Mathematical and Computer Modelling 44 (2006) 43–48www.elsevier.com/locate/mcm

Modeling dynamic sealed-offer k-double auctions using a mixture ofdistributions

Christopher Keller

Operations and Decision Technology Department, Kelley School of Business, Indiana University, 1309 East 10th Street, Bloomington,IN 47405-1701, United States

Received 21 October 2004; accepted 9 December 2004

Abstract

This paper develops a novel market-price statistic for a sealed-offer k-double auction. The auction considered is for ahomogenous product with multiple buyers and multiple sellers. The market-price statistic is developed from modeling price asa mixture of statistical distributions for buyers’ prices and for sellers’ prices. Ex post classical efficiency and ex ante Bayesianincentive efficiency of the method are considered.c© 2005 Elsevier Ltd. All rights reserved.

Keywords: Auctions; Markets; Equilibrium behavior; Price; Probabilistic models; Mixture distribution; Buyer; Seller; Distribution function;Estimation; Analytic solution

1. Introduction

Auctions are methods for matching buyers and sellers of products and services and simultaneously settingthe trading price of such transactions. Some theoretical and empirical insights to traditional auctions are wellknown; see [1]. But the recent technological increases in network communications allow auction mechanisms tobe implemented on a greater scale than ever before.

Web sites like ebay.com and priceline.com may be familiar as business-to-consumer online auctions. But networkedauctions can have a significant impact on business profitability; see [2] or [3]. Business-to-business (B2B) onlineauctions have demonstrated significant potential, from $109 billion worth of transactions in 1999 to an expected $2.7trillion worth of transactions in 2004 [4]. Online auctions may save businesses both time and money in purchasingor selling supplies; see [5]. Furthermore, the easy accessibility of online auctions permits greater market growth indepth and quality of market players; see generally [6–8]. This opportunity motivates the present research regardingthe automation of auction operations.

This paper considers auctions in which there are many buyers and many sellers. Both buyers and sellers submitsealed-bids of price for one unit of a homogenous good. Buyers or sellers may not simultaneously submit offers to buyand offers to sell. The bids are sealed, in the sense that buyers and sellers know only their own bid and not the bids ofother buyers or sellers. Buyers or sellers may purchase or sell multiple units by submitting multiple sealed-bids each

E-mail address: chmkelle@indiana.edu.

0895-7177/$ - see front matter c© 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2004.12.013

44 C. Keller / Mathematical and Computer Modelling 44 (2006) 43–48

for a single unit. Alternatively, one can consider unitary sealed-offers involving multiple units at the same price asmultiple sealed-offers for a single unit at identical prices. Specifically, “all-or-nothing” block sales of multiple unitsof the good are not considered. The possibility of a buyer or seller being able to submit multiple bids also admitsthe possibility of game-theoretic strategic behavior on the part of the buyer or seller in attempting to manipulate finaltransaction price. “Shills” is a game-theory analogue of such bidding behavior.

Ex post classical efficiency occurs in any k-double auction where the buyers and sellers each bid their respectivereservation value. However, if bids are viewed as functions of respective reservation price, admitting game-theoreticstrategic behavior, then buyers have an incentive to bid below their private reservation values and sellers have anincentive to ask above their private reservation values; see [9]. A Bayesian game of strategy can be formulatedwhere the asks and bids are functions of respective reservation prices [10]. The existence and nature of differentiableequilibrium strategies for any fixed value k were considered by [11]. A relevant discussion of ex ante efficiency isprovided in [12].

With private reservation values and individual incentives, there is no fixed-k trading mechanism that is ex anteefficient [13]. However, this paper considers the case when k is not fixed, but rather is determined partially from thedata and partially from ex ante presumptive knowledge of buyers’ and sellers’ price distributions. In this limited case,the variable-k double auction pricing mechanism considered is not only ex post classically efficient, but also ex anteBayesian incentive efficient.

This paper is organized as follows: Section 2 describes the standard reverse-sorting solution to determining marketprice in a double-auction market; Section 3 presents a novel model of a variable k-double auction and provides sometheoretical results; Section 4 demonstrates the ex ante Bayesian incentive compatability of the solution; Section 5 is afurther discussion and conclusion.

2. The double-auction market

Buyers and sellers submit sealed-bids of price for one unit of a homogenous good. Sealed-bids for buyers andfor sellers are organized as separate series. Buyers or sellers may not simultaneously submit offers to buy and offersto sell. A buyer’s sealed offer is termed a bid, and denoted B. A bid is a function of a buyer’s reservation value, aprice beyond which the buyer would be better off without the transaction. A seller’s sealed offer is termed an ask,and denoted A. An ask is a function of a seller’s reservation value, a price below which the seller would be better offwithout the transaction.

