Consider the following parlor game between two players.

It begins when a referee flips a coin, notes whether it comes up heads or tails, and then shows this result to player 1 only. Player 1 may then (1) pass and thereby pay $5 to player 2 or (2) bet. If player 1 passes, the game is terminated. However, if he bets, the game continues, in which case player 2 may then either (1) pass and thereby pay $5 to player 1 or (2) call. If player 2 calls, the referee then shows him the coin; if it came up heads, player 2 pays

$10 to player 1; if it came up tails, player 2 receives $10 from player 1.

(a) Give the pure strategies for each player. (Hint: Player 1 will (b)
have four pure strategies, each one specifying how he would respond to each of the two results the referee can show him;
player 2 will have two pure strategies, each one specifying how he will respond if player 1 bets.)

(b) Develop the payoff table for this game, using expected values for the entries when necessary. Then identify and eliminate any dominated strategies.

(c) Show that none of the entries in the resulting payoff table are a saddle point. Then explain why any fixed choice of a pure strategy for each of the two players must be an unstable solution, so mixed strategies should be used instead.

(d) Write an expression for the expected payoff in terms of the probabilities of the two players using their respective pure strategies. Then show what this expression reduces to for the following three cases: (i) Player 2 definitely uses his first strategy, (ii) player 2 definitely uses his second strategy, (iii) player 2 assigns equal probabilities to using his two strategies.