The price of democracy in coalition formation

Whenever rational agents form coalitions to execute tasks, doing so via a decentralized negotiation process—while more robust and democratic—may lead to a loss of efficiency compared to a centralized solution. To quantify this loss, we introduce the notion of the Price of Democracy (PoD), which measures the amount of resources needlessly committed to the task(s) at hand. After defining this concept for general coalitional games, we instantiate it in the setting of weighted voting games, a simple but expressive class of coalitional games that can be used to model resource allocation in multiagent scenarios. We approach the problem of forming winning coalitions in this setting from a non-cooperative perspective, and put forward an intuitive deterministic bargaining process, which exhibits no delay of agreement (i.e., the agents are guaranteed to form a winning coalition in round one) and allows for efficient computation of bargaining strategies. We show a tight bound of 3/2 on the PoD of our process if two rounds of bargaining are allowed, and demonstrate that this bound cannot improve with more rounds. We then generalize our bargaining process to settings where multiple coalitions are allowed to be formed, show that this generalization also exhibits no delay of agreement, and discuss the PoD in such settings.