Game Theory - an Introduction

A post over on Rachel's blog has reminded me of something I read a while ago about Game Theory, War Gaming and the justification of MAD in one of my favourite books - 'Human Instinct' by Robert Winston.

It's a bit of an aside from the specific discussion of Unit 2.3 so I have avoided posting this on the boards but nethertheless, if you look past the references to evolutionary biology and genetics (both very much Winston's 'bag'), you will see the striking relevance to our own study subject. Anyway, this is some of what Professor Winston says on Game Theory in his book:


In the early 1970s, the famous evolutionary biologist John Maynard Smith set out to tackle [the] problem [of accepting violence as a tactic], borrowing a technique used by mathematicians, economists and military analysts - game theory. Game theory was best known for its use in 'war games', or simulations of military conflict. When, after the end of the Second World War, it became clear that the Soviet Union had built the atom bomb, military analysts scrutinised the strategies open to the West. The possibility of Armageddon presented a rather thorny problem, but eventually everyone realized that mutually assured destruction, or MAD, was the optimum solution. MAD entailed pointing an enormous number of warheads at your enemy; everyone would die, no matter who fired the first missile. (Game theory does however, depend on your opponent acting rationally. Had Stalin not died in 1953 and continued to become confused and delusional, things might well have worked out quite differently.)
Surprisingly, game theory also proved to be an extremely useful way of modelling the evolution of animal behaviour. Such models are bound to be simplified; there is no pretence in simulating real-life situations with all the nuances and complexities they entail. Instead, game theory presents a stripped-down, quantitative analysis - an approximation that can inform our theories of animal behaviour. It has produced some interesting results.
Maynard Smith thought up a game in which two kinds of animal are competing for territory. This is a situation that could be found among thousands of different species in the real world, but our version is simplified and the rules are as follows. There are two kinds of player, the Hawks and the Doves. The first kind, the Hawks, are willing to fight for the prize, which in this case is territory. The Doves, as the name suggests, are peace-loving and cowardly.
I should emphasise here that these are not two different types of animal, they are two different strategies. Therefore no animal can know whether its opponent is a Hawk or a Dove before the fight, in the same way as you may not know if someone is a berserker [(a particularly nasty breed of Viking warrior - explained ibid)] when he's walking down the street.
Maynard Smith then assigned numerical values to the rewards and costs involved in the game. The winner of the territory gets +50 points and the loser gets zero points. If one of the players in injured, that will cost them -100 points. If one of the players begins a confrontation and then walks away, as Doves are prone to do, the cost of the confrontation is -10 points, whether or not they win or lose the territory. (This cost represents time and energy involved in the ritual, and in the animal world one often sees two animals facing off, baring teeth, brandishing antlers or making threatening noises.) If a Hawk squares up against a Hawk, there will always be a fight. We can assume that all Hawks are equivalent in strength and speed, so each has a 50 per cent chance of winning, therefore, will get the territory - +50 points; the loser, who is injured, is deducted 100 points. If a Hawk confronts a Dove, the Dove will always walk away and let the Hawk win. The payoff for the Hawk will be +50. The Dove has to bear the costs of the face-off and is charged -10 points. No-one is injured. If two Doves challenge each other, neither will get around to fighting. Again each Dove has to pay -10 points for the face-off, and the chances of winning are again fifty-fifty. Therefore, one Dove will get 40 points, and the losing Dove will be charged -10.
So, what happens if we begin with a population made up entirely of the peace-loving Doves? Each fight will be decided non-violently after a face-off. If the game is repeated many times, with all members of the population fighting random opponents, the average payoff for each player will be the total payoff per fight, 40 minus 10 divided by two (the number of players), which equals +15.
So far, so good. The average payoff is positive, which bodes well for the Doves. But what if we bring into play the rule that every so often there will be a mutation, and one player switches strategy? What happens if one solitary Dove changes into a Hawk? The Hawk will obviously win every fight, gaining 50 points per confrontation, compared to the Doves' average of +15. This numerical success, which can be thought of as its evolutionary fitness, means that the solitary Hawk will thrive, constantly winning fights, allowing it to live long and prosper. Hawk genes will therefore spread through the population at the expense of Dive genes.
Now, suppose the population consists entirely of Hawks. Every single fight will end in violent conflict. The average payoff for each player, in the long term, is -25 (the total payoff per fight is -50, divided by two). In this environment, a solitary Dove will do well: the Dove's average payoff will be -10 (it will walk away every time from a Hawk), which is substantially better than the Hawk's average of -25. Therefore, the Dove's genes will prosper and spread at the expense of the Hawk's genes.
The bottom line is that neither an all-Hawk population nor an all-Dove population is stable. Maynard Smith translated the game into the equations of game theory and realized that the only stable populations would have to be a mixture of Hawks and Doves; this he called an evolutionary stable strategy (ESS). It would describe a population in which any mutation would not knock the population off balance; it would always gravitate back towards the ESS. In this case, with the points system I have described, the ESS turns out to be 58 per cent Hawk and 42 per cent Dove. The mixture is not necessarily the best possible outcome, but it is stable, and that is what counts.
Thus, two very different strategies can live along side each other; in this case, no other strategy or set of strategies could replace it, no matter how quickly individuals mutate from Dove to Hawk or vice versa. Similarly, [t]he evolutionary stable strategy applied to this scenario: if you or I were a player in the game, we could do no better - for our own genetic fitness - than to play Hawk 58 per cent of the time and Dove 42 per cent of the time, assuming we do not know who our opponent is in each confrontation.



Robert Winston (2002), pp 290-293.