A clutter on a set X is a simple hypergraph with pairwise not-comparable hyperedges, hence in particular any set of Von Neumann-Morgenstern (VNM) -stable sets of an irreflexive simple digraph is a clutter. A clutter (X,E) is representable by VNM-stable sets or VNM if there exists an irreflexive simple digraph (X, ?) such that E is a set of VNM-stable sets of (X, ?). The class of VNM clutters on a set X is characterizedVNM-stable sets, kernels, clutters, Sperner systems

Stefano Vannucci

English

Research Papers in Economics

     
 
 
UNIVERSITÀ DEGLI STUDI DI SIENA 
 
 
 
 
 
QUADERNI DEL DIPARTIMENTO 
DI ECONOMIA POLITICA 
  
   
 
Stefano Vannucci 
 
 
 
 
 
 
Von Neumann-Morgenstern Clutters 
 
 
 
 
 
 
 
n. 592 –  Aprile  2010 
  
      
   
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
        
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Abstract  - A clutter on a set  X is a simple hypergraph with pairwise not-comparable 
hyperedges, hence in particular any set of Von Neumann-Morgenstern (VNM) -stable sets of an 
irreflexive simple digraph is a clutter. A clutter (X, E) is representable by VNM-stable sets or 
VNM if there exists an irreflexive simple digraph (X, ∆) such that E is a set of VNM-stable sets 
of (X, ∆). The class of VNM clutters on a set X is characterized. 
 
Jel Classification: C70, C71 
 
Keywords: VNM-stable sets, kernels, clutters, Sperner systems 
 
 
 
 
 
