The Interval Shapley Value: An Axiomatization S.Z. Alparslan G¨ok
∗
R. Branzei
†
S. Tijs
‡§
Abstract The Shapley value, one of the most widespread concepts in Operations Research applications of cooperative game theory, was defined and axiomatically characterized in different game-theoretic models. Recently much research work has been done in order to extend OR models and methods, in particular cooperative game theory, for situations with interval data. This paper focuses on the Shapley value for cooperative games where the set of players is finite and the coalition values are compact intervals of real numbers. The interval Shapley value is characterized with the aid of the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory. Keywords: cooperative games, interval data, the Shapley value, Operations Research JEL Classification: C 71 ∗
S¨ uleyman Demirel University, Faculty of Arts and Sciences, Department of Mathematics, 32 260 Isparta, Turkey and Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey, e-mail:
[email protected] † Faculty of Computer Science, “Alexandru Ioan Cuza” University, Ia¸si, Romania, email:
[email protected] ‡ CentER and Department of Econometrics and OR, Tilburg University, The Netherlands, e-mail:
[email protected] § The authors gratefully thank two anonymous referees whose detailed remarks and suggestions improved the presentation substantially, and acknowledge the support of TUBITAK (Turkish Scientific and Technical Research Council).
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1
Introduction
The Shapley value (Shapley, 1953) is one of the most interesting single-valued solution concepts in cooperative game theory. Introduced and characterized for cooperative games with transferable utility (TU games) with a finite player set and where coalition values are real numbers, it has captured much attention being extended in new game theoretic models and widely applied for solving reward/cost sharing problems in Operations Research (OR) and economic situations, sociology, computer science, etc. The Shapley value assigns to each cooperative TU game one payoff vector whose components are real numbers. Important contributions concerning the Shapley value and its characterizations are: Shapley (1953), Owen (1972), Young (1985), Hart and Mas-Colell (1989), Monderer, Samet and Shapley (1992). The Shapley value has been also treated for cooperative TU games whose player set is countably infinite (Artstein (1971), Pallaschke and Rosenm¨ uller (1997), Rosenm¨ uller (2000)) and infinite (Shapley (1962), Aumann and Shapley (1974), Hart and Mas-Colell (1996)). Models in cooperative game theory where there is a finite player set but players consider cooperation in some more sophisticated manner (as, for example, multi-choice coalitions or fuzzy coalitions, or under communication restrictions), and various models of cooperative games where the domain of the characteristic function is different from the set of real numbers, treat the Shapley value accordingly. We also refer the reader to Roth (1988) and Aumann and Hart (2002) to have an idea about the abundance of literature concerning the Shapley value and its variations. Uncertainty on coalition values, which is a challenge of the modern world, led to new models of cooperative games and corresponding Shapley-like values (Branzei, Dimitrov and Tijs (2003), Timmer, Borm and Tijs (2003), Alparslan G¨ok, Branzei and Tijs (2008)). This paper focuses on properties of the interval Shapley value (Alparslan G¨ok, Branzei and Tijs, 2008) and axiomatically characterizes it on an additive cone of cooperative interval games. A cooperative interval game (Alparslan G¨ok, Miquel and Tijs, 2009) is an ordered pair < N, w > where N = {1, . . . , n} is the set of players, and w : 2N → I(R) is the characteristic function which assigns to each coalition S ∈ 2N a closed and bounded interval w(S) ∈ I(R), such that w(∅) = [0, 0]. The interval Shapley value associates to each cooperative interval game a payoff vector whose components are compact intervals of real numbers. Cooperative games in the additive cone on which we characterize the interval Shapley value arise from several 2
