paradox proof decision rules in weighted voting - Springer Link

M.J. Holler (ed.): The Logic of Multiparty Systems. Copyright 1987 by Martinus Nijhoff Publishers, Dordrecht, The Netherlands

PARADOX PROOF DECISION RULES IN WEIGHTED VOTING

Manfred. J. HOLLER· University of Aarhus

Abstract: The mapping of the distribution of voting weights into the indices of Banzhaf and Shapley - Shubik, measuring a priori voting power, is not monotonic. Corresponding paradoxes imply a potential to increase an agent's voting power by decreasing his voting weight. Paradox proofness of a voting body is achieved. if the distribution of voting power is strictly proportional to the vote distribution. In this paper, a randomized decision rule is proposed to bring about paradox proofness.

1. Introduction It is well known among game theorists and political scientists that the distribution of voting weights in a voting body, e.g., the seat distribution in a parliament, is generally a poor proxy for the distribution of voting power. This view gains support by non - monotonicity effects (so - called paradoxes) which occur if a priori voting power is measured. by the power indices of Banzhaf and Shapley - Shubik 1 and effects of changes in the seat distribution are analysed by these measures. These non - monotonicity effects challenge the idea of political representation inasmuch as an increase in votes (and seats in the representational voting body) decreases the voting power of the corresponding agent. In this paper, a family of paradox proof decision rules will be presented. which rule out paradoxical effects as described.. The randomization of the

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decision rules is the heart of the proposed solution. The concept of paradox proof decision rules (the PPDR solution) is suggested as a tool for comparing voting systems with respect to notions of stability and efficiency. Moreover, it is proposed as a yardstick to the construction of new representational bodies. In the final section of this paper, two interpretations are given with respect to the application of this concept. First, however, we will take a look at the "Paradox of Redistribution" and the "Paradox of New Members", in order to illustrate the paradoxical effects with respect to the vote - power relation. In Section 3, borrowing from Berg and Holler (1986) and Holler (1982a, 1985), the concept of strict proportional power (SPP) is defined. Several solutions to SPP are discussed in Section 4. In Section 5, a simple model of randomized decision is elaborated for the three - player case which permits satisfaction of various political and institutional restrictions without distorting SPP. In the concluding Section 6, a frequency interpretation as well as a probability interpretation are given for the randomized decision rule.

2. Voting Paradoxes

In a series of articles by Dreyer and Schotter (1980), Fischer and Schotter (1978) and Schotter (1979, 1982) a surprising phenomenon was found within the analysis of voting power: the "Paradox of Redistribution". The paradox exists when, "given a vote distribution w = (w 1 , ... ,w n ) and a voting rule d, we can construct a new vote distribution w' - (w~, ... ,w;) which gives at least one voter a smaller proportion of the votes yet gives him more power, when power is measured by either the Banzhaf or the Shapley - Shubik power index" (Fischer and Schotter, 1978, p.52; for a discussion see Holler, 1982c). A voting or decision rule dj is defined as follows: A subset S of N is called a winning coalition with respect to a decision rule dj , if the sum of the voting weights IW j of players i E S is equal or larger than dj . The "Paradox of New Members", which has been introduced by Brams (1975, p.178) and Brams and Affuso (1976), is a specialization of the ·Paradox of Redistribution". Fischer and Schotter (1978, p.66) illustrate the relationship between the two voting power paradoxes by comparing a voting body v - (d,w) to a voting body v' - (d,w'), assuming the decision rule d - 0.7, w - (0.6,0.4,0.0) and w' - (0.5,0.25,0.25). A change from w to w' implies an increase of the weight of voter 3 from 0.0 to 0.25. Therefore, voter 3 represents the idea of a new member. The Banzhaf index relating to v is (0.5,0.5,0.0), while the Banzhaf index relating to v' is (0.6,0.2,0.2). The corresponding Shapley - Shubik indices 426

