_{1}

Characterizations of the classes of all choice functions that select the cores or the externally stable cores induced by an underlying revealed dominance digraph are provided. Relying on such characterizations, the basic order-theoretic structure of the corresponding sets of revealed cores is also analyzed. In particular, it is shown that the poset of all revealed cores ordered by set inclusion is a median meet semilattice: therefore, any profile of revealed cores may be aggregated by means of the simple majority rule.

The core of a game is the set of its undominated outcomes, with respect to a suitably defined dominance irreflexive relation, or loopless digraph. Now, consider the ongoing operation of a multi-agent system, e.g. an organization or indeed any decision-making unit whose outputs are aptly modeled as the outcomes of a game. Let us then assume that the set of available options does in fact change at a faster pace than the behavioural attitudes of the relevant players and the latter interact as predicted by the core of that game. It follows that the corresponding choice behaviour of the given interaction system as recorded by its choice function should be constrained in some way by its game-theoretic structure and thus somehow reveal that fact. But then, what are the characteristic “fingerprints” of such a choice function, namely the testable behavioural predictions of the core as a solution concept? Or more simply, which choice functions defined over arbitrary subsets of an “universal” outcome set may be regarded as revealed cores? Let us call that issue, for ease of reference, the (full domain) core revelation problem.

Apparently, such a problem has never been addressed in its full generality in the extant literature. To be sure, parts of the massive body of literature on “revealed preference” provide partial answers addressing the case of nonempty cores, i.e. of acyclic revealed dominance digraphs (see e.g. [

The present paper is aimed at filling this gap in the literature by addressing the general core revelation problem with full domain as formulated above. It contributes to the extant literature in the following ways:

・ it provides characterizations of all choice functions with full domain―proper or not―that represent revealed cores,

・ under several variants of the notion of core (Theorems 7, 10, and 14).

Moreover,

・ A study of the basic order-theoretic structure of the corresponding classes of revealed core-solutions as canonically ordered by set-inclusion is also provided (Theorems 17, 20, 21 and 22). In particular, it is shown that the class of all revealed cores (as opposed to, say, the class of nonempty-valued revealed cores) is a meet sub-semilattice of the lattice of all choice functions, and in fact a median meet semilattice (see Theorem 17). A remarkable consequence of that fact is that any profile of revealed cores is amenable to aggregation by the simple majority rule.

Thus, it turns out that each revealed core embodies a considerable part of standard maximizing choice, while the global structure of (full domain) revealed cores retains the order-theoretic properties of the space of all (full domain) choice functions that is most significant from the point of view of simple majority aggregation.

A further generalization of the core revelation problem to the case of choice functions with an arbitrary domain (along the lines of [

The paper is organized as follows: Section 2 includes a presentation of the model and the main characterization results; Section 3 provides some basic results concerning the order-theoretic properties of the classes of revealed core-solutions previously characterized; Section 4 consists of a few concluding remarks.

Let X be a set denoting the “universal” outcome set, with cardinality_{X} the set of all choice functions on X, and _{c}-is the set of all subsets of X with a nonempty-va- lued choice set i.e.

Let

For any

The a-core of

The core (a-core) of

A dominance digraph

Remark 1. It should be emphasized here that any dominance digraph may arise in a natural way from an underlying game in coalitional form and from a related game in strategic form. Indeed, the dominance digraph

For any

Two binary relations_{c} induced by a choice function

A choice function

The following choice functions provide some remarkable examples―and non-examples―of revealed cores. In particular, the first one will also play a role in the proofs of some results in Section 3, while the second one is a version of the well-known―and widely studied―“satisficing behavior”.

Example 2. Notice that digraph

Example 3. Take

Example 4. By way of contrast, take again

The main objective of this article is precisely to provide a characterization of all revealed cores in

To begin with, let us consider two requirements concerning local existence of nonempty choice sets.

No-dummy property (ND):

2-Properness (2-PR):

It is easily checked that ND is satisfied by all revealed cores, while 2-PR is only violated by core solutions when the underlying dominance digraph is not asymmetric. A stronger property that obviously entails both ND and 2-PR is:

Properness (PR):

The following properties of a choice function

Chernoff Contraction-consistency (C): for any

Concordance (CO): for any

Superset consistency (SS): for any

Property C is a contraction-consistency condition for choice sets in that it requires that any outcome chosen out of a certain set should also be chosen out of any subset of the former: essentially, it says that any good reason to choose a certain option out of a given menu should retain its strength in every submenu of the former containing that option.