OR and economic situations with interval data. For example, peer group games (Branzei, Mallozzi and Tijs, 2008) are useful in modeling sequencial production situations, auctions and flow situations. Interval games in the subclass we consider arise from airport situations with interval data (Alparslan G¨ok, Branzei and Tijs, 2009), and the interval Shapley value of an airport interval game coincides with the interval Baker-Thompson allocation (inspired by Baker (1965) and Thompson (1971)). Also, some connection problems where the cost of the connections are affected by interval uncertainty (Moretti et al., 2008), lot sizing problems with uncertain demands (Drechsel and Kimms, 2008)), and sequencing problems where parameters are compact intervals of real numbers (Alparslan G¨ok et al., 2008) can be solved using interval games in the additive cone under consideration. We refer the reader to Alparslan G¨ok (2009) and Branzei, Tijs and Alparslan G¨ok (2008) for an overview of recent developments in the theory of cooperative interval games and its potential applications. Game Theory is not the only branch of Operations Research which takes into account in its models interval uncertainty that people and firms are facing on an every day basis. The need to deal with interval data led to the development of interval calculus (Moore (1979, 2008)) and of variations of models and methods of Operations Research which are based on interval data (Ramadan (1996), Janiak and Kasperski (2008)). The rest of the paper is organized as follows. In Section 2 we recall basic notions and facts from the theory of cooperative interval games. We study the properties of the interval Shapley value on the class of size monotonic interval games in Section 3. Section 4 gives an axiomatic characterization of the interval Shapley value on a special subclass of cooperative interval games. A concluding remarks section discusses mainly topics for further research.
2
Preliminaries
In this section some preliminaries from interval calculus and the theory of cooperative interval games are given (Alparslan G¨ok, Branzei and Tijs (2008), Alparslan G¨ok, Miquel and Tijs (2009)). A cooperative interval game is an ordered pair < N, w > where N = {1, . . . , n} is the set of players, and w : 2N → I(R) is the characteristic function such that w(∅) = [0, 0], where I(R) is the set of all nonempty, compact intervals in R. For each S ∈ 2N , the worth set (or worth interval) w(S) 3
of the coalition S in the interval game < N, w > is of the form [w(S), w(S)], where w(S) is the minimal reward which coalition S could receive on its own and w(S) is the maximal reward which coalition S could get. The family of all interval games with player by IGN . £ denoted ¤ £ ¤set N is Let I, J ∈ I(R) with I = I, I , J = J, J , |I| = I − I and α ∈ R+ . Then, £ ¤ (i) I + J = I + J, I + J ; ¤ £ (ii) αI = αI, αI .
