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Defender/Offender Game With Defender Learning

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Classical Game Theory Hawk-Dove Game Hawk-Dove Game Evolutionary Stable Evolutionary Stable Strategy (ESS) 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 In classic HD game neither strategy is an ESS: hawks will invade a population of doves in vise versa

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Classical Game Theory What if Hawks are not always Hawks, but only if they own a resource they defend? (“Bourgeois” strategy). 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 anti- Bourgeois strategies can be ESS Maynard Smith and Parker, 1976; Maynard Smith, 1982: both Bourgeois anti- Bourgeois strategies can be ESS If defense is not 100% failure proof anti-Bourgeois (Offenders) are often the only ESS If defense is not 100% failure proof anti-Bourgeois (Offenders) are often the only ESS

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Conditional strategy What happens to a Bourgeois (Defender) if it fails to find a resource to own and defend? 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 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. 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.

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Our Model Goal: 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) 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 Investigate how the ESS depends on population size, competition intensity and learning ability Assumptions Assumptions Two pure strategies: Natural Born Offenders and Conditional Defenders. Defense is not 100% failure-proof. 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. 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. NBOs don’t seek to own a resource and always play the Offender role. Poisson distribution of individuals into patches of resources Poisson distribution of individuals into patches of resources Offenders divide gain equally Offenders divide gain equally Defenders learn to defend their patch more efficiently when attacked often Defenders learn to defend their patch more efficiently when attacked often

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Our Model Variables Variables n = # individuals in the population n = # individuals in the population k = # patches (n/k is the intensity of competition) k = # patches (n/k is the intensity of competition) f 0 = probability of defense failing by a “naïve” (unlearned) Defender f 0 = probability of defense failing by a “naïve” (unlearned) Defender r = Defender’s learning rate r = Defender’s learning rate Methods Methods Analytical model (in Maple) Analytical model (in Maple) Individual based model (work in progress) Individual based model (work in progress)

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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): 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): where. Actual number of Offenders (Born Offenders plus unlucky Defenders), Actual number of Offenders (Born Offenders plus unlucky Defenders), where p is the frequency of Defenders

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Our Model Defenders’ learning (f = probability of defense failure): exponential decay of failure rate with learning. Defenders’ learning (f = probability of defense failure): exponential decay of failure rate with learning. Defender’s gain (each of N O offenders steals (1- f) portion of resources): Defender’s gain (each of N O offenders steals (1- f) portion of resources):

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Our Model Offender’s fitness (stolen from Defenders + gained from undefended patches): Offender’s fitness (stolen from Defenders + gained from undefended patches): Defender’s fitness (G D if P 1, W O otherwise) Defender’s fitness (G D if P 1, W O otherwise) Equilibrium: solve for p Equilibrium: solve for p

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Results If defense is failure-proof (f 0 = 0), Defense is the only ESS (even without any learning): Δ W p = frequency of Defenders n = 100 k = 100 r = 0 f 0 = 0

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Results If (f 0 > 0) and no learning: Low f 0 : both are ESS Δ W High f 0 : Offense if the only ESS p = frequency of Defenders n = 100 k = 100 r = 0 f 0 = 0.01

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Results If (f 0 > 0) and learning: Low f 0: Defense is the only ESS and two equilibria exist: one stable and one unstable Δ W p = frequency of Defenders n = 100 k = 100 r = 0.25 f 0 = 0.01

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Results If (f 0 > 0) and learning: High f 0: Neither is an ESS and a stable equilibrium exists Δ W Δ W p = frequency of Defenders n = 100 k = 100 r = 0.25 f 0 = 0.1

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Results Effect of f 0 and population size (n) on the location of stable equilibrium Decreases with f 0 and with population size

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Results Effect of competition intensity (n/k) on the location of stable equilibrium: Increases with n/k

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Conclusions Learning ability in Defenders can lead to Defense becoming the ESS 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). 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. 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. This can explain polymorphism and/or intermediate strategies of resource defense, territoriality and mate guarding in animals.

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