Eric Danan
email: eric.danan@u-cergy.fr.

Thibault Gajdos
email: thibault.gajdos@univmed.fr

Jean-Marc Tallon
email: jean-marc.tallon@univ-paris1.fr.

Philippe Mongin

Aggregating sets of von Neumann-Morgenstern utilities

Keywords: Aggregation, vNM utility, indeterminacy, neutrality, utilitarianism. JEL Classification. D71, D81

published or not. The documents may come

Introduction

A common feature of social choice theory is the assumption that individual have completely determined preferences or utilities, that are fully identified by the social planner. Such an assumption might, however, seem very demanding. First, individuals may envision more than one utility function, either because their preferences are incomplete (Aumann [2], Bewley [START_REF] Bewley | Knightian decision theory: Part I[END_REF], Dubra et al. [START_REF] Dubra | Expected utility theory without the completeness axiom[END_REF], Evren and Ok [START_REF] Evren | On the multi-utility representation of preference relations[END_REF]), or because they are uncertain about their tastes (Cerreia-Vioglio [START_REF] Cerreia-Vioglio | Maxmin expected utility over a subjective state space: Convex preferences under risk[END_REF], Dekel et al. [START_REF] Dekel | Representing preferences with a unique subjective state space[END_REF], Koopmans [START_REF] Koopmans | On the flexibility of future preferences[END_REF], Kreps [START_REF] Kreps | A representation theorem for "preference for flexibility[END_REF]), or because they are driven by several "selves" or "rationales" (Ambrus and Rozen [START_REF] Ambrus | Rationalizing choice with multi-self models[END_REF], Green and Hojman [START_REF] Green | Choice, rationality and welfare measurement[END_REF], Kalai et al. [START_REF] Kalai | Rationalizing choice functions by multiple rationales[END_REF], May [START_REF] May | Intransitivity, utility, and the aggregation of preference patterns[END_REF]). Second, even if all individuals have single, fully determined utility functions, fully identifying them may be either infeasible or too costly, so that the social planner may be unable or unwilling to assign a precise utility function to each individual (C.Manski [START_REF] Manski | Social choice with partial knowledge of treatment responses[END_REF][START_REF] Manski | Policy choice with partial knowledge of policy effectivemess[END_REF]).

This paper proposes a first step towards a social choice theory allowing for situations in which individuals have only partially determined preferences or utilities. In order to account for such situations, we endow individuals with sets of utility functions. Such a set represents the possible utility functions the individual may have, according to the social planner. The particular case where this set is a singleton then corresponds to the standard setting in which the individual has a single, fully determined utility function. We shall say that the utility function is determinate in this case and indeterminate otherwise, to summarize the different situations mentioned above.

We consider risky alternatives and study social welfare functions mapping profiles of individual von Neumann-Morgenstern (henceforth vNM) utility sets to social vNM utility sets. In this setting, when utility functions are determinate, classical Pareto and Independence of Irrelevant Alternatives axioms lead to a very specific and tractable form of the social welfare function: utilitarianism (Coulhon and Mongin [START_REF] Coulhon | Social choice theory in the case of von neumann-morgenstern utilities[END_REF]). Specifically, the Pareto and Independence of Irrelevant Alternatives axioms together yield a classical neutrality property: the social utility level is a function of the individual utility levels only. Linearity of the vNM utility function then straightforwardly implies linearity of the social utility level in the individual utility levels. Hence, the social utility function is a linear combination of the individual utility functions.

As it turns out, moving from determinate to indeterminate utilities makes things much more complex even in this simple setting. First, in general, there is not a single utility level, but a set (in fact, an interval) of possible utility levels for each alternative, so that the classical neutrality property cannot be used. Natural generalizations of the Pareto and Independence of Irrelevant Alternatives axioms nevertheless yield a generalized "interval-neutrality" property: the social utility interval is a function of the individual utility intervals only. Second, linearity of vNM utility functions does not imply any linearity property for utility intervals. It however yields weaker convexity properties which, together with other regularity properties, allow us to express the social utility interval as a union of linear combinations of individual utility intervals. Third, pinning down the utility intervals for all alternatives is not enough to fully pin down the social utility set. Despite this, the generalized axioms are powerful enough to yield full-fledged characterization results.

The remainder of the paper is built as follows. Section 2 describes the setup and the generalized Pareto and Independence of Irrelevant Alternatives axioms. In Section 3 the interval-neutrality property is defined and shown to be implied by the Pareto and Independence of Irrelevant Alternatives axioms, and a characterization of interval-neutral social welfare functions is provided. This characterization of interval neutrality, although falling short of fully pinning down the social utility set, is the key technical step towards full-fledged characterization results, which are obtained in Section 4. In particular, the classes of utilitarian and "multi-utilitarian" social welfare functions are characterized, and their degree of "social indeterminacy" is analyzed. Section 5 provides generalizations of the preceding results to more general alternatives and domains. Section 6 concludes. Proofs are gathered in the Appendix.

Setup

Let X be a non-empty set of social alternatives. We assume that X is a set of probability measures (lotteries) over a non-empty set Z of social (sure) outcomes and that, furthermore, X is convex and contains all probability measures on Z with finite support (simple lotteries). Thus we may for instance take X to be the set of all probability measures on Z with finite, countable, or (if a topology on Z is given) compact support. We will maintain this assumption on X until Section 5, in which we will consider more general sets of alternatives. Given two alternatives x, y ∈ X and a number λ ∈ [0, 1], we let xλy denote the convex combination λx + (1 -λ)y ∈ X.

A utility function u on X associates to each alternative x ∈ X a utility level u(x) ∈ R.

A utility function u on X is a vNM utility function if it is affine, i.e. if u(xλy) = λu(x) + (1 -λ)u(y) for all x, y ∈ X and all λ ∈ [0, 1]. Let P ⊆ R X denote the set of all vNM utility functions on X. P is a linear subspace of R X and contains all constant functions.

Given a real number γ ∈ R, we abuse notation by also letting γ denote the corresponding constant function in P .

We consider non-empty sets of utility functions, i.e. non-empty subsets of R X . Our interpretation of such a set is that the utility function may possibly be any member of the set, without further information being available. The utility function is determinate if the set is a singleton and indeterminate otherwise. We restrict attention to sets of vNM utility functions. More precisely, let P denote the set of all non-empty, compact, and convex subsets of P , where P is endowed with the subspace topology and R X with the product topology. Note that P contains in particular all convex hulls of finite sets of vNM utility functions on X and, hence, all singletons.

A vNM utility set U ∈ P can be seen as representing an underlying preference relation over alternatives in X, in the sense that for all x, y ∈ X,

x y ⇔ [∀u ∈ U, u(x) ≥ u(y)] . (1)
The special case where U is a singleton then corresponds to standard expected utility preferences (von Neumann and Morgenstern [START_REF] Neumann | Theory of games and economic behavior[END_REF]). When U is not a singleton, on the other hand, the preference relation is incomplete, i.e. there are alternatives x, y ∈ X such that neither x y nor y x (unless all elements of U are positive affine transformations of one another), but satisfies all the other standard vNM axioms. Shapley and Baucells [START_REF] Shapley | A theory of multiperson utility[END_REF] axiomatize the existence of a closed and convex subset U of P satisfying (1), whereas Dubra et al. [START_REF] Dubra | Expected utility theory without the completeness axiom[END_REF], assuming X is the set of all Borel probability measures on a compact metric space Z, axiomatize the existence of a closed and convex set U of continuous functions in P satisfying (1). 1 Let I be a non-empty and finite set of individuals. Given a non-empty domain D ⊆ P I , a social welfare function F on D associates to each profile (U i ) i∈I ∈ D of individual (vNM) utility sets a social (vNM) utility set U = F ((U i ) i∈I ) ∈ P. We will maintain the assumption that D = P I from now on, and will relax it in Section 5.

The most classical axioms for social welfare functions with determinate utilities are the Pareto and Independence of Irrelevant Alternatives axioms. Both of them can be generalized to indeterminate utilities. The Pareto Indifference (resp. Pareto Weak Preference) axiom requires that if all individuals are indifferent between alternatives x and y (resp. weakly prefer x to y) then so does society. To the extent that a utility set U ∈ P can be seen as representing an underlying preference relation in the sense of [START_REF] Ambrus | Rationalizing choice with multi-self models[END_REF], it is natural to generalize the Pareto axioms to utility sets as follows.

Axiom 1 (Pareto Indifference). For all (U i ) i∈I ∈ D and all x, y ∈ X, if u i (x) = u i (y) for all u i ∈ U i and all i ∈ I then u(x) = u(y) for all u ∈ F ((U i ) i∈I ).

Axiom 2 (Pareto Weak Preference). For all (U i ) i∈I ∈ D and all x, y ∈ X, if u i (x) ≥ u i (y) for all u i ∈ U i and all i ∈ I then u(x) ≥ u(y) for all u ∈ F ((U i ) i∈I ).

Independence of Irrelevant Alternatives essentially requires the social ranking between alternatives x and y to depend only on the individual rankings between x and y. Again, to the extent that a utility set U ∈ P can be seen as representing an underlying preference relation in the sense of (1), it is natural to view the ranking between x and y as determined by the set {(u(x), u(y)) : u ∈ U}. Let us introduce the following notation: given a subset Y of X and a utility set U ∈ P, let U| Y denote the restriction of U to Y , i.e. 1 Neither of these two results fully characterize the existence of a utility set U ∈ P satisfying (1). Indeed, the set U in the former result cannot be taken to be compact in general. However, a sufficient condition for the existence of a compact U is that all functions in U be bounded (since we can then scale all these functions down to range within a given bounded interval and then take the closure of the convex hull of these scaled functions), and this sufficient condition will obtain in particular if either Z is finite or X contains all probability measures on Z with countable support. The set U in the latter result, on the other hand, can always be taken to be compact (because all utility functions in this set are necessarily bounded), but the additional continuity property rules out some members of P, unless Z is finite.

U| Y = {u| Y : u ∈ U}. 2 We thus generalize the Independence of Irrelevant Alternatives axiom to utility sets as follows.

Axiom 3 (Independence of Irrelevant Alternatives). For all (U

i ) i∈I , (U ′ i ) i∈I ∈ D and all x, y ∈ X, if U i | {x,y} = U ′ i | {x,y} for all i ∈ I then F ((U i ) i∈I )| {x,y} = F ((U ′ i ) i∈I )| {x,y} .
For a social welfare function with determinate utilities, Independence of Irrelevant Alternatives implies that the restriction of the social utility function to any finite subset of alternatives depends only on the restrictions of the individual utility functions to this subset of alternatives (Blau [4], D'Aspremont and Gevers [START_REF] D'aspremont | Equity and the informational basis of collective choice[END_REF]). This is simply because any function is fully determined by its restrictions to all pairs of elements of its domain. This argument does not hold for indeterminate utilities: a set of functions on a common domain is generally not fully determined by the corresponding sets of restrictions to all pairs of elements of the domain, because there is generally more than one way of "gluing" together these sets of restrictions. Nevertheless, because we restrict attention to vNM utility functions, it can in fact be shown that Independence of Irrelevant Alternatives still implies that the restriction of the social utility set to any finite subset of alternatives depends only on the restrictions of the individual utility sets to this subset of alternatives in our setting (see Lemma 15 in the Appendix).

Interval neutrality

In the classical Arrovian framework, neutrality refers to the fact that if alternatives x and y are ranked by individuals in a given situation (i.e. for a given profile of individual preferences) exactly as alternatives z and w are in another situation, then the social ranking between x and y in the first situation is the same as the social ranking between z and w in the second one. In the richer framework of social welfare functions with determinate utilities, neutrality means that if the individual utility levels of alternatives x and y in a given situation are the same as the individual utility levels of alternatives z and w in another situation, then the social ranking between x and y in the first situation is the same as the social ranking between z and w in the second one (Sen [22]). If profiles of individual utility functions are mapped to social utility functions rather than social preference relations, then neutrality can be simply stated by requiring that if the individual utility levels of alternative x in a given situation are the same as the individual utility levels of alternative y in another situation, then the social utility level of x in the first situation is the same as the social utility level of y in the second situation (Coulhon and Mongin [START_REF] Coulhon | Social choice theory in the case of von neumann-morgenstern utilities[END_REF]).

When the utility function is indeterminate, however, an alternative does not in general have a single utility level but rather a set (in fact, a non-empty and compact interval) of possible utility levels. More precisely, given a utility set U ∈ P, let U(x) = U| {x} denote the utility interval of an alternative x ∈ X. We generalize the neutrality property by requiring that if the individual utility intervals of alternative x in a given situation are the same as the individual utility intervals of alternative y in another situation, then the social utility interval of x in the first situation is the same as the social utility interval of y in the second situation.

Definition 1 (Interval-neutrality). A social welfare function F on a domain

D ⊆ P I is interval-neutral if for all (U i ) i∈I , (U ′ i ) i∈I ∈ D and all x, y ∈ X, if U i (x) = U ′ i (y) for all i ∈ I then F ((U i ) i∈I )(x) = F ((U ′ i ) i∈I )(y).
For determinate utilities, neutrality is equivalent to the conjunction of Independence of Irrelevant Alternatives and Pareto Indifference. This equivalence does not hold in our framework, but, it is still the case that Independence of Irrelevant Alternatives and Pareto Indifference together imply interval-neutrality.

Lemma 1. If a social welfare function F on D = P I satisfies Independence of Irrelevant Alternatives and Pareto Indifference then F is interval-neutral.

