Conflict resolution

• Evolutionary stable strategies (ESS)– Game theory– Assumptions

• Evolution of display– Hawks and doves– Arbitrary asymmetry

Assessment in red deer

What is an ESS?

• Strategy = the behavioral response of an individual

• ESS = a strategy which if adopted by all members of a population cannot be invaded by any alternative strategy

• The ESS is found using game theory. Game theory is needed when the fitness consequences of a behavior depend on what others are doing, i.e. is frequency dependent

ESS assumptions

• Infinite population

• Asexual (haploid) reproduction

• All strategies are specified

• Either pairwise contests occur or one individual competes against a group

Evolution of display: Hawks vs Doves

• Possible behaviors: – Display– Fight but risk injury– Retreat

• Possible strategies:– Hawk: fight until injured or opponent retreats– Dove: display initially but retreat if opponent

attacks

Payoff matrix

Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury

Opponent: Hawk Dove

Actor: HawkDove

(V-C)/2

Payoff matrix

Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury

Opponent: Hawk Dove

Actor: HawkDove

(V-C)/2 V

Payoff matrix

Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury

Opponent: Hawk Dove

Actor: HawkDove

(V-C)/2 V 0

Payoff matrix

Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury

Opponent: Hawk Dove

Actor: HawkDove

(V-C)/2 V 0 V/2

Pure ESS

Resource > cost; V = 2; C = 1

Opponent: Hawk Dove

Actor: HawkDove

1/2 2 0 1

1/2 > 0, so Hawks resist invasion by doves2 > 1, so Hawks can invade dovesESS = all Hawks => pure ESS

Mixed ESSResource < cost; V = 1; C = 2

Opponent: Hawk Dove

Actor: HawkDove

-1/2 1 0 1/2

-1/2 < 0, so Doves can invade Hawks1 > 1/2, so Hawks can invade dovesESS = mix of Hawks and Doves => mixed ESSIf the frequency of Hawks is p, and Doves is 1-p and at the ESSthe fitness of Hawks = the fitness of Doves, then(-1/2)p + (1-p) = (0)p + (1/2)(1-p)1 - 3p/2= 1/2 - p/21/2 = p

Mixed ESS mechanisms

• Stable strategy set in which a single individual sometimes performs one strategy and sometimes another with probability p

• Stable polymorphic state in which a fraction, p, of the population adopts one strategy while the remainder, 1-p, adopts the other

• Note that highest payoff is not the ESS

ESS SolutionsOpponent: A B

Actor: AB

� � A is a pure ESS

Opponent: A B

Actor: AB

� �

Stablemixed ESS

Opponent: A B

Actor: AB

� �

Unstable mixed ESS

� = highest payoff in column

Opponent: Hawk Dove Bourgeois

Actor: HawkDoveBourgeois

Uncorrelated asymmetry

• Opponents differ, but not with regard to fighting ability• Example: hawk - dove - bourgeois

– Bourgeois strategy: if owner play hawk, if intruder play dove

– Assume that owner and intruder are equally frequent and get equal payoffs

(V-C)/2 V 0 V/2

Opponent: Hawk Dove Bourgeois

Actor: HawkDoveBourgeois

Uncorrelated asymmetry

• Opponents differ, but not with regard to fighting ability• Example: hawk - dove - bourgeois

– Bourgeois strategy: if owner play hawk, if intruder play dove

– Assume that owner and intruder are equally frequent and get equal payoffs

(V-C)/2 V 0.5H:H + 0.5H:D 0 V/2 0.5D:H + 0.5D:D 0.5H:H + 0.5H:D + 0.5H:D +0.5D:H 0.5D:D 0.5D:H

Opponent: Hawk Dove Bourgeois

Actor: HawkDoveBourgeois

Uncorrelated asymmetry

• Opponents differ, but not with regard to fighting ability• Example: hawk - dove - bourgeois

– Bourgeois strategy: if owner play hawk, if intruder play dove

– Assume that owner and intruder are equally frequent and get equal payoffs:

(V-C)/2 V 3V/4-C/4 0 V/2 V/4(V-C)/4 3V/4 V/2

Hawk-Dove-Bourgeois

Therefore, arbitrary asymmetries should resolve conflicts

Opponent: Hawk Dove Bourgeois

Actor: HawkDoveBourgeois

1/2 2 5/4 0 1 1/2 1/4 3/2 1

If V > C (V = 2, C = 1), then H is pure ESS

Opponent: Hawk Dove Bourgeois

Actor: HawkDoveBourgeois

-1/2 1 1/4 0 1/2 1/4 -1/4 3/4 1/2

If V < C (V = 1, C = 2), then B is pure ESS

Residency in speckled wood butterflies

Note, however, that this effect has been found to be due to the bodytemperature of the resident (Stutt and Wilmer 1998)

Finding the ESS by simulation

If you have a Mac computer, you can download the game theorySimulation from Keith Goodnight at http://gsoft.smu.edu/GSoft.html