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Exercise 4.15

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(a)

Let

X

be the earnings by player

B

. Suppose

B

guesses a number

j

with probability

b_{j}

. Then,

E (X) = \sum_{j = 1}^{100} j p_{j} b_{j}

To maximize $E (X)$ then, $B$ should set $b_{j} = 1$ for the $j$ for which $j p_{j}$ is maximal. Since $p_{j}$ are known, this quantity is known.

(b)

Suppose player

P (A = k) = \frac{c_{A}}{k}

, and

P (B = k) = b_{k}

. Then,

E (X) = \sum_{k = 1}^{100} (k \frac{c_{A}}{k} b_{k}) = c_{A}

Thus, irrespective of what strategy $B$ adopts, their expected earnings are the same, so $B$ has no incentive to change strategies. Similar argument can be made for $A$ .

(c)

part

b

answers this part as well.

fifthist

2021-12-05 00:00

Exercise 4.15

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