The prediction value

Abstract

We introduce the prediction value (PV) of player i as the difference between the conditional expectations of v(S) when i cooperates or not in a probabilistic TU game. The latter combines a standard TU game and a probability distribution over the set of coalitions. The PV reflects the importance of information about a given player’s behavior for predicting, e.g., committee decisions that are subject to opinion interdependencies. The PV is characterized by anonymity, linearity, a consistency requirement and two normalization conditions. Every multinomial probabilistic value, hence every binomial semivalue, coincides with the PV for a particular family of probability distributions. So the PV can 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.

1 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 (2008b). The values differ in how marginal contributions to distinct coalitions are weighted.

With an appropriate rescaling, weights on specific marginal contributions can always 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 which is drawn from $2^{N{\setminus } i}$ according to a value-specific probability distribution $P_{i}$ and the worth of the same coalition when i joins. That is, a probabilistic value equals ${\mathbb {E}}_{P_{i}}[v(S\cup i)-v(S)]$ for a fixed family of distributions $\{P_{i}\}_{i\in N}$.¹

Unless the probabilistic value in question is also a (multibinary or) multinomial probabilistic value (cf. Puente 2000; Freixas and Puente 2002; Carreras and Puente 2015a), the presence of player i in the realized coalition S can statistically depend on whether player $j\ne i$ belongs to it or not. This may plausibly be the case, for example, when voting is preceded by a process of information transmission or opinion formation.² Unfortunately the expectation ${\mathbb {E}}_{P_{i}}[v(S\cup i)-v(S)]$, interpreted as power or importance of player i, can behave in strange ways when the family of distributions $\{P_{i}\}_{i\in N}$ implicates correlated behavior. The following example illustrates the conceptual problem.³

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 coalitions. Irrespective of whether we derive $P_i$ by projecting P to $2^{N{\setminus } i}$ or by conditioning P on $i\not \in S$, the probabilistic value ${\mathbb {E}} _{P_{i}}[v(S\cup i)-v(S)]$ is zero for all players. However, that no member of this decision body should have any voting power or importance is somewhat counterintuitive.

Here, the expectation of a difference is uninformative since non-zero marginal contributions $v(S\cup i)-v(S)>0$ do not count when the underlying probability distribution P treats $S\cup i$ and S as null events. More generally, the problem with the expected marginal contribution ${\mathbb {E}}_{P_{i}}[v(S\cup i)-v(S)]$ is that it treats 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.

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 the same role as $\{P_{i}\}_{i\in N}$ does for probabilistic values or corresponding families $\{P_{i}^{v}\}_{i\in N}$ do 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.

We thus depart from most of the previous literature in two respects: first, similar to Laruelle and Valenciano (2008a), we explicitly 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 expectations. Specifically, we define the prediction value (PV) of any given player $i\in N$ as the difference in v’s expected value when the distribution P|i which conditions P on the event $\{i\in S\}$ and the distribution $P|\lnot i$ which conditions on $\{i\notin S\}$ are applied. In other words, we suggest to evaluate ${\mathbb {E}}_{P|i}[v(S)] - {{\mathbb {E}}}_{P|\lnot i}[v(S)]$ instead of ${{\mathbb {E}}} _{P_i}[v(S \cup i)-v(S)]$. The two coincide in interesting special cases, but not in general. In particular, our approach is similar in spirit but factually differs from the evaluation of conditional decisiveness as in Laruelle and Valenciano (2005, 2008a).

The margin 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).

The existing literature—the references above are a small selection—obviously does not suffer from a shortage of solution concepts in general, nor of ones targeted at quantifying a player’s power in TU games. But as Aumann (1987) argued: “Different solution concepts are like different indicators of an economy; different methods for calculating a price index; different maps (road, topo, political, geologic, etc., not to speak of scale, projection, etc.); ...They depict or illuminate the situation from different angles; each one stresses certain aspects at the expense of others.” Taking up Aumann’s metaphor, we do not suggest another scale or projection of a player’s vector of marginal contributions here. We rather propose to look at a slightly different kind of map. For some situations—involving statistically independent cooperation decisions—the picture may look identical to that generated by, say, the Banzhaf value (giving the latter additional force/basis/clout); in others, such as Example 1, it will provide a new perspective.

Example

1 (continued) In the considered assembly, the probability of coalition S was $P(S)=0$ for $|S|=2$ or $|S|=3$ and $P(S)=1/12$ otherwise. The respective conditional probabilities follow from Bayes’ rule. In particular, knowing that player i cooperates (i.e., is part of S) gives rise to

$$\begin{aligned} P|i(S)= {\left\{ \begin{array}{ll} \frac{1}{6} &{} \text {if }S=\{i\}\,\text { or } \,S=N\backslash j,j\ne i \,\text { or }\,S=N, \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

(1)

Similarly, we obtain

$$\begin{aligned} P|\lnot i(S)= {\left\{ \begin{array}{ll} \frac{1}{6} &{} \text {if }S=\varnothing \,\text { or }\,S=\{j\},j\ne i\,\text { or }\,S=N \backslash i, \\ 0 &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

(2)

in case i is known not to cooperate. The expected value of v(S), which corresponds to the probability that the assembly passes the proposal in question, is changed by the observation that, say, voter 1 supports rather than opposes it by

(3)

The information that player 1 votes in a particular way is not enough to predict an assembly decision perfectly. (That would be the case if ${{\mathbb {E}}}_{P|1}[v(S)]-{{\mathbb {E}}}_{P|\lnot 1}[v(S)]=1$). But it changes the odds significantly. Player 1 may not have voting power in the traditional sense, but his or her vote is important from an informational perspective. It may be quite valuable to investors or speculators, for instance. The same applies here to the other members of the assembly to identical degrees. Asymmetric voting weights or asymmetric roles in opinion dynamics and coalition formation would naturally give rise to different PV numbers for the respective assembly members.

A null player who has a voting weight that cannot matter for matching a required majority 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, just like traditional indices. But 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 leader. The proposed change of perspective—from, 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 introduce and investigate properties of the prediction value. We formally define it in Sect. 3, after collecting some preliminaries in Sect. 2. We describe a set of characteristic properties in Sect. 4 and relate the PV to traditional probabilistic values in Sect. 5. 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, etc. We briefly conduct such a posteriori analysis with the PV in an application to the Dutch Parliament in Sect. 6, and we conclude in Sect. 7. All mathematical proofs are contained in the Appendix.

2 Preliminaries

A TU game is an ordered pair (N, v) where $N\subset {\mathbb {N}}$ represents a non-empty, finite set of players and $v:2^N\rightarrow {\mathbb {R}}$ is the characteristic function which specifies the worth v(S) of any subset or coalition $S\subseteq N$ and satisfies $v(\varnothing )=0$. The set of all TU games is denoted by ${\mathcal {G}}$, and the set of all TU games with player set N by ${\mathcal {G}} ^N$.

