You are playing a game where there are 9 boxes laid out in a row, numbered 1 through 9 from left to right. Randomly placed in one of the boxes is a slip of paper that says "GAME OVER". The other eight boxes each contain $1000.
You are to pick the boxes one at a time. If you pick a box with $1000, you keep the money and you must pick another box. If at any point you select the box that says "GAME OVER", the game ends and you leave with the prize money you've accumulated to that point. The only catch is, the host of the game show will tell you which direction the "GAME OVER" box is in, and you must guess your next box in that direction (it's like a guessing game where you have to guess a number from 1 to 9, and after each guess the host tells you "higher" or "lower" until you finally guess the number he is thinking of). However, the goal of this game is not to land on the "GAME OVER" box (since you eventually will), but to maximize the number of guesses you take (and thus your profit) before you land on it.
Question 1: Is there an optimal strategy for this game? If so, what is it and what is your expected profit? If not, why not?
Question 2: What if you were the host, and instead of randomly placing the "GAME OVER" box, you could choose where it went - is there a strategy that would minimize the expected profit of the contestant?
(In reply to Observation re Question 2
by Steve Herman)
The question states that you must choose a box in the same direction as
the host indicates the game over box to be in. This invalidates
the strategy used to exceed $4000 expected earnings.
There is no mathematical strategy that the host can use against a
contestant that picks the boxes optimally. The contestant can
reduce their expectation however with poor box selection.
There is a method that the host can use to reduce the expected
earnings, but this would be based on psychology, not mathematics, and
would require several trials to optimze. Note though, that with
any one particular trial within this process, the worst case scenario
has the host expecting to pay out $4000, so the actual answer, given
the ability to refine the strategy based on observed contestant
behavior, has an upper bound of $4000, but precise value uncalculable.