The Prediction Value

We introduce the prediction value (PV) as a measure of players’ informational importance in probabilistic TU games. The latter combine a standard TU game and a probability distribution over the set of coalitions. Player i’s prediction value equals the difference between the conditional expectations of v(S) when i cooperates or not. We characterize the prediction value as a special member of the class of (extended) values which satisfy anonymity, linearity and a consistency property. Every n-player binomial semivalue coincides with the PV for a particular family of probability distributions over coalitions. The PV can thus be regarded as a power index in specific cases. Conversely, some semivalues – including the Banzhaf but not the Shapley value – can be interpreted in terms of informational importance.


INTRODUCTION
Concepts of power and importance in models of cooperation are central to numerous studies in sociology, political science, mathematics, and economics. Much of the literature applies values or power indices which attribute fixed roles -often perfectly symmetric -to all players in the underlying coalition formation process and then focus on their marginal contributions. Most prominent examples are the Shapley value and Banzhaf value (Shapley 1953;Banzhaf 1965); others can be found in Roth (1988), Owen (1995), Felsenthal and Machover (1998) or Laruelle and Valenciano (2008).
A player who makes a positive marginal contribution, i.e., who can raise some coalitions' worth by joining in, or lowering it by leaving, is considered as important and powerful. Others who never affect a coalition's worth v(S) are referred to as dummy or null players. The powerful ones are attributed a positive share of the decision body's aggregate ability to implement collective decisions or to create surplus; the indicated value concepts differ just in how marginal contributions to distinct coalitions are weighted. For instance, the Shapley value weights a player i's marginal contribution to a coalition S i according to the total number of ordered divisions of the reduced player set N \ i into members of S and its complement; the Banzhaf value weights i's marginal contributions equally for all S ⊆ N \ i. 1 With an appropriate rescaling, weights on specific marginal contributions can be interpreted as a probability distribution. So Shapley value, Banzhaf value, and more generally probabilistic values (Weber 1988) correspond to the expectation of a difference. This difference is between the worth of a random coalition S that is drawn from 2 N\i according to a value-specific probability distribution P i and the worth of the same coalition when i joins, i.e., a probabilistic value equals E P i [v(S ∪ i) − v(S)] for a fixed family of distributions {P i } i∈N . 2 The expectation of a difference, however, can behave in strange ways when the family of distributions {P i } i∈N implicate correlated voting behavior. This can be the case, for example, when voting is preceded by a process of information transmission or opinion formation. 3 The following example, which we owe to Moshé Machover, illustrates the conceptual problem.
Example 1. Consider the canonical simple majority decision rule with an assembly of 5 voters. Let P be the probability distribution that assigns probability 0 to the 20 coalitions containing exactly two or exactly three voters; and equal probability of 1/12 to each of the remaining 12 divisions. Here, the probabilistic value E P i [v(S ∪ i) − v(S)] is zero for all players. That no member of this decision body should have any voting power or importance is somewhat counterintuitive however. This paper proposes an alternative approach: namely, to consider the difference of two expectations. These expectations will be derived from a given probabilistic description P of coalition formation. The latter plays a similar role as {P i } i∈N for probabilistic values or corresponding families {P v i } i∈N for values that evaluate marginal contributions in game v-specific ways. 4 However, we take P as a primitive of the collective decision situation under investigation, rather than of the solution concept. 1 We adopt the usual notational simplifications like writing S \ i or S ∪ ij instead of S \ {i} or S ∪ {i, j}.
2 More precisely, a probabilistic value draws on a family of families of distributions, parameterized by the player set N. One may equivalently consider suitable probability distributions P i on {S ∈ 2 N : i ∈ S} and then evaluate See, for example, the seminal opinion formation model of DeGroot (1974): individuals start with initial opinions (beliefs) on a subject represented by an n-dimensional vector of probabilities, and repeatedly update their individual opinion based on the current opinions of their peers. Different structures of consensus formation can be captured by different network topologies. 4 This is, for instance, the case when positive probability is only attached to minimal winning coalitions (see, e.g. Holler 1982 andHoller andLi 1995).
We thus depart from the literature in two respects: first, we consider probabilistic games (N, v, P) where (N, v) is a standard TU game and P is a probability distribution on N's power set 2 N . Second, we introduce a new value that reflects the difference between two conditional expected values. Specifically, we define the prediction value (PV) of any given player i ∈ N as the difference in v's expected value when the distribution P|i which conditions P on the event {i ∈ S} and the distribution P|¬i which conditions on {i / ∈ S} are applied. In other words, we suggest to evaluate The two coincide in interesting special cases, but not in general.
