This chapter studies the ordinal content of supermodularity on lattices. This chapter is a generalization of the famous study of binary relations over finite Boolean algebras obtained by Wong, Yao and Lingras. We study the implications of various types of supermodularity for preferences over finite lattices. We prove that preferences on a finite lattice merely respecting the lattice order cannot disentangle these usual economic assumptions of supermodularity and infinite supermodularity. More precisely, the existence of a supermodular representation is equivalent to the existence of an infinitely supermodular representation. In addition, the strict increasingness of a complete preorder on a finite lattice is equivalent to the existence of a strictly increasing and infinitely supermodular representation. For wide classes of binary relations, the ordinal contents of quasisupermodularity, supermodularity and infinite supermodularity are exactly the same. In the end, we extend our results from finite lattices to infinite lattices.
- JEL Classifications: D11
The aim of this chapter is mainly twofold. It intends first to emphasize that on finite lattices, preferences merely respecting the lattice order cannot disentangle the usual economic hypothesis of supermodularity representations from the much stronger supermodular representations. Thus, we complement the work of Chambers and Echenique [1, 2] who nicely prove under the assumption of weak monotonicity that supermodularity is equivalent to the notion of quasisupermodularity introduced by Milgrom and Shannon .
Second, we aim at offering simple constructive proofs for the existence of -supermodular representations on finite lattices, hence generalizing to finite lattice the characterization obtained on finite Boolean Algebras by Wong, Yao and Lingras  of complete preorders representable by belief functions.
It is well known that supermodularity is a concept widely used in relation with economies of scale. It indicates a synergy relationship of subsystems, so that the marginal returns to the marginal element are closely related to the size of the existing elements. This creates nonlinear expectations that could have wide potential applications in social sciences. For example, we might see their applications in nonlinear pricing models as well as product bundling models.
This chapter is organized as follows. Section 2 introduces several notions of supermodularity. Then we propose our main result Theorem 1 over -supermodularity representations and underline through Proposition 1 that, in a finite lattice, quasisupermodularity is a very weak assumption, since for weakly increasing preference relations which are complete preorders, it cannot be distinguished from weak quasisupermodularity but also from what we call strong quasisupermodularity. Section 3 finally shows that complete preorders merely requiring strong monotonicity for preferences lead to the existence of supermodular representations.
2. Infinite Supermodularity
2.1. supermodularity and preference
Definition 1 Let be a finite lattice and a preference relation on X (i.e. a binary relation on with asymmetric part and symmetric part ).
is said to be weakly increasing if .
is said to be strictly increasing if it is weakly increasing and .
is said to be weakly quasisupermodular if .
is said to be quasisupermodular if and .
is said to be strongly quasisupermodular if .
Note that what we call strong quasisupermodularity is dual (with instead of of what Chambers and Echenique  called modularity, a property referred to as Generalized Kreps by Epstein and Marinacci .
Definition 2 A function is said to be quasisupermodular if, for any , implies and implies . It is said to be supermodular if, for any , .1
Definition 3 is said to be supermodular if it allows a supermodular representation , i.e. there exists supermodular such that: for all , if then and if then . Furthermore, the representation is weakly increasing if .
Definition 4 Let be a finite lattice then is said to be supermodular if
The following simple example illustrates that indeed supermodularity and supermodularity are two different notions for weakly increasing functions on a finite lattice
Example 5 let and consider where consists of the following partitions of
and is defined by if partition is a refinement of partition It is straightforward to see that and for Furthermore let be defined by and clearly is supermodular but not supermodular as just proved now. In actual fact, for a function defined by , then is supermodular if and only if .
Chambers and Echenique  have shown that a preference relation on a lattice has a weakly increasing supermodular representation if and only if it has a weakly increasing and quasisupermodular representation. Now, we show that this is also equivalent to allowing a weakly increasing supermodular representation.
Theorem 1: A binary relation on X has a weakly increasing and quasisupermodular representation if and only if it has a weakly increasing and supermodular representation.
Proof. If part: Since an supermodular representation is always super modular, the if part is immediate.
Only if: This part of the proof is highly inspired by the paper of David Kreps, A representation theorem for preference for flexibility, and especially by the proof of Lemma 3, p.572.
From Theorem 1 of Chambers and Echenique , we know that there is a weakly increasing supermodular which represents .
Let be the total preorder induced on by , i.e. . Let be the asymmetric part of . Clearly, agrees with , i.e. and . Therefore, the proof will be complete if one shows that can be represented by a -supermodular function which, by construction, will be necessarily weakly increasing.
Henceforth, to simplify the exposition, we will abuse of notation, letting be denoted again by . Note that this newis again monotone, i.e. , since .
Let denote the equivalence class of any There is a finite number of equivalence classes, . Note that since is supermodular: . This will be very useful later on. At last, for any , define . The following lemma will be crucial [8, 9].
