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The War of Attrition game in Papilio zelicaon


Game theory: The war of attrition game
John Maynard Smith Scientist
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I developed these conditions for an evolutionary stable strategy, and then I applied it to various games. And one of the things that came out of the - at first, I have to say, I thought it was pretty trivial, any idiot could do it in an afternoon. I couldn't see it was important, didn't know that it was going to lead anywhere very much. But I did remember, I went through the following series of rather odd notions. I invented what's known as the War of Attrition Game. The idea is that you and I are going to play the war of attrition game. There is a prize of $10 - it was in America, you understand, it had to be $10 and not 10 ECUs - so we put down $10 between us. And we each of us - well, the way I initially saw it, we'll go on displaying at one another, saying, I want it, I want it, I want it, and you'll do the same. We go on doing this, and whoever gives up first, the other guy gets the $10. Well, obviously, we'd go on forever, except that it costs us. The longer we go on, the more it's costing us. So the question really comes, how much am I prepared to pay for $10? Well, obviously not more than $10, you'd think. But, you know, I've been doing this for a bit, do I stop now, or do I go on, or what do I do? What is, in my terms, evolutionary stable strategy playing this war of attrition game? It turns out, and this is a little trickier to prove, but it can be proved, that what we ought to do is, I should choose a length of time for which I'm to go on, time T, which must not be a fixed time, it must be a time drawn at random from a distribution, and the distribution is a negative exponential with mean $10, and you should do the same. And that's evolutionary stable strategy, that's what you have to do. But the result is, of course, on average you don't make any money, on average you pay $10 to get the $10 bill, it seems absurd. The average pay off to a player playing the war of attrition game is zero. I'm reasonably sure this is a correct mathematical solution, but it's also absurd. If you and I were actually confronted with this argument about the $10, what would we do? We would say, 'Well, let's toss a coin for it, for God's sake.' You know, we'll toss a coin, and if I win, I'll have the $10, if you win, you have the $10, it won't cost us anything and one of us will make $10 and the other one will make nothing, but on average, we'll make $5 a contest. And I thought, well, animals don't do this because they can't toss a coin. But actually, what's the coin doing? All the coin is doing is introducing an asymmetry between you and me. Up to then we're completely in the same situation, but now we're not, I'm heads, you're tails, we're different. Erm, it's a purely symbolic, conventional asymmetry, but it is an asymmetry. So I thought, well, perhaps animals could use a conventional asymmetry to settle contests, and the obvious one to use is who got there first. It's what became known as the Bourgeois Principle, for obvious reasons, that in a contest between two animals, the first animal to find the resource is the 'owner', the second guy is the 'intruder', and the owner gets it because it introduces an asymmetry, not because it's more valuable to him, not because he's better positioned to win the fight, but just because there is an asymmetry and the 'owner' gets it, 'intruder' backs down, it turns out to be an ESS. And I thought, well, this sounds all right, but it has to be nonsense. I mean, it sounds so simple, it can't be true. And anyway, animals haven't read Adam Smith, they don't know about ownership, that can't be right.

The late British biologist John Maynard Smith (1920-2004) is famous for applying game theory to the study of natural selection. At Eton College, inspired by the work of old Etonian JBS Haldane, Maynard Smith developed an interest in Darwinian evolutionary theory and mathematics. Then he entered University College London (UCL) to study fruit fly genetics under Haldane. In 1973 Maynard Smith formalised a central concept in game theory called the evolutionarily stable strategy (ESS). His ideas, presented in books such as 'Evolution and the Theory of Games', were enormously influential and led to a more rigorous scientific analysis and understanding of interactions between living things.

Listeners: Richard Dawkins

Richard Dawkins was educated at Oxford University and has taught zoology at the universities of California and Oxford. He is a fellow of New College, Oxford and the Charles Simonyi Professor of the Public Understanding of Science at Oxford University. Dawkins is one of the leading thinkers in modern evolutionary biology. He is also one of the best read and most popular writers on the subject: his books about evolution and science include "The Selfish Gene", "The Extended Phenotype", "The Blind Watchmaker", "River Out of Eden", "Climbing Mount Improbable", and most recently, "Unweaving the Rainbow".

Tags: Evolutionary Stable Strategy, USA, Game Theory, Adam Smith

Duration: 4 minutes, 17 seconds

Date story recorded: April 1997

Date story went live: 24 January 2008