Chapter 29 Game Theory

1. Chapter 29 Game Theory • Key Concept: Nash equilibrium and Subgame Perfect Nash equilibrium (SPNE)

2. Chapters 29, 30 Game Theory, Game Applications • A good time to talk about game theory since we have actually seen some types of equilibria last time.

3. Game theory is concerned with the general analysis of strategic interaction. • It can be used to study parlor games, political negotiation, and economic behaviors.

4. Normal form games: Prisoner’s dilemma.

5. This game has a strictly dominant strategy. • A dominant strategy is one which is optimal no matter what the other player does. • Once a player has a strictly dominant strategy, the task of predicting what other will do (the most interesting part in game) becomes unnecessary.

6. We may not be this lucky to have a strictly dominant strategy all the time. • But we might have a strictly dominated strategy. • So at least we can say we won’t play strictly dominated strategies.

8. One way to “solve” the game above is to use iterated elimination of strictly dominated strategies.

9. So our story may go like, 1 is rational, so 1 will not play M. Given everyone knows this, 2 is rational, 2 will not play R. Given everyone knows this, 1 is rational, 1 will not play U. So (D, L) becomes our prediction. This process uses some common knowledge of rationality.

10. But, are you really a person who will play according to this logic of iterated elimination of strictly dominated strategies? • Will you get the highest payoff if you do? • Talk a bit about the games we played before and the cognitive hierarchy theory.

11. Despite of this, theorists still feel that iterated elimination does not give sharp predictions.

13. In battle of sexes game, iterated elimination does not help. • A stronger solution, called Nash equilibrium, is often used.

14. The concept of Nash equilibrium is this. • A strategy profile (so every player has a part) is a Nash equilibrium, if given your opponent plays his equilibrium strategy, you play yours is a best response to that. • In other words, players are best responding to each other.

15. Equivalently, we can say that given your opponent is playing equilibrium, you have no profitable deviation.

16. That is exactly why we are looking at the reaction function in Cournot competition. • A reaction function is simply a function that gives you the best response to every possibly plays of your opponent. • We then intersect the two reaction functions to get Nash equilibrium where both are best responding to each other.

17. In the battle of sexes game, there are two pure Nash equilibria, (ballet, ballet) and (football, football).

18. Notice that in Nash equilibrium, a player’s is checking that he is best responding to his opponent’s play. • In other words, implicitly we are assuming that a player’s belief is correct, or consistent with opponent’s play. • So Nash equilibrium has two important components: best responding and correct beliefs.

19. There are some interesting empirical works supporting Nash equilibrium. • For instance, football players in penalty kicks.

20. Yet there are also reasons sometimes people don’t feel comfortable playing Nash equilibrium. • Or sometimes we can even coordinate better than Nash. • Look at the coordination games we played.

22. In the literature, there is a well-recognized “problem” with Nash equilibrium about empty threats.

23. Suppose in the first class meeting I came in and announced: “I want total dedication from my students, so you should drop all the other courses and just take my course. This will guarantee you spend all the efforts in my class. If you do not do so, I will not let you take the final and you will fail.”

24. Now it is a sequential game. You first decide whether you will drop other courses and then if you do not, I decide whether I will bar you from the final.

25. My threat of not letting you take the final seems not credible (an empty threat) since at the final, what is the point of baring you from exam if my purpose is to increase your efforts in the course and efforts are done?

26. Moreover, NTU is a serious institution. So you expect that at the final, I will not carry out the threats and hence you will not drop other courses. • That is a NE.

27. Now another Nash equilibrium looks like this. You take my threats seriously and you drop all other courses. Since you drop all courses I don’t need to carry out my threats of barring you from the exam. So my empty threat is never tested. I am very happy with the outcome. And you, given your pessimistic belief, are also doing what is the best for yourself. This does not seem very reasonable. So we want to rule out NE of this kind.

28. This brings us to Subgame Perfect Nash equilibrium (SPNE). • A subgame is just like a game, so let us talk a bit about game trees.

29. Drop ---- (2, 5) Don’t Drop -----Bar (0, -5) -----Don’t Bar (4, 2) • So for SPNE, we solve backwards.

30. Talk about the sheep-lion story. • The Stackelberg equilibrium is an SPNE.

31. We are now equipped with NE and SPNE, it seems enough so what else can theorists do? • There are still lots of things. • For one, look at finite repetition of PD game. Defect all the time is the unique SPNE. But do you really believe in that?

32. One potential answer is to appeal to infinite horizon. • Last time when we talked about repeated Cournot. With repeated interactions, as long as the future is important enough, we might be afraid of future punishment, which makes current cooperation possible. • This is the result of Folk theorem.

33. Another interesting direction is, do we really believe that people are so rational?

34. Can not-so-rational players play out equilibrium? • Talk a bit about traffic and social conventions.

36. Some points from Chapter 30 • Mixed Strategies • We have learned that the equilibrium is in mixed strategies in the penalty kick games in soccer.

37. Now we can formalize how to find a mixed equilibrium. • For illustration purpose, let us solve for the equilibria of the battle of the sexes.

40. There are many interesting games capturing different strategic interaction situations. • Let us just look at some.

41. The Hawk-dove game • If two wild dogs come across a piece of food, they can fight or share. Fighting is hawkish, sharing is dovish.

43. Obviously it cannot be an equilibrium if everyone plays hawk. And if all play dove, someone wants to deviate and play hawk. • If we have a population of wild dogs, could we expect some mixture of hawk types and dove types in equilibrium?

44. Suppose the fraction playing hawk is p. Then the expected payoff of a hawk isH=-2p+4(1-p). • The expected payoff of a dove is 2(1-p). • So there is an equilibrium in which • -2p+4(1-p)=2(1-p) or p=1/2.

45. This equilibrium is sort of stable because when p>1/2, there are many hawks, it is better to play dove. On the other hand, when p<1/2, there are many doves, it is better to play hawk. • This is called an evolutionary stable strategy.

46. Dictator game • Alice and Bob have 1 dollar to divide between them. Alice proposes a division. • Bob has no say. What Alice proposes will be carried out. • How much will you propose if you were Alice?

47. Ultimatum game • Alice and Bob have 1 dollar to divide between them. Alice proposes a division. • If Bob agrees, the division is carried out. If Bob disagrees, the game ends. • How much will you propose if you were Alice?

48. Chapter 29 Game Theory • Key Concept: Nash equilibrium and Subgame Perfect Nash equilibrium (SPNE)