TOPIC 6 REPEATED GAMES The same players play the same game G period after period. Before playing in one period they perfectly observe the actions chosen in the previous periods. (perfect observability). G is the stage-game (or constituent game). G(T) is the repeated game with horizon T (that is, played T times). The strategic behaviour in G(T) can be very different from the behaviour in G (played once). We will focus on a repeated Prisoners´ Dilemma (PD): under what conditions can cooperation be obtained between selfish players?

REPEATED GAMES: SOME PREVIOUS ISSUES. How do players evaluate a flow or stream of payoffs? By the sum of their present or discounted value, that is, its value in t = 1 euros (present euros). The discount factor of a player, d, is the present value (t = 1 euros) of one euro obtained tomorrow (t = 2). With an interest rate r > 0, d =1/(1+r). Therefore, 0 < d < 1. It measures the relative importance of the present versus the future. If d is close to 1 means that the future is very important. If it is close to 0, future payoffs are then relatively unimportant. Besides the interest rate, some other factors influence d, such as the tastes, the probability of continuing active in the game….

REPEATED GAMES: SOME PREVIOUS ISSUES. The horizon of a repeated game. - finite horizon: there is a last period (a deadline) which is common knowledge among the players. - infinite horizon: when the relationship does not have a predetermined lenght. For instance, in each period there is a positive probability p of playing again in the next period and a probability (1 – p) of ending the game. This situation is equivalent to an infinite horizon repeated game, where the discount factor of the players is d´ = p.d, where d is the real discount factor.

REPEATED GAMES: SOME PREVIOUS ISSUES. A strategy of a player in G(T) is a complete plan that specifies in every period what action should be chosen as a function of each possible previous history of the game. The history of the game in period t is just the sequence of vectors of actions observed until period t – 1. Unconditional or uncontigent strategies: for instance in a repeated PD, “NC after any history (no matter what your rival has done in the past)” or “C after any history”.

REPEATED GAMES Strategies of a conditional cooperator (reciprocity): play cooperatively as long as your rival does so, but any defection on his part triggers a period of punishment. TRIGGER STRATEGY: “Begin by cooperating (C) in t = 1, then C if everybody has cooperated at every previous period. But, if any player does not cooperate, then switch to the strategy NC after any history” This is an unforgiven strategy. The lenght of the punishment does not depend on the behaviour of your opponent during the punishment. TIT-FOR-TAT STRATEGY: “Cooperate at the start and then, play in each period as your opponent did in the previous period.”

COLLUSION Two firms set prices simultaneously. The good produced by every firm is homogeneous and its aggregate market demand is given by a function D(p). Both firms have an identical constant average (and marginal) cost c >0. They play repeatedly this game and have identical discount factor d > 0. Find the conditions for collusion (in the monopoly price) to be sustainable in the repeated game. What happens if the number of firms increases? (does collusion depend on market concentration?)

COLLUSION Two firms with identical discount factor d = 0,65, compete in prices in two identical and independent markets. In market 1, it takes one period to observe the opponent´s price, but in market 2, it takes two periods to observe the opponent´s price and then react to it (there is an information lag). Discuss if collusion is sustainable in each market separatedly. Asume now that both firms link collusion to its maintenance in both markets (multimarket contact).

COLLUSION Two firms with identical discount factor, compete in prices. The demand function at date t is μ t-1 D(p t ), where μ.d < 1. Derive the set of discount factors such that full collusion is sustainable in the repeated game. What would this model predict about the relative ease of sustaining collusion in expanding and declining industries?

QUALITY AND REPUTATION. A consumer chooses whether or not to purchase a service from a firm. If the consumer does not purchase, then both players receive a payoff of 0. If the consumer decides to purchase, then the firm must decide whether to produce high or low quality. In the former case, both players have a payoff of 1. In the latter case, the firm´s payoff is 2 and the consumer´s payoff is -1. A) Suppose this is the stage-game of an infinitely repeated game. Find an equilibrium in which high quality is provided every period. B) Suppose now that a long-lived firm (B) plays against a sequence of short-lived consumers (A t ) who only live one period but perfectly observe how B behaved in the past. C) What problems might appear if the consumers have to incur a cost of checking B´s history? What would happen if there is imperfect observation? For instance, suppose that even if B intends to provide high quality, there is a small probability of making a mistake and providing low quality.

THE FIRM AS A DEPOSITORY OF REPUTATION. Suppose now that firms (B t ) are also short-lived. For example, they only live for one period. What would be the equilibrium in this case? Suppose that each B t lives two periods. In period t, B t plays the quality game and in period t + 1 retires and lives of his savings. B 1 creates the firm B-Honest Co. (BH) with a good reputation of providing high quality. Describe an equilibrium of the repeated game in which each B t acquires BH and maintains its reputation. Assume for example, that the market price of BH (with reputation) is 10 (and it would be 0 without reputation).

THE FIRM AS A DEPOSITORY OF REPUTATION. The consumers´ strategy: “A t trusts B t if B t owns BH and no previous owner of BH has provided low quality in the past. In other case, A t does not purchase.” Given this strategy and if B t can sell BH in period t + 1 by 10 monetary units, his optimal strategy is to acquire BH and keep its good reputation providing high quality. Each owner of BH is willing to maintain its reputation in order to recover its acquisition cost (his consumption in the second period depends on this fact).

FINITELY REPEATED GAMES The only perfect equilibrium outcome in a finitely repeated prisoners´dilemma is “always do not cooperate”, that is, the repetition every period of the equilibrium of the stage game. If the stage game has a multiplicity of inefficient Nash equilibria (NE), then the previous result does not hold. That is, it might exist a perfect equilibrium of the finitely repeated game in which there is cooperation in all except for the last period. Intuition: it is possible to punish a player for deviating in the next-to-last period by specifying that if he does not deviate the NE he prefers will occur in the last period, and that deviations lead to the NE he likes less.

A GAME WITH OVERLAPPING GENERATIONS OF PLAYERS. Consider a repeated game in which overlapping generations of players live for 10 periods, so that at each date t there is one player of age 10 who is playing his last round, one player of age 9 who has two rounds still to play, and so on down to the new player who will play 10 times. Each period, the 10 players simultaneously choose whether to work hard (at a cost 1) or to shirk (at a cost zero), and their choices are revealed at the end of the period; players share equally in the resulting output, which is twice the number who choose to work. Payoffs in the repeated game are the sum of the per-period utilities. Find a Nash equilibrium of the repeated game in which everybody, except the player of age 10 cooperates (works hard).

A GAME WITH OVERLAPPING GENERATIONS OF PLAYERS. Strategy: “Age-10 players always shirk. So long as no player has ever shirked when his age is less than 10, all players of age less than 10 work. If a player has ever shirked when his age is less than 10, then all players shirk.” Notice that the “cooperative” equilibrium we have derived remains an equilibrium if we suppose that workers observe only the total number of shirkers but not their identities.