Repeated Games This week we examine the effect of repetition on strategic behavior in games with perfect information If a game is played repeatedly with the same players the players may behave very differently than if the game is played just once a one shot game e g repeatedly borrow a friend s car versus renting a car Two types of repeated games Finitely repeated the game is played for a finite and known number of rounds for example 2 rounds repetitions Infinitely or Indefinitely repeated the game has no predetermined length players act as though it will be played indefinitely or it ends only with some probability Finitely Repeated Games Writing down the strategy space for repeated games is difficult even if the game is repeated just 2 rounds For example consider the finitely repeated game strategies for L R the following 2x2 game played just twice U For a row player D U1 or D1 Two possible moves in round 1 subscript 1 For each first round history pick whether to go U2 or D2 The histories are U1 L1 U1 R1 D1 L1 D1 R1 2 2 2 2 8 possible strategies 16 strategy profiles Strategic Form of a 2 Round Finitely Repeated Game This quickly gets messy L2 L2 R2 U2 U2 L1 D2 L2 U2 D2 R2 R2 R1 D2 U1 D1 L2 U2 D2 R2 Finite Repetition of a Game with a Unique Equilibrium Fortunately we may be able to determine how to play a finitely repeated game by looking at the equilibrium or equilibria in the one shot or stage game version of the game For example consider a 2x2 game with a unique equilibrium e g the Prisoner s Dilemma higher numbers years in prison are worse Does the equilibrium change if this game is played just 2 rounds A Game with a Unique Equilibrium Played Finitely Many Times Always Has the Same Subgame Perfect Equilibrium Outcome To see this apply backward induction to the finitely repeated game to obtain the subgame perfect Nash equilibrium spne In the last round round 2 both players know that the game will not continue further They will therefore both play their dominant strategy of Confess Knowing the results of round 2 are Confess Confess there is no benefit to playing Don t Confess in round 1 Hence both players play Confess in round 1 as well As long as there is a known finite end there will be no change in the equilibrium outcome of a game with a unique equilibrium This is also true for zero or constant sum games Finite Repetition of a Sequential Move Game Recall the incumbent rival game In the one shot sequential move game there is a unique subgame perfect equilibrium where the rival enters and the incumbent accommodates Does finite repetition of this game change the equilibrium Should it The Chain Store Paradox Selten 1978 proposed a finitely repeated version of the incumbent rival entry game in which the incumbent firm is a monopolist with a chain of stores in 20 different locations He imagined that in each location the chain store monopolist was challenged by a local rival firm indexed by f 1 2 20 The game is played sequentially firm 1 decides whether to enter or not at location 1 chain store decides to fight accommodate then firm 2 etc Consider the last rival firm 20 Since the incumbent gains nothing by fighting this last firm and does better by accommodating he will accommodate and firm 20 will therefore choose to enter But if the incumbent will accommodate firm 20 there is nothing he gains from fighting firm 19 etc By backward induction each firm f 1 2 20 chooses Enter and the Incumbent always chooses Accommodate This game theoretic solution is what Selten calls the induction hypothesis What about Deterrence Selten noted that while the induction argument is the logically correct gametheoretic solution assuming rationality and common knowledge of the structure of the game it does not seem empirically plausible why Under the enter accommodate equilibrium the incumbent earns a payoff of 2x20 40 But perhaps he can do better for instance suppose the incumbent chooses to fight the first 15 rivals and accommodate the last 5 If this strategy this is common knowledge then the first 15 stay out and earn a payoff of 1 each while the incumbent earns 5x15 2x5 85 40 Even if some of the first 15 rivals choose to enter anyway say 2 5ths 6 the incumbent can still be better off in that case he gets 5x 15 6 2x5 55 40 This contradiction between the game theoretic solution and an empirically plausible deterrence hypothesis is what Selten labeled the chain store paradox The paradox results from the game theoretic assumption that all players presume one another to be perfectly rational and know via common knowledge the structure of the game They are thus led to conclude that the incumbent will never ever fight By this standard fighting would be an irrational move and would never be observed Finite Repetition of a Simultaneous Move Game with Multiple Equilibria The Game of Chicken Consider 2 firms playing the following one stage Chicken game The two firms play the game N 1 times where N is known What are the possible subgame perfect equilibria In the one shot stage game there are 3 equilibria Ab Ba and a mixed strategy where row plays A and column plays a with probability and the expected payoff to each firm is 2 Games with Multiple Equilibria Played Finitely Many Times Have Many Subgame Perfect Equilibria Some subgame perfect equilibrium of the finitely repeated version of the stage game are 1 Ba Ba N times 2 Ab Ab N times 3 Ab Ba Ab Ba N times 4 Aa Ab Ba N 3 rounds Strategies Supporting these Subgame Perfect Equilibria 1 Ba Ba Row Firm first move Play B Avg Payoffs 4 1 Second move After every possible history play B Column Firm first move Play a Second move After every possible history play a 2 Ab Ab Row Firm first move Play A Avg Payoffs 1 4 Second move After every possible history play A Column Firm first move Play b Second move After every possible history play b 3 Ab Ba Ab Ba Row Firm first round move Play A Avg Payoffs 5 2 5 2 Even rounds After every possible history play B Odd rounds After every possible history play A Column Firm first round move Play b Even rounds After every possible history play a Odd rounds After every possible history play b What About that 3 Round S P Equilibrium 4 Aa Ab Ba 3 Rounds only can be supported by the strategies Row Firm first move Play A Second move If history is A a or B b play A and play B in round 3 unconditionally If history is A b play B and play B in round 3 unconditionally If history is B a play A and play A in round 3 unconditionally Column Firm