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Solving for Mixed Strategies: 

Solving for Mixed Strategies Principles of International Politics Third Edition Chapter 8

What are Mixed Strategies?: 

What are Mixed Strategies? Earlier, we learned about pure strategy equilibria: A player plays his best response strategy with certainty. Review Chapter 4’s Walk-through if needed. Mixed strategy equilibria involve selecting moves probabilistically. More specifically, players mix their strategy across all of their choices in a probability ratio that makes the other player indifferent between his moves.

When to Solve for Mixed Strategy Equilibria: 

When to Solve for Mixed Strategy Equilibria Mixed strategy equilibria are primarily found in strategic form games. All games must have an odd number of Nash equilibria. We solve for mixed strategy equilibria: When the game has no equilibria in pure strategies, or When the game has an even number of equilibria in pure strategies. We must solve for a mixed strategy equilibrium for such common games as chicken, battle of the sexes, and coordination.

Finding Mixed Strategies: 

Finding Mixed Strategies My goal in playing a mixed strategy is to make you indifferent between the payoffs from your strategies. Here, Rose wants to pick her moves with a probability p that makes Colin indifferent between playing left or right. Do any pure strategy equilibria exist?

Computing Mixed Strategies: 

Computing Mixed Strategies First, calculate the expected utility of playing left and of playing right for Colin, using Rose’s probability values (p and 1 – p). EU(L) = p(7) + (1 – p)(1) EU(L) = 7p + 1 – 1p = 6p + 1 EU(R) = p(4) + (1 – p)(8) EU(R) = 4p + 8 – 8p = –4p + 8 Be sure you understand where we got these numbers.

Computing, Continued: 

Computing, Continued Then set these two expressions equal to each other to find the value of p that makes Colin indifferent between his choices. EU(L) = EU(R) 6p + 1 = –4p + 8 2p = 7 p = 2/7 Rose should play up with probability 2/7 (and down with probability 5/7) to make Colin indifferent between playing left and right.

Finding Colin’s Mixed Strategy: 

Finding Colin’s Mixed Strategy Colin too wants to pick his moves to make Rose indifferent between playing up or down. Solve for Colin’s mixed strategy. Find EU(up), EU(down), and then solve for q. Scroll up to Rose’s strategy if you need guidance.

Colin’s Mixed Strategy: 

Colin’s Mixed Strategy For Colin, EU(U) = q(6) + (1 – q)(3) EU(U) = 6q + 3 – 3q = 3q + 3 EU(D) = q(2) + (1 – q)(5) EU(D) = 2q + 5 – 5q = –3q + 5 EU(U) = EU(D) 3q + 3 = –3q + 5 6q = 2 q = 1/3

The Complete Equilibrium: 

The Complete Equilibrium A game’s complete equilibrium includes all pure strategy equilibria, if they exist, and the mixed strategy equilibrium, if one exists. For Rose and Colin, the complete equilibrium is, “up, left; down, right; p = 2/7, q = 1/3”

Pop Quizzes: 

Pop Quizzes Professor Ross want the class to study regularly, not just the night before the test. One way to do this is to give pop quizzes with some frequency (p) in hopes that students will always be prepared for a quiz. Each player (R or C, class) picks a move without knowing the other’s move.

Preferences and Payoffs: 

Preferences and Payoffs Prof Ross prefers: no quiz, students prepared > quiz, students prepared > quiz, students not prepared > no quiz students not prepared. The class prefers no quiz to quiz, but if there is a quiz they prefer being prepared: no quiz, students not prepared > quiz, students prepared > no quiz, students prepared > quiz, students not prepared. Do any pure strategy equilibria exist?

Prof. Ross’s Strategy: 

Prof. Ross’s Strategy Satisfy yourself that there are no pure strategy equilibria for this game. Solve for Prof. Ross’s strategy mix. Calculate EU(study) and EU(~study) [~ means “not”], using p and the class’s payoffs. Set those two expressions equal to each other and solve for the value of p that makes the class indifferent between studying and not. Click to advance and check your work.

Mixing for Prof. Ross: 

Mixing for Prof. Ross Find EU(study): EU(study) = p(3) + (1 – p)(2) EU(study) = 3p + 2 – 2p = p + 2 Find EU(~study): EU(~s) = p(0) + (1 – p)(4) EU(~s) = 0p + 4 – 4p = –4p + 4 Set these equal and solve: p + 2 = –4p + 4 5p = 2  p = 2/5 If the term has fifteen classes, how many pop quizzes must Prof. Ross commit to giving to achieve the goal of regular student studying?

The Class’s Choice: 

The Class’s Choice Compute the strategy mix for the class. Solve for EU(quiz) and EU(~ quiz), using q and Prof. Ross’s payoffs Set these equal to each other. Click to advance and check your work.

The Class’s Solution: 

The Class’s Solution EU(quiz): EU(quiz) = q(3) + (1 – q)(2) EU(quiz) = 3q + 2 – 2q = q + 2 EU(~quiz): EU(~quiz) = q(4) + (1 – q)(1) EU(~quiz) = 4q + 1 – q = 3q + 1 Set these equal and solve: q + 2 = 3q + 1 2q = 1  q = 1/2 To play its mixed strategy, the class would want to credibly commit to studying half of the time. This makes Prof. Ross indifferent between preparing a quiz and not preparing a quiz.

More Practice: 

More Practice The next slide contains four games in which you can practice solving for mixed strategy equilibria. For more practice, try Chapter 4 in James Morrow’s Game Theory for Political Scientists (Princeton University Press, 1994). Your library or your instructor may have a copy. Your instructor may also recommend other resources.

Practice Games: 

Practice Games