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Complement Submodular Information Measures for Balanced and Robust Data Selection A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring An identity for second Eulerian numbers via lattice-point counting $t$-tone edge coloring of graphs Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Universality theorems for generalized splines Framing Triangulations for Arbitrary Integer Flow Polytopes On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs The complexity of frugal digraph homomorphisms Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux Enumerating Pattern Avoiding Parking Functions Incidence toric ideals and three-point functions Unique Winning Opening Move in Three-Row Chomp Strong majority colorings of graphs A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Spectral radius and edge-disjoint connected factors of graphs New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256 Flexible DP-4-coloring of planar graphs without 4-cycles and intersecting triangles Balanced intersection size distributions in projective planes List Reconstruction Problem with List Size Two Is Dimensionality a Barrier for Retrieval Models? The INIEP: Irreducible and Positive Realizations The number of Pfaffian orientations on punctured polygonally cellulated surfaces Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Finite-state enumeration of adjacency-constrained 132-avoiding permutations AMDS and quantum AMDS Constacyclic codes of length $4p^ς$ over $\mathbb{F}_{{p}^{m}}$
A counterexample to conjecture "Catch 22"
Bogdan Butyrin, Vladimir Gurvich, Anton Lutsenko, Mariya Naumova · 2024-06-19 · via math.CO updates on arXiv.org

We construct a finite deterministic graphical (DG) game without Nash equilibria in pure stationary strategies. This game has 3 players $I=\{1,2,3\}$ and 5 outcomes: 2 terminal $a_1$ and $a_2$ and 3 cyclic. Furthermore, for 2 players a terminal outcome is the best: $a_1$ for player 3 and $a_2$ for player 1. Hence, the rank vector $r$ is at most $(1,2,1)$. Here $r_i$ is the number of terminal outcomes that are worse than some cyclic outcome for the player $i \in I$. This is a counterexample to conjecture ``Catch 22" from the paper ``On Nash-solvability of finite $n$-person DG games, Catch 22" (2021) arXiv:2111.06278, according to which, at least 2 entries of $r$ are at least 2 for any NE-free game. However, Catch 22 remains still open for the games with a unique cyclic outcome, not to mention a weaker (and more important) conjecture claiming that an $n$-person finite DG game has a Nash equilibrium (in pure stationary strategies) when $r = (0^n)$, that is, all $n$ entries of $r$ are 0; in other words, when the following condition holds: $\qquad\bullet$ ($C_0$) any terminal outcome is better than every cyclic one for each player. A game is play-once if each player controls a unique position. It is known that any play-once game satisfying ($C_0$) has a Nash equilibrium. We give a new and very short proof of this statement. Yet, not only conjunction but already disjunction of the above two conditions may be sufficient for Nash-solvability. This is still open.