Evolutionary Game Theory
Amit Bahl
CIS620
Outline
EGT versus CGT
Evolutionary Stable Strategies
Concepts and Examples
Replicator Dynamics
Concepts and Examples
Overview of 2 papers
Selection methods, finite populations
EGT v. Conventional Game
Theory
Models used to study interactive decision
making.
Equilibrium is still at heart of the model.
Key difference is in the notion of
rationality of agents.
Agent Rationality
In GT, one assumes that agents are
perfectly rational.
In EGT, trial and error process gives
strategies that can be selected for by
some force (evolution - biological,
cultural, etc…).
This lack of rationality is the point of
departure between EGT and GT.
Evolution
When in biological sense, natural selection
is mode of evolution.
Strategies that increase Darwinian fitness
are preferable.
Frequency dependent selection.
Evolutionary Game Theory
(EGT)
Has origins in work of R.A. Fisher [The Genetic Theory of
Natural Selection (1930)].
•Fisher studied why sex ratio is approximately equal in many
species.
•Maynard Smith and Price introduce concept of an ESS [The
Logic of Animal Conflict (1973)].
•Taylor, Zeeman, Jonker (1978-1979) provide continuous
dynamics for EGT (replicator dynamics).
ESS Approach
ESS = Nash Equilibrium + Stability
Condition
Notion of stability applies only to isolated
bursts of mutations.
Selection will tend to lead to an ESS, once
at an ESS selection keeps us there.
ESS - Definition
•Consider a 2 player symmetric game with ESS given by I
with payoff matrix E.
•Let p be a small percentage of population playing mutant
strategy JI.
•Fitness given by
W(I) = W0 + (1-p)E(I,I) + pE(I,J)
W(J) = W0 + (1-p)E(J,I) + pE(J,J)
•Require that W(I) > W(J)
ESS - Definition
Standard Definition for ESS (Maynard
Smith).
I is an ESS if for all J  I,
E(I,I)  E(J,I) and
E(I,I) = E(J,I)  E(I,J) > E(J,J)
where E is the payoff function .
ESS - Definition
Assumptions:
1) Pairwise, symmetric contests
2) Asexual inheritance
3) Infinite population
4) Complete mixing
ESS - Existence
Let G be a two-payer symmetric game
with 2 pure strategies such that
E(s1,s1)  E(s2,s1) AND
E(s1,s2)  E(s2,s2)
then G has an ESS.
ESS Existence
If a > c, then s1 is ESS.
If d > b, then s2 is ESS.
s1
s2
s1
a
c
s2
b
d
Otherwise, ESS given by playing s1 with
probability equal to (b-d)/[(b-d)+(a-c)].
ESS - Example 1
Consider the Hawk-Dove game with
payoff matrix
H
D
H
"-25,-25"
0,50
D
50,0
15,15
Nash equilibrium given by (7/12,5/12).
This is also an ESS.
ESS - Example 1
Bishop-Cannings Theorem:
If I is a mixed ESS with support a,b,c,…,
then E(a,I) = E(b,I) = … = E(I,I).
At a stable polymorphic state, the fitness
of Hawks and Doves must be the same.
W(H) = W(D) => The ESS given is a
stable polymorphism.
Stable Polymorphic State
ESS - Example 2
Consider the Rock-Scissors-Paper Game.
Payoff matrix is given by
R S P
R -e 1 -1
S -1 -e 1
P 1 -1 -e
Then I = (1/3,1/3,1/3) is an ESS but
stable polymorphic population
1/3R,1/3P,1/3S is not stable.
ESS - Example 3
Payoff matrix :
s1
s2
s3
s1
1,1
"-2,2"
2,-2
s2
2,-2
1,1
"-2,2"
s3
"-2,2"
2,-2
1,1
Then I = (1/3,1/3,1/3) is the unique NE,
but not an ESS since E(I,s1)=E(s1,s1)= 1.
Sex Ratios
Recall Fisher’s analysis of the sex ratio.
Why are there approximately equal
numbers of males and females in a
population?
Greatest production of offspring would be
achieved if there were many times more
females than males.
Sex Ratios
Let sex ratio be s males and (1-s)
females.
W(s,s’) = fitness of playing s in population
of s’
Fitness is the number of grandchildren
W(s,s’) = N2[(1-s) + s(1-s’)/s’]
W(s’,s’) = 2N2(1-s’)
Need s* s.t. s W(s*,s*)  W(s,s*)
Dynamics Approach
Aims to study actual evolutionary process.
One Approach is Replicator Dynamics.
Replicator dynamics are a set of
deterministic difference or differential
equations.
