Dominant and Dominated Strategies

Transcription

1 Dominant and Dominated Strategies Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign May 29th, 2015 C. Hurtado (UIUC - Economics) Game Theory

2 On the Agenda 1 Formalizing the Game 2 Dominant and Dominated Strategies 3 Iterated Delation of Strictly Dominated Strategies 4 Iterated Delation of Dominated Strategies 5 Exercises C. Hurtado (UIUC - Economics) Game Theory

3 Formalizing the Game On the Agenda 1 Formalizing the Game 2 Dominant and Dominated Strategies 3 Iterated Delation of Strictly Dominated Strategies 4 Iterated Delation of Dominated Strategies 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 1 / 22

4 Formalizing the Game Formalizing the Game Up to this point we defined game without been formal. Let me introduce some Notation: - set of players: I = {1, 2,, N} - set of actions: i I, a i A i, where each player i has a set of actions A i. - strategies for each player: i I, s i S i, where each player i has a set of pure strategies S i available to him. A strategy is a complete contingent plan for playing the game, which specifies a feasible action of a player s information sets in the game. - profile of pure strategies: s = (s 1, s 2,, s N ) N i=1 Si. Note: let s i = (s 1, s 2,, s i 1, s i+1,, s N ) S i, we will denote s = (s i, s i) (S i, S i). - Payoff function: u i : N Si R, denoted by ui(si, s i) i=1 C. Hurtado (UIUC - Economics) Game Theory 2 / 22

5 Formalizing the Game Formalizing the Game Definition Now we can denote game with pure strategies and complete information in normal form by: Γ N = {I, {S i} i, {u i} i}. What about the games with mix strategies? We have taken it that when a player acts at any information set, he deterministically picks an action from the set of available actions. But there is no fundamental reason why this has to be case. A mixed strategy for player i is a function σ i : S i [0, 1], which assigns a probability σ i(s i) 0 to each pure strategy s i S i, satisfying s i S i σ i(s i) = 1. We denote the set of mixed strategies by (S i). Note that a pure strategy can be viewed as a special case of a mixed strategy in which the probability distribution over the elements of S i is degenerate. C. Hurtado (UIUC - Economics) Game Theory 3 / 22

6 Formalizing the Game Example Meeting in New York: - Players: Two players, 1 and 2 - Rules: The two players can not communicate. They are suppose to meet in NYC at noon to have lunch but they have not specify where. Each must decide where to go (only one choice). - Outcomes: If they meet each other, they enjoy other s company. Otherwise, they eat alone. - Payoffs: They attach a monetary value of 100 USD to other s company and 0 USD to eat alone. player 1 player 2 A B C A 100,100 0,0 0,0 B 0,0 100,100 0,0 C. Hurtado (UIUC - Economics) Game Theory 4 / 22

7 Formalizing the Game Example Meeting in New York: - set of players: I = {1, 2} - set of actions: A 1 = {A, B}, and A 2 = {A, B, C} - strategies for each player: S 1 = A 1, and S 2 = A 2 (Why?) - Payoff function: u i : 2 Si R, denoted by ui(si, s i) i=1 { 100 if s i = s i u(s i, s i) = 0 if s i s i Player 2 - pure strategies: S 2 = {A, B, C}. Player 2 has 3 pure strategies. - mixed strategies: (S 2) = {(σ 2 1, σ 2 2, σ 2 3) R 3 σ 2 m 0 m = 1, 2, 3 and 3 m=1 σ2 m = 1} C. Hurtado (UIUC - Economics) Game Theory 5 / 22

8 Dominant and Dominated Strategies On the Agenda 1 Formalizing the Game 2 Dominant and Dominated Strategies 3 Iterated Delation of Strictly Dominated Strategies 4 Iterated Delation of Dominated Strategies 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 6 / 22

9 Dominant and Dominated Strategies Dominant and Dominated Strategies Now we turn to the central question of game theory: What should be expected to observe in a game played by rational agents who are fully knowledgeable about the structure of the game and each others rationality? To keep matters simple we initially ignore the possibility that players might randomize in their strategy choices. The prisoner s dilemma: * Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. * The prosecutors do not have enough evidence to convict the pair on the principal charge. They hope to get both sentenced to a year in prison on a lesser charge. * Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. * Here is the offer: - If A and B each betray the other, each of them serves 2 years in prison - If A betrays B but B remains silent, A will be set free and B will serve 3 years in prison (and vice versa) - If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge) C. Hurtado (UIUC - Economics) Game Theory 7 / 22

10 Dominant and Dominated Strategies Dominant and Dominated Strategies Let me put prisoner s dilemma as a game of trust: player 1 player 2 trust cheat trust 5,5 1,10 cheat 10,1 2,2 Observe that regardless of what her opponent does, player i is strictly better off playing Cheat rather than Trust. This is precisely what is meant by a strictly dominant strategy. Player 2 plays Trust. Player 1 knows that 10 > 5, better to Cheat. Player 2 plays Cheat. Player 1 knows that 2 > 1, better to Cheat. Regardless of the other s strategies, it is always better to Cheat. Note that both would be better off if they both play trust. Lesson: self-interested behavior in games may not lead to socially optimal outcomes. C. Hurtado (UIUC - Economics) Game Theory 8 / 22

11 Dominant and Dominated Strategies Dominant and Dominated Strategies Definition A strategy s i S i is a strictly dominant strategy for player i if for all s i s i and all s i S i, u i (s i, s i ) > u i ( s i, s i ). A strictly dominant strategy for i uniquely maximizes her payoff for any strategy profile of all other players. If such a strategy exists, it is highly reasonable to expect a player to play it. In a sense, this is a consequence of a player s rationality. C. Hurtado (UIUC - Economics) Game Theory 9 / 22