To show that the converse of Lemma 1 does not hold, let Z = {z 1 , z 2 , z 3 }, let X be the set of all probability measures on Z, and define the social welfare function F on D = P I by for all (U i ) i∈I ∈ D, F ((U i ) i∈I ) = i∈I U i if there exist i ∈ I, u i ∈ U i and z, z ′ ∈ Z such that u i (z) = u i (z ′ ) and F ((U i ) i∈I ) = {u ∈ P : ∀x ∈ X, u(x) ∈ i∈I U i (x)} otherwise.

Then F ((U i ) i∈I )(x) = i∈I U i (x) for all (U i ) i∈I ∈ D and all x ∈ X, so F is intervalneutral. Now, let U = conv({0, 1}) ∈ P and U ′ = conv({0, u ′ }) ∈ P, where u

′ ∈ P is defined by u ′ (z 1 ) = u ′ (z 2 ) = 1 and u ′ (z 3 ) = 0. Then F ((U) i∈I )| {z 1 ,z 2 } = [0, 1] 2 , so F does not satisfy Pareto Indifference. Moreover, U ′ | {z 1 ,z 2 } = U| {z 1 ,z 2 } and F ((U ′ ) i∈I )| {z 1 ,z 2 } = conv({(0, 0), (1, 1)}) = F ((U) i∈I )| {z 1 ,z 2 }
, so F does not satisfy Independence of Irrelevant Alternatives either. For determinate utilities, neutrality plays a key role by carrying linearity of vNM utility functions into linearity of the social welfare function. For indeterminate utilities this is no longer the case, because vNM utility intervals do not enjoy the linearity property that vNM utility levels do. But vNM utility intervals still exhibit some (weaker) structure and, in particular, convexity properties. Lemma 2. (a) For all U ∈ P, x, y ∈ X, and λ ∈ [0, 1], U(xλy) ⊆ λU(x) + (1 -λ)U(y).

(b) For all U ∈ P, x, y ∈ X, and λ ∈ [0, 1], λ max U(x) + (1 -λ) min U(y) ∈ U(xλy).

To illustrate Lemma 2, consider the example represented in Figure 1. The set of outcomes is Z = {z 1 , z 2 } and the set X of alternatives is the set of all probability measures on the shaded area on the graph is not convex, it still contains all line segments joining the maximum of a utility interval with the minimum of another utility interval, i.e.

u 1 u 2 u 3 u 4 z 1 1 2 4 4 z 2 3 1 2 1 z 1 z 2 x u(z 1 ) u(z 2 ) u 1 3 u 2 4 u 3 u 4 1 4 2 1 2 3 u(x)
λ max U(x) + (1 -λ) min U(y) ∈ U(xλy)
. This establishes a relationship between the two functions just defined. Armed with this lemma, we obtain the following characterization of interval-neutrality.

Theorem 1. Assume Z contains at least two elements. Then a social welfare function F on D = P I is interval-neutral if and only if there exists a non-empty, compact, and

convex set Φ ⊂ (R I + ) 2 × R such that, for all (U i ) i∈I ∈ D and all x ∈ X, F ((U i ) i∈I )(x) = (α,β,γ)∈Φ i∈I α i U i (x) - i∈I β i U i (x) + γ . (2)
Moreover, Φ can be taken such that (2) holds for some non-empty, compact, and convex set

Φ ′ ⊂ (R I + ) 2 × R if and only if Φ ⊆ Φ ′ ⊆ {(α -η, β -η, γ) : (α, β, γ) ∈ Φ, η ∈ R I + }.
Thus a social welfare function F is interval-neutral if and only if the social utility interval is the union of a set of affine transformations of all individual utility intervals, with individual i's utility interval entering twice in each affine transformation, once with a non-negative coefficient α i and once with a non-positive coefficient -β i . Moreover, the set Φ of weight-constant vectors (α, β, γ) is "almost" unique in the sense that, first, there exists a unique Φ that is minimal with respect to set inclusion and, second, another set Φ ′ satisfies (2) if and only if it consists in the minimal Φ to which are added weightconstant vectors that are always irrelevant to the social utility interval. Indeed, it is easily checked that, for all utility interval

U i (x), if α i ≥ α i -η i ≥ 0 and β i ≥ β i -η i ≥ 0 then (α i -η i )U i (x) -(β i -η i )U i (x) ⊆ α i U i (x) -β i U i (x).
It is interesting to compare the characterization (2) of interval-neutrality we obtain for indeterminate utilities with a set-valued version of the characterization of neutrality obtained by Coulhon and Mongin [START_REF] Coulhon | Social choice theory in the case of von neumann-morgenstern utilities[END_REF] for determinate utilities, namely:

F ((U i ) i∈I ) = i∈I θ i U i + γ, (3)
for some θ ∈ R I and γ ∈ R. There are three dimensions along which ( 2) is more general than

(i) In ( 2) the social utility interval is made of several affine combinations of all individual utility intervals, rather than a single one as in ( 3).

(ii) In ( 2) each individual i's utility interval enters twice in each affine transformation rather than once (so (3) corresponds to the particular case where either α i = 0 or

β i = 0).
To illustrate this point, let I = {1, 2} and consider the two social welfare functions

F 1 (U 1 , U 2 ) = U 1 + U 2 and F 2 (U 1 , U 2 ) = 2(U 1 + U 2 ) -(U 1 + U 2 ), which
obviously satisfy (2). F 1 uses only one weight per individual whereas F 2 uses two. These two functions agree if both U 1 (x) and U 2 (x) are singletons, but otherwise F 1 yields a smaller utility interval than F 2 . For instance,

F 1 ([0, 1], [0, 1]) = [0, 2] ⊂ [-2, 4] = F 2 ([0, 1], [0, 1]
). More generally, for any social welfare function F satisfying (2), we have

F ((U i ) i∈I )(x) = (α,β,γ)∈Φ i∈I (α i u i (x) -β i v i (x)) + γ : u i , v i ∈ U i , i ∈ I ⊇ (α,β,γ)∈Φ i∈I (α i -β i ) u i (x) + γ : u i ∈ U i , i ∈ I = (α,β,γ)∈Φ i∈I (α i -β i ) U i (x) + γ ,
where, in general, equality holds if and only if U i (x) is a singleton for all i ∈ I.

(iii) Whereas (3) fully pins down the social set of utility functions, (2) only pins down all social utility intervals. To illustrate why a utility set is not fully pinned down by all its utility intervals in general, let us go back to the example depicted in Figure 1, and note that the set U ′ = conv({u 1 , u 2 , u 3 , u 4 }) of utility functions fills the same shaded area on the graph than U does, so that we have U ′ (x) = U(x) for all x ∈ X, although we clearly have U ′ = U since u 4 / ∈ U. This is because U and U ′ only differ in terms of "utility correlations": in both sets of utility functions it is possible that the utility level of z 1 be equal to 4 and it is also possible that the utility level of z 2 be equal to 1, but in U ′ these two possibilities may arise from the same utility function whereas in U they cannot.

Finally, let us briefly sketch the proof of the "only if" part of Theorem 1 (the "if" part is straightforward). First note that, from interval-neutrality, we know that

F ((U i ) i∈I )(x) is a function of (U i (x)) i∈I . Equivalently, both max F ((U i ) i∈I )(x) and min F ((U i ) i∈I )(x)
are functions of (max U i (x), min U i (x)) i∈I . Now, these two functions are not necessarily affine but Lemma 2 does impose some structure on them. Most importantly, max F ((U i ) i∈I )(x) is convex, non-decreasing in each max U i (x), and non-increasing in each min U i (x), whereas min F ((U i ) i∈I )(x) is concave, non-decreasing in each min U i (x), and non-increasing in each max U i (x). These two functions are also Lipschitzian and (up to a change of variable) share the same recession function. These properties enable us to construct, using standard results on conjugate convex functions, a common, compact set

Φ ⊂ (R I + ) 2 × R such that max F ((U i ) i∈I )(x) = max (α,β,γ)∈Φ i∈I α i max U i (x) - i∈I β i min U i (x) + γ , min F ((U i ) i∈I )(x) = min (α,β,γ)∈Φ i∈I α i min U i (x) - i∈I β i max U i (x) + γ ,
which is equivalent to (2).

Utilitarianism

As we have seen, Theorem 1 falls short of fully pinning down the social utility set (and hence, social preferences) because, roughly speaking, utility intervals do not keep track of "utility correlations" between alternatives. Unlike interval-neutrality, Independence of Irrelevant Alternatives and Pareto Indifference both impose some structure on these correlations. This structure turns out to be sufficient to obtain a full-fledged characterization of social welfare functions satisfying these two axioms.

Theorem 2. Assume Z contains at least three elements. Then a social welfare function F on D = P I satisfies Independence of Irrelevant Alternatives and Pareto Indifference if and only if there exists a non-empty, compact, and convex set Φ

⊂ (R I + ) 2 × R such that, for all (U i ) i∈I ∈ D, F ((U i ) i∈I ) = (α,β,γ)∈Φ i∈I α i U i - i∈I β i U i + γ . (4)
Moreover, Φ can be taken such that (2) holds for some non-empty, compact, and convex

set Φ ′ ⊂ (R I + ) 2 × R if and only if Φ ⊆ Φ ′ ⊆ {(α -η, β -η, γ) : (α, β, γ) ∈ Φ, η ∈ R I + }.
Unlike the characterization (2) of interval-neutral social welfare functions, (4) fully pins down the social utility set and not only the utility intervals. Compared with the characterization (3) obtained by Coulhon and Mongin [START_REF] Coulhon | Social choice theory in the case of von neumann-morgenstern utilities[END_REF] for determinate utilities, (4) remains more general in the dimensions (i) and (ii) described above.

Regarding dimension (i), let us recall that in (3) where each individual is assigned one (positive, negative, or null) weight, strengthening Pareto Indifference to Pareto Weak Preference ensures that all weights are non-negative, thereby making the social welfare function utilitarian. 4 Similarly, in (4) where each individual is assigned one non-negative and one non-positive weight, Pareto Weak Preference ensures that all non-positive weights are equal to zero, so we obtain the following characterization.

⊂ R I + × R such that, for all (U i ) i∈I ∈ D, F ((U i ) i∈I ) = (θ,γ)∈Ω i∈I θ i U i + γ . (5)
Moreover, Ω is unique.

Such social welfare functions may be called multi-utilitarian, as they are unions of utilitarian rules. Each individual is thus assigned a set of weights rather than a single one (note that this makes the set of weight-constant vectors fully unique). Utilitarian social welfare functions then correspond to the particular case in which the set of weightconstant vectors is a singleton. To characterize them we introduce the following axiom.

Axiom 4 (Determinacy Preservation). For all ({u i }) i∈I ∈ D, there exists u ∈ P such that F (({u i }) i∈I ) = {u}.

Determinacy Preservation simply requires social utility to be determinate whenever individual utilities are. We obtain the following characterization of utilitarianism for utility sets.

Corollary 2. Assume Z contains at least three elements. A social welfare function F on D = P I satisfies Independence of Irrelevant Alternatives, Pareto Weak Preference, and Determinacy Preservation if and only if there exist a vector θ ∈ R I + and a number γ ∈ R such that, for all (U i ) i∈I ∈ D,

F ((U i ) i∈I ) = i∈I θ i U i + γ. (6)
Moreover, θ and γ are unique.

Note that social welfare functions satisfying (4) (and, hence, those satisfying ( 5) or ( 6)) enjoy the following properties:

(i) For all (U i ) i∈I , (U ′ i ) i∈I ∈ D,
if there exist a strictly positive real number a and a collection (b i ) i∈I of real numbers such that U ′ i = aU i +b i for all i ∈ I then there exists a real number b such that F ((

U ′ i ) i∈i ) = aF (U i ) i∈I ) + b.
This invariance property, called "cardinal measurability, unit-comparability", is already known to be satisfied by utilitarian rules in the particular case of determinate utilities (D'Aspremont and Gevers [START_REF] D'aspremont | Equity and the informational basis of collective choice[END_REF]).

(ii) For all (U i ) i∈I ∈ D, if F ((U i ) i∈I ) is a singleton then, for all i ∈ I, either U i is a singleton or α i = β i = 0 for all (α, β, γ) ∈ Φ. An individual i ∈ I for which α i = β i = 0 for all (α, β, γ) ∈ Φ is completely "irrelevant" to society, in the sense that for all (U j ) j∈I , (U ′ j ) j∈I ∈ D, F ((U j ) j∈I ) = F ((U ′ j ) j∈I ) whenever U j = U ′ j for all j ∈ I \ {i}. Thus, if all individuals are "relevant" then social utility can only be determinate if all individual utilities are themselves determinate: society cannot "resolve" individual indeterminacy. For example, one may find it desirable for society to select two profiles (u i ) i∈I , (v i ) i∈I of individual utility functions out of each profile (U i ) i∈I of individual utility sets and use some affine transformation i∈I α i u i -β i v i + γ of the selected individual utility functions as the (determinate) social utility function, but this is incompatible with the axioms of Theorem 2 (and, in fact, with interval-neutrality).

(iii) For all (U i ) i∈I , (U ′ i ) i∈I ∈ D, if U i ⊆ U ′ i for all i ∈ I then F ((U i ) i∈I ) ⊆ F ((U ′ i ) i∈I
). Thus, more indeterminacy at the individual level translates to more indeterminacy at the social level.

