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

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

Mixed ESS Resource < cost; V = 1; C = 2 Opponent:HawkDove Actor:Hawk Dove -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 - 3p/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 Opponent:AB Actor:A B � A is a pure ESS Opponent:AB Actor:A B � Stable mixed ESS Opponent:AB Actor:A B � �� � Unstable mixed ESS � = highest payoff in column

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

Opponent:HawkDoveBourgeois Actor:Hawk Dove 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 (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:HawkDoveBourgeois Actor:Hawk Dove 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: (V-C)/2 V3V/4-C/4 0 V/2 V/4 (V-C)/43V/4 V/2

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