By (i) and (ii) we see that I(R) has a cone structure. In this paper we also need a partial subtraction operator. We define I − J, £ ¤ only if |I| ≥ |J|, by I − J = I − J, I − J . Note that I − J ≤ I − J. We recall that I is weakly better than J, which we denote by I < J, if and only if I ≥ J and I ≥ J. We also use the reverse notation I 4 J, if and only if I ≤ J and I ≤ J. For w1 , w2 ∈ IGN we say that w1 4 w2 if w1 (S) 4 w2 (S), for each S ∈ 2N . For w1 , w2 ∈ IGN and λ ∈ R+ we define < N, w1 + w2 > and < N, λw > by (w1 + w2 )(S) = w1 (S) + w2 (S) and (λw)(S) = λ · w(S) for each S ∈ 2N . So, we conclude that IGN endowed with 4 is a partially ordered set and has a cone structure with respect to addition and multiplication with non-negative scalars described above. For w1 , w2 ∈ IGN with |w1 (S)| ≥ |w2 (S)| for each S ∈ 2N , < N, w1 − w2 > is defined by (w1 − w2 )(S) = w1 (S) − w2 (S). The model of interval cooperative games is an extension of the model of classical T U -games. We recall that a classical T U -game < N, v > is defined by v : 2N → R, v(∅) = 0. We denote the size of a coalition S ⊂ N by |S|, the family of such games by GN , and recall that GN is a (2|N | − 1)dimensional linear space for which unanimity games form an interesting basis. Let S ∈ 2N \ {∅}. The unanimity game based on S, uS : 2N → R is defined by ½ 1, S ⊂ T uS (T ) = 0, otherwise. The reader is referred to Part I in Branzei, Dimitrov and Tijs (2008) for a survey on classical T U -games. Interval solutions are useful to solve reward/cost sharing problems with interval data using cooperative interval games as a tool. Building blocks for interval solutions are interval payoff vectors, i.e. vectors whose components belong to I(R). We denote by I(R)N the set of all such interval payoff vectors. 4
Now, we recall that the interval imputation set I(w) of the interval game w, is defined by ( ) X I(w) = (I1 , . . . , In ) ∈ I(R)N | Ii = w(N ), Ii < w(i), for all i ∈ N , i∈N
and the interval core C(w) of the interval game w, is defined by ( ) X C(w) = (I1 , . . . , In ) ∈ I(w)| Ii < w(S), for all S ∈ 2N \ {∅} . i∈S
We notice that both I(w) and P C(w) consist of efficient interval payoff vectors, N i.e. (I1 , . . . , In ) ∈ I(R) with i∈N Ii = w(N ), satisfying specific rationality conditions. We call a game < N, w > size monotonic if < N, |w| > is monotonic, i.e., |w| (S) ≤ |w| (T ) for all S, T ∈ 2N with S ⊂ T . For further use we denote by SM IGN the class of size monotonic interval games with player set N . The interval marginal operators and the interval Shapley value were defined on SM IGN in Alparslan G¨ok, Branzei and Tijs (2008) as follows. Denote by Π(N ) the set of permutations σ : N → N of N . The interval marginal operator mσ : SM IGN → I(R)N corresponding to σ, associates with each w ∈ SM IGN the interval marginal vector mσ (w) of w with respect to σ defined by mσi (w) = w(P σ (i) ∪ {i}) − w(P σ (i)) for each i ∈ N , where P σ (i) := {r ∈ N |σ −1 (r) < σ −1 (i)}, and σ −1 (i) denotes the entrance number of player i. For size monotonic games < N, w >, w(T ) − w(S) is defined for all S, T ∈ 2N with S ⊂ T since |w(T )| = |w| (T ) ≥ |w| (S) = |w(S)|. Now, we notice that for each w ∈ SM IGN the interval marginal vectors mσ (w) are defined for each σ ∈ Π(N ), because the monotonicity of |w| implies w(S ∪ {i}) − w(S ∪ {i}) ≥ w(S) − w(S), which can be rewritten as w(S ∪ {i}) − w(S) ≥ w(S ∪ {i}) − w(S). So, w(S ∪ {i}) − w(S) is defined for each S ⊂ N and i ∈ / S. We notice that all the interval marginal vectors of a size monotonic game are efficient interval payoff vectors. The interval Shapley value Φ : SM IGN → I(R)N is defined by 1 X Φ(w) := mσ (w), for each w ∈ SM IGN . (1) n! σ∈Π(N )
We can write (1) as follows 1 X Φi (w) = (w(P σ (i) ∪ {i}) − w(P σ (i))). (2) n! σ∈Π(N )
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The terms after the summation sign in (2) are of the form w(S ∪ {i}) − w(S), where S is a subset of N not containing i. Note that there are exactly |S|!(n − 1 − |S|)! orderings for which one has P σ ({i}) = S. The first factor, |S|!, corresponds to the number of orderings of S and the second factor, (n − 1 − |S|)!, is just the number of orderings of N \ (S ∪ {i}). Using this, we can rewrite (2) as Φi (w) =
X |S|!(n − 1 − |S|)! (w(S ∪ {i}) − w(S)). (3) n!
S:i∈S /
Note that
X |S|!(n − 1 − |S|)! = 1. (4) n!
S:i∈S /
3
Properties of the interval Shapley value
In this section we study some properties of the interval Shapley value on the class of size monotonic interval games. Proposition 3.1. The interval Shapley value Φ : SM IGN → I(R)N is additive. Proof. First, we show that for each σ ∈ Π(N ) the interval marginal operator mσ : SM IGN → I(R)N is additive, i.e., for all w1 , w2 ∈ SM IGN , mσ (w1 + w2 ) = mσ (w1 ) + mσ (w2 ). Let σ ∈ Π(N ) and k ∈ N . Then, mσσ(k) (w1 + w2 ) = − = + =
(w1 + w2 )({σ(1), . . . , σ(k)}) (w1 + w2 )({σ(1), . . . , σ(k − 1)}) w1 ({σ(1), . . . , σ(k)}) − w1 ({σ(1), . . . , σ(k − 1)}) w2 ({σ(1), . . . , σ(k)}) − w2 ({σ(1), . . . , σ(k − 1)}) mσσ(k) (w1 ) + mσσ(k) (w2 ).
Now, using the additivity property of interval marginal operators we obtain
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that Φ : SM IGN → I(R)N is an additive map, i.e., Φ(w1 + w2 ) =
1 X mσ (w1 + w2 ) n! σ∈Π(N )
1 X 1 X mσ (w1 ) + mσ (w2 ) = n! n! σ∈Π(N )
σ∈Π(N )
= Φ(w1 ) + Φ(w2 ),
for all w1 , w2 ∈ SM IGN . Let w ∈ SM IGN and i, j ∈ N . Then, i and j are called symmetric players, if w(S ∪ {j}) − w(S) = w(S ∪ {i}) − w(S), for each S with i, j ∈ / S. We leave the proof of the following proposition to the reader. Proposition 3.2. Let i, j ∈ N be symmetric players in w ∈ SM IGN . Then, Φi (w) = Φj (w). Let w ∈ SM IGN and i ∈ N . Then, i is called a dummy player if w(S ∪ {i}) = w(S) + w({i}), for each S ∈ 2N \{i} . Proposition 3.3. The interval Shapley value Φ : SM IGN → I(R)N has the dummy player property, i.e. Φi (w) = w({i}) for all w ∈ SM IGN and for all dummy players i in w. Proof. This follows from (3) by taking (4) into account. PropositionP 3.4. The interval Shapley value Φ : SM IGN → I(R)N is efficient, i.e., i∈N Φi (w) = w(N ).
Proof. First, we show that for each σ ∈ P Π(N ) the interval marginal operator σ N N m : SM IG → I(R) is efficient, i.e. i∈N mσi (w) = w(N ). Let w ∈ SM IGN and σ ∈ Π(N ). Then, X i∈N
mσi (w)
=
N X
mσσ(k) (w)
k=1
= w(σ(1)) +
n X
w(σ(1), . . . , σ(k)) − w(σ(1), . . . , σ(k − 1))
k=2
= w(σ(1)) + w(σ(1), . . . , σ(n)) − w(σ(1)) = w(N ). 7
Now, using the efficiency of interval marginal operators, we obtain that Φ : SM IGN → I(R)N is an efficient map, i.e., X i∈N
Φi (w) =
1 X X 1 X X σ 1 mσi (w) = mi (w) = n!w(N ) = w(N ), n! i∈N n! n! i∈N σ∈Π(N )
σ∈Π(N )
for each w ∈ SM IGN .
4
An axiomatization of the interval Shapley value on a cone of interval games
Let S ∈ 2N \ {∅}, I ∈ I(R) and let uS be the unanimity game based on S. The cooperative interval game < N, IuS > is defined by (IuS )(T ) = uS (T )I for each T ∈ 2N \ {∅}, and its Shapley value is given by ½ I/ |S| , i ∈ S Φi (IuS ) = [0, 0] , i ∈ / S. In the sequel such interval games will play a central role. We denote by KIGN the additive cone generated by the set © ª K = IS uS |S ∈ 2N \ {∅} , IS ∈ I(R) .
So, each element of the cone is a finite sum of elements of K. We notice that KIGN ⊂ SM IGN , and axiomatically characterize the restriction of the interval Shapley value to the cone KIGN . Theorem 4.1. There is a unique solution Ψ : KIGN → I(R)N satisfying the properties of additivity, efficiency, dummy-player and symmetry. This solution is the interval Shapley value. Proof. From Propositions 3.1, 3.2, 3.3 and 3.4 and KIGN ⊂ SM IGN we obtain that Φ satisfies the four properties on KIGN . Conversely, let Ψ be an interval value satisfiying the four properties on N KIGN . We have to Pshow that Ψ = Φ. Take w ∈ KIG N. Notice that w can be written as w = S∈2N \{∅} IS uS . Then, for each S ∈ 2 \{∅} and IS ∈ I(R) P P P we have Ψ( S∈2N \{∅} IS uS ) = S∈2N \{∅} Ψ(IS uS ) and Φ( S∈2N \{∅} IS uS ) = P S∈2N \{∅} Φ(IS uS ) by additivity. Therefore we need to show that for each 8
S ∈ 2N \ {∅} and IS ∈ I(R), Ψ(IS uS ) = Φ(IS uS ). Take S ∈ 2N \ {∅} and IS ∈ I(R). Note that for all i ∈ N \ S, (IS uS )(T ∪ {i}) − (IS uS )(T ) = [0, 0] = (IS uS )({i}). By the dummy player property, we have Ψi (IS uS ) = Φi (IS uS ) = [0, 0], for all i ∈ N \ S. (5) Now, suppose that i, j ∈ S, i 6= j. Then the symmetry property implies that Φi (IS uS ) = Φj (IS uS ) for all i, j ∈ S, (6) and, similarly, Ψi (IS uS ) = Ψj (IS uS ) for all i, j ∈ S. By efficieny, (5) and (6) we obtain for any S ∈ 2N \ {∅} and IS ∈ I(R) Ψi (IS uS ) = Φi (IS uS ) =
IS . (7) |S|
Hence, Ψ(w) = Φ(w) for all w ∈ KIGN by (5) and (7).
5
Final Remarks
Our main contribution in this paper is the axiomatic characterization of the interval Shapley value on an additive cone of cooperative interval games. Inspiring was Shapley’s axiomatic characterization (see Proposition 3.5 in Branzei, Dimitrov and Tijs, 2008). We notice that whereas the Shapley value (Shapley, 1953) is defined and axiomatically characterized for arbitrary cooperative TU games, the interval Shapley value is defined only for a subclass of cooperative interval games, called size monotonic games, and is axiomatically characterized only on a strict subset of size monotonic games. The restriction to the class of size monotonic games was imposed by the need to establish efficiency of interval marginal vectors, and consequently of the interval Shapley value. In fact, the partial subtraction operator, introduced in Alparslan G¨ok, Branzei and Tijs (2008), is an essential tool in interval game theory because by using the Moore’s subtraction operator 1 efficiency 1
The Moore’s subtraction operator (Moore, 1979) is defined by I ⊖ J = [I − J, I − J].
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of interval marginal contribution vectors (on which the definition of the interval Shapley value is based) might be lacking. However, the fact that our subtraction operator is partially defined has a negative impact on extending the characterization of the interval Shapley value on KIGN to the whole class SM IGN , besides the fact that the set of interval unanimity games is infinite. Therefore, to characterize the interval Shapley value on the class of size monotonic games is a topic for further research, for which alternative characterizations of the Shapley value might be helpful. Now, we notice that the definition of the interval Shapley value is a generalization of the first formulation of the Shapley value (Shapley, 1953) which gives to each player his/her expected marginal worth with respect to the uniform distribution on the set of permutations of a finite player set. An alternative formula for the Shapley value in terms of dividends (Harsanyi, 1959) could be easily extended to the interval setting on a suitable class of cooperative interval games which does not necessarily coincide with the class SM IGN (because of the partially defined subtraction operator in the definition of the interval dividends). To characterize the Shapley-like value defined in terms of interval dividends is another interesting topic for further research.
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