The difference between the respective conditional expectations can be interpreted as the importance of a player in the probabilistic game (N, v, P) in several ways. Most generally, it captures the informational or predictive value of knowing i's decision in advance of the process which divides N into some final coalition S and its complement. Moreover, in case i's membership of the coalition which supports a specific bill or cooperates in a joint venture is statistically independent of others, the PV provides a measure of i's influence on the outcome of collective decision making, or of i's power in (N, v, P).
A null player who, say, has a voting weight that cannot matter for matching a required threshold and whose behavior is uncorrelated with the remaining players has a PV of zero. Endowing the same player with greater voting weight will at some point translate into a positive value -reflecting the difference that her vote can now make for the outcome. Leaving initial voting weights unchanged, the PV will also ascribe positive importance to the null player if interdependencies make its cooperation a predictor of whether a proposal is passed.
Plausible causes for dependencies abound and, for instance, include the possibility that the player in question is actually without vote but 'followed' by the official voters (as, say, their paramount or supreme leader). The proposed change of perspectivefrom, traditionally, the expected difference that a player would make by an ad-hoc change of coalition membership towards the difference in expectations for the collective outcome which is associated with that player's cooperation -opens the route to studying voting and coalition formation as the result of social interaction. Final votes may be determined by whether i is initially a supporter or opponent even if i is a null player of (N, v), and this is arguably a source of power just like official voting weight. We believe that evaluating changes in conditional expectations can help to quantify this in future research.
Here, we primarily want to introduce and investigate the prediction value. We formally define it in Section 2. We describe a set of characteristic properties in Section 3 and relate the PV to traditional probabilistic values in Section 4. The considered distributions P could embody the a prioristic presumptions of traditional power measures, i.e., be the uniform distribution on 2 N or the space of permutations on N. (Interestingly, the latter does not make PV and Shapley value coincide.) But P could equally well be based on empirical data -say, observations of past voting behavior in a decision making body like the US Congress, EU Council of Ministers, etc. We briefly conduct such a posteriori analysis with the PV in an application to the Dutch Parliament in Section 5 and conclude in Section 6.

PROBABILISTIC GAMES AND THE PREDICTION VALUE
A TU game is an ordered pair (N, v) where N ⊂ N represents a non-empty, finite set of players and v : 2 N → R is the characteristic function which specifies the worth v(S) of any subset or coalition S ⊆ N and satisfies v(∅) = 0. The set of all TU games is denoted by G, and the set of all TU games with player set N by G N . The cardinality of a finite set S ⊂ N is denoted |S|.
Given any non-empty coalition S ⊆ N, the so-called unanimity game u S is defined by u S (T ) = 1 if S ⊆ T and u S (T ) = 0 otherwise. Note that we will drop the player set N from our notation when it is clear from the context; so u S is shorthand for (N, u S ). Moreover, we refer to u {i} simply as u i .
A probabilistic game is an ordered triple (N, v, P), where (N, v) is a TU game and P is a probability distribution on the power set of N, 2 N . The set of all probabilistic games is denoted by PG; and PG N is the restriction to the class of probabilistic games with player set N.
A TU value is a function which assigns a real number to all elements of N for any given TU game. An extended value is a mapping ϕ that assigns to each probabilistic game (N, v, P) a vector ϕ(N, v, P) ∈ R |N| . ϕ i (N, v, P) will be interpreted as a measure of the 'difference', in an abstract sense, that player i makes for the probabilistic game (N, v, P). It might, for instance, relate to the average of marginal contributions v(S ∪ i) − v(S) that are made by i to coalitions S ∈ N \ i, to the difference that i makes to a potential function (i.e., a mapping from PG to R) when i is added to the player set N such that N ∪ i = N, or to any other indicator of how important the behavior or presence of player i might be to the members of N or an outside observer.
TU values and extended values are defined on two distinct domains, G and PG. Extended values can be regarded as technically the more general concept because any given TU value can be turned into an extended value simply by ignoring the distribution P that is specified as part of probabilistic game (N, v, P). For instance, the in this way 'generalized' Shapley value is defined by 5 When the considered set of players N is clear from the context, we simplify notation by writing S i instead of S⊆N: i/ ∈S , or S i instead of S⊆N: i∈S . and similarly the (generalized) Banzhaf value can be defined by (2) Both the original Shapley TU value and the Banzhaf TU value (which was at first restricted to simple games, and later extended to general TU games by Owen 1975) are special instances of probabilistic values, as introduced by Weber (1988), with either such that each element Q i of the collection Q = {Q i } i∈N denotes a probability distribution on {S ⊆ 2 N : i ∈ S}, or such that Q i denotes a probability distribution on 2 N\i . For instance, Laruelle and Valenciano (2005) have proposed two probabilistic values, Φ + and Φ − , which respectively take Q i (S) and Q i (S) to denote the probability of coalition S being realized conditional on i voting no and conditional on i voting yes. For a given probabilistic game (N, v, P) this suggests to work with the conditional probability distributions P|i and P|¬i as follows: for all S ⊆ N and, similarly, One might then consider and with n = |N| and s = |S| (see Laruelle and Valenciano 2005, Prop. 3). 6 We, however, suggest an altogether different approach to assessing the importance of N's members in a probabilistic games (N, v, P). It is not based on probabilistic values, nor marginal contributions in general.
