(Introduction to Probability, Blitzstein and Nwang)
Player A chooses a random integer between 1 and 100, with probability pj of choosing
j (for j = 1, 2, . . . , 100). Player B guesses the number that player A picked, and receives
that amount in dollars if the guess is correct (and 0 otherwise).(a) Suppose for this part that player B knows the values of pj . What is player B’s
optimal strategy (to maximize expected earnings)?(b) Show that if both players choose their numbers so that the probability of picking j is
proportional to 1/j, then neither player has an incentive to change strategies, assuming
the opponent’s strategy is fixed. (In game theory terminology, this says that we have
found a Nash equilibrium.)
16.(c) Find the expected earnings of player B when following the strategy from (b). Express
your answer both as a sum of simple terms and as a numerical approximation. Does the
value depend on what strategy player A uses?
a)
Pick the integer for which $p_j \cdot j$ is highest.
b) I dont't get the question.
c) Let $X^A$ be the number that player A picks, and $X^B$ the one B picks. Let $W$ be the earning.Then
\begin{align}
E(W) &= \sum_{i=1}^{100} \sum_{j=1}^{100} E(W|X^A=i, X^B=j) \cdot P(X^A=i, X^B=j)\\
&= \sum_{i=1}^{100} i \frac{c^2}{i^2} = c^2\sum_{i=1}^{100} \frac{1}{i} = \frac{1}{H_n} = 0.192775
\end{align}
Is this correct? How to answer part b) ? Maybe I don't understand the problem right.
Best Answer
For (b), notice that if A has the strategy of picking $i$ with probability $c/i$, then B's expected winnings is $c$ whatever strategy B uses, and so there is no incentive to change.
Similiarly, if B has the strategy of picking $i$ with probability $c/i$, then B's expected winnings is $c$ whatever strategy A uses, and so there is no incentive to change there either.
This also answers the last part of (c).