Interval measures of power

ELSEVIER Mathematical Social Sciences 33 (1997) 23-74

mathematical social

sciences

Interval measures of power

Alan Taylor, William Zwicker* Department of Mathematics, Union College, Schenectady, NY 12308, USA

Received 1 May 1994; revised 1 February 1996

Abstract

Standard power indices, introduced by Shapley and Shubik, Banzhaf, Johnston, and others, assign real numbers to the players in a simple game as a quantitative measure of their influence in the voting situation represented by the game. We consider instead two 'interval notions of power' that assign intervals of real numbers to the players. The first of these notions originates in the observation that although a particular choice of weights in a weighted game may not provide an accurate measure of power, the interval of all possible weights for a player is, in fact, a reasonable reflection of the player's influence. The resulting weight interval is extended to non-weighted games via the observation that every simple game is an intersection of weighted games. For example, we have shown (Taylor and Zwicker, Games and Economic behavior, 1993, 5,170-181) that the president's weight interval in the (non-weighted) federal system is (15/115, 17/117), suggesting that the president has between 13% and 15% of the power. The development of a combinatorial equivalent, the dispersal interval, yields a uniform method for calculating weight intervals. Our second interval index, the market interval, is based on a modification of Peyton Young's idea that the influence of players can be measured by the sizes of the bribes they can demand when they sell their votes. We show, for example, that the president's market interval in the federal system is (0, 0 .2537, . . . ) ; he can reasonably insist on between 0% and 25% of the total bribe. An iterated version of the market interval, based on rationality assumptions about other players' actions and linked with the notion of a bribery equilibrium, produces potentially smaller intervals that, in some cases, shrink to single points in the limit.

Keywords: Voting; Power; Power index; Simple game; Weighted voting game

1. Introduction

The p rob lem of designing fair vot ing systems is central to social choice theory . In this paper we are concerned with vot ing systems in which an al ternative (such

* Corresponding author.

0165-4896/97/$17.00 (~ 1997 Elsevier Science B.V. All rights reserved PI1 S0165-4896(96)00818-9

24 A. Taylor, W. Zwicker / Mathematical Social Sciences 33 (1997) 23-74

as a bill or an amendment) is pitted against the status quo (no change in the body of the law). Such systems are conveniently formalized by simple games.

For our purposes, a simple game G is a structure (N, W) in which N is the set { 1 , 2 , . . . , n} and W is an arbitrary collection of subsets of N (see Shapley, 1962; yon Neumann and Morgenstern, 1944). The idea is that the bill being voted on passes if the set of those in favor belongs to W. For this reason, elements of N are called people or players, sets in W are called winning coalitions, and subsets of N not in W are called losing coalitions. Within this paper we assume that all games are monotonic (every superset of a winning coalition is winning) and have the property that N is a winning coalition and ~J is a losing coalition. We place no other restrictions on G.

Interesting problems arise when attempting to design fair voting systems for circumstances in which it is appropriate for different players to have different amounts of influence. For example, suppose that each of the four townships in a certain county sends one representative to the county legislature. If the percentage breakdown of the county's population among the townships is 34%, 34%, 17%, and 15%, how should the representatives' votes be counted? The naive solution is to weight the representatives' votes so that a 'yes' by the first representative counts as +0.34, etc. and to declare that the proposal passes if the total weight in favor is 0.51 or more. The results is an example of a weighted game; more generally G -- (N, W) is weighted if there exists a function w : N---~ R and a quota q E ~ such that a coalition X is winning iff ,~ {w(x) : x E X}/> q.

The observation that the naive solution can badly fail to reflect the ideal of one person, one vote, and, more generally, that a particular choice of weights for a weighted game may be greatly disproportionate to the distribution of influence, goes back at least to Shapley and Shubik (1954). In the above example it is easy to see that if we use the naive solution, the representative from the 15% township has no influence (that representative is a dummy- meaning that a change in his or her vote can never affect the outcome) while the other three have equal influence (any permutation among them leaves the set W fixed).

Given this failure of the naive solution we might next ask how to choose, from among all possible simple games G that might be employed as voting systems for our county, the one that comes closest to equalizing the influence of individual voters from different townships. The conventional solution entails defining an index that assigns to each player in G a number measuring that player's amount of influence, or 'power' . The numbers add to one and may be thought of as percentages of the whole. The game whose power distribution most closely matches the distribution of the population then has a claim that it is the most fair.

A number of such indices have been proposed since the original by Shapley and Shubik in 1954. A court case has even led to the enshrinement of the Banzhaf index (Banzhaf, 1965) in New York state law (Lucas, 1982). Each of these indices

A. Taylor, W. Zwicker / Mathematical Social Sciences 33 (1997) 23-74 25

measures power in a plausible way, and a number of these plausibilities can be made rigorous via characterization theorems (see Dubey, 1975; Dubey and Shapley, 1975; Straffin, 1982; Owen, 1978a, 1978b, 1982; or Lambert, 1988 for example). However, the different indices can produce wildly different measurements (see Brams et al. 1989; Straffin, 1982; Taylor 1995), and later comments on the power of the president) and none has emerged as being clearly more appropriate than the others. Felsenthal and Machover (1995) opine that "from a conceptual point of view the present state of the subject is rather unsatisfactory".

It certainly appears that anyone attempting to design fair voting systems must confront the problem of how to measure power. But there seems to be no reason why such a measure need assign a single number, or 'point', to each player (as do all of the published indices). In this paper we introduce two interval indices that assign ranges of values, rather than points, as measurements of power. Each is based on a reasonably plausible way of thinking about power. In each case, if we buy the plausibility argument, we must also believe both that power is 'fuzzy', and that the degree of fuzziness (width of the intervals) can vary considerably from game to game.

In Section 2 we introduce the first of these indices. It is based on the premise that the interval of weights possible for a player in a weighted game is, in fact, a reasonable representation of his or her power, even though it is clear that a particular choice of weights may not be representative. The resulting weight

interval can be extended to non-weighted games by making the observation (Taylor and Zwicker, 1993) that every simple game can be written as an intersection of weighted games. As an example, we note that in the federal system (which is not weighted) the president's weight interval is (15/115, 17/117) (Taylor and Zwicker, 1993), suggesting that the president has between 13% and 15% of the power. Table 1 provides values for the original European Economic Com- munity, for the general majority game (in which s/> 1 is fixed and the winning coalitions are those containing any s or more of the t players), and for a game that seems to be a reasonable candidate for a fair voting system in the four-township county described earlier. Notice (Table 1) that the president's weight interval contains none of the values generated by standard indices (but barely misses for the Shapley-Shubik value).

The introduction, in Section 3, of a Seemingly unrelated power index called 'dispersal power', and the proof (Theorem 2.1) that it is equal to the weight interval, yield several benefits. Dispersal power has its own independent form of plausibility, which is inherited by the weight interval. It has not been 'rigged' to produce intervals rather than point values, so the fact that it never yields points can be interpreted as evidence for the legitimacy of measuring power with intervals. Finally, the proof of Theorem 2.1 produces a uniform method for computing weight intervals. In Section 4 we use this 'dispersal method' to


Table 1 Values of weight interval (W1), market interval (MI), iterated market interval (MI*), Shapley-Shubik (SSI), Banzhaf (B1), and Johnston (Jl) power indices for several voting games

Game/Person WI M1 MI* SS1 BI Jl

G4.5/any (0,1/3) [1/5,1/2] {1/5} 1/5 1/5 1/5 player

G . . . . /any (0, 2/(t - 1)) [ l / t , 1/2] {l/t} 1/t 1/t 1/t player

Gv.,1/any (0, 1/6) [0, 1/5] {1/11} 1/11 1/11 player

Gs. , (s<-t-2) / (0, 2/(t + 1)) [O, 1 / ( t - s + l ) ] (l/ t} 1/t 1/t any player

EEC/Lux- [0, 1/9) (0} {0} embourg

EEC/France, Germany, or Italy

0 0 0

1/11

1/t

(1/7,1/3) [0,1/2] [1/4,3/8] 14/60 5/21 1/4

EEC/Belgium (0, 1/4) [0, 1/2] [0, 1/4] 9/60 3/21 1/8 or Netherlands

(1/5,1/2) [1/3,1/2] [1/3,1/2] 1/3 1/3 11/30 acounty [ either 'big' township

G ... . . y/ [0, 1/2) [0, 1/2] [0, 1/3] 1/6 1/6 4/30 either 'little' township

GFed/the (0.130, 0.146) [0, 0.2537] a [0, 0.2537]" 0.16 0.038 0.77 president

a Incorporates the vice-president's tie-breaking power. A good reference for the point valued indices is Brains et al. (1989).

recalculate the president's weight interval in the US federal system (simplified by suppressing the vice president's tie-breaking role in the Senate).

Section 5 introduces the market interval index. Young (1978) defined a point power index based on the notion that the influence of a politician is most appropriately judged by the size of the bribe he or she can command on the open market. Young quotes Thomas Hobbes (Leviathan, part I, ch. 10): "The 'value' or 'worth' of a man is, as of all other things, his price, that is to say, so much as would be given for the use o f his power."

Starting with the same basic intuition, we take a different approach from that in Young (1978) and construct the market interval index, MI, by arguing that certain


demands by player p are clearly too high (because no lobbyist could possibly minimize his or her expenses by spending that high a percentage of total funds available on p 's vote alone) while others are clearly too low (for an analogous reason). This leaves an interval M I ( p ) of reasonable demands. An interesting extension of the concept is provided by an iteration of the bidding, in which each politician assumes, at stage n + 1, that all his or her colleagues will behave rationally by demanding bribes from within their own interval of reasonable demands, as determined at stage n. The completely iterated version, MI*, thus incorporates precisely that infinite ramification inherent in the use of the word 'rational' whenever it is used by a game theorist. That is, each player will act rationally, knows that everyone else do likewise, knows that everyone knows that everyone will do likewise, etc. ad infinitum.

We have calculated M I and MI* for the same examples as in the previous section. The results appear in Table 1. Section 6 contains the statements and proofs of the main results on the behavior of MI. In Section 7 we calculate MI*

for the general majority game and show, in particular, that with s < t - 1, each iteration further shrinks the interval, with convergence at infinity to the one-point interval {l/t}. This provides an intriguing example of a situation in which the infinite ramification in the rationality hypothesis truly makes a difference (most previous applications of rationality appear to require only a bounded ramification). Section 8 introduces the notion of a bribery equilibrium and shows, via a generalization of the ideas in Section 7, that MI* is well defined for a broad class of simple games. The arguments raise some intriguing questions about the relationship between bribery equilibria and homogeneously weighted games. In Section 9, we calculate the president's market interval in the US federal system.

Do interval indices of power deserve consideration as contenders for the 'right' way to measure influence in voting systems? In Sections 10 and 11 we take up some more general arguments. The first of these considers the possibility that, at least in some games, power is inherently fuzzy, and thus appropriately measured by an interval. The second considers the issue of the additivity of power under the formation of voting blocs. We point out that it is impossible for any point measure to be additive, but that interval measures can have a suitably phrased additivity property.

2. The weight interval

Let us begin by assuming that G is a monotonic weighted game. In this section we consider only non-negative weightings for which the sum of the weights is one. We refer to any such weighting as an N N weighting; the two 'N"s stand for 'non-negative' and 'normalized'. Note that there is little loss of generality in this restriction since we can take any weighting for a monotonic game, change


negative weights to sufficiently small positive ones, and then scale, to arrive at an NN weighting. After this modification, each person's weight equals his or her fraction of the total weight.

Definition. Let G be a monotonic game. If G is weighted, then for any person p, p 's weight interval, WI(p) , is the set of all possible values of w(p) as w ranges over all NN weightings for G. Equivalently, WI(p) is the smallest interval containing this set.

The equivalence above follows from the observation that whenever w I and w 2 are NN weightings for G, so is the weighted average a * w 1 + b * w 2, where a and b are any two non-negative numbers summing to one. Consequently, the possible values of w(p) form an interval.

Why should anyone feel that the weight interval represents, in any way, a measure of voting power? For one thing, the weight interval appeals directly to the intuition, from Section 1, behind the naive argument in favor of weights that are proportional to the populations of constituencies, while sidestepping the flaw in the original version. Picture a very large number of voters, say 1000 of them, each having a single vote. Suppose we group them into three batches containing 331, 335, and 334 voters, respectively, and have each batch vote as a bloc (together and the same way, as if each batch were a single voter) while setting the quota at 501. The resulting system is equivalent to three players voting according to majority rule.

This suggests that each of the three players in the batched system has 'about ' one-third of the total voting power. But the very same voting system with three players may be induced by setting the batch size at 251, 251, and 498; the word 'about ' must be construed so loosely that 0.251 is 'about ' one-third. If we calculate just how much play exists in this sort of power allocation, and allow an arbitrarily large number of voters in place of the original 1000, then we arrive again at the weight interval. This is not difficult to establish for the general case - it follows from the observation that any weighting can be approximated as closely as we wish by employing rational weights that, after having their denominators cleared, may be thought of as integer weights corresponding to batch sizes. Thus, comparing the weight interval of two players for a weighted game is roughly the same as comparing the relative numbers of interchangeable individual voters who would be needed to create voting blocs that were equivalent, in their effect on outcomes, to these players (that is, it is the same as comparing the players' 'constituencies').

