1. Mixed Strategies

2. Overview Principles of mixed strategy equilibria Wars of attrition All-pay auctions

3. Tennis Anyone S R

4. Serving S R

5. S R

6. The Game of Tennis Server chooses to serve either left or right Receiver defends either left or right Better chance to get a good return if you defend in the area the server is serving to

7. Game Table Receiver Server LeftRight Left¼¾ Right¾¼

8. Game Table Receiver Server LeftRight Left¼¾ Right¾¼ For server: Best response to defend left is to serve right Best response to defend right is to serve left For receiver:Just the opposite

9. Nash Equilibrium Notice that there are no mutual best responses in this game. This means there are no Nash equilibria in pure strategies But games like this always have at least one Nash equilibrium What are we missing?

10. Extended Game Suppose we allow each player to choose randomizing strategies For example, the server might serve left half the time and right half the time. In general, suppose the server serves left a fraction p of the time What is the receiver’s best response?

11. Calculating Best Responses Clearly if p = 1, then the receiver should defend to the left If p = 0, the receiver should defend to the right. The expected payoff to the receiver is: p x ¾ + (1 – p) x ¼ if defending left p x ¼ + (1 – p) x ¾ if defending right Therefore, she should defend left if p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾

12. When to Defend Left We said to defend left whenever: p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾ Rewriting p > 1 – p Or p > ½

13. Receiver’s Best Response p Left Right ½

14. Server’s Best Response Suppose that the receiver goes left with probability q. Clearly, if q = 1, the server should serve right If q = 0, the server should serve left. More generally, serve left if ¼ x q + ¾ x (1 – q) > ¾ x q + ¼ x (1 – q) Simplifying, he should serve left if q < ½

15. Server’s Best Response LeftRight ½ q

16. Putting Things Together ½ q p S’s best response 1/2 R’s best response

17. Equilibrium ½ q p S’s best response 1/2 R’s best response Mutual best responses

18. Mixed Strategy Equilibrium A mixed strategy equilibrium is a pair of mixed strategies that are mutual best responses In the tennis example, this occurred when each player chose a 50-50 mixture of left and right.

19. General Properties of Mixed Strategy Equilibria A player chooses his strategy so as to make his rival indifferent A player earns the same expected payoff for each pure strategy chosen with positive probability Funny property: When a player’s own payoff from a pure strategy goes up (or down), his mixture does not change

20. Generalized Tennis Receiver Server LeftRight Lefta, 1-ab, 1-b Rightc, 1-cd, 1-d Suppose c > a, b > d Suppose 1 – a > 1 – b, 1 - d > 1 – c (equivalently: b > a, c > d)

21. Receiver’s Best Response Suppose the sender plays left with probability p, then receiver should play left provided: (1-a)p + (1-c)(1-p) > (1-b)p + (1-d)(1-p) Or: p >= (c – d)/(c – d + b – a)

22. Sender’s Best Response Same exercise only where the receiver plays left with probability q. The sender should serve left if aq + b(1 – q) > cq + d(1 – q) Or: q <= (b – d)/(b – d + a – b)

23. Equilibrium In equilibrium, both sides are indifferent therefore: p = (c – d)/(c – d + b – a) q = (b – d)/(b – d + a – b)

24. Minmax Equilibrium Tennis is a constant sum game In such games, the mixed strategy equilibrium is also a minmax strategy That is, each player plays assuming his opponent is out to mimimize his payoff (which he is) and therefore, the best response is to maximize this minimum.

25. Does Game Theory Work? Walker and Wooders (2002) Ten grand slam tennis finals Coded serves as left or right Determined who won each point Tests: Equal probability of winning Pass Serial independence of choices Fail

26. Battle of the Sexes Chris Pat OperaFights Opera3,10,0 Fights0,01,3

27. Hawk-Dove Krushchev Kennedy HawkDove Hawk0, 04, 1 Dove1, 42, 2

28. Wars of Attrition Two sides are engaged in a costly conflict As long as neither side concedes, it costs each side 1 per period Once one side concedes, the other wins a prize worth V. V is a common value and is commonly known by both parties What advice can you give for this game?

