Revealed cores: characterizations and structure

Characterizations of the 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. JEL classification: C70, C71, D01, D03


Introduction
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.[1]- [4]).Moreover, there is also some work covering the case of possibly empty sets of undominated outcomes for an arbitrary-i.e.possibly not irreflexive-binary relation R, hence putting aside the original game-theoretic interpretation of R as a dominance relation (see e.g.[5], and [6]).But of course the dominance relation of a game in its usual meaning has to be irreflexive (no outcome dominates itself), and the core of a game may well be empty, because its revealed dominance digraph may have cycles.Here, we are interested precisely in the general version of the core revelation problem for the full domain, namely in a characterization of all revealed cores as solutions for a certain "universal" outcome set and all of its subsets, including (locally) empty-valued cores.
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 [6]) would be most helpful.That task is left as a topic for another paper.
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.

Choice Functions and Revealed Cores
Let X be a set denoting the "universal" outcome set, with cardinality # 3 X ≥ , and ( ) X  its power set.It is also assumed for the sake of convenience that X is finite (but it should be remarked that the bulk of the ensuing analysis is easily lifted with suitable minor adaptations to the case of an infinite outcome set).A choice function on X (with full domain) is a deflationary operator on ( ) for any A X ⊆ (empty choice sets are allowed).A choice function c is proper if ( ) c A ≠ ∅ whenever A X ∅ ≠ ⊆ .We denote C X the set of all choice functions on X, and X C  the subset of all proper choice functions on X.The proper subdomain of X c C ∈ -written D c -is the set of all subsets of X with a nonempty-valued choice set i.e.
, and any Y X ⊆ , a  and s  denote the asymmetric and symmetric components of  , respectively, while ( ) T  is the smallest transitive R ⊇  .Moreover,  is strictly acyclic iff its transitive closure is irreflexive, and a strict partial order iff it is both asymmetric and transitive.
Let X X ∆ ⊆ × be an irreflexive binary relation on X, denoting a suitably defined dominance relation: ( ) and ( ) is the induced dominance subdigraph on Y. Broadly speaking, the core of ( ) The a-core of ( ) The core (a-core) of ( ) A dominance digraph ( ) , X ∆ is also said to be core-perfect or strictly acyclic (acyclic, respectively) if ( ) , respectively) for any Y X ⊆ .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 ( ) ∆ defined by the following rule can be attached in a natural way to any coalitional game x A E S ∈ ∈ and i z y  for all i S ∈ and z A ∈ (see [7] for further details).
, respectively) for any Y X ⊆ .Then, we also say that c is core-rationalizable (a-core-rationalizable, ES-core-rationalizable, ES-a-core-rationalizable respectively) by the dominance digraph ( ) , X ∆ .Clearly, ES (a-)core-solutions are refinements of (a-)core solutions.Revealed cores will also be used as a generic label to denote all the foregoing choice functions.
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 3. Take G X ∅ ⊆ ⊂ and consider the nonempty valued dichotomic choice function as defined by the "lax" satisficing rule ( ) The main objective of this article is precisely to provide a characterization of all revealed cores in X C , and study their basic order-theoretic structure.
To begin with, let us consider two requirements concerning local existence of nonempty choice sets.
No-dummy property (ND): 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: . 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 γ or Generalized Condorcet-consistency) is an expan- sion-consistency condition for choice sets, requiring that an outcome chosen out of a certain set and of a second one should also be chosen out of the larger set given by the union of those two sets: it says that any good reason to choose a certain option out of two given menus should retain its strength in the larger menu obtained by merging those two menus.
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 R X X ⊆ × be any (binary) relation on X, and define R X X ∆ ⊆ × by the following rule: for any , x y X  [8], while Theorem 8 of [3] only includes a specialized version of the same point.

. 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 [1] and [2].
Theorem 7. Let X c C ∈ .Then, the following statements are equivalent: ( ) by definition, and ( ) this is an extension to arbitrary choice functions of the proof of the same result for proper choice functions due to [2]). (iv , hence c satisfies ND.Moreover, for any Y Z X ⊆ ⊆ and any , it must also be the case that ( ) and CO is satisfied as well.■ Remark 8. Notice that the equivalence between statements (ii) and (iii) of Theorem 7 above might in fact be credited to [1] because it is strictly related (indeed, essentially equivalent) to a full-domain specialized version of Theorem 3 of that paper, though the latter concerns nonempty core-solutions over an arbitrary domain hence, strictly speaking, is a statement about a class of proper choice functions on arbitrary domains.On the other hand, [9] has a similar result (see its Theorem 2.5), namely a characterization by the conjunction of C and CO of the choice functions selecting the outcomes "permitted" by all outcomes-or "not prohibited" by any outcome-according to an arbitrary "permission" or "prohibition" binary relation.A characterization of "sums" of revealed cores or "multi-criteria choice functions" by the conjunction of ND and C is suggested in [10].
Remark 9.The foregoing characterization result is tight.To check that, consider the following examples.1) Let I X c C ∈ be defined as follows: for any A X ⊆ ,

