Unanimity, Pareto optimality and strategy-proofness on connected domains

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1 Unanimity, Pareto optimality and strategy-proofness on connected domains Souvik Roy and Ton Storcken y February 2016 Abstract This is a preliminary version please do not quote 1 Extended Abstract This paper contributes to the restricted domain literature in Social Choice. In reaction to the seminal impossibility results of Arrow[1978], Gibbard[1973] and Satterthwaite[1975] restrictions of the set of admissible preferences are investigated on whether these form so called possibility or impossibility domains. Where a possibility domain refers to a set of admissible preferences allowing for non-dictatorial collective choice rules which satisfy the conditions such as imposed by Arrow, Gibbard or Satterthwaite. On an impossibility domain, however, such rules are necessarily dictatorial and therefore strengthen (at least logically) these impossibility results. This literature is elaborate and goes at least back to Black[1948] on single peaked preference domains. Descriptions of possibility domains often su er from transparency, see e.g. Sen[1969] or Kalai[1977,1980]. Also it appears that whether a domain is a possibility or impossibility domain is rather a interplay of the admissible preferences than the number of these. For instance it has been found that for any number of alternative larger than or equal to three there exists an impossibility domain containing six preferences, see e.g. Sanver[2007] or Storcken[1984], whereas a possibility domain based on an inseparable top-pair may be as large as m! (m 1)! + (m 2)!; where m is the number of alternatives. However, in the frame work of collective choice rules the idea of top-connectedness seems fruitful. Two alternatives are directly top-connected if there are preferences at which they switch best and second best position. At these preferences the tail need not be the same. The idea of direct top-connectedness is at least practical in formal studies on restricted domains. It can for instance be visualized by a graph which connects those alternatives that are directly topconnected. In Aswal[2003] it is shown that if su cient directly top-connections Economic Research Unit, Indian Statistical Institute, Kolkata y School of Business and Economics, Maastricht University 1

2 are present then the domain is an impossibility domain. Here we elaborate on this approach. In that we try to characterize domains that posses certain types of top-connections. The notion of an inseparable top pair is in all the cases we study the characterizing property. For preference rules the notion of inseparable pair can at least be traced back to Kalai[1980]. A pair of alternatives (a; b) is said to be inseparable if in all preferences where a is ordered above b no other alternative is ordered strictly between these two. Of course to have a meaning the existence of such preferences in which a is ordered above b is assumed. In this set up we study collective choice rules where upper parts of preferences have a major in uence on the outcomes. Therefore this inseparability is only needed as long as a is a best alternative at some order. Inseparability of a pair therefore translates into (a; b) being an inseparable top-pair if at all orders where a is best b is second best, see also e.g. Aswal[2003] and Bochet[2005]. In order to deduce the results listed hereafter we needed some richness conditions: top-richness and tail-richness. Top-richness means that if for some admissible preference a is at the top and b comes at at second place, then there are admissible preferences where b is top and a is second best. Tail-richness means that given that a is best and b is second best at some admissible preference, then for x and y disjoint from a and b there are such preferences where a is best b is second best and x is preferred to y: It means that preference on any pair disjoint from a and b is not constant over the admissible preferences with a best and b second best. Further we some times need the explicit presence of a Condorcet cycle. Corollary 1 states that top-rich and top-connected domains with a Condorcet cycle are possibility domains if and only if there is an inseparable toppair. Here being a possibility domain means the existence of Pareto optimal, strategy-proof and non-dictatorial choice rule. Corollary 2 states that tail-rich and top-connected domains are possibility domains if and only if there is an inseparable top-pair. In this case a possibility domain means the existence of a unanimous, strategy-proof and non-dictatorial choice rule. Theorem 3 is on domains which are not top-connected. It states that top-rich and tail-rich domains are possibility domains if and only if there is an inseparable top-pair. Here possibility domain has the same interpretation as in the latter mentioned Corollary 2. We also show that some equivalences do not hold with fewer assumptions. To nalize the paper we need to nd some more examples which show that all combinations of fewer assumptions a ect these equivalences. 2 Introduction To be written 2

