scholarly journals Algorithms for Closed Under Rational Behavior (CURB) Sets

2010 ◽  
Vol 38 ◽  
pp. 513-534 ◽  
Author(s):  
M. Benisch ◽  
G. B. Davis ◽  
T. Sandholm
Keyword(s):  
Nash Equilibrium ◽  
Polynomial Time ◽  
Solution Concept ◽  
Rational Behavior ◽  
Strategy Space ◽  
Normal Form Games ◽  
Polynomial Time Algorithms ◽  
Game Theoretic ◽  
Theoretic Solution ◽  
Best Responses

We provide a series of algorithms demonstrating that solutions according to the fundamental game-theoretic solution concept of closed under rational behavior (CURB) sets in two-player, normal-form games can be computed in polynomial time (we also discuss extensions to n-player games). First, we describe an algorithm that identifies all of a player’s best responses conditioned on the belief that the other player will play from within a given subset of its strategy space. This algorithm serves as a subroutine in a series of polynomial-time algorithms for finding all minimal CURB sets, one minimal CURB set, and the smallest minimal CURB set in a game. We then show that the complexity of finding a Nash equilibrium can be exponential only in the size of a game’s smallest CURB set. Related to this, we show that the smallest CURB set can be an arbitrarily small portion of the game, but it can also be arbitrarily larger than the supports of its only enclosed Nash equilibrium. We test our algorithms empirically and find that most commonly studied academic games tend to have either very large or very small minimal CURB sets.

scholarly journals Efficient Computation of Semivalues for Game-Theoretic Network Centrality

2018 ◽  
Vol 63 ◽  
pp. 145-189 ◽  
Author(s):  
Mateusz K. Tarkowski ◽  
Piotr L. Szczepański ◽  
Tomasz P. Michalak ◽  
Paul Harrenstein ◽  
Michael Wooldridge
Keyword(s):  
Polynomial Time ◽  
Shapley Value ◽  
Centrality Measures ◽  
Efficient Computation ◽  
Solution Concepts ◽  
Banzhaf Index ◽  
The Shapley Value ◽  
Game Theoretic ◽  
Theoretic Solution ◽  
Network Centralities

Some game-theoretic solution concepts such as the Shapley value and the Banzhaf index have recently gained popularity as measures of node centrality in networks. While this direction of research is promising, the computational problems that surround it are challenging and have largely been left open. To date there are only a few positive results in the literature, which show that some game-theoretic extensions of degree-, closeness- and betweenness-centrality measures are computable in polynomial time, i.e., without the need to enumerate the exponential number of all possible coalitions. In this article, we show that these results can be extended to a much larger class of centrality measures that are based on a family of solution concepts known as semivalues. The family of semivalues includes, among others, the Shapley value and the Banzhaf index. To this end, we present a generic framework for defining game-theoretic network centralities and prove that all centrality measures that can be expressed in this framework are computable in polynomial time. Using our framework, we present a number of new and polynomial-time computable game-theoretic centrality measures.

Games with Unawareness

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yossi Feinberg
Keyword(s):  
Nash Equilibrium ◽  
Incomplete Information ◽  
Equilibrium Solution ◽  
Solution Concept ◽  
The Other ◽  
Extensive Form ◽  
Normal Form Games ◽  
Incomplete Information Games ◽  
Extended Nash Equilibrium ◽  
Information Games

AbstractWe provide a tool to model and solve strategic situations where players’ perceptions are limited, as well as situations where players realize that other players’ perceptions may be limited and so on. We define normal, repeated, incomplete information, and extensive form games with unawareness using a unified methodology. A game with unawareness is defined as a collection of standard games (of the corresponding form). The collection specifies how each player views the game, how she views the other players’ perceptions of the game and so on. The modeler’s description of perceptions, the players’ description of other players’ perceptions, etc. are shown to have consistent representations. We extend solution concepts such as rationalizability and Nash equilibrium to these games and study their properties. It is shown that while unawareness in normal form games can be mapped to incomplete information games, the extended Nash equilibrium solution is not mapped to a known solution concept in the equivalent incomplete information games, implying that games with unawareness generate novel types of behavior.