RD - Example 1
Assumptions: Discrete time model, nonoverlapping generations.
xi(t) = proportion playing i at time t
(i,x(t)) = E(number of replacement for
agent playing i at time t)
 j {xj(t) (j,x(t))} = v(x(t))
xi(t+1) = [xi(t) (i,x(t))]/ v(x(t))
RD - Example 1
Assumptions: Discrete time model, nonoverlapping generations.
xi(t+1) - xi(t) = xi(t) [(i,x(t)) - v(x(t))]
v(x(t))
RD - Example 2
Assumptions : overlapping generations,
discrete time model.
In time period of length , let fraction 
give birth to agents also playing same
strategy.
j xj(t)[1 +  (j,x(t))] = v(x(t))
xi(t+) = xi(t)[1 +  (i,x(t))]
v(x(t))
RD - Example 2
Assumptions : overlapping generations,
discrete time model.
xi(t+) - xi(t)= xi(t)[ (i,x(t)) -  v(x(t))]
1+  v(x(t))
RD - Example 3
Assumptions: Continuous time model,
overlapping generations.
Let   0, then
dxi /dt
=
xi(t)[(i,x(t)) - v(x(t))]
Stability
Let x(x0,t): S X R S be the unique
solution to the replicator dynamic.
A state x  S is stationary if dx/dt = 0.
A state x is stable if it is stationary and for
every neighborhood V of x, there exists a
U  V s.t. x0  U,  t
x(x0,t)  V.
Propostions for RD
If (x,x) is a NE, then x is a stationary
state of the RD.
 dxi /dt
=
xi(t)[(i,x(t)) - v(x(t))]
What about the converse?
Consider population of all doves.
Propostions for RD
If x is a stable state of the RD, then (x,x)
is a NE .
Consider any perturbation that introduces a
better reply.
What about the converse? Consider:
s1
s2
s1
1,1
0,0
s2
0,0
0,0
Stronger notion of Stability
A state x is asymptotically stable if it is
stable and there exists a neighborhood V
of x s.t. x0  V, limt x(x0,t) = x.
ESS and RD
In general, every ESS is asymptotically
stable.
What about the converse?
ESS and RD
Consider the following game:
s1
s2
s3
s1
0,0
"-2,1"
1,1
s2
1,-2
0,0
1,4
s3
1,1
4,1
0,0
Unique NE given by x* = (1/3,1/3,1/3).
If x = (0,1/2,1/2), then
E(x,x*)=E(x*,x*)=2/3 but
E(x,x)=5/4 > 7/6=E(x*,x).
ESS and RD
s1
s2
s3
x*
x
s1
0,0
"-2,1"
1,1
s2
1,-2
0,0
1,4
x*
2/3,2/3
2/3,7/6
s3
1,1
4,1
0,0
x
7/6,2/3
5/4,5/4
In 2X2 games, x is an ESS if and only if x is
asymptotically stable.
A Game-Theoretic Investigation of
Selection Methods Used in Evolutionary Algorithms
Ficici, Melnik, Pollack
Selection Methods
How do common selection methods used
in evolutionary algorithms function in
EGT?
Dynamics and fixed points of the game.
Selection function
xi(t+1) = S(F(xi(t)),xi(t))
where S is the selection function,
F is the fitness function, and
xi(t) is the proportion of population
playing i at time t.
Fitness Dependent
Selection
f’ = (p X f)/(p • f)
{x(x0,t): t R} = orbit passing through x0.
Truncation Selection
1) Sort by
fitness
2) Replace
k% of lowest
by k% of
highest
Truncation Selection
Consider the Hawk-Dove game with ESS
given by (7/12 H, 5/12 D)
If .5 < xH(t) < 7/12, then xH(t+1) = 1.
Truncation Selection
Map Diagram:
(, )-ES Selection
 Given a population of  offspring, the best  are chosen
to parent the next generation.
 More extreme than truncation selection.
Linear Rank Selection
Agents sorted according to fitness.
Assigned new fitness values according to
their rank.
Causes fitness to change linearly with
rank.
Causes cycles around ESS.
Linear Rank Selection
Map Diagram:
Boltzman Selection
Inspired by simulated annealing.
Selection pressure slowly increased over
time to focus search.
In some cases, BS is able to retain the
dynamics and equilibria in EGT.
Boltzman Selection
Map Diagram:
Effects of Finite Populations on Evolutionary Stable
Strategies.
Ficici, Pollack
Finite Populations
Effects of finite population on EGT.
Begin at ESS (7/12,5/12) and test
n=60,120,300,600, and 900 for 2000
generations.
100 replicates of each experiment.
Finite Populations
Results:
Convergence
For a n player name, consider the MC with
states given by #of hawks.
Define transition matrix P.
bt = b0Pt
E(xH(t)) = (1/n)
ni=0 bHt * i
limt E(xH(t)) = bH
Estimate | E(xH(t)) - bH |
Convergence Simulation
Results: