Cooperative Games With a Simplicial Core

Rodica Brânzei and Stef Tijs

Abstract In this paper n-person cooperative games having the property that the core is a subsimplex of the imputation set are characterized. Also a characterization of games where the core is a subsimplex of the dual imputation set is given by using some duality relations for games. We also give a geometric characterization of games with a non–empty core, which follows easily from the well-known Bondareva–Shapley theorem.

Mathematics Subject Classification: 91A12 Key words: cooperative games, imputation set, core

1 Introduction

A cooperative n-person game is a pair < N, v >, where N = {1, 2, ..., n} is the set of players and v : 2^{N} → IR is the characteristic function with domain the family of subsets of N. Such subsets are called coalitions and v(S) is called the value of coalition S ∈ 2^{N}. Such a game models a situation where a group N of persons can cooperate and also subgroups. For each subgroup S the value v(S) indicates the amount of money which they can obtain when cooperating. There is only one restriction on the characteristic function, namely v(ϕ) = 0, the value of the empty coalition is 0. This implies that the set of characteristic functions of n-person games forms with the obvious operations a (2^{n} − 1)-dimensional linear space G^{N}. Often v and < N, v > will be identified. The question: ‘How to divide v(N), if all the players in N are cooperating?’ has given rise to many proposals called solution concepts. Of the onepoint solution concepts we mention only the Shapley value [6], the τ-value [9] and the nucleolus [5]. Sometimes subsets of payoff distributions of v(N) are assigned to games as solutions; such a subset consists of points which are from a certain point of view better than the points outside. Three of such subsets, namely the imputation set, the dual imputation set and the core [4] will play a role in this paper and we describe them now.

The imputation set I(v) of a game < N, v > is defined by

I(v) = {x ∈ IR^{n} | ∑_{i=1}^{n}x_{i} = v(N), x_{i} ≥ v({i}) for each i ∈ N}

Balkan Journal of Geometry and Its Applications, Vol.6, No.2, 2001, pp. 7-15 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

and consists of those payoff distributions of v(N), which are individual rational i.e. player i obtains an amount x_{i} which is at least as large as his individual value v({i}), which he can obtain by staying alone. From the geometric point of view, the imputation set I(v) is equal to the intersection of the efficiency hyperplane

{x ∈ IR^{n} | ∑_{i=1}^{n}x_{i} = v(N)} and the orthant {x ∈ IR^{n} | x ≥ i(v)} of individual ratio-

nal payoff vectors.

The imputation set is non–empty iff v(N) ≥ ∑_{i=1}^{n}v({i}). If v(N) > ∑_{i=1}^{n}v({i}), then

I(v) is an (n − 1)-dimensional simplex with extreme points f^{1}(v), f^{2}(v), ..., f^{n}(v), where the k-th coordinate (f^{i}(v))_{k} of f^{i}(v) equals v({k}) if k ≠ i and (f^{i}(v))_{i} =

v(N) − ∑_{k∈N\{i}}v({k}).

For an n-person game < N.v > and S ∈ 2^{N} we define the dual value v^{∗}(S) of S as v^{∗}(S) = v(N) − v(N \ S).

The amount v^{∗}(S) can be seen as the marginal contribution of S to the grand coalition or also as a sort of blocking power of S. The dual imputation set I^{∗}(v) of the game v is given by

I^{∗}(v) = {x ∈ IR^{n} | ∑_{i=1}^{n}x_{i} = v(N), x_{i} ≤ v^{∗}({i}) for each i ∈ N} .

It consists of distributions of v(N), where no player gets more than his marginal contribution to the grand coalition. From the geometric point of view I^{∗}(v) is equal to the intersection of the efficiency hyperplane and the orthant {x ∈ IR^{n} | x ≤ u(v)} of subutopic vectors.

Note that I^{∗}(v) ≠ ∅ iff ∑_{i=1}^{n}v^{∗}({i}) ≥ v(N). In the case of strict inequality, I^{∗}(v)

is an (n − 1)-dimensional simplex with extreme points g^{1}(v), g^{2}(v), ...g^{n}(v), where

(g^{i}(v))_{k} = v^{∗}({k}) if k ≠ i, and (g^{i}(v))_{i} = v(N) − ∑_{k∈N\{i}}v^{∗}({k}).

The core C(v) of a game < N, v > is a subset of I(v) ∩ I^{∗}(v) defined by

C(v) = {x ∈ IR^{n} | ∑_{i=1}^{n}x_{i} = v(N), ∑_{i∈S}x_{i} ≥ v(S) for all S ∈ 2^{N}} .

Note that v({i}) ≤ x_{i} ≤ v^{∗}({i}) for all i ∈ N and x ∈ C(v). So, the core is the bounded solution set of a set of linear inequalities, which means that the core is a polytope i.e. the convex hull of a finite set of vectors in IR^{n}. When proposing a core element for the division of v(N) among the players, no subgroup S ⊂ N will have an incentive to split off. However, the core may be empty. Independently, Bondareva in [2] and Shapley in [8] gave necessary and sufficient conditions for the non-emptyness of the core: let < N, v > be an n-person game; then C(v) ≠ ∅ iff

v(N) ≥ ∑_{S∈2^{N} \{∅}}λ_{S}v(S) in case λ_{S} ≥ 0 for all S ∈ 2^{N} \ {∅} and ∑_{S∈2^{N} \{∅}}λ_{S}e_{S} = e^{N}.

Here e^{S} ∈ IR^{N} is the characteristic vector of the coalition S, with (e^{S})_{i} = 0 if i ∉ S, and (e^{S})_{i} = 1 otherwise.

In Section 2 we like to reformulate this Bondareva–Shapley result in geometric terms. In Section 3 we characterize n-person simplex games where the core is a subsimplex of the imputation set, and in Section 4 duality results for games lead to a characterization of dual simplex games, where the core is a subsimplex of the dual imputation set. Section 5 concludes.

2 Geometric characterization of games with a nonempty core

We define the per capita value v̄(S) of coalition S ≠ ∅ by \frac{1}{|S|}v(S) where |S| is the

cardinality of S. Further, let for a subsimplex Δ(S, v) = conv{f^{i}(v) | i ∈ S} of

I(v) = Δ(N, v), the barycenter \frac{1}{|S|}∑_{i∈S}f^{i}(v) be denoted by b(S, v). Then we obtain

the following characterization of games with a non-empty core.

Theorem 2.1. The game < N, v > has a non-empty core iff ∑_{S∈2^{N} \{∅}}μ_{S}b(S, v) =

b(N, v) with μ_{S} ≥ 0, ∑_{S∈2^{N} \{∅}}μ_{S} = 1, implies that ∑_{S∈2^{N} \{∅}}μ_{S}v̄(S) ≤ v̄(N).

The theorem tells that < N, v > has a non-empty core iff for each way of writing the barycenter of the imputation set as a convex combination of barycenters of subsimplices, the per capita value of N is at least as large as the corresponding convex combination of per capita values of the subcoalitions.

Proof of Theorem 2.1. For λ = (λ_{S})_{S∈2^{N} \{∅}}, let μ = (μ_{S})_{S∈2^{N} \{∅}} be defined by μ_{S} = n^{−1}|S|λ_{S}. Then

λ_{S} ≥ 0, ∑_{S∈2^{N} \\{∅\}}λ_{S}e^{S} = e^{N} iff ∑_{S∈2^{N} \\{∅\}}μ_{S} \frac{e^{S}}{|S|} = \frac{e^{N}}{|N|}, μ_{S} ≥ 0, ∑_{S∈2^{N} \\{∅\}}μ_{S} = 1.

This implies

(i) λ_{S} ≥ 0, ∑_{S∈2^{N} \{∅}}λ_{S}e^{S} = e^{N} iff ∑_{S∈2^{N} \{∅}}μ_{S}b(S, v) = b(N, v)

since b(S, v) = (v({1}), v({2}), ..., v({n})) + α|S|^{−1}e^{S} for each S ∈ 2^{N} \ {∅}

where α = v(N) − ∑_{i∈N}v({i}).

