Linearity of the core correspondence
Denes Palvolgyi
∗
Hans Peters
†
Dries Vermeulen
‡
July 22, 2016
Abstract
We characterize the sets of balanced TU-games on which the core correspondence is
linear—first introduced by Bloch and de Clippel (2010)—by finitely many linear equalities
and inequalities. Thus, the core is piecewise linear.
1
Introduction
Since the graph of the core correspondence is defined by finitely many linear inequalities,
this graph is a polyhedral set. It is therefore to be expected that the core correspondence
is piecewise linear. Bloch and de Clippel (2010) introduced an equivalence relation based on
the extreme points of the core to identify the classes of balanced games on which the core
is additive. In this paper, using a result of Kuipers, Vermeulen and Voorneveld (2010), we
characterize those equivalence classes by finitely many linear inequalities. It follows that the
core is indeed piecewise linear.
Notation.
For subsets X and Y of Rn and α, β ∈ R, αX + βY = {αx + βy | x ∈ X, y ∈ Y }.
We write N = {1, . . . , n}. For S ⊆ N , eS ∈ RN is defined by
(
1 if i ∈ S
S
ei =
0 if i ∈
/ S.
For a convex set X ⊆ Rk , let H be the smallest affine subspace that contains X. The relative
interior relint (X) of X is the collection of points x ∈ X for which there is an open set U ⊆ Rk
such that U ∩ H is a subset of X. The set of extreme points of X is denoted by ext (X).
∗ Dept. of Mathematical Economics and Economic Analysis, Corvinus University of Budapest, Budapest,
Hungary.
† Department of Quantitative Economics, Maastricht University, P.O.Box 616, 6200 MD Maastricht, The
Netherlands.
‡ Department of Quantitative Economics, Maastricht University, P.O.Box 616, 6200 MD Maastricht, The
Netherlands.
1
Linearity of the core correspondence
2
2
Preliminaries
A transferable utility game, or game, is a pair (N, v) where N = {1, . . . , n} is the set of players
and v is a function that assigns to each coalition S ⊆ N its worth v(S) ∈ R. The worth v(∅)
of the empty set is zero. Throughout this paper we keep the player set N fixed. To simplify
notation we write v instead of (N, v) to denote a game.
A vector x ∈ Rn is an allocation. The ith coordinate xi of the allocation x represents the payoff
P
to player i ∈ N . For coalition S ⊆ N , the aggregate payoff i∈S xi is denoted by x(S). An
allocation x is efficient for v if it distributes the worth of the grand coalition among the players
of the game v, i.e., if x(N ) = v(N ). An efficient allocation x is a core allocation for v if
x(S) ≥ v(S) for all S ⊆ N .
The set of core allocations, denoted by C(v), is called the core of v. A game whose core is not
empty is balanced (Bondareva, 1962; Shapley, 1967).
It is well-known and easy to prove that for any two balanced games v and w and 0 ≤ α ≤ 1
we have αC(v) + (1 − α)C(w) ⊆ C(αv + (1 − α)w). Here, αv + (1 − α)w is the game defined
by (αv + (1 − α)w)(S) = αv(S) + (1 − α)w(S) for every coalition S.
3
Limit games for balanced games
For an allocation x ∈ Rn , a nonempty coalition S with x(S) = v(S) is called tight at x in v.
The collection of coalitions that are tight at x in v is denoted by T (v, x). We say that w is a
limit game for v if for every extreme point x ∈ C(v) there is an extreme point y ∈ C(w) with
T (v, x) ⊆ T (w, y). The collection of limit games for v is denoted by L(v). For later reference
we note that if w ∈ L(v) and w0 ∈ L(w), then w0 ∈ L(v).
For S ⊆ N , eS ∈ Rn is the vector with eSi = 1 if i ∈ S and eSi = 0 otherwise. A set of coalitions
B is a basis if {eS | S ∈ B} is a basis for Rn .
Let v be a balanced game, let the set B of coalitions be a basis, and let T be a coalition with
T ∈
/ B 1 . We say that B is feasible for v if there is an extreme point x ∈ C(v) such that
B ⊆ T (v, x). We say that T is a neighbor of B if there exists a basis C that is feasible for v
such that T ∈ C and |B ∩ C| = n − 1. We say that T is spanned by B if the unique numbers
P
λ(S) with eT = S∈B λ(S)eS are all nonnegative.
