Cooperative interval games: a survey R. Branzei
∗
S. Tijs
†
S.Z. Alparslan G¨ok
‡§
November 25, 2008
Abstract Natural questions for people or businesses that face interval uncertainty in their data when dealing with cooperation are: Which coalitions should form? How to distribute the collective gains or costs? The theory of cooperative interval games is a suitable tool for answering these questions. This survey aims to briefly present the state-of-theart in this booming field of research and its applications. Keywords: cooperative games, interval data, interval cores, convex games, big boss games JEL Classification: C71
1
Introduction
The model of cooperative interval games fits all the situations where participants consider cooperation and know with certainty only the lower and upper bounds of all potential revenues or costs generated via cooperation. A cooperative interval game is a pair consisting of a set of players and a worth ∗
Faculty of Computer Science, ”Alexandru Ioan Cuza” University, Ia¸si, Romania, email:
[email protected] † CentER and Department of Econometrics and OR, Tilburg University, Tilburg, The Netherlands and Department of Mathematics, University of Genoa, Italy, e-mail:
[email protected] ‡ Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey and S¨ uleyman Demirel University, Faculty of Arts and Sciences, Department of Mathematics, 32 260 Isparta, Turkey, e-mail:
[email protected] § This author acknowledges the support of TUBITAK (Turkish Scientific and Technical Research Council).
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function which associates to each coalition a closed interval in R such that the empty coalition receives [0, 0]. Cooperative interval games are introduced in Branzei, Dimitrov and Tijs (2003) to handle bankruptcy situations where the estate is known with certainty while claims belong to known intervals of real numbers. They defined two Shapley-like values each of which associated to each game with interval data a vector of intervals, and studied their interrelations using the arithmetic of intervals (Moore (1979)). Carpente et al. (2005) considered games in strategic form and constructed related cooperative interval games similarly with the procedure used by von Neumann (1928) and von Neumann and Morgenstern (1944). Alparslan G¨ok, Miquel and Tijs (2008) considered cooperative interval games and looked at selections of such games which are classical cooperative games. Based on solutions for selections such as the core and the Shapley value they defined straightforwardly solutions for cooperative interval games. For example, the core of a cooperative interval game was defined as the union of the cores of all its selections. The focus in this survey is on interval solution concepts, i.e. solutions which associate to each cooperative interval game a set consisting of vectors of intervals. The arithmetic of intervals and the classical cooperative game theory play here important roles. First, we recall here basic interval calculus. For further use we denote by I(R) the set£ of all in R. ¤ closed £ intervals ¤ 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 substraction 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 J 4 I, if and only if J ≤ I and J ≤ I. We say that I is better than J, which we denote by I ≻ J, if and only if I < J and I 6= J. Let A and B be subsets of RN , and let a = (a1 , . . . , an ) ∈ A, b = (b1 , . . . , bn ) ∈ B with a ≤ b. We define a¤b by ([a1 , b1 ], . . . , [an , bn ]) and A¤B by {a¤b|a ∈ A, b ∈ B, a ≤ b} .
A cooperative game in coalitional form is an ordered pair < N, v >, where N = {1, 2, ..., n} is the set of players, and v : 2N → R is a map, assigning 2
to each coalition S ∈ 2N a real number, such that v(∅) = 0. Often, we also refer to such a game as a TU (transferable utility) game. We denote by GN the family of all (classical) cooperative games with player set N . The core C (Gillies (1959)) and the Shapley value φ (Shapley (1953)) are central solution concepts defined on GN . For a game v ∈ GN and a coalition T ∈ 2N \ {∅}, the subgame with player set T is the game vT defined by vT (S) = v(S) for all S ∈ 2T . In the sequel we denote such subgames by < T, v >. A game v ∈ GN is called convex (or supermodular) if v(S ∪ T ) + v(S ∩ T ) ≥ v(S) + v(T ) for each S, T ∈ 2N holds true. A game < N, v > is called a big boss game with n as a big boss (Muto et al. (1988), Tijs (1990)) if the following conditions are satisfied: (i) v ∈ GN is monotonic, i.e. v(S) ≤ v(T ) if for each S, T ∈ 2N with S ⊂ T; (ii) v(S) = 0 if n ∈ / S; P (iii) v(N ) − v(S) ≥ i∈N \S (v(N ) − v(N \ {i})) for all S, T with n ∈ S ⊂ N .
