Mixed Nash Equilibria and Lotka-Volterra Fixed Points
This is a continuation of the last chapter, and still basically technical. But it is less demanding and you might summon your energy again and challege.
We look at the general Lotka-Volterra equation obtained from the game matrix of previous chapter in the specific case of N = 2.
dx1/dt = b1 x1 + a11 x12 + a12 x1 x2
dx2/dt = b2 x2 + a21 x1 x2 + a22 x22
When we solve this set of equations, we obtain x1 、x2 that vary over time. But in some special choice of their initial values, it could stay constant keeping those values forever. If that happens, that set of special initial values (x1, x2) is termed Fixed Pointとof Lotka-volterra system. We assign capital letters (X1,X2) to indicate fixed point. by definition, since they keep their values, we have dX1/dt = dX2/dt = 0, that lead to
0 = X1 ( b1 + a11 X1 + a12 X2 )
0 = X2 ( b2 + a21 X1 + a22 X2 )
In general, there are four set of solution to this set of second-order algebraic equation. We concentrate on
0 = b1 + a11 X1 + a12 X2
0 = b2 + a21 X1 + a22 X2
We get a set (X1,X2) with nonzero value for X1 and X2. For example, in our previous example orbits attracted to the Center of Absorbing Votices, this fixed point correspond exactly to the location of the center of vortex. In original game theoretical language, above equation can be rewriten as
dPyou(s,x)/dsi = 0
which is exactly the condition for Nash equilibrium. So, we obsrve the identity
Nash Equibria of Game <-> Fixed Points of Lotka-Volterra
when we represent the game theory in terms of Lotka-Volterra equation thruogh the replicator dynamics.
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