- 69 Views
- Uploaded on
- Presentation posted in: General

Defender/Offender Game

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Defender/Offender Game

With Defender Learning

- 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

- 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

- 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.

- 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

- 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)

- 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

- 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):

- Offender’s fitness (stolen from Defenders + gained from undefended patches):
- Defender’s fitness (GD if P1, WO otherwise)
- Equilibrium: solve for p

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

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

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

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

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

Decreases with f0and with population size

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

Increases with n/k

- 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.