In the case of one buyer and one seller, trade only occurs if A ≤ B, and the market price is a value in the interval[A, B]. In order to generalize this result for multiple buyers and multiple sellers, let m and n represent the number ofsealed-bids for buyers (bids) and for sellers (asks), respectively. The asks and bids are independently ordered:

Ai i = 1, n such that A1 ≤ A2 · · · ≤ An

B j j = 1, m such that B1 ≥ B2 · · · ≥ Bm .

In order to maximize economic surplus, ex post classical efficiency, the optimal allocation of buyer’s bids to seller’sbids requires:

determine index l such that Al ≤ Bl and Al+1 > Bl+1.

The optimal price is in the interval [Al , Bl ]. For a simple introduction, see [14, Chapter 7]; many technicalvariations are noted in [15]:

An ExampleIndex 1 2 3 4 5 6

Asks (sellers) $10 $12 $15 $18 $25 $30Bids (buyers) $35 $30 $25 $22 $20 $15

There are four sales (indices 1–4). The optimal market price is in the interval [$18, $22]. If the market price isset at $20, then the buyers’ surpluses would be, respectively, $15, $10, $5, and $2; the sellers’ surpluses would be,respectively, $10, $8, $5, and $2. The total economic surplus is then $57. No single price outside of the market-price interval will yield a total economic surplus greater than or equal to the total economic surplus generated by anyprice within this market-price interval. Furthermore, any price within this market-price interval yields the same totaleconomic surplus.

C. Keller / Mathematical and Computer Modelling 44 (2006) 43–48 45

Although every price within this market-price interval yields the same total economic surplus, it is not the case thatthis economic surplus is allocated evenly between the buyers and the sellers. In the above example, if the price is setat $18, then the buyers’ surpluses are $17, $12, $7, and $4, for a total of $40 collectively for the buyers; the sellers’surpluses are $8, $6, $3, and $0, for a total of $17 collectively for the sellers. Total economic surplus remains at $57.Suppose instead that the price is set at the upper range of the interval, $22. In this case the buyers’ surpluses are $13,$8, $3, and $0, for a total of $24; the sellers’ surpluses are $12, $10, $7, and $4, for a total of $33. Total economicsurplus remains at $57.

3. The k-double auction

The k-double auction is a one-parameter set of rules for setting the price of trade between multiple buyers andmultiple sellers for a homogenous good. Buyers or sellers may not simultaneously submit offers to buy and offers tosell. In a sealed-offer k-double auction for one buyer and one seller for a homogenous good, the price of trade is set atk ∗ A+(1−k)∗ B, k ∈ [0, 1]. Novel results derived from modeling the double-auction using a mixture of distributionsare presented below.

The reverse-sorting algorithm of Section 2 that produces the optimal market-price interval is modeled upon a setof discrete bids. In order to generalize this result and to develop a variable k-double auction, bids are modeled as adiscrete sample from a mixture of statistical distributions:

a. Seller’s bids are a sample from a statistical distribution, F(x).b. Buyer’s bids are a sample from a statistical distribution, G(x).c. The proportion of seller’s bids to total bids is α, 0 < α < 1.d. The proportion of buyer’s bids to total bids is then 1 − α.e. A mixture of these distributions represents the price distribution:

α ∗ F(x) + (1 − α) ∗ G(x).

The variable k-double action market-price statistic is determined by solving a statistical formulation of the reverse-sorting algorithm:

Determine x such that α ∗ F(x) = (1 − α) ∗ (1 − G(x)). (1)

Let x∗ denote the market price that is determined. The left-hand side of Eq. (1) is the probability of a seller’s bidbeing accepted. F(x∗) is the unconditional probability of a seller’s bid being less than the market price. In order tobalance the number of sample sellers’ bids with the number of sample buyers’ bids, the sellers’ bids distribution isweighted, α ∗ F(x∗). Thus, the left-hand side is the probability of a seller’s bid being less than the market priceconditioned on the proportion of sellers’ bids to total bids in the market.

The right-hand side of the Eq. (1) is the probability of a buyer’s bid being accepted. 1 − G(x∗) is the unconditionalprobability of a buyer’s bid being greater than the market price. In order to balance the number of sample sellers’ bidswith the number of sample buyers’ bids, the buyer’s bids distribution is weighted, (1 − α) ∗ (1 − G(x∗)). Thus, theright-hand side is the probability of a buyer’s bid being greater than the market price conditioned on the proportion ofbuyers’ bids to total bids in the market.