Note that big boss games form a cone in GN . Further, a game < N, v > is a total big boss game with big boss n if and only if < T, v > is a big boss game for each T ∈ 2N with n ∈ T . For classical cooperative game theory we refer the reader to Branzei, Dimitrov and Tijs (2005, 2008) and Tijs (2003). The paper is organized as follows. Section 2 presents formally the model of a cooperative interval game, recalls basic arithmetic of interval games and gives definitions of special interval games. In Section 3 the focus is on interval solution concepts and their properties on particular classes of cooperative interval games. Section 4 discusses some applications of cooperative interval games in economic and OR situations.
2
The model
Formally, a cooperative interval game in coalitional form (Alparslan G¨ok, Miquel and Tijs (2008)) is an ordered pair < N, w > where N = {1, 2, . . . , n} is the set of players, and w : 2N → I(R) is the characteristic function such that w(∅) = [0, 0]. For each S ∈ 2N , the worth set (or worth interval) w(S) of the coalition S in the interval game < N, w > is of the form [w(S), w(S)], 3
where w(S) is the lower bound and w(S) is the upper bound of w(S). We denote by IGN the family of all interval games with player set N . Note that if all the worth intervals are degenerate intervals, i.e. w(S) = w(S) for each S ∈ 2N , then the interval game < N, w > corresponds in a natural way to the classical cooperative game < N, v > where v(S) = w(S). Some classical T U -games associated with an interval game w ∈ IGN will play a key role, namely the border games < N, w >, < N, w > and the length game < N, |w| >, where |w| (S) = w(S) − w(S) for each S ∈ 2N . Note that w = w + |w|. 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). For a game w ∈ IGN and a coalition S ∈ 2N \ {∅}, the interval subgame with player set T is the game wT defined by wT (S) = w(S) for all S ∈ 2T , i.e. wT is the restriction of w to the set 2T . Next, we refer to such subgames by < T, w >. An interval game w ∈ IGNPis called I-balanced if for each balanced map λ : 2N \ {∅} → R+ we have S∈2N \{∅} λ(S)w(S) 4 w(N ). Recall that a map P λ : 2N \ {∅} → R+ is called a balanced map if S∈2N \{∅} λ(S)eS = eN . The class of I-balanced games is denoted by IBIGN . We call a game < N, w > supermodular if w(S) + w(T ) 4 w(S ∪ T ) + w(S ∩ T ) for all S, T ∈ 2N . We call a game w ∈ IGN convex if < N, w > is supermodular and its length game < N, |w| > is also supermodular. We denote by CIGN the class of convex interval games with player set N and notice that the nonempty set CIGN is a subcone of IGN . The reader can find in Alparslan G¨ok, Branzei and Tijs (2008b) and Branzei, Tijs and Alparslan G¨ok (2008b) several characterizations of convex interval games. 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 . 4
For size monotonic games < N, w >, w(T ) − w(S) is well defined for all S, T ∈ 2N with S ⊂ T . Note that the fact that < N, |w| > is supermodular implies that < N, |w| > is monotonic. As a by-product we obtain that each game w ∈ CIGN is size monotonic. Now, we define for each w ∈ SM IGN and each i ∈ N , the marginal contribution of i in the game w by Mi (w) = w(N ) − w(N \ {i}). Let w ∈ IGN and let < N, |w| > be the corresponding length game. Then, we call a game < N, w > a big boss interval game if its border game < N, w > and the game < N, |w| > are classical (total) big boss games. We denote by BBIGN the set of all big boss interval games with player set N (without loss of generality we denote the big boss by n). Note that BBIGN is a subcone of IGN . The reader is referred to Alparslan G¨ok, Branzei and Tijs (2008c) and Branzei, Tijs and Alparslan G¨ok (2008b) for characterizations of big boss interval games.
3
Interval solution concepts
The interval imputation set I(N, w) of the interval game w, is defined by ( ) X N I(N, w) = (I1 , . . . , In ) ∈ I(R) | Ii = w(N ), w(i) 4 Ii , for all i ∈ N . i∈N
The interval core C(w) of the interval game w, is defined by ( ) X C(w) = (I1 , . . . , In ) ∈ I(N, w)| Ii < w(S), for all S ∈ 2N \ {∅} . i∈S
P
P Here, i∈N Ii = w(N ) is the efficiency condition and i∈S Ii < w(S), S ∈ 2N \ {∅}, are the stability conditions of the interval vectors. It is proved (Theorem 3.1 in Alparslan G¨ok, Branzei and Tijs (2008a)) that the interval core of a cooperative interval game is nonempty if and only if the game is I-balanced. Convex interval games and big boss interval games are I-balanced games. The interval core of a big boss interval game can be explicitly described as the set K(T, w) = {(I1 , . . . , In ) ∈ I(T, w) |[0, 0] 4 Ii 4 Mi (T, w) ∀i ∈ T \ {n}} . 5
Two special elements of the interval core of a big boss interval game are the big boss interval point B(T, w) defined by ½ [0, 0], j ∈ T \ {n} Bj (T, w) = w(T ), j = n, and the union interval point U(T, w) defined by ½ Mj (T, w), j ∈ T \ {n} P Uj (T, w) = w(T ) − i∈T \n Mi (T, w), j = n.
