e-Lecture : Evolutionary Games and Ecosystems

Taksu Cheon

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Evolutionary Games and Ecosystems (6c)

Vector-Matrix Natations of Games and Replicator Dynamics

In this chapter, we show that the game theory with mixed strategy can be cast into the form of Lotka-Volterra equation. This one would be very highly technical, so general readers are advised to skip.
To consider the mixed Nash equilibrium, it is useful to formulate the game theory in terms of matrix, vector, and "expectation values" in the manner reminiscent of quantum mechanics. We consider a game played by two players who have M+1 strategy to choose from. payoff of a player is given by the Game Matrix

@@@@@A = {Aij} Ai,j = 0..M

Let's assume that there is a group of N people, you among them. You repeatedly play this game with randomely changing opponent. We represent your strategy by a vector s :

@@@@@s = {s0}{ s1} ...{ sM}@G@s0 = 1-s1- ... -sM

This is meant to be a column vector whose tranposed couterpart is the row vector s+ = {s0, s1, ..., sM}. We can replace your random oppenent by a single "average player" having "average strategy" of the whole group.which is repesented by vector x :

@@@@@x = {x0}{x1} ... {xM}@G@x0 = 1-x1- ... -xM

Your expected payoff is

@@@@@Pyou(s,x) = <s|A|x> = s+ A x = ƒ°ij si Aij xj = s+ p(A, x)

The last equality defines p as p(A, x) = A x which is the "partial" payoff vector made up of payoff pi for each strategy "i". You tweek s to increase Pyou . Suppose in you make slight change in the probability si for the strategy "i" byƒÂsi to make it si +ƒÂsi . The change in the payoff is given by

@@@@@ƒÂ<s|A|x> = Ý<s|A|x>/Ýsi EƒÂsi

It is reasonable to postulate that you adopt that change with a probability proportional to make this change in the strategy "i". Nowassuming that all individuals in the group behave just like you do, the rate of change in xi is given by the above expression with postmusidentification s = x . Thus we have

@@@@@1/xi . dxi/dt = Ý<s|A|x>/Ýsibs=x = pi(A, x) - p0(A, x)

This is so-called replicator equation, which can be further rewritten as

@@@@dxi/dt = xi . [ bi +ƒ°'j aij xj ]

where the coefficients are given by bi = Ai0 -A00 , aij = Aij -Ai0 -A0j +A00 and the sum ƒ°'j indicate that of j = 1..M . This is nothing but the M-dimensional generalization of Lotka-Volterra equation.

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