4.1.14 problem 15

(a)

Let $X$ be the earnings by player $B$. Suppose $B$ guesses a number $j$ with probability $b_j$. Then,

$$\text{E}\left(X\right)=\sum _{j=1}^{100}j{p}_j{b}_j$$

To maximize $\text{E}\left(X\right)$ 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,

$$\text{E}\left(X\right)=\sum _{k=1}^{100}\left(k\frac{c_A}{k}{b}_k\right)=c_A$$

.

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.