Games for the Strategic Influence of Expectations

We introduce a new class of games where each player's aim is to randomise her strategic choices in order to affect the other players' expectations aside from her own. The way each player intends to exert this influence is expressed through a Boolean combination of polynomial equalities and inequalities with rational coefficients. We offer a logical representation of these games as well as a computational study of the existence of equilibria.Comment: In Proceedings SR 2014, arXiv:1404.041

The Nucleolus, the Kernel, and the Bargaining Set: An Update

One of David Schmeidler’s many important contributions in his distinguished career was the introduction of the nucleolus, one of the central single-valued solution concepts in cooperative game theory. This paper is an updated survey on the nucleolus and its two related supersolutions, i.e., the kernel and the bargaining set. As a first approach to these concepts, we refer the reader to the great survey by Maschler (1992); see also the relevant chapters in Peleg and Sudholter (2003). Building on the notes of four lectures on the nucleolus and the kernel delivered by one of the authors at the Hebrew University of Jerusalem in 1999, we have updated Maschler’s survey by adding more recent contributions to the literature. Following a similar structure, we have also added a new section that covers the bargaining set. The nucleolus has a number of desirable properties, including nonemptiness, uniqueness, core selection, and consistency. The first way to understand it is based on an egalitarian principle among coalitions. However, by going over the axioms that characterize it, what comes across as important is its connection with coalitional stability, as formalized in the notion of the core. Indeed, if one likes a single-valued version of core stability that always yields a prediction, one should consider the nucleolus as a recommendation. The kernel, which contains the nucleolus, is based on the idea of “bilateral equilibrium” for every pair of players. And the bargaining set, which contains the kernel, checks for the credibility of objections coming from coalitions. In this paper, section 2 presents preliminaries, section 3 is devoted to the nucleolus, section 4 to the kernel, and section 5 to the bargaining set.Iñarra acknowledges research support from the Spanish Government grant ECO2015-67519-P, and Shimomura from Grant-in-Aid for Scientific Research (A)18H03641 and (C)19K01558

Hacia la solución de juegos matriciales con incertidumbre difusa Tipo-2 a través de optimización lineal

This paper presents some theoretical and computing considerations about how to deal with fuzzy uncertainty in the parameters of the classical games model. Indeed, when multiple experts are involved in a game situation, then their opinions lead to have uncertainty since most of the times they are not agree to each others. This kind of uncertainty can be modeled using Type-2 fuzzy sets, which implies a specialized methods and sub-models.Some considerations about the use of Type-2 fuzzy sets and what does this imply when computing solutions, are presented. A general model which includes this kind of uncertainty is defi ned on the base of the extension principle and α-cuts representation theorem. A possible way for solving this model is glimpsed and put down for discussion and implementation.Este artículo presenta algunas consideraciones computacionales y teóricas acerca de cómo incluír incertidumbre difusa en los parámetros de un problema clásico de juegos. De hecho, cuando varios expertos están involucrados en un problema de juegos, todas sus opiniones llevan a pensar en una fuente incertidumbre, ya que muchas veces esos expertos no están de acuerdo entre sí. Ese tipo de incertidumbre puede modelarse mediante conjuntos difusos Tipo-2, lo que implica usar modelos y métodos especiales para llegar a una respuesta adecuada.Se presentan algunos aspectos importantes acerca del cálculo de soluciones en presencia de este tipo de incertidumbre. Un modelo general que incluye incertidumbre difusa Tipo-2 es presentado, el cual se basa en el principio de extensión y el teorema de representación de α-cortes. Un posible método de solución es puesto a consideración para discusión e implementación

Application of Fuzzy State Aggregation and Policy Hill Climbing to Multi-Agent Systems in Stochastic Environments

Reinforcement learning is one of the more attractive machine learning technologies, due to its unsupervised learning structure and ability to continually even as the operating environment changes. Applying this learning to multiple cooperative software agents (a multi-agent system) not only allows each individual agent to learn from its own experience, but also opens up the opportunity for the individual agents to learn from the other agents in the system, thus accelerating the rate of learning. This research presents the novel use of fuzzy state aggregation, as the means of function approximation, combined with the policy hill climbing methods of Win or Lose Fast (WoLF) and policy-dynamics based WoLF (PD-WoLF). The combination of fast policy hill climbing (PHC) and fuzzy state aggregation (FSA) function approximation is tested in two stochastic environments; Tileworld and the robot soccer domain, RoboCup. The Tileworld results demonstrate that a single agent using the combination of FSA and PHC learns quicker and performs better than combined fuzzy state aggregation and Q-learning lone. Results from the RoboCup domain again illustrate that the policy hill climbing algorithms perform better than Q-learning alone in a multi-agent environment. The learning is further enhanced by allowing the agents to share their experience through a weighted strategy sharing

Logics for Non-Cooperative Games with Expectations

We introduce the logics E(G) for reasoning about probabilistic expectation over classes G of games with discrete polynomial payoff functions represented by finite-valued Lukasiewicz formulas and provide completeness and complexity results. In addition, we introduce a new class of games where players' expected payoff functions are encoded by E(G)-formulas. In these games each player's aim is to randomise her strategic choices in order to affect the other players' expectations over an outcome as well as their own. We offer a logical and computational characterisation of this new class of games.Godo acknowledges support from the Spanish projects EdeTRI (TIN2012-39348-C02-01) and AT (CONSOLIDER CSD 2007-0022). Marchioni acknowledges support from the Marie Curie Project NAAMSI (FP7-PEOPLE-2011-IEF).Peer Reviewe

Game Theory Relaunched

The game is on. Do you know how to play? Game theory sets out to explore what can be said about making decisions which go beyond accepting the rules of a game. Since 1942, a well elaborated mathematical apparatus has been developed to do so; but there is more. During the last three decades game theoretic reasoning has popped up in many other fields as well - from engineering to biology and psychology. New simulation tools and network analysis have made game theory omnipresent these days. This book collects recent research papers in game theory, which come from diverse scientific communities all across the world; they combine many different fields like economics, politics, history, engineering, mathematics, physics, and psychology. All of them have as a common denominator some method of game theory. Enjoy