Boolean Game Research Articles

Optimal Strategy Synthesis for Stochastic Boolean Games

In this study, we regard a classification of winning strategy and payoff function over infinite plays. The central goal is to examine the values of the game and then to determine the existence of optimal (∈-optimal) strategy. Furthermore, we are interested in the subject of what sort of optimal (∈-optimal) strategy exists. We first review on generalised reachability games and then we introduce a new game, called Boolean games and concentrate on games with a Boolean combination of the reachability games. The primary contribution is on the existence of ∈-optimal finite memory strategy of each player for any ∈ > 0. We also prove every player has no ∈-optimal memoryless strategy for some Boolean game.

Expressiveness and Nash Equilibrium in Iterated 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.

Incentive-based search for equilibria in boolean games

Search for equilibria in games is a hard problem and many games do not have a pure Nash equilibrium (PNE). Incentive mechanisms have been shown to secure a PNE in certain families of games. The present study utilizes the similarity between Asymmetric Distributed Constraints Optimization Problems (ADCOPs) and games, to construct search algorithms for finding outcomes and incentives that secure a pure Nash equilibrium in Boolean games. The set of values returned by the search algorithm for a chosen incentive mechanism is termed an incentive plan. The two incentive mechanisms that are used by the present study are taxation and side-payments. Both are described and their performance on PNE search in Boolean games is evaluated. An incentive plan for the taxation mechanism consists of the values of imposed tax, while an incentive plan for the side payments mechanism consists of values for a set of transfer functions. The distributed search algorithms address two different requirements. One is to find an incentive plan that enables to secure a PNE. The other requirement is that the algorithms return a PNE that satisfies some global objective, such as a PNE that maximizes social welfare. The Boolean game is first transformed into an ADCOP. Then, a designated distributed search algorithm is applied to find the desired outcome. Two distributed search algorithms are described, incorporating k-ary constraints as well as soft constraints that relate to global objectives. The new and innovative algorithm - Concurrent Asymmetric Branch and Bound - is found experimentally to be much faster than the former algorithm. An extensive experimental evaluation on several types of social-network-based Boolean games is presented. The degree of intervention in the game is found to be small for both incentive mechanisms. In other words, the overall tax or the total amount of side-payments is a small fraction of the general utility. The density of the networks has a substantial effect on the solution quality as well as on the computational effort to find them.

Regulation of game result for n‐person random evolutionary Boolean games

AbstractThis paper investigates the regulation of game result for n‐person random evolutionary Boolean game (REBG). Using the semi‐tensor product of matrices, the n‐person REBG together with the reference evolutionary Boolean game are converted into an augmented system. Based on the augmented system, a necessary and sufficient condition is presented for the solvability of the regulation of game result of n‐person REBG by constructing a series of stochastic reachable sets. The study of an illustrative example shows the effectiveness of the obtained new results.

From Boolean game to potential game

Abstract Using semi-tensor product of matrices, the vector space structure of Boolean games and their some specified subsets are proposed. By resorting to the vector space structure and potential equation, we give an alternative proof for the fact that a symmetric Boolean game is a potential game. The two advantages of this new approach are revealed as follows: (1) It can provide the corresponding potential function; (2) It can be used to explore new potential Boolean games. The corresponding formula is provided to demonstrate the first advantage. As for the second one, the renaming symmetric Boolean games and the weighted symmetric Boolean games are also proved to be potential and weighted potential respectively. Moreover, as a non-symmetric game, the flipped symmetry Boolean game has been constructed and proved to be potential.

Stochastic set stabilisation of n ‐person random evolutionary Boolean games and its applications

This study studies the stochastic set stabilisation of an n-person random evolutionary Boolean game (REBG). Firstly, the definition of stochastic set stabilisation has been proposed, and the n-person REBG is converted to an equivalent algebraic form by using the semi-tensor product of matrices. Secondly, by defining a series of stochastic reachable sets, a necessary and sufficient condition is presented for the state feedback stochastic set stabilisation of n-person REBG. Thirdly, based on the concept of uniformly one-step reachability with probability one, a constructive procedure is proposed for the calculation of stochastic reachable sets. Finally, as two applications of stochastic set stabilisation, the output tracking control and the strategy consensus of n-person REBG are investigated, respectively, and two necessary and sufficient conditions are presented.

