The lattice structure of the S-Lorenz core

For any TU game and any ranking of players, the set of all preimputations compatible with the ranking, equipped with the Lorenz order, is a bounded join semi-lattice. Furthermore, the set admits as sublattice the S-Lorenz core intersected with the region compatible with the ranking. This result uncovers a new property about the structure of the S-Lorenz core. As immediate corollaries, we obtain complementary results to the findings of Dutta and Ray (Games Econ Behav, 3(4):403–422, 1991), by showing that any S-constrained egalitarian allocation is the (unique) Lorenz greatest element of the S-Lorenz core on the rank-preserving region the allocation belongs to. Besides, our results suggest that the comparison between W- and S-constrained egalitarian allocations is more puzzling than what is usually admitted in the literature.


Introduction
The Lorenz criterion is widely accepted to compare profiles of revenues on the ground of egalitarianism. A profile Lorenz dominates another if its cumulative distribution, from rich to poor, is lower than the other one. But, the use of that rule as a social value may be conflicting with the maximizing behavior of agents. The cooperative game theory provides an appropriate framework to assess whether individual interest V. Iehlé (B) LEDa & CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France e-mail: vincent.iehle@dauphine.fr and egalitarianism can be accommodated together (see, e.g., Arin and Feltkamp 2002;Arin and Inarra 2001;Dutta 1990;Dutta andRay 1989, 1991;Hougaard et al. 2001;Jaffray and Mongin 2003;Llerena et al. 2008;Roth et al. 2005) and complements the classical analysis of inequality measurement (see, e.g., Atkinson and Bourguignon 1982;Dasgupta et al. 1973;Kolm 1977). 1 We consider here the solution concepts defined by Dutta andRay (1989, 1991) in cooperative games with transferable utility (TU games). The root concept is given by the notion of Lorenz core, defined recursively, that embodies egalitarianism and maximizing behavior together with the property of robustness against credible multilateral deviations of coalitions of players. The Lorenz core is defined in two manners depending whether the weak or strong domination relation is chosen in the definition of deviation (W-Lorenz core and S-Lorenz core in the remainder). The central solution in the analysis of Dutta andRay (1989, 1991) is given by the Constrained Egalitarian Allocation, also defined in two manners according to the chosen relation. A WCEA (resp. SCEA) is a Lorenz-undominated allocation in the W-Lorenz core (resp. S-Lorenz core). Hence, the distinction between the two definitions of a constrained egalitarian allocation is tight. However, it leads to contrasted results.
The sets of WCEAs and SCEAs are not comparable in general (see Dutta and Ray 1991,for a list of examples). Both solutions have also different qualitative properties. For the WCEA, Dutta and Ray (1989) find that the solution is unique, while its existence is not granted, except for convex games. In addition, they show that the solution is the Lorenz greatest allocation of the core in convex games, but not necessarily the greatest of the W-Lorenz core. For the SCEA, Dutta and Ray (1991) find that the solution exists under the mild assumption of weak superadditivity but is not unique in general. The outcome is, therefore, very sensitive to the choice of the domination criterion, and, in practice, the trade-off between uniqueness and existence may lead to a dilemma.
We complement some of the findings of Dutta and Ray (1991). Our conclusions derive from an abstract (though not difficult!) result which underpins the structure of the S-Lorenz core. For any TU game and any ranking of players, the set of all preimputations compatible with the ranking, equipped with the Lorenz order, is a bounded join semi-lattice. Furthermore, the set admits as sublattice the S-Lorenz core intersected with the region compatible with the ranking (Theorem 2). The result allows to compare easily S-Lorenz core allocations and to construct new allocations that Lorenz dominate others in the S-Lorenz core. As an immediate corollary, we obtain that the set of SCEAs is either empty or a singleton on each rank-preserving region (Corollary 1). As a by-product result, we also obtain that the set of SCEAs is finite, and what is more, the cardinality of the set can be made more precise according to the location of some SCEAs (Corollary 2). The underlying property beyond these results is that each SCEA is the Lorenz greatest element of the S-Lorenz core on the rank-preserving region of the allocation.
The note is organized as follows: Sect. 2 is devoted to notations and basic definitions; Sect. 3 presents the lattice structure result and its corollaries; in Sect. 4, we discuss briefly the implications of the lattice structure property; the proof of our main result, Theorem 2, is postponed to Appendix.

