Chapter 24

Here is a description of the simplest poker game. There are two players and only two cards in the deck, an Ace (A) and a King (K). First, the deck is shuffled and one of the two cards is dealt to player 1 . That is, nature chooses the card for player 1 . It is the Ace with probability \(1 / 2\) and the King with probability \(1 / 2\). Player 2 does not receive a card. Player 1 observes his card and then chooses whether to bid (B) or fold (F). If he folds, then the game ends with player 1 getting a payoff of \(-1\) and player 2 getting a payoff of 1 (that is, player 1 loses his ante to player 2 ). If player 1 bids, then player 2 must decide whether to bid or fold. When player 2 makes this decision, she knows that player 1 bid, but she has not observed player 1's card. The game ends after player 2's action. If player 2 folds, then the payoff vector is \((1,-1)\), meaning player 1 gets 1 and player 2 gets \(-1\). If player 2 bids, then the payoff depends on player 1 's card; if player 1 holds the Ace, then the payoff vector is \((2,-2)\); if player 1 holds the King, then the payoff vector is \((-2,2)\). Represent this game in the extensive form and in the Bayesian normal form.

Suppose Andy and Brian play a guessing game. There are two slips of paper; one is black and the other is white. Each player has one of the slips of paper pinned to his back. Neither of the players observes which of the two slips is pinned to his own back. (Assume that nature puts the black slip on each player's back with probability \(1 / 2\).) The players are arranged so that Andy can see the slip on Brian's back, but Brian sees neither his own slip nor Andy's slip. After nature's decision, the players interact as follows. First, Andy chooses between \(\mathrm{Y}\) and \(\mathrm{N}\). If he selects \(\mathrm{Y}\), then the game ends; in this case, Brian gets a payoff of 0 , and Andy obtains 10 if Andy's slip is black and \(-10\) if his slip is white. If Andy selects \(\mathrm{N}\), then it becomes Brian's turn to move. Brian chooses between \(\mathrm{Y}\) and \(\mathrm{N}\), ending the game. If Brian says \(\mathrm{Y}\) and Brian's slip is black, then he obtains 10 and Andy obtains 0 . If Brian chooses \(Y\) and the white slip is on his back, then he gets \(-10\) and Andy gets 0 . If Brian chooses \(\mathrm{N}\), then both players obtain 0 . (a) Represent this game in the extensive form. (b) Draw the Bayesian normal-form matrix of this game.

Draw the extensive-form representation of the following three-card poker game. There are three cards in the deck: an Ace, a King, and a Queen. The deck is shuffled, one card is dealt to player 1 , and one card is dealt to player 2 . Assume the deck is perfectly shuffled so that the six different ways in which the cards may be dealt are equally likely. Each player observes his own card but not the card dealt to the other player. Player 1 then decides whether to fold or bid. If he folds, then the game ends and the payoff vector is \((-1,1)\); that is, player 1 obtains \(-1\) (he loses his ante), and player 2 obtains 1 (gaining player 1 's ante). If player 1 bids, then player 2 must decide whether to fold or bid. If player 2 folds, then the game ends with payoff vector \((1,-1)\). If player 2 decides to bid, then the players reveal their cards. The player with the higher card wins, obtaining a payoff of 2 ; the player with the lower card loses, getting a payoff of \(-2\). Note that when both players bid, the stakes are raised. By the way, the Ace is considered the highest card, whereas the Queen is the lowest card.