Game Theory - Lecture #8

Transcription

1 Game Theory - Lecture #8 Outline: Randomized actions vnm & Bernoulli payoff functions Mixed strategies & Nash equilibrium Hawk/Dove & Mixed strategies

2 Random models Goal: Would like a formulation in which players choose randomly Matching pennies: Randomization desirable? H T H 1, 1 1, 1 T 1, 1 1, 1 Domination: Best response to randomized strategy? L R T 0, 3, M 2, 2, B 3, 0, M is not a best response to either L or R If opponent plays L/R, then M is a best response Imperfect information: Available information is only correlated to truth, e.g., Automobile engine light Noisy measurements in prisoner s dilemma Demand given price 1

3 Randomized action profiles Original strategic setup: Set of players, {1, 2,..., n} For each player, a set of actions A i For each player, preferences on action profiles characterized by a payoff function: U i : A R Question: How do we extend preferences to lotteries over action profiles? Extension: Strategic game with vnm (Von Neumann and Morgenstern) preferences Set of players For each player, a set of actions A i For each player, preferences on lotteries on action profiles characterized by a (vnm) payoff function: U i ( (A)) R Notation: (Set) denotes probability distributions over a Set of outcomes Important special case: vnm preferences given by expected utility over action profiles (Bernoulli payoff) Key observation: Payoff values define preferences over distributions Original setting: preferences payoffs over profiles Extension: preferences payoffs over profiles preferences over distributions Concern: Moving further away from true preferences 2

4 Example Original setting: preferences payoffs over profiles Fact: Several payoff functions reflect preferences C D C 2, 2 0, 3 D 3, 0 1, 1 C D C 3, 3 0, 4 D 4, 0 1, 1 These are the same game (Prisoner s dilemma) in terms of original ordinal preference Extension: preferences payoffs over profiles preferences over distributions These are different games in terms of probability preferences Player 1 vnm utility depends on probabilities of {CC, CD, DC, DD}: Left game: Right game: Similar for Player 2 Compare following probability distributions: U 1 (p) = p CC 2 + p CD 0 + p DC 3 + p DD 1 U 1 (p) = p CC 3 + p CD 0 + p DC 4 + p DD 1 (2/5, 3/5, 0, 0) vs (0, 0, 0, 1) Payoff values take on heightened importance in extended setting. Dependence on payoff values can result in peculiar outcomes. 3

5 Expected payoff peculiarities In the new framework, the preferences are over probability distributions Issue: Are expected payoffs reasonable? Example: Allais paradox Consider the following two lotteries: A: $2 million with certainty a: $10 million with probability 0.1, $2 million with probability 0.89, $0 with probability 0.01 Which do you prefer? Consider two more lotteries: B: $10 million with probability 0.1 and $0 with probability 0.9 b: $2 million with probability 0.11 and $0 with probability 0.89 Which do you prefer? Most people prefer (A > a) and (B > b). Make sense? Problem: Preferences cannot be represented by an expected payoff function over u(10), u(2), u(0). Preference evaluation for (A > a) u(2) > 0.1u(10) u(2) u(0). Subtract 0.89u(2) and add 0.89u(0) to each side 0.11u(2) u(0) > 0.1u(10) + 0.9u(0) This implies that the expected payoff of lottery b exceeds that of lottery B! Conclusion: Decision maker s preferences cannot always be represented by an expected payoff function. Nonetheless, we will make use of expected payoffs. 4

6 Mixed strategies A mixed strategy is a probability distribution over a player s actions. Specifically, a player selects α i (A i ) Consequences: Joint action probabilities are products of player probabilities Bernoulli payoff becomes expected utility with independent players New notation: U i (α i, α i ) Continuing previous example: Player 1 chooses α 1 = (α 1C, α 1D ) Player 2 chooses α 2 = (α 2C, α 2D ) Resulting probability distribution over joint actions is (p CC, p CD, p DC, p DD ) = (α 1C α 2C, α 1C α 2D, α 1D α 2C, α 1D α 2D ) Inherited expected utilities: Left game: Right game: (Likewise for U 2 ( )) U 1 (α 1, α 2 ) = 2 α 1C α 2C + 0 α 1C α 2D + 3 α 1D α 2C + 1 α 1D α 2D U 1 (α 1, α 2 ) = 3 α 1C α 2C + 0 α 1C α 2D + 4 α 1D α 2C + 1 α 1D α 2D Reconciled viewpoint: New setup is same as old setup with Set of players New set of actions α i (A i ) New payoff functions U i (α i, α i ) which is expected value of original payoff functions assuming independent players 5