3 3 Top-connected sets of preferences Let A and N denote a non-empty and nite set of alternatives and agents respectively. All agents have the same domain D of individual preferences. This domain D is a subset of the set of linear orders L on A: We consider choice functions ' from D N to A assigning to each pro le p in D N an alternative '(p) in A: The following notions and notations on preferences are used throughout the paper. Let R 1 and R 2 be two linear orders in L let x and y be two di erent alternatives in A: To save parentheses we write ab 2 R 1 instead of (a; b) 2 R 1 ; which has the usual interpretation of a being (strictly) preferred to b at R 1 : We denote :::ab::: = R 1 if a is consecutively preferred above b at R 1 : In line with this ab::: = R 1 means that a is best and b is second best at R 1 : Further, :::a:::b::: = R 1 ; a::: = R 1 and :::a = R 1 have self explanatory interpretations. The restriction of R 1 to a subset X of A is denoted by R 1 j X = fab 2 R 1 : a; b 2 Xg: The upper contour set of R 1 at a is de ned as usual by Up(a; R 1 ) = fx 2 A : xa 2 R 1 g: Here this set will contain alternative a. Next we discuss some structural aspects of D. Let x and y be alternatives in A: Then D x = fr 2 D : x::: = Rg is the set of linear orders in D having x as the best alternative and D xy = fr 2 D : xy::: = Rg is the set of linear orders in D having x as best and y as second best alternative. Further, B = fx 2 A : D x 6= ;g is the subset of alternatives x such that x is the best alternative for at least one preference in D: We say that D x or x is directly top-connected to D y or y respectively if there are xy::: = R 1 in D x and yx::: = R 2 in D y : Notation x! y: We point out here that R 1 and R 2 may di er on Anfx; yg: We say that subset C of B is a top-connected component if for all x and y in C, with x 6= y; there is a positive integer k and there are alternatives z 1 ; z 2 ; :::; z k in C such that z 1 = x; z k = y and z i! z i+1 for all i 2 f1; 2; :::; k 1g: For di erent alternatives x, y and v in B we say that v separates x from y if there is a preference R 2 D such that a:::v:::b::: = R: Let C be a non-trivial subset of B containing at least two alternatives. Then C an inseparable set if for all di erent a and b in C there is no v 2 BnC which separates a from b: For di erent alternatives a and b in B pair ab is called inseparable if there are R in D a with ab::: = Rj B and there is no c 2 Bnfa; bg which separates a from b: So, ab is an inseparable pair if D a is non-empty and b is ordered second best at every ordering in D a : We say that D contains an inseparable pair or set if we can respectively nd such alternatives a and b or subset C. A Condorcet cycle is a triple of alternatives, say a; b and c; together with a triple of preferences, say R abc ; R bca and R cab in D, such that a:::b:::c::: = R abc ; b:::c:::a::: = R bca and c:::a:::b::: = R cab and Up(b; R abc )\ Up(c; R bca ) \ Up(a; R cab ) = ;: Remark 1 Note that three alternatives x; y and z in B form a Condorcet cycle if x! y! z as then xy:::z::: = R xyz 2 D, yz:::x::: = R yzx 2 D and z:::x:::y::: = R zxy 2 D and further Up(y; R xyz )\ Up(z; R yzx )\Up(x; R zxy ) = ;: The latter follows because Up(y; R xyz )\ Up(z; R yzx ) = fyg and y =2 Up(x; R zxy ): For the results below some richness conditions for the domains of admissible 3

4 preferences are needed. These conditions impose that su ciently variation in preference between alternatives is guaranteed. Here richness comes in two types. 1. Domain D is called tail-rich if for all di erent alternatives a, b; x and y in B, with D ab 6= ;; there are R 2 D ab such that :::x:::y::: = R. 2. Domain D is called top-rich if for all di erent alternatives a and b nonemptiness of D ab implies that D ba : Remark 2 Tail-richness imposes that whenever there are preferences where a is at the top and b is second best D a is not constant in preference between alternatives x and y disjoint from a and b: Top richness means that if there is a preference in D where a is best and b is second best then there are preferences in D where b is best and a is second best. That is if a and b are directly topconnected whenever there are preferences in D where the top two alternatives are a and b: Next we state some notions on pro les. Let p and q be some pro les in D N and j an agent in N: Let S be a subset of N: For preferences R 1 and R 2 in D the N-tuple ((R 1 ) S ; (R 2 ) NnS ) denotes the pro le p such that p(i) = R 1 for all agents i in S and p(i) = R 2 for all agents i in NnS: In case S is