Conflict resolution Evolutionary stable strategies ESS Game theory

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 Opponent: Hawk Actor: Hawk Dove (V-C)/2 Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury

Payoff matrix Opponent: Hawk Actor: Hawk Dove (V-C)/2 Dove V Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury

Payoff matrix Opponent: Hawk Actor: Hawk Dove (V-C)/2 0 Dove V Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury

Payoff matrix Opponent: Hawk Actor: Hawk Dove (V-C)/2 0 Dove V V/2 Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury

Pure ESS Resource > cost; V = 2; C = 1 Opponent: Hawk Actor: Hawk Dove 1/2 0 1/2 > 0, so Hawks resist invasion by doves 2 > 1, so Hawks can invade doves ESS = all Hawks => pure ESS 2 1

Mixed ESS Resource < cost; V = 1; C = 2 Opponent: Hawk Actor: Hawk Dove -1/2 0 Dove 1 1/2 -1/2 < 0, so Doves can invade Hawks 1 > 1/2, so Hawks can invade doves ESS = mix of Hawks and Doves => mixed ESS If the frequency of Hawks is p, and Doves is 1 -p and at the ESS the fitness of Hawks = the fitness of Doves, then (-1/2)p + (1 -p) = (0)p + (1/2)(1 -p) 1 - 3 p/2= 1/2 - p/2 1/2 = 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 Solutions Actor: Opponent: A B �= highest payoff in column � � A is a pure ESS A B Opponent: A B � � Opponent: A Actor: A B Stable mixed ESS B � � Unstable mixed ESS

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 Opponent: Actor: Hawk (V-C)/2 Hawk 0 Dove Bourgeois Dove V V/2 Bourgeois

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 Opponent: Actor: Hawk (V-C)/2 Hawk 0 Dove Bourgeois 0. 5 H: H + 0. 5 D: H Dove V V/2 Bourgeois 0. 5 H: H + 0. 5 H: D 0. 5 D: H + 0. 5 D: D 0. 5 H: D + 0. 5 D: D 0. 5 D: H

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: Opponent: Actor: Hawk (V-C)/2 Hawk 0 Dove Bourgeois (V-C)/4 Dove V V/2 3 V/4 Bourgeois 3 V/4 -C/4 V/2

Hawk-Dove-Bourgeois If V > C (V = 2, C = 1), then H is pure ESS Opponent: Actor: Hawk Dove Bourgeois 1/2 0 1/4 2 1 3/2 5/4 1/2 1 If V < C (V = 1, C = 2), then B is pure ESS Opponent: Actor: Hawk Dove Bourgeois Hawk -1/2 0 -1/4 Dove 1 1/2 3/4 Therefore, arbitrary asymmetries should resolve conflicts Bourgeois 1/4 1/2

Residency in speckled wood butterflies Note, however, that this effect has been found to be due to the body temperature of the resident (Stutt and Wilmer 1998)

Finding the ESS by simulation If you have a Mac computer, you can download the game theory Simulation from Keith Goodnight at http: //gsoft. smu. edu/GSoft. html