$(N,v)\in {\mathcal {G}}$ is a simple game if v is a monotone Boolean function, i.e., $v(S)\in \{0,1\}$ and $v(S)\le v(S^{\prime })$ for all $S\subseteq S^{\prime }\subseteq N$, such that $v(\varnothing )=0$ and $v(N)=1$. A coalition S with $v(S)=1$ is then called winning. Given any non-empty coalition $S\subseteq N$, the so-called unanimity game $u_S$ is defined by $u_S(T)=1$ if $S\subseteq T$ and $u_S(T)=0$ otherwise. 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 $u_i$.

Player i is called a dummy player in (N, v) if $v(S\cup i)-v(S)=v(i)$ for all $S\subseteq N{\setminus } i$. Every player i with $v(i)=0$ is said to be dependent. A dummy player who is dependent is also known as a null player. If (N, v) is a simple game and $v(S)=1$ if and only if $i\in S$, then player i is called a dictator.

TU games (N, v) have explicitly been combined with probability distributions P over coalitions $S\subseteq N$ since Owen (1972). P is typically mentioned in order to provide probabilistic motivation or foundations for a particular solution concept, not as a characteristic of a collective decision making situation. We want to emphasize its role as a primitive and define a probabilistic game as 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 ${\mathcal {PG}}$; and ${\mathcal {PG}}^N$ 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 $\varphi$ that assigns to each probabilistic game (N, v, P) a vector $\varphi (N,v,P)\in {\mathbb {R}} ^{|N|}$. $\varphi _i(N,v,P)$ 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 $v(S\cup i)-v(S)$ that are made by i to coalitions $S\in N{\setminus } i$, to the difference that i makes to a potential function (i.e., a mapping from ${\mathcal {PG}}$ to ${\mathbb {R}}$) when i is added to the player set $N^{\prime }$ such that $N^{\prime }\cup i=N$, 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, ${\mathcal {G}}$ and ${\mathcal {PG}}$. 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 (N, v, P). For instance,

$$\begin{aligned} \phi _i(N,v,P)=\sum _{S\not \ni i}\frac{|S|!(|N|-|S|-1)!}{|N|!}(v(S\cup i)-v(S)) \end{aligned}$$

(4)

correspondingly ‘extends’ the Shapley value and the (extended) Banzhaf value can be defined by⁵

$$\begin{aligned} \beta _i(N,v,P)=\frac{1}{2^{n-1}}\sum _{S\not \ni i}(v(S\cup i)-v(S)). \end{aligned}$$

(5)

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).⁶ They are defined by

$$\begin{aligned} \Psi _i(N,v,Q)=\sum _{S\ni i}Q_i(S)(v(S)-v(S{\setminus } i))={{\mathbb {E}}}_{Q_i}[v(S)-v(S{\setminus } i)] \end{aligned}$$

(6)

such that each element $Q_i$ of the collection $Q=\{Q_i\}_{i\in N}$ denotes a probability distribution on $\{S\subseteq 2^N:i\in S\}$, or

$$\begin{aligned} \Psi _i(N,v,Q^{\prime })=\sum _{S\not \ni i}Q^{\prime }_i(S)(v(S\cup i)-v(S))={{\mathbb {E}}}_{Q^{\prime }_i}[v(S\cup i)-v(S)] \end{aligned}$$

(7)

such that $Q^{\prime }_i$ denotes a probability distribution on $2^{N{\setminus } i}$.⁷

Several subclasses of probabilistic values have received special attention. Semivalues satisfy (6) and (7) for weights $Q_i(S)$ and $Q_i'(S)$, respectively, which are identical for all $i\in N$ and depend on S only via its cardinality |S| (Dubey et al. 1981). Then for all $i\in N$

$$\begin{aligned} \Psi _i(N,v,Q')=f_i^q(N,v)=\sum \limits _{S\subseteq N\backslash i}q_{|S|}\cdot (v(S\cup i)-v(S)) \end{aligned}$$

(8)

for a vector of n non-negative numbers $q=(q_0,\dots ,q_{n-1})\ne 0$ with

$$\begin{aligned} \sum \limits _{k=0}^{n-1}{\left( {\begin{array}{c}{n-1}\\ k\end{array}}\right) }q_k=1. \end{aligned}$$

(9)

The Shapley value arises by setting $q_k=\frac{1}{n{\left( {\begin{array}{c}{n-1}\\ k\end{array}}\right) }}$; the Banzhaf index for $q_k=\frac{1}{2^{n-1}}$. The latter—but not the former—is also a binomial semivalue: there exists $0<p<1$ such that (8) holds for

$$\begin{aligned} q_k=p^k(1-p)^{n-k-1}\quad \text { for }k=0, \ldots , n-1. \end{aligned}$$

(10)

See Dubey et al. (1981), Carreras and Freixas (2008), and Carreras and Puente (2012).

Multinomial values have been introduced by Puente (2000) and are obtained from (6) or (7) by requiring that each player j is part of the formed coalition with probability $\tilde{p}_j$ independently of any other player.⁸ Let $g_i^{\tilde{p}}(N,v)$ denote the multinomial value of player i for a fixed vector $\tilde{p}=(\tilde{p}_1, \ldots , \tilde{p}_n)\in [0,1]^n$. Then for all $i\in N$

$$\begin{aligned} \Psi _i(N,v,Q')\!=\!g_i^{\tilde{p}}(N,v)\!=\!\sum \limits _{S\subseteq N\backslash i}\ \prod _{j\in S}\tilde{p}_j \prod _{j\in N{\setminus } (S\cup i)} (1\!-\!\tilde{p}_j) \cdot (v(S\cup i)-v(S)).\nonumber \\ \end{aligned}$$

(11)

3 The prediction value

For a given probabilistic game (N, v, P) define the conditional probability distributions P|i and $P|\lnot i$ as follows: for all $S\subseteq N$

$$\begin{aligned} P| i(S) = {\left\{ \begin{array}{ll} \frac{P(S)}{\sum \limits _{T\ni i}P(T)} &{} \text {if }i\in S\text { and } \sum \limits _{T\ni i}P(T)\ne 0, \\ 0 &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

(12)

and, similarly,

$$\begin{aligned} P|\lnot i(S) = {\left\{ \begin{array}{ll} \frac{P(S)}{\sum \limits _{T\not \ni i}P(T)} &{} \text {if }i\notin S\text { and } \sum \limits _{T\not \ni i}P(T)\ne 0, \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

(13)

Laruelle and Valenciano (2005) have suggested to consider the conditional decisiveness measures

$$\begin{aligned} \Phi ^+_i(N,v,P)={{\mathbb {E}}}_{P|i}[v({ S})- v({ S}{\setminus } i)] \end{aligned}$$

(14)

and

$$\begin{aligned} \Phi ^-_i(N,v,P)={{\mathbb {E}}}_{P|\lnot i}[v({S}\cup i)- v({S})], \end{aligned}$$

