We prove a general possibility result for collective decision problems where individual allocations are one-dimensional, preferences are single-peaked (strictly convex), and feasible allocation pro les cover a closed convex set. Special cases include the celebrated median voter theorem (, ) and the division of a non disposable commodity by the uniform rationing rule (). We construct a canonical peak-only rule equalizing in the leximin sense individual gains from an arbitrary benchmark allocation: it is ef cient, group-strategyproof, fair, and (for most problems) continuous. These properties leave room for many other rules, except for symmetric non disposable division problems.
The combinatorial clock auction (CCA) has frequently been used in recent spectrum auctions. It combines a dynamic clock phase and a one‐off supplementary round. The winning allocation and the corresponding prices are determined by the Vickrey–Clarke–Groves rules. These rules should encourage truthful bidding, whereas the clock phase is intended to reveal information. We inquire into the role of the clock when bidders have lexicographic preferences for raising rivals' costs. We show that in an efficient equilibrium, the clock cannot fully reveal bidders' types. In the spirit of the ratchet effect, in the supplementary round competitors extract surplus from strong bidders whose type is revealed. We also show that if there is substantial room for information revelation, that is, if the uncertainty about the final allocation is large, all equilibria of the CCA are inefficient. Qualitative features of our equilibria are in line with evidence concerning bidding behavior in some recent CCAs.
Predictions under common knowledge of payoffs may differ from those under arbitrarily, but finitely, many orders of mutual knowledge; Rubinstein's (1989) Email game is a seminal example. Weinstein and Yildiz (2007) showed that the discontinuity in the example generalizes: for all types with multiple rationalizable (ICR) actions, there exist similar types with unique rationalizable action. This paper studies how a wide class of departures from common belief in rationality impact Weinstein and Yildiz's discontinuity. We weaken ICR to ICR-lambda, where lambda is a sequence whose n-th term is the probability players attach to (n-1)-th-order belief in rationality. We find that Weinstein and Yildiz's discontinuity remains when lambda_n is above an appropriate threshold for all n, but fails when lambda converges to 0. That is, if players' confidence in mutual rationality persists at high orders, the discontinuity persists, but if confidence vanishes at high orders, the discontinuity vanishes.