The extension of the weight interval to the case of non-weighted games relies on the fact that any monotonic game can be written as the intersection of monotonic weighted games. We shall define the monotonic dimension of G = (N, W) to be the smallest k for which there exist monotonic weighted games


(N, W1), (N, W2) . . . . . (N, Wk) with W= N {W/: i = 1, 2 . . . . . k}. Dimension was introduced for graphs in the 1970s, extended to hypergraphs by Jereslow (1975), and discussed for simple games in Taylor and Zwicker (1993, 1997).

Definition. If G = (N, W) is any monotonic game of monotonic dimension k, and p is any person, then WI(p) will be the smallest interval I guaranteeing that for every choice of k monotonic weighted games whose intersection is G, together with a choice of NN weighting for each, all k of p's weights lie within I.

For example, suppose that some simple game G has a monotonic dimension 2 and that there is only one way to choose two weighted games that intersect to give G ('the dimensional representation is unique'). If player p has WI(p) equal to the open interval (0.2, 0.4) in one of these weighted games and WI(p)'= (0.3, 0.5) in the other, then the above definition sets WI(p) in G to be the union of the intervals (0.2, 0.5). Note that it is a result of the overlap in the first two intervals that the smallest interval containing both is simply their union. The intuitive rationale behind using (0.2, 0.5) is as follows: the two weighted games making up G agree that p's influence is greater than 0.2 and agree that it is less than 0.5. Perhaps a stronger statement is defensible, but surely we can agree that p's influence in G lies in the (0.2, 0.5) range.

In the case of the US federal system, discussed in Theorem 2.1, not only is the dimensional representation unique but one of the two weight intervals nests inside the other (Taylor and Zwicker, 1992). It is examples such as the federal system that make the extension of the weight interval to at least some non-weighted games seem most intuitive.

However, there are other dimension 2 examples that yield disjoint intervals (such as (0, 2/101) and (1/4, 1/2)) and there are examples for which dimensional representation is not unique. In the former case our definition chooses the smallest interval containing the union (here, (0, 1/2)) and in the latter it chooses the smallest interval containing all the intervals arising from all the games appearing in the various representations. While this final interval still represents the strongest restriction on p's power that all games participating in dimensional representations can agree with, we share the referee's sense that any claim that this interval represents p's power is less than compelling.

As of this writing we do not understand why some games have unique dimensional representation, or what it means when the weight intervals from intersecting games overlap or nest. Clearly, we need to known more before deciding which non-weighted games, if any, are appropriate for the extension of the weight interval.

If p is any person in a monotonic game G containing at least one other person, and w is an NN weighting for G, then, unless w(p) has one of the extreme values zero or one, it is always possible to shift some sufficiently small amount of weight


to w(p) or away from w(p) without destroying w's legitimacy as an NN weighting. This tells us that WI(p) is an open interval in the relative topology of the [0, 1] interval, and it is straightforward to see that this conclusion holds for non-weighted games as well.

Even without the method we introduce below, it is fairly easy to use ad hoc methods to compute the weight interval in some simple examples. For example, with three players and majority rule, W I ( p ) = (0, 1/2) for each of the three players. This example might lead us to suspect that the weight interval will typically be too large to comment much on the issue of power. Moreover, if the game is not weighted, then the weight interval tends to be even larger than in the weighted case. So it is gratifying, and somewhat surprising, to discover that in at least one important example the weight interval is so tight that it is 'almost a point'.

Theorem 2.1 (Taylor and Zwicker, 1993). In the US federal system with abstentions and the tie-breaking power of the vice president ignored, the president's weight interval is (15/115, 17/117).

The voting system referred to consists of the 435 members of the House, the 100 senators, and the president. The winning coalitions are those containing the president together with a majority of both chambers, and also those containing at least two-thirds of each chamber (veto override if the president does not belong to the coalition). A complete proof of the theorem is given in Taylor and Zwicker (1993), where it is also shown that the system is of monotonic dimension 2, even with the tie-breaking role (for Senate majorities) of the vice president factored in. Thus, the theorem tells us that for any way (but there is only one way in this case) of realizing this game as the intersection of two weighted systems via two NN weightings, the fraction of weight held by the president must be between 13% and 15% in both weightings. We rework part of the calculation using the dispersal method (see Section 3) in Section 4.

The other examples of voting systems discussed are all weighted. We employ the standard notation, i.e.

[q; Wl, w2, w3 . . . . . Wn],

to denote the weighted game with quota q and weights w 1, w 2, w 3 , . . . , w n. In particular, we are concerned with

• The 'at least s out of t' game Gs, t (general form of majority rule) in which s and t are integers with 1 ~< s < t:

t

[s; 1, 1 . . . . . 1]


• The original European Economic Community (EEC):

[ 1 2 ; 4 , 4 , 4 , 2 , 2 , 1 ]

in which France, Germany, and Italy each had four votes, Belgium and the Netherlands each had two votes, Luxembourg had one vote, and passage required 12 out of 17 votes. Note that Luxembourg was a dummy (Brams, 1985).

• The game we will denote G . . . . t ry:

[4; 2, 2, 1, 1],

which may be thought of as the two-out-of-three system in which the third voter has been split in two. This is one of the simplest voting systems for which influence is not equal; it may well be the fairest system for the mythical county discussed earlier.

3. Dispersals

There is an alternate characterization of the weight interval, given in terms of what we call left and right 'dispersals', and its development pays off in several ways. We begin with an informal presentation of the underlying idea, which is quite simple.

Let G be a simple monotonic game, p be a particular player, and j and k be positive integers. Imagine that we have some evidence that j 'yes' votes by p can be worth more than k 'yes' votes apiece by each of the other players. If we wish to see what this tells us about power, we might reason informally as follows. Let p 's fraction of the total power equal x, and the fraction held by all the others (combined) equal 1 - x. Then

jx > k ( 1 - x) ,

and so x > k / ( j + k) . I f j = 6 and k = 1, as is the case in the example that follows, then our conclusion is that whatever is p's fraction of power, it is greater than 1/7.

For a concrete example of the type of evidence that we have in mind, we turn to the original EEC voting system, described in Section 1, to show that a batch containing six votes by France (F) can be worth more than a batch containing one vote apiece by Italy (I), Germany (G), Belgium (B), the Netherlands (N), and Luxembourg (L). These two batches will each be added to a common third batch, or 'base', consisting of five 'yes' votes by I; five 'yes' votes by G; Two 'yes' votes by each of B and N, and 0 'yes' votes by each of F and L.

This base is a 'multiset', so called because we distinguish the number of times each member is in the set. In Table 2(a) the base has been rearranged, in the form


Table 2 A left dispersal showing that France's voting power in the original EEC system is greater than 1/7

(a) Before the dispersal, the left column is an arrangement of a multiset base into a sequence of six coalitions. The right column is a different arrangement of the same base. The base multiplicities are 1=5, G=5 , B=2, N=2, F=0, andL=0.

Left Right

I ,G I,B B,N,G I ,G I ,G G,N I ,G I ,G I ,G I ,G,B B,N,I I ,G,N

(b) In the dispersal itself, France (F) has been added , six times, to the coalitions on the left, resulting in six winning coalitions (total weight of 12 or more). Each of the other EEC countries have been added, once each, to the coalitions on the right, yielding six losing coalitions.

Left (winning) Right (losing)

I , G , F I ,B,G B ,N ,G ,F I ,G,B I , G , F G,N, I I , G , F I ,G ,N I , G , F I ,G ,B ,L B ,N , I ,F I ,G,N

of the lef t -hand column, into a sequence of six convent ional sets or coalit ions; they are convent ional in that no multiple membersh ip is al lowed for any single

coalit ion. This sequence has the same collective multiplicities as the base, so that

G, for example , is a m e m b e r of five o f the coalitions. The r ight-hand co lumn is a

different r ea r rangement of the same base into a sequence of six coalitions.

Intui t ively the two sequences, arising as they do f rom the c o m m o n base, represent

the same collective voting power.

In Table 2(b) we have added F to each of the coalit ions on the left and added I,

G, B, N, and L to those on the right, being careful never to add a m e m b e r to a

coal i t ion to which it a l ready belongs. The result is an enlarged left co lumn of six

winning coalit ions (each has a weight of at least 12, the quota) and an enlarged

right co lumn of six losing coalitions. Clearly, six Frances were more helpful to the left co lumn than were the o ther countries to the right ' equivalent ' c o l u m n - evidence that six votes by F can, somet imes, be wor th more than one vote by each o f the others.

M a d e precise, such a construct ion is an example o f what we will call a ' left dispersal ' , while similar construct ions that place upper limits on power are

re fer red to as ' r ight dispersals ' . O the r examples o f dispersals appear in Section 4. The index called 'dispersal power ' codifies a natural extension of the earlier

reasoning that x must be greater than 1/7, and is defined by setting D I ( p ) to be


the value(s) of x determined by all inequalities induced by left and right dispersals. Such an index raises some questions: • When is DI(p) well defined? (It would be ill defined if the set of possible x

values is empty because of inconsistent inequalities such as x > 1 / 2 and x < 1/3.)

• When is DI(p) a point (because infinitely many inequalities pin down a unique possible x value)? We answer these questions below, showing that DI(p) is well defined if and

only if G is weighted, and that for weighted games D l ( p ) = WI(p). A consequence is that for weighted games DI(p) is never a point when G has more than one player, since there is always some 'wiggle room' in assigning NN weights in a weighted game. First, however we must make these notions precise.

Definition. Let G = (N, W) be a monotonic simple game. A coalition sequence X is an arbitrary finite sequence

. . . . . x o

of subsets of N, whose length n we denote by IX[. Given two such sequences, X and X', the relationship X - X' is said to hold if both

(i) IX I = I x ' l , and (ii) For each p E N , I(i: p EXi}] = 1{i: p ~X;}]

Comment. It is worth considering interpretations for conditions (i) and (ii) above. By a trade between two coalitions, we mean an exchange in which a batch of players in the first coalition, none of whom belong to the second, move from the first to the second, while a similar movement (by a batch of possibly dissimilar size) takes place in the reverse direction. Imagine that the coalitions in X engage in a sequence of such trades. After the trades, the newly transformed sequence X' will satisfy X - X'. Furthermore, any X' satisfying X ~ X' can be obtained from X via trades (see Taylor and Zwicker, 1995, for details).

Alternatively, consider a multiset Z made up of elements of N appearing with possible multiplicities of 0, 1, 2, 3 . . . . . From such a multiset we can form, in more than one way, a sequence of ordinary sets whose union, counting multiplicities, is Z. Now X and X' satisfy (ii) if and only if they can be so formed from the same multiset Z, as in our example of France in the EEC. Other interpretations are discussed in Taylor and Zwicker (1997).

Definition. Let X be a coalition sequence, a be any subset of N, and k/> 0 be any non-negative integer. Then a dispersal of a over X of multiplicity k consists of a sequence ¥ of subsets of a, satisfying

• I¥1 = I x l . • For each p E a , [{i: p E Y/}[ = k .


• For each i ~< IX[, Y,. A X i = tt. The idea is that k copies of a are being dispersed among the sets in the

coalition, expanding each X i to become X i tO Y~. We will use X U Y to denote the sequence X 1 U Y1, X2 U II2 . . . . , X, U Y,.

Definition. Let G = (N, W) be a monotonic simple game, p a player in N, and j and k be any integers satisfying j > 0 and k ~> 0. Then ~3 = (X U Y, X' U Y') is a left dispersal of p, of multiplicity j, k if • X - - X ' .

• Y is a dispersal of {p} over X of multiplicity j. • Y' is a dispersal of N - {p} over X' of multiplicity k. • Each set X i U Y~ is winning, and each set X I U Y~ is losing.

Thus a left dispersal of p of multiplicity j, k is constructed by starting with two coalition sequences, X = (X 1 . . . . , X n ) and X' = (X ' 1 . . . . . X '} such that X' arises from X via a sequence of trades, then adding p to some j of the X~ (which did not already include p) and, for each q other than p, adding q to some k of the X I (which did not already include q), in such a way that each expanded Xi is winning and each expanded X I is losing.

The length [U[ of U is defined to be the length of either of its two coordinates. A left dispersal is evidence that j votes by p can be worth more than k votes apiece by each of the other players. For this reason, we define the inequality induced by U to be

x > k / ( j + k ) ,

where x is a numerical variable whose ultimate value corresponds to p ' s dispersal power.

A right dispersal V of p of multiplicity m, n is defined similarly, except that we require rn > 0 and n ~> 0, that ~" is a dispersal of N - {p} of multiplicity rn over X, that Y' is a dispersal of {p} of multiplicity n over X', and that the induced inequality is

x < m / ( n + m ) .

To keep all this straight, it may be helpful to picture the 'unprimed' sequence as a column running down the left-hand side of the page, with the 'primed' sequence running down the right-hand side. For both types of dispersals the left column is winning and the right is losing; what distinguishes left dispersals is that the additional copies of the individual p have been incorporated into the left column. The restrictions on j, k, m, and n allow induced inequalities to include x > 0 but n o t x > l , a n d x < l but n o t x < 0 .

Finally, p 's dispersal power, DI(p) , is defined to be the interval of values of x that satisfy all inequalities induced by all of p 's left and right dispersals. DI is considered to be well defined when the interval DI(p) is non-empty for each p.


Theorem 3.1. Let G be any monotonic simple game. Then: (i) I f G is weighted, then WI(p) = Dl(p) for each player p.

(ii) G is weighted if and only if DI is well defined and if and only if DI(p) is non-empty for some p.

