Chapter 32 Power and stability in politics

Chapter 32

POWER

AND STABILITY IN POLITICS

PHILIP D. STRAFFINJr. Beloit Colleoe

Contents

1. The Shapley Shubik and Banzhaf power indices 2. Structural applications of the power indices 3. Comparison of the power indices 4. Dynamic applications of the power indices 5. A spatial index of voting power 6. Spatial models of voting outcomes Bibliography

Handbook of Garne Theory, Volume 2, Edited by R.J. Aumann and S. Hart © Elsevier Science B.V., 1994. All rights reserved

1128 1130 1133 1138 1140 1144 1150

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"Political science, as an empirical discipline, is the study of the shaping and sharing of power" - Lasswell and Kaplan, Power and Soeiety. In this chapter we will treat applications of cooperative game theory to political scienee. Our focus will be on the idea of power. We will start with the use of the Shapley and Banzhaf values for simple garnes to measure the power of political actors in voting situations, with a number of iUustrative applications. The applications will point to the need to compare these two widely used power indices, and we will give two eharacterizations of them. We will then consider several ways in which political dynamics might be analyzed in terms of power. Finally, we will consider the role of political ideology and show how it can be modeled geometrically. For these spatial models of voting, we will consider both the problem of measuring the power of voters, and the problem of prescribing rational voting outcomes. For other directions in which game theory has influenced political science, see Chapters 30 (Voting procedures) and 31 (Social choice).

I.

The Shapley-Shubik and Banzhaf power indices

A voting situation can be modeled as a cooperative game in characteristic function form in which we assign the value 1 to any coalition which can pass a bill, and 0 to any coalition which cannot. The resulting game is known as a simple 9ame rShapley (1962)]. We call the coalitions which can pass bills winnin9 eoalitions, and note that the garne is completely determined by its set of winning coalitions.

Definition. A simple game (or voting game) is a set N and a collection ~ subsets of N such that

of

(i) ~ ¢ ~]F. (ii) N e ~8/. (iii) Se~¢v and S ~ T=~ Te"JV. A coalition S is called a minimal winnin9 coalition if Se~/U, but no proper subset of S is in ~ . The "monotonicity" property (iii) implies that the winning coalitions of any simple game can be described as the supersets of its minimal winning coalitions. A simple game is proper if Se ~tU~ N \ S ¢ ~J[/~,or equivalently if there are not two disjoint winning coalitions. A simple garne is stron9 if S ¢ ~ ~ N \ S e ~ , so that "deadlocks" are not possible. A common form of simple garne is a weighted votin9 9ame [q; w» w2 ..... wù]. Here there are n voters, the /th voter casts wl votes, and q is the quota of rotes needed to pass a bill. In other words, Se~V'~~i~s wi/> q. A weighted voting game will be proper if q > w/2, where w is .the sum of the w~. It will also be strong if w

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1129

is an odd integer and q = (w + 1)/2. Weighted voting games appear in many contexts: stockholder voting in corporations, New York State county boards, United Nations agencies, the United States Electoral College, and multi-party legislatures where parties engage in bloc voting. See Lucas (1983) for many examples. On the other hand, not all politically important simple garnes are weighted voting games. A familiar example is the United States legislative scheme, in which a winning coalition must contain the President and a majority of both the Senate and the House of Representatives, or two-thirds of both the Senate and the House. Given a simple game, we desire a measure of the power of individual voters in the game. Shapley and Shubik (1954) proposed using the Shapley value as such a measure: "There is a group of individuals all willing to vote for some bill. They vote in order. As soon as enough members have voted for it, it is declared passed, and the member who voted last is given credit for having passed it. Let us choose the voting order of members randomly. Then we may compute how often a given individual is pivotal. This latter number serves to give us our index." [Shapley and Shubik (1954)] In other words, the Shapley-Shubik power index of voter i is the number of orders in which voter i is pivotal, n! where n! is the total number of possible orderings of the voters. For example, in the weighted voting garne [-7; 4, 3, 2, 1] ABCD the orderings, with pivotal voters underlined, are ABCD ABDC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB CBAD CBDA CDAB CDBA DABC DACB DBAC DBCA DCAB DCBA and the Shapley-Shubik power indices are (~A

=

14

__ 2 q~D - - ~ .

To get a combinatorial formula for the Shapley-Shubik power index, define i to be a swing voter for coalition S if S~~ff, S\{i}¢'fff. Then

4~,= Z (s-1)!(~-s)! i swings for S

/1!

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P.D. Straffin Jr.

where s = IS l, the number of voters in S. This follows from the observation that voter i is pivotal for an ordering if and only if i is a swing voter for the coalition S of i and all voters who precede i. There are ( s - 1)! ways in which the voters before i could be ordered, and (n - s)! ways in which the voters who follow i could be ordered. A second game theoretic power index was proposed by Banzhaf (1965): "The appropriate measure of a legislator's power is simply the number of different situations in which he is able to determine the outcome. More explicitly, in a case in which there are n legislators, each acting independently and each capable of influencing the outcome only by means of his votes, the ratio of the power of legislator X to the power of legislator Y is the same as the ratio of the number of possible voting combinations of the entire lCgislature in which X can alter the outcome by changing his vote, to the number of combinations in which Y can alter the outcome by changing his vote." [Banzhaf (1965)]. In other words, voter i's power should be proportional to the number of coalitions for which i is a swing voter. It is convenient to divide this number by the total number of coalitions containing voter i, obtaining the unnormalized Banzhafindex number ofswings for voter i

/~; =

2ù_ 1

The standard Banzhaf index fl is this index normalized to make the indices of all voters add to 1. In the weighted voting game above the winning coalitions, with swing voters underlined, are AB ABC ABD ACD ABCD and the Banzhaf indices are

BA= I~Ö flB=~ÖO flC=flD-- I~Ö. Although the Shapley-Shubik and Banzhaf indices were first proposed in the second half of the twentieth century, Riker (1986) has found a similar combinatorial discussion of voting power in the works of Luther Martin, a delegate to the United Stares Constitution Convention in 1787. For methods of computing the ShapleyShubik and Banzhaf indices, see Lucas (1983).

2.

