Assume a one-shot game G is repeated T times. Let G(T) denote this repeated game. This expands the strategy space for each player. Player i’s strategy would be a sequence of moves, one move for each round. More precisely, a player’s strategy specifies the action to be taken in each stage for each possible history of play through the previous round. This introduces the possibility of reputations, threats, and rewards.
The method for analyzing repeated games, as well as sequential games, is backward induction – look forward and reason backward.
1) Fixed Repetitions:
If T is fixed, how would the solution toG(T) be related to the solution to G? Would the equilibrium change? Suppose T = 2. Clearly, the solution in the second round must be the solution to the one-shot game. Now step back to the first round. Each player looks forward to see that the solution to the second round will be the solution to the one-shot game. That is, payoffs in the second round will not depend on play in the first round. Therefore, best responses in the first round are the one-shot game best responses. So, the game unravels.
Result: The solution to a repeated game with fixed repetitions is the sequence of solutions to the one-shot game, if the solution to the one-shot game is unique.
This is sometimes called the end-game problem.
If the solution to the one-shot game is not unique, play in the first round can influence which of the multiple equilibria is played in the second round.
2) Indefinite Repetitions:
If the horizon were infinite – or the game were repeated a random number of times – the repeated game would not unravel, and richer strategies might comprise the equilibrium.
Consider a trigger strategy: if I play Nice until you play Nasty, then I play Nasty forever. Both players following a trigger strategy is a Nash equilibrium – and each would play Nice in each round – but would it really be in my interest to follow through on my threat if you do play Nasty? It turns out that this isn’t a problem. The Folk Theorem (Friedman 1971) guarantees that a trigger strategy works in supporting a large number of outcomes as long as the players do not discount the future too heavily.
Otherwise I might play Tit-for-Tat- penalize your Nasty-playing opponent for only one round; that is, your strategy in a round is your opponent’s strategy in the previous round. Experimental proof suggests that tit-for-tat is hard to beat. But it’s an awful strategy if there’s a possibility of mistakenly accusing the other player of playing Nasty.
3) Cartel Enforcement:
Enforcing a collusive agreement is more likely to be successful in long-term relationships. In the one-shot game, as we saw above, cheating is the dominant strategy. If the cartel has a known horizon, again every member cheats. Since penalties in an infinitely repeated game can support the cartel equilibrium, cheating can be deterred. However, the members must not discount the future too heavily; in particular, the probability that G(T) ends can’t be too high. Also of course precise detection is assumed.
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