Lemma 1. For any , there exists a unique such that .
Proof. Let us first show uniqueness. Suppose that and satisfy the property of Lemma 1. Then, and so .
Since is finite, there exists at least one minimal element for in , which is denoted by . The proof will be completed if we show that .
Note that . Actually, if and , then weak increasingness of implies , hence since .
It remains to prove that , or, equally, that , , implies . So let us show that if , , , then not() is impossible.
If not (), then . Actually, one has always and , so if , then we would get , a contradiction.
Let us see now that, from the definition of , implies . Actually, if , since and is monotone, it turns out that , which entails . Therefore, , but, since , this contradicts the fact that is a minimal element of for .
So and, by supermodularity, . But and , hence . So, by monotonicity, . Therefore, , a contradiction, which completes the proof of Lemma 1. ■
We can now turn to finishing the proof of Theorem 1. We intend to define, for any , in a consistent way such that the function , defined by for any , represents .
Let be the minimal element of for . Since for any and implies , one has . For any , let . Therefore, for any , .
Let us now show by induction that the ‘s can be defined in such a way that there exists , satisfying , and , where .
This is true for . Suppose that this has been done up to , and let us prove the result for index .
Let . Let us first show that we can suitably obtain , for . Note that .
Since by monotonicity implies , hence , it comes from the definition of that . Therefore, at step , is already defined, so since there is a finite number of ‘s, one can choose such that and such that be positive. For such an , we consequently get .
It thus remains to see that we can choose suitably the values of the remaining ‘s satisfying for . So for any given , , we need to show that it is possible to get , where . Let be such that , then by monotonicity . If , then , and if , then necessarily and, therefore, by definition of , necessarily , indeed implies . So , , , . Since any such that has not yet been attributed a value , we can state for such ‘s. It comes that , that is .
So finally we get ‘s satisfying the required condition of representation: with , . It remains to show that defined this way is indeed -supermodular. While we might involve Möbius inversion as in the seminal book of Rota ), we choose for sake of self completion to propose the following direct proof. Let , and let us prove that
For let , then:
But since .
Hence, is an supermodular and weakly increasing by construction. ■
The following corollary shows that, as soon as the binary relation on is a complete preorder, that is, reflexive, transitive and complete (i.e. , or or both), one can obtain a much more general result.
Corollary 1: For a complete preorder on a lattice , the following assertions are equivalent:
is weakly increasing and quasisupermodular.
has a weakly increasing and quasisupermodular representation.
has a weakly increasing and supermodular representation.
Proof. (i) (iii): Starting the proof of only if of Theorem 1 at the point where we were considering the finite equivalent classes of for gives the result, taking into account the fact that by hypothesis is monotone, or, in other words, weakly increasing, and that quasisupermodular implies: . (iii) (ii) is immediate. (ii) (i): Let be a weakly increasing and quasisupermodular representation of . Let , then weakly increasing implies and representation of the complete preorder implies , therefore is weakly increasing. It remains to prove that is quasisupermodular. Since is weakly increasing and since and , one gets and so indeed . Let us now show : and represents implies , quasisupermodular then implies , and represents the complete preorder finally implies , which completes the proof of the corollary.■
Remark and example: The following example illustrates that indeed even a complete preorder on a finite lattice may possess both a weakly increasing super modular representation which is not supermodular and also a weakly increasing supermodular representation.
Consider a finite set and the finite lattice . Let be defined by if the cardinal of , denoted , equals , if , if and if .
As proved in Chateauneuf and Jaffray  in Example 4, is supermodular but not supermodular. Let be the complete preorder on defined by . Clearly, is a weakly increasing supermodular representation of which is not supermodular. Hence, has a weakly increasing and quasisupermodular representation, and therefore from Corollary 1 has a weakly increasing and supermodular representation.
For instance, setting , if and if , defining does the job, since if , if , if and if . Hence, is a weakly increasing and supermodular representation of .
2.2. Weak quasisupermodularity, quasisupermodularity and strong quasisupermodularity
Now we show, for a weakly increasing complete preorder on a finite lattice , the equivalence of the different notions of quasisupermodularity defined in Definition 1.
Proposition 1: Let be a finite lattice, if is a weakly increasing complete preorder, then the following statements are equivalent.
⪰ is weakly quasisupermodular.
⪰ is quasisupermodular.
is strongly quasisupermodular.
Proof. (2) (1): We need to show Suppose not, then by monotonicity, we must have quasisupermodularity implies , a contradiction. (1) (3): Suppose we need to show that Since thus if then hence . By weak quasisupermodularity, we have i.e.(3) (2): We need to show that and Since is weakly increasing it is clear that is always true. Now we prove the second statement: First, is weakly increasing implies thus if it is not then it must be the case that strong quasisupermodularity implies , which says contradicting ■
Thus, we have shown that, for a weakly increasing complete preorder over a finite lattice , supermodularity.
