On the inverse problem for a subclass of linear, symmetric and efficient values of cooperative TU games

Operations Research Letters 44 (2016) 618–621

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Operations Research Letters journal homepage: www.elsevier.com/locate/orl

On the inverse problem for a subclass of linear, symmetric and efficient values of cooperative TU games Jony Rojas Rojas ∗ , Francisco Sanchez Sanchez Department of Basic Mathematics, Mathematics Research Center, A.C., Jalisco S/N, Col. Valenciana CP: 36023 Guanajuato, Gto, Mexico

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Article history: Received 26 April 2016 Received in revised form 8 July 2016 Accepted 9 July 2016 Available online 15 July 2016

abstract In this paper we solve the inverse problem for each linear, symmetric, efficient and regular value (or LSER value for short). That is, given a payoff vector and a LSER value, we find all TU games such that LSER value for them is equal to the given payoff vector. Also, we characterize those linear, symmetric and efficient values that further satisfy the inessential game property. © 2016 Published by Elsevier B.V.

Keywords: The linear, symmetric and efficient value The inverse problem The null space of a linear operator

1. Introduction The study of the properties of values for TU games is one of the topics of cooperative game theory since the seminal axiomatic contribution of Shapley in 1953. This is due to the desire to understand the different contexts in which the cooperative game theory can be applied. For instance, there are several studies where the original axioms proposed by Shapley in 1953 are substituted by weak versions or by different axioms that fit certain applications. Other work considers the family of solutions that emerge by taking out one or two of the original axioms. The most popular family of values consist of values that are linear, symmetric, and efficient. This family of values was initially presented by Ruiz et al. [14]. In this paper we study the inverse problem of each linear, symmetric, efficient and regular value (or LSER value for short). By a regular value we mean that each of its parameter is nonzero. This is, given an n-vector of payoffs, say x, to find out all the games in GN , such that for these TU games the LSER value equals x. Also, we characterize the linear, symmetric and efficient values that satisfy the inessential game property. Dragan [5] solved the inverse problem for the Shapley value. He proposed a different system of weights, a similar problem was solved for the more general case of the weighted Shapley values, based on some results from linear algebra and the potential of these values due to Hart and Mas-Colell [8,9]. In this case, the solution was successful because he discovered a new algebraic basis for the

∗

Corresponding author. E-mail addresses: [email protected] (J.R. Rojas), [email protected] (F.S. Sanchez).

http://dx.doi.org/10.1016/j.orl.2016.07.009 0167-6377/© 2016 Published by Elsevier B.V.

null space of the linear operator, that he called the potential basis. Later, Dragan [6] uses a similar algebraic basis to solve the inverse problem for semivalues. However, in the case of LSER values such a basis is missing, so that in the present paper, we have to use another approach to solve the inverse problem for these values. There are several situations where it may be interesting to know the totality of cooperative games that have a common payoff vector. For example, suppose there is a certain agreement between companies to distribute their costs according to a certain linear, symmetric and efficient value. Moreover, assume such companies wish to diminish such costs for the next semester. Hypothetically, if these agents can for-see such costs (i.e. the payoff vector) for the next period, then it would be useful to know which ‘‘game’’ will produce such a vector. Once they know the collection of those games sharing the given payoff, they will have information that will help them to make decisions (e.g. reinforce certain coalitions). On the other hand, in cooperative game theory literature there are many different values that have been used in certain concrete situations. On many occasions, such values are linear, symmetric and efficient (LSE), to name a few: Shapley value, solidarity value or the least square prenucleolus. Moreover, the study of the inverse problem for a LSE value could, potentially, be used to give a useful characterization of the value. For example, Béal et al. [2] obtain a characterization of the average tree solution (see Herings et al. [10]) by looking at the inverse problem for this value. Béal et al. [3] provide an alternative characterization of the class of LSER values, and also, they examine the inverse problem for the Shapley value and the equal division value. Furthermore, Béal et al. [4] and Yokote et al. [17] also deal with the inverse problem for the Banzhaf

