98 2 8 1.41 1.90 0 10 6.59 5.32 32 1 0.22Minute 101 2.60 1.29 2 8 1.48 1.96 0 10 4.86 4.34 20 1 0.27
(For groups with positive bids only.)Proportion: proportion of cross bidders.
T-statistics for the proportion of hourly sample and minute sample is:
T-statistics for the proportion of dailyly sample and minute sample is:
T-statistics for the proportion of hourly sample and daily sample is:
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Table 1.3 B
Percentage of Cross Bidders in All Bidders
Percentage of cross bidder Percentage of true cross bidder I Percentage of true cross bidder II
Daily 0.23(700/3007) 0.21(627/3007) 0.22(648/3007)Hourly 0.21(363/1700) 0.18(312/1700) 0.19(324/1700)Minute 0.30(149/491) 0.25(121/491) 0.26(128/491)
True cross bidder I: bidders who bid across competing auctions and who are never the highest bidder of more than one auctions at any
moment.
True cross bidder II: bidders who bid across competing auctions and who are never the highest bidder of more than one auctions at any
moment in the last day.
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Table 1.4
Statistics of Cross Bidders for the Last Day
Groups Auctions per group Last day cross bidder per group Last day bidder per group Proportion
Mean Std Min Max Mean Std Min Max Mean Std Max Min Mean StdDaily 419 2.28 0.87 2 9 0.63 1.02 0 6 3.61 2.44 12 1 0.17 0.28
Hourly 235 2.37 0.97 2 8 0.71 1.11 0 6 3.62 2.55 12 1 0.20 0.31Minute 88 2.60 1.26 2 8 0.93 1.23 0 6 3.06 2.19 10 1 0.27 0.35
(For groups with positive bids in the last day only.)
Proportion: proportion of cross bidders.
T-statistics for the proportion of hourly sample and minute sample is:
T-statistics for the proportion of daily sample and minute sample is:
T-statistics for the proportion of hourly sample and daily sample is:
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Table 1.5
Result on Bidding on the Auction with the Lowest Standing Bid(for all bidders)
GroupsBids on the auction with lowest
standing bid Total bids Proportion
Mean Std Max Min Mean Std Max Min Mean StdDaily 458 8.95 7.68 37 0 13.29 12.51 79 1 0.79 0.20
Hourly 258 8.45 7.85 37 0 13.26 12.97 79 1 0.77 0.25Minute 101 7.86 8.32 36 1 10.89 13.17 63 1 0.87 0.16
(Groups with positive bids only.)Proportion: proportion of bids submitted on auction with the lowest standing bid.
T-test for proportion of minute and hourly sample:
T-test for proportion of minute and hourly sample:
T-test for proportion of hourly and daily sample:
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Table 6
Result on Bidding on the Auction with the Lowest Standing Bid(For groups with cross bidders only)
GroupsBids on the lowest
(cross bidder)Total bids
(cross bidder)Proportion
(Cross BidderBids on lowest
(all bidders)Bids (all Bidders)
Proportion
(all bids)Mean Std Max Min Mean Std Max Min
Daily 286 5.97 4.94 31 0 8.78 7.76 47 2 0.76 8.78 18.60 0.74Hourly 138 6.54 5.50 27 0 9.56 8.51 41 2 0.77 12.42 18.36 0.76Minute 55 7.69 6.54 31 1 10.75 10.19 47 2 0.81 12.44 17.82 0.78
Proportion: proportion of the bids on lowest standing bid by cross bidders among all bids by cross bidders
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Table 1.7
Result on Bidding on Group with Zero Bid Auctions
Groups with positive bids and with zero bid auctions
Groups with auctions receiving more than 1 bids
Proportion
Daily 74 20 30%Hourly 27 6 22%Minute 20 3 15%
Table 1.8
Average Number of Bids Submitted in a Group by a Bidder
For all bidders For cross bidders only1 day 2.02 3.581 hour 2.01 3.63
1 minute 2.24 3.97
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Table 1.9
Price Paid by Cross Bidders and Non-Cross Bidders
Price paid by cross bidder/price paid by non cross bidderMean Standard deviation
Daily (obs =103) 0.91 0.17Hourly (obs =41) 1.08 0.95Minute (obs =17) 0.89 0.19
Cross bidders: defined as those cross bidders who are never the highest bidder of more than one auctions (type I)
ratio<1.
For minute data:
For hourly data:
For daily data: At 0.01 level, for minute data and daily data, winners who bid across competing auctions pay lower price than other bidders. Those winners who cross bid pay a price that is 10% lower. For hourly data, the price paid by winners who cross bid and who do not cross bid are not significantly different.
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Chapter 2
Liability Rules and Evolutionary Dynamics11
(Published inJournal of Institutional and Theoretical Economics, 157 (4) Dec 2001.)
Abstract
We consider the convergence properties of behavior under a comparative negligence
rule (CN) and under a rule of negligence with contributory negligence (NCN), assuming
bilateral care with three care levels. Using an evolutionary model, we show that CN reduces
the proportion of the population using low care more rapidly than does NCN. However NCN
increases the proportion of the population using high (efficient) care more rapidly than does
CN. As a result, the mean care level increases more rapidly and the mean social cost falls
more rapidly under CN than under NCN. (JEL: K 13, C 79)
11 I am very grateful to Donald N. Dewees for extensive help and support. Without his help the paper could never be the present form. Myrna Wooders introduced me to the evolutionary game theory and provided many support. Joanna Robert and Michael Peters gave me valuable comments and encouragements. I am also very grateful to two anonymous referees for their very detailed comments that greatly improve the paper.
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1 Introduction
An economic analysis of law considers tort liability as a tool that can induce injurers
to internalize the costs they impose on others. An efficient liability rule should
provide incentives for a causative contributor to an accident to minimize the sum of
accident and avoidance costs by taking cost-justified precautions. Many authors have
discussed the equilibrium and efficiency of different liability rules. (BROWN
[1973], LANDES AND POSNER [1987], SHAVELL [1987], ARLEN [1992]).
An important issue is how to choose the best liability rule. In recent years the
comparative negligence rule (CN hereafter) has spread widely, replacing the rule of
negligence with contributory negligence (NCN hereafter). Eight US states had
adopted comparative negligence by 1971, but an additional 34 adopted it between
1971 and 1985. CN was argued to be inferior to NCN because the court must decide
on the degree of the negligence by both parties (WHITE [1988]). POSNER [1992]
stated that “the modern movement to substitute comparative for contributory
negligence” is one of the three “most important counter examples to the efficiency
theory of law”. WHITE [1989] tested empirically whether the incentive to take care
to avoid accidents is stronger under NCN than under CN.
The above literature only considered whether the rules provide an efficient incentive
to take care. WITTMAN, FRIEDMAN, CREVIER AND BRASKIN [1997]
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addressed another consideration in choosing among liability rules: the speed of
convergence to equilibrium levels of care. When behavior is not at equilibrium, Nash
equilibrium is seldom achieved instantaneously. WITTMAN, FRIEDMAN,
CREVIER AND BRASKIN [1997] undertook an experimental test of convergence
to equilibrium under different liability rules. In their laboratory experiment,
convergence to equilibrium (measured by mean care level) is much more rapid under
comparative negligence than under contributory negligence. They gave no
theoretical explanation for their result.
From time to time, society is away from the equilibrium level of care for different
reasons. Individuals may not be fully rational. In auto accidents, there may be new
drivers who do not correctly perceive the risks and costs, such as when a
demographic bulge hits driving age, or a wave of immigration produces a stock of
new drivers. At any time some individuals may experiment with new strategies.
When a court or legislature changes the rules, it will take time before drivers adapt
themselves to that change. Changing technology can change the risk and cost of
driving significantly12. Any of these factors can require drivers to adjust their
behavior to a new optimum, incurring high social costs if the adjustment is slow.
The main contribution of this paper is that we use an evolutionary approach to
analyze the speed of convergence under different liability rules, using a simple
setting of bilateral care with three care levels. Homogenous drivers decide whether to
12 For example, anti-lock brakes and air bags may reduce the cost of aggressive driving. The increase in the proportion of large sport-utility vehicles and pickup trucks may reduce the benefits of careful driving of their owners and increase the benefits of careful driving for owners of small cars.