(iv) F 1 ((U i ) i∈I ) ⊆ F 2 ((U i ) i∈I ) for all (U i ) i∈I ∈ D if and only if Φ 1 ⊆ Φ 2 . Thus F 1 is
"socially more determinate" than F 2 if and only if F 1 has a smaller set of weightconstant vectors than F 2 . Hence, in particular, utilitarian social welfare functions Finally, let us briefly sketch the proof of the "only if" part of Theorem 2 (the proof of the "if" part is straightforward). Beforehand, it is useful to understand in more detail why a set of utility functions is only partially pinned down by all utility intervals. To this end, let us go back to the example that we considered in Figure 1 in Section 3. The definition of the four utility functions u 1 , u 2 , u 3 , u 4 ∈ P on the set Z = {z 1 , z 2 } are recalled in the left-hand side table of Figure 2. The right-hand side graph depicts these utility functions but this time in R Z rather than R X (this is possible, of course, since a vNM utility function u ∈ P is fully determined by the vector (u(z 1 ), u(z 2 )) ∈ R Z ). The set U = conv({u 1 , u 2 , u 3 }) of utility functions now corresponds to the shaded triangle u 1 u 2 u 3 . The utility interval U(x) of alternative x ∈ X (the set X of alternatives is represented by the thick segment on the graph) can now be visualized as follows: max U(x) corresponds to the hyperplane supporting U in the (normal) direction x whereas min U(x) corresponds to the hyperplane supporting U in the direction -x.

u 1 u 2 u 3 u 4 z 1 1 2 4 4 z 2 3 1 2 1 u(z 1 ) u(z 2 ) u 4 u 1 u 2 x u 3 0 1 2 3
Being essentially a compact and convex subset of R Z , a set of utility functions is fully determined by its supporting hyperplanes in all directions of R Z . The utility intervals of all alternatives, however, only determine these supporting hyperplanes in the nonnegative and non-positive directions and, hence, do not fully pin down the set of utility functions. Thus, the set U ′ = conv({u 1 , u 2 , u 3 , u 4 }) of utility functions (corresponding to the quadrilateral u 1 u 2 u 3 u 4 ), although strictly larger than U, only differs from U in directions with both positive and negative components and, hence, yields the same utility intervals as U for all alternatives. Now for the proof sketch, first note that by Theorem 1, we know there exists a nonempty, compact, and convex set Φ ⊂ (R I + ) 2 ×R such that (2) holds. In order to strengthen (2) to (4), it is sufficient to show that two social welfare functions F and F ′ on D = P I satisfying Independence of Irrelevant Alternatives and Pareto Indifference that always yield the same social utility interval must always yield the same social utility set. Given our topological assumptions and since we restrict attention to vNM utilities, it turns out to be sufficient to show that F and F ′ must always yield the same restriction of the social utility set to any pair of alternatives.

So suppose there is a profile

(U i ) i∈I ∈ D such that F ((U i ) i∈I )(x) = F ′ (U i ) i∈I (x) for all x ∈ X but F ((U i ) i∈I )| {z 1 ,z 2 } = F ′ (U i ) i∈I | {z 1 ,z 2 } for two outcomes z 1 , z 2 ∈ Z. As we just explained, F ((U i ) i∈I )| {z 1 ,z 2 } and F ′ (U i ) i∈I | {z 1 ,z 2 }
must then differ in some direction of R {z 1 ,z 2 } with both positive and negative components. For instance, F ((

U i ) i∈I )| {z 1 ,z 2 } and F ′ (U i ) i∈I | {z 1 ,z 2 }
, may be the sets U and U ′ of the previous example, which differ in the direction (2, -1) (among others), as 2u 4 (z 1 ) -u 4 (z 2 ) > 2u(z 1 ) -u(z 2 ) for all u ∈ U. Take then an outcome z 3 ∈ Z \ {z 1 , z 2 } and a profile ( Ũi ) i∈I ∈ D that agrees with (U i ) i∈I on {z 1 , z 2 } and such that ũi (z 3 ) = 2ũ i (z 1 ) -ũi (z 2 ) for all ũi ∈ Ũi and all i ∈ I. By Independence of Irrelevant Alternatives, we must then have

F (( Ũi ) i∈I )| {z 1 ,z 2 } = U and F ′ ( Ũi ) i∈I | {z 1 ,z 2 } = U ′
and, by Pareto Indifference, we must also have ũ(z 3 ) = 2ũ(z 1 )-ũ(z 2 ) for all ũ ∈ F ((U i ) i∈I ) ∪ F ′ ((U i ) i∈I ). Hence the above inequality yields F (( Ũi ) i∈I )(z 3 ) = [START_REF] Bewley | Knightian decision theory: Part I[END_REF][START_REF] Manski | Social choice with partial knowledge of treatment responses[END_REF] and 7 ∈ F ′ (( Ũi ) i∈I )(z 3 ), a contradiction.

General alternatives and domains

Up to now we have maintained two assumptions in order to simplify the exposition. First, the set of alternatives is a convex set of probability measures over some set Z of outcomes containing all probability measures on Z with finite support. Second, the domain of the social welfare function is the set of all profiles of (vNM) utility sets on X. Our results, however, also hold for other alternatives and domains. In this section we state general properties of the set of alternatives and the domain of the social welfare functions that are sufficient for our results and we discuss some particular settings in which these properties are satisfied.

First, we assume X is a mixture space, i.e. any set endowed with a mixing operation [0, 1] × X × X → X, (λ, x, y) → xλy, such that for all x, y ∈ X and all λ, µ ∈ [0, 1], x1y = x, xλy = y(1 -λ)x, and (xλy)µy = x(λµ)y. Note that all definitions and axioms introduced so far have been stated with this general notation and, hence, apply to any mixture space.

Any convex set of probability measures over some set Z of outcomes is a mixture space, whether it contains all probability measures with finite support or not. Thus, besides all instances mentioned in Section 2, convex sets of probability measures with continuous or differentiable density are also mixture spaces. A somewhat different example, that has received recent attention in the literature, is the set of all compact and convex subsets of any of the above instances of mixture spaces, interpreted as menus (opportunity sets).

More generally, whenever X is a convex subset of some linear space, defining the mixing operation xλy = λx + (1 -λ)y by means of the vector addition and scalar multiplication operations turns it into a mixture space. Such a mixture space, in addition, has the particular property that any pair of {x, y} ⊆ X of distinct alternatives is separated, i.e. for all r, s ∈ R there exists u ∈ P such that u(x) = r = u(y) = s. In fact, Mongin [START_REF] Mongin | A note on mixture sets in decision theory[END_REF] shows that all pairs of distinct alternatives in X are separated if and only if there exists a mixture preserving bijection from X into a convex subset of some linear space (and that, moreover, the affine dimension of this subset is unique), and provides examples of mixture spaces in which no, or some but not all, pairs of distinct alternatives are separated. Let us say, more generally, that a set Y ⊆ X is separated if, for all v ∈ R Y , there exists u ∈ P such that u| Y = v. 5Second, we introduce the following richness conditions on the domain of the social welfare function:

Definition 2. A domain D ⊆ P I is • Indifference-rich if, for all x, y ∈ X and all (U i ) i∈I ∈ D such that U i (x) = U i (y) for all i ∈ I, there exist z ∈ X and (U ′ i ) i∈I , (U ′′ i ) i∈I ∈ D such that U ′ i | {x,z} = U ′′ i | {y,z} = {(γ, γ) : γ ∈ U i (x) = U i (y)} for all i ∈ I.
• Polygon-rich if, for all non-empty and finite (V i ) i∈I ⊂ (R 2 ) I , there exist x, y ∈ X, x = y, and (U i ) i∈I ∈ D such that U i | {x,y} = conv(V i ) for all i ∈ I.

• Calibration-rich if, for all x, y ∈ X, all (U i ) i∈I ∈ D and all λ ∈ (0, 1), there exist z ∈ X and (U ′ i ) i∈I ∈ D such that, for all i ∈ I, U ′ | {x,y} = U| {x,y} and u

′ i (x) = u ′ i (yλz) for all u ′ i ∈ U ′ i .
In words, D is indifference-rich if, starting from any profile in D in which each individual has the same utility intervals for two alternatives x and y, we can always find a third alternative z as well as a second and a third profile in D in which each individual has the same utility interval for z as for x or y in the first profile and, furthermore, is indifferent between x and z in the second profile as well as between y and z in the third profile. D is polygon-rich if every profile of (convex hulls of) finite sets of pairs of utility levels (polygon) corresponds to the restriction of some profile in D to some pair of distinct alternatives. Finally, D is calibration-rich if, starting from any two alternatives x and y, any profile in D, and any coefficient λ strictly between 0 and 1, we can always find a third alternative z and a second profile in D having the same restriction to the pair {x, y} as the first profile and in which each individual is indifferent between x and yλz (by an appropriate calibration of the utility level of z).

Note that polygon-richness requires X to contain a separated pair of alternatives, and calibration-richness requires X to contain a separated triple of alternatives. If X is a non-empty and convex subset of some linear space (i.e. if all pairs of distinct alternatives in X are separated), then the full domain D is always indifference-rich, is polygon-rich whenever X contains a separated pair of alternatives (i.e. is of affine dimension at least one), and is calibration-rich whenever X contains a separated triple of alternatives (i.e. is of affine dimension at least two).

Technically, indifference-regularity is necessary for Independence of Irrelevant Alternatives and Pareto Indifference to imply interval-neutrality (Lemma 1). Polygon-richness ensures that any profile of real intervals corresponds to the utility intervals of some alternative in some profile in the domain, and also allows the construction of some particular profiles in the proof of Theorem 1. Calibration-richness is needed for the last step in the proof sketch of Theorem (4). Our results thus generalize as follows.

Proposition 1. Lemma 1 holds for any non-empty mixture space and any indifferencerich domain. Lemma 2 holds for any non-empty mixture space. Theorem 1 holds for mixture space containing a separated pair of alternatives and any polygon-rich domain. Theorem 2, Corollary 1, and Corollary 2 hold for any mixture space containing a separated triple of alternatives and any indifference-rich, polygon-rich, and calibration-rich domain.

Conclusion

This paper constitutes a first step towards extending social choice theory to situations in which individuals may have only partially determined preferences or utilities. Specifically, we adopt a standard multi-profile setting in which individuals are endowed with vNM utility functions and introduce indeterminacy by replacing vNM utility functions by sets of vNM utility functions, such a set representing the possible utility functions the individual may have, according to the social planner. Whereas it is known that for determinate utilities, basic axioms preclude all but utilitarian aggregation rules, it turns out that the class of rules satisfying natural generalizations of these axioms is much richer for indeterminate utilities. This class, and several subclasses (including the utilitarian class) can be characterized, although moving from determinate to indeterminate utilities raises several technical and conceptual issues.

One implication of our results is that in this settings, basic axioms entail some degree of social indeterminacy: as soon as one individual has indeterminate utility, so must have society. In other words, there is no possibility for the social planner to "resolve" individual indeterminacies, as could seem desirable. One possible way out of this impossibility could be to relax the vNM assumption on social preferences (allowing for a max-min criterion, for instance). More generally, it seems worthwhile investigating whether similar results obtain when introducing indeterminacy in other social choice settings.

Appendix: proofs

We directly state the proofs in the general setting of Section 5.

Proof of Lemma 1

Assume X is a non-empty mixture space, D is an indifference-rich domain, and F satisfies Independence of Irrelevant Alternatives and Pareto Indifference.

Let (U i ) i∈I , (U ′ i ) i∈I ∈ D and x, y ∈ X such that U i (x) = U ′ i (y) for all i ∈ I. Since D is indifference-rich, there exist z ∈ X and (U ′ i ) i∈I , (U ′′ i ) i∈I ∈ D such that U ′ i | {x,z} = U ′′ i | {y,z} = {(γ, γ) : γ ∈ U i (x) = U i (y)} for all i ∈ I. Hence F ((U i ) i∈I )(x) = F ′ ((U i ) i∈I )(x) = F ′ ((U i ) i∈I )(z) = F ′′ ((U i ) i∈I )(z) = F ′′ ((U i ) i∈I )(y)
, where the first and third inequalities follow from Independence of Irrelevant Alternatives and the second and fourth ones from Pareto Indifference.

Proof of Lemma 2

Assume X is a non-empty mixture space.

∈ U such that u(x) = max U(x) and u ′ (y) = min U(y). Suppose λ max U(x) + (1 -λ) min U(y) > max U(xλy). Then λ max U(x)+(1-λ) min U(y) > u(xλy) = λu(x)+(1-λ)u(y). Hence, since u(x) = max U(x), it must be that u(y) < min U(y), a contradiction since u ∈ U. Similarly, suppose λ max U(x) + (1 -λ) min U(y) < min U(xλy). Then λ max U(x) + (1 - λ) min U(y) < u ′ (xλy) = λu ′ (x) + (1 -λ)u ′ (y). Hence, since u ′ (y) = min U(y), it must be that u ′ (x) > max U(x), a contradiction since u ′ ∈ U.

