50 problems

Let $S$ be a subtraction set, and consider any two tie-breaking conventions in the associated self-interest cumulative subtraction game. For a heap size $h$, let the discrepancy be…

For every integer $m\ge 1$ and every real algebraic number $\alpha$ of degree exactly $m$ over $\mathbb{Q}$, there exists a rational $m\times m$ absorbing two-player zero-sum game…

Determine the minimax-optimal alternating-regret rates for online learning: (i) for online linear optimization over the probability simplex $\Delta_d$ over a time horizon $T$, dete…

Let $B=K_{\aleph_0}$, and let $G_1,\dots,G_k\subseteq B$ be finitely many infinite subgraphs. In the color-preserving Maker–Breaker game on $B$, Maker and Breaker alternately claim…

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…

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…

Ostrovski's conjecture. Under these conditions, the Nash equilibrium does not Pareto dominate typical fictitious-play orbits. This conjecture gives a sufficient transition-combinat…

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…

Let $\mathbf{A}$ be a game matrix and let $\alpha$ be a learning rate. Let $I_G(t)$ denote the relevant maximal-set quantity in the imitation dynamics, and let $\alpha^*$ be a lear…

Let $\mathbf{A}$ be a game matrix and let $\alpha$ be a learning rate. Suppose that the maximal set of $I_G(t)$ is constant for all $t\geq t_0$. Let $\mathcal{X}^*$ denote the maxi…

Let $(\epsilon_X,\epsilon_Y)$ be a pair of game parameters, and let $Z^a,Z^b,Z^c,Z^d,Z^e,Z^f$ denote the equilibria appearing in the cyclic competition bimatrix game. A heteroclini…

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…

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.

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…

Let $\lambda_{\rm \max}:(0,1]^2\to[1,\infty)$ be the function measuring the maximum value associated with the game. Maximizer conjecture. The maximum value of…

For each integer $n>1$, let $(\epsilon_n,\delta_n)$ be the unique solution in $[0,1]\times[0,1]$ of … with $\delta_n>\epsilon_n$. Strict-monotonicity conjecture. The sequences…

Semicontinuity conjecture. The displayed inequality holds for every generic two-action game and every $d\in\{0,1,\ldots,m\}$.

Bi-SP Conjecture. If $G$ has an $(s,V_T)$-path, then $P_1$ and $P_2$ have such a path in common, that is,

Let $G$ be a strongly connected planar graph, and let $g$ denote its undirected girth. A static equilibrium is an equilibrium of the static type, while a $2$-chase equilibrium and…

Let an $n$-person deterministic graphical game be play-once if every player makes at most one move during a play. The game is Nash-solvable if it has a Nash equilibrium in pure sta…

Consider the semi-restricted Rock, Paper, Scissors game with score $S_n$, and let $S_n^{\mathrm{op}}$ denote the score when both players use optimal strategies. Let $W$ and…

Let $G$ be the undirected graphical model modelling a generic $n$-player game $X$ with binary choices in normal form. Let $\mathcal{C}=\operatorname{global}(G)$ and let…

Henning's conjecture. For every tree $T$ of order $n$, $I_s(T) \geq \frac{3}{7}n$. The paper disproves this conjecture by exhibiting arbitrarily large trees for which the Sweller-s…

Let $n$ and $m$ satisfy $2^{n-1}+1\leq m<2^n$, and set … For a nonempty subset $J\subseteq[n]$, a $J$-cube is the set of binary vectors obtained by fixing all coordinates outside…