Structural applications of the power indices

Nassau County Board, 1964. This example prompted Banzhaf's original investigation of voting power (1965). The players were the county board representatives from

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Hempstead #1, Hempstead #2, North Hempstead, Oyster Bay, Glen Cove and Long Beach. The weighted voting garne was [-58; 31, 31, 21, 28, 2, 2]. H1H2N

O

GL

Notice that whether a bill passes or not is completely determined by voters H1, H2, and O: if two of them vote for the bill it will pass, while if two of them vote against the bill it will fail. Voters N, G and L are dummies in this game. They can never affect the outcome by their votes. The Shapley-Shubik power indices are ¢ù1

=

¢ù~

= ~bo -- !3

and

~bN= ~/)G = q~L = 0

and the Banzhaf indices are the same.

United Nations Security Council. Shapley and Shubik (1954) analyzed the Security Council, which then had eleven members. A winning coalition needed seven members, including all five of the Council's permanent members, who each had veto power proposed actions. Denote the players by P P P P P N N N N N N , where we will not distinguish among permanent members, or among non-permanent members, since they played symmetric roles. There were (151)=462 possible orderings. Of these, a non-permanent member pivots only in an ordering which looks like (PPPPPN)_N(NNNN), where the parenthesis notation simply means that the pivotal N is preceded by five Ps and one N, and followed by four Ns. The number of such orderings is (1)(o) 6 4 = 6. Hence the six non-permanent members together held only 6/462 = 0.013 of the Shapley-Shubik power, with the remaining 0.987 held by the five permanent members. In 1965 the Security Council was expanded to include ten non-permanent and five permanent members. A winning coalition now needs nine members, still including all five permanent members. There are now (½5)= 3003 orderings of P P P P P N N N N N N N N N N , with a non-permanent member pivoting in orderings 8 6 which look like (PPPPPNNN)N_ (NNNNNN), of which there are (3)(0) = 56. Hence the expansion increased the proportion of Shapley-Shubik power held by the non-permanent members just slightly, to 56/3003 = 0.019. Slightly harder calculations for the Banzhaf index give the total proportion of power held by the non-permanent members as 0.095 before 1965 and 0.165 after 1965. See Brams (1975). Notice that the difference between the two power indices is significant in this case. Council of the European Economic Community. Table 1 gives the Shapley-Shubik indices of representatives in the Council of the EEC in 1958, and after the expansion in 1973. The Banzhaf indices are similar - see Brams (1976). The figures illustrate several interesting phenomena. First, note that Luxembourg was a dummy in the 1958 Council. It was not in 1973, although new members had joined and Luxembourg's share of the votes decreased from1958 to 1973. Brams (1976) calls

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Table 1 Power indices for the Couneil of the European Economic Community Member

1958 Weight

France Germany Italy Belgium Netherlands Luxembourg Denmark Ireland United Kingdora Quota

4 4 4 2 2 1 12 of 17

1973

Shapley Shubik index

0.233 0.233 0.233 0.150 0.150 0.000 -

Weight

10 10 10 5 5 2 3 3 10 41 of 58

Shapley-Shubik index

0.179 0.179 0.179 0.081 0.081 0.010 0.057 0.057 0.179

this the Paradox of New Members: adding new members to a voting body may actually increase the voting power of some old members. See Brams and Affuso (1985) for other voting power paradoxes illustrated by the addition of new members to the EEC in the 1980s.

United States Electoral College. In the United States Electoral College, the Presidential electors from each state have traditionally voted as a bloc, although they are not constitutionally required to do so. Hence we can view the Electoral College as a weighted voting garne with the states as players. Since electors are not assigned in proportion to population, it is of interest to know whether voting power is roughly proportional to population. In particular, is there a systematic bias in favor of either large states or small states? Banzhaf (1968), M a n n and Shapley (1962), Owen (1975) and Rabinowitz and M a c D o n a l d (1986) have used the Banzhaf and Shapley-Shubik power indices to study this question. The Shapley-Shubik index indicates that power is roughly proportional to the number of electoral votes (not population), with a slight bias in favor of large states. The large state bias is more extreme with the Banzhaf index. We will consider Rabinowitz and MacDonald's work in Section 5. Canadian Constitutional amendment scheme. An interesting example of a voting game which is not a weighted voting game was a Canadian Constitutional amendment scheme proposed in the Victoria Conference of 1971. An amendment would have to be approved by Ontario, Quebec, two of the four Atlantic provinces (New Brunswick, N o v a Scotia, Prince Edward Island and Newfoundland), and either British Columbia and two prairie provinces (Alberta, Saskatchewan and Manitoba) or all three prairie provinces. Table 2 shows the Shapley-Shubik and Banzhafindices for this scheme, from Miller (1973) and Straffin (1977a). First of

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1133

Table 2 Power indices for Canadian Constitutional a m e n d m e n t schemes Province

1980 Population

Victoria scheme (1971)

Adopted scheme (1982)

(Vo) Ontario Quebec British Columbia Alberta Manitoba Saskatchewan Nova Scotia New Brunswick Newfoundland Prince Edward Island

Shapley-Shubik

Banzhaf

Shapley-Shubik

Banzhaf

31.55 31.55 12.50 4.17 4.17 4.17 2.98 2.98 2.98 2.98

21.78 21.78 16.34 5.45 5.45 5.45 5.94 5.94 5.94 5.94

14.44 12.86 10.28 9.09 9.09 9.09 9.09 8.69 8.69 8.69

12.34 11.32 10.31 9.54 9.54 9.54 9.54 9.29 9.29 9.29

35.53 26.52 11.31 9.22 4.23 3.99 3.49 2.87 2.34 0.51

all, notice how weil Shapley-Shubik power approximates provincial populations, even though the scheme was not constructed with knowledge of the index. Second, notice that the Banzhaf index gives quite different results, even differing in the order of power: it says that the Atlantic provinces are more powerful than the prairie provinces, while the Shapley-Shubik index says the opposite. Unfortunately, this clever and equitable scheme was not adopted. The last two columns of the table give the Shapley-Shubik and Banzhaf indices for the considerably more egalitarian scheme of Canada's Constitution Act, as approved in 1982 [Kilgour (1983), Kilgour and Levesque (1984)].

3.