empty or equal to N we have of course a unanimous pro le (R 2 ) N or respectively (R 1 ) N : The restriction of pro le p to subset S of N is denoted by pj S : Notations like ((R 1 ) S ; qj NnS ) have obvious interpretations. We call such a pro le S-unanimous. Pro le ((R 1 ) S ; (R 2 ) NnS ) is said to be a (S; NnS)-unanimous pro le. We call p and q an S-deviation if there are preferences R 1 and R 2 in D such that p = ((R 1 ) S ; qj NnS ) and q = ((R 2 ) S ; qj NnS ): In case S equals a singleton, say fjg; we also write j-deviation. We end this section with a list of well-known conditions for choice functions. Let ' be a choice function from D N to A: Then ' is called Pareto optimal if for all pro les p the collective choice '(p) is not Pareto dominated, i.e. there is no alternative x di erent from '(p) which is preferred to '(p) by all agents. Choice function ' is unanimous if '(R N ) = R for all N-unanimous pro les R N : Choice function ' is called dictatorial, with dictator i; if for all pro les p the outcome '(p) is the best alternative of agent i; i.e. '(p)::: = p(i): Choice function ' is strategy-proof if for all agents i and all i-deviations p and q either '(p) = '(q) or :::'(p):::'(q)::: = p(i): Choice function ' is intermediate strategy-proof if for all subsets S of N all R in D and all pro les p and q such that p = (R S ; qj NnS ) either '(p) = '(q) or :::'(p):::'(q)::: = R: Choice function ' is Maskin monotone if '(p) = '(q) for all pro les p and q such that Up('(p); q(i)) Up('(p); p(i)): Remark 3 It is elementary that Pareto optimality implies unanimity. Further, it is well-known that on every subdomain of linear orderings strategy-proofness and intermediate strategy-proofness are equivalent and that each of these implies Maskin monotonicity. These facts will be used without further reference or remark. 4

5 Let S be a subset of N and V be a set of S-unanimous pro les in D N : Given a choice function ' we say that coalition S is decisive on V, if '(R S ; pj NnS ) = x for all (R S ; pj NnS ) 2 V where x is the best alternative at R: Further, S is decisive if it is decisive on the set of all S-unanimous pro les in D N : 4 Decisiveness under Pareto optimality In proofs of impossibility theorems a major step is on showing that decisiveness of coalitions at some pro les expands "epidemically" to all possible pro les. The unrestricted domain assumption in those studies enables this epidemic spread without great e orts. In the restricted case at hand a more subtle technique like in Aswal e.a. [2003] is needed. This shows that decisiveness of a coalition disseminates along a path of top-connected alternatives. To avoid repeated enumerations in this section assume that ' is a strategyproof and Pareto optimal choice function. First we show that if a coalition is decisive at all unanimous oppositions, then it is decisive that is also when the opposition is not unanimous. In an unrestricted situation this follows directly from Maskin monotonicity. Here some extra lines are needed. Lemma 1 Let subset S of N be decisive on all (S; NnS)-unanimous pro les. The S is decisive. Proof. In order to prove that S is decisive let p be an S-unanimous pro le such that p(i) = R for all i in S; where R 2 D a : It is su cient to prove that '(p) = a: We prove this by induction on k the number of di erent preferences in pj NnS : The basis k = 0 or k = 1 follows by Pareto optimality or because S is decisive on all (S; N ns)-unanimous pro les. Next we prove the induction step. Let T 1 ; T 2 ; :::; T k partition NnS such that all agents in T l have the same preference say R l : Let q be the pro le such that q(i) = p(i) if i =2 T 1 and q(i) = R 2 if i 2 T 1 : The induction hypothesis now yields that '(q) = a: Strategy-proofness yields that a'(p) 2 R 2 and '(p)a 2 R 1 : In case '(p) = a we are done. To the contrary assume that '(p) 6= a: Now consider pro le r such that q(i) = p(i) if i =2 T 2 and q(i) = R 1 if i 2 T 2 : The induction hypothesis yields that '(r) = a: Strategyproofness now implies that '(p)a 2 R 2 and a'(p) 2 R 1 : This is contradicting earlier ndings. Hence, '(p) = a: The following three Lemmas show how decisive spreads along a path of topconnected alternatives. The proof is based on induction of which the basis is dealt with in the following two Lemmas. Lemma 2 Let S N be decisive on f(r S ; R NnS )g; where a and b are different, ab::: = R and b::: = R: Then S is decisive on (D a ) S (D b ) NnS = f((r 1 ) S ; (R 2 ) NnS ) : R 1 2 D a and R 2 2 D b g: Proof. Consider a pro le ((R 1 ) S ; (R 2 ) NnS ) 2 (D a ) S (D b ) NnS : It is suf- cient to prove that '((R 1 ) S ; (R 2 ) NnS ) = a: Pareto optimality implies that 5

