Inspired by a [question about bisection](https://mathoverflow.net/questions/77442/) 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). 



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> What is the optimal strategy? 
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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 $Y$.



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> What are the (respective) optimal strategies in this case?
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It is still not optimal for $Y$ to use bisection. The exact same strategy as above has the exact same payoff odds for X. 



A competitive root finding game
Possibly nothing simple, even when there is only one turn left. Suppose that there is just one turn left, the intervals are of length 4 and 6 (just to scale up) and the moves must create intervals of length $\frac{j}{4}.$ Then by my calculations, the player with the shorter interval should guess $$\frac54, \frac64 \text{ or }\frac74 \text{ with probabilities } \frac{4462\ }{6005\ }, \frac{53}{6005}, \frac{1490}{6005}$$ 
while the other player should choose $$\frac54, \frac74 \text{ or } \frac84 \text{ with probabilities } \frac{392}{1201} ,\frac{200}{1201}, \frac{609}{1201}.$$


