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.