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.

For part (a) since you will eventually get the N dollars, your only motivation is to have as few guesses as possible.

I managed to find a formula for the value of a game where the number is selected on the interval 1 to 2^n - 2 (so chosen because you can always pick the middle number.)

The break even point in terms of K is
[(2^n - 1)(2^(n-1) - 1)]/[(n-1)2^n - 1]

For the 0 to 62 game n=6 this is 1953/321 ≈ $6.08
For the 0 to 126 game n=7 this is 8001/796 ≈ $10.40

Although you can't really do this
For the 0 to 100 game n=log(102)/log(2) and K≈8.713095094

I'll be interested to see how close this is.

Posted by Jer on 2014-05-16 16:04:57