We decide to play the following game: An integer N will be randomly selected from the interval 0 - 100, inclusive. You try to guess N. After each guess, I tell you whether N is higher or lower than your guess.

If you successfully guess the integer, you win N dollars. Each guess costs you K dollars.

For each of the variants (a) and (b) below, what is the maximum value of K for which you'd be willing to play this game? Which strategy would you use to try to maximize your winnings in the long run?

(a) Once you start a round, you must continue until you guess N exactly.

(b) You may stop playing a round if you determine that N is too small to keep paying more money to guess N exactly. The money you've already spent on guesses is lost, but you may then start a new round with a new N.

(In reply to

re(4): Additional thoughts by tomarken)

Tomarken:

Well, if you want to pursue this, see my first part(b) post, where I give

Expected winnings = P(X) = (X+100)*(101 - X)/202.

Expected Additional cost = K*(1+E(100-X)*(100-X)/101).

After after determining that the maximum winnings is 68, for instance, the game changes to

Expected winnings = P(X) = (X+68)*(69 - X)/138.

Expected Additional cost = K*(1+E(68-X)*(68-X)/69).

You and I both expect that (X+68)*(69 - X)/138 - K*(1+E(68-X)*(68-X)/69)

is negative for all X <= 69, where

K = ((X+100)*(101 - X)/202) / (1+E(100-X)*(100-X)/101)