Comparison of the power indices

We have seen that the Shapley-Shubik and Banzhaf indices can give quite different results. The Banzhaf index, for example, gave considerably more power to the non-permanent members of the U.N. Security Council, and to the smaller Canadian provinces. The asymptotic behavior of the two indices for large voting bodies is also very different. Consider three examples: (1) In the game [5; 3, 1, 1, 1, 1, 1, 1] call the large voter X. Then 4)x = 0.429, while fix = 0.455. However, if we keep X having one-third of the votes and keep the quota at a majority, but let the number of small voters go to infinity, then q~x~½, while fix ~ 1.1 11ntuitively, what is happening hefe is that for the Shapley-Shubik index, X will pivot if he votes after ¼ of the small voters and before 3 of them, i.e. half the time. For the Banzhaf index, X will be a swing voter for any coalition with between ½ and 65of the votes, whereas a small voter will be a swing only for coalitions with "exactly half" of the votes. The latter are very rare compared to the former.

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(2) In the game [5; 3, 2, 1, 1, 1, 1] call the large voters X and Y. Then ~bx = 0.400 and qSr = 0.200, while fix = 0.393 and flr = 0.179. However, if we keep X having one-third of the votes, Y having two-ninths, and the quota at a majority, but let the number of small voters go to infinity, then ~bx~0.391 and ~br~0.141, while fix ~ 1 and fly-"~O. 2 (3) In their original paper Shapley and Shubik (1954) considered a legislative scheme in which a bill must be approved by a president P and by a majority in each of a three-member senate (SSS) and a five-member house ( H H H H H ) . Shapley-Shubik power indices are q~p=0.381, ~bs=0.107, ~bH=0.059. Banzhafindices are tip = 0.228, fls = 0.114, flH = 0.086. The discrepancy grows as the size of the senate and house grows. For the United States legislative scheme with a senate of 101 (including the vice-president) and house of 435, we have ~bF = 0.5000, q5s = 0.0025, qSH= 0.0006 but tip = 0.0313, fls = 0.0031, BH = 0.0015. The difference in the power of the president is particularly striking. Given these differences, it is important to characterize the Shapley-Shubik and Banzhafindices weil enough to understand which index would be most appropriate in which situations. We will consider here two approaches: an axiomatic characterization due to Dubey [Dubey (1975) and Dubey and Shapley (1979)], and a probability characterization due to Straftin (1977a). First, some terminology. A power index is a function K which assigns to each player i in a simple game G a real number Ki(G ) >~O. Recall that a voter i is a dummy in a simple game (N, ~ ) if S ~ ~ implies S\{i}~~l¢r. The unanimity oame U s with player set N is the game whose only winning coalition is the grand coalition N of all players. Finally, given two simple games G1 = (Nx, ~t¢/1) and G2 = (N2, ~¢/~2), with N~ and N2 not necessarily disjoint Definition. if S ~ N I ~ ~

G1 A G 2 is the simple game with N = N~ u N 2 and S~~#/~ if and only 1 and S~N2~3¢1/" 2.

Definition. G 1 V G 2 is the simple game with N = N1 u N 2 and S E ~ if and only if Sc~Nle~W1 or Sc~N2~~ir2 . Thus to win in G 1 A G 2 a coalition must win in both G~ and G2, whereas to win in G~ v G 2 it must win either in G~ o r G 2. Consider the following axioms for a power index K: Axiom 1.

Ki(G ) = 0 if and only if i is a d u m m y in G.

ZFor the "oceanic garne" where the number of small voters becomes a continuum, the limit values are attained exactly both hefe and in (1). See Milnor and Shapley (1978), Dubey and Shapley (1979), Straffin (1983).

1135

Ch. 32: Power and Stability in Politics

Axiom 2.

Ki(G1) + Ki(G2) = Kl(G1 ^ G2) + Kl(G1 v G2).

Axiom 3.

If i is a player in U N, then Ki(UN) = 1/[NI.

Axiom 3'.

If i is a player in U N, then Ki(UN) = 1/2 INI- 1

It is clear that the Shapley-Shubik index satisfies Axioms 1 and 3, and the unnormalized Banzhaf index satisfies Axioms 1 and 3'. Both indices also satisfy Axiom 2. (The normalized Banzhaf index fl does not satisfy Axiom 2, which is why fl' is more convenient.) Dubey proved: Theorem 3.1 [Dubey (1975)].

The Shapley- Shubik index is the unique power index which satisfies Axioms 1, 2 and 3.

Theorem 3.2 [Dubey and Shapley (1979)].

The unnormalized Banzhaf index is the unique power index which satisfies Axioms 1, 2 and 3'.

We will show that a power index satisfying Axioms 1 and 2 is determined by its vahes on unanimity games. Note that any simple game G can be written as G = Us, v Us2 v ... v Us,ù, where $ 1 , . . , Sm are the minimal winning coalitions of G. Our proof is by induction on m, the number of minimal winning coalitions. If m = 1, G is a unanimity game with dummies and K is determined. If m > 1, then by Axiom 2 we have Proofs.

Ki(G ) = Ki(Usl ) + Ki(Us2 v ... v Usm ) - Ki(Usl ^ (Us~ v ... v Usm)) = Kl(Us1 ) + Ki(Us~ v ... v U s J - Ki(Us~~s~ v ... v Us,~s~))

and the games on the right have fewer minimal winning coalitions than G, so that K z is determined inductively. [] For example, consider the game G = [ 7 ; 4, 3, 2, 1] ABCD of Section 1. The minimal winning coalitions are AB and ACD. Hence Ki(G ) = Ki(UAn ) + Ki(UAcD) -- Ki(UABcD )

so that B(G) = ( 1» »10 , 0 ) + {1_ i] 1_ 1_~ -- (~2, 1~, 1-12, 1~)

[1 1- 1- 1_~ ~,4,4,4,4J

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and B'(G) = (~,~,0,0) ~ 1 1 O ! ! ~ _ r ,! 8 , ±8 , s±, 8L~ + (~,v,4,4, , a,a,a» Because Axiom 2 plays such a powerful role in this characterization of the Shapley-Shubik and Banzhaf indices, using the characterization to understand the difference between the indices requires having a good intuitive feel for why Axiom 2 should hold. However, we can at least say, looking at Axioms 3 and 3', that which of the two indices you use might depend on how powerful you think players are in unanimity games of different sizes. Adding a new player to a unanimity game UN cuts fl' of each old player by a factor of two, but only lowers ~b of each old player from 1/INI to 1/(INI + 1). For an alternative characterization of q~ and fl' in probabilistic terms, consider a voting model in which each voter i's probability p~ of voting "yes" on a bill is a random variable. Each voter asks the question of individual effect: What is the probability that my vote will make a difference to the outcome? That is, what is the probability that other voters will cast their rotes for a bill in such a way that the bill will pass if I vote for it, but fail if I vote against it? Of course, the answer to this question depends not only on the voting game, but also on the probability distributions of the pi's. Two possible assumptions are the

Independence assumption. Each p~ is chosen independently from the uniform distribution on [0, 13. Homogeneity assumption. A random variable p is chosen from the uniform distribution on [0, 1], and Pi = P for all i. We then have [Straffin (1977a)] Theorem 3.3. The answer to roter i's question of individual effect under the independence assumption is il'i.

