Homo Oecon DOI 10.1007/s41412-016-0031-2 RESEARCH PAPER

A Well-Behaved Index of a Priori P-Power for Simple N-Person Games Dan S. Felsenthal1,2

Received: 1 March 2016 / Accepted: 1 October 2016  Springer International Publishing Switzerland 2016

Abstract As far as we know, all previously suggested indices of a priori P-power for simple n-person games violate one or more postulates that are considered compelling for a reasonable P-power index. A new index is proposed in this paper which satisfies all these postulates. Keywords Indices of voting power  N-person simple games  I-power  P-power  Postulates for a reasonable P-power index  Winning coalitions of least size JEL Classification Codes C71  C78

1 Introduction A simple voting game—briefly, SVG—is a collection W of subsets of a finite set N, satisfying the following three conditions: 1. 2. 3. 4.

N [ W; [ 62 W; Monotonicity: whenever X ( Y ( N and X [ W then also Y [ W. W is said to be a proper SVG if, in addition, it satisfies the condition. Whenever X [ W and Y [ W then X \ Y = [. Otherwise, W is said to be improper.

& Dan S. Felsenthal [email protected] 1

School of Political Sciences, University of Haifa, 199 Abba Hooshi Boulevard, Mount Carmel, 3498838 Haifa, Israel

2

Voting Power and Procedures Programme, London School of Economics and Political Science, London, England, UK

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We shall refer to N, the largest set in W, as the latter’s assembly. The members of N are the voters of W. A set of voters (that is, a subset of N) is called a coalition of W. A coalition S is said to be a winning or losing coalition, according as S [ W or S 62 W. The bargaining problem in simple n-person cooperative voting games can be phrased as follows: given an assembly of n persons (n [ 2) bargaining over the division of a fixed (divisible) prize worth, say, 1 unit of transferable utility (TU) (e.g., money), what is the expected share of each of the n persons if: • •

•

The sole purpose of every person is to maximize his or her expected share. Any group of persons can divide the prize among its members if and only if the group constitutes a winning coalition and if the members of the group reached a (prior) agreement as to how to divide the prize among themselves. The possible winning coalitions are known in advance but only one of them can actually form, i.e., the game is proper.1

To date the n-person bargaining problem has not been conclusively solved because no definitive and realistic general theory of bargaining for cooperative games with more than two players exists. Therefore the various indices proposed to date for estimating the expected share in the fixed prize of the members in an nperson cooperative game—which, following Felsenthal and Machover (1998, ch. 6), we shall call P-power indices—must be assessed by examining which postulates— i.e., intuitively compelling conditions—they satisfy. Failure to satisfy these conditions is a suspect counter-intuitive behavior, which can be regarded as paradoxical or, in extreme cases, pathological, and may indicate that a P-power index guilty of it must be discarded. In the next section we list six postulates which seem to us to be compelling that a reasonable P-power index should satisfy. In the third section we propose a P-power index that satisfies all these postulates. In the fourth section we list several indices that were considered in the literature as possible P-power indices and indicate for each of them which of the six postulates they fail to satisfy. The fifth section concludes.

2 Postulates for a Reasonable a Priori P-Power Index Following Felsenthal and Machover (1995, 1998, ch. 7), as well as Felsenthal et al. (1998), a reasonable P-power index should satisfy the following six postulates:

1 The winning coalitions are either listed—in which case the SVG is not weighted, or it may be possible to assign to every person i a non-negative integer weight, wi, and in order for a group of persons to constitute a winning coalition its members’ weights must sum to at least some quota q—which is called the decision rule. In this case we have a weighted voting game—briefly WVG—and in order to preclude the possibility that two or more disjoint winning coalitions can form the quota q is set such that n P q  1=2ð wi Þ þ 1.

