Transfinite Version of Welter’s Game

We study the transfinite version of Welter’s Game, a combinatorial game played on a belt divided into squares numbered with general ordinal numbers. In particular, we give a solution for the game, based on those of the transfinite version of Nim and the original version of Welter’s Game using Cantor normal form

Constructive comparison in bidding combinatorial games

A class of discrete Bidding Combinatorial Games that generalize alternating normal play was introduced by Kant, Larsson, Rai, and Upasany (2022). The major questions concerning optimal outcomes were resolved. By generalizing standard game comparison techniques from alternating normal play, we propose an algorithmic play-solution to the problem of game comparison for bidding games. We demonstrate some consequences of this result that generalize classical results in alternating play (from Winning Ways 1982 and On Numbers and Games 1976). In particular, integers, dyadics and numbers have many nice properties, such as group structures, but on the other hand the game * is non-invertible. We state a couple of thrilling conjectures and open problems for readers to dive into this promising path of bidding combinatorial games.Comment: 23 pages, 1 figur

Asymmetric Bertrand duopoly: game complete analysis by algebra system Maxima

In this paper we apply the Complete Analysis of Differentiable Games (introduced by D. Carfì in [3], [6], [8] and [9]) and already employed by himself and others in [4], [5], [7]) and some new algorithms employing the software wxMaxima 11.04.0 in order to reach a total knowledge of the classic Bertrand Duopoly (1883), viewed as a complex interaction between two competitive subjects, in a particularly difficult asymmetric case. The software wxMaxima is an interface for the computer algebra system Maxima. Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, systems of linear equations, polynomials, and sets, vectors, matrices. Maxima yields high precision numeric results by using exact fractions, arbitrary precision integers, and variable precision floating point numbers. Maxima can plot functions and data in two and three dimensions. The Bertrand Duopoly is a classic oligopolistic market in which there are two enterprises producing the same commodity and selling it in the same market. In this classic model, in a competitive background, the two enterprises employ as possible strategies the unit prices of their products, contrary to the Cournot duopoly, in which the enterprises decide to use the quantities of the commodity produced as strategies. The main solutions proposed in literature for this kind of duopoly (as in the case of Cournot duopoly) are the Nash equilibrium and the Collusive Optimum, without any subsequent critical exam about these two kinds of solutions. The absence of any critical quantitative analysis is due to the relevant lack of knowledge regarding the set of all possible outcomes of this strategic interaction. On the contrary, by considering the Bertrand Duopoly as a differentiable game (games with differentiable payoff functions) and studying it by the new topological methodologies introduced by D. Carfì, we obtain an exhaustive and complete vision of the entire payoff space of the Bertrand game (this also in asymmetric cases with the help of wxMaxima 11.04.0) and this total view allows us to analyze critically the classic solutions and to find other ways of action to select Pareto strategies, in the asymmetric cases too. In order to illustrate the application of this topological methodology to the considered infinite game we show how the complete study gives a real extremely extended comprehension of the classic model.Asymmetric Bertrand Duopoly; Normal-form games; software algorithms in microeconomic policy; Complete Analysis of a normal-form complex interaction; Pareto optima; valuation of Nash equilibriums; bargaining solutions

Normal Form Structures in Extensive Form Games

Mailaith, Samuelson, and Swinkels (1992) introduce the normal form information set. Normal form information sets capture situations in which players can make certain decisions as if they knew their opponents' had chosen from a particular subset of their strategies. In this paper, we say that an extensive form game represents a normal form game if, for each situation, the corresponding choice in the extnesive form is made with the player knowing that the opponents have chosen from the relevant subset. We show that normal form games exist that cannot be represented. We develop an algorithm that generates a representation whenever one exists and present a necessary and sufficient condition for a normal form game to be representable. Keywords: representation, extensive form game, normal form game, information set, extensive form--normal form equivalence. JEL Classification Numbers: C70, C72.

Two Games on Arithmetic Functions: SALIQUANT and NONTOTIENT

We investigate the Sprague-Grundy sequences for two normal-play impartial games based on arithmetic functions, first described by Iannucci and Larsson in \cite{sum}. In each game, the set of positions is N (natural numbers). In saliquant, the options are to subtract a non-divisor. Here we obtain several nice number theoretic lemmas, a fundamental theorem, and two conjectures about the eventual density of Sprague-Grundy values. In nontotient, the only option is to subtract the number of relatively prime residues. Here are able to calculate certain Sprague-Grundy values, and start to understand an appropriate class function.Comment: 8 pages, 1 figur

Lack of Consensus Among Scholars on the Issue of Video Game “Addiction”

Whether pathological video game overuse constitutes a distinct mental disorder remains an issue of controversy among scholars. Both empirical data and scholarly opinions differ regarding the status of pathological gaming and whether “addiction” is the best frame by which to understand video game use. The current study sought to examine the status of scholarly opinions in a survey of 214 scholars to examine their opinion of possible behavioral effects of games. Results indicated a variance of opinions. About 60.8% of scholars agreed pathological gaming could be a mental health problem, whereas 30.4% were skeptical. However, only 49.7% believed the DSM criteria for “internet gaming disorder” were valid, with slightly higher numbers, 56.5%, supporting the World Health Organization “gaming disorder” diagnosis. More scholars worried about both the DSM and WHO criteria over-pathologizing normal youth than weren’t worried about this. Scholars were likewise split over whether the DSM/WHO had precipitated moral panics over video games. Belief in pathological gaming was positively predicted by hostile attitudes toward children and negatively by participants’ experience with games. Overall results indicated continued significant disagreements among scholars related to pathological gaming. Claims of consensus are, at this juncture, likely prematur