Simple games for under 3

In voting theory and social choice theory, decision systems can be represented as simple games, i.e., cooperative games defined through their players or voters and their set of winning coalitions. The weighted voting games form a well-known strict subclass of simple games, where each player has a voting weight so that a coalition wins if the sum of weights of their members exceeds a given quota. Since the number of winning coalitions can be exponential in the number of players, simple games can be represented much more compactly as intersections or unions of weighted voting games. A simple game’s dimension (codimension) is the minimum number of weighted voting games such that their intersection (union) is the given game. It is known there are voting systems with a high (co)dimension. This work introduces the multidimension as the minimum size of an expression with intersections and unions on weighted voting games necessary to obtain the considered simple game. We generalize this notion to subclasses of weighted voting games and analyze the generative properties of these subclasses. We also characterize the simple games with finite generalized multidimension over the set of weighted voting games without dummy players. We provide a comprehensive classification for simple games up to a certain number of players. These results complement similar classification results for generalized (co)dimensions. Our results show how generalized multidimension allows representing more simple games and more compactly, even for a small number of players and for subclasses.

In many multiagent systems, small changes in individual-level rules may lead to very large changes at the group-level. This phenomenon is striking in the “aggressor-defender game,” a simple participative game in which each participant randomly selects two others from the group (A and B). In the aggressor game, everyone tries to position themselves so that A is always between themselves and B. In the defender game everyone tries to position themselves between A and B. Despite these exceedingly simple rules and the seemingly small difference between them, the two games exhibit very different dynamics. The aggressor game produces a highly dynamic group that rapidly expands over time whereas the defender game quickly collapses to a tight knot. I analyze these games and provide some insight as to how these two group level behaviors arise, thereby linking the micro- and macro-levels. I also introduce and analyze a new, related and simpler game, the stalker game, in which each participant selects and pursues a single participant, and which also produces a collapsing group. It is suggested that such a geometrical analysis may be applicable for other multiagent systems such as insect societies and collective robotics.

Consider a situation with n agents or players, where some of the players form a coalition with a certain collective objective. Simple games are used to model systems that can decide whether coalitions are successful (winning) or not (losing). A simple game can be viewed as a monotone boolean function. The dimension of a simple game is the smallest positive integer d such that the simple game can be expressed as the intersection of d threshold functions, where each threshold function uses a threshold and n weights. Taylor and Zwicker have shown that d is bounded from above by the number of maximal losing coalitions. We present two new upper bounds both containing the Taylor-Zwicker bound as a special case. The Taylor-Zwicker bound implies an upper bound of (n choose n/2). We improve this upper bound significantly by showing constructively that d is bounded from above by the cardinality of any binary covering code with length n and covering radius 1. This result supplements a recent result where Olsen et al. showed how to construct simple games with dimension |C| for any binary constant weight SECDED code C with length n. Our result represents a major step in the attempt to close the dimensionality gap for simple games.

Communication is present among human societies. The philosophy of communication studies the foundational aspects of communication. In this paper, the intention is to study how knowledge, information and ideas can be shared in communication. As a starting point, a simple game of communication is characterised. The simple game shows that it is possible to share knowledge, information and ideas. However, in communication, several different linguistic means of expressions are used. So, we must extend the communication game to include these different kinds of linguistic expressions. Metaphors offer an important class of expressions, which open new lines of thoughts. Metaphors are ‘poetically or rhetorically ambitious use of words,’ where, according to Davidson, the use of them is based on some kind of ‘artistic success’. However, this kind of characterisation does not help us to find a fruitful, semantical analysis for them. Possible worlds semantics offer a natural semantical tool which explains these peculiarities. Especially the identity of individuals becomes relativised to the methods of identification, which can be generalised to all modal contexts. This allows us to generalise the simple communication game. So, this paper explicates one aspect of the very important philosophical topic.

A characterization of completeness of blockability relations with respect to unanimity

Forms of representation for simple games: Sizes, conversions and equivalences

Discussion

We present an overview of research on a certain model of influence in a social network. Each agent has to make an acceptance/rejection decision, and he has an inclination to choose either the yes-action or the no-action. The agents are embedded in a social network which models influence between them. Due to the influence, a decision of an agent may differ from his preliminary inclination. Such a transformation between the agents’ inclinations and their decisions are represented by an influence function. Follower functions encode the players who constantly follow the opinion of a given unanimous coalition. We examine properties of the influence and follower functions and study the relation between them. The model of influence is also compared to the framework of command games in which a simple game is built for each agent. We study the relation between command games and influence functions. We also define influence indices and determine the relations between these indices and some well-known power indices. Furthermore, we enlarge the set of possible yes/no actions to multi-choice games and investigate the analogous tools related to influence in the multi-choice model.

This paper regards some measures for sharing (public) goods or budgets among members with different participation quotas in a binary decision-making process. The main characteristic of such measures is that they should have elements of solidarity with those who have a weak quota of participation in the process. These measures seem appropriate for deals that require solidarity, which contrasts with the classical power indices such as the Shapley and Shubik index or the Banzhaf index. Moreover, we provide a new representation for two power indices—the Public Help Index ξ (proposed by Bertini and Stach in 2015) and the particularization of the solidarity value proposed by Nowak and Radzik in 1994—the ψ index—in a simple game using null player free winning coalitions. As it is known, a set of null player free winning coalitions unequivocally determines a simple game. Finally, we compare considered power indices considering some properties in simple games.

This study aims to explore the application of artificial intelligence algorithms in the development of a simple 2D game called Cuypy using the Python programming language. Cuypy is designed to provide an interactive, educational, and challenging gameplay experience. Artificial intelligence is implemented to enhance enemy intelligence, game adaptability to player behavior, and overall system responsiveness. The development focuses on the integration of pathfinding algorithms and machine learning techniques to improve enemy strategy, enabling non-player characters to learn from player actions and adjust their behavior dynamically to increase the level of challenge over time. Additionally, this study emphasizes the use of reinforcement learning mechanisms that allow enemies to predict player movement patterns and automatically adapt their strategies. Pygame is utilized as the graphical library to build the user interface and animations, while Python serves as the main programming language supporting efficient AI implementation. The novelty of this research lies in the real-time behavioral adaptation of game enemies based on player interactions, a feature that is rarely explored in simple Python-based games. This study also highlights the integration of pathfinding algorithms to ensure more realistic, adaptive, and engaging enemy movements. Overall, the research contributes to the development of interactive Python-based games and provides insights into applying AI to simple educational games with automatically adjustable difficulty levels

Understanding the impact of interaction mechanics on the user’s emotional state can aid in shaping the user experience. For eliciting the emotional state of a user, designers and researchers typically employ subjective or expert assessment. Yet these methods are typically applied after the user has finished the interaction, causing a delay between stimulus and assessment. Physiological measures potentially offer more reliable indication of a user’s affective state in real-time. We present an experiment to increase our understanding of the relation of certain stimuli and valence of induced emotions in games. For this we designed a simple game to induce negative and positive emotions in the player. The results show a high correspondence between our classification of participants’ physiological signals and subjective assessment. However, creating a clear causality between game elements and emotions is a daunting task, and our designs offer room for improvement.

A simple game (N,v) is given by a set N of n players and a partition of 2N into a set L of losing coalitions L' with value v(L')=0 that is closed under taking subsets and a set W of winning coalitions W' with v(W')=1. Simple games with ∝= minp≥0max w∈W,L∈L} p(L')/p(W') 0.

Power measures derived from the sequential query process

Symmetry of simple games and permission of voters

An approach to simple bargaining games and related problems