PREFERENCE DOMAINS AND THE MONOTONICITY OF CONDORCET EXTENSIONS

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1 PREFERECE DOMAIS AD THE MOOTOICITY OF CODORCET EXTESIOS PAUL J. HEALY AD MICHAEL PERESS ABSTRACT. An alternatve s a Condorcet wnner f t beats all other alternatves n a parwse majorty vote. A socal choce correspondence s a Condorcet extenson f t selects the Condorcet wnners and nothng else whenever a Condorcet wnner exsts. It s well known that Condorcet extensons are not monotonc (hence, not ash mplementable) when all preferences are admssble, but are mplementable when restrcted to a doman n whch Condorcet wnners always exst. We fll the gap by studyng the ntermedate domans, and fnd that monotoncty s volated on all such domans. Keywords: Condorcet wnner; ash mplementaton; votng rules. JEL Classfcaton: D02, D71, D ITRODUCTIO An alternatve s a Condorcet wnner f does not lose to any alternatve n a parwse majorty vote. A socal choce correspondence (SCC) s a Condorcet extenson f t selects exactly the set of Condorcet wnners, whenever a Condorcet wnner exsts. o restrctons are made when no Condorcet wnner exsts, except that the socal choce correspondence must pck somethng at every preference profle. We say that we can (ash) mplement a Condorcet extenson f there s a game form (or, mechansm) whose ash equlbrum outcomes always exactly concde wth the SCC. In ths paper we study the preference domans on whch we can mplement Condorcet extensons. It s well known that Condorcet extensons are not ash mplementable when all strct or all weak preferences are admssble (Jackson, 2001; Sajo, 1987, e.g.). Maskn (1999) shows that Condorcet extensons are ash mplementable when only The authors thank Yaron Azrel for hs helpful comments. Dept. of Economcs, The Oho State Unversty, 1945 orth Hgh street, Columbus, Oho 43210, U.S.A.; healy.52@osu.edu. Dept. of Poltcal Scence, Stony Brook Unversty, Stony Brook,.Y , U.S.A.; mchael.peress@stonybrook.edu. 1

2 2 HEALY AD PERESS preferences that admt Condorcet wnners are admssble. 1 Our paper explores the ntermedate cases, where more preferences may be admssble than just those that admt Condorcet wnners, but t s not assumed that all profles are admssble. Our fndng s that f the doman of admssble preference profles s any strct superset of those profles that admt Condorcet wnners, then no Condorcet extenson on that doman s Maskn monotonc. Therefore, t wll be nether strategy-proof nor ash mplementable by any mechansm. 2. OTATIO & EVIROMET A set of = {1,..., n} of n 3 agents are to select an outcome from a fnte set of alternatves X, where X (the number of elements of X) s at least two. Denote by X the set of non-empty subsets of X. Each agent has complete, reflexve, and transtve preferences R X X, where xr y denotes that x s weakly preferred to y. Let P denote the asymmetrc part of R ( strct preferences ). We denote the profle of all agents preferences by R = (R 1,...,R n ). Let R be the space of all possble preferences over X and P be the space of all strct preferences over X. Gven a set of admssble preference profles D R n, a socal choce correspondence (SCC) s a mappng f : D X that selects a set of alternatves for each profle R. ote that, by defnton, f (R ) for all R D. If f s sngle valued, we refer to t as a socal choce functon (SCF). For any subset A D, let f A be the restrcton of f to profles n A. Defne (x, y;r ) = { : xp y} to be the number of agents who strctly prefer x over y at profle R. An alternatve x s sad to be a weak Condorcet wnner at R f, for every y X \{x}, (x, y;r ) (y, x;r ). In other words, a weak Condorcet wnner x does not lose a parwse pluralty vote aganst any other alternatve, assumng ndfferent voters abstan. There are many preference profles for whch no weak Condorcet wnner exsts. Let W = {R R n : ( x X) ( y X), (x, y;r ) (y, x;r )} be the set of preference profles that admt a weak Condorcet wnner. Defne f W : W X to be the SCC that selects all weak Condorcet wnners for any R W. A SCC f s sad to be a weak Condorcet extenson f f (R )= f W (R ) whenever R W, and weak Condorcet consstent f f (R ) f W (R ) whenever R W. o restrctons are placed on weak Condorcet extensons or weak Condorcet consstent SCCs outsde of W. There can be multple weak Condorcet wnners at a gven preference profle. For example, f all agents are ndfferent over all alternatves, then all alternatves are weak 1 The doman restrcton n Maskn s paper s not dscussed explctly, but clear from the proof.

