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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Replicator Dynamics

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Replicator dynamics

Replicator Dynamics


Replicator dynamics

Nash makes sense (arguably) if…

-Uber-rational

-Calculating


Such as auctions

Such as Auctions…


Or oligopolies

Or Oligopolies…


Replicator dynamics

But why would game theory matter for our puzzles?


Replicator dynamics

Norms/rights/morality are not chosen;rather…

We believewe have rights!

We feel angry when uses service but doesn’t pay


Replicator dynamics

But…

From where do these feelings/beliefs come?


Replicator dynamics

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


Replicator dynamics

We consider a large population, ,of players

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


Replicator dynamics

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


Replicator dynamics

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


Replicator dynamics

Note:

Rationality and consciousness don’t enter the picture.


Replicator dynamics

Suppose there are two strategies, and .

We start with:

Some number, , of players assigned strategy

and some number, , of players assigned strategy


Replicator dynamics

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


Replicator dynamics

The state of the population is given by

where , , and


Replicator dynamics

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.


Replicator dynamics

For example, consider

the coordination game:

And the following starting frequencies:


Replicator dynamics

Payoff for player who is playing is

Since depends on and we write


Replicator dynamics

We interpret payoffs as rate of reproduction (fitness).


Replicator dynamics

The average fitness, , of a population is the

weighted average of the two fitness values.


Replicator dynamics

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…


Replicator dynamics

This is the replicator equation:

Current frequency of strategy

Own fitness relative

to the average


Replicator dynamics

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


Replicator dynamics

If:

: The proportion of As is non-zero

: The fitness of A is above average

Then:

: A will be increasing in the population


Replicator dynamics

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


Replicator dynamics

 The steady states are

such that


Replicator dynamics

A

B

Recall the payoffs of our (coordination) game:

b, c

a, a

A

B

c, b

d, d

a > c

b < d


Replicator dynamics

= “asymptotically stable” steady states

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


Replicator dynamics

What were the pure Nash equilibria of the coordination game?


Replicator dynamics

A

B

b, c

a, a

A

B

c, b

d, d


Replicator dynamics

1

0


Replicator dynamics

And the mixed strategy equilibrium is:


Replicator dynamics

Replicator teaches us:

We end up at Nash

(…if we end)

AND not just any Nash

(e.g. not mixed Nash in coordination)


Replicator dynamics

Let’s generalize this to three strategies:


Replicator dynamics

Now…

is the number playing

is the number playing

is the number playing


Replicator dynamics

Now…

is the proportion playing

is the proportion playing

is the proportion playing


Replicator dynamics

The state of population is

where , , ,

and


Replicator dynamics

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 dynamics

Fitness for player playing is


Replicator dynamics

In general, fitness for players with strategy R is:


Replicator dynamics

The average fitness, f, of the population is:


Replicator dynamics

Replicator is still:

Current frequency of strategy

Own fitness relative

to the average


Replicator dynamics

Notice not asymptotically stable

It cycles


Replicator dynamics

R

P

S

R

0

-1

2

P

2

0

-1

S

-1

2

0


Replicator dynamics

Note now is asymptotically stable


Replicator dynamics

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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