Conversely, property CO (also variously denoted as

Property SS is also an expansion-consistency requirement for choice sets: it rules out the possibility that the choice set of a certain menu be nonempty and strictly included in the choice sets of a smaller menu.

We are now ready to prove the main results of this paper. Let us start from the following simple.

Claim 5. Let

(i)

(ii) for any

(iii) R is reflexive iff

(iv) R is total iff

(v) R is quasi-transitive iff

Proof. (i) For any

(ii) Let

(iii) Indeed, by definition for any

(iv) Suppose

(v) Suppose that R is quasi-transitive, and that both

and xRy), and (not zRy and yRz) i.e.

Remark 6. The content of the previous Claim is certainly not unknown, but I have been unable to find a reference in print to it except for the statement of point (iv) in [

The following Theorem extends and/or supplements some previous characterization results for revealed cores due to [

Theorem 7. Let

(i) c satisfies ND, C and CO;

(ii) there exists an irreflexive

(iii) there exists a reflexive relation

(iv)

Proof. (i) Þ (iv): Let

it might be the case that

(ii) Û (iii) (see [

(iii) Þ (iv): See [

(iv) Þ (iii): Trivial.

(iii) Þ (i): Suppose that there exists a reflexive relation

Remark 8. Notice that the equivalence between statements (ii) and (iii) of Theorem 7 above might in fact be credited to [

Remark 9. The foregoing characterization result is tight. To check that, consider the following examples.

1) Let

2) Let

3) Let

Next, we have a similar characterization result for revealed a-cores which is also an extension to the general case of possibly non-proper choice functions of previous results as discussed below (see Remark 13).

Theorem 10. Let

(i) c satisfies ND, 2-PR, C and CO;

(ii) there exists an irreflexive relation

(iii) there exists a total relation

(iv)

Proof. (i) Þ (iii): Let

(ii) Û (iii): Suppose that there exists a total relation

Then, as recorded by Claim 5 (ii)

is asymmetric since R is total, hence in particular

(ii) Þ (i): Suppose that there exists a dominance digraph

otherwise

Also, for any

Moreover, let

(iii) Û (iv): See the proof of Theorem 7 above. ■

Remark 11. The foregoing characterization result is also tight. To see this, consider the following examples.

1) Let

2) Let

3) Let

4) Let

Corollary 12. (see also [

(i) c satisfies C and CO;

(ii) there exists a strictly acyclic dominance digraph

(iii) there exists a total relation

(iv) there exists a relation

(v)

Proof. (i) Þ (ii): Since

In particular,

indeed strictly acyclic and

(ii) Þ (i): See the proof of Theorem 7 above.

(i) Û (iii): Obvious, by Theorem 10 above, since, again,

(iii) Û (iv): Suppose there exists

(iii) Û (v): See the proof of Theorem 6 above, and of course [

Remark 13. Actually, it is well-known that a proper c satisfies both C and CO if and only if there exists a binary relation R on X such that

Let us now turn to characterizations of revealed externally stable core-solutions. Since externally stable cores (of nonempty sets) are nonempty the corresponding choice functions are proper: thus, given the traditional focus on proper choice functions, this subclass of revealed cores is the most widely studied, and best known (thanks again to [

Theorem 14. Let

(i) c satisfies PR, C, CO and SS;

(ii) there exists a quasi-transitive relation

(iii) there exists a total and quasi-transitive relation

(iv)

(v) there exists a reflexive and negatively transitive relation

(vi) there exists a negatively transitive relation

(vii) there exists an irreflexive relation

(viii) there exists an irreflexive and transitive relation

(ix) there exists a a strict partial order

Proof. (i) Þ (ii) ( [

(ii) Þ (i) ( [

(ii) Û (iii): Let be

(iii) Û (iv): See the proof of Theorem 7 above.

(iii) Û (v): Let ^{a}x and zR^{a}y. By quasi-transitivity, it follows that zR^{a}x, whence in particular not xRz i.e. R is negatively transitive. Moreover, totality implies reflexivity of R. Conversely, let ^{a}y and yR^{a}z. Then, in particular, not yRx and not zRy. It follows that, by negative transitivity, not zRx whence, by totality, xRz. Thus, xR^{a}z i.e. R is quasi-transitive as well.

