QR 38, 3/1/07 and 3/8/07 Mixed strategies Keeping the opponent indifferent Finding a mixed strategy equilibrium Odds rat

1. QR 38, 3/1/07 and 3/8/07 Mixed strategies Keeping the opponent indifferent Finding a mixed strategy equilibrium Odds ratios Mixing in practice

2. I. Keeping the opponent indifferent • Some games have no pure-strategy equilibrium. Instead, choices will keep cycling. • Example: Attack-defend situations, where one player prefers to coordinate and the other to pick different options.

3. Lack of pure strategy equilibrium • Another example: tennis. Walker-Wooders reading. • If serve is predictable (forehand or backhand), will lose. This is true for many sports. • So, you can be exploited if you behave predictably; knowing how the other will behave is always a benefit.

4. Randomization • To prevent exploitation, have to keep your opponent guessing. • So, when no pure strategy equilibrium, have to randomize; use a mixed strategy.

5. Mixed strategies • Mixed strategy: your actual move will be chosen randomly from the set of pure strategies with a specified probability. • I.e., you use each pure strategy a certain percent of the time. • Actually need to randomize to get to an equilibrium; can’t just alternate.

6. Mixed strategies • Every simultaneous-move game has a Nash equilibrium in pure or mixed strategies. • D&S present method for finding a mixed-strategy equilibrium in zero-sum games; we will just go to the general case. • Limit focus to 2x2 games.

7. II. Finding a mixed-strategy equilibrium • Consider zero-sum game. Solve for p and q.

8. Steps in finding equilibrium • Payoffs used to find mixed strategies must be cardinal, not just a ranking (ordinal). • Elements of finding a mixed-strategy equilibrium: • Expected values • Best-response curves • Upper envelope • Value of game = payoff in equilibrium

9. Calculating p and q • Row plays up with probability p, down with probability 1-p. • Column plays left with probability q, right with probability 1-q. • Equilibrium means that your opponent has no incentive to change his strategy • That means he is indifferent between his two pure strategies.

10. Calculating p and q • So, Row chooses p so that Column is indifferent between left and right. • Column chooses q so that Row is indifferent between up and down. • So to find p and q, calculate Column’s expected payoff as a function of p, and Row’s expected payoff as a function of q.

11. Calculating p and q • C’s payoff for choosing left = 3p+(1-p) • C’s payoff for choosing right = 2p+4(1-p) • For R to make C indifferent, set these equal to one another and solve for p: • 3p+1-p = 2p+4-4p • 2p+1=4-2p • 4p=3 • p=3/4 • If R chooses p=3/4, C will be indifferent between left and right.

12. Calculating p and q • R’s payoff for choosing up = 2q+3(1-q) • R’s payoff for choosing down = 4q+(1-q) • C will choose q to make R indifferent between these two pure strategies: • 2q+3-3q = 4q+1-q • 3-q = 3q+1 • 4q=2 • q=1/2 • So, to indicate the equilibrium: R will play up with probability ¾, down with probability ¼; C will play left with probability ½ and right with probability ½.

13. Calculating p and q • What we have done is to calculate each player’s expected payoff of each pure strategy as a function of the other player’s probability choice • Then set the expected payoff of the pure strategies equal to one another • And solve for the other player’s probability choice (p and q)

14. Best response curves • Can show the same logic by looking at best-response curves • Show each player’s payoff to each pure strategy as a function of the other player’s probability choice • This will produce two lines • The “upper envelope” created by these two lines shows the best response to the other player’s choice

15. Best response curves • Take the two players’ best response curves and plot them on the same graph. • Where the two lines intersect is the equilibrium; each player’s strategy here is a best response to the other’s. • This will give the same value of p and q as above.

16. Mixed strategy equilibrium • A mixed-strategy equilibrium involves keeping your opponent indifferent between using her pure strategies. • If she weren’t indifferent, she would have an incentive to shift her strategy, so it wouldn’t be an equilibrium. • So, to find a mixed-strategy equilibrium, set the expected values of different pure strategies to be equal, and solve for the probabilities of each move.