3 MOOTOICITY OF CODORCET EXTESIOS 3 Condorcet wnners. Defne x X to be a strong Condorcet wnner at R f, for every y X \ {x}, (x, y;r ) > (y, x;r ). Strong Condorcet wnners must be unque when they exst. Let S R n be the preference profles for whch a strong Condorcet wnner exsts, and f S : S X to be the SCF that selects the strong Condorcet wnner at every R S. A SCC f s a strong Condorcet extenson (or, equvalently, strong Condorcet consstent) f f (R ) = f S (R ) whenever R S. Obvously, S W. 2 Furthermore, every weak Condorcet extenson s a weak Condorcet consstent SCC, and every weak Condorcet consstent SCC s a strong Condorcet extenson, but the opposte relatons do not hold. To defne monotoncty, we frst say that an alternatve x X mantans poston from R to R f, for every, xr y mples xr y. In other words, x mantans poston f, for every, everythng x was beatng under R contnues to be beaten by x under R. A SCC f s monotonc f, whenever x f (R ) and x mantans poston from R to R, then x f (R ). A mechansmγ=(s, g) conssts of a strategy space S= n =1 S and an outcome functon g : S X. A strategy profle s s a (pure strategy) ash equlbrum ofγat R f, for every and s S, g(s )R g(s, s ). For any R, let µ Γ (R ) dentfy the set of pure-strategy ash equlbra ofγat R. The mechansmγash mplements a SCC f f, for every R, g(µ Γ (R ))= f (R ). In that case we say that f s ash mplementable. The followng theorem, due to Maskn (1999), shows that monotoncty s an mportant necessary condton for a SCC to be ash mplementable. Theorem (Maskn, 1999). If a SCC f : D X s not monotonc, then t s not ash mplementable. Maskn (1999) also proves that, wth at least three agents, monotoncty s suffcent for ash mplementaton when the o Veto Power axom s added. Formally, f satsfes o Veto Power f { :xr y y X} n 1 mples x f (R ). Theorem (Maskn, 1999). If n 3 and f : D X satsfes monotoncty and o Veto Power then f s ash mplementable. We state and prove our result n the followng secton. 2 The operator s strct, meanng S W.

4 4 HEALY AD PERESS Theorem. If D S W, then 3. THE MAI RESULT (1) all strong Condorcet extensons are monotonc and ash mplementable, (2) all weak Condorcet consstent SCCs are monotonc and ash mplementable, and (3) all weak Condorcet extensons are monotonc and ash mplementable. If D W, then (4) strong Condorcet extensons may or may not be monotonc, (5) weak Condorcet consstent SCCs may or may not be monotonc, and (6) all weak Condorcet extensons are monotonc and ash mplementable. If S W D, then (7) no strong Condorcet extenson s monotonc, (8) no weak Condorcet consstent SCC s monotonc, and (9) no weak Condorcet extenson s monotonc. Maskn (1999) essentally proves results 1 3 and 6. The novelty of our theorem s n the mpossblty of ash mplementaton when W D (results 7 9). The argument s smple: Followng result 9, suppose f s a weak Condorcet extenson. Take any R W, where no weak Condorcet wnner exsts (e.g., panel A of Table I). For every x f (R ), there s some y that strctly beats x n a parwse majorty vote at R. ow consder the profle R {x,y} that s dentcal to R, except x and y are floated to the top of each agent s preference rankng (panel B of Table I). Here, y s the unque (strong) Condorcet wnner. Ths means R {x,y} S D, so f s defned at R{x,y}. In fact, we know f (R{x,y})={y} snce f s a weak Condorcet extenson. ow, can f be monotonc? By constructon, x (whch s n f (R )) mantans poston from R to R {x,y}, so monotoncty would requre x f (R{x,y} ). But we ve just shown that f (R {x,y} )={y}, so f cannot be monotonc. A smlar example proves results 7 and 8. Arguments such as ths are common n past work; see Amorós (2009), for example. Detals are provded n the followng proof. Proof of the Theorem. For results 1 3, consder f f S D, whch s the unque strong Condorcet extenson, the unque weak Condorcet consstent SCC, and the unque weak Condorcet extenson on D. Pck any R,R D S. If the element x such that f (R )= {x} mantans poston from R to R, then (x, y;r ) (x, y;r ) (y, x;r ) (y, x;r ) for all y. Thus, f (R ) = {x}, as requred by monotoncty. Ths SCC also