(15)

as indicators of player i’s importance in (N, v, P).⁹ It is easy to see that $\Phi ^+(N,v,P)=\Phi ^-(N,v,P)=\beta (N,v,P)$ if and only if $P(S)\equiv 2^{-|N|}$ for all $S\subseteq N$, and one can similarly obtain identity with the (extended) Shapley value. Namely,

$$\begin{aligned} \Phi ^+(N,v,P)=\phi (N,v,P)\Longleftrightarrow & {} P(\varnothing )=0,\ P(S)=\frac{1}{s \left( {\begin{array}{c}n\\ s\end{array}}\right) \sum _{t=1}^{n}\frac{1}{t}} \text { if } S\ne \varnothing ,\nonumber \\ \end{aligned}$$

(16)

$$\begin{aligned} \Phi ^-(N,v,P)=\phi (N,v,P)\Longleftrightarrow & {} P(N)=0,\ P(S)=\frac{1}{(n-s) \left( {\begin{array}{c}n\\ s\end{array}}\right) \sum _{t=1}^{n}\frac{1}{t}} \text { if }S\ne N \nonumber \\ \end{aligned}$$

(17)

with $n=|N|$ and $s=|S|$ (see Laruelle and Valenciano 2005, Prop. 3).¹⁰

We propose an altogether different approach to assessing the importance of N’s members in a probabilistic game (N, v, P). It is not based on probabilistic values, nor marginal contributions in general. Weighted marginal contributions may misrepresent player i’s importance in (N, v, P) in 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 non-zero marginal contributions $v(S\cup i)-v(S)>0$ do not count when the underlying probability distribution P treats both events $S\cup i$ and S as null events. Adding up marginal contributions also leads to strange conclusions if only one of the coalitions S and $S\cup i$ has positive probability.

Example 2

Consider an assembly of 3 voters in which coalitions $\{1,3\}$, $\{2,3\}$ and $\{1,2,3\}$ are winning. Assume voters 2 and 3 are enemies and always vote contrary to each other. Here, coalition $S=\{1,2\}$ might have positive probability under P and $P|\lnot 3$, while $P(N)=0$. A problem for measures like $\Phi ^-(N,v,P)$ is then that they are strictly increased by a contribution which 3 makes in the null event of joining $S=\{1,2\}$.

One thing that outside observers, members $j\ne i$ of N or i herself might still care about in the examples 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. This 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 ${{\mathbb {E}}} _{P|i}[v(S)]- {{\mathbb {E}}}_{P}[v(S)]$ as a way of quantifying i’s effect on the outcome. And it is arguably 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., ${{\mathbb {E}}} _{P}[v(S)]- {{\mathbb {E}}}_{P|\lnot i}[v(S)]$. Combining these two evaluations of how knowledge of i’s decision changes the expectation of the game by summing them, we obtain:

Definition 1

The prediction value (PV) of player i in the probabilistic game (N, v, P) is defined as

(18)

Remark 1

In case that coalition membership is statistically independent for every $i\ne j$, i.e., if P is a product measure on $2^N$, the equality holds whenever $i\in S$. Then equations (14), (15), and (18) all evaluate to the same number—for instance, to the Banzhaf value if $P(S)\equiv 2^{-|N|}$. That the “expectation of a difference” in (14) or (15) coincides with the “difference between two expectations” in (18), however, fails to hold in general. In particular, we will show in Corollary 2 that there is no probability distribution P which would allow the Shapley value to be interpreted as measuring informational importance.

Remark 2

The restriction of the prediction value to simple games has been identified by Häggström et al. (2006) as playing a key role in extending the Condorcet jury theorem (on asymptotically correct simple majority decisions by n statistically independent voters) to weighted majority decisions with arbitrary joint vote distributions.¹¹ Häggström et al. call the difference between expectations based on P|i and $P|\lnot i$ the “effect” of voter i. Their observation that player i’s effect can be interpreted as a normalized form of the correlation between i’s vote and the random jury outcome extends straightforwardly to probabilistic TU games. Namely, writing $\mathbf {1}_{\{i\in S\}}$ for the indicator function of event $\{i\in S\}$ and $p_i=P(\{S:i\in S\})$, the covariance of v(S) and $\mathbf {1}_{\{i\in S\}}$ is

$$\begin{aligned} \text {Cov}(v(S),\mathbf {1}_{\{i\in S\}})= & {} {{\mathbb {E}}}_{P}[[v(S)\cdot \mathbf {1}_{\{i\in S\}}]-{{\mathbb {E}}}_{P}[v(S)]\cdot {{\mathbb {E}}}_{P}[\mathbf {1}_{\{i\in S\}}] \end{aligned}$$

(19)

$$\begin{aligned}= & {} {{\mathbb {E}}}_{P}[v(S)\cdot (\mathbf {1}_{\{i\in S\}}-{{\mathbb {E}}}_{P}[\mathbf {1}_{\{i\in S\}}])] \nonumber \\= & {} p_i {{\mathbb {E}}}_{P}[v(S)\cdot (1-p_i )| i\in S] +(1-p_i){{\mathbb {E}}}_{P}[-p_i v(S)| i\notin S] \nonumber \\= & {} p_i(1-p_i)\cdot \xi _i(N,v,P). \end{aligned}$$

(20)

Dividing by standard deviations $\sigma _{v(S)}$ and $\sigma _{\mathbf {1}_{\{i\in S\}}}=\sqrt{p_i(1-p_i)}$, the PV and the correlation coefficient $\text {Corr}(v(S),\mathbf {1}_{\{i\in S\}})$ can be seen to satisfy

$$\begin{aligned} \xi _i(N,v,P)= \text {Corr}(v(S),\mathbf {1}_{\{i\in S\}})\cdot \frac{\sigma _{v(S)}}{\sqrt{p_i(1-p_i)}}. \end{aligned}$$

(21)

4 Characterizing the prediction value

We will now provide 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 $G^{\prime }=(N^{\prime },v^{\prime },P^{\prime })$ related through a bijection $\pi :N\rightarrow N^{\prime }$ such that for all $S\subseteq N$, $v(S)=v^{\prime }(\pi S)$ and $P(S)=P^{\prime }(\pi S)$ where $\pi S:=\{\pi (i)|i\in S\}$. An extended value $\varphi$ is anonymous if for every such G and $G^{\prime }\in {\mathcal {PG}}$

$$\begin{aligned} \varphi _i(N,v,P)=\varphi _{\pi (i)}(N^{\prime },v^{\prime },P^{\prime }) \quad \text { for all }i\in N. \end{aligned}$$

(22)

Definition 3

An extended value $\varphi$ is linear if for all $(N,v,P),(N,v^{\prime },P)\in {\mathcal {PG}}$ and real constants $\alpha ,\beta$

$$\begin{aligned} \varphi (N, \alpha v+\beta v^{\prime },P) =\alpha \varphi \left( N, v,P\right) +\beta \varphi (N, v^{\prime },P) . \end{aligned}$$

(23)

Linearity combines two properties, scale invariance and additivity. Especially the latter is far from being innocuous.¹² 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 $G^{\prime }$ compare when one can be viewed as a reduced form of the other.