J.R. Rojas, F.S. Sanchez / Operations Research Letters 44 (2016) 618–621

value and the Shapley value, respectively. Other examples may be found in Yokote [16] and Béal et al. [1]. Both articles find bases for the kernel of the corresponding value, and a certain invariance axiom is proposed that permits them to characterize the solution. The inverse problem is also investigated in OR for measuring voting power via simple games. Kurz and Napel [13] provide a recent survey on this question, and Kurz [12] studies the inverse problem of power indices in order to design voting rules for a committee such that a given desired power distribution is met as closely as possible. This paper is divided in five sections. In Section 2 we give the notation and definitions necessary to explain our work. In Section 3 we present our results and in Section 4 we characterize all LSE values that satisfy the inessential game property. Section 5 has our conclusions. 2. Preliminaries We recall briefly some concepts of linear algebra. Given a real vector space V its neutral element will be denoted by 0V and its dimension by dim(V ). If the vector space V is the direct sum of the 2 1 2 1 2 vector spaces V 1 and V 2, that is, V = V + V and V ∩ V = 0V , we will write V = V 1 V . If g : V → W is a linear transformation from the vector space V to W , then we will denote by null(g ) its null space and by g (V ) its range. Let X be a subset of V , span(X ) will denote the smallest vector subspace containing X . Let N be a fixed finite set, with |N | = n, a set of n players. A cooperative game with transferable utility or simply a TU game on N is a function v : 2N → R such that v(∅) = 0. The set of TU games v on N, denoted by GN , forms a linear space such that dim(GN ) = 2n − 1. For each coalition S ⊆ N, v(S ) describes the worth of the coalition S when its members cooperate. For any two TU games v and w in GN and any α ∈ R, the TU game αv + w is defined as follows: for each S ⊆ N, (αv + w)(S ) = αv(S ) + w(S ). Given a payoff vector z ∈ Rn and S ⊆ N we denote z (S ) by z (S ) = i∈S zi . Shapley [15] proposes an algebraic basis for GN composed by the collection of unanimity TU games {uS }S ∈2N \∅ , where for all T ⊆ N uS (T ) =



1 0

if S ⊆ T , otherwise.

A TU game v ∈ GN is called inessential if for each ∅ ̸= S ⊆ N, v(S ) = i∈S v({i}). The set of inessential TU games, denoted by A, is a subspace of GN with dim(A) = n. An algebraic basis for A is the collection of unanimity TU games {u{i} }i∈N . By a value in GN we mean a function Φ : GN → Rn . Let v ∈ GN be a TU game, and let i ∈ N be a player, then Φi (v) is called the payoff of player i in v according to Φ . We consider the following properties for a value Φ in GN :

• Efficiency: i∈N Φi (v) = v(N ), for all v ∈ GN . • Linearity: Φ (αv + w) = α Φ (v) + Φ (w), for all α ∈ R and all pairs of TU games v, w ∈ GN . • Symmetry: Φ (θ ∗ v) = θ · Φ (v), for all permutations θ of N. Recall that the TU game θ ∗ v is defined as S → v(θ −1 S ) and θ · (y1 , . . . , yn ) = (yθ(1) , . . . , yθ(n) ), for all vectors (y1 , . . . , yn ) ∈ Rn . 

Ruiz et al. [14] provide a characterization of all linear, symmetric and efficient values (LSE). This family of values was rediscovered later by Hernández-Lamoneda et al. [11]. In the latter reference a 1 − 1 correspondence is established between (n − 1)— tuples b = (b1 , . . . , bn−1 ) ∈ Rn−1 and the LSE values for n players. The correspondence is also obtained in Driessen and Radzik [7].

619

The correspondence is given by b → ϕ b , where

v(N )

ϕib (v) =

n

+



b|S |

 n−1  v(S ) −

S ∋i:S ̸=N

 b|S | n−1 v(S ). S ̸∋i

|S |−1

(1)

|S |

Expression (1) has the following interpretation: first, v(N ) is divided in an equal way among the players, that is, each player v(N ) receives n . Then, we need to make compensations between agents according to their different participation in the game. This compensation is made via transfers among the players such that the amount v(N ) is preserved. These transfers are made following a simple rule: for each coalition S ̸= N, the players in S receive from the players in N \ S, an equal division of

nb|S | v(S )

(|nS |)

