On fixed-path rationing methods

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(1)CAHIER 24-2001. ON FIXED-PATH RATIONING METHODS Lars EHLERS.

(2) CAHIER 24-2001. ON FIXED-PATH RATIONING METHODS Lars EHLERS1. 1. Centre de recherche et développement en économique (C.R.D.E.) and Département de sciences économiques, Université de Montréal. October 2001. _______________________ The author wishes to thank William Thomson and especially an associate editor for their useful comments and suggestions..

(3) RÉSUMÉ Moulin (1999) caractérise les méthodes de rationnement suivant un sentier fixe par l’efficacité, la non-manipulation, l’homogénéité et la monotonicité de ressources. Nous donnons ici une preuve simple de son résultat.. Mots clés : méthodes de rationnement suivant un sentier fixe, préférences unimodales. ABSTRACT. Moulin (1999) characterizes the fixed-path rationing methods by efficiency, strategyproofness, consistency, and resource-monotonicity. In this note, we give a straightforward proof of his result.. Key words : fixed-path rationing methods, single-peaked preferences.

(4) 1 Introduction We consider the problem of allocating a commodity among a group of agents with single-peaked preferences. For example, the commodity is a project requiring a certain number of hours of labor. One hour of labor is either in

(5) nitely divisible (the continuous model) or indivisible (the discrete model). Sprumont (1991) is the paper that originated a number of axiomatic studies in the continuous model over the past ten years. Recently, Moulin (1999) introduces a new class of rules, called

(6) xed-path rationing methods, and characterizes them by eciency, strategy-proofness, consistency, and resource-monotonicity. His theorem applies to both the continuous and the discrete model. The purpose of this note is to give a straightforward proof of his result. 1. 2 The Model and the Result Our formulation allows variations in the population and in the collective endowment. There is a

(7) nite set N = f1; : : : ; n g of potential agents. Let Z denote the set of all (possible) endowments. The set Z is either R or N [ f0g. When Z = R , we speak of the continuous model, and when Z = N [ f0g, we speak of the discrete model. For each agent i 2 N there is an a priori

(8) xed maximal consumption, denoted by X 2 Z nf0g. Given X 2 Z nf0g, agent i's consumption set is [0; X ] \ Z . From now on, given b 2 Z , we write [0; b] instead of [0; b] \ Z . Thus, [0; X ] denotes agent i's consumption set. The vector (X ) 2N of maximal consumption is

(9) xed throughout. P Given N N , let X 2 X . Each agent i 2 N is equipped with a preference relation R over [0; X ]. Let P denote the strict preference relation associated with R . The preference relation R is single-peaked if there is a number p(R ) 2 [0; X ], called the peak of R , such that for all x ; y 2 [0; X ], if y < x p(R ) or p(R ) x < y , then x P y . Let R denote the set of all single-peaked preferences over [0; X ]. A collective endowment of a commodity has to be allocated among a

(10) nite set of agents. We allow the set of agents to be any (

(11) nite) subset N N . Let 2 Z denote 2. 0. +. i. +. i. i. i. i. N. i. i. i. N. i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. 1 In the continuous model, the

(12) rst studies of consistency and resource-monotonicity, respectively,. are Thomson (1994a,b). 2 All results remain valid when. N. is a countable, in

(13) nite set.. 1.

(14) the endowment. Given N N , a (preference) pro

(15) le R is a list (R ) 2 such that for all i 2 N , R 2 R . Let R denote the set of all pro

(16) les for N . Given R 2 R , let p(R) (p(R )) 2 . Given S N N and R 2 R , let R (R ) 2 denote the restriction of R to S . Given N N , an economy is a pair (R; ) 2 R [0; X ]. Let E R [0; X ]. The economy (R; ) 2 E is in excess demand if P 2 p(R ), P and it is in excess supply if > 2 p(R ). For all N N , an allocation for (R; ) 2 E is a vector z 2 Z such that for P all i 2 N , z 2 [0; X ], and 2 z = , i.e., we do not allow free disposal. Let Z (R; ) denote the set of all allocations for (R; ). An allocation rule, or simply a rule, associates with each economy an allocation. Formally, a rule ' is a mapping ' : [ N E ! [ N Z , such that for all N N and all (R; ) 2 E , '(R; ) 2 Z (R; ). A