Definition 4

Given $G=(N,v,P)\in {\mathcal {PG}}$ and a dependent player $i\in N$, the probabilistic game $G_{-i}=(N_{-i},v_{-i},P_{-i})\in {\mathcal {PG}}$ is a reduced game derived from G by removal of i if $N_{-i}=N{\setminus } i$, and for all $S\subseteq N{\setminus } i$ ¹³

$$\begin{aligned} P_{-i}(S)&=P(S)+P(S\cup i) \end{aligned}$$

(24)

$$\begin{aligned} v_{-i}\left( S\right)&= {\left\{ \begin{array}{ll} \frac{P(S)}{P(S)+P(S\cup i)}\cdot v( S) +\frac{P(S\cup i)}{P(S)+P(S\cup i)} v( S\cup i) &{} \text {if }P_{-i}(S) >0,\\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(25)

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 $S\subseteq N_{-i}$ is the convex combination of the associated old worths, v(S) and $v(S\cup i)$, weighted according to the respective probabilities under P. We will require that the extended value of any player $j\in N_{-i}$ stays unaffected by the removal of i.¹⁴

Definition 5

An extended value $\varphi$ is consistent if for all $G=(N,v,P)\in {\mathcal {PG}}$ and all dependent players $i\in N$ in v, we have $\varphi _j(G)=\varphi _j(G_{-i})$ for all $j\in N{\setminus } i$.

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 $G^{\prime }=(N^{\prime }, v^{\prime }, P^{\prime })$, where the presence of a lobbyist i has been neglected. The more accurate model would include the lobbyist and be $G=(N^{\prime }\cup i, v, P)$. 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 $S\cup i$ 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 $P^{\prime }$ and value $v^{\prime }$ of each coalition $T\subseteq N^{\prime }$ in the ‘misspecified’ game without i are defined in a probabilistically correct way, i.e., if the misspecified game $G^{\prime }$ equals $G_{-i}$, then the assessment of any actor $j\ne i$ 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.

Proposition 1 is not enough to fully characterize the PV. For instance, for every $a,b\in {\mathbb {R}}$, the extended value ${\varphi }_i^{(a,b)}(N,v,P)= a\cdot {{\mathbb {E}}}_{P|i}[v(S)]+ b\cdot {{\mathbb {E}}}_{P|\lnot i}[v(S)]$ satisfies anonymity, linearity and consistency.¹⁵

Our characterization in Theorem 1 below will use that if an extended value is linear and consistent, it is fully determined by its image for the subclass of all 2-player probabilistic games.¹⁶ The question then is how the extended values of 2-player probabilistic games should suitably be restricted.

Before giving an answer 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 if the behavior of $N{\setminus } i$ and of i are not statistically independent. In case of independence, i.e., if the presence of $i\in N$ 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. 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:¹⁷

Definition 6

An extended value $\varphi$ satisfies the informational dummy-dictator property (IDDP) if for $i\in N$ and $|N|=2$

$$\begin{aligned} \varphi _{i}(\{i,j\},u_{j},P)=P|i(ij)-P|\lnot i(j). \end{aligned}$$

(26)

Regarding dictators themselves it makes sense to impose the following for 1-player probabilistic games:

Definition 7

An extended value $\varphi$ satisfies full control if $\varphi _i(\{i\},u_i,P)=1$ for all i, P where $P(\{i\})>0$, and $\varphi _i(\{i\},u_i,P)=0$ otherwise.

This formalizes that if N consists of just a single player $i\in {\mathbb {N}}$ 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. Immediately from the definition of the PV we obtain.

Proposition 2

The prediction value satisfies full control and IDDP.

Remark 3

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 $\phi$ will not coincide because the Shapley value of null players is zero.

We have the following characterization result:

Theorem 1

There is a unique extended value $\varphi$ which satisfies linearity, consistency, full control and IDDP. It is anonymous and $\varphi \equiv \xi$.

We note that the four properties in Theorem 1 are independent and non-redundant.

Lemma 1

• (i) The extended value

  $$\begin{aligned} \varphi _{i}^{1}(N,v,P)=\sum _{ S\subseteq N, i\in S} \alpha _{S}\cdot {{\mathbb {E}}}_{P|i}\left( u_{S}\right) , \end{aligned}$$

  (27)

  with $v=\sum _{S\subseteq N}\alpha _{S}\cdot u_{S}$ being the unique decomposition of v into unanimity games, satisfies linearity, consistency, full control but not IDDP.

• (ii) The extended value

  $$\begin{aligned} \varphi _{i}^{2}(N,v,P)=\xi _{i}(N,v,P)-\varphi _{i}^{1}(N,v,P) \end{aligned}$$

  (28)

  satisfies linearity, consistency, IDDP   but not full control.

• (iii) The extended value

  $$\begin{aligned} \varphi _{i}^{3}(N,v,P)=\sum _{\begin{array}{c} S \subset N, \\ i \in S,|S|\le 2 \end{array}}v(S)\cdot P|i(S)-\sum _{\begin{array}{c} T \subset N, \\ i \notin T,|T|\le 2 \end{array}}v(T)\cdot P|\lnot i(T) \end{aligned}$$

  (29)

  satisfies linearity, full control, IDDP but not consistency.

• (iv) Let $|N|\ge 3$ and $v=\sum _{S\subseteq N}\alpha _{S}\cdot u_{S}$. The extended value

  $$\begin{aligned} \varphi _{i}^{4}(N,v,P)=\sum \limits _{S\subseteq N,\alpha _{S}\ne 0}\xi _{i}(N,u_{S},P) \end{aligned}$$

  (30)

  satisfies consistency, full control, IDDP but not linearity.

5 Relation between prediction value and probabilistic values

The example values discussed earlier (like $\phi ,\beta ,\Phi ^+, \Phi ^-$) all are 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 $P(S)\equiv 2^{-|N|}$. We now study the relationship between (extended) probabilistic values and the prediction value somewhat more generally.

Following Weber (1988), the class of probabilistic values is characterized by (i) linearity, (ii) positivity: $\varphi (N,v,P)\ge 0$ if v is monotonic and (iii) the dummy player property: $\varphi _i(N,v,P)=v(i)$ if i is a dummy. The PV is not a probabilistic value. It is linear (see Proposition 1) but violates (ii) and (iii).¹⁸

Theorem 2

Let $\Psi$ be a probabilistic value as defined in (6). The identity $\Psi (\cdot ,Q)\equiv \xi (\cdot ,P)$ holds for $n>1$ if and only if there exist probabilities $0< \tilde{p}_i<1$ for each player such that

$$\begin{aligned} Q_i(S\cup i)=P|i(S\cup i)=\prod _{j\in S}\tilde{p}_j\cdot \prod _{j\in N\backslash (S \cup i)} (1-\tilde{p}_j) \end{aligned}$$

(31)

holds for all $S\subseteq N\backslash i$, $i\in N$ and

$$\begin{aligned} P(S)=\prod _{j\in S}\tilde{p}_j\cdot \prod _{j\in N\backslash S} (1-\tilde{ p}_j) \end{aligned}$$

(32)

holds for all $S\subseteq N$.

