The Better Angels of Our Nature: Why Vio - By Steven Pinker Page 0,133

conflagration, a destabilizing event in the simulation of states can set off either a skirmish or a world war.

In these simulations, the destructiveness of a war depends mainly on the territorial size of the combatants and their alliances. But in the real world, variations in destructiveness also depend on the resolve of the two parties to keep a war going, with each hoping that the other will collapse first. Some of the bloodiest conflicts in modern history, such as the American Civil War, World War I, the Vietnam War, and the Iran-Iraq War, were wars of attrition, where both sides kept shoveling men and matériel into the maw of the war machine hoping that the other side would exhaust itself first.

John Maynard Smith, the biologist who first applied game theory to evolution, modeled this kind of standoff as a War of Attrition game.64 Each of two contestants competes for a valuable resource by trying to outlast the other, steadily accumulating costs as he waits. In the original scenario, they might be heavily armored animals competing for a territory who stare at each other until one of them leaves; the costs are the time and energy the animals waste in the standoff, which they could otherwise use in catching food or pursuing mates. A game of attrition is mathematically equivalent to an auction in which the highest bidder wins the prize and both sides have to pay the loser’s low bid. And of course it can be analogized to a war in which the expenditure is reckoned in the lives of soldiers.

The War of Attrition is one of those paradoxical scenarios in game theory (like the Prisoner’s Dilemma, the Tragedy of the Commons, and the Dollar Auction) in which a set of rational actors pursuing their interests end up worse off than if they had put their heads together and come to a collective and binding agreement. One might think that in an attrition game each side should do what bidders on eBay are advised to do: decide how much the contested resource is worth and bid only up to that limit. The problem is that this strategy can be gamed by another bidder. All he has to do is bid one more dollar (or wait just a bit longer, or commit another surge of soldiers), and he wins. He gets the prize for close to the amount you think it is worth, while you have to forfeit that amount too, without getting anything in return. You would be crazy to let that happen, so you are tempted to use the strategy “Always outbid him by a dollar,” which he is tempted to adopt as well. You can see where this leads. Thanks to the perverse logic of an attrition game, in which the loser pays too, the bidders may keep bidding after the point at which the expenditure exceeds the value of the prize. They can no longer win, but each side hopes not to lose as much. The technical term for this outcome in game theory is “a ruinous situation.” It is also called a “Pyrrhic victory”; the military analogy is profound.

One strategy that can evolve in a War of Attrition game (where the expenditure, recall, is in time) is for each player to wait a random amount of time, with an average wait time that is equivalent in value to what the resource is worth to them. In the long run, each player gets good value for his expenditure, but because the waiting times are random, neither is able to predict the surrender time of the other and reliably outlast him. In other words, they follow the rule: At every instant throw a pair of dice, and if they come up (say) 4, concede; if not, throw them again. This is, of course, like a Poisson process, and by now you know that it leads to an exponential distribution of wait times (since a longer and longer wait depends on a less and less probable run of tosses). Since the contest ends when the first side throws in the towel, the contest durations will also be exponentially distributed. Returning to our model where the expenditures are in soldiers rather than seconds, if real wars of attrition were like the “War of Attrition” modeled in game theory, and if all else were equal, then wars of attrition would fall into an exponential distribution of magnitudes.

Of course, real wars fall into a power-law distribution, which has a thicker tail than