Proof of Theorem 1

Assume X is a mixture space containing a separated pair of alternatives and D is an indifference-rich and polygon-rich domain. Clearly, if there exists a non-empty, compact, and convex set Φ ⊂ (R I + ) 2 × R such that (2) holds then F is interval-neutral. Conversely, assume F is interval-neutral. Note that for a given non-empty, compact, and convex set Φ ⊂ (R I + ) 2 × R, (2) holds if and only if, for all (U i ) i∈I ∈ D and all x ∈ X,

max F ((U i ) i∈I )(x) = max (α,β,γ)∈Φ i∈I α i max U i (x) - i∈I β i min U i (x) + γ , min F ((U i ) i∈I )(x) = min (α,β,γ)∈Φ i∈I α i min U i (x) - i∈I β i max U i (x) + γ . ( 7
)
Let K denote the set of all non-empty and compact real intervals. Since D is polygonregular, for all (K i ) i∈I ∈ K I , there exist x ∈ X and

(U i ) i∈I ∈ D such that U i (x) = K i for all i ∈ I. Hence {(U i (x)) i∈I : (U i ) i∈I ∈ D,
x ∈ X} = K I so, by interval-neutrality, there exists a unique function G :

K I → K such that, for all (U i ) i∈I ∈ D and all x ∈ X, F ((U i ) i∈I )(x) = G((U i (x)) i∈I ). Let T = {(r, s) ∈ (R I ) 2 : ∀i ∈ I, r i + s i ≥ 0}, and define the functions G, G : (R I ) 2 → R ∪ {-∞, +∞} by, for all (r, s) ∈ (R I ) 2 , G(r, s) =    max G(([-s i , r i ]) i∈I ) if (r, s) ∈ T, +∞ otherwise, G(r, s) =    -min G(([-r i , s i ]) i∈I ) if (r, s) ∈ T, +∞ otherwise.
Clearly, dom(G) = dom(G) = T . 6 Moreover, G(r, s) > -∞ and G(r, s) > -∞ for all (r, s) ∈ (R I ) 2 , so G and G are proper. Also note that, for all (r, s) ∈ T ,

G(r, s) + G(s, r) = max G(([-s i , r i ]) i∈I ) -min G(([-s i , r i ]) i∈I ) ≥ 0.
Finally, for all (U i ) i∈I ∈ D and all x ∈ X, we have

max F ((U i ) i∈I )(x) = max G((U i (x)) i∈I ) = G((max U i (x), -min U i (x)) i∈I ), min F ((U i ) i∈I )(x) = min G((U i (x)) i∈I ) = -G((-min U i (x), max U i (x)) i∈I ),
so for a given non-empty, compact, and convex set Φ ⊂ (R I + ) 2 × R, (7) holds if and only if, for all (r, s) ∈ T ,

G(r, s) = max (α,β,γ)∈Φ (αr + βs + γ), G(r, s) = max (α,β,γ)∈Φ (αr + βs -γ). (8)
Lemma 3. G and G are convex.

Proof. We only state the proof for G, the argument for G is similar. Let (r, s), (r ′ , s ′ ) ∈ T . Since D is polygon-rich, there exist x, y ∈ X, x = y, and (U i ) i∈I ∈ D such that

U i | {x,y} = conv({w i , w ′ i }
) for all i ∈ I, where w i , w ′ i ∈ R {x,y} are defined by

w i (x) = -s i , w ′ i (x) = r i , w i (y) = -s ′ i , w ′ i (y) = r ′ i .
Note that for all i ∈ I and all λ ∈ [0, 1], we have

U i (xλy) = [-λs i -(1-λ)s ′ i , λr i +(1-λ)r ′ i ]. Hence, for all λ ∈ [0, 1], G(λ(r, s) + (1 -λ)(r ′ , s ′ )) = G(λr + (1 -λ)r ′ , λs + (1 -λ)s ′ ) = max G(([-λs i -(1 -λ)s ′ i , λr i + (1 -λ)r ′ i ]) i∈I ) = max G((U i (xλy)) i∈I ) = max F ((U i ) i∈I )(xλy) ≤ λ max F ((U i ) i∈I )(x) + (1 -λ) max F ((U i ) i∈I )(y) = λ max G((U i (x)) i∈I ) + (1 -λ) max G((U i (y)) i∈I ) = λ max G(([-s i , r i ]) i∈I ) + (1 -λ) max G(([-s ′ i , r ′ i ]) i∈I ) = λG(r, s) + (1 -λ)G(r ′ , s ′ ),
where the inequality follows from Lemma 2(a). Proof. We only state the proof for G, the argument for G is similar. It is sufficient to prove that for all i ∈ I, G is non-decreasing in (r i , s i ). So fix some i ∈ I and let (r, s), (r ′ , s ′ ) ∈ T such that (r ′ i , s ′ i ) ≥ (r i , s i ) and (r ′ j , s ′ j ) = (r j , s j ) for all j ∈ I \ {i}. We proceed in two steps. First, we prove that G(r ′ , s ′ ) ≥ G(r, s) whenever r i +s i > 0. To this end, we assume without loss of generality that r ′ i +s ′ i ≤ 2(r i +s i ). Since D is polygon-rich, there exist x, y ∈ X, x = y, and (U j ) j∈I ∈ D such that U j | {x,y} = conv({w j , w ′ j , w ′′ j , w ′′′ j }) for all j ∈ I, where w j , w ′ j , w ′′ j , w ′′′ j ∈ R {x,y} are defined by

w j (x) = -s ′ j , w ′ j (x) = s ′ j -2s j , w ′′ j (x) = r ′ j , w ′′′ j (x) = 2r j -r ′ j , w j (y) = s ′ j -2s j , w ′ j (y) = -s ′ j , w ′′ j (y) = 2r j -r ′ j , w ′′′ j (y) = r ′ j .
Note that for all j ∈ I, we have

U j (x) = U j (y) = [-s ′ j , r ′ j ] and U j (x 1 2 y) = [-s j , r j ]. Hence, G(r, s) = max G(([-s j , r j ]) j∈I ) = max G((U j (x 1 2 y)) j∈I ) = max F ((U j ) j∈I )(x 1 2 y) ≤ 1 2 max F ((U j ) j∈I )(x) + 1 2 max F ((U j ) j∈I )(y) = 1 2 max G((U j (x)) j∈I ) + 1 2 max G((U j (y)) j∈I ) = 1 2 max G(([-s ′ j , r ′ j ]) j∈I ) + 1 2 max G(([-s ′ j , r ′ j ]) j∈I ) = G(r ′ , s ′ ),
where the inequality follows from Lemma 2(a).

Second, we prove that G(r ′ , s ′ ) ≥ G(r, s) whenever r i + s i = 0. To this end, suppose

G(r ′ , s ′ ) < G(r, s).
Clearly, it must then be that r ′ i + s ′ i > 0. Since D is polygon-rich, there exist x, y ∈ X, x = y, and (U j ) j∈I ∈ D such that U j | {x,y} = conv({w j , w ′ j }) for all j ∈ I, where w j , w ′ j ∈ R {x,y} are defined by

w j (x) = -s j , w ′ j (x) = r j , w j (y) = -s ′ j , w ′ j (y) = r ′ j .
Note that for all j ∈ I and all λ ∈ [0, 1], we have

U j (xλy) = [-λs j -(1 -λ)s ′ j , λr j + (1 - λ)r ′ j ]. Hence, max F ((U j ) j∈I )(x) = max G((U j (x)) j∈I ) = max G([-s j , r j ]) j∈I ) = G(r, s) > G(r ′ , s ′ ) = max G([-s ′ j , r ′ j ]) j∈I ) = max G((U j (y)) j∈I ) = max F ((U j ) j∈I )(y), so there exists λ ∈ (0, 1) such that λ max F ((U j ) j∈I )(x) + (1 -λ) min F ((U i ) i∈I )(y) > max F ((U j ) j∈I )(y). Moreover, max F ((U j ) j∈I )(xλy) = max G((U j (xλy)) j∈I ) = max G([-λs j -(1 -λ)s ′ j , λr j + (1 -λ)r ′ j ]) j∈I ) = G(λ(r, s) + (1 -λ)(r ′ , s ′ )) ≤ G(r ′ , s ′ ) = max F ((U j ) j∈I )(y),
where the inequality follows from the first step. It follows that λ max F ((U j ) j∈I )(x) + (1 -λ) min F ((U j ) j∈I )(y) > max F ((U j ) j∈I )(xλy), a contradiction by Lemma 2(b). Proof. We only state the proof for G, the argument for G is similar. By Lemma 3, G is upper semi-continuous since T is a polyhedral cone (Rockafellar [20, Theorem 10.2, Theorem 20.5]), so it is sufficient to prove that G is lower semi-continuous, i.e. that cl G = G. 7 By definition, cl G ≤ G. Conversely, let (r, s)

∈ (R I ) 2 . If (r, s) / ∈ T then cl G(r, s) = G(r, s) = +∞ since T is closed. If (r, s) ∈ T then let (r ′ , s ′ ) ∈ (R I ) 2 such that r ′
i > r i and s ′ i > s i for all i ∈ I. Then (r ′ , s ′ ) belongs to the relative interior of T and, hence, cl G(r, s) = lim λ→0 + G((1 -λ)(r, s) + λ(r ′ , s ′ )) (Rockafellar [START_REF] Rockafellar | Convex analysis[END_REF]Theorem 7.5]). By Lemma 4, we have G(r, s) ≤ G((1 -λ)(r, s) + λ(r ′ , s ′ )) for all λ ∈ [0, 1] and, hence, G(r, s) ≤ cl G(r, s). Lemma 6. For all (r, s) ∈ T and all µ > 1, G(µ(r, s)) + G(µ(r, s)) ≤ µ(G(r, s) + G(r, s)).

Proof. Let (r, s) ∈ T and µ > 1. Since D is polygon-rich, there exist x, y ∈ X, x = y, and

(U i ) i∈I ∈ D such that U i | {x,y} = conv({w i , w ′ i }
) for all i ∈ I, where w i , w ′ i ∈ R {x,y} are defined by

w i (x) = -µs i , w ′ i (x) = µr i , w i (y) = µs i , w ′ i (y) = -µr i .
Note that for all i ∈ I, we have

U i (x) = [-µs i , µr i ], U i (y) = [-µr i , µs i ], U i (x 1 2 y) = {0}, U i (x 1 µ (x 1 2 y)) = [-s i , r i ],
and

U i (y 1 µ (x 1 2 y)) = [-r i , s i ]. Hence, µ+1 2µ G(µ(r, s)) -µ-1 2µ G(µ(r, s)) = µ+1 2µ max F ((U i ) i∈I )(x) + µ-1 2µ min F ((U i ) i∈I )(y) ≤ max F ((U i ) i∈I )(x µ+1 2µ y) = max F ((U i ) i∈I )(x 1 µ (x 1 2 y)) = G(r, s),
where the inequality follows from Lemma 2(b). Similarly,

µ-1 2µ G(µ(r, s)) -µ+1 2µ G(µ(r, s)) = µ-1 2µ max F ((U i ) i∈I )(x) + µ+1 2µ min F ((U i ) i∈I )(y) ≥ min F ((U i ) i∈I )(x µ-1 2µ y) = min F ((U i ) i∈I )(y 1 µ (x 1 2 y)) = -G(r, s),
where the inequality follows again from Lemma 2(b). Hence,

G(r, s) + G(r, s) ≥ µ+1 2µ G(µ(r, s)) -µ-1 2µ G(µ(r, s)) -µ-1 2µ G(µ(r, s)) + µ+1 2µ G(µ(r, s)) ≤ G   -(λ min{-r i , s i } + (1 -λ) min{-r ′ i , s ′ i }), λ max{-r i , s i } + (1 -λ) max{-r ′ i , s ′ i } i∈I   = G λ(-min{-r i , s i }, max{-r i , s i }) i∈I +(1 -λ)(-min{-r ′ i , s ′ i }, max{-r ′ i , s ′ i }) i∈I ≤ λG((-min{-r i , s i }, max{-r i , s i }) i∈I ) +(1 -λ)G((-min{-r ′ i , s ′ i }, max{-r ′ i , s ′ i }) i∈I ) = λg(r, s) + (1 -λ)g(r ′ , s ′ ),
where the first inequality follows from Lemma 4 and the second one from Lemma 3. Now, suppose G is not Lipschitzian. Then g is not Lipschitzian either. Hence, since g is finite and convex, there must exist (r, s) ∈ (R I ) 2 such that g0 + (r, s) = +∞ (Rockafellar [20, Theorem 10.5]), i.e. G0 + (T (r, s)) = +∞, a contradiction by Lemma 7. Finally, note that for all (r, s) ∈ (R I ) 2 , g0 + (r, s) = G0 + (T (r, s)) < +∞ by Lemma 7. Hence g0 + is Lipschitzian since it is its own recession function (Rockafellar [20,Theorem 10.5]), so

G0 + is Lipschitzian. Lemma 9. For all (r, s), (r ′ , s ′ ) ∈ (R I ) 2 , if G0 + ′ ((r, s), (r ′ , s ′ )) = -G0 + ′ ((r, s), -(r ′ , s ′ )) then 8 G0 + ′ ((r, s), (r ′ , s ′ )) = lim µ→+∞ G ′ (µ(r, s), (r ′ , s ′ )) = lim µ→+∞ -G ′ (µ(r, s), -(r ′ , s ′ )) = lim µ→+∞ G ′ (µ(r, s), (r ′ , s ′ )) = lim µ→+∞ -G ′ (µ(r, s), -(r ′ , s ′ )).
Proof. We only state the proof for G, the argument for G is similar, given Lemma 7. Let (r, s), (r ′ , s ′ ) ∈ (R I ) 2 . If (r, s) / ∈ T then the lemma is trivially true (all directional derivatives are equal to +∞), so assume (r, s) ∈ T . Then, by definition of G0 + ′ , for all ε > 0, there exists δ 0 > 0 such that

G0 + ((r, s) + δ 0 (r ′ , s ′ )) -G0 + (r, s) δ 0 ≤ G0 + ′ ((r, s), (r ′ , s ′ )) + ε
and, hence, lim µ→+∞ G(µ(r, s) + µδ 0 (r ′ , s ′ )) -G(µ(r, s)) µδ 0 ≤ G0 + ′ ((r, s), (r ′ , s ′ )) + ε.