(ii) ∑_{S∈2^{N} \{∅}}λ_{S}v(S) ≤ v(N) iff ∑_{S∈2^{N} \{∅}}μ_{S}v̄(S) ≤ v̄(N).

From these observations the proof of Theorem 2.1 follows easily. □

3 Characterization of simplex games

Let us call a game <N, v > a T -simplex game, where ϕ≠T ⊂N, if v(N)>∑_{i=1}^{n}v({i}) are

C(v) = conv{f^{i}(v) | i ∈ T } = Δ(T, v). Note that for a T -simplex game the imputation set is an (n − 1)-dimensional simplex with f^{1}(v), f^{2}(v), ..., f^{n}(v) as extreme points and the core is a (|T | − 1)-dimensional subsimplex. In [1] N-simplex games (and also dual N-simplex games) were introduced and the family of these games was denoted by SI^{N} (and SI_{∗}^{N}). The main results obtained were:

(i) SI^{N} = {v ∈ G^{N} | v(S) ≤ ∑_{i∈S}v({i}) for all S ≠ N, ∑_{i∈N}v({i}) < v(N)} is a cone

and the core correspondence is additive of SI^{N};

(ii) For v ∈ SI^{N} : CIS(v) = ENSR(v) = τ(v), CC(v) = C(v), where CIS(v) is the center of the imputation set and ENSR(v) is the center of the dual imputation set. For the definition of the τ-value we refer to [9], [1] or to [3].

In the next Theorem 3.1 we give some properties for games v ∈ SI^{T}, the set of T -simplex games. In Theorem 3.2 it turns out that these properties are characterizing properties. Then we show in Example 3.4 that SI^{T} is not necessarily a cone of games if T ≠ N.

For a game < N, v > the zero-normalization is the game < N, 0 > with

v_{0}(S) = v(S) − ∑_{i∈S}v({i}) for each S ∈ 2^{N}.

Theorem 3.1. Let v ∈ SI^{T} for ∅ ≠ T ⊂ N and let v_{0} be the corresponding zero– normalization. Then

(i) (Losing property) v_{0}(S) ≤ 0 for each S ∈ 2^{N} with T \ S ≠ ∅

(ii) (Veto player property) T = ∩{S ∈ 2^{N} | v_{0}(S) = v_{0}(N)}

(iii) (N, 0)-monotonicity property) v_{0}(S) ≤ v_{0}(N) for all S ∈ 2^{N}.

Remarks. In the spirit of [7] we call S with v_{0}(S) ≤ 0 losing–coalitions and those with v_{0}(S) = v_{0}(N) winning coalitions. Players who are in each winning coalition are called veto players. Property (ii) says then that the set of veto players is equal to T.

Proof of Theorem 3.1.

(i) Take S ∈ 2^{N} such that there is a k ∈ T \ S. Then f^{k}(v) ∈ C(v) which implies

that v(S) ≤ ∑_{i∈S}(f^{k}(v))_{i} = ∑_{i∈S}v({i}), v_{0}(S) ≤ 0.

(iii) By (i), for S ∈ 2^{N} with T \ S ≠ ∅ : v_{0}(S) ≤ 0 ≤ v_{0}(N). For S ∈ 2^{N} with T ⊂ S and each x ∈ C(v) = conv{f^{i}(v) | i ∈ T } we have

v(S) ≤ ∑_{i∈S\T}x_{i} + ∑_{i∈T}x_{i} = ∑_{i∈S\T}v({i}) + (v(N) − ∑_{i∈N\T}v({i})) ,

which is equivalent with v_{0}(S) ≤ v_{0}(N).

(ii) From (i) it follows that v_{0}(S) = v_{0}(N) > 0 implies that T \ S= ∅, T ⊂ S. So T ⊂ ∩{S ∈ 2^{N} | v_{0}(S) = v_{0}(N)}. For the converse inclusion we have to prove that

(∩{S ∈ 2^{N} | v_{0}(S) = v_{0}(N)}) \ T = ∅.