i¼1

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2.1 Dummy, Ordinary Voter, Blocker (aka Vetoer) and Dictator A reasonable P-power index should award no power (0) to a dummy—i.e., to a voter who does not belong to any minimal winning coalition (MWC), and it should award the entire power (1) to a dictator,—i.e., to a voter who constitutes the sole MWC. A voter who belongs to every MWC is a blocker (aka vetoer) because no coalition can win without him or her; hence the P-power of a blocker ought to be equal to or larger than that of an ordinary voter,—i.e., a voter who belongs to some, but not all MWCs,– but smaller than that of a dictator.2 2.2 Monotonicity The postulate of monotonicity requires that in a weighted voting game (WVG) the voting powers of any two voters must not be in reverse order to their weights: if voter b has greater weight than a, then b must not have less voting power than a. The justification for this, as far as a priori P-power is concerned, is intuitively obvious: b having greater weight implies that b’s bargaining position is at least as good as a’s. 2.3 Donation Consider two weighted voting games, U and V, with the same voters and the same quota, and weights that differ in only one respect: the weight of voter a in V is greater by some positive amount than in U, whereas the weight of voter b in V is smaller by the same amount than b’s weight in U. In other words, V is obtained from U by a ‘donation’ (or transfer) of some of b’s weight to a. The donation postulate stipulates that a’s voting power in V should not be smaller than in U. Again, the justification is obvious: receiving a donation of a quantum of weight should not be detrimental to the recipient’s bargaining position. 2.4 Annexation A postulate weaker than the donation postulate concerns a special case of the donation postulate: we now assume that b is not a dummy in the WVG U and the donation consists of the whole of b’s weight. So in V b becomes a dummy—and can therefore be ignored3—and a’s weight is the sum of the weights that a and b had in U. In other words, a has annexed the whole of b’s voting rights (or weight). The 2

Note that, contrary to what Felsenthal and Machover (1998, ch. 3) called a reasonable I-power index, we are not requiring that under a reasonable P-power index the voting power of a blocker ought to be always larger than that of an ordinary voter, or that the voting power of an ordinary voter should always be positive. This is so because we recognize the possibility that under certain types of weighted voting games (WVGs) some ordinary voters may be excluded a priori from being members in a winning coalition and hence become de facto dummies, while other ordinary voters may become, as a result, de facto blockers. We discuss and exemplify this possibility in the next section.

3

All measures and indices of voting power considered here, or proposed in the literature, have the property that removing a dummy voter from a SVG leaves the values assigned to the other voters unchanged.

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bloc postulate, as applied to WVGs, says that a’s voting power in V (following the annexation) must be at least as great as the power that voter had in U (prior to the annexation).4 2.5 Blocker’s Share An index of relative voting power satisfies the blocker’s share postulate (BSP) if whenever v is a blocker (aka vetoer) in a simple voting game (SVG) U and the least size of a winning coalition of U is k,5 then the value that the index assigns to v is at least 1/k. The BSP seems compelling for an index of P-power for the following reason. A blocker v has bargaining power second to none, because no winning coalition can form without v. If a winning coalition S of least size, k, is about to form, v can surely insist on a payoff at least as large as that of any other member, hence at least 1/ k. Also v has no reason to allow a winning coalition T whose size is larger than k to form, unless the payoff it guarantees to v is no worse than s/he would have received from S. 2.6 Added Blocker Let U be a SVG with assembly N. Let v be a new voter, not a member of N. We say that the SVG V is obtained from U by adding v as a blocker if the assembly of V is N [ {v}, obtained from N by adding v as a new member; and the winning coalitions of V are obtained from those of U by adding v to each of them. Thus a winning coalition of V is of the form A [ {v}, where A is a winning coalition of U. Clearly, v is a blocker in V. The added blocker postulate (ABP) stipulates that the voting powers of the old voters (members of N) in V should be proportional to their voting powers in U. In other words, if a and b are any two old voters, the ratio between their voting powers in V must be the same as in U. The argument in favor of this requirement is that there is nothing to imply that the addition of the new blocker v is of greater relative advantage to the bargaining strength of some of the voters of U than to others. Of course, the addition of v will certainly mean that the powers of all voters who are not dummies in U must be reduced compared to what it was in U, because v will now take a share of the total power. But they should all be reduced in the same proportion. Note that v is in an extremely strong bargaining position under V: no winning coalition can form without v. So v can demand a certain share, say s, of the total payoff (which is always set as 1), and leave it to the old voters to form a coalition A that would be winning under U, which v would join to form a coalition A [ {v} that wins under V. As far as v is concerned, which A is formed is immaterial, as s/he 4

There is a more general formulation of the bloc postulate, which applies also to SVGs that are not weighted; see Felsenthal and Machover (1998: 255).

5

A winning coalition of least size must obviously be a minimal winning coalition (MWC), i.e., a coalition such that the defection of any member from it renders it to be a losing coalition.

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can insist on getting the same cut, s. Then the old voters, in trying to form A, have the remaining payoff, 1 - s, to bargain about and share among themselves. In doing so, they are in exactly the same relative position to one another regarding the leftover 1 - s as they would have been under U in bargaining for the entire payoff, 1, and deciding how to share it. In the next section we propose a plausible index of P-power which is wellbehaved in the sense that it satisfies all the aforementioned six postulates.