These points play an important role in characterizing big boss games using the description of the interval core (see Theorem 3.3 in Alparslan G¨ok, Branzei and Tijs (2008c)) and in defining the T -value on this class of games. Let w ∈ BBIGN . The T -value, T : BBIGN → I(R)N , is defined for each w ∈ BBIGN by 1 T (N, w) = (U(N, w) + B(N, w)). 2 Let w ∈ IGN , I, J ∈ I(N, w) and S ∈ 2N \ {∅}. We say that I dominates J via coalition S, and denote it by I domS J, if (i) Ii ≻ Ji for all i ∈ S, P (ii) i∈S Ii 4 w(S).
For I, J ∈ I(N, w), we say that I dominates J and denote it by I dom J if there is an S ∈ 2N \ {∅} such that I domS J. I is called undominated if there does not exist J and a coalition S such that J domS I. The interval dominance core DC(w) of an interval game w ∈ IGN consists of all undominated elements in I(N, w). For w ∈ IGN a subset A of I(N, w) is a stable set if the following conditions hold: (i) (Internal stability) There does not exist I, J ∈ A such that I dom J or J dom I. (ii) (External stability) For each I ∈ / A there exist J ∈ A such that J dom I.
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It is proved in Alparslan G¨ok, Branzei and Tijs (2008a) that if w ∈ IGN and A is a stable set of w then, C(w) ⊂ DC(w) ⊂ A. Next, we define on the class of size monotonic interval games interval marginal operators, the interval Shapley value and the interval Weber set. Denote by Π(N ) the set of permutations σ : N → N . Let w ∈ SM IGN . We introduce the notions of interval marginal operator corresponding to σ, denoted by mσ , and of interval marginal vector of w with respect to σ, denoted by mσ (w). The marginal vector mσ (w) corresponds to a situation, where the players enter a room one by one in the order σ(1), σ(2), . . . , σ(n) and each player is given the marginal contribution he/she creates by entering. We denote by Pσ (i) the set {r ∈ N |σ −1 (r) < σ −1 (i)} of predecessors of i in σ, where σ −1 (i) denotes the entrance number of player i and define mσi (w) = w(Pσ (i) ∪ {i}) − w(Pσ (i)). The interval Weber set W on the class of size monotonic interval games is defined by W(w) = conv {mσ (w)|σ ∈ Π(N )} for each w ∈ SM IGN . We notice that for traditional TU-games we have W (v) 6= ∅ for all v ∈ GN , while for interval games it might happen that W(w) = ∅ (in case none of the interval marginal vectors mσ (w) is defined). Clearly, W(w) 6= ∅ for all w ∈ SM IGN . Furthermore, whereas for traditional convex games the core and the Weber set coincide, if w ∈ CIGN then W(w) ⊂ C(w). The interval Shapley value Φ : SM IGN → I(R)N is defined by Φ(w) =
1 X mσ (w), for each w ∈ SM IGN . n! σ∈Π(N )
Since Φ(w) ∈ W(w) for each w ∈ SM IGN and convex interval games are size monotonic interval games we have Φ(w) ∈ C(w) for each w ∈ CIGN (see Proposition 4.1 in Alparslan G¨ok, Branzei and Tijs (2008b)). The Shapley value Φ on the class of convex interval games satisfies the properties of additivity, efficiency, symmetry and dummy player. With the use of the ¤ operator, classical solutions and multi-solutions on subsets of GN , can be easily extended to interval solutions and interval multi-solutions on suitable subsets of IGN . In Alparslan G¨ok, Branzei and Tijs (2008b) the square core is defined by C ¤(w) = C(w)¤C(w) for each w ∈ IBIGN , the square Weber set is defined by W ¤(w) = W (w)¤W (w) for each w ∈ SM IGN , the square marginal vectors are defined by m¤,σ(w) = 7
mσ (w)¤mσ (w) for each w ∈ IGN be such that |w| ∈ GN is supermodular (or convex) and σ ∈ Π(N ) and the square Shapley value is defined by Φ¤(w) = φ(w)¤φ(w) for each w ∈ IGN be such that |w| ∈ GN is supermodular (or convex). It is proved that the interval core and the square core coincide on the class IBIGN , the square core coincides with the square Weber set on the class CIGN , the interval Shapley value and the square Shapley value coincide for each w ∈ IGN be such that |w| ∈ GN is supermodular (or convex). The notion of interval bi-monotonic allocation schemes was defined for big boss interval games in Alparslan G¨ok, Branzei and Tijs (2008c) as follows. Let w ∈ BBIGN with n as a big boss and denote by Pn the set {S ⊂ N |n ∈ S} of all coalitions containing the big boss. We