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A note on finding the Nucleolus of an n-person cooperative game by a single linear program J. Puerto Facultad de Matemáticas. Universidad de Sevilla. January 15, 2008 Abstract Kohlberg

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A note on finding the Nucleolus of an n-person cooperative game by a single linear program J. Puerto Facultad de Matemáticas. Universidad de Sevilla. January 15, 2008 Abstract Kohlberg (1972) proved that the nucleolus of a cooperative n-person game may be obtained by solving a single minimization linear program with O(2 n!) constraints. Later, Owen (1974) proposes a smaller size minimization linear program with O(n 4 ) constraints and large constraint coefficients. As an alternative, Maschler et al. (1979) found the nucleolus by solving a sequence of O(4 n ) minimization linear programs. More recently, Sankaran (1991) proved that the nucleolus can be found by solving a sequence of only O(2 n ) minimization linear programs whose constraint coefficients are 1, 0, 1. In this paper, we show a new method to find the nucleolus by solving a unique minimization linear program with O(4 n ) constraints whose coefficients are 1, 0, 1. 1 Introduction This paper presents new advances on computing the nucleolus of a cooperative game with side payments as defined by Schmeidler (1969)[9]. Kohlberg (1972) [2] proved that the nucleolus can be found by solving a unique linear program of extremely large size with O(2 n!) constraints. Owen (1974) [6] showed how this program can be reduced to a more tractable size of O(4 n ) constraints, although the constraint coefficients are of very of large size. On the other hand, Maschler, Peleg and Shapley (1979) [3] gave another method of finding the nucleolus by giving a constructive definition of the lexicographic center of a cooperative game and showing the equivalence of this concept and the nucleolus. In their approach they require to 1 2 BACKGROUND 2 solve a sequence of O(4 n ) minimization linear programs with 1, 0, 1 constraint coefficients. This approach was improved by Sankaran (1991) [8] who gave a method for computing the nucleolus solving a sequence of only O(2 n ) minimization linear programs with 1, 0, 1 constraint coefficients. Later Potters et al.(1996) [7] describe a fast algorith to find the nucleolus of any game with non-empty imputation set. This algorithm is based on solving a prolonged simplex algorithm. It requieres solving n 1 linear programs with at most 2 n + n 1 rows and 2 n 1 columns. Since then, one can find some improvements on the computation of the nucleolus in particular classes of games, but nothing has been done on the general case. In this paper, we show an alternative method, based on a simple reformulation, to compute the nucleolus of a n-person cooperative game by solving a unique minimization linear program with O(4 n ) constraints and with 1, 0, 1 constraint coefficients. Although the complexity of the new problem is similar to Owen s one (see [6]), the advantage of the new proposal is that all constraint coefficients are 1, 0, 1, whereas in Owen s formulation some coefficients are extremely large. Thus, our formulation gives a computationally more stable method. This special form of the program has proven to be specially suitable for other optimization problems, as for the convex order median location problem (see [4]). 2 Background Given the set of players N = {1,..., n}, a coalition of N is any S N. The set of all possible coalitions of N shall be denoted by 2 N. For a game with set of players N = {1,..., n}, we shall define its characteristic function as the map v : 2 N R, (1) defined for every coalition S N as the maximum profit that the coalition S can make by acting on its own, without taking into account what the other players in N \ S can do. So, v(n) is the best payoff that the coalition formed by all the players can obtain. This coalition, N, is called the grand coalition. Therefore, a cooperative game can be represented by Γ = (N, v) where N is its set of players {1, 2,..., n} and v is its characteristic function v : 2 N R S v(s) (2) 3 THE MAIN RESULT 3 having v( ) = 0 and v(s) the maximum profit that the coalition S can make without the help of any of the other players. The nucleolus of a cooperative game was introduced by Schmeidler in [9] and for its definition requieres the concept of excess vector for each allocation of the game. The vector of excesses of x is the vector θ(x) R 2n 1 θ(x) = (e(s, x)), with e(s, x) = v(s) i S x i S N. (3) The nucleolus is the unique vector that minimizes lexicographically the non-decreasing sorted vector of excesses. In what follows we restrict ourselves to the nucleolus of a cooperative game defined on the set of imputations, namely the set I = {x R N : x i v(i), i, and n x i = v(n)}. (4) i=1 Needless to say that the result clearly extends to any polytope. For the sake of readability, we embed the problem in R 2n 1 R n, a space of large dimension where the first 2 n 1 coordinates correspond to the excesses (where we assume any ordering on the subsets of N) and the remaining n to players allocations. In this space, we deal with the polytope P = {(θ, x) R 2n 1 R n : i S x i + θ S v(s), S N, x i v(i), i, and n x i = v(n)}. i=1 This way we identify simultaneously the nucleolus (x ) and its excesses (θ ). 