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take a high, medium or low levels of care. The high level of care is the social
optimum. In this setting of three care levels, the Nash equilibrium under both CN
and NCN is the social optimum.
The evolutionary approach assumes that a strategy that does well is imitated, while a
strategy that does badly is rejected. We assume that in every period an individual
reflects on the payoff from his strategy and shares strategy and payoff information
with others. In every period a fraction of those individuals with a lower payoff
change their current strategies to more profitable strategies. This eventually leads to
the convergence to the equilibrium strategy. Because of inertia and uncertainty the
fraction of individuals that change their strategy in any period is relatively small.
The greater is the payoff difference, the greater is the incentive to change the
strategy.
We show that CN reduces the proportion of the population using low care more
rapidly than does NCN. However NCN increases the proportion of the population
using high (efficient) care more rapidly than does CN. As a result, the mean care
level increases more rapidly and the mean social cost falls more rapidly under CN
than under NCN during the early periods. This is consistent with the result in
WITTMAN, FRIEDMAN, CREVIER AND BRASKIN [1997].
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The paper is organized as follows: section 2 states the basic assumptions and the
Nash equilibrium under different liability rules, section 3 considers evolutionary
dynamics and section 4 concludes.
2 Liability Rules and Nash Equilibrium
We consider a framework of bilateral care with three care levels. Suppose the
population has infinitely many drivers. There are only three levels of care for each
driver, high ( ), medium ( ) and low ( ), with the costs of taking care as , , and
respectively. For simplicity, we also use to denote the amount of care under
three care levels, . Once an accident occurs, a total damage is
incurred.
A driver does not know what kind of other drivers he will encounter during the day.
We model these encounters as random matches. The probability of an accident in a
match between two drivers is: and , where is the probability
of an accident when a driver taking care meets another driver taking care
. We assume that:
Assumption 1. Social cost is minimized if all drivers take high care. Social cost is
maximized if all drivers take low care. Thus:
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. (2.1)
From this assumption, we have: , and
other similar inequalities.
We further assume that:
Assumption 2. An increase in the care level reduces more effectively the probability
of an accident when the care level of the other driver is relatively low.
This implies that and other similar inequalities.
Because taking high care minimizes social costs, any driver in an accident who does
not take high care is negligent (or contributorily negligent).
Under CN, both drivers share the losses according to their relative negligence, or
care shortfall. For example, if an accident happens between a driver taking medium
care and a driver taking low care, then the driver taking medium care will bear
portion of the total loss13, while the other party will bear the rest of the
loss. Since , we have . The following is the payoff matrix of the game.
Table 2.1
13 This is a typical assumption in comparative negligence, see, e.g., COOTER and ULEN (1997). It may take a more general form as a function of the care levels and (or) the cost of care.
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Payoff Under CN
H M L
H
M
L
Under NCN, the parties share the liability in the same way as under CN except for
the cases when both parties are negligent. In those cases, they both incur half of the
total damage. The payoff of the game is the same form as that for CN, with .
Under NCN drivers taking low care and medium care are considered equally
negligent.
Proposition 2.1. If taking high care is socially efficient, then under both CN and
NCN, taking high care is a Nash equilibrium.
Proof: Given the assumption about the parameter values, it is easy to check from the
payoff matrix that taking high care is a Nash equilibrium.
As we can see from the payoff matrix, the only difference between CN and NCN is
the value of . When there is an accident between a party taking low care and a
party taking medium care, the party who takes low care has to incur a larger fraction
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of the cost of the accident under CN than under NCN. This provides a greater
incentive to abandon low care under CN, as we will discuss in the next section.
3 Evolutionary Dynamics
In reality it is rare that Nash equilibrium is achieved instantaneously. Nash
equilibrium requires that players are rational and know the payoff functions of all
players, that they know their opponents are rational and know the payoff functions,
that they know their opponents know, etc. In actual life, these requirements may not
be met14.
This poses a problem: will the Nash equilibrium always be closely approximated at
least in the long run? If this is the case, does the outcome converge to the Nash
equilibrium rapidly and what is the path of the convergence?
To answer this question, we use an evolutionary approach (MAYNARD SMITH
[1982]). The idea of evolutionary games began with the idea that animals are
genetically programmed to play different pure strategies, and that the genes whose
strategies are more successful will have higher reproductive fitness. The population
fractions of strategies whose payoff against the current distribution of opponents'
play is relative high will tend to grow at a faster rate, and any steady state must be
14 REA [1987] points out that individual may be judgment proof or they could misperceive both the risks and costs. They may not choose the Nash equilibrium care level (they are unresponsive). The author suggests that, in comparison to the negligence rule with contributory negligence, the negligence rule with comparative negligence is more robust to the presence of these unresponsive individuals.
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Nash equilibrium. There is no need for the strong requirement of rationality and
common knowledge among players.
Evolution can be taken as a metaphor for learning in economics. Individuals respond
to different payoffs by modifying their strategies. If we assume inertia in human
behavior and costs associated with switching strategies, then the proportion of the
population choosing each strategy changes smoothly. In the following, the
proportion of drivers taking care is subject to evolutionary pressure over time. The
fraction of the population using better performing strategies will increase relative to
those using lower payoff strategies. Our main focus will not be on the steady state of
evolution, but on the relative speed and the path of convergence to the steady state
under different liability rules.
We denote and the proportion of drivers taking high, medium and
low level of care at time , . Given the population composition ( ,
), a driver will meet drivers taking high care with probability and will meet drivers
taking medium and low care with probability . Under CN (and also NCN,
which corresponds to ), at any time , the expected payoff for a driver who
takes high care is:
. (3.1)
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The expected payoff for a driver who takes medium care is:
. (3.2)
The expected payoff for a driver who takes low care is:
. (3.3)
The mean payoff of the population is:
. (3.4)
The value of is the social cost.
When , some drivers will find it profitable to switch from the strategy of taking
low or medium care to the strategy of taking high care, and will increase. The
payoff differential exerts evolutionary pressure on the population composition. The
standard model of the movement of the composition of the population in
evolutionary game theory is that of the replicator dynamics15, (TAYLOR AND
JONKER [1978]), defined as:
, (3.5)
where is the time derivative of , and is a constant. The rate of the
growth (decline) of the proportion of the population using a strategy is proportional
15 This is only for simplicity of the analysis. As we will see later, most of the analysis is still true if we use more general evolutionary dynamics such as a growth monotone dynamics.
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to the amount by which that strategy's payoff exceeds (falls below of) the average
payoff of the whole population. The standard replicator dynamics can be derived
from different models of individual learning behavior (for example, NACHBAR
[1990])16. The NACHBAR [1990] model can be reasonably used in the driving
environment.
Evolutionary game theory is widely used in economics, for example in choosing the
most likely equilibrium from all possible Nash equilibriums. Replicator dynamics
allows us to compare the convergence properties under different liability rules in a
given social environment and with a same learning pattern.
Simple calculations using expressions (3.1)-(3.4) show that the dynamics of a
population taking high care is exactly the same17 under both CN and NCN, and can
be written as:
,
with
16 In this model, individuals meet randomly somebody else to exchange information about each other's strategy and payoff. The individual with a lower payoff switches his strategy if the switching cost is less than the payoff difference. Assuming the switching cost is independently determined across individuals and is uniformly distributed on [0,M], we get the exact form of replicator dynamics as in (3.5).17 Notice that at any given population composition is always the same under CN and NCN.
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(3.6)
Under Assumption 1 and Assumption 2, we can check that each term in the above
equation is always positive, i.e., at any population composition with . The
proportion of drivers taking efficient cares always strictly increases.
For the dynamics of a population taking medium care, under CN (also under NCN,
which corresponds to ),
(3.7)
Since , , (by Assumption
2), if , we have:
(3.8)
As increases to a certain extent, becomes small and we have , and
will begin to decrease monotonically.
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Similarly, will decrease monotonically as increases to a certain extent.