17. Mixed strategy equilibrium • Note that since p and q are probabilities, their values must be between 0 and 1. If they don’t fall in this range, check your math – or maybe there are only pure strategy equilibria.

18. Value of the game • The value of the game to each player is their expected payoff in equilibrium. • C’s expected payoff = 3pq+1(1-p)q+ • 2p(1-q)+4(1-p)(1-q) • = 3(3/4)(1/2)+1(1/4)(1/2)+2(3/4)(1/2)+ • 4(1/4)(1/2) • = 9/8+1/8+6/8+4/8 • = 2 ½ • Value of the game for C is 2 ½

19. Value of the game • Value of the game for R = 2pq+4(1-p)q+3p(1-q)+1(1-p)(1-q) • =2(3/4)(1/2)+4(1/4)(1/2)+3(3/4)(1/2) • +1(1/4)(1/2) • =6/8+4/8+9/8+1/8 • =2 ½

20. USTR-Japan trade game

21. Equilibrium in US-Japan game • Calculate expected payoff to US for J’s choice: • If US chooses permit: p(.5)+(1-p)0 = .5p • If US chooses restrict: p(0)+(1-p)1 = 1-p • J will choose p to make US indifferent between these two: .5p=1-p • 1.5p=1 • p=1/1.5=2/3

22. Equilibrium in US-Japan game • Calculate expected payoff to J for US choice: • If J chooses J: q(1)+(1-q)0=q • If J chooses US: q(0)+(1-q)(.5)=.5-.5q • US will choose q to make J indifferent between these two pure strategies: • q=.5-.5q • 1.5q=.5 • q=.5/1.5=1/3

23. Best response curves • Graph payoffs as function of other player’s choice • Upper envelope = best response • Equilibrium at the intersection of the best response curves. • Note that there are 3 points of intersection: 0, 0; 1,1; and 2/3, 1/3 • This is because there are also two pure strategy equilibria in the game

24. US-Japan game • Best-response curves show that pure strategies are the best response except at the point where these two curves intersect; this is the mixed-strategy equilibrium. • If coordinated randomization were possible (e.g., alternating), could improve US expected payoff from 1/3 to 2/3. • Value of the game = 1/3 for US, 1/3 for Japan (on board)

25. III. Odds method • Odds method: a general formula for calculating mixed strategies. • Odds = the chances of one of two events occurring, for example odds win:lose. • For some people, thinking in terms of odds is more intuitive • The odds method has to give the same answer as the indifference method!

26. Odds method • If the odds of winning:losing are 2:1, then the probability of winning is 2/3. • Add the two “oddments” together to find the denominator. • Odds W:L; win W/(W+L) proportion of the time, lose L/(W+L)

27. Odds method

28. Odds method • Row’s expected value for up: qa+(1-q)b • R’s expected value for down: qc+(1-q)d • Using the indifference method, set these equal to one another: • qa+(1-q)b=qc+(1-q)d • Translate this expression to the odds of q:(1-q) • q(a-c)=(1-q)(d-b); q/(1-q)=(d-b)/(a-c) • q:(1-q)=(d-b):(a-c)

29. Odds method • C’s expected payoff for choosing left = pA+(1-p)C • C’s expected payoff for choosing right = pB+(1-p)D • Set these equal to one another: • pA+(1-p)C = pB+(1-p)D • Translate to the odds of p:(1-p) : • p(A-B)=(1-p)(D-C); p/(1-p)=(D-C)/(A-B) • p:(1-p)=(D-C):(A-B)

30. Applying odds method to US-Japan game • p:(1-p) = (1-0):(.5-0) • = 1:.5 • = 2:1 (so p=2/3) • q:(1-q) = (.5-0):(1-0) • = .5:1 • = 1:2 (so q=1/3)

31. Odds method • Note that p and q are probabilities, and so must be positive numbers. • This means that (D-C) must have the same sign as (A-B); and (d-b) must have the same sign as (a-c). • If this isn’t the case, no mixed strategy equilibrium; there must be a pure strategy equilibrium.

