Optimal Voting Rules

We study dominant strategy incentive compatible (DIC) and deterministic mechanisms in a social choice setting with several alternatives. The agents are privately informed about their preferences, and have single-crossing utility functions. Monetary transfers are not feasible. We use an equivalence between deterministic, DIC mechanisms and generalized median voter schemes to construct the constrained-efficient, optimal mechanism for an utilitarian planner. Optimal schemes for other welfare criteria such as, say, a Rawlsian maximin can be analogously obtained

Revenue maximization in the dynamic knapsack problem

We analyze maximization of revenue in the dynamic and stochastic knapsack problem where a given capacity needs to be allocated by a given deadline to sequentially arriving agents. Each agent is described by a two-dimensional type that reflects his capacity requirement and his willingness to pay per unit of capacity. Types are private information. We first characterize implementable policies. Then we solve the revenue maximization problem for the special case where there is private information about per-unit values, but capacity needs are observable. After that we derive two sets of additional conditions on the joint distribution of values and weights under which the revenue maximizing policy for the case with observable weights is implementable, and thus optimal also for the case with two-dimensional private information. In particular, we investigate the role of concave continuation revenues for implementation. We also construct a simple policy for which per-unit prices vary with requested weight but not with time, and prove that it is asymptotically revenue maximizing when available capacity/ time to the deadline both go to infinity. This highlights the importance of nonlinear as opposed to dynamic pricing.Knapsack, revenue maximization, dynamic mechanism design

Bayesian and Dominant Strategy Implementation Revisited

We consider a standard social choice environment with linear utility and one-dimensional types. We show by counterexample that, when there are at least three physical alternatives, Bayes-Nash Incentive Compatibility (BIC) and Dominant Strategy Incentive Compatibility (DIC) need no longer be equivalent. The example with three alternatives is minimal since we do obtain a general equivalence result for settings with only two social alternatives. Our negative result does not mathematically contradict the Manelli and Vincent (2010) equivalence obtained in a one-object auction setting, but it shows that BIC-DIC equivalence is only valid in restrictive environments. Our insights are based on mathematical results about the existence of monotone measures with given monotone marginals.Bayesian Implementation, Dominant Strategy Implementation, Equivalence

On the Equivalence of Bayesian and Dominant Strategy Implementation

We consider a standard social choice environment with linear utilities and independent, one-dimensional, private types. We prove that for any Bayesian incentive compatible mechanism there exists an equivalent dominant strategy incentive compatible mechanism that delivers the same interim expected utilities for all agents and the same ex ante expected social surplus. The short proof is based on an extension of an elegant result due to Gutmann et al. (Annals of Probability, 1991). We also show that the equivalence between Bayesian and dominant strategy implementation generally breaks down when the main assumptions underlying the social choice model are relaxed, or when the equivalence concept is strengthened to apply to interim expected allocations.Bayesian Implementation, Dominant Strategy Implementation, Equivalence

Powers of paths in tournaments

In this short note we prove that every tournament contains the $k$-th power of a directed path of linear length. This improves upon recent results of Yuster and of Gir\~ao. We also give a complete solution for this problem when $k=2$, showing that there is always a square of a directed path of length $\lceil 2n/3 \rceil-1$, which is best possible.Comment: 6 pages; updated affiliations; accepted at CP