Analysing the Impact of Multiple Constraints on the Solution Point in Game Theory: A Theoretical Study into Problems of Type (m×2) or (2×n)

Abstract

This paper addresses an atypical challenge in solving game-theoretic problems via the graphical method when the payoff matrix is of dimensions (m×2) or (2×n), implying that one player is limited to at most two strategies. The graphical method represents the two-strategy player’s options as the axes of a plane, while the second player’s strategies manifest as linear constraints. The optimal solution for the first player is then determined by the intersection of these constraints, discarding any that do not influence the solution point. The focus of this study is on the special scenarios where the solution point for the first player is defined by more than two constraints, which contrasts with the standard case, in which the solution point arises from the intersection of exactly two constraints. Such instances are non generic and require careful handling due to the presence of additional constraints. The paper aims to analyze this case using linear programming, with a focus on how to identify and exclude constraints when solving the second player’s problem via the graphical method, It also seeks to examine the impact of excluding these constraints on both players in order to ensure the attainment of the correct optimal solution. Moreover, it contributes to a deeper understanding of the solution procedure of the graphical method; especially in non-standard cases that pose additional challenges for the use of the graphical method. Through this analysis, the active constraints that must be used in the graphical solution process are identified, which implies excluding inactive constraints and considering them as redundant, thereby not to be utilized in the graphical solution procedure.