Question 1

Consider the game given by

Strategy 1 Strategy 2
Strategy 1 \((2,1)\) \((-1,-1)\)
Strategy 2 \((-1,-1)\) \((1,2)\)

We saw in lectures that there is a mixed Nash equilibrium given by \[\underline{x}=(3/5,2/5)\quad\quad\underline{y}=(2/5,3/5).\] By considering pure strategies, verify that this is indeed a Nash equilibrum.

Question 2

Consider the game given by

Strategy 1 Strategy 2
Strategy 1 \((-4,1)\) \((2,0)\)
Strategy 2 \((2,2)\) \((1,3)\)

  1. Find the safety values and maxmin strategies for each player.

  2. Suppose that there is a mixed Nash equilibrium. Use the Equality of Payoffs Theorem to determine this equilibrium.

  3. Verify that the mixed Nash equilibrium is individually rational.

Question 3

Consider the game given by

Strategy 1 Strategy 2
Strategy 1 \((5,4)\) \((3,6)\)
Strategy 2 \((6,3)\) \((1,1)\)

  1. Find the safety values and maxmin strategies for each player.

  2. Suppose that there is a mixed Nash equilibrium. Use the Equality of Payoffs Theorem to determine this equilibrium.

  3. Verify that the mixed Nash equilibrium is individually rational.

Question 4

Consider the game given by

Strategy 1 Strategy 2
Strategy 1 \((2,2)\) \((3,1)\)
Strategy 2 \((1,3)\) \((4,4)\)

Graph the rational reaction sets and find all the Nash equilibria. Which if any of these are Pareto optimal?

Question 5

Consider the game given by

Strategy 1 Strategy 2
Strategy 1 \((-10,5)\) \((2,-2)\)
Strategy 2 \((1,-1)\) \((-1,1)\)

Graph the rational reaction sets and find all the Nash equilibria. Which if any of these are Pareto optimal?