Related papers: Game values of arithmetic functions
We survey recent developments in the theory of impartial combinatorial games in misere play, focusing on how the Sprague-Grundy theory of normal-play impartial games generalizes to misere play via the indistinguishability quotient…
We introduce operational semantics into games. And based on the operational semantics, we establish a full algebra of games, including basic algebra of games, algebra of concurrent games, recursion and abstraction. The algebra can be used…
Euclid is a well known two-player impartial combinatorial game. A position in Euclid is a pair of positive integers and the players move alternately by subtracting a positive integer multiple of one of the integers from the other integer…
The theory of combinatorial game (like board games) and the theory of social games (where one looks for Nash equilibria) are normally considered as two separate theories. Here we shall see what comes out of combining the ideas. The central…
We explain why numbers occurring in the classification of polygon spaces coincide with numbers of self-dual equivalence classes of threshold functions, or of regular Boolean functions, or of decisive weighted majority games.
In a recent arXiv-manuscript Fox studies infinite subtraction games with a finite (ternary) and aperiodic Sprague-Grundy function. Here we provide an elementary example of a game with the given properties, namely the game given by the…
We apply the Sprague-Grundy Theorem to LCTR, a new impartial game on partitions in which players take turns removing either the Left Column or the Top Row of the corresponding Young diagram. We establish that the Sprague-Grundy value of any…
The concept of nimbers--a.k.a. Grundy-values or nim-values--is fundamental to combinatorial game theory. Nimbers provide a complete characterization of strategic interactions among impartial games in their disjunctive sums as well as the…
The Sprague-Grundy (SG) theory reduces the sum of impartial games to the classical game of $NIM$. We generalize the concept of sum and introduce $\cH$-combinations of impartial games for any hypergraph $\cH$. In particular, we introduce the…
Evolutionary game theory is a common framework to study the evolution of cooperation, where it is usually assumed that the same game is played in all interactions. Here, we investigate a model where the game that is played by two…
Shapley's discounted stochastic games, Everett's recursive games and Gillette's undiscounted stochastic games are classical models of game theory describing two-player zero-sum games of potentially infinite duration. We describe algorithms…
We begin by reviewing and proving the basic facts of combinatorial game theory. We then consider scoring games (also known as Milnor games or positional games), focusing on the "fixed-length" games for which all sequences of play terminate…
We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the…
Combinatorial Game Theory(CGT)is a branch of Game Theory that has developed largely independently of Economic Game Theory (EGT), and is concerned with deep mathematical properties of two-player zero-sum games recursively defined over…
We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd…
Here, we consider numerical methods for stationary mean-field games (MFG) and investigate two classes of algorithms. The first one is a gradient-flow method based on the variational characterization of certain MFG. The second one uses…
We present a general way of defining various reduction games on \omega\ which "represent" corresponding topologically defined classes of functions. In particular, we will show how to construct games for piecewise defined functions, for…
Genetic programming is the practice of evolving formulas using crossover and mutation of genes representing functional operations. Motivated by genetic evolution we develop and solve two combinatorial games, and we demonstrate some…
We provide a characterization of the set of real-valued functions that can be the value function of some polynomial game. Specifically, we prove that a function $u : \dR \to \dR$ is the value function of some polynomial game if and only if…
We provide game-theoretic proofs of some well-known existence theorems of Friedberg numberings for the class of all partial computable functions, including (1) the existence of two incomparable Friedberg numberings; (2) the existence of a…