Player 1 (the "hider") and player 2 (the "seeker") play the following game. There are four boxes with lids, arranged in a straight line. For convenience, the boxes are labeled A, B, C, and D. The administrator of the game gives player 1 a $100 bill, and player 1 must hide it in one of the four boxes. Player 2 does not observe where player 1 hides the $100 bill. Once player 1 has hidden the bill, player 2 must open one (and only one) of the boxes. If the money is in the box that player 2 opens, then player 2 keeps the $100. If it is not, player 1 gets to keep the $100.
(a) Does this game have a pure-strategy Nash equilibrium?
(b) Find the mixed-strategy Nash equilibrium.
(c) Suppose it is common knowledge that player 1 likes the letter "A" and would get extra satisfaction from putting the money in box A. Let this satisfaction be equivalent to receiving $20. Assume this is in addition to any money received in the game. How does player 1's preference for the letter "A" affect the equilibrium mixing probabilities? Calculate the new equilibrium strategy profile if you can.
(d) Describe the equilibria of the game in which player 1's extra satisfaction from selecting box A is equivalent to receiving $120.