7 problems

Optimal-strategy conjecture. After observing $I_{k}=1$, the player whose turn it is should give up his turn to his opponent if and only if $p_{i,k}<\frac{1}{2}$ for all $i>k$. This…

In the two-sided secretary game, suppose there are $U$ men and $U$ women, each player meets $N$ partners over $N$ rounds, and preferences satisfy universal rank symmetry: if a man…

In the cooperative two-sided secretary game, let $N$ be the number of rounds and let $R_N(1)$ denote the expected $N$-rank of a player entering the game, under an optimal common st…

Let $\{F_n\}$, $\{G_n\}$ and $\{H_n\}$ be the sequences of real functions and let $g_n$, $h_n$, $g$, $h$ and $\Theta$ be as in the preceding pointwise convergence conjecture. Assum…

Let $\{F_n\}$, $\{G_n\}$ and $\{H_n\}$ be sequences of real functions on $\{1,\ldots,n\}$ satisfying … Define $g_n(x):=nG_n(\lfloor nx\rfloor)$ and…

Let $U_n(x)$ be the expected duration of owning the relatively best object whose rank remains within the two when the time to go is $n$, conditional on accepting the relatively bes…

Let $C_n$ denote the spider constant for a spider with $n$ legs, so that the explicitly computed values are $C_0=1$, $C_1=\sqrt{2}$, and $C_2=\sqrt{3}$. Failure of the square-root…