8 Given a function f and a point x ∈ dom(f ), f ′ (x, y) denotes the (one-sided) directional derivative of f at x in direction y, i.e. f ′ (x, y) = lim δ→0+ f (x+δy)-f (x) δ if x ∈ dom(f ) and f ′ (x, y) = +∞ otherwise.

Hence, for all ε > 0, there exist δ 0 > 0 and µ 0 > 1 such that, for all µ ≥ µ 0 , G(µ(r, s) + µδ 0 (r ′ , s ′ )) -G(µ(r, s)) µδ 0 ≤ G0 + ′ ((r, s), (r ′ , s ′ )) + ε.

Moreover, since µ(r, s) + δ 0 (r ′ , s ′ ) = (1 -1 µ )(µ(r, s)) + 1 µ (µ(rs) + µδ 0 (r ′ , s ′ )), we have

G(µ(r, s) + δ 0 (r ′ , s ′ )) ≤ (1 -1 µ )G(µ(r, s)) + 1 µ G(µ(rs) + µδ 0 (r ′ , s ′ ))
by Lemma 3 and, hence,

G(µ(r, s) + δ 0 (r ′ , s ′ )) -G(µ(r, s)) δ 0 ≤ G(µ(r, s) + µδ 0 (r ′ , s ′ )) -G(µ(r, s)) µδ 0 .
Hence, for all ε > 0, there exist δ 0 > 0 and µ 0 > 1 such that, for all µ ≥ µ 0 , G(µ(r, s) + δ 0 (r ′ , s ′ )) -G(µ(r, s)) δ 0 ≤ G0 + ′ ((r, s), (r ′ , s ′ )) + ε and, hence, for all δ ∈ (0, δ 0 ),

G(µ(r, s) + δ(r ′ , s ′ )) -G(µ(r, s)) δ ≤ G0 + ′ ((r, s), (r ′ , s ′ )) + ε
by Lemma 3 (Rockafellar [START_REF] Rockafellar | Convex analysis[END_REF]Theorem 23.1]). Hence, by definition of G ′ , for all ε > 0, there exists µ 0 > 1 such that G ′ (µ(r, s), (r ′ , s ′ )) ≤ G0 + ′ ((r, s), (r ′ , s ′ )) + ε for all µ ≥ µ 0 . Now, assume G0 + ′ ((r, s), (r ′ , s ′ )) = -G0 + ′ ((r, s), -(r ′ , s ′ )). Then, by the preceding paragraph, for all ε > 0, there exists µ 0 > 1 such that, for all µ ≥ µ 0 ,

G ′ (µ(r, s), (r ′ , s ′ )) -ε ≤ G0 + ′ ((r, s), (r ′ , s ′ )) = -G0 + ′ ((r, s), -(r ′ , s ′ )) ≤ -G ′ (µ(r, s), -(r ′ , s ′ )) + ε ≤ G(µ(r, s), (r ′ , s ′ )) + ε
(where the last inequality follows from Rockafellar [START_REF] Rockafellar | Convex analysis[END_REF]Theorem 23.1]), so we obtain

lim µ→+∞ G ′ (µ(r, s), (r ′ , s ′ )) = lim µ→+∞ -G ′ (µ(r, s), -(r ′ , s ′ )) = G0 + ′ ((
Let C = {(-η, -η) : η ∈ R I + } ⊂ (R I -) 2 .
Clearly, C is a non-empty, closed, and convex cone containing no line and, moreover, we have C = T • .9 Let L ⊂ (R I ) 2 be the set of points where G0 + is differentiable. Clearly, cl(L) = T (Rockafellar [START_REF] Rockafellar | Convex analysis[END_REF]Theorem 25.5]), so L is non-empty. 10 Let E = {∇G0 + (r, s) : (r, s) ∈ L} and M = cl(conv(E)). By Lemmas 4 and 8, E is a non-empty and bounded subset of (R I + ) 2 and, hence, M is a non-empty, compact, and convex subset of (R I + ) 2 .

Lemma 10. For all (α, β)

∈ M + C, G * (α, β) + G * (α, β) ≤ 0. Proof. We claim that G * (α, β) + G * (α, β) ≤ 0 for all (α, β) ∈ E. If this is true then we have G * (α, β) + G * (α, β) ≤ 0 for all (α, β) ∈ M since G * + G * is closed (Rockafellar [20, Theorem 9.3]). Moreover, for all η ∈ R I + , we have G * (α -η, β -η) = sup (r,s)∈T (αr + βs - η(r+s)-G(r, s)) ≤ G * (α, β) and G * (α-η, β-η) = sup (r,s)∈T (αr+βs-η(r+s)-G(r, s)) ≤ G * (α, β) since r + s ≥ 0 for all (r, s) ∈ T . Hence G * (α, β) + G * (α, β) ≤ 0 for all (α, β) ∈ M + C. To prove the claim, let (α, β) ∈ E. Since G * and G * are closed, it is sufficient to find two sequences (α µ , β µ ) µ≥1 and (α µ , β µ ) µ≥1 in (R I ) 2 such that lim µ→+∞ (α µ , β µ ) = lim µ→+∞ (α µ , β µ ) = (α, β) and lim µ→+∞ G * (α µ , β µ ) + lim µ→+∞ G * (α µ , β µ ) ≤ 0. Moreover, since G * and G * are bounded below, a sufficient condition for lim µ→+∞ G * (α µ , β µ ) + lim µ→+∞ G * (α µ , β µ ) ≤ 0 is that G * (α µ , β µ ) + G * (α µ , β µ ) ≤ 0 for all µ ≥ 1.
In order to construct these sequences, note that since (α, β) ∈ E, we have (α, β) = ∇G0 + (r, s) for some (r, s) ∈ L by definition. Clearly, (r, s) must then belong to the interior of T (Rockafellar [START_REF] Rockafellar | Convex analysis[END_REF]Corollary 25.1.1]). Hence, for all µ ≥ 1, µ(r, s) also belongs to the interior of T , so ∂G(µ(r, s)) is non-empty and compact (Rockafellar [20,Theorem 23.4]).11 It follows that there exists (α µ , β µ ) ∈ ∂G(µ(r, s)) such that G

′ (µ(r, s), -(r, s)) = -α µ r -β µ s since G ′ (µ(r, s), •)
G * (α µ , β µ ) = -G(µ(r, s)) -µG ′ (µ(r, s), -(r, s)) = -G(µ(r, s)) -µ lim ε↓0 G((µ -ε)(r, s)) -G(µ(r, s)) ε = lim ε↓0 (µ -ε)G(µ(r, s)) -µG((µ -ε)(r, s)) ε .
Similarly, there exists (α µ , β µ ) ∈ ∂G(µ(r, s)) such that

G * (α µ , β µ ) = lim ε↓0 (µ -ε)G(µ(r, s)) -µG((µ -ε)(r, s)) ε .
Hence,

G * (α µ , β µ ) + G * (α µ , β µ ) = lim ε↓0 (µ -ε)(G(µ(r, s)) + G(µ(r, s))) -µ(G((µ -ε)(r, s)) + G((µ -ε)(r, s))) ε ≤ 0 by Lemma 6. Moreover, G0 + ′ ((r, s), •) is linear since (r, s) ∈ L (Rockafellar [
(R I + ) 2 × R such that M ′ ⊆ Φ ⊆ M ′ + C ′ .
For the converse, first assume that Φ ⊆ M ′ + C ′ , i.e. that there exists (α 0 , β 0 , γ 0 ) ∈ Φ\(M ′ +C ′ ). Then, since M ′ +C ′ is closed and convex and contains no line, there exists (r, s, t) ∈ (R I ) 2 ×R with t = 0 such that α 0 r +β 0 s+γ 0 t > αr +βs+γt for all (α, β, γ) ∈ M ′ + C ′ . Clearly, it must then be that (r, s, t)

∈ C ′• = T ×R -. Hence, if we let (r ′ , s ′ ) = ( r |t| , s |t| ) ∈ T , we have α 0 r ′ +β 0 s ′ -γ 0 > αr ′ +βs ′ -γ for all (α, β, γ) ∈ M ′ +C ′ , so sup (α,β,γ)∈Φ (αr ′ + βs ′ -γ) > sup (α,β,γ)∈M ′ +C ′ (αr ′ + βs ′ -γ) = G(r ′ , s ′ ). Now, assume that Φ ⊆ M ′ + C ′ but M ′ ⊆ Φ, i.e
. that there exists (α 0 , β 0 , γ 0 ) ∈ M ′ \ Φ. Then there exists (α 0 , β 0 , γ 0 ) ∈ E ′ \ Φ (Rockafellar [START_REF] Rockafellar | Convex analysis[END_REF]Theorem 18.6]) and, hence, there exists (r, s, t) ∈ (R I ) 2 × R with t = 0 such that α 0 r + β 0 s + γ 0 t > αr + βs + γt for 14 Given a real-valued function f , epi(f ) denotes the epigraph of f , i.e. epi(f ) = {(x, γ) : [START_REF] Rockafellar | Convex analysis[END_REF]Corollary 25.1.2]). Clearly, it must then be that (r, s, t) Finally, let Φ = conv(M ′ ∪ M ′′ ). Then (8) holds, as well as the uniqueness result, by Lemma 13.

γ ≥ f (x)}. all (α, β, γ) ∈ Φ + C ′ ⊆ M ′ + C ′ (Rockafellar
∈ C ′• = T × R -. Hence, if we let (r ′ , s ′ ) = ( r |t| , s |t| ) ∈ T , we have α 0 r ′ + β 0 s ′ -γ 0 > αr ′ + βs ′ -γ for all (α, β, γ) ∈ Φ, so sup (α,β,γ)∈Φ (αr ′ + βs ′ -γ) < sup (α,β,γ)∈M ′ +C ′ (αr ′ + βs ′ -γ) = G(r ′ , s ′ ). Let M ′′ = {(α, β, -γ) : (α, β, γ ∈ M ′ } and C ′′ = C × R -.
′ ∪ M ′′ ) ⊆ Φ ⊆ (M ′ + C ′ ) ∩ (M ′′ + C ′′ ). Lemma 13. (M ′ + C ′ ) ∩ (M ′′ + C ′′ ) = conv(M ′ ∪ M ′′ ) + (C × {0}). Proof. Since 0 ∈ C ′ ∩ C ′′ , we have M ′ ⊆ M ′ + C ′ and M ′′ ⊆ M ′′ + C ′′ . Moreover, M ′ ⊆ M ′′ +C ′′

Proof of Theorem 2

Assume X is a mixture space containing a separated triple of alternatives and D is an indifference-rich, polygon-rich, and calibration-rich domain. Clearly, if there exists a non-empty, compact, and convex set Φ ⊂ (R I + ) 2 × R such that (4) holds then F satisfies Independence of Irrelevant Alternatives and Pareto Indifference. Conversely, assume F satisfies these two axioms. By Lemma 1 and Theorem 1, since D is indifferencerich and polygon-rich, there then exists a unique non-empty, compact, and convex set Φ ⊂ (R I + ) 2 × R such that (2) and the associated uniqueness result hold. Let F ′ denote the social welfare function defined by (4) with the same domain D and the same set Φ of weight-constant vectors. Then F ′ satisfies Pareto Indifference and Independence of Irrelevant Alternatives and, for all (U i ) i∈I ∈ D and all x ∈ X, we have F ((U i ) i∈I )(x) = F ′ ((U i ) i∈I )(x). In order to complete the proof of the theorem (including the uniqueness part), it is sufficient to show that F = F ′ . Lemma 14. Let F and F ′ be two social welfare functions on D satisfying Pareto Indifference and Independence of Irrelevant Alternatives. If F ((U i ) i∈I )(x) = F ′ ((U i ) i∈I )(x) for all (U i ) i∈I ∈ D and all x ∈ X then F ((U i ) i∈I )| {x 1 ,x 2 } = F ′ ((U i ) i∈I )| {x 1 ,x 2 } for all (U i ) i∈I ∈ D and all x 1 , x 2 ∈ X.

Proof. Assume there exist (U i ) i∈I ∈ D and x 1 , x 2 ∈ X such that F ((U i ) i∈I )| {x 1 ,x 2 } = F ′ ((U i ) i∈I )| {x 1 ,x 2 } , so (without loss of generality), there exists u ∈ F ((U i ) i∈I ) such that

u| {x 1 ,x 2 } / ∈ F ′ ((U i ) i∈I )| {x 1 ,x 2 } .
It is sufficient to find some ( Ũi ) i∈I ∈ D and x ∈ X such that F (( Ũi ) i∈I )(x) = F ′ (( Ũi ) i∈I )(x).

