commonplace book redux

John Maynard Smith, the biologist who first applied game theory to evolution, modeled this kind of standoff as a War of Attrition game. 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 an exponential (in this case, a greater number of severe wars). But an exponential can be transformed into a power law if the values are modulated by a second exponential process pushing in the opposite direction. And attrition games have a twist that might do just that. If one side in an attrition game were to leak its intention to concede in the next instant by, say, twitching or blanching or showing some other sign of nervousness, its opponent could capitalize on the “tell” by waiting just a bit longer, and it would win the prize every time. As Richard Dawkins has put it, in a species that often takes part in wars of attrition, one expects the evolution of a poker face.

Now, one also might have guessed that organisms would capitalize on the opposite kind of signal, asign of continuing resolve rather than impending surrender. If a contestant could adopt some defiant posture that means “I’ll stand my ground; I won’t back down,” that would make it rational for his opposite number to give up and cut its losses rather than escalate to mutual ruin. But there’s a reason we call it “posturing.” Any coward can cross his arms and glower, but the other side can simply call his bluff. Only if a signal is costly—if the defiant party holds his hand over a candle, or cuts his arm with a knife—can he show that he means business. (Of course, paying a self-imposed cost would be worthwhile only if the prize is especially valuable to him, or if he had reason to believe that he could prevail over his opponent if the contest escalated.)

In the case of a war of attrition, one can imagine a leader who has a changing willingness to suffer a cost over time, increasing as the conflict proceeds and his resolve toughens. His motto would be: “We fight on so that our boys shall not have died in vain.” This mindset, known as loss aversion, the sunk-cost fallacy, and throwing good money after bad, is patently irrational, but it is surprisingly pervasive in human decision-making. People stay in an abusive marriage because of the years they have already put into it, or sit through a bad movie because they have already paid for the ticket, or try to reverse a gambling loss by doubling their next bet, or pour money into a boondoggle because they’ve already poured so much money into it. Though psychologists don’t fully understand why people are suckers for sunk costs, a common explanation is that it signals a public commitment. The person is announcing: “When I make a decision, I’m not so weak, stupid, or indecisive that I can be easily talked out of it.” In a contest of resolve like an attrition game, loss aversion could serve as a costly and hence credible signal that the contestant is not about to concede, preempting his opponent’s strategy of outlasting him just one more round.

I already mentioned some evidence from Richardson’s dataset which suggests that combatants do fight longer when a war is more lethal: small wars show a higher probability of coming to an end with each succeeding year than do large wars. The magnitude numbers in the Correlates of War Dataset also show signs of escalating commitment: wars that are longer in duration are not just costlier in fatalities; they are costlier than one would expect from their durations alone. If we pop back from the statistics of war to the conduct of actual wars, we can see the mechanism at work. Many of the bloodiest wars in history owe their destructiveness to leaders on one or both sides pursuing a blatantly irrational loss-aversion strategy. Hitler fought the last months of World War II with a maniacal fury well past the point when defeat was all but certain, as did Japan. Lyndon Johnson’s repeated escalations of the Vietnam War inspired a protest song that has served as a summary of people’s understanding of that destructive war: “We were waist-deep in the Big Muddy; The big fool said to push on.”

The systems biologist Jean-Baptiste Michel has pointed out to me how escalating commitments in a war of attrition could produce a power-law distribution. All we need to assume is that leaders keep escalating as a constant proportion of their past commitment—the size of each surge is, say, 10 percent of the number of soldiers that have fought so far. A constant proportional increase would be consistent with the well-known discovery in psychology called Weber’s Law: for an increase in intensity to be noticeable, it must be a constant proportion of the existing intensity. (If a room is illuminated by ten lightbulbs, you’ll notice a brightening when an eleventh is switched on, but if it is illuminated by a hundred lightbulbs, you won’t notice the hundred and first; someone would have to switch on another ten bulbs before you noticed the brightening.) Richardson observed that people perceive lost lives in the same way: “Contrast for example the many days of newspaper-sympathy over the loss of the British submarine Thetis in time of peace with the terse announcement of similar losses during the war. This contrast may be regarded as an example of the Weber-Fechner doctrine that an increment is judged relative to the previous amount.” The psychologist Paul Slovic has recently reviewed several experiments that support this observation. The quotation falsely attributed to Stalin, “One death is a tragedy; a million deaths is a statistic,” gets the numbers wrong but captures a real fact about human psychology.

If escalations are proportional to past commitments (and a constant proportion of soldiers sent to the battlefield are killed in battle), then losses will increase exponentially as a war drags on, like compound interest. And if wars are attrition games, their durations will also be distributed exponentially. Recall the mathematical law that a variable will fall into a power-law distribution if it is an exponential function of a second variable that is distributed exponentially. My own guess is that the combination of escalation and attrition is the best explanation for the power-law distribution of war magnitudes.

Steven Pinker, The Better Angels of our Nature, Chapter 5

Though conventional terrorism, as John Kerry gaffed, is a nuisance to be policed rather than a threat to the fabric of life, terrorism with weapons of mass destruction would be something else entirely. The prospect of an attack that would kill millions of people is not just theoretically possible but consistent with the statistics of terrorism. The computer scientists Aaron Clauset and Maxwell Young and the political scientist Kristian Gleditsch plotted the death tolls of eleven thousand terrorist attacks on log-log paper and saw them fall into a neat straight line.261 Terrorist attacks obey a power-law distribution, which means they are generated by mechanisms that make extreme events unlikely, but not astronomically unlikely.

The trio suggested a simple model that is a bit like the one that Jean-Baptiste Michel and I proposed for wars, invoking nothing fancier than a combination of exponentials. As terrorists invest more time into plotting their attack, the death toll can go up exponentially: a plot that takes twice as long to plan can kill, say, four times as many people. To be concrete, an attack by a single suicide bomber, which usually kills in the single digits, can be planned in a few days or weeks. The 2004 Madrid train bombings, which killed around two hundred, took six months to plan, and 9/11, which killed three thousand, took two years. But terrorists live on borrowed time: every day that a plot drags on brings the possibility that it will be disrupted, aborted, or executed prematurely. If the probability is constant, the plot durations will be distributed exponentially. (Cronin, recall, showed that terrorist organizations drop like flies over time, falling into an exponential curve.) Combine exponentially growing damage with an exponentially shrinking chance of success, and you get a power law, with its disconcertingly thick tail. Given the presence of weapons of mass destruction in the real world, and religious fanatics willing to wreak untold damage for a higher cause, a lengthy conspiracy producing a horrendous death toll is within the realm of thinkable probabilities.

A statistical model, of course, is not a crystal ball. Even if we could extrapolate the line of existing data points, the massive terrorist attacks in the tail are still extremely (albeit not astronomically) unlikely. More to the point, we can’t extrapolate it. In practice, as you get to the tail of a power-law distribution, the data points start to misbehave, scattering around the line or warping it downward to very low probabilities. The statistical spectrum of terrorist damage reminds us not to dismiss the worst-case scenarios, but it doesn’t tell us how likely they are.

Steven Pinker, The Better Angels of our Nature, Chapter 6