Suppose that your parents are deciding where to hide your (insert non-offensive, not religiously affiliated holiday here) presents while you’re trying to decide where to look for them. The payoff to your parents from successfully hiding your presents is 7 while their payoff from you finding them is 1. Your payoff from successfully finding your presents is 10 and your payoff from not finding them is 3. Your parents are considering hiding the presents in one of three places: under the bed, in the closet, or under the floor boards. They have left the house, but only briefly, so you must decide where to look with your limited time. If you look in the same place that your parents have hidden the presents you find them. If not, not.
(a) Suppose that you have two pure strategies: you can either look under the bed and in the closet (since those don’t take too long) or you can pry up the floor boards and look under them (that’s harder so it takes longer). Find a mixed-strategy Nash equilibrium for this game if it is possible. (b) Now suppose that you have three options. You can look under the bed and in the closet, you can look under the bed and under the floor boards, or you can look in the closet and under the floor boards. Once again, find a mixed-strategy Nash equilibrium if one exists.