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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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Presentation Transcript
slide2

Nash makes sense (arguably) if…

-Uber-rational

-Calculating

slide6

Norms/rights/morality are not chosen;rather…

We believewe have rights!

We feel angry when uses service but doesn’t pay

slide7

But…

From where do these feelings/beliefs come?

slide8

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

slide9

We consider a large population, ,of players

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

slide11

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
slide12

Note:

Rationality and consciousness don’t enter the picture.

slide13

Suppose there are two strategies, and .

We start with:

Some number, , of players assigned strategy

and some number, , of players assigned strategy

slide16

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.

slide17

For example, consider

the coordination game:

And the following starting frequencies:

slide18

Payoff for player who is playing is

Since depends on and we write

slide20

The average fitness, , of a population is the

weighted average of the two fitness values.

slide21

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…

slide22

This is the replicator equation:

Current frequency of strategy

Own fitness relative

to the average

slide23

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

slide24

If:

: The proportion of As is non-zero

: The fitness of A is above average

Then:

: A will be increasing in the population

slide25

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

slide27

A

B

Recall the payoffs of our (coordination) game:

b, c

a, a

A

B

c, b

d, d

a > c

b < d

slide28

= “asymptotically stable” steady states

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

slide30

A

B

b, c

a, a

A

B

c, b

d, d

slide31

1

0

slide34

Replicator teaches us:

We end up at Nash

(…if we end)

AND not just any Nash

(e.g. not mixed Nash in coordination)

slide36

Now…

is the number playing

is the number playing

is the number playing

slide37

Now…

is the proportion playing

is the proportion playing

is the proportion playing

slide39

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

slide43

Replicator is still:

Current frequency of strategy

Own fitness relative

to the average

slide48

R

P

S

R

0

-1

2

P

2

0

-1

S

-1

2

0

slide51

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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