CHAPTER 9 Nash Equilibrium 1-1

Transcription

1 . CHAPTER 9 Nash Equilibrium 1-1

2 Rationalizability & Strategic Uncertainty In the Battle of Sexes, uncertainty about other s strategy can lead to poor payoffs, even if both players rational Rationalizability does not require that each player s belief (about others strategies) is consistent with others strategies. All it requires is that (1) players form beliefs, (2) best-respond to these beliefs, and (3) all players know (1),(2) Rationalizability, as a prediction, may be appropriate in settings where players have little chance to coordinate beliefs Eg., guessing meeting place upon getting lost at Disney 1-2

3 Resolving Strategic Uncertainty Strategic uncertainty may be resolved in many ways: - social norms (pedestrians veering to right prior to colliding) -rules(driving on right side of road, to avoid collision) - prior communication ( Let s meet at water fountain, if lost) - mediators (Plaques advising location of Lost and Found ) -prior history of play Let us update the concept of rationality to settings where strategic uncertainty is absent or limited. 1-3

4 Nash Equilibrium Fix a nfg. Definition. A profile s = (s 1,, s n ) is a Nash equilibrium for every player i, s i εbr i (s -i ), i.e. u i (z,s -i ) u i (s i,s -i ) for all zεs i. Notes: -Says players strategies are best responses to one another s - Full strategic certainty as each player thinks of others stats. -There is no explanation why all players focus on this profile -One interpr n: Mediator prescribes strats., then in best interest of each player to obey, if he believes other will obey 1-4

5 Examples I Circles indicate a NE (which is a profile, not a payoff) Matching Pennies has no NE (in pure strats.) Prisoner s Dilemma has only one NE (it is inefficient) Battle of the Sexes has two NE (strategic certainty does not eliminate problem of coordination) 1-5

6 Examples II Again, multiple NE. The concept of NE does not imply uniqueness of equilibrium beliefs. 1-6

7 Nash Equilibrium X Rationalizability Fact A: If a sole profiles survives the iterated deletion of dominated strategies, then there is a Nash equilibrium, it is unique, and it is the surviving profile. Fact B: A Nash equilibrium (there may be multiple) always survives the iterated deletion of dominated strategies. Fact C: In a nfg with finitely many players and strategies, there exists a NE in possibly mixed strategies. (Eg, Matching Pennies, ((½,½), (½,½)) is a NE) See Robert Gibbons book, page

8 Computing NE From Matrices Find b.r. of 1 to each strat. of 2 mark 1 s payoff, row by row Find b.r. of 2 to each strat. of 1 mark 2 s payoff, col. by col. Any cell with two marksindicates a NE Three (pure) NE: (K,X),(L,Z),(M,Y) Matrices good only for games with finitely many strategies. 1-8

9 Computing NE By Calculus Recall Partnership Game. Used calculus to compute the b.r. of each player to her belief about other s effort: - x = BR 1 (y*) =1+cy* and y = BR 2 (x*) =1+cx* (* is for mean) NE is a profile (x,y) such that x is a br to y, y is a br to x That is, NE is a solution (x,y) to above system. Solve: x = 1 + cy, y = 1+ cx 1 x = = y 1 c 1-9

10 Cournot Duopoly by Calculus Two firms set, simultaneously and independently, their prices Consumers demand 10 p 1 +p 2 of firm 1 s good 10 p 2 + p 1 of firm 2 s good Production costless, so i s profit is (10 p i + p j )p i = 10p i p 2 i+ p j p i To find b.r. of one to other s p j, optimize quadratic: Derivative 10 + p j 2p i = 0 solved by p i = 5 + p i /2 (Note 2 nd der<0, so this is a global maximum) Solve system p 1 = 5 + p 2 /2, p 2 = 5 + p 1 /2 to get p 1 = 10 = p 2 This is a NE by definition the only one! 1-10

11 Why can NE be inefficient? In the Prisoner s Dilemma, the sole NE is inefficient because of conflict between individual interests(squeal is dominant for each player) and group interests(mum is jointly better) Lack of coordination is another explanation, e.g. QWERTY keyboards (between end-users and manufacturers) QWERTY illustrates that inefficiency is possible even if players have the same preferences and no strategic uncertainty 1-11

12 Congruous Sets At one extreme, rationalizability does not require consistency/congruence between beliefs about others s strategies and their strategies. At other extreme, NE involves all players believing/focusing on a strategy profile that is actually played. Intermediate notion. Consider set of profiles X = X 1 x x X n with X i subset of S i. X is congruousif, for each player i, strategy s i belongs in X i iffthere is a belief θ -i εδx -i to which θ -i εδx -i. Weakly congruousif, for each i, strategy s i belongs in X i if 1-12

13 Example X = {w,y}x{k,l} is congruous -w is a br to k for sure, y is a br to l for sure -k is a br to y for sure, l is a br to w for sure Generally, a profile s is a NE iff {s} is weakly congruous. 1-13