(iii) l f p is any player, DI(p) is non-empty, and if G has more than one player, then DI(p) is a 'proper' interval containing infinitely many points.

Proof. It is straightforward to show that when G is weighted, WI(p) C Dl(p): the informal argument that x > 1/7 actually shows that if w is any NN weighting for G, then w(p)> 1/7. By the same token, w(p) must satisfy all of the induced inequalities, from which it follows that w(p) ~ DI(p). This 'easy' direction for (i) immediately yields an easy direction for (ii), for if G is weighted, then DI(p) cannot be empty - it contains WI(p). We present proofs of the 'hard' directions of (i) and (ii) in the appendix. Also, note that (iii) is a corollary of the first two parts, because if DI(p) contains at least one number, then it follows from (ii) that G is weighted, from which we know that WI(p) C DI(p). The result then follows from the observation that when a weighted game has more than one player, there is always some 'wiggle room' in the values of the weights, so that WI(p) contains infinitely many values. []

Corollary (of the proof) 3.2. The weight interval of any player in a weighted game is completely determined by the inequalities induced by the finitely many left and right dispersals of length less than 3 41ul .

Proof. Let us use 'DI*' to refer to the version of DI that is calculated using only dispersals of length bounded as above. Then the proof in the appendix actually shows that DI* C WI. Thus, we obtain the chain DI* C WI C DI C DI*, from which WI = DI*. []

Note that because DI*(p) is determined by finitely many strict inequalities, it cannot possibly consist of a single point. This alternate proof of Theorem 3.1 (iii) tells us why dispersal power is inherently fuzzy - x ' s value cannot be pinned down precisely, because the concept underlying this way of measuring power can, in fact, only yield finitely many limitations on this value (even though it appears at first to offer the possibility of yielding infinitely many such limitations).

As a consequence of the above theorem and its corollary, we obtain two related methods for calculating Wl(p) when p is any player in a weighted game G:

The first weight algorithm. First, list all objects of the appropriate 'format' to be left or right dispersals of length 3 4fNf o r less. Next, list all induced inequalities for


those which, upon checking, turn out to be legitimate dispersals; from these, the interval WI(p) is clear.

The second weight 'algorithm' (dispersal method). In step one, guess the left and right dispersals which are extremal, i.e. those that determine, through their induced inequalities, the tightest possible interval I. The second step is a check. Assume that e is an arbitrary element of this interval I, and construct an NN weighting w for G with the property that w ( p ) = e, thus establishing the correctness of the guess.

The first algorithm comes equipped with a gua ran tee - it will work every time. However, because of the unreasonable length of the search, this algorithm is not of practical use. The second method is not exacfly an algorithm, relying instead on developing a 'feel' for how dispersals work, and its accompanying guarantee (discussed below) is weaker, but it appears to be a practical and effective method. In particular, it does not seem to be difficult to find the extremal dispersals, or to carry out the check, for the games we are interested in, such as those voting examples considered in this paper (see the sample calculation in the next section).

Suppose that, in some given instance, we have successfully carried out both steps of this second method. Can we be sure that the answer is correct? Yes, and only the 'easy' direction of Theorem 3 . i0 ) is needed to establish this certainty. However, if we wish to be sure that, in all cases, a judicious guess in step one exists (one which does, in fact, produce the weight interval), we must rely on the 'hard' direction of Theorem 3.1(i).

Thus, the two methods possess complementary virtues. This observation leads quite naturally to a question:

Question 3.3. Does there exist a method for calculating the weight interval that is computationally feasible and is also a true algorithm?

We do not know the answer to the question. However, Daniel Velleman has pointed out that the problem of determining, given a weight function and quota for the set N, and a second weight function and quota for N, whether or not they determine the same collection of winning coalitions, is NP-complete. This result (which is quite similar to a variety of others in the area of NP-completeness) can be used to show that if the concept of 'canonical weights' for a weighted game could be made precise, then the task of computing the canonical weights from a given set of weights must remain computationally infeasible until that day, should it ever come, when someone proves P = NP.

As phrased above, Question 3.3 is ill-posed. Suppose that we make it precise as follows. Is there a computationally feasible algorithm that determines, given a weight function and a quota for the set N, the values of WI(p) for the game

A. Taylor, W. Zwicker I Mathematical Social Sciences 33 (1997) 23-74 37

induced by that function and quota? If so, we would be in the somewhat surprising position of being able to feasibly compute, given both a weight function and quota for the set N and a second weight function and quota for N, whether or not their induced games determine the same W I values but unable (unless P = N P ) to feasibly compute whether or not these games were the same game.

4. The president's weight interval via the dispersal method

Before calculating the president's weight interval in the federal system, we work a miniaturized example with a nine-member Senate. The smaller size allows us to fully write out the dispersals.

Our voters will consist of nine senators, So, S ~ , . . . ,Ss, and the president, P (there is no House in this version). Winning coalitions will be those containing the president together with at least five senators, as well as those containing at least eight senators (veto override). Table 3 consists of, respectively, a left dispersal (Table 3(a)) of multiplicity 9, 2 (with induced inequality x > 2/[9 + 2]), and a right dispersal (Table 3(b)) of multiplicity 9, 4 (x < 4 / [ 9 + 4]). We also give a brief

Table 3 (a) A left dispersal of multiplicity 9, 2, for the nine-member senate

Winning Losing

X~ U Y, X; U Y;

{sl,s2,s3,s ,,ss} o{P} {s2,s3,s4,ss,s6} t0{P} {S3,S4,S5,S6,S7}to{P } {S4,ss,s6, s7,ss}to{P} {ss,s~,s.,s~,so}to{P} {s6,s7,s8, so, s l}U{P } {S7,S8,So,SI,S2}U{P } {s~,so, S,,S~,S~}U{P} {so, sl,s2,s3,s,}U{P}

{s,,s~,s~,s.,s,}U{s~,s;) {s,,s~,s.,s,,s6}U{s.s~} {s3,s.,s~,s6,s~}U(s~,so} {s. ,s , ,s6,s.s ,}U{so,sl} {s~,s~,s.s,,so}U{s,,s~} {s~,s,,s,,So, S,}u{s~,s3} (s,,ss, So, S,,s~}U{s~,s,} (s,,so, S.S~,s~}U{s.,s~) {So, S.S~,S~,s~}U{s~,s~}

(b) A right dispersal of multiplicity 9, 4 for the nine-member senate

Winning Losing

X~ U Y~ X; U Y;

(s,,s~,s3,s,}U{s,,s6,s7,ss} {s~,s3,s.,s,}U{s~,s7,s~,so} {s3,s.,s~,s6}U{s.s,,So,Sl} {s,,s,,s6, s~}u(s~,So,S~,S2} (Ss,S6, S7,Ss}O{So, SI,S2,S3} {S6, S7,S8, So}O{sI,S2,S3,S4} {s7,ss,So, Sl}U{s2,s3,s4, s5} {S8, So,Sl,S2}U{s3,s4,sf,s6} {So,Sl,s2,s3}U{s4,Ss,S6, s7}

{st,s2,s3,s4} U {P} {s2, s 3, s4, ss} U {P} {s s,s 4,ss,s6} O {P} {s4, s~, s6, sT} U {e} {s,, s6, sT, s,} U {e} {s6, s,, s., So) U {e} {sT,ss, So,S,} U {P} {ss, So, s,, s2} U {P} {So, Sl, s2, s3} U {P}


explanation of where these numbers came from, and a check that any e in the interval (2/11, 4/13) can be made to work as the president's weight in some NN weighting for this system. This establishes WI(P) = (2/11, 4/13) for this miniature Senate.

Note that in the left dispersal, P appears 9 times among the Y~, while each other player appears twice among the Y~. By using 9 as the dispersal's length we can rotate each senator into any coalition sequence, such as the Y~, an equal number of times. We choose 5 for the size of each X~ and 2 for that of the Y~ to construct minimally winning and maximally losing unions, expected in extremal dispersals. The particular rotation pattern is one that scales up easily to apply to the real, 100-member Senate.

For the 'check', let e be any number in the interval (2/11, 4/13), set the president's weight to be e and the weight of each senator to be (1 - e)/9. To see that this weighting works, it suffices tO confirm that each losing coalition weighs less than each winning coalition, leaving room for the insertion of a quota.

In particular, we need to confirm that losing coalitions of the form 4S + P (four senators and the president) weight less than winning coalitions of the form 8S: • i.e. that 4( (1- e)/9) + e < 8 ( ( 1 - e)/9); • i.e. that 4 / 9 - 4 / 9 e + e < 8 / 9 - 8/9e; • i.e. that e<4 /13 , which is so. Similarly, it is straightforward to confirm that all forms of maximal losing coalitions weigh less than all forms of minimal winning coalitions.

We present below the extremal left and right dispersals that determine the president's weight interval in the US federal system, as simplified by dropping the vice-president's tie-breaking role in the Senate. In Taylor and Zwicker (1993) we show that this interval is (15/115, 17/117), but use ad hoc methods rather than the dispersal method described in Section 3. However, the calculation in Taylor and Zwicker (1993) employs the same 'check' (the second step in the dispersal method) to show that the dispersals below are indeed extremal, so we do not duplicate that step here.

As above, let 'H' stand for 'member(s) of the House', 'S' stand for 'senator(s)', and 'P' stand for 'the president'. Let NEe a denote the set containing the 435H, 100S, and P, and GEe a = (NEe d, W) be the monotonic simple game representing the simplified federal system. In Taylor and Zwicker (1993) we show that this game is of monotonic dimension 2, and that its representation as an intersection of two (monotonic) weighted games is unique.

These two weighted games, G 1 =(NFed, W1) and G z = ( N F e d , W 2 ) , are as follows: for G 1 the minimal winning coalitions are of two types: • 51S+P, and • 67S.

For G2, the two types of minimal winning coalitions are:


• 218H + P, and • 290H.

It is straightforward to see that G1 and G 2 are weighted, and that W = W1 M W 2. Thus, it suffices to present extremal dispersals showing that WI(P)= (15/115, 17/117) for G~, and WI(P)= (71/506, 73/508) for G2; the president's weight interval in GEe ~ is, by definition, the smallest containing both of t h e s e - but the interval for G 2 is entirely contained in that for G 1, so the final answer is (15/115, 17/117).

Extremal dispersals for G 1. Our left dispersal,

(xur, x 'uv ' )

is of multiplicity 100, 15 and length 100 in which X - X ' , Y is a dispersal of {P) over X of multiplicity 100, and Y' is a dispersal of NFe d --{P} over X' of multiplicity 15. Specifically, each X i is a set of 51S and each individual S is a member of 51 of the X r This can be done, for example, by numbering the individual senators as so, Sl, . . . , S99 , and setting, for each i, Xi = (sti 1, sll+i 1 . . . . , sts0+~l}, where [k] denotes the remainder when k is divided by 100, so that s~ is a member of the coalitions Xls0+~l, Xt51+~l . . . . , Xtt00+~ 1. In this case we can take, for each i , X I = X i ,Y i={P} , and Y i = s o m e set of 15S (together with some H) in such a way that each individual S and H is a member of exactly 15 of the Y'~ and such that X'~ M Y~ = 0. This can be done by setting Y ~ - - { s [ 5 1 + i ] , s [ 5 2 + i ] , . . . , s [65+ i l } U ~ . ( s o that s~ is a member of the coalitions y t l • , !

[35.~1, Y[36+~1, • • Y[49+il), where ~ is an easy-to-specify set of House members. It is easy to see that each coalition X~ U Y~ is winning, and each coalition X I U YI is losing, in G 1 .

The just-described left dispersal could be quickly reconstructed from the following (highly elliptical) chart:

Winning Losing

f 5 1 s + P 51s+15S 100J51s + P 51s + 15S / i

1 5 1 s + P 51s+15S

Next, the right dispersal with induced inequality ' x < 17/117' can be similarly reconstructed from this chart:

Losing Winning

f 5 0 s + P 50s+17S 1 0 0 J 5 0 s + P 50s+17S / -

( 5 0 s + P 50s+17S


Extremal dispersals for G 2. The extremal left dispersal, of multiplicity 435, 71, can be reconstructed from:

Winning Losing

['218h + P 218h + 71H 100 ~218h + P 218h +: 71H

L218h'+ P 218h + 71H

The extremal right dispersal, of multiplicity 435, 73, can be reconstructed from:

Losing Winning

1"217h + P 217h + 73H X00J217h + P 217h + 73H

/ : L217h'+ P 217h + 73H

This completes our calculation of the president's weight interval in the federal system.

5. Market interval and the vote-selling game

Imagine that a lobbyist has been charged with bribing a certain legislature to pass some particular bill. Each legislator would tend to vote against the bill, but is willing to be bribed. Each legislator makes an offer to the lobbyist, who then must decide whose votes to buy, at the prices offered. Being thrifty, he or she will spend the bare minimum necessary to buy a winning coalition. Supposing that each legislator would like to sell his or her vote for as much as possible, what amounts should they charge?

It seems that the question is most easily analyzed in terms of the percentage of the sum of all offers that a given legislator's offer represents. If this percentage is sufficiently low, then his vote will certainly be bought, but for less than he could have successfully demanded, and if it is too high he will not be bought at all. When the percentage is within some intermediate range, whether the vote is bought or not depends on the relative values of the other offers.