3 A Newly Proposed P-Power Index Our proposed P-power index constitutes a slight modification of the Deegan–Packel (DP) index (1978, 1982) by replacing in its underlying assumptions ‘MWC’ by ‘WCLS’ thus: • • •

Only a winning coalition of least possible size (WCLS) will be formed.6 All possible WCLS have equal probability of being formed. All members of the victorious WCLS receive equal shares of the ‘spoils’.

The first of these assumptions seems plausible from a P-power viewpoint: a noncritical member of a winning coalition is not needed for its victory and hence is not entitled to share in the spoils, and will arguably be excluded from the coalition during the bargaining leading to its formation. Moreover, a winning coalition where each of its members is critical will also strive to be of least size, i.e., that the number of its members will be as small as possible so that the share of each member in the ‘spoils’ will be as large as possible. It is also usually faster and more convenient to conduct negotiations and reach an agreement among members of a small group than among members of a large group. In case there are several possible WCLS—which must, by definition, be of equal size—there is no a priori reason to assume that their formation is not equiprobable, so we assume it is equiprobable. As the defection of any member from an WCLS is critical, it follows that, by definition, within every WCLS every member is equally important for its victory— and hence it is plausible to assume that a victorious WCLS will share the ‘spoils’ equally among its members. It should be emphasized that we are looking for an a priori index of P-power, that is a measure for estimating the average (or expected) payoff that a player in an nperson cooperative game can expect before playing the game rather than the probable final result when the game is actually played. 6

To avoid confusion with Riker’s ‘Size Principle’ (see Riker 1962 pp. 32ff)—which predicts that in a WVG the (winning) coalition that will be formed will be such that the sum of its members’ weights is as small as possible, WCLS in our proposed index means a MWC with fewest members, regardless of their weights. Thus, for example, in the WVG [51; 50, 46, 3, 2] Riker‘s ‘Size Principle’ would predict that the MWC {46,3,2} would form because the sum of its members’ weights is smallest, whereas our proposed index states that as the three MWCs in this WVG {50,46}, {50,3} and {50,2} have the same (smallest) number of members, they all constitute WCLS and hence have the same probability of being formed.

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The proposed index, PI, of a voter v under a simple voting game W, which we shall denote by PIv[W], is defined as the expected value (or probabilistic mean) of v’s payoff. PIv[W] is obtained as follows. Suppose W has m WCLS S1,S2,…,Sm, whose (equal) size is g. By assumption, the probability of the event that Si is formed equals 1/m. Moreover, in this event v’s payoff is 1/g, or 0, according as v belongs or does not belong to Si. So in order to obtain the expected value PIv[W] we must add up the products of the probability 1/m and each of the 1/g possible payoffs. Thus, omitting the terms with payoff 0, we have X PIv ½W  :¼ 1=m 1=g v2Si

Thus, for example, in the WVG [W] = [5; 3,2,1,1] the only WCLS is {3,2} so the expected payoff of each of its two members is 1/2 and that of each of the remaining two members (who are not members of the WCLS) is 0—this despite the fact that they are not strictly dummies. However, one may argue that as the voter with weight 3 is a blocker his/her expected share should be larger than 1/2 because as no winning coalition can form without this voter it is possible for him/ her, apparently, to resort to a tactic of ‘divide and rule’, i.e., to threaten the voter whose weight is 2 that unless s/he agrees to accept only a small token of the ‘spoils’ s/he (the blocker) will form a winning coalition with the two voters whose weight is 1, so that the voter whose weight is 2 will get nothing. A similar threat can be directed, apparently, by the blocker to the two voters whose weight is 1 if s/he decides to approach them first. But is such a tactic necessarily effective and hence should lead us to expect that the blocker’s expected share should be larger than 1/2 in this WVG? We do not think so for a simple reason: in order to try and neutralize the ability of the blocker to ‘divide and rule’, the remaining voters can react by forming a firm alliance among themselves and confront the blocker with a choice, say, to divide the ‘spoils’ equally among all (four) voters or to forgo any ‘spoils’. In view of such possible development we believe that the voter with weight 3 is likely to decide that it would be worthwhile for him/her to quickly form the WCLS {3,2} and agree to split the ‘spoils’ equally with the voter whose weight is 2. Now let us look at the following two WVGs: [Y] = [7; 5,4,2,1] and [Z] = [12; 6,5,4,3]. In Y there are two WCLS, {5,4} and {5,2}. Although the voter whose weight is 5 is not strictly a blocker (because the coalition {4,2,1} is a MWC), we assume that, as explained above, a MWC which is not also an WCLS will not form, hence in Y the voter with weight 5 is a de facto blocker because s/he is a member of both WCLSs, {5,4} and {5,2}; hence his/her expected payoff in Y is 1/2(1/2 ? 1/2) = 1/ 2. Since each of the members whose weight is 4 and 2 belong to only one of the two WCLS their expected payoff is 1/2  1/2 = 1/4, whereas the expected payoff of the member with weight 1 is 0 because although s/he is not strictly a dummy s/he is considered to be a de facto dummy because s/he does not belong to any WCLS.