12 Dominant and Dominated Strategies Dominant and Dominated Strategies What about if a strictly dominant strategy doesn t exist? player 1 player 2 a b c A 5,5 0,10 3,4 B 3,0 2,2 4,5 You can easily convince yourself that there are no strictly dominant strategies here for either player. Notice that regardless of whether Player 1 plays A or B, Player 2 does strictly better by playing b rather than a. That is, a is strictly dominated by b. C. Hurtado (UIUC - Economics) Game Theory 10 / 22

13 Dominant and Dominated Strategies Dominant and Dominated Strategies Definition A strategy s i S i is strictly dominated for player i if there exists a strategy s i S i such that for all s i S i, u i ( s i, s i ) > u i (s i, s i ). In this case, we say that s i strictly dominates s i. In words, s i strictly dominates s i if it yields a strictly higher payoff regardless of what (pure) strategy rivals use. Note that the definition would also permits us to use mixed strategies Using this terminology, we can restate the definition of strictly dominant: A strategy s i is strictly dominant if it strictly dominates all other strategies. It is reasonable that a player will not play a strictly dominated strategy, a consequence of rationality, again. C. Hurtado (UIUC - Economics) Game Theory 11 / 22

14 Iterated Delation of Strictly Dominated Strategies On the Agenda 1 Formalizing the Game 2 Dominant and Dominated Strategies 3 Iterated Delation of Strictly Dominated Strategies 4 Iterated Delation of Dominated Strategies 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 12 / 22

15 Iterated Delation of Strictly Dominated Strategies Iterated Delation of Strictly Dominated Strategies player 1 player 2 a b c A 5,5 0,10 3,4 B 3,0 2,2 4,5 We argued that a is strictly dominated (by b) for Player 2; hence rationality of Player 2 dictates she won t play it. We can push the logic further: if Player 1 knows that Player 2 is rational, he should realize that Player 2 will not play strategy a. Notice that we are now moving from the rationality of each player to the mutual knowledge of each player s rationality. Once Player 1 realizes that 2 will not play a and deletes this strategy from the strategy space, then strategy A becomes strictly dominated by strategy B for Player 2. If we iterate the knowledge of rationality once again, then Player 2 realizes that 1 will not play A, and hence deletes A. Player 2 should play c. We have arrived at a solution. C. Hurtado (UIUC - Economics) Game Theory 13 / 22

16 Iterated Delation of Strictly Dominated Strategies Iterated Delation of Strictly Dominated Strategies Definition A game is strict-dominance solvable if iterated deletion of strictly dominated strategies results in a unique strategy profile. Since in principle we might have to iterate numerous times in order to solve a strict-dominance solvable game, the process can effectively can only be justified by common knowledge of rationality. As with strictly dominant strategies, it is also true that most games are not strict-dominance solvable. You might worry whether the order in which we delete strategies iteratively matters. Insofar as we are working with strictly dominated strategies so far, it does not. C. Hurtado (UIUC - Economics) Game Theory 14 / 22

17 Iterated Delation of Dominated Strategies On the Agenda 1 Formalizing the Game 2 Dominant and Dominated Strategies 3 Iterated Delation of Strictly Dominated Strategies 4 Iterated Delation of Dominated Strategies 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 15 / 22

18 Iterated Delation of Dominated Strategies Iterated Delation of Dominated Strategies Definition A strategy s i S i is a weakly dominant strategy for player i if for all s i s i and all s i S i, u i (s i, s i ) u i ( s i, s i ), and for at least one choice of s i the inequality is strict. Definition A strategy s i S i is weakly dominated for player i if there exists a strategy s i S i such that for all s i S i, u i ( s i, s i ) u i (s i, s i ), and for at least one choice of s i the inequality is strict. In this case, we say that s i weakly dominates s i. Definition A game is weakly-dominance solvable if iterated deletion of weakly dominated strategies results in a unique strategy profile. C. Hurtado (UIUC - Economics) Game Theory 16 / 22

19 Iterated Delation of Dominated Strategies Iterated Delation of Dominated Strategies Using this terminology, we can restate the definition of weakly dominant: A strategy s i is weakly dominant if it weakly dominates all other strategies. You might worry whether the order in which we delete strategies iteratively matters. Delation of dominated strategies could leave to different outcomes. P2 L R U 5,1 4,0 P1 M 6,0 3,1 D 6,4 4,4 P2 P2 L R L R P1 U 5,1 4,0 D 6,4 4,4 P1 M 6,0 3,1 D 6,4 4,4 C. Hurtado (UIUC - Economics) Game Theory 17 / 22

20 Exercises On the Agenda 1 Formalizing the Game 2 Dominant and Dominated Strategies 3 Iterated Delation of Strictly Dominated Strategies 4 Iterated Delation of Dominated Strategies 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 18 / 22

21 Exercises Exercises Exercise 1. Prove that a player can have at most one strictly dominant strategy. Exercise 2. Apply the iterated elimination of strictly dominated strategies to the following normal form games. Note that in some cases there may remain more that one strategy for each player. Say exactly in what order you eliminated rows and columns. Exercise 3. Apply the iterated elimination of dominated strategies to the following normal form games. Note that in some cases there may remain more that one strategy for each player. Say exactly in what order you eliminated rows and columns. C. Hurtado (UIUC - Economics) Game Theory 19 / 22

22 Exercises Exercises Exercise 2 (cont.). C. Hurtado (UIUC - Economics) Game Theory 20 / 22

23 Exercises Exercises Exercise 2 (cont.). C. Hurtado (UIUC - Economics) Game Theory 21 / 22

24 Exercises Exercises Exercise 2 (cont.). C. Hurtado (UIUC - Economics) Game Theory 22 / 22