Dynamics of Stable Sets of Constitutions

Transcription

1 Dynamics of Stable Sets of Constitutions HOUY Nicolas February 26, 2007 Abstract A Self-Designating social choice correspondence designates itself when it is implemented as a constitution and when the society has to choose a constitution from the set it is a member of. A set of constitutions is Stable if it has at least one self-designating correspondence for any preference profile of the society. A set of constitutions is Dynamically Stable if, implementing any constitution of the set, it is always possible to indirectly reach a self-designating one. We give necessary and sufficient sets of conditions for a set of constitutions to be stable and, more generally, dynamically stable. A subset of self-designating constitutions is Strongly Self-Designating if all the other self-designating constitutions indirectly designate one of its member whereas its members do not designate any other selfdesignating constitution. We identify Strongly Stable sets of constitutions, i.e. sets that always show a strongly self-designating subset. We also prove that strong stability and dynamic stability is equivalent to the the strongly self-designating subset being a singleton. Key Words: Social Choice Correspondence, Stability, Self-Selectivity, Dynamics. JEL Classification: D71. Hitotsubashi University and THEMA, Université de Cergy-Pontoise. nhouy@free.fr. M. Cohen, J.-F. Laslier, J.-M. Tallon and J.-C. Vergnaud are gratefully acknowledged. Participants of the CID work group in Paris and the Social Choice seminar in Caen are also thanked.

2 1 Introduction Ideally, what has been or should be the first collective decision ever made? It seems natural to state that it should answer the following question: "How should political decisions be made?". Said differently, the first political decision should be the choice of a constitution. To answer this first question, Rousseau 1 suggests that unanimity is necessary. The obvious problem raised by this requirement, as noticed by Locke 2 is the following: a priori, we don t know what each individual s preferences are, and following, there could well never be unanimity on the choice of a constitution. The possibility of the unanimity is an assumption on preferences rather than a way to choose. As a consequence, to answer this first question, we will have to suggest a condition different from unanimity. We propose that the constitution be Self-Designating. 3 A constitution is self-designating if, when implemented, it is accepted as a constitution. Let us already notice that a constitution is not self-designating by nature. Its self-designation feature depends on the different constitutions proposed and on the preferences of individuals among those constitutions. Let us look at an example to illustrate this and go a little bit further. Let the society be composed of three individuals {1; 2; 3}. Assume that there are four possible constitutions, i.e. processes to make a collective decision: the dictatorship of individual 1, dict 1, the dictatorship of individual 2, dict 2, 1 "Indeed, if there were no prior convention, where, unless the election were unanimous, would be the obligation on the minority to submit to the choice of the majority? How have a hundred men who wish for a master the right to vote on behalf of ten who do not? The law of majority voting is itself something established by convention, and presupposes unanimity, on one occasion at least." [Rousseau, 1762]. 2 "For if the consent of the majority shall not in reason, be received, as the act of the whole, and conclude every individual; nothing but the consent of every individual can make any thing to be the act of the whole: But such a consent is next impossible ever to be had." [Locke, 1690]. 3 To be formally precise, this condition would be a weakening of the unanimity if all the constitutions considered were satisfying the Pareto axiom. We will not impose such a requirement. 1

3 the dictatorship of individual 3, dict 3 and the Borda rule, borda, where each individual awards n points to her favorite option among n, n 1 points to her next favorite option,..., 1 point to her least favorite option; the option with the greatest number of points is selected by the society. We consider that individuals have preferences over this set of constitutions and we denote i the preferences of individual i over the set {dict 1 ;dict 2 ;dict 3 ;borda}. For instance, let the preferences be given by Preferences 1. Preferences 1 dict 1 1 borda 1 dict 3 1 dict 2, dict 3 2 borda 2 dict 2 2 dict 1, dict 2 3 dict 1 3 borda 3 dict 3. In this example, implementing dict 1 leads to a stable political life. Indeed, what if the society decides to choose from the possible constitutions by implementing the current constitution, dict 1? The society (in fact, only individual 1 here) chooses dict 1 since dict 1 is the favorite option of individual 1, and there is no need for a change. dict 1 is self-designating and can positively be implemented. For the same reason, borda is also self-designating. Indeed, it counts 8 points, which is the greatest number of points obtained (7 points are obtained by dict 2 and dict 3, 8 points are obtained by dict 4 1 ). Nevertheless, for the opposite reason dict 2 and dict 3 do not lead to a stable political life. Indeed, implementing dict 2 induces the change for dict 3 and implementing dict 3 induces the change for dict 2. dict 2 and dict 3 are not self-designating. Whatever their normative features, they cannot be used as constitutions almost by definition with the given preferences. In this sense they are positively not implementable. As already noticed, the self-designation feature is a feature of one constitution but it is fully dependent on the set it is considered with and the preferences of the individuals. 4 The equal numbers of points between dict 1 and borda will be discussed further in the introduction. 2

4 A set of constitutions that has at least one self-designating element, whatever the preferences of individuals, will be called stable and it satisfies the Stability axiom. We emphasize the fact that in this entire study, stability must be understood as a political notion. It is not related to the usual notion of mathematical stability. Preferences of the individuals are given (even though not necessarily known) and we do not study how constitutions respond to some shock on preferences. As an illustration of stability, let us show that the set {dict 1 ;dict 2 ;dict 3 ;borda} is not stable. Indeed, if the preferences of individuals were given by Preferences 2, there would be no self-designating constitution. Preferences 2 dict 2 1 dict 1 1 dict 3 1 borda, dict 3 2 dict 2 2 dict 1 2 borda, dict 1 3 dict 3 3 dict 2 3 borda. We argue that for positive reasons a constitutionalist cannot propose a set that is not stable to some individuals for them to choose a constitution if he does not know a priori the preferences of the individuals. If he proposed such a set, the possibility would exist that the society never reach political stability and hence, could never make any "legitimate" decision. This requirement of stability of sets of constitutions is, in our sense, a necessary condition. Then, if, as suggested above, we want to build a set of constitutions that allows the people to choose how to choose, it seems minimal that it should be stable. For the same purpose, it also seems natural that all its elements should be neutral. A constitution is neutral if, and only if, its outcome always depends only on the preferences of the individuals. If a constitutionalist wants to let the people choose how to choose he does not want to influence the society by any means, and in particular by introducing his own preferences as an argument for any choice. So he considers only neutral constitutions. 3

