Optim Lett

DOI 10.1007/s11590-014-0840-9

ORIGINAL PAPER

The least square nucleolus is a normalized Banzhaf value

J. M. Alonso-Meijide • M. Álvarez-Mozos • M. G. Fiestras-Janeiro

Received: 13 May 2014 / Accepted: 4 December 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract In this note we study a truncated additive normalization of the Banzhaf value. We are able to show that it corresponds to the least square nucleolus (LS-nucleolus), which was originally introduced as the solution of a constrained optimization problem [4]. Thus, the main result provides an explicit expression that eases the computation and contributes to the understanding of the LS-nucleolus. Lastly, the result is extended to the broader family of individually rational least square values [6].

Keywords Coalitional games • Banzhaf value • Efficiency • Least square nucleolus

A cooperative game with transferable utility (just TU game from now on) is a pair (N, v) where N is a finite set of players and v, the characteristic function, is a real valued function on 2n with v(0) = 0. Consider a TU game (N, v). A vector x g Rn is called an allocation. For each allocation x g Rn and each nonempty coalition S ç N,the excess of coalition S at allocation x is given by e(S, x) = v(S) — x (S) with x (S) = Xigs xi. The average excess at x is given by e(v) = 2n—i X0=sçn e(S, x). An allocation, x, is efficient whenever x (N) = v(N). The set of imputations of (N, v) is defined as the set of all efficient allocations that are individually rational, i.e., I (N, v) = {x g Rn : x(N) = v(N), xt > v({i}), for every i g N}. We denote by G

J. M. Alonso-Meijide

Departamento de Estatística e Investigación Operativa, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

M. Álvarez-Mozos (B)

Departament de Matemática Económica, Financera i Actuarial, Universitat de Barcelona,

Barcelona, Spain

e-mail: mikel.alvarez@ub.edu

M. G. Fiestras-Janeiro

Departamento de Estatística e Investigación Operativa, Universidade de Vigo, Vigo, Spain

Published online: 16 December 2014

Springer

the set of all TU games with a nonempty imputation set. Let H Ç g,a value on H is a map, f, that associates an allocation to every TU game in H, i.e., for every (N, v) e H, f (N, v) e RN . One of these values is the Banzhaf value [3] that we denote by B. For every (N, v) e G and i e N, the Banzhaf value is defined by1

Bi (N,v) = —j £ (v(S U i) - v(S)).

SÇN\i

It is well known that the Banzhaf value is not, in general, an efficient allocation. Hammer and Holzman [2] proposed two efficient values based on the Banzhaf value that we call the multiplicative normalization of the Banzhaf value and the additive normalization of the Banzhaf value. Let H Ç G be the set of TU games satisfying XjeN Bj (N, v) = 0. The multiplicative normalization of the Banzhaf value, Bm, assigns to every (N, v) e H an allocation which is proportional to B(N, v). Formally, Bm is the value on H defined for every (N, v) e H and i e N by

Bm(N, v) v(N)-Bi(N, v).

iK ' ' Z Bj (N ,v) iy ' ;

Given a TU game, the additive normalization of the Banzhaf value, Ba, is obtained by adding the same amount to every agent's Banzhaf value. Formally, Ba is the value on G defined for every (N, v) e G and i e N by

Ba(N, v) = Bi (N, v)+ j j v(N) - £ Bj (N, v) n j e N

Ruiz et al. [4] proved the equivalence between the additive normalization of the Banzhaf value and the Least Square prenucleolus, LS-prenucleolus. The LS-prenucleolus is defined for every (N, v ) e G as the optimal solution of the optimization problem

min ^ (e(S, x) -e(v))2 s.t. x(N) = v(N).

It is clear that, in general, the optimal solution of this problem is not an imputation. In order to solve this drawback, Ruiz et al. [4] defined the Least Square nucleolus, LS-nucleolus, for every ( N, v) e G as the optimal solution of the optimization problem

1 We will write S U i instead S U{i} and S\i instead S\{i} to simplify the notation.

Given a finite set S, we denote by lowercase s its number of elements.

min ^ (e(S, x) - e(v))2

s.t. x(N) = v(N)

Xi > v({i}), for every i e N.

Since both the Banzhaf value and the LS-nucleolus satisfy strategic equivalence,2 from now on we assume that for every i e N, v({i}) = 0. If the LS-prenucleolus is an imputation, it coincides with the LS-nucleolus, but in general, both concepts provide different allocations. Ruiz et al. [4] proposed the following algorithm to obtain the LS-nucleolus.

Algorithm 1

Step 1. Take k = 1, x1 = Ba(N, v) and M1 = {i e N : x}(N, v) < 0}. Step 2. Take k = k + 1. For every j e N,

i xk-1 , xk-1(Mk-1) if ■ , Mk-1

xk I xj + n-mk-1 if j e M [ 0 otherwise

and Mk = Mk-1 U {i e N : xk(N, v) < 0}. Step 3. If Mk = Mk-1, xk is the LS-nucleolus. Otherwise, go to Step 2.