5 MOOTOICITY OF CODORCET EXTESIOS 5 R 1 R 2 R 3 z x y y z x x y z R {x,y} 1 R {x,y} 2 R {x,y} 3 y x y x y x z z z (A) (B) TABLE I. Example preferences llustratng the proof. satsfes o Veto Power snce n 1 agents rankng x as top-ranked at R guarantees that x s the strong Condorcet wnner at R, and therefore f (R )={x}. Thus, f s ash mplementable. For results 4 and 5, consder an example n whch X = {x, y} and D = S {R,R }, where R W s any profle such that (x, y;r )= (y, x;r ) and R W s the profle where all agents are ndfferent between x and y. ote that f W (R )= f W (R )={x, y}. Suppose f S f S, f (R )={x}, and f (R )={y}. Then f s a strong Condorcet extenson and a weak Condorcet consstent SCC, but s not a weak Condorcet extenson. It s also not monotonc: x s chosen at R and x mantans poston from R to R, but x f (R ). Thus, monotoncty fals for some strong Condorcet extensons and some weak Condorcet consstent SCCs. To show there exsts monotonc strong Condorcet extensons and weak Condorcet consstent SCCs, let f f W D. Here, f x f (R ) for some R D W then x s a weak Condorcet wnner at R. If x mantans poston from R to R D, then (x, y;r ) (x, y;r ) (y, x;r ) (y, x;r ) for all y. Thus, x s a weak Condorcet wnner at R and so x f (R ), as requred by monotoncty. All weak Condorcet extensons on such a doman must satsfy f f W D, so ths also proves result 6. To show results 7 9, recall that assume W D. Pck any R D \ W and any x f (R ). Snce there s no weak Condorcet wnner at R, then there s some y X such that (y, x;r ) > (x, y;r ). ow consder the preference relaton R {x,y} where, for each, (1) xr y xr {x,y} (2) yr x yr {x,y} (3) wr z wr {x,y} yp {x,y} z for every z {x, y}, z for every z {x, y}, and xp {x,y} z for all w, z {x, y}. In other words, R {x,y} s dentcal to R except the par {x, y} s bubbled up to the top of R. ote that (y, x;r {x,y} ) > (x, y;r{x,y} ) snce (y, x;r ) > (x, y;r ), and also n = (y, z;r ) > (z, y;r ) = 0 for all z {x, y}. Thus, the unque weak Condorcet wnner at R {x,y} s y, whch also verfes that R {x,y} S D. If f s a weak Condorcet

