A dynamic algorithm for computing Multiweighted Shapley Values of cooperative tu games Текст научной статьи по специальности «Математика»

A Dynamic Algorithm for Computing Multiweighted Shapley Values of Cooperative TU Games

Irinel Dragan

University of Texas, Mathematics, Arlington, Texas 76019-0408, USA

Abstract In an earlier paper of the author, (Dragan, 1994), the Multi-weighted Shapley Values have been introduced as linear operators on the space of TU games, which satisfy the efficiency and the dummy player axioms. This is the class of values, which for different systems of weights includes among others, the Shapley Value, the Weighted Shapley Value, the Random Order Values, the Harsanyi payoff vectors, the Owen coalition structure values, etc. An early dynamic algorithm for computing the Shap-ley Value is due to late M. Maschler (1982). This algorithm is building a sequence of allocations, corresponding to a sequence of games, ending with the Shapley Value, corresponding to the null game. In the present work, we show a similar algorithm for computing the Multiweighted Shapley Values. The algorithm is illustrated by applying it to a MWSV which is neither a Harsanyi payoff vector, nor a Random Order Value. As the algorithm is using results from our earlier paper, to make the paper self contained, the basic definitions and notations are given in the first section, while the characterizations of the MWSVs are further given in the second section and the algorithm is presented in the last section. Notice that in fact our algorithm contains a class of algorithms, in which by taking various systems of weights, the algorithm will compute different values. In this class, the well known weights of the Shapley Value will generate the Maschler algorithm for computing the value.

Keywords: Standard and Unanimity bases, linearity, efficiency, dummy player axiom, Multiweighted Shapley Values.

1. Introduction

Let N be a finite set, the set of players, \N\ = n, and GN be the space of cooperative TU games with the set of players N. This space GN has a standard basis and also a very popular basis, the unanimity basis, used by L.S.Shapley (1953a) in deriving a formula, for his well known solution for the cooperative TU games. Denote by D = {DS G GN : S C N,S = 0}, the standard basis, where the basic games are DS(T) = 1, for T = S, and DS(T) = 0, otherwise; denote by U = {US G GN : S C N,S = 0},the unanimity basis, where US(T) = 1, VT D S, and US(T) = 0, otherwise. It is well known that any v G GN has representations in the two bases,

namely

v = ^^ v(S)DS, and v

SCN

SCN

X) (S)Us■

(1)

Between the components in the two bases the relationships are

v(S Av (T), and (S) = J2(-1)S-tv(T), VS C N,S = (2)

TCS TCS

where v(S) is called the worth of v for coalition S, and the number Av (S) is called the (Harsanyi) dividend of v for coalition S. These relationships will be used later for deriving from the properties of a value relative to one representation the properties relative to the other representation. Of course, (2) can be written in matrix form as

v = MA, and 4 = M-1v, (3)

where v and A are 2n — 1 dimensional vectors of the components in the two bases and M is the Mobius matrix of vectors of the unanimity basis. From (2) follows that for any player i and all coalitions T containing i, we have

v(T) — v(T — {i])= £ Av (S u{i}), (4)

SCT-{i

equalities to be used for characterizing any dummy player i in terms of the dividends of the coalitions containing the player.

A value & is an operator from GN to Rn, where for the game v G GN the components &i(v), yi G N, are the wins of players in the game. The value & is a linear value, if we have &(av + [w) = a&(v) + [&(w), for all real numbers a and 3, and all games v G GN and w G GN. If & is a linear operator and v G GN, then from (1) we obtain

&(v) =^2 v(S)&(DS) = rv, and &(v) = ^ Av (S)&(US) = AA, (5)

SCN SCN

respectively, where both r and A are n x (2n — 1) matrices with = &i(DS), and \S = &i(US), for all i G N and S C N,S = 9. Of course, we have A = rM and r = AM-1. By using terms of Linear Algebra, it is clear that r and A are matrix representations of the linear operator & relative to the bases D and U, respectively.

They allow the computation of & when the game is given in coalitional form, or in dividend form. Let us take the following

Example 1. Consider a game in coalitional form v G G{1,2,3} and a linear operator & : G{i,2,3} R3 defined by

3 1 1 3

<pl{v) = -v(l) - ~[v(l, 2) - «(2)] - ~[v(l, 3) - «(3)] + -Kl, 2, 3) - «(2, 3)],

1 3 3 1

$2(v) = -v(2) + - Kl, 2) - «(1)] + -K2, 3) - «(3)] + -Kl, 2, 3) - v(l, 3)],

1 3 3 1

Mv) = g«(3) + 3 [v(l, 3) - «(1)] + -K2, 3) - «(2)] + -Kl, 2, 3) - v(l, 2)].