Suppose that this set is non–empty and that r is an element of it. We will deduce a contradiction. For each U ∈ 2^{N} with r ∉ U, U is not winning. This implies that max{v_{0}(U) | r ∉ U} < v_{0}(N). Take ε ∈ (0, 1) such that (1−ε)v_{0}(N) > v_{0}(U) for all U with r ∉ U. Then we claim that for each t ∈ T the element z = (1 − ε)f^{t}(v) + εf^{r}(v) is a core element, which is in contradiction with the fact that C(v) = Δ(T, v) = conv{f^{i}(v) | i ∈ T }. To prove the claim note that for S ∈ 2^{N} with

(a) t ∉ S, r ∉ S : ∑_{i∈S}z_{i} = (1 − ε)∑_{i∈S}v({i}) + ε∑_{i∈S}v({i}) ≥ v(S) by (i).

(b) t ∉ S, r ∈ S : ∑_{i∈S}z_{i} = (1 − ε)∑_{i∈S}v({i}) + ε (∑_{i∈S}v({i}) + v_{0}(N)) > ∑_{i∈S}v({i}) ≥

v(S) by (i).

(c) t ∈ S, r ∉ S : ∑_{i∈S}z_{i} = (1−ε)∑_{i∈S}v({i})+(1−ε)v_{0}(N)+ε∑_{i∈S}v({i}) = ∑_{i∈S}v({i})+

(1 − ε)v_{0}(N) > ∑_{i∈S}v({i}) + v_{0}(S) = v(S) in view of the choice of ε.

(d) t ∈ S, r ∈ S : ∑_{i∈S}z_{i} = ∑_{i∈S}v({i}) + v_{0}(N) ≥ ∑_{i∈S}v({i}) + v_{0}(S) = v(S), where the

inequality follows from (iii). □

Example 3.1. For T ⊂ N, the unanimity game u_{T} : 2^{N} → IR is defined by u_{T} (S) = 1 if T ⊂ S and u_{T} (S) = 0, otherwise. The game u_{T} is a T -simplex game with C(u_{T}) = conv{e^{i} | i ∈ T } and I(u_{T}) = conv{e^{i} | i ∈ N}, where e^{i} is the i-th standard basis element in IR^{n}.

Example 3.2. A game is called simple [7] if v(S) ∈ {0, 1} for all S ∈ 2^{N} and v(N) = 1. The United Nations Security Council Game < N, v > with N = {1, 2, ..., 15} and

v(S) = 1 if {1, 2, 3, 4, 5} ⊂ S and |S| ≥ 9,
v(S) = 0 otherwise

is a {1, 2, 3, 4, 5}-simplex game.

It corresponds to the situation when a bill can pass only if at least nine members agree with, among them the five veto players 1,2,3,4 and 5 are. In fact, all simple games with a non–empty core and with v({i}) = 0 for each i ∈ N are T –simplex games (see Corollary 3.1), where T is the non–empty set of veto players.

Example 3.3. Let N = \{1, 2, 3, 4\}, v(\{1, 2\}) = v(\{1, 2, 3\}) = v(N) = 1, v(\{1, 2, 4\}) = \frac{1}{2},

v(S)=0 otherwise. Then < N, v > is a {1, 2}-simplex game, with C(v)=conv{e^{1}, e^{2}}.

Example 3.4. Now we show that T -simplex games do not necessarily form a cone by considering the two 5-person {1, 2}-simplex games < N, v > and < N, w > with

v({1, 2, 3}) = v({1, 2, 4}) = 1 = v(N), v(S) = 0 otherwise
w({1, 2, 3}) = w({1, 2, 5}) = 1 = w(N), w(S) = 0 otherwise.

Then

\begin{matrix}C(v) = conv\{f^{1}(v), f^{2}(v)\} = conv\{e^{1}, e^{2}\}, \\ C(w) = conv\{f^{1}(w), f^{2}(w)\} = conv\{e^{1}, e^{2}\}. \end{matrix}

For the sum game u = v + w we have

u({1, 2, 3}) = 2 = u(N), u({1, 2, 4}) = u({1, 2, 5}) = 1, u(S) = 0 otherwise.

Note that u is not a simplex game, so it is certainly not an element of SI^{{1,2}}.