The reason why weighted marginal contributions may misrepresent (N, v, P) is that they implicitly treat i's decision, say, to change her no vote into a yes (or vice versa) as being fully detached from the respective probabilities of observing the considered two coalitions with and without i. Example 1 already highlighted the effect that nonzero marginal contributions v(S ∪ i) − v(S) > 0 simply don't count at all when the underlying probability distribution P treats both events S∪i and S as null events. Adding up weighted marginal contributions also leads to strange conclusions if only one of the coalitions S and S ∪ i has positive probability, as illustrated in the following example. One thing that outside observers, members j = i of N, or i herself might still care about is the informational gain that comes with the knowledge: "i will (not) be part of the eventually formed coalition". Knowing this might imply that j cannot (or must) be amongst the members of the coalition. And it may have ramifications for the expected surplus that is created or the passage probability of the bill being debated. In other words, it may be useful to base one's evaluation of collective decision making as described by (N, v, P) on P|i rather than P when i is known to support the decision. This suggests looking at the difference E P|i [v(S)] − E P [v(S)] as a way of quantifying i's effect on the outcome. And, of course, it is of similar interest -and may yield a rather different quantification of the difference that i's decision makes -not to look at how much i's support increases the expected worth v(S) but at how much i's opposition lowers it, i.e., Combining these two evaluations of how knowledge of i's decision changes the expectation of the game by summing them, we obtain: 6 Note that, as emphasized by Laruelle and Valenciano, the respective distribution P which needs to be assumed in order to obtain the Shapley value as the expected marginal contribution conditional on i being a member of the random coalition S and, alternatively, conditional on i not being a member, differ.
Definition 1. The prediction value (PV) of player i in the probabilistic game (N, v, P) is defined as Example 3 (Example 1 revisited). Consider again the canonical simple majority decision rule with an assembly of 5 voters with P(S) = 0 for |S| = 2 or |S| = 3 and P(S) = 1/12 otherwise. The conditional probabilities are given by and, similarly, The prediction value follows as Remark 1. In case that coalition membership is statistically independent for every i = j, i.e., if P is a product measure on 2 N , the equality P | i(S) = P | ¬i(S \ i) holds whenever i ∈ S. Then equations (7), (8), and (11) all evaluate to the same number -to the Banzhaf value, for instance, if P(S) ≡ 2 −|N| . That the "expectation of a difference" in (7) or (8) coincides with the "difference between two expectations" in (11), however, fails to hold in general. In particular, we will show in Corollary 1 that there is no probability distribution P which would allow the Shapley value to be interpreted as measuring informational importance.

CHARACTERIZING THE PREDICTION VALUE
This section provides an axiomatic characterization of the prediction value. We begin with two classical conditions that are part of many axiomatic systems in the literature on TU values. The first is anonymity, which requires that the indicated difference to the game that is ascribed to any player by an extended value does not depend on the labeling of the players. The second is linearity, which demands of an extended value that it is linear in the characteristic function component v of probabilistic games.
Definition 2. Consider two probabilistic games G = (N, v, P) and G = (N , v , P ) related through a bijection π : N → N such that for all S ⊆ N, v(S) = v (πS) and P(S) = P (πS) where πS := {π(i)|i ∈ S}. An extended value ϕ is anonymous if for every such G and G ∈ PG Definition 3. An extended value ϕ is linear if for all (N, v, P), (N, v , P) ∈ PG and real constants α, β Linearity combines two properties, scale invariance and additivity. Especially the latter is far from being innocuous. 7 But linearity is frequently imposed on solution concepts for TU games; and the PV, as the difference of two expectations, embraces it rather naturally.
The third characteristic property of the PV concerns the way how the respective extended values of two games G and G compare when one can be viewed as a reduced form of the other. We first formalize this reduction relation between two games, and afterwards define a consistency property which connects the extended values of correspondingly related games.
So, when one moves from a given probabilistic game G to the reduced game G −i , first, player i is removed from the set of players; second, the probabilities of all coalitions in G which only differ concerning i's presence are aggregated; and, third, the corresponding new worth v −i (S) of coalitions S ⊆ N −i is the convex combination of the associated old worths, v(S) and v(S ∪ i), weighted according to their respective probabilities under P. The following property requires that the extended value of any player j ∈ N −i stays unaffected by the removal of i. 9 See, e.g., Felsenthal and Machover (1998, 6.2.26) and Luce and Raiffa (1957, p. 248). 8 Note that if i were not a dependent player, i.e., v(i) = 0, then v −i would not be a properly defined TU game because v −i (∅) = 0 in this case. 9 The condition is vaguely reminiscent of the amalgamation properties considered by Lehrer (1988) or Casajus (2012).
