Question: In a certain card game, player 1 holds two Kings and one Ace. He discards one card, lays the other two face down on the table, and announces either 2 Kings or Ace King. Ace King is a better hand than 2 Kings. Player 2 must either Fold or Bet on player 1's announcement. The hand is then shown and the payoffs are as follows:
1. If player 1 announces the hand truthfully and player 2 Folds, player 1 wins $1 from player 2.
2. If player 1 lies that the hand is better than it is and player 2 Folds, player 1 wins $2 from player 2.
3. If player 1 lies that the hand is worse than it is and player 2 Folds, player 2 wins $2 from player 1.
4. If player 2 Bets player 1 is lying and if player 1 is actually lying the above payoffs are doubled and reversed. If player 1 is not lying, player 1 wins $2 from player 2.
(a) Use Gambit to draw a game tree.
(b) Give a complete list of the pure strategies for each player as given by Gambit and write down the game matrix.
(c) Find the value of the game and the optimal strategies for each player.
(d) Modify the game so that player 1 first chooses one of the three cards randomly and tosses it. Player 1 is left with two cards-either an A and K, or two K's. Player 1 knows the cards he has but player 2 does not and the game proceeds as before. Solve the game.
(e) As in the previous part but assume that player 1 does not know the outcome of the random discard. That is, he does not actually know if he has 1A1K, or 2K. The payoffs remain as before replacing lying with incorrect and truthfully with correct.