Maximal Domains for Strategy-Proof or Maskin Monotonic Choice Rules

Domains of individual preferences for which the well-known impossibility theorems of Gibbard-Satterthwaite and Muller-Satterthwaite do not hold are studied. To comprehend the limitations these results imply for social choice rules, we search for the largest domains that are possible. Here, we restrict the domain of individual preferences of precisely one individual. It turns out that, for such domains, the conditions of inseparable pair and of inseparable set yield the only maximal domains on which there exist non-dictatorial, Pareto-optimal and strategy-proof social choice rules. Next, we charaterize the maximal domains which allow for Maskin monotone, non-dictatorial and Pareto-optimal social choice rules.


Introduction
The two most negative results on decentralization of social choice functions are, respectively, the Gibbard-Satterthwaite (Gibbard (1973) and Satterthwaite (1975)) and the Muller-Satterthwaite (Muller and Satterthwaite, 1977) theorems. The Gibbard-Satterthwaite theorem states that over an unrestricted domain of linear orderings-and with at least three alternatives-any surjective and strategy-proof social choice function is dictatorial. On the other hand, the Muller-Satterthwaite theorem, by establishing the connections between strategy-proofness and Maskin monotonicity, states that any unanimous and Maskin monotonic social choice function is dictatorial.
These two theorems have a damaging impact on decentralization. Strategyproofness and Maskin monotonicity are, respectively, necessary conditions for dominant strategy implementation and Nash implementation. If a planner restricts himself to institutions corresponding to normal game forms having a unique equilibrium at every preference profile, then only trivial social choice functions can be decentralized.
However, the two results strongly rely on the assumption of an unrestricted preference domain. Restricted domains have delivered possibility results on strategy-proof and Maskin monotonic social choice functions. 1 For instance, if preferences are quasi-linear with respect to a numeraire good, then Clark-Groves mechanisms are strategy-proof. 2 If preferences are singlepeaked, then generalized median voting rules are strategy-proof (see e.g. Moulin, 1980). Based on this observation, we would like to address two fundamental questions. The first one is to know when a preference domain escapes the negative conclusions of the aforementioned theorems with non-trivial social choice functions. Stated differently, we want to know when a preference domain is a strategy-proof/Maskin monotonicity possibility domain. The non-trivial social choice functions we see as admissible are functions that satisfy Pareto optimality, non-dictatorship and strategy-proofness/Maskin monotonicity. The two necessary and sufficient conditions we uncover are respectively strategyproof admissibility and Maskin admissibility. They can be readily checked for any preference domain at hand. Next, the second question is to check how much restrictions are needed from the unrestricted domain to recover a possibility domain. Another way to tackle this question is the following: how many preference profiles should be removed from the unrestricted domain in order to get a possibility domain? Notice that the number of preference profile is depends both on the number of agents and on the number of al-ternatives. Therefore, we look for such domains which are maximal -i.e. there are no supersets of this domain on which such social choice rules exist. Such maximal domains indicate a minimal necessary restriction of the set of profiles and herewith the impact of the so-called impossibility theorems mentioned above: the smaller these maximal domains are the more restrictive the properties of the social choice rules of these impossibility theorems are. The necessary and sufficient conditions we uncover are -at least for the strategy-proof case-easy to interpret and check.
The question we address is not new -at least for the strategy-proof casebut we depart from the "standard" approach to maximal domains. In the literature, the approach that is usually followed is to take a possibility domains and to find the maximal enlargement of this domain so that the possibility result still hold. For instance, this is the case of Barberà, Sonnenschein and Zhou (1994); Barberà, Gul and Stacchetti (1994); Serizawa (1995); Serizawa and Ching (1998); Berga and Serizawa (2000), or Masso and Neme (2001). Each paper deals with entire restricted domain of preferences. Moreover, as far as we are aware of, Puppe and Tasnadi (2006) is the only paper that deals with maximal domains for non-trivial Maskin monotonic social choice functions.
The key difference between our approach on maximality and the papers cited above can be loosely summarized as follows. The papers using restricted domains follow a "bottom-to-top" approach. A restricted possibility domain is identified and the question of its maximal enlargement is investigated. On the other hand, we follow a "top-to-bottom" approach. We start from Gibbard-Sattherthwaite and Muller-Satterthwaite theorems and study the minimal restrictions on (unrestricted) preferences domains that allow an escape from impossibility results: how much do we need to restrict the domain in order to get a possibility result. Closely related but different questions were posed by Kalai and Muller (1977) and Kalai and Ritz (1980). They studied the general conditions for domains which admit the existence of non-dictatorial Arrow-type social welfare functions. 3 A different approach of the question we address here can be found in Maus, Peters and Storcken (2005,...).
