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# Replicator Dynamics - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Replicator Dynamics' - vicky

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Presentation Transcript

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

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

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

• Players “imitate” others’ strategies

Rationality and consciousness don’t enter the picture.

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.

the coordination game:

And the following starting frequencies:

Since depends on and we write

We interpret payoffs as rate of reproduction (fitness).

weighted average of the two fitness values.

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…

Current frequency of strategy

Own fitness relative

to the average

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

: The proportion of As is non-zero

: The fitness of A is above average

Then:

: A will be increasing in the population

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

We will call such points steady states.

0

such that

B

Recall the payoffs of our (coordination) game:

b, c

a, a

A

B

c, b

d, d

a > c

b < d

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

What were the pure Nash equilibria of the coordination game?

B

b, c

a, a

A

B

c, b

d, d

0

We end up at Nash

(…if we end)

AND not just any Nash

(e.g. not mixed Nash in coordination)

is the number playing

is the number playing

is the number playing

is the proportion playing

is the proportion playing

is the proportion playing

where , , ,

and

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

It cycles

P

S

R

0

-1

2

P

2

0

-1

S

-1

2

0

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