Replicator Dynamics
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Replicator Dynamics. Nash makes sense ( arguably ) if… - Uber - rational - Calculating. Such as Auctions…. Or Oligopolies…. But why would game theory matter for our puzzles ?. Norms/rights/morality are not chosen ; rather… We believe we have rights!

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Nash makes sense (arguably) if…

-Uber-rational

-Calculating





Norms/rights/morality are not chosen;rather…

We believewe have rights!

We feel angry when uses service but doesn’t pay


But…

From where do these feelings/beliefs come?


In this lecture, we will introduce replicator dynamics

The replicator dynamic is a simple model of evolution and prestige-biased learning in games

Today, we will show that replicator leads to Nash


We consider a large population, ,of players

Each period, a player is randomly matched with another player and they play a two-player game


Each player is assigned a strategy. Players cannot choose their strategies


We can think of this in a few ways, e.g.:

  • Players are “born with” their mother’s strategy (ignore sexual reproduction)

  • Players “imitate” others’ strategies


Note:

Rationality and consciousness don’t enter the picture.


Suppose there are two strategies, and .

We start with:

Some number, , of players assigned strategy

and some number, , of players assigned strategy


We denote the proportion of the population playing strategy as , so:


The state of the population is given by

where , , and


Since players interact with another randomly chosen player in the population, a player’s EXPECTEDpayoff is determined by the payoff matrix and the proportion of each strategy in the population.


For example, consider

the coordination game:

And the following starting frequencies:


Payoff for player who is playing is

Since depends on and we write


We interpret payoffs as rate of reproduction (fitness).


The average fitness, , of a population is the

weighted average of the two fitness values.


How fast do and grow?

Recall

First, we need to know how fast grows

Let

Each individual reproduces at a rate , and there are of them. So:

Next we need to know how fast grows. By the same logic:

By the quotient rule, and with a little simplification…


This is the replicator equation:

Current frequency of strategy

Own fitness relative

to the average


Growth rate of A

Current frequency of strategy

Because that’s how many As can reproduce

Own fitness relative

to the average

This is our key property.

More successful strategies grow faster


If:

: The proportion of As is non-zero

: The fitness of A is above average

Then:

: A will be increasing in the population


We are interested in states where

since these are places where the proportions of A and B are stationary.

We will call such points steady states.

0


The steady states are

such that


A

B

Recall the payoffs of our (coordination) game:

b, c

a, a

A

B

c, b

d, d

a > c

b < d


= “asymptotically stable” steady states

i.e., steady states s.t.the dynamics point toward it


What were the pure Nash equilibria of the coordination game?


A

B

b, c

a, a

A

B

c, b

d, d


1

0



Replicator teaches us:

We end up at Nash

(…if we end)

AND not just any Nash

(e.g. not mixed Nash in coordination)



Now…

is the number playing

is the number playing

is the number playing


Now…

is the proportion playing

is the proportion playing

is the proportion playing



For example,

Consider the

Rock-Paper-Scissors

Game:

With starting frequencies:

R

P

S

R

0

-1

1

P

1

0

-1

S

-1

1

0





Replicator is still:

Current frequency of strategy

Own fitness relative

to the average



R

P

S

R

0

-1

2

P

2

0

-1

S

-1

2

0



For further readings, see:

Nowak Evolutionary Dynamics Ch. 4

WeibullEvolutionary Game Theory Ch. 3

Some notes:

Can be extended to any number of strategies

Doesn’t always converge, but when does converges to Nash

Thus, dynamics “justify Nash” but also give more info


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