Journal of Economic Theory 84, 4172 (1999) Rationing a Commodity along Fixed Paths* Herve Moulin Department of Economics, Duke University, Box 90097, Durham, North Carolina 27708 moulinecon.duke.edu Received January 21, 1998; revised July 11, 1998 A commodity is divided among agents with single-peaked preferences. The commodity is either infinitely divisible or comes in indivisible units. A rationing method elicits individual peaks (demands); if the commodity is overdemanded (resp. underdemanded), no agent receives more (resp. less) than his peak. A fixed-path rationing method allocates an overdemanded ``good'' along a path independent of individual demands, except that an agent receives exactly his demand if it is below the path-generated share. An underdemanded ``bad'' is allocated similarly along another path. The four properties efficiency, strategyproofness, resource monoto- nicity, and consistency characterize the set of fixed path rationing methods. Journal of Economic Literature Classification Numbers: D63, D70.  1999 Academic Press Key Words: rationing method; fair division; single peaked preferences; strategy proofness; consistency. 1. INTRODUCTION We consider the problem of rationing a single commodity when a given set of agents formulate a ``demand,'' i.e., request a certain amount of the commodity, but the actual quantity of commodity that must be shared among these agents does not match the sum of individual demands. Say that agent i request the amount x i and that t units must be allocated to these agents. If t < i x i , the commodity is overdemanded (we speak of a ``good''), and each agent receives a share y i such that y i x i and  i y i =t; if  i x i <t, the commodity is underdemanded (we speak of a ``bad'') and agent i receives y i such that x i y i and  i y i =t. Under the slightly esoteric name of ``fair division under single-peaked preferences,'' the above problem has received a great deal of attention by the recent mechanism design literature. Examples of the commodity being allocated include a task requiring a certain number of hours of labor (say, Article ID jeth.1998.2468, available online at http:www.idealibrary.com on 41 0022-053199 30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved. * Critical comments by seminar participants in the Universities of Michigan, Oregon, Paris- Dauphine, and Barcelona are gratefully acknowledged. Special thanks go to two very careful referees of this journal.