Given a monotonic simple game (N, W) we make these considerations precise by describing a new 'game' called the vote-selling game, as follows:

A play is a function f : N---> [0, 1] satisfying • {f(p) : p E N} = 1. (Thus, what we are calling a 'play' is usually called an imputation.) The cost o f a coalition Z C N is defined to be ~ {f(p) : p E Z}. The winning coalitions whose cost is minimal (compared with that of other winning coalitions) are called f ' s cheapest winning coalitions; there may be one or several. A strategy for a player p is a real number c E [0, 1] and a play of this strategy is a play f with f (p) = c.

A play f of the strategy c for p will be called certainly successful for p if p is in


each of f ' s cheapest winning coalitions, and it will be called certainly unsuccessful for p if p is in none of f ' s cheapest winning coalitions.

For a fixed player p E N, we define three sets of strategies, denoted L (for ' low'), H (for 'high'), and R (for ' reasonable') as follows: • c ~ L iff every play f of c is certainly successful for p. • c ~ H iff every play f of c is certainly unsuccessful for p. • c ~ R i f f c E [ O , 1 ] - ( L U H ) .

It turns out that if L is neither [0, 1] nor 0, then L = [0, a) for some a satisfying O<a ~ 1/2. If H is not empty then H = (b, 1] for some b satisfying 0~<b < 1. Neither of these assertions is completely obvious; the proofs involve the compactness of the set of plays and appear in Section 6. If p is a vetoer (a member of every winning coalition), then L = [0, 1] and R is empty, while in all other cases R is a non-empty closed interval [a, b] referred to as the interval of reasonable prices for p, or the market interval for p, Ml(p). MI is therefore a well-defined interval index of power for any game that contains no vetoers.

The intuition behind using the word 'reasonable' is as follows. A price c for player p is obviously too low iff for some d > c it is the case that every play of d is cer ta inly successful for p. This turns out to be true iff c ~ L (because L is open on the right). On the other hand, a price c for player p is obviously too high iff every play of c is certainly unsuccessful. This is true iff c E H. Hence, the price c for player p is in R iff it is neither obviously t o o low nor obviously too high.

The vote-selling game described above yields an interval R(p) of prices that are reasonable for player p to 'charge' for his or her vote. Notice that the interval R(p) is arrived at without any knowledge of - or assumptions about - what prices are being charged by the other players. However, once the vote-selling game has been played (or at least analyzed by all the players), then the standard rationality assumption allows everyone to assume that, say, player q will choose his or her price in R(q). This suggests that we generalize the vote-selling game in a way that lends itself to iteration. One such generalization is as follows.

Given a game (N, W) and intervals 1 1 , . . . , I n C [0, 1], we consider a new game, called the vote-selling game with respect to I 1 , • • • , In, that is played exactly as is the vote-selling game described above, except that a play is now a function f : N--~ [0, 1] satisfying ~ { f (p ) : p ~ N} = 1 and f ( j ) Elj for every j E N.

The iterated market interval, MI*(p), is obtained by repeating (potentially, ad infinitum) the analysis that yields the market interval, as follows. First, we consider the vote-selling game described at the beginning of this section. The analysis yields a sequence MIl(Pl ) . . . . ,MIl(pn ) of intervals. If we have completed the kth analysis and obtained intervals MIk(pl ) . . . . . MIk(Pn), then the k + l s t analysis is that of the vote-selling game with respect to MIk(pl ) . . . . . Mlk(pn ). We define the iterated market interval for p, MI*(p), to be f-I {MIk(p): k = 1,2, 3 , . . . ) .

We have determined, for several games of interest, the values of the market


and iterated market indices. These appear in Table 1 in Section 2. In particular, in the federal system, the market interval and iterated market interval agree, and yield an interval of approximately [0, 1/4] for the president (the calculation is in Section 9).

6. Structure of the market intervals

In this section we examine the structure of the intervals L, H, and R that arise in the vote-selling game and show, in particular, that L is open on the right and H is open on the left. This is what allows us to conclude that the interval R of reasonable prices is, in fact, a closed interval. Also, we consider the extreme cases and their relationship to the existence of vetoers in either the reduced game or subgame determined by the player p.

Let G = (N, W) be a simple game.

Theorem 6.1. For any person p ~ N, L is an interval open on the right relative to

[0, 1], and H is an interval open on the left relative to [0, 1].

Proof. First we establish that L is an interval by showing that if c ~ L and 0 ~< e < c, then e E L. Let f be any play of e, and let f ' be any play satisfying f ' ( p ) = c and f ' ( q ) < ~ f ( q ) for each q E N - { p } . Now every cheapest winning coalition under f ' contains p, every coalition omitting p costs at least as much under f as under f ' , and every coalition containing p costs at most as much under f as under f ' . Hence, every cheapest winning coalition under f includes p, as desired.

Next we show that L is open on the right relative to [0, 1]. Assume c E L and let c~ denote the set of all plays f for which f ( p ) = c. Of course, the set of all functions mapping {1 . . . . , n} to [0, 1] is the product of n copies of [0, 1], and so it is a compact set. Moreover, c¢ is simply the closed subspace of all such functions whose values sum to one and for which f ( p ) = c. Hence ~ is also compact. Since c ~ L, every f in ~ is certainly successful for p, and so we can consider the function that associates to each f ~ q¢ the difference between the cheapest winning coalition not containing p and the cheapest winning coalition that does contain p. This is a continuous function on a compact set and so it achieves its minimum at some f E q¢. Let e be the difference between the cheapest winning coalition containing p and the cheapest winning coalition not containing p for this play f. Let d = c + e/3.

We claim that d E L. Let g be any play of d, and g' be any play satisfying g ' ( p ) = c and g ' ( q ) ~ g ( q ) for each q @ N - { p } . Under g', every cheapest winning coalition containing p costs less, by an amount equal to e or more, than any coalition cheapest among those omitting p. Also, while the cost of a coalition


containing p may be greater under g than under g', the difference is at most e/3, and while the cost of a coalition omitting p may be less under g than under g', the difference is at most e/3. Since e/3 + e/3 < e, it must be that under g the cheapest winning coalitions all contain p, which means that d @ L.

The proof that H is an interval proceeds, similarly, by showing that if c E H and c < e <~ 1, then e E H, and the proof that H is open on the left parallels the corresponding argument for L. []

It follows immediately that R is either empty or is a closed interval. When is R empty? Clearly, the sets H and L are disjoint, so R is only empty when L = [0, 1] or H = [0, 1]. However , it is straightforward to see that a play of 0 cannot be certainly unsuccessful for p, so that H is never [0, 1]. It is also easy to show (see below) that L is [0, 1] if and only if p is a vetoer, and that if p is not a vetoer then H D (1/2, 1]. A simple game containing at least one vetoer is referred to as weak. A consequence, then, is the following corollary:

Corollary 6.2. If p is not a vetoer, then the market interval for p, MI(p), is a non-empty closed interval contained in [0, 1/2]. In particular, the market interval is a well-defined power index for every non-weak game.

In addition to vetoers, other special players such as 'weak vetoers ' , 'passers', and 'dummies' play a role in determining the extreme possibilities for MI(p). Also, if we wish to determine when a price of zero is certainly successful for p regardless of the prices of the other players (i.e. when 0 is not in MI(p)) or when a price of one-half is certainly unsuccessful (i.e. when MI(p) C [0, 1/2)), then the 'weakness of the reduced game for p ' and the 'weakness of the subgame for p ' become important. Hence, before turning to these results, we will recall the necessary concepts.

Definition. A player p will be called a vetoer if he or she belongs to every winning coalition, a weak vetoer if he or she belongs to every winning coalition except N - {p} (which may or may not be winning), a passer (or 'master player ') if {p} is a winning coalition, and a dummy if every coalition containing p is winning if and only if it is winning after p is removed (p ' s vote never makes a difference).

The notion of a weak game goes back at least to Shapley (1962); the consideration of vetoers allows for a slightly finer analysis than does the consideration of weakness. In keeping with the spirit of the present paper, it is worth noting that a game G is non-weak iff every play of the associated vote-selling game yields a winning coalition of cost less than one.

The following are special cases of two concepts that are well known in the li terature (see, for example, Einy, 1985).


Definition. If G--(N, W) is a simple game and p E N, then we define two new games called the reduced game RGp for p and the subgame SGp for p as follows:

g G p = ( N - { p } , W * ) , where W * = { X C N - { p } : X U { p } E W } ,

S G p = ( N - { p } , W * * ) , where W * * = { X C N - { p } : X ~ W } .

With these at hand, we can state the main results of this section, and thus answer some of the questions above. The situation for vetoers and dummies is quite simple, as indicated by the following two propositions. The proofs are straightforward, and have been omitted.

Proposition 6.3. The following are equivalent: (a) p is a vetoer. (b) L = [0, 1]. (c) a / 2 E L (equivalently, L ~ [ O , 1/2]). (d) H = ~.

(e) MI(p) = fl (i.e. R = ~)).

Proposition 6.4. The following are equivalent: (a) p is a dummy. (b) /4 -- (0, 11. (c) Ml(p) = [0, O] = {0}.

The following theorem provides much of what we need to describe the situation for players who are neither vetoers nor dummies.

Theorem 6.5. If p is not a vetoer, then: (1) Every play f of zero is certainly successful for p (equivalently, Ml(p) C

(0, 1]) iff p is a weak vetoer and the reduced game RGp is not weak. (2) Every play f of one-half is certainly unsuccessful for p (equivalently,

MI(p) C [0, 1/2)) iff every player q who is a vetoer for the subgame SGp is also a vetoer for the whole game G.

Proof. (1) First, note that since 0 is never in H, MI(p) omits 0 if and only if 0 E L. Assume that every play of 0 is certainly successful for p. We first show that p is a weak vetoer. Suppose not and choose a winning coalition Y such that Y C N - {p, q}. Set f (p) = O, f (q) = 1, and f(r) = 0 for every other r. Then Y is one of f ' s cheapest winning coalitions (costing 0) and p , ~ Y. Hence, f is not certainly successful for p. We now show that the reduced game RGp is not weak. Suppose it is weak, and let q be a vetoer for RGp. Set f (p) = 0, f (q) = 1, and f(r) = 0 for every other r. Since p is a weak vetoer but not a vetoer, we know that N - {p} E W and the cost of N - {p} is one unit. Moreover , if X is a winning


coalition with p ~ X and X ' = X - {p}, then X ' is winning in RGp and so q E X'. Hence , X' , and thus X, costs one unit also. It follows that p does not belong to every cheapest winning coalition (i.e. N - {p} is such) and so f is not certainly successful for p.

Conversely, suppose that p is a weak vetoer and the reduced game RGp is not weak. Let f be a play of 0 for p. Choose q so that f(q) ~ O. Since q is not a vetoer for RGp we can choose X C N - {p} so that X is winning in RGp but q~X. Thus X U (p} E W and the cost of X U {p} is less than one since q,~X U {p}. Now, since p is a weak vetoer, the only winning coalition not containing p is N - {p}, which costs one. Hence, p belongs to every cheapest winning coalition.

(2) First, note that because p is not a vetoer, 1/2g~ L. Hence MI(p) omits 1/2 if and only if 1/2EH if and only if Ml(p) C [0, 1/2). Assume that every play of 1/2 is certainly unsuccessful for p. Assume q is not a vetoer for G. We will show that q is not a vetoer for the subgame SGp. Choose X E W so that q,~X. Consider the play of 1/2 that also makes f(q)= 1/2. Then p belongs to the winning coalition X U {p} of cost 1/2 and so there must exist a winning coalition Y of cost strictly less that 1/2. But then p , ~ Y and q ~ Y, and so Y shows that q is not a vetoer for the subgame.

Conversely, assume that every player q who is a vetoer for the subgame SGp is also a vetoer for the whole game. Assume f is a play of 1/2. Since p is not a vetoer , there exists X E W so that p,~X. This X costs at most 1/2. Choose q E N - { p } so that f (q ) > 0.

Case 1. q is not a vetoer for the subgame SGp.

In this case we can choose Y C N - {p, q} so that Y E W. Then Y costs less than 1/2, so f is certainly unsuccessful for p.

Case 2. q is a vetoer for the subgame SGp.

Then q is also a vetoer for the whole game (by assumption) and so any winning coalition containing p also contains q and thus costs strictly more that 1/2. Thus N - { p } is a winning coalition cheaper than any containing p, and so 1/2 is certainly unsuccessful for p. []

The above results justify the tree diagram of Fig. 1, with the exception of a few points referenced by numbers in the figure and discussed in what follows.

(1) Since p is neither a dummy nor a vetoer, there exists some winning coalition C that is minimal (properly contains no other winning coalition), contains p as an element, and omits at least one other player. It is easy to show that if e > 0 is sufficiently small, then the play f for which f (q ) = e if q E C and f(q) = 12 otherwise (where 12 is chosen to make the values of f add to unity)

46 A. Taylor, W. Zwicker / Mathematical Social Sciences 33 (1997) 23 -74

Is p a vetoer?

If so, R = D" If not , is p a d u m m y ?

If so, R = (01. If not , is p a

w e a k vetoer?

If so, is p

a passer?

I f so, R = {1 /2 ) (~1 I f not, is

RGp w e a k ?

I f no t , R -

[0, b], where b >0 (1). Both

b < I / 2 and b = I / 2 a r e

possible (2).