One reason for why this consistency property could be desirable is the following. Suppose that the considered model is misspecified in the sense that a player of interest in the game is not taken into account by the rest of the players (or an outside observer). For instance, consider the situation of a voting game G = (N , v , P ), where the presence of a lobbyist i has been neglected. The more accurate model would include the lobbyist and be G = (N ∪ i, v, P). The effect of the lobbyist endorsing a proposal or opposing it would explicitly be captured by the probability distribution P: for example, voters with strong ties to i may be likely to vote the same way, while others behave oppositely. Coalitions S and S ∪ i which differ only in i's presence will consequently have very different P-probabilities depending on whether S includes i's fellow travelers or opponents. But if the probability P and value v of each coalition T ⊆ N in the 'misspecified' game without i are defined in a probabilistically correct way, i.e., if the misspecified game G equals G −i , then the assessment of any actor j = i should be unaffected by whether one considers G or G −i .
Consistency can thus be seen as formalizing robustness to probabilistically correct misspecifications.
Proposition 1. The prediction value is anonymous, linear, and consistent.
Proof. Anonymity and linearity of ξ are obvious from Definition 1. To prove consis- In the alternative case, both sides are zero by definition. So in either case Analogously, one can check that when S ⊆ N \ ij. By using the definition of v −i and invoking equality (21) one can verify that Similarly, by definition of v −i together with (22), we get One can then infer where the second equality uses (23) and (24), and the third one follows by shifting the corresponding terms from inside the square brackets to the outer summation.
Proposition 1 is not enough to fully characterize the PV. For example, Φ + i (N, v, P) satisfies anonymity, linearity and consistency, too (see Lemma 1 below). Theorem 1 will provide a unique characterization of the PV. In a nutshell, the underlying argument will be as follows: if extended values are linear and consistent, they are determined by their image for the subclass of 2-player probabilistic games. It is then a question of how the extended values of 2-player probabilistic games should suitably be restricted.
For that purpose it is worth recalling two implications of i being part of the formed coalition: first, i's presence means that i contributes to the formed coalition her voting weight, productivity, etc. This reveals information about the expected worth directly. But, second, i's presence also affects the expected worth indirectly because it reveals information about the presence and contributions of other players, at least if the behavior of N \ i and of i are not statistically independent. In case of independence, i.e., if the presence of i ∈ N presence has no informational value according to P, and if moreover i is a null player in the TU-game (N, v), then a reasonable extended value can be expected to assign zero to i. If, in contrast, knowledge of the behavior of null player i does change the odds of a proposal being passed, then i has positive informational value.
For illustration, consider a voting game in which j is a dictator according to the rules formalized by v (i.e., v(S) = 1 ⇔ j ∈ S). Let the voting behavior of j be perfectly correlated with that of some other player i (formally a null player). Now note that it is not part of the model (N, v, P), which mathematically describes the rules of the collective decision body involving i and j and the random outcomes of coalition formation processes, why the votes of i and j always coincide. 'Null player' i might simply follow 'dictator' j in all his decisions. Alternatively, player i could be irrelevant merely from a formal perspective, i.e., have no say de jure; while it is her who imposes all her wishes on j -that is, she rules de facto. In either case the informational values of i and j are identical. They are also maximal (and could plausibly be normalized to, say, 1) in the sense that the outcome can be predicted perfectly when knowing that i or j votes yes or no.
We combine the requirement that an independent null player i should be assigned an extended value of zero with the requirement that i has a value of one in the considered perfect correlation case as follows: 10 Definition 6. An extended value ϕ satisfies the informational dummy-dictator property (IDDP) if for i ∈ N and |N| = 2 Regarding dictators themselves it makes sense to impose the following for 1-player probabilistic games: This formalizes that if N consists of just a single player i ∈ N with v(i) = u i (i) = 1 then i's importance or the difference that i makes to this game should plausibly be evaluated as unity. 11 Immediately from the definition of the PV we obtain Proposition 2. The prediction value satisfies full control and (IDDP).
Remark 2. We remark that (IDDP) implies a positive extended value for a null player i even if i's behavior is imperfectly but still positively correlated with that of a dictator j. This is, e.g., the case when a yes-vote by i is made more likely by most other players voting yes, i.e., for the implicit probabilistic model behind the Shapley value. For a probabilistic game with a dictator where P reflects any Shapley value-like probabilistic assumptions, this means that PV and Shapley value ϕ will not coincide: the Shapley value satisfies the traditional null player axiom, i.e., it assigns zero to any player i who does not directly affect the worth of any coalition S.