Theorem 3.4.

The answer to roter i's question of individual effect under the homogeneity assumption is q~i. Proofs. The answer to voter i's question of individual effect is the probability that a bill passes if we assume i votes for it, but would fail if i voted against it. This is exactly the probability that i will be a swing for the coalition S of "yes" voters, assuming i votes yes. This probability is

isw~ings(je~s_ipj)(je~N_S(1--pj)). for

S

(1)

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If we set all pj = p and take the average value of (1) over p in [0, 1] we get

Z

P~-I(1-P)"-~dP =

i swings for S

F~

Y',

pS- 1(1 _ p ) , - S d p

i swings for S

(s -

1)!(n - s)! _ q~,.

i swings for S

/~]

In the penultimate step we used the "beta function identity" ~ò x a ( 1 - x)bdx = a!b!/(a + b + 1)!. This proves Theorem 3.4. On the other hand, averaging as each pj ranges independently over [0, 1] is equivalent to setting each pj = ½. If we do this for (1) we get

i swings for S

1 1 2s-12ù-s -

1 ~ 2ù-1 i swings

number of swings for i fl, 2ù-1 = i"

for S

This proves Theorem 3.3.

[]

One way to interpret these theorems practically is to think of Pi as the "acceptability" of a given bill to voter i. The independence assumption says that the acceptability of a bill to voter i is independent of its acceptability to any other voter j. Under the homogeneity assumption the acceptability of any given bill is the same to all voters: voters judge bills by common standards. If we believe that voters in a certain body have such common standards, the Shapley-Shubik index is applicable; if we believe voters behave independently, the Banzhaf index is the instrument of choice. For the United States legislative scheme, the Shapley-Shubik index might be most appropriate, while for the diverse Canadian provinces, we might prefer the Banzhaf index. The model also gives insight into the strange behavior of the Banzhaf index in examples 1 and 2 at the beginning of this section. If a large number of small voters vote independently with pj chosen from the uniform distribution on [0, 1], the law of large numbers says that about half of them will vote yes. In that case, the outcome will be determined by how the largest voter X votes. Alternatively, it is possible to use this model to define "partial homogeneity assumptions" tailored to particul~r voting situations. Certain groups of vt~ters would choose their pi homogeneously, independent from other groups of voters. We can even build in an elementary form of ideological opposition by having some groups of voters choose Pi opposite to other groups (p~ = 1 - p j). See Straffin (1977a) and Straffin et al. (1981) for examples. In Section 5 we will see another, more powerful approach to measuring voting power when ideological considerations are important.

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4.

Dynamic applications of the power indices

If politics is the shaping of power, politieal actors might act to increase their power, and the rational ehoice assumption that they do so might have some explanatory efficacy in political dynamies. We will consider three possible situations ofthis type. First, notice that voters can sometimes increase their power by forming coalitions. For example, consider [3; 1, 1, 1, 1, 1]. If the first three voters form a coalition and act as a voting bloc, the resulting garne is [3; 3, 1, 1], in which the bloc has all the voting power. Each of the voters in the bloc has increased his share of the 1 to ½. Strategy of this k i n d - that voters should strive to form a power from -~ minimal winning coalition to maximize their power - has been formalized by Riker as the size principle [Riker (1962), Riker and Ordeshook (1973)]. However, forming eoalitions does not always increase power, since other players may also be forming coalitions. For example, if both the first two voters and the last two voters in the five person majority garne form coalitions, the resulting game is {-3;2, 1, 2]. Each of the two-voter blocs has power ½, which is less than the 52-its members had before. An example of this phenomenon is considered in [Straffin (1977b)]. The county board of Rock County, Wisconsin is a 40-player majority garne. There are two large cities in the county, Beloit with 11 board members, and Janesville with 14 board members. Bloc voting by board members from the two cities has not emerged, but has occasionally been urged upon the board members from Beloit. Indeed, if the board members from Beloit did form a bloc, it would have (by the Shapley-Shubik index) ~11-0.367 öof the power, instead of the ~11ö -_ 0.275 its members have without bloc voting. However, if the Janesville board members also formed a bloc in response to Beloit bloc voting, the resulting garne would be [21; 11, 14, 1, ..., 1], B J 15 others and in this game q~ß = 0.180, with Beloit board members considerably worse oft than without bloc voting. Perhaps the Beloit board members are being politically canny in resisting urgings to vote as a bloc. Straffin (1977b) gives general conditions for when bloc voting by each of two blocs helps both blocs, helps orte bloc but hurts the other, or hurts both bloes. If one way for two voters to change their power is to form a coalition and agree to cast their votes together, another way is to quarrel, i.e. "agree" to cast their votes differently. Kilgour (1974) first analyzed the power effects of quarreling voters. For a simple example, eonsider [3; 2, 1, 1]. ABC To calculate the Shapley-Shubik indices we list the orderings and underline the

