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## PowerPoint Slideshow about ' Replicator Dynamics' - vicky

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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

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

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.

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.

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

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

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

= “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?

0

We end up at Nash

(…if we end)

AND not just any Nash

(e.g. not mixed Nash in coordination)

Consider the

Rock-Paper-Scissors

Game:

With starting frequencies:

R

P

S

R

0

-1

1

P

1

0

-1

S

-1

1

0

Notice not asymptotically stable

It cycles

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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