QR 38 3/20/07, More on repeated games in IR I.Folk theorem II.Other solutions to the PD III.Repeated PDs in practice

I. Folk theorem Two players using TFT can sometimes maintain cooperation in an infinitely-repeated PD. However, TFT is not always the best strategy; the best depends on what the other player is doing. That is, TFT is not the only equilibrium strategy. In fact, many equilibria exist in the repeated PD. Almost anything can happen, if we allow for mixed strategies; there is an infinite number of equilibria. This proposition is known as the folk theorem.

The folk theorem CooperateDefect Cooperate5, 50, 6 Defect6, 02, 2

The folk theorem The folk theorem says that strategies that guarantee players at least what they could achieve by playing the one-shot equilibrium can be sustained by some kind of punishment strategy (if discount factors are high enough). Here, this means that outcomes with payoffs above 2, 2 can be sustained.

The folk theorem Outcomes in the region bordered by white are the feasible per-round payoffs in this game. The red lines indicate the minimum payoff that must be maintained in equilibrium. Any outcome in the resulting area ((2, 2), A, (5, 5), B) can be an equilibrium. 0, 0 6, 0 0, 6 2, 2 5, 5 A B

The folk theorem In IR, the folk theorem is usually seen as a “positive” result, since it means that cooperation can emerge. But among game theorists, more “negative” because it means that predicting the outcome of a repeated game is impossible.

II. Other solutions to the PD Are there ways other than repetition to solve the PD? Many have been suggested, but most rely on changing the payoffs so that the game is no longer a PD: Asymmetric players External enforcement External changes to payoffs (rewards or punishments)

Asymmetric players – alliance example Consider states cooperating in a military alliance. If states are about the same size, they could be in a PD, each preferring that the other spend more on defense. But if one is much larger than the other, the larger will gain most of the benefits from having an adequate defense.

Alliance example So, the payoff to the large state of cooperating (spending more) is higher than the payoff to the small state of cooperating. So the large state no longer has PD payoffs. Equilibrium outcome is exploitation of the large by the small.

Alliance example Symmet- ric alliance High spending Low spending High spending 3, 31, 4 Low spending 4, 12, 2 Asym- metric alliance High spend- ing Low spend- ing High spend- ing 4, 33, 4 Low spend- ing 2, 11, 2 Large Small

Alliance example Could generalize this example to multiple players. Application to NATO: expect the U.S. to spend disproportionately on defense, and it does.

External enforcement If external enforcement is available, can overcome the PD. But this is rarely an option in IR.

External changes to payoffs Could incorporate additional punishments for defection, such as trade sanctions. –This is an example of issue linkage, common and important in IR. Or, could use rewards rather than punishments. Provide additional foreign aid; this is common in military coalitions.

External changes to payoffs The idea of either rewards or punishments is to make defection no longer a dominant strategy. –Rewards increase the payoff to cooperation, punishments decrease the payoff to defection. But both suffer from some difficulties: –Linked payoffs must be contingent, only punish for D, reward for C –So C and D have to be clearly observable, no contaminated by accidents.

Problems with external changes to payoffs –It must be possible to stop the flow of rewards and punishments at any time; not one-shot. These are problematic. For example, if Jordan was getting increased aid to keep it from aiding Iraq, have to be able to observe whether smuggling is occurring; aid must be ongoing, not one-shot.

Problems with external changes to payoffs Credibility: punishment in the repeated PD is credible, you just revert back to the Nash equilibrium of the one-shot game. But linkages may not be an equilibrium – are they subgame perfect? U.S. may renege on promises of aid; or not be willing to bear the costs of carrying out a threat.

III. Applications to IR Analysis of the repeated PD has provided the foundation for studies of cooperation in IR. Advantages of this approach: Provides unifying framework for analysis Relies on self-enforcement and reciprocity, appropriate assumptions for IR Structure of PD widely appropriate for situations where cheating is a temptation Highlights importance of good information, use of effective punishment strategies, credibility of threats and promises

Implications of repeated games for IR A longer shadow of the future should encourage cooperation, because more weight on the future means that cheating becomes less attractive. This only holds if the underlying game is PD

Repeated games in IR Results of case studies: PD not as prevalent as many assume –Many cases look more like harmony, deadlock, or battle of the sexes; maybe chicken. Then repetition doesn’t encourage cooperation as in the PD. –Example: alliance partners often continue to cooperate after a war, because they believe they are playing an assurance game. But this dissipates over time, they move to a PD or worse.

Repeated games in IR But, fairly strong support for general idea that the shadow of the future encourages cooperation. –In 1914, leaders felt they were vulnerable and cared only about the short term. –Contrast this to debt negotiations, where bankers and debtors have long-term relationships and generally find a way to cooperate.

Limitations of repeated game analysis Where this framework falls short: Neglects domestic politics Reciprocity can backfire: “echo effects” –Reagan and the collapse of U.S.-Soviet détente –Bargaining over tariffs