Game Theory Lecture 7 Refinements: Perfect equilibrium

Transcription

1 Game Theory Lecture 7 Refinements: Perfect equilibrium Christoph Schottmüller University of Copenhagen October 23, / 18

2 Outline 1 Perfect equilibrium in strategic form games 2 / 18

3 Refinements I multiplicity of NE some NE feel a bit unreasonable refinement : introduce some criterium that some NE satisfy and others do not most useful in extensive form games we will return to refinement later Example L R T 1,1 0,0 B 0,0 0,0 Table: (B, R) is NE; intuitive outcome? 3 / 18

4 Perfect equilibrium I problem with equilibrium (B, R): other player might make a mistake say, with some small probability η a player trembles then strictly better for player 1 (2) to choose T (L). 4 / 18

5 Perfect equilibrium II formally: G = N, (A i ), (u i ) m i : number of actions of player i tremble : vector η R m1+ +mn ++ where ηi k is probability that action ai k A i is chosen accidentally by player i define G (η) as game G but with strategy space such that S i (η) = {α i A i : α k i η k i } i.e. actions have to be completely mixed and put minimum probability ηi k on action ai k a mixed NE exists in G (η) (proof similar to the Nash theorem) 5 / 18

6 Perfect equilibrium III Definition (perfect equilibrium) A Nash equilibrium in G is (trembling hand) perfect if there is a sequence (η t ) t=1 of trembles converging to zero and a sequence of Nash equilibria α(η t ) in G (η t ) for each t, such that the sequence (α(η t )) t=1 converges to the Nash equilibrium in G. 6 / 18

7 Perfect equilibrium IV previous example: L R T 1,1 0,0 B 0,0 0,0 unique equilibrium of any G (η) is (T, L) only (T, L) is perfect equilibrium 7 / 18

8 Perfect equilibrium V Theorem Every finite strategic game has a perfect equilibrium. Proof. (sketch) 8 / 18

9 Perfect equilibrium VI Example L R T 1,1 0,0 B 1,0 2,1 Show that (T,L) is not a perfect equilibrium and (B,R) is a perfect equilibrium 9 / 18

10 Perfect equilibrium VII Definition (ε-perfect equilibrium) A mixed strategy profile α ε is an ε-perfect equilibrium if it is completely mixed and U i (α ε i, a i) > U i (α ε i, a i ) for any a i, a i A i implies α ε i (a i ) ε. in short: completely mixed but only best responses are played with more than ε probability 10 / 18

11 Perfect equilibrium VIII Theorem Let α be a mixed strategy profile in G. The following statements are equivalent 1 α is a perfect equilibrium 2 α is a limit of ε-perfect equilibria for ε going to zero 3 α is the limit of a sequence (α(ε)) ε 0 of completely mixed strategies such that α is a best response to α(ε) for ε small enough. Proof. (1) implies (2): If α is a perfect NE, there is a sequence α(η t ) of NE in G (η t ) converging to α.... (2) implies (3): 11 / 18

12 Perfect equilibrium IX let ai k be played with positive probability under α... (3) implies (1): Cut off if necessary the first few elements of the sequence of α(ε) such that for the remaining sequence α is a best reply to all α(ε). ηi k(ε) = α(ε)k i if α assigns zero probability to ai k ηi k(ε) = min lα(ε) l i otherwise / 18

13 Perfect equilibrium X Some comments: the definition says: there is a sequence ; requiring that for any sequence of trembles converging to zero there is a sequence of equilibria leads to strict perfect equilibria. However, those do often not exist: Example L M R T 1,1 1,0 0,0 B 1,1 0,0 1,0 Table: no strict perfect equilibrium L is a strictly dominant strategy Check that (T, L) and (B, L) are both perfect equilibria but not strictly perfect 13 / 18

14 Perfect equilibrium XI like most refinements perfection is sensitive to the formulation of the game: Example L C R T 1,1 0,0-1,-2 M 0,0 0,0 0,-2 B -2,-1-2,0-2,-2 Table: sensitive to additional irrelevant actions same game as first example plus strictly dominated actions B and R Check that (M,C) is also perfect equilibrium in this game (of course, (T,L) remains a perfect equilibrium as well) 14 / 18

15 Perfect equilibrium XII three player game: perfect equilibrium eliminates more than just equilibria in weekly dominated actions L R T 1,1,1 1,0,1 D 1,1,1 0,0,1 A L R T 1,0,0 0,1,2 D 0,1,0 1,0,0 Table: a NE that is not perfect despite not being in weakly dominated strategies B (D,L,A) is a NE and none of the actions is weakly dominated not perfect: for small enough trembles of P2 and P3 T will always be better than D for P1; 15 / 18

16 Perfect equilibrium XIII A Nash equilibrium in completely mixed strategies is also a perfect equilibrium. Why? Example H T H 1,-1-1,1 T -1,1 1,-1 Table: Matching Pennies 16 / 18

17 Review Questions What is the idea behind refinements? (why might we need them?) What is the idea behind perfect equilibrium? How is perfect equilibrium defined? What problems does perfect equilibrium have? reading: lecture notes by Hans Keiding p. 1-6; MSZ p (or OR ) *reading: lecture notes by Hans Keiding p / 18

18 Exercises 1 Show in the game below that (T,L) is not a perfect equilibrium and (B,R) is a perfect equilibrium. L R T 2,1 1,1 M 1,2 0,2 B 0,0 3,1 2 Show that (D,L,A) is a Nash equilibrium but not a perfect equilibrium in the following 3 player game: L R T 1,0,0 1,0,1 D 1,1,1 0,1,1 A L R T 0,1,1 0,0,0 D 1,0,0 1,1,0 3 Show that no player uses a weakly dominated strategy in any perfect equilibrium. B 18 / 18