6 '(R S ; (R 2 ) NnS ) 2 fa; bg : As '(R S ; R NnS ) = a; strategy-proofness implies that '(R S ; (R 2 ) NnS ) 6= b: So, '(R S ; (R 2 ) NnS ) = a: Strategy-proofness now implies '((R 1 ) S ; (R 2 ) NnS ) = a: Lemma 3 Let S N be decisive on (D a ) S (D b ) NnS. Let a! b and b! c, where a; b and c are three di erent alternatives. Then S is decisive on (D b ) S (D c ) NnS : Proof. Let ab::: = R ab 2 D a ; ba::: = R ba 2 D b, bc::: = R bc 2 D b and cb::: = R cb 2 D c : In view of Lemma 2 it is su cient to prove that '((R bc ) S ; (R cb ) NnS ) = b: As S is decisive on (D a ) S (D b ) NnS we have '((R ab ) S ; (R bc ) NnS ) = a: Consider pro le ((R ab ) S ; (R cb ) NnS ): Pareto optimality implies '((R ab ) S ; (R cb ) NnS ) 2 fa; b; cg. Having '((R ab ) S ; (R bc ) NnS ) = a strategy-proofness implies '((R ab ) S ; (R cb ) NnS ) =2 fb; cg: So, '((R ab ) S ; (R cb ) NnS ) = a: Consider pro le ((R ba ) S ; (R cb ) NnS ): Pareto optimality implies '((R ba ) S ; (R cb ) NnS ) 2 fb; cg: By '((R ab ) S ; (R cb ) NnS ) = a strategy-proofness implies '((R ba ) S ; (R cb ) NnS ) 6= c: So, '((R ba ) S ; (R cb ) NnS ) = b. By this strategy-proofness yields '((R bc ) S ; (R cb ) NnS ) = b: The following Lemma essentially shows the induction step on this dissemination of decisiveness along a path of top-connected alternatives. Lemma 4 Let S N be decisive on (D x1 ) S (D x2 ) NnS : For k 2 let x 1 ; x 2 ; x 3 ; ::: ; x k be alternatives such that 1. #fx t 1 ; x t ; x t+1 g = 3 for all t 2 f2; 3; :::; k 1g 2. x t! x t+1 for all t 2 f1; 2; :::; k 1g: Then S is decisive on D xt D xs for all t < s and t; s 2 f1; :::; kg. Proof. Let S N be decisive on (D x1 ) S (D x2 ) NnS : We may assume that k 3 as for k = 2 the statement obviously holds. By induction on m 2 we prove S is decisive on D yt D ys for all t < s and t; s 2 f1; :::; mg for y 1 ; y 2 ; :::y m being a subsequence of consecutive elements of x 1 ; x 2 ; x 3 ; :::; x k ; i.e. there is a number l such that x l = y 1 ; x l+1 = y 2 ; :::x l+m 1 = y m. As #fx t 1 ; x t ; x t+1 g = 3 for all t 2 f2; 3; :::; k 1g the induction basis follows readily from Lemma 3. To prove the induction step let S be decisive on D yt D ys for all t < s and t; s 2 f1; :::; mg for x l = y 1 ; y 2 ; :::y m such a subsequence of x 1 ; x 2 ; x 3 ; :::; x k. Let y m+1 = x l+m+1 ; such that l + m + 1 k: It is su cient to prove that S is decisive on (D y1 ) S (D ym+1 ) NnS : Let (R S ; R NnS ) 2 (D y1 ) S (D ym+1 ) NnS ; such that y 1 y 2 ::: = R: It is su cient to prove that '(R S ; R NnS ) = y 1 : Further let y 2 y 1 ::: = e R 2 D y2 : The induction hypothesis applied to y 2 ; y 3 ; :::; y m+1 yields that '( e R S ; R NnS ) = y 2 : As y 1 y 2 ::: = R, strategy-proofness implies that '(R S ; R NnS ) 2 fy 1 ; y 2 g: If :::y 1 :::y 2 ::: = R; then '(R S ; R NnS ) 6= y 2 by Pareto optimality. If :::y 2 :::y 1 ::: = R; then '(R S ; R NnS ) 6= y 2 by strategy-proofness because by the induction hypothesis '(R S ; ( e R) NnS ) = y 1, where y 2 y 1 ::: = e R. So, '(R S ; R NnS ) = y 1 : 6

7 Remark 4 For alternatives x 1 ; x 2 ; x 3 ; ::: ; x k consider cycles such as x 1! x 2! x 3! :::! x 1 ; where the alternative appearing in a consecutive triple are di erent. Let S be decisive on x 1 x 2 ; meaning S is decisive on (D x1 ) S (D x2 ) NnS. Then Lemma 3 implies S is decisive on x 2 x 3 : The latter now implies with Lemma 4 that S is decisive on x 2 x 1 : So, S is decisive on x 1 x 2 and x 2 x 1 : Now Lemma 4 implies that S is decisive on x i x j for all i; j 2 f1; 2; :::; kg: 5 Top-connected and top-rich domains Here we study the case in which all alternatives are top-connected. It means that there is precisely one top-connected component. There are several well-known examples of such domains. On the one hand we have the non-restricted domain on which only dictatorial choice functions are Pareto optimal and strategy-proof. On the other hand we have restricted domains, for instance those consisting of all single peaked preferences with respect to a given linear order of the alternatives. On these domains there exist anonymous, Pareto optimal and strategy-proof choice functions. The unrestricted domain has no inseparable pair where a domain of single peaked preferences has several separable pairs. The following example clari es that a domain which has an inseparable pair allows for nondictatorial, Pareto optimal and strategy-proof choice functions. Although these choice functions are far from anonymous as there is one agent who is decisive on all but one pair of alternatives, in fact the inseparable pair. Example 1 Inseparable pair Let D be a domain such that ab is an inseparable pair at D, where a; b 2 B and B consists of at least three alternatives. So, there are preferences ab::: = R ab and b::: = R b in D. And if R is a preference in D, with best alternative a; then alternative b is second best at R: Now de ne ' ab for an arbitrary pro le in D N as follows ' ab (p) = x if p(1) 2 D x and x 6= a = a if p(1) 2 D a and ab 2 p(2) = b if p(1) 2 D b and ba 2 p(2): Then ' ab