( ) max
( ) It is immediately checked that III c satisfies ND and CO, but violates C since e.g.
be defined as follows: for any A X ⊆ ,

( ) max
where L is a linear order on X.It is easily seen that III c satisfies ND and C, but violates CO.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 X c C ∈ .Then, the following statements are equivalent: Then, as recorded by Claim 5 (ii) ( ) { } suppose that there exists a dominance digraph ( ) , for any Y X ⊆ .Then, as recorded by Claim 5 (ii) (ii) ⇒ (i): Suppose that there exists a dominance digraph ( ) and ND is therefore satisfied by c.Furthermore, for any , ) (ii) there exists a strictly acyclic dominance digraph ( ) is proper hence in particular it also satisfies ND and 2-PR.Therefore, by Theorem 10 (ii) above, there exists a dominance digraph ( ) for any Y X ⊆ .Moreover, since by hypothesis c is proper, ( ) x y ∆ ≠ ∅  for any , x y X ∈ , therefore ∆ is asymmetric as well.Thus, ( ) The reverse implication is trivial.
(iii) ⇔ (v): See the proof of Theorem 6 above, and of course [2].■ Remark 13.Actually, it is well-known that a proper c satisfies both C and CO if and only if there exists a bi- as defined above -indeed, ( ) c R c R = for any choice function that satisfies C (see e.g.[2] [4]).Also notice that the equivalence between (ii) and (iii) is due to [3].Thus, Corollary 12 is-essentially-a restatement of the Sen-Plott-Suzumura characterization of revealed "rational" (proper) choice functions or, equivalently, revealed non-empty core solutions.
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 [1] and [4]; it should also be recalled here that externally stable cores are in particular a subclass of unique Von Neumann-Morgenstern stable sets).Therefore, for the sake of convenience, we collect in the following Theorem a few notable characterizations of revealed externally stable cores (to the best of the author's knowledge, only some of them are already known and available in print, namely those recorded in [4] which correspond to the first equivalence of the following Theorem, as mentioned explicitly in its proof below). Theorem ]): By Theorem 2.6 of [4], if c satisfies PR, C, CO and SS then there exists a (reflexive (iii) ⇔ (iv): See the proof of Theorem 7 above.
(iii) ⇔ (v): Let R X X ⊆ × be total and quasi-transitive, and , , x y z X ∈ such that not xRy and not yRz.Hence, yRx and zRy since R is total.Therefore, by definition, yR 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 R X X ⊆ × be reflexive and negatively transitive.Suppose there exist , x y X ∈ such that not xRy and not yRx: then, by negative transitivity, not xRx, a contradiction since R is reflexive.Thus, R is also total.Moreover, let xR 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.
(viii) ⇔ (iii): Suppose that there exists a dominance digraph ( ) for any nonempty Y X ⊆ .Moreover, by Claim 5 (v), R ∆ is quasi-transitive.Also, notice that since by hypothesis ∆ is both irreflexive and transitive, it must be asymmetric as well.Therefore, by Claim 5 (iv), R ∆ is total.Conversely, suppose that there exists a total and quasi-transitive relation R X X ⊆ × for any nonempty Y X ⊆ .Moreover, by Claim 5 (iii), (v), and in view of quasi-transitivity and totality of R, R ∆ is both quasi-transitive and asymmetric, hence transitive as well, and such that (viii) ⇔ (ix): Suppose that there exists a dominance digraph ( ) x y z x y = . Clearly, IV c satisfies PR, C and SS.
However, IV c fails to satisfy CO since . Clearly, V c satisfies PR, C and CO but fails to satisfy SS since Notice again that Theorem 14 above is essentially a refinement of well-known results due to Suzumura (see e.g.[4], Theorems 2.8 and 2.10) and [3], whose Theorems 3, 4, and 7 amount essentially to the equivalence between statements (iii), (iv) and (vii).It should also be mentioned here that the conjunction of C and SS turns out to be equivalent (see e.g.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-looking 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 [4], p. 21) does not materialize within the space of (total) choice functions and is therefore strictly confined to the realm of partial choice functions; (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.