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1139

pivot in each: ABC*

ACB

BAC*

BCA* +

CAB

CBA* +

so that ~ba = z and ~bB= qSc = I. Now suppose A and B quarrel. Kilgour interprets this to mean that orderings which have both A and B at or before the pivot are no longer feasible. With the orderings marked by the asterisk ruled out, and the remaining orderings considered equally likely, the Shapley-Shubik indices become qS~ = ½, ~ß~ = 0, and qSc°- ½. Both A and B have lost power because of their quarrel. The Banzhaf index, with the quarreling interpretation that winning coalitions which contain both A and B a r e not feasible, gives a similar result. If it were always true that quarreling members lose power, we would have support for the moral dictum that nastiness can only hurt you. However, in some situations quarreling can increase the power of the quarrelers. If B and C quarrel in the above example, the orderings marked by + are infeasible and the resulting Shapley-Shubik indices are ~ba ° = ½, ~b~ = ~bcO-~-1 B and C have gained power at the expense of A. Once again, the Banzhaf index gives a similar result. Brams (1976) called this the "paradox of quarreling members," and pointed out that It would appear, therefore, that power considerations - independently of ideological considerations - may inspire conflicts among members of a voting body simply because such conflicts enhance the quarreling members' voting power. EBrams (1976, p. 190)]. For other possibilities and examples, see Straffin (1983). Brams and Riker (1972) and Straffin (1977c) have suggested that power dynamics might be a way to explain the onset of a "bandwagon effect" at a critical point in the building of rival coalitions. As an example, consider the 1940 Republican Presidential nominating convention in the United States. On the fifth ballot Wendell Willkie had 429 votes, Robert Taft 377, and the remaining 194 of the 1000 votes were split among other candidates. Suppose we model this situation as 1-501; 429, 377, 1, W

T

..-,

1].

194 Others

The Shapley-Shubik power indices work out to be ~bw =0.4023, q~T=0.1356 and q~o = 0.0024. Consider a single "O" deciding whether to remain uncommitted to either major candidate, or tO support one of them. If she commits to Taft, the game becomes [501; 429, 378, 1, W

T

..., 1],

193 O's

with q5w = 0.3999, ~bT = 0.1370. The commitment to T has raised T's power by 0.0014, which is less than the power our O would retain by remaining uncommitted. If power is thought of as a marketable commodity, which can be paid for by

P.D. Straffin Jr.

1140

concessions or political promises, commitment to T is a losing proposition. On the other hand, commitment to W would produce [501; 430, 377, 1, W

T

..., 1]

193 O's

with ~bw = 0.4064, q~~ =0.1333. This commitment raises W's power by 0.0041, considerably more than O's uncommitted power. Hence our focal O might be wise to bargain for commitment to W on the next ballot. Since this is true for all O's, we might predict a bandwagon rush to commit to W. Willkie did win handily on the hext ballot. In general, as long as rival blocs are fairly evenly matched and far from the winning quota, an uncommitted voter has more power than that voter would add to either bloc by joining that bloc. However, as the blocs slowly build strength, there comes a time when it is power-advantageous for any uncommitted voter (hence all uncommitted voters) to join the larger bloc. The precise form for this bandwagon threshold for large games is derive~d in Straffin (1977c), where other examples from U.S. no~jnating conventions are considered. The claim, of course, is not that delegates to~nominating conventions calculate Shapley-Shubik or Banzhaf power indices and make decisions accordingly, but that a well-developed political sense might be tuned to shifts of power in a form not unrelated to what these indices measure.

5.

A spatial index of voting power

The Shapley-Shubik index measures the probability that a voter will be pivotal as a coalition forms, assuming that all orders of coalition formation are equally likely. It is clear that in any actual voting body, all orders of eoalition formation are not equally likely. One major distorting influence is the factor of ideology, which has traditionally been modeled geometrically. A familiar example is the one-dimensional left-right liberal-conservative spatial model of voter ideology. For a more sophisticated model, we could place voters in a higher dimensional ideologieal space, where distance between voters represents ideological difference. Owen (1971) and Shapley (1977) proposed adapting the Shapley-Shubik index to such spatially placed voters. Figure 1 shows a two-dimensional placement of the voters A, B, C, D, E in a symmetrie majority game [3; 1, 1, 1, 1, 1]. The Shapley Shubik index, ignoring ideology, would give each voter ½of the voting power. On the other hand, common political wisdom would expect the "centrist" voter D to have more power than the other voters. In this context Shapley pictures a forming coalition as a line [in the general k-dimensional case, a (k - 1)-dimensionat hyperplane] sweeping through the plane in a fixed direction given by some unit vector u, picking up voters as it moves. The vector u could be thought of as describing the "ideological direction"

Ch. 32: Power and Stability in Politics

1141

E~

/'" c

A

D

E

D

B

B

A

C

D

Figure 1

of the bill under consideration - the particular mix of ideological factors embodied in the bill. In Figure 1, a coalition forming in direction Ul picks up voters in order CEDAB, with D as pivot, while a coalition forming in direction u 2 picks up voters in order C D E B A with E as pivot. In this way, each unit vector, hence each point on the unit circle [in general the unit ( k - 1)-sphere] is labeled by a voter. Voter

1142

P.D. Straffin Jr.

i's Shapley-Owen power index ¢i is the proportion of the unit circle labeled by that voter. In other words, it is probability that voter i is pivotal, assuming that all directions of coalition formation are equally likely. In the example of Figure 1 the indices are

~/A

=

20o/360o = 0.06,

~kD= 204°/360 ° = 0.57,

~PB= 62o/360o = 0.17,

ffC = 22°/360° = 0.06,

OE = 52°/360° = 0.14.

Voter D is indeed most powerful. In two dimensions, the Shapley-Owen index is efficiently calculable by a rotation al#orithm, which I will describe for the case of a simple majority garne in which the total number of votes is odd (although the algorithm is more generally applicable). Define a median line to be a line L in the plane R 2 such that each open half plane H 1 and H2 of R 2 \ L contains less than a majority of votes. It follows that (i) every median line contains at least one voter point, and each closed half plane H 1 w L and H2 w L contains a majority of votes, (il) there is exactly one median line in each direction, and (iii) voter i's Shapley-Owen power index is the sum of the angles swept out by median lines passing through point i, divided by 180 ° (thinking of lines as undirected).

Figure 2

Ch. 32: Power and Stability in Politics

1143

Hence to calculate the indices, start with any median line, say one passing through A in Figure 2. Imagine rotating it counterclockwise about A until it passes through another voter point (C in this example), assigning the angle of rotation to voter A. As we rotate past AC, the median line switches to passing through C. Continue rotating, always assigning the angle of rotation to the voter through whose point the median line passes, until after 180 ° rotation the line returns to its original position. In practice, of course, the algorithm is discrete, since the median line can only change its voter point at those directions, finite in number, when it passes through two or more voter points. The Shaplëy-Owen indices of the voters in our example are the angles marked in Figure 2, divided by 180 °. Figure 3 shows a two-dimensional spatial plot of states in the United States Electoral College, from Rabinowitz and Macdonald (1986). It was constructed from a principal component analysis of Presidential election data in the period 1944-1980. • MA

,RI

IDEOLOGY 64

72

68

/

MN

ME" 1 WV

w~ \IV

•VT

ON • NJ° «

]]~ . . . . .

