Potential, Value, and Coalition Formation

Potential, Value, and Coalition Formation Annick Laruelley

Federico Valencianoz

Universite de Caen Basse Normandie

Universidad del Pa s Vasco

September 4, 2012

Abstract In this paper a simple probabilistic model of coalition formation provides a uni ed interpretation for several extensions of the Shapley value. Weighted Shapley values, semivalues, and weak (weighted or not) semivalues, and the Shapley value itself appear as variations of this model. Moreover, some notions that have been introduced in the search of alternatives to Shapley's seminal characterization, as 'balanced contributions' and the 'potential' are reinterpreted from this point of view. The natural relationships of these conditions with some the mentioned families of 'values' are shown. These reinterpretations strongly suggest that these conditions are more naturally interpreted in terms of coalition formation than in terms of the classical notion of 'value'. Keywords: Coalitional games, value, potential, probabilistic models. JEL classi cation number: C71, D84

We want to thank Emilio Calvo, Sergiu Hart and Andreu Mas-Colell for their comments. This research has been supported by the Spanish Ministerio de Ciencia y Tecnolog a under project BEC2000-0875, by the Universidad del Pa s Vasco under project UPV/EHU00031.321-HA-7918/2000, by the Generalitat Valenciana (grupo 3086), and by the IVIE. The rst author also acknowledges nancial support from the Spanish Ministerio de Ciencia y Tecnolog a under the Ramon y Cajal Program. Part of this paper was written while the second author was visiting the Departamento de Fundamentos del Analisis Economico in the Universidad de Alicante, whose hospitality is gratefully acknowledged. y z

Faculte des Sciences economiques et de Gestion, Esplanade de la Paix, F-14000 Caen, France (Corresponding author) Departamento de Econom a Aplicada IV, Universidad del Pa s Vasco, Avenida

Lehendakari Aguirre, 83, E-48015 Bilbao, Spain. (Phone: 34-946013696; Fax: 34-946017028; e-mail: [email protected])

1

Introduction

From Shapley's (1953b) seminal paper a copious family of 'solutions' for transferable utility (TU) games has grown in di erent directions with many rami cations. Among others, weighted Shapley values (Shapley (1953a), Kalai and Samet (1987, 1988), Weber (1988)), probabilistic values (Weber, 1979, 1988), semivalues (Weber (1979, 1988), Dubey, Neyman and Weber (1981), Einy (1987)), weak semivalues and weighted weak semivalues (Calvo and Santos, 2000), as well as nontransferable utility (NTU) and non atomic extensions of some of these notions, or their restriction to special subdomains as, e.g., simple games. Most of these extensions have been born out of axiomatic explorations: dropping axioms, nding weaker or more appealing ones, recombining already existing ones, etc. This leads to heterogeneous families of objects, of which the Shapley value is an omnipresent particular case. One aim of this paper is to provide a simple model encompassing some of these notions. Since the very seminal paper, the probabilistic interpretation in terms of expected marginal contributions has been the main source of models for the di erent extensions of the Shapley value (see, e.g., Owen (1975, 1982), Stra n's (1977, 1982, 1988), Weber (1979, 1988), Dubey, Neyman and Weber (1981)). As is well-known, the Shapley value and some of its extensions admit a more or less clear interpretation in probabilistic terms. From this point of view, Weber's (1979, 1988) very general concept of 'probabilistic value', or, better, of 'group value', covers a wide range of solutions, including the notions formerly alluded with which we deal in this paper. A group value is a vector of probabilistic values, that is, a vector of expected marginal contributions of every player to the worth of a coalition based on n di erent points of view. In other words, each player has his/her own prior or assessment of the probability of joining di erent coalitions. Nevertheless the possibility of accommodating di erent extensions into this very general notion is achieved at the cost of a quite di erent probabilistic story for every case: the formation of the grand coalition in a certain order, all orders being equally probable or not, or the probabilistic assessments of the di erent players of the coalition they will join satisfying di erent 'consistency' requirements. The same can be said about di erent notions, as for instance the potential (Hart and Mas-Colell, 1989), for which it is possible to give an ad hoc probabilistic interpretation. In this paper we focus on the probabilistic point of view and propose a more speci c probabilistic model to account for several families of 'solutions': The expected marginal contribution of every player to the coalition that will form (conditional to that player belonging to it) for a given probability distribution over coalitions. With respect to previous models a novelty is that it is assumed that only one coalition will form, and the players' expectation are based on a common prior. This model generates the whole family 1

of weighted weak semivalues (and nothing else). While di erent speci cations over the probability distribution yield the other mentioned families, weak semivalues, semivalues and weighted Shapley values, supported in the literature by di erent axiomatic combinations. That is, in incomplete information terms, all these values turn out to be 'interim' expectations of the di erent players based on a common prior about the coalition that will form. Moreover, some notions that have been introduced in the search of alternatives to Shapley's seminal characterization, as 'balanced contributions' (Myerson, 1980) and the 'potential' (Hart and Mas-Colell, 1989), signi cantly related to these families, are reinterpreted from in terms of this model, and their meaning and relationships with these values become straightforward. The rest of the paper is organized as follows. In section 2 the probabilistic model is introduced, and in section 3, the ex ante and interim expected marginal contributions are formulated. In section 4 it is shown how the Shapley value and its above mentioned extensions t into the proposed probabilistic model. In section 5 'balanced contributions' (Myerson, 1980), and the 'potential' (Hart and Mas-Colell, 1989) are reinterpreted and discussed in terms of our probabilistic model. Finally, section 6 summarizes the main conclusions.