3. -Supermodular representation for strictly monotone preference on a lattice
Definition 6 A function is strictly increasing if
Theorem 2 below shows that if the preference relation on i a complete preorder, then strict monotonicity of is not only necessary but also sufficient in order to get a strictly increasing supermodular function representing . Moreover, the proof offers a simple constructive way to build such a representation.
Theorem 2: Let be a complete preorder on , then the following statements are equivalent:
is strictly increasing.
has a strictly increasing and quasisupermodular representation.
has a strictly increasing and supermodular representation.
Proof. (i) (iii): Let denote the equivalence class of for , and let us consider the finite number of equivalence classes . As in the proof of Theorem 1, it is enough to show that there exist such that, setting , one gets if and only if . Actually, such an will indeed represent and be supermodular. Moreover, since is strictly increasing, implies and we will get implies so will be strictly increasing. So let us define inductively the ‘s in order that the function defined by represents .
Let stands for the minimal element in . Note that . Actually, , , so if , indeed by reflexivity of , and if , then since is strictly increasing. It turns out that, letting (eventually ), one gets .
Let us now consider . For any , implies by strict monotonicity of . So for any given , one gets if and only if . Actually: and, conversely, given , is impossible, and, since , one gets . Therefore, defining , one gets for that where .
Consider now . The same reasoning as before shows that for any , , , and implies . Since the ‘s belonging to are finite, let be such that . Note that this quantity is well defined since has already been defined for . Choose sufficiently great in order that . Choose now the remaining ‘s where such that . Then, necessarily . So we get .
Indeed, this process applies step by step along increasing rank of the classes, and thus gives the searched for result.
(iii) (ii) is immediate. (ii) (i) is immediate since because represents the complete preorder. ■
As an immediate consequence, we obtain a stronger form of Corollary 5 of Chambers and Echenique .
Corollary 2 Let be a finite lattice. If a binary relation on has a strictly increasing representation, then it has a strictly increasing supermodular representation and even a strictly increasing supermodular representation.
Proof. Let be a strictly increasing representation of and define the complete preorder on by . Then, and not(). Hence, is a strictly increasing complete preorder on . From Theorem 2, , hence , has a strictly increasing supermodular representation and, therefore, has a strictly increasing supermodular representation. This indeed implies that has a supermodular representation as it is proved in Corollary 5 of Chambers and Echenique . ■
4. Extensions to infinite lattices
We shall extend our major result to infinite lattices.
First it should noted that when we consider infinite lattices, we would need a separability in order represent the given preference. The following is a counter example.
Example 7 Let L be the standard Borel algebra on the std. Lebesgue measure, be a distribution that with mass points on all the rational numbers of . We define an order on L to be x < y if or this is the induced lexicographic order on L. It is well known that the lexicographic order is not separable and does not allow a representation, thus the defined order would not allow any representation.
Definition 8 A preference is said to be lower finitely separable if there is a countable set C such that 1. is finite for all 2. .
A differential operator on a ordered lattice can be introduced in the following manner.
Definition 9 The difference operator on lattice is defined recursively as
The following proposition is well known for supermodular functions on finite lattices, one can see for example the work by J. P. Barthelemy (2000) p. 199–200.
Proposition 10 an increasing function is -supermodular if and only if .
Now we are ready to extend our result that strictly increasing preference must allow strictly increasing -supermodular representation on any infinite lattices with lower finite separability.
Proposition 11 Let be a lattice with lower finite separability then the following two statements are equivalent.
is a strictly increasing preference relation.
There exists an strictly increasing supermodular representing .
Proof. Recall that a subset is said to separate if such that .
Let be a chain relative to separating the preference since is strictly increasing, we know that C will also separate the lattice order Denote the maximal element of It is well defined due to lower finiteness.
Given any weight function assigned to the separating chain Denote if and otherwise.
Clearly given any positive function the function represents
Now we claim that we can properly choose in such a way that is infinitely supermodular.
Actually, we choose and note this this is possible because is a finite lattice.
We claim if we choose as in above then will infinitely supermodular.
In fact, the first difference: if and otherwise.
if or by our choice of and otherwise.
It can be easily checked that if there is some if not, then
This completes the proof. ■
In this chapter, we have explored the ordinal content of supermodularity on lattices. We studied the implications of various types of supermodularity for preferences over lattices. Especially we show that preferences on a lattice merely respecting the lattice order cannot disentangle these usual economic assumptions of supermodularity and infinite supermodularity. In addition, the strict increasingness of a complete preorder on a lattice is equivalent to the existence of a strictly increasing and infinitely supermodular representation. For wide classes of binary relations, the ordinal contents of quasisupermodularity, supermodularity and infinite supermodularity are exactly the same.
- Clearly, u supermodular implies u quasisupermodular.