Inspired by a question about bisection I wondered about the following: The are two players X and Y and a moderator Z who knows two (random,independent, uniformly chosen) hidden reals $x$ and $y$ from (0,1). The game has $t$ turns. At the beginning of each turn player X knows an open interval containing $x$, she names an internal point and learns which sub-interval contains $x$. Simultaneously, Y does the same for $y$. The player with the shorter interval wins one dollar from the other player (in case of a tie, no one wins).
What is the optimal strategy?
For $t=1$ turn it is obviously to choose the midpoint $0.5$. But for $t=2$ the following strategy beats a player who has decided on bisection: on turn 1 choose $0.45$ and on turn 2 choose either $0.225$ or $0.675$.
Then the probabilities of winning are $0.45,0.225$ and $0.325 $ for $+1,0$ and $-1$ respectively.
Also: Consider the variation where X wins only if her interval is $90 \%$ or less that of player $Y$.
What are the (respective) optimal strategies in this case?
It is still not optimal for $Y$ to use bisection. The exact same strategy as above has the exact same payoff odds for X.