Dominance

1. Dominance

2. 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.

3. An Example – Prisoner’s Dilemma • 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

4. An Example – Prisoner’s Dilemma • 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

5. 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.

6. Solving Using Dominance • 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.

7. 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.

8. 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.

9. 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

10. 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

11. 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

12. 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?

13. 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.

14. 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?

15. 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.

16. 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.

17. 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

18. 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

19. 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.

20. 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?

21. 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

22. 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

23. Graphically – Bid Shading Profit If I shade down my bid, this is my profit profile v My bid Highest rival bid v

24. Graphically – Bidding Above Value Profit If I shade up my bid, this is my profit profile v My bid Highest rival bid v

25. Graphically – Bid = Value Profit If bid=value, this is my profit profile v Highest rival bid v My bid

26. 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

27. 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

28. 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.

29. 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)

30. 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!

31. 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.

32. 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?

33. 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?

34. 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