Our approach is different. Like in Kalai and Muller (1977) and in Kalai and Ritz (1980) we are interested whether on a give domain there exist social choice functions satisfying some given properties, e.g. Pareto optimality, non-dictatorship and strategy-proofness or Maskin monotonicity. On top of that we look for such domains which are maximal, i.e. there are no super sets of this domain on which such social choice functions exist. Maximal domains indicate a minimal necessary restriction of the set of profiles to allow for such social choice functions and herewith the impact of the so called impossibility theorems mentioned above: the smaller these maximal domains are the more restrictive the properties of the social choice functions of these impossibility theorems seem to be. For example our results imply, which is in line with e.g. a result of Aswal, Chatterji and Sen (2003), that in the three alternatives case by excluding only one preference from precisely one agent yields a domain which allows for social choice functions having these properties. Of course this restricted domain is a maximal domain.
One way by which domain restrictions escape from the impossibility results is that the restriction prevents the spreading of decisiveness power. In many proofs of impossibility results decisiveness of a coalition on one pair of alternatives spreads to all pairs of alternatives. By deleting specific preferences this spreading of power is stopped. Restricting the sets of preferences for different agents may lead to very technical descriptions as Examples 4 and 5 might indicate. Here the possibility is obtained because of an interdependency between the restrictions of the two sets of admissible preferences. Herewith strategy-proofness has no consequences on the decisiveness granted to the two agents by the social choice function. To the best of our knowledge there is not a complete result characterizing all those restricted domains which admit Pareto optimal, non-dictatorial and strategy-proof or Maskin monotonic social choice functions even if we disregard the maximality of those restrictions. In view of Examples 4 and 5 finding such characterizations look at least very complicated. Therefore we concentrate on a partial result and hope that this will help to solve the general case. Instead of allowing that any agent's preference domain can be restricted we restrict the preference domain of precisely one agent, say agent 1. It will appear that in this way decisiveness power of the coalition from which only this agent 1 is excluded is prevented to spread over all pairs. Let N denote the set of all agents. Then N − {1} denotes this coalition. Apart from a simplification we consider this subclass interesting on its own. It answers the question what we have to know about one agent independently of the others in order to obtain a domain on which the impossibility theorems have no bite. Note that as our results are on maximal possibility domains these results provide sufficient conditions in case more than one agent preference set is restricted. Clearly this is an asymmetric approach, but the non-dictatorship conditions allows for asymmetric allocation of decisiveness. A more symmetric approach at which all the admissible sets of preferences are the same intuitively seems to fit better to a case where instead of non-dictatorship anonymity is invoked upon the social choice function.
We identify two conditions that are both necessary and sufficient for the existence of non-trivial social choice functions on domains where only one agent's preference set is restricted. The social choice function K we identify is a hierarchical choice function. Agent 1 with restricted set of admissible preferences almost always gets his preferred alternative. In that sense, the social choice function is almost dictatorial. The conditions we identify are respectively strategy-proof admissibility and Maskin admissibility. Essentially these conditions indicate how decisiveness of coalition N − {1} spreads under strategy-proofness and under Maskin monotonicity.
We then examine the maximality of this domain restrictions. We study, in turn, strategy-proofness and Maskin monotonicity since the conditions we obtain are different. For strategy-proofness, we find that a domain of preferences is a maximal possibility domain if the restricted set of preferences of agent 1 has one inseparable pair or one inseparable set. In case of an inseparable pair decisiveness of coalition N −{1} is restricted to the "reverse" of this pair where in case of an inseparable set it is restricted to any pair of alternatives belonging to this set. The notion of inseparable pair is wellknown in relation with non-dictatorial Arrow-type welfare functions. See e.g. Kalai and Ritz (1980). Just to fix idea, let us briefly discuss the notion of inseparable pair and inseparable set. We say that an agent has an inseparable pair if there exist two alternatives x and y such that whenever x is ranked best, then y is second-best. The pair (x, y) is then called inseparable. The definition of inseparable pair used by Kalai and Ritz (1980) is stronger: it says that all preferences where x is preferred to y these two are ranked adjacent to each other. Suppose for instance that a board of managers has two vacancies. Current members of the board are contemplating several candidates. Among those are b, a bossy individual, and w a wimp that is afraid of b. Then, an agent i could rank b at the top and w second because he expects w to be obedient and to copy b s decisions-the power of b would then be increased 4 .
Clearly, agent i has an inseparable pair (b, w). Also a set of single peaked preferences on a finite set of alternatives possesses an inseparable pair 5 .
An agent has an inseparable set if there exist a set of alternatives Bof cardinality at least equal to three-such that all preferences with a best alternative in B ranks the alternatives in B adjacent to each other (while the ranking of alternatives both within B and the complement of B can be reversed from one profile to another) 6 . For instance if B stand for similar alternatives/products of different brands and an agent who is highly interested in this type of product will order B on top above all the rest although the actual ordering in B might be dependent on his mood or earlier experience. Or, suppose that there are two parties, left and right, involved in an election where several positions have to be distributed (e.g. prime minister versus lower ranked ministers). Indeed, each party has several candidates. Agents have to vote for one of the two parties and have to rank candidates within each party according to the position they would like them to have. There, it is natural to assume that agents rank individuals from the same party adjacent to each other. Then, every agent has an inseparable set. 7 On the other hand, in order to escape the Gibbard-Satterthwaite theorem, we only need one agent to satisfy the domain restriction imposed by inseparable pair or inseparable set. The notion of inseparable set goes at least back to Storcken (1989).
