Cooperative weakest link games

Abstract

We introduce Weakest Link Games (WLGs), a cooperative game modeling domains where a team's value is determined by its weakest member. The game is represented as an edge-weighted graph with designated source and target vertices, where agents are the edges. The quality of a path between the source vertex and target vertex is the minimal edge weight along the path; the value of a coalition of edges is the quality of the best path contained in the coalition, and zero if the coalition contains no such path. WLGs model joint projects where the overall achievement depends on the weakest component, such as multiple-option package deals, or transport domains where each road has a different allowable maximum load.We provide methods for computing revenue sharing solutions in WLGs, including polynomial algorithms for calculating the value of a coalition, the core, and the least-core. We also examine optimal team formation in WLGs. Though we show that finding the optimal coalition structure is NP-hard, we provide an O(log n)-approximation. Finally, we examine the agents' resistance to cooperation through the Cost of Stability.