From the above analysis, we have:
Corollary 3.1. Under evolutionary dynamics, both CN and NCN lead to
convergence to the social optimum in the long run.
Proof: As in the above analysis, always strictly increases, and will decrease
as increases to a certain extent. Therefore, the population composition (
must converge to (1,0,0), which is the social optimum.
Given any population composition, the difference between the payoff of individuals
taking low care and the average payoff of the population is bigger under CN than
under NCN. Therefore under CN, the proportion of individuals taking low care
decreases faster under CN at any given population composition (the proportion of
individuals taking medium care decreases more slowly). We have the following
lemma:
Lemma 3.2. At any population composition ( ), the instantaneous growth
rate of the proportion of individuals taking high care is the same under CN and NCN.
The instantaneous rate of decrease of the proportion of individuals taking low care is
greater under CN than under NCN by . The instantaneous rate of
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decrease of the proportion of individuals taking medium care is greater under NCN
than under CN by .
Proof: It is easy to check by using expressions (3.1)-(3.4), and by the definition of
replicator dynamics (3.5).
Unfortunately, the above lemma is only a local property and it assumes that we are at
the same population composition under CN and NCN. Once the system begins to
evolve, the evolution of the system will follow different paths under the two liability
rules and the local comparison becomes meaningless. Now we begin to discuss the
global convergence property and the convergence path under the two liability rules.
We can look at the path of the dynamics of population composition by looking at the
( ) plane (since ). From the same starting point at time , since the
increase rate of is the same under both liability rules and the decrease rate of is
slower under a comparative negligence rule, we have:
Lemma 3.3. Starting from the same point, the path (when is small) under CN is
above the path under NCN.
We may wonder if this is always true, or the two paths may cross at a later point.
Lemma 3.4. After starting from the same point, the path under CN will always be
above the path under NCN.
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Proof: Suppose that at a later point, the two paths reach a same composition .
Starting from this point, the path under CN will again be above the path under NCN.
So the two paths can never cross.
Therefore the path of convergence under CN in the plane is always above the
one under NCN. See an example in figure 3 of the simulation.
Lemma 3.5. For two population compositions and
, if and , then the increase rate of at point
under CN is less than the increase rate of at point under NCN.
Proof: The rate of change of has exact the same expression under both CN and
NCN at any given population composition. From the form of the dynamics and the
fact that the path under CN is always above the path under NCN in the plane,
it is sufficient to prove that for a fixed , (which is now a function of only
because of the constraint ) is a decreasing function of .
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Since , , and
(by Assumption 1 and Assumption 2), we have:
Intuitively, when increases (so decreases), the mean payoff increases.
Therefore, there is less evolutionary pressure for individuals to take a high level of
care. According to the lemma, at any level of , the increase rate of is larger
under NCN than under CN. We have:
Proposition 3.6. Globally, the proportion of population taking the efficient level of
care increases faster under NCN than under CN.
Locally, under CN there is a stronger evolutionary pressure on the individuals who
take low care; while under NCN there is a stronger evolutionary pressure on the
individuals who take medium care. CN is more effective in reducing the number of
individuals taking low care while NCN is more effective in reducing the number of
individuals taking medium care. Such a difference is significant when the proportion
of individuals taking medium or low care is high. When society is close to the social
optimum, this difference almost disappears. Globally, since at any given level of
there are more individuals taking low care and the mean payoff of the population is
lower under NCN, increases at a faster rate with NCN.
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Which rule is better? It seems that a model of replicator dynamics suggests that NCN
is better, since under NCN more individuals take the efficient care at any time. But
that is not the whole story. Society cares about minimizing the total present value of
social cost. For our case, the social cost at a given time can be measured by
and the present value of social cost is:
(3.9)
which depends not only on how many individuals take efficient care, but also on the
care taken by other individuals. Under NCN more individuals take efficient care, but
at any level of more individuals take low care, which might be more costly to the
society.
Another measure used in WITTMAN, FRIEDMAN, CREVIER AND BRASKIN
[1997] is the mean care level of the population. In our case, the mean care level is:
. Though this measure ignores the social cost associated with each
population composition, it is better than looking at only one component of the
population composition.
We can compare the speed of the change of social cost and of the change of
the mean care level of the population under CN and NCN.
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Proposition 3.7. At a given population composition, the social cost falls more
rapidly and the mean care level of the population increases more rapidly under CN
than under NCN.
Proof: Under both CN and NCN, we have:
(3.10)
Therefore,
(3.11)
At a given population composition , is the same under both CN and
NCN. By equations (3.2) and (3.3), the difference in payoff between taking medium
care under CN and under NCN is: , and the difference in payoff
between taking low care under CN and under NCN is: .
Using the replicator dynamics, we have that the difference of the time derivative
of the mean social cost between under CN and under NCN is:
(3.12)
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By Assumption 1 and Assumption 2, , and:
.
Therefore, increases at a greater rate under NC than under NCN, i.e., the social
cost decreases at a greater rate under NC than under NCN.
Similarly, for the mean care level , we have:
. (3.13)
The difference of the time derivative of the mean care level between under CN
and under NCN is:
.
Therefore, the mean care level increases at a greater rate under CN than under
NCN.
Since CN is more effective in reducing the number of individuals taking low care
and NCN is more effective in reducing the number of individuals taking medium
care, and taking low care is more costly for the society, at a given population
composition, CN minimizes the social cost more effectively (at least for a short
period of time). Also for a period of time the mean care level of the society increases
more quickly under CN.
We cannot generalize the result in proposition (3.7) (which is a local property) to a
global property. After a very long period of time, the result might not be true any
more. However in the real world, the inefficiency is caused by periodic shocks, so
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we may not observe an undisturbed convergence for very long period. The result in
proposition 3.7 can be applied in most cases.
We use simulations to further illustrate the difference between CN and NCN. The
parameter values are chosen to make it socially optimal for all individuals to take
high care under both CN and NCN. The values of the parameters are:
and . We take 9 and 1. We use the discrete counterpart of the
replicator dynamics, that is:
(3.14)
Table 2 displays the time path of the composition of population, the social cost and
the mean care level under CN and NCN with an assumed initial condition
0.7, . In this dynamics, both liability rules lead to the
social optimum. The convergence paths display the features as we discussed above.
The proportion taking high care increases faster under NCN than under CN (figure
1). The proportion taking low care decreases faster under CN. The proportion taking
medium care decreases more slowly under CN (figure 2). See figure 3 for the path of
( .
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Table 2.2
Simulation Results
Comparative Negligence Negligence with Contributory Negligenceh m l hsocial cost mean care h m l social cost mean care
0 0.700 0.150 0.150-3.7000 -4.4403 8.500 0.700 0.150 0.150 -4.4403 8.5001 0.752 0.146 0.103-3.7518 -4.3108 8.932 0.752 0.138 0.110 -4.3290 8.8702 0.794 0.138 0.068-3.7938 -4.2159 9.248 0.795 0.126 0.079 -4.2411 9.1633 0.827 0.128 0.045-3.8273 -4.1492 9.471 0.831 0.114 0.055 -4.1743 9.3874 0.854 0.117 0.029-3.8540 -4.1036 9.624 0.859 0.103 0.038 -4.1250 9.5535 0.875 0.106 0.018-3.8753 -4.0729 9.728 0.882 0.092 0.026 -4.0896 9.6746 0.893 0.096 0.012-3.8926 -4.0524 9.799 0.900 0.082 0.018 -4.0644 9.7607 0.907 0.086 0.007-3.9069 -4.0385 9.847 0.915 0.073 0.012 -4.0468 9.8228 0.919 0.076 0.005-3.9188 -4.0291 9.881 0.927 0.065 0.008 -4.0344 9.8659 0.929 0.068 0.003-3.9289 -4.0225 9.905 0.937 0.058 0.005 -4.0257 9.896
10 0.938 0.060 0.002-3.9376 -4.0179 9.922 0.945 0.051 0.003 -4.0196 9.91911 0.945 0.054 0.001-3.9451 -4.0145 9.935 0.952 0.045 0.002 -4.0153 9.93512 0.952 0.048 0.001-3.9517 -4.0120 9.946 0.958 0.040 0.001 -4.0121 9.94713 0.957 0.042 0.000-3.9574 -4.0101 9.954 0.964 0.036 0.001 -4.0098 9.95614 0.962 0.037 0.000-3.9625 -4.0085 9.960 0.968 0.031 0.001 -4.0081 9.96315 0.967 0.033 0.000-3.9669 -4.0073 9.965 0.972 0.028 0.000 -4.0067 9.969
Present value of social cost: -33.7903 Present Value of social cost -33.8837
h, m, l: proportion of population taking high (10), medium (9), low (1) level of careDiscount factor =0.9
Figure 2.1
Proportion of Individuals Taking High Care
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Figure 2.2
Proportion of Individuals Taking Medium Care
Figure 2.3
The Composition of the Population in the Simulation
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We also give the value of mean social cost at each period. Until period 12 the social
cost is smaller under CN than under NCN for each period. After that the mean social
cost becomes greater under CN for each period. The mean care level is also greater
under CN until period 11. After that, the trend also reversed. In the simulation, as
time goes on, the proportion of individuals taking low care is very small, so NCN
outperforms CN (even though not significantly) because it reduce the proportion
taking medium care more effectively.