For Maskin monotonicity, the characterization of maximal domains is more intricate and relies on the existence of disjoint subsets Y and Z of the sets of alternative and the existence of an asymmetric and transitive relation P on the set of alternatives. Now coalition N − {1} is decisive on all pairs in Y and all pairs (z, y) such that z is in Z and y is in Y . To achieve that this decisiveness does not spread further it is on the one hand required that at all admissible preferences for agent 1, with the best alternative in Y , all alternatives in subset Y ∪ Z are preferred to all those not in this union. Actually this requirement follows from condition (a) of Theorem 3 5 Let m be the number of alternatives. It is not difficult to see that the set of single peaked preferences contains 2 (m−1) preferences and the maximal set of preferences which has this inseparable pair contains (m − 1) · ((m − 1)!) + ((m − 2)!) preferences. So, the former set of preferences is only a small fraction of the latter set of preferences. 6 Observe that if the cardinality of B is equal to 2, then we have in fact two inseparable pairs. 7 Thus, the strategy-proof and Pareto-efficient rules that one may identify there will be less hierarchical than the one we identify. and part 1 of condition (d) of this Theorem 8 . This condition is similar to the separable set condition. As this is the only requirement on the set Y it is not necessarily "separated" from Z. That is there may be alternatives z in Z which are preferred to an alternative in Y at a preference with its best element in Y . Therefore on the other hand it is required that decisiveness on a pair (z, y) or (y , y) cannot spread to a pair (z, z ) or to a pair (y, z ) for alternatives z and z in Z and y and y in Y . Now part 2 of condition (d) in Theorem 3 essentially takes care that in these situations Maskin monotonicity has no bite.
The plan of the paper is as follows. In section 2, we introduce the model and the necessary definitions useful for the paper. In Section 3, we introduce necessary and sufficient conditions for the existence of possibility results. In Section 5 we characterize the maximal domain for strategy-proof, efficient and non-dictatorial social choice functions. Next, in Section 6 we characterize the maximal domain for Maskin monotonic, efficient and non-dictatorial social choice functions. Finally, we offer some concluding remarks in Section 7.

The model
There is a set of alternatives A = {1, ..., m}, with m 3 and a set of agents N = {1, ..., n} with n 2.
Every agent i is endowed with a preference relation p(i) over the alternatives of A that is (strongly) complete, anti-symmetric and transitive; that is a preference relation is a linear order over alternatives. Here for a preference relation, say p(i), we take the usual assumptions that p(i) is a subset of A×A, the two fold Carthesian product set of A, that for alternatives x and y we interpret (x, y) ∈ p(i) as agent i weakly prefers x to y where (x, y) ∈ p(i) and (y, x) / ∈ p(i) is interpreted as agent i strictly prefers x to y. Furthermore, if B is a non-empty subset of alternatives then best(p(i)| B ) denotes the best alternative in B according to agent i's preference p(i), i.e. the alternative b in B such that for all x in B either b = x or b is strictly preferred to x by agent i. Let L(A) denote the set of all these preferences and L (A) N be the set of possible preference profiles 9 . For different alternatives x and y, x... = p(i) means that x is the best alternative at p(i), ...x...y... = p(i) means that x is strictly preferred to y at p(i), ...xy... = p(i) means that x is strictly preferred to y at p(i) and there is no alternative in between these alternatives x and y and x...y = p(i) means that x is the best alternative and y is the worst alternative at p(i). Let L x (A) denote the set of linear orderings that order x best.
For an arbitrary relation R on A, i.e. R ⊆ A × A the upper contour of an alternative x at R is defined as up(x, R) = {y ∈ A : (y, x) ∈ R} and the lower contour of x at R is defined as To model restrictions of domains of individual preferences let ∅ = L i ⊆ L(A) be the domain of individual preferences of agent i ∈ N . From now on, we assume that be the set of alternatives that agent 1 can order best in at least one of his admissible preferences. Furthermore, let 10 A social choice function K is a function from L N to A. For coalitions M and profiles p, q ∈ L N , the preference profile p is said to be a M -deviation of a profile q if p| N −M = q| N −M .
Except for intermediate strategy-proofness which can at least be traced back to Peters e.a. (1991) the following conditions for social choice functions are well-known. We just rephrase these in the notation at hand.
Pareto optimality: The social choice function K is Pareto optimal if for each (x, y) ∈ A 2 and each p ∈ L N such that for all agents i ∈ N , Strategy-proofness:The social choice function K is strategy-proof if for each agent i ∈ N and each p, q ∈ L N such that q is an {i}-deviation of p, we ... × L(A) because the restiction of a function to a subdomain used in the definition Mdeviation is a well-known concept and therefore needs no further explanation. 10 We have that either K(p) = K(q) or ...K(p)...K(q)... = p(i).