 
Stefano Vannucci, Dipartimento di Economia Politica, Università degli Studi di Siena. 
e-mail:vannucci@unisi.it Let H = (X;E) be a simple hypergraph i.e. X is a set, and E ￿ P(X),
where P(X) is the power set of X: X will also be referred to as the set of
vertices of H, and E as the set of hyperedges of H. Then, H is a clutter i⁄E
is a ￿-antichain, namely E * E0 for any two distinct E;E0 2 E. In particular
the ￿ boundary￿cases with E = f;g, or E = ; are allowed. [Notice that such a
terminology is slightly at variance with that of other authors, who sometimes
denote as ￿ simple hypergraphs￿ , ￿ Sperner systems￿or ￿ Sperner families￿those
clutters (X;E) -as de￿ned above- such that ; = 2 E and
[
E = X: see
e.g. Berge (1989)]. A simple digraph is a pair D = (X;￿) where X is a
set and ￿ ￿ X ￿ X; D is irre￿exive -or loopless- i⁄ (x;x) = 2 ￿ for any
x 2 X. A Von Neumann-Morgenstern stable set (henceforth VNM-stable
set) of an irre￿ exive simple digraph (X;￿) is a set S ￿ X such that (i)
internal stability i.e. [(x;y) = 2 ￿ for any x;y 2 S], and (ii) external stability
i.e. [for any z 2 X r S there exists x 2 S such that (x;z) 2 ￿] hold (see
e.g. Von Neumann and Morgenstern (1953), Schmidt and Str￿hlein (1985),
and Ghoshal, Laskar and Pillone (1998)). Let S(X;￿) denote the set of all
VNM-stable sets of (X;￿). Clearly, S1 * S2 for any two distinct S1, S2 2
S(X;￿) (otherwise, properties (i) and (ii) of S1 turn out to be mutually
inconsistent). Thus, for any irre￿ exive simple digraph (X;￿), (X;S(X;￿))
is a clutter. This observation motivates the following
De￿nition 1 (VNM clutters) A clutter H = (X;E) is representable by
VNM-stable sets or VNM i⁄ there exists an irre￿exive simple digraph (X;￿)
such that E ￿ S(X;￿).
In general, a clutter may or may not be VNM, as made clear by the
following examples
Example 2 Consider X = fx;y;zg with x 6= y 6= z 6= x, and
E = ffx;yg;fx;zg;fy;zgg.
It is easily checked that (X;E) is a clutter, but not a VNM one. Indeed,
suppose to the contrary that there exists an irre￿exive simple digraph (X;￿)
such that E ￿ S(X;￿). Then, by property (ii) of VNM-stable sets as applied
to fx;yg, ￿\f(x;z);(y;z)g 6= ?, which contradicts the assumption that both
fx;zg and fy;zg satisfy property (i) of VNM-stable sets.
Example 3 Consider now X as de￿ned above in the previous example, E0 =
ffx;yg;fx;zgg, and ￿0 = f(y;z);(z;y)g. Clearly, E0 = S(X;￿0) i.e. (X;E0)
is indeed a VNM clutter.
1Example 4 Let (X;E) be a clutter such that E \ E0 = ? for any pair of
distinct E;E0 2 E.
Then, de￿ne ￿ = f(x;y) 2 X ￿ X: fx;yg * E for all E 2 Eg.
Clearly, for any E 2 E and any x;y 2 E, (x;y) = 2 ￿ by de￿nition.
Moreover, suppose there exists z 2 X rE such that (x;z) = 2 ￿ for all x 2 E.
Then, for any x 2 E there exists E0 2 E such that fx;zg ￿ E0. Since by
construction E 6= E0 and E\E0 6= ?, the existence of such an E0 contradicts
our starting hypothesis. It follows that E is indeed a VNM-stable set of
(X;￿). Therefore, (X;E) is a VNM clutter.
Remark 5 (Kernel-representable clutters) Let (X;￿) be an irre￿exive sim-
ple digraph, and ￿￿1 ￿ X ￿ X the inverse of ￿, namely, for any x;y 2 X,
(x;y) 2 ￿￿1 i⁄ (y;x) 2 ￿. Of course, (X;￿￿1) is also an irre￿exive sim-
ple digraph. A subset of vertices K ￿ X is a kernel of (X;￿) - written
K 2 K(X;￿) - i⁄ K 2 S(X;￿￿1). Now, consider any clutter (X;E) and
declare it kernel-representable (KR) i⁄ there exists an irre￿exive simple di-
graph such that E ￿ K(X;￿). It is immediately checked that, in view of the
foregoing observations, a clutter is KR i⁄ it is VNM.
The foregoing observations and examples raise the issue of identifying
