Download A note on finding the Nucleolus of an n-Person

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.
e-mail: [email protected]
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(2n !) constraints. Later,
Owen (1974) proposes a smaller size minimization linear program with O(n4 ) constraints
and large constraint coefficients. As an alternative, Maschler et al. (1979) found the
nucleolus by solving a sequence of O(4n ) minimization linear programs. More recently,
Sankaran (1991) proved that the nucleolus can be found by solving a sequence of only
O(2n ) 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(4n ) 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(2n !)
constraints. Owen (1974) [6] showed how this program can be reduced to a more tractable
size of O(4n ) 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(4n ) 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(2n ) 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 2n + n − 1 rows
and 2n − 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(4n ) 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 2N .
For a game with set of players N = {1, . . . , n}, we shall define its characteristic function
as the map
v : 2N → 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 : 2N
S
−→ R
−→ 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
n −1
of excesses of x is the vector θ(x) ∈ R2
θ(x) = (e(S, x)),
with e(S, x) = v(S) −
X
xi ∀ S ⊂ N.
(3)
i∈S
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 ∈ RN : xi ≥ v(i), ∀i, and
n
X
xi = v(N )}.
(4)
i=1
Needless to say that the result clearly extends to any polytope.
n −1
For the sake of readability, we embed the problem in R2
×Rn , a space of large dimension
where the first 2n − 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
n
×R :
X
xi + θS ≥ v(S), S ⊂ N, xi ≥ v(i), ∀i, and
i∈S
n
X
xi = 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(2n ) variables and O(4n ) 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, . . . , 2n − 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
n −2
the permutation σ, the nucleolus (θ∗ , x∗ ) is the unique minimum of (1, δ, δ 2 , . . . , δ 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, . . . , 2n − 1} be such that θk∗ = zk for k < r and θk∗ > zk for r ≤ k < 2n . For any
δ > 0 we have that
(1, δ, δ 2 , . . . , δ 2
n −2
, 0, . n. ., 0)[(θ∗ , x∗ )t − z] = δ r−1 (θr∗ − zr ) +
P2n −2
= δ r−1 [(θr∗ − zr ) +
k=r
δ k (θk+1 − zk+1 )
P2n −2
k=r
δ k−r+1 (θk+1 − zk+1 )]
= δ r−1 K(δ).
(5)
Note that limδ→0 K(δ) = (θr∗ − zr ) > 0. This implies that the above scalar product is positive
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
n −2
(1, δ, δ 2 , . . . , δ 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 2
n −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
P2n −2
i=0
δ i θ(i)
s.t.: θ(0) ≥ θ(1) ≥ · · · ≥ θ(2n −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
P2n −2
i=0
(δ i − δ i+1 )(iti +
s.t.: θk − ti ≥ 0 ∀ i, k.
(θ, x) ∈ P.
P2n −2
k=0
(θk − ti ))
(9)
REFERENCES
5
The objective function and the first group of constraints represent the ordered weight sum
Pn
of the values 2i=0−2 δ i θ(i) , where θ(0) ≥ θ(1) ≥ · · · ≥ θ(2n −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 2
n −2
. This formulation, together with the fact that for the
n −2
permutation σ, (θ∗ , x∗ ) is the unique minimum of (1, δ, δ 2 , . . . , δ 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(2n ) variables and O(4n ) constraints.
References
[1] P.C. Fishburn, Utility Theory for Decision Making. Wiley, New York, 1970.
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Math., 23 (1972), 34-39.
[3] M. Maschler, B. Peleg, L.S. Shapley, Geometric properties of the Kernel, Nucleolus and
related solution concepts, Mathematics of Operations Research, 4 (1979), 303-338.
[4] S. Nickel and J. Puerto, Location theory — a unified approach, Springer, 2005.
[5] W. Ogryczak and A. Tamir, Minimizing the sum of the k largest functions in linear time,
Information Processing Letters 85 (2003), 117-122.
[6] G. Owen, A note on the Nucleolus, International Journal of Game Theory, 3 (1974),
101-103.
[7] J.A.M. Potters, J.H. Reijnierse, M. Ansing, Computing the nucleolus by solving a prolonged simplex algorithm, Math. Operations Research, 21, (1996), 757-768.
[8] J.K. Sankaran, On finding the Nucleolus of an N -person cooperative game, International
Journal of Game Theory, 19 (1991), 329-338.
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17 (1969), 1163-1170.
[10] S. Tijs, , CEnter Discussion Paper No. 20006-89.