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Infinitely Repeated Games. In an infinitely repeated game, the application of subgame perfection is different - after any possible history, the continuation must be a NE - but after any history each subgame looks like the original game We cannot use generalized backward induction, because there is no last period - the trick is to recognize that each subgame is identical to the whole game - this simplifies things since only need to consider the initial game A technical issue: we must discount future payoffs using a discount factor ( 0<δ<1) - without discounting the payoffs are not finite

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Repeating the one-shot NE is always an SPNE. If (2) plays D always - then a NE can come from (1) playing D - everyone playing D always is a SPNE In the finite repeated prisoner’s dilemma game a unique SPNE is {(D, D), (D, D)…(D, D)} In the infinitely repeated version, there are multiple SPNE as long as players are sufficiently patient - one SPNE is always {(D, D), (D, D)…(D, D)} - the existence of other SPNE implies that cooperation is possible - Cooperation can yield a higher long-run payoff 2121 CD C3, 31, 4 D 4, 1 2, 2

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One-shot: unique NE is {D, D} Finite repeated: unique SPNE {(D, D), …, (D, D)} Infinitely repeated: Claim: if players are sufficiently patient (if the discount factor is sufficiently high), then cooperation can be sustained in a SPNE in the infinitely repeated game. Nash Reversion - consider the following “trigger strategy” Play C if you have always seen C, otherwise play D. - if both players follow this strategy, we always observe (C, C) Infinite prisoner’s dilemma 2121 CD C3, 31, 4 D 4, 1 2, 2

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“trigger strategy” Under what conditions is there a SPNE in trigger strategies Is the trigger strategy a best response for (1) if (2) uses the trigger strategy? For the trigger strategy to be a BR for (1): CD C3, 31, 4 D 4, 1 2, 2 Patience matters in infinitely repeated games If players do not value the future at all, the analysis of repeated games is the analysis of repeated one-shot games

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Using the “trigger strategy” we can get many different paths Path of cooperation in even periods and non-cooperation in odd periods - - then a trigger that goes to D if they ever observe anything different from this and we would get DD forever Path of alternating Or any other path you can think of Even if the stage game has an unique equilibrium, there may be SPNE in the infinitely repeated game in which no stage’s outcome is a NE of a stage game The Folk Theorem: take a game, play it infinitely often -if players are patient enough you can get a wide variety of subgame paths -some paths might require a very high discount rate – a lot of patience -you need further refinements in order to predict behavior The Folk Theorem.

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2 firms: they play Bertrand each period for an infinite # of periods - the firms discount future payoffs Recall that in the one-shot game with two firms P 1 = P 2 = C is the unique NE. - Any NE in the one shot game is a NE in the repeated game If firm (2) plays P 2 = C every period Then firm (1) gets 0 profits no matter what, so P 1 = C i s a BR 1 P1 = P2 = C is also a SPNE In the infinitely repeated Bertrand game, many other price paths are possible - there are other SPNEs ( a lot of them) - we will focus on the one in which both players choose the monopoly price P M Infinitely repeated Bertrand.

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Dynamic Bertrand with tacit collusion.

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Collusion as N rises.

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This is the folk theorem A reasonable focal point – too many possibilities - firms choose symmetric strategies - on the frontier so each gets Anything can happen Because there are so many equilibria Using SPNE alone we cannot predict what is to happen in infinitely repeated games Any price between the marginal cost and the monopoly price can be sustained Any payoffs in the triangle can be the average per period payoffs in a SPNE - if players are sufficiently patient they can use a Nash reversion strategy - anything beats getting zero forever - However, with Nash reversion, players cannot do worse than the NE Too many possibilities exist.

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Cournot with Nash reversion. In Cournot, there are also SPNE with trigger strategies - firms can tacitly collude - what keeps it up is the prospect of future gains Average per period payoffs, that can occur in SPNE in infinitely repeated game, using Nash revision, if sufficiently patient.

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Minimax payoffs.

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Cournot with Non-Nash reversion. What does non-Nash reversion strategy look like? The Folk Theorem states that the only lower bound on payoffs in an infinitely repeated game when players are sufficiently patient is given by the minimax payoffs (rather than the Nash eqm. Payoffs) One player can force the other to the minimax payoffs In Cournot, the minimax payoff is zero Infinitely repeated games pose problems for analysis - because of the infinite # of equilibria it is difficult to predict the path of play - it is difficult to perform comparative statics - we can explain anything; the analysis does not add much value

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Using infinitely repeated games. One response is to ignore repeated game considerations (1) focus on simple dynamic games (2) assume players repeat a one-shot NE when it has intuitive properties Another response is introduces a state space - assumes players’ strategies are a function of the current state, not of the history - value functions with Markov perfect equilibria - this removes history dependence A third response is to use the insights to explain conditions under which cooperation can occur -.make standard assumptions such as players are rational - find restriction on parameters that make cooperation possible

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Tit for Tat. This repeated PD strategy has only one period of memory; players cannot carry a grudge Strategy (TFT):t=1 Cooperate t>1 Cooperate if opponent played C in period t – 1 Defect if opponent played D in period t – 1 This is not equilibrium analysis There are some good properties of TFT 1) nice – starts out cooperating, never initiates defection 2) simple – easy to follow, easy for opponent to understand 3) forgiving – after D, it is willing to cooperate again if opponent does 4) provocable – never lets cheating go unpunished Axelrod’s experiments: he collected strategies for computerized Prisoner Dilemma games - In a round-robin tournament each strategy played every other strategy. - TFT was the winner. - Nevertheless, a simple defect strategy will always beat TFT, but provides a low payoff

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