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.