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.
| Driver 2 | |||
|---|---|---|---|
| Swerve | Not Swerve | ||
| Driver 1 | Swerve | (3, 3) | (2, 4) |
| Not Swerve | (4, 2) | (1, 1) | |
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.
Chicken DEFINITION
Chicken is a two-person variable-sum game in which each player has two strategies: to swerve to avoid a collision or not to swerve and possibly cause a collision. Neither player has a dominant strategy. The compromise outcome, in which both players swerve, and the disaster outcome, in which both players do not, are not Nash equilibria. The other two outcomes, in which one player swerves and the other does not, are Nash equilibria.
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.