Math 486.  February 11, 2003. Midterm 1.
6 problems, 11 pts each. Name________________________
Return this page.  The test and solutions will be posted at web at 9:30 am.
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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
Find an equilibrium and the corresponding payoff.

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.