I f so, R = [0, 1/2] . (4) I f not , R = [a, 1/2],

where 0 < a < 1/2 . (5)

Fig . 1. Conditions governing the value of the market interval, R = MI(p ) , o f a p l a y e r p .

makes C the sole cheapest winning coalition, so e is not in the set H, from which b > 0 .

(2) If N = {p , q, r} and the only minimal winning coalitions are { p } and {q, r} , then b = 1/2, while if the minimal winning coalitions are { p } , {q} and {r}, then b = 1/3.

(3) This is easy to verify. (4) and (5) Because p is a weak vetoer, every player q ¢ p is a vetoer in SGp,

but not every player is a vetoer, since p is an e lement of some winning coalition C with the properties described in (1) above. Then, by Theorem 6.5, b = 1/2.

To see that a < 1/2 in (5), let e be a small positive number a n d f be the play for which f ( p ) = 1/2 - e and f ( q ) = O for each q ~ p (where O is chosen to make the values of f add to unity). Then, since p is not a passer, every winning coalit ion containing p costs at least 1 /2 - e + ~ . If e is sufficiently small, then this is more than 1/2 + e, the cost of the coalition N - { p } (which is winning, since p is not a vetoer). This shows that not every play of 1/2 - e is definitely successful for p, so that a ~< 1/2 - e < 1/2.


7. The iterated market interval: An example

We consider the general majority game G,., (in which 1 ~< s < t are integers and the winning coalitions are those containing any s or more of the t people) and show that iteration of the vote-selling game yields a sequence of intervals that shrink to a point, with M I * ( p ) = {l/t} for each of the t players. If s = 1 or s = t - 1, then this happens by the first or second stage. However , for intermediate value of s the intervals only close in to a point at the limit.

If we set l 0 = 0, r 0 = 1, and define l, and r n by setting

M l n ( p ) = [In, rn],

where MIn(p ) is defined as in Section 5, then we begin with the following lemma.

Lemma 7.1. For the game G~, t and any player p: (a) 1 / t ~ [In, r~] for every n ( f r o m which it is clear that I i ~ r j for each i and j ) . (b) I f we define sequences a i and b i inductively by setting a o = O, b o = 1, and

a~+~ = [1/(s + 1 ) ] - [ ( t - s - 1)/(s + 1)]b~,

or 0 when this quantity is negative,

and

bn+ 1 = [ 1 / ( t - s + 1)] - [(s - 1 ) / ( t - s + 1)Jan

for every n, then [ln, r~] = [a n, bn].

Proof. First, we establish (a) via induction. The base step is clear, so we assume that 1/t ~ [l~, r~] and consider the vote-selling game with respect to the intervals [l~, rn] (i.e. this same interval assigned to each player). Consider the play

1/t, 1/t, 1/ t . . . . . 1/t

(by which we mean the function whose values on the players Pl , P2 . . . . are enumera ted by this list). Note that under this play, no player is certainly successful and no player is certainly unsuccessful. It follows that 1/t is neither in Ln+ 1 nor in Hn+ 1, from which 1/t E Ml~+l, as desired.

Part (b) also has an inductive proof. Assume that [In, r,] = [a, ,b~]. Then [In+ 1, rn+l] = [an+ i, b,+a] follows immediately from Claims 1-6 below.

Claim 1. an+ 1 >~0 and bn+ 1 E l .

Proof. That an+ 1 >10 is immediate from the ' . . . or 0 when' clause in the inductive definition. That bn+ 1 ~< 1 follows easily from b,+l'S definition together with a~ >I 0.

Claim 2. an+ 1 <~b~ and bn+ 1 ~ a , .

48 A. Taylor, W. Zwicker / Mathematical Social Sciences 33 (1997) 23 -74

Proof. Since b n = r n >I 1 / t , an÷ l ~< [1/(s + 1)] - [ ( t - s - 1)/(s + 1) ] ( l / t ) = 1 / t<~

b n. Similarly, bn÷ 1 ~>a n.

Claim 3. an+lj~'Ln+ 1.

Proof. Consider the play

s + l t - s - 1 A

an+ 1,an+ 1 . . . . ,an÷ 1 , b n, b . . . . . , b n ,

in which the player p under consideration is one of those assigned the value an+ 1. Note that the total cost is 1 and that this play of a n +1 is possibly successful for p since his value is as low as any (see previous claim).

Claim 4. ] f c < a n + l , t h e n c ~ L ~ + I.

Proof. Assume, by way of contradiction, that f is a play of s o m e C<an+ 1 for p that is possibly unsuccessful. Then at least s players other than p must have been assigned values of c or less. With a total of at least s + 1 players assigned values

below an+l, by comparing f with the play in Claim 2, we see that the remaining t - s - 1 players are forced to have an average assigned value strictly above

b n = r n, which is impossible since every value assigned by f lies in the interval

[In, rn].

Claim 5. bn+tfi~.an+ 1.

Proof. Consider the play

t - s + l s - 1

b n + l , b n + l , . . . , b n + l , a n , a n , . . . , a n ,

in which the player p under consideration is one of those assigned value bn+ I. Reason as in Claim 2.

Claim 6. I f c > b , + l , t h e n C~nn+ 1.

Proof. Assume, by way of contradiction, that f is a play of some c > bn+ 1 for p that is possibly successful. Reason as in Claim 4. []

Next we consider the consequences of this l emma for different values of s and t. Recall that 10 = 0 and r 0 = 1.

Case 1. Assume s = t - 1. Substituting t - 1 for s in the inductive definitions above, we obtain:


In+ 1 = ( l / t ) -- (O)(r . ) and rn . 1 = (1 /2 ) - ( [ t - 21/2)( l . )

so tha t 11 = 1/ t and r I = (1 /2 ) - ([t - 2 ] /2)0 = 1/2, f rom which we get l 2 = 1/ t and r 2 - - - - - ( 1 / 2 ) - ( [ t - 2 ] / 2 ) ( 1 / t ) = 1/t. Clearly, M l n ( p ) remains equal to { l / t} for n > 2, so we 's tabil ize at s tage 2' (actually, s tage 1 if t = 2).

Case 2. A s s u m e s = 1 < t - 1. Subst i tut ing 1 for s in the induct ive definit ions above , we obtain:

l.+a = (1 /2 ) - (It- 2 ] / 2 ) ( r . ) and rn+ 1 ---- ( l / t ) - (0) ( l . )

so tha t l 1 = (1 /2 ) - ([t - 2 ] /2 ) (1 ) = (1 - t + 2 ) / (2 ) - but this quant i ty is e i ther 0 or

nega t ive , so tha t actually l~ = 0 via the ' . . . or 0 when ' clause. Also, r~ = 1/t. Thus

12 = (1 /2 ) - ([t - 2 ] / 2 ) ( 1 / t ) = 1/t , and the process once again stabilizes at s tage 2,

Case 3. A s s u m e 1 < s < t - 1.

C la im. lim=[l.] = 1/t.

I f we assume tha t n /> 1, and momen ta r i l y ignore the ' . . . or 0 when ' clause in the

f o r m u l a fo r ln+l, we obtain:

l ,+ 1 = ( 1 / [ s + 1]) - ( [ t - s - 1]/[s + 1 ] ) ( / , )

=(1/[s+ l l ) - ( [ t - s - l ] / [ s+ l])((1/[t-s+ 11) - (Is - l ] / [ t - s + 1 ] ) ( / . _ 1 ) )

= (2 / ( I s + 1][t - s + 1]) + ( ( I t - s - 1]is - 1]) / ( is + l l t / - s + 11) ) ( l ,_~)

A B

= A + B ( I , _ I ) ,

where A and B are quant i t ies in the large pa ren theses in the prev ious express ion.

Since A and B are each (strictly), be tween 0 and 1 in value , this final express ion does yield a posi t ive value for l~ + 1, showing that it was safe to d rop the ' . . . or 0 w h e n ' clause af ter s tage 1.

N o w , since I 0 = 11 = 0, 12~+z = A + A B + . . . + A B ~, and ! i m [ / 2 n + 2 ] = A / ( 1 -

B ) = 1/ t , which establ ishes the claim. For the r ight -hand endpoin t ,

l i ra [r.] = l i r a [ ( 1 / i t - s + 1]) - ([s - 1 ] / [ t - s + 111(1._1) ]

= [ ( 1 / [ t - s + 1]) - ([s - 1 ] / [ t - s + 1])] [ l ira [ / . - d ]

= [ ( 1 / [ t - s + 11) - ([s - l l / [ t - s + a l ) ] [ 1 / t ]

= 1 / t .


This completes the proof of the main theorem on iterated market intervals for the general majority games.

Theorem 7.2. For the game Gs, , (with any choice of integers 1 ~<s < t), M l * ( p ) =

{l/t} for each player p.

8. Bribery equilibria

If the iterated market interval, MI*(p) , is non-empty for each p, then we will say that MI* is well defined. The equation of which games have a well-defined MI* is related to the concept of a 'bribery equilibrium':

Definition. A play f of the game G is a bribery equilibrium, or b.e., if every player is a member of at least one cheapest winning coalition and is a non- member of at least one cheapest winning coalition.

Such an f is an equilibrium in that any slight increase in the value o f f (p ) (while f ' s values on the other players decrease slightly, maintaining their relative proportions to each other) would mean that p is no longer in any cheapest winning coalition, so that p would be smart to move his or her asking price back down again. Any slight decrease (again, maintaining relative proportions among the others) would mean that p is in every cheapest winning coalition (thus guaranteeing that p is bribed), but any even smaller decrease would yield the same guarantee with greater payoff, so again p is subject to 'restorative forces' pushing his or her asking price back towards f (p ) . We show, below, that for a broad class of games bribery equilibria always exist.

Of course, one might argue that the above notion does not truly represent an equilibrium, since there may be some incentive for p to move down from f ( p ) in order to get the guarantee. Indeed, our definition is weaker than that adopted in Young (1978). However, Young's equilibrium notion requires, if equilibria are to exist, that the basic game be augmented both by a choice of a floor price p0 (a minimum price player p would accept) for each player and by a tie-breaking rule 9- (saying which coalition is bought when several are equally cheap). The equilibria then depend on the particular choice of floor prices and of 3-, rather than being intrinsic to the game.

Note that the constant function f ( p ) = 1/t is a b.e. for Gs. ,, and that the argument in Lemma 7.1(a) generalizes to show that if G is any simple game and f is any b.e. for G, then f ( p ) E M I n ( p ) for each n, so f 'witnesses the well- definedness of MI*'. More generally, if we define ~ ( p ) to be the range of all bribery equilibria values on p (i.e. ( f (p ) : f is a b.e.}), then ~ 9 ~ ( p ) C Ml*(p)


always holds. (The chart at the end of Section 10 shows that the reverse containment can fail.) This leads to:

Theorem 8.1. For any simple game G = (N, W) having no vetoers, the iterated market interval MI* is well defined.

Proof. By the above remarks, it suffices to show that G has at least one b.e. Let us define a play f to be maximal if the cost of a cheapest winning coalition under f, CCWC(f) , is maximal among CCWC(g) for all plays g, and let ~t(G) denote CCWC(f) for any maximal f. The idea of maximality is due to Young (1978). Every game has a maximal f, by a simple compactness argument: if ~ --- [0, 1] lel is given the standard product topology, and 92 is the subspace consisting of all plays f (i.e. all imputations) of G, then 92 is compact, so the continuous function on 92 given by f ~ CCWC(f) achieves its maximum. Thus, it suffices to show that in a game lacking vetoers, any maximal f is a b.e. A b.e. need not be maximal, as we show later.

We will handle the contrapositive: let f be any play that is not a b.e. We construct a play g satisfying CCWC(g)> CCWC(f).

Case 1. Assume that some player p appears in none of f ' s cheapest winning coalitions. Note that f(p) > 0. Choose O > 0 such that the cost of each winning coalition containing p exceeds, by at least 8, the cost of each of f ' s cheapest winning coalitions. Let g be any play such that f ( p )>g(p )>0 , with f ( p ) - g(p) ~ 8/3, distributing the excess lost by p so that g(q)>f(q) for each q ~ p . The cost of each winning coalition omitting p goes strictly up under g, by at most 0/3, while the cost of each winning coalition containing p goes down by at most 8/3, so after the change p is still in none of g's cheapest winning coalitions. Thus CCWC(g) > CCWC(f).

Case 2. Assume that some player p appears in all of f ' s cheapest winning coalitions. Note that since p is not a vetoer, f(p) < 1. Choose 8 > 0 such that the cost of each winning coalition omitting p exceeds, by at least a, the cost of each of f ' s cheapest winning coalitions. Let g be any play such that f(p) < g(p) < 1, with g ( p ) - f ( p ) <~ 8/3, distributing the loss thus necessitated so that g(q)<f(q) for each q ~ p. The cost of each winning coalition containing p goes strictly up under g, by at most 8/3, while the cost of each winning coalition omitting p goes down by at most 8/3, so after the change p is still in all of g's cheapest winning coalitions. Thus CCWC(g) > CCWC(f). []

We end by pointing out connections with two other areas of game theory: homogeneously weighted games and aspirations. If w works as a weighting for the weighted game G, then w is said to be a homogeneous weighting if every minimal winning coalition has the same weight, and G is homogeneous if a homogeneous


weighting for G exists. Not every weighted game is homogeneous, but for those that are the homogeneous weighting is unique up to scaling (see, for example, Von Neumann and Morgenstern, 1944; Ostmann, 1987; or Rosenmiiller, 1984). Clearly, a homogeneous weighting w for a game lacking vetoers is (after scaling) a b.e.