2

A common prior on the coalition formation

An N -coalitional game is a pair (N; v), where N = f1; :::; ng is a set of players and v is a map that assigns to each subset or coalition S

N a real number or worth v(S), such

that v(;) = 0. In the usual interpretation the worth of a coalition represents what this coalition can guarantee for itself if it forms. This 'worth' can be the amount of something objectively measurable as money, or some other physical entity. A second possibility is interpreting it as 'utility'. In the rst interpretation nothing about the players or their preferences is put into the model beyond the implicit assumption that for all of them 'the more the better'. In the second, these preferences are included at the cost of assuming a rather improbable con guration of players' preferences, equivalent to assuming that there exists some in nitely divisible good in which the players' utility is linear. GN will denote the set of all coalitional games with N as the set of players. When this set is clear from the context we will refer by v to game (N; v)... We will drop i's brackets in Snfig and S [ fig.

In the standard interpretation of a coalitional game (N; v), this pair is all what enters

the model. Here we see a coalitional game as only one of the ingredients in a very simple coalition formation model. In accordance with the usual interpretation v(S) is seen as what coalition S would get if it forms. The di erence is this: we assume that only a 2

coalition will form. Thus a second ingredient enters the model: a probability distribution over all coalitions that is exogenous and independent of the game. Let us denote p the distribution of probability that associates with each coalition S its probability of forming p(S). Any such a probability distribution can be represented by a P map p : 2N ! R, where 0 p(S) 1 for any S N; and p(S) = 1: We assume that S N

the probability of any player belonging to the resulting coalition is strictly positive, that is, for all i 2 N Prob (the coalition that forms contains i) = Prob (S 3 i) =

X

p(T ) > 0:

T :i2T

This rules out the case of players who will never belong to the resulting coalition1 . Let PN denote the set of all such probability distributions. A few remarks are worth here. First note the possibility of no coalition forming (if p(;) > 0) exists, but the trivial case p(;) = 1 is excluded. Second, the events 'i belongs to the resulting coalition' and 'j belongs to the resulting coalition' are not necessarily independent in general, this is only a particular case. Third, this probability distribution can be interpreted as an objective or external probabilistic assessment, possibly based on whatever available data. Alternatively, it can be interpreted in incomplete information terms as a common prior about the players' types, where each player is of one of two possible types: either a 'joiner' or a 'standing alone'. A context in which this basic model makes a clear sense is provided by simple games interpreted as voting rules (see Laruelle and Valenciano, 2005). In section 5, given a pair (v; p) 2 GN

a player i, thus yielding a new pair

PN , we will consider eliminating or "ignoring"

(v N ni ; pN ni )

2 GN ni

PN ni . Namely, v N ni denotes

the game de ned by v N ni (S) := v jN ni (S) = v(S) for any S

N ni. Similarly, if p

describes N 's behavior and i is ignored, so that only the information concerning N ni matters, coalitions S and S [ i are indistinguishable. Thus, we de ne pN ni 2 PN ni by2 pN ni (S) := p(S) + p(S [ i) It is easy to check from (1) that for all j 2 N n i; X X Probp (S 3 j) = p(T ) = T :j2T N

for all S

T :j2T N ni

N ni:

pN ni (T ) = ProbpN ni (S 3 j) :

(1)

(2)

That is to say, for any player in N n i the probability of belonging to the coalition that will form in N according to p, and in N n i according to pN ni , are the same. 1

Such cases can be dealt with by eliminating this player and reformulating appropriately the coalitional

game and the probability distribution for the remaining n 1 players. 2 Observe this condition yields as a particular case the well-known condition of consistency for semivalues n+1 (Dubey, Neyman and Weber, 1981) pn + pn+1 s = ps s+1 :

3

3

Ex ante and interim marginal contributions

Given (v; p) 2 GN

PN , the ex ante expected marginal contribution of any player i to

the coalition that will form is given by Ep [v(S)

v(Sni)] =

X

p(S)(v(S)

v(Sni)):

(3)

S:i2S

Similarly, the interim expected marginal contribution of a player i to the coalition that will form, conditional to such player belonging to it, denoted by i (v; p), is given by X Ep [v(S) v(Sni)] p(S) P = (v(S) v(Sni)): i (v; p) := Ep [v(S) v(Sni) j S 3 i] = Prob (S 3 i) p(T ) S:i2S

T :i2T

(4)

Or, denoting by

pi

the probability distribution over coalitions resulting from p condi-

tional to 'i 2 S' (i.e., the coalition that will form contains i), that is, 8 < Pp(S) if i 2 S; p(T ) T :i2T pi (S) := Prob (S j S 3 i) = : 0 otherwise,

(4) can be rewritten as i (v; p)