Since F ′ ((U i ) i∈I )| {x 1 ,x 2 } is a non-empty, compact, and convex subset of R 2 , there exists t ∈ R 2 such that t 1 + t 2 = 0 and t 1 u(x 1 ) + t 2 u(x 2 ) > t 1 u ′ (x 1 ) + t 2 u ′ (x 2 ) for all u ′ ∈ F ′ ((U i ) i∈I ). Three cases may occur: either t > 0, or t < 0, or t 1 > 0 and t 2 < 0 (there is a fourth case, t 1 < 0 and t 2 > 0, but it can be brought back to the third case by permuting x 1 and x 2 ). If t > 0 then, letting x = x 1 t 1 t 1 +t 2 x 2 ∈ X, we have u(x) > u ′ (x) for all u ′ ∈ F ′ ((U i ) i∈I ), so that F ((U i ) i∈I )(x) = F ′ ((U i ) i∈I )(x). Similarly, if t < 0 then, letting again x = x 1 t 1 t 1 +t 2 x 2 ∈ X, we have u(x) < u ′ (x) for all u ′ ∈ F ′ ((U i ) i∈I ), so that F ((U i ) i∈I )(x) = F ′ ((U i ) i∈I )(x). Finally, if t 1 > 0 and t 2 < 0 then two cases may occur: either t 1 + t 2 > 0 or t 1 + t 2 < 0.

If t 1 + t 2 > 0 then, since D is calibration-rich, there exist x ∈ X and ( Ũi ) i∈I ∈ D such that, for all i ∈ I, Ũi | {x 1 ,x 2 } = U i | {x 1 ,x 2 } and, for all ũi ∈ Ũi , ũi (x 1 ) = ũi (x 2 (-t 2 t 1 )x). Hence F (( Ũi ) i∈I )| {x 1 ,x 2 } = F ((U i ) i∈I )| {x 1 ,x 2 } and F ′ (( Ũi ) i∈I )| {x 1 ,x 2 } = F ′ ((U i ) i∈I )| {x 1 ,x 2 } since F and F ′ satisfy Independence of Irrelevant Alternatives, so there exists ũ ∈ F (( Ũi ) i∈I ) such that t 1 ũ(x 1 ) + t 2 ũ(x 2 ) > t 1 ũ′ (x 1 ) + t 2 ũ′ (x 2 ) for all ũ′ ∈ F ′ (( Ũi ) i∈I ). Moreover, since F and F ′ satisfy Pareto Indifference, we have ũ(x 1 ) = ũ(x 2 (-t 2 t 1 )x) = -t 2 t 1 ũ(x 2 )+ t 1 +t 2 t 1 ũ(x) for all ũ ∈ F (( Ũi ) i∈I ) and ũ′ (x 1 ) = ũ′ (x 2 (-t 2 t 1 )x) = -t 2 t 1 ũ′ (x 2 ) + t 1 +t 2 t 1 ũ′ (x) for all ũ′ ∈ F (( Ũ′ i ) i∈I ). Hence the above inequality is equivalent to ũ(x) > ũ′ (x) for all ũ′ ∈ F ′ (( Ũi ) i∈I ), so that F (( Ũi ) i∈I )(x) = F ′ (( Ũi ) i∈I )(x).

Similarly, if t 1 +t 2 < 0 then, since D is calibration-rich, there exist x ∈ X and ( Ũi ) i∈I ∈ D such that, for all i ∈ I, Ũi | {x 1 ,x 2 } = U i | {x 1 ,x 2 } and, for all ũi ∈ Ũi , ũi (x 2 ) = ũi (x 1 (-

t 1 t 2 )x). Hence F (( Ũi ) i∈I )| {x 1 ,x 2 } = F ((U i ) i∈I )| {x 1 ,x 2 } and F ′ (( Ũi ) i∈I )| {x 1 ,x 2 } = F ′ ((U i ) i∈I )| {x 1 ,x 2 }
since F and F ′ satisfy Independence of Irrelevant Alternatives, so there exists ũ ∈ F (( Ũi ) i∈I ) such that t 1 ũ(x 1 ) + t 2 ũ(x 2 ) > t 1 ũ′ (x 1 ) + t 2 ũ′ (x 2 ) for all ũ′ ∈ F ′ (( Ũi ) i∈I ). Moreover, since F and F ′ satisfy Pareto Indifference, we have ũ(x 2 ) = ũ(x 1 (-t 1 t 2 )x) = -t 1 t 2 ũ(x 1 ) + t 1 +t 2 t 2 ũ(x) for all ũ ∈ F (( Ũi ) i∈I ) and ũ′ (x 2 ) = ũ′ (x 1 (-

Figure 1 :

 1 Figure 1: Example of set of utility functions

Corollary 1 .

 1 Assume Z contains at least three elements. A social welfare function F on D = P I satisfies Independence of Irrelevant Alternatives and Pareto Weak Preference if and only if there exist a non-empty, compact, and convex set Ω

Figure 2 :

 2 Figure 2: Example of set of utility functions (continued)

  (a) Let U ∈ P, x, y ∈ X, and λ ∈ [0, 1]. Then by definition, U(xλy) = {u(xλy) : u ∈ U} = {λu(x) + (1 -λ)u(y) : u ∈ U} ⊆ {λu(x) + (1 -λ)v(y) : u, v ∈ U} = λU(x) + (1 -λ)U(y). (b) Let U ∈ P, x, y ∈ X, and λ ∈ [0, 1]. Then by definition, there exist u, u ′

Lemma 4 .

 4 G and G are non-decreasing.

Lemma 5 .

 5 G and G are continuous.

  r, s), (r ′ , s ′ )) by passing to the limit as ε→ 0 + . Let G * , G * : (R I ) 2 → R ∪ {-∞, +∞}denote the conjugate functions of G and G, respectively, i.e. for all (α, β) ∈ (R I ) 2 , G * (α, β) = sup (r,s)∈T (αr + βs -G(r, s)), G * (α, β) = sup (r,s)∈T (αr + βs -G(r, s)). Since G and G are closed proper convex functions, so are G * and G * (Rockafellar [20, Theorem 12.2]). Moreover, G * and G * are clearly bounded below since, for all (α, β) ∈ (R I ) 2 , G * (α, β) ≥ -G(0, 0) and G * (α, β) ≥ -G(0, 0).

  is the support function of ∂G(µ(r, s)) (Rockafellar [20, Theorem 23.2, Theorem 23.4]). Hence we have G(µ(r, s)) + G * (α µ , β µ ) = µ(α µ r + β µ s) (Rockafellar [20, Theorem 23.5]) and, hence,

  and M ′′ ⊆ M ′ +C ′ by Lemma 10. Hence conv(M′ ∪M ′′ ) ⊆ (M ′ +C ′ )∩(M ′′ +C ′′ )since the set on the right hand side is convex. Hence conv(M ′ ∪ M ′′ ) + (C × {0}) ⊆ (M ′ +C ′ )∩(M ′′ +C ′′ ) since (C ×{0}) = C ′ ∩C ′′ . Conversely, let ν ∈ (M ′ +C ′ )∩(M ′′ +C ′′ ).By definition, there exist (α,β, γ) ∈ M ′ , (α, β, γ) ∈ M ′′ , η, η ∈ R I + , and τ , τ ∈ R + such that ν = (α, β, γ) + (-η, -η, τ ) = (α, β, γ) + (-η, -η, -τ ).Hence, setting τ τ +τ = τ τ +τ = 1 2 in case τ = τ = 0, we haveν = τ τ +τ α, β, γ + -η, -η, τ + τ τ +τ α, β, γ + (-η, -η, -τ ) = τ τ +τ α, β, γ + τ τ +τ α, β, γ + -τ τ +τ η + τ τ +τ η , -τ τ +τ η + τ τ +τ η , 0 , so ν ∈ conv(M ′ ∪ M ′′ ) + (C × {0}).

Given a function f on a set S and a subset T of S, f | T denotes the function on T defined by f | T (s) = f (s) for all s ∈ T .

Given a set S, conv(S) denotes the convex hull of S.

We call utilitarian any social welfare function in which social utility is a non-negative affine combination of individual utilities, even if individuals do not have equal weights.

More precisely, Mongin[START_REF] Mongin | A note on mixture sets in decision theory[END_REF] defines a pair {x, y} to be separated if there exists u ∈ P such that u(x) = u(y). Our definition is equivalent to his for pairs but is more demanding for larger sets.

Given a function f , dom(f ) denotes the effective domain of f .

Given a function f , cl f denotes the closure of f .

Given a cone S, S • denotes the polar cone of S, i.e. S • = {y : ∀s ∈ S, xy ≤ 0}.

Given a set S, cl(S) denotes the closure of S.

Given a function f and a point x, ∂f (x) denotes the subdifferential of f at x, i.e. ∂f (x) = {x * : ∀y, f (y) ≥ f (x) + (yx)x * }.

and D-

= 1 µ (G(µ(r, s)) + G(µ(r, s))), so G(µ(r, s)) + G(µ(r, s)) ≤ µ(G(r, s) + G(r, s)).

Let G0 + , G0 + : (R I ) 2 → R ∪ {-∞, +∞} denote the recession functions of G and G, respectively, i.e. (Rockafellar [20,Theorem 8.5]) for all (r, s) ∈ (R I ) 2 , G0 + (r, s) = lim µ→+∞ G(µ(r, s)) µ , G0 + (r, s) = lim µ→+∞ G(µ(r, s)) µ .

G0 + and G0 + are positively homogenous, proper convex functions by definition, and are closed since G and G are closed. Also note that G0 + and G0 + are non-decreasing by Lemma 4.

Lemma 7. dom(G0 + ) = T and < +∞. Moreover, for all µ > 1, defining x, y, U i , w i , and w ′ i as in the proof of Lemma 6 yields

.

Lemma 8. G, G, G0 + , and G0 + are Lipschitzian.

Proof. We only state the proof for G and G0 + , the argument for G and G0 + is similar.

First, for all (r, s)

non-empty, compact, and convex subsets of (R I + ) 2 × R.

Lemma 12. For all non-empty, compact, and convex set Φ

Similarly, for all non-empty, compact, and convex set Φ

Proof. We only state the proof for G, the argument for G is similar. Since G is closed, it is the conjugate function of G * (Rockafellar [20, Theorem 12.2]), i.e. for all (r, s) ∈ T , 14

Lemma 15. Let U, U ′ ∈ P, let n ∈ N, n > 2, and let y 1 , . . . , y n ∈ X.

Proof. Assume U| {y 1 ,...,yn} = U ′ | {y 1 ,...,yn} , so (without loss of generality), there exists u ∈ U such that u| {y 1 ,...,yn} / ∈ U ′ | {y 1 ,...,yn} . It suffices to find some x = y 1 λ 1 (. . .

is a non-empty, compact, and convex subset of R n , there exists t ∈ R n such that t 1 , . . . , t n = 0 and t 1 u(y 1 ) + . . . + t n u(y n ) > t 1 u ′ (y 1 ) + . . . + t n u ′ (y n ) for all u ′ ∈ U ′ . Three cases may occur: either t > 0, or t < 0, or neither t > 0 nor t < 0. If t > 0 then, letting x = x ′ = y 1 t n-1 +tn y n )) ∈ X, there exists no u ′ ∈ U ′ such that u(x) = u ′ (x) and u(

Proof. Assume U = U ′ , so (without loss of generality) there exists u


Emmanuel Militon
email: emma-nuel.militon@math.polytechnique.fr

Actions of groups of homeomorphisms on one-manifolds

In this article, we describe all the group morphisms from the group of compactly-supported homeomorphisms isotopic to the identity of a manifold to the group of homeomorphisms of the real line or of the circle. MSC: 37C85.

Introduction

Fix a connected manifold M (without boundary). For an integer r ≥ 0, we denote by Diff r (M ) the group of C r -diffeomorphisms of M . When r = 0, this group will also be denoted by Homeo(M ). For a homeomorphism f of M , the support of f is the closure of the set:

{x ∈ M, f (x) = x} .
We denote by Diff r 0 (M ) (Homeo 0 (M ) if r = 0) the identity component of the group of compactly supported C r -diffeomorphisms of M (for the strong topology). If r = dim(M ) + 1, these groups are simple by a well-known and difficult theorem (see [START_REF] Banyaga | The structure of classical diffeomorphism groups, Mathematics and its Applications[END_REF], [START_REF]Bounemoura Simplicité des groupes de transformations de surface[END_REF], [START_REF] Fischer | On the group of all homeomorphisms of a manifold[END_REF], [START_REF] Mather | Commutators of diffeomorphisms I[END_REF], [START_REF] Mather | Commutators of diffeomorphisms II[END_REF]).

In [START_REF] Ghys | Prolongements des difféomorphismes de la sphère[END_REF], Étienne Ghys asked whether the following statement was true: if M and N are two closed manifolds and if there exists a non-trivial morphism Diff ∞ 0 (M ) → Diff ∞ 0 (N ), then dim(M ) ≥ dim(N ). In [START_REF] Mann | Homomorphisms between diffeomorphism groups[END_REF], Kathryn Mann proved the following theorem. Take a connected manifold M of dimension greater than 1 and a onedimensional connected manifold N . Then any morphism Diff ∞ 0 (M ) → Diff ∞ 0 (N ) is trivial: she answers Ghys's question in the case where the manifold N is onedimensional. Mann also describes all the group morphisms Diff r 0 (M ) → Diff r 0 (N )

for r ≥ 3 when M as well as N are one-dimensional. The techniques involved in the proofs of these theorems are Kopell's lemma (see [START_REF] Navas | Groups of Circle Diffeomorphisms[END_REF] Theorem 4.1.1) and Szekeres's theorem (see [START_REF] Navas | Groups of Circle Diffeomorphisms[END_REF] Theorem 4. 1.11). These theorems are valid only for a regularity at least C 2 . In this article, we prove similar results in the case of a C 0 regularity. The techniques used are different.

