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