The theorem shows that a probabilistic value $\Psi$ is a PV if and only if it is a multinomial value $g^{\tilde{p}}$, i.e., satisfies (11), for an interior vector of probabilities $\tilde{p}=(\tilde{p}_1,\ldots ,\tilde{p}_n)$. This also clarifies the connection between semivalues and the prediction value. Namely, a semivalue $f^q$ can be interpreted as a PV if and only if $f^q$ is also a multinomial value $g^{\tilde{p}}$. Identity $f^q\equiv g^{\tilde{p}}$ and the fact that weightings $Q_i(S)$ can depend only on |S| for semivalues imply (i) $\tilde{p}_i=\tilde{p}_j=:p$ for all $i,j\in N$ in Theorem 2 and (ii) q satisfies (10) for p. So we obtain:

Corollary 1

For a given semivalue $f^q$ and $n>1$ there exists P such that $f^q(\cdot )\equiv \xi (\cdot , P)$ on ${\mathcal {G}}^N$ if and only if $f^q$ is a binomial semivalue, i.e., q satisfies (10) for some $p\in (0,1)$.¹⁹

Because the Shapley value $\phi$ is no binomial semivalue and differs from any such value on ${\mathcal {G}}^N$ when $|N|\ge 3$, we have:

Corollary 2

For $n\ge 3$ there exists no P such that $\phi (\cdot )\equiv \xi (\cdot ,P)$ on ${\mathcal {G}}^N$.

6 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.

Table 1 Values in the Dutch Parliament

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 $\beta$ and $\phi$.

We used the parliamentary information system Parlis ²⁰ 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.²¹ Defining P by these empirical frequencies, we calculated the corresponding prediction values $\xi _i$ of the parties as well as their positive and negative conditional decisiveness values $\Phi ^+_i$ and $\Phi ^-_i$ defined in (14) and (15). A summary of the results is given in the bottom part of Table 1.

The PV-scores $\xi _i$ of Dutch parties tend to be higher than their respective traditional Banzhaf or Shapley power measures $\beta _i$ and $\phi _i$, and even the decisiveness measures $\Phi ^+_i$ and $\Phi ^-_i$ 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.

Table 2 Correlation coefficients for 2008–2010 votes in Dutch Parliament

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 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 far right-wing, the latter as a far 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).

7 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 formalized by Theorem 2, 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 formal fact of 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.²²

We note that the prediction value does not distinguish between correlation and causation in cases of interdependence. 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 $j\ne i$ ‘follow’ i and cast their weight as their supreme leader i would if he had any, (ii) $i\ne k$ and all players $j\ne k$ 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 inclination.²³ Since knowing i’s decision—rather than i’s initial inclination—will always fully reveal the realized outcome, we regard finding $\xi _i=1$ in all three scenarios a feature, not a flaw.

However, this example points to an interesting extension of the described “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) the given proposal 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,²⁴ and plan to pursue this extension in future research.

Appendix

1.1 Proof of Proposition 1

Anonymity and linearity of $\xi$ are obvious from Definition 1. To prove consistency, consider $(N,v,P)\in {\mathcal {PG}}$ and let $i\in N$ be dependent in v. Let $j\in N{\setminus } i$ and $S\subseteq N{\setminus } ij$. Equations (12) and (24) imply that

$$\begin{aligned} P_{-i}|j( S\cup j)= P|j( S\cup j) +P|j(S\cup i\cup j) \end{aligned}$$

(33)

and

$$\begin{aligned} P_{-i}|\lnot j( S)= P|\lnot j( S) +P|\lnot j( S\cup i). \end{aligned}$$

(34)

By using the definition of $v_{-i}$ and invoking equality (33) one can verify that

$$\begin{aligned} P_{-i}|j( S\cup j) v_{-i}( S\cup j) \!=\! P|j(S\cup i\cup j) v( S\cup i\cup j) \!+\! P|j( S\cup j) v( S\!\cup \!j). \end{aligned}$$

(35)

Similarly, by definition of $v_{-i}$ together with (34), we get

$$\begin{aligned} P_{-i}|\lnot j\left( S\right) v_{-i}\left( S\right) = P|\lnot j\left( S\cup i\right) v\left( S\cup i\right) + P|\lnot j\left( S\right) v\left( S\right) . \end{aligned}$$

(36)

One can then infer

$$\begin{aligned} \xi _j(N_{-i},v_{-i},P_{-i})&=\sum _{S\subseteq N{\setminus } ij}\big \{ P_{-i}|j\left( S\cup j\right) v_{-i}\left( S\cup j\right) - P_{-i}|\lnot j\left( S\right) v_{-i}\left( S\right) \big \} \\&=\sum _{S\subseteq N{\setminus } ij}[ \{ P|j\left( S\cup i\cup j\right) v\left( S\cup i\cup j\right) +P|j\left( S\cup j\right) v\left( S\cup j\right) \}\\&\qquad -\big \{ P|\lnot j\left( S\cup i\right) v\left( S\cup i\right) +P|\lnot j\left( S\right) v\left( S\right) \big \} ] \\&=\sum _{S\subseteq N{\setminus } j}\big \{ P|j\left( S\cup j\right) v\left( S\cup j\right) -P|\lnot j\left( S\right) v\left( S\right) \big \} \\&= \xi _j(N,v,P), \end{aligned}$$

where the second equality uses (35) and (36), and the third one follows by shifting the corresponding terms from inside the square brackets to the outer summation.

1.2 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 2-player probabilistic games and hence probabilistic games in general.

Lemma 2

Consider an extended value $\varphi$ that is linear on the space of all 2-player probabilistic games and consistent. For any set N with $|N|=2$, the mapping $(N,v,P)\mapsto \varphi (N,v,P)$ is fully determined by the numbers

$$\begin{aligned} x_{ij}:=\varphi _i(N,u_j,P)\quad \text { for }\ i,j\in N. \end{aligned}$$

(37)

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 $(N,v)\in {\mathcal {G}}^N$ there are constants $\alpha _i,\alpha _j,\alpha _{ij}$ such that

$$\begin{aligned} v\equiv \alpha _iu_i+\alpha _ju_j+\alpha _{ij}u_{ij}. \end{aligned}$$

(38)

And thus, for arbitrary P and $i\in N$, $\varphi$’s linearity implies

$$\begin{aligned} \varphi _i(N,v,P)=\alpha _i \underbrace{\varphi _i(N,u_i,P)}_{:=x_{ii}}+\alpha _j \underbrace{\varphi _i(N,u_j,P)}_{:=x_{ij}} +\alpha _{ij}\underbrace{ \varphi _i(N,u_{ij},P)}_{:=x_{i,ij}}. \end{aligned}$$