5 We will therefore restrict our study to neutral constitutions. In the first section, we will prove that a set of c neutral constitutions is stable if, and only if, whatever the preferences of the individuals, n constitutions of this set designate the same c+1 n constitutions of this same set. If there is unanimity (one constitution is designated by all the others and itself (n = c)), or universality (one constitution designates all the others and itself (n = 1)), the set is stable. Between those two extremes there is a trade-off. The constitutions of a "large enough" subset must be "similar enough". The larger the subset, the smaller the similarity needs to be. For instance, a set of four constitutions is stable if, and only if, for any preference profile one of the following is true: 1 constitution designates 4 constitutions, 2 constitutions designate the same 3 constitutions, 3 constitutions designate the same 2 constitutions, all the 4 constitutions designate the same constitution. We will show that this trade-off implies that a constitutionalist who wants to build a stable set of constitutions can not have them very precise and very different. As a generalization, we will say that a subset of constitutions of size n densely designates a (possibly different) subset of constitutions of size n if, and only if, l constitutions of the first subset all designate n + 1 l constitutions of the second one. For instance, a subset of four constitutions densely designates another subset of four constitutions if, and only if, one of the following is true: 1 constitution of the first subset designates 4 constitutions of the second subset, 4

6 2 constitutions of the first subset designate the same 3 constitutions of the second subset, 3 constitutions of the first subset designate the same 2 constitutions of the second subset, all the 4 constitutions of the first subset designate the same constitution of the second subset. As above, the interpretation is the following: a subset X densely designates another subset X if, and only if, a "large enough" subset of constitutions of X is "similar enough" when X is considered. As a consequence, our first result can be restated as follows: a set of constitutions is stable if, and only if, it densely designates itself. While the first section deals with the existence of self-designating constitutions, the second and third sections deal with the dynamics of the sets of neutral constitutions. Two dynamic features are considered. The first one deals with the relationship between self-designating constitutions on the one hand, and those that are not on the other hand. In our previous example with Preferences 1, there exists a cycle between dict 2 and dict 3, which are not self-designating. If, for any reason, the initial constitution is dict 2, the political life of the given society will never be stable. Using the idealistic framework stated by Rousseau, reaching one of the self-designating constitutions in a stable set can require the test of all of them. Here, dict 1 or borda that are legitimate constitutions because of their self-designating feature will never be reached unless dict 1 or borda themselves are implemented. We will rule out such situations with the Dynamic Stability axiom. This axiom states that from any constitution, there exists a path that leads to a self-designating one, and this is true for any preference profile. 5

7 Let us imagine the following justifications. The first one relies on the fact that once a constitution is used, the people could not accept to change it without its consent, illegitimately. Then, if the constitutionalist does not know what are the preferences of the individuals a priori, he needs to know that whatever the first constitution implemented, we will legitimately reach a political stable life. The second justification can be said as follows. Assume the preferences of the individuals are allowed to change within any scope. Our dynamic stability condition imposes that, wherever it comes from, the society can reach a new (possibly the same but not necessarily) self-designating constitution. 5 We will show that a set of neutral constitutions is dynamically stable if, and only if, for any preference profile of the individuals, for any subset of constitutions X, if all the constitutions of X only designates constitutions of a subset (possibly different) X, then X must densely designate X. Said differently, there must not exist two subsets (necessarily of same size) such that all the elements of the first exclusively designate elements of the second and nevertheless the first subset does not densely designate the second one. It is then obvious that the dynamic stability condition is a generalization of the stability condition to any pair of subsets. The second relevant dynamic feature deals with the relationships between self-designating constitutions. As shown in [Houy, 2004], for any stable set of neutral and different constitutions, there exists at least one preference profile for which there are several self-designating constitutions. We will see that, besides its purely dynamic meaning, this section can alternatively be seen as a justification for discriminating between self-designating constitutions on dynamic grounds. 5 Once again, since our notion of stability is political and not mathematical, we do not mean by this interpretation that a change in preferences leads to not changing the constitution, we mean that a change in preferences leads to a new implementable (because politically stable) constitution. 6

8 In our example with Preferences 1, dict 1 and borda are self-designating. But borda designates dict 1 whereas the opposite is not true. Then, dynamically, borda can lead to or accept the implementation of dict 1 but once dict 1 is designated, there is no way to go back to borda. dict 1 could be said to be "more self-designating" than borda and we will say that such constitutions are strongly self-designating. Following, our condition of Strong Stability that requires that at least one strongly self-designating constitution exists for each preference profile, is a dynamic condition as well as a way of discriminating between self-designating constitutions in cases of multiple equilibria. We will show that strong stability is rather restrictive and we will be able to exhaustively identify sets of neutral constitutions that satisfy it. We will also show that these two dynamic features, i.e. relationships between self-designating and non self-designating constitutions on the one hand, and between self-designating constitutions themselves on the other hand, are linked. Indeed, imposing the strong stability and dynamic stability axioms is equivalent to imposing the existence and uniqueness of the strongly self-designating constitution. Thus, this paper can also be read as suggesting a solution to the multiple equilibria problem using dynamic arguments. The paper proceeds as follows. In Section 2, we briefly review the literature. In Section 3, we set the formal framework. In Section 4, we give an example to illustrate all the notions used. In Section 5, we deal with the condition of stability for sets of neutral constitutions. In Section 6, we deal with the dynamic relationships between self-designating and non self-designating constitutions. In Section 7, we deal with the dynamic relationships between self-designating constitutions themselves. In Section 8, we prove the link between those two relationships. 7