VNM clutters ( in the same vein, a representation problem concerning ￿nite
lattices and stable matchings is addressed by Blair (1984)).
The aim of this note is in fact to provide a simple characterization of
VNM clutters. In order to state our result in a most concise manner, let us
￿rst introduce some further auxiliary notions.
De￿nition 6 (Conjugation relation of a simple hypergraph) Let H = (X;E)
be a simple hypergraph. Then the conjugation relation IH ￿ X ￿ X of H is
de￿ned as follows: for any x;y 2 X, (x;y) 2 IH i⁄ there exists an E 2 E
such that fx;yg ￿ E.
De￿nition 7 (Complete bigraph) Let Y;Z be two nonempty sets. A (sim-
ple, nontrivial) bigraph on (Y;Z) is a triple (Y;Z;R) where R ￿ Y ￿ Z. A
bigraph (Y;Z;R) is complete if R = Y ￿ Z.
De￿nition 8 Let (X;￿) be a simple digraph. A ￿-basis of a complete bi-
graph is a B ￿ X such that (B;X r B;￿ [ ￿￿1) is a complete bigraph.
2De￿nition 9 (Complete bigraph generator (CBG)) An hyperedge E 2 E of
a clutter H = (X;E) is a complete bigraph generator i⁄ it is an IH-basis of
a complete bigraph.
A clutter H = (X;E) is CBG-free i⁄ no E 2 E is a complete bigraph
generator. Moreover, clutter H is conjugation-saturated i⁄ it is CBG-free
and IH 6= IH0 for any CBG-free clutter H0 = (X;E0) such that E ￿ E0.
Then, we have the following
Theorem 10 Let H = (X;E) be a clutter. Then, H is VNM i⁄ it is CBG-
free.
Proof. Observe that, by de￿nition, H is not CBG i⁄for any E 2 E and any
z 2 X r E there exists xz 2 E such that (z;xz) = 2 IH.
Then, suppose H = (X;E) is VNM, and let (X;￿) be an irre￿ exive
simple digraph such that E ￿ S(X;￿). Now, assume that there exist an
hyperedge E 2 E and a vertex z 2 XrE such that (z;x) 2 IH for all x 2 E.
Thus, for any x 2 E there exists an Ezx 2 E such that fx;zg ￿ Ezx whence,
by internal stability of Ezx, (x;z ) = 2 ￿: but then, external stability of E is
violated. It follows that H is not VNM, a contradiction.
Conversely, suppose that H = (X;E) is such that for any E 2 E and any
z 2 X rE there exists xz 2 E with (z;xz) = 2 IH. Then, de￿ne ￿H ￿ X ￿X
by the following rule: for any x;y 2 X, f(x;y);(y;x)g ￿ ￿H if (x;y) = 2 IH,
and f(x;y);(y;x)g\￿H = ? if (x;y) 2 IH: notice that (X;￿H) is irre￿ exive.
Therefore, for any E 2 E and any x;y 2 E, (x;y) = 2 IH hence (x;y) = 2 ￿H,
and E satis￿es internal stability with respect to (X;￿H). Moreover, by
hypothesis, for any z 2 X r E there exists xz 2 E with (z;xz) = 2 IH i.e. in
particular (xz;z) 2 ￿H, by de￿nition of ￿, hence E also satis￿es external
stability with respect to (X;￿H). Thus, E ￿ S(X;￿H) as required.
Remark 11 Notice that, as it is easily checked, the trivial clutters (;;f;g)
and (X;f;g) with a nonempty X are both VNM. Moreover, it is worth empha-
sizing that the foregoing Theorem also implies that clutter (X;E) of Example
2 is not VNM because it is clearly not CBG-free (indeed, each of its hyper-
edges is a complete bigraph generator). On the contrary, its odd-ciclycity is
not key to its being not VNM. To see this, consider clutter (fx;y;z;u;vg;E)
with E = ffx;yg;fy;zg;fz;ug;fu;vg;fv;xgg,
which is also odd-cyclic (namely, there exist a positive integer k, and
2k + 1 distinct hyperedges Ei 2 E and vertices xi, i = 1;:::;2k + 1 such
3that fxi;xi+1g ￿ Ei, i = 1;:::;2k and fx1;x2k+1g ￿ E2k+1). However,
(fx;y;z;u;vg;E) is CBG-free hence by Theorem 10 is indeed a VNM clutter:
to con￿rm the latter statement it is only to be checked that there is no Ei 2 E
comprising one of the following pairs: fx;zg;fx;ug;fy;vg;fy;ug;fz;vg.
Remark 12 In view of Theorem 10 it is easily checked that a remarkable
class of clutters which are not VNM is provided by (nontrivial) Steiner triple