32. IV. Mixing in practice • What does mixing mean in practice? Have to have a device that randomizes for you. • Walker-Wooders, distribution of serves in tennis. Should be random. Do see equal probabilities of winning to each side, as expected. But players “switch” too often to be truly random.

33. Mixing in practice • Schelling uses the example of a threat that leaves something to chance; brinksmanship. • But he doesn’t really look at a mixed-strategy equilibrium. • When mixing in a coordination game, you run the risk of arriving at an uncoordinated outcome. So the payoff is generally less than from a pure strategy equilibrium. • But you have to take this chance of an occasional poor outcome to get the highest expected value; if your opponent is playing a mix, your best response is to play a mix as well.

34. Linkage strategy • What if, in the US-J game, US were able to use a linkage strategy? • A linkage strategy changes the payoffs to Japan if it builds in Japan. • Show this by subtracting x from J’s payoffs. • This changes the mixed-strategy equilibrium in a counterintuitive way: the United States now permits more imports. Why?

35. U.S.-Japan game with linkage

36. Linkage game, US equilibrium strategy • q:1-q= (d-b):(a-c) • = .5-(-x):(1-x)-0 • = .5+x:1-x • If x=.1, then q:1-q=.6:.9; 2:3 • So q=2/5. • Above q=1/3; it has now increased. U.S. is more likely to permit imports.

37. Linkage game • Why is U.S. now more likely to permit imports? • Remember that the point of a mixed strategy is to keep the other player indifferent. • Now Japan gets a lower payoff from building in Japan, so to keep Japan indifferent U.S. has to be more likely to allow imports.

38. Linkage game • Understanding how linkage changes the equilibrium U.S. strategy requires that U.S. anticipate how Japan will react. • Japan is now more likely to do what the U.S. wants, so U.S. doesn’t have to restrict as much to keep Japan indifferent.

39. Value of the linkage game • U.S. value = (2/3)(2/5)(1/2) + (1/3)(3/5)(1) • = 2/15 + 3/15 • = 1/3 (no change) • Japan value = (2/3)(2/5)(9/10) - • (1/10)(2/3)(3/5) + (1/2)(1/3)(3/5) • = 6/25-1/25+1/10 • = 15/50 • = 3/10 (<1/3, decreased)

40. Value of the linkage game • So value to US hasn’t changed, value to Japan has decreased even though US is more likely to allow imports • This illustrates that in order to determine whether a change in payoffs benefits a player, we have to look at changes in strategy and how this affects the value of the game.

41. Mixed strategies in IR • Would a state ever really play a mixed strategy? • What does this mean in practice – do states really randomize? • Schelling suggests that they do, even if not consciously, because the process of decisionmaking leaves something to chance. • Techniques like tripwires can also leave something to chance.

42. Analyzing power • Power is defined as the ability to make another player do something he wouldn’t do, and doesn’t want to do. • Portray power as changing the payoffs associated with a particular strategy.

43. Analyzing coordination and distribution • Coordination games have multiple equilibria • With mixed strategies added to the set of strategies, can see that multiple equilibria will often emerge; and some equilibria leave all players unhappy.

44. Coordination and distribution • Distributional conflict arises when players disagree over which of the equilibria they prefer (battle of the sexes, chicken). • Then, need a signal like a focal point to identify a particular equilibrium. • Focal points create common conjectures.

45. Coordination and distribution • Where do focal points come from? • BdM: international institutions; especially effective in pure coordination situations like traffic rules. • Note BdM’s discussion of power and motivation: interesting but an example of decision-theoretic, not game-theoretic, reasoning.