Posets and Semilattices of Revealed Cores
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 ( )  where P is a set and  is a reflexive, transitive and an-tisymmetric binary relation on P (i.e. for any x P ∈ , x x  and for any , , x y z P ∈ , x z  whenever x y  and y z  , and x y = whenever x y  and y z  ).For any x P  with a top element or maximum 1 P is any j P ∈ which is covered by 1 P -written 1 P j  -i.e. 1 P j < and l j = for any l P ∈ such that 1 P j l <  . The set of all coatoms of P is denoted P A * .Dually, an P is any j P ∈ which is an upper cover of 0 P -written 0 P j  -i.e.0 P j < and l j = for any l P ∈ such that 0 P l j <  .The set of all atoms of P is denoted P A .A poset ( ) is a meet semilattice (join semilattice, respectively) if for any , x y P ∈ the  -greatest lower bound x y ∧ (the  -least upper bound x y ∨ , respectively) of { } , x y does exist.Moreover, P is lattice if it is both a meet semilattice and a join semilattice.
A lattice ( ) is bounded if there exist both a bottom element 0 P and a top element 1 P (hence in particular a finite lattice is also bounded), distributive iff ( ) ( ) ( ) for any , , x y z P ∈ , complemented if it is bounded and for any x P ∈ there exists x P ′ ∈ such that 1 P x x′ ∨ = and 0 P x x′ ∧ =, and Boolean iff it is both distributive and complemented.
A meet semilattice ( ) is a distributive lattice for any x P ∈ , and has the coronation (or join-Kelly) property if-for any , , x y z ∨ ∨ exists in P whenever , x y x z ∨ ∨ and y z ∨ also exist.A meet semilattice is median if it is lower distributive and has the coronation property.
The set X C of all choice functions on X can be endowed in a natural way with the point-wise set inclusion partial order  by positing, for any , for each A X ⊆ .Clearly, the identity operator id c is its top element, and the constant empty-valued choice function c ∅ its bottom element.
It is well-known, and easily checked, that ( )  is in fact a Boolean lattice with join ∨ = ∪ (i.e.set-union) and meet ∧ = ∩ (i.e.set-intersection), both defined in the obvious component-wise manner: see e.g.[11].
For any ,  c is defined by the following rule: for any x X denote the set of all revealed core-solutions on X, the set of all revealed nonempty-valued core-solutions, and es X C * the set of all revealed externally stable core-solutions on X, respectively).We also denote with a slight abuse of notation ( ) C *  the corresponding subposets of ( ) , respectively).We have the following.
Theorem 17.The poset ( ) , X C *  of revealed core-solutions is a sub-meet-semilattice of ( ) itself as its top element, but not a sub-join-semilattice of ( ) , X C  .It also satisfies the coronation property hence it is a median meet semilattice.The bottom element of ( ) Moreover, the set of coatoms of ( ) hence c c′ ∩ satisfies C. Finally, since c and c′ satisfy CO, for any , Thus, we only have to check that ( ) does also satisfy CO.In order to check this last point, consider any , A B X ⊆ , and for some , 1, 2, 3 i j = . Hence, in particular, it also follows that ( )( ) ( )( ) for some , 1, 2, 3 i j = .Now, by hypothesis, ( ) ) and ( ) As a consequence, ( ) ) C *  has the join-Kelly property and is therefore a median meet-semilattice as claimed.
It is easily checked that id c , the top element of ( )  , X ∆ such that ( ) ( ) x y ′ ⊇ : a contradiction again because ( ) To check that each xy c C  c x y = ∅ .Thus, for any Conversely, assume that c is an atom of ( ) Remark 19.Since ( ) , X C *  is a semilattice with a top element (and indeed a finite one, under finiteness of X), it follows that it is also a lattice with meet = ∩ and join of a pair given by the meet of the (nonempty) set of upper bounds of that pair (see e.g.[12]), which is however not a sublattice of ( ) , X C  .Thus, the poset of revealed core-solutions enjoys the remarkably regular structure of a median meet-semilattice.Notice that an important consequence of that fact is the following: any profile of revealed cores admits medians and the latter with the simple majority 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.[11] for some results on posets and lattices of other classes of choice functions and related aggregation rules in the same vein).
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: C *  it is only to be observed-in view of Theorem 7-that id c does in fact also satisfy 2-PR.Similarly-in view of Theorem 7 and of the proof of Theorem 17 provided above-to see that C + is the set of coatoms of ( ) The last statement about minimal elements of ( ) C *  are the single-valued choice functions that satisfy C and CO.Proof.First, notice that by definition id c is proper, hence id X c C * ∈  since as previously shown it is a core-solution.Also, it is immediately checked that, by definition, any xy c + is proper.Therefore, the proof of Theorem 17 also establishes that C + is the set of coatoms of ( ) , X C *   .In the same vein, it is immediately checked that , , , c c c c -as defined above in the proofs of the two previous Theorems-are also proper.It

