Economic and Game Theory
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Editors: David K. Levine, Wolfgang Pesendorfer
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| 01/12/2000 08:55 AM by David K. Levine; When can cooperation be sustained in the Prisoner's Dilemma game? |
A typical prisoners dilemma game has payoffs
cooperate
defect
cooperate
3,3
0,4
defect
4,0
1,1
If both players
cooperate, they get the greatest possible per player payoff of three: if one
defects the per player payoff is two; if both defect it is one. However,
regardless of how the other player plays it is always better to defect since it
increases the payoff, in this example by one. This is called a dominant
strategy equilibrium because the best strategy does not depend on what the other player is doing.
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.
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