6 problems, 11 pts each. Name________________________

Return this page. The test and solutions will be posted at web at 9:30 am.

Put your name on the class list.

1. Nim. Last move loses. Player may take 1, 5, or 8 stones in
a move. Initial position: 1 pile, 1000 stones.

Who wins and how?

2. Blackjack. P has 10 and 7. Dealer shows 10. Cards left: 6,
6, 6,8, 8. Bet is $10.

(a) P is the only player.

(b) P is the second player, and the first player draws a card.

Find an optimal strategy and the corresponding payoff.

3. Matrix game.

3 | 4 | 3 | 0 | 3 |

5 | 5 | 5 | 5 | 6 |

5 | 4 | 4 | 6 | 3 |

7 | 5 | 3 | 6 | 4 |

4. Extensive form, 3 players, A B, C.

initial position

chance move

/
\

0.1
0.9

/
\

A
B

/
\
/ \

B
C
B C

/ \
/ \
/ \ /
\

1,2,3 0,0,0 -1,-2,-3
1,0,1 0,1,0
-2,0,2

Find an equilibrium and the corresponding payoff.

5. Roulete. P bets $x on black in roulette (with
0 and 00) using a coupon which gives him additional $y in case of winning.
Compute P's expected payoff.

6. Blackjack insurance. Dealer's face-up card is Ace. The house
bets 2 to 1 up to $100 that

the dealer will not get a blackjack. Should the player take the
bet?

(a) There are many decks in the shoe.

(b) One deck is used, and the player has 10+10.