(17) xed-path rationing method (Moulin, 1999) relies on two

(18) xed monotonic paths in the box 2N [0; X ]. For economies in excess demand, individual consumptions are computed along the

(19) rst path, except that an agent whose demand is below his pathconsumption receives exactly his demand. We apply a similar procedure for economies in excess supply by using the other path. We refer to Moulin (1999) for additional discussion. Formally, given N N , an N -path is a mapping g(N ) : [0; X ] ! Z such that i. i. i. i. i. i. N. N. N. N. N. S. i. i. S. N. N. N. N. i. i. N. i. N. i. N. N. i. N. N. i. i. i. N. i. N. N. N. N. N. i. P. 3. N. N. (a) for all 2 [0; X ], 2 g (N; ) = , and for all i 2 N , g (N; ) X ; and (b) for all ; ~ 2 [0; X ] such that ~, for all i 2 N , g (N; ) g (N; ~). N. i. i. N. i. N. i. i. i. In the above de

(20) nition, (a) is feasibility of an N -path and (b) is monotonicity of an N -path. For an N -path g (N ), let (g (N )) denote the range of g (N ), i.e., (g (N )) fg(N; ) j 2 [0; X ]g. A full path g speci

(21) es for each set N N an N -path g(N ) (i.e. g (g(N )) N ) such that (c) for all N N~ N , proj [ (g(N~ ))] = (g(N )). N. N. 4. N. Condition (c) says that the projection of the range of the N~ -path g(N~ ) on Z is the range of the N -path g(N ). Let G denote the family of all full paths. N. 3 Abusing notation, for 2 [0; XN ] we write g(N; ) instead of g(N )(). 4 Here, for a set B Z N~ , we denote by projN [B ] the projection of B on Z N .. 2.

(22) Fixed-Path Rationing Method,. (. g. (g + ;g ; ). + ;g; ). : Given two paths. g +; g ;. 2 G , the N and.

(23) xed-path rationing method is de

(24) ned as follows. For all N P all (R; ) 2 E , (i) when 2 p(R ), there exists 2 Z such that for all + ; P i 2 N, (R; ) minfp(R ); g (N; )g, and 2 minfp(R ); g (N; )g = ; P and (ii) when 2 p(R ), there exists 2 Z such that for all i 2 N , + ; P (R; ) maxfp(R ); g;(N; )g, and 2 maxfp(R ); g;(N; )g = . N. j. (g. ;g. ). i. i. j. (g. ;g. j. N. +. i. i. i. N. +. i. j. N. ). i. i. i. i. N. 5. i. i. Moulin (1999) characterized the class of

(25) xed-path rationing methods by the following four axioms. First, a rule only selects ecient allocations. Second, no agent can gain by misrepresenting his preference relation. Third, when some agents leave with their allotments, then the rule allocates the remaining amount to the agents who did not leave in the same way as before. Fourth, the amount assigned to each agent weakly increases whenever the collective endowment increases. 6. 7. Eciency: For all N N and all (R; ) 2 E , if P 2 p(R ), then '(R; ) P 2 p(R ), then '(R; ) p(R). p(R), and if N. i. i. i. N. i. N. Strategy-Proofness: For all N N , all i 2 N , and all (R; ); (R0; ) 2 E such N. that R nf g = R0 nf g , ' (R; )R ' (R0; ). N. i. N. i. i. Consistency: For all ' (R 0 ; i. N. P2 j. N. i. N0. . i. N. 0 'j (R; )) = 'i (R; ).. N , all (R; ) 2 E , and all N. i. 2. N 0,. Resource-Monotonicity: For all N N and all (R; ); (R; 0) 2 E , if 0 , then '(R; ) '(R; 0). N. Theorem 2.1 (Moulin, 1999) A rule satis

(26) es eciency, strategy-proofness, con-. sistency, and resource-monotonicity if and only if it is a

(27) xed-path rationing method.. 3 Proof of Suciency Throughout let ' be a rule satisfying the properties of Theorem 2.1.. P. 5 Note that in (i) and (ii) is unique if i2N p(Ri ) 6= . 6 Sprumont (1991) pointed out that eciency is equivalent to same-sidedness. Below we use. same-sidedness in de

(28) ning eciency. 7 When it is unambiguous, we sometimes use. . 3. and. . to denote the vector partial ordering..