= Epi [v(S)

v(Sni)] =

X

pi (S)(v(S)

v(Sni)):

(5)

S:i2S

What is the interest of formulae (3) and (4) or (5)? If a coalition is forming it is ingenuous for players to expect receiving their marginal contribution. They should know this will possibly be too much or too little to ask for. Nevertheless, the expected marginal contribution of each player to the coalition which forms seems a reasonable assessment of the relevance of this player. This is exactly what these formulae give either ex ante (unconditionally) or interim (conditionally) based on the two inputs. In the next section we show how some 'values' are particular cases of the equivalent formulae (4) and (5). Note that di erent distributions of probability can lead to the same interim marginal contribution. The reason is that the probability of the empty coalition does not a ect the expected marginal contribution of player i conditional to the coalition that forms contains i, as far as the probability of the nonempty coalitions remains proportional. Therefore one can modify the probability of the empty coalition and re-scale proportionally the probability of the others without modifying the expected conditional marginal contributions. But this is the only degree of freedom. Proposition 1 Let p,p0 2 PN , then

i (v; p)

=

i (v; p

0)

for any player i and any coali-

tional game (N; v); if and only if

p0 (S) p(S) = 1 p(;) 1 p0 (;) 4

for all S 6= ;:

Recall that the trivial case p(;) = 1 is excluded. On the other hand, this indeterminacy would disappear if it is required p(;) = 0... But this would rule out, for instance, the case in which every player independently joins the coalition that will form with a certain probability.

4

Some cooperative 'solutions' revisited

In this section it is shown how this simple probabilistic coalition formation model gives a conceptual common basis to reinterpret from a uni ed point of view some families 'solutions' as particular cases of (4) or its equivalent (5), for particular probability distributions. But previously let us brie y compare our model with the notion of probabilistic value3 . For any i 2 N; a probabilistic value (Weber, 1979, 1988) is a map that associates with

each game (N; v) a real number given by i (v)

=

X

S:i2S

for some piS such that 0

piS

1 and

piS (v(S) P

v(Sni));

(6)

piS = 1: The piS are interpreted as player

S:i2S

i's subjective distribution of probability over the coalitions containing this player, based on his/her own subjective assessment of the probability of joining di erent coalitions4 . Thus a probabilistic value involves and concerns only one player. Weber characterized these real valued maps by the linearity, dummy and positivity axioms. From formula (5) it is immediate to check that for any distribution of probability p 2 PN , and any

player i,

i(

; p) : GN ! R is a 'probabilistic value'. Thus, formally speaking, for any

( ; p) : GN ! RN ; is a vector of probabilistic values. Nevertheless the interpretation

p,

underlying either concept is di erent. A probabilistic value is usually interpreted as a subjective evaluation of a player's marginal contribution from the point of view of that

particular player. The only vectors of probabilistic values, or 'group values' considered by Weber are the semivalues and the random-order values. But our vectors of probabilistic values are not to be interpreted as vectors of subjective evaluations based on n independent points of view. On the contrary, they should be interpreted as assessments of every player's 3

We will not discuss here Stra n's (1977, 1982 and 1988) probabilistic model, extended by Dubey,

Neyman and Weber (1981) to all semivalues. In Laruelle and Valenciano (2004) it is shown how this sophisticated model ts within the one considered here. 4 In fact, Weber's notation was slightly di erent though equivalent. He writes i (v)

=

X

S:i2S =

piS (v(S [ i)

v(S)); with 0

piS

1; and

X

S:i2S =

5

piS = 1:

marginal contribution based on a single (and same for all players) distribution of probability over coalitions. But of course not all vectors of probabilistic values can be interpreted as particularizations of (4)-(5). In fact one point of this paper is to show how some solution concepts which

t rather slackly into the probabilistic notion of group value, do

t tightly and

naturally into the model speci ed in sections 2 and 3.

4.1

The Shapley value

For a game (N; v) the Shapley (1953b) value of player i is given by Shi (v) =

X (n

S:S N

s)!(s n!

1)!

(v(S)

v(Sni)):

(7)

There are several interpretations of the map Sh : GN ! RN , characterized by the well-

known conditions of e ciency, anonymity, null player and additivity. Shapley's original interpretation is that of a 'value' for TU games in the sense of von Neumann and Morgenstern's (1944) notion of value of a two-person zero sum games, and that of Nash's (1950) solution to the bargaining problem. That is to say, the value that would mean for a rational player the prospect of engaging into any situation of the class considered, TU games in this case, with other rational players. An alternative interpretation is as an allocation rule, or a way of distributing the worth of the grand coalition among the n players in any situation that can be described as a coalitional game. A third interpretation is as an expected payo of each player for a speci c random procedure to form the grand coalition sequentially and assigning to each player his/her marginal contribution to the worth of the coalition formed when incorporating. A fourth interpretation is as a utility function '(i; v) = Shi (v) representing von Neumann-Morgenstern preferences over 'roles' or positions (i; v) in coalitional games and lotteries over them5 . In the present coalition formation model the Shapley value corresponds to interim expectations for a rather peculiar prior. Namely, we have the following result whose simple proof is omitted. Proposition 2 such that

(v; p) = Sh(v) for any coalitional game (N; v) if and only if p 2 PN is p(S) = (1

p(;))

1 s n P

t=1 5

1 t

n s

1

;

for any S 6= ;:

This interpretation is the motivating idea in Shapley (1953b), although his treatment does not involve

preferences explicitly. For a more explicit study from this point of view see Roth (1977a), and also Roth (1977b) and Laruelle and Valenciano (2002) in the context of simple games.