is obviously Pareto optimal, non-dictatorial. It is also strategy-proof. Clear agents i 3 can not manipulate as they have in uence on the outcome at any pro le. Agent 2 can determine the outcome only at those pro les where a is rank at the top in agent 1 s preference. At those pro les the choice function o ers agent 2 a choice between a and b: Therefore 2 cannot manipulate either. The choice function assigns either agents 1 best or second best alternative. If this second choice is assigned by deviating agent 1 can only alter the outcome to an alternative which is ordered below the second best, because the pair ab is inseparable. So, agent 1 cannot manipulate as well. As the foregoing example applies to top-connected domains it shows that the existence of an inseparable pair is a su cient condition allow for strategyproof, Pareto optimal and non-dictatorial choice functions on such domains. 7

8 We will now show that it is also a necessary condition. To be precise we show that if a top-rich and top-connected domain with a Condorcet cycle has no inseparable pair, then every strategy-proof and Pareto optimal choice function on that domain is dictatorial. (See also Corollary 2) So, let ' be a strategy-proof and Pareto optimal choice function on toprich and top-connected domain D. And let D have a Condorcet cycle but no inseparable pair. Note that this means that there are at least three alternatives in B: We prove that ' is dictatorial. As the domain is rich and there is no inseparable pair every alternative in B is directly top-connected to two other alternatives in B: Because B is nite we therefore can form a sequence, say x 1 ; x 2 ; x 3 ; :::; x k ; such that 1. #fx t 1(mod k) ; x t ; x t+1(mod k) g = 3 for all t 2 f1; 2; 3; :::; kg; 2. x t! x t+1(mod k) for all t 2 f1; 2; :::; kg; 3. fx 1 ; x 2 ; x 3 ; :::; x k g = B 4. x 1 = x k The following Lemma s show that in the absence of inseparable pairs topconnected rich domains are so called impossibility domains, that is on these domains only dictatorial choice functions are strategy-proof and Pareto optimal. The proof structure is standard that is we show that the family of decisive coalitions forms an ultra- lter. Finiteness of the set of agents therewith implies the existence of a decisive singleton agent coalition. This agent is then the dictator by Lemma 1. Lemma 5 Let S N: Then either S is decisive or NnS is decisive. Proof. Let R ij denote a linear order with x i x j ::: = R ij : To ease notation let j = i + 1: Because x i! x j there are R ij and R ji in D which only di er on x i and x j. Consider the pro le p ij = ((R ij ) S ; (R ji ) NnS )): Pareto optimality and strategy-proofness imply that '(p ij ) 2 fx i ; x j g. Case 1 '(p ij ) = x i : Now like in Remark 4 Lemma 4 implies that S is decisive on all (D a ) S (D b ) NnS for all a and b in B: With this Lemma 1 yields that S is decisive. Case 2 '(p ij ) = x j : Then similarly considering the reverse cycle x k ; x k 1 ; x k 3 ; :::; x 1 it follows that NnS is decisive. Lemma 6 Let S and T be decisive subsets of N: Then S \ T is decisive. Proof. As a super-set of a decisive set is decisive we may assume that S[T = N: Because domain D is rich it contains a Condorcet cycle. So, there are a; b and c in C and there are R abc ; R bca and R cab in D such that a:::b:::c::: = R abc ; b:::c:::a::: = R bca and c:::a:::b::: = R cab and Up(b; R abc )\ Up(c; R bca ) \ Up(a; R cab ) = ;. Consider the pro le p = ((R abc ) S\T ; (R bca ) SnT ; (R cab ) T ns ). As S is decisive '((R bca ) S\T ; (R bca ) SnT ; (R cab ) T ns ) = b: So, strategy-proofness implies 8

9 '(p) 2 Up(b; R abc ): As T is decisive '((R abc ) S\T ; (R bca ) SnT ; (R abc ) T ns ) = a. Hence, strategy-proofness implies that '(p) 2 Up(a; R cab ): If S \ T is not decisive, then (NnS)[(NnT ) is decisive by Lemma 5 and '((R abc ) S\T ; (R cab ) SnT ; (R cab ) T ns ) = c: By which strategy-proofness would imply that '(p) 2 Up(c; R bca ): With the above this would mean that '(p) 2 Up(b; R abc )\ Up(c; R bca )\Up(a; R cab ): As this cannot be it follows that S \ T is decisive. The previous two Lemmas 5 and 6 imply the desired result. Theorem 1 Let D be a top-connected and top-rich domain with Condorcet cycles but without an inseparable pair. Then every strategy-proof and unanimous choice function ' from D N to A is dictatorial. Theorem 1 and Remark 1 have the following immediate consequence. Corollary 1 Let D be a top-connected and top-rich domain with Condorcet cycles domain. There exist non-dictatorial, strategy-proof and Pareto optimal choice functions with domain D N if and only if D has an inseparable pair. Proof. Theorem 1 covers the only-if part. In view of Example 1 to prove the if-part it is su cient to consider the case of at most two alternatives in B: Because D is non-empty B is non-empty. By top-richness there are at least two alternatives. If B has precisely two alternatives, then indeed D has inseparable pairs. Moreover, as any Pareto optimal and monotone voting rule on these two alternatives is strategy-proof the domain is a possibility domain. 