Preliminaries
Let N = {1, . . . , n} be a finite set of players and v be a real-valued function defined over the nonempty subsets of N . The pair (N , v) is called a cooperative game with transferable utility (TU game for short). A preimputation for S is a vector x ∈ R S such that i∈S x i = v(S). Let F S be the set of preimputations of S (we use F instead of F N and preimputation instead of preimputation of N ). For each coalition S ⊆ N , |S| is the equal division payoff of S. Player i is at least as desirable as j in the Let be the set of all bijective functions from N to itself (an element σ ∈ is a permutation of N ). We use the notation x σ to define the transformation of x according to σ , i.e., For any x ∈ A,x is the vector obtained by permuting the indices such thatx 1 ≥x 2 ... ≥x t . The allocation y ∈ A weakly Lorenz dominates and Lorenz dominates x ∈ A, y L x, if the inequalities hold with at least one strict inequality for some j. Let E A be the set of allocations x ∈ A such that there is no y ∈ A that Lorenz dominates x.

Structure of the S-Lorenz core and SCEAs
The Lorenz core and constrained egalitarian allocations are defined recursively by Dutta andRay (1989, 1991). An important feature of their solution is that the Lorenz criterion is part of the objectives of the whole population of players and also of any subgroup of the population taken separately. Given this consensus among the players, every coalition can still weigh on the final outcome via the threat of credible blocking (which accounts for egalitarianism). Thus, it worths pointing out that the solution differs from a narrower view of egalitarianism within the core that considers simply as solution concept the Lorenz-undominated allocations of the core, and where, con-sequently, blocking does not account for any egalitarian requirement (see, e.g., Arin and Feltkamp 2002;Arin and Inarra 2001;Hougaard et al. 2001).
The S-Lorenz core and SCEAs are defined as follows.

Definition 1 The S-Lorenz core of a singleton coalition {i} is
The S-Lorenz core of a coalition of S is then defined by: The set of SCEAs is E L * (N ). 2 Dutta and Ray (1991) show that the SCEA is not always unique, contrary to the set of WCEAs, which is either empty or a singleton as shown by Dutta and Ray (1989). Nevertheless, they alleviate this drawback by producing a set of sufficient conditions under which the locations of SCEAs can be inferred precisely (Dutta and Ray 1991, Theorem 3). Let us recall their result. The conclusion of Theorem 1 has two interesting implications. First, two different SCEAs, say x and y, cannot produce the same ranking of players with respect to their payoffs since two different allocations x and y with the same Lorenz curve satisfy necessarily x i > x j and y i < y j for some i, j = 1, . . . , n. Second, the set of SCEAs is necessarily finite. We obtain those two properties, without any assumption on the TU game, as immediate corollaries of our main result.
The following example illustrates how one can even strengthen the conclusion of Theorem 1.
Example 1 Consider the following example taken from Dutta and Ray (1991, Example 1) = 0 (clearly the game satisfies the assumptions of Theorem 1). Then, In other words, each SCEA is the (unique) greatest Lorenz element of the S-Lorenz core on the region that preserves the ranking of players it belongs to. Again, this property holds in general for any TU game and can be deduced from our main result.
We turn now to the formal statement of the announced results. First, we make precise what we mean by rank-preserving regions.
Definition 2 Given a permutation σ ∈ , the rank-preserving region R σ (according to σ ) is the subset of R N defined by Two vectors x and y belong to the same region if, and only if, x, y ∈ R σ for some σ ∈ . 5 Let R = {R σ | σ ∈ } be the family of all rank-preserving regions. First, it is an easy matter to verify that there is a bijection between R and . Second, it holds that ∪ R R = R N , |R| = n! and (x, . . . , x) ∈ ∩ R R for every x ∈ R. Next, given R ∈ R, the relation L defined over F ∩ R is a partial order, in particular L is antisymmetric on F ∩ R. 6 A set P equipped with a partial order defined on P is said to be a partially ordered set (poset for short). A poset (P, ) is a join semi-lattice if for any x, y ∈ P, the least upper bound of {x, y}, denoted by x ∨ y, exists. A join sublattice of (P, ) is (M, ), with ∅ = M ⊆ P, such that x ∨ y ∈ M for any x, y ∈ M. 7 We state now the main result of the paper. Its proof is postponed to Appendix. The proof consists in providing the algebraic definition of the least upper bound operator ∨. This operator will be defined as follows. Let x, y be two preimputations that belong to the same rank-preserving region R σ * ; define z 1 := min{x 1 ,ȳ 1 } and, for each i = 2, . . . , n,