6

Thus, the Shapley value of player i gives the expected marginal contribution of this player to the coalition that will form conditional to this player belonging to it if the coalition is chosen as follows. With probability p(;) < 1 the empty coalition is chosen; with probability 1

p(;) proceed as follows: a coalition's size (from 1 to n) is chosen

with probability inversely proportional to the size, then one coalition of size s is chosen at random, all of them being equally probable. Note this probabilistic model is completely di erent from the usual one. It is worth remarking that from the point of view of the current model the probability that yields the Shapley value has no special appeal or naturalness. We will come back to this later.

4.2

Semivalues, weak semivalues and weighted weak semivalues

We deal now with some solution concepts originated by imposing di erent conditions on a vector of probabilistic values relating its components. The family of semivalues (Weber (1979), see also Weber (1988), Dubey, Neyman and Weber (1981), and Einy (1987)) emerges by imposing anonymity on a vector of probawhich assigns to each game (N; v) a vector in Rn ,

bilistic values. A semivalue is a map given by i (v)

X

=

ps (v(S)

v(Sni))

S:i2S

for some ps such that 0

ps

P

1 for s = 1; :::; n; and

(8)

ps = 1.

S:i2S

In Calvo and Santos (2000) two classes of 'solutions' emerge axiomatically by imposing to a vector of probabilistic values either balanced contributions or the weaker w-balanced contributions (both conditions are discussed in section 5). Weak semivalues emerge by imposing the rst condition. A weak semivalue is a map each game (N; v) a vector (

i (v))i2N i (v)

=

in Rn , given by X

pS (v(S)

: GN ! RN which assigns to

v(Sni));

(9)

S:i2S

for some pS such that 0

pS

1; and

P

S:i2S

pS = 1; for every i 2 N:

Weighted weak semivalues emerge when the second condition is imposed. A weighted weak semivalue is a map which assigns to each game (N; v) a vector ( RN ,

given by pN ;wN (v) i

=

X

wi pS (v(S)

pN ;wN (v))i2N i

v(Sni));

in

(10)

S:i2S

where wN = (wi )i2N is an N -weight vector s.t., wi > 0; for all i 2 N , and pN = P (pS )S N;S6=; is a family of coe cients such that pS 0 for all S 6= ;; and wi pS = 1 S:i2S

7

for every i 2 N: No clear interpretation of this family, born axiomatically, nor of the coef-

cients pS and the 'weights', is given in Calvo and Santos (2000). They interpret weighted

weak semivalues just as a subclass of vectors of probabilistic values in which the players' beliefs (in Weber's terms) satisfy a weak form of 'consistency'. Namely, for some system of weights w, they satisfy

piS pjS

=

wi wj ,

for all S

N and all i; j 2 S: While weak semivalues

form the subfamily in which all players have the same weight.

The following result provides a precise probabilistic interpretation of each of these nested families, showing that all of them t into formulae (4)-(5). ( ; p) : GN ! RN ; generated by formula (5) for

Proposition 3 (i) The family of maps

di erent probability distributions p in PN ; is the family of weighted weak semivalues. (ii) The subfamily of maps

( ; p) associated with probability distributions p 2 PN such

that for any two players the probability of entering the coalition which will form is the same, is the family of weak semivalues. (iii) The subfamily of maps

( ; p) associated with probability distributions p 2 PN for

which the probability of a coalition to form depends only on its size, is the family of semivalues. Proof. (i) Confronting formula (10) with formulae (4)-(5), it is clear that for any probability distribution p 2 PN , de ning pN = (pS )S pS := p(S);

and wi :=

N;S6=; ;

and wN = (wi )i2N by

1 1 ; = P p(T ) Prob (S 3 i)

(11)

T :i2T

it holds i (v; p)

pN ;wN (v) i

=

for any player i and any game (N; v). Thus, for all p,

( ; p) is a weighted weak semivalue.

Reciprocally, any weighted weak semivalue can be generated in this way. Let

pN ;wN

be a weighted weak semivalue given by (10), with associated coe cients pS and weights wi . Mind there is a degree of indeterminacy both in the weights and the coe cients: the weights can be all multiplied by any positive constant and the coe cients divided by this P same constant so that the constraint wi pS = 1 is satis ed and the associated weighted S:i2S

weak semivalue remains the same. Thus, de ning p0N = (p0S )S by p0S :=

p PS

pT

and wi0 := wi

X

N;S6=; ,

T :T N;T 6=;

T :T N;T 6=;

8

pT ;

and w0N = (wi0 )i2N ,

we have

pN ;wN

=

p0N ;w0N .