1 Note
that, for every x ∈ ext (C(v)), the set T (v, x) contains a basis.
3
Linearity of the core correspondence
A feasible pair for v is a pair (B, T ) where B is a basis that is feasible for v, and coalition
T ∈
/ B is a neighbor of B or spanned by B.
For a balanced game v, we construct a system of linear inequalities of which the solution set
will equal the polyhedral cone L(v). Let (B, T ) be a feasible pair for v. Let λ(S) for S ∈ B be
the unique real numbers such that
eT =
X
λ(S)eS .
S∈B
The linear inequality generated by the pair (B, T ) is
w(T ) ≤
X
λ(S)w(S).
S∈B
We denote the set of all inequalities generated in this way by I(v). The following result is
proved in Kuipers, Vermeulen and Voorneveld (2010).
Theorem 3.1 Let v be a balanced game. Then a game w satisfies the inequalities in I(v) if
and only if w ∈ L(v).
This result implies that L(v) is a polyhedral cone in the space of all games.
4
Linearity of the core correspondence
Let B be the collection of balanced games. The core correspondence C : B RN is the
correspondence that assigns to each balanced game v ∈ B the set C(v).
Let L be a convex subset of B. The core correspondence C is linear on L when for all v, w ∈ L
and every 0 < α < 1 we have
C(αv + (1 − α)w) = αC(v) + (1 − α)C(w).
(1)
We identify the sets of games on which the core correspondence is linear. Our first result,
Theorem 4.2, gives a sufficient condition for (1) to hold. This identification was in fact already
established in Bloch and de Clippel (2010).
In the proof of Theorem 4.2 we use the following Lemma. The lemma shows that in a limit
game for a balanced game, no additional extreme core points can emerge.
Lemma 4.1 Let v, w ∈ B and w ∈ L(v). Then, for every y ∈ ext (C(w)) there is an x ∈
ext (C(v)) such that T (v, x) ⊆ T (w, y).
Linearity of the core correspondence
4
Proof. For a game v ∈ B and x ∈ ext (C(v)), define Q(v, x) as the convex hull of vectors eS
with S ∈ T (v, x). Kuipers and Derks (2002) showed that the sets Q(v, x) are of full dimension,
their union over all x ∈ ext (C(v)) equals the unit cube of dimension n, and intersections of
sets Q(v, x) have empty interior.
Now notice that T (v, x) ⊆ T (w, y) implies that Q(v, x) ⊆ Q(w, y). Thus, since w ∈ L(v), the
union of all sets Q(w, y) over those y ∈ ext (C(w)) for which there is an x ∈ ext (C(v)) with
T (v, x) ⊆ T (w, y) already equals the unit cube of dimension n. Hence, for every y ∈ ext (C(w))
there is an x ∈ ext (C(v)) such that T (v, x) ⊆ T (w, y).
The following proposition says that for each balanced game the core correspondence is linear
on the set of its limit games.
Theorem 4.2 Let v ∈ B. Then C(·) is linear on L(v).
Proof. Let w, w0 ∈ L(v) and 0 < α < 1. Let w̃ = αw + (1 − α)w0 and let z ∈ ext (C(w̃)). By
Lemma 4.1 there is an x ∈ ext (C(v)) such that T (v, x) ⊆ T (w̃, z). By definition of L(v) there
are y ∈ ext (C(w)) and y 0 ∈ ext (C(w0 )) such that T (v, x) ⊆ T (w, y) and T (v, x) ⊆ T (w0 , y 0 ).
We claim that z = αy + (1 − α)y 0 , which completes the proof of the proposition. To show this
claim, note that for all S ∈ T (v, x) we have
z(S) = w̃(S) = αw(S) + (1 − α)w0 (S) = αy(S) + (1 − α)y 0 (S).
Since T (v, x) contains a basis, this implies z = αy + (1 − α)y 0 .
5
Maximal convex sets
Call a convex subset L of B maximal if the core correspondence is linear on L but not linear
on any subset L0 of B with L ⊆ L0 and L 6= L0 . In the remainder of this paper we consider the
question which convex sets L ⊆ B are maximal.
Lemma 5.1 Let L be a convex subset of B and let the core correspondence be linear on L.
Then L ⊆ L(v) for every v ∈ relint (L).
Proof. Take a game v in the relative interior of L. Clearly, v ∈ L(v). Take any w ∈ L \ {v}.