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In Z there are four WCLS—{6,5,4}, {6,5,3}, {6,4,3}, and {5,4,3}. As each of the four voters belongs to three of the four WCLS their expected payoff is the same, i.e., it is equal to 1/4  (1/3 ? 1/3 ? 1/3) = 1/4. Let us look now at a non-weighted SVG7 with nine players, a - i, whose seven MWCs are: fa; c; ig; fa; d; e; f ; ig; fa; d; e; g; ig; fb; c; d; ig; fb; c; e; ig; fb; d; f ; ig; fb; e; f ; g; h; ig: This SVG is not weighted because it is impossible to assign to each of the voters some weight w and fix a quota q such that it would be possible to reproduce the above seven MWCs—and only them. This impossibility can be explained briefly as follows: given that {a, c, i} is a winning coalition and {a, b, i} is a losing coalition it follows that if it were possible to assign weights to the various players then c’s weight should have been larger than b’s. However, given that {b, d, f, i} is a winning coalition but {c, d, f, i} is a losing coalition it follows that if it were possible to assign weights to the various players then b’s weight should have been larger than c’s—which is a contradiction! Since in the above list of seven MWCs there is only a single WCLS, i.e., {a, c, i}, it follows that according to the proposed index the expected payoff of each of players a, c, i is 1/3 while the expected payoff of each of the remaining six players is 0 despite the fact that none of them is strictly a dummy. The proposed index satisfies all the six postulates stated in Sect. 2. Proof Under the proposed index only WCLS are assumed to be formed. As a dictator is, by definition, the sole WCLS (composed of a single member), the dictator’s expected share in the ‘spoils’ is 1, as postulated. As a dummy, by definition, does not belong to any WCLS, a dummy’s expected share in the ‘spoils’ is 0, as postulated. As a blocker belongs, by definition, to all WCLS but is not a dictator, the expected share of a blocker is positive, but smaller than 1, as postulated. Since all WCLS are, by definition, of equal size, it is impossible for a voter whose weight is smaller to belong to a larger number of WCLS than a voter whose weight is larger. Hence the expected share in the ‘spoils’ of a voter whose weight is smaller cannot be larger than that of a voter whose weight is larger, thereby satisfying the monotonicity postulate. As under the proposed index only WCLS are assumed to be formed, a voter to whom some weight is ‘donated’ from some other voter cannot be worse off in terms of his expected share in the ‘spoils’ after receiving the extra weight. This is so because if the recipient has been a member of a WCLS before receiving the extra weight then s/he will, a fortiori, remain a member of a WCLS which may even be smaller than the previous one—and thus increase his/her expected share in the ‘spoils’; and if the recipient has not been a member of a WCLS before receiving the extra weight then either the receipt of the extra weight may turn him/her into a member of a WCLS—in which case the recipient’s expected share increases, or the receipt of the extra weight would still not be sufficient to make him/her a member of a WCLS—in which case the recipient’s expected share in the ‘spoils’ would stay the 7 SVGs that are not weighted are possible when n C 4. All SVGs where n = 3 can be represented as WVGs.

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same as before. Thus, as the proposed index must satisfy the donation postulate, it must also satisfy, a fortiori, the weaker annexation postulate. As it is assumed under the proposed index that only WCLS are formed, that all WCLS are equiprobable and that all members of the victorious WCLS receive equal shares of the ‘spoils’, then it follows that a blocker’s expected share must be equal to 1/k (where k is the size of each WCLS), thus satisfying the blocker’s share postulate. Finally, the proposed index satisfies the added blocker postulate. This is so because, by assumption, as only WCLS are formed then the addition of a blocker does not change the number of the WCLS that existed prior to the addition of the extra blocker but only the size of each of them. Since it is also assumed that all WCLS are equiprobable and that all members of the victorious WCLS receive equal shares of the ‘spoils’, then it follows that the ratio between the expected shares in the ‘spoils’ of any of the other members of the WCLS must remain the same as it was before the addition of the extra blocker. h

