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(2): Additional thoughts
While I appreciate anybody who tries to defend my insights, and I also appreciate very much your recent contributions to this site, I feel obliged to point out (since you asked) that there are a number of problems with your comments about part (b).
Problem 1) If you initially guess x, and N is lower than x, than your expected additional gain/loss is not p(x) = (x-1)/2 - E(x)K. This is because you still have the option of stopping the game after your 2nd guess, if N is lower than your 2nd guess, and this option improves your expected part(b) profit.
Problem 2) You talk about $8.60, but that number is no longer the right number to think about. $8.60 is the breakeven for the part(a) game where N is known to be between 0 and 100. After discovering that N is less than 69, the breakeven for going forward had we been playing the part (a) game is given by
((69-1)/2) / E(69) = 34/(363/69) = $6.46. This is a 25% drop!
Problem 3) But, in any event, the breakeven for the part(a) game is not relevant. What matters is the breakeven for the part (b) game. It has presumably also gone down by about 25%. But K has not gone down. That is why, if K is anywhere near the maximum breakeven, it does not make sense to take a 2nd guess.
By the way, it would still make sense to take a second guess if K is well below the maximum breakeven. For instance, if K was just a penny, then it would make sense to keep guessing unless we had already made several guesses and knew that N was under 1. This observation is not relevant to my calculation of the maximum breakeven, which is based on the strategy assuming that K was the maximum breakeven.