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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

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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