The matrix representations are obtained as follows: we collect the coefficients of all values of the characteristic function to get

/3 11 _ l _ 1 _3 3\

4 4 4 4 4 4 4

_3 1 _3 3 _1 3 1 8 8 8 8 8 8 8

_3 _3 1 _1 3 3 1 \ 8 88 88 88/

and A = rM, where in M the coalitions are ordered in the same way as in r. We

get

/1 00 1/2 1/2 0 3/4 \ A = I 0 10 1/2 0 1/2 1/8 I .

\0 0 1 0 1/2 1/2 1/8J

A basic idea in the following is that the properties of the linear operator can be translated into properties of the matrix representations. To obtain such properties we should be able first to characterize a linear operator depending on a set of weights, that will be called a MWSV, by a system of algebraic conditions. This will allow us to discover for example whether or not, the above operator is a MWSV. Then, we shall use the same conditions to obtain similar results for classes of well known values. The following definitions give well known concepts. A player i is a dummy player in v G GN, if we have v(S) — v(S — {i}) = v({i}), for all coalitions S containing player i. A linear operator <P has the dummy player property if for any dummy player i we have the equality ^i(v) = v({i}). Further, the linear operator <P is efficient if we have the equality &i.(v) = v(N). In the next section, we

ieN

give characterizations of linear operators satisfying the efficiency axiom, then those satisfying the dummy player axiom and by putting these results together we get a characterization of linear operators satisfying both axioms, that is MWSVs. This will be done for games in dividend form; then, for games in coalitional form we shall translate the results obtained for the dividend form, by using the relationships between the components in the two bases.

2. Multiweighted Shapley Values.

There are already many papers devoted to the axiomatization of Shapley Value and the values obtained by removing some axiom from the group of axioms characterizing the Shapley Value. The list of our references is not an exhaustive list of them, however in the list, the paper by Weber (1998) is doing quite an extensive job. What we shall be trying to do is to remove the symmetry axiom, and obtain explicit expressions of the corresponding values, for dividend form games and coalitional form games; we shall call this value a Multiweighted Shapley Value, briefly MWSV. After that, in the next section we shall be deriving an algorithm for computing this value. The paper by Naumova (2005) does also eliminate the symmetry, but is replacing also the other ones by axioms connected to consistency.

Definition 1. A linear operator from GN to Rn is a Multiweighted Shapley Value if it has the dummy player property and it is efficient.

This definition is suggested by the fact that a Multiweighted Shapley Value, or MWSV, is a generalization of the Shapley Value.

Theorem 1. A linear operator <P : GN ^ Rn has the dummy player property, if and only if we have

Xf = 0, Vi /S, VS C N,S = 9, X{} = 1, Vi e N, where for all S C N,S = 9, we denoted Xf = $i(Us), Vi e N.

Proof. If & has the dummy player property, then taking into account that in the game Uf € GN any player i / S is a dummy, we have Xf = &i(Uf) = Uf({i}) = 0, that is we got the first equalities (6); in U{i} the player i is also a dummy, so that we have X{i} = U{i}({i}) = 1, hence (6) hold. To prove the converse, we should notice that a player i is a dummy player in a game v € GN if and only if we have

Av(S)=0,VS C N,i€S, Av({i}) = v({i}),Vi/N. (7)

Now, if (6) hold, then for any game v € GN in the sum of the second formula (5) we have only terms for coalitions S containing the player i; if i is a dummy, the sum reduces to &i(v) = X{i}Av({i}), from which by (6) we obtain &i(v) = v({i}), that is v has the dummy player property. □

Corollary 2. A linear operator & : GN ^ Rn has the dummy player property if and only if for each coalition S with |S | = 1, there exist numbers Xs , Vi € S, such that & can be represented as

&i(v) = v({i})+ Xf Av (S), Vi e N. (8)

S:ieS,|S| = i

Theorem 3. A linear operator & : GN ^ Rn is efficient if and only if the sum of entries in each column of the matrix A equals one.

Proof. If & is efficient, then for v = Uf we should have ^ &i(US) = US(N) = 1,

ieN

or J2 Xs = 1. Conversely, from the second formula (5), we compute

ieN

E &i(v)= E (E Xf )Av(S)= E Av(S)= v(N), (9)

ieN SCN ieN SCN

where the hypothesis and the second formula (1) have been used for S = N; hence & is efficient. □

From Definition 1, Theorems 1 and 3, it follows a characterization of Multi-weighted Shapley Values:

Theorem 4. A linear operator & : GN ^ Rn is a Multiweighted Shapley Value if and only if its matrix representation A relative to the unanimity basis in GN satisfies for all coalitions S C N, S = ty, the equalities

Xf = 0, Vi/S, E Xf = 1. (10)

ief

In this case, & can be represented by (8), where the coefficients should satisfy the second conditions (10).