systems i.e. clutters H = (X;E) such that #X ￿ 4, E ￿ fY ￿ X : #Y = 3g
and for any two distinct x;y 2 X there exists precisely one E 2 E with
fx;yg ￿ E. Indeed, any hyperedge E of such a clutter is a CBG: to check
the latter statement, take any E 2 E and observe that there exist x 2 X rE
and y 2 E, and for any such x;y there exists by assumption an E0 2 E with
fx;yg ￿ E0.
Let us denote a clutter H = (X;E) as (strictly) VNM-complete if it is rep-
resentable as the set of all VNM-stable sets of (X;￿H) i.e. E = S(X;￿H),
where ￿H = (
[
E ￿
[
E) r IH as de￿ned above in the proof of Theorem
10. Then, we have the following straightforward corollary of the previous
theorem, namely
Corollary 13 A clutter H = (X;E) is VNM-complete i⁄ it is conjugation-
saturated. In particular, that requires
[
E = X.
Proof. Assume that H = (X;E) is VNM-complete and there exists E0 ￿ E
such that H0 = (X;E0) is CBG-free and IH = IH0. But then, by de￿nition
S(X;￿H0) = S(X;￿H), and by Theorem 10, E0 ￿ S(X;￿H0) = S(X;￿H),
a contradiction since by hypothesis E = S(X;￿H).
Conversely, let H = (X;E) be conjugation-saturated, and suppose that
it is not VNM-complete, namely E ￿ S(X;￿H). Then, by Theorem 10 there
exists a CBG-free H0 = (X;E0) such that E ￿ E0 = S(X;￿H). Since H is
conjugation-saturated, IH 6= IH0 hence by de￿nition IH ￿ IH0. But then,
there exist E0 2 S(X;￿H)rE and x;y 2 E0r
[
E whence -by de￿nition of
￿H- x￿Hy which contradicts internal stability of E0 with respect to (X;￿H).
Finally, observe that if Xr
[
E 6= ; then (x;y) = 2 IH for any x 2 Xr
[
E
and y 2 X rfxg whence by de￿nition fxg 2 S(X;￿H)rE and thus (X;E)
is not VNM-complete.
4As an example, observe that clutter H = (fx;y;zg;ffx;ygg ) where
x 6= y 6= z 6= x is clearly CBG-free but not conjugation-saturated since
clutter H0 = (fx;y;zg;ffx;yg;fzgg) is such that IH0 = f(x;y);(y;x)g = IH.
Thus, the foregoing Corollary entails that H0 is not VNM-complete.
On the contrary, for any set X with #X ￿ 3 and any x 2 X and positive
integer 1 < r < #X take a ￿ minimal￿r-uniform star-clutter with centre x,
i.e. a clutter H￿
x = (X;E) such that
[
E = X, #E = r for every E 2 E,
and for any two distinct E;E0 2 E, E \ E0 = fxg: by construction, H￿
x
is CBG-free since for any two distinct E;E0 2 E and any y 2 E r fxg,
z 2 E0rfxg it turns out that f(y;z);(z;y)g\IH￿
x = ;. Moreover, H￿
x is also
a conjugation-saturated clutter since for any CBG-free clutter H0 = (X;E0)
with E0 ￿ E, and any E0 2 E0 r E there exist two distinct E1;E2 2 E and
y 2 E1 r E2, z 2 E2 r E1 such that fy;zg ￿ E0, hence f(y;z);(z;y)g ￿ IH0
while f(y;z);(z;y)g \ IH￿
x = ; by construction of H￿
x. Therefore, Corollary
13 implies that H￿
x is indeed VNM-complete.
References
[1] Berge C. (1989): Hypergraphs. North Holland: Amsterdam.
[2] Blair C. (1984): Every ￿nite distributive lattice is a set of stable match-
ings, Journal of Combinatorial Theory (Series A) 37, 353-356.
[3] Ghoshal J., Laskar R., Pillone D. (1998): Topics on domination in di-
rected graphs, in T.W. Haynes, S.T. Hedetniemi, P.J. Slater (eds.): Dom-
ination in Graphs. Advanced Topics. M.Dekker: New York.
[4] Schmidt G., Str￿hlein T. (1985): On kernels of graphs and solutions of
games: a synopsis based on relations and ￿xpoints, SIAM Journal of
Algebraic and Discrete Methods 6, 54-65.
[5] Von Neumann J., Morgenstern O. (1953): Theory of Games and Economic
Behavior (third edition). Princeton University Press: Princeton.
5

Von Neumann-Morgenstern Clutters

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Von Neumann-Morgenstern Clutters

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