We have the following characterization result: Theorem 1. There is a unique extended value ϕ which satisfies linearity, consistency, full control and (IDDP). It is anonymous and ϕ ≡ ξ.
The full proof is provided in the appendix, together with proof of the following lemma. It certifies that none of the four axioms in Theorem 1 is redundant.

10
The case of independence corresponds to P|i(ij) = P|¬i(j), while the correlated dictator case amounts to P|i(ij) = 1 and P|¬i(j) = 0.

11
One might actually debate whether this should also be required in case that P(∅) = 1. satisfies consistency, full control, (IDDP) but not linearity.

RELATION BETWEEN PREDICTION VALUE AND PROBABILISTIC VALUES
The example values discussed earlier (like ϕ, β, Φ + , Φ − ) all are members of the class of probabilistic values, i.e., they have in common that they weight marginal contributions of a player by some probability measure. We already noted in Remark 1 that the natural extension of the Banzhaf value agrees with the prediction value if P(S) ≡ 2 −|N| . We now study the relationship between such members of the class of extended probabilistic values and the prediction value somewhat more generally.
Recall that Weber (1988) has shown that the class of probabilistic values is characterized by linearity, positivity, and the null player axiom. The PV is linear but satisfies neither positivity nor the null player axiom. 12 Like the prediction value, probabilistic values generally do not satisfy symmetry. That property formalizes the idea that any symmetric players i, j ∈ N in a TU game (N, v) should have the same value; it will only be satisfied by a probabilistic value Ψ or the PV ξ if the respective probability measures Q from (3) or P from (11) are fully symmetric regarding i and j.
The following result characterizes the connection between probabilistic values and the prediction value.
Theorem 2. The identity Ψ(·, Q) ≡ ξ(·, P) holds for n > 1 if and only if there exist probabilities 0 <p i < 1 for each player such that 12 See Remark 2 concerning null players. Positivity is violated, e.g., for a probabilistic game (N, v, P) where v is positive and there is a player i such that P|i ≡ 0.
holds for all S ⊆ N\i, i ∈ N and holds for all S ⊆ N.
The proof can be found in the appendix. An important subclass of probabilistic values has the symmetry property: semivalues are defined by (3) and weights Q i (S) that depend on S only via |S| (Dubey et al. 1981). 13 They are defined by for a vector of n non-negative numbers q = (q 0 , . . . , q n−1 ) = 0 with n−1 k=0 n − 1 k q k = 1.
The Shapley value arises by setting q k = 1 n( n−1 k ) ; the Banzhaf index for q k = 1 2 n−1 . The following result characterizes the connection between semivalues and the prediction value by answering the question: for which q can one find P such that f q (N, v) = ξ(N, v, P) for all (N, v) ∈ G N ? This identifies all semivalues which can be interpreted as the prediction value for specific P.
Proposition 3. For a given semivalue f q and n > 1 there exists P such that f q (·) ≡ ξ(·, P) on G N if and only if there is an α > 0 with q k = q 0 α k > 0 for all 0 k n−1, where q −1 0 = n−1 k=0 n−1 k α k . Proof. Assume f q (·) ≡ ξ(·, P). Then, anticipating Lemma 4 (see the appendix), we conclude q |S|−1 = P|i(S) for all {i} ⊆ S ⊆ N and all i ∈ N. Applying Theorem 2 it follows that there existp j ∈ (0, 1) for all j ∈ N such that q |S\i| = j∈Sp j · j∈N\(S∪i) (1 −p j ). From P|i(S) = q |S|−1 = P|j(S), where i, j ∈ S, one obtains p i =p j for all i, j ∈ N. Setting α =p 1 1−p 1 we can write q k = α k · (1 −p 1 ) n−1 = p k 1 (1 −p 1 ) n−k−1 for all 0 k n − 1. We observe that q satisfies equation (29) if we choose q −1 0 = n−1 k=0 n−1 k α k = (1 −p 1 ) −n+1 . Sincep 1 →p 1 1−p 1 is a bijection from (0, 1) to (0, ∞), we directly obtainp 1 from a given α > 0 and can then easily check that f q defined by q k = q o α k is indeed identical to ξ(·, P) with P defined byp j =p 1 for all j ∈ N and by equation (27). 13 See, e.g., Calvo and Santos (2000) for a discussion of other subclasses like weighted Shapley values, weak semivalues (Q i (S) depends only on S), or weighted weak semivalues (Q i (S) is decomposable as w i · p S where w i depends only on i and p S depends only on S).
