Mechanism Design without Money for Common Goods

We initiate the study of mechanism design without money for common goods. Our model captures a variation of the well-known one-dimensional facility location problem if the facility is assumed to have a capacity constraint $k<n$ where $n$ is the population size. This new model introduces a richer game-theoretic context compared to the classical facility location, or public goods, problem. Our key result contributes a novel perspective relating to the "major open question" (Barbar\`a et al., 1998) posed by Border and Jordan (1983) by showing the equivalence of dominant strategy incentive compatible (DIC) mechanisms for common goods and the family of Generalized Median Mechanisms (GMMs). This equivalence does not hold in the public goods setting and, by situating GMMs in this broader game-theoretic context, is the first complete characterization of GMMs in terms of purely strategic properties. We then characterize lower bounds of the welfare approximation ratio across all DIC mechanisms and identify a DIC mechanism which attains this lower bound when $k<\lceil (n+1)/2\rceil$ and $k=n$. Finally, we analyze the approximation ratio when the property of DIC is weakened to ex post incentive compatibility.