scholarly journals Sampling Individually Fundamental Simplexes as Sets of Players’ Mixed Strategies in Finite Noncooperative Game for Applicable Approximate Nash Equilibrium Situations with Possible Concessions

2016 ◽  
Vol 40 (1) ◽  
pp. 105-143
Author(s):  
Vadim Romanuke
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Finite Lattice ◽  
Noncooperative Game ◽  
Mixed Strategies ◽  
Approximate Equilibrium ◽  
Approximate Nash Equilibrium ◽  
Universal Approach ◽  
Finite Lattices

In finite noncooperative game, a method for finding approximate Nash equilibrium situations is developed. The method is prior-based on sampling fundamental simplexes being the sets of players’ mixed strategies. Whereas the sampling is exercised, the sets of players’ mixed strategies are mapped into finite lattices. Sampling steps are envisaged dissimilar. Thus, each player within every dimension of its simplex selects and controls one’s sampling individually. For preventing approximation low quality, however, sampling steps are restricted. According to the restricted sampling steps, a player acting singly with minimal spacing over its lattice cannot change payoff of any player more than by some predetermined magnitude, being specific for each player. The finite lattice is explicitly built by the represented routine, where the player’s mixed strategies are calculated and arranged. The product of all the players’ finite lattices approximates the product of continuous fundamental simplexes. This re-defines the finite noncooperative game in its finite mixed extension on the finite lattices’ product. In such a finite-mixed-extension-defined game, the set of Nash equilibrium situations may be empty. Therefore, approximate Nash equilibrium situations are defined by the introduced possible payoff concessions. A routine for finding approximate equilibrium situations is represented. Approximate strong Nash equilibria with possible concessions are defined, and a routine for finding them is represented as well. Acceleration of finding approximate equilibria is argued also. Finally, the developed method is discussed to be a basis in stating a universal approach for the finite noncooperative game solution approximation implying unification of the game solvability, applicability, realizability, and adaptability.

scholarly journals On Sparse Discretization for Graphical Games

2020 ◽  
Vol 69 ◽  
pp. 67-84
Author(s):  
Luis Ortiz
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Sufficient Conditions ◽  
Research Note ◽  
Graphical Games ◽  
Approximate Nash Equilibrium ◽  
Approximate Nash Equilibria ◽  
Polymatrix Games ◽  
The Impact ◽  
Game Representation

Graphical games are one of the earliest examples of the impact that the general field of graphical models have had in other areas, and in this particular case, in classical mathematical models in game theory. Graphical multi-hypermatrix games, a concept formally introduced in this research note, generalize graphical games while allowing the possibility of further space savings in model representation to that of standard graphical games. The main focus of this research note is discretization schemes for computing approximate Nash equilibria, with emphasis on graphical games, but also briefly touching on normal-form and polymatrix games. The main technical contribution is a theorem that establishes sufficient conditions for a discretization of the players’ space of mixed strategies to contain an approximate Nash equilibrium. The result is actually stronger because every exact Nash equilibrium has a nearby approximate Nash equilibrium on the grid induced by the discretization. The sufficient conditions are weaker than those of previous results. In particular, a uniform discretization of size linear in the inverse of the approximation error and in the natural game-representation parameters suffices. The theorem holds for a generalization of graphical games, introduced here. The result has already been useful in the design and analysis of tractable algorithms for graphical games with parametric payoff functions and certain game-graph structures. For standard graphical games, under natural conditions, the discretization is logarithmic in the game-representation size, a substantial improvement over the linear dependency previously required. Combining the improved discretization result with old results on constraint networks in AI simplifies the derivation and analysis of algorithms for computing approximate Nash equilibria in graphical games.

AN APPROACH FOR SOLVING FUZZY GAMES

2006 ◽  
Vol 14 (03) ◽  
pp. 277-292 ◽  
Author(s):  
WEI YI LIU ◽  
JIN LI ◽  
KUN YUE ◽  
NING SONG ◽  
HONG YAO
Keyword(s):  
Genetic Algorithm ◽  
Nash Equilibrium ◽  
Mixed Strategies ◽  
Fuzzy Environment ◽  
Approximate Nash Equilibrium ◽  
Fuzzy Games ◽  
The Relationship

This paper is to compute a Nash equilibrium in a fuzzy environment, which is represented by a fuzzy approximate Nash equilibrium in a space of discrete mixed strategies. For discrete mixed strategies, the relationship between the discrete degree and the approximate degree is discussed. Based on the fuzzy regret degree, a genetic algorithm for computing a fuzzy Nash equilibrium is given.

