Defender/Offender Game With Defender Learning Classical Game Theory • Hawk-Dove Game • Evolutionary Stable Strategy (ESS) strategy, which is the best response to any other strategy, including itself; cannot be invaded by any new strategy • In classic HD game neither strategy is an ESS: hawks will invade a population of doves in vise versa Classical Game Theory • What if Hawks are not always Hawks, but only if they own a resource they defend? (“Bourgeois” strategy). • Maynard Smith and Parker, 1976; Maynard Smith, 1982: both Bourgeois antiBourgeois strategies can be ESS • If defense is not 100% failure proof anti-Bourgeois (Offenders) are often the only ESS Conditional strategy • What happens to a Bourgeois (Defender) if it fails to find a resource to own and defend? • If this is the end of the story (cannot play Offense, no resource to defend = 0 fitness), then Offenders dominate • Here we consider a “Conditional Defense” strategy: if a player owns a resource, he defends it. If it fails to own one, it switches to Offense. “Natural Born Offenders” offend no matter what. Our Model • Goal: • Find the ESS(s) when Defenders (Bourgeois) are able to learn to defend their turf more efficiently (one way of making the life of the Offender more difficult) • Investigate how the ESS depends on population size, competition intensity and learning ability • Assumptions • Two pure strategies: Natural Born Offenders and Conditional Defenders. Defense is not 100% failure-proof. • CDs defend their turf if they are the first to arrive on it. If they fail to own such resource, they become offenders. • NBOs don’t seek to own a resource and always play the Offender role. • Poisson distribution of individuals into patches of resources • Offenders divide gain equally • Defenders learn to defend their patch more efficiently when attacked often Our Model • Variables • n = # individuals in the population • k = # patches (n/k is the intensity of competition) • f0 = probability of defense failing by a “naïve” (unlearned) Defender • r = Defender’s learning rate • Methods • Analytical model (in Maple) • Individual based model (work in progress) Our Model • Probability of being the first on a patch (the number of individuals per patch is distributed by Poisson; one of them will be the first to arrive): P1 where n k 1 e i i 1 i i! . • Actual number of Offenders (Born Offenders plus unlucky Defenders), N O n (1 pP1 ) where p is the frequency of Defenders Our Model • Defenders’ learning (f = probability of defense failure): exponential decay of failure rate with learning. f f 0 exp( r NO ND ) • Defender’s gain (each of NO offenders steals (1- f) portion of resources): G D (1 f ) NO Our Model • Offender’s fitness (stolen from Defenders + gained from undefended patches): WO N D (1 G D ) k (1 p) NO • Defender’s fitness (GD if P1, WO otherwise) W D P1G D (1 P1 )W O • Equilibrium: solve W W D W O 0 for p Results If defense is failure-proof (f0 = 0), Defense is the only ESS (even without any learning): ΔW p = frequency of Defenders n = 100 k = 100 r=0 f0 = 0 Results If (f0 > 0) and no learning: Low f0 : both are ESS p = frequency of Defenders ΔW n = 100 k = 100 r=0 f0 = 0.01 High f0 : Offense if the only ESS Results If (f0 > 0) and learning: Low f0: Defense is the only ESS and two equilibria exist: one stable and one unstable p = frequency ΔW of Defenders n = 100 k = 100 r = 0.25 f0 = 0.01 Results If (f0 > 0) and learning: High f0: Neither is an ESS and a stable equilibrium exists p = frequency ΔW of Defenders n = 100 k = 100 r = 0.25 f0 = 0.1 Results Effect of f0 and population size (n) on the location of stable equilibrium Decreases with f0 and with population size Results Effect of competition intensity (n/k) on the location of stable equilibrium: Increases with n/k Conclusions • Learning ability in Defenders can lead to Defense becoming the ESS • In case of high defense failure rate, learning ability in Defenders result in neither strategy being an ESS, i.e., in a stable equilibrium of the two pure strategies (or an ESS mixed strategy). • The equilibrium frequency of Defenders decreases with defense failure rate and population size and increases with competition intensity. • This can explain polymorphism and/or intermediate strategies of resource defense, territoriality and mate guarding in animals.