Economic and Game Theory
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A typical prisoners dilemma game has payoffs
If the game is repeated for a fixed number of periods, then the previous analysis is still valid in the final period. Working backward, in the next to last period players should realize their opponent is going to defect in the final period regardless, so defection is the best strategy in the next to last period too. Continuing on, we see the defection is the equilibrium in all periods.
If the game is repeated indefinitely, the situation changes. Suppose that there is discounting so that future payoffs are worth less than current payoffs. Suppose the interest rate is r. The strategies the provide the most incentive for the other player to cooperate are trigger strategies: cooperate as long as the other player cooperates, but defect if they have defected even once. Suppose the other player is playing a trigger strategy. Is it best to cooperate or defect? Cooperation yields a payoff of 3 in every period. That is a present value of 3/r. Defecting yields a higher payoff of 4 in the first period, but at most 1 in each subsequent period. Equivalently, this is a payoff of 1 every period including the first, plus a bonus of 3 in the first period. The present value of 1 every period is 1/r. The present value of the bonus is 3, so the total present value of defecting in the first period is 3+(1/r). Defection is optimal if 3+(1/r)>3/r, or r>2/3. In other words, if r is no larger than 2/3 then cooperation is acceptable.
The overall conclusion is that if the players are sufficiently patient (the interest rate is low enough) it is an equilibrium for both players to use trigger strategies, resulting in cooperation in every period.[Manage messages]