The market-price statistic always has a solution. Rewrite Eq. (1) as

α ∗ F(x) − (1 − α) ∗ (1 − G(x)) = 0. (2)

For x = 0, the function in Eq. (2) has a value of α − 1 < 0. As x → ∞, the function approaches a value of α > 0.Since the function is non-decreasing, then there must exist an intermediate value x ∈ [0, ∞], such that Eq. (2) issatisfied.

If one further supposes that at least one of the statistical distributions is strictly increasing, then the market-pricestatistic solution is unique. Computationally, the monotonic nature of the equation admits that any local optimizationalgorithm (hill-climber) may be used to determine the solution.

Example results with the uniform distribution

If we assume that F(x) and G(x) are uniform distributions, then a closed form solution for the market-price statisticis given by:

46 C. Keller / Mathematical and Computer Modelling 44 (2006) 43–48

x∗=

αa(d − c) + (1 − α)d(b − a)

α(d − c) + (1 − α)(b − a)

where F(x) ∼ U [a, b] and G(x) ∼ U [c, d]. If the traded good is a commodity, then arbitrage may exist ifF(x) 6= G(x). If instead F(x) = G(x), then the solution reduces to a transparent form of the usual k-double auctionresult, x∗

= α ∗ a + (1 − α) ∗ b. If the additional assumption of equality of proportion between buyers and sellers isadded, then the general solution reduces to x∗

= (a + b)/2. This result is consistent with a model solution in [16].

4. Ex-ante Bayesian incentive efficiency

With private reservation values and individual incentives, there is no fixed k-trading mechanism that is ex anteefficient [13]. This paper considers the case when k is not fixed, but rather is determined partially from the data andpartially from ex ante presumptive knowledge of the buyer’s and seller’s price distributions. Buyers or sellers maynot simultaneously submit offers to buy and offers to sell. In this situation the k-double auction is not only ex postclassically efficient, but also ex ante Bayesian incentive efficient.

Consider the example with a single uniform distribution, U [a, b]. The market-price solution is x∗= α ∗ a +

(1 − α) ∗ b. If α is estimated at the number of asks (seller’s bids), nS, divided by the total number of asks and bids(seller’s and buyer’s bids), T , then a = nS/T . If any seller offers an additional ask, then a = (nS + 1)/(T + 1). As aconsequence, the new market price, x∗′, is no greater than the previous market price, x∗. That is,

x∗=

nS

Ta +

(1 −

nS

T

)b ≥

nS + 1T + 1

a +

(1 −

nS + 1T + 1

)b = x∗′.

Every offer by a seller can only reduce the market price. Similarly and symmetrically, every offer by a buyer canonly increase the market price. This result remains true if the single uniform distribution assumption is relaxed infavor of different uniform distributions for buyers and for sellers. That is,

x∗=

nST a(d − c) + (1 −

nST )d(b − a)

nST (d − c) + (1 −

nST )(b − a)

≥

nS+1T +1 a(d − c) + (1 −

nS+1T +1 )d(b − a)

nS+1T +1 (d − c) + (1 −

nS+1T +1 )(b − a)

= x∗′.

New seller’s offers reduce or leave unchanged market price. Similarly, and symmetrically, new buyer’s offersincrease or leave unchanged market price. This result is not merely an anomaly of the uniform distribution assumption.It is true generally. The market price is given for the commodity-based single distribution,

Determine x such that α ∗ F(x) = (1 − α) ∗ (1 − F(x))

which is equivalent to,

Determine x such that F(x) = 1 − α.

If α (the proportion of a seller’s offers to total offers) is estimated such that an additional seller’s offer increasesα, as it should, then 1 − α = F(x) must decrease. Since F(x) is a non-decreasing statistical distribution, thenthe new solution, x∗′, must also be reduced. So, no new offer by a seller can increase market price. Similarly, andsymmetrically, no new offer by a buyer can decrease market price.

Finally, the result remains true with different distributions for buyers and for sellers. The market price is given as

Determine x such that α ∗ F(x) = (1 − α) ∗ (1 − G(x)).

Collecting the α terms, this is equivalent to,

Determine x such that α(F(x) − G(x) + 1) = 1 − G(x).

If α (the proportion of a seller’s offers to total offers) is estimated such that a new seller’s offer increases α, thenthe left-hand side of the above inequality must increase. Thus, 1 − G(x) must increase, which implies G(x) mustdecrease. Since G(x) is a non-decreasing statistical distribution, x∗′ must also be reduced. So, no new offer by a sellercan increase market price. Similarly, and symmetrically, no new offer by a buyer can decrease market price.