We calculate the present value of the expected social cost with the discount factor
:
(3.15)
In 16 periods, the present value of the expected total social cost is -33.7903 under
CN and –33.8837 under NCN. The social cost is about 3% lower under CN. With the
parameters we have randomly chosen, the cost saving is not very significant.
In reality, we may not see such monotonic convergence. From time to time there will
be random shocks to the composition of the population, either because of new
entrants or some other factors that drive society away from the equilibrium. The
shocks may be very small or relatively large. The evolutionary pressure through
imitation and learning leads society to the Nash equilibrium. The constant
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appearance of shocks makes the rate of convergence and the path of convergence
very important, as it leads to different social cost.
Our results are consistent with the experimental results in WITTMAN, FRIEDMAN,
CREVIER AND BRASKIN [1997]. The experiments in that paper showed the
convergence of the mean care level to the Nash equilibrium under both CN and
NCN, and they showed that CN promotes a faster convergence to the Nash
equilibrium than NCN. In our analysis, starting from a same population composition,
the mean care level increases more rapidly under CN than under NCN for a period of
time. The result may not be maintained in the long run without further shocks, but it
is usually true in reality when inefficiency is often caused by periodical shocks.
In WITTMAN, FRIEDMAN, CREVIER AND BRASKIN [1997], individuals are
assumed to choose their best response, given the behavior of other individuals in the
population. They considered an adjustment dynamics using model
, where is the state at time , is the best response function,
and is a forecast of state at time using information at time ( is a proxy
for all other influences that vary slowly). They consider several possible models by
choosing different and . One of their findings is that when estimating the
population care level ( as the population mean care level), the model in which
players give their best response to the mean care level of the last period ( )
provides a good fit of the data (with ). The model in which players give
their best response to the distribution of previous care levels also provides a good fit
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for the data. Our evolution model considers the distribution of care levels at time as
a function of last period's distribution of care levels, which is more reasonable for a
large population in a social environment.
Instead of using the simple replicator dynamics, we can use a more general
evolutionary dynamics model such as a growth monotonic system (see VEGA-
REDONDO [1996]):
, (3.16)
with the condition: if , then .
Intuitively, in a growth monotonic system, strategies that “do better” grow faster.
Under a growth monotonic system, most of our results persist. After starting from the
same point, the path in the ( plane under CN is always above the path under
NCN, using the same reasoning as in Lemma 3.4. (Therefore, at any given there
will be fewer individuals taking low level of care under CN than under NCN). If we
further assume that 18 at any given , we can get the
result that grows faster under NCN as in proposition 3.6. The results about the
comparison of mean social cost and mean care level require more restriction on the
form of the evolutionary system.
18 The condition is quite natural: at a given , if increases a little bit (so decreases a little bit), the average payoff of the population will increase. The benefit of switching to taking high level of care will decrease and the increasing rate of will decrease.
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The method we used can also be applied to more than three care levels, and it is still
true that CN is more effective in reducing the proportion of individuals taking the
most inefficient care levels and NCN is more effective in reducing proportion of
individuals taking the medium care levels. This is beyond the scope of this paper.
4 Conclusion
In this paper we use a simple replicator dynamics model to study the evolutionary
dynamics of comparative negligence and negligence with contributory negligence.
We compare analytically the convergence properties under these two rules, finding
that the proportion of the population taking high care increases more slowly under
CN than under NCN. However CN reduces the proportion of the population using
low care more rapidly than does NCN. At a given population composition, the mean
social cost falls more rapidly and the mean care level increase more rapidly under
CN than under NCN. This may explain the modern movements to substitute
comparative for contributory negligence.
Intuitively, the advantage of CN is that this rule is more effective in inducing the
very careless individuals to abandon their current strategy and to take a more
efficient strategy. Very careless individuals impose more cost on society than those
slightly careless individuals do. Under comparative negligence the cost is shared
according to relative negligence. Very careless individuals always have to incur
more cost of an accident under comparative negligence rule than under negligence
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with contributory negligence rule. This is why comparative negligence reduces social
cost more effectively and is better than negligence under contributory negligence.
It is in society's interest that liability rules lead to optimal care and that the optimum
be approached rapidly when out of equilibrium. Therefore, when comparing the
effects of different liability rules, in addition to considering the efficiency of
equilibrium, it is also necessary to consider how rapidly the society converges to that
equilibrium.
References
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COOTER, R., AND T. ULEN [1997], Law and Economics, Reading, Mass.:
Addison-Wesley.
LANDES, W. M., AND R. A. POSNER [1987], The Economic Structure of Tort
Law, Harvard University Press: Cambridge, MA.
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NACHBAR, J. H. [1990], “Evolutionary Selection in Dynamic Games,”
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MAYNARD SMITH, J. [1982], Evolution and the Theory of Games, Cambridge
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POSNER, R. A. [1992], Economic Analysis of Law, 4th ed., Little, Brown.
REA, S. [1987], “The Economics of Comparative Negligence,” International Review
of Law and Economics, 7, 149-162.
SHAVELL, S. [1987], Economic Analysis of Accident Law, Harvard University
Press: Cambridge, MA.
TAYLOR, P., AND L. JONKER [1978], “Evolutionarily Stable Strategies and
Game Dynamics,” Mathematical Biosciences, 40(2), 145-56.
VEGA-REDONDO F. [1996], Evolution, Games, and Economic Behavior, Oxford
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WHITE, M. J. [1988], “The Economics of Accident Law,” Michigan Law Review,
86, 1217-1231.
- - [1989], “An Empirical Test of the Comparative and Contributory Negligence
Rules in Accident Law,” Rand Journal of Economics, 20, 308-330.
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WITTMAN, D. AND D. FRIEDMAN AND S. CREVIER AND A.BRASKIN
[1997], “Learning Liability Rules,” Journal of Legal Studies, 26, 145-164.
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Chapter 3
Rational Legal Decision-Making, Value Judgment and the Efficient
Precaution in Tort Law
Abstract
In this paper, we provide a model for rational legal decision-making by considering the
consistency of social decision-making on resource allocations. We emphasize that legal
decision-making is not based on individual utilities. The purpose of law is described by a
subjective social welfare function, with a social value judgment and an attitude towards
distributional inequality as parameters of the social preference. Thus a legal decision-making
always involves a social value judgment and an attitude toward distributional inequality.
Social efficiency can be different from the maximization of social wealth. In tort law, the
determination of the efficient precaution level and the award of compensation depend on the
social value judgment, and the compensation scheme implicitly transfers wealth from the
less socially valued party to the more socially valued party. We provide another explanation
for the award of punitive damage and for the arbitrariness of the dollar value of the punitive
damage award.
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1. Introduction
In this paper, we provide a model for rational legal decision-making. In the
model, the efficiency is described by a subjective social welfare function, with a
social value judgment and a social attitude toward distributional inequality as
parameters of the social preference. As an application of the model, we analyze the
determination of the due precaution levels and of the damage compensations in tort
law, and provide an explanation for the award of punitive damage and for the
arbitrariness in the dollar value of punitive damage award.