. Similarly,

each player in N \ S, pays out an equal division of the previous amount. Thus, the final payment that each agent receives is given by (1). Example 1. We consider a TU game v ∈ G{1,2,3} and a LSE value ϕ b , with b = (1, 1), as follows: 1

ϕ1b (v) = v({1}) − v({2, 3}) + [v({1, 2}) + v({1, 3})] 2

−

1 2

1

[v({2}) + v({3})] + v({1, 2, 3}), 3

1

ϕ (v) = v({2}) − v({1, 3}) + [v({1, 2}) + v({2, 3})] b 2

2

−

1 2

1

[v({1}) + v({3})] + v({1, 2, 3}), 3

1

ϕ (v) = v({3}) − v({1, 2}) + [v({1, 3}) + v({2, 3})] b 3

2

−

1 2

1

[v({1}) + v({2})] + v({1, 2, 3}). 3

If we calculate the value for the following TU games v({1}) = v({2}) = 0, v({3}) = 2, v({1, 2}) = v({1, 3}) = v({2, 3}) = v({1, 2, 3}) = 3, and if 3 ∈ S and S ̸= {1, 2, 3}, if S = {1, 2, 3}, otherwise,

1



w(S ) = 3 0

we obtain that ϕ b (v) = ϕ b (w) = (0, 0, 3). The above example shows two TU games such that for these TU games the value ϕ b equals (0, 0, 3). In the next section we aim to find all games v ∈ GN such that ϕ b (v) = x, where x is a vector of payments previously given and bl ̸= 0 for all l = 1, . . . , n − 1. We will say that ϕ b is a regular value if bl ̸= 0 for all l = 1, . . . , n − 1. In addition, we will write LSER value instead of linear, symmetric, efficient and regular value. 3. The null space of LSER value In this section, we find an algebraic basis of the null space of each LSER value ϕ b . To do this, for each b ∈ Rn−1 , with bl ̸= 0 for all l = 1, . . . , n − 1, we define the following set of games W b , W∗b ⊆ GN as follows: W b = {wSb : S ⊆ N , 2 ≤ |S |}, with

w (T ) = b N



1 0

if |T | = 1, otherwise,

(2)

and for all S ⊆ N such that 2 ≤ |S | ≤ n − 1,

 1   b n−2 1 b wS (T ) =  b|S | |S | − 1  0

if |T | = 1 and T ̸⊆ S , if T = S , otherwise.

(3)

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J.R. Rojas, F.S. Sanchez / Operations Research Letters 44 (2016) 618–621

And W∗b = {w{bi} : i ∈ N } ⊆ GN , where for all i ∈ N

 1  

if i ∈ T and T ̸= N ,

w (T ) = b|T |  n 0 b {i}

Proposition 4. For each LSER value ϕ b : GN → Rn , it holds that W b is an algebraic basis for null(ϕ b ). (4)

if T = N , otherwise.

Now, we are ready to state the main theorem: Theorem 1. Let ϕ b : GN → Rn be a LSER value and x ∈ Rn be a payoff vector. Then, ϕ b (v) = x if and only if

v=

 xi n

i∈N

w{bi} +



aS wSb ,

S ⊆N |S |≥2

In the proof of this theorem we need some results. Given x ∈ Rn we define the TU game xt , for each t = 1, . . . , n, as follows

 x (S ) =

xi

i∈S

0

+

t −1

 zk bt

=

n n−1

bt zi +





b t t −2

t −1

−

zk bt

k∈N

bt

n−2 

n−1 − n−t −11

zk

k̸=i

n−1

k̸=i

n

+

t −1

t

1

(n − |S | − 1)b1 b1 |S |b|S | − × = 0. n−1 b| S | n−1

This implies that wSb ∈ null(ϕ b ). As wSb was chosen arbitrarily, then W b ⊆ null(ϕ b ). By Proposition 3 the dim(null(ϕ b )) = 2n − n − 1. Combining the previous equality with dim(span(W b )) = 2n − n − 1 and that W b ⊆ null(ϕ b ), we have null(ϕ b ) = span(W b ).  N n Lemma 5. Let Y ⊆ GN such that dim(Y ) = n and let Φ : G → R be a linear value in GN . Then, GN = Y null(Φ ) if and only if Φ (Y ) = Rn .