Theorem 3.2. Let < N, v > be a game with v_{0}(N) > 0. Suppose that

(i) v_{0}(S) ≤ v_{0}(N) for each S ∈ 2^{N}

(ii) T := ∩{S | v_{0}(S) = v_{0}(N)}≠ ∅

(iii) v_{0}(S) ≤ 0 for all S with T \ S≠ ∅.

Then C(v) = Δ(T, v), v ∈ SI^{T} .

Proof. We have to show that C(v) = Δ(T, v).

(a) Suppose x ∈ Δ(T, v). Then for each i ∈ N there is α_{i} ≥ 0 such that x_{i} =

v({i}) + α_{i}v_{0}(N) and ∑_{i=1}^{n}α_{i} = 1, α_{i} = 0 for i ∈ N \ T. Then for

S ∈ 2^{N} : ∑_{i∈S}x_{i} = ∑_{i∈S}v({i}) + v_{0}(N)∑_{i∈S}α_{i} = ∑_{i∈S}v({i}) + v_{0}(N) ∑_{i∈T ∩S}α_{i}.

In case

T ⊂ S : ∑_{i∈S∩T}α_{i} = ∑_{i∈T}α_{i} = ∑_{i∈N}α_{i} = 1,

so

∑_{i∈S}x_{i} = ∑_{i∈S}v({i}) + v_{0}(N) ≥ ∑_{i∈S}v({i}) + v_{0}(S) = v(S),

where the inequality follows from (i).

In case

T \ S≠ ∅ : ∑_{i∈S}x_{i} = ∑_{i∈S}v({i}) + v_{0}(N) ∑_{i∈T ∩S}α_{i} ≥ ∑_{i∈S}v({i}) ≥ v(S),

where the last inequality follows from (iii). So x ∈ C(v). We have proved that Δ(T, v) ⊂ C(v).

(b) For the converse inclusion, we show that x ∈ I(v) \ Δ(T, v) implies that x ∉ C(v). Take x ∈ I(v) \ Δ(T, v). Then there is a k ∈ N \ T with x_{k} = v({k}) + ε and ε > 0. Further x_{i} ≥ v({i}) for all i ∈ N. By (ii) there is an S with v_{0}(S) = v_{0}(N) and k ∉ S. This implies

x(S) = v(N) − ∑_{i∈N\S}x_{i} ≤ v(N) − ∑_{i∈N\S}(v{i}) − ε =

= v_{0}(N) + ∑_{i∈S}v({i}) − ε = v_{0}(S) + ∑_{i∈S}v({i}) − ε = v(S) − ε.

So we have proved that ∑_{i∈S}x_{i} ≤ v(S) − ε, hence x ∉ C(v). □

As a corollary of Theorem 3.2 we obtain the following well–known fact about simple games.

Corollary 3.1. Let < N, v > be a game with the properties:

(i) v(S) ∈ {0, 1} for each S ∈ 2^{N},

(ii) Δ(N, v) = I(v) = conv{e^{1}, e^{2}, ..., e^{n}},

(iii) The set of veto players T = ∩{S | v(S) = 1} is non–empty.

Then C(v) = Δ(T, v).

4 Characterization of dual simplex games

Now, we focus on characterizing all games with the property that the core is a non– empty subsimplex of the dual imputation set I^{∗}(v). Let us denote by SI_{∗}^{T} the set of

n-person games with ∅ ≠ T ⊂ N, v^{∗}(N) < ∑_{i=1}^{n}v^{∗}(i) and C(v)=conv{g^{i}(v) | i∈T }=

Δ^{∗}(T, v).

Example 4.1. Let < N, v > be the 3-person game with v({i}) = 0 for each i ∈ N v({1, 2}) = 1, v({1, 3}) = 2, v({2, 3}) = v(N) = 6. Then v^{∗}(N \ {i}) = v^{∗}(N) = 6 for each i ∈ N, v^{∗}({1}) = 0, v^{∗}({2}) = 4 and v^{∗}({3}) = 5. Here C(v) = I(v) ∩ I^{∗}(v) = conv{6e^{1}, 6e^{2}, 6e^{3}} ∩ conv{(−3, 4, 5), (0, 1, 5), (0, 4, 2)} = conv{(0, 1, 5), (0, 4, 2)} = Δ^{∗}({2, 3}, v), so v ∈ SI_{∗}^{{2,3}}.