6 Top-connected and tail-rich domains Like on top-connected and top-rich domains with Condorcet cycles we show that for top-connected and tail-rich domains the inseparable pair existence is a characterizing condition for such domains to be a possibility domains. Here, however, we consider strategy-proof and unanimous choice functions. An example of such a possibility domains is a domain of all single dipped preferences with respect to some given base relation, see e.g. Inada[1964]. So, in this section assume that ' is a strategy-proof and unanimous choice function. Let D be a top-connected and tail-rich domain without an inseparable pair. Note that is implies that B contains at least three alternatives. Remark 5 Note that because D is tail-rich and has no inseparable pair we have for any three alternatives x; y; and z in B that there are orders R z 2 D z with z:::x:::y::: = R z : Indeed this holds if x is second best at some order in D z. If x is not second best at any such a z-best, then as D has no inseparable pair for some such z-best orders y is not second best and the result follows by tail-richness. In particular by Remark 1 this means that if x! y! z then the domains contains a Condorcet cycle. We proof that ' is dictatorial in the same classical way as before. we start a little di erent though. First we show that on a two coalition con ict on a top pair say a and b the winner is decisive over that pair. 9

10 Lemma 7 Let S N and let R and R be orders in D such that ab::: = R and ba::: = R for some top connected alternatives a and b: Then S is decisive on (D a ) S (D b ) NnS or NnS is decisive on (D b ) NnS (D a ) S : Proof. As by unanimity '(R N ) = a it follows by strategy-proofness that '(R S ; R NnS ) 2 fa; bg: We distinguish that following two cases. Case 1 '(R S ; R NnS ) = a: We prove that in this case S is decisive on (D a ) S (D b ) NnS : Therefore consider a pro le ((R 1 ) S ; (R 2 ) NnS ) 2 (D a ) S (D b ) NnS : It is su cient to prove that '((R 1 ) S ; (R 2 ) NnS ) = a: Unanimity implies '((R 2 ) N ) = b: So, strategy-proofness therewith yields that '(R S ; (R 2 ) NnS ) 2 fa; bg. As '(R S ; R NnS ) = a; strategy-proofness implies that '(R S ; (R 2 ) NnS ) 6= b: So, '(R S ; (R 2 ) NnS ) = a: Strategy-proofness now implies '((R 1 ) S ; (R 2 ) NnS ) = a: Case 2 '(R S ; R NnS ) = b: Likewise now it follows that NnS is decisive on (D b ) NnS (D a ) S : Next we show that if a coalition is decisive over a pair say ab then it is over any pair ac: Lemma 8 Let a; b; c 2 B be three alternatives. Let S be decisive on (D a ) S (D b ) NnS : Then S is decisive on (D a ) S (D c ) NnS : Proof. Let R a 2 D a and R c 2 D c : It is su cient to prove that '((R a ) S ; (R c ) NnS ) = a: To the contrary let '((R a ) S ; (R c ) NnS ) = x and x 6= a: We end the proof by deducing a contradiction. As there is no inseparable pair in D alternative a cannot be always the second best at preferences where b is best. By Remark 5 there are preference R b 2 D b such that b:::x:::a::: = R b : As S is decisive on (D a ) S (D b ) NnS we have in particular that '((R a ) S ; (R b ) NnS ) = a: But as NnS prefers x to a at R b it can manipulate at pro le ((R a ) S ; (R b ) NnS ) as '((R a ) S ; (R c ) NnS ) = x: This however violates the strategy-proofness assumption on ': The proceeding step shows for directly top-connected pairs of alternatives, say a and b; that if S is decisive on pair ab it is also decisive on pair ba: Lemma 9 Let a; b 2 B be di erent alternatives, with a! b. Let S be decisive on (D a ) S (D b ) NnS. Then S is decisive on (D b ) S (D a ) NnS : Proof. As there is no inseparable pair there are alternatives x in Bnfa; bg and R bx 2 D bx : Further by Remark 5 there are Rba x = x:::b:::a::: and Rx ab = x:::a:::b::: in D x : As a! b there are R ab in D ab and R ba in D ba : In view of Lemma 7 it is su cient to show that '((R ba ) S ; (R ab ) NnS ) = b: By Lemma 8 Coalition S is decisive on (D a ) S (D x ) NnS : So, '((R ab ) S ; (Rba x )NnS ) = a: Therewith, as a is second best at R ba strategy-proofness implies '((R ba ) S ; (Rba x )NnS ) 2 fa; bg. But as '((R ba ) N ) = b by unanimity, strategy-proofness implies that '((R ba ) S ; (Rba x )NnS ) 6= a: So, '((R ba ) S ; (Rba x )NnS ) = b: Therefore by strategyproofness we have '((R bx ) S ; (Rba x )NnS ) = b: Consider ((R bx ) S ; (Rab x )NnS ): As by unanimity '((Rab x )N ) = x; it follows by strategy-proofness and x is second best at R bx that '((R bx ) S ; (Rab x )NnS ) 2 10