|S |

 n−2



n−1



S ̸∋i

|S |−1



t −1

n−1

b|S |

b S

|S |−1

 b| S |  b|S |  n−1  z t (S ) − n−1 z t (S )

= n−1 ×

=

b i

ϕib (wSb ) = b1 −

bt z i

 n −2 

otherwise,

S ∋i

= bt zi +

then δN = 0. It follows that W b is linearly independent, hence, dim(span(W b )) = 2n − n − 1. Now, we prove that span(W b ) ⊆ null(ϕ b ). To do this, let wSb ∈ W b and i ∈ N. If i ∈ S, then

If i ̸∈ S, then

Proof. Let z ∈ Rn with z (N ) = 0, i ∈ N and t ∈ {1, . . . , n − 1}, n

S ⊆N \{j} 2≤|S |≤n−1

if |S | = t ,

Proposition 2. Let ϕ b : GN → Rn be a LSE value. If z ∈ Rn is such n that z (N ) = 0, then ϕ b (z t ) = n− b z for all t = 1, . . . , n − 1. 1 t z t (N )

S ⊆N :|S |≥2

b1 |S |−1 (n − |S |)b1 = 0. − ϕ (w ) =  n−1  n−1 b| S |

for each S ⊆ N.

ϕib (z t ) =



S ⊆N :|S |≥2

(5)

where the aS are arbitrary constants.

t

δS wSb = 0GN and let T ⊆ N such  b that 2 ≤ |T | ≤ n − 1. If we evaluate S ⊆N :|S |≥2 δS wS in T , then  we obtain that S ⊆N :|S |≥2 δS wSb (T ) = δT . Combining the previous  equality and that S ⊆N :|S |≥2 δS wSb evaluated in any coalition is zero we have δS = 0 for all S ⊆ N with 2 ≤ |S | ≤ n − 1. Given that     b 0= δS wS ({j}) = δS + δN , with j ∈ N , Proof. Suppose that

t

Proof. We assume that GN = Y null(Φ ). Given that dim(Y ) = n, then dim(null(Φ )) = 2n − n − 1. Hence, Φ is surjective. This implies that Φ (Y ) = Rn . Reciprocally, if Φ (Y ) = Rn , then Φ is surjective. Therefore, dim(null(Φ )) = 2n − n − 1. As dim(Y ) = n, then Φ is an isomorphism if it is restricted to vector space Y . Let v ∈ Y ∩ null(Φ ), then Φ (v) = 0Rn . Given that Φ is an isomorphism if it is restricted to Y and v ∈ Y we have v = 0GN . This implies that Y ∩ null(Φ ) = {0GN }. n The previous equality and that dim(null(Φ )) + dim(Y ) = 2 − 1 N implies that G = Y null(Φ ). 





n−1

−1 n−1

bt zi .

As i ∈ N and t ∈ {1, . . . , n − 1} are arbitrary, then ϕ b (z t ) = nb−t n1 z for all t = 1, . . . , n − 1.  In the next proposition we characterize the LSE value, ϕ b , whose null space has dimension 2n − n − 1.

Proposition 6. For value ϕ b : GN → Rn , it holds that  each LSER N b b G = span(W∗ ) span(W ) and



n 0

if i = j, otherwise.

Proposition 3. Let ϕ b : GN → Rn be a LSE value such that b ̸= 0Rn . Then dim(null(ϕ b )) = 2n − n − 1.

ϕib (w{bj} ) =

Proof. Suppose that ϕ b : GN → Rn is such that b ̸= 0Rn . Let x ∈ Rn and l ∈ {1, . . . , n − 1} such that bl ̸= 0. We consider the game n x(N ) x(N ) n −1 l xˆ = n t =1 tct + λz , where λ = nbl , zi = xi − n and ct is defined as ct (S ) = 1 if |S | = t and ct (S ) = 0 if |S | ̸= t. Given that the games ct are symmetric, then by Proposition 2 and that ϕ b is linear, symmetric and efficient we have

Proof. Let i, j ∈ N. If i = j, then

ϕ b (ˆx) =

x(N ) n

ι+

nλ n−1

b {i}

where ι = (1, . . . , 1) ∈ R . Hence, x ∈ R is arbitrary, then ϕ is surjective and therefore, dim(null(ϕ b )) = 2n − n − 1.  n

1

n 

|S |−1

l=1

 n −1  =

1

n−1

If i ̸= j, then

 S ∋i S ∋j

b

In the next proposition we show a generator set of the null space of each LSER value ϕ b .