Example 4.2. Let < N, v > be the 3-person unanimity game based on {1, 2}, so v({1, 2}) = v({1, 2, 3}) = 1, v(S) = 0 otherwise. Then v^{∗}({3}) = 0 and v^{∗}(S) = 1 otherwise. The core C(v) equals conv{e^{1}, e^{2}} = conv{f^{1}(v), f^{2}(v)} = conv{g^{2}(v), g^{1}(v)}. So C(v) = Δ({1, 2}, v) = Δ^{∗}({1, 2}, v), hence v ∈ SI^{{1,2}} and v ∈ SI_{∗}^{{1,2}}.

To solve the characterization problem for dual simplex games, we can use our characterization result in Section 3 for simplex games. For that purpose we develop some duality relations for cooperative games in the next lemma.

Lemma 4.1. For each v ∈ G^{N} and all k ∈ N, T ⊂ N, T ≠ ∅ we have 14 R.Brânzei and S.Tijs

(i) (v^{∗})^{∗} = v (ii) −f^{k}(v) = g^{k}(−v^{∗}) (iii) Δ^{∗}(T, v) = −Δ(T, −v^{∗}) (iv) C(−v^{∗}) = −C(v)

(v) C(−v^{∗}) = Δ(T, −v^{∗}) iff C(v) = Δ^{∗}(T, v), which is equivalent to −v^{∗} ∈ SI^{T} iff v ∈ SI_{∗}^{T} .

Proof. We only prove (iv) and leave the other proofs to the readers.

C(−v^{∗}) =

= {x ∈ IR^{n} | ∑_{i=1}^{n}x_{i} = −v^{∗}(N), ∑_{i∈S}x_{i} ≥ −v^{∗}(S) for each S ∈ 2^{N}} =

= {x ∈ IR^{n} | ∑_{i=1}^{n}x_{i} = −v(N), ∑_{i∈N\S}x_{i} ≤ −v(N \ S) for each S ∈ 2^{N} } =

= − {y ∈ IR^{n} | ∑_{i=1}^{n}y_{i} = v(N), ∑_{i∈N\S}y_{i} ≥ v(N \ S) for each S ∈ 2^{N} } =

= − {y ∈ IR^{n} | ∑_{i=1}^{n}y_{i} = v(N), ∑_{i∈T}y_{i} ≥ v(T) for each T ∈ 2^{N}} =

= −C(v).

□

The key of finding the characterization of dual simplex games lies now in Lemma 4.1 (v): v ∈ SI_{∗}^{T} iff −v^{∗} ∈ SI^{T} . So we can use the characterization of v ∈ SI^{T} of Section 3 but with −v^{∗} in the role of v and obtain Theorem 4.2. Let ∅ ≠ T ⊂ N and let v_{0}(N) > 0. Then v ∈ SI_{∗}^{T} iff the following three properties hold:

(i) Dual (N, 0)-monotonicity property: (v^{∗})_{0}(S) ≥ (v^{∗})_{0}(N) for all S ∈ 2^{N} (ii) Dual veto player property: ∩{S ∈ 2^{N} | (v^{∗})_{0}(S) = (v^{∗})_{0}(N)} = T ≠ ∅ (iii) Dual losing property: (v^{∗})_{0}(S) ≥ 0 for all S ∈ 2^{N} with T \ S≠ ∅.

5 Concluding remark

It could be interesting to study for simplex games and also for dual simplex games the relations between different existing solution concepts such as the τ-value, the nucleolus, the Shapley value, CIS etc.

References

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Rodica Brânzei Faculty of Computer Science, ”Al.I. Cuza” University 11, Carol I Bd., 6600 Iaşi, Romania E-mail address: branzeir@infoiasi.ro

Stef Tijs CentER and Department of Econometrics and Operations Research Tilburg University P.O. Box 90153, 5000 LE Tilburg, The Netherlands E–mail address: S.H.Tijs@kub.nl