scholarly journals A Procedural Characterization of Solution Concepts in Games

2014 ◽  
Vol 49 ◽  
pp. 143-170 ◽  
Author(s):  
J. Y. Halpern ◽  
Y. Moses
Keyword(s):  
Nash Equilibrium ◽  
Correlated Equilibrium ◽  
Sequential Equilibrium ◽  
Solution Concepts ◽  
Knowledge Based ◽  
Precise Sense ◽  
Uniform Definition ◽  
Game Theoretic ◽  
Theoretic Solution

We show how game-theoretic solution concepts such as Nash equilibrium, correlated equilibrium, rationalizability, and sequential equilibrium can be given a uniform definition in terms of a knowledge-based program with counterfactual semantics. In a precise sense, this program can be viewed as providing a procedural characterization of rationality.

scholarly journals On Non-Cooperativeness in Social Distance Games

2019 ◽  
Vol 66 ◽  
pp. 625-653
Author(s):  
Alkida Balliu ◽  
Michele Flammini ◽  
Giovanna Melideo ◽  
Dennis Olivetti
Keyword(s):  
Nash Equilibrium ◽  
Polynomial Time ◽  
Social Distance ◽  
Price Of Anarchy ◽  
Cluster Formation ◽  
Solution Concept ◽  
Price Of Stability ◽  
The Social ◽  
Np Complete ◽  
Inverse Distance

We consider Social Distance Games (SDGs), that is cluster formation games in which the utility of each agent only depends on the composition of the cluster she belongs to, proportionally to her harmonic centrality, i.e., to the average inverse distance from the other agents in the cluster. Under a non-cooperative perspective, we adopt Nash stable outcomes, in which no agent can improve her utility by unilaterally changing her coalition, as the target solution concept. Although a Nash equilibrium for a SDG can always be computed in polynomial time, we obtain a negative result concerning the game convergence and we prove that computing a Nash equilibrium that maximizes the social welfare is NP-hard by a polynomial time reduction from the NP-complete Restricted Exact Cover by 3-Sets problem. We then focus on the performance of Nash equilibria and provide matching upper bound and lower bounds on the price of anarchy of Θ(n), where n is the number of nodes of the underlying graph. Moreover, we show that there exists a class of SDGs having a lower bound on the price of stability of 6/5 − ε, for any ε > 0. Finally, we characterize the price of stability 5 of SDGs for graphs with girth 4 and girth at least 5, the girth being the length of the shortest cycle in the graph.

Equilibria in Dynamic Multicriteria Games

2017 ◽  
Vol 19 (01) ◽  
pp. 1750002 ◽  
Author(s):  
Anna Rettieva
Keyword(s):  
Nash Equilibrium ◽  
Discrete Time ◽  
Solution Concept ◽  
Status Quo ◽  
Nash Bargaining ◽  
Game Model ◽  
Multicriteria Games ◽  
Common Resource ◽  
Game Theoretic

Mathematical models involving more than one objective seem more adherent to real problems. Often players have more than one goal which are often not comparable. These situations are typical for game-theoretic models in economic and ecology. In this paper, new approaches to construct equilibria in dynamic multicriteria games are constructed. We consider a dynamic, discrete-time, game model where the players use a common resource and have different criteria to optimize. First, we construct the guaranteed payoffs in a several ways. Then, we find an equilibrium as a solution of a Nash bargaining scheme with the guaranteed payoffs playing the role of status quo points. The obtained equilibrium, called a multicriteria Nash equilibrium, gives a possible solution concept for dynamic multicriteria games.

scholarly journals A Game theoretic approach for competition over visibility in social networks

