Replicator Dynamics 1
Nash makes sense (arguably) if -Uber-rational -Calculating 2
Such as Auctions 3
Or Oligopolies Image courtesy of afagen on Flickr. CC BY NC-SA Image courtesy of longislandwins on Flickr. CC-BY 4
But why would game theory matter for our puzzles? 5
Norms/rights/morality are not chosen; rather We believe we have rights! We feel angry when uses service but doesn t pay 6
But From where do these feelings/beliefs come? 7
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 8
We consider a large population, N, of players Each period, a player is randomly matched with another player and they play a two-player game 9
Each player is assigned a strategy. Players cannot choose their strategies 10
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 11
Note: Rationality and consciousness don t enter the picture. 12
Suppose there are two strategies, A and B. We start with: Some number, N A, of players assigned strategy A And some number, N B, of players assigned strategy B 13
We denote the proportion of the population playing strategy A as X A, so: x A = N A /N x B = N B /N 14
The state of the population is given by (x A, x B ) where x A 0, x B 0, and x A + x B = 1. 15
Since players interacts 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. 16
For example, consider the coordination game: a > c b < d A B A B a, a b, c c, b d, d And the following starting frequencies: x A =.75 x B =.25 17
Payoff for player who is playing A is f A Since f A depends on x A and x B we write f A (x A, x B ) f A (x A, x B ) = (probability of interacting with A player)*u A (A,A) + (probability of interacting with B player)*u A (A,B) = x A *a + x B *b =.75*a +.25*b 18
We interpret payoff as rate of reproduction (fitness). 19
The average fitness, f, of a population is the weighted average of the two fitness values. f(x A, x B ) = x A *f A (x A, x B ) + x B *f B (x A, x B ) 20
How fast do x A and x B grow? Recall x A = N A / N First, we need to know how fast does N A grows Let N A = dn A /dt Each individual reproduces at a rate f A, and there are N A of them. So: N A = N A * f A (x A, x B ) Next we need to know how fast N grows. By the same logic: N = N * f(x A, x B ) By the quotient rule, and with a little simplification 21
This is the replicator equation: x A = x A * (f A (x A, x B ) f(x A, x B )) Current frequency of strategy Own fitness relative to the average 22
Growth rate of A x A = x A * (f A (x A, x B ) f(x A, x B )) 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 23
x A = x A * (f A (x A, x B ) f(x A, x B )) If: Then: x A > 0: The proportion of As is non-zero f A > f: The fitness of A is above average x A > 0: A will be increasing in the population 24
The steady states are x A = 0 x A = 1 x A such that f A (x A, x B ) = f B (x A, x B ) 25
Recall the payoffs of our (coordination) game: A B A a, a b, c B c, b d, d a > c b < d 26
= asymptotically stable steady states i.e., steady states s.t. the dynamics point toward it f A (x A, x B ) f B (x A, x B ) x A = 0 x A = 1 27
What were the pure Nash equilibria of the coordination game? 28
A B A a, a b, c B c, b d, d 29
0 1 30
And the mixed strategy equilibrium is: x A = (d b) / (d b + a c) 31
0 (d b) / (d b + a c) 1 32
Replicator teaches us: We end up at Nash ( if we end) AND not just any Nash (e.g. not mixed Nash in coordination) 33
Let s generalize this to three strategies: R P S 34
Now N R is the number playing R N P is the number playing P N S is the number playing S 35
Now x R is the proportion playing R x P is the proportion playing P x S is the proportion playing S 36
The state of population is (x R, x S, x P ) where x R 0, x P 0, x S 0, and x R + x S + x P = 1 37
For example, Consider the Rock-Paper-Scissors Game: R R P S 0-1 1 P 1 0-1 S -1 1 0 With starting frequencies: x R =.25 x P =.25 x S =.5 38
Fitness for player playing R is f R f R (x R,x P,x S ) = (probability of interacting with R player)*u R (R,R) + (probability of interacting with P player)*u R (R,P) + (probability of interacting with S player)*u R (R,S) =.25*0 +.25*-1 +.5*1 =.25 39
In general, fitness for players with strategy R is: f R (x R,x P,x S ) = x R *0 + x P *-1 + x S *1 40
The average fitness, f, of the population is: f(x R,x P,x S ) = x R *f R (x R,x P,x S ) + x P *f P (x R,x P,x S ) + x S *f S (x R,x P,x S ) 41
Replicator is still: x R = x R * (f R (x R,x P,x S ) f(x R,x P,x S )) Current frequency of strategy Own fitness relative to the average 42
x P =1 x R =.5, x P =.5 x S =.5, x P =.5 (x R =.33, xp=.33,x S =.33) x R =1 x R =.5, x S =.5 x S =1 43
A B C Image by MIT OpenCourseWare. 44
Notice not asymptotically stable It cycles Will show this in HW 45
R P S R 0-1 2 P 2 0-1 S -1 2 0 46
A B C Image by MIT OpenCourseWare. 47
Note now is asymptotically stable Will solve for Nash and show this is what dynamics look like in HW 48
For further readings, see: Nowak Evolutionary Dynamics Ch. 4 Weibull Evolutionary Game Theory Ch. 3 Some notes: Can be extended to any number of strategies Doesn t always converge, but when does converges to Nash We will later use this to provide evidence that dynamics predict behavior better than Nash 49
MIT OpenCourseWare http://ocw.mit.edu 14.11 Insights from Game Theory into Social Behavior Fall 2013 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.