4 A Brief Review of the Main P-Power Indices Proposed to Date in the Literature In this section we list briefly the main P-power indices that were proposed to date in the literature indicating for each of them which of the six postulates listed in Sect. 2 they violate. For each index we cite only the original source in which it was first proposed, as well as where it is possible to find examples or proofs that the index violates some postulate(s). The indices are listed according to the year they first appeared in the literature. 4.1 The Shapley–Shubik index (Shapley and Shubik 1954) The Shapley–Shubik index of a voter v under a SVG W is defined as the Shapley value (Shapley 1953) of v as player in the SVG W. It is denoted by SSv[W] or simply SSv if the SVG in question is obvious from the context. Shapley and Shubik (1954) proposed SSv [W] as a measure of v’s voting power under the SVG W. (In fact, they claimed that this is the only tenable way of measuring voting power.) The value of SSv is determined as follows. Imagine the voters forming a queue, all lining up to vote for a bill. The pivotal voter in this queue is the voter v such that the set of voters lined up before v constitute a losing coalition, but they together with v constitute a winning coalition. In other words, as the voters in the queue cast their votes—all voting ‘yes’—the pivotal voter is the one who tips the balance. (It is easy to see that in every possible queue of the voters there is a unique pivotal voter). Now, to obtain SSv we form all possible queues of the voters, count in how many of them our voter v is pivotal, and divide this number by the total number of queues. In probabilistic terms, SSv can be defined as the probability that v is

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the pivotal voter in a randomly formed queue, assuming that all possible queues are equiprobable.8 We wish to stress that the Shapley–Shubik index relies for its justification on the axiomatic derivation of the Shapley value (of which it is a special case), not on any model of voting protocol, bargaining or coalition formation. In particular, the queue formation procedure of voting we have just described is merely a heuristic device for calculating the values of SS. It is not intended as a justification of the Shapley– Shubik index, and is certainly not to be taken seriously as a description of how voting actually takes place.9 The number of ways in which a set of m objects can be ordered, denoted by m!, is equal to m(m - 1)(m - 2) …2 1. (In particular, 0! = 1 because the empty set has just one, vacuous ordering.) So for an assembly N of size n, the number of queues that can be formed by the voters is n!. In order to calculate SSv we need to count the number of those queues, out of the total n!, in which v is pivotal. To do so directly, by examining individually each of the n! queues, can be quite tedious, since n! is quite large even for modest values of n, and grows very rapidly as n increases. Fortunately, we can use a shortcut that simplifies the counting. Consider a queue in which v is pivotal. We depict it schematically as follows: m  1 voters h h h f

S

v

n  m voters } } }

g

Here v is the mth voter in the queue; the m - 1 boxes represent the voters that precede v in the queue, and the n-m diamonds represent those who follow v. S is the set of m voters up to v (inclusive of v). The crucial point to note is that our assumption that v is pivotal in this queue is equivalent to v being a critical member of the coalition S: the ‘yes’ votes of the boxes are insufficient to pass a bill, but the added ‘yes’ of v tips the balance. Thus we have associated with the given queue in which v is pivotal a coalition S in which v is a critical member. Next we observe that this coalition S in which v is critical is associated with exactly (m - 1)!(n - m)! distinct queues in which v is pivotal, because the boxes can be lined up in (m-1)! different ways and the diamonds can be lined up in (n - m)! different ways, without affecting the pivotal status of v. Hence we have: Shortcut In order to count the number of queues in which v is pivotal, find the coalitions in which v is a critical member. Each such coalition, say S, contributes (m - 1)!(n - m)! to the count, where m is the size of S. In order to find SSv, divide the total count by n! The Shapley–Shubik index satisfies five of the six postulates listed in Sect. 2. The only postulate it violates is the added blocker postulate. A flagrant example of this 8

An alternative presentation of the Shapley–Shubik index imagines a roll-call of the voters, who line up to vote for or against a proposed bill. The pivotal voter in a roll-call tips the balance for approval or rejection of the bill; and SSv is the probability that v is the pivotal voter in a randomly formed roll-call. For details see Felsenthal and Machover (1996) or Felsenthal and Machover (1998: 187–189). 9

See Felsenthal and Machover (1998: 200–206) for a discussion of the widespread misapprehension regarding this point.

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violation can be found in Felsenthal and Machover (1998, Example 7.9.16, p. 272).10 4.2 The Banzhaf (Relative) Index (Banzhaf 1965) The Banzhaf index for any voter v, denoted bv, is obtained as follows: X g bv :¼ gv = x2N x where gv is the number of divisions of the voters other than v in which v’s vote is decisive. This index violates three of the six postulates listed in Sect. 2: it violates the donation, bloc, and blocker’s share postulates. For examples of these violations see Felsenthal and Machover (1998, Examples 7.8.5, 7.8.6, p. 253; Examples 7.8.14, 7.8.16 pp. 256–257; and Example 7.8.5, p. 253, respectively).11 4.3 The Deegan–Packel Index (Deegan and Packel 1978, 1982) In 1978 Deegan and Packel (1978) proposed a new voting-power index. As is evident from the very title of the 1982 version of their paper (Deegan and Packel 1982)—‘‘To the (minimal winning) victors go the (equally divided) spoils …’’—this index is explicitly based on the notion of P-power; but, unlike the Shapley–Shubik index, whose justification is axiomatic, it relies for its justification on a specific bargaining model. Deegan and Packel cast voting as a TU cooperative game, and take voting behavior to be motivated by office seeking.12 In each play of the game, bargaining leads to the formation of a single winning coalition, which thereupon gains its ‘spoil’: 1 unit of TU. This is distributed among the members of the winning coalition as their individual payoffs. All non-members of this coalition get 0 payoff. Bargaining is assumed to be a random rather than deterministic process. Moreover, Deegan and Packel base their bargaining model on three assumptions: 10 The violation was called by Felsenthal and Machover ‘flagrant’ because not only the ratio between the old voters’ payoffs was not maintained after a new blocker joined the assembly, but the payoffs of two of the old voters were reversed: in the original situation player a was stronger than b according to the Shapley–Shubik index, whereas after the additional blocker joined the assembly player b became stronger than a according to this index. 11 The Banzhaf index is merely a normalization of the Penrose measure (Penrose 1946, 1952). As the Penrose measure (also called ‘the absolute Banzhaf index’) does not suffer from any pathology, it has been recognized by Felsenthal and Machover (1998) as the only adequate I-Power measure, and hence they recognized the Banzhaf index (which is a normalization of the Penrose measure) as an adequate I-Power index, but because it violates the above-mentioned postulates they disqualified it as a reasonable P-Power index. For the distinction between I-power and P-power see Felsenthal and Machover (1998, chs. 3,6). 12

The formation of the winning coalition as well as the distribution of the ‘spoils’ among its members are consequent upon a process of bargaining. The motivation of voting behavior that this view assumes has been called ‘office seeking’ by political scientists. The alternative motivation is called ‘policy seeking’ where the main concern is not the division of ‘spoils’ among the winners, but controlling the action of the collectivity by passing some bill or resolution. For a discussion of these alternative motivations in a political context see Coleman (1971: 272), Laver and Schofield (1990, ch. 3), as well as Felsenthal and Machover (1998: 17–19).

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• • •

Only minimum winning coalitions (MWCs) are formed. All MWCs have equal probability of being formed. All members of the victorious MWC receive equal shares of the ‘spoils’.

The Deegan-Packel index of voter v under a SVG W, which we shall denote by DPv[W], is defined as the expected value (aka probabilistic mean) of v’s payoff. DPv[W] is obtained as follows. Suppose W has m MWCs, S1, S2, …, Sm, whose respective sizes are s1, s2, …, sm. By assumption, the probability of the event that Si is formed equals 1/m. Moreover, in this event v’s payoff is 1/si or 0, according as v belongs or does not belong to Si. So in order to obtain the expected value DPv[W] we must add up the products of the probability 1/m and each of the m possible payoffs. Thus, omitting the terms with payoff 0, we have DPv :¼ 1=m

X v2Si

1=Si :

Here the summation is over all i such that v [ Si. If v is a dummy, there is no MWC containing v, so the summation in the above equation has no non-zero terms, hence DPv[W] = 0. The Deegan-Packel index violates five of the six postulates listed in Sect. 2: it violates the monotonicity, donation, bloc, blocker’s share, and the added blocker postulates. For examples of these violations see Felsenthal and Machover (1998, proof of Theorem 7.6.9 pp. 245-246; Examples 7.8.5 and 7.8.6, p. 253; Example 7.8.15, p. 257; Example 7.8.5, p. 253; and Example 7.9.12, p. 271, respectively). 4.4 The Johnston Index (Johnston 1978) The Johnston count, which we shall denote by JC, is obtained as follows. Any coalition S in which voter v is a critical member contributes 1/c(S) to JCv, where c(S) is the number of critical members of S. The idea is that the worth of S is divided equally among its critical members. The Johnston index, which we shall denote by JI, is obtained from JC by normalization, so that the values of JI for all members of the assembly N add up to 1: X JIv ½W  ¼ JCv ½W = JCx ½W : x2N This index violates four of the six postulates listed in Sect. 2: it violates the donation, bloc, blocker’s share, and the added blocker postulates. For examples of these violations see Felsenthal and Machover (1998, Example 7.8.6, p. 253; Example 7.8.16, p. 257; Example 7.8.6, p. 253; and Example 7.9.12, p. 271, respectively). 4.5 The Public Good Index (Holler 1978, 1982;) The Public Good index (PGI), like the Deegan–Packel index, is based on the assumption that only MWCs are formed. But the prize of victory according to the PGI index is regarded not as a unit of TU, a private good, to be divided among the

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members of a victorious MWC, but as a public good enjoyed in its entirety by all members of this coalition (but only by them!). So the Public Good count of a voter v, which we shall denote by PGCv, is simply the number of MWCs to which v belongs. And the Public Good index, which we shall denote by PGI, is obtained from PGC by normalization, so that the values of PGI for all members of the assembly N add up to 1: X PGIv ½W  ¼ PGCx ½W : x2N This index violates four of the six postulates listed in Sect. 2: it violates the monotonicity, donation, bloc, and blocker’s share postulates. For examples of these violations see the same examples in Felsenthal and Machover (1998) that demonstrate the violation of these postulates by the Deegan–Packel index. The monotonicity failure of this index is also critically discussed in Holler et al. (2001), in Holler and Napel (2004) and in Freixas and Kurz (2016). 4.6 The Shift Minimal Winning Coalition index (Alonso-Meijide and Freixas 2010; Alonso-Meijide et al. 2012) Contrary to the Deegan–Packel and Public Good indices, which are based on all MWCs, the Shift index is based on a smaller number of MWCs, i.e., on those MWCs in which it is not possible to replace any player for a weaker one and still maintain the condition of a winning coalition. To determine which MWCs are included in the reduced (shift) set of MWCs, one must order the players according to their desirability relation (Isbell 1958) as follows: •

•

•

In a SVG W player i is said to be equally desirable to player j (as a coalitional partner), denoted i * j, if for any coalition S such that i 62 S and j 62 S, S [ {i} [ W $ S [ {j} [ W. Player i is said to be (strictly) more desirable than player j, denoted i  j, if the following two conditions are fulfilled: (1) For every coalition S such that i 62 S and j 62 S, S [ {j} [ W ? S [ {i} [ W. (2) There exists a coalition T such that i 62 T and j 62 T, T [ {i} [ W and T [ {j}62 W. In a WVG if wi C wj then player i is said to be at least as desirable as player j, denoted i C j.

Let us now look, as an example, at a WVG with five players a,b,c,d,e, whose weights are 4,3,1,1,1, respectively, and where the quota is 5, (which can be denoted as [5; 4,3,1,1,1]). In this WVG there are seven MWCs, as follows: {a,b}, {a,c}, {a,d}, {a,e}, {b,c,d}, {b,c,e}, {b,d,e}. Accordingly, the desirability relations among the five players are a  b  c  d  e. Although {a,b} is a MWC it is not shift minimal because player b is strictly more desirable than each of players c,d,e while player a can still form a MWC with each of these three players, so one can reduce the number of MWCs by deleting the MWC {a,b} thereby ending up with six shift minimal MWCs.13 13 See Taylor and Zwicker (1999 Ch. 3) for references and history on the mathematical use of the shift ordering.

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Given these shift minimal MWCs, the power of the various players belonging to them can be computed either according to the Deegan-Packel index—in which case this Shift Deegan-Packel power index violates all the five postulates that are violated by the Deegan-Packel index, or according to the Public Good index—in which case this Shift power index violates all the four postulates that are violated by the Public Good index.14 4.7 The Minimum Sum Representation (MSR) Index (Freixas and Kaniovsky 2014) This index is applicable only to weighted voting games (WVGs). Given a WVG with n players of the form [q; w1,w2, …,wn] where wi is the weight of player i and q is the quota needed to pass a resolution, the P-power of every member i according to the MSR index is equal to his share in the sum of weights of the minimum sum representation, i.e., to the sum of all members’ weights in an isomorphic WVG where the quota and weights are represented by the smallest integers. Thus, for example, for the WVG [51; 48, 48, 2] the smallest isomorphic WVG is [2; 1,1,1]. As the sum of weights of the players in this smallest isomorphic WVG is 3, the power of each of the players according to the MSR index in this WVG is 1/3. As the proposers of this index explicitly admit (see Freixas and Kaniovsky 2014, p. 744), the MSR index violates the donation and bloc postulates. Moreover, from the list of smallest WVGs with up to four voters displayed in Table B.1 in this article, it can be seen that the MSR index also violates the blocker’s share postulate. Thus in the SVG with four voters, a,b,c,d, whose MWCs are {a,b}, {a,c}, {a,d} and whose smallest WVG representation is [4; 2,1,1,1], voter a (with weight 2) is a sole blocker whose P-power according to the blocker’s share postulate should be at least 1/2 in this WVG, but according to the MSR index it is only 2/5.