 S ∋i:S ̸=N

ϕib (w{bj} ) =

bl z = x, n

ϕ (w ) = 1 + b i

(6)

=

 S ∋i S ∋j

= 0.

1

 n −1  − |S |−1

1

 n −1  − |S |−1

1



n−1 S ⊆N \{i} S ∋j

 S ∋i S ∋j

|S |

1

 n−1  |S |−1

l−1





n−1 l−1

= n.

J.R. Rojas, F.S. Sanchez / Operations Research Letters 44 (2016) 618–621

It is clear that the set of games W∗b is linearly independent, and therefore, dim(span(W∗b )) = n. Let x ∈ Rn and we define v x ∈ GN  xi b b as v x = i∈N n w{i} . From the linearity of ϕ and (6) we obtain

that ϕ b (v x ) = x. This equality plus that x ∈ Rn was chosen arbitrarily implies that ϕ b (W∗b ) = Rn . By Lemma 5 we obtain that  GN = span(W∗b ) null(ϕ b ). By Proposition 3 we have null(ϕ b ) = b span(W ). The above equality implies that G = span(W∗ ) N

b



span(W ). 

621

1 Proof. We suppose that ϕ b : GN → Rn is such that l=1 bl = n− . n  n Let w ∈ A and x ∈ R such that w = i∈N xi u{i} . It is easy to verify

n−1

x(N )

t that w = xˆ , where xˆ = n t =1 tct + t =1 z with z and ct as in the proof of Proposition 3. Given that the games ct are symmetric, then by Proposition 2 and that ϕ b is a LSE value we obtain that

ϕ b (w) = ϕ b (ˆx) =

x(N )

b

n

ι+

 n −1

n

n

n −1 

n − 1 l =1

bl z ,

where ι = (1, . . . , 1) ∈ Rn . From (8) we obtain the results.

Proof of Theorem 1. We consider v ∈ G . By Proposition 6 there exist v1 ∈ span(W∗b ) and v2 ∈ span(W b ) = null(ϕ b ) such that v is uniquely expressed as v = v1 +v2 . From the linearity of ϕ b we have ϕ b (v) = ϕ b (v1 ) + ϕ b (v2 ) = ϕ b (v1 ). Given that, there exist unique b constants a{i} such that v1 = i∈N a{i} w{i} , then by Proposition 6 we conclude that

(8) 

N

ϕ b (v) =



a{i} ϕ b (w{bi} ) = (na{1} , . . . , na{n} ).

(7)

i∈N

From (7) it follows ϕ b (v) = x if and only if v =



S ⊆N :|S |≥2



aS wSb , where the aS are arbitrary constants.

xi i∈N n

w{bi} +



Example 2. To illustrate the results, we return to Example 1 to solve the inverse problem of value ϕ b with b = (1, 1). We consider the payoff vector x = (0, 0, 3). We want to find all games for which the value ϕ b gives x as a result. Computing the algebraic basis of the null space formed by the games wSb ∈ W b , with S ⊆ N, 2 ≤ |S | ≤ n, and using formula (5), we obtain the solution of the inverse problem

v({1}) = a{2,3} + a{1,2,3} , v({2}) = a{1,3} + a{1,2,3} , v({3}) = 1 + a{1,2} + a{1,2,3} , v({1, 2}) = a{1,2} , v({1, 3}) = 1 + a{1,3} , v({2, 3}) = 1 + a{2,3} , and v({1, 2, 3}) = 3; the set depends on four parameters, and in general on 2n − n − 1 parameters. It is easy to verify that the games from Example 1 are obtained for the following value of the parameters:

• a{1,2} = 3, a{2,3} = a{1,3} = 2 and a{1,2,3} = −2, • a{1,2} = a{2,3} = a{1,3} = a{1,2,3} = 0. 4. Inessential game property N  Now, assume that we have a game v ∈ G such that v(S ) = v({ i }) for all S ⊆ N, that is, the worth of any coalition is equal i∈S

to the sum that each member of the coalition can get by himself. It is reasonable to assume that the payment of any player in the game must be precisely what he can get by himself. This property is known in the literature as inessential game property. In this section we characterize all LSE values, ϕ b , that satisfy this property. Inessential game property. Φ (v) = (v({1}), . . . , v({n})), for all v ∈ A. Proposition 7. Let ϕ b : GN → Rn be a LSE value. ϕ b satisfies the n−1 n −1 inessential game property if and only if l=1 bl = n .

5. Conclusions In this paper we provide an algebraic basis of the null space of the LSE value ϕ b , for all b ∈ Rn−1 with bl ̸= 0 for all l = 1, . . . , n−1. This allows us to decompose the space of games, GN , as a direct sum of the null space of ϕ b and its respective complement. Using this decomposition we obtain the solution of the inverse problem of ϕ b . Acknowledgments The authors want to thank our anonymous referees for their helpful comments. Also, we want to thank Luis Hernández Lamoneda for helpful comments. Financial support was received by CONACyT grant 167924. References [1] S. Béal, E. Rémila, P. Solal, Veto players, the kernel of the Shapley value and its characterization, working paper, Université de Franche-Comté, Besancon, France, 2014. [2] S. Béal, E. Rémila, P. Solal, Characterization of the average tree solution and its kernel, J. Math. Econom. 60 (2015) 159–165. [3] S. Béal, E. Rémila, P. Solal, A decomposition of the space of TU- games using addition and transfer invariance, Discrete Appl. Math. 184 (2015) 1–13. [4] S. Béal, E. Rémila, P. Solal, Decomposition of the space of TU- games, strong transfer invariance and the Banzhaf value, Oper. Res. Lett. 43 (2015) 123–125. [5] I.C. Dragan, The potential basis and the weighted Shapley value, Libertas Math. 11 (1991) 139–150. [6] I.C. Dragan, On the inverse problem for semivalues of cooperative t.u. games, Int. J. Pure Appl. Math. 22 (2005) 545–561. [7] T.S.H. Driessen, T. Radzik, A weighted pseudo-potential approach to values for TU-games, Int. Trans. Oper. Res. 9 (2002) 303–320. [8] S. Hart, A. Mas-Colell, The potential basis of the shapley value, in: The Shapley Value, Essays in Honor of Lloyd S. Shapley, Cambridge Univ. Press, Cambridge, 1988, pp. 139–150. [9] S. Hart, A. Mas-Colell, Potential, value, and consistency, Econometrica 57 (1989) 589–614. [10] P.J.-J. Herings, G. van der Laan, D. Talman, The average tree solution for cycle free games, Games Econom. Behav. 62 (2008) 77–92. [11] L. Hernández-Lamoneda, R. Juarez, F. Sánchez-Sánchez, Dissection of solutions in cooperative game theory using representation techniques, Internat. J. Game Theory 35 (2007) 395–426. [12] S. Kurz, The inverse problem for power distributions in committees, Soc. Choice Welf. 45 (2016) 65–88. [13] S. Kurz, S. Napel, Heuristic and exact solutions to the inverse power index problem for small voting bodies, Ann. Oper. Res. 215 (2014) 137–163. [14] L.M. Ruiz, F. Valenciano, J.M. Zarzuelo, The family of least square values for transferable utility games, Games Econom. Behav. 24 (1998) 109–130. [15] L.S. Shapley, A value for n-person games, in: H.W. Kuhn, A.W. Tucker (Eds.), Contributions to the Theory of Games, Vol. II, Princeton University Press, Princeton, 1953, pp. 307–317. [16] K. Yokote, Weak addition invariance and axiomatization of the weighted Shapley value, Internat. J. Game Theory 44 (2015) 275–293. [17] K. Yokote, Y. Funaki, Y. Kamijo, A new basis and the Shapley value, Math. Social Sci. 80 (2016) 21–24.