P

Moreover, as

1; taking p 2 PN , given by p(;) := 0; and (v; p) =

p0S = 1 and

S:S N;S6=; p(S) := p0S

p0N ;w0N

(v) =

P

S:i2S

wi0 p0S =

for S 6= ;; we obtain pN ;wN

P

wi pS =

S:i2S

(v);

for any game (N; v). Therefore, any weighted weak semivalue is a particular case of ( ; p): (ii) In view of (i), confronting formulae (4) and (9), it is clear that when the probability of being a member of the coalition that will form is the same for all players, what emerges from the model presented in section 4 is the family of weak semivalues. (iii) Again confronting (4) and (8), and in view of the previous equivalences, the new one is straightforward. Thus, weighted weak semivalues are the vectors of players' interim expected marginal contributions to the coalition that will form (i.e., conditional to the player belonging to it), for arbitrary probability distributions in PN . Therefore, both the coe cients pS and the 'weights' wi in (10) acquire a precise meaning given by (11). That is, after eliminating the indeterminacy factor as in the second part of the proof of (i) in Proposition 3, pS becomes the probability of S (6= ;) being the coalition formed, while the 'weight' wi of player i becomes the inverse of the probability of such player entering the coalition that will

form. In other words, the products wi pS in (10) become conditional probabilities. Thus 'weights' disappear in this model as just an 'optical e ect'. Weak semivalues correspond to probability distributions over coalitions for which the probability of any player entering the coalition which will form is the same. Finally, semivalues correspond to the particular case in which the probability of a coalition forming only depends on its size. Observe also that under the point of view provided by this model the nature of the 'consistency' alluded by Calvo and Santos becomes clear: the players' assessments are based on a single and same probability distribution over coalitions (i.e., a common prior). Thus from this point of view it turns out arti cial to speak of a vector of assessments resulting from n points of view6 .

4.3

Weighted Shapley values

Weighted Shapley values result by eliminating symmetry in Shapley's (1953b) characterizing system. This extension was introduced by Shapley himself in (1953a) associating a 6

In contrast with the usual interpretation, in Laruelle and Valenciano (2003) semivalues, in the context

of simple games and within the axiomatic approach, are interpreted as assessments of the relative capacity to in uence the outcome of a vote attached to di erent roles in di erent decision rules from a single point of view.

9

positive weight wi with each player and distributing payo s in unanimity games proportionally to these weights. In this way, keeping the other conditions (linearity, e ciency and null-player), a unique payo vector for every game is determined7 . Hart and Mas-Colell (1987) characterized weighted Shapley values by means of e ciency and the existence of w-potential (see section 5). Then, in view of Calvo and Santos's (2000) characterization of weighted weak semivalues (linearity, positivity, dummy player and w-balanced contributions), and the equivalence of w-potential and w-balanced contributions (Calvo and Santos, 1997), it follows that weighted Shapley values are weighted weak semivalues, and consequently also t in this probabilistic model. Moreover, weighted Shapley values are the e cient weighted weak semivalues. The point is: which probability distributions generate these values? The following lemma, similar to Theorem 11 in Weber (1988) that we prove for completeness, answers the question in terms of the marginal probabilities in formula (5). Lemma 1 A map ( ; p) : GN ! RN ; generated by formula (5) satis es 'e ciency', (i.e., P i (v; p) = v(N ) for all v), if and only if the probability distribution p 2 PN ; veri es

i2N

the following conditions: (i)

X

pi (N ) = 1

and

X

(ii)

i2N

X

pi (S) =

i2S

pj (S [ j)

j2N nS

(8S

N ):

Proof. (() Assume p 2 PN veri es conditions (i) and (ii). Then for any game (N; v); X

i (v; p)

i2N

=

= v(N )

X

pi (N ) +

i2N

X X

pi (S)(v(S)

v(Sni))

i2N S:i2S

S N

X

=

0

v(S) @ X

S N

X

pi (S)

i2S

0

v(S) @

X

j2N nS

X

pi (S)

i2S

1

pj (S [ j)A

X

j2N nS

1

pj (S [ j)A = v(N )

Where the last equality follows form (i) and (ii). P S ()) Now assume i (v; p) = v(N ); for any game (N; v). Then let (N; u ) denote for i2N

any S 7

N , the unanimity game such that uS (T ) = 1 if T

S and uS (T ) = 0 otherwise.

Kalai and Samet (1987) extended the notion to 'weight systems' enabling a weight zero for some players,

and characterized the resulting family of 'random order' values, which in Weber (1988) are characterized as the unique vectors of probabilistic values that satisfy e ciency. As we will see only the weighted Shapley values within this family t in (4)-(5).