The purpose of law and the role of moral value in legal decision-making are
always in hot debate. Most of the debates are based on philosophical foundations19.
Many legal commentators and morality theorists emphasize the role of value
judgment20. Moral theorists suppose that there is universal justice and fairness that is
based on exogenously preset social norms21. For example, utilitarianism is based on a
social welfare function, which is the (weighted) average of individual utilities. As
pointed out by Posner (1997), this approach has its defects: the weights in the social
welfare function are arbitrary, and the aggregation of individual utility will be
19 See a debate in Harvard Law Review (May 1998) by Posner et al and the work by Posner (1979) and Deworkin (1980).20 An interesting example is Calabresi's analysis of tort law. In Calabresi (1972), it is argued that liability should be placed in such a way to minimize accident avoidance costs and thus maximize wealth. However, in Calabresi (1985), he argues: “who is the cheapest avoider of a cost, depends on the valuations put on acts, activities, and beliefs by the whole of our law and not on some objective or scientific notion”. “What is efficient, or passes a cost-benefit test, is not a ‘scientific’ notion separated from beliefs and attitudes, and always must respond to the question of who we wish to make richer or poorer”.21 In Rawls' justice theory, a society cares about the welfare of the worst individual.
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vulnerable to the abuse in the measurement and aggregation of these utilities. The
concepts of justice, fairness and the social value judgment have not yet played any
role in the standard law and economic analysis. The economic analysis of law ruled
out the role of value judgment by assuming that the purpose of law is to maximize
total social wealth (or to minimize total social cost). According to Posner (1984),
wealth maximization is the only social value that courts can do much to promote.
Courts should take the distribution of wealth for granted, as a responsibility that the
political system has allocated elsewhere, and should seek to maximize wealth in their
judgments. Dworkin (1980) criticizes that wealth is neither a value nor an instrument
of social value22.
In actual life, legal decision-making is not based on individual utilities. In
eminent domain cases, which is defined as ‘the power to take private property for
public use by a state, municipality, or private person or corporation authorized to
exercise functions of public character, following the payment of just compensation to
the owner of that property’, the compensation is not determined by the individual
utilities. The measure of damages is the fair market value of the property harmed or
taken for public use. The market value is commonly defined as the price that could
22 Mercuro and Samuels (1986) also challenged Posner on the empirical assertion: “The issue remains that of whether courts affect the distribution of rights and of wealth. We affirm that in a slow gradual, incremental manner, the courts, through the precedential and appellate system are making law and that in making law they are determining who will have what rights and that this constitutes determining the distribution of wealth in society. Surely Posner does not think that with respect to rights litigants are indifferent as to what courts decide. The only question then is whether court decisions affect prices through affecting the distributional basis of prices, namely rights. We affirm that they do.”
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have reasonably resulted from negotiations between an owner who was willing to
sell and a purchaser who desired to buy23.
In this paper, we do not base the social decision-making on individual utility.
Individual decision-making on consumption and risk taking is based on the
individual utility. On the contrary, social decision, as in many cases of legal decision,
deals with activities involving trading and exchange. Social preferences cannot be
measured by individual preferences. Collectively determined exchange values of
commodities (usually their market values) are imposed in legal decision. Marx
recognized quite early that social preferences couldn’t be measured in terms of
individual preferences. One of the foundations of Marxist analysis is the separation
of use-value and exchange-value. Commodities can fulfill human needs. The
concrete and qualitative character of need in relation to each commodity is reflected
in its use-value. But once in the exchange, commodities lose their qualitative
differences and become quantitatively equivalent to other commodities, thus
masking their use-value. In social decision-making, the exchange-value is involved,
as in the eminent domain case. Therefore social welfare is not the aggregation of
individual utilities. Marxist studies contrast sharply with the liberalist view about the
political and social structure.
23 ? For example, to construct a park on a track of land owned by many owners, individual bargaining will make it impossible for construction because of the hold-out problem. Society will decide the value of land collectively, usually the market value (if such value exists) and impose this value on the individuals.
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In legal decision-making, the society values the commodity by its exchange
value (market value), and not by the value derived from individual utilities.
Therefore it is natural to consider social decision-making on the base of the market
value of resources owned by individuals, or on the resource allocation among
individuals. This contrasts to the public choice theory, in which the society is merely
a mechanism for combining private preferences into a social decision-making. Even
though many researches in public choice reach results against the rationality of social
decision-making, the incoherence predicted by public choice theory is not reflected
in the actual behavior of the society. Legislature is not chaotic; outcomes in legal
decision-making are predictable and stable; and law must treat equals equally.
Especially, consistency is a basic requirement for legal decision-making, as required
by justice and fairness.
Therefore, the rationality in social decision-making can be better described
by the consistency in resource allocations among social members. We assume very
intuitive assumptions about social rationality about resource allocation among
individuals: a social preference is transitive and complete; the society prefers more
than less; mixing two resource allocations with any third one in a specific way
preserves the order of preference. Together with other technical conditions, we show
that any social decision-making is described by maximizing a subjective social
welfare function, with a social value judgment and a social attitude towards
distributional inequality as parameters of the social preferences. The result is
obtained by using the advance in Savage’s subjective utility theory in individual
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decision-making with uncertainty and finite states, in which an individual can assign
a subjective probability to the states of the world and a value to each outcome.
However, the states of the world in subjective utility theory are replaced by the set of
individuals in the society, and the outcome is replaced by the resource of each
individual. The subjective social welfare function represents the subjective
perception of the society; and it expresses the social ethics. When making social
decision, the society puts more weight on the interest of the party with more social
value; the attitude toward distributional inequality reflects the social ideology on
distributional inequality.
The value judgment and the attitude toward distributional inequality are
internal to a society. They are endogenous to the social life and can be changed by
social and political activities. Social preferences vary across different societies, so do
the value judgment and the notion of justice and fairness. If lump sum transfers are
possible, a society always makes decisions that maximize total social resources and
then distributes them among all parties. Otherwise, either because of the property
law or other constraints, the bias to the interest of some individuals (or group of
individuals) will lead the social decision-making away from wealth maximization.
Therefore, the purpose of law can be different from the maximization of social
wealth.
As an application, we consider the determination of the due precaution level
and the award of damage compensation in tort law. In the economic analysis of law,
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the due precaution level is determined by maximizing total social wealth, and the
award of damage compensation is always equal to the actual damage. The only
reason for punitive damage is that there is a probability of avoiding the compensation
by injurers (Polinsky and Shavell (1998)). Other theories for punitive damage
include that punitive damage serves as deterrence and (or) retribution24. Cooter and
Ulen (1988) argued that punitive damage is necessary because some illegal benefits
should not be counted in the social welfare function25. The view of retribution
suggests that the awards of punitive damage express social outrage at some form of
behavior (Chapman and Trebilcock (1989)), which implies that there is a moral
judgment consideration behind the awards of punitive damage.
From our results, in a rational legal decision-making setting, if the injurer in a
tort case is valued less than the victim by the society, the society imposes a due
precaution level that is stricter than the one maximizing total social wealth. To
induce such a precaution, punitive damage must be awarded. Similarly, if the victim
is valued less than the injurer by society, the society imposes a due precaution level
that is less strict than the one maximizing total social wealth. Correspondingly, the
compensation is less than the actual damage. When punitive damage awards become
necessary, the exact amount of the award of punitive damage award is very sensitive
to the estimation bias of social value judgment. This can explain why punitive
damage seems arbitrary, as observed in the empirical evidence (Daniels and Martin
(1990)).24 There is a large literature on the analysis of punitive damage. See Biggard (1995) for some of the citations.25 As pointed out in Klevorick (1985), it is not clear why there is such a domain, as illicit, and what activity is illicit.
