### Game Theory

```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,
by any new strategy
• In classic HD game neither
strategy is an ESS: hawks
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 antiBourgeois 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):

P1 
where  
n
k
1 e
i
i 1
i

i!
.
• Actual number of Offenders (Born Offenders plus unlucky
Defenders),


N O  n (1  pP1 )
where p is the frequency of Defenders
Our Model
• Defenders’ learning (f = probability of defense
failure): exponential decay of failure rate with
learning.
f  f 0 exp(  r
NO
ND
)
• Defender’s gain (each of NO offenders steals (1- f)
portion of resources):

G D  (1  f )
NO
Our Model
• Offender’s fitness (stolen from Defenders + gained
from undefended patches):
WO 
N D (1  G D )  k (1  p)
NO
• Defender’s fitness (GD if P1, WO otherwise)

W D  P1G D  (1  P1 )W O
• Equilibrium: solve  W  W D  W O  0
for p
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
Results
If (f0 > 0) and no learning:
Low f0 : both are ESS
p = frequency
of Defenders
ΔW
n = 100
k = 100
r=0
f0 = 0.01
High f0 : Offense if the only ESS
Results
If (f0 > 0) and learning:
Low f0: Defense is the only ESS and two equilibria
exist: one stable and one unstable
p = frequency
ΔW
of Defenders
n = 100
k = 100
r = 0.25
f0 = 0.01
Results
If (f0 > 0) and learning:
High f0: Neither is an ESS and a stable equilibrium
exists
p = frequency
ΔW
of Defenders
n = 100
k = 100
r = 0.25
f0 = 0.1
Results
Effect of f0 and population size (n) on the location of stable
equilibrium
Decreases with f0 and with population size
Results
Effect 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.
```