Evolutionary Games and Population Dynamics
| Josef Hofbauer and Karl Sigmund, 1998 |
Introduction for game theorists
Consider the following game, usually called Chicken: Jonny and Oscar have the option to escalate a brawl or to give in. If both give in, they get nothing. If only one player gives in, he pays 1 dollar to the other. But if both escalate the fight, each has an expected loss of 10 dollars, say, for medical treatment.
Clearly, by giving in, Jonny can maximize his minimal payoff. Oscar is in the same position: he also could maximize his minimal payoff by giving in. But will both players give in? Hardly so. If they guess that the other will give in, they will certainly escalate. But if both escalate, both are worse off.
If x and y are the probabilities that John and Oscar escalate, then the expected payoff of Jonny is -10xy+x-y. If Oscar escalates with a probability larger than 1/10, Jonny should quit. If Oscar escalates with a probability smaller than 1/10, Jonny should escalate. If both Jonny and Oscar escalate with a probability of exactly 1/10, they are in a Nash equilibrium: neither of them has anything to gain by deviating unilaterally from his equilibrium. But neither Jonny nor Oscar has any reason not to deviate from 1/10 either, as long as the other sticks to 1/10. Oscar has no reason to care one way or the other, as long as Jonny escalates with a probability of 1/10. But if Jonny has any reason for believing that Oscar escalates with a higher probability, he should never escalate; and if Oscar suspects that Jonny has such a reason, then he should certainly escalate. What should Oscar do, for instance, if Jonny has escalated twice in the first five rounds? Should he conclude that this was a statistical fluke? Even if it was such a fluke, Jonny could reasonably suspect that Oscar would attribute it to a higher propensity to escalate. Again, the argument for 1/10 look rather spurious.
Oddly enough, it was a biologist who offered a convincing explanation. John Maynard Smith, who was studying animal contests at the time, viewed the Chicken game in a population-dynamical setting. There were no longer just Jonny and Oscar engaged in the game, but a large number of players meeting randomly in contests where they had do decide whether to escalate it would obviously pay to escalate less often, and vice versa. In this sense, self-regulation leads to the value of 1/10 - self-regulation, not between two players, but within a population. (p. xvi-xvii)