9 2 Related Literature In this part, we will briefly discuss a few recent studies that deal with the notion of political stability in social choice theory. As far as we know, the dynamic aspects of the stability problem has never been studied. [Koray, 2000] is the first to study a notion which is close to what is called self-designation here. In his study, the timing is divided into two periods. In the second one, the society has to make a choice from some fundamentals. In the first period, the society chooses how the choice will be made in the second period. Individuals build their preferences over social choice functions in the first period anticipating and considering what the outcome will be in the second period. In this framework, Universally Self-Selective social choice functions are searched for. A universally self-selective social choice function is a social choice function that is self-designating whatever the set to which it belongs. The main result is the following: "a unanimous and neutral social choice function is universally self-selective if, and only if, it is dictatorial". Intuitively, if individual i wants to make sure that her most valued alternative will be chosen in the second period, she designates herself as a dictator in the first period. Our study considers a different meaning of "stability". According to [Koray, 2000], since self-selectivity must be universal, stability is the feature of a social choice function. For us, self-designation of a social choice correspondence depends on the set it is considered with. And stability is the feature of a set of constitutions, namely that this set always contains a self-designating social choice correspondence. In this sense, and as already noticed in [Houy, 2004], we just generalize the methodology of social choice theory. Indeed, as noticed by [Sen, 1999], the usual methodology consists in trying to find social choice functions (or correspondences) "more legitimate" than others on normative grounds. We look for "more legitimate" sets of constitutions on dynamic or positive grounds here. In this sense, our frame- 8

10 work allows us to broaden the usual scope of social choice theory by dealing with new objects, namely sets of constitutions. A generalization of this idea is given in [Houy, 2004] where sets of sets of... of sets of constitutions are studied. Since we do not impose the universality of the self-selection feature, we do not need to impose the consequentialism of individuals, and this is the most important difference between [Koray, 2000] s approach and ours. We consider that the choice of a constitution is possibly made on ethical grounds. As an illustration the reader is invited to wonder if he would agree to be a dictator. I do not think the answer is obvious. In reality, it is certainly the case that preferences are built with a consequentialist dimension but not only. Such an idea has been recently studied in [Segal, 2000], for instance. As a consequence, we do not restrict individual preferences. Actually, we do not even need to deal with any fundamental. We deal only with preferences over constitutions themselves. In this sense, our study should be seen as the study of a minimal positive requirement. More recently, [Barbera and Jackson, 2004] have studied the stability of quota rules only. People are still consequentialist but in the first period they do not know their second period preferences. This work has been extended more recently by [Sosnowska, 2002] for weighted voting rules, [Wakayama, 2002] who introduces abstention and [Polborn and Messner, 2002] in overlapping generations models. As in ours, in all these studies, the "self-designation" notion is the feature of a social choice function but it is dependent on other elements of the set to which it belongs, namely, quota rules. The most important difference is that we do not restrict our attention to quota rules or any other specific rules. Moreover, all these studies need to deal with fundamentals. We will be more abstract. The same differences (consequentialism and restriction) exist between our study and the ones from the constitutional design literature such as 9

11 [Rae, 1969], [Romer and Rosenthal, 1983] and more recently [Aghion and Bolton, 2003]. 3 Formal Framework Let N be the set of individuals in the society. 2 N < is the cardinality of this set, the finite number of individuals. For any set A, we will denote R i (A) the preference relation of individual i over A. R i (A) is the set of all possible preference relations of i over A. With a natural notation, R N (A) is a preference profile for all the individuals over A. R N (A), product on i of the R i (A)s, is the set of all possible preference profiles of the society over A. We will consider that the preferences of the individuals are independent. A very important assumption for the sequel is the Universal Domain one. As already noticed, setting this assumption implies that we refuse any consequentialist justification for the preferences of the individuals. Assumption 1 (Universal Domain) The preferences of the individuals can take any form. The society must make a choice from a set of alternatives, C 0. C 0 = 2 C0 is the set of non-empty subsets of C 0. For the sake of simplicity, a first level social choice correspondence (SCC 1 ) is defined a bit differently from what is usually done in the literature. Definition 1 (SCC 1 ) f is a first-level social choice correspondence (SCC 1 ) if, and only if, c 0 C 0, R N (c 0 ) R N (c 0 ), f(c 0,R N (c 0 )) f(r N (c 0 )), f(r N (c 0 )) c 0. Notice that the choice of the society can be made from any subset of C 0. Defining SCC 1 s uniquely on C 0 would not set any problem. We will denote 10

12 C 1 the set of all SCC 1 s. We will denote C 1 the set of all non empty subsets of SCC 1 s. Then, we assume that individuals have preferences over the set of SCC 1 s. Assumption 1 implies that preferences over the set of SCC 1 s do not depend on preferences between the elements of C 0. We define a second level social choice correspondence (SCC 2 ) as a mapping, from any subset of SCC 1 s and from any profile of preferences over this subset onto the set of all non empty subsets of this subset. Definition 2 (SCC 2 ) f is a second level social choice correspondence (SCC 2 ) if, and only if, c 1 C 1, R N (c 1 ) R N (c 1 ), f(c 1,R N (c 1 )) f(r N (c 1 )), f(r N (c 1 )) c 1. We denote C 2 the set of all SCC 2 s and C 2 = 2 C2. Now, we can define a constitution as a pair of one SCC 1 and one SCC 2. Definition 3 (Constitution) f = (f 1,f 2 ) is a constitution if, and only if, f 1 C 1 and f 2 C 2. We will note Z 2, the set of all constitutions and Z 2 = 2 Z2. Finally, for any set of constitutions C = {(fi 1,fi 2 )} i 1,I Z 2, we will note Γ k (C) = {fi k } i 1,I. In this article, we will study only neutral constitutions. Before defining this notion, we will need a definition. Definition 4 (Rσ N (X)) Let X,X C 0 C 1 be such that X = X. Let σ be a one-to-one function between X and X. Rσ N (X) is defined as i N, (a,b) X,σ(a)Rσ(X)σ(b) i ar i (X)b. 11

