Despite the prevalence of weighted voting in the real world, there has been relatively little work studying real people's behavior in such settings. This paper proposes a new negotiation game, based on the weighted voting paradigm in cooperative games, where players need to form coalitions and agree on how to share the gains. We show that solution concepts from cooperative game theory (in particular, an extension of the Deegan-Packel Index) provide a good prediction of people's decisions to join a given coalition. With this insight in mind, we design an agent that combines predictive analytics with decision theory to make offers to people in the game. We show that the agent was able to obtain higher shares from coalitions than did people playing other people, without reducing the acceptance rate of its offers. These results demonstrate the potential of incorporating concepts from cooperative game theory in the design of negotiating agents.