Theorem 1.1. Let M be a connected manifold of dimension greater than 2 and let N be a connected one-manifold. Then any group morphism Homeo 0 (M ) → Homeo(N ) is trivial.

The case where the manifold M is one-dimensional is also well-understood.

Theorem 1.2. Let N be a connected one-manifold. For any group morphism ϕ : Homeo 0 (R) → Homeo(N ), there exists a closed set K ⊂ N such that:

1. The set K is pointwise fixed under any homeomorphism in ϕ(Homeo 0 (R)).

2. For any connected component I of N -K, there exists a homeomorphism h I : R → I such that:

∀f ∈ Homeo 0 (R), ϕ(f ) |I = h I f h -1 I .
Remark: By a theorem by Matsumoto (see [START_REF] Matsumoto | Numerical invariants for semiconjugacy of homeomorphisms of the circle[END_REF] Theorem 5.3), every group morphism Homeo 0 (S 1 ) → Homeo 0 (S 1 ) is a conjugation by a homeomorphism of the circle. Moreover, any group morphism Homeo 0 (S 1 ) → Homeo(R) is trivial. Indeed, as the group Homeo 0 (S 1 ) is simple, such a group morphism is either one-toone or trivial. However, the group Homeo 0 (S 1 ) contains torsion elements whereas the group Homeo(R) does not: such a morphism cannot be one-to-one.

2 Proofs of Theorems 1.1 and 1.2

Fix an integer d ≥ 1. For a point p in R d , we denote by G d p the group Homeo 0 (R d -{p}). This group is seen as a subgroup of Homeo 0 (R d ) consisting of homeomorphisms which pointwise fix a neighbourhood of the point p. We will call embedded

(d -1)-dimensional ball of R d the image of the closed unit ball of R d-1 = R d-1 × {0} ⊂ R d under a homeomorphism of R d . For an embedded (d -1)-dimensional ball D ⊂ R d (which is a single point if d = 1), we denote by H d
D the subgroup of Homeo 0 (R d ) consisting of homeomorphisms which pointwise fix a neighbourhood of the embedded ball D. Finally, if G denotes a subgroup of Homeo(R d ), a point p ∈ R d is said to be fixed under the group G if it is fixed under all the elements of this group. We denote by Fix(G) the (closed) set of fixed points of G.

The theorems will be deduced from the following propositions. The two first propositions will be proved respectively in Sections 3 and 4.

Proposition 2.1. Let ϕ : Homeo 0 (R d ) → Homeo(R) be a group morphism. Suppose that no point of the real line is fixed under the group ϕ(Homeo 0 (R d )). Then, for any embedded (d -1)-dimensional ball D ⊂ R d , the group ϕ(H d D ) admits at most one fixed point. Proposition 2.2. Let ϕ : Homeo 0 (R d ) → Homeo(R) be a group morphism. Then, for any point p in R d , the group ϕ(G d p ) admits at least one fixed point. Proposition 2.3. For any group morphism ψ : Homeo 0 (R d ) → Homeo(S 1 ), the group ψ(Homeo 0 (R d )) has a fixed point.

Proof of Proposition 2.3. Recall that the group Homeo 0 (R d ) is infinite and simple and that the group Homeo(S 1 )/Homeo 0 (S 1 ) is isomorphic to Z/2Z. Hence any morphism Homeo 0 (R d ) → Homeo(S 1 )/Homeo 0 (S 1 ) is trivial. Therefore, the image of a morphism Homeo 0 (R d ) → Homeo(S 1 ) is contained in Homeo 0 (S 1 ).

For some background about the bounded cohomology of groups and the bounded Euler class of a group acting on a circle, see Section 6 in [START_REF] Ghys | Groups acting on the circle[END_REF]. By [START_REF] Mather | The vanishing of the homology of certain groups of homeomorphisms[END_REF] and [START_REF] Matsumoto | Bounded cohomology of certain groups of homeomorphisms[END_REF]:

H 2 b (Homeo 0 (R d ), Z) = {0}
. Therefore, the bounded Euler class of a morphism Homeo 0 (R d ) → Homeo 0 (S 1 ) vanishes: this action has a fixed point.

Proof of Theorem 1.1. Let d = dim(M ). The theorem will be deduced from the following lemma. Let us see why this lemma implies the theorem. Consider a morphism Homeo 0 (M ) → Homeo 0 (N ). Take an open set U ⊂ M homeomorphic to R d and let us denote by Homeo 0 (U ) the subgroup of Homeo 0 (M ) consisting of homeomorphisms supported in U . By Lemma 2.4 and Proposition 2.3, the restriction of this morphism to the subgroup Homeo 0 (U ) is trivial. Moreover, as the group Homeo 0 (M ) is simple, such a group morphism is either one-to-one or trivial: it is necessarily trivial in this case.

Proof of Lemma 2.4. Take a group morphism ϕ : Homeo 0 (R d ) → Homeo(R). Suppose by contradiction that this morphism is nontrivial. Replacing if necessary R with a connected component of the complement of the closed set Fix(ϕ(Homeo 0 (R d ))), we can suppose that the group ϕ(Homeo 0 (R d )) has no fixed points.

Let us prove that, for any points

p 1 = p 2 in R d : Fix(ϕ(G d p1 )) ∩ Fix(ϕ(G d p2 )) = ∅.
The proof of this fact requires the following lemma.

Lemma 2.5. Let d ′ ≥ 1 be an integer. Let p 1 = p 2 be two distinct points in R d ′ . Then, for any homeomorphism f in Homeo 0 (R d ′ ), there exist homeomorphisms

f 1 , f 3 in G d ′ p1 and f 2 in G d ′ p2 such that: f = f 1 f 2 f 3 . Proof. Take a homeomorphism f in Homeo 0 (R d ′ ). Let f 1 be a homeomorphism in G d ′ p1 such that f -1 1 sends the point f (p 1 ) to a point which lies in the same connected component of R d ′ -{p 2 } as the point p 1 . Let f 2 be a homeomorphism in G d ′ p2 which is equal to f -1 1 f in a neighbourhood of the point p 1 .
The existence of the homeomorphism f 2 is easy to prove when d ′ = 1, is a consequence of the Schönflies Theorem when d ′ = 2 and of the annulus theorem by Kirby and Quinn when d ′ ≥ 3 (see [START_REF] Kirby | Stable homeomorphisms and the annulus conjecture[END_REF] and [START_REF] Quinn | Ends of maps III : Dimensions 4 and 5[END_REF]). Changing if necessary the homeomorphism f 2 into the composition of the homeomorphism f 2 with a homeomorphism supported in a small neighbourhood of the point p 1 , the homeomorphism

f 3 = f -1 2 f -1 1 f belongs to G d ′ p1 .
Take two points p 1 and Proof of Theorem 1.2. Let ϕ : Homeo 0 (R) → Homeo(N ) be a nontrivial group morphism. By Proposition 2.3, we can suppose that the manifold N is the real line R. Replacing R with a connected component of the complement of the closed set Fix(ϕ(Homeo 0 (R))) if necessary, we can suppose that the group ϕ(Homeo 0 (R)) has no fixed point. Recall that the group Homeo 0 (R) is simple. Hence any morphism Homeo 0 (R) → Z/2Z is trivial. Thus, any element of the group ϕ(Homeo 0 (R)) preserves the orientation of R.

p 2 in R d . Suppose by contradiction that Fix(ϕ(G d p1 )) ∩ Fix(ϕ(G d p2 )) = ∅. By Lemma 2.
By Propositions 2.1 and 2.2, for any real number x, the group ϕ(G 1

x ) has a unique fixed point h(x). Take a homeomorphism f in Homeo 0 (R) which sends a point

x in R to a point y in R. Then f G 1 x f -1 = G 1 y and, taking the image under ϕ, ϕ(f )ϕ(G 1 x )ϕ(f ) -1 = ϕ(G 1 y ). Hence ϕ(f )(Fix(ϕ(G 1 x ))) = Fix(ϕ(G 1 y )). Therefore, for any homeomorphism f in Homeo 0 (R), ϕ(f )h = hf .
Let us prove that the map h is one-to-one. Suppose by contradiction that there exist real numbers x = y such that h(x) = h(y). The point h(x) is fixed under the groups ϕ(G 1

x ) and ϕ(G 1 y ). However, the groups G 1

x and G 1 y generate the group Homeo 0 (R) by Lemma 2.5. Therefore, the point h(x) is fixed under the group ϕ(Homeo 0 (R)), a contradiction. Now we prove that the map h is either strictly increasing or strictly decreasing. Fix two points x 0 < y 0 of the real line. For any two points x < y of the real line, let us consider a homeomorphism f x,y in Homeo 0 (R) such that f x,y (x 0 ) = x and f x,y (y 0 ) = y. As ϕ(f x,y )h = hf x,y , the homeomorphism ϕ(f x,y ) sends the ordered pair (h(x 0 ), h(y 0 )) to the ordered pair (h(x), h(y)). As the homeomorphism ϕ(f x,y ) is strictly increasing:

h(x) < h(y) ⇔ h(x 0 ) < h(y 0 ) and h(x) > h(y) ⇔ h(x 0 ) > h(y 0 ).
Hence the map h is either strictly increasing or strictly decreasing. Now, it remains to prove that the map h is onto to complete the proof. Suppose by contradiction that the map h is not onto. Notice that the set h(R) is preserved under the group ϕ(Homeo 0 (R)). If this set had a lower bound or an upper bound, then the supremum of this set or the infimum of this set would provide a fixed point for the group ϕ(Homeo 0 (R)), a contradiction. This set has neither upper bound nor lower bound. Let C be a connected component of the complement of the set h(R). To simplify the exposition of the proof, we suppose that the map h is increasing. Let us denote by x 0 the supremum of the set of points x such that the real number h(x) is lower than any point in the interval C. Then the point h(x 0 ) is necessarily in the closure of C: otherwise, there would exist an interval in the complementary of h(R) which strictly contains the interval C. We suppose for instance that the point h(x 0 ) is the supremum of the interval C. Choose, for each couple (z 1 , z 2 ) of real numbers, a homeomorphism g z1,z2 in Homeo 0 (R) which sends the point z 1 to the point z 2 . Then the sets g x0,x (C), for x in R, are pairwise disjoint: they are pairwise distinct as their suprema are pairwise distinct (the supremum of the set g x0,x (C) is the point h(x)). Moreover, those sets do not contain any point of h(R) and the infima of those sets are accumulated by points in h(R). Hence, these sets are pairwise disjoint. Then the set C has necessarily an empty interior as the topological space R is second-countable. Therefore C = {h(x 0 )}, which is not possible.

Proof of Proposition 2.1

The proof of this proposition is similar to the proofs of Lemmas 3.6 and 3.7 in [START_REF] Militon | Continuous actions of the group of homeomorphisms of the circle on surfaces[END_REF]. We need the following lemma. The proof of this lemma is almost identical to the proof of Lemma 2.5 and is omitted.

h 3 in H d D and h 2 in H d D ′ such that h = h 1 h 2 h 3 .
For such an embedded (d -1)-dimensional ball D, let

F D = Fix(ϕ(H d D )
). Let us prove that these sets are pairwise homeomorphic. Take two embedded (d -1)-dimensional balls D an D ′ and take a homeomorphism h in Homeo 0 (R d ) which sends the set D onto D ′ . Observe that

hH d D h -1 = H d D ′ and that ϕ(h)ϕ(H d D )ϕ(h) -1 = ϕ(H d D ′ ). Therefore: ϕ(h)(F D ) = F D ′ .
In the case where these sets are all empty, there is nothing to prove. We suppose in what follows that they are not empty.

Given two disjoint embedded (d -1)-dimensional balls D and D ′ , Lemma 3.1 implies, as in the proof of Lemma 2.4: Proof. Let (a 1 , a 2 ) be a connected component of the complement of F D0 . It is possible that either a 1 = -∞ or a 2 = +∞. Consider a homeomorphism e : R d-1 × R → R d such that e(B d-1 × {0}) = D 0 , where B d-1 denotes the unit closed ball in R d-1 . For any real number x, let D x = e(B d-1 × {x}). Given two real x = y, take a homeomorphism η x,y in Homeo 0 (R) which sends the point x to the point y. Consider a homeomorphism h x,y such that the following property is satisfied. The restriction of eh x,y e -1 to B d-1 × R is equal to the map:

F D ∩ F D ′ = ∅.
B d-1 × R → R d-1 × R (p, z) → (p, η x,y (z))
Notice that, for any real numbers x and y, h x,y (D x ) = D y Let us prove by contradiction that there exists a real number x = 0 such that F Dx ∩ (a 1 , a 2 ) = ∅. Suppose that, for any such embedded ball D x , F Dx ∩ (a 1 , a 2 ) = ∅. We claim that the open sets ϕ(h 0,x )((a 1 , a 2 )) are pairwise disjoint. It is not possible as there would be uncountably many pairwise disjoint open intervals in R.

