Analysis of the Pareto equilibrium in multi-objective games using semi-tensor product

Abstract

Multi-objective games (MOGs) have received much attention in recent years as a class of games with vector payoffs. Based on the semi-tensor product (STP), this paper discusses the MOG, including the existence, finite-step reachability, and finite-step controllability of Pareto equilibrium of this model, from both static and dynamic perspectives. First, the MOG concept is presented using multi-layer graphs, and STP is used to convert the payoff function into its algebraic form. Then, from the static perspective, two necessary and sufficient conditions are proposed to verify whether all players can meet their expectations and whether the strategy profile is a Pareto equilibrium, separately. Furthermore, from the dynamic perspective, a strategy updating rule is designed to investigate the finite-step reachability of the evolutionary MOG. Finally, the finite-step controllability of the evolutionary MOG is analyzed by adding pseudo-players, and a backward search algorithm is provided to find the shortest evolutionary process and control sequence.

1. Introduction

As one of the important tools in researching interaction mechanisms and behavior among individuals, game theory has a wide range of applications in areas such as cybernetics, biology, and economics. It has also become a popular interdisciplinary field. The payoff function is one of the important components of finite games and can be categorized into scalar and vector forms. Most of the existing references focus on the cases where payoffs are scalar. However, games with vector payoffs, which are commonly known as multi-objective games (MOGs) or multi-criteria games, are ubiquitous in reality. Games with vector payoffs offer a noteworthy study topic when internal motivation is considered (Puerto and Perea, 2018). For instance, considering tobacco consumers as players, as discussed by Ismaili (2018), every player’s payoff vector contains three components: the pleasure of smoking, the cost of cigarettes, and the impact on life expectancy. Consequently, the payoff can be modeled in the form of a vector, the numerical components of which demonstrate commodities (Hamel and Löhne, 2018), and the ultimate payoff depends on each component of the payoff vector. For example, Rădulescu et al. (2020) applied this type of game to MOG multi-agent systems, and every component of the payoff vector represented the performance on a different objective.

Equilibrium analysis is an important issue in game theory. For games with scalar payoffs, the existence of equilibria has been investigated, such as pure Nash equilibrium (Shah-Mansouri and Wong, 2018), Bayesian Nash equilibrium (He et al., 2024), fuzzy strong Nash equilibrium (Huang and Liu, 2022), and stationary Nash equilibrium (Jaśkiewicz and Nowak, 2023). Therefore, it is reasonable and practical to explore the equilibrium of MOGs. Because there is more than one objective for every player in an MOG, an accepted equilibrium is the Pareto equilibrium (Qu et al., 2015). Pareto equilibrium is a strategy profile in which no player can be better without making the others worse (Smith et al., 2014; Yan and Hayakawa, 2024). Based on methods such as Ky Fan’s minimax inequality and fixed-point theorem (Wang SY, 1993; Lu, 2008), the stochastic maximum principle with Markov jumps and Poisson jumps and the Lagrangian multiplier technique (Zhang et al., 2018; Hu et al., 2024), and the theorem of quasi-equilibrium existence (Ding, 2000a, 2000b), the existence of Pareto equilibrium has been investigated. To date, there has been no reference to finite-step reachability or finite-step controllability of Pareto equilibria in MOGs.

Specifically, for the Pareto equilibrium problem in MOGs, it is necessary to discuss the individual expectations of players, because cooperation and competition complement each other in the game process. In particular, for MOGs, if we examine only Pareto equilibrium which is cooperation-oriented, then the equilibrium points may be altruistic but not egoistic, which is not true in most real scenarios. Equally important, if we examine only Nash equilibrium, then such strictly competitive games do not reach an equilibrium point (Choobineh and Mohagheghi, 2019; Peng et al., 2022), because it is nearly impossible for all objectives to simultaneously achieve the best equilibrium. Consequently, by setting the payoff vector and ultimate payoff expectations for every player in an MOG and then researching the Pareto equilibrium, we can achieve more flexible simulation of various problems, where the ultimate payoff can be expressed using the weighted sum, weighted average, polynomial function, and so on.