In this note we prove that the LS-nucleolus of a TU game with a non-empty imputation set is also a normalization of the Banzhaf value. The truncated normalization of the Banzhaf value, B*, is the value on G defined for every (N, v) e G as follows:

1. If for every l e N, ^jeN Bj (N, v) - v(N^ /n < Bl(N, v), then for every i e N,

B*(N, v) = B, (N, v) + 1 ^v(N) - £ Bj (N, v)j . (1)

2. If there is some l e N with jeN Bj (N, v) - v(N)) /n > Bi(N, v), then for every i e N,

Bi (N, v) = Bi (N, v) - min(B; (N, v), c} (2)

where c > 0 such that XieN min(Bj (N, v), c} = jeN Bj (N, v) - v(N).

The truncated normalization of the Banzhaf value above emerged while looking for an additive normalization of the Banzhaf value that satisfies individual rationality. The two cases considered distinguish games where Ba satisfies individual rationality and games where it fails to do so. In the first case B* selects the allocation given by Ba. In the second case, there is some player with a negative payoff according to Ba. In this case, the payoffs according to the Banzhaf value are reduced in a fixed amount subject

2 Avalueon G, f,satisfies strategic equivalence iffor every (N, v) e G, a > 0,and fi e RN, f(N ,av = af(N, v) + f, where (N, av + f) is defined for every S c N by (av + f)(S) = av(S) + f(S).

to no one receiving a negative payoff. The solution in this second case is inspired by the CEL bankruptcy rule [1].

Next we show that the truncated normalization of the Banzhaf value is, in fact, the LS-nucleolus.

Proposition 1 For every (N, v) e G, the allocation given by the truncated normalization of the Banzhaf value is the LS-nucleolus of (N, v).

Proof Take A = |i e N : eN Bj (N, v) — v(N)) /n < Bi (N, v)J .If A = N then Bl (N, v) = Ba(N, v) and it is an imputation. Then, by [4] it is the LS-nucleolus. Now, assume that there is some l e N with ^jeN Bj (N, v) — v (N) /n >

Bi (N, v). We show that Algorithm 1 ends up at allocation Bl(N, v). Take k = 1, x1 = Ba(N, v), and M1 = {i e N : x}(N, v) < 0}. Then, M1 = 0. Besides, M1 = N because the imputation set I (N, v) = 0 and v(N) > 0. Take k = 2 and we obtain x2 following Step 2 in Algorithm 1. Then, x2 = 0, for every j e M1 and

2 1 x 1 ( M1 ) a 1 a

x2 = x 1 + = Ba(N, v) +-r V Ba(N, v),

J J n — m1 J n — m1 t—1

for every j e M1. Taking into account that

——f V Ba(N, v) = —1—j V Bi (N, v) + m 1 (v(N) — V Bi (N,v) ) - m1 ^ n — m1 ¿-1 (n — mi)n\ ¿-1 I

ieM1 i eM1 V i eN /

----k ( v(N) — V Bi (N,v)

n(n — m1) 1

\ i eN\M1 /

n — m1 x—'

+ "7-K Z Bi (N, v),

n(n — m1)

we obtain

x 2 = B j (N ,v) + 11 v(N) — > 'Bi (N, v)

1 ( v(N) — V Bi (N, v) J

n V ieN '

+ , m1 u (v(N) — V Bi (N,v) n(n — m1) 1 ¿-1

\ i eN\M1 )

+ V Bi(N,,) = Bj(N..) + U — V b.^

n(n — m1) n — m1 1 ¿-1

i e M 1 i e N\ M1

Next, M2 = | j e N : x2 < 0j. If M2 = M\ we finish. Otherwise, we repeat Step 2 in Algorithm 1. In the end we obtain xk = 0, for every j e Mk-1 and

xk = b (N,v)+(v(N) - £ B(N

\ i eN\Mk-1 J

for every j e N\Mk-1, where Mk-1 = |j e N : xk_1 < 0J. This allocation is

precisely B*(N, v). □

Let us examine the LS-nucleolus in some examples. The first two are instances of weighted majority games. A weighted majority game is determined by a weight wi > 0 for every player i e N and a quota q > 0 that determines the minimum joint weight that a coalition must reach in order to be a winning coalition. The worth of a coalition is 1 if itis winning and 0 otherwise. We denote a weighted majority game by [q; w\,..., wn ].

Example 1 Let (N, v) be the weighted majority game with N = {1, 2, 3, 4, 5} and [5; 3, 2, 2, 2, 1]. Its Banzhaf value is B(N, v) = (0.5625, 0.3125, 0.3125, 0.3125, 0.1875). Then, ZieN B, (N,v) = 1.6875 > v(N) = 1. Thus, B\N,v) = (0.425, 0.175, 0.175, 0.175, 0.05) = Ba(N, v). In this case each player's Banzhaf value is reduced in 0.1375.

