Fuzzy Optimization in Matrix Game Theory

Matrix game theory provides a fundamental framework for modelling strategic interactions between rational decision-makers. Traditional formulations assume precise payoffs and often rely on linear programming to identify equilibrium strategies. However, real-world environments are permeated by ambiguity and imprecision in outcomes. Fuzzy optimisation extends classical matrix games by incorporating membership functions that capture the vagueness of payoffs. Through fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets, scholars have developed methods to represent uncertainty more faithfully. Rankings based on linear or accuracy functions, decomposition techniques and multi-objective programming algorithms have been employed to transform fuzzy matrix games into crisp optimisation problems. Key advances include the use of goal programming to handle multiple objectives, the adoption of belief structures to aggregate evidence, and the integration of ordered weighted aggregation to reflect decision-makers’ preferences. Such approaches enhance computational tractability and provide robust strategy recommendations under uncertainty. Applications span sensor selection and intrusion detection in networked systems, corporate environmental decision-making, telemedicine resource allocation and market share competition. By blending fuzzy logic with game-theoretic concepts, researchers achieve a more nuanced account of strategic behaviour when payoffs cannot be specified precisely.

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One recent study developed a method for solving general intuitionistic fuzzy bi-matrix games arising in corporate environmental behaviour analysis. By defining a new class of asymmetric intuitionistic fuzzy numbers and establishing order relations, the problem was converted into a multi-objective quadratic programme. A goal programming approach then yielded optimal strategies under varying acceptance degrees, demonstrating superior decision quality in environmental management scenarios.

Another investigation applied hesitant fuzzy sets to transform a multiple-attribute decision-making problem into a two-person matrix game. Using an ordered weighted aggregation-based TOPSIS procedure alongside linear programming, researchers identified strategy profiles that maximise players’ aggregated preference scores. Numerical examples in investment selection illustrated how hesitation in membership assignments can be systematically managed to yield robust outcomes.

A further contribution addressed multi-criteria zero-sum matrix games with intuitionistic fuzzy goals by introducing indeterminacy-resolving functions. This framework generates a pair of multi-objective linear programmes whose Pareto-optimal solutions correspond to security strategies for both players. Comparative simulations underscored the method’s ability to reconcile conflicting criteria and handle degrees of membership and non-membership in strategic contexts.

Fuzzy Optimization in Matrix Game Theory publication trend

The graph below shows the total number of articles in fuzzy optimization in matrix game theory across all publications each year (not limited to Nature Index journals).

Technical terms

Fuzzy set: A mathematical construct that assigns each element a degree of membership between zero and one, representing gradual belonging.

Matrix game: A game-theoretic model where players choose strategies that correspond to rows and columns of a payoff matrix.

Zero-sum game: A special matrix game in which one player’s gain equals the other player’s loss.

Intuitionistic fuzzy set: An extension of fuzzy sets characterised by separate degrees of membership and non-membership, allowing for an indeterminacy margin.

Hesitant fuzzy set: A form of fuzzy set permitting multiple potential membership values to express hesitation in assigning a single degree.

Multi-objective programming: An optimisation approach involving two or more objective functions, typically seeking Pareto-optimal trade-offs.

Dempster–Shafer belief structure: A framework for combining evidence and representing uncertainty through belief and plausibility measures.

References

1. Zero-Sum Matrix Game with Payoffs of Dempster-Shafer Belief Structures and Its Applications on Sensors. Sensors (2017).
2. Resolving Indeterminacy Approach to Solve Multi-Criteria Zero-Sum Matrix Games with Intuitionistic Fuzzy Goals. Mathematics (2020).
3. Two-person game with hesitant fuzzy payoff: An application in MADM. RAIRO - Operations Research (2021).
4. Bi-Matrix Games with General Intuitionistic Fuzzy Payoffs and Application in Corporate Environmental Behavior. Symmetry (2022).