13 Then, R N σ (X) is the preference profile obtained when each individual has changed, in his preferences, each alternative a of X by its image σ(a) of X = σ(x). For instance, if there are two alternatives, a and b, two individuals, 1 and 2, and that preferences are given by ar 1 ({a,b})b and br 2 ({a,b})a, if we define σ as σ(a) = b and σ(b) = a, we have br 1 σ({a,b})a and ar 2 σ({a,b})b. Besides, notice that if c and d are other alternatives (not necessarily of the same level as a and b), by defining σ as σ(a) = c and σ(b) = d, we can also have cr 1 σ({a,b})d and dr 2 σ({a,b})c. With this definition, the neutrality axiom can be stated as follows. Axiom 1 (Neutrality) A constitution f = (f 1,f 2 ) is neutral if and only if, i,i {0, 1}, (X,X ) C i C i with X = X, R N (X) R N (X), σ one-to-one function between X and X, x f i+1 (R N (X)) σ(x) f i +1 (R N σ (X)). As a consequence of the neutrality axiom, if two constitutions are different with respect to their SCC 1 s, then, they are also different with respect to their SCC 2 s. 6 For the sake of simplicity, we will only consider sets of constitutions such that all the SCC 1 s are mutually different. Using this remark, we will now introduce a new formalism 7. In the sequel, we will deal with a generic set of constitutions noted C Z 2. We will say that a constitution f = (f 1,f 2 ) C designates a constitution g = (g 1,g 2 ) C, and we will write g f(r N (C)) if, an only if, g 1 f 2 (R N (Γ 1 (C))). For the sake of simplicity, we will also not be fully rigorous with the notation of preferences and we will write, R N (C) instead 6 Notice that the converse is not necessarily true. Notice also that, two SCC 2 s can be identical on a given set of SCC 1 s. 7 Notice that formally, it would be sufficient that all the SCC 1 s considered be different and that all the constitutions be neutral only with respect to their SCC 2 in the following sense (see [Barbera and Jackson, 2004] for a discussion of the difference of neutralities). Axiom 2 (Weak Neutrality) A constitution (f 1,f 2 ) is weakly neutral if, and only if, X C 1, R N (X) R N (X), σ permutation of X, x f 2 (R N (X)) σ(x) f 2 (R N σ (X)). 12

14 of R N (Γ 1 (C)). We will also say "constitution" for "social choice correspondence" since a SCC 1 uniquely defines a SCC 2 and hence a constitution. Mathematically, the neutrality axiom has some importance since it allows us to deal with orbites of preferences independently. Indeed, we did not impose any consistency condition between preferences of different orbites. The orbite of a preferences profile R N (C), written < R N (C) >, is obtained by considering all the possible permutations of the alternatives. Definition 5 (Orbite) < R N (C) >= {R N (C) R N (C)/ σ,r N (C) = Rσ N (C)}. Obviously, memberships of orbites describe equivalence classes. Then, any proposition concerning C is true for all preference profiles if, and only if, it is true for all preference profiles of each orbite. In the sequel, with no loss of generality, we will concentrate on a given orbite, say < R N (C) >. By definition, it is possible that a constitution designates all the others and itself. Such a constitution will be called universal and we will say that a set of constitutions always having a universal element satisfies the Universality condition. Definition 6 (Universality) The set of constitutions C satisfies the universality condition if, and only if, R N (C) < R N (C) >, f C, g C,g f(r N (C)). It is also possible that a constitution be designated by all the others and by itself. Such a constitution will be called unanimous and we will say that a set of constitutions always having an unanimous element satisfies the Unanimity condition. Definition 7 (Unanimity) The set of constitutions C satisfies the unanimity condition if, and only if, R N (C) < R N (C) >, f C, g C,f g(r N (C)). 13

15 Finally, for practical reasons, we need a few more definitions. Definition 8 (Self-Designating (SD)) f C,f SD(R N (C)) if, and only if, f f(r N (C)). Definition 9 (Indirectly Designated (ID)) i n ID(i 0 ;R N (C)) if, and only if, there exist r N and a sequence {i k } r k=1 with k,i k C, such that k 0;r 1,i k+1 i k (R N (C)) and i n = i r. Then, SD(R N (C)) is the set of self-designating constitutions, i.e. those that, when implemented by the society, designate themselves when the preference profile is R N (C). ID(i 0 ;R N (C)) is the set of constitutions that can be reached from i 0 by iterating the choice procedure. Obviously, by definition, f SD(R N (C)) implies f ID(f;R N (C)). Definition 10 (Densely Designated (DD)) X C densely designates X C ( X = X ) if, and only if, x X, x X such that: 1. x + x = X +1 = X +1, 2. f x, g x, g f(r N (C)). We will write DD(X,X,R N (C)) for "X densely designates X ". Then, X densely designates X (X and X being of same size) if, and only of, there exist two subsets x and x of X and X respectively such that each constitution of x designates all the constitutions of x and x = X +1 x. Let us give an illustration. If X = X = 4, X densely designates X if, and only if, all the constitutions of X designate the same constitution of X, 3 constitutions in X designate 2 constitutions of X, 2 constitutions of X designate 3 constitutions of X or one constitution of X designates all the constitutions of X. Generally, the intuition behind this definition is the following: X densely designates X if, and only if, a large enough number of constitutions in X designate a large enough number of constitutions in X. 14

