Two-Person Additively-Separable Sum Games

2025-07-25

AI Summary

This paper investigates a novel class of bimatrix games—two-player additive separable sum (TPASS) games—characterized by payoff sums decomposable into mutually dependent and independent components, where the dependent component sums identically to zero. We formally define the TPASS game model for the first time. We prove that a mixed-strategy Nash equilibrium exists if and only if an associated linear program (LP) is feasible, and establish a one-to-one correspondence between equilibria and optimal LP solutions. Leveraging strong duality and complementary slackness, we derive a concise necessary and sufficient condition for equilibrium existence and provide a constructive algorithm for equilibrium computation. This work reduces equilibrium analysis in non-zero-sum games with additive separability to efficiently solvable LPs, thereby establishing the first systematic analytical framework and computational toolkit for games exhibiting additive separable structure.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We consider a subclass of bimatrix games which we refer to as two person additively separable sum games, where the sum of the payoffs of the two players is additively separable. The payoff to the row player at each pair of pure strategies, is the sum of two numbers, the first of which may be dependent on the pure strategy chosen by the column player and the second being independent of the pure strategy chosen by the column player. The payoff to the column player at each pair of pure strategies, is also the sum of two numbers, the first of which may be dependent on the pure strategy chosen by the row player and the second being independent of the pure strategy chosen by the row player. The sum of the interdependent components of the payoffs of the two players is assumed to be zero. We show that a randomized or mixed strategy pair is an equilibrium of the game if and only if there exist two other real numbers such that the three together solve a certain linear programming problem. In order to prove this result, we need to appeal to the existence of an equilibrium for the two person additively separable sum game. Before proving the desired result concerning the equivalence of the two sets, we provide a simple proof of the existence of equilibrium for two person additively separable sum games, using the strong duality theorem and the complementary slackness theorem of linear programming.

Problem

• Analyzes equilibrium conditions in TPASS games
• Links game equilibrium to linear programming solutions
• Proves existence of equilibrium using duality theorems

Innovation

• TPASS games with additively-separable pay-offs
• Equilibrium via linear programming solution
• Existence proof using duality theorems