Related papers: Polytopality of simple games
We introduce several methods of decomposition for two player normal form games. Viewing the set of all games as a vector space, we exhibit explicit orthonormal bases for the subspaces of potential games, zero-sum games, and their orthogonal…
We introduce the $k$-stellated spheres and compare and contrast them with $k$-stacked spheres. It is shown that for $d \geq 2k$, any $k$-stellated sphere of dimension $d$ bounds a unique and canonically defined $k$-stacked ball. In…
We present an index theory of equilibria for extensive form games. This requires developing an index theory for games where the strategy sets of players are general polytopes and their payoff functions are multiaffine in the product of…
We solve the problem of automatically computing a new class of environment assumptions in two-player turn-based finite graph games which characterize an ``adequate cooperation'' needed from the environment to allow the system player to win.…
Starting by a simple game $Q $ as a combinatorial data, we build up a cell complex $M(Q)$, whose construction resembles combinatorics of the permutohedron. The cell complex proves to be a combinatorial manifold; we call it the \textit{…
We initiate the study of the phantom version of Maker-Breaker positional games. In a phantom game, the moves of one of the players are hidden from the other player, who still has the complete information. We look at the biased $(a:b)$…
The matching complex $\mathsf{M}(G)$ of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. These complexes appear in various places and found applications in many areas of mathematics including computational geometry,…
The domination game is an optimization game played by two players, Dominator and Staller, who alternately select vertices in a graph $G$. A vertex is said to be dominated if it has been selected or is adjacent to a selected vertex. Each…
A new representation of Game Theory is developed in this paper. State of players is represented by a density matrix, and payoff function is a set of hermitian operators, which when applied onto the density matrix give the payoff of players.…
Let $\Gamma$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\Gamma$ on the circle is either trivial or semi-conjugate to a unique minimal action on the so-called simple circle.
We give a simple proof that some iterated derived subdivision of every PL sphere is combinatorially equivalent to the boundary of a simplicial polytope, thereby resolving a problem of Billera (personal communication).
We study stochastic evolution of optional games on simple graphs. There are two strategies, A and B, whose interaction is described by a general payoff matrix. In addition there are one or several possibilities to opt out from the game by…
We study infinitely repeated games in settings of imperfect monitoring. We first prove a family of theorems that show that when the signals observed by the players satisfy a condition known as $(\epsilon, \gamma)$-differential privacy, that…
In this paper we answer a question posed by V.V. Batyrev. The question asked if there exists a complete regular fan with more than quadratically many primitive collections. We construct a smooth projective toric variety associated to a…
Games on graphs provide a natural model for reactive non-terminating systems. In such games, the interaction of two players on an arena results in an infinite path that describes a run of the system. Different settings are used to model…
Let $\gamma_g(G)$ be the game domination number of a graph $G$. It is proved that if ${\rm diam}(G) = 2$, then $\gamma_g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil- \left\lfloor \frac{n(G)}{11}\right\rfloor$. The bound is attained: if…
The canonical double cover $\D(\Gamma)$ of a graph $\Gamma$ is the direct product of $\Gamma$ and $K_2$. If $\Aut(\D(\Gamma))\cong\Aut(\Gamma)\times\ZZ_2$ then $\Gamma$ is called stable; otherwise $\Gamma$ is called unstable. An unstable…
Strategic games admit a multi-graph representation, in which two kinds of relations, accessibility, and preferences, are used to describe how the players compare the possible outcomes. A category of games with a fixed set of players…
We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability $p_{F}$ for the existence of Maker's strategy to claim a member of $F$ in the unbiased game played on…
The canonical pair of a proof system $P$ is the pair of disjoint NP sets where one set is the set of all satisfiable CNF formulas and the other is the set of CNF formulas that have $P$-proofs bounded by some polynomial. We give a…