Coalitions on the Fly in Cooperative Games

TL;DR

This paper studies an online cooperative game where players arrive sequentially, proposing a policy that incentivizes coalition formation to maximize social welfare with near-optimal competitive ratio bounds.

Contribution

It introduces a new online value distribution policy for dynamic cooperative games, achieving near-optimal competitive ratios under certain conditions.

Findings

Established an upper bound of 3*min/max on competitive ratio for irrevocable policies.

Proposed a policy with a competitive ratio of min{1/2, 3*min/max}.

Analyzed bounds for non-irrevocable policies with limited players.

Abstract

In this work, we examine a sequential setting of a cooperative game in which players arrive dynamically to form coalitions and complete tasks either together or individually, depending on the value created. Upon arrival, a new player as a decision maker faces two options: forming a new coalition or joining an existing one. We assume that players are greedy, i.e., they aim to maximize their rewards based on the information available at their arrival. The objective is to design an online value distribution policy that incentivizes players to form a coalition structure that maximizes social welfare. We focus on monotone and bounded cooperative games. Our main result establishes an upper bound of $\frac{3\mathsf{min}}{\mathsf{max}}$ on the competitive ratio for any irrevocable policy (i.e., one without redistribution), and proposes a policy that achieves a near-optimal competitive ratio of $\min\left\{\frac{1}{2}, \frac{3\mathsf{min}}{\mathsf{max}}\right\}$, where $\mathsf{min}$ and $\mathsf{max}$ denote the smallest and largest marginal contribution of any sub-coalition of players respectively. Finally, we also consider non-irrevocable policies, with alternative bounds only when the number of players is limited.