Intermediate strategy-proofness: The social choice function K is intermediate strategy-proof if for each coalition M ⊆ N and for each profile This condition means that the social choice function is not beneficial for coalitional deviations from profile p to profile q where at profile p all members of the deviating coalition have the same preference. In Theorem 1 it is shown that intermediate strategy-proofness is equivalent to strategy-proofness on any (restricted) domain of profiles over linear orderings. It is used as a handy consequence of strategy-proofness in several proofs.

Maskin monotonicity: The social choice function
Next we introduce four notions by which we can formulate the main results of this paper. Decisiveness of coalitions appear also in our setting as a powerful tool to analyze the problem at hand.
If K is Pareto optimal, then it follows immediately that I A ⊆ D K (M ). It appears that the social choice functions which simultaneously satisfy Pareto optimality, non-dictatorship and either strategy-proofness or Maskin monotonicity on the domains at hand are almost dictatorial.
The social choice functions by which it is proved that a domain is a possibility domain are essentially hierarchical where agent 2 is decisive on any pair in D and agent 1 is decisive on the reversed remaining pairs in Hierarchical social choice function: Define the hierarchical social choice function K D corresponding to D for every profile p ∈ L N as follows and alternative y ∈ A such that up(y, D) = ∅ and p(1) ∈ L 1 y as follows Now it is straight forward to see that if the pair (y, x) is in D then agent 2 is decisive on the pair (x, y). Because of D being unequal to both I A and A × A, it follows that K D is non-dictatorial.

Necessary and sufficient conditions for strategyproofness and Maskin monotonicity
We are interested in determining conditions on the domain of preferences that ensure the existence of social choice functions satisfying Pareto optimality, non-dictatorship and strategy-proofness/Maskin monotonicity. The necessary and sufficient conditions we uncover have two direct consequences. The first one is that we offer a possibility to directly check whether a given preference domain is respectively a strategy-proof or a Maskin monotonic possibility domain. 11 The second one is that in order to escape the impossibilities stated in the Gibbard-Satterthwaite and the Muller-Satterthwaite Theorems, it is enough to destroy the product structure of the unrestricted domain by restricting the set of admissible preference relations of only one agent, say agent 1. The specific way in which the preferences of agent 1 must be restricted will be the object of the next section where we turn our attention to the study of maximal domains. Before proceeding to the results, we introduce in turn our central definitions for this section and we illustrate them with examples.
Strategy-proof admissibility: Let L 1 be the set of admissible preferences of agent 1 and let Let strategy(D, L 1 ) stands for the strategy-proof admissibility of the pair (D, L 1 ).
Before going to the definition of Maskin admissibility, it is instructive to recall the notion of a linked domain introduced in Aswal, Chatterji and Sen (2003).
Connectedness: Fix a domain of preferences R. A pair of alternatives x and y are connected, denoted x C y, if there exists p(i), p (i) ∈ R such that p(i) = xy..., and p (i) = yx....
Linkedness: Fix a domain of preferences R. Let B ⊂ A and x / ∈ B. Then x is linked to B if there exists y, z ∈ B such that x C y and x C z.
We compare next the notion of linked domain and of strategy-proof admissibility. Whereas these two notions are antinomic in nature, it is instructive to grasp some of the differences they entail on preference domains since both operate on domain restrictions.
Example 1 Let A = {x, y, z} and let the preference domain be L(A). Consider p(i) = xyz. Then x C y since there exists p (i) = yxz, and z is linked to {x, y}. This implies that L(A) is linked. Moreover, notice that when m = 3 there exists no subdomain of L(A) that is linked. Obviously, L(A) cannot be a strategy-proof possibility domain. Suppose (y, x) ∈ D. Applying the definition of strategy-proof admissibility, we get first that (z, x) ∈ D and (y, z) ∈ D. It is then easy to see that the decisiveness on pairs cannot be stopped so that D = A × A.
Example 2 shows that, when the number of alternatives is three, we only need to remove one preference relation to recover a possibility result. We will show in the next section that this observation generalizes: given a set of alternatives of cardinality m, one needs only to remove (m − 1)! − (m − 2)! preference relations to obtain a strategy-proof possibility domain. Since Aswal, Chatterji and Sen (2003) and our paper show ways to restrict domains and get, respectively, an impossibility and a possibility result, we show in the next example that both restrictions can lead to the same number of preference relations being removed. In that sense, the conclusion that the Gibbard-Satterthwaite Theorem is far more robust than suggested by the conditions of the theorem itself should be taken with caution. In our opinion, both their work and ours show that the robustness of the theorem is linked to the specific ways in which restriction operations are performed. In the previous two definitions of strategy-proof admissibility and Maskin monotonic admissibility, (1) implies that K D is non-dictatorial, Condition (2) imposes some rationality on the decisiveness of agent 2 and together with (3) it guarantees that K D is strategy-proof or Maskin monotonic. Paretooptimality follows from (3) part one.
