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MATH/MTHE 339: Evolutionary Game Theory

Practice Final (Optional Assignment 5)

2022

1 . (5 marks total ) Consider the following 2 player normal form game:

Nicole

	L	R
U	− 2 , 1	x , 2
D	− 1, y	3 , 2

a . (1 mark ) For which values of x and y is (D,L) the only pure Nash equilibrium?

b . (1 mark ) Let σ A = (1/3, 2/3) and σ N = (3/4, 1/4) be mixed strategies for Andrew and Nicole, respectively. For which values of x and y is the strategy profile (σA ,σN ) a mixed Nash equilibrium?

c . (3 marks ) Imagine that there is a third party who will send strategy recommendations to the players.

With probability 1/4, Andrew is told to play U and Nicole L. With probability 1/2, Andrew is told to play

U and Nicole R. With probability 1/4, Andrew is told to play D and Nicole L. For which values of x and y is this a correlated equilibrium?

2 . (5 marks total ) Laura and Fiona are trying to decide between doing their gardening chores (C), shirking their gardening chores (S ), or selling their ownership in the garden. If either sell their ownership, both receive a guaranteed payoff of 3/2. Laura decides first where to sell her ownership (O) or note sell her ownership (N), after which Fiona makes the same decision.

If neither sell their ownership, then, without knowing what the other is going to do, they must choose

between doing their chores or shirking their chores. The payoffs to each are given in the matrix below:

Fiona

	C	S
C	1 2	1 , 3
S	2 2	0, 0

a . (1 mark ) Draw a game tree for this game. Clearly label strategies, nodes, information sets and payoffs.

b . (4 mark ) Find all subgame perfect Nash equilibria.

3 . (5 marks total ) Consider the following 2 player normal form game: for b > c > 0.

Mark

	S	D
S	b − c /2, b − c /2	b − c,b
D	b , b − c	0, 0

a . (2 mark ) Find all Nash equilibria.

b . (3 mark ) Suppose that after each round the game may be played again with probability 1/4. Consider the trigger strategy T in which one plays S initially, but will play D forever if at any point either player plays D. Find conditions on b and c such that (T,T) is a subgame perfect Nash equilibrium.

4 . (10 marks total ) Consider the following payoff matrix:

Π =

1 1 − 1 .

a . (4 marks ) Find all the equilibria of the replicator equation for this game, and show which of these

are ESSes and which are not.

b . (4 marks ) Characterize the stability of all equilibria by linearizing and evaluating the eigenvalues.

c . (2 marks ) Sketch the phase portrait for the replicator dynamics.

5 . (10 marks total ) Consider the following bimatrix game:

Π = , Ψ = ,

where Π is the payoff to x playing y and Ψ is the payoff to y playing x.

a . (4 marks ) Find all the equilibria of the 2 type replicator equation for these payoff matrices.

b . (4 marks ) Characterize the stability of these equilibria by linearizing and evaluating the eigenvalues.

c . (2 marks ) Sketch the phase portrait.

6 . (5 marks total ) Consider the following payoff matrix:

Π = ,

with parameter γ for the replicator equation. Draw a bifurcation diagram for γ ∈ [ − 2, 2].

7 . (5 marks total ) Consider the resident population governed by the equation: n ˙ r = r2nr − .

Consider a rare mutant whose dynamics are governed by:

n ˙ m = m2nm − − (1 + m − r) .

a . (1 mark ) What is the invasion exponent Sr(m)?

b . (1 mark ) Find the evolutionary singular strategy.

c . (1 mark ) Is it convergence stable? Justify your answer.

d . (1 mark ) Is it an ESS? Justify your answer.

e . (1 mark ) What does this analysis say about the evolution of the trait r?