Lecture IX: Evolution
Markus M. M¨obius
March 10, 2004
Learning and evolution are the second set of topics which are not discussed
in the two main texts. I tried to make the lecture notes selfcontained.
•
Fudenberg and Levine (1998), The Theory of Learning in Games, Chap
ter 1 and 2
1
Introduction
For models of learning we typically assume a fixed number of players who
find out about each other’s intentions over time. In evolutionary models the
process of learning is not explicitly modeled. Instead, we assume that strate
gies which do better on average are played more often in the population over
time. The biological explanation for this is that individuals are genetically
programmed to play one strategy and their reproduction rate depends on
their fitness, i.e. the average payoff they obtain in the game. The economic
explanation is that there is social learning going on in the background 
people find out gradually which strategies do better and adjust accordingly.
However, that adjustment process is slower than the rate at which agents
play the game.
We will focus initially on models with random matching: there are
N
agents who are randomly matched against each other over time to play a
certain game. Frequently, we assume that
N
as infinite. We have discussed
in the last lecture that random matching gives rise to myopic play because
there are no repeated game concerns (I’m unlikely to ever encounter my
current opponent again).
We will focus on symmetric
n
by
n
games for the purpose of this course.
In a symmetric game each player has the same strategy set and the payoff
1
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matrix satisfies
u
i
(
s
i
, s
j
) =
u
j
(
s
j
, s
i
) for each player
i
and
j
and strategies
s
i
, s
j
∈
S
i
=
S
j
=
{
s
1
, .., s
n
}
.
Many games we encountered so far in the
course are symmetric such as the Prisoner’s Dilemma, Battle of the Sexes,
Chicken and all coordination games. In symmetric games both players face
exactly the same problem and their optimal strategies do not depend on
whether they play the role of player 1 or player 2.
An important assumption in evolutionary models is that each agent plays
a fixed pure strategy until she dies, or has an opportunity to learn and about
her belief.
The game is fully specified if we know the fraction of agents
who play strategy
s
1
, s
2
, .., s
n
which we denote with
x
1
, x
2
, .., x
n
such that
∑
n
i
=1
x
i
= 1.
2
Mutations and Selection
Every model of evolution relies on two key concepts  a
mutation mechanism
and a
selection mechanism
. We have already discussed selection  strategies
spread if they give above average payoffs. This captures social learning in a
reduced form.
Mutations are important to add ’noise’ to the system (i.e.
ensure that
x
i
>
0 at all times) and prevent it from getting ’stuck’.
For example, in
a world where players are randomly matched to play a Prisoner’s Dilemma
mutations make sure that it will never be the case that all agents cooperate or
all agents defect because there will be random mutations pushing the system
away from the two extremes.
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 Spring '10
 Dingli
 Steady State, Game Theory, replicator dynamics

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