Example 2 Let (N, v) be the weighted majority game with N = {1, 2, 3, 4} and [3; 1, 1, 1, 0]. It easy to check that B(N, v) = (0.25, 0.25, 0.25, 0). Then, ZieN Bi(N, v) = 0.75 < v(N) = 1. Thus, B*(N,v) = (0.3125, 0.3125, 0.3125, 0.0625) = Ba(N, v). In this case the Banzhaf value of every player can be increased in 0. 0625, keeping individual rationality. Although player 4 is a null player, he receives the minimum amount.

Lastly, we revisit the example given in [4].

Example 3 Let (N, v) be the TU game given by N = {1, 2, 3, 4, 5} and the characteristic function defined as v(3,4) = 1, v(3, 5) = 1, v(4, 5) = 1, v(1, 3,4) = 1, v( 1, 3, 5) = 1, v(1, 4, 5) = 1, v(2, 3, 4) = 1.4, v(2, 3, 5) = 1, v(2, 4, 5) = 1, v(3, 4, 5) = 1.75, v( 1, 2, 3, 4) = 1.75, v(1, 2, 3, 5) = 1, v(1, 2, 4, 5) = 1, v(1, 3, 4, 5) = 2, v(2, 3, 4, 5) = 2, v(N) = 2. The characteristic function v assigns 0 to all the remaining coalitions. Then,

B(N, v) = (0.0375, 0.0875, 0.80625, 0.80625, 0.6625), Ba(N, v) = (-0.0425, 0.0075, 0.72625, 0.7265, 0.5825), and B*(N, v) & (0, 0, 0.714583, 0.714583, 0.570833).

According to the Banzhaf value, we observe that there are two weak (but not symmetric) players, 1 and 2, and three strong players, 3, 4, and 5. In this case, the additive normalization of the Banzhaf value proposes a payoff below the first agent's standalone worth. Consequently, B* differs from Ba. Note that the truncation takes place not only on the payoff of player 1 but also on the payoff of player 2, who get both 0.

Next, we briefly describe how the above procedure can be generalized to the class of individually rational least square values [5]. A map w from {1,...,n} on R+ is called a symmetric weight function. This map is used to weight the excess vector at efficient allocations. Note that all coalitions of the same size have the same weight. Take a symmetric weight function w. The IRLSw value is defined for every (N, v) e G as the optimal solution of the optimization problem

min ^ (e(S, x) — e(v))2w(s)

s.t. x(N) = v(N)

xi > v({i}), for every i e N.

If we take w(s) = 1, for every s = 1,..., n, the IRLSw value of the TU game (N, v) coincides with its LS-nucleolus. In case the feasible region of this optimization problem is only the set of efficient allocations, the optimal solution is called a least square value, LSw. Ruiz et al. [5] proposed a procedure to obtain the IRLSw value quite similar to Algorithm 1. In fact, both procedures differ only in the starting point, being LSw(N, v) the initial allocation used to obtain the IRLSw value.

Ruiz et al. [5] show that for every (N, v) e G the allocation LSw(N, v) is given by

LSw( N, v) = ^^ + — ( nai (v) -Y aj (v) \ (3)

n an \ ' 1

where ai (v) = £ v(S)w(s), for every i e N and

n-1 ( 2\

a = X w(s)(n - J•

s=1 v '

It is easy to check, after some reorganization of the terms in expression (3), that LSW(N, v) = 1 V (v(S) — v(S\i)) w(s)

+ - ( v(N) - £ a £ (v(S) - v(S\j)) w(s) \ •

j eN Sbj

Following the same reasoning as the one done in Proposition 1, we provide an explicit expression for the IRLSw value. Take (N, v) e G and A = | j e N : LSw(N, v) > oj. If A = N, then for every i e N

IRLSf (N, v) = LSW(N, v).

If A = N, then for every i e N

IRLSW (N, v) = 1 V (v(S) - v(S\i)) w(s) a

- min j (v(S) - v(S\i)) w(s), c

where c > 0 is such that

-Y(v(S) - v(S\j))w(s), c

= - X Z (v(S) - v(S\i)) w(s) - v(N).

j eN Ssi

Finally, Ruiz et al. [6] pointed out that the additive normalization of a semivalue corresponds to a certain least square value. Hence, we can say that the truncated normalization of a semivalue corresponds to a certain individually rational least square value.

Acknowledgments Authors acknowledge the financial support of Ministerio de Ciencia e Innovación through projects MTM2011-27731-C02 and MTM2011-27731-C03, and of Generalitat de Catalunya through project 2014SGR40. Last but not least, we would like to thank the associated editor and the referees for their comments and suggestions which helped improve a previous version of the manuscript. Finally, the usual disclaimer applies.

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