Mixed Strategies

Mixed Strategies

Mixed Strategies Definition: A mixed strategy of a player in a simultaneous move game is a probability distribution over the player’s actions In matching pennies a mixed strategy will be ai = (ai(H), ai(T)), where 0 ≤ ai(.) ≤ 1

Mixed Strategies Where 0  q 1

Player 2 q 1 - q Head Tail Expected Payoff p Player 1 1-p Head 1, -1 -1, 1 2q - 1 Tail -1, 1 1, -1 1 – 2q Mixed Strategies Expected 1 - 2p 2p-1 Payoff

Mixed Strategies 1 – 2q > 2q – 1 if and only if q < ½ • player 1’s Best pure-strategy response is: • Tail if q < ½ • Head if q > ½ • Indifferent between H and T if q = ½

Mixed Strategies Where 0  p  1

Mixed Strategies E1(Payoff) = pq*1 + p(1 - q)*(-1) + (1 – p)q*(-1) + (1 – p)(1 – q) * 1 = (1 – 2q) + p(4q – 2) Maximize E1(Payoff) choosing p. If 4q – 2 < 0 [q < ½]  p = 0 (Tail) is best response If 4q – 2 > 0 [q > ½]  p = 1 (Head) is best response If 4q – 2 = 0 [q = ½]  any p in [0, 1] is a best response

Mixed Strategies E2(Payoff) = pq*(-1) + p(1 - q)*1 + (1 – p)q*1 + (1 – p)(1 – q) *(-1) = (2p - 1) + q(2 – 4p) Maximize E2(Payoff) choosing q. If 2 - 4p < 0 [p > ½]  q = 0 (Tail) is best response If 2 - 4p > 0 [p < ½]  q = 1 (Head) is best response If 2 - 4p = 0 [p = ½]  any q in [0, 1] is a best response

Mixed Strategies q b2(p) 1 b1(q) 1/2 The unique Nash equilibrium is in mixed-strategy: (a1, a2) = ((1/2,1/2), (1/2,1/2)) p 1 1/2

Mixed Strategies Definition: The mixed strategy profile a* in a simultaneous-move game with VNM preferences is a mixed strategy Nash equilibrium if, for each player i and every mixed strategy ai of player i, the expected payoff to player i of a* is at least as large as the expected payoff to player i of (ai, a*-i) according to a payoff function whose expected value represents player i’s preferences over lotteries.

Mixed Strategies Equivalently, for each player i, Ui(a*) ≥ Ui (ai, a*-i) for every mixed strategy profile ai of player i, Where Ui(a) is player i’s expected payoff to the mixed strategy profile a

Mixed Strategies Alternative definition: The mixed strategy profile a* is a mixed strategy Nash equilibrium if and only if a*i is in Bi(a*-i) for every player i.

Mixed Strategies A player’s expected payoff to the mixed strategy profile a is a weighted average of her expected payoffs to all mixed strategy profiles of the type (ai, a-i), where the weight attached to (ai, a-i) is the probability ai(ai) assigned to ai by player i’s mixed strategy ai Where Ai is player i’s set of actions (pure strategies)

Mixed Strategies MSNE Proposition: A mixed strategy profile a* in a strategic game in which each player has finitely many actions is a mixed strategy Nash equilibrium if and only if, for each player i, • The expected payoff, given a*-i, to every action to which a*i assigns positive probability is the same, • The expected payoff, given a*-i, to every action to which a*i assigns zero probability is at most the expected payoff to any action to which a*i assigns positive probability. (See page 116 in Osborne.) • So actions which the player is mixing between must yield the same expected payoff. Those that are not being mixed, must not yield a higher expected payoff than those that are.