The following Lemma shows that the condition of strategy-proof admissibility and Maskin admissibility are sufficient to guarantee that K D is strategyproof or Maskin monotonic respectively and therewith explains the names of these two requirements on D and L 1 .
Lemma 1 Consider the hierarchical social choice function K D : L N → A, corresponding to D. Then 1. strategy(D, L 1 ) implies that K D is non-dictatorial, Pareto optimal and strategy-proof; 2. Maskin(D, L 1 ) implies that K D is non-dictatorial, Pareto optimal and Maskin monotone.
We now prove that the conditions of strategy-proof admissibility and Maskin admissibility are in fact also necessary to guarantee the existence of social choice functions that are Pareto efficient, non-dictatorial and strategyproof/Maskin monotonic.
For the rest of this section, let K be a Pareto optimal and non-dictatorial social choice function. We will show that whenever K is strategy-proof or Maskin monotonic, then A × A 1 is transitive and that the pair (D K (N − {1}), L 1 ) is respectively strategy-proof admissible or Maskin admissible.
To avoid needless repetitions, assume that K is at least Maskin monotonic. So, cases at which K is strategy-proof are spelled out explicitly.
The following Lemma formulates a condition when decisiveness at a specific profile spreads to decisiveness on a specific pair. Actually this condition coincides with the linking condition of Aswal, Chatterji and Sen (2003) introduced at the beginning of the section.

Proof. By the assumptions on
It is sufficient to prove that K(q) = x. Let r ∈ L N be an (N − M )-deviation of p and an M -deviation of q. Now, Pareto optimality implies that K(r) ∈ {x, y}. If K(r) = y, then Maskin monotonicity would imply the contradiction that K(p) = y. Therefore, K(r) = x. But then Maskin monotonicity implies that K(q) = x.
To the contrary, suppose that I A 1 = D K (N − {1}). We show that this leads to the contradiction that agent 1 is a dictator. Consider R ∈ L 1 and a profile p ∈ L N , with xy... = R = p(1) for some x and y. To deduce the contradiction, it is sufficient to prove that K(p) = x. Consider the profiles q and r-both N − {1}-deviations of psuch that q(i) = yx... and r(i) = y...x for each i 2. Because of Pareto optimality, it follows that K(q) ∈ {x, y}. Now, by Lemma 2, it follows that K(q) = y, otherwise (y, x) ∈ (D K (N − {1}) − I A 1 ). But then K(q) = x and (x, y) ∈ D K ({1}). So, K(r) = x. By Maskin monotonicity, we obtain that K(p) = x.
We show that this leads to the contradiction that K is dictatorial. For R ∈ L 1 , consider the social choice function K R defined by K R (p) = K(R, p) for each profile p ∈ L(A) N −{1} . It is clear that K R is surjective-and even unanimous-and Maskin monotonic. By Theorem 1, it is strategy-proof. Hence, by the Gibbard and Satterthwaite Theorem, it follows that K R (p) is dictatorial, say by agent i R 2.
Consider two preferences R and R in L 1 . In order to prove that K is dictatorial, it is sufficient to show that i R = i R . To the contrary, suppose that i R = i R . We deduce a contradiction and are done. Obviously there are different alternatives x, y, z 1 , z 2 , ...z k−1 and z k , where k may be zero, such that z 1 z 2 ...z k x... = R and z 1 z 2 ...z k y... = R . So x and y are the first alternatives on which the preferences R and R differ. Consider profiles p and q which are {1}-deviations such that p(1) = R, q(1) = R , p( i R ) = q(i R ) = y... and p(i R ) = q(i R ) = x.... Then since i R is a dictator at K R , it follows that K(p) = y and because i R is a dictator at K R , it follows that K(q) = x. Finally, because low(y, R) ⊆ low(y, R ), we have a contradiction with Maskin monotonicity of K.
(Proof of transitivity) Let (x, y), (y, z) ∈ D K (N − {1}). It is sufficient to prove that (x, z) ∈ D K (N − {1}). This is trivially the case when x, y and z are not three different alternatives. So, let these three alternatives be different. Let R = z... ∈ L 1 arbitrary. Consider p ∈ L N such that p(1) = R and p(i) = xz...
Combining Lemma's 1, 2, 3 and 4 yields the following corollaries which are the central results of this section.

Maximal domains for strategy-proofness
In the preceeding section, we introduced ways to check whether a given domain is respectively a strategy-proof or a Maskin monotonic domain. We now study the specific way to make a domain a strategy-proof possibility domain. Moreover, we look for domain restrictions that are minimal. That is the preference domain obtained is a maximal domain: any enlargement would make the domain to be dictatorial. As emphasized in the introduction, our approach can thus be seen as the opposite of Aswal, Chatterji and Sen (2003). Since we already know from the previous section that the preferences of only one agent need to be restricted in order to obtain a possibility domain, we assume that L 1 ⊂ L(A) while L i = L(A) for each i = 1.
Maximal strategy-proof possibility domains are characterized by the following two inseparability notions.