11 fb; xg: As '((R bx ) S ; (R x ba )NnS ) = b, strategy-proofness now implies that '((R bx ) S ; (R x ab )NnS ) = b: By this strategy-proofness implies that '((R ba ) S ; (R x ab )NnS ) = b: But then Maskin monotonicity implies '((R ba ) S ; (R ab ) NnS ) = b: These ndings above now yield the desired impossibility result. Theorem 2 Let D be a top-connected and tail-rich domain without an inseparable pair. Then every strategy-proof and unanimous choice function ' from D N to A is dictatorial. Proof. It su ces to prove that the family of decisive coalitions is an ultra- lter. Note that as D be a top-connected and without an inseparable pair B has at least three alternatives. The top-connectedness of D and Lemmas 8 and 9 straight forwardly yield that for an arbitrary coalition S N either S is decisive or NnS is decisive. By remark 5 domain D has Condorcet cycles. Note that in the proof of Lemma 6 only strategy-proofness of ' is used. So, we may conclude that also here if coalitions S N and T N are decisive then S \T is decisive as well. By this we have shown that the family of decisive coalitions is an ultra- lter. As N is nite this implies that there is an agent which is decisive and therefore by Lemma 1 ' is dictatorial. Theorem 2 and Example 1 essentially yield the following immediate consequence. Corollary 2 Let D be a top-connected and tail-rich domain. Then there exist non-dictatorial, strategy-proof and unanimous choice functions on D N if and only if D has an inseparable pair and B contains at least two alternatives. Proof. Theorem 2 covers the only-if part and the observation that if B is a singleton by unanimity ' is dictatorial. In view of Example 1 to prove the if-part it is su cient to consider the case at which B has precisely two alternatives. Then any unanimous and monotone voting rule on these two alternatives is strategy-proof. Therefore the domain is a possibility domain. 7 Top-rich and Tail-rich domains In this section we do not demand domain D to be top-connected. Therefore directly top-connectedness induces top-connected components, say C 1 ; C 2 ; :::; C l, which partition B: In such a component all alternatives are top-connected to each other and only to those in that component. Recall that alternatives a and b are directly top-connected only if neither D ab nor D ba is empty. So, in the absence of top-richness there might be alternative say x and y in di erent components such that D xy is non-empty. As we assumed that x and y are in di erent components, hence not connected, this can only happen if D yx is empty. More subtle conditions than the inseparable condition are needed to characterize for instance tail-rich domains, which are not top-connected. A similar conclusion may be draw when considering top-rich domains. We consider results in these 11

12 directions useful and interesting but for the moment not within the direct scope direct scope of this research. Top and tail-rich domains are possibility domains only if there is an inseparable pair. As this result is closely related to the previous two characterizations and moreover easily derived it is discussed here. Theorem 3 Let D be a top-rich and tail-rich domain. Then there exist nondictatorial, strategy-proof and unanimous choice functions on D N if and only if D has an inseparable pair. Proof. The if-part is obvious by Example 1. To prove the only-if-part let D have no inseparable pair. Let ' be a unanimous and strategy-proof choice function on D: It is su cient to prove that ' is dictatorial. Because D is toprich it follows that every component C t consists of at least three alternatives. Let D t = [fd c : c 2 C t g be the set of preferences in D with a top element in C t : Clearly, D t is a connected tail-rich domain without an inseparable pair. Therefore as C t consists of at least three alternatives, by Theorem 2 we have that 'j Dt is dictatorial, say with dictator i t : Consider the proof of Lemma 8. It is independent of the top-connected condition imposed the Section of Top-connected and tail-rich domains. Therefore we may conclude that fi t g is decisive on (D x ) fitg (D y ) Nnfitg for all x 2 C t and all y 2 B: Hence, there is a unique agent, say i; such that i = i t for all t 2 f1; 2; 3; :::; lg: But then fi t g is decisive on (D x ) fig (D y ) Nnfig for all x 2 B and all y 2 B. This implies by Lemma 1 that ' is dictatorial. 8 Concluding Remarks Indeed Corollaries 2 and?? state necessary and su cient conditions for certain "rich" domains to allow for non-dictatorial, strategy-proof and Pareto optimal choice functions. The characterizing conditions being respectively inseparable top-pair. The following example shows that the equivalences in the Corollary 1 may fail at not top-connected domains as well as in Theorem 3 at top-rich domains which are not necessarily tail-rich. Example 2 Inseparable set Let C be a non-trivial subset of B containing at least two alternatives and let C be an inseparable set. That is for all c and d in C and all x in BnC and all R c 2 D c we have c:::d:::x::: = R c : Consider the following choice function ' C de ned at an arbitrary pro le p as follows ' C (p) = x if x =2 C and p(1) 2 D x = c if p(1) 2 D x ; x 2 C and c::: = p(2)j C : So, at a pro le p choice function ' C chooses the best alternative of agent 1 unless this is an alternative in C: In that case ' C (p) is the best alternative among C of agent 2: Choice function ' C is Pareto optimal, because choice ' C (p) at an 12