7 Mixed strategy best response Define the best response function, B i ( ), as B i (α i ) = {α i : U i (α i, α i ) U i (α i, α i ) for all α i (A i )} Note that the best response function is actually a set This definition is exactly as before except: Player actions are replaced with mixed strategies Player utilities are replaced with expected utilities assuming independent players Example: Generic two player/two action game L R T a, A b, B B c, C d, D Assume mixed strategies are α 1 = (p, 1 p) for row player and α 2 = (q, 1 q) for column player Player 1 must maximize over p [0, 1] ( ) ( ) p q a + (1 q) b + (1 p) q c + (1 q) d Fact: 1 (q a + (1 q) b) > (q c + (1 q) d) B row (q) = 0 (q a + (1 q) b) < (q c + (1 q) d) [0, 1] (q a + (1 q) b) = (q c + (1 q) d) Similar analysis to derive B col (p) 6

8 Mixed strategy Nash equilibrium The mixed strategy profile α = (α 1,..., α n) is a mixed strategy Nash equilibrium if for every player i, α i B i (α i) Celebrated Nash theorem: Every strategic game with vnm preferences in which each player has finitely many actions has a mixed strategy Nash equilibrium. Example: Matching pennies Sketch best response for row player Sketch best response for column player Observe lack of pure strategy Nash equilibrium Observe mixed strategy Nash equilibrium Note indifference phenomenon Nash result due to (advanced) fixed point theory Want to find (α 1,..., α n) such that α (B 1 ( ),..., B n ( )) α Illustration: A continuous function on the closed interval [0,1] must have a fixed point, i.e., an x [0, 1] such that x = f(x) 7

9 Hawk/Dove Setup: H D H 0, 0 6, 1 D 1, 6 3, 3 H: hawk = aggressive D: dove = passive Model of game of chicken or traffic intersection First look: What are the pure (i.e., non-randomized) action NE? Best response function for row player: B row (H) = D & B row (D) = H Symmetric for column player NE: (H, D) and (D, H) 8

10 Hawk/Dove: Mixed strategies H D H 0, 0 6, 1 D 1, 6 3, 3 Second look: What are the mixed strategy NE? As before, we construct best response function, but for mixed strategies Row: Pr (H) = p and Pr (D) = 1 p Column: Pr (H) = q and Pr (D) = 1 q Players select {H, D} independently Best response for row player: Need to maximize expected payoff, i.e., ( ) ( ) max p 0 q + 6 (1 q) + (1 p) 1 q + 3 (1 q) 0 p 1 1 B row (q) = [0, 1] 0 ( ) ( ) 0 q + 6 (1 q) > 1 q + 3 (1 q) ( ) ( ) 0 q + 6 (1 q) = 1 q + 3 (1 q) ( ) ( ) 0 q + 6 (1 q) < 1 q + 3 (1 q) Conclusion: 1 q < 3/4 B row (q) = [0, 1] q = 3/4 0 q > 3/4 1 p < 3/4 & B col (p) = [0, 1] p = 3/4 0 p > 3/4 9

11 H/D: Best response plots p q NE occur at intersection of best response plots NE of original pure strategy game are still present New mixed strategy NE: (p, q ) = (3/4, 3/4) Peculiarity: At mixed strategy NE, players are indifferent, i.e., B row (3/4) = [0, 1] & B col (3/4) = [0, 1] i.e., at NE, best response is to play (H, D) with any probability combination. The mixed strategy NE makes both players indifferent Question: Are there other outcome that could lead to more desirable behavior? 10