Defender offender game
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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

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Defender/Offender Game

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Defender offender game

Defender/Offender Game

With Defender Learning


Classical game theory

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 theory1

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 anti-Bourgeois strategies can be ESS

  • If defense is not 100% failure proof anti-Bourgeois (Offenders) are often the only ESS


Conditional strategy

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

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 model1

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 model2

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):

    where .

  • Actual number of Offenders (Born Offenders plus unlucky Defenders),

    where p is the frequency of Defenders


Our model3

Our Model

  • Defenders’ learning (f = probability of defense failure): exponential decay of failure rate with learning.

  • Defender’s gain (each of NO offenders steals (1- f) portion of resources):


Our model4

Our Model

  • Offender’s fitness (stolen from Defenders + gained from undefended patches):

  • Defender’s fitness (GD if P1, WO otherwise)

  • Equilibrium: solve for p


Results

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


Results1

Results

If (f0 > 0) and no learning:

Low f0 : both are ESS

ΔW

High f0 : Offense if the only ESS

p = frequency of Defenders

n = 100

k = 100

r = 0

f0 = 0.01


Results2

Results

If (f0 > 0) and learning:

Low f0: Defense is the only ESS and two equilibria exist: one stable and one unstable

ΔW

p = frequency of Defenders

n = 100

k = 100

r = 0.25

f0 = 0.01


Results3

Results

If (f0 > 0) and learning:

High f0: Neither is an ESS and a stable equilibrium exists

ΔW

p = frequency of Defenders

n = 100

k = 100

r = 0.25

f0 = 0.1


Results4

Results

Effect of f0and population size (n) on the location of stable equilibrium

Decreases with f0and with population size


Results5

Results

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

Increases with n/k


Conclusions

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.


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