We show that w ∈ L(v). Take any x ∈ ext (C(v)). We show that there exists a y ∈ ext (C(w))
with T (v, x) ⊆ T (w, y).
Linearity of the core correspondence
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Since v is an element of the relative interior of L, there is a w0 ∈ L \ {v} and a 0 < α < 1 such
that v = αw + (1 − α)w0 . We first show that there are y ∈ ext (C(w)) and z ∈ ext (C(w0 )) such
that x = αy + (1 − α)z.
By (1) there are y ∈ C(w) and z ∈ C(w0 ) such that x = αy + (1 − α)z. Suppose that
y∈
/ ext (C(w)). Take p, q ∈ C(w) and λ ∈ (0, 1) with λp+(1−λ)q = y. Then s = αp+(1−α)z ∈
C(v), t = αq + (1 − α)z ∈ C(v), and x = λs + (1 − λ)t. Since s 6= x and t 6= x, this contradicts
the assumption that x ∈ ext (C(v)).
So, indeed y ∈ ext (C(w)) and z ∈ ext (C(w0 )). Let S ∈ T (v, x). Then
v(S) = x(S) = αy(S) + (1 − α)z(S) ≥ αw(S) + (1 − α)w0 (S) = v(S).
Hence, y(S) = w(S) and z(S) = w0 (S), so that S ∈ T (w, y) and S ∈ T (w0 , z). It follows that
T (v, x) ⊆ T (w, y).
The following corollary is an immediate consequence.
Corollary 5.2 Let L ⊆ B be a maximal subset of B. Then L = L(v) for every v ∈ relint (L).
Proof. By Lemma 5.1, L ⊆ L(v) for any v ∈ relint (L). Since C is linear on L(v) by Proposition
4.2 and L is maximal, it follows that L = L(v).
Is any set L(v) maximal? The answer to this question is negative. For instance, let A be the
set of all additive games. Then it is easy to see that A = L(a) for any a ∈ A. Take any
v ∈ B \ A. Then the core correspondence is linear on the convex hull of A ∪ {v}, so that A is
not maximal.
In the remainder of this paper we characterize the class of balanced games v for which L(v) is
maximal. We first prove the following.
Theorem 5.3 Let v, w ∈ B. Equivalent are
[1] L(w) ⊆ L(v).
[2] I(v) ⊆ I(w).
Proof. A.
Suppose that L(w) ⊆ L(v). Take a feasible pair (B, T ) for v. We show that
(B, T ) is also feasible for w, so that I(v) ⊆ I(w). Since (B, T ) is feasible for v, there is an
x ∈ ext (C(v)) with B ⊆ T (v, x), and T ∈
/ B is a neighbor of B or spanned by B.
Linearity of the core correspondence
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Since w ∈ L(v), by definition there is a y ∈ ext (C(w)) with T (v, x) ⊆ T (w, y). Then also
B ⊆ T (w, y), and T ∈
/ B is a neighbor of B or spanned by B. Hence, (B, T ) is feasible for w.
It follows that I(v) ⊆ I(w).
B. Suppose that I(v) ⊆ I(w). Take w0 ∈ L(w). Then, by Theorem 3.1, w0 satisfies the
inequalities in I(w). Thus, since I(v) ⊆ I(w), w0 also satisfies the inequalities in I(v). Then,
by Theorem 3.1, w0 ∈ L(v). It follows that L(w) ⊆ L(v).
We say that I(v) is minimal if there is no game w ∈ B with I(w) ⊆ I(v) and I(w) 6= I(v).
Theorem 5.4 Let v ∈ B. Equivalent are
[1] L(v) is maximal.
[2] I(v) is minimal.
Proof. A.
Suppose that L(v) is not maximal. Then, due to Lemma 5.1 and Theorem 4.2
there is a w ∈ B with L(v) ⊆ L(w) and L(v) 6= L(w). It follows from Theorem 5.3 that
I(w) ⊆ I(v) and I(w) 6= I(v). Hence, I(v) is not minimal.
B. Suppose that I(v) is not minimal. Take a game w ∈ B with I(w) ⊆ I(v) and I(w) 6= I(v).
Then, by Theorem 5.3, L(v) ⊆ L(w) and L(v) 6= L(w). Hence, L(v) is not maximal.
It follows from Theorem 7 and Corollary 8 in Kuipers, Vermeulen and Voorneveld (2010) that
the class of all convex games is a maximal set on which the core correspondence is linear.
References
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Linearity of the core correspondence
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