10

Then

P

i (u

N ; p)

=

i2N

P

pi (N ); and (i) follows: And for any S

N , let u ^S the game such

i2N

that u ^S (T ) = 1 if T ! S and u ^S (T ) = 0 otherwise. Then we have ( pi (S) if i 2 S; S S (u ; p) (^ u ; p) = i i pi (S [ i) if i 2 N n S. Thus, from (5) and the e ciency of ( ; p), X X X S uS ; p) = pi (S) i (u ; p) i (^ i2N

i2N

i2S

X

j2N nS

pj (S [ j) = 0:

Hence (ii) follows too. The following result, an easy corollary of Lemma 1, shows the connection between the 'weights' and the associated probability distribution for weighted Shapley values. Proposition 4 A map

( ; p) : GN ! RN ; generated by formula (4) satis es 'e ciency'

n, and coincides with the w-weighted Shapley value for a vector of positive weights w 2 R+

if and only if the probability distribution p 2 PN ; veri es the following conditions: P p(S) wi S:T S P = P (8T N; 8i 2 T ): p(R) wj R:i2R

As one can take wi =

1 Prob(S3i) ,

j2T

it is possible to eliminate wi in the equations of

the above system. Thus, Proposition 4 yields an alternative characterization of weighted Shapley values. Corollary 1 A map

( ; p) : GN ! RN ; generated by formula (5) satis es 'e ciency' if

and only if the probability distribution p 2 PN ; veri es the following conditions: X 1 1 = (8T N ): Prob (S T ) Prob (S 3 i) i2T

As it happened with the Shapley value, note the lack of a clear interpretation of the conditions on the probability distributions established in Lemma 1, in Proposition 4, or in its corollary. This only shows how unnatural is requiring or expecting 'e ciency' (i.e., interim expectations adding up to v(N )) in the setting provided by this simple coalition formation model. Only for very special (not to say ad hoc) probability distributions over coalitions will this be so. In Weber (1988), referring to the random order values, it is stated that "a collection of individual probabilistic values is e cient for all games in its domain precisely when the players' probabilistic views of the world are consistent; that is, only when the various.. [individual probabilistic views] arise from a single distribution [over the n! possible orderings of the players]" (italics and brackets are ours). But it should be noted that this type of consistency is to be found as we have shown in all the 'values' given by

( ; p) for arbitrary p's with and without e ciency. 11

5

Balanced contributions and the potential reexamined

In this section we examine and reinterpret the 'balanced contributions' condition (Myerson, 1980), and the notion and existence of 'potential' (Hart and Mas-Colell, 1989) from the point of view provided by this coalition formation framework.

5.1

Balanced contributions

In Myerson (1980) the Shapley value is characterized by means of e ciency and Balanced Contributions (BC): For any (N; v) 2 GN ; and all i; j 2 N; i (v)

i (v

N nj

)=

j (v)

j (v

N ni

):

(12)

The usual interpretation is that for any two players the bene t of each of them from the participation of the other is the same. Calvo and Santos (2000) show that the weak semivalues are the group values that satisfy this condition, while the weighted weak semivalues are those that satisfy a weaker version of this property in which every player's bene ts are weighted according to a 'weight system' w = (wi )i2N . Namely: w-Balanced Contributions (w-BC): For any game (N; v) and all i; j 2 N; 1 ( wi

i (v)

i (v

N nj

)) =

1 ( wj

j (v)

j (v

N ni

)):

(13)

In our framework, the BC condition can be reformulated as follows: For all i; j i (v; p)

i (v

N nj

; pN nj ) =

j (v; p)

j (v

But this condition does not hold for all p's, but only when

N ni

; pN ni ):

P

(14)

p(T ) =

T :i2T

P

p(T ) for

T :j2T

any two players as we will presently show. The condition that holds for any p is this: the mutual contributions of any two players to each other's expected marginal contribution are the same, as the following Proposition shows. Proposition 5 For all (N; v) 2 GN , all p 2 PN , and all i; j 2 N; it holds Ep [v(S)

v(Sni)]

EpN nj [v N nj (S)

v N nj (Sni)]

= Ep [v(S)

v(Snj)]

EpN ni [v N ni (S)

v N ni (Snj)]:

(15)

Proof. Taking into account (1), we have

=

Ep [v(S) X

S:i2S N

v(Sni)] p(S)(v(S)

EpN nj [v N nj (S) v N nj (Sni)] X v(Sni)) pN nj (S)(v N nj (S) S:i2S N nj

12

v N nj (Sni))

=

X

p(S)(v(S)

v(Sni))

S N

=

X

X

S N nj

p(S) (v(S)

v(Sni)

(p(S) + p(S [ j)) (v(S)

v(Sni))

v(Snj) + v(Sn fi; jg):

S N

Then (15) follows from the symmetry of the last expression with respect to i and j. Note that (15) can be equivalently rewritten as (

X

N nj

p(T )) (

i (v; p)

i (v

p(T )) (

j (v; p)

j (v

; pN nj ))

(16)

T :i2T

= (

X

N ni

; pN ni )):

T :j2T P1

Thus calling again wi :=

p(T ) ;

as in (11) in subsection 4.2, makes clear that (15)-

T :i2T

(16) is the probabilistic counterpart in our framework of the w-balanced contributions condition (13). Again, in exact correspondence with what happened with 'weighted' weak semivalues, 'weights' disappear as a 'misunderstanding'. It can be then concluded that in our setting condition (15), the counterpart of 'weighted' balanced contributions in the usual setting , is the really general condition satis ed by all evaluations

( ; p), while from

(16) is clear that the counterpart of balanced contributions (14) holds only in especial cases. In sum we have: Proposition 6 The evaluation given by the map (i) satis es condition (15)-(16) for any p 2 PN :

( ; p) : GN ! RN

(ii) satis es (14), if and only for any two players the probability of belonging to the coalition that will form is the same.