(v) Û (vi): Let

(iii)Þ (vii): Let be

(vii) Þ (i): Suppose that there exists a dominance digraph

(viii) Û (iii): Suppose that there exists a dominance digraph

(viii) Û (ix): Suppose that there exists a dominance digraph

Remark 15. Observe that the characterization result of revealed externally stable cores in terms of properties of choice functions included in Theorem 14 is also tight. To see this, consider the following examples.

1) Let

2) Let

3) Let

4) Let

to satisfy SS since

Remark 16. Notice again that Theorem 14 above is essentially a refinement of well-known results due to Suzumura (see e.g. [

Path Independence (PI): for any

Thus, the equivalent statements of Theorem 14 are also equivalent to the statement “

It should be remarked that the characterizations provided above are in general quite straightforward extensions to arbitrary choice functions (with full domain) of previously known results concerning proper choice functions (with full domain). Indeed, the gist of the results offered in the present section may be summarized as follows:

(i) remarkably, the characterizations of general revealed cores and a-cores considered here consist of the very same properties used to characterize their nonempty-valued counterparts as supplemented with very mild-look- ing local nonemptiness requirements for choice sets of singleton and two-valued subsets, respectively;

(ii) the exact correspondence between revealed core-solutions and maximizing “rational” choice functions is confirmed to hold within the general space of arbitrary choice functions: the alleged extra-generality of the latter subclass that has sometimes been alluded to in the literature (as e.g. in [

(iii) finally, and most notably, the class of general revealed cores turns out to inherit some of the supplementary order-theoretic structure enjoyed by its larger ambient space as compared to the smaller and less regular space of proper choice functions: that is precisely the topic of the next section.

Let us now turn to a global description of the order-theoretic structure of the class of all revealed core-solutions (a-core-solutions, nonempty-valued core-solutions, externally stable core-solutions, respectively).

A partially ordered set or poset is a pair

A poset

A lattice

A meet semilattice

has the coronation (or join-Kelly) property if―for any

The set

For any

The minimum ND choice function

Now, let

notation

denotes

Theorem 17. The poset

Proof. Let

Moreover, for any

hence

Finally, since c and

and CO also holds for

Furthermore, let us suppose that

Thus, we only have to check that

By definition, it follows that

It is easily checked that

Now, consider

Next, take any

Moreover, by definition

Let

Suppose first that

It follows that if

Conversely, let c be a coatom of

To check that each

then either

Conversely, assume that c is an atom of

To check that

Now, posit

Remark 18. Notice that finiteness of X has been used in the proof above in order to show that the set of coatoms of

CO*: for any family

Remark 19. Since

Thus, the poset of revealed core-solutions enjoys the remarkably regular structure of a median meet-semilat- tice. Notice that an important consequence of that fact is the following: any profile of revealed cores admits medians and the latter coincide with the simple majority consensus revealed core if the profile consists of an odd list of revealed cores. Therefore, in case several revealed cores are to be considered for aggregation, due perhaps to locally missing or unreliable data and/or plurality of information sources, an amalgamation process by means of the simple majority aggregation rule is available (see e.g. [

The posets of revealed a-core-solutions, nonempty-valued core-solutions, and externally stable core-solutions are considerably less regular, as recorded by the following results, namely:

Theorem 20. The poset

Proof. To check that

The proof of Theorem 17 already establishes that

Next, consider

However,

Therefore,

The last statement about minimal elements of

Theorem 21. The poset

Proof. First, notice that by definition

Theorem 22. The poset

Proof. Observe that for any

Finally, it is immediately checked by direct inspection that

Thus, while only the poset of revealed core-solutions is a (meet) sub-semilattice of

Choice functions with full domain which may be regarded as core-solutions or externally stable core solutions of an underlying dominance digraph

An obvious extension of the present paper should address the characterization problem for revealed cores on arbitrary domains. That open issue is left as a topic for further research.

StefanoVannucci, (2015) Revealed Cores: Characterizations and Structure. Applied Mathematics,06,2279-2291. doi: 10.4236/am.2015.614200