Game theory models strategic interactions among agents, focusing on situations of conflict and offering various solution concepts, known as equilibria. Bayesian games consider situations in which players have private types and act and receive payoffs based on those types. The problem of computing Bayesian Nash equilibria has been thoroughly investigated, and solutions for various Bayesian games have been applied to many real-world applications. From a designer’s point of view, the inverse problem of constructing a game that has a particular equilibrium value, yielding a certain payoff for each player and each of its possible types, is also of interest as it can provide methods for incentivizing agents towards certain decisions. In this paper, we address this problem in Bayesian games by using a genetic programming approach to evolve payoff functions that construct a Bayesian game given a specified equilibrium configuration and corresponding payoff values.

• existence of barren plateaux is one of the challenges in the practical use of variational quantum classifiers; • a noise-based mechanism that shifts training data during optimization, helping escape barren plateaux, is proposed; • the approach is tested with a variational quantum classifier modeling BET index changes using other indices from Europe and the United States; • simulations use the Pennylane framework. The barren plateaux phenomenon has been identified as a significant challenge for variational quantum algorithms, particularly for classification tasks. In this article, we propose a novel approach to mitigating this problem for variational quantum classifiers during the optimization phase. The noisy optimization mechanism shifts the training data by adding a small amount of uniform noise, thereby inducing changes in the parameters being searched. The effectiveness of the method is evaluated using real financial data, modeling the evolution of the BET index in relation to well-known indices from neighboring Central and Eastern European countries, as well as from Western Europe and the United States. The results demonstrate that this approach significantly improves upon the corresponding baseline quantum classifier and provides results comparable to those of established classical methods.

Abstract Decision trees rank among the most popular and efficient classification methods. They are used to represent rules for recursively partitioning the data space into regions from which reliable predictions regarding classes can be made. These regions are usually delimited by axis-parallel or oblique hyperplanes. Axis-parallel hyperplanes are intuitively appealing and have been widely studied. However, there is still room for exploring different approaches. In this paper, a splitting rule that constructs axis-parallel hyperplanes by computing the Nash equilibrium of a game played at the node level is used to induct a Nash Equilibrium Decision Tree for binary classification. Numerical experiments are used to illustrate the behavior of the proposed method.

Multiclass imbalanced Classification problems are some of the most challenging tasks in machine learning, that are usually related to sensitive real-world applications. A Game-Theoretic Equilibria Classifier (GTEC) that enhances the One-Versus-Rest (OvR) approach for imbalanced data sets is proposed. GTEC innovatively applies a discrete-time differential game strategy to train OvR classifiers on strategically selected data sub-samples guided by the game-theoretic framework. The proposed method is tested on a set of synthetically generated datasets. GTEC’s behavior on the Cardiotocography dataset illustrates the approach on a real benchmark. Numerical examples are used to illustrate the efficacy of the approach.

Community structure detection algorithms are used to identify groups of nodes that are more connected to each other than to the rest of the network. Multipartite networks are a special type of network in which nodes are divided into partitions such that there are no links between nodes in the same partition. However, such nodes may belong to the same community, making the identification of the community structure of a multipartite network computationally challenging. In this paper, we propose a new fitness function that takes into account the information induced by existing links in the network by considering shadowed connections between nodes that have a common neighbor. The existence of a correct fitness function, i.e. one whose optimum values correspond to the community structure of the network, enables the design and use of optimization-based heuristics for solving this problem. We use numerical experiments performed on artificial benchmarks to illustrate the effectiveness of this function used within an extremal optimization based algorithm and compared to existing approaches. As a direct application, a multipartite network constructed from a direct marketing database is analyzed.

Game theory models strategic and conflicting situations and offers several solution concepts that are known as game equilibria, among which the Nash equilibrium is probably the most popular one. A less known equilibrium, called the (k,t)-robust equilibrium, has recently been used in the context of distributed computing. The (k,t)-robust equilibrium combines the concepts of k-resiliency and t-immunity: a strategy profile is k-resilient if there is no coalition of k players that can benefit from improving their payoffs by collective deviation, and it is t-immune if any action of any t players does not decrease the payoffs of the others. A strategy profile is (k,t)-robust if it is both k-resilient and t-immune. In this paper an evolutionary approach of approximating (k,t)-robust equilibria is proposed. Numerical experiments are performed on a game that models node behavior in a distributed system.

