Question 1

Consider the example of the matrix game involving two beetle sizes from lectures:

Small Large
Small \(5\) \(1\)
Large \(8\) \(3\)

  1. Show from the definition of an evolutionary stable strategy (ESS) that Small is not an ESS, and Large is an ESS.

  2. Use the Equilibrium and Stability conditions to give an alternative verification of these results.

Question 2

Consider the following (grossly simplified) model of two currencies. Members of the population want to trade, but can only do so in a common currency. We model this as a game with payoff matrix

Euros Dollars
Euros \(1\) \(0\)
Dollars \(0\) \(1\)

  1. Determine the Nash equilibria of this game, and using the definition of an ESS find those which are ESSs and the value of the uniform invasion threshold.

  2. Use the Equilbrium and Stability conditions to give an alternative verification of which Nash equilibria are ESSs.

Question 3

Consider the version of the Prisoners’ Dilemma given by the game with payoff matrix

Confess Deny
Confess \(4\) \(1\)
Deny \(6\) \(1\)

  1. Determine the Nash equilibria of this game, and using the definition of an ESS find those which are ESSs and the value of the uniform invasion threshold.

  2. Use the Equilbrium and Stability conditions to give an alternative verification of which Nash equilibria are ESSs.

Question 4

In the Hawk-Dove game players compete for food. They can either act like a Hawk and fight for the food, or like a Dove and submit to the other player. The food is worth \(v>0\), and if both players are doves then they share it equally. If both players are hawks then they fight, and losing the fight costs \(c>0\), while the winner gaines \(v\). Assuming there is a 50/50 chance of winning a fight the payoff matrix for this game is

Hawk Dove
Hawk \(\frac{v-c}{2}\) \(v\)
Dove \(0\) \(\frac{v}{2}\)

Determine the ESSs in each of the following three cases:

  1. \(v>c\).

  2. \(v=c\).

  3. \(v<c\).

Question 5

Consider the game with payoff matrix

A B C D
A \(0\) \(1\) \(2\) \(-2\)
B \(-1\) \(0\) \(3\) \(-1\)
C \(-3\) \(2\) \(0\) \(-1\)
D \(-2\) \(1\) \(1\) \(0\)

  1. Determine the pure ESSs (if any).

  2. You may assume that there is a unique symmetric Nash equilibrium with \(S(\underline{x})=\{B,C\}\) given by \(\underline{x}=(0,3/5,2/5,0)\). Show that this is not an ESS.

  3. Use the Bishop-Cannings Theorem to classify the ESSs for this game.

Question 6

Consider the game with payoff matrix

A B C
A \(0\) \(0\) \(1\)
B \(1\) \(0\) \(0\)
C \(0\) \(1\) \(0\)

Show that the strategy \((1/3,1/3,1/3)\) is an ESS.