Math 486.  February 11, 2003. Midterm 1.
6 problems, 11 pts each. Name________________________
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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?

Solution.
We mark positions by W or L  if it is winning  or loosing positions for the player who starts.
Fow winning positions we write  winning moves (the number of stones to take).

 0 1 2 3 4 5 6 7 8 9 W L W L W L W L W W terminal 1 1 1,5 1,5 8
 10 11 12 13 14 15 16 17 18 19 W W W W L W L W L W 5 8 5 8 1,8 1 1,5

So the winning positions repeat with period 13.
The first player wins at initial positions of the forms
13n,  13n+2, 13n+4, 13n+6, 13n+8, 13n+9,13n+10, 13n+11, and  13n+12.
The second player wins
if the first player starts  at the other positions, namely, 13n+1, 13n+3, 13n+5, 13n+7..  In particular,  1000=13*76+12  is 12  modulo 13, so the first player wins
by  taking 5 stones.
Any strategy of the second player is optimal.

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.

Solution.
P goes over if draws, so  P stands.
In the case (a), P wins with probability 3/5 and loses with probability
2/5. The expected payoff is  \$2.
An extensive form is
chance move (\$2)
/                             \
3/5                             2/5
/                                       \
\$10                                    -\$10

In the case (b),  an extensive form is

chance move (\$2)
/                             \
3/5                             2/5        (first player draws)
/                                        \
6,6,8,8 left    (\$0)                 6,6,6,8 left   (\$5)         (cards left; lifted payoff)
/           \                                     /           \
1/2        1/2                              3/4          1/4                                               (dealer's face-down card 6 or 8)
/                   \                            /                   \
\$10                  -\$10                 \$10               -\$10                                         (P's payoff)

The expected payoff is  \$2.

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.

Solution.
 3 4 3 0' 3 5' 5*' 5*' 5' 6* 5 4 4 6* 3' 7* 5* 3' 6* 4
* maximal in its column, ' minimal in its row.  The second row & second or third column give a saddle point. The corresponding payoff  (the value of game) is 5 (for the row player).

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.

Solution.
initial position     0.1, 1.1, 0.3
chance move
/                                     \
0.1                                        0.9
/                                                 \
A     1,2,3                                      B  0,1,0
//          \                                               //            \
B  1,2,3     C 0,0,0                    0,1,0     B           C  -2,0,2
/ /      \      //          \                                 /     \ \     /       \ \
1,2,3   0,0,0    -1,-2,-3              1,0,1       0,1,0       -2,0,2

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.

Solution.
The payoff is  (x+y)(9/19)-x(10/19)= (9y - x)/19.

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.

Solution. (a)  The player wins (insurance pays) with probability 4/13. So if his bet is \$x, then his expected payoff
is  \$(4/13)2x-(9/13)x= -\$x/8. He should not buy insurance (x=0).
(b) The player wins (insurance pays) with probability 14/49.So if his bet is \$x, then his expected payoff
is  \$(14/49)2x-(35/49)x= -\$x/7. He should not buy insurance.