e-Lecture : Evolutionary Games and Ecosystems

Taksu Cheon
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Evolutionary Games and Ecosystems (10a)

Technical Details of Knight & Peasants Game

We target only those who are confident to follow a complicated arguments in this chapter. Please read the Chapter (6c) again, and write down its last equation on a sheet of paper.
We write the payoff matrix A, its transposition A+ , and the strategy vector x as

　
[ 0
0
0]
A= [ b b-a
b-R]
[-d fR-d
-d]
[ 0
b
-d]
A+= [ 0 b-a
fR-d]
[ 0 b-R
-d]
[1-x-y]
x= [ x ]
[ y ]

The replicator dynamics derived from the last equation of (6c) with the matrix A is described by

　　　　　dx/dt = b x - a x2 - R x y
　　　　　dy/dt = -d y + f R x y

This is in fact nothing but a good old Lotka-Volterra equationd of prey-predator type. We next introduce altruism into this game. Since the altruistic game matrix is given bythe transposed matrix A+ , a game plan specified by the mixture of egoism : altruism = κ : (1-κ) is decribed using a modified game matrix

　　　　　Aκ = κ A + (1-κ) A+

Replicator dynamics of this game matrix iscalculated as

　　　　　dx/dt = (1-κ)b x - a x2 - (1-κ- κf)R x y
　　　　　dy/dt = -(1-κ)d y + (f-κ- κf) R x y

The satble fixed points of this set of differential equation vary with the value of k. By defining the quantity

　　　　　κ* = (f-ad/bR) / (1+f)

we can classify them as follows. In case of κ > κ*, a stable fixed point is given by

　　　　　X = (1-κ)/(f-κ- κf)・(d/R)
　　　　　Y = (1-κ)/(1-κ- κf)・{ bR - ad/(f-κ- κf) }/R2 ,

while in case of κ < κ* , this point becomes unstable (namely, the neighboring linearized map becomes hyperbolic), and new stable point with extinct Y specified by

　　　　　X = (1-κ)b/a
　　　　　Y = 0

appears. We shall display this results graphically and give explanations.
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