3 The main result Theorem 3.1 The nucleolus of a n-person cooperative game can be calculated by solving a minimization linear program with O(2 n ) variables and O(4 n ) constraints whose constraint coefficients are 1, 0, 1. Proof: The Nucleolus (θ, x ) corresponds to the lexicographical minimization of the non-decreasingly sorted vectors of excesses. Therefore, there exists a permutation σ, of (1,..., 2 n 1), that sorts the elements of the θ-variables such that (θ, x ) is the lexicographical minimum with respect to the θ-variables (excesses). 3 THE MAIN RESULT 4 First of all, it is a folklore result that on compact domains lexicographical minimization is equivalent to linear programming. This can be traced back (at least for finite sets) in [1, p.70]) and one explicit proof can be found in the recent discussion paper [10]. In any case and for the sake of completeness, we prove that after sorting the θ-variables according with the permutation σ, the nucleolus (θ, x ) is the unique minimum of (1, δ, δ 2,..., δ 2n 2, 0,. n.., 0)(θ, x) t on P for some δ 1. Take z ext(p ) {(θ, x )}, where ext(p ) denotes the set of extreme points of P, and let r {1, 2,..., 2 n 1} be such that θ k = z k for k r and θ k z k for r k 2 n. For any δ 0 we have that (1, δ, δ 2,..., δ 2n 2, 0,. n.., 0)[(θ, x ) t z] = δ r 1 (θr z r ) + 2 n 2 k=r δk (θ k+1 z k+1 ) = δ r 1 [(θ r z r ) + 2 n 2 k=r δk r+1 (θ k+1 z k+1 )] = δ r 1 K(δ). Note that lim δ 0 K(δ) = (θ r z r ) 0. This implies that the above scalar product is positive (5) for all δ δ(z) (δ(z) depends on the extreme point z). Consider δ = min{δ(z) : z ext(p ) {(θ, x )}}. Hence, for all δ δ one has that (1, δ, δ 2,..., δ 2n 2, 0,. n.., 0)[(θ, x ) t z] 0 z ext(p ) {(θ, x )}. (6) However, for z = (θ, x ) t, it attains the null value. Thus, (θ, x ) = arg min{(1, δ, δ 2,..., δ 2n 2, 0,. n.., 0)(θ, x) t : (θ, x) ext(p )} δ δ. (7) Thus, once the permutation that gives the lexicographic ordering in the optimum is known, finding the nucleolus reduces to solve a linear program. Nevertheless, in order to apply the above argument we need to prove that the problem that gives the lex-minimum, namely min 2 n 2 i=0 δ i θ (i) s.t.: θ (0) θ (1) θ (2 n 2) (8) (θ, x) P can be written as a linear programming problem. This formulation is doable using the result in [5]. Consider the following linear programming problem min 2 n 2 i=0 (δi δ i+1 )(it i + 2 n 2 k=0 (θ k t i )) s.t.: θ k t i 0 i, k. (9) (θ, x) P. REFERENCES 5 The objective function and the first group of constraints represent the ordered weight sum of the values 2 n 2 i=0 δ i θ (i), where θ (0) θ (1) θ (2 n 2). Notice that this formulation results from the reformulation of the ordered median problem that appears in [5, Section 3]. It is clearly applicable here because we consider the convex case of the weighted ordered average, i.e δ 0 δ 1 δ 2n 2. This formulation, together with the fact that for the permutation σ, (θ, x ) is the unique minimum of (1, δ, δ 2,..., δ 2n 2, 0. n.. 0)(θ, x) t on P, proves that computing the nucleolus of a n-person cooperative game is equivalent to solving the continuous linear program (9) with O(2 n ) variables and O(4 n ) constraints. References [1] P.C. Fishburn, Utility Theory for Decision Making. Wiley, New York, [2] E. Kohlberg, The nucleolus as a solution of a minimization problem, SIAM J. Appl. Math., 23 (1972), [3] M. Maschler, B. Peleg, L.S. Shapley, Geometric properties of the Kernel, Nucleolus and related solution concepts, Mathematics of Operations Research, 4 (1979), [4] S. Nickel and J. Puerto, Location theory a unified approach, Springer, [5] W. Ogryczak and A. Tamir, Minimizing the sum of the k largest functions in linear time, Information Processing Letters 85 (2003), [6] G. Owen, A note on the Nucleolus, International Journal of Game Theory, 3 (1974), [7] J.A.M. Potters, J.H. Reijnierse, M. Ansing, Computing the nucleolus by solving a prolonged simplex algorithm, Math. Operations Research, 21, (1996), [8] J.K. Sankaran, On finding the Nucleolus of an N-person cooperative game, International Journal of Game Theory, 19 (1991), [9] D. Schmeidler, The Nucleolus of a characteristic function game, SIAM J. Appl. Math. 17 (1969), [10] S. Tijs,, CEnter Discussion Paper No

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