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Entries in St. Vincent (1)

Thursday
Nov032011

Game Theory II: Hawks & Doves & Others

In game theory, a game called “Hawks & Doves” imagines a showdown between two birds, both of which act on instinct and with a survival tactic to win a resource, say food or a breeding opportunity.

Every bird’s tactic is one of two options: he either fights (hawk) or fakes like he will fight and flees if the other bird doesn’t back down (dove).

If two birds in conflict both be Hawks, they fight to the point of injury, and ceteris paribus each has a 50% chance of winning and a 50% chance of losing. It’s a probability coin toss.

Being nearly fatally injured is worse for the loser than winning is beneficial for the winner. You may win the fight and eat or breed, which is good, but that guy almost died, which is really bad.

We might score the fight +50 for the winner and –80 for the loser. Because all fights according to the conditions of the game are always 50/50 odds, the average payout for the fight is (50 minus 80) divided by 2 = –15, which means over the long run and after enough fights, all hawks average a loss.

If two birds in conflict both be Doves, then they both pretend to want to fight until one or the other eventually backs down. This could take a second, or it could last for days, but neither of them ever intend to fight, only display as though they will.

As with two hawk fights, the winner in a dove standoff gets a resource, which we valued at +50. The losing dove gets no resources, but he also doesn’t get injured, so that’s a zero payoff rather than a big negative.

However, both doves spend energy pretending they are going to fight, so that must count for something negative. Say they both spend the same energy posing like fighters and expend –10 of energy each, which means the winning dove nets +40 and the losing dove, –10.

Unlike with the hawks, the average payout for a two-dove showdown is a gain, (40 – 10)/2 which equals +15, which means over the long run and after enough chest-bump standoffs, every dove averages a win.

So why isn’t everyone a non-fighter for a positive average gain? Because when you’re a hawk, the payoff is risk-free when you encounter a dove. You take it all, +50, and, at the first little peck in the head the dove backs down and gets a zero.

In the game, the dove doesn’t even waste standoff energy when he sees he’s against a hawk. Hawks beat doves. Every time. In the game. It’s a more complex version of the game "Chicken."

Music break:

In "Hawks & Doves," there are NOT two kinds of birds. There is one species of bird with different tendencies to express either hawk or dove behavior. The species expresses hawk and dove behavior in proportion to how the payoffs for survival condition that behavior.

If most everyone expresses dove behavior, then a hawk can make a killing with little risk.

If most everyone expresses hawk behavior, then it’s going to get bloody with a lot of injured losers. And though he wins against doves, every hawk’s going to lose half the time when he fights another hawk.

If you crunch the numbers for payouts of +50 for winning, –80 for losing fights, and –10 for bluffing, then the average payouts for hawks and doves look like this:

Hawks: –15p + 50(1 – p) = 50 – 65p
Doves: 0p + 15(1 – p) = 15 – 15p

where p is the proportion of the population playing hawk, and (1 – p), the proportion playing dove.

In stable populations, the average payoffs between hawks and doves are equal, which makes p = 0.7 in the example: a world with 70% hawks and 30% doves.

  • If that world started far from stability, say with 1 hawk and 99 doves, it would many generations later have 70% hawks and 30% doves (The world would “approach” that proportion).
  • If that world started with 99 hawks and 1 dove, it would also generations later have 70% hawks and 30% doves.
  • If that world started with 50 hawks and 50 doves, it would still, generations later, have 70% hawks and 30% doves.

This all assumes the average payoffs stay the same through the generations. This also assumes the behavior stays the same: a species of bird exhibiting hawk and dove behavior, stuck in a timeless struggle against one another for an evolutionary stable strategy.

But that’s not what happens. Mutation happens. New behaviors begin to show hawk and dove behavior blended in the same strategy. Perhaps there's a new kind of bird that beats up on doves, but won’t fight any hawks. Bully!

Perhaps it's a bird that will sometimes fight hawks and win, but knows when to be a dove and take a dive rather than lose. Perhaps it's a hawk all its life and then, one day, switches, out of the blue.

In game theory, the hawk and dove behavior isn't choosen. The birds play out probabilities of possible phenotypes and successful mutations arise to either live or not live. They are automatons.

In the human world, we’d like to think we can choose. Suppose the birds choose, too?

Cooperation is the best choice for maximizing resources. Guaranteed. It's been proven mathematically and morally. If the Hawks & Doves and the crafty Mutations amongst them would all share, then we wouldn't have so much conflict. Perhaps that's true, depending upon how much of the resource was available and for how long.

What usually develops during cooperation is the simple game theory game, Prisoner's Dilemma, where the hawk may agree not to fight, but he's afraid the other bird is a hawk, too, and won't live up to his promise not to fight. So the cooperative hawk feels like he's going to get taken. He's afraid violence will disturb the peace.

In the 1950s, a contest was held to solve the problem of the Prisoner's Dilemma. What's the best strategy for hawks willing to be cooperative but distrustful of other bird's commitment to peace? Computer scientists, philosophers, mathematicians: answers came from all around. Organizers pitted strategies against one another. Some strategies involved extremely complicated math and decision matrices.

The winningest strategy was simple: Tit-for-Tat, "do unto others" until they do you wrong, then do to them as they did to you. Over enough games played, Tit-for-Tat won.

What a boring way to win.