18 problems

Let Algorithm be the algorithm defined in the paper, let $M$ denote its termination parameter, and let $\epsilon>0$. Termination conjecture. Algorithm will terminate with…

Monotonicity conjecture. If the integer triplet $(n,d,k)$ is accurate, then the integer triplet $(n',d',k')$ is also accurate whenever $n'\geq n$, $d'\leq d$, and $k'\leq k$. This…

Conjecture on accurate triplets. If $n$, $d$, and $k$ are integers and $n\ge d(k-1)+1$, then $(n,d,k)$ is accurate. The paper presents this as a strengthening of a conjecture of Pá…

Csóka's conjecture. If $n dk$ and $k$ is a positive real number, then

Consider a competitive search game with finite state set $S$ and initial probability distribution $p\in\Delta(S)$. For each state $s\in S$, let $A_s$ be the set of initial distribu…

Let $P$ be the transition matrix of a time-homogeneous competitive search game, and let $\pi$ be an invariant distribution of $P$. The value for player 1 is denoted by $v_1(\pi)$.…

Consider a booby trap search game played on a hypergraph whose hyperedges $S$ have positive rewards $r(S)$. A Searcher strategy assigns probabilities $p(S)$ to choosing hyperedges.…

Equalizing-property conjecture. The Hider's optimal strategy always has the equalizing property.

Largest-weight-support conjecture. Rank all pure Hider strategies by these weights; then an optimal Hider mixed strategy uses $d$ pure strategies having the largest weights, for so…

Set-aside and equalizing conjecture. In the general case, an optimal Hider strategy would have this form.

Consider a caching game in which a Hider places objects in locations and a Searcher adaptively searches for them. After the Searcher finds the first object, the remaining objects i…

Consider the caching game with $k=j\ge2$, in which the hider caches $k$ nuts and the searcher aims to find all of them. Integral-depth multi-nut conjecture. If … then the value of…

Assume the two-nut caching game and let the discrete-limit theorem provide its stated upper bound for $h$. Discrete-limit sharpness conjecture. If … then the best upper bound suppl…

Assume the two-nut caching game with parameters $n$ and $h$, and let the value denote the optimal winning probability. Large-depth value conjecture. If … then the value of the game…

Assume the two-nut case $k=j=2$. Let the value of the game be bounded above by the quantity in the source theorem. Two-nut sharpness conjecture. The bound … is sharp if … and eithe…

Assume $k=j=2$, so the hider caches two nuts and the searcher aims to find both. A hiding strategy is represented by pairs $(y_1,y_2)$ of depths, with the hole placements randomize…

Let $v(k,j,n,h)$ be the value of the discrete caching game and let $v^*(k,j,h)$ be the corresponding limit-game value. The source states the upper bound … For $k=j=2$, the transfor…

Consider an optimal hider strategy in the extremal double-limit caching game with two nuts, and let $\mu$ be the probability measure describing the depth of a random nut. The suppo…