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Section 2 uses a result analogous to the existence of subjective expected
utility theory on a finite state space (Nakamura (1990), Chew and Karni (1994)) to
discuss rational social decision-making for resource allocation among individuals (or
among groups of individuals). The social welfare function and the value judgment in
legal decision, as well as the efficiency in legal decision-making are discussed in this
section. Section 3 discusses the due precaution level and the award of damage
compensation in tort law. We also provide another explanation for the award of
punitive damage. Section 4 concludes the paper.
2. Rationality, subjective social welfare function and value
judgment
Social decision-makings involving the exchange of commodities are not
based on individual utilities of the commodity; they are based on the exchange
values (or market values) of these commodities. For a given kind of activity, each
social decision leads to a resource allocation among the involved parties. The society
compares all possible resource allocations and makes decisions based on such
comparison. We use the similar notation as in subjective expected utility theory as in
Nakamura (1990), substituting the states of the world by parties involved in a social
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decision, and the set of consequences by the set of resources received by each
individual.
Assume that there are parties ( individuals or groups of individuals) in
an activity, . All subsets of , , are the subgroups of
these parties. The resource that party receives is an element in the space ,
which can be one dimensional (as profit or loss) or multidimensional. A resource
allocation is denoted by , in which is the resource of party .
Assume also that there is a nature ordering on (such as the nature ordering of
profit on ). The set of all possible allocations is denoted by .
Social preference is a binary relation on .
For two allocations in , means that allocation is at least as good as
allocation from the social point of view.
Several additional notations and definitions as defined in Nakamura (1990) will be
useful in stating the assumption about social rationality. For any resource allocations
, any individual resource allocation , and any subgroups ,
means that in allocation and allocation , all parties in the subgroup
have the same resource: for all ; means that in allocation , all
parties in subgroups get resource that is equal to . A binary allocation is an
allocation with and , in which each parties in subgroup get resource
, while each of the rest of the parties get resource . Such binary allocation is
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denoted by . An equal allocation is an allocation such that all parties have
the same resource , and every can be identified with an equal allocation.
A partition of is a sequence of non-empty subgroups that are mutually disjoint and
whose union equals ; it divides the parties into exclusive subgroups. For a k-
partition and , if for all , that is, in allocation , if
each parties in subgroup have resource for all , then the allocation is written
as .
A null subgroup is a subgroup such that for all , :
allocations for parties in a null subgroup’s have no effect on the social preference.
On the contrary, a universal subgroup is a subgroup such that for all
, : allocations for parties in a universal subgroup determine the
social preference. The social preference is bounded if, for each allocation , there are
such that : for any allocation, there is a more preferred equal
allocation, and there is a less preferred equal allocation. A standard sequence is a set
for which there exists such that , and either
and for all , or , and
for all .
Given the above notation, we now state the axioms concerning the rationality in
social decision over resource allocations among parties. These axioms apply to all
allocation , all individual resources , and all subgroups :
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A1. The binary relation on is bounded, complete and transitive.
A2. If , then for some .
A3. If is not null, then iff ; if is not universal, then iff
.
A4. If and , then .
A5. Every strictly bounded standard sequence is finite.
Since on is bounded, and by A2, for each allocation , for some 26: for
every allocation, there is an equivalent equal allocation. We denote the equal
allocation corresponding to as . For binary allocation , the
corresponding equal allocation is denoted by .
The following is the key axiom for the representation theorem:
A6. For any partition of , , ,
, , denote allocation as the allocation in which each
party in has resource , as the allocation in which each party in
has resource , and as the equal allocation for the binary allocation
, then:
26 Since on is bounded, for each allocation , there are such that . By definition of an allocation, and . Therefore, by A2, we have for some
.
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A1 implies that the society has a complete ranking of all resource allocations; any
pair of allocations can be compared and such ranking does not lead to cyclical
results. A2 is a restricted solvability axiom, which implies that every allocation has
an equivalent equal allocation among all parties. A3 and A4 are monotonic axioms.
In A3, if equal allocation is preferred than equal allocation (for example, is
larger than in quantity), then for any equal allocation and any subgroup ,
suppose we change the equal allocations to such that those parties in have
resource , and we change the equal allocation to such that those parties in
have resource , then allocation is preferred to . More specifically, for any
equal allocation, if we increase the resource for some parties, then the society is
better off.
In A4, if equal allocation is preferred than equal allocation (for example, is less
than in quantity), and if we change the equal allocation to the allocation such
that those parties in have resource , which is more than , and we change the
equal allocation to the allocation such that those parties in a subset of have
resource , then the allocation is preferred to the allocation . For any equal
allocation, if we increase the resource for more parties, the society is better off.
A5 is an Archimedean axiom, which guarantees a good form of utility function in the representation.
A6 is the key axioms and there are different forms of this axiom as in Nakamura
(1990) and in Chew and Karni (1994). The form here is taken from Chew and Karni
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(1994). It is analog to the independent axiom of Savage. The analog to the
independence axioms can be seen as follows: Take arbitrary allocations and , any
subgroup , and any third allocation . Consider the allocation constructed from
, and by requiring that the resource of each individual to be the resource ,
such that the equal allocation is equivalent to the allocation in which each parties
in B has the resource that party has as in and all other parties has the resource
that party has as in (i.e., and A2 assure that such equivalent equal
resource allocation exists). Similarly, construct from , and as above. Then if
is preferred to , then is preferred to . As in the independent axiom, the
order of preference is preserved when and are “mixed” with the same allocation
in the sense above. However, because of the finiteness mixture and the specific way
of mixing, this axiom is weaker than the independent axiom in the original Savage
Axioms.
Using the result of Nakamura (1990), Chew and Karni (1994), we have the following
theorem:
Theorem 2.1. Suppose social preference relation satisfies A1-A6. Then there
exists a unique (relative) weight with , and a real valued
function , unique up to a positive linear transformation, such that for any
allocations , , we have:
(2.1)
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Therefore, a rational social preference can be represented by a subjective social
welfare function:
(2.2)
It is important to notice that the social welfare function and the parameters
and are all derived from the revealed social preference, and they are subjective to
the society, representing the internal belief and perception of the society. We call the
social welfare function as the subjective social welfare function.
This theorem is only a restatement of the finite state Savage's theorem of
individual decision-making with uncertainty in Nakamura (1990), Chew and Karni
(1994), where interested readers can find the detailed proof of the theorem. The
subjective social welfare function obtained has the same form as an individual
expected utility27. For individual expected utility, represents the belief over the
probability of the states of the world, while represents the individual attitude
towards risk. We will see that these parameters have different meanings in our
setting, as is illustrated by the following example of pure redistribute policy.
27 There are two different approaches for individual decision-making. One is Ascumbe-Aumann (A-A) approach, which is based on the objective lottery of some prospectus, where the probability is exogenously given. Another approach is Savage's subjective utility approach based on the preference of alternative actions, where the probability is endogenous to the preference.
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We consider a problem of pure social income redistribution among parties without
cost and distortion. The initial income is . Redistribution is implemented through
lump sum taxes and subsidies.
The society maximizes the social welfare function under the resource constraint:
(2.3)
Without loss of generality, we suppose . From the well-known results in
optimization problem, if is strictly convex, the optimal allocation for society is to
give all the resources to the first party and nothing to the rest of the parties. If is
strictly concave, the first order condition becomes
(2.4)
If , from condition (2.4), the society allocates more resource to party than to
party . Society emphasizes party 's interest more than that of party 's. Therefore
we can define as the social value of party . These social values can appear, for
instance, as the social perception that party is more important for the society or
more valuable for the future of the society.
Function represents the social attitude toward distributional inequality. If is
convex, the society is inequality loving and all resources are concentrated to the most
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influential party. If is concave, the society is inequality averse. As in individual
utility theory, the curvature of , as defined in the Arrow-Pratt coefficient ,
reflects the extent of distributional inequality averse.
Corollary 2.2. In the subjective social welfare function, the weight represents the
social value of party , and the function represents the social attitude toward
distributional inequality. Parties with larger social value get more resources. Arrow-
Pratt coefficient describes the society's extent of inequality aversion (or willingness
of wealth transfer).
Therefore, the subjective social welfare function includes two important
aspect of social ethics: the social values assigned to members of the society and the
social attitude toward distributional inequality, which are not any general principle
imposed on the society from outside, but internal and subjective to the society.