Rational Verification in Iterated Electric Boolean Games

Electric boolean games are compact representations of games where the players have qualitative objectives described by LTL formulae and have limited resources. We study the complexity of several decision problems related to the analysis of rationality in electric boolean games with LTL objectives. In particular, we report that the problem of deciding whether a profile is a Nash equilibrium in an iterated electric boolean game is no harder than in iterated boolean games without resource bounds. We show that it is a PSPACE-complete problem. As a corollary, we obtain that both rational elimination and rational construction of Nash equilibria by a supervising authority are PSPACE-complete problems.

Learning to Win in Evolutionary Two-person Boolean Game with Fixed Strategy Updating Rule

This paper introduces an algorithm to learn the strategy updating rule for a two- person Boolean game using the records of the history strategies and game results. The two-person game in this paper is introduced as a zero-sum game along with a Boolean strategy set, and the strategies are governed by fixed Boolean functions whose arguments are the history strategies and game results with additive binary noise, which can be modeled as a stochastic Boolean dynamic system. However, for this easy-to-play game, there is no effective convenient methods to win more often. To achieve this goal, a learning algorithm based on Boolean regression and maximum-likelihood estimation is put forward to learn the strategy updating rule and the noise property using the records of the history strategies and game results. In addition, extensive simulations via actual examples have illustrated the effectiveness of the proposed learning algorithm.

Effectivity functions and efficient coalitions in Boolean games

Boolean games are a logical setting for representing strategic games in a succinct way, taking advantage of the expressive power and conciseness of propositional logic. A Boolean game consists of a set of players, each of which controls a set of propositional variables and has a specific goal expressed by a propositional formula. We show here that Boolean games are a very simple setting, yet sophisticated enough, for analysing the formation of coalitions. Due to the fact that players have dichotomous preferences, the following notion emerges naturally: a coalition in a Boolean game is efficient if it has the power to guarantee that all goals of the members of the coalition are satisfied. We study the properties of efficient coalitions.

Dependencies between players in Boolean games

Boolean games are a logical setting for representing static games in a succinct way, taking advantage of the expressive power and succinctness of propositional logic. A Boolean game consists of a set of players, each of them controlling a set of propositional variables and having a specific goal expressed by a propositional formula, or more generally a specification of the player’s preference relation in some logical language for compact preference representation, such as prioritized goals. There is a lot of graphical structure hidden in a Boolean game: the satisfaction of each player’s goal depends on players whose actions have an influence on her goals. Exploiting this dependency structure facilitates the computation of pure Nash equilibria, by partly decomposing a game into several sub-games that are only loosely related.

Boolean game on scale-free networks

Inspired by the local minority game, we propose a network Boolean game and investigate its dynamical properties on scale-free networks. The system can self-organize to a stable state with better performance than the random choice game, although only the local information is available to each agent. By introducing the heterogeneity of local interactions, we find that the system will achieve the best performance when each agent's interaction frequency is linearly correlated with its information capacity. Generally, the agents with more information gain more than those with less information, while in the optimal case, each agent almost has the same average profit. In addition, we investigate the role of irrational factor and find an interesting symmetrical behavior.

Memory-Based Boolean Game and Self-Organized Phenomena onNetworks

We study a memory-based Boolean game (MBBG) taking place on aregular ring, wherein each agent acts according to its localoptimal states of the last M time steps recorded in memory, andthe agents in the minority are rewarded. One free parameter pbetween 0 and 1 is introduced to denote the strength of theagent willing to make a decision according to its memory. It isfound that giving proper willing strength p, the MBBG system canspontaneously evolve to a state of performance better than therandom game; while for larger p, the herd behaviour emerges toreduce the system profit. By analysing the dependence ofdynamics of the system on the memory capacity M, we find that ahigher memory capacity favours the emergence of the betterperformance state, and effectively restrains the herd behaviour,thus increases the system profit. Considering the high costof long-time memory, the enhancement of memory capacity forrestraining the herd behaviour is also discussed, and M = 5 issuggested to be a good choice.

Self-organized Boolean game on networks

A model of a Boolean game with only one free parameter p that denotes the strength of local interaction is proposed wherein each agent acts according to the information obtained from his neighbors in the network, and those in the minority are rewarded. The simulation results indicate that the dynamic of the system is sensitive to network topology, whereby the network of larger degree variance, i.e., the system of greater information heterogeneity, leads to less system profit. The system can self-organize to a stable state and perform better than the random choice game, although only the local information is available to the agents. In addition, in heterogeneity networks, the agents with more information gain more than those with less information for a wide extent of interaction strength p.

σ‐short Boolean algebras

AbstractWe introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)