5 Conclusion As far as we know, of the several P-power indices proposed to date only the P-power index proposed in Sect. 3 of this article satisfies all the six postulates listed in Sect. 2—which to us seem compelling for any reasonable P-power index. We do not claim that these six postulates constitute a complete set. Additional reasonable postulates can no doubt be proposed. As suggested in Felsenthal and Machover (1995, p. 223) one direction in which one may seek such postulates is the limiting behavior of a power index as the number of voters tends to infinity.15 Felsenthal and 14 In the above 5-voter WVG the shift Deegan-Packel index is 1/4, 1/6, 7/36, 7/36, and 7/36, for voters a,b,c,d,e, respectively —thereby demonstrating the non-monotonicity of this index because although b’s weight is larger than that of c,d, or e, this index awards b a smaller P-power than that awarded to c,d, or e. The shift Public Good index in this example awards the same P-power (1/5) to each of the five voters as each of them belongs to the same number (3) of shift MWCs. 15

For example, if we define a voter b as strong blocker in case all MWCs in a SVG W are of the form {b,x} where x is any voter other than b, then perhaps it may be reasonable to postulate that b’s power should tend to 1 with an increase in the number of voters. The Deegan–Packel and the Public Good indices, as well as our proposed index, would violate this postulate because according to these indices the power of a strong blocker is always 1/2 (which does satisfy the blocker’s share postulate).

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Machover (1998, p. 278) also suggested that although we cannot expect P-power to be strictly proportional to I-power, it may perhaps be reasonable to expect that the two ought to be positively correlated.16 We are aware that it may be argued that a reasonable P-power index must not only satisfy a set of compelling postulates, but that, ideally, the normative bargaining model underlying this index must also be convincing—and that the one underlying the index proposed by us may not be really convincing. Thus, some critics of our proposed index have argued that although the first two assumptions underlying our proposed P-power index, (i.e., that only WCLS are formed and that their formation is equiprobable) are indeed reasonable, the third assumption (i.e., that the members of the victorious WCLS will divide the ‘spoils’ among them equally) seems to them to be rather arbitrary. However, regardless of whether any of the readers agrees with this critique, the trouble is that where there are more than two voters no such conclusively convincing bargaining model is as yet available,17 nor is there any known way to obtain such a normative model. We are also aware that our proposed index may not be unique, i.e., that another P-power index satisfying the same postulates but assigning, ceteris paribus, different expected powers to the various types of players, may be proposed. In this case the choice between these two indices would have to be determined according to the reasonableness of their (different) underlying bargaining models.18 Moreover, it is always possible that an additional compelling postulate may be suggested which neither our proposed index nor any of the previously proposed indices satisfies—in which case we may be facing an impossibility theorem regarding the existence of a reasonable P-power index. So a critical reader of our proposed P-power index may be currently left with a hard choice: to adopt our proposed P-power index if only because it satisfies all postulates that we could think of that a reasonable P-power index ought to satisfy, or to continue using one of the previously proposed P-power indices despite the fact that each of them violates one or more of the postulates that a reasonable P-power index should satisfy, and whose underlying bargaining model, to put it mildly, does not seem to be more convincing than the one underlying our proposed index.

16 It should be noted that the Banzhaf index—which is recognized by Felsenthal and Machover as the only acceptable I-power index, is not co-monotone with the Deegan–Packel, Public Good and Shapley– Shubik indices. 17

For one or two voters the problem is trivial: either there is a dictator, who always gets the entire ‘spoils’ and hence all the P-power; or there are two symmetric voters, whose expected shares must be equal. 18 Felsenthal and Machover (1998, p. 278) mention the possibility that a P-power index of a voter v in a SVG W which most probably satisfies all the six postulates mentioned in Sect. 2 is an index, b(k) v , obtained by normalizing the k-th powers of the Banzhaf scores of the voters, and if k is sufficiently large. This index can be formally represented as

bðvkÞ :¼ ðgv Þk =

X x2N

ðgx Þk

However, inasmuch as this index does indeed satisfy all the six postulates listed in Sect. 2, the bargaining model underlying it seems to us much less reasonable than that underlying our proposed index.

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Homo Oecon Acknowledgments I am grateful to one of the referees whose numerous comments and suggestions helped me to improve this paper.

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