C. Keller / Mathematical and Computer Modelling 44 (2006) 43–48 47

In addressing ex ante Bayesian efficiency, the above results demonstrate that neither a buyer nor a seller have anyincentive to make additional spurious bids other than actual bids. What remains to be shown is that the offers made bybuyers and sellers are not merely functions of their respective reservation prices, but the reservation prices themselves.

The ex ante expected profit for any seller’s reservation price is conditional on the strike price, x∗. If a transactionis realized, profit is the difference between the strike price and seller’s reservation price. In particular, the amount ofprofit is independent of the seller’s asking price. That is, under the present analysis, the seller’s asking price affectsonly the probability of obtaining a profit and not the amount of profit itself. Since any increase over the reservationprice by the seller decreases the probability that such an ask bid is less than the strike price, then the expected valuefor overbidding by sellers is not greater than zero. Thus, there is no ex ante incentive for seller’s to ask higher prices.Similarly, and symmetrically, there is no ex ante incentive for buyers to bid prices lower than their true reservationvalue. This result follows from the primary assumption that the price distributions for buyers and for sellers is knownex ante. This assumption is a significant restriction. In this limited case, this novel model demonstrates an originalcontribution to the standard fixed k-double auction market solution.

If the model is to be developed to the case when F(x) and G(x) must also be estimated from the data, then furthertheoretical development is necessary, but some possible extensions of this original research are discussed. The double-auction mechanism is extremely flexible and may be extended easily to repeated auctions. In the examples given inthe paper, the sealed-offers were of a set whose limiting inclusion was not specified. In practice, in a repeated auction,this limiting inclusion could be any number of conditions, including, for example, offers received during a specifiedtime or until a triggering number is reached. Thus, although not addressed directly, the model presented is inherentlygeneralizable to repeated auctions.

If the buyers’ and sellers’ distributions are estimated from sample data, this estimation re-introduces the possibilityof strategic bidding by both buyers and by sellers. Suppose, for example, that the distributions are assumed to be of atype of distribution which may be specified by the two parameters, µ and σ (the mean and standard deviation). In thiscase, any sealed offer then affects the proportion of buyers and sellers in the same way as before. But inclusion of ahigh-value offer in the parameter estimates will tend to increase the estimated value for both µ and σ . As is clear fromthe model, changes in distributional parameters ultimately will affect the strike price. Thus, ex ante buyers and sellersmay have an incentive to bid at prices different to their reservation values. As a practical matter, in current major stockmarkets, for example, order book information containing asks and bids information is proprietary and generally notsubject to disclosure, in part ameliorating strategic bidding.

Once the possibility of false-bidding incentive is created, then a second problem is already introduced, especiallyin the network context. The second problem then is the overwhelming submission of large numbers of strategic bidsdesigned only to manipulate the trading price. For example, trades are matched up for sellers beginning with the lowestbid and proceeding to the highest bid. Spurious sales bids may be introduced “offering” to sell only at extremely highvalues. These spurious bids move the sellers’ distribution towards a higher mean and, consequentially, the strike pricetowards a higher value. Yet, because of the very high value of the sales bid, it is the case that the probability of itever being matched to a corresponding value is essentially zero. Therefore, there is no feature presented in the presentmodel constraining disincentive to limit such behavior. Such strategic behavior, however, might be addressed byeither “cut-off” values for bids or trading as, for example, the volatility trading stops implemented in current markets.Alternatively, if there is a fixed cost associated with making offers, or perhaps unaccepted offers, then, at some levelof volume, the marginal effect on strike price from a spurious bid is outweighed by the offer’s cost. Thus, in a thickmarket, such strategic behavior may be constrained.

The implementation issues briefly discussed above present abundant opportunities for future research. The actualoperation of the variable-k double-auction mechanism is being considered in current experimental economicsresearch; see [17].

5. Discussion and conclusion

The market-price statistic as described avoids a fixed k-double auction in favor of a variable k-double auction. Thisvariability avoids the arbitrariness of fixed k-double auctions and also provides a Bayesian-incentive compatibilityfor reservation prices in the limited case when the reservation price distributions are known. A general form for thedistribution of the market-price statistic is unknown. Furthermore, the distribution of the market-price statistic is likelydifferent, depending upon the particular bid distributions.

48 C. Keller / Mathematical and Computer Modelling 44 (2006) 43–48

This paper has demonstrated a novel market-price statistic that is useful in modeling the solution obtained by thereverse-sorting auction prices. This statistic is predicated on the assumptions that buyers’ prices and sellers’ pricesarise from separate distributions and that market price may be modeled as a mixture of the buyers’ and the sellers’price distributions. No general form for the distribution of the market-price statistic is available, since the distributionmay depend upon the underlying modeling assumptions of buyers’ prices and sellers’ prices.

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