~ , . P 80

~~~~°°

j.........~~ 7 6

SD *lA

NHMT • ND

CO' NV

"

• AK

1 NM

"IN

~52

• KS

' 48

TN -VA • WY

•

°FL

NE ,ID

*AZ

56

NC" -LA

44

' OK

,SC

"UT

GA" PARTY *AL

MS Figure

3

1144

P.D. Straffin Jr.

The first component (vertical, explaining 48~o of the variance) is identifiable as liberal-conservative. The second component (horizontal, explaining an additional 31~ of the variance) has to do with traditional Republican-Democratic party identification. Since the third component explains only an additional 7~ of the variance, the two-dimensional spatial plot captures most of the ideological information in the election data. Notice that the analysis also gives the projection of the ten Presidential election axes onto the principal component plane. In calculating the Shapley-Owen indices of the states in Figure 3, Rabinowitz and Macdonald noted that the directions for all of the Presidential elections 1964-1980 lie within a sector of about 90°. Hence to get a measure of electoral power applicable to the modern period, they used only directions in this sector. The results are shown in Table 3. The modified Shapley-Owen indices are given in column 3, and are translated into effective electoral votes in column 5. Column 6 gives the classical Shapley-Shubik indices in the same form. The difference between these figures, in column 7, is a measure of the extent to which a state's ideological position increases or decreases its effective electoral power. Notice that the states which benefit most heavily in proportion to their number of votesCalifornia, Texas, Illinois, Ohio and Washington- are all centrally located in Figure 3. Stares which lose most because of ideology are the outlying states Massachusetts, Rhode Island, Idaho, Nebraska and Utah. Rapoport and Golan (1985) carried out an interesting comparative study of the Shapley-Shubik, Banzhaf, and Deegan-Packel (1979) power indices and their corresponding spatial generalizations. They computed all six indices for parties in the Israeli Knesset in 1981, and compared the results to the perceived distribution of power as rated by three groups of political experts. The spatial corrections improved the match (as measured by mean absolute deviation) between the power indices and perceived power, for all three indices. The resulting match was quite good for Banzhaf, not quite as good for Shapley-Owen, and poor for Deegan-Packel.

6. Spatial models of voting outcomes When we represent voters by points in an ideological space, we can also represent voting alternatives by points in the same space. For example, the two dimensions of R 2 as an ideological space might represent yearly expenditures for military purposes and for social welfare purposes. A voter is represented by his ideal point, whose coordinates are the amounts he would most like to see spent for these two purposes. A budget bill is represented by the point whose coordinates are its expenditures for the two purposes. The assumption is that in a vote between two rival budget bills each voter, at least when voting is sincere, will vote for the bill closest to his ideal point. It is most common to use the standard Euclidean measure of distance, although in some situations other distance measures might be more appropriate.

Table 3 Power in the modern sector [Rabinowitz and Macdonald (1986)] Rank

State

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

California Texas New York Illinois Ohio Pennsylvania Michigan New Jersey Florida North Carolina Missouri Wisconsin Washington Tennessee Indiana Maryland Kentucky Virginia Louisiana Connecticut Iowa Oregon Colorado Georgia Minnesota South Carolina Alabama Arkansas New Mexico Oklahoma West Virginia New Hampshire Montana Mississippi Nevada Maine Delaware Kansas Alaska Arizona South D a k o t a Hawaii Vermont North D a k o t a Massachusetts Utah Wyoming Nebraska Idaho Rhode Island

Power (9/o)~

Electoral votes

Power electoral votes b

Shapley electoral votes

PowerShapley °

12.02 6.99 6.70 5.71 5.47 4.96 3.97 3.66 3.36 2.53 2.41 2.34 2.33 2.22 2.19 2.16 2.02 1.91 1.85 1.60 1.56 1.53 1.48 1.47 1.32 1.26 1.26 1.07 1.06 0.95 0.85 0.83 0.82 0.78 0.76 0.73 0.65 0.63 0.62 0.62 0.57 0.52 0.44 0.37 0.33 0.32 0.27 0.26 0.19 0.12

47 29 36 24 23 25 20 16 21 13 11 11 10 11 12 10 9 12 10 8 8 7 8 12 10 8 9 6 5 8 6 4 4 7 4 4 3 7 3 7 3 4 3 3 13 5 3 5 4 4

64.64 37.60 36.02 30.74 29.42 26.68 21.37 19.68 18.09 13.63 12.95 12.57 12.52 11.93 11.78 11.62 10.86 10.29 9.95 8.62 8.37 8.22 7.93 7.89 7.10 6.78 6.77 5.77 5.69 5.08 4.57 4.45 4.39 4.17 4.06 3.91 3.51 3.41 3.34 3.33 3.07 2.77 2.39 2.00 1.78 1.75 1.46 1.42 1.02 0.64

49.91 29.60 37.29 24.25 23.19 25.31 20.05 15.92 21.09 12.86 10.84 10.84 9.84 10.84 11.85 9.84 8.84 11.85 9.84 7.84 7.84 6.85 7.84 11.85 9.84 7.84 8.84 5.86 4.87 7.84 5.86 3.89 3.89 6.85 3.89 3.89 2.91 6.85 2.91 6.85 2.91 3.89 2.91 2.91 12.86 4.87 2.91 4.87 3.89 3.89

+ 14.74 + 8.00 1.27 +6.50 + 6.23 + 1.37 + 1.33 + 3.77 - 3.00 +0.77 +2.11 + 1.73 +2.68 + 1.08 - 0.07 + 1.78 + 2.02 + 1.56 -0.11 + 0.78 + 0.52 + 1.37 +0.10 -3.96 - 2.74 1.06 - 2.07 - 0.09 + 0.82 -2.76 -

1.29

+0.56 + 0.50 - 2.68 +0.17 + 0.02 + 0.60 - 3.44 + 0.43 - 3.52 +0.16 1.12 --:'~.53 - 0.92 -

11.08

-3.12 -

1.46

3.45

--2.87 -- 3.25

aPercentage of times the state occupies the pivotal position. b Percentage of times the stare occupies the pivotal position multiplied by the total number of electoral votes (538). ¢Difference between power measures.

P.D. Straffin Jr.

1146