A Theory of Patronized Goods: Inefficient and Efficient Equilibria

2011 ◽  
pp. 65-87 ◽  
Author(s):  
A. Rubinstein
Keyword(s):  
Nash Equilibrium ◽  
Social Policy ◽  
Nash Equilibria ◽  
Institutional Environment ◽  
Social Motivation ◽  
Pareto Optimal ◽  
Market Failures ◽  
Equilibrium Conditions ◽  
Policy Issues

The article considers some aspects of the patronized goods theory with respect to efficient and inefficient equilibria. The author analyzes specific features of patronized goods as well as their connection with market failures, and conjectures that they are related to the emergence of Pareto-inefficient Nash equilibria. The key problem is the analysis of the opportunities for transforming inefficient Nash equilibrium into Pareto-optimal Nash equilibrium for patronized goods by modifying the institutional environment. The paper analyzes social motivation for institutional modernization and equilibrium conditions in the generalized Wicksell-Lindahl model for patronized goods. The author also considers some applications of patronized goods theory to social policy issues.

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.

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

scholarly journals Semidefinite Programming and Nash Equilibria in Bimatrix Games

2020 ◽  
Author(s):  
Amir Ali Ahmadi ◽  
Jeffrey Zhang
Keyword(s):  
Nash Equilibrium ◽  
Semidefinite Programming ◽  
Nash Equilibria ◽  
Valid Inequalities ◽  
Bimatrix Games ◽  
Competitive Game ◽  
Approximate Nash Equilibria ◽  
Hard Problems ◽  
Rank 2

We explore the power of semidefinite programming (SDP) for finding additive ɛ-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact Nash equilibrium can be recovered. We show that, for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on the iterative linearization of smooth nonconvex objective functions whose global minima by design coincide with rank-1 solutions. Empirically, we demonstrate that these algorithms often recover solutions of rank at most 2 and ɛ close to zero. Furthermore, we prove that if a rank-2 solution to our SDP is found, then a [Formula: see text]-Nash equilibrium can be recovered for any game, or a [Formula: see text]-Nash equilibrium for a symmetric game. We then show how our SDP approach can address two (NP-hard) problems of economic interest: finding the maximum welfare achievable under any Nash equilibrium, and testing whether there exists a Nash equilibrium where a particular set of strategies is not played. Finally, we show the connection between our SDP and the first level of the Lasserre/sum of squares hierarchy.

scholarly journals Limit Points of Endogenous Misspecified Learning

Econometrica ◽  
2021 ◽  
Vol 89 (3) ◽  
pp. 1065-1098
Author(s):  
Drew Fudenberg ◽  
Giacomo Lanzani ◽  
Philipp Strack
Keyword(s):  
Nash Equilibrium ◽  
High Probability ◽  
Nash Equilibria ◽  
Prior Belief ◽  
Positive Probability ◽  
Initial Belief ◽  
Long Run ◽  
Limit Points

We study how an agent learns from endogenous data when their prior belief is misspecified. We show that only uniform Berk–Nash equilibria can be long‐run outcomes, and that all uniformly strict Berk–Nash equilibria have an arbitrarily high probability of being the long‐run outcome for some initial beliefs. When the agent believes the outcome distribution is exogenous, every uniformly strict Berk–Nash equilibrium has positive probability of being the long‐run outcome for any initial belief. We generalize these results to settings where the agent observes a signal before acting.

scholarly journals Dinamicke igre ulaska na trziste

2005 ◽  
Vol 50 (165) ◽  
pp. 121-144
Author(s):  
Bozo Stojanovic
Keyword(s):  
Nash Equilibrium ◽  
Dynamic Games ◽  
Nash Equilibria ◽  
Dynamic Game ◽  
Backward Induction ◽  
Perfect Equilibrium ◽  
Equilibrium Concept ◽  
Reasonable Solution ◽  
Multiple Nash Equilibria ◽  
Entry Games

Market processes can be analyzed by means of dynamic games. In a number of dynamic games multiple Nash equilibria appear. These equilibria often involve no credible threats the implementation of which is not in the interests of the players making them. The concept of sub game perfect equilibrium rules out these situations by stating that a reasonable solution to a game cannot involve players believing and acting upon noncredible threats or promises. A simple way of finding the sub game perfect Nash equilibrium of a dynamic game is by using the principle of backward induction. To explain how this equilibrium concept is applied, we analyze the dynamic entry games.

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