Inseparable pair: The set of preferences L i has an inseparable pair So, the ordered pair of alternatives (x, y) is an inseparable pair of the set of admissible preferences of agent i if y is ordered second best in all preferences where x is ordered best. It is already mentioned in the introduction that on a finite set of alternatives the set of single peaked preferences has an inseparable pair. To be more specific on this let the set of alternatives be {a 1 , a 2 , ..., a m }. Consider all single peaked preferences with respect to the basic order a 1 a 2 ...a m . That is at such a preference say R there is an alternative say a t such that a t ..  So, for all preferences R in L i , if the best alternative at R is in B, then B is preferred to (A − B) at R. Note that this condition trivially holds for the empty set, any singleton set and the set A itself. Therefore these are excluded. Furthermore, if B consists of precisely two alternatives, then having an inseparable set means having two inseparable pairs, which explains why sets with cardinality 2 are excluded in the definition of inseparable set.
The notion of inseparable pair is well-known see, e.g. Kalai and Ritz (1980), although there it is a slightly stronger condition. That is L i has an inseparable pair (x, y) if for all R ∈ L i if ...x...y... = R, then ...xy... = R.
Here we only need this inseparability if x is top alternative, because the almost dictatorial choice function depends mainly on the top alternatives of agent 1, the agent with the restricted preference set. In Aswal, Chatterji and Sen (2003) it is called the unique seconds property, to emphasize its origin we stick to the name as used here. A similar remark as for inseparable pair holds for inseparable set as defined in Storcken (1989).
Corollary 1 characterizes strategy-proof possibility domains in terms of a set of pairs of alternatives on which the coalition of agents whose sets of preferences are not restricted are decisive. Consider a Pareto optimal, strategy-proof and non-dictatorial social choice function K, such that for all strategy-proof possibility domains L N with L N ⊆ L N , we have that L N = L N . So, L N is a maximal strategy-proof possibility domain. We shall prove in Theorem 1 that the conditions on D K (N − {1}) under this maximality property yield that L 1 either has one inseparable pair or one inseparable set. Furthermore, we deduce that these separabilities are not only necessary but also sufficient. The following example shows why these inseparabilities imply that a domain at hand is a strategy-proof possibility domain.

Example 6
In case L 1 has an inseparable pair (y, x) or an inseparable set B, it follows straightforwardly that strategy(D, L 1 ), where D = {(x, y)} ∪ I A or D = (B × B) ∪ I A respectively. So, Lemma 1 implies that in these situations, the hierarchical social choice function K D is non-dictatorial, Pareto optimal and strategy-proof. Thus, in that case, L N is a strategy-proof possibility domain.
The following Lemmas are needed for the proof of Theorem 1.
Proof. To the contrary let L 1 x = ∅. Take L 1 = L 1 ∪ {R ∈ L(A) : xy... = R} for some fixed y ∈ A − {x}. Clearly, by taking L i = L(A) for i > 1, we obtain that L N L N . By lemma 1, the latter is a strategy-proof possibility domain. This yields a contradiction with L N being maximal.
Intuitively it is reasonable that the smaller the set of decisive pairs of N − {1}, the larger L 1 can be taken. The following Lemma shows that we may shrink the set of decisive pairs of N − {1}.
Lemma 7 There are disjoint subsets Y and Z of A such that 1. Y is non-empty, Y ∪ Z = A, and #(Y ∪ Z) 2; Proof. In Lemma 6, we proved that there are disjoint Y and Z such that Also, by definition, Y is non-empty and because of x * , y * ∈ Y ∪ Z, with It remains to prove that Y ∪ Z = A. In order to do so, suppose that Y ∪ Z = A. We prove that we may take Z = ∅ and that Y = A. Consider y ∈ Y and R = y...a in L 1 where a ∈ A. Because of strategy(((Y ∪ Z) × Y ) ∪ A × A, it follows that Z = ∅. But then Z = {a}. As the previous holds for every a ∈ A for which there are y ∈ Y , R ∈ L 1 y with y...a = R and Z is a singleton, it follows that for each y ∈ Y and each R ∈ L 1 y that y...a = R. Hence, strategy((Y × Y ) ∪ I A , L 1 ) and as Z is a singleton Y = A. By taking Z = ∅ this shows the existence of such Y and Z.

There is a non
y for some y ∈ Y , then for all a ∈ Y and all b ∈ A − Y we have ...a...b... = R} which means that L 1 has an inseparable set Y .
