Replicator Dynamics
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Transcript of Replicator Dynamics
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Replicator Dynamics
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Nash makes sense (arguably) if…
-Uber-rational
-Calculating
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Such as Auctions…
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Or Oligopolies…
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But why would game theory matter for our puzzles?
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Norms/rights/morality are not chosen; rather…
We believe we have rights!
We feel angry when uses service but doesn’t pay
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But…
From where do these feelings/beliefs come?
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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
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We consider a large population, , of players
Each period, a player is randomly matched with another player and they play a two-player game
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Each player is assigned a strategy. Players cannot choose their strategies
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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
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Note:
Rationality and consciousness don’t enter the picture.
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Suppose there are two strategies, and .
We start with:
Some number, , of players assigned strategyand some number, , of players assigned
strategy
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We denote the proportion of the population playing strategy as , so:
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The state of the population is given by where , , and
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Since players interact with another randomly chosen player in the population, a player’s EXPECTED payoff is determined by the payoff matrix and the proportion of each strategy in the population.
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For example, consider the coordination game:
And the following starting frequencies:
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Payoff for player who is playing is
Since depends on and we write
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We interpret payoffs as rate of reproduction (fitness).
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The average fitness, , of a population is theweighted average of the two fitness values.
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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…
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Current frequency of strategy
Own fitness relative to the average
This is the replicator equation:
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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
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If: : The proportion of As is non-zero : The fitness of A is above average
Then: : A will be increasing in the population
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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
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The steady states are
such that
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a, a b, c
c, b d, d
A
B
A BRecall the payoffs of our (coordination) game:
a > cb < d
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= “asymptotically stable” steady statesi.e., steady states s.t. the dynamics point toward it
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What were the pure Nash equilibria of the coordination game?
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a, a b, c
c, b d, d
A
B
A B
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0 1
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And the mixed strategy equilibrium is:
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Replicator teaches us:
We end up at Nash(…if we end)
AND not just any Nash(e.g. not mixed Nash in coordination)
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Let’s generalize this to three strategies:
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Now…
is the number playing is the number playing is the number playing
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Now…
is the proportion playing is the proportion playing is the proportion playing
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The state of population is where , , ,and
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R P S
P
R
S
0
1
-1
-1
-1
1
1
0
0
For example, Consider the Rock-Paper-Scissors Game:
With starting frequencies:
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Fitness for player playing is
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In general, fitness for players with strategy R is:
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The average fitness, f, of the population is:
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Current frequency of strategy
Own fitness relative to the average
Replicator is still:
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Notice not asymptotically stableIt cycles
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R P S
P
R
S
0
2
-1
-1
-1
2
2
0
0
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Note now is asymptotically stable
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For further readings, see: Nowak Evolutionary Dynamics Ch. 4Weibull Evolutionary Game Theory Ch. 3
Some notes:Can be extended to any number of strategiesDoesn’t always converge, but when does converges
to NashThus, dynamics “justify Nash” but also give more info