Contrast to standard social choice theory, in which the social welfare is a
function of individual utilities, this subjective social welfare function has no relation
with any specific individual utility. Both and are society level variables. Since
the social ethics is internal to the society, different society has different ethics; social
preference can be changed through social activities (the change of social preference
through interest group competition is out the scope of the present paper28), and ethics
in a society can be changed. Social preferences can change without any changes in 28 In Zheng (2001), we relate the social preference to democratic voting equilibrium, and discuss how interest groups can influence social preference.
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individual preferences, and vice versa. The social values and the attitude toward
distributional inequality reflect the social and political power, social norms and
institutional constraints.
Social preference can be independent from each individual preference;
however, society uses its forces to impose its ethics on its members by changing the
incentive of individuals through legal and other social decision-makings. Inevitably,
social preferences often conflict with individual preferences and have to be imposed
by force, and collective decisions are used sometimes as instruments to keep some
groups’ advantageous social position.
We can explicitly include the product activities in social decision-making.
Each productive activity leads to a certain kind of resource distribution. For
production activities with significant externalities (either positive or negative) and
high transaction costs, some kinds of social decisions have to be made. In some
cases, some parties are benefited from the activity while other parties incur a loss.
For example, in tort cases, society has to choose the due precaution levels. Each
social decision leads to a resource allocation among all parties involved in the
production.
Suppose that the productive activity that the society can choose from is .
If the society chooses activity , then the resource obtained by group is . The
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total social wealth is . From the above analysis, a rational social
decision is determined by maximizing a certain social welfare function:
If lump sum transfers are available and there is no cost for wealth transfer, it
is easy to see that the society always chooses the activity that maximizes total social
wealth. Even with the consideration of the social ethics, economic efficiency is still
achieved.
In reality, lump sum transfers are not always possible and wealth
maximization is no longer the objective of the society. The society cannot transfer
freely the individual wealth, either because of property law or other considerations.
For simplicity, we assume that the society does not care about the distribution of
wealth for legal decisions, as distribution objective is not important for many legal
decisions and such objective should be left to other public policies, such as tax-
transfer policy. Under such simplification, function in the subjective social welfare
function is linear. Social welfare function in determining the optimal activity is
represented by:
.
The society does not choose the resource allocation; rather it only chooses an
optimal action . If all involving parties have the same social value, i.e., is
constant for all , then the social welfare function becomes total social wealth (up to
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a constant). The society chooses the action that maximizes total social wealth. This is
exactly the argument in law and economic analysis: the purpose of law is to
maximize the total social wealth.
However, if parties have different social values, i.e., the i ’s are not constant,
then maximizing the social welfare is different from maximizing total social wealth.
With different social values, the society chooses the actions that are biased to the
interests of its more valued parties; the more diverse of the social values, the less
likely total social wealth is maximized. As we will see in tort cases, economic
efficiency (social wealth maximization) can be achieved only if the injurers and the
victims have the same social value. Otherwise, the due precaution level is either too
strict (when the victim is more influential) or too loose (when the injurer is more
influential).
An activity that creates considerable wealth may still be socially undesirable
if it hurts a more valuable party. An activity that reduces the wealth could be socially
desirable if it benefits the party with more social value and hurts some other parties.
Different societies have their own social preferences, so the value judgment
in legal decision-making can be different, and the notion of justice and fairness can
be different. There is no universal concept of justice or fairness, as the critical legal
studies emphasizes, see Fitapatrick and al. (1987). Changing social environments,
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such as technology progresses and interest group activities can change social
preference and thus social value judgments.
The lobbying activities in the process of law making are aimed at changing
social preferences, making the law decision beneficial to some parties. Judges and
juries, considered as “with characteristically judicial virtues of rationality, neutrality
and respect for the institution”, are trying to find the social preference in legal cases.
Activities of lawyers are also aimed at influencing social preferences.
Proposition 2.3. All legal decisions involve social value judgment. The
purpose of law is for social efficiency, which is the maximization of a social welfare
function, with the social value judgment and the attitude toward distributional
inequality as the parameters of social preference. When lump sum transfers are
always possible, legal efficiency leads to social wealth maximization. Otherwise,
legal efficiency is different from social wealth maximization if the involved parties
have different social values, and the interests of the parties with more social value
judgment are more emphasized by the society.
Posner's work (for example, Posner (1979)) claims that efficiency in
economic analysis of law is wealth maximizing. He suggests that wealth
maximization seems a more defendable principle than utilitarianism. Dworkin (1980)
criticizes that wealth is neither a value nor an instrument of social value. The above
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result shows that the wealth maximization can be the purpose of law, but only in
some special cases.
Individuals only care about their own interests, trying to get more resources
from society and to maximize their own utilities. There are conflicts between social
preferences and individual preferences. Society has to use coercive power to impose
social values on individuals. The imposition of compensation in tort law is aimed at
inducing individuals to act in the social interest (to maximize the social welfare
function) and to take the social optimal level of precaution.
3. The determination of efficient precaution level and the
damage compensation in tort law
As an application of the above analysis, we consider the determination of due
precaution levels and the award of compensation in torts. For simplicity, we consider
circumstances involving only two parties: the injurer and the victim, and only the
injurer takes action, which is characterized by the level of precaution. The injurer
chooses precaution level , leading to a resource allocation between injurer and
victim: , with . Legal decisions cannot make
lump sum transfer because of the property law constraint, therefore, to achieve the
desired resource allocation, the society imposes a due precaution level by imposing
some compensation schemes.
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Legal decisions maximize a subjective social welfare function, with its social
values and attitude toward distributional inequality as parameters. We assume that
the society is inequality neutral in legal decision-making (i.e., is linear). This
assumption is reasonable if the society does not want to use tort law for
redistribution of wealth. The social welfare function associated with an action of
precaution level is29:
(3.1)
If both parties have the same social value, , maximizing social
welfare is equivalent to maximizing total social wealth. The social efficiency
becomes the wealth maximization, as claimed by Posner (1979). When ,
maximization of the social welfare function is different from the maximization of
social wealth. In the extreme case, when is approaching 1 or 0, the wealth of one
party is almost not counted by the society, which is the case described by Cooter and
Ulen (1988): some illegal benefits should not be counted in the social welfare
function.
It is easy to check that even when is not linear, if is concave and
, maximizing social welfare still leads to maximizing the total social
wealth.
29 We assume, without loss of generality, that the social value is constant for such behavior. The precaution level has not any effect on the social value. For cases that the social values are functions of the precaution level, the analysis is the same.
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For most contract and tort cases, members of the society have equal chance to
be the injurer and the victim; the injurer and the victim have the same social value. In
these cases, wealth maximization is the motive of society. Legal decision tries to
impose the actions that maximize social wealth. This can explain why in most cases
of contract and tort law, standard economic analyses of law without considering
value judgment is very successful. Full compensation induces the socially optimal
precaution.
However, for some other cases, by history or interest group activities, the
group of injurer and the group of victims has very different social value. Wealth
maximization will be no longer true.
Suppose the social values of the injurer and the victim are and . For precaution
level , the revenue and cost function of the injurer are and , with
property . More precaution reduces profit of the action. At the
same time, the action causes a fixed damage to a potential injurer with probability
. More precaution will reduce the probability of an accident but at a decreasing
rate: . The action with precaution level results in an expected
allocation , with . Social welfare associated
with precaution level is:
.
(3.2)
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The efficient level of precaution for the society is determined by:
.
Therefore, the social efficient precaution level is determined by:
, (3.3)
The wealth maximizing precaution level is determined by:
(3.4)
Without the imposition of the compensation, the injurer does not consider the
externality that he imposes on the victim. The preferred precaution level of the
injurer is 0. Society must provide incentives for the injurer to increase the precaution
level by imposing an award of damage on the injurer, forcing him to internalize
the externality. The injurer's profit after the compensation is:
.
