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 re: solution for question 2
If the host chooses randomly, all strategies of the contestant which
chooses either first or last at every step is equally optimal. That's
why, when the host can choose his strategy, he has to give equal
probability to all such strategies. Of course, it will be more likely
that a contestant will always choose rightmost or leftmost box, or
switch at every step (left-right-left-right-...), but investigating the
probabilities would require large experiments :-), so I just assumed
the probabilities to be equal.
To Jer: yes, but then we could add another step for the host: if the
host knows the contestant assumes the host will minimize payoff ....
and continue like this. I think the question 2 was meant as: the
contestant assumes the game to be fair (all boxes equally likely), the
host however is cheating.
However, I think Steve Herman has a better point.