The social dilemmas described above [Deadlock, Stag Hunt] are symmetric games in which both players share the same preferences. Preferences need not match, however. It is possible for one player’s preferences to be those of a prisoner’s dilemma and the other player’s preferences to be those of chicken or stag hunt or something else. Some of these hybrid games have been suggested as models for human conflicts.
The game of bully is a cross between chicken and deadlock. One player has the same preferences as a chicken player. He would like to defect, but fears mutual defection. The other player has the deadlock preferences and prefers to defect no matter what (preferably with the other player cooperating). These two sets of preferences describe a game that looks like this:
One instance of bully is the biblical tale demonstrating the wisdom of King Solomon (1 Kings 3:16-28). Two women claim the same child as their own. One is the real mother, and the other is an impostor. Solomon proposes splitting the child in two. Hearing this horrific suggestion, one woman abandons her claim to the child. Solomon awards the child to that woman. The real mother would love her child so much she would give it up to save its life.
In other words, the true mother has the preferences of a chicken player. The knife is poised over the child. The dilemma is whether to stand firm (defect) or give in (cooperate). The real mother most wants to prevail – to stand firm on her claim while the impostor backs down. The worst outcome from the real mother’s standpoint is for neither woman to give in. Then the child is cut in two.
The impostor has deadlock preferences. She evidently would rather see the child killed than have it go to her rival. The name “bully” tells what happens. The deadlock player can be a bully and defect. The chicken player is powerless to prevent this. The only thing she can do is cut her losses by cooperating. Ergo, the woman who gives in is the real mother.
Bully is a model for military confrontations in which one nation prefers to start a war, and the other views the war as a catastrophe to be avoided at all costs. To the extent this is an accurate model, the conclusion is gloomy. The belligerent nation will likely get its way, while the conciliatory nation is exploited to maintain the peace. The prognosis is probably worse yet. Real nations’ preferences are fluid, and a nation that feels it has been exploited may decide that war isn’t so bad after all.
William Poundstone, Prisoner’s Dilemma, Doubleday, NY 1992, pp. 221-222.