Partial-Conflict Games DEFINITION

A partial-conflict game is one in which both players can benefit by cooperation, but they may have strong incentives not to cooperate.

Variable-Sum Games DEFINITION

Games of partial conflict are variable-sum games, in which the sum of payoffs to the players at the different outcomes varies.

EXAMPLE 7 Prisoners’ Dilemma

The Prisoners’ Dilemma is a two-person variable-sum game. It provides a simple explanation of the forces at work behind arms races, price wars, and agreeing to environmental regulations. In these and other similar situations, the players can do better by cooperating. But there may be no compelling reasons for them to do so unless there are credible threats of retaliation for not cooperating. The name Prisoners’ Dilemma was first given to this game by Princeton mathematician Albert W. Tucker (1905–1994) in 1950.

Before defining the formal game, we introduce it through a story.

The game of Prisoners’ Dilemma, as we already noted, has many applications, but we will use it here to model a recurrent problem in international relations: arms races between antagonistic countries, which earlier included the superpowers of the United States and the Soviet Union but more recently have included such countries as India and Pakistan and Israel and some of its Arab neighbors. Other countries, such as Iran, may be antagonistic to more than one other country (e.g., Israel and the United States).

For simplicity, assume there are two nations, Red and Blue. Each can independently select one of two policies:

There are four possible outcomes:

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This situation can be modeled by means of the matrix in Table 15.8, which gives the possible outcomes. Here, Red’s choice involves picking one of the two rows, whereas Blue’s choice involves picking one of the two columns.

We assume that the players can rank the four outcomes from best to worst, where , , , and ; thus, the higher the number, the greater the payoff. The resulting game is an ordinal game: It indicates an ordering of outcomes from best to worst but says nothing about the degree to which a player prefers one outcome over another. To illustrate, if a player despises the outcome that he or she ranks 1 but sees little difference among the outcomes ranked 4, 3, and 2, the “payoff distance” between 4 and 2 will be less than that between 2 and 1, even though the numerical difference between 4 and 2 is greater.

The ordinal payoffs to the players for choosing their strategies of and are shown in Table 15.9, where the first number in the pair indicates the payoff to the row player (Red), and the second number the payoff to the column player (Blue). Thus, for example, the pair (1, 4) in the second row and first column signifies a payoff of 1 (worst outcome) to Red and a payoff of 4 (best outcome) to Blue. This outcome occurs when Red unilaterally disarms while Blue continues to arm, making Blue, in a sense, the winner and Red the loser.

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Let’s examine this strategic situation more closely. Should Red select Strategy or ? There are two cases to consider, which depend on what Blue does:

In both cases, Red’s first strategy () gives it a more desirable outcome than its second strategy (). Consequently, we say that A is Red’s dominant strategy because it is always advantageous for Red to choose over .

In the Prisoners’ Dilemma, dominates for Red, so we presume that a rational Red would choose . A similar argument leads Blue to choose A as well—that is, to pursue a policy of arming. Thus, when each nation strives to maximize its own payoffs independently, the pair is driven to the outcome (), with payoffs of (2, 2). The better outcome for both, (), with payoffs of (3, 3), appears unobtainable when this game is played noncooperatively.

The outcome (), which is the product of dominant strategy choices by both players in the Prisoners’ Dilemma, is called a Nash equilibrium.

Note that in the Prisoners’ Dilemma, if either player departs from (), the payoff for the departing player who switches to drops from 2 to 1 at () and (). Not only is there no benefit from departing, but there is actually a loss, with the player selecting being punished with its worst payoff of 1. These losses would presumably deter each nation from moving away from the Nash equilibrium of (), assuming that the other nation sticks to .

Even if both nations agreed in advance jointly to pursue the socially beneficial outcome, (), in which both nations disarm and receive payoff 3, the result is called unstable: If either nation alone reneges on the agreement and secretly arms (as North Korea did when it developed nuclear weapons), it will benefit, obtaining its best payoff of 4. Consequently, each nation would be tempted to go back on its word and select . Nations with no great confidence in the trustworthiness of their opponents have good reason to try to protect themselves against the other side’s defection from an agreement by arming.

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Note that if 4, 3, 2, and 1 in the Prisoners’ Dilemma were not just ranks but numerical payoffs, their sum would be at the mutual-defection outcome and at the mutual-cooperation outcome. At the other two outcomes, the sum, , is still different, illustrating why the Prisoners’ Dilemma is a variable-sum game.