Note that the operator considered in Example 1, where the matrix A has been computed, satisfies the conditions (10), hence it is a MWSV. Similary, we get the following:

Corollary 5. The Shapley Value, the Weighted Shapley Value, the Owen coalitional structure value and the Harsanyi payoff vectors, are MWSV's.

Proof. The Shapley Value is obtained in (5) for Af = ^Vi G S, and Af = 0 otherwise (see (Owen, 1995)), hence (10) hold. The Weighted Shapley Value (1953b) is defined in (5) by means of a weight vector, say ¡j, G + , as Af = G S,

and Xs = 0, otherwise, (see (Kalai et al., 1987), and (Vasiliev, 2007)), so that (10) are holding. The Owen coalitional structure value is defined in (5) as follows: let {Si,..., Sm} be an ordered partition of N; for any coalition S, denote by J(S) = {j G {1,..., m} : SHSj = 9}, that is the set of indices of those blocks which contain players in S. Then, we take Xf = \J(S)| / \S n Sj \, yi G S n Sj, and Xf = 0, otherwise. Note that after using an axiomatic approach to define a coalition structure value, to show its uniqueness Owen has computed the i— th component &i(US), and he got exactly the just mentioned values of X's, (Owen, 1977). Obviously, the conditions (10) are satisfied. The Harsanyi payoff vectors are defined in (5) by weights which beside (10) satisfy Xf > 0, yi G S, (see (Harsanyi, 1963), and (Vasiliev et al., 1980, and 2002)). Therefore, they are MWSVs; the same is true for the Random order values (see (Weber, 1988), and (Derks et al., 2006)). □

Note that the operator considered above in Example 1 does not belong to any of the classes of values discussed in Corollary 5, but it is a MWSV. Note also that the Aumann-Dreze coalitional value is not a MWSV, (see (Aumann et al., 1974)). Indeed, for a partition {S1,..., Sm} of N, this value is obtained by taking in (5) the weights Xf = if S C Sj for some j and i G S, and Xf = 0 otherwise, so that if S is not included in any block, we get Xf = 0, yi G N, and the last conditions (10) do not hold.

Note that we characterized the MWSVs by an explicit formula, (8), in case of a game given in dividend form. Now, we intend to do the same in case of a game in coalitional form. Obviously, we shall be using the relationships between the dividend form and the coalitional form shown in (2) and (3). To get such a result we need the following

Lemma 6. Let & : GN ^ Rn be a linear operator, and denote

YS = &(DS), Xs = &(US), yS C N,S = 9, (11)

where the games Ds G GN form the standard basis, and the games Us G Gn form the unanimity basis for GN. Then, for every coalition S we have

xf = £ YT, yi G N, YS = E ( — 1)t-SXf, yi G N. (12)

TDS TDS

Proof. This follows easily from (1). Based on Lemma 6, we obtain:

□

Theorem 7. The matrix representation A of a linear operator & relative to the unanimity basis U for GN satisfies

(a) Xf = 0, Vi/S, (b) E Xf = 1, VS C N,S = ty, (13)

ief

if and only if the matrix representation r relative to the standard basis D for GN will satisfy

(a) = -Ji , Vi € S, (b) E (E j) = 1 VS C N,S = ty, (14)

T:TDS jeS

where we use for convenience yf = —Yi^, Vi € N.

Proof. For any pair i and S, with i € S, from (13)(a) we have that XT = 0 for all coalitions T D S — {i} which do not satisfy T D S, because i € T; hence by the second formula (12), we get (14) (a). On the other hand, from (13)(b) and the first formulas (12), we get (14)(b). Conversely, (11) and (14)(b), imply (13)(b). Also, notice that if i / S, then for each coalition T D S and contains i, the coalition T — {i} is including S, either. Therefore, if i / S then we can pair all terms of the sum in the first formula (12) to write it as Xf = (yt + yJ {i}). Now,

T :T DS,ieT

(14)(a) shows that (13)(a) holds. □

From Theorems 4 and 7 follows:

Theorem 8. A linear operator & : GN if and only if its matrix representation satisfies (14).

In this case, & can be represented by

^ Rn is a Multiweighted Shapley Value r relative to the standard basis in GN,

&i(v)= £ Yf [v(S) — v(S — {i})], Vi € N. (15)

f:ief

Note that (15) do not contain Yf with i / S, and this is also true for (14)(b). Note also that (14)(b) are equivalent to Weber's conditions (Weber, 1988, Thm 11), so that our theorem 8 is essentially Weber's result obtained in a different way. In our earlier paper (Dragan, 1994), we considered MWSVs with monotonicity properties, in order to make connections with Weber's Random Order Values, or Harsanyi payoff vectors. It was proved that the Random Order Values are the MWSVs with Yf > 0, Vi € S, in (15), As the computational algorithm to be given below works for any MWSV, it works also for Ramdom Order Values. Of course, a reader who has studied the Weber's paper may wonder whether, or not, in our paper the MWSVs with monotonicity properties have been studied. The above mentioned paper contains the answer. Note that the operator considered in Example 1 does not satisfy such conditions, hence it is not a Random Order Value. In fact, it is easy to see that this operator is not monotonic, because for the monotonic

game v(1) = v(2) = v(3) = 0,v(1, 2) = v(1, 3) = v(2, 3) = v(1, 2, 3) = 1, we get <Pi(v) = = j,<Ps(v) = not all components are nonnegative.