(39)

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

$$\begin{aligned} \begin{array}{ll} P_{-j}(\varnothing )=P(\varnothing )+P(j), &{} \quad P_{-j}(i)=P(i)+P(ij), \\ (u_{ij})_{-j}(\varnothing )=0, &{} \quad (u_{ij})_{-j}(i)= {\left\{ \begin{array}{ll} \frac{P(ij)}{P(i)+P(ij)} &{} \text { if }P(i)+P(ij)>0, \\ 0 &{} \text { otherwise.} \end{array}\right. } \end{array} \end{aligned}$$

(40)

In case $P(i)+P(ij)>0$, we have

$$\begin{aligned} \varphi _{i}(N,u_{ij},P)&= \varphi _i\big (\{i\},\tfrac{P(ij)}{P(i)+P(ij)} \cdot u_{i} ,P_{-j}\big ) \nonumber \\&= \tfrac{P(ij)}{P(i)+P(ij)}\cdot \varphi _i(\{i\},u_{i},P_{-j}) \nonumber \\&= \tfrac{P(ij)}{P(i)+P(ij)}\cdot \varphi _{i}(N,u_{i},P) = \tfrac{P(ij)}{ P(i) +P(ij)}\cdot x_{ii}, \end{aligned}$$

(41)

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 $\varphi _{i}(N,u_{ij},P)=0$ because in this case $(u_{ij})_{-j}(\{i\})=0$ by Definition 4, so that $(u_{ij})_{-j}$ is the all-zero game $\mathbf {0}$ in that case. Consistency requires $\varphi _{i}(N,u_{ij},P)=\varphi _{i}(\{i\},(u_{ij})_{-j},P_{-j})= \varphi _{i}(\{i\},\mathbf {0},P_{-j})=0$ due to linearity.

In summary,

$$\begin{aligned} x_{i,ij}= \left\{ \begin{array}{ll} \tfrac{P(ij)}{P(i)+P(ij)}\cdot x_{ii} &{} \quad \text {if }P(i)+P(ij)>0, \\ 0 &{}\quad \text {otherwise.} \end{array}\right. \end{aligned}$$

(42)

And in a similar fashion one obtains

$$\begin{aligned} x_{j,ij}= {\left\{ \begin{array}{ll} \tfrac{P(ij)}{P(j)+P(ij)}\cdot x_{jj} &{} \text {if }P(j)+(Pij)>0, \\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(43)

$\square$

For any $v\equiv \alpha _iu_i+\alpha _ju_j+\alpha _{ij}u_{ij}$ we have

$$\begin{aligned} \varphi _i(N,v,P)= {\left\{ \begin{array}{ll} \alpha _j\cdot x_{ij}+\left( \alpha _i+\tfrac{\alpha _{ij}\cdot P(ij)}{P(i)+P(ij) }\right) \cdot x_{ii} &{}\quad \text {if }P(i)+P(ij)>0, \\ \alpha _j\cdot x_{ij}+\alpha _i\cdot x_{ii} &{}\quad \text {otherwise} \end{array}\right. } \end{aligned}$$

(44)

and an analogous expression for $\varphi _j(N,v,P)$. This finding can be generalized from just two players to arbitrary N:

Lemma 3

Let $\varphi$ be a consistent and linear extended value. Then the mapping $(N,v,P)\mapsto \varphi (N,v,P)$ is fully specified by the parameters in (37).

Proof

Using the n-player unanimity games as a basis for ${\mathcal {PG}}^N$ one can always write

$$\begin{aligned} v\equiv \sum \limits _{\varnothing \subsetneq T\subseteq N} \alpha _Tu_T. \end{aligned}$$

(45)

Letting $i\in N$ be an arbitrary but fixed player, we will use induction on n in order to prove the following

Claim: There exist $\beta _{ij}$, depending on the $\alpha _T$ and P, such that

$$\begin{aligned} \varphi _i(N,v,P)=\sum _{j=1}^{n} \beta _{ij}x_{ij} \quad \text { where } x_{ij}:=\varphi _i(N,u_j,P). \end{aligned}$$

(46)

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 $|N|\ge 3$, $|T|\ge 2$ and assume that the statement is true for all player sets N of cardinality $n-1$. Let $j\in N{\setminus } i$ be a player, which must be dependent in $u_{T}$ because $|T|\ge 2$. Now we consider the reduced game $(N_{-j},(u_{T})_{-j},P_{-j})$. From consistency we conclude

$$\begin{aligned} \varphi _{i}(N,u_{T},P)=\varphi _{i}(N_{-j},(u_{T})_{-j},P_{-j}). \end{aligned}$$

Applying the induction hypothesis implies the existence of $\beta _{ik}^{\prime }$, which depend on $P_{-j}$ and hence on P, such that

$$\begin{aligned} \varphi _{i}(N,u_{T},P)=\sum _{k=1,k\ne j}^{n}\beta _{ik}^{\prime }\varphi _{i}(N_{-j},u_{k},P_{-j}). \end{aligned}$$

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 $1\le k\le n$ with $j\ne k$. Inserting $\varphi _{i}(N_{-j},u_{k},P_{-j})=\varphi _{i}(N,u_{k},P)=x_{ik}$ then proves the claim . $\square$

We remark that the coefficients $\beta _{ij}$ referred to in the above proof get quite complicated for increasing n. In the following we will use only the fact that they are well-defined given v and P.

Proof of Theorem 1

To complete the proof we now show how the values $x_{ii}=\varphi _i(N,u_i,P)$ and $x_{ij}= \varphi _i(N,u_j,P)$ can be computed from the corresponding values for the player set $N^{\prime }=\{i,j\}$. Since $\left( u_i\right) _{-j}=u_i$ for all $i\ne j$ we can recursively conclude from consistency

$$\begin{aligned} \varphi _i(N,u_i,P)= & {} \varphi _i(\{i,j\},u_i,P^\star ) \text { and} \end{aligned}$$

(47)

$$\begin{aligned} \varphi _i(N,u_j,P)= & {} \varphi _i(\{i,j\},u_j,P^\star ) , \end{aligned}$$

(48)

where

$$\begin{aligned} P^\star (S)=\sum _{T\subseteq N{\setminus } ij} P(S\cup T)\text { for any } S\subseteq \{i,j\} . \end{aligned}$$

(49)

Using equation (49) and similarly defining

$$\begin{aligned} P^{\prime }(S)=\sum _{T\subseteq N{\setminus } i} P(S\cup T)\text { for any } S\subseteq \{i\}, \end{aligned}$$

(50)

we conclude $\varphi _i(\{i\},u_i,P^{\prime })=\varphi _i(\{i,j\},u_i,P^\star )$ 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 $\varphi$ satisfies IDDP   the values of $x_{ij}$ are determined, and hence $\varphi$ is determined on the class of 2-player probabilistic games. Then $\varphi \equiv \xi$ 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.