5.2

The potential

In Hart and Mas-Colell (1989) the Shapley value is characterized as the unique e cient solution that admits a 'potential'. A solution

: GN ! RN admits a potential if there

exists a map P : GN ! R; called then the 'potential' of i (v)

= P (v)

P (v N ni )

, such that

(for all i 2 N ):

(17)

They show that the Shapley value's potential is given by8 P (v) =

X (s

S N

1)!(n n!

s)!

v(S) = Ep [

jN j v(S)]; jSj

(18)

where p is a 'standard' though somewhat ad hoc probability distribution over coalitions. 8

They require P (;; v) = 0.

13

Calvo and Santos (2000) prove that semivalues and weak semivalues also admit a potential. But this property needs to be relaxed to cover other solutions, as is the case of the weighted Shapley values, that for a vector of weights w admit a 'w-potential' (Hart and Mas-Colell, 1989). In fact Hart and Mas-Colell's w-potential is a combination of e ciency and something else. The amalgamation of these two conditions allows them to characterize the weighted Shapley values with it. Calvo and Santos (1997) disentangle these two : GN ! R N

conditions and formulate the condition in the following terms. A solution

admits a w-potential if there exists a map P : GN ! R; and a vector of positive weights w; such that

i (v)

= wi (P (v)

P (v N ni ))

(for all i 2 N ):

(19)

The same authors in Calvo and Santos (2000) prove that weighted weak semivalues are the only vectors of probabilistic values that admit a w-potential. For a precise interpretation of the former notions and results in our setting, we introduce a general concept of 'potential' with an obvious meaning in our model (so much that it makes super uous that term). For any (N; v) 2 GN and any p 2 PN ; let P(v; p) := Ep [v(S)];

(20)

that is to say, P(v; p) is the expected worth of the coalition that will form, given the game (N; v) and the probability distribution over coalitions p. Then, the consistency relation (1) to restrict p 2 PN to N ni, yields the following relation. Proposition 7 For any (N; v) 2 GN , and all p 2 PN , Ep [v(S)

v(Sni)] = Ep [v(S)]

EpN ni [v N ni (S)]:

(21)

Proof. According to (1), from (4) we have EpN ni [v N ni (S)] =

Ep [v(S)]

X

p(S) v(S)

S N

=

X

p(S) v(S)

S N

=

X

X

X

pN ni (S) v N ni (S)

S N ni

(p(S) + p(S [ i)) v(S)

S N ni

p(S) (v(S)

v(Sni)) = Ep [v(S)

v(Sni)]:

S:i2S N

That is to say, the expected marginal contribution of every player to the worth of the coalition which will form, coincides with the marginal contribution of his/her participation to the expected worth. Then, combining (4) and (21), we have i (v; p)

=

Ep [v(S) v(Sni)] Ep [v(S)] = Prob (S 3 i) Prob (S 3 i) 14

EpN ni [v N ni (S)] Prob (S 3 i)

:

(22)

It should be remarked that formula (22), equivalent to (21), includes as particular cases all results involving the di erent variations of the potential notion. First notice that for any p 2 PN such that for any player the probability of belonging to the coalition that will form is the same (as for weak semivalues), the denominators in (22) for di erent i's become 'hidden' in the sense that they do not depend on i. Thus, calling := Prob (S 3 i) ; and denoting P (v) :=

P(v; p)

=

Ep [v(S)]

= Ep [v(S) j i 2 S];

a mysterious function (the usual 'potential' in fact) emerges, so that (22) can be rewritten in this particular case for any i as i (v; p)

=

Ep [v(S)]

EpN ni [v N ni (S)]

= P (v)

P (v N ni ):

That is, yielding Hart and Mas-Colell's condition (17), which is thus guaranteed only for weak semivalues (as established by Calvo and Santos (2000)). But note that in general Prob(S 3 i) depends on i and consequently the above formula not even makes any sense. While (22) makes sense and holds for any p: And notice that again denoting by wi the inverse of Prob(S 3 i) ; that is, wi := i (v; p)

= wi (Ep [v(S)]

P1

p(T ) ;

condition (22) becomes

T :i2T

EpN ni [v(S)]) = wi (P(v; p)

P(v N ni ; pN ni )):

In other words, the general condition satis ed by all evaluations of the form

(23) ( ; p) is

(22), or its equivalent (23), whose counterpart in the usual setting is 'weighted' potential. Again, as before, the 'weighted' character of the notion evaporates9 . Similarly, formula (20) seems that of a general, natural and transparent notion of potential, even if that name seems super uous in this context, given its obvious meaning: the expected worth of the coalition which will form.

6

Concluding remarks

Within Weber's very general notion of 'group value', we have provided a more speci c model with some unifying power in which a variety of notions acquire a precise probabilistic interpretation. Several families of 'solutions' born out of di erent departures from 9

This is not that surprising after having seen what happens with balanced contributions, given the

equivalence of w-balanced contributions and the existence of a w-potential in the usual setting (Calvo and Santos, 1997)). But we have preferred discussing separately both notions and their meaning in our setting, for we are mainly concerned in this paper with the 'meaning' of notions and results.

