Five Games Beyond the Prisoner's Dilemma

In the classical prisoner’s dilemma, each player, acting rationally, is incentivized to defect. There are clear payoff rankings: the highest payoff occurs when you defect while the other player cooperates; the next highest when both players cooperate; the next when both defect; and the lowest when you cooperate while the other defects. Other games involve different payoff rankings.

Stag Hunt

The Stag Hunt involves two hunters seeking food. They can either hunt together for a stag, which provides a large shared meal, or each go after a hare alone, giving a smaller but guaranteed meal. If one hunter goes for the stag while the other defects to hunt a hare, the stag hunter ends up with nothing. If both go for the stag, they achieve the greatest reward. If both go for hares, they receive a smaller reward. The highest payoff for a player comes from cooperation, unlike the prisoner’s dilemma, where the greatest payoff occurs when one defects while the other cooperates. There is no incentive to defect when the other hunter cooperates. There exist two Nash equilibria: both hunters seeking a stag, and both hunters going for a hare. Neither player can improve their payoff by changing strategy alone in either case. An interesting question is whether the Stag Hunt or the classical prisoner’s dilemma is a better model for security cooperation and arms control. Does defection while the other cooperates have a better payoff than mutual cooperation, or not?

Chicken

Chicken involves two drivers speeding toward each other. Each can either swerve or go straight. Swerving is mildly humiliating, so it gives a lower payoff than going straight, and the payoff is even lower if only one driver swerves. The highest payoff is achieved by the driver who goes straight while the other swerves. The second highest comes from both drivers swerving. The lowest occurs when both continue straight, resulting in a crash. Unlike the prisoner’s dilemma, the payoff is higher for a driver who cooperates while the other defects than for both defecting. The two equilibria are asymmetric: each driver is incentivized to do the opposite of the other. A player who can credibly signal commitment to defection makes it rational for the other to cooperate, since mutual defection is the most costly outcome.

In geopolitics, it is interesting to consider how the side that can make the other believe it will not cooperate gets the other to cooperate, so the weaker country that can credibly signal its unwillingness to cooperate is not necessarily the one that loses. The Iranian regime may consider the cost of cooperation with the United States to be equal to the cost of defection. Cooperation would be catastrophic, as it would undermine the regime’s legitimacy and potentially trigger a coup or revolution, while defection would be catastrophic, since the regime could collapse from war with the US. If Iran manages to credibly signal a decision to defect, then the US can either defect, which would mean continuing the war, depleting resources, and losing political legitimacy, or cooperate, with a comparatively better payoff. The Chicken game offers a plausible explanation of how Iran, a far smaller and weaker nation, can successfully extract a concession from the United States.

Battle of the Sexes

The Battle of the Sexes involves two partners who want to spend the evening together but disagree on what to do. One prefers the opera and one wants to watch a sports match, but both would rather be together at either event than be apart. The two equilibria are both partners at the opera and both at the match, and both are stable. Unlike the Stag Hunt, where both players know they want to coordinate on a specific outcome, both partners agree they want to coordinate but disagree about which equilibrium to coordinate on. This is a strong model for setting standards and creating institutions: setting a technical standard or forming an alliance. Each party knows it wants to coordinate, since the payoff is always higher for cooperation than defection, yet each wants the shared standard to be its own. It is interesting to explore how the Battle of the Sexes connects to Schelling points as a way to coordinate on a shared framework. As with Chicken, bargaining power can come from the ability to move first or to commit to a certain action, forcing the other player to choose between cooperation, which gives a higher payoff, and defection, which gives a lower one.

The Ultimatum Game

Unlike the simultaneous games above, the Ultimatum Game is sequential. There is a pot, say ten dollars. The Proposer offers a split. The Responder either accepts, in which case both get the proposed amounts, or rejects, in which case both get nothing. The Responder, faced with any positive offer, should accept, since any positive amount beats nothing. Given this, the Proposer should offer the smallest possible positive amount and keep the rest. However, in many human experiments, offers below around twenty to thirty percent tend to be rejected, since people are reluctant to accept a humiliating split. People are willing to pay a cost to punish unfairness, demonstrating that emotions like fairness, spite, and reputation ought to be considered as parts of the payoff function. In geopolitics, many countries appear to exhibit the same behaviour. A weaker state, such as Venezuela, will sometimes reject what it considers an unfair or coercive deal at a cost to itself. It is also worth considering the rational benefits of rejecting an unfair deal: it helps develop a reputation of not being pushed, and might incentivize a fairer proposal next time. Reputation becomes a critical component of the payoff function, shifting the incentives.

Public Goods

Finally, the Public Goods game is the generalization of the prisoner’s dilemma to N players. Each player starts with an endowment and chooses how much to put into a common pot. The pot is multiplied by some factor r between 1 and N, and the result is divided equally among everyone, including both contributors and non-contributors. Every token you contribute therefore returns r / N to you personally, which is less than 1 because r is smaller than N. So you lose money on every token you put into the pot. The dominant strategy is to contribute nothing and free-ride on everyone else’s contributions. Yet because the pot is multiplied by a factor greater than 1, it is in each individual’s interest for the pot to be maximized, which would mean everyone donating their tokens. The Nash equilibrium and the social optimum rest in opposite directions, the same dynamic as the prisoner’s dilemma, but now in a group. The Public Goods game is a far better model for international cooperation than the prisoner’s dilemma, since two countries are rarely acting together in isolation. It is interesting to consider what happens when there are varying contributions and payoffs for different players, which is arguably a more accurate model of geopolitical cooperation or the allocation of state resources. Group size and dynamics are likely to be significant considerations in such a case.