The injurer chooses a precaution level to maximize his utility. If the
injurer's utility is only a function of his profit from the activity, with a form as
, then injurer maximizes his profit. The first order
condition is:
(3.5)
To induce the socially optimal precaution , by comparing condition (3.3)
and condition (3.5), the compensation should be set to:
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(3.6)
Under a negligence rule, the injurer pays compensation only if his
precaution level is less than the due care level . Under a strict liability rule, injurer
pays compensation once damage occurs. As in standard economic analysis of law,
both a negligence rule and a strict liability rules induce the socially optimal
precaution.
There are three different cases:
1. . The group of injurer has higher social value. According to (3.3)
and (3.6), the due precaution level under a negligence rule is less strict than the
precaution level maximizing total social wealth, . The compensation to be paid
is also lower than full compensation, .
2. . The group of victim has higher social value. The due precaution
level under a negligence rule is more strict than the precaution level maximizing total
social wealth, . The compensation to be paid is greater than the actual damage,
. The injurer has to pay a punitive damage.
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3. . This is the case extensively studied in law and economics. The due
precaution level under a negligence rule is equal to the precaution level maximizing
total social wealth, . The compensation is equal to the actual damage, .
If the injurer has higher social value, the injurer is less likely to be found
negligent and he pays less for compensation when he is found negligent. The injurer
takes less precaution than the precaution level maximization total social wealth and
gets more profit than under a full compensation rule. If the victim has higher social
value, the society imposes a stricter due precaution level such that the victim has
more chance to get compensation given the injurer's action. The compensation is
greater than the actual damage. The injurer gets less profit from his action than under
the wealth maximization precaution level .
In standard law and economic analysis, punitive damages can be awarded
only if there is a possibility that injurer can avoid compensation with some
probability. The compensation should be the harm multiplied by the reciprocal of the
probability that the defendant can escape from compensation (see Polinsky and
Shavell (1998)). However, there are many cases in which the probability of detection
and full compensation is very high and punitive damages are still rewarded (for
instance in the case of certain assaults, pollution, etc). Our analysis shows that the
award of punitive damage is possible even without possibility of avoiding
compensation, and the award of punitive damage is rather a consequence of different
social value of the injurer and the victim. An injurer with very small social value is
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the one morally repugnant, thus the award of punitive damage always associates with
a moral judgment.
Most literature on tort compensation focuses on the award of full damage and
punitive damage, neglecting the case of awarding less than full damage30. In the case
of less than full damage, the action of the injurer is often regarded as accidental or
inevitable.
We now look at the expected post-compensation resource allocation. Under a
strict liability rule, the payoff of the injurer is proportional to the social welfare:
.
The payoff of the victim is proportional to the actual damage:
.
Under a negligence rule, the injurer's payoff is:
if ,
and
if .
The payoff of victim is
if ,
and30 Dobbs (1989) considered the case of under-compensation.
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if .
The payoff of the injurer is greater under a negligence rule than under a strict
liability rule and the payoff of the victim is less under a negligence rule than under a
strict liability rule.
The payoffs of the victim and of the injurer depend on their relative social
values. If victim has less social value than the injurer, , he always ends up
with a negative payoff, even under a strict liability rule. In case where both parties
have the same social value, only strict liability will make the victim whole; a
negligence rule will make the victim poorer than he would be. In both cases, the
society sacrifices the victim's interest to the injurer's benefit.
If the victim has higher social value than the injurer, , then under a
strict liability rule, the victim gets a positive payoff from the action. He may get
positive payoff even under a negligence rule. Therefore, the choice of compensation
schemes implicitly transfers wealth from the party with less social value to the party
with more social value. The choice of due care level also reflects “who we wish to
make richer or poorer” (Calabresi (1985)).
The common view is that it is sufficient for the injurer to make the victim
whole. The victim plays only a passive role. In some states in US, even in case where
the punitive damage is awarded, victims only get full compensation and the rest of
the compensation goes to the state treasury. However, under a negligence rule,
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without the award of punitive damage to the victim, the victim cannot be made
whole in general.
Proposition 3.2. Tort law maximizes a subjective social welfare and induces
a socially optimal precaution level by the imposition of compensation in case of
damage. The optimal precaution level is more or less strict than the precaution level
maximizing total social wealth and the compensation is greater or less than the actual
damage, depending on the relative social value of the injurer and the victim. Tort law
implicitly transfers wealth from the less socially valued party to the more socially
valued party.
One of the characteristics of punitive damage award is its arbitrariness.
Whatever the purpose of the punitive damage award, it is criticized of being
unpredictable, even out of control31. A recent paper of Sunstein et al. (1998) studies
the source of such arbitrariness. They choose 899 jury-eligible citizens and estimate
the results of deliberation consisting of different composition of juries. They find that
people's moral judgment are remarkably widely shared, (and punitive damages are
largely determined by value judgment), but people have a great deal of difficulty in
mapping their moral judgment on to an unbounded scale of dollars. They accredit
such difficulty to the human behavior factor.
31 One study of 47 counties in US over a several-year period, median verdicts ranged from less than $10,000 in some area to as much as $204,000 in San Diego. See Daniels and Martin (1990).
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Such arbitrariness can be explained by the excessive sensibility of the dollar
value of a punitive damage award to social value estimation bias in deciding the
punitive damage. When deliberating a case, the juries are attempting to assess the
social preference in that case, the “sense of community”. In our setting, juries are
trying to find the exact value of . The bias in the estimation is inevitable. From
expression (9), the award of damage is determined by: . For relative
large , a small estimation bias will not affect greatly the value of . However,
when is very small (thus punitive damage should be awarded), the value of
becomes extremely sensitive to the small bias of estimation of .
For a numerical example, we consider a case: the damage caused by the
injurer's activity is . The jury has to find the value of relative social value
judgment. Suppose there is an estimate bias . For any value of , the estimation
can be in the range , with . Therefore the estimation
of the award of damage lies in the range , with .
We take three values of : 0.9, 0.5, and 0.1. When =0.9, the victim is
under-compensated. When =0.1, punitive damage is awarded. We give a range of
the award of damage when there is a small estimation bias, =0.05.
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Table 3.1.
Sensitivity of the Award of the Punitive Damage to the Estimation Bias of
Range of award of damage
0.9 0.053D to 0.176D
0.5 0.818D to 1.220D
0.1 5.667D to 19.00D
The range of award of damage is from 0.053D to 0.176D when ;
when , the range of award of damage is from 0.818D to 1.220D; and when
, the range of award of damage is from 5.667D to 19.00D. When
and punitive damage is necessary, the possible value of the award of damage can
range from about 6 times to 19 times of the actual damage, demonstrating a
significant arbitrariness.
This kind of excessive sensibility of the dollar value of the punitive damage
award to the estimation of social value requires the procedure considerations of
punitive damage, as in the criminal law case. The award of punitive damage should
treat this problem seriously. A strict procedure and some very detailed judicial rules
should be proposed to solve the problem (Ellis (1989)).
4. Summary
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In this paper, we establish a subjective social welfare function in legal decision-
making, with a social value judgment and a social attitude toward distributional
inequality as parameters of the social preference. The starting point is the coherence
and the rationality of legal decision-making; and the social preference is not based on
individual utilities, but based on the resource allocation among social members.
Contrast to the view of a general concept of justice and morality, the social ethics is
subjective to the society.
The existence of a social value judgment also implies that social wealth
maximization is the purpose of law only if all parties involved in a legal decision-
making have the same social value. Otherwise, legal decisions emphasize the interest
of the parties with more social values, and total social wealth is not maximized.
As an application, in tort law, the determination of the due precaution level and the
award of damage compensation depend on the social value judgment. If the injurer
has a higher social value, the due precaution level is less strict than the precaution
level maximizing total social wealth, and the compensation to be paid is lower than
the full compensation. If the victim has a higher social value, the due precaution
level is stricter than the precaution level maximizing total social wealth, and the
compensation to be paid is greater than the actual damage: the injurer has to pay a
punitive damage. By comparing the after-compensation resource allocation, we find
that tort law implicitly transfers wealth from the less socially valued parties to the
more socially valued parties. Our paper provides another explanation of the award of
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punitive damage, and provides an explanation of the arbitrariness of the dollar value
of the punitive damage award.
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