155 problems

In mathematics, the Hales–Jewett theorem is a fundamental combinatorial result of Ramsey theory, named after Alfred W. Hales and Robert I. Jewett, that concerns the degree to which…

Matsumoto's conjecture. For any graph $G$ with $n\geq 4$ vertices,

Mertens's conjectures. Every stochastic game with signals has a limit value. Moreover, when Player 1 is more informed than Player 2, the uniform maxmin and the limit value coincide…

A finite deterministic graphical game (DG game) with a unique cyclic outcome is considered, together with the rank vector $r$ whose entry $r_i$ counts the terminal outcomes that ar…

Let $Q$ be a tree network, let $\alpha>0$ be the attack duration, let $E$ denote the set used to define the $E$-patrolling strategy, let $\mu$ denote the measure of $Q$, and let…

The $1/e$-strategy waits until time $1/e$ and then accepts the first candidate who is best so far. In certain two-person games, a decision maker faces an adversary trying to minimi…

Let a finite game be a weighted potential game, and consider a fictitious-play process in that game. Harris's conjecture. The rate of convergence of fictitious play is exponential…

Equal-SINR conjecture. The Pareto-optimal solution occurs when all users achieve the same SINRs, $\SINR[opt]$.

Blackwell determinacy conjecture. The game $\Gamma(S)$ should be determined, that is,

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 $RH(r,s,A)$ be a Robin Hood game with resource functions $r,s$ and index set $A$. A random strategy is independent of the index set $A$ if its choices do not depend on $A$. Old…

Let $RH(1,s,[0,1])$ be the Robin Hood game with one item available to Robin at each step, constant Sheriff's memory parameter $s$, and index set $A=[0,1]$. The second random strate…

Consider a continuum population game with a near-tight distribution of agent characteristics. A finitely additive joint distribution on characteristics and actions is an equilibriu…

Let $\Gamma=(A_i,u_i)_{i\in I}$ be a game in which each $A_i$ is a compact Hausdorff space and each utility function $u_i:A\to[-B,+B]$ is continuous in the product topology on $A$.…

Let Player 2's mixed strategy space be the feasible polytope determined by the game, and let the corresponding facet be the facet associated with the index $k$ and the candidate st…

Transition-computability conjecture. The function mapping a game configuration together with a legal move to the consequent configuration in the Yu-Gi-Oh! TCG is computable.

Let a potential mean field game be one for which the optimizers of an additional mean field control problem are also solutions to the mean field game, although some mean field game…

Controller's alpha-Rational Stackelberg equilibrium conjecture. This choice of initial action profile should imply that (\nu^*, i^{H}_*,\mu^R_*_{a^*_H}) is an $\alpha$-Rational S…

Consider the given imbalanced $(m,2k+1)$-RPS game and a Nash equilibrium $N$. Let $P_i(S)$ be the probability that player $i$ plays $S$, and let $P_i(T_p)$ be the probability that…

For $t,k\in\mathbb{N}$, let $r_1,\dots,r_{k-1},s$ be variables in $(0,1)$, and let $E_i(R)$, $E_i(P)$, $E_i(S)$, $E_s(P)$, $E_s(S)$, $E_p(R)$, $E_p(P)$, and $E_p(S)$ be the expecte…

Let $m\text{ in }\mathbb{N}$ denote the number of players, and let $E_i(R)$, $E_i(P)$, and $E_i(S)$ denote the expected payoffs associated with the strategies used in the paper. No…

The imbalanced $(m,2n+1)$-RPS is constructed from the paper's imbalanced $(m,3)$-RPS construction by iterated blow-ups. A game is strongly playable when it has the property defined…

Let $G$ be the graph encoding conditional-independence constraints for a generic binary game with $n$ players, let $\mathcal{M}_G$ denote the discrete conditional independence mode…

An $m$-player $(2n+1)$-$RPS$ game is a game with $2n+1$ objects and $m$ players interacting through the paper's $RPS$ payoff structure. A blow-up of a game $G_1$ at an object repla…

Consider an ecological game in which species or groups interact through dominance relations. An ecosystem is asymmetric yet stable when its dominance structure is imbalanced but it…