Indeed, suppose by contradiction that there exists real numbers x = y such that ϕ(h 0,x )((a 1 , a 2 )) ∩ ϕ(h 0,y )((a 1 , a 2 )) = ∅. Notice that the homeomorphism h -1 0,x h 0,y and h -1 0,y h 0,x send respectively the set D 0 to sets of the form D z and D z ′ , where z, z ′ ∈ R. Hence, for i = 1, 2, the homeomorphisms ϕ(h -1 0,x h 0,y ) (respectively ϕ(h -1 0,y h 0,x )) sends the point a i ∈ F D0 to a point in F Dz (respectively in F D z ′ ). By hypothesis, these points do not belong to (a 1 , a 2 ). Therefore

ϕ(h -1 0,y h 0,x )(a 1 , a 2 ) = (a 1 , a 2 ) or ϕ(h 0,x )(a 1 , a 2 ) = ϕ(h 0,y )(a 1 , a 2 ).
But this last equality cannot hold as the real endpoints of the interval on the left-hand side belong to F Dx and the real endpoints point of the interval on the right-hand side belongs to F Dy . Moreover, we saw that these two closed sets were disjoint, a contradiction. ). Take a real number r < p 1 . Then, for any homeomorphisms

g 1 in G D , g 2 in G D ′ and g 3 in G D , ϕ(g 1 ) • ϕ(g 2 ) • ϕ(g 3 )(r) < p 2 .
By Lemma 3.1, this implies that the following inclusion holds:

ϕ(g)(r), g ∈ Homeo 0 (R d ) ⊂ (-∞, p 2 ].
The supremum of the left-hand set provides a fixed point for the action ϕ, a contradiction.

Proof of Proposition 2.2

This proof uses the following lemmas. For a subgroup G of Homeo 0 (R d ), we define the support Supp(G) of G as the closure of the set:

x ∈ R d , ∃g ∈ G, gx = x . Let Homeo Z (R) = {f ∈ Homeo(R), ∀x ∈ R, f (x + 1) = f (x) + 1} .
To prove Proposition 2.2, we need the following lemmas. Lemma 4.1. Let G and G ′ be subgroups of the group Homeo + (R) of orientationpreserving homeomorphisms of the circle. Suppose that the following conditions are satisfied.

1. The groups G and G ′ are isomorphic to the group Homeo Z (R).

The subgroups G and G

′ of Homeo + (R) commute: ∀g ∈ G, g ′ ∈ G ′ , gg ′ = g ′ g.
Then Supp(G) ⊂ Fix(G ′ ) and Supp(G ′ ) ⊂ Fix(G). Lemma 4.1 will be proved in the next section. We now provide a proof of Lemma 4.2.

Proof of Lemma 4.2. Take a closed ball B contained in U . Changing coordinates if necessary, we can suppose that B is the unit closed ball in R d . Take an orientationpreserving homeomorphism h : R → (-1, 1). For any orientation-preserving homeomorphism f : R → R, we define the homeomorphism λ h (f ) : R d → R d in the following way.

1. The homeomorphism λ h (f ) is equal to the identity outside the interior of the ball B.

For any

(x 1 , x ′ ) ∈ R × R d-1 ∩ int(B): λ h (f )(x 1 , x ′ ) = ( 1 -x ′ 2 h • f • h -1 ( x 1 1 -x ′ 2
), x ′ ).

The map λ h defines an embedding of the group Homeo + (R) into the group

Homeo 0 (R d ). The image under λ h of the group Homeo Z (R) is a subgroup of Homeo 0 (R d ) which is supported in U .
Let us complete now the proof of Proposition 2. Take a subgroup G 1 of Homeo 0 (R d ) which is isomorphic to Homeo Z (R) and supported in B. Such a subgroup exists by Lemma 4.2. This subgroup commutes with any subgroup G 2 of Homeo 0 (R d ) which is isomorphic to Homeo Z (R) and supported outside B.

If the group ϕ(Homeo 0 (R d )) admits a fixed point, there is nothing to prove. Suppose that this group has no fixed point. As the group Homeo 0 (R d ) is simple, the morphism ϕ is one-to-one. Moreover, any morphism Homeo 0 (R d ) → Z/2Z is trivial: the morphism ϕ takes values in Homeo + (R). Hence the subgroups ϕ(G 1 ) and ϕ(G 2 ) of Homeo(R) satisfy the hypothesis of Lemma 4.1. By this lemma:

∅ = Supp(ϕ(G 1 )) ⊂ Fix(ϕ(G 2 )).
We claim that the group G d B is generated by the union of its subgroups isomorphic to Homeo Z (R). This claim implies that

∅ = Supp(ϕ(G 1 )) ⊂ Fix(ϕ(G d B )).
1. The set F is pointwise fixed under any element in Homeo Z (R).

2. For any connected component K of the complement of F , there exists a homeomorphism h K : R → K such that:

∀f ∈ Homeo Z (R), ∀x ∈ K, ψ(f )(x) = h K f h -1 K . Lemma 5.2. Any group morphism Homeo Z (R) → Z is trivial.
Proof of Lemma 5.2. Actually, any element in Homeo Z (R) can be written as a product of commutators, i.e. elements of the form aba -1 b -1 , with a, b ∈ Homeo Z (R). For an explicit construction of such a decomposition, see Section 2 in [START_REF] Eisenbud | Transverse foliations of Seifert bundles and self-homeomorphisms of the circle[END_REF].

Observe that the center of the group Homeo Z (R) is the subgroup generated by the translation x → x + 1. Let α (respectively α ′ ) be a generator of the center of

G (respectively of G ′ ). Let A α = R -Fix(α) and A α ′ = R -Fix(α ′ ).
As the homeomorphisms α and α ′ commute:

α ′ (A α ) = A α α(A α ′ ) = A α ′ .
Take any connected component I of A α and any connected component I ′ of A α ′ . Then, either I is contained in I ′ , or I ′ is contained in I, or I and I ′ are disjoint.

We now prove that only the latter case can occur. Suppose by contradiction that the interval I is strictly contained in the interval I ′ . Let ∼ be the equivalence relation defined on I ′ by x ∼ y ⇔ (∃k ∈ Z, x = α ′k (y)).

The topological space I ′ / ∼ is homeomorphic to a circle. By Lemma 5.1, the group G ′ preserves the interval I ′ . Notice that the group G ′ / < α ′ >≈ Homeo 0 (S 1 ) acts on the circle I ′ / ∼. As the group G ′ commutes with the homeomorphism α, this action preserves the nonempty set (A α ∩ I ′ )/ ∼. As α ′ (A α ) = A α , the endpoints of the interval I are sent to points in the complement of A α under the iterates of the homeomorphism α ′ . Hence the set (A α ∩ I ′ )/ ∼ is not equal to the whole circle I ′ / ∼. However, by Theorem 5.3 in [START_REF] Matsumoto | Numerical invariants for semiconjugacy of homeomorphisms of the circle[END_REF] (see the remark below Theorem 1.2), any non-trivial action of the group Homeo 0 (S 1 ) on a circle is transitive. Hence, the group G ′ / < α ′ > acts trivially on the circle I ′ / ∼: for any element β ′ of G ′ , and any point x ∈ I ′ , there exists an integer k(x, β ′ ) ∈ Z such that β ′ (x) = α ′k(x,β ′ ) (x). Fixing such a point x, we see that the map

G ′ → Z β ′ → k(x, β ′ )
is a group morphism. Such a group morphism is trivial by Lemma 5.2. Therefore, the group G ′ acts trivially on the interval I ′ , a contradiction.

Of course, the case where the interval I ′ is strictly contained in I is symmetric and cannot occur.

Suppose now that I = I ′ . Take any element β ′ in G ′ . As the homeomorphism β ′ commutes with α, by Lemma 5.1, the homeomorphism β ′ is equal to some element of G on I. As the homeomorphism β ′ commute with any element of G, there exists a unique integer k

(β ′ ) such that β ′ |I = α k(β ′ ) |I . The map k : G → Z
(R)). Moreover Fix(ψ(Homeo Z (R))) = Fix(ψ(t)).
Claim 2. Any action of the group Homeo Z (R) on R without fixed points is conjugate to the standard action.

It is clear that these two claims imply Lemma 5.1.

First, let us prove Claim 1. The set Fix(ψ(t)) is preserved under any element in ψ(Homeo Z (R)), because any element of this group commutes with the homeomorphism ψ(t). Moreover, any element in ψ(Homeo Z (R)) preserves the orientation. Hence any connected component of the complement of Fix(ψ(t)) with infinite length is preserved under the action of the group ψ(Homeo Z (R)). We suppose now that any connected component of the complement of Fix(ψ(t)) has a finite length. Let us denote by ≡ the equivalence relation on R such that x ≡ y if and only if the points x and y belong to the same connected component of Supp(ψ(t)). The morphism ψ induces an action of the group Homeo Z (R)/ < t >≈ Homeo 0 (S 1 ) on the quotient topological space R/ ≡ which is homeomorphic to R if the set Fix(ψ(t)) has a nonempty interior. However, such an action is trivial by the remark below Theorem 1.2. Hence, any connected component of Supp(ψ(t)) is preserved by the action. Restricting to one of these connected components if necessary, we can suppose that the closed set Fix(ψ(t)) contains only isolated points. If this set is empty, there is nothing to prove. Otherwise, let Fix(ψ(t)) = {x i , i ∈ A} , where A is a set contained in Z and contains 0 and where the sequence (x i ) i∈A is strictly increasing. Then, for any element f in Homeo Z (R), there exists an integer i(f ) such that ψ(f )(x 0 ) = x i(f ) . The map i : Homeo Z (R) → Z is a group morphism: it is trivial by Lemma 5.2. Claim 1 is proved.

By Claim 1, restricting if necessary the action on a connected component of the complement of Fix(ψ(t)), we can suppose that the homeomorphism ψ(t) has no fixed point. Changing coordinates if necessary, we can suppose that the homeomorphism ψ(t) is the translation x → x+1. The morphism ψ induces an action ψ of the group Homeo Z (R)/ < t >≈ Homeo 0 (S 1 ) on the circle R/Z. This action is nontrivial: otherwise, there would exist a nontrivial group morphism Homeo 0 (S 1 ) → Z. By the remark below Theorem 1.2, there exists a homeomorphism h of the circle R/Z such that, for any homeomorphism f in Homeo Z (R)/ < t > (which can be canonically identified with Homeo 0 (R/Z)):

ψ(f ) = hf h -1 .
Take a lift h : R → R of h. For any integer n, denote by T n : R → R the translation x → x + n. For any homeomorphism f in Homeo Z (R), there exists an integer n(f ) such that ψ(f ) = T n(f ) hf h-1 .

However, the map n : Homeo Z (R) → Z is a group morphism: it is trivial by Lemma 5.2. This completes the proof of Claim 2.

Lemma 2 . 4 .

 24 Any group morphism Homeo 0 (R d ) → Homeo(R) is trivial.

  5, a point in this set is a fixed point of the group ϕ(Homeo 0 (R d )), a contradiction. By Proposition 2.2, the sets Fix(ϕ(G d p )), for p ∈ R d are nonempty. We just saw that they are pairwise disjoint. Recall that, for any embedded (d -1)-dimensional ball D, the set Fix(ϕ(H d D )) contains the union of the sets Fix(ϕ(G d p )) over the points p in the closed set D. Hence, this set has infinitely many points as d ≥ 2, a contradiction with Proposition 2.1.

Lemma 3 . 1 .

 31 Take two disjoint embedded (d -1)-dimensional balls D and D ′ in R d . For any homeomorphism f in Homeo 0 (R d ), there exist homeomorphisms h 1 ,

Lemma 3 . 2 .

 32 Fix an embedded (d -1)-dimensional ball D 0 of R d . Then any connected component C of the complement of F D0 meets one of the sets F D , where D is an embedded (d -1)-dimensional ball disjoint from D 0 .

Lemma 3 . 3 .

 33 Each set F D contains only one point. Proof. Suppose that there exists an embedded (d-1)-dimensional ball D such that the set F D contains two points p 1 < p 2 . By Lemma 3.2, there exists an embedded (d -1)-dimensional ball D ′ disjoint from D such that the set F D ′ has a common point with the open interval (p 1 , p 2

Lemma 4 . 2 .

 42 Take any nonempty open subset U of R d . Then there exists a subgroup of Homeo 0 (R d ) isomorphic to Homeo Z (R) which is supported in U .

2 .

 2 Proof of Proposition 2.2. Fix a point p in R d . Take a closed ball B ⊂ R d which is centered at p. Let G d B be the subgroup of G d p consisting of homeomorphisms which pointwise fix a neighbourhood of the ball B. Let us prove that Fix(G d B ) = ∅.

  is a nontrivial group morphism. But such a map cannot exist by Lemma 5.2. Lemma 4.1 is proved. Proof of Lemma 5.1. Denote by t a generator of the center of the group Homeo Z (R). The connected components of the complement of Fix(ψ(t)) are each preserved by the group ψ(Homeo Z

	It remains to prove Lemma 5.1.
	Claim 1.

)) which tends to +∞. Take positive integers n 1 , n 2 and n 3 such that a n1 < b n2 < a n3 . Fix x 0 < a n1 . We notice then that for any homeomorphisms

, the following inequality is satisfied:

However, any element g in Homeo 0 (R d ) can be written as a product

where g 1 and g 3 belong to G d B and g 2 belongs to G d B ′ . The proof of this fact is similar to that of Lemma 2.5. Therefore:

The greatest element of the left-hand set is a fixed point of the image of ϕ: this is not possible as this image was supposed to have no fixed point.

Observe that the group ϕ(G d p ) is the union of its subgroup of the form ϕ(G d B ′ ), with B ′ varying over the set B p of closed balls centered at the point p. By compactness, the set

is nonempty. Proposition 2.2 is proved.

5 Proof of Lemma 4.1

We need the following lemmas. The first one will be proved afterwards.

Lemma 5.1. Let ψ : Homeo Z (R) → Homeo + (R) be a group morphism. Then there exists a closed set F ⊂ R such that: