Dominance. Overview In this unit, we explore the notion of dominant strategies Dominance often...
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Transcript of Dominance. Overview In this unit, we explore the notion of dominant strategies Dominance often...
Dominance
Overview
In this unit, we explore the notion of dominant strategies
Dominance often requires weaker views of rationality than does standard equilibrium play
These weaker rationality requirements support choice of equilibria satisfying dominance over other equilibria.
An Example – Prisoner’s Dilemma
Cooperate Defect
Cooperate 3, 3 0, 4
Defect 4, 0 1, 1
In this game, it pays to defect regardless of the rival’s strategy
Defect is a best response to cooperate
Defect is a best response to defect
An Example – Prisoner’s Dilemma
Cooperate Defect
Cooperate 3, 3 0, 4
Defect 4, 0 1, 1
In the language of dominance: The cooperate strategy
is strictly dominated by defect
This means that the defect strategy gives strictly higher payoffs for Rowena than does cooperate
Rationality
Rationality axiom 1: Never play a strictly dominated strategy regardless of your opponent
Why? Even if you have serious doubts about the
rationality of the other player. A dominated strategy does strictly worse than
some other strategy… regardless of your rival’s play So it should be avoided.
Solving Using Dominance
Cooperate Defect
Cooperate 3, 3 0, 4
Defect 4, 0 1, 1
In the prisoner’s dilemma, we can solve the game purely by eliminating dominated strategies
Since this elimination leaves each side only one undominated strategy, this pair constitutes an equilibrium.
Team Production
Both the design and the production departments are required to produce some saleable output.
The quality of the output determines the price for which it can be sold.
For each unit of effort undertaken by either team, up to 10 units, profits increase by $1.5million/unit. After that, it does not increase.
Costs of Effort
It costs $1million per unit of effort in either department
Effort is unobservable by management To compensate design and production,
management has instituted a profit sharing plan whereby production and design each get one-third of the profits as compensation.
Optimal Effort
From the perspective of the firm as a whole, each unit of effort up to 10 taken by design and production costs only $1m and has a return of 50% Therefore from the firm’s perspective each
department should exert 10 units of effort
Equilibrium Effort
Notice that the design team needs to determine its level of effort not knowing the choice of the production team.
What are its profits if design chooses effort e1 and production chooses e2?
Profit1 = Profit share – Cost of effort
Profit1 = (1/3)(1.5e1 + 1.5e2) – e1
Equilibrium Effort Continued
Profit1 = (1/3)(1.5e1 + 1.5e2) – e1
Notice that regardless of e2, Profit1 is decreasing in e1
So any choice e1 > 0 is dominated by e1 = 0. Hence design exerts no special effort despite
the profit sharing incentives The situation for production is analogous The conclusion is that both production and
design will try to free ride off the efforts of the other and no effort will occur
Solving the Free Rider Problem
Free rider problems appear in numerous settings
Devising incentive schemes to solve these problems is critical
What was wrong with the profit sharing scheme?
Bonuses
Suppose that instead of doing a straight profit sharing arrangement, the firm uses a bonus system to compensate design and production.
Recall that if production were efficient, profits would be $30m and the profit share gave away 2/3rds of this amount or $20m.
Instead, suppose that the firm pays each team a bonus of $10m + $1 if they reach the profit target of $30m.
Equilibrium Analysis
Suppose that design expects production to work all-out to meet the target.
To receive the bonus, design has to work all-out too.
If it doesn’t, then the analysis is as it was before but without even the profit sharing incentive---therefore design either works all-out or not at all.
How do these situations compare?
Design Choices
If design doesn’t work, it earns zero If they works all-out, profits equal the bonus
less the cost of effort, which nets design $1. Thus, it is better to work all-out than not at all,
so a best response to production’s working all-out is for design to do likewise
Bottom line: The structure of incentive schemes (as well as the total amount) can have a big effect on free-rider problems.
Iterative Elimination
Recall that rationality axiom #1 prescribed that it was never a good idea to play a dominated strategy
If you have some confidence of your rival’s rationality, you might be willing to assume that she follows this axiom as well.