3. An algorithm for computing the Multiweighted Shapley Values

Consider a game v G GN and an nx(2n — 1) matrix A satisfying the conditions (10) of Theorem 4. This matrix defines a MWSV by formula (8), if the game is in dividend form, or the matrix r = AM-1 and formula (15) define a MWSV if the game is in coalitional form. The algorithm needs the matrix A even though the TU game is in coalitional form; obviously, if r is available, then A = rM. If v is the null game, then &(v) = 0. Otherwise, there is a coalition S for which v(S) = 0, and the following iterative procedure can start. We assume that all coalitions of sizes smaller than \S\ have zero worth. Initially, x1 =0 G Rn, and v1 = v G GN. Suppose that in step k > 1 we have an allocation xk G Rn and a game vk G GN available. The step k is asking for the computation of the allocation and game to be needed in the next step, when a coalition Sk with vk(Sk) = 0 has been selected:

xk+1 = xk + vk(Sk).xsk, vk+1 = vk — vk(Sk).USk. (16)

Obviously, the sequences obtained depend on the sequence of coalitions selected; however, we have the following result:

Theorem 9. For any sequence of coalitions {Sk} in any step k we have

xk + &(vk)= &(v), k = 1, 2,... (17)

Proof. For k = 1, as x1 = 0 and v1 = v, (17) holds. Assume that (17) holds for all k = 1, ...,p; then, from (16), the induction assumption, the linearity of &, and &(USk) = Xs , it follows that (17) holds for k = p +1, hence (17) holds for any value of k. □

Note that by Theorem 9 the procedure stops if vk+1 = 0, and in this case we have xk+1 = &(v), that is the MWSV has been computed. The algorithm is justified by the convergence theorem:

Theorem 10. For any sequence of coalitions in which a coalition Sk with vk(Sk) = 0 is introduced only if vk(S) = 0 for all coalitions with \S\ < |Sk | — 1, the algorithm described by formulas (16) converges to &(v) in a finite number of steps.

Proof. Notice that by the second formulas (16), the worth of coalitions with a smaller size than Sk, as well as those for which the worth has been made equal to zero in the previous steps, is kept equal to zero. Also, we get vk+1(Sk) = 0. As the number of coalitions of the same size is finite, the coalitions of size | Sk | will be exhausted in a finite number of steps and the algorithm will go to coalitions of

higher sizes. In a finite number of steps the coalitions of all sizes will be exhausted.

□

Example 2. Return to the constant sum game and the operator already considered in Example 1, for which we computed the matrix A. By using the iterative procedure described by formulas (16) we compute this operator for our game in four steps shown in the following auto explanatory table

Step k Coice xk xk xk 12 13 23 123

1 1, 2 1/2 1/2 0 0 1 1 0

2 1, 3 1 1/2 1/2 0 0 1 -1

3 2, 3 1 1 1 0 0 0 -2

4 1, 2, 3 -1/2 3/4 3/4 0 0 0 0

k

v

where &(v) is on the last row. Remarks:

a) The MWSV can be defined by two axioms only, the linearity and the carrier axiom, because for a linear operator the dummy player and the efficiency axioms together are equivalent to the carrier axiom (Dragan, 1994),

b) A quasi-value is a random order value (Gilboa et al., 1991), so that the algorithm described above can be used for the computation of quasi-values. In fact, it could also be used for the computation of Semivalues (Dubey et al., 1981) and Least Square Values, (Ruiz et al., 1998), because each of them is a Shapley Value as it has been recently proved by the author (Dragan, 2005).

c) Notice that for the weights of the Shapley Value, substituted in the first formulas (16) we shall obtain the formulas given by Maschler in his algorithm for computing the Shapley Value (1982); it follows that our algorithm is an extension of Maschler's algorithm to Multiweighted Shapley Values.

d) Naumova (2005) discussed also values for TU games without the symmetry axiom, but used a set of axioms different than the ones considered above.

Aknowledgement. This paper is published as a reminder of the work done by

late Michael Maschler, who attended its presentation and intended to contribute,

by extending the algorithm for computing the values of Nontransferable utilities

games.

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