2019 ◽  
Vol 8 (2) ◽  
pp. 674-682
Author(s):  
Khadija Touya ◽  
Mohamed Baslam ◽  
Rachid El Ayachi ◽  
Mostafa Jourhmane
Keyword(s):  
Social Networks ◽  
Nash Equilibrium ◽  
Cooperative Game ◽  
Theoretic Approach ◽  
Current Strategy ◽  
Frequency Space ◽  
Specific Frequency ◽  
Game Theoretic ◽  
Best Responses ◽  
Game Theoretic Approach

Social Networks have known an important evolution in the last few years. These structures, made up of individuals who are tied by one or more specific types of interdependency, constitute the window for members to express their opinions and thoughts by sending posts to their own walls or others' timelines. Actually, when a content arrives, it's located on the top of the timeline pushing away older messages. This situation causes a permanent competition over visibility among subscribers who jump on opponents to promote conflict. Our study presents this competition as a non-cooperative game; each source has to choose frequencies which assure its visibility. We model it, exploring the theory of concave games, to reach a situation of equilibrium; a situation where no player has the ultimate ability to deviate from its current strategy. We formulate the named game, then we analyze it and prove that there is exactly one Nash equilibrium which is the convergence of all players' best responses. We finally provide some numerical results, taking into consideration a system of two sources with a specific frequency space, and analyze the effect of different parameters on sources' visibility on the walls of social networks.

scholarly journals Expressiveness and Nash Equilibrium in Iterated Boolean Games

2021 ◽  
Vol 22 (2) ◽  
pp. 1-38
Author(s):  
Julian Gutierrez ◽  
Paul Harrenstein ◽  
Giuseppe Perelli ◽  
Michael Wooldridge
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Infinite Sequence ◽  
Multi Agent Systems ◽  
Temporal Logics ◽  
Agent Systems ◽  
Temporal Properties ◽  
Multi Agent ◽  
Game Theoretic ◽  
Boolean Games

We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model of multi-agent systems [Gutierrez et al. 2013, 2015a]. In such a game, each agent  has a goal  , represented using (a fragment of) Linear Temporal Logic ( ) . The goal  captures agent  ’s preferences, in the sense that the models of  represent system behaviours that would satisfy  . Each player controls a subset of Boolean variables , and at each round in the game, player is at liberty to choose values for variables in any way that she sees fit. Play continues for an infinite sequence of rounds, and so as players act they collectively trace out a model for , which for every player will either satisfy or fail to satisfy their goal. Players are assumed to act strategically, taking into account the goals of other players, in an attempt to bring about computations satisfying their goal. In this setting, we apply the standard game-theoretic concept of (pure) Nash equilibria. The (possibly empty) set of Nash equilibria of an iterated Boolean game can be understood as inducing a set of computations, each computation representing one way the system could evolve if players chose strategies that together constitute a Nash equilibrium. Such a set of equilibrium computations expresses a temporal property—which may or may not be expressible within a particular fragment. The new notion of expressiveness that we formally define and investigate is then as follows: What temporal properties are characterised by the Nash equilibria of games in which agent goals are expressed in specific fragments of  ? We formally define and investigate this notion of expressiveness for a range of fragments. For example, a very natural question is the following: Suppose we have an iterated Boolean game in which every goal is represented using a particular fragment of : is it then always the case that the equilibria of the game can be characterised within ? We show that this is not true in general.

scholarly journals Holding a group together: non-game-theory vs. game-theory

2021 ◽  
Author(s):  
Michael Richter ◽  
Ariel Rubinstein
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Competitive Equilibrium ◽  
Solution Concept ◽  
Equilibrium Concept

Abstract Each member of a group chooses a position and has preferences regarding his chosen position. The group’s harmony depends on the profile of chosen positions meeting a specific condition. We analyse a solution concept (Richter and Rubinstein, 2020) based on a permissible set of individual positions, which plays a role analogous to that of prices in competitive equilibrium. Given the permissible set, members choose their most preferred position. The set is tightened if the chosen positions are inharmonious and relaxed if the restrictions are unnecessary. This new equilibrium concept yields more attractive outcomes than does Nash equilibrium in the corresponding game.

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