15

Shapley's seminal characterization appear as particular cases of a simple story: The interim evaluation of the expected marginal contribution of every player to the coalition that will form conditional to that player belonging to it, based on a common prior on the coalition formation. The weighted weak semivalues appear as the family of 'values' associated with this model, while the other are narrower circles around the seminal concept that correspond to particular cases of this model. Weak semivalues and semivalues are but subfamilies associated to speci c simple conditions on the probabilities, while weighted Shapley values, and the Shapley value itself, appear associated with rather peculiar conditions on these probabilities. But this is only natural, given that 'e ciency' is a rather unnatural condition to be required from the point of view provided by this model. On the contrary, the existence of potential and the balanced contributions condition, appear as obvious properties of some particular cases within this general model. While the 'weighted' variants of both notions turn out to be the true general and transparent properties with a clear meaning within this simple coalition formation model. The tight

tting of a variety of notions born out of axiomatic explorations into so

simple a model deserves some re ection. Some of the solutions considered in this paper (weak semivalues, weighted or not, as well as semivalues) appear more directly apprehensible through the 'glasses' provided by this simple model than through the axioms that characterize them. Moreover, the model itself, of which these families are variations, gives a clear meaning to some characterizing properties, as the existence of potential and the balanced contributions condition, and their satisfaction by these 'solutions'. These reinterpretations strongly suggest that these conditions are more naturally interpreted in terms of (and as related to) coalition formation than in terms of (or as related to) the classical notion of 'value' from which they originated.

References [1] Calvo, E., and J. C. Santos, 1997, Potentials in cooperative TU-games, Mathematical Social Science 34, 175-190. [2] Calvo, E., and J. C. Santos, 2000, Weighted Weak Semivalues, International Journal of Game Theory 29, 1-9. [3] Dubey, P., A. Neyman, and R. J. Weber, 1981, Value Theory without E ciency, Mathematics of Operations Research 6, 122-128. [4] Einy, E., 1987, Semivalues of Simple Games, Mathematics of Operations Research 12, 185-192.

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[5] Hart, S., and A. Mas Colell, 1989, Potential, Value and Consistency, Econometrica 57, 589-614. [6] Kalai, E., and D. Samet, 1987, On Weighted Shapley values, International Journal of Game Theory 16, 205-222. [7] Kalai, E., and D. Samet, 1987, Weighted Shapley values, In Roth, A. E., ed., (1988), 83-99. [8] Laruelle, A., and F. Valenciano, 2002, Power Indices and the Veil of Ignorance, International Journal of Game Theory 31, 331-339. [9] Laruelle, A., and F. Valenciano, 2003, Semivalues and Voting Power, International Game Theory Review 5, 41-61. [10] Laruelle, A., and F. Valenciano, 2005, Assessing success and decisiveness in voting situations, Social Choice and Welfare 24, 171-197. [11] Myerson, R., 1980, Conference Structures and Fair Allocation Rules, International Journal of Game Theory 9, 169-182. [12] Nash, J. F., 1950, The Bargaining Problem, Econometrica 18, 155-162. [13] Owen, G., 1975, Multilinear extensions and the Banzhaf value, Naval Research Logistics Quaterly, 741-750. [14] Owen, G., 1982, Game Theory, Academic Press, 2nd edition, New York. [15] Roth, A., 1977a, The Shapley Value as a von Neumann-Morgenstern Utility, Econometrica 45, 657-664. [16] Roth, A., 1977b, Utility Functions for Simple Games, Journal of Economic Theory 16, 481-489. [17] Roth, A. E., 1988, ed., The Shapley Value. Essays in Honor of Lloyd S. Shapley, Cambridge University Press, Cambridge. [18] Shapley, L. S., 1953a, Additive and Non-additive Set Functions, Ph.D. Thesis, Department of Mathematics, Princeton University. [19] Shapley, L. S., 1953b, A Value for n-Person Games, Annals of Mathematical Studies 28, 307-317. Reprint in Roth, A. E., ed., (1988), 31-40. [20] Stra n, P. D., 1977, Homogeneity, Independence and Power Indices, Public Choice 30, 107-118. 17

[21] Stra n, P. D., 1982, Power Indices in Politics. In: Political and Related Models, Brams, S. J., W. F. Lucas, and P. D. Stra n, eds, New York, Springer-Verlag, 256321. [22] Stra n, P. D., 1988, The Shapley-Shubik and Banzhaf Power Indices as Probabilities. In Roth, A. E., ed., (1988), 71-81. [23] von Neumann, J. and O. Morgenstern, 1944, 1947, 1953, Theory of Games and Economic Behavior. Princeton: Princeton University Press. [24] Weber, R. J., 1979, Subjectivity in the Valuation of Games. In: Game Theory and Related Topics, Moeschlin, O. and D. Pallaschke, eds, North Holland Publishing Co, Amsterdam, 129-136. [25] Weber, R. J., 1988, Probabilistic Values for Games. In Roth, A. E., ed., (1988), 101-119.

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