6 6 HEALY AD PERESS extenson, weak Condorcet consstent, or a strong Condorcet extenson, t must be that f (R {x,y} )={y}. ow we ask whether f can be monotonc. By constructon, x f (R ) mantans poston from R to R {x,y}, so monotoncty would requre that x f (R{x,y}). But we have just derved that f (R {x,y} )={y}, a contradcton, so f cannot be monotonc. Remark 1. If D does not contan all of W but does contan some profles outsde of W then the mpossblty result remans true as proven as long as, for every R D \W, there s some R that plays the same role as R{x,y} n the proof. Remark 2. We assume n 3. If n = 2 then every preference profle admts a weak Condorcet wnner (W = R 2 ), but there are weak Condorcet extensons that are not ash mplementable when D s large enough. Examples can be constructed easly usng the necessary condtons from Dutta and Sen (1991). 4. RELATED WORK As stated n the ntroducton, the mpossblty result under the specal cases of D = P n and D = R n s well-studed. Assumng X 3 and D P n, (all strct preferences are admssble), Muller and Satterthwate (1977) prove that f f s monotonc and satsfes ctzens soveregnty (the range of f equals X) then f must be dctatoral. Snce a weak Condorcet extenson on P n satsfes ctzen soveregnty and s not dctatoral, ths proves that t cannot be monotonc. Amorós (2009) defnes the unequvocal majorty of a socal choce functon to be the mnmal number of agents such that f that many agents rank an alternatve x at the top of the preferences, then the socal choce functon pcks x. For weak Condorcet extensons the unequvocal majorty s n/2. Assumng all strct preferences are admssble (D = P n ), Amorós shows that f the unequvocal majorty of a socal choce correspondence s less than n (n 1)/m (where m s the number of alternatves) then t cannot be Maskn monotonc. Snce n/2 + 1 < n (n 1)/m, ths also shows that weak Condorcet extensons are not monotonc when all strct preferences are admssble. Ozkal-Sanver and Sanver (2007) also show the mpossblty of mplementng weak Condorcet extensons when all preferences are admssble. Ther approach dffers n that they take the (non-transtve) majorty relaton as prmtve, but the result s equvalent.

7 MOOTOICITY OF CODORCET EXTESIOS 7 All of these results requre that all strct preferences be admssble, and are therefore more restrctve (more lkely to generate an mpossblty result) than any of the cases covered n our theorem. If the complete ndfference profle s admssble, Sajo (1987) shows that Maskn monotoncty of a (sngle-valued) SCF mples that the SCF s constant. Thus, f D contans the ndfference profle and two other profles n W that have non-overlappng sets of weak Condorcet wnners, then any weak Condorcet consstent SCF defned on D would be non-constant and therefore not monotonc. Such a doman could be a subset of W, n whch case Sajo s result s a specal case of (5) above. Or t could be that D s nether a superset nor a subset of W, n whch case Sajo s mpossblty result s not covered by our theorem. Postve results on mplementaton can be obtaned f a stronger equlbrum concept s used (see Palfrey and Srvastava, 1991; Peress, 2008; and Bag et al., 2009, for example). If preferences are sngle-peaked (Mouln, 1980, e.g.) then mplementaton s also no problem, because D S. REFERECES Amorós, P., Unequvocal majorty and maskn monotoncty. Socal Choce and Welfare 33, Bag, P. K., Sabouran, H., Wnter, E., Mult-stage votng, sequental elmnaton and condorcet consstency. Journal of Economc Theory 124, Dutta, B., Sen, A., A necessary and suffcent condton for two-person nash mplementaton. Revew of Economc Studes 58, Jackson, M. O., A crash course n mplementaton theory. Socal Choce and Welfare 18, Maskn, E., ash equlbrum and welfare optmalty. Revew of Economc Studes 66, Mouln, H., On strategy-proofness and sngle-peakedness. Publc Choce 35, Muller, E., Satterthwate, M. A., The equvalence of strong postve assocaton and strategy-proofness. Journal of Economc Theory 14, Ozkal-Sanver, I., Sanver, M. R., February ash mplementablty of tournament solutons, Istanbul Blg Unversty Workng Paper. Palfrey, T. R., Srvastava, S., ash mplementaton usng undomnated strateges. Econometrca 59,

8 8 HEALY AD PERESS Peress, M., Selectng the condorcet wnner: Sngle-stage versus mult-stage votng rules. Publc Choce 137, Sajo, T., On constant maskn monotonc socal choce functons. Journal of Economc Theory 42 (2),