This suggests that you should eliminate her dominated strategies in thinking about the game.
Dominance Solvable Games
To use dominance to solve a game: Delete dominated strategies for each of the
players Look at the smaller game with these strategies
eliminated Now delete dominated strategies for each side
from the smaller game Continue this process until no further deletion
is possible If only single strategies remain, the game is
dominance solvable
More on Dominance Solutions
Not all games are dominance solvable If after elimination, a small set of strategies
remain for each player These strategies survive iterative dominance
and are relatively more robust than others
Weak Dominance
To eliminate a strategy as being dominated, we required that some other strategy always be better no matter the rival’s move
Suppose we weaken this: A strategy is weakly dominated if, no matter
what the rival does, there is some strategy that does equally well and sometimes strictly better.
Auctions
eBay and a number of other online auctions use “proxy bidding” rules
Under a proxy bid, you enter a bid amount, but what you pay is determined by the second highest bid plus a small increment.
Suppose that you know your willingness to pay for an item for sale on eBay.
What should you bid?
A Model of eBay
There’s a lot of “sniping” on eBay Sniping is where bidders wait for the last possible
instant to bid In that case, there is little feedback about other bids at
the time you place your bid Think of the following version of the eBay game
There are an unknown number of potential bidders You know your value, but know little about other
bidders (including their rationality or their valuations) All bidders choose bids simultaneously
High bid wins Pays second highest bid
Bidder’s Problem
How should you bid in this auction? It turns out that eliminating weakly dominated
strategies provides an answer regardless of your rival’s choice
Graphically – Bid Shading
Highest rival bidv
Profit
v
My bid
If I shade down my bid, this is my profit profile
Graphically – Bidding Above Value
Highest rival bidv
Profit
v
My bid
If I shade up my bid, this is my profit profile
Graphically – Bid = Value
Highest rival bidv
Profit
v
My bid
If bid=value, this is my profit profile
Comments
Notice that when bid = value I win in all the cases when bid < value And in some cases where I lost earlier. Moreover, these cases are profitable
Notice that when bid = value I win in fewer cases than when bid > value But I made losses in all the cases where I won
when bid > value Therefore I’m better off losing then
Weak Dominance
Therefore: Bid = value
Does at least as well as all other strategies in many cases
And strictly better in some cases So all other strategies are weakly dominated by
bid = value
So we can use weak dominance (one round of deletion) to find the best strategy in this auction
Case Study: Tender Offers
A frequent strategy among corporate raiders in the 80s was the two-tiered tender offer. Suppose the initial stock price is $100. In the event that a firm is taken private,
shareholders get $90 per share. Campeau will buy shares a $105 for the first
50%, and $90 for the remainder.
Tenders...
All shares are bought at the blended price of totals tendered. For instance, if z%>50% of shares are
tendered, then the price is P =$105 x (50/z) + $90 x ((z-50)/z) P = $90 + $15 x (50/z)
Details
Notice that the tender is a binding agreement to purchase shares regardless of the success of the takeover.
Second, notice that if everyone tenders, the raider pays: P = $90 + $15 x (50/100) = $97.50
which is cheaper than the initial price of the stock!
Dominance of the tender
What is less obvious is that it is a dominant strategy to accept the tender: Three cases to consider:
z>50. Then P = $90 + $15 x (50/z) > 90 z<50. Then P = $105 > 100 z=50. Then P = $105 > 100 or 90
So it is a dominant strategy to sell your shares.
A White Knight
Suppose Warren Buffet offers to buy all shares at $102 conditional on getting a majority.
Does this undo the two-tiered offer strategy?
Dominance revisited
Again, consider the 3 cases: z < 50. P = $105 vs $100 or $102. z > 50. P = 97.50 vs $90 z = 50. P = 105 vs $100 or $102.
Is there any way to undermine the two-tiered deal?
Summary
Rationality Axiom: Don’t play dominated strategies
As your confidence about the rationality of your opponent grows, can iteratively delete dominated strategies to arrive at a good plan
Deletion of weakly dominated strategies can give clarity in even complicated situations