It follows that semivalues which allow for the interpretation as a prediction value form a special subclass of semivalues. They are known as binomial semivalues (see Dubey et al. 1981;Carreras and Freixas 2008; Puente 2012 -an axiomatic characterization has been given by Amer and Giménez 2007). Specifically, a p-binomial semivalue is defined by Setting α = p 1−p matches the parametrization in Proposition 3, where q 0 can be determined from equation (29). 14 For each given α > 0 we obtain a unique semivalue. The Banzhaf value corresponds to α = 1 and q 0 = 1 2 n−1 . For n = 1 each α > 0 yields the same value given by q = 1. For n = 2 we set α = q 1 q 0 . For n 3 it depends on the specific semivalue whether it can be viewed as a restriction of the PV or not. As already suggested by our Remark 2 on correlated decisions and null players, a negative result obtains for the Shapley value ϕ. It illustrates the fundamental difference between traditional semivalues and the new value concept proposed in this paper: Corollary 1. For n 3 there exists no P such that ϕ(·) ≡ ξ(·, P) on G N .

PREDICTION VALUES IN THE DUTCH PARLIAMENT 2008-2010
As illustration of the prediction value's practical applicability and of how its informational importance indications can be very different from power ascriptions by traditional values, we consider the seat distribution and voting behavior in the Dutch Parliament between 2008 and 2010. This was the period of the left-centered Balkenende IV government, which consisted of Christian democrats from the CDA and Christen Unie parties and the social democratic PvdA.  In some definitions in the literature the extreme cases p = 0 and p = 1 are allowed, too, with the convention 0 0 = 1. For p = 0 we would get the dictatorial index and for p = 1 the marginal index. See Owen (1978) for details. However, note that neither p = 0 nor p = 1 satisfy the conditions from Proposition 3.
The distribution of the 150 seats in parliament between its eleven parties is displayed in the top part of Table 1. The three government parties held a majority of 80 out of 150 seats. When voting on non-constitutional propositions, the Dutch Parliament applies simple majority rule. It is straightforward to define a voting game with this information, and to calculate the corresponding a priori Banzhaf and Shapley values β and ϕ.
We used the parliamentary information system Parlis 15 in order to extract information on members, meetings, votes and decisions on propositions in the 2008-2010 period. From the records of regular plenary voting rounds, where parties vote as blocks, we derived the empirical frequencies of the 2 11 conceivable divisions into yes and no-camps from 2720 observations. 16 Defining P by these empirical frequencies, we calculated the corresponding prediction values ξ i of the parties as well as their positive and negative conditional decisiveness values Φ + i and Φ − i defined in (7) and (8). A summary of the results is given in the bottom part of Table 1.
The PV-scores ξ i of Dutch parties tend to be higher than their respective traditional Banzhaf or Shapley power measures β i and ϕ i , and even the decisiveness measures Φ + i and Φ − i which incorporate the same empirical estimate of P. In particular, the prediction value ascribes rather substantial numbers also to small parties like D66, SGP, or Verdonk.  This reflects specificities of the political situation in the Netherlands and that the PV picks up corresponding correlations between the voting behavior of different parties. Varying majorities at calls are quite common in the Dutch Parliament. The member parties of the government do not necessarily vote the same way; some are frequently 15 The data is available through http://data.appsvoordemocratie.nl 16 We pooled all regular plenary votes in order to illustrate the simplest way in which data can be used to infer interdependencies in a voting body -one might want to split the data with respect to topics, or weight distinct calls by their importance, in actual political analysis. Note that the Dutch Parliament's chairperson assumes that parties vote as blocks unless some MP demands voting by call. Only then can members of the same party vote differently. We excluded such cases of 'non-coherent voting' from our analysis. supported by smaller opposition parties. The correlation coefficients reported in Table 2 indicate, for instance, that SGP and D66 quite commonly voted the same way as CU and PvdA. Their PV numbers hence differ much less than their seat shares.
Verdonk and SP constitute an extreme case in this respect. The former is commonly considered as right-wing, the latter as a left-wing party; still both voted the same way at each call in the data set (presumably having different reasons). Perfect correlation of their votes implies that both have the same prediction value -despite SP having 25 seats and Verdonk but one: knowing either's vote in advance would have been equally valuable for predictive purposes. Measures based on marginal contributions, in contrast, clearly favor SP over Verdonk (though less so if the a posteriori correlation between SP's and Verdonk's votes is ignored). Interestingly, the GL party has the second-highest prediction value: despite it not being in government and having only the sixth-largest seat share, support by GL was a better predictor of a bill's success than support by any except the biggest party (CDA).

CONCLUDING REMARKS
Traditional semivalues like the Shapley or Banzhaf values and the prediction value provide two qualitatively distinct perspectives on the importance of the members of a collective decision body. One highlights the difference that an ad-hoc change of a given player i's membership in the coalition which eventually forms would make from an ex ante perspective; the other stresses the difference that the change of a player's presumed membership makes for one's ex ante assessment of realized worth. As the figures in Table 1 illustrate, both can differ widely in case players' behavior exhibits interdependencies. But, as captured by Proposition 3, they coincide in case of statistical independence. The latter is presumed by the behavioral model underlying, e.g., the Banzhaf value, but incompatible with that underlying the Shapley value.
