Bridging game theory and the knapsack problem: a theoretical formulation

This paper presents an approach to solving the knapsack problem in which the solution can be derived based on a treatment of the classical bargaining problem. In spatial game theory, all the players form an \(n\)-polyhedron in space, and the bargaining set of items is positioned such that the geometrical distance of each item from every player reference vertex point is inversely proportional to its utility to the player. The game-theoretical-based distance of an item to a player is defined as the ratio of the geometrical distance referenced from the player’s vertex position to the sum of distances from all the \(n\)-player vertices of the polyhedron, and its game moment is derived from the product of utility and this distance. Pareto optimality can then be achieved by balancing the effective game-moment contributions from \(n\)-player subsets of items at equilibrium. The Pareto-optimal solution is defined such that for a given set of consolidated items, further addition of items to the knapsack will result in diminishing returns in their payoffs or profits attained together with the corresponding unwanted increases in constraints or burdens to cause destabilization of this equilibrium. This game-theoretical approach is employed by having the game player entities work cooperatively to maximize profit and minimize burdens in order to arrive at a solution and is the first spatial game representation of the knapsack problem.