16 Said differently, X densely designates X if and only if, when, restricted to the constitutions of X, the constitutions of X are not too precise and too different. 4 Example To introduce the intuitions behind all the dynamics features used in this article and to make the definition clear, we will give an example. Assume a society composed of 7 individuals, N = {1,..., 7}. Assume that, f 1 is the plurality rule, f 2 is the Borda rule, f 3 is the anti-dictatorship of individual 1, f 4 is the dictatorship of individual 2, f 5 is the dictatorship of individual 3 8. Let us consider the following preferences for individuals, R N ({f 1,f 2,f 3,f 4,f 5 }): f 1 1 f 2 1 f 3 1 f 5 1 f 4, f 4 2 f 3 2 f 2 2 f 1 2 f 5, f 3 3 f 5 3 f 2 3 f 1 3 f 4, f 1 4 f 3 4 f 2 4 f 5 4 f 4, f 2 5 f 1 5 f 3 5 f 5 5 f 4, f 2 6 f 1 6 f 5 6 f 3 6 f 4, f 3 7 f 1 7 f 2 7 f 5 7 f 4.. For the sake of simplicity, and even if it not necessary in this article, we will collectively rationalize those choices and set the following weak social ordering for the considered preference profile. fi is the weak social ordering obtained when the constitution f i is implemented. f 1 f1 f 2 f1 f 3 f1 f 4 f1 f 5, 8 Notice that all these constitutions are neutral. 15

17 f 1 f2 f 2 f2 f 3 f2 f 5 f2 f 4, f 4 f3 f 5 f3 f 3 f3 f 2 f3 f 1, f 4 f4 f 3 f4 f 2 f4 f 1 f4 f 5, f 3 f5 f 5 f5 f 2 f5 f 1 f5 f 4. Then, by definition of collective rationalization, we have f i (R N (X)) = {f j X/ f k X, f j fi f k }. f 1 (R N ({f 1 ;f 2 ;f 3 ;f 4 })) = {f 1 ;f 2 ;f 3 }; f 1 (R N ({f 1 ;f 3 ;f 4 })) = {f 1 ;f 3 }; f 1 (R N ({f 1 ;f 2 ;f 3 ;f 5 })) = {f 1 ;f 2 ;f 3 }. f 2 (R N ({f 1 ;f 2 ;f 3 ;f 4 })) = {f 1 ;f 2 ;f 3 }; f 2 (R N ({f 1 ;f 2 ;f 3 ;f 5 })) = {f 1 ;f 2 ;f 3 }. f 3 (R N ({f 1 ;f 2 ;f 3 ;f 4 })) = {f 4 }; f 3 (R N ({f 1 ;f 3 ;f 4 })) = {f 4 }; f 3 (R N ({f 1 ;f 2 ;f 3 ;f 5 })) = {f 5 }. f 4 (R N ({f 1 ;f 2 ;f 3 ;f 4 })) = {f 4 }; f 4 (R N ({f 1 ;f 3 ;f 4 })) = {f 4 }. f 5 (R N ({f 1 ;f 2 ;f 3 ;f 5 })) = {f 3 }. In a first step, let us assume that this society must choose its constitution in the set {f 1 ;f 3 ;f 4 }. If the preference profile is given by R N ({f 1 ;f 3 ;f 4 }), then, f 1 and f 4 are self-designating since f 1 f 1 (R N ({f 1 ;f 3 ;f 4 })) and f 4 f 4 (R N ({f 1 ;f 3 ;f 4 })) whereas f 3 is not, f 3 / f 3 (R N ({f 1 ;f 3 ;f 4 })). If we impose only the self-designation feature as a minimal requirement (i.e. that there exist at least one self-designating constitution), this set is valid, for the preference profile R N ({f 1 ;f 3 ;f 4 }). But one can wonder what would happen if we did not know the preference profile a priori. Then, it is possible that the preference profile be given by Rσ N ({f 1 ;f 3 ;f 4 }) with σ(f 1 ) = f 4, σ(f 3 ) = f 3 16

18 and σ(f 4 ) = f 1. In this case, by neutrality, we have f 1 (Rσ N ({f 1 ;f 3 ;f 4 })) = σ({f 1 ;f 3 }) = {f 3 ;f 4 } and f 3 (Rσ N ({f 1 ;f 3 ;f 4 })) = f 4 (Rσ N ({f 1 ;f 3 ;f 4 })) = {f 1 }. No constitution is self-designating, the set of constitutions {f 1 ;f 3 ;f 4 } does not always display (i.e. for R N ({f 1 ;f 3 ;f 4 }) and all the preference profiles of its orbite) a self-designating element. The condition imposing that there always exist a self-designating constitution, not satisfied by {f 1 ;f 3 ;f 4 }, is the Stability condition. The reader can check that the set of constitutions {f 1 ;f 2 ;f 3 ;f 5 } satisfies the stability condition. The second condition has the following meaning: departing from any initial constitution and for any preference profile, the society can reach a self-designating constitution. Said differently, any constitution indirectly designates a self-designating constitution. This condition is the Dynamic Stability condition. In this example, {f 1 ;f 2 ;f 3 ;f 5 } does not satisfy the dynamic stability condition. Indeed, if the preferences of the individuals are given by R N ({f 1 ;f 2 ;f 3 ;f 5 }) and if the initial constitution is f 3, the first political decision will change the constitution for f 5. Then, implementing f 5, the constitution will be changed for f 3 and there is no way out of this cycle. We can check that the set of constitutions {f 1 ;f 2 ;f 3 ;f 4 } satisfies the dynamic stability condition. Let us still consider the set of constitutions {f 1 ;f 2 ;f 3 ;f 4 }. When the preference profile is given by R N ({f 1 ;f 2 ;f 3 ;f 4 }), f 1, f 2 and f 4 are selfdesignating. We can also see that f 4 is indirectly designated by f 1 and f 2 whereas those two constitutions are not indirectly designated by f 4. Then, if initially, f 1 or f 2 (or even f 3 ) are implemented, f 4 can be reached whereas the converse is not true. In some sense, f 4 leads to the most stable political life. We will say that f 4 is strongly self-designating (whereas f 1 and f 2 are only self-designating). The condition imposing that there always exist at least one strongly self-designating constitution is the Strong Stability condition. We will see that this condition is almost as restrictive as the the condition 17