Theorem 2 Let (N , v) be a TU game and R
Let x ∨ y be the vector z σ * −1 . We show through a sequence of short claims that ∨ is the least upper bound operator for the poset L := (F ∩ R, L ). In addition, we show that if x, y ∈ L * (N ) then x ∨ y also belongs to L * (N ), that is L * (N ) ∩ R forms a join 5 Note that x belongs to the same region as y is not a transitive relation. 6 Lorenz relation L is only a partial preorder on F. 7 See Davey and Priestley (2002) for more details. sublattice of L if L * (N ) ∩ R is nonempty. These facts will lead us to the statements 1 and 2 of Theorem 2. The next example illustrates better how natural is the construction of the least upper bound x ∨ y, in a simple case. Table 1 where the ordered profilesx andȳ of seven players have been reported together with the profile z. The allocations x = (4, 0.5, 5, 4, 12, 1.5, 9) and y = (3,0,9,6,9,0,9) belong to a same rank-reserving region but are not Lorenz comparable. It can be read immediately in Table 1 that the resulting allocation z σ * −1 Lorenz dominates both x and y.

Example 2 Consider
Remark 1 (only) From a theoretical point of view, finding the greatest lower bound operator would be of interest to define fully a lattice structure. Here, the natural candidate is the following operator x ∧ y := z σ * −1 , where z 1 := max{x 1 ,ȳ 1 } and, for each i = 2, . . . , n, But, x ∧ y does not belong to F ∩ R as shown in the example reported in Table 2 (with σ = Id).
As a direct consequence of Theorem 2, two Lorenz-undominated allocations x, y of the S-Lorenz core cannot coexist in the same rank-preserving region. For otherwise, z constructed as above would Lorenz dominate x and y and belong to the S-Lorenz core, which cannot be the case. What is more, the unique Lorenz-undominated allocation is necessarily the Lorenz greatest element of the S-Lorenz core on the region it belongs to (by means of a similar argument). This leads to the following straightforward corollaries, which provide new features on the SCEAs and complements Theorem 1. (N , v) be a TU game. For every rank-preserving region R ∈ R, if there is an SCEA x ∈ R then x is the greatest Lorenz element of L * (N ) ∩ R.

Corollary 1 Let
Hence, the set of SCEAs is finite and contains at most n! elements (the number of rank-preserving regions), and even more precisely at most #{R ∈ R : L * (N )∩ R = ∅} elements. 9 The previous statement can be reinforced if an SCEA belongs to several adjacent regions (typically if some components of the vector are equal). (N , v) be a TU game. If there is an SCEA x such that x ∈ ∩ T R for some subfamily T ⊆ R then x is the greatest Lorenz element of L * (N ) ∩ (∪ T R).

Corollary 2 Let
If the above condition is fulfilled, there are at most n!−|T |+1 SCEAs. For the polar case T = R, the set of SCEAs is a singleton, equal to the vector with components a(N ).
Corollaries 1 and 2 are probably the most eloquent results to be deduced from Theorem 2; so, we do not state any further variations in the same spirit.
Finally, note that above properties are not satisfied by the W-Lorenz core (i.e., with a weak domination relation) and the WCEAs. Dutta and Ray (1989, Example 5) construct a convex game where there exists a W-Lorenz core allocation, in the same region R as the unique WCEA, that is not Lorenz dominated by the WCEA (and thus, this W-Lorenz core allocation must be Lorenz dominated by another W-Lorenz core allocation). The example is striking at the first glance, but it simply says that the W-Lorenz core intersected with R, and equipped with the Lorenz order, is not a sublattice of (F ∩ R, L ).