Proof. (Only-if-part) Suppose L 1 is a maximal strategy-proof possibility domain. By the previous Lemma there are disjoint subsets Y and Z of A such that follows by the maximality of L 1 that V ⊆ L 1 . Now V is defined such that it contains all sets of preferences which have an inseparable pair (a, b). Therefore L 1 ⊆ V . So, L 1 = V . Now suppose strategy((Y × Y ) ∪ I A , L 1 ). Then it follows that L 1 has an inseparable set Y . Consider the set W . By proving that L 1 = W, we end the proof of the only-if-part. Because obviously strategy((Y × Y ) ∪ I A , W ) and therefore obviously strategy((Y × Y ) ∪ I A , L 1 ∪ W ), it follows by the maximality of L 1 that W ⊆ L 1 . But W is defined such that is contains all sets of preferences which have an inseparable set Y . Therefore (If-part) By example 2 it is clear that if L 1 equals either V or W , then L N is a strategy-proof possibility domain. It remains to prove the maximality of it. Suppose L i ⊆ L i for all agents i and L N is a maximal strategy-proof possibility domain. It is sufficient to prove that L N = L N . By the only-if-part, it follows that L 1 has either an inseparable set say Y or an inseparable pair say ( a, b). Because L 1 ⊆ L 1 and is such that it contains all sets of preferences which either have an inseparable set Y or an inseparable pair (a, b), it follows that the inseparable sets or pairs are equal and that L 1 = L 1 .

Maximal domains for Maskin monotonicity
In this section we characterize the maximal Maskin monotonic possibility domains for the case that precisely one agent's set of preferences is restricted. Our next Theorem spells out a characterization of Maskin monotonic possibility domains in terms of a set of pairs of alternatives on which the coalition of agents whom sets of preferences are not restricted are decisive.
Remark 2 Consider Lemmas 5', 6' and 7' obtained from Lemmas 5, 6 and 7 by replacing the word "strategy" by the word "Maskin" respectively . The proofs of these Lemma's follow likewise by the same substitution in the proofs of the original Lemma's. To avoid obvious repetitions neither the Lemma's 5', 6' and 7' nor their proofs are written out here. c there is no partition X 1 , X 2 of Y ∪ Z with #X 1 2 and X 1 × X 2 ⊆ P and Proof. (Only-if-part) Let L N be a maximal Maskin monotonic possibility domain. We will show the existence of sets Y , Z and relation P satisfying the conditions formulated in the theorem. By Lemma 7' we have Y , . For different alternatives a and b, define (a, b) ∈ P if for all R ∈ L 1 y and all y ∈ Y we have that ...a...b... = R. By definition, P is asymmetric and because the preferences in L 1 are transitive, P is transitive.
Because of the definition of P and Lemma 5 ' it follows that ]. This proves part (a).
We have to prove the following implications for x, y, z ∈ A and R, R ∈ V .
The proof of the first implication follows immediately from the first condition in the definition of V and [(Y ∪ Z) × (A − (Y ∪ Z))] ⊆ P . To prove the second implication, let x, y, z, R and R be as in the premises of implication 2. It is sufficient to prove that low(x, R) low(x, R ) or (x, z) ∈ (Y ∪Z)×Y . Clearly by implication 1, it follows that z ∈ (Y ∪ Z). If z ∈ Y , then evidently (x, z) ∈ (Y ∪ Z) × Y . So, suppose z / ∈ Y , which implies that z ∈ Z. Now because of P ⊆ R and the definition of P , it follows that (x, z) / ∈ P . Of course This completes the proof of part (b).
In order to prove (c) let X 1 and X 2 be a partition of Y ∪ Z with #X 1 2 and X 1 ×X 2 ⊆ P . It is sufficient to prove that this contradicts the maximality of L N . Then A × A and I A ∪ [X 1 × (Y ∩ X 1 )] is transitive.Bythe definition of P it follows that X 1 ∩ Y = ∅. Consider y ∈ X 1 ∩ Y and x ∈ X 2 . Then (x, y) ∈ (Y ∪ Z) × Y . Let R ∈ L 1 y and let R ∈ L(A) be such that R| A−{x} = R | A−{x} and y...x = R . Then R ∈ L 1 and Maskin(( which contradicts the maximality of L N . Next we prove (d).
First we prove L 1 ⊆ V . Let R ∈ L 1 . Then (1) follows because of the definition of P . In order to show that also (2) is satisfied let z... = R. Let T = (non-up(z, P ) − {z}) ∩ (Y ∪ Z). Take x = best(R| T ). So x ∈ T and for some y ∈ Y there are R in L 1 with y...z...x... = R . Because of z ∈ Z, it follows that ( R). So, there are t ∈ low(x, R ) and x ∈ low(t, R).
Because of the definition of x and the fact that x ∈ low(t, R), it follows that t / ∈ T . But then, because t ∈ low(x, R ), it follows that t ∈ nonup(z, P ) − {z}. Now, because t / ∈ T, it implies that t / ∈ (Y ∪ Z). So, t ∈ A − (Y ∪ Z) and T ⊆ low(t, R) which in turn yields (2).
Next we prove that V ⊆ L 1 . Because of L 1 ⊆ V and the maximality of L 1 , it is sufficient to prove that V is a Maskin monotonic possibility domain. there is no partition X 1 , X 2 of Y ∪ Z with #X 1 2, X 1 × X 2 ⊆ P and W = {R ∈ L(A) : Then there are z ∈ (Y ∪ Z ) − {x} and R ∈ L x (A) = W x such that x...z = R. Now by the assumptions on Y ,Z ,P and W we have that As there is no partition X 1 and X 2 of Y ∪Z such that X 1 × X 2 ⊆ P and #X 1 ≥ 2 we have Y = {y} and Z = {x}. Hence (y, z) is an inseparable pair in W and therewith as well in V . This however leads by (b) to the contradiction Y = Y = {y} and Z = Z = {x}.