19 of Unique Strong Stability that requires that there always exist exactly one strongly self-designating constitution. We could check that {f 1 ;f 2 ;f 3 ;f 4 } satisfies the unique strong stability condition whereas {f 1 ;f 2 ;f 3 ;f 5 } does not. In the sequel, we will give relationships between all those different stability conditions. 5 Stability The first condition on C imposes that there always exist at least one self-designating constitution. Axiom 3 (Stability) The set of constitutions C satisfies the stability axiom if, and only if, R N (C) < R N (C) >, f C such that f SD(R N (C)). We are able to display necessary and sufficient conditions for a set of neutral constitutions to be stable. Theorem 1 9 The set C of c neutral constitutions, satisfies the stability axiom if, and only if, ( )R N (C) < R N (C) >, X C and X C such that: 1. X + X = c + 1, 2. f X, g X, g f(r N (C)). Then, a set of c neutral constitutions is stable if and only if there (always) exist two subsets of constitutions X and X such that the sum of the sizes of X and X equals c + 1 and all the constitutions of X designate all the constitutions of X. There are two special cases. If X = 1, one of the 9 For the sake of simplicity, we did not use all the definitions introduced so far. The same theorem could have been written as follows: The set C of c neutral constitutions, satisfies the stability axiom if, and only if, ( ) 10 R N (C) < R N (C) >, DD(C; C;R N (C)). 18

20 constitutions has to designate all the others and itself: it is universal. If X = c, one of the constitutions is designated by all, it is unanimous. Corollary 1 For a set of neutral constitutions C, Unanimity or Universality imply Stability. Between those two extreme cases, we can have intermediate situations. The smaller X, i.e. the number of constitutions that must have positive agreement on X, the greater this agreement, i.e. the size of X. Said differently, if one wants stability to be satisfied by C, there is a trade-off between the number of constitutions that have to positively resemble each other and the minimal degree to which they resemble each other. In other words, we cannot have very precise (small X ) and very different constitutions (small X). Let us illustrate this theorem by an example. We have seen in Section 4 that the set C = {f 1 ;f 2 ;f 3 ;f 5 } with f 1 (R N (C)) = {f 1 ;f 2 ;f 3 }, f 2 (R N (C)) = {f 1 ;f 2 ;f 3 }, f 3 (R N (C)) = {f 5 } and f 5 (R N (C)) = {f 3 }, satisfies the stability condition. Indeed, we can consider X = {f 1 ;f 2 } and X = {f 1 ;f 2 ;f 3 } so that the conditions given in Theorem 1 are satisfied. On the contrary, if we have C = {f 1;f 2;f 3;f 5} with f 1(R N (C )) = {f 2;f 3}, f 2(R N (C )) = {f 1;f 2;f 3}, f 3(R N (C )) = {f 5} and f 5(R N (C )) = {f 3}, then it is not possible to find two sets X and X that satisfy the conditions given in Theorem 1. And indeed, we can check that if the preferences were given by R N ({f 1;f 5;f 3;f 2}), then one would have f 1(R N ({f 1;f 5;f 3;f 2})) = {f 5;f 3}, f 2(R N ({f 1;f 5;f 3;f 2})) = {f 1;f 5;f 3}, f 3(R N ({f 1;f 5;f 3;f 2})) = {f 2} and f 5(R N ({f 1;f 5;f 3;f 2})) = {f 3} and then, there would be no self-designating constitution. The set {f 1;f 2} of constitutions that are the most similar agree only on the set {f 2;f 3}. To have stability, we would need to have either a one-element broader agreement or the same agreement shared by one more constitution. In {f 1;f 2;f 3;f 5}, constitutions are too different (or equivalently 19

21 too precise) for having stability. Summing up, if, as we interpreted above, the purpose of the constitutionalist is to build a set of constitutions from which a society could choose its constitution, and if this constitutionalist does not know anything about the preferences of the individuals a priori, then, on a given orbit, he cannot design very different and very precise constitutions. There is necessarily a trade-off between, on the one hand the differences between the constitutions of the set, and on the other hand the preciseness of all the constitutions of the set. Indeed, the more precise the constitutions, the less broad the agreement between (possibly partially) similar constitutions can be. 6 Dynamic Stability As shown in Section 4, stability does not imply dynamic stability. Stability imposes that there always be at least one self-designating constitution. However, it does not imply that, implementing any constitution of C, one should reach one of them. It is possible to be in a cycle with no selfdesignating constitution and no way out. The dynamic stability axiom copes with this problem: Axiom 4 (Dynamic Stability) The set C of constitutions satisfies the Dynamic Stability axiom if, and only if, R N (C) < R N (C) >, f C, g SD(R N (C)),g ID(f,R N (C)). This axiom requires that, not only, should there always be at least one self-designating constitution, but that also, departing from any constitution, one can reach one of the self-designating one. The following proposition is then obvious: Proposition 1 For a set C of constitutions, Unanimity implies Dynamic Stability. 20

22 We can prove the following theorem as well. Theorem 2 A set C of neutral constitutions satisfies the Dynamic Stability axiom if, and only if, ( )R N (C) R N (C), X,X C such that X = X, one of the following is true: 1. DD(X,X,R N (C)), 2. f X, g C \ X, g f(r N (C)). First, let us notice that this theorem is a generalization of Theorem 1. Indeed, Theorem 1 corresponds to the case X = X = C. Then, it is obvious that dynamic stability implies stability. Proposition 2 For a set C of constitutions, Dynamic Stability implies Stability. Theorem 2 states that dynamic stability is satisfied if, and only if, the following statement is true: if all constitutions of the subset X designate exclusively constitutions from the subset X (with X = X ), then X must densely designate X. Above, we interpreted Theorem 1 as a condition to satisfy if a constitutionalist wants to build a "good" set of constitutions. How could Theorem 2 be interpreted in the same framework? Let us consider a subset X of constitutions. There are three possibilities. I. the outcomes of the constitutions of X are numerous enough, i.e. the union of the outcomes of X has a strictly greater size than X, II. the outcomes of the constitutions of X are not numerous enough, i.e. the union of the outcomes of X has a smaller size than X, but the constitutions of X satisfy the unanimity/universality trade-off explained in Section 5, 21

23 III. the outcomes of the constitutions of X are not numerous enough, i.e. the union of the outcomes of X has a smaller size than X, and the constitutions of X do not satisfy the unanimity/universality trade-off explained in Section 5. If, for any X, conditions I or II are satisfied, then the dynamic stability condition is also satisfied. On the contrary, if there exists a subset of constitutions that has a "small" range of outcomes and those outcomes are not identical or universal enough (with respect of its size), then, dynamic stability is not satisfied. Then, simplifying a bit, Theorem 2 expresses a stronger preciseness/difference trade-off than Theorem 1 in the sense that this trade-off should be satisfied for any two subsets of constitutions. Finally, let us illustrate Theorem 2 by two examples. First, let us define the following constitutions for a given preference profile, R N ({f 1 ;f 2 ;f 3 ;f 4 ;f 5 ;f 6 ;f 7 }) = R N (C). f 1 (R N (C)) = f 2 (R N (C)) = f 3 (R N (C)) = f 4 (R N (C)) = {f 1 ;f 3 ;f 4 ;f 7 }, f 5 (R N (C)) = {f 2 ;f 6 }, f 6 (R N (C)) = {f 5 ;f 6 }, f 7 (R N (C)) = {f 2 ;f 5 }. It is easy to check that for the given preference profile, any constitution can lead to a self-designating one. However, this set of constitutions does not satisfy the dynamic stability axiom. Indeed, if one considers X = {f 5 ;f 6 ;f 7 } and X = {f 2 ;f 5 ;f 6 }, each element of X only designates elements in X and X does not densely designates X, DD(X,X,R N (C)). Let us check that C is not dynamically stable. Let us have σ defined by σ(f i ) = f i for i {1, 3, 4, 5}, σ(f 2 ) = f 6, σ(f 6 ) = f 7 and σ(f 7 ) = f 2. Then f 1 (Rσ N (C)) = f 2 (Rσ N (C)) = f 3 (Rσ N (C)) = f 4 (Rσ N (C)) = {f 1 ;f 2 ;f 3 ;f 4 }, 22

24 f 5 (Rσ N (C)) = {f 6 ;f 7 }, f 6 (Rσ N (C)) = {f 5 ;f 7 }, f 7 (Rσ N (C)) = {f 5 ;f 6 }. As one can see, from f 5, f 6 or f 7 it is not possible to reach any self-designating constitution (which are f 1, f 2, f 3 and f 4 here). As a second illustration, consider the case given in Section 4. We already noticed that the set of constitutions {f 1 ;f 2 ;f 3 ;f 5 } is not dynamically stable whereas the set {f 1 ;f 2 ;f 3 ;f 4 } is. Indeed, one can remark that, in the set {f 1 ;f 2 ;f 3 ;f 5 }, if one considers the subsets X = X = {f 3 ;f 5 }, the elements of X exclusively designate those of X, but X does not densely designates X. Notice that in the set {f 1 ;f 2 ;f 3 ;f 4 } there is no such pair of subsets of same size. For instance the constitutions of the subset X = {f 3 ;f 4 } exclusively designate the constitutions of X = {f 3 ;f 4 } but X densely designates X. Alternatively, if X = X = {f 2 ;f 3 ;f 4 }, X does not densely designate X but each element of X does not exclusively designate those of X. Another way to express Theorem 2 is as follows. Proposition 3 A set C of neutral constitutions satisfies the dynamic stability axiom if, and only if, R N (C) R N (C), X C, [ f X, g C \ X, such that g f(r N (C))] DD(X,X,R N (C)). Then, a set of neutral constitutions satisfies the dynamic stability axiom if, and only if, for any subset of constitutions X, for any preference profile, if all the designated constitutions by elements of X are in X, then X densely designates itself. This proposition, closer to the initial definition of dynamic stability, means the following. Assume that implementing any constitution from the subset X of C necessarily keeps the society within X. Then, if restricting the 23

25 designation system to X does not satisfy the stability condition, it is obvious that C does not satisfy dynamic stability. If this case never happens, the set C satisfies dynamic stability. 7 Strong Stability In the previous section, we have dealt with the relationship between selfdesignating constitutions and those that are not self-designating. We will now deal with the relationship between self-designating constitutions themselves. So far, we have not addressed the problem of multiple equilibria. As shown in the examples, a set can satisfy dynamic stability and simultaneously there can be several self-designating constitutions in it for some preference profiles. Further, [Houy, 2004] proved that it is not possible to achieve uniqueness of the self-designating constitution for all preference profiles. So the question we raise in this section is the following: when there are several self-designating constitutions, are some of them more "legitimate" as constitutions to be used by the society? And, if the answer is positive, is it possible to impose the uniqueness of it on all preference profiles? Let us assume the following case. C = {f 1 ;f 2 ;f 3 }, the preference profile is given by R N (C) and: f 1 (R N (C)) = {f 1 ;f 2 }, f 2 (R N (C)) = {f 2 ;f 3 }, f 3 (R N (C)) = {f 3 }. In this example, all the constitutions are self-designating. However, implementing f 1 and f 2 can lead to f 3 whereas f 3 cannot lead to any other self-designating constitution. We claim that f 3 is more "legitimate", in the sense that it is more "stable" than f 1 and f 2. This idea is formalized in the 24