Concluding remarks
Our main result deals with the structure of the S-Lorenz core. We did not use here any formal result of lattice theory, but we have shown that embedding the S-Lorenz core into lattices leads to a better understanding of the location of these allocations. To the best of our knowledge, such feature has never been suggested before for any core-like solutions in cooperative games, except of course in the specific setting of two-sided matching models where the lattice structure of stable matchings plays a great role (see, e.g., Knuth 1981;Roth and Sotomayor 1990). 10 Our approach provides also a handy framework to derive qualitative properties of the SCEAs and identify better the differences between the WCEAs and SCEAS. If we follow an admitted view by comparing them in terms of qualitative properties-existence (SCEA) versus uniqueness (WCEA)-our results show that the comparison is more puzzling than expected. 11 Indeed, on any given rank-preserving region, we have shown that the SCEA is the unique greatest Lorenz element of the S-Lorenz core, while it is known that the unique WCEA does not necessarily Lorenz dominate every other element of the W-Lorenz core (Dutta and Ray 1989, Example 5). 9 Using a different strategy, Llerena et al. (2008) show that the set of SCEAs is finite for any TU game. Dutta and Ray (1989) obtain their main positive results for convex games à la Shapley (1971). Recall that convexity in a cooperative game simply refers to the supermodularity of the characteristic function of the game. Besides, supermodularity is also known to be the baseline for comparative statics in lattice-embedded environments (see, e.g., Shannon 1994, 1996). It is, therefore, tempting to ask how supermodularity operates with respect to the lattice framework based on the Lorenz order.

Claim 3
If neither x L y nor y L x then x ∨ y L x and x ∨ y L y.
Proof From the definition of ∨: for all i = 1, . . . , n, i j=1 (x ∨ y) j ≤ i j=1 x j and i j=1 (x ∨ y) i ≤ i j=1 y j . Since x L y does not hold, there is k such that k j=1x j > k j=1 y j . It follows that x ∨ y L x. The symmetric reasoning leads to the same conclusion for y.
Claim 4 x L y iff x ∨ y = x.
Proof Obvious from the definition of Lorenz dominance and the construction of z.
Claim 5 if z L x and z L y then z x ∨ y.
Proof Suppose not then i j=1 (x ∨ y) i = min{ i j=1 x j ; i j=1 y j } < i j=1 z j for some i = 1, . . . , n, which contradicts that z L x and z L y.
The above 5 claims prove statement 1. of Theorem 2 (and observe that a N L x for every x ∈ F).
Proof It is true for i = 1, for i ≥ 2 it suffices to remark that from the definition of (x ∨ y) i and (x ∨ y) i−1 , it holds that i j=1 (x ∨ y) j = min{ i j=1 x j ; i j=1 y j } ≥ min{ i−1 j=1 x j ; i−1 j=1 y j } + min{x i ; y i } = i−1 j=1 (x ∨ y) j + min{x i ; y i }. It follows that (x ∨ y) i ≥ min{x i ; y i }.
We show now the statement 2. of Theorem 2. Suppose by way of contradiction that x, y ∈ L * (N ) ∩ R and x ∨ y / ∈ L * (N ). Dutta and Ray (1991, Theorem 1, p. 411) prove the following: L * (N ) = {x ∈ F | for no S ⊆ N : a(S) > x i ∀i ∈ S} Hence, there exists S ⊆ N such that a(S) > (x ∨ y) i for all i ∈ S. Since min{x i ; y i } ≤ (x ∨ y) i for all i ∈ N (from Claim 6), it holds that min{x i * ; y i * } ≤ (x ∨ y) i * for some i * such that i * ≤ j for any j ∈ S. Suppose that min{x i * ; y i * } = x i * . Then, it holds that x i * ≤ (x ∨ y) i * < a(S). Since σ * = Id, it also holds that x i < a(S) for all i ∈ S (recall that the components of the vector x are nonincreasing). It follows that x does not belong to L * (N ), which is a contradiction.