Suppose voters in a voting garne are positioned in R k and consider points in R k as alternatives from which they are to make a collective choice. Could we say from garne theoretical considerations which point should be the outcome or, if that is asking too much, at least specify a set of points within which the outcome should lie? From among a number of proposed answers to this question, we will consider three. For simplicity of presentation, we will assume that k ~<2, and that the voting garne is a simple majority garne with an odd number of voters. The eore Definition. An alternative x is in the core of a spatial voting game if and only if there is no alternative y such that y defeats x. If the voting game is one-dimensional, then for a simple majority game with an odd number of voters, the core consists of exactly one point, the ideal point of the median voter. Hence in particular the core is always non-empty. For higher dimensions, the core will be non-empty if and only if the voter ideal points satisfy a restrictive symmetry condition first given by Plott (1967). To derive this condition for k = 2, recall the idea of a median line from the last section. Notice that if L is a median line and x is a point not on L, then x will lose under majority voting to x', which is the reflection of x across L, moved slightly towards L. This is because x' will be preferred to x by all voters in L w H 2, where H 2 is the half plane containing x', and this is a majority of voters. Since x can be beaten if there is some median line which does not contain x, an alternative x can only be in the core if it lies on all median lines. We have proved Theorem 6.la [Davis et al. (1972), Feld and Grofman (1987)]. The core of a simple majority garne in R 2 will be non-empty if and only if all median lines pass through a single point. I f t h e y do, that point, which taust be a roter ideal point, is the unique point in the core. A bit of geometry shows that this is equivalent to Theorem 6.1b [Plott (1967)]. The core of a simple majority garne in R 2 will be non-empty if and only if the roter ideal points all lie on a collection oflines L i süch that (i) the lines L~ all pass through the ideal point of one roter A, and (ii) A is the median roter on each of the lines L i. This condition is clearly structurally u n s t a b l e - small perturbations of voter ideal points will destroy it. Hence Theorem 6.le. empty core.

Generically, simple majority garnes in dimensions two or higher have

Ch. 32: Power and Stability in Politics

1147

The top cycle set. If the core is empty, then no alternative can directly defeat all other alternatives. However, we might consider the set of alternatives x which at least can defeat all other alternatives in a finite number of steps. Definition. An alternative x is in the top cycle set of a spatial voting game if for any other alternative y, there is a finite sequence ofalternatives x = Xo, x l , . . . , xm = y such that x i defeats x~+l for all i = 0 . . . . . m - 1. It is surprising how large the top cycle set is:

Theorem 6.2 [McKelvey (1976)]. I f the core of a simple majority garne in R « is empty, then the top cycle set is all of R k. To see what this theorem means, suppose we have three voters A, B, C as in Figure 4, and suppose we start with any alternative w. Choose any other point in the plane, perhaps one far away like z. McKelvey's theorem says that there must be a chain of alternatives starting with w and ending with z, such that each alternative in the chain is preferred to the preceding alternative by a majority of the voters. If we control the voting agenda and present the alternatives in the

Figure 4

1148

P.D. Straffin Jr.

chain in order, we can get our small society to accept z, or any other alternative in the plane. The agenda controller, in the generic case where the tore is empty, has complete power. Figure 4 illustrates how to construct a McKelvey chain for this example. Notice that AB, AC and BC are all median lines. If we reflect w across AC and move it in slightly to get x, then x beats w. Now reflect x across BC and move it in slightly to get y, which beats x. Now reflect y across AB and move it in slightly to get z. We have found the required chain w, x, y, z. This simple example has in it the proof of the general theorem for k = 2. For first suppose that, in this example, we were trying to get to some other point t. Just keep repeating the three reflections in order, obtaining points x l , y l , z l , x2,Y2, z2 .... as far away from A, B and C as we please. When we get some z,ù far enough away beyond t, t will be unanimously preferred to z m and we will have our chain. Second, suppose we have any configuration of any number of voters, but empty tore. Then by Theorem 6.la there must be three median lines which do not pass through a common point, and hence bound a triangle. Let this triangle play the role of ABC in the above argument. McKelvey's result, with its implications of the inherent instability of majority rule and the power of agenda control, has generated a large literature. For a thoughtful discussion, see Riker (1980). The strong point. If the core is empty, then every alternative is defeated by some other alternative. However, we might look for the alternative which is defeated by the fewest possible other alternatives. Definition.

For a point x in R k, def(x) is the set of all points in R k which defeat x.

Definition. The strong point of a simple majority game in R k is the point x in R k for which k-volume of def(x) is as small as possible. [Grofman et al. (1987)]. It is easy to show that the strong point exists and is unique. However, it might seem that the strong point would be difficult to compute. This is not true, at least for k = 2, by a recent result of Owen and Shapley, which connects voting outcomes to voting power. Theorem 6.3 [Owen and Shapley (1989)]. The strong point in a two-dimensional majority voting game is the wei9hted average of the roter ideal points, where the weights are the Shapley-Owen power indices of the voters. In other words, the strong point is located exactly at the "center of power". I refer you to Owen and Shapley (1989) for the general proof of this theorem, but I would like to illustrate in a simple case the elegant geometric insight on which it is based. In Figure 5 we would like to find a formula for the area of def(X),

1149

Ch. 32: Power and Stability in Politics

Figure 5

which in this case is the union of six segments of circles, one of which, X S T U , is shaded in the figure. The area of this segment is twice the difference between the circular sector B S X and the triangle B UX. Adding six such areas together, we get

2

2

'

where «, fl and 7 are measured in radians. Since the area of triangle A B C is a constant not depending on X, to minimize the area of der(X) we must choose X to minimize « A X 2 + f l B X 2 + 7 C X 2. It is well known that this weighted sum of squared distances is minimized by taking X = ŒA + f l B + T C _ ~ A Œ+fl+y ~