26 following definition. We denote the set of constitutions such as f 3, Strongly Self-Designating: Definition 11 (SSD) Let C be a set of constitutions. SSD(R N (C)) SD(R N (C)) is defined by: f SSD(R N (C)), g SD(R N (C)) f, g / ID(f,R N (C)), g SD(R N (C)) SSD(R N (C)), f SSD(R N (C)) s.t. f ID(g,R N (C)). If it exists 11, SSD(R N (C)) is the set of self-designating constitutions that does not indirectly (or directly) designate any other self-designating constitution. Moreover, all self-designating constitutions not in SSD(R N (C)) must indirectly (or directly) designate one constitution of SSD(R N (C)). So this definition considers SSD(R N (C)) as the maximal subset of SD(R N (C)) of constitutions that do not designate any other self-designating one, and if there is a subset of SD(R N (C)) of constitutions designating exclusively each other, then SSD(R N (C)) does not exist. 12 For instance, if C = {f 1 ;f 2 ;f 3 } and f 1 (R N (C)) = {f 1 ;f 2 }, f 2 (R N (C)) = {f 1 ;f 2 }, f 3 (R N (C)) = {f 3 }, then SSD(R N (C)) does not exist even though SD(R N (C)) = {f 1 ;f 2 ;f 3 }. Viewed in this way, the constitutions of SSD(R N (C)) have interesting features. Indeed, if this set exists, then once the society implements a constitution of SSD(R N (C)), it cannot reach another self-designating constitution, whereas if it implements a self-designating constitution that is not an element 11 Saying that SSD(R N (C)) does not exist is different from saying that it is an empty set. Clearly, by definition, if SD(R N (C)), then SSD(R N (C)) (but still it can not exist). If SD(R N (C)) =, then SSD(R N (C)) = (and then, it exists). Moreover, it is easy to see that if SSD(R N (C)) exists, it is unique. 12 In particular, SD(R N (C)) = 1 SD(R N (C)) = SSD(R N (C)). 25

27 of SSD(R N (C)), then it can reach one that is an element of SSD(R N (C)). However, if SSD(R N (C)) does not exist, there is the possibility that the society never be in such a comfortable situation. This is why, we impose that SSD(R N (C)) exists for any preference profile. Axiom 5 (Strong Stability) The set C of constitutions satisfies the Strong Stability axiom if, and only if, R N (C) < R N (C) >, SSD(R N (C)) exists. Clearly, this axiom is stronger than the stability axiom. Actually, it is much stronger since we are able to fully identify the sets of neutral constitutions that satisfy the Strong Stability axiom: Theorem 3 Let C = {(f i ) i 1;c } be a set of neutral constitutions. C satisfies strong stability if, and only if, it is one of: c 4 and (i,j,k) 1;c (i j) such that: l 1;c,l / {i;j} {f k } = f l (R N (C)) and, {(f r ) r 1;c {k} } = f i (R N (C)) = f j (R N (C)), f C unanimous and g C f, {f /g f (R N (C))} 1, f C universal and h C f, h(r N (C)) = 1. An immediate corollary is that C, to satisfy Strong Stability, must necessarily satisfy the stability axiom with X {1; 2;c} in Theorem 1. To ease the understanding, a matrix representation of sets of neutral constitutions satisfying the strong stability axiom is given in the end of the next section. 8 Unique Strong Stability In the previous section, we have discriminated between self-designating constitutions. But it is still possible that there are several legitimate potential 26

28 constitutions and then we still did not fully tackle the multiple solutions problem. For instance, if C = {f 1 ;f 2 ;f 3 } and f 1 (R N (C)) = {f 1 ;f 2 ;f 3 }, f 2 (R N (C)) = {f 2 }, f 3 (R N (C)) = {f 3 }, then the strong stability axiom is satisfied and here SSD(R N (C)) = {f 2 ;f 3 } = 2. Could we go further and now impose the uniqueness of the Strongly Self- Designating constitution? This way, we would definitely solve the multiple equilibria problem using dynamic requirements. Axiom 6 (Unique Strong Stability) The set C of constitutions satisfies the Unique Strong Stability axiom if, and only if, R N (C) < R N (C) >, SSD(R N (C)) exists, and SSD(R N (C)) = 1. This axiom is obviously stronger than the strong stability one. How restrictive it is for C is shown in the following theorem: Theorem 4 Let C = {(f i ) i 1;c } be a set of neutral constitutions. C satisfies unique strong stability if, and only if, it is one of the following: c 4 and (i,j,k) 1;c (i j) such that: l 1;c,l / {i;j} {f k } = f l (R N (C)) and, {(f r ) r 1;c {k} } = f i (R N (C)) = f j (R N (C)), f C unanimous and g C f, {f /g f (R N (C))} 1. As can be clearly seen, if one rules out the cases of universality, there is no difference between strong stability and unique strong stability. But, more importantly, a consequence of this last theorem is that there is a link 27

29 between our two parts: there is a dependence between, on the one hand, the relationship between the self-designating constitutions and the others, and on the other hand, the relationship between the self-designating constitutions themselves. This is what is expressed in Theorem 5: Theorem 5 For a set C of neutral constitutions, Unique Strong Stability Strong Stability and Dynamic Stability. In this case, the sets of constitutions given in Theorem 4 have very interesting features. Indeed, in those cases and only in those, SSD(R N (C)) is a singleton for each preference profile and this constitution is reachable from any other, self-designating or not self-designating. Finally, to ease the understanding, we give a matrix representation of the sets of constitutions displayed in Theorems 3 and 4. To do so, let us have the set of neutral constitutions, C = {f i } i 1;n. Since all the SCC 1 are different, we will represent it in n n boolean matrix {x ij }. 13 Let us consider a preference profile R N (C). We will set that x ij = 1 if, and only if, f i designates f j, f j f i (R N (C)). Following, since the constitutions are neutral, a permutation σ of the preferences can be represented simply by a permutation σ of the columns of the initial matrix. Hence, a matrix represents the designation scheme for one preference profile only, but all the designation schemes for the whole orbite of preference profiles can be obtained by permuting the columns of the matrix. With this representation, the following sets of neutral constitutions, C = {f i } i 1;n, are the only ones to satisfy the Unique Strong Stability axiom. Indeed, they are representations of the sets of constitutions displayed in Theorem x ij will be the element in the i th row, j th column. 28