In order to prove Y ⊆ Y let y ∈ Y . To the contrary assume y / ∈ Y . Then by the inclusion Y ∪ Z ⊆ Y ∪ Z we may conclude that y ∈ Z and that there . By the proof of the previous inclusion, we may assume that both First we show that Y = Y . To the contrary let b ∈ Y − Y . Now for all c ∈ Y ⊆ Y, there are preferences R and R in V such that cb... = R and bc... = R . If b / ∈ (Y ∪ Z ), then R ∈ V ⊆ W violates condition 1 of W . So, b ∈ (Y ∪ Z ) which means that b ∈ Z . Because of R ∈ W, it follows that (x, b) / ∈ P for all x ∈ A − {c, b}. Now, because R ∈ W ,in view of condition 2 of W it follows that (c, b) ∈ P . Note that for all x ∈ Y − {b, c}, there are preferences R = xbc... in V ⊆ W . So, because of (c, b) ∈ P , it follows that Y = {c}. Since {c, b} ⊆ Y , it follows that (c, b) is not an inseparable pair in V . Therefore, there are x ∈ A − {c, b} and preferences Because of R ∈ V and c ∈ Y , it follows that (y, x) / ∈ P for all y ∈ A − {c, x}. But then, there are preferences R = bcx... in V ⊆ W . Note that although b, x ∈ Z and (x, b) / ∈ P , there is no t ∈ A − (Y ∪ Z ) such that b...t ...x... = R contradicting condition 2 of W . In consequence, Y = Y .
So, Y = Y and Z = Z . Next we show that P = P which then by the definition of V and W yields the desired result that V = W . First we show that P ⊆ P . To the contrary, suppose that (a, b) ∈ P and (a, b) / ∈ P . Because P ∪ P ⊆ [(Y ∪Z)×(A−Y )] and ((Y ∪Z)×(A−(Y ∪Z))) ⊆ P ∩P , it follows that a ∈ Y ∪ Z and b ∈ Z. Now because of the definition of V there are R ∈ V with bat... = R for some t ∈ (A − (Y ∪ Z)). Since V ⊆ W , this clearly yields a contradiction with condition 2 of W . So, P ⊆ P .
Next we prove P ⊆ P . Suppose (a, b) / ∈ P . We prove that (a, b) / ∈ P . Because (a, b) / ∈ P it follows by the definition of V that there are R ∈ V such that both y... = R for some y ∈ Y and ...b...a... = R. As R ∈ V ⊆ W . This shows that (a, b) / ∈ P .

Conclusion
By restricting the domain of only one agent, we showed that it is possible to escape the negative conclusions of the Gibbard-Satterthwaite and the Muller-Satterthwaite theorems. Obviously, the social choice functions we construct that are strategy-proof, Pareto optimal and non-dictatorial or Maskin monotonic, Pareto optimal and non-dictatorial, indeed have a dictatorship flavor since, at many preference profiles, the agent with a restricted set of preferences gets his top alternative. A question of interest would be to study how these social choice functions evolve as we restrict the preferences of more than one agent. We leave this question open for future research.
We close the discussion with two examples. The first example shows that if we restrict the sets of preferences of exactly two agents, then neither the condition of inseparable pair nor the condition of inseparable set is a necessary condition for the maximality of a strategy-proof possibility domain. So at K agent 2 is decisive on the pairs (x, y) and (x, w) whereas agent 1 is decisive on the remaining pairs. So K is not dictatorial. As K(p) ∈ {best(p(1)), best(p(2))} we have that K is Pareto optimal. To see that K is strategy-proof consider profiles p with p(2) = x.... Then the outcome for any {1}-deviation of p is either x or z depending on whether agent 1 prefers x to z or z to x respectively. A similar reasoning holds for {2}-deviation of profile p with p(1) = y... or p(1) = w.... It is straight forward to see that for both agents i the set L i does not have an inseparable pair or set. By which we may conclude that these inseparabilities are not necessary conditions for maximal strategy-proof possibility domains in case the set of preferences of more than one agent is restricted. Note further that K is tops-only.
The following and last example show a strategy-proof possibility domain for two agents with the same set of admissible preferences. Also in this case these sets are not containing an inseparable pair or set. Moreover the social choice rule which is discussed in this example is not tops-only. Since this social choice function treats agents symmetrically, it is non-dictatorial. Given the special position of w, K is also Pareto optimal. In general such imputation social choice correspondences on unrestricted domains are strategyproof if the outcome sets are compared on their best alternatives only. But then as K yields a singleton at each of the profiles in the restricted domain at hand, we have as a consequence that it is strategy-proof. Notice that this set of admissible preferences does neither have an inseparable pair nor an inseparable set.