The Evolutionary Games The Evolutionary Games We PlayWe Play

Psychology 3107Psychology 3107

IntroductionIntroduction

Animals tend to behave in ways that Animals tend to behave in ways that maximize their inclusive fitnessmaximize their inclusive fitness

Usually pretty straightforwardUsually pretty straightforward

But, sometimes we must know what others But, sometimes we must know what others are doing before we adopt a strategyare doing before we adopt a strategy

What if your mating call is drowned out by What if your mating call is drowned out by others’ calls, what to do, ahh what to do…others’ calls, what to do, ahh what to do…

Fitness and StrategiesFitness and Strategies

In certain cases payoffs, and hence fitness In certain cases payoffs, and hence fitness maximization, depend on what other populations maximization, depend on what other populations are doingare doing

When the payoff to one individual depends on When the payoff to one individual depends on the behaviour of others we cannot use the the behaviour of others we cannot use the principle of fitness maximization until we know:principle of fitness maximization until we know: What the alternatives areWhat the alternatives are P(encountering alternatives)P(encountering alternatives) Consequences of encounterConsequences of encounter

Game TheoryGame Theory

Think of it like a gameThink of it like a gameEach individual’s behaviour is its strategy, Each individual’s behaviour is its strategy, payoffs are in units of fitnesspayoffs are in units of fitnessPlayers produce more players (offspring)Players produce more players (offspring)Changes in fitness are directly proportional Changes in fitness are directly proportional to payoffsto payoffsAn evolutionary Stable Strategy is one An evolutionary Stable Strategy is one that, when adopted by enough individuals, that, when adopted by enough individuals, maximizes payoffmaximizes payoff

Pure StrategyPure Strategy

One that cannot be replacedOne that cannot be replaced

Food storingFood storing

Recover your own seeds (Anderssen and Recover your own seeds (Anderssen and Krebs, 1978)Krebs, 1978)

If they recovered communally, a selfish If they recovered communally, a selfish hoarder would replace the communals hoarder would replace the communals damned qucklydamned quckly

Mixed StrategiesMixed Strategies

Hawks and DovesHawks and Doves Not real hawks or doves, strategiesNot real hawks or doves, strategies

Always fight, or always give upAlways fight, or always give up

Look at the payoffsLook at the payoffs

Look at the costsLook at the costs

Determine what proportion should be Determine what proportion should be hawks and should be doveshawks and should be doves

Hawks and DovesHawks and Doves

Say its all DovesSay its all DovesHawk shows up, wins resourceHawk shows up, wins resourceSpreads genesSpreads genesNow more hawksNow more hawksOh oh, now you are fighting, P(injury) = .5Oh oh, now you are fighting, P(injury) = .5Now being a dove paysNow being a dove paysEither strategy good when rare, bad when Either strategy good when rare, bad when commoncommon

Doves and HawksDoves and Hawks

V =V alue of resource for winnerV =V alue of resource for winner

W = cost of a woundW = cost of a wound

T = cost of display (no fighting)T = cost of display (no fighting)

(John Maynard Smith, 1978)(John Maynard Smith, 1978)

Whoa, I know Kung FuWhoa, I know Kung Fu

Set up a payoff MatrixSet up a payoff Matrix

Opponent in the contestOpponent in the contest

HawkHawk DoveDove

PayoffPayoffHawkHawk ½(V-W)½(V-W) VV

ReceivedReceived

ByBy DoveDove 00 ½V-T½V-T

ESS as easy as 123ESS as easy as 123

If W > V then there can be no pure ESSIf W > V then there can be no pure ESS In a population of hawks, a small number of In a population of hawks, a small number of

doves do better than hawksdoves do better than hawks

E(dove,hawk) > E(hawk, hawk)E(dove,hawk) > E(hawk, hawk)

E(dove, hawk) = 0E(dove, hawk) = 0

E(hawk, hawk) = ½(V-W)E(hawk, hawk) = ½(V-W) W > V, therefore ½(V-W) < 0W > V, therefore ½(V-W) < 0

Pure Doves don’t do it eitherPure Doves don’t do it either

Payoff to Hawk is VPayoff to Hawk is V

Payoff to doves is less than thatPayoff to doves is less than that (½W – T)(½W – T)

HmmmHmmm

So, what proportion of hawks and doves So, what proportion of hawks and doves balances it out?balances it out?

What is theoretical population What is theoretical population biologist to do?biologist to do?

Find the proportion (p) of hawks of hawks Find the proportion (p) of hawks of hawks such that the following equation balances:such that the following equation balances:

p ½(V-W) = (1-p) V = p (0) + (1-p) (½V– T)p ½(V-W) = (1-p) V = p (0) + (1-p) (½V– T)

Simply (?) solve for pSimply (?) solve for p

p = (V+2T) / (W+ 2T)p = (V+2T) / (W+ 2T)

Apply it, sort ofApply it, sort of

Say V = 10Say V = 10W = 20W = 20T = 3T = 3

Opponent in the contestOpponent in the contestHawkHawk DoveDove

PayoffPayoffHawkHawk -5-5 1010ReceivedReceivedByBy DoveDove 00 22

Now, sub that back into the formulaNow, sub that back into the formula

P = 16/26 or 8/13P = 16/26 or 8/138/13ths of the population, with these payoff 8/13ths of the population, with these payoff values, must be hawksvalues, must be hawksThe values are not that important really, the The values are not that important really, the point is that you can determine the point at point is that you can determine the point at which a strategy can coexist with another which a strategy can coexist with another strategy as an ESSstrategy as an ESSCould be percentage of population, or Could be percentage of population, or percentage of time each animal adopts a given percentage of time each animal adopts a given strategystrategy

So?So?

It is actually It is actually applicable that’s soapplicable that’s so

Toads looking for Toads looking for breeding grounds breeding grounds (Davies and (Davies and Hallaway, 1979)Hallaway, 1979)

Payoffs determinedPayoffs determined

Another soAnother so

DungfliesDungflies

Should a male hang around poo as it gets older?Should a male hang around poo as it gets older?

ConclusionsConclusions

This is a very brief intro to game theoryThis is a very brief intro to game theory

This stuff is way powerfulThis stuff is way powerful

You have to sit and think some about the You have to sit and think some about the payoffs and costspayoffs and costs

Dynamic programming models are Dynamic programming models are becoming more popularbecoming more popular