Subset Weight Maximization with Two Competing Agents

Abstract

We consider a game of two agents competing to add items into a solution set. Each agent owns a set of weighted items and seeks to maximize the sum of their weights in the solution set. In each round each agent submits one item for inclusion in the solution. We study two natural rules to decide the winner of each round: Rule 1 picks among the two submitted items the item with larger weight, Rule 2 the item with smaller weight. The winning item is put into the solution set, the losing item is discarded.

For both rules we study the structure and the number of efficient solutions, i.e. Pareto optimal solutions. For Rule 1 they can be characterized easily, while the corresponding decision problem is NP-complete under Rule 2. We also show that there exist no Nash equlibria. Furthermore, we study the best-worst ratio, i.e. the ratio between the efficient solution with largest and smallest total weight, and show that it is bounded by two for Rule 1 but can be arbitrarily high for Rule 2. Finally, we consider preventive or maximin strategies, which maximize the objective function of one agent in the worst case, and best response strategies for one agent, if the items submitted by the other agent are known before either in each round (on-line) or for the whole game (off-line).