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MATH/MTHE 339: Evolutionary Game Theory

Homework Assignment 4

2022

1 . (7 marks total ) Consider the following payoff matrix:

Π 1 =

2 − 1 0 .

a . (6 marks ) Analyze the replicator equation for this payoff matrix by finding all of the equilibria and

characterizing their stability by linearizing.

b . (1 marks ) Roughly sketch the phase portrait. You do not have to be exact in the placement of equi- libria. Simply show the qualitative nature of the equilibria.

2 . (7 marks total ) Consider the following payoff matrix:

Π 2 =

− 3 1 0 .

a . (6 marks ) Analyze the replicator equation for this payoff matrix by finding all of the equilibria and

characterizing their stability by linearizing.

b . (1 marks ) Roughly sketch the phase portrait. You do not have to be exact in the placement of equi- libria. Simply show the qualitative nature of the equilibria.

3 . (7 marks total ) Consider the following payoff matrix:

Π 3 =

− 3 − 1 1 .

a . (6 marks ) Analyze the replicator equation for this payoff matrix by finding all of the equilibria and

characterizing their stability by linearizing.

b . (1 marks ) Roughly sketch the phase portrait. You do not have to be exact in the placement of equi- libria. Simply show the qualitative nature of the equilibria.

4. (4 marks total ) Consider the following payoff matrix:

Π 4 = ,

with parameter γ for the replicator equation. Draw a bifurcation diagram for γ ∈ [0, 3].