+ fl B + T C. n n

These coefficients of A, B and C are exactly the Shapley-Owen power indices of voters A, B and C.

1150

P.D. Straffin Jr.

T h e c o n c l u s i o n of T h e o r e m 6.3 d o e s not, u n f o r t u n a t e l y , h o l d in d i m e n s i o n s k > 2. F o r o t h e r s o l u t i o n c o n c e p t s for the p r o b l e m of v o t i n g o u t c o m e s , a n d for a g e n e r a l i n t r o d u c t i o n to o t h e r spatial v o t i n g ideas, see E n e l o w a n d H i n i c h (1984). F o r a s u r v e y o f c o a l i t i o n a l q u e s t i o n s in spatial v o t i n g , see Straffin a n d G r o f m a n (1984).

Bibliography Brams, S.J. (1975) Garne Theory and Politics. New York: Free Press. Brams, S.J. (1976) Paradoxes in Politics. New York: Free Press. Brams, S.J. and P. Affuso (1985) 'New paradoxes of voting power on the EC Council of Ministers', Electoral Studies, 4:135 139 & 290. Brams, S.J. and W.H. Riker (1972) 'Models of coalition formation in voting bodies', in: Herndon and Bernd, eds., Mathematical Applications in Political Science VI. Charlottesville: University of Virginia Press. Banzhaf, J. (1965) 'Weighted voting doesn't work: a mathematical analysis', Rutgers Law Review, 19: 317-343. Banzhaf, J. (1968) 'One man, 3.312 rotes: a mathematical analysis of the electoral college', Villanova Law Review, 13: 304-332. Davis, O., M. Degroot and M. Hinich (1972) 'Social preference orderings and majority rule', Econometrica, 40: 147-157. Deegan, J. and E. Packel (1979) 'A new index of power for simple n-person garnes', International Journal of Garne Theory, 7: 113-123. Dubey, P. (1975) 'On the uniqueness of the Shapley value', International Journal of Garne Theory, 4: 131-139. Dubey, P. and L.S. Shapley (1979) 'Mathematical properties of the Banzhaf index', Mathematics of Operations Research, 4: 99-131. Enelow, J. and M. Hinich (1984) "The Spatial Theory of Votin9: An Introduction. Cambridge: Cambridge University Press. Feld, S. and B. Grofman (1987) 'Necessary and sufficient conditions for a majority winner in n-dimensional spatial voting games: an intuitive geometric approach', American Journal of Political Science, 31: 709-728. Grofman, B., G. Owen, N. Noviello and G. Glazer (1987) 'Stability and centrality of legislative choice in the spatial context', American Political Science Review, 81:539 552. Kilgour, D.M. (1974) 'A Shapley value analysis for cooperative games with quarreling', in Rapoport, A., Garne Theory as a Theory of Conflict Resolution. Dordrecht: Reidel. Kilgour, D.M. (1983) 'A formal analysis of the amending formula of Canada's Constitution Act, 1982', Canadian Journal of Political Science, 16: 771-777. Kilgour, D.M. and T.J. Levesque (1984) 'The Canadian constitutional amending formula: bargaining in the past and future', Public Choice, 44: 457-480. Lucas, W.F. (1983) 'Measuring power in weighted voting systems', in: S.J. Brams, W. Lucas and P.D. Straffin, eds., Political and Related Models. New York: Springer Verlag. Mann, I. and L.S. Shapley (1962) 'The a priori voting strength of the electoral college', in Shubik, M., Garne Theory and Related Approaches to Social Behavior. New York: Wiley. McKelvey, R. (1976) 'Intransitivities in multidimensional voting models and some implications for agenda control', Journal ofEconomic Theory, 12:472 482. Miller, D. (1973) 'A Shapley value analysis of the proposed Canadian constitutional amendment scheme', Canadian Journal of Political Science, 4: 140-143. Milnor, J. and L.S. Shapley (1978) 'Values of large garnes II: oceanic garnes', Mathematics of Operations Research, 3: 290-307. Owen, G. (1971) 'Political games', Naval Research Logistics Quarterly, 18: 345-355. Owen, G. (1975) 'Evaluation of a Presidential election game', American Political Science Review, 69: 947-953.

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Owen, G. and L.S. Shapley (1989) 'Optimal location of candidates in ideological space', International Journal of Garne Theory, 18: 339-356. Ordeshook, P. (1978) Garne Theory and Political Science. New York: New York University Press. Plott, C. (1967) 'A notion of equilibrium and its possibility under majority rule', American Economic Review, 57: 787-806. Rabinowitz, G. and S. Macdonald (1986) 'The power of the states in U.S. Presidential elections', American Political Science Review, 80: 65-87. Rapoport, A. and E. Golan (1985) 'Assessment of political power in the Israeli Knesset', American Political Seience Review, 79: 673-692. Riker, W.H. (1962) The Theory of Political Coalitions. New Haven: Yale University Press. Riker, W.H. (1980) 'Implications from the disequilibrium of majority rule for the study of institutions', American Political Science Review, 74: 432-446. Riker, W.H. (1986) 'The first power index', Social Choice and Welfare, 3: 293-295. Riker, W.H. and P. Ordeshook (1973) An Introduction to Positive Political Theory. Englewood Cliffs: Prentice Hall. Roth, A. (1988) 'The Shapley Value: Essays in Honor of Lloyd S. Shapley'. Cambridge: Cambridge University Press. Shapley, L.S. (1962) 'Simple games: an outline of the descriptive theory', Behavioral Science, 7: 59-66. Shapley, L.S. (1977) 'A comparison of power indices and a non-symmetric generelization', RAND Paper P-5872, Rand Corporation Santa Monica. Shapley, L.S. and M. Shubik (1954)'A method for evaluating the distribution of power in a committee system', American Political Science Review, 48: 787-792. Straffin, P.D. (1977a) 'Homogeneity, independence and power indices', Public Choice, 30: 107-118. Straffin, P.D. (1977b) 'The power of voting blocs: an example', Mathematics Magazine, 50: 22-24. Straffin, P.D. (1977c) 'The bandwagon curve', American Journal ofPolitical Science, 21:695 709. Straffin, P.D. (1980) Topics in the Theory of Voting. Boston: Birkhauser. Straffin, P.D. (1983) 'Power indices in politics', in: S.J. Brams, W. Lucas and P.D. Straffin, eds., Political and Related Models. New York: Springer Verlag. Straffin, P.D., M. Davis and S.J. Brams (1981) 'Power and satisfaction in an ideologically divided voting body', in: M. Holler ed., Power, Voting and Voting Power. Wurzburg: Physica Verlag. Straffin, P.D. and B. Grofman (1984) 'Parliamentary coalitions: a tour of models', Mathematics Magazine, 57:259 274.