For the examples that we have examined, this w is not the unique b.e., but is

the unique maximal play. For example, consider the homogeneously weighted game

[3; 2, 1, 1, 11,

for which the weighting enumerated above is homogeneous. It can be shown that, after normalizing to make all plays sum to 5, every b.e. enumerates values that fit the pattern

2 + e , 1 - 2 e , 1 - 2 e , l + 3 e ,

where 0~<e~<l/2. Notice that only if e = 0 (duplicating the homogeneous weighting) is this play maximal. Perhaps it is true for every homogeneous game that the unique homogeneous weighting is also the unique maximal play.

We are indebted to Roger Myerson for pointing out to us that the equilibrium notions studied here are related to the solution concept known as aspirations and explored by Cross (1967), Bennett (1983), and others. Briefly, if we begin with any play f : N-+ [0, 1] for some simple game and then we rescale f ' s outputs by dividing them by the constant CCWC (f ) , then the resulting function- call it f * - i s normalized in a way different from the way f is. The original f satisfied E {f(p): p ~ N} = 1, while f* satisfies CCWC(f*) = 1. Let us refer to any g satisfying CCWC(g) = 1 as a pre-aspirat ion (our terminology). Similarly, we can rescale any pre-aspiration to obtain a corresponding play. Thus, plays are in bijective correspondence with pre-aspirations.

It then turns out that the play f satisfies that every player is a member of at least one winning coalition (recall that this is half of what we ask for in a bribery equilibrium) if and only if the corresponding pre-aspiration f* is an 'aspiration' (our terminology here is from Bennett, 1983). Also, the play f is maximal if and only if the corresponding pre-aspiration is a 'balanced aspiration'.

9. Calculating the president's market interval

We show below that M I ( P ) = [0, 17/67], where P denotes the president and the simple game is GEe a. (which represents the US federal system, ignoring abstentions but including the vice president's tie-breaking power). It is clear that the left endpoint is zero, since P is neither a vetoer nor a weak vetoer (see Fig. 1 in


Section 6). To show that the right endpoint is 17/67 (approximately 0.2537) it suffices to establish the two lemmas below.

Lemma 9.1. Every play o f 17/67 + e is certainly unsuccessful for P (where e > 0 is arbitrary).

Proof. Let f be any play of 17/67 + e for P. Denote the values o f f on the House members by

h~ ~ h 2 ~ . . . ~ h 4 3 5 ,

the values on the senators by

S 1 ~ S 2 ~ . . . ~ S l 0 0 ,

and the value on the vice president by vp. Also, let X denote the cost of a cheapest winning coalition containing P, and Y denote the cost of a cheapest winning coalition omitting P.

Case I. Assume up >i s51. Then

X > 17/67 + [h 1 + h 2 + . . . + h218] + [s I + s 2 + . . . + s s1 ]

and

y = [ h 1 + h 2 + . . . + h290] + [s I --[- s 2 - ~ - - . . . --[-- s67 ] .

Assume, for the sake of contradiction, that f is possibly successful for P. Then X <~ Y, from which

17/67 < [h219 + h220 -t- . . . q- h290] + [$52 -+- $53 --[- . . . --I- $67 ] .

Case H. Assume vp < s51. Then

X > 17/67 + [ h I + h 2 + . . . + h 2 1 s ] + Is 1 -~ S 2 - ~ . . . "~ $50 At- vp],

and Y is as before. In this case, X ~< Y yields

17/67 < [h219 -4- h220 + . . . + h290] + [(Ss1 - - vp) + s52 + s53 + . . . + $67 ] .

In both cases, then, if f is possibly successful for P,

17/67 < [h219 q- h2z o + . . . + h290] -1- [s51 + s52 - - ~ - . . . --[- s67 ] .

Hence the total cost of the House equals

h 1 + h 2 + . . . + h435 > h219 d- h220 q- . . . + h435

i> (217/72)[h219 + h22o + . . . + hz90].

Similarly, the total cost of the Senate is a t least

( 5 0 / 1 7 ) [ S 5 1 + S52 + . . . + S67 ] .

Thus the total combined cost of the House, Senate, and vice-president is at least


(217/72)[h2t 9 + h220 + . . . + h290] + (50/17)[s51 + s52 + . . . + $67 ]

>i (50/17)[[h219 + h220 + . . . + h290] + [ss1 + s52 + . . . + $67]] ( * )

> (50/17)(17/67) = 50/67.

Note that inequality (* ) follows from the fact that 50 /17<217 /72 , and this explains why the final fraction uses numbers from the Senate rather than the House (but if the vice-president's tie-breaking power in the Senate is removed, the right endpoint, 17/67, of MI(P) changes to 72/289, a fraction using numbers from the House).

Finally, this means that the total cost of the federal system (House, Senate, president, and vice-president) exceeds (50 /67)+ (17 /67 )= 1, a contradiction since the values of f must add to unity. []

Lemma 9.2. Some play of 17/67 is possibly successful for the president.

Proof. Let f be the play for which f(P)=17/67, vp=O, h i = 0 for each i,

s I = s 2 = . . . = ss0 = 0, and s51 = s52 = . . . = Sl00 = 1/67. Then the cost of a cheapest winning coalition containing P is

17/67 + [h a + h 2 + . . . + h218] + [s I + s 2 + . . . +ss0 ] + vp = 17/67,

and the cost of a cheapest winning coalition omitting P is

[h 1 + h 2 + . . . + h290] 4- Is I + s 2 + . . . 4- s67 ] = 17/67,

so the president is a member of a cheapest winning coalition. []

10. Why use intervals?

It is critical to the nature and progress of mathematics that vague, intuitive notions be ultimately pinned down by precise definitions. But how can we know whether the spirit of the original notion has been successfully captured by some proposed precise definition?

One test of success is whether the precise definition allows us to prove much of what we had hoped for. Another is whether it agrees with our intuition for some important examples. These 'telling examples' help to illuminate what may be wrong with some proposal for a precise version of an idea.

For example, any definition of 'limit' that is posed in terms of an eventual monotone approach ('gets closer and closer') stumbles over the example f(x) = x sin(1/x) as x approaches 0. However , this is only a 'telling' example for the limit concept because our original intuitions are sharp enough to convince us that the limit of f(x) is 0 even though f wiggles a lot. Thus, the way mathematicians use telling examples suggests that they function as 'operational P la tonis ts ' - they act,


while closing in on the 'right' definition, as if they believe from the start that there must be a single, exact idea behind some originally vague notion.

Let us turn to the question of telling examples in the case of the vague idea of a point index of power. (Here, the term 'point' serves to distinguish traditional indices from the 'interval' indices we consider.) Of the various attempts (Shapley- Shubik, Banzhaf, etc.) to make this notion precise, none has yet emerged as being clearly correct. Why is this so, and what does it say to us about the appro- priateness of 'operational Platonism' for power indices?

Consider the weighted simple games of the form

k

H k =[2; 1, 1,~l/k, 1 / k , . . . , I/k] ' .

When k = 1, this is the generalized majority rule game, G2. 3 = [2 ; 1, 1, 1]. For larger k values, we may think of the third player, q3, of G2, 3 a s having been 'split' into k players Pl, P 2 , • • • , Pk"

Alternatively, we could start with H k and derive G2, 3 by 'fusing' pl , P2 . . . . , Pk into a 'voting bloc' (see, for example, Straffin, 1980); this means that the pi are replaced by a single player q3, that coalitions in the fused game correspond to those coalitions in the original game for which the pi vote en masse, and that a coalition in the fused game is winning if and only if the corresponding coalition is winning in the original game. The process of deriving a game G' from G by forming one or more voting blocs is well defined (for any simple game G) by the above remarks (although splitting is not).

How should the cumulative power of the voters being fused compare with that of the fused bloc? This additivity question is central to the problem of measuring power.

Observation 10.1. Any complete answer to the additivity question would determine the correct power index for all weighted games.

Here, by a complete answer we mean a rule that determines power in G' from power in G, whenever G' arise from G via the formation of a voting bloc. The proof of observation 10.1 is based on the assumption that each of the t players in Gs.t should get power 1/t in any reasonable point index of power. The observation then follows immediately from the fact that any weighted game arises from some Gs, , via a sequence of voting bloc formations.

The H k games, then, represent one simple set of examples for testing how various indices deal with additivity. In Table 4, the column headed 'Ind' gives the power of an individual Pi in the H k game, while 'Cum' gives the sum of the power of all the Pr A totally additive index would make Cum -- 1/3, which is the power of the voting block into which the Pi could fuse.

Thus, for k -- 2, these indices appear to be additive, since fusing PI and P2 does


Table 4 Banzhaf and Shapley-Shubik power in the H k games

Banzhaf Shapley-Shubik

Ind Cum Ind Cum

General k l l (k + 2 k) kl(k + 2 k) 21(k + 2)(k + 1) 2kl(k + 2)(k + 1) k = 1 1/3 1/3 1/3 1/3 k = 2 1/6 1/3 1/6 1/3 k =3 1/11 3/11 1/10 3/10 k = 100 10 -3°a 10 -:8~ 2/10,000" 1150 a

aApproximate value.

not enhance (or diminish) their cumulative power. At k = 3, the indices begin to diverge from each other and we judge that fusing does enhance power. Of course, the probabilistic interpretations (see Straffin, 1982, for example) of these indices provide plausible explanations for such enhancement. In these examples the enhancement is (roughly) linear in k for Shapley and Shubik and exponential for Banzhaf.

Are any of the H k 'telling'? For example, is it clear, on intuitive grounds whether power should grow linearly or exponentially in k, or whether, when k = 3, either 1/10 or 1/11 is correct as the voting power o fp i ? Although H 3 is a very s i m p l e simple game, our intuitions are apparently not sharp enough to use this game to defeat Banzhaf or Shapley-Shubik as a candidate for the precise definition of voting power.

In fact, there is a notable lack of telling examples in the subject, which suggests that our intuitive notion of voting power is, at least at the time of this writing, too vague to be made precise. Some possible reasons for the vagueness (and their consequences) are:

(1) Our experience with the informal notion of voting power is still too limited. (With the passage of time, intuitions will improve and some particular power index will emerge as a clear choice.)

(2) Our intuitions cover several precise ideas. (Once sorted out, they will lead to several distinct indices that measure different types of power. This idea is developed further in Felsenthal et al., 1994.)

(3) Our intuitions have misled us. (There is no precise idea of voting power, and the study of this nonentity will eventually die.)

That is to say, the ultimate number of precise ideas may be one, several, or none, Yet we should at least consider the following variant on (3):

(4) As currently formulated, our intuitive notion of voting power leads to no precise definition; there are simple voting systems for which our intuitions will never be sharp enough to pin down a single number that correctly measures a player's influence. Yet our intuitions are good enough to eliminate certain values as being clearly inappropriate, which leaves us with a range of possibilities.

A. Taylor, W. Zwicker / Mathematical Social Sciences 33 (1997) 23-74

Table 5 Interval power in the H k Games (for any k >/2)

57

ql for i= 1,2 p, for i= 1,2 . . . . ,k

wt (1/3, 1/2) [0, 1/2)

M1 [1/3, 1/21 [0, 1/21

Mlj for any j/>2 (or MI*) [1/3, 1/2] [0, 1/3] ~N~ {1/3} [0, 1/3]

If this fourth reason is correct, then interval measures are appropriate, and the apparently greater precision of point measures is illusory.

Table 5 provides the values assigned by the interval indices WI, M I , M I * , and ~ 3 ~ , to each of the two types of players (qi for i = 1 , 2 , and pi for i = 1, 2 , . . . , k) in H k for k/> 2 (only the individual power is listed, since it is unclear how to add intervals). Surprisingly (or perhaps not, for these indices) power is independent of k.

11. Additivity

If the lack of agreement on the proper index of political power is actually due to the existence of several distinct types of power, so that the appropriate index depends on the nature of the application, what principle might distinguish among the types? Perhaps the various applications call for different answers to the 'additivity question' mentioned in the previous section. Certainly, Table 4 suggests that the standard point indices are highly non-additive (albeit in different ways). Some see this as a virtue, because it roughly reflects intuitions from some real-world voting situations. It is possible, however, that in some contexts (or when voting on certain types of issues- see the discussion of 'P-power' in Felsenthal et al., 1994) a high degree of additivity is appropriate.

In fact, we will show, with an easy argument known to many (Shapley, for example), that no reasonable point index can be additive. A straightforward extension of the additivity notion to interval measures reveals that the weight interval is additive (in this sense) on the class of weighted games. Finally, a plausibility argument, not included in this paper, suggests that the market-related indices will frequently tend to be additive, in particular in those situations for which relatively few players form the bloc and for which the players in the bloc are isomorphic to each other and also to a large number of unfused players (standard point indices are not additive in these situations).

We start with a few definitions. If ~ is a class of monotonic simple games we will say that q~ is adequate if it includes all the generalized majority games (Gs. t

58 A. Taylor, W. Zwickei" / Mathematical Social Sciences 33 (1997) 23-74

for all t > s t> 1) and is closed under the formation of voting blocs (whenever G ~ ~ and G ' is obtained from G via the formation of a voting bloc, G' E ~ ) .

Observation 11.1. The class 74/" o f all monotonic weighted simple games is adequate, and is contained in every other adequate class.

The proof is left to the reader. By a point power index we will mean a function that assigns to each game G in some adequate class qg an imputation (an

assignment of a non-negative real number to each player in G in such a way that the numbers sum to one). An interi~al imputation will be an assignment of a non-empty subinterval of [0, 1] to each player in some game, in such a way that there is at least one way to generate an imputation by choosing one number from each interval, and an interval power index will be a function assigning an interval imputation to each game in some adequate class. It is straightforward to check that WI is an interval power index on the class of all monotonic simple games. While MI, MI*, and ~ always assign interval imputations to those monotonic simple games without vetoers, any adequate class must contain some games with vetoers (and for such games these indices fail to assign interval imputations).

It seems reasonable to require that any isomorphism of games preserves power. We will say that a power index is anonymous if it has this property, and note that an anonymous index necessarily assigns the same number or interval to two players in a given game G if there exists an automorphism of G that exchanges the players. A point index of power is additive on the class q¢ if the powers

assigned to any group of players Pl , P2 . . . . . Pk from any game G E qg sum to the power assigned to the voting bloc q in the game G ' formed by fusing these players. An interval index is additive if it always possible to choose numbers from the intervals assigned to these players Pl , P2 . . . . . . Pk in such a way that their sum lies in the interval assigned to the bloc q.

Proposition 11.2. Let ~ be any anonymous point index o f power on any adequate class q¢. Then ~ is not additive.

Proof. Let ~ be such an index, and consider the weighted game G3, 5 = [3; 1, 1, 1, 1, 1]. If we form one voting bloc from the first two players and another from the third and fourth players, then we obtain the weighted game K = [3; 2, 2, 1], which is isomorphic to G2, 3 = [2, 1, 1, 1]. Anonymity requires that each player in G3, 5 be assigned power of 1/5, so if ~ was additive, then players in K would be assigned powers of 2/5, 2/5, 1/5, respectively. Yet these same players must each be assigned power 1/3 by virtue of the isomorphism with G2,3, and this is impossible. []

A. Taylor, IV. Zwicker / Mathematical Social Sciences 33 (1997) 23-74 59

Proposition 11.3. The weight interval is additive on the class °W of all weighted simple games.

Proof. Let Pl, P2, • • •, Pk be any group of players from some weighted simple game G, which are fused to become the player q in G'. If w is any NN weighting realizing G, then w certainly chooses numbers from the intervals assigned to the Pi, and the sum of these numbers can be used as the weight of q in the obvious induced weighting for G'. []

The above results suggest that the following three phenomena are essentially the same:

(1) There is always 'wiggle room' when choosing weight for a weighted game. (2) Forming voting blocs in different ways can yield isomorphic games. (3) Point indices cannot be additive. We conjecture that WI assigns the tightest possible intervals among additive

and anonymous power indices on ~V. If so, this conjecture would be a stronger manifestation of the above idea, sewing Propositions 11.2 and 11.3 into a single result.

Acknowledgements

Both authors were supported by NSF grant DMS-9101830. We wish to thank the referee for several suggestions that improved the exposition.

Appendix

This appendix contains a proof of the 'difficult' containment in Theorem 3.1, part (i), followed by a proof of the difficult direction of part (ii). These arguments are closely related to the characterization theorem, in Taylor and Zwicker (1992), stating that an arbitrary simple game is weighted if and only if it is trade robust (a sequence of trades among winning coalitions cannot render them all losing). Our proof of Theorem 3.1(i) both quotes this result, and uses a similar, if more elaborate, outline. Certainly, a familiarity with Taylor and Zwicker (1992) would be helpful.

Theorem. Let G = (N, W ) be any monotonic, weighted game, and p be any player in N. Then DI (p ) C WI(p) .

Proof. Let A be an arbitrary subset of N, and consider a function f : A--~ [0, 1]. Such a function can be considered to be a 'partial weighting', which may or may


not have an extension to all of N that is an NN weighting for G. We define, below a combinatorial property - NN trade robustness - and prove, in Lemma A, that a partial weighting f extends to an NN weighting for G if and only if f is NN trade robust.

In particular, this is so in the special case A = {p}, f = {(p, r)}. In Lemma B we prove that, in this special case, if r is in D I ( p ) , then f is NN trade robust. The theorem clearly follows from the two lemmas.

The characterization theorem in Taylor and Zwicker (1992) makes use of a property called ' trade robustness for A' , which is proved to hold for a partial weighting f if and only if f extends to a weighting for G. There are two differences in context between Taylor and Zwicker (1992) and the current paper; these differences explain the added complication in the definition of 'NN trade robustness' as well as some of the changes in the argument. First, the current context assumes that G is monotonic and, correspondingly, that the values of f and of its potential extension are non-negative. This shows up most clearly in the proof of Claim 4 of Lemma A. Secondly, since NN weights must sum to unity, a partial weighting with domain A C N also makes an implicit commitment as to the weights of the as-yet-unweighted players in N - A - their weights must sum to 1 - ¢¢(f), where c¢(f) is the total 'committed' weight of the players in A. Thus, it is possible that a particular choice of weights for the players in A gets us into trouble indirectly, because there is the wrong amount of weight remaining, proportionately, to later distribute among the players in N - A. This concern is embodied in the sets Z i and Z I that appear in the definition of 'NN trade robustness' , below, but not in that of ' trade robustness for A'.

D e f i n i t i o n . Suppose that G = (N, W) is a monotonic simple game, A C N, and f : A ----> [0, 1]. We will say that f is N N trade robust if the following holds for every a//:

I f all = ( ~ , S¢' ) , where 6~ = X 1 U II1 U Z~ , X2 U II2 U Z2 . . . . , X k U Y~ U Z k and .Se' = X ' 1 U Y~ U Z ' 1, X~ U Y~ U Z ~ , . . . , X~ U Y~ U Z~ are coalition sequences of

length j ~< 3 41N-AI satisfying the conditions

(i) for every q E N , 1{i: q ~ S i } [ < ~ l { i : q ~S ' i} l ; (ii) for each i, Y~ and Y~ are subsets of A and are disjoint from Xi and X ' i,

respectively; (iii) there exist non-negative integers b and d such that the sequence Z is a

dispersal of multiplicity b of the set N - A over the sequence X U Y and the sequence Z ' is a dispersal of multiplicity d of the set N - A over the sequence X' U Y'; and

(iv) letting cg(f) = ~ {f (q) : q E A} denote the total 'committed weight', ~ = E {( ,~{f(q): q ~ Y i } ) : i<~k} stand for the total weight of all the Yi and ~ = E {(,~ {f(q) : q E Y~}) : i ~< k} be the total weight of all the Y'i, c(1 - cg(f)) + ~ ~< d(1 - ~ ( f ) ) + 5~;


Then, it is not the case that X/U Y~ U Z i is winning for each i and also that X' i U YI U Z I is losing for each i.

Any oR satisfying the 'If' in the above will be referred to as an N N scheme for f (of length [oR] equal to the length of either of oR's coalition sequences), and any NN scheme satisfying the 'Then' will be said to be balanced. Thus f is NN trade robust precisely when every NN scheme is balanced.

Comment. Roughly speaking, clauses (i) and (iii) state that the coalitions X~ U Y~ U Z I in the right-hand column have, on average, more people in the X parts and more weight in the Y and Z parts, than do their counterpart unprimed coalitions in the left-hand column. If this is so, then f certainly seems to be in trouble if all the 'more powerful' coalitions do worse!

Lemma A. Let G = (N, W ) be a simple monotonic game, A be a subset o f N, and

f : A----~ [0, 1]. Then f can be extended to an N N weighting w for G i f and only i f f is N N trade robust.

Proof. First, assume that such an extension w exists and consider any NN scheme oR for f. We will use w to weight the coalitions appearing in oR. Clause (i) ensures that the total weight of the X portions of the primed coalitions is at least as great as that of the X portions of the unprimed coalitions, while the other clauses say that the same holds for the Y and Z portions. Hence, the average weight of the primed coalitions X I U Y~ U Z~ is not less than that of the unprimed coalitions. Thus, it cannot be that each primed coalition weighs strictly less than each unprimed coalition, and this tells us that oR is balanced. So, f is NN trade robust.

Throughout the proof of Lemma A we assume that f is NN trade robust, with total committed weight of ~ ( f ) , and we let q be any player in N - A. We prove below that there exists some number r E [0, 1 - ~ ( f ) ] such that the extension, f ' = f U {(q,r)}, of f to the domain A U {q} remains NN robust. Through iteration, f can therefore be extended to an NN trade robust function f* whose domain is all of N. A separate argument shows that at the last stage of the iteration, when the domain o f f is N - {q}, f ' ( q ) can be set equal to 1 - ~ ( f ) , guaranteeing that f* ' s weights add to 1. It is straightforward to check that f* must weight each winning coalition E more heavily than each losing coalition F (if not, consider the NN scheme oR whose sequences both have length 1, with YI = E, Y[ = F and X 1 = X ' I = Z 1 =Z '~=0) . Then f* is the desired NN weighting extending f.

It remains to show that f can be extended to an NN trade robust f ' by assigning to q some weight r E [0, 1 - c~(f)]. The argument follows the broad outline established in Taylor and Zwicker (1992), in that we define a notion of 'high failure' and of 'low failure' (corresponding, intuitively, to 'r seems to be too large'


and to 'r seems to be too small', respectively), and show that some r values remain when these failures are discarded.

More precisely, we establish, after defining the terms, the following series of claims:

Claim 1. Every failure is either high or low.

Claim 2. No failure is both high and low.

Claim 3. I f r is a low failure, and 0 <~ s < r then s is a low failure. Also, i f t is a high failure and t < u <- 1, then u is a high failure.

Claim 4. The interval o f high failures does not include 0, nor does the interval o f low failures include 1 - ~ ( f ) .

Claim 5. The interval o f low failures includes its supremum, and the interval o f

high failures includes its infimum.

Together the claims imply that the set of elements of [0, 1 - c¢(f)] that are not failures forms a non-empty subinterval, as desired.

Definition. A number r will be called a failure if there exists an imbalanced NN scheme v// for f ' = f U {(q, r)}, i.e. a witness to the fact that f ' is not NN trade robust.

Now, we wish to classify failures by classifying such witnesses.

Definition. The type [0-//] (of an NN scheme a// f o r f ' = f U {(q, r)} witnessing that r is a failure) is a 2 x 2 matrix, whose entries

are given by

a=[{i: q e Y/}I, b =l{i: q~Y:}l, c =l{i: u ~ Zi}[, for any (equivalently, every)

element u of N - (A U {q}), and d = l { i : u ~ Z~}I, for any (equivalently, every)

element u of N - (A U {q}).

Next, we define some of these matrices to be high types, and others to be low

A. Taylor, W. Zwicker / Mathematical Social Sciences 33 (1997) 23-74

Table 6 Classification table of high and low types

63

Low types High types

l[ca ~]wherea>c* I[~ db.] whereb>d *

II[o O] II[0 c ~]

llI[~ bd*]whered>b* l l l [ f ~]wherec>a* Note: The starred and unstarred letters follow the notational convention introduced below.

types. There is a further classification of each into three subtypes, given by roman numerals in Table 6. Note that not every type appears in the table.

Remark on nota t ion . One of the key features that distinguishes subtypes from one another is the set of locations of zero entries. For that reason, we will adopt the convention that, from this point on, unstarred letter entries in 2 × 2 matrices always stand for strictly positive integers, while starred entries stand for non- negative integers. When a starred letter has, in the course of an argument, been assumed or shown to be non-zero, the star & dropped.

Definition. The number r is a low failure if there exists an imbalanced NN scheme of low type that witnesses r's failure and is a high failure if there similarly exists a witness of high type.

Recall that our immediate goal is to establish that every failure is high or low. The first step is to show that any imbalanced coalition scheme 0//can be modified so as to satisfy both

(i) at least one entry in [°//]'s first row is 0, and (ii) at least one entry in [~//]'s second row is 0.

Definition. Such a 0// will be called reduced.

The idea behind reducing 0// is to shift some elements appearing in the Y or Z portions over to the X portions. Suppose the first row of [~/] is [a b] and that a >t b. Shift each of the b occurrences of q among the Y~ over to the corresponding X~ and also shift any b of the a occurrences of q among the Y~ over to the corresponding X~. After this operation, q/will remain an imbalanced coalition scheme if it was one to start with, and will satisfy (i) above. Proceed similarly if b ~ a .


If [q/]'s second row is [c dJ and c ~> d, then shift all elements of each Z~ over to the corresponding X~. Also, identify, for each player p in the set N - (A U {q)) , d occurrences of p among the Z i, and shift each of these over to the corresponding X i. Again, (ii) is made true while ~ ' s essential features are preserved, and the case d/> c is handled similarly.

Ei ther of these operations (or both together) are known as X absorption, and we will employ notation as in the example below. If

then ~V~--~ , where x

Of course, this is but one example of X absorption. Now, the types appearing in Table 6 do not quite exhaust the possibilities

among reduced types, so to show that every failure is high or low, we need to demonstrate that imbalanced e//'s having any of the remaining reduced types cannot occur. The reason is that they contradict the NN trade robustness of the original partial weighting f, before it is extended by assigning weight to q - this violates the hypotheses.

The simplest of these impossible types is

because, taken together, all the zeros tell us that q/ does not really exploit the value of f ' on the newly weighted q . So 6// can be considered an imbalanced coalition scheme for f, and no such ~//can exist. Note that 0//meets the restriction on lengths of NN schemes for f, since it satisfies the more stringent limitation on NN schemes for f ' .