(29) Lemma 3.1 ' satis

(30) es peaks-onliness, i.e., for all N N and all (R; ); (R0; ) 2 E , if p(R) = p(R0 ), then '(R; ) = '(R0; ). N. Proof. Let N N , i 2 N , and (R; ); (R0; ) 2 E be such that p(R ) = p(R0 ) and N. i. R nf g = R0 nf g .. i. By repeating the argument for pro

(31) les that di er only in one agent's preference, it suces to show that '(R; ) = '(R0 ; ). By eciency and strategyP P proofness, ' (R; ) = ' (R0; ). Thus, 2 nf g ' (R; ) = 2 nf g ' (R0; ) and R nf g = R0 nf g . Hence, by jN j 2 f1; 2g or consistency, '(R; ) = '(R0 ; ). N. i. N. i. i. N. i. N. i. j. N. j. i. j. N. j. i. i. Let R 2 RN be such that for all i 2 N , p(R ) = X . For all N N and all 2 [0; X ], let g (N; ) '(R ; ). Let g (g (N )) N . Let R 2 RN be such that for all i 2 N , p(R ) = 0. For all N N and all 2 [0; X ], let g ;(N; ) '(R ; ). Let g ; (g ;(N )) N . The following lemma applies to any two-agent population. X. X. i. i. +. N. +. X. +. N. N. 0. 0. i. 0. N. Lemma 3.2 g ;(f1; 2g).. N. N. is a

(32) xed-path method for f1; 2g with f1; 2g-paths. '. g +(f1; 2g). and. Proof. We only prove the lemma for the case of excess demand. The case of excess. supply is symmetric. First, we show that g (f1; 2g) is a f1; 2g-path. Feasibility follows from the de

(33) nition of ', and monotonicity from resource-monotonicity of '. Thus, g (f1; 2g) satis

(34) es (a) and (b). Finally, we show for all (R; ) 2 E f g such that p(R ) + p(R ), there exists 2 [0; Xf g ] such that +. +. 1;2. 1. 2. 1;2. '(R; ) = (minfp(R1 ); g1+(f1; 2g; )g; minfp(R2 ); g2+(f1; 2g; )g):. (1). If p(R) '(Rf g ; ), then by strategy-proofness, '(R; ) = '(Rf g; ) and (1) holds for = . Without loss of generality, suppose that p(R ) < ' (Rf g ; ). Then by eciency, ' ((R ; R ); ) p(R ). If ' ((R ; R ); ) < p(R ), then let R0 2 R be such that p(R0 ) = p(R ) and ' (Rf g ; )P 0 ' ((R ; R ); ). Since by peaks-onliness, ' ((R0 ; R ); ) = ' ((R ; R ); ), the previous relation contradicts strategy-proofness. Thus, ' ((R ; R ); ) = p(R ) and ' ((R ; R ); ) p(R ). Hence, by strategy-proofness, '(R; ) = '((R ; R ); ). Monotonicity of g (f1; 2g) X. X. 1;2. 1;2. 1. 1. 1. X. 1. 1. 1. 2. 1. 1. 1. 1. X. 1. 1. 2. 1. 1. 1. X. 1;2. 1. X. 1. 1;2. 1. 2. 1. X. X. 1. 1. 2. 2. 1. 2. X. 1. 2. X. 1. 2. 1. 4. X. 2. X. 2. +.