29. Pure Strategy Equilibria Suppose that player 1 will concede after t 1 periods and player 2 after t 2 periods Where 0 < t 1 < t 2 Is this an equilibrium? No: 1 should concede immediately in that case This is true of any equilibrium of this type

30. More Pure Strategy Equilibria Suppose 1 concedes immediately Suppose 2 never concedes This is an equilibrium though 2’s strategy is not credible

31. Symmetric Pure Strategy Equilibria Suppose 1 and 2 will concede at time t. Is this an equilibrium? No – either can make more by waiting a split second longer to concede Or, if t is a really long time, better to concede immediately

32. Symmetric Equilibrium There is a symmetric equilibrium in this game, but it is in mixed strategies Suppose each party concedes with probability p in each period For this to be an equilibrium, it must leave the other side indifferent between conceding and not

33. When to concede Suppose up to time t, no one has conceded: If I concede now, I earn –t If I wait a split second to concede, I earn: V – t –  if my rival concedes – t –  if not Notice the –t term is irrelevant Indifference: (V –  x (f/(1 – F)) = -  x (1 – f/(1-F)) f/(1 – F) = 1/V

34. Hazard Rates The term f/(1 – F) is called the hazard rate of a distribution In words, this is the probability that an event will happen in the next moment given that it has not happened up until that point Used a lot operations research to optimize fail/repair rates on processes

35. Mixed Strategy Equilibrium The mixed strategy equilibrium says that the distribution of the probability of concession for each player has a constant hazard rate, 1/V There is only one distribution with this “memoryless” property of hazard rates That is the exponential distribution. Therefore, we conclude that concessions will come exponentially with parameter V.

36. Observations Exponential distributions have no upper bound---in principle the war of attrition could go on forever Conditional on the war lasting until time t, the future expected duration of the war is exactly as long as it was when the war started The larger are the stakes (V), the longer the expected duration of the war

37. Economic Costs of Wars of Attrition The expectation of an exponential distribution with parameter V is V. Since both firms pay their bids, it would seem that the economic costs of the war would be 2V Twice the value of the item???? But this neglects the fact that the winner only has to pay until the loser concedes. One can show that the expected total cost if equal to V.

38. Big Lesson There are no economic profits to be had in a war of attrition with a symmetric rival. Look for the warning signs of wars of attrition

39. Wars of Attrition in Practice Patent races R&D races Browser wars Costly negotiations Brinkmanship

40. All-Pay Auctions Next consider a situation where expenditures must be decided up front No one gets back expenditures Biggest spender wins a price worth V. How much to spend?

41. Pure Strategies Suppose you project that your rival will spend exactly b < V. Then you should bid just a bit higher Suppose you expect your rival will bid b >=V Then you should stay out of the auction But then it was not in the rival’s interest to bid b >= v in the first place Therefore, there is no equilibrium in pure strategies

42. Mixed Strategies Suppose that I expect my rival will bid according to the distribution F. Then my expected payoffs when I bid B are V x Pr(Win) – B I win when B > rival’s bid That is, Pr(Win) = F(B)

43. Best Responding My expected payoff is then: VF(B) – B Since I’m supposed to be indifferent over all B, then VF(B) – B = k For some constant k>=0. This means F(B) = (B + k)/V

44. Equilibrium Mixed Strategy Recall F(B) = (B + k)/V For this to be a real randomization, we need it to be zero at the bottom and 1 at the top. Zero at the bottom: F(0) = k/V, which means k = 0 One at the top: F(B 1 ) = B 1 /V = 1 So B 1 = V

45. Putting Things Together F(B) = B/V on [0, V]. In words, this means that each side chooses its bid with equal probability from 0 to V.

46. Properties of the All-Pay Auction The more valuable the prize, the higher the average bid The more valuable the prize, the more diffuse the bids More rivals leads to less aggressive bidding There is no economic surplus to firms competing in this auction Easy to see: Average bid = V/2 Two firms each pay their bid Therefore, expected payment = V, the total value of the prize.

47. Big Lesson Wars of attrition and all-pay auctions are a kind of disguised form of Bertrand competition With equally matched opponents, they compete away all the economic surplus from the contest On the flipside, if selling an item or setting up competition among suppliers, wars of attrition and all-pay auctions are extremely attractive.