13 arbitrary pro le p is either the best alternative of agent 1 or it is the best among C of agent 2 and agent 1 orders all alternatives of C at the top. So, ' C (p) is not Pareto dominated at p. It is also non-dictatorial as C contains at least two alternatives and is itself non-trivial. To see that ' C is strategy-proof rst consider agent 2: If agent 1 has C at the top agent 2 has option set C. However in this case the outcome is the agent 2 s best among C; so at these situations he cannot manipulate. If agent 1 has a best alternative which is not in C then agent 2 has no in uence on the outcome. So, all in all agent 2 cannot manipulate at ' C : Next we show that agent 1 cannot manipulate. This is obvious if agent 1 s best choice is the outcome. So, we only have to consider situations where agent 1 has C at top. Now the outcome is in C and determined by agent 2: So, by unilateral deviations of 1 he cannot get an other alternative in C. However by such deviations an element in AnC might become the outcome. The latter is however less preferred by 1 as C is an inseparable set. So, agent 1 cannot manipulate either. All other agents have no in uence on the outcome and therefore cannot manipulate at ' C : The following example shows that the existence of a Condorcet cycle in Corollary 1 and that of tail-richness in Corollary 2 is essential for the two equivalences. Example 3 Choosing corner points of a square Let a; b; c and d be the corners of a square in R 2 : Let for instance have a coordinates ( 1; 1); b coordinates (1; 1); c coordinates (1; 1) and d coordinates ( 1; 1): Consider Euclidean preferences with a peak point, say x and preference declines with Euclidean distance to x: So, indi erence classes form concentric circles with mid point x: Now let the set of all admissible preferences D such consist of all such Euclidean preferences where x is not on a perpendicular bisector of any pair of these corners a; b; c and d: That is at such preferences there are only strict preferences between these four point. This domain D is top-connected, has no Condorcet cycle and is not tail-rich. By Lemma 5 there are no Pareto optimal and strategy-proof choice rules for in case there are two agents. Choosing independently on the one hand between top and bottom and on the other between left and right yields non-dictatorial strategy-proof and Pareto-optimal rules in case of three and therewith any higher number of alternatives. 9 References 1. Aswal A., S. Chatterji, A. Sen, 2003 Dictatorial domains, Economic Theory, 22, p Arrow K.J.,1978, Social choice and individual values, Yale University Press (19 th edition). 3. Black D., 1948, On the rationale of group decision making, Journal of Political Economy,56, p

14 4. Bochet O. and Storcken T., 2005, Maximal domains for strategy-proof or Maskin monotone choice rules, METEOR research memorandum, Maastricht. 5. Gaertner W., 2002, Restricted domains, in: K.J. Arrow, A.K. Sen and K. Suzumura (ed.), Handbook of Social Choice, 1, chapter 3, p , Elsevier. 6. Gibbard A., 1973, Manipulation of voting schemes: A general result, Econometrica, 41, p Inada K., 1964, A note on the simple majority decision rule, Econometrica, 32, p Kalai E. and Muller E., Characterization of domains admitting nondictatorial social welfare functions and nonmanipulable voting procedures, Journal of Economic Theory, 16, p Kalai E. and Ritz Z., Characterization of the private alternatives domains admitting Arrow social welfare functions, Journal of Economic Theory, 22, p Moulin H., 1980, On strategy-proofness and single peakedness, Public Choice, 35, p Ozdemir U, M.R. Sanver, 2007, Dictatorial domains in preference aggregation, Social Choice and Welfare, 28, p Ritz. Z., 1985, Restricted domains, Arrow social welfare functions and noncorruptable and nonmanipulable social choice correspondences: The case of private alternatives, Mathematical Social Science, 4, p Satterthwaite M.A., 1975, Strategy-proofness and Arrow s conditions: Existence and correspondence theorem for voting procedures and social welfare functions, Journal of Economic Theory, 10, p Sen A.K. and Pattanaik P.K., 1969, Necessary and su cient conditions for rational choice under majority decision, Journal of Economic Theory, 1, p Storcken T., 1985, Societies with a priori information and anonymous social welfare functions, Methods of Operations Research, 54, p