In real life, of course, people often manage to escape the noncooperative Nash equilibrium in the Prisoners’ Dilemma. Either the game is played within a larger context, in which other incentives are at work, such as cultural norms that prescribe cooperation [though this is just another way of saying that defection from (, ) is not rational, rendering the game not the Prisoners’ Dilemma], or the game is played on a repeated basis—it is not a one-shot affair—so players can induce cooperation by setting a pattern of rewards for cooperation and penalties for noncooperation.

In a repeated game, factors like reputation and trust may play a role. Realizing the mutual advantages of cooperation in costly arms races, players may inch toward the cooperative outcome by slowly phasing down their acquisition of weapons over time, or even destroying them. (The United States and Russia have been doing exactly this.) They may also initiate other productive measures, such as improving their communication channels, making inspection procedures more reliable, writing agreements that are truly enforceable, or imposing penalties for violators when their violations are detected (as has occurred through reconnaissance or spy satellites).

The Prisoners’ Dilemma illustrates the intractable nature of certain competitive situations that blend conflict and cooperation. The standoff that results at the Nash equilibrium of (2, 2) is obviously not as good for the players as that which they could achieve by cooperating—but they risk a good deal if the other player defects.

While saddlepoints are Nash equilibria in total-conflict games, they can never be worse for both players than some other outcome (as in partial-conflict games like the Prisoners’ Dilemma). The reason is that if one player does worse in a total-conflict or zero-sum game, the other player must do better.

The fact that the players must forsake their dominant strategies to achieve the (3, 3) cooperative outcome (see Table 15.9) makes this outcome a difficult one to sustain in one-shot play. on the other hand, assume that the players can threaten each other with a policy of tit-for-tat in repeated play: “I’ll cooperate on each round unless you defect, in which case I will defect until you start cooperating again.” If these threats are credible, the players may well shun their defect strategies and try to establish a pattern of cooperation in early rounds, thereby fostering the choice of (3, 3) in the future.

EXAMPLE 8 Chicken

In Chicken, two drivers approach each other at high speed. Each must decide at the last minute whether to swerve to the right (to avoid crashing) or to not swerve. This scenario occurs with tractors in the movie Footloose (1984) and with buses in the 2011 remake, and a variation occurs when two cars approach the edge of a cliff in the classic James Dean movie A Rebel without a Cause. Here are the possible consequences of their actions:

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These outcomes and their associated strategies are summarized in Table 15.10.

If both drivers persist in their attempts to “win” with a payoff of 4 by not swerving, the resulting outcome will be mutual disaster, giving each driver his or her worst payoff of 1. Clearly, it is better for both drivers to back down and each obtain 3 by swerving, but neither wants to be in the position of being intimidated into swerving (payoff of 2) when the other does not (payoff of 4).

Notice that neither player in Chicken has a dominant strategy. His or her better strategy depends on what the other player does: Swerve if the other does not, don’t swerve if the other player swerves, making this game’s choices highly interdependent, which is characteristic of many games. The Nash equilibria in Chicken, moreover, are (4, 2) and (2, 4), suggesting that the compromise of (3, 3) will not be easy to achieve because both players will have an incentive to deviate in order to try to be the winner. This game is referred to as an anticoordination game because the Nash equilibria of (4, 2) and (2, 4) require the players to play different strategies.

In fact, there is a third Nash equilibrium in Chicken, but it is in mixed strategies, which can be computed only if the payoffs are not ranks, as we have assumed here, but numerical values. Even if the payoffs were numerical, however, it can be shown that this equilibrium is always worse for both players than the cooperative (3, 3) outcome. Moreover, it is implausible that players would sometimes swerve and sometimes not—randomizing according to particular probabilities—in the actual play of this game, compared with either trying to win outright or reaching a compromise.

The two pure-strategy Nash equilibria in Chicken suggest that, insofar as there is a “solution” to this game, it is that one player will succeed when the other caves in to avoid the mutual-disaster outcome. But there are certainly real-life cases in which a major confrontation was defused and a compromise of sorts was achieved in a Chicken-type game. This fact suggests that the one-sided solutions given by the two pure-strategy Nash equilibria may not be the only pure-strategy solutions, especially if the players are farsighted and think about the possible untoward consequences of their actions.

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International crises, labor-management disputes, and other conflicts in which escalating demands may end in wars, strikes, and other catastrophic outcomes have been modeled by the game of Chicken (see Spotlight 15.2 on page 646 for more on game theorists who have analyzed these and other games). But it can be shown that Chicken, like the Prisoners’ Dilemma, is only one of the 78 essentially different ordinal games in which each player can rank the four possible outcomes from best to worst.