Mechanism Design on Discrete Structures

We study strategyproof (SP) mechanisms for the location of a facility on a discrete graph.We give a full characterization of SP mechanisms on lines and on sufficiently large cycles. In-terestingly, the characterization deviates from the one given by Schummer and Vohra [21] for thecontinuous case. In particular, it is shown that an SP mechanism on a cycle is close to dictatorial,but all agents can affect the outcome, in contrast to the continuous case. Our characterization isalso used to derive a lower bound on the approximation ratio with respect to the social cost thatcan be achieved by an SP mechanism on certain graphs. Finally, we show how the representationof such graphs as subsets of the binary cube reveals common properties of SP mechanisms andenables one to extend the lower bound to related domains.