We now have met two dilemmas of fundamental importance in human affairs. Are there others?

In 1966, Melvin J. Guyer and Anatol Rapoport, then both at the University of Michigan, catalogued all the simple games. The simplest true games have two players making a choice between two alternatives. It is reasonable to think that these “2 × 2” games ought to be particularly important and common. The prisoner’s dilemma and chicken are 2 × 2 games, of course. There are seventy-eight distinct 2 × 2 games when payoffs are simply ranked rather than assigned numerical values.

A symmetric game is one where the payoffs are the same for each player under comparable circumstances – where neither player is in a privileged position. If player A is the lone cooperator, his payoff is the same as player B’s would be if he was the lone cooperator, and so forth. Symmetric games are the easiest to understand, and probably the most important in social interactions. People are much the same the world over. No conflicts are so common or so bitter as those between people wanting the same thing. So let’s look at the symmetric games.

There are just four payoffs to worry about in a symmetric 2 × 2 game. Let “CC” be the payoff, to each of the two players, when both cooperate. “DD” is the payoff when both defect. When one player cooperates and the other defects, the payoff to the lone cooperator will be called “CD,” and the payoff to the lone defector, “DC.”

All the variety of 2 × 2 symmetric games rests in the relative values of the four payoffs CC, DD, CD, and DC. As usual, let’s have the players rank them in order of preference (and they must agree on the ranking, it being a symmetric game). Let’s further assume that there are no “ties,” that there is always a distinct preference between any two payoffs.

Any given preference ordering of the four payoffs defines a game. For instance, when

– meaning that the DC outcome is preferred to CC, which is preferred to DD, which is preferred to CD – the game is a prisoner’s dilemma. (The usual further requirement, that the average of the DC and CD payoffs be less than the CC payoff, applies only when players hare assigned numerical values to the payoffs. Here we’re just ranking them.)

There are twenty-four possible rankings of four payoffs, and thus twenty-four symmetric 2 × 2 games. Not all are dilemmas. In most, the correct strategy is obvious.

The disturbing thing about the prisoner’s dilemma and chicken is the way that the common good is subverted by individual rationality. Each player desires the other’s cooperation, yet is tempted to defect himself.

Let’s see what this means in general terms. The payoff CC must be preferred to CD. That means that you’re better off when your partner returns your cooperation. Also, DC must be better than DD. You still hope the other player cooperates, even when you yourself defect.

Of the twenty-four possible orderings of four payoffs, exactly half have the CC payoff higher than the CD payoff. Likewise, exactly half have DC preferred to DD. Just six of the possible orderings simultaneously meet both requirements. The six look like this:

Not all of these are troublesome. If defection is bad all around, everyone will avoid it. For a true dilemma to exist, there must be a temptation to defect – otherwise, why defect?

In the prisoner’s dilemma, there is every temptation to defect. No matter what the other player does, you are better off defecting. The temptation need not be that acute to pose a dilemma. You may have a hunch what the other player is going to do, and know it is in your advantage to defect provided the hunch is right. This could cause you to defect, even though there may be no incentive to defect if your hunch is wrong.

We require, then, that either of two conditions be met. Either there is an incentive to defect when the other player cooperates (DC>CC), or there is an incentive to defect when the other player defects (DD>CD) – or both.

This rules out two of the games above. With the payoffs ordered CC>CD>DC>DD or CC>DC>CD>DD, there is no incentive to defect at all. Not only is mutual cooperation the best possible outcome, but a player is guaranteed to do better by cooperating, no matter what the other player does.

Crossing these two games off the list leaves just four games. Each is important enough to rate a name.

All four are common games in real-life interactions. For that reason they are called “social dilemmas.” All are closely related, too. Each of the other three games can be derived from the prisoner’s dilemma by switching two of the payoffs in the order of preference. You can think of the prisoner’s dilemma as a center of gravity around which the others orbit. Chicken is a prisoner’s dilemma with punishment and sucker payoffs reversed. The stag hunt is the prisoner’s dilemma with the preferences of the reward and temptation payoffs switched. Deadlock is a prisoner’s dilemma with the reward and punishment payoffs switched.