Infinitely Repeated Games Econ 171 Finitely Repeated Game Take any game play it then play it again for a specified number of times The game that is repeated is known as the stage game Let players observe all previous play For every history that you have observed you could have a different response Prisoners dilemma 3 times How many strategies that vary with other players s actions on previous move First move C or D Second move C always C if C D if D D if C C if D D always Third move There are 4 possible histories of other guy s moves For each move by other guy there are two things you can do on your next move That gives you 2x2x2x2 16 possible 3d move strategies Then you have 2 possible first move strategies 4 possible 2d move strategies and 16 possible third move strategies That is 2x4x16 128 strategies that depend on observed behavior of other guy In a repeated game that consists of four repetitions of a stage game that has a unique Nash equilibrium A There are four subgame perfect Nash equilibria B There are 24 16 subgame perfect Nash equilibria C There is only one subgame perfect Nash equilibrium D The number of subgame perfect Nash equilibria varies depending on the details of the game Twice Repeated Prisoners Dilemma Two players play two rounds of Prisoners dilemma Before second round each knows what other did on the first round Payoff is the sum of earnings on the two rounds Prisoners Dilemma Player 2 Cooperate P L A Cooperate y E R 1 Defect Defect R R S T T S P P T R P S Temptation Reward Punishment Sucker Two Stage Prisoners Dilemma Working back Player 1 Cooperate Defect Player 2 Cooperate Player 1 C C 2R 2R Playe Pl 2 r1 D D C Cooperate Defect Player 1 D C Player 1 C Pl 2 D C D R S R T R P S R R T R S R P T R Defect D Player 1 C D Pl 2 C D C D Etc etc C Pl 2 D C D C D P R P S T P P R P T P S 2P 2P Two Stage Prisoners Dilemma Working back further Player 1 Cooperate Defect Player 2 Cooperate Player 1 C C Playe Pl 2 r1 D 10 21 D C Cooperate Defect Player 1 D C Player 1 C Pl 2 D C D C D C 21 11 10 0 11 1 10 11 21 22 11 12 Defect Player 1 D C D Pl 2 D C Pl 2 D C 11 22 12 11 0 1 D C 11 2 12 11 12 1 D Longer Game What is the subgame perfect outcome if Prisoners dilemma is repeated 100 times Work backwards In last round nothing you do affects future so you play the dominant strategy for stage game defect Since last round is determined nothing you do in next to last round affects future so you play dominant strategy for stage game defect Work your way back Only subgame perfect outcome is Defect always More generally In a subgame perfect equilibrium for a finitely repeated game where the stage game has a unique N E the moves in the last stage are determined for each person s strategy Given that the moves in the last stage don t depend on anything that happened before the Nash equilibrium in previous stage is uniquely determined to be the stage game equilibrium And so it goes All the way back to the beginning Infinitely repeated game Wouldn t make sense to add payoffs You would be comparing infinities Usual trick Discounted sums Just like in calculating present values We will see that cooperative outcomes can often be sustained as Nash equilibria in infinitely repeated games Why consider infinite games We only have finite lives Many games do not have known end time Just like many human relationships Simple example A favorite of game theorists After each time the stage game is played there is some probability d 1 that it will be played again and probability 1 d that play will stop Expected payoff discounts payoffs in later rounds because game is less likely to last until then Cleaning house as a Repeated Prisoners Dilemma Maybe a finite game if you have a fixed lease and don t expect to see roommate again after lease expires Most relationships don t have a known last time Usually some room for residual good will In a repeated game after each round of play a fair coin is tossed If it comes up heads the game continues to another round If it comes up tails the game stops What is the probability that the game is played for at least three rounds A B C D E 1 3 2 3 1 4 1 2 1 8 Calculating sums In a repeated game with probability d of continuation after each round the probability that the game is still going at round k is dk 1 Calculate expected winnings if you receive R so long as the game continues R dR d2R d3R d4R R 1 d d2 d3 d4 What is this infinite sum Adding forever The series 1 d d2 d3 d4 Is known as a geometric series When d 1 this series converges That is to say the limit as n approaches infinity of 1 d d2 d3 d4 dn exists Let S 1 d d2 d3 d4 Then dS d d2 d3 d4 And S dS 1 So S 1 d 1 S 1 1 d Infinitely repeated prisoners dilemma and the Grim Trigger Strategy Suppose 2 players play repeated prisoners dilemma where the probability is d 1 that you will play another round after the end of each round The grim trigger strategy is to play cooperate on the first round and play cooperate on every round so long as the other doesn t defect If the other defects the grim trigger strategy plays defect on all future rounds When is there a symmetric SPNE where all play Grim Trigger Suppose that the other player is playing Grim Trigger If you play Grim Trigger as well then you will cooperate as long as the game continues and and you will receive a payoff of R Your expected payoff from playing Grim Trigger if the other guy is playing Grim Trigger is therefore R 1 d d2 d3 d4 R 1 d What if you defect against Grim Trigger If you defect and the other guy is playing Grim Trigger you will get a payoff of T R the first time that you defect But after this the other guy will always play defect The best you can do then is to always defect as well Your expected payoff from defecting is therefore T P d d2 d3 d4 T Pd 1 d Cooperate vs Defect If other guy is playing Grim trigger and nobody has yet defected your expected payoff from playing cooperate is R 1 d If other guy …