call a scheme B = (BiS )i∈S,S∈Pn an (interval) allocation scheme (bi-mas) for w if (BiS )i∈S is an interval core element of the subgame < S, w > for each coalition S ∈ Pn . Such an allocation scheme B = (BiS )i∈S,S∈Pn is called a bi-monotonic (interval) allocation scheme (bi-mas) for w if for all S, T ∈ Pn with S ⊂ T we have BiS < BiT for all i ∈ S \ {n}, and BnS 4 BnT . In a bi-mas the big boss is weakly better off in larger coalitions, while the other players are weakly worse off. We say that for a game w ∈ BBIGN with n as a big boss an imputation I = (I1 , . . . , In ) ∈ I(N, w) is bi-mas extendable if there exists a bi-mas B = (BiS )i∈S,S∈Pn such that BiN = Ii for each i ∈ N . Each element of the interval core of a big boss interval game is extendable to a bi-mas (see Theorem 4.1 in Alparslan G¨ok, Branzei and Tijs (2008c)).
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Applications
Cooperative interval games are a useful tool for modeling various economic and OR situations where payoffs for people or businesses are affected by interval uncertainty and decisions regarding cooperation as well as estimations of potential shares of achieved collective gains have to be made ex-ante. We mention here minimal spanning tree networks, management applications such as funds’ allocation of firms among their divisions and cost allocation and/or surplus sharing in joint projects, sequencing situations, conflict resolution and bankruptcy situations, assignment of taxes, when there is interval uncertainty regarding the homogeneous good at stake. Some of these situations are modeled as cooperative interval games in Alparslan G¨ok, Branzei and Tijs (2008d) and Branzei and Alparslan G¨ok (2008). Several applications for 8
the model of cooperative interval games and discussions about potential applications can be also found in literature. For example Drechsel and Kimms (2008) consider a lot sizing problem with uncertain demand, providing also an algorithm for computing elements in the interval core. Of particular interest for applications of cooperative interval games are the hints on using interval solutions (Branzei, Tijs and Alparslan G¨ok (2008a)). The recent developments in the field of cooperative interval games offer new opportunities for game practice.
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[6] Branzei R. and Alparslan G¨ok S.Z., “Bankruptcy problems with interval uncertanity”, Economics Bulletin, Vol. 3, no. 56 (2008) pp. 1-10. [7] Branzei R., Dimitrov D. and Tijs S., “Shapley-like values for interval bankruptcy games”, Economics Bulletin 3 (2003) 1-8. [8] Branzei R., Dimitrov D. and Tijs S., “Models in Cooperative Game Theory: Crisp, Fuzzy and Multi-Choice Games”, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin Vol. 556 (2005). [9] Branzei R., Dimitrov D. and Tijs S., “Models in Cooperative Game Theory, Springer, Game Theory and Mathematical Methods” (2008). [10] Branzei R., Tijs S. and Alparslan G¨ok S.Z., “How to handle interval solutions for cooperative interval games”, preprint no. 110, Institute of Applied Mathematics, METU (2008a). [11] Branzei R., Tijs S. and Alparslan G¨ok S.Z., “Some characterizations of convex interval games”, preprint no. 106, Institute of Applied Mathematics, METU and Tilburg University, Center for Economic Research, The Netherlands, CentER DP 55 (2008b) (to appear in SING4 special issue of AUCO Czech Economic Review, Issue 1, 2009). [12] Carpente L., Casas-M´endez B., Garc´ıa-Jurado I. and van den Nouweland A., “Interval values for strategic games in which players cooperate”, University of Oregon Economics Department Working Papers 16 (2005). [13] Drechsel J. and Kimms A., “An algorithmic approach to cooperative interval-valued games and interpretation problems”, Working paper, University of Duisburg-Essen (2008). [14] Gillies D. B., “Solutions to general non-zero-sum games.” In: Tucker, A.W. and Luce, R.D. (Eds.), Contributions to the theory of games IV, Annals of Mathematical Studies 40. Princeton University Press, Princeton (1959) pp. 47-85. [15] Moore R., “Methods and applications of interval analysis”, SIAM, Philadelphia (1979).
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