An Introduction to Evolutionary Game Theory: Lecture 2 - School of ...

Some Properties of the Replicator Dynamics (I)Replicator equations (REs):ẋ i = x i [(Ax) i − x.Ax]Set of coupled cubic equations (when x.Ax ≠ 0)Let x ∗ = (x1 ∗,...,x Q ∗ ) be a fixed point (steady state) of the REsx ∗ can be (Lyapunov-) stable, unstable, attractive (i.e. there isbasin of attraction), asymptotically stable=attractor ( =stable +attractive), globally stable (basin of attraction is S Q )Only possible interior fixed point satisfies (there is either 1 or 0):(Ax ∗ ) 1 = (Ax ∗ ) 2 = ... = (Ax ∗ ) Q = x ∗ .Ax ∗x 1 + ... + x Q = 1Same dynamics if one adds a constant[c j to the payoff matrixA = (A ij ): ẋ i = x i [(Ax) i − x.Ax] = x i (Ãx) i −], x.Ãx whereà = (A ij + c j )Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2