| In group formation games, our objective is to find a partition of agents that maximizes the total utility received by the agents. However, if the agents are selfish and only care to maximize their own individual utilities, then they may not act as prescribed in the optimal state. Agents will change groups until they reach a Nash equilibrium, which is a stable state in which no agent is able to increase her payoff by a unilateral deviation. We are interested in determining the price of anarchy, which is the ratio of the lowest quality Nash equilibrium and the quality of the optimal state.;We describe several classes of games in which agents must be partitioned into a fixed number of groups. The agents want to form small groups with their friends, while avoiding being grouped with their enemies. We demonstrate that the agents form Nash equilibria that are close to optimal. We describe another class of group formation games in which agents must work together on projects, but want to minimize the size of their groups in order to maximize their share of the profits, while still having enough members to successfully complete their projects. We show that the stable states produced by selfish agents can be of very low quality, unless the agents have enforceable contracts that ensure agents cannot join or leave projects too easily.;Finally, we examine the quality of social choice functions, which map agents' orderings over a set of alternatives to a single winning alternative. We assume that every agent has costs over all of the alternatives and that these costs form a metric space. However, the social choice mechanisms may not have access to these costs, especially since they may be unknown to the agents themselves. As a result, we cannot simply select the alternative that minimizes the total agent cost. Instead, we assume that the social choice functions only have access to the agents' ordinal preferences that are induced by the metric space. We examine the distortion, which is the worst-case ratio of the quality of the alternative selected by the social choice mechanism and the quality of the optimal alternative. Bounding the distortion acts as a method for determining the quality of the social choice function itself. We provide distortion bounds for a variety of deterministic and randomized mechanisms, such as plurality, Copeland, and randomized dictatorship. |