The network community detection problem consists in identifying groups of nodes that are more densely connected to each other than to the rest of the network. The lack of a formal definition for the notion of community led to the design of various solution concepts and computational approaches to this problem, among which those based on optimization and, more recently, on game theory, received a special attention from the heuristic community. The former ones define the community structure as an optimum value of a fitness function, while the latter as a game equilibrium. Both are appealing as they allowed the design and use of various heuristics. This paper analyses the behavior of such a heuristic that is based on extremal optimization, when used either as an optimizer or within a game theoretic setting. Numerical results, while significantly better than those provided by other state-of-art methods, for some networks show that differences between tested scenarios do not indicate any superior behavior when using game theoretic concepts; moreover, those obtained without using any selection for survival suggest that the search is actually guided by the inner mechanism of the extremal optimization method and by the fitness function used to evaluate and compare components within an individual.

Discrete-time dynamic games capture changes within game parameters (payoffs), representing thus a more realistic real-world decision model. A new approach to equilibria detection and tracking in a dynamic game setting, called Dynamic Equilibrium Tracking Differential Evolution, is proposed. A discrete-time dynamic asymmetric Cournot oligopoly is introduced and used to evaluate the proposed method by means of numerical experiments for instances up to 70 players. Results indicate the effectiveness and the potential of the proposed method.

Minimax or worst-case optimization is concerned with the minimization of the maximum output in all scenarios of a given problem. The minimax problem can be transformed in a two players zero-sum game considering the fact that the Nash equilibria of this game would represent the solution of the original problem. Using the Nash ascendancy relation the equilibria of the game can be directly computed using a differential evolution algorithm. Results obtained by using this approach are compared with best known results from literature on six minimax benchmark problems.

Generative relations represent a powerful algebraic tool for characterizing and detecting game equilibria. Generative relations are particularly useful for defining new classes of equilibria, for example joint equilibria. In order to avoid difficulties for games involving many players (n ≥ 2) a new generative relation for fuzzy Nash-Pareto equilibrium is introduced. An evolutionary procedure relying on this generative relation is used for detecting several types of fuzzy equilibria. Experimental results indicate the effectiveness and robustness of the proposed approach.

One of the main issues in computational game theory is equilibria detection in multi player games. This problem is approached using a generative relation for strategy profiles and two different search operators: crowding based differential evolution and a simple stepping stone reinforcing search algorithm. A probabilistic generative relation is also derived in order to tackle a higher number of players. Each approach has advantages and disadvantages, illustrated by numerical experiments for games involving from two to a hundred players.

A generative relation for Aumann equilibrium is proposed. An evolutionary procedure based on nondomination with respect to the generative relation is used for detecting Aumann equilibrium.

Based on a study of how people play the centipede game, an equilibrium configuration that models the human behavior is detected. This configuration is a joint equilibrium obtained as a fuzzy combination of Nash and Pareto equilibria. In this way a connection between normative theory, computational game theory and behavioral games is established.

A general technique for detecting equilibria in finite non cooperative games is proposed. Fundamental idea is that every equilibrium is characterized by a binary relation on the game strategies. This relation - called generative relation -- induces an appropriate domination concept. Game equilibrium is described as the set of non dominated strategies with respect to the generative relation. Slight generalizations of some well known equilibrium concepts are proposed. A population of strategies is evolved according to a domination-based ranking in oder to produce better and better equilibrium approximations. Eventually the process converges towards the game equilibrium. The proposed technique opens an way for qualitative approach of game equilibria. In order to illustrate the proposed evolutionary technique different equilibria for different continuous games are studied. Numerical experiments indicate the potential of the proposed concepts and technique.

A new hybrid approach to optimization in dynamic environments called collaborative evolutionary-swarm optimization (CESO) is presented. CESO is a simple method for tracking moving optima in a dynamic environment by combining the search abilities of an evolutionary algorithm for multimodal optimization and a particle swarm optimization algorithm. A collaborative mechanism is designed for the two methods. Numerical experiments indicate CESO to be an efficient method for the selected test problems compared with other evolutionary approaches.

Multimodal optimization problems consist in detecting all global and local optima of a problem. A new evolutionary approach to multimodal optimization called Roaming technique (RO) is presented. Roaming uses two original concepts in order to detect multiple optima: a stability measure for subpopulations and an external population called archive to store detected optima. Individuals in the archive are refined by evolving them independently. Performance of Roaming is compared by means of numerical experiments with two other evolutionary techniques.

A new evolutionary technique for multicriteria optimization called Guiding Hyper-plane Evolutionary Algorithm (GHEA) is proposed. The originality of the approach consists in the fact that the fitness assignment is realized by using a guiding hyperplane and a new non Pareto optimality concept. Numerical experiments illustrate the performance of GHEA compared with the popular NSGA-II and SPEA2.