(35) implies in the continuous model that g (f1; 2g) is continuous with respect to . Now in the continuous model (by the previous fact, g (f1; 2g; 0) = 0, g (f1; 2g; Xf g) = X , and the intermediate value theorem) and in the discrete model (by monotonicity of g (f1; 2g)), there exists 0 2 [0; Xf g] such that g (f1; 2g; 0) = ' (R; ). By monotonicity of g (f1; 2g) and ' (R; ) ' (R ; ), we have 0 and g (f1; 2g; 0) g (f1; 2g; ) > p(R ). Hence, (1) holds for = 0 . + 2. 8. + 2. + 2. 1;2. 2. +. + 2. 1 ;2. +. + 1. 2. + 1. 2. 2. X. 1. Lemma 3.3 g is a full path. Proof. It is easy to check that for all N N , g (N ) satis

(36) es (a) and (b). Let N N~ N . By consistency of ' and the de

(37) nition of g , proj [ (g (N~ ))] (g (N )). Let 0 2 [0; X ]. Monotonicity of g (N~ ) implies in the continuous model +. +. +. +. N. +. that. P2. +. N. g +(N~ ). is continuous with respect to . Similarly to Lemma 3.2, then P 2 g (N;~ X ) = X , and monotonicity of g (N~ ) imply that g (N~ ; 0) = 0, P ~ 00) = 0. Thus, by consistency of ' there exists 00 2 [0; X ] such that 2 g (N; ~ 00) = ' (R ; 00) = ' (R ; 0) = and the de

(38) nition of g , we have for all i 2 N g (N; g (N; 0 ). Thus, g (N; 0 ) 2 proj [ (g (N~ ))] and proj [ (g (N~ ))] (g (N )). Hence, g satis

(39) es (c) and g is a full path. . P2 i. i. N. i. +. N. +. i. i. ~ N. i. N. +. N. +. ~ N. i. i. N. +. +. +. +. X. ~ N. i. i. +. +. N. +. N. i. X. i. +. N. +. Similarly it can be shown that g; is a full path.. Lemma 3.4 ' = . (g. + ;g; ). .. Proof. We only prove the lemma for the case of excess demand. Suppose that P there exist N N and (R; ) 2 E such that 2 p(R ) and '(R; ) 6= + ; (R; ). Then there exist i; j 2 N such that N. i. (g. ;g. ' (R; ) < ( i. g. + ;g; ). (R; ) and ' (R; ) > j. i. i. (g. ;g. ). (g. + ;g; ). i;j. i. ;g. (R; ):. (R; ) and ' (Rf g; 00) = . ). ;g. + ;g; ). j. + ;g; ). j. (g. . + ;g; ). (g. (R; ) + ; (R; ) and. Let 0 ' (R; ) + ' (R; ) and 00 + ; sistency of + ; , (Rf g; 00) = + ; (R; ). Thus, by Lemma 3.2, (g. i. N. ). i. (g. . i. + ;g. (g j. ). . (g j. (2). (R; ). By con(Rf g; 00) =. + ;g; ). i;j. ). j. ' (Rf g ; 00 ) = ( i. i;j. i. g. + ;g; ). (g. j. i;j. j. (R; ):. (3). 8 Thomson (1994b, Proof of Theorem 2, Part (i)) formally shows that if ' is ecient and resourcemonotonic, then for all N N and all R 2 RN , '(R; ) is continuous with respect to (and therefore g2+ (f1; 2g) is continuous with respect to ).. 5.

(40) By consistency of ', ' (Rf g; 0) = ' (R; ) and ' (Rf g; 0 ) = ' (R; ). Now the previous fact combined with (2) and (3) contradicts resource-monotonicity of '. i. i;j. i. j. i;j. j. References Moulin, H.: Rationing a Commodity along Fixed Paths, J. Econ. Theory 84 (1999), 41{72. Sprumont, Y.: The Division Problem with Single-Peaked Preferences: A Characterization of the Uniform Allocation Rule, Econometrica 59 (1991), 509{519. Thomson, W.: Consistent Solutions to the Problem of Fair Division when Preferences are Single-peaked, J. Econ. Theory 63 (1994a), 219{245. Thomson, W.: Resource-Monotonic Solutions to the Problem of Fair Division when Preferences are Single-Peaked, Soc. Choice Welfare 11 (1994b), 205{223.. 6.

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