Game Theory. Wolfgang Frimmel. Dominance

Transcription

1 Game Theory Wolfgang Frimmel Dominance 1 / 13

2 Example: Prisoners dilemma Consider the following game in normal-form: There are two players who both have the options cooperate (C) and defect (D) Both players decide simultaneously Assume that c > a > d > b C D C a,a b,c D c,b d,d What will be the outcome of this game? Is the outcome pareto-optimal? 2 / 13

3 Strictly dominant strategies A strategy is a strictly dominant strategy for a player if it is strictly better than any other available strategy regardless of other players strategy choices. A rational player will always choose a strictly dominant strategy No strategic interaction and no information about other player s characteristics necessary Formal definition of strict dominance A pure strategy si strictly dominates s i if and only if u i (si, s i)>u i (s i, s i ) s i There is no belief under which the player would play s i, for s i would always yield a higher expected payoff than s i no matter what player i believes about the other players. 3 / 13

4 Weakly dominant strategies A player s action weakly dominates another action if the first action is at least as good as the second action, no matter what the other players are doing. Formal definition of weak dominance A pure strategy si weakly dominates s i if and only if u i (si, s i) u i (s i, s i ) s i and at least one of the inequalities is strict. If a player is rational, then she will not play a weakly dominated strategy in a dominant strategy equilibrium. 4 / 13

5 Example Consider the following game (for simplicity, we ignore payoffs for player 2): L R T 1 0 M 2 0 B 2 1 M weakly dominates T B weakly dominates M B strictly dominates T 5 / 13

6 Dominant-strategy equilibrium Definition A strategy s i is a dominant if and only if s i weakly dominates every other strategy s i. A strategy profile s is a dominant-strategy equilibrium if and only if s i is a dominant strategy for each player i. Cooperate Defect Cooperate 5,5 0,6 Defect 6,0 1,1 Defect is a dominant strategy for both players, therefore the strategy profile s (s1, s 2 ) = (Defect, Defect) is a dominant strategy equilibrium. Note, that dominant strategy equilibrium only requires weak dominance. 6 / 13

7 Strictly dominated strategies Following this, a rational player will never play a strategy s i that is strictly dominated by another strategy s i Formal definition of strictly dominated strategies A pure strategy s i is strictly dominated by strategy s i if and only if u i (s i, s i)<u i (s i, s i ) s i Is there a strictly dominant strategy in the following game? L M R U 1,0 1,2 0,1 D 0,3 0,1 2,0 Is there a strictly dominated strategy? 7 / 13

8 Rationalizability Dominance captures well the implications of rationality Games in which dominance alone leads to a precise prediction are rare and not very interesting for game theory Also common knowledge about rational players is an important assumption Rationalizability captures the implications of the common knowledge assumption Rationalizability is equal to iterative deletion of strictly dominated strategies 8 / 13

9 Iterative deletion of strictly dominated strategies We can eliminate strictly dominated strategies from the set of available strategies, since it will never be played (not rationalizable). All players simultaneously eliminate all strictly dominated strategies from their available strategy set One must delete the strategies that are strictly dominated by mixed strategies (but not necessarily by pure strategies) If one player still has one strictly dominated strategy left in the reduced game, the procedure continues until no strictly dominated strategy is left. Iterative deletion of dominated strategies may lead to a unique prediction for a game (but not necessarily) 9 / 13

10 k-times iterative deletion of strictly dominated strategies Implications If every player is rational, every player knows that every player is rational, every player knows that every player knows that every player is rational,..., then every player must play a strategy that survives k-times iterated deletion of strictly dominated strategies. A strategy is rationalizable iff it survives iterated deletion of strictly dominated strategies. If it is common knowledge that every player is rational, then every player must play a rationalizable strategy. Any rationalizable strategy is consistent with common knowledge of rationality. 10 / 13

11 Example Consider again the following game: L M R U 1,0 1,2 0,1 D 0,3 0,1 2,0 For player 2 (column player), R is strictly dominated by M - if she knows the game and is rational, she will never play R If player 1 (row player) knows that player 2 is rational and knows the game, he will play as if the game was L M U 1,0 1,2 D 0,3 0,1 11 / 13

12 Example (cont.) Now, for player 1, strategy D is strictly dominated by strategy U - he will never player D if he is rational and knows the game. If player 2 knows that player 1 is rational and knows the game, she will play as if the game was L M U 1,0 1,2 For player 2, L is strictly dominated by M - if she knows the game and is rational, she will never play L The unique strategy profile which survived iterative deletion of strictly dominated strategies is s (s 1, s 2 ) = (U, M ) M U 1,2 The only rationalizable strategies are U for player 1 and M for player 2 12 / 13

13 Example: Pricing strategies Consider two competing firms that need to decide about the prices of their products. Payoffs are profits in millions. Answer the following questions: 1 Is there a dominant strategy in this game? 2 Is there a dominated strategy? 3 Find all rationalizable strategies. 4 Can you find a prediction for the outcome of the game? e 0.95 e 1.30 e 1.55 e ,6 7,1 10,4 e ,1 8,2 14,7 e ,0 6,2 8,5 13 / 13