For independent individual voting decisions, the conditioning on different votes of player i adds no behavioral information to the numerical one about i's weight contribution to either the yes or no camp. Then i's informational importance and i's voting power or influence -reflected by sensitivity of the collective decision to a last-minute change of i's behavior -are aligned. 17 It might be criticized that in cases of interdependence, the prediction value fails to distinguish correlation and causation. For illustration, consider decisions by a weighted voting body in which some player i has zero weight but all other players' decisions are perfectly correlated with that of i. Player i's prediction value is then one irrespective of whether (i) players j = i 'follow' i as, say, their guru or supreme leader and cast their weight as i would if he had any, (ii) i = k and all players j = k follow a specific other player k, or (iii) all players debate the merit of a proposal based on different initial inclinations and collective opinion dynamics converge to, for instance, the majority 17 In the case of the Banzhaf value, coincidence between voter i's influence as picked up by i's average marginal contribution and the informational effect of knowing i's vote has been hinted at by Felsenthal and Machover (1998, 3.2.12-15). inclination. 18 But since knowing i's decision -rather than i's initial inclination -will always fully reveal the realized outcome, ξ i = 1 can be regarded more as a feature than a flaw.
This example points to an interesting extension of the proposed "difference of conditional expected values"-approach to measuring importance. Namely, start with a given description (N, v, P) of a decision body where P corresponds to, say, the Banzhaf uniform distribution and augment it by the formal description of a social opinion formation process which defines a mapping from players' binary initial voting inclinations to a distribution over final ones after social interaction. One can then capture a player i's combined social and formal influence in the decision body by answering the question: how much does knowing that i's initial inclination is in favor (or against) modify the final outcome which is to be expected? We conjecture that this approach actually has advantages over extending marginal contribution-based analysis to social interaction, 19 and plan to pursue this extension in future research. 18 See Grabisch and Rusinowska's (2010) related work on possibilities to aggregate individual influence in command structures. 19 See, for instance, the power scores derived from swings in societies with opinion leaders by van den Brink et al. (2013).

APPENDIX
Proof of Theorem 1. The proof proceeds in three steps. First, in Lemma 2 we prove for |N| = 2 that linearity and consistency imply that an extended value is determined by unanimity games. Second, we generalize this to all probabilistic games in Lemma 3. Finally, we show that the full control property and (IDDP) characterize the PV for 2player probabilistic games and hence probabilistic games in general.
Lemma 2. Consider an extended value ϕ that is linear on the space of all 2-player probabilistic games and consistent. For any set N with |N| = 2, the mapping (N, v, P) → ϕ(N, v, P) is fully determined by the numbers Proof. Let P be a fixed probability distribution on 2 N with N = {i, j}. The set of unanimity games {u i , u j , u ij } forms a basis for the space of all TU games on N. In particular, for any (N, v) And thus, for arbitrary P and i ∈ N, ϕ's linearity implies We need to show that x i,ij and x j,ij are fully determined by x ii and x ij . To see this, notice first that both players are dependent in (N, u ij , P). So we may consider the reduced game obtained by j's removal, which involves N −j = {i} and P −j (∅) = P(∅) + P(j), P −j (i) = P(i) + P(ij), In case P(i) + P(ij) > 0, we have where the first equality invokes consistency, the second linearity, and the third one exploits that ({i}, u i , P −j ) is the reduction of (N, u i , P) by player j and again consistency. When P(i) = P(ij) = 0 we have ϕ i (N, u ij , P) = 0 because in this case (u ij ) −j ({i}) = 0 by Definition 4, so that (u ij ) −j is the all-zero game 0 in that case. Consistency requires ϕ i (N, u ij , P) = ϕ i ({1}, (u ij ) −j , P −j ) = ϕ i ({i}, 0, P −j ) = 0 due to linearity.
In summary, And in a similar fashion one obtains For any v ≡ α i u i + α j u j + α ij u ij we have and an analogous expression for ϕ j (N, v, P). This finding can be generalized from just two players to arbitrary N: Lemma 3. Let ϕ be a consistent and linear extended value. Then the mapping (N, v, P) → ϕ(N, v, P) is fully specified by the parameters in (31).
Proof. Using the n-player unanimity games as a basis for PG N one can always write Letting i ∈ N be an arbitrary but fixed player, we will use induction on n in order to prove the following Claim: There exist β ij , depending on the α T and P, such that The claim is obvious for a single player and was proven for |N| = 2 in Lemma 2. In view of linearity, it suffices to prove the statement for unanimity games u T , where nothing needs to be shown when the cardinality of T is one. So we consider |N| 3, |T | 2 and assume that the statement is true for all player sets N of cardinality n − 1. Let j ∈ N \ i be a player, which must be dependent in u T because |T | 2. Now we consider the reduced game (N −j , (u T ) −j , P −j ). From consistency we conclude Applying the induction hypothesis implies the existence of β ik , which depend on P −j and hence on P, such that Since (u k ) −j = u k the reduced game of (N, u k , P) is given by (N −j , u k , P −j ) for all 1 k n with j = k. Inserting ϕ i (N −j , u k , P −j ) = ϕ i (N, u k , P) = x ik then proves the claim, and the theorem.
