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Social Choice and Welfare

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01-04-2015

The Banzhaf value in the presence of externalities

Authors: M. Álvarez-Mozos, O. Tejada

Published in: Social Choice and Welfare | Issue 4/2015

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Abstract

We propose two generalizations of the Banzhaf value for partition function form games. In both cases our approach is based on probability distributions over the set of coalition structures that may arise for any given set of players. First, we introduce a family of values, one for each collection of these latter probability distributions, defined as the Banzhaf value of a coalitional game obtained as the expectation taken according to the given probability distributions of the original partition function form game. For each value of the family we provide two characterization results within the set of all partition function form games. Both results rely on a property of neutrality with respect to the amalgamation of players. Second, we propose another family of values that differ from the previous ones in that the latter values take into account only the information about the most likely coalition structure that may arise according to the given probability distributions. Each value of the second family is also characterized in two results by means of a collusion neutrality property. Unlike the characterizations of the first approach, these characterizations can be restricted to the set of simple games in partition function form.

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For instance, this has recently been the case in the formation process of the regional governments in the Basque Country and Catalonia.

The difference between these two properties has been studied in Alonso-Meijide et al. (2012).

The asymmetry in the delegation agreement is discussed in more detail in Sect. 3.

We restrict our attention to non-singleton universes of players.

We point out that there are some authors (e.g. Casajus 2012) that use “simple games” to refer to a class of coalitional games where no monotonicity condition is imposed.

We note that \(\mathbb {R}_{+}=[0,\infty )\).

In Sect. 4 we do not impose the consistency requirement given by condition (b).

We opt for consistency, as the property considered relates the probability that players in \(N_{-j} \setminus S\) are organized according to \(P_{-S}\) within two different nested scenarios: one where \(j\) is present, one where \(j\) is absent.

Note that we actually associate a coalitional game with every game in partition function form.

See Example 2.2.

Recall that \(\emptyset \) is a member of any partition.

Note that given \(S\subseteq N\) and \((S,P)\in EC^{N}\), the reference to \(S\) as “active” coalition when we look at \(v(S,P)\) actually originates from an interpretation of the coalition structure \(P\) per se, with no reference to the game whatsoever. Accordingly, it can be thought that \((S,P)\) represents a situation in which players in \(S\) agree to actively participate in \(S\), while players in \(N\setminus S\) simply organize themselves into \(P_{-S}\).

The same is trivially true for one-player games.

We denote by \(\mathsf {Ba}\) the Banzhaf value of coalitional games. We note that it can be defined for every \((N,v)\in \mathcal {CG}\) by \(\mathsf {Ba}(N,v)=\mathsf {Ba}^\Lambda (N,v)\), where \(\Lambda \in \mathcal {L}\) is arbitrary. The \(\{ij\}\)-reduced game for coalitional games is then independent of \(\Lambda \) and coincides with the one used in Casajus (2012).

By definition the summation over the empty set is zero.

For the sake of the notation any reference to \(\Lambda \) has been omitted.

This part is based on the proof of Theorem 7 in Casajus (2012).

This part is based on Lehrer (1988).

The independence of the axioms can be proved analogously as in Casajus (2012).

The independence of the axioms is trivial.

All the results contained in this section remain valid if we dispense with condition (iii) and hence do not impose any monotonicity condition in the definition of \(\mathcal {SG}\).

For instance, consider the following example. There are three parties in a parliament, \(N=\{1,2,3\}\), and the seats are distributed as follows: Parties 1 and 2 have both 20 seats and Party 3 has 15 seats. Obviously, no party has an absolute majority. If absolute majority is required, then the three parties are in fairly symmetric position. This is indeed captured by the simple coalitional game \((N,v)\) where \(v(\{k\})=0\) and \(v(N_{-k})=v(N)=1\) for all \(k\in N\). However, if only a relative majority is needed and we consider \(\Lambda ^{**}\)—i.e. no parties make any agreement when they are in the opposition—then clearly Parties 1 and 2 are in a better position than Party 3. The only way to capture this by means of a game where the values of embedded coalitions are either \(0\) or \(1\), is by letting \(v(\{1\},**)=v(\{2\},**)=1\) and \(v(\{3\},**)=0\). That is, \(\mathcal {SG}\) must necessarily contain games where there are two or more winning coalitions in the same partition.

Without conditions (c) and (d) the interpretation is similar but uniqueness of the most likely configurations is not guaranteed.

Taking the minimum is relevant only when the set \(\left\{ \mathop {\hbox {arg max}}\limits \limits _{T'\in P_{-S}} \lambda ^{N}(S,P_{-T'} \cup \{T'_{+j}\})\right\} \) is not a singleton. This occurs, for instance, in the \(\Lambda \) considered in Examples 2.1 and 2.2.

See Theorem 1 in Casajus (2012). A value on \(\mathcal {SG}, \mathsf {f}\), is symmetric if for every \((N,v)\in \mathcal {SG}, v(S_{+i})=v(S_{+j})\) for all \(S\subseteq N \setminus \{i,j\}\) implies \(\mathsf {f}_i (N,v)=\mathsf {f}_j(N,v)\).

The independence of the axioms is trivial.

We stress that \(\mathcal {L}\cap \tilde{\mathcal {L}} \ne \emptyset \) as both sets contain the \(\Lambda \) defined in Examples 2.1—except if \(p=\frac{1}{2}\)—and 2.2, so a proper comparison between values belonging to the two families can be established. Note that while \(\mathsf {Ba}^\Lambda \) and \(\widetilde{\mathsf {Ba}}^\Lambda \) coincide for \(\Lambda \in \{\Lambda ^{*},\Lambda ^{**}\}\), both values differ in general for \(\Lambda ^{p}\).

The independence of the axioms is trivial.

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Title: The Banzhaf value in the presence of externalities
Authors: M. Álvarez-Mozos
O. Tejada
Publication date: 01-04-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 4/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-014-0861-4

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