The other impossible types are those reduced types that have either equal entries in the first column together with zero entries in the second column, or vice versa. These are impossible because the process of Z absorption, described below, converts an imbalanced ~// of such a type into one that is an imbalanced NN scheme for f. This kind of absorption proceeds in the same way regardless of which column is non-zero, so we will assume that

[a 0] and proceed by shifting each of the a occurrences of q among the Y~ over to the corresponding Z i. The reader may check that the modified 0-// satisfies all the requirements of an imbalanced NN scheme for f. (In particular, the Z i were originally a dispersal of N - (A U {q)) of multiplicity a, so shifting a copies of q into the Zi turns them into a dispersal of N - A, still of multiplicity a.)


Recapping, if r is any failure, and q / a n y witness, then we may assume, without loss of generality, that a// is both reduced and not of an impossible type. All possibilities for [a//] now appear in Table 6, so we have established:

Claim 1. Every failure is either high or low.

To establish that no failure is both high and low, it suffices show that, from a high type °l/and a low type ~F, which both witness the failure of the same number r, we can construct an impossible failure Y for r. In each case Y is built from and ~//" via X absorption and 'linear combination'. First, we introduce some terminology.

If ~ = (5¢, b °') is any NN scheme of length k, and j any integer, jSe denotes the coalition sequence of length j k obtained by repeating 5¢ j times (so that the k + 3rd coalition in 25¢ is X 3 t.J Y3 U Z 3, for example) and j0-// denotes ( j5¢, jow' ). Given a second scheme ~ = (~ , J ' ), q / + ~ is defined to equal (5¢ + J , 5¢' + J ' ), where O ° + J is formed by following 5 e by J . Now it should be clear what is meant by 'the linear combination rna//+ nO//'', and a moment 's thought shows that a linear combination of NN schemes is an NN scheme, which is imbalanced if both schemes it is built from are imbalanced. Also, notice that [m°// + n°U] = m [ ~ ] + n [~] , with the right-hand side computed via the standard scalar multipli- cation and addition of matrices.

Claim 2. No failure is both high and low.

Proof of claim. The general outline is as described two paragraphs previously. There are nine cases to consider, since there are three possible low subtypes for 0//, and three possible high subtypes for ~. We abbreviate subtype assumptions by using the Roman numerals from Table 6; ' I I - I I I ' stands for 'assume that ~ is a low type II witness to r 's failure and ~F is a high type III witness', for example.

Recall that impossible failures are contradictory because they violate the assumed NN trade robustness of f. Thus, it is necessary to check that if ~// and ~F meet the restriction on lengths of NN schemes for f ' , the Y built from o// and meets the less stringent restriction on lengths of NN schemes for f. In particular, if s denotes the bound on ]q/I and I~l, then s = 34k, where k = I N - (A U {q})l t>0. So IN-AI = k + 1 and we must show that ~<34(k+1)" In practice, we go through this lengthy check in the case I - I and leave the other (similar) checks to the reader.

I - I . Let

[a o] ~ = c*

and

66 A. Taylor, W. Zwicker I Mathematical Social Sciences 33 (1997) 23-74

where a > c*, and b > d*. Set off. = b a / / + aO//., so that

[°/4/']= [/~cb, a~/b*]

and let ~/-~__~o~, where [o~] is equal to x

[°e o o] or [0 o Of] (or

[~ o] which is impossible) . In the first instance, choose integers m and n such tha t ma = mc + ne, and set :Z equal to m q / + n ~ , so that

[~]=[mcrn+ane 0 ] ,

which is impossible . In the second instance, choose m and n to satisfy mb = m d + ne, set :Z = m ° / / + n ~ , and p roceed similarly.

For the length check, obse rve that a , b < ~ s , so I~l-<Z~2. Also, rn can be chosen to be e and n can be chosen to be a - c*, so m, n ~< s hold when m and n a r e minimized . This m a k e s I~1 -<s2 + s . 2, 2 -< 3 s 3 = 3" (34k) 3 = 3" (33"4k) ~ (34"4k) ,

which equals 34~k÷~), as desired.

I - I I . Le t

[a o] [~ 0q a / /= c* and ~ = ,

whe re a > c*. Set off. = b ~ / + a°//, so that

= a0q I~l [~ca~+,~ and let ° / ¢ r - -~ , where [ ~ ] is equal to

x

[o o]. Finish as in I - I , first instance.

I - I I I . Le t

c* and o//= ,

where a > c*, and g > h*. Choose integers m and n to satisfy m(a - c * ) = n ( g - h*) , so that ma + nh* = mc* +ng . Then set ~ = m ° / / + n ~ , so


[ma+nh* O] [°/4/'] = [_me* + ng '

which is impossible.

II-I. Let

o / /=[~ O] and ~ = [ 0 0 b , ] ,

where b > g*. Set ~W = bq /+ a~" so that

[aob ab ] [°W] = bd + ag* ;

let °W>--->~, where [~] is equal to x

[ 0 0 e ] "

Finish as in the second instance of I-I.

II-II. Let

o / /=[~ O] and ~ = [ 0 c b ] ;

Set 0/4/" 1 = b°//+ a ~ so that

and let ~1 ~->~i, where [~1] is equal to

[o c o ] Similarly, set °W 2 = cq/+ dO//', so that

[ bd] [ ~ 2 1 = a c cd cd

and let ~W 2 ~-~2, where [~2] is equal to

[aoC Finally, let ~ f = °~ 1 + °~ 2 and note that [~f] equals

ac ac bd '

which is impossible (reduce first, if this is not clear).


II-III . Let

~ = [ 0 0 ] and ~v=[gc ~ ] '

where c > g*. Set ~V = ca//+ d~V so that

[~]= [ac + dg* 0]. cd cd '

let ~V~--~, where [~/] is equal to X

[; 0] Choose integers m and n satisfying that rag*+ ne = mc , and let Y = m~V + n~. Then [:~] equals

[m '+ne X] m c


III-I . Let

~ = [ 0 0 ~ ; ] and °v=[O 0 hg,],

where d > b* and g > h*. Choose integers m and n satisfying m ( d - b*) = n ( g -

h*), so that rob* + ng = m d + nh*, and observe that

+nOV]=[O mb* + n g ] [mO LO m d + nh* J '


III-II . Let

~ t = [ O 0 b*] and 7 / '= [0 c o g ] ,

where d > b*. Set ~kV = cO?/+ d°V so that

[ 0 b * c + d g ] [°W]= cd cd

and let °W~--~, where [~] is equal to X

[0 ;] Choose integers m and n satisfying that m b + ne = rod, and set Y = m°-tt + n~. Then [:~] equals

A. Taylor, W. Zwicker I Mathematical Social Sciences 33 (1997) 23-74

[~ m b * + n e J md


69

I I I - I I I . Let

~ = [ 0 0 ~d] and ~ = [ " ; 00],

where d > b * and h > g * . Set °W= h a t / + d ~ so that

and let °W>--->og, where [~] is equal to x

(or

which is impossible). In the first instance, finish as in I I - I I I , and in the second instance finish as in I I I - I I .

This completes the proof of Claim 2.

Claim 3. I f r is a low failure, and 0 <~ s < r, then s is a low failure. Also, if t is a high failure and t < u <~ 1, then u is a high failure.

Proof. Let r be a low failure, as witnessed by 07/, and s be any real number satisfying O ~< s < r. Then °7/ witnesses that s is a low failure as well, because for any ~/ of low type, satisfaction of the clause (iv) inequality,

b(1 - ~7(f)) + ~ ~< d(1 - g ( f ) ) + ~ ,

is 'aided' by decreasing q's weight (this can most easily be seen by breaking into subcases). The second part of the claim follows similarly.

Claim 4. The interval of high failures does not include 0, nor does the interval of low failures include 1 - ~ ( f ) .

Proof. First, assume by contradiction that ~/ is a witness of high type to the failure of r = O. We break into cases, depending on 07/,s subtype.

Case I. let

D},


where b > d*. Delete enough q's from the Y~ (without shifting them anywhere else) to transform OR into ~, where

Note that ~3 is impossible, still satisfies the clause (iv) inequality because r = 0, and remains imbalanced thanks to G's monotonicity.

For the remaining cases we provide only the recipe for constructing the NN scheme of impossible types from the hypothetical counterexample. In each case the fact that the impossible ~d is imbalanced follows from monotonicity, and clause (iv) is preserved either because r = 0 or, in the case r = 1 - ~ ( f ) , because

1 - ~ ( f ' ) = 0. Case H. Let

Delete all q's appearing among the Y~ to create OR' with

0] Then to create ~ with

t l:[ c 0°], first shift the q's from the Xi's to Y~'s when possible, then (if the 'count ' cannot be brought up to c in this manner) simply add more q's to some Y,. for which qflgX~ U Y,..

Case IlL Let

where c > a*. By the same method as in the previous case, create 0~ with

t. __ic 0] Next, assume by contradiction that °R is a witness of low type to the failure of

r = 1 - ~ ( f ) . Again, we break into cases depending on OR's subtype. Case I. Let

0] where a > c*. By similar methods to those in Case III of the first half, create with

A. Taylor, W. Zwicker I Mathematical Social Sciences 33 (1997) 23-74 71

This time, however, shift elements of N - (A U {q}) into some Zi, and /o r add them to some Zi.

Case H. let

o] Create ~ with

[; o] by deleting elements of some Z~. Finish as in the immediately preceding case. Case III. Let

where d > b*. Create °-3# with

by deleting elements of some Z'~.

Claim 5. The interval o f low failures includes its supremum, and the interval o f high failures includes its infimum.

Proof. The interval of low failures for which some particular all serves as a witness includes its supremum because the inequality in clause (iv) is 'weak' (~<). The interval of low failures is thus the union, taken over the finitely many 0//of length ~<3 4. (where k = IN - (A U {q})l) of intervals that are closed on the right, and so this finite union is also closed on the right. The second argument goes similarly.

This completes the proof that we can extend the original f to an NN trade robust f* with domain A. However , for f* to be an NN weighting, its weights must add to 1. Claim 6, established below, shows that this can be arranged, and completes our proof of Lemma A.

Claim 6. I f f is N N trade robust with domain N - {q}, then f* = f U ( q, 1 - q~(f)) is N N trade robust.

Proof. We have already established that 1 - c~(f) is not a low failure, so we need only show that it is not a high failure. Notice that in a reduced NN scheme q/ for f * there is nothing left to put into the Z i or the Z I so that

° ;] if ~ 's type is high. Now if such a ~ witnessed that 1 - cO(f) is a failure, we could


shift each occurrence of q among the Y, into the corresponding Zi, and have a witness that f is not NN trade robust.

Lemma B. Let G be a weighted game and p be any player. I f r is any number in DI (p ) , then f = {(p, r)} is N N trade robust.

Proof. If oR is any imbalanced NN scheme for this f and

o,], then oR can be thought of as a left dispersal. The induced inequality for this dispersal is X > d*/(a + d*), and it is satisfied by those X values for which oR does not witness that X is a low failure. Similar comments apply to right dispersals and the type

Thus, if r is any number in DI(p) , then r is not a failure via witnesses having either of the two 'dispersal types' from the preceding paragraph. To see that r is not a failure at all (so that f is NN trade robust), we need to establish that witnesses of high or low type which are not dispersal types can safely be ignored because of the special case f = {(p, r)}. This is accomplished by the next two claims.

Claim 1. I f oR is any imbalanced N N scheme for f = {(p, r)}, then

Claim 2. I f oR is any imbalanced N N scheme for f = {(p, r)}, and

[a* O] [OR]= c* '

then exactly one o f a*, c* is zero.

Proof of claim 1. By way of contradiction, suppose some imbalanced OR was of this type. Shift all q occurring among the Y'i, and all elements of all Z ' i, over to the corresponding X I. The resulting ~ remains an NN scheme, and is imbalanced because the clause (iv) inequality for 0~ becomes '0 ~< 0'. But ~t witnesses that the empty function is not NN trade robust, and it follows that G is not trade robust in the sense of Taylor and Zwicker (1992). This is a contradiction, since if G is not trade robust, G is not weighted (see Taylor and Zwicker (1992)]).

Proof of claim 2. Let OR be an imbalanced NN scheme of the given type, and consider the clause (iv) inequality. Observe that the Y~ are all empty (since the


on ly po t en t i a l m e m b e r was p ) , so the r igh t -hand side of the inequa l i ty is zero , and

the inequa l i t y can be wr i t t en

c * ( 1 - r) + a*r<~O.

This canno t ho ld unless at leas t one of c*, a* is zero, If they are bo th zero , og

w o u l d wi tness the fa i lure of the e m p t y funct ion to be N N t r a d e robus t .

This c o m p l e t e s the p r o o f of the ' h a r d ' d i rec t ion of T h e o r e m 3 .1( i ) ; D I ( p ) C

W l ( p ) for m o n o t o n i c weight games . []

T h e ha rd d i rec t ion of T h e o r e m 3.1(i i ) now fol lows read i ly . If D I ( p ) is well

de f ined for some p l aye r p in a m o n o t o n i c s imple game G, choose r to be any

e l e m e n t of D I ( p ) . Since f = { (p , r)} is N N t r a d e robus t (by L e m m a B) , it ex tends

to an N N weigh t ing for G (by L e m m a A) ; in pa r t i cu la r , G is we igh ted . []

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Interval measures of power

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