1.3 Proof of Lemma 1

(i) Linearity is inherited from the expected value ${{{\mathbb {E}}}}_{P|i}$.

To see consistency we first verify that ${ {{{\mathbb {E}}}}}_{P_{-j}|i}( [ {\small { {u_{S}}}}]_{_{-j}}) = { {{{\mathbb {E}}}}}_{P|i}\left( {\small { {u_{S}}}} \right)$ for any probability measure P. If $\sum _{\begin{array}{c} U\subseteq N\backslash j, \\ i\in U \end{array}}P_{-j}(U)=0$, then both sides of the equation are equal to zero. So, we may assume $\sum _{\begin{array}{c} U\subseteq N\backslash j, \\ i\in U \end{array}}P_{-j}(U)>0$ in the following. For the reduced game $G_{-j}=(N_{-j},v_{-j},P_{-j})$ we have

Inserting (25) provides

$$\begin{aligned} { {{{\mathbb {E}}}}}_{P_{-j}|i}[ [ {\small { {u_{S}}}}] _{_{-j}}]= & {} \frac{1}{\sum _{\begin{array}{c} U\subseteq N, \\ i\in U \end{array}}P(U)}\sum _{\begin{array}{c} T\subseteq N\backslash j, \\ i\in T \end{array}} \big \{P(T)\cdot {\small { {u_{S}}}} (T)+P(T\cup j)\cdot {\small { {u_{S}}}}(T\cup j)\big \} \\= & {} \frac{1}{\sum _{\begin{array}{c} U\subseteq N, \\ i\in U \end{array}}P(U)}\sum _{i\in T\subseteq N}P(T)\cdot {\small { {u_{S}}}}(T)= { {{{\mathbb {E}}}}}_{P|i}\left( {\small { {u_{S}}}} \right) . \end{aligned}$$

Note that $v_{-j}=\sum _{S\subseteq N}\alpha _{S}\cdot \left[ {\small { {u_{S}}}}\right] _{_{-j}}$ and hence

$$\begin{aligned} \varphi _{i}^{1} {(N_{-j},v_{-j},P_{-j})}= & {} {\sum _{\begin{array}{c} S\subseteq N\backslash j, \\ i\in S \end{array}}\alpha _{S}\cdot {{{\mathbb {E}}}_{P_{-j}|i}}( \left[ { {u_{S} }}\right] _{_{-j}}) }= { {}\sum _{\begin{array}{c} S\subseteq N\backslash j, \\ i\in S \end{array}}\alpha _{S}\cdot {{{\mathbb {E}}}_{P|i}}\left( { {u_{S}}}\right) } \\= & {} \varphi _{i}^{1} {(N,v,P)} \end{aligned}$$

which confirms consistency of $\varphi ^{1}$.

Full control follows from $\varphi _i^1(\{i\},u_i,P)={{\mathbb {E}}}_{P|i}[u_i(S)] =P|i(\{i\})$. This is 1 if $P(\{i\})>0$ and otherwise 0 by the definition of P|i.

In remains to be shown that $\varphi ^{1}$ does not satisfy IDDP. For the unanimity game $u_{j}$ note that $u_{j}=\sum _{S\subseteq N}\alpha _{S}\cdot u_{S}$ provides

$$\begin{aligned} \alpha _{S}= {\left\{ \begin{array}{ll} 1&{} \text { for }S=\left\{ j\right\} , \\ 0&{} \text { otherwise.} \end{array}\right. } \end{aligned}$$

However, the summation in (27) is over all S with $i\in S$ such that

$$\begin{aligned} \varphi _{i}^{1}(\{i,j\},u_{j},P)=0 \end{aligned}$$

(51)

in contradiction to (26).

(ii) $\varphi ^{2}$ inherits linearity and consistency from $\xi$ and $\varphi ^{1}$. Inserting (51) into (28) provides

$$\begin{aligned} \varphi _{i}^{2}(\{i,j\},u_{j},P)=\xi _{i}(\{i,j\},u_{j},P) \end{aligned}$$

such that $\varphi _{i}^{2}$ inherits IDDP from $\xi _{i}$. As both $\xi$ and $\varphi ^{1}$ satisfy full control we get

$$\begin{aligned} \varphi _{i}^{2}(\{i\},u_{i},P)=0, \end{aligned}$$

in contradiction to Definition 7.

(iii) Linearity is obvious. For $|N|\le 2$ the extended value $\varphi ^{3}$ 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 $\left| N\right| =3$ and perfect correlation $P(N)=1/2=P(\varnothing )$. Here,

$$\begin{aligned} \varphi _{i}^{3}(N,v,P)=0 \quad \text { for all }i\in N. \end{aligned}$$

(52)

However, for the reduced game $G_{-j}=(N_{-j},v_{-j},P_{-j})$ we get

$$\begin{aligned} N_{-j}= & {} N{\setminus } j, \\ P_{-j}(S)= & {} P(S)+P(S\cup j)\text { for all }S\subseteq N{\setminus } j \\= & {} {\left\{ \begin{array}{ll} 1/2 &{} \text { for }S\in \left\{ N\backslash j,\varnothing \right\} , \\ 0 &{} \text { otherwise,} \end{array}\right. } \\&\\ v_{-j}\left( S\right)= & {} {\left\{ \begin{array}{ll} v(N) &{} \text {for }S=N\backslash j, \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

For $\varphi ^{3}$ follows

$$\begin{aligned} \varphi _{i}^{3}(N_{-j},v_{-j},P_{-j})=v(N)-v(\varnothing )=v(N)\text { for all } i\in N \end{aligned}$$

which contradicts (52).

(iv) Consider the reduced game $G_{-j}=(N_{-j},v_{-j},P_{-j})$. PV is consistent and therefore

$$\begin{aligned} \xi _{i}(N_{-j},(u_{S})_{-j},P_{-j})=\xi _{i}(N,u_{S},P) \quad \text { for all }i\in N{\setminus } j\text {.} \end{aligned}$$

We conclude

$$\begin{aligned} \varphi _{i}^{4}(N_{-j},v_{-j},P_{-j})= & {} \sum \limits _{S\subseteq N,\alpha _{S}\ne 0}\xi _{i}(N_{-j},(u_{S})_{-j},P_{-j}) \\= & {} \sum \limits _{S\subseteq N,\alpha _{S}\ne 0}\xi _{i}(N,u_{S},P)=\varphi _{i}^{4}(N,v,P) \quad \text { for all }i\in N{\setminus } j \end{aligned}$$

which confirms consistency of $\varphi _{i}^{4}$.

Full control and IDDP   follows from $\varphi _{i}^{4}(\{i\},u_{i},P)=\xi _{i}(\{i\},u_{i},P)$ and $\varphi _{i}^{4}(\{i,j\},u_{j},P)=\xi _{i}(\{i,j\},u_{j},P)$.

