Entropy Games and Matrix Multiplication Games (original) (raw)

Games with Imperfect Information

The MIT Press eBooks, 2014

Games have been extensively studied, either in computer science, mathematics or even economy. Nevertheless, each discipline has its own interest in using this formalism. Computer science for instance is attached to calculability issues. These results have some direct applications in model checking or compilation. Recently, a new type of game has been introduced: games with imperfect information. They allow the modeling of more sophisticated systems, but bring also new calculability problems. In this document, we introduce a general method to prove the determinacy of any type of game. This method is used several times, and allow us to solve some open problems. This document introduces also several examples of important games stating for important properties. Then, a new type of game unifying the concepts of concurrency and imperfect information is presented. Finally, we discuss of the extension on infinite arenas.

Matrix Games with Uncertain Entries

International Journal for Research in Applied Science and Engineering Technology, 2019

In classical game theory, a conflict of two opponents can be modelled as an equilibrium-based matrix game. We assume a conflict of two non-cooperative antagonistic opponents with a finite number of strategies with zero-sum or constant sum pay-offs. At the same time, we suppose that the elements of the payoff matrix describing the game are not fixed and are allowed to change within a specified interval. Supposing that some of the elements of the payoff matrix are uncertain, it is evident that this would influence the utilities of both players at the same time and moreover, such entropy of the model would eventually influence the position of equilibria or its very existence. We propose a modelling approach that allows one to find a solution of the game with either pure or mixed strategies of opponents with the guaranteed payoffs under the assumption that a specified number of unspecified entries would attain different values than expected. The chosen robust approach is presented briefly as well as the necessary circumstances of matrix game solutions. Our novelty approach follows and is accompanied by an explanatory example in the end of the paper.

On the complexity of problems on simple games

RAIRO - Operations Research, 2011

Simple games cover voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. A simple game or a yes-no voting system is a set of rules that specifies exactly which collections of "yea" votes yield passage of the issue at hand, each of these collections of "yea" voters forms a winning coalition. We are interested in performing a complexity analysis on problems defined on such families of games. This analysis as usual depends on the game representation used as input. We consider four natural explicit representations: winning, losing, minimal winning, and maximal losing. We first analyze the complexity of testing whether a game is simple and testing whether a game is weighted. We show that, for the four types of representations, both problems can be solved in polynomial time. Finally, we provide results on the complexity of testing whether a simple game or a weighted game is of a special type. We analyze strongness, properness, decisiveness and homogeneity, which are desirable properties to be fulfilled for a simple game. We finalize with some considerations on the possibility of representing a game in a more succinct representation showing a natural representation in which the recognition problem is hard.

On the Complexity of Computing Values of Restricted Games

International Journal of Foundations of Computer Science, 2002

The aim of this paper is to compute Shapley's and Banzhaf's values of cooperative games restricted by a combinatorial structure. There have been previous models developed to study the problem of games with partial cooperation. Games restricted by a communication graph were introduced by Myerson and Owen. Another type of combinatorial structure introduced by Gilles, Owen and van den Brink is equivalent to a subclass of antimatroids. Cooperative games in which the set of players is a partially ordered set, that is, games on distributive lattices was investigated by Faigle and Kern. We introduce a new combinatorial structure called augmenting system which is a generaligation of the antimatroid structure and the system of connected subgraphs of graph. We present new algorithmic procedures for computing values of games under augmenting systems restrictions and we show that there exist problems with polynomial algorithm complexity.

Multi-player games with LDL goals over finite traces

Information and Computation, 2020

Linear Dynamic Logic on finite traces (LDL F) is a powerful logic for reasoning about the behaviour of concurrent and multi-agent systems. In this paper, we investigate techniques for both the characterisation and verification of equilibria in multi-player games with goals/objectives expressed using logics based on LDL F. This study builds upon a generalisation of Boolean games, a logic-based game model of multi-agent systems where players have goals succinctly represented in a logical way. Because LDL F goals are considered, in the settings we study-Reactive Modules games and iterated Boolean games with goals over finite traces-players' goals can be defined to be regular properties while achieved in a finite, but arbitrarily large, trace. In particular, using alternating automata, the paper investigates automata-theoretic approaches to the characterisation and verification of (pure strategy Nash) equilibria, shows that the set of Nash equilibria in multi-player games with LDL F objectives is regular, and provides complexity results for the associated automata constructions.

On the computational complexity of Nash equilibria for bimatrix games

Information Processing Letters, 2005

The computational complexity of finding a Nash equilibrium in a nonzero sum bimatrix game is an important open question. We put forward the notion of (0, 1)-bimatrix games, and show that some associated computational problems are as hard as in the general case.

On the complexity of deciding degeneracy in a bimatrix game with sparse payoff matrix

Theoretical Computer Science, 2013

In this paper, we study the problem of deciding degeneracy in a bimatrix game with sparse payoff matrix. We show that it is NP-Complete to decide whether a bimatrix game is degenerate even if its payoff matrix is sparse. However, for a win-lose bimatrix game, it is in P to decide whether it is degenerate.

Decision algorithms for multiplayer noncooperative games of incomplete information

Computers & Mathematics with Applications, 2002

Extending the complexity results of Reif [1,2] for two player games of incomplete information, this paper (see also [3]) presents algorithms for deciding the outcome for various classes of multiplayer games of incomplete information, i.e., deciding whether or not a team has a winning strategy for a particular game. Our companion paper, [4] shows that these algorithms are indeed asymptotically optimal by providing matching lower bounds. The classes of games to which our algorithms are applicable include games which were not previously known to be decidable. We apply our algorithms to provide alternative upper bounds, and new time-space trade-offs on the complexity of multiperson alternating Turing machines [3]. We analyze the algorithms to characterize the space complexity of multiplayer games in terms of the complexity of deterministic computation on Turing machines. In hierarchical multiplayer games, each additional clique (subset of players with the same information) increases the complexity of the outcome problem by a further exponential. We show that an S(n) space bounded k-player game of incomplete information has a deterministic time upper bound of k + 1 repeated exponentials of S(n). Furthermore, S(n) space bounded k-player blindfold games have a deterministic space upper bound of k repeated exponentials of S(n). This paper proves that this exponential blow-up can occur. We also show that time bounded games do not exhibit such hierarchy. A T(n) time bounded blindfold multiplayer game, as well as a T(n) time bounded multiplayer game of incomplete information, has a deterministic space bound of T(n). (~

On the complexity of deciding bimatrix games similarity

Theoretical Computer Science, 2008

In this paper, we show that it is NP-complete to decide whether two bimatrix games share a common Nash equilibrium. Furthermore, it is co-NP-hard to decide whether two bimatrix games have exactly the same set of Nash equilibria.

On the Construction of High Dimensional Simple Games

2016

Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i.e., "yes" and "no", every voting system can be described by a (monotone) Boolean function Ļ‡{0,1}^nā†’{0,1}. However, its naive encoding needs 2^n bits. The subclass of threshold functions, which is sufficient for homogeneous agents, allows a more succinct representation using n weights and one threshold. For heterogeneous agents, one can represent Ļ‡ as an intersection of k threshold functions. Taylor and Zwicker have constructed a sequence of examples requiring k> 2^n/2-1 and provided a construction guaranteeing k<n n/2āˆˆ 2^n-o(n). The magnitude of the worst-case situation was thought to be determined by Elkind et al. in 2008, but the analysis unfortunately turned out to be wrong. Here we uncover a relation to coding theory that allows the determination of the minimum number k for a subclass of voting systems. As an ...