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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
• 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 anti-Bourgeois 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):
where .
• Actual number of Offenders (Born Offenders plus unlucky Defenders),
where p is the frequency of Defenders
P1 1
i
ie
i!i1
NO n(1 pP1)
n
k
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):
f f0 exp( rNOND)
GD (1 f )NO
Our Model
• Offender’s fitness (stolen from Defenders + gained from undefended patches):
• Defender’s fitness (GD if P1, WO otherwise)
• Equilibrium: solve for p
WO ND (1 GD ) k(1 p)
NO
WD P1GD (1 P1)WO
W WD WO 0
Results
If defense is failure-proof (f0 = 0), Defense is the only ESS (even without any learning):
ΔW
p = frequency of Defenders
n = 100k = 100r = 0
f0 = 0
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 = 100k = 100r = 0
f0 = 0.01
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 = 100k = 100r = 0.25
f0 = 0.01
Results
If (f0 > 0) and learning:
High f0: Neither is an ESS and a stable equilibrium exists
ΔW
p = frequency of Defenders
n = 100k = 100r = 0.25
f0 = 0.1
ResultsEffect of f0 and population size (n) on the location of
stable equilibrium
Decreases with f0 and with population size
ResultsEffect 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.