Question 1

Consider the game

1 2 3
T \((10,0)\) \((0,10)\) \((3,3)\)
M \((2,10)\) \((10,2)\) \((6,4)\)
B \((3,3)\) \((4,6)\) \((6,6)\)

  1. Suppose that player 1 believes that player 2 will used the mixed strategy \((\frac{1}{3},\frac{1}{3},\frac{1}{3})\). What is their best response?

  2. Suppose that player 2 believes that player 1 will used the mixed strategy \((0,\frac{1}{3},\frac{2}{3})\). What is their best response?

Question 2

Consider the zero-sum game where player 1 has matrix \[A=\left(\begin{array}{ll} 0 & z\\ 1 & 2\end{array}\right)\] where \(z\) is an unknown real number. Find the gain floor and loss ceiling and hence determine whether there is a pure Nash equilibrium.

Question 3

Suppose that we fix some \(x>0\) and consider the game

L M R
U \((x,x)\) \((x,0)\) \((x,0)\)
C \((0,x)\) \((2,0)\) \((0,2)\)
D \((0,x)\) \((0,2)\) \((2,0)\)

  1. If they exist, determine all pure Nash equilibria.

  2. Show that if \(0<x<1\) then there is a mixed Nash equilibrium given by \[\underline{x}=\underline{y}=(0,\frac{1}{2},\frac{1}{2})\] and another given by \[\underline{x}=\underline{y}=(1-x,\frac{x}{2},\frac{x}{2})\] but that neither of these are Nash equilibria if \(x>1\).

Question 4

Consider the two-player zero-sum game with payoff matrix \(A\) given by \[A=\left(\begin{array}{rccc} 3 & 4 & 1-t & 2\\ -2 & t & 1 & 5\\ 2 & 1 & 1 & 2\end{array}\right)\] where \(t\in\mathbb R\).

  1. For which values of \(t\) does there exist a pure Nash equilibrium?

  2. Now suppose that \(t=0\). Is it possible to have a mixed Nash equilibrum of the form \(\underline{x}=(a,b,b)\), \(\underline{y}=(k,l,m,n)\) where the payoff to player 1 is \(2\)? Give a reason for your answer.