Self Check Answers

1. The row minima for rows 1, 2, and 3 are 2, 5, and 1, respectively. Because 5 is the maximum of the minimum values, row 2 is the associated minimax strategy.
2. The column maxima for columns 1, 2, and 3 are 9, 8, and 5, respectively. Because 5 is the minimum of the maximum values, column 3 is the associated maximin strategy.
3. The game has a saddlepoint because the maximum of the minimum row values is equal to the minimum of the maximum column values. The value of the game is 5, which is the payoff when the players follow the maximin and minimax strategies.

4. The probabilities must add to 1. Because $p_1+p_2+p_3=1/2+1/3+p_3=1$, then $p_3=1/6$. The expected value $E=3$ because $E=p_1{s}_1+p_2{s}_2+p_3{s}_3=(1/2)4+(1/3)6+(1/6)(-6)=2+2+-1)=3$.

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5. Solving for $q$ in the equation $E_1=6q-1=4-9q=E_2$ yields $15q=5$ or $q=5/15$, which reduces to $q=1/3$.

6. Table 15.3 is rewritten with ordinal payoffs from Mark’s perspective in the payoff matrix to the left and from Lisa’s perspective in the payoff matrix to the right:

   2	1
   4	5
   6	3

   5	6
   3	2
   1	4

   For a fixed row and a fixed column, the entries of the payoff matrices sum to 7.

7. When Anneliese, Binh, and Charlie have bids of $150, $301, and $100, respectively, then Binh gets the item, pays $150 (the second highest bid), and achieves a $\$200-\$150=\$50$ payoff. Because Anneliese and Charlie don’t get the item and don’t pay anything, each receives a payoff of 0.

   If Anneliese bids anything less than $301, then she will not get the item; this means that she has a payoff of 0. If she bids $301 or more, then she could receive the item and pay more than what it is worth to her. This is a negative payoff. Because she cannot increase her payoff, she has no desire to change her bid. Binh already gets the item, so bids above $150 do not change who gets the item or how much Binh pays. If Binh bids less than $150, then he doesn’t get the item and his payoff is 0. Binh doesn’t wish to risk bidding $150 and possibly not getting the item. Binh has no incentive to change his bid. Like Anneliese, if Charlie bids more than $301, he will win the stamp but pay too much for it. This result could occur if he bids $301, too. For bids less than $301, Charlie still doesn’t get the item and pays nothing. Charlie has no reason to change his bid, either.

8. Player $B$ would play $r$ for both of his decision nodes. For the left node, $B$ prefers $r$ to $l$ because $r$gives 3 as opposed to 1 from $l$. Similarly, for the right node, $B$ prefers $r$ to $l$ because $r$ gives 4 as opposed to 2 from $l$. Because $A$ can look ahead and anticipates $B$ playing $r$ regardless of her decision, she will prefer to play $L$ (because she gets 2 instead of 1).