Mixed Strategies • Suppose both airlines mix between both strategies. • United’s expected payoff from entering and staying out must be the same: • -50q +100(1-q) = 0q + 0(1-q) --> q = 2/3 • American’s expected payoff from entering and staying out must be the same: • -50p +100(1-p) = 0p + 0(1-p) --> p = 2/3 • Symmetric expected payoffs are thus: • -50(2/3)(2/3) +100(2/3)(1/3) + 0(1/3)(2/3)+0(1/3)(1/3) = 0 • Note that equalizing the conditional expected payoffs gives you the interior solution (if it exists) while maximizing the unconditional expected payoffs will give you ALL NE. • ALL NE are thus {((1,0),(0,1)); ((0,1),(1,0)); ((2/3,1/3),(2/3,1/3)) }

Mixed Strategies Proposition: Every simultaneous-move game with vNM preferences and a finite number of players in which each player has finitely many actions has at least one Nash equilibrium, possibly involving mixed strategies.

American q 1 - q Enter Stay out United p Enter -50, -50 150, 0 1 – p Stay out 0, 100 0, 0 Mixed StrategiesAsymmetric game

Asymmetric United/American Solution • Consider the unconditional expected payoff of United: • E[UU] = -50pq + 150p(1-q) + 0(1-p)q + 0(1-p)(1-q) • = -200pq + 150p = p(150-200q) • So United’s Best Response correspondence is: • If 150-200q > 0 <=> q < 3/4 ==> p=1. • If 150-200q < 0 <=> q > 3/4 ==> p=0. • If 150-200q = 0 <=> q = 3/4 ==> p  [0,1]. • Consider the unconditional expected payoff of American: • E[UA] = -50pq + 100q(1-p) + 0(1-q)p + 0(1-p)(1-q) • = -150pq + 100q = q(100-150p) • So American’s Best Response correspondence is: • If 100-150p > 0 <=> p < 2/3 ==> p=1. • If 100-150p < 0 <=> p > 2/3 ==> p=0. • If 100-150p = 0 <=> p = 2/3 ==> p  [0,1]. • Graph the BR correspondences (in p,q space) to find ALL NE.

Mixed StrategiesAsymmetric game • Pure-strategy Nash equilibrium: (Enter, Stay out) (Stay out, Enter) • Mixed-strategy Nash equilibrium: (aU, aA) = ((2/3,1/3), (3/4,1/4))

Mixed Strategies Definition: In a strategic game with vNM preferences, player i’s mixed strategy aistrictly dominates her action a’i if Ui(ai, a-i) > ui(a’i, a-i) for every a-i

L R T 1, . 1, . M 4, . 0, . B 0, . 3, . Mixed Strategies Does this game have any dominated pure strategies? No, but if the row player mixes equally between M and B, then if the column player plays L, row gets 4(1/2)+0(1/2) = 2 if she mixes while just 1 if she plays T. If column plays R, row gets 0(1/2)+3(1/2) = 3/2 if she mixes, while again just 1 by playing T. Thus T is strictly dominated by a mixed strategy.

L C R T 5, 5 20, 10 25, 3 M 10, 15 10, 10 15, 10 B 3, 25 15, 10 20, 15 Mixed Strategies What are the NE (pure and mixed) of this game?

Method of finding all mixed-strategy Nash equilibrium • For each player i, choose a subset Si of her set Ai of actions. • Check whether there exists a mixed strategy profile a such that (1) the set of actions to which each strategy ai assigns positive probability is Si and (2)a satisfies the conditions in proposition 116.2 in Osborne. • Repeat the analysis for every collection of subsets of the players’ sets of actions

B S X B 4, 2 0, 0 0,1 S 0, 0 2, 4 1, 3 Mixed Strategies

Mixed Strategies • Potential types of equilibria: • 1) Player one plays 1 strategy, Player two plays 1 strategy. • These are pure strategy NE. • 2) Player one plays 1 strategy, Player two plays 2 strategies. • One plays a pure strategy, Two mixes on BS, BX, or SX • 3) Player one plays 1 strategy, Player two plays 3 strategies. • One plays a pure strategy, Two mixes on BSX • 4) Player one plays 2 strategies, Player two plays 1 strategy. • One mixes on BS, Two plays a pure strategy • 5) Player one plays 2 strategies, Player two plays 2 strategies. • One mixes on BS, Two plays BS, BX, or SX • 6) Player one plays 2 strategies, Player two plays 3 strategies. • One mixes on BS, Two mixes on BSX

Mixed Strategies