∈
by definition, x Y ∈ and for any y Y ∈ there exists y Y such that .Thus, ( ) R c is reflexive, as required.Moreover, if( )xR c y then by C it must also be the case that = ∅where L is a linear order on X and x * is its bottom element.Clearly, I c violates ND, but satisfies C and CO; See the proof of Theorem 7 above.(i) ⇔ (iii): Obvious, by Theorem 10 above, since, again, X c C ∈  entails that c satisfies ND and 2-PR.
(v) ⇔ (vi): Let R X X ⊆ × be a negatively transitive relation such that ( ) max Y c Y R = ≠ ∅ for any nonempty Y X ⊆ .Then in particular, for any x X ∈ , hence R is reflexive as well.The reverse implication is trivial.(iii)⇒(vii): Let be R X X ⊆ × total, quasi-transitive and such that ( ) [4]) to another well-known and widely used property, namely: Path Independence (PI): for any , equivalent statements of Theorem 14 are also equivalent to the statement " minimum ND choice function[ ]

1
and the set of its atoms is C − .Proof.Let , c′ satisfy ND: hence c c′ ∩ does also satisfy ND.Moreover, for any A B X ⊆ ⊆ , since c and c′ both satisfy C, ( and CO also holds for c c′ ∩ .It follows that, by Theorem 7 above, in particular, it follows that ( ) , X C *  is lower distributive.Furthermore, let us suppose that 1 and C, by construction.
either A is a singleton or A B = .Thus, in any case, if A B ⊆

,XC 3 B
*  and c C − ∉ .Then, by definition of C − , ( ) c A = ∅ for any A such that # 2 A = , and there exists B X ⊆ such that # ≥ and ( ) c B ≠ ∅ .It follows that, for any( Notice that finiteness of X has been used in the proof above in order to show that the set of coatoms of ( ), X C *  is contained in C + .The latter statement clearly holds for an infinite X as well provided CO is replaced with the following stronger version of "Concordance" .CO*: for any family { } i i IA ∈ of subsets of X,( )

Theorem 20 .
The poset(  )    , a X C *  of revealed a-core-solutions has a top element, id c , and C + is the set of its coatoms, but it is neither a sub-meet-semilattice nor a sub-join-semilattice of ( ) , 2-PR, C, CO and such that (a) any c′ that satisfies ND, 2-PR, C and CO.Proof.To check that id c is indeed the top element of ( )

C
*  it is only to be checked that any xy c C + + ∈ does also satisfy 2-PR (which is clearly the case, by definition).The proof of Theorem 17 already establishes that ( ) , a X C *  is not a sub-join-semilattice of ( ) , X C  since, as it is easily checked, I c and II c as defined there do belong to a X C * .Next, consider III c and IV c defined as follows: assume without loss of generality  of nonempty-valued core-solutions has a top element, id c , and C + is the set of its coatoms, but it is neither a sub-meet-semilattice nor a sub-join-semilattice of ( )

which is also externally stable). Example 4. By way of contrast, take
2-PR, C, and CO.Then, by ND, C and CO (and in view of CO is satisfied by c. (iii) ⇔ (iv): See the proof of Theorem 7 above.■ Remark 11.The foregoing characterization result is also tight.To see this, consider the following examples.
X c C ∈ as defined above (see Remark 9).It is easily seen that III c satisfies ND, 2-PR and C, but violates CO.Corollary 12. (see also [2] [4]) Let X c C ∈  .Then, the following statements are equivalent: (i) c satisfies C and CO; any nonempty Y X ⊆ .Again, irreflexivity and transitivity asymmetry of ∆ , which therefore a strict partial order.The reverse implication is trivial.■ 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.
does also satisfy ND, C and CO hence as observed above as defined above: it satisfies ND, by definition, and, being nonempty-valued precisely on singletons, it trivially satisfies C and CO as well.Thus,[ ] 1 c .Clearly, by Theorem 7, c satisfies ND, C and CO.If