1. Computation of Mixed Strategy Equilibria in Repeated Games Juho Timonen 17.6.2015 Instructor: PhD (Tech) Kimmo Berg Supervisor: Prof. Harri Ehtamo Saving and publishing this work on Aalto University website is allowed. Otherwise all rights reserved.

2. Background A Nash equilibrium of a stage game is an action profile where no player can gain by changing only his own action In a repeated game, the same stage game is played infinitely again and again –Players discount their payoffs from each round with a factor δ –Repeating the game allows players to reach payoffs that are not equilibria in a single-shot game

3. Background A strategy profile –determines players’ actions after each possible history –is a Nash equilibrium if all players are best responding to others’ strategies No-one has incentive to deviate since it will be punished an results in a smaller total payoff –is a subgame-perfect equilibrium if it is a Nash equilibrium after every possible history Equilibrium even after some-one has deviated

4. Background If players are allowed to randomize over their pure strategies, the set of equilibria grows remarkably Finding equilibrium payoffs (and strategies) is computationally a difficult problem What does the payoff set look like? ?

5. Goals Produce an algorithm that calculates (all) equilibria in 2x2 bimatrix games Investigate how the players’ discount factors affect equilibria Study differences between different types of 2x2 games –Prisoner’s Dilemma, Stag Hunt, Hawk-Dove, Coordination, Battle of Sexes etc. Compare mixed strategy equilibrium payoff sets with those of pure and correlated strategies –How much bigger the set is when mixing is allowed? –Are these additional payoffs Pareto-efficient?

6. Possible methods/tools Algorithm that tries to find boxes and line segments by setting needed continuation payoffs so that player(s) are indifferent between their actions Fixed-point characterization of equilibria Self-supporting sets Seeking equilibria using different classifications of 2x2 bimatrix games Matlab

7. Restrictions Limiting to two-player games Each player has only two pure strategies between which they are allowed to randomize Only non-correlated strategies are allowed –No public correlation devices Perfect monitoring Players observe only the realized actions and not the actual propabilities that others place on their pure actions

8. Source bibliography Mailath & Samuelson, Repeated Games and Reputations (2006) Berg & Schoenmakers, Construction of randomized subgame-perfect equilibria in repeated games (2015) Borm, A classification of 2x2 bimatrix games (1987)