To verify that $\varphi _{i}^{4}$ is not linear put $w=\sum \nolimits _{S\subseteq N}\beta _{S}\cdot u_{S}$.

$$\begin{aligned} \varphi _{i}^{4}(N,v+w,P)=\sum \limits _{S\subseteq N,\alpha _{S}+\beta _{S}\ne 0}\xi _{i}(N,u_{S},P) \end{aligned}$$

which is in general not equal to

$$\begin{aligned} \sum \limits _{S\subseteq N,\alpha _{S}\ne 0}\xi _{i}(N,u_{S},P)+\sum \limits _{S\subseteq N,\beta _{S}\ne 0}\xi _{i}(N,u_{S},P). \end{aligned}$$

$\square$

1.4 Proof of Theorem 2

The proof is based on three insights, stated in Lemmas 4–6.

Lemma 4

From $\Psi (\cdot ,Q)\equiv \xi (\cdot ,P)$ follows $Q_i(S)=P|i(S)$ for all $\{i\}\subseteq S\subseteq N$.

Proof

For an arbitrary subset $\{i\}\subseteq S\subseteq N$ we consider the unanimity game $u_S$ and obtain the formulas

and

$$\begin{aligned} \Psi _{i}(u_S,Q)=\sum \limits _{\{i\}\subseteq T\subseteq N}Q_i(T)[ u_S(T)-u_S(T\backslash i)]=\sum \limits _{T:S\subseteq T} Q_i(T). \end{aligned}$$

Now we prove the proposed statement by induction on the subsets S in decreasing order of their cardinalities using the assumption $\xi _i(u_S,P)=\Psi _{i}(u_S,Q)$. For the induction start $S=N$ we have $P|i(N)=Q_i(N)$. Using the induction hypothesis for all $S^{\prime }\subseteq N$ with $|S^{\prime }|>|S|$ yields $P|i(S)=Q_i(S)$. $\square$

Lemma 5

From $\Psi (\cdot ,Q)\equiv \xi (\cdot ,P)$ follows $P|i(U)=P|\lnot i(U\backslash i)$ for all $\{i\}\subseteq U\subseteq N$ with $|U|\ge 2$.

Proof

We set $U=N\backslash S\cup i$ so that we have to prove $P|i(N\backslash S\cup i)=P|\lnot i(N\backslash S)$ for all subsets $\{i\}\subseteq S\subsetneq N$.

For fixed S we consider the unanimity game $u_{N\backslash S}$ and obtain the formulas

and

$$\begin{aligned} \Psi _{i}(u_{N\backslash S},Q)=\sum \limits _{T\ni i}Q_i(T)\Big [u_{N\backslash S}(T)-u_{N\backslash S}(T\backslash i)\Big ] =0. \end{aligned}$$

Now we prove the proposed statement by induction on the subsets S in increasing order of their cardinalities using the assumption $\xi _i(u_S,P)=\Psi (u_S,Q)$. For the induction start $S=\{i\}$ we have $P|i(N)-P|\lnot i(N\backslash i)=0$, which is equivalent to $P|i(N)=P|\lnot i(N\backslash \{i\})$. Using the induction hypothesis for all $S^{\prime }\subseteq N$ with $|S^{\prime }|<|S|$ yields $P|i(N\backslash S\cup i)=P|\lnot i(N\backslash S)$. $\square$

Put $p_i:=\sum _{T\ni i }P(T)\in [0,1]$ for all $i\in N$. Whenever $p_i>0$ we have $P|i(S)=\frac{P(S) }{p_i}$ for all $\{i\}\subseteq S\subseteq N$ and $P|i(S)=0$ in all other cases. The next lemma excludes the case $p_i=1$ for at least two players.

Lemma 6

If $\Psi (\cdot ,Q)\equiv \xi (\cdot ,P)$ and if there exists an index $i\in N$ with $p_i=1$, then $n=1$.

Proof

From $p_i=\sum _{T\ni i}P(T)=1$ we conclude $P(S)=0$ for all $S\subseteq N\backslash i$. Thus we have $P|i(T)=0$ for all $T\ni i$ with $|T|\ge 2$ due to Lemma 5. This yields $P(\{i\})=$ $Q_i(\{i\})=1$ and $Q_j(S)=0$ for all $S\ni j$, where $(S,j)\ne (\{i\},i)$, and all $j\in N$ due to Lemma 4. For each $j\in N\backslash i$ we then have $\sum _{S\ni j} Q_j(S)=0\ne 1$ – a contradiction. $\square$

Proof of Theorem 2

From Lemma 6 we conclude

$$\begin{aligned} 0\le p_i:=\sum \limits _{T\ni i}P(T)<1 \end{aligned}$$

for all $i\in N$. If $p_i=0$ for an index $i\in N$, 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 choose $\tilde{p} _i=p_i$.

We have

$$\begin{aligned} P(S)=\frac{p_i}{1-p_i}\cdot P(S\backslash i) \end{aligned}$$

for all $S\ni i$ with $|S|\ge 2$ due to Lemma 5 and $p_i>0$.

Thus inductively we obtain

$$\begin{aligned} P(S)=\prod _{j\in S\backslash i} \frac{p_j}{1-p_j}\cdot P(\{i\}) \end{aligned}$$

for all $i\in N$ and all subsets $S\ni i$ of N.

Inserting the previous equations into $p_i=\sum \limits _{S\ni i}P(S)$ yields

$$\begin{aligned} p_i=P(\{i\})\cdot \sum \limits _{S\not \ni i} \prod _{j\in S} \frac{p_j}{1-p_j}=P(\{i\})\cdot \prod _{j\in N\backslash i} \left( \frac{p_j }{1-p_j}+1\right) =P(\{i\})\cdot \prod _{j\in N\backslash i}\frac{1}{1-p_j}. \end{aligned}$$

Thus we have

$$\begin{aligned} P(\{i\})=p_i\,\cdot \,\prod _{j\in N\backslash i}(1-p_j), \end{aligned}$$

which then yields

$$\begin{aligned} P(S)=\prod _{j\in S} p_j\,\cdot \,\prod _{j\in N\backslash S}(1-p_j) \end{aligned}$$

(53)

for all $\varnothing \ne S\subseteq N$. By using $\sum _{S\subseteq N}P(S)=1$ we conclude that equation (53) 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 $i\in N$ and define

$$\begin{aligned} P(S)=\prod _{j\in S}p_j\cdot \prod _{j\in N\backslash S} \left( 1-p_j\right) , \end{aligned}$$

i.e., P is a product measure. Next set

$$\begin{aligned} Q_i(S\cup i)=P|i(S\cup i)=\prod _{j\in S}p_j\cdot \prod _{j\in N\backslash (S\cup i)} \left( 1-p_j\right) , \end{aligned}$$

for $i\in N\backslash S$, i.e. the $Q_i(S\cup i)$ derive from the same product measure. We can easily verify $P|i(S)=P|\lnot i(S \backslash i)$ for all $S\ni i$ and all $i\in N$. Inserting this into the definition of the prediction value provides $\xi (\cdot ,P)=\Psi (\cdot ,Q)$. $\square$