We remark that the coefficients β ij referred to in the above proof get quite complicated for increasing n. In the following we will use only the fact that they are welldefined given v and P.
Proof of Theorem 1. To complete the proof we now show how the values x ii = ϕ i (N, u i , P) and x ij = ϕ i (N, u j , P) can be computed from the corresponding values for the player set N = {i, j}. Since (u i ) −j = u i for all i = j we can recursively conclude from consistency where Using equation (43) and similarly defining we conclude ϕ i ({i}, u i , P ) = ϕ i ({i, j}, u i , P ) from consistency. Thus, the full control property, in connection with consistency and linearity, implies x ii = 1 for all player sets N (containing player i). If ϕ satisfies (IDDP) the values of x ij are determined, and hence ϕ is determined on the class of 2-player probabilistic games. Then ϕ ≡ ξ follows from Lemma 3. Finally note that the full control property and (IDDP) do not depend on the labeling of the players, which implies anonymity.
Proof of Lemma 1.
(i) Linearity of Φ + follows from (7). For notational convenience putP = P|i. For the which is equal to one for v = u i and P ({i}) > 0 and equal to zero if P ({i}) = 0. To see that Φ + does not satisfy (IDDP) note that For the unanimity game u j follows (ii) Ψ 2 i (N, v, P) inherits linearity and consistency from ξ and Φ + . From (47) follows Ψ 2 i ({i, j}, u j , P) = ξ i ({i, j}, u j , P) and therefore (IDDP). From Proposition 2 and (45) we know that both ξ and Φ + satisfy full control such that Ψ 2 i ({i}, u i , P) = 0, contrary to Definition 7.
(iii) Linearity is obvious. For |N| 2 the extended value Ψ 3 is identical to the PV and the latter satisfies full control and (IDDP). For a counterexample to consistency consider a game G −j = (N, v, P) with |N| = 3 and perfect correlation P(N) = 1/2 = P(∅). Here, However, for the reduced game G −j = (N −j , v −j , P −j ) we get for all i ∈ N which does not coincide with (48).
(iv) Consider the reduced game G −j = (N −j , v −j , P −j ). PV is consistent and therefore ξ i (N −j , (u S ) −j , P −j ) = ξ i (N, u S , P) for all i ∈ N \ j.
We conclude Proof of Theorem 2. The proof is based on three insights, stated in Lemmas 4-6.
Proof. For an arbitrary subset {i} ⊆ S ⊆ N we consider the unanimity game u S and obtain the formulas Lemma 6. If Ψ(·, Q) ≡ ξ(·, P) and if there exists an index i ∈ N with p i = 1, then n = 1.
Proof. From p i = T i P(T ) = 1 we conclude P(S) = 0 for all S ⊆ N\i. Thus we have P|i(T ) = 0 for all T i with |T | 2 due to Lemma 5. This yields P({i}) = Q i ({i}) = 1 and Q j (S) = 0 for all S j, where (S, j) = ({i}, i), and all j ∈ N due to Lemma 4. For each j ∈ N\i we then have S j Q j (S) = 0 = 1 -a contradiction.
Proof of Theorem 2. From Lemma 6 we conclude 0 p i := T i P(T ) < 1 for all i ∈ N. If p i = 0 for an index i ∈ N, then we have Q i (S) = 0 due to Lemma 4, which contradicts the definition of the Q i (S). Thus we have 0 < p i < 1. Later on it will turn out that indeed we can choosep i = p i . We have P(S) = p i 1 − p i · P(S\i) for all S i with |S| 2 due to Lemma 5 and p i > 0. Thus inductively we obtain P(S) = for all ∅ = S ⊆ N. By using S⊆N P(S) = 1 we conclude that equation (49) is also valid for the empty set and thus for all subsets of N. Lemma 4 and a short calculation gives also the first formula of the proposed statement.
To verify that the converse holds as well let 0 < p i < 1 be given for all i ∈ N and define P(S) = j∈S p j ·

j∈N\S
(1 − p j ) , i.e., P is a product measure. Next set (1 − p j ) , for i ∈ N\S, i.e. the Q i (S ∪ i) derive from the same product measure. We can easily verify P|i(S) = P|¬i(S\i) for all S i and all i ∈ N. Inserting this into the definition of the prediction value provides ξ(·, P) = Ψ(·, Q).