are (0.5,0.5,0.0) for v and (2/3,116,116) for v'. Both indices indicate that the power of voter 1 has increased in spite of the fact that his (or her) voting weight decreased. (Formulas for the indices are given in Section 3 of this paper.) Fischer and Schotter (1978) show that, given a decision rule d, no paradox proof vote distribution exists for any voting body with n ~ 6 (n ~ 7) members if the voting power is measured by the Banzhaf index (Shapley - Shubik index). Given a vote distribution w, we can always find a vote distribution w' which will increase some member's power while decreasing his (or her) voting weight. If we restrict the choice of the decision rule to majority voting, the "critical size" of the voting body lowers from n ~ 6 (n ~ 7, respectively) to n ~ 4. Therefore, the voting paradoxes are potential for a large number of multiparty systems which we find in the political and the economic sectors of democratic societies. In a recent publication, Rapoport and Cohen (1984) show that (a) the relative frequency of the occurrence of the "Paradox of New Members" generally decreases for both power indices as the size of the original voting body and the number of new members increases, (b) the Banzhaf and Shapley - Shubik indices cannot be distinguished from each other on the basis of the relative frequency of the "Paradox of New Members", (c) for m ~ 4 the relative frequency of the paradox generally decreases as the decision rule d increases, and (d) for m ~ 5, the mean size of the paradox (Le., in general terms, the average power measure distortion experienced by the voters for whom the paradox occurs) is sufficiendy small to render its practical importance questionable. (m is the number of voters in the orignal voting body, Le., before a new member enters.) However, for voting bodies of m ~ 4 members the relative frequency of occurrence of the paradox and its mean size seem to be quite substantial and are not to be neglected if the rules of the voting game are to be implemented "rationally" . The described voting power paradoxes imply various possibilities of manipulation. In accordance with the "Paradox of Redistribution", a member of the voting body can increase his voting power by voluntarily reducing his voting weights. Inasmuch as the voting body is determined by the outcome of general elections and the voting weights are positively related to the vote shares gained in these elections, a loss in vote shares (and, thus, a decrease in voting weight) can increase the a priori voting power of the representing agent in the voting body under consideration. If applied to a parliamentary party system, this implication of the voting power paradoxes dodges the Schumpeterian idea of democracy as a competition of the politicians via winning elections. The "will of the people" (i.e., of the voters) does not fmd its expression in the workings of the body in a faithful way if the described paradoxes occur and the manipulation of the seat - power relation can be used to influence the 427

political outcome contrary to the voters' support. If we abstract from the distortion between vote shares and seat shares 2 , caused by electoral formulas (Le., apportionment methods) such as aRondt, Imperial, and St. Lague (see, e.g., Rokkan, 1968; Lijphardt and Gibbard, 1977; Laakso and Taagepera, 1982), the problem of paradox proof decision rules in weighted voting can be solved by designing decision rules that - given the distribution of weights (and support) lead to a distribution of voting power which is identical with the distribution of voting weights (see Nurmi, 1982). For a large range of voting bodies, however, the substitution of the decision rule d by any alternative decision rule d' does not bring about full concurrence of voting weights and power shares, i.e., strict proportional power as defined in detail in the following section. Before describing a solution to this problem, we suggest the following theorem and lemma for n ~ 3. J Theorem 1: A decision rule in .weighted voting is paradox proof if, and only if, it guarantees strict proportional power (SPP).

Lemma 1: A decision rule in weighted voting is strategy proof (i.e., increasing the vote share is the dominating strategy) if, and only if, it guarantees strict proportional representation (SPP).

3. The Concept of Strict Proportional Power The concept of strict proportional power (SPP) implies equality of the voter's weight and his (or her) voting power. Given "one man, one vote" and a proportional electoral formula, the voter's weight is represented in the aggregate by the share of seats his representatives occupy in the reduced voting body. Thus, the voting power of voters who cast identical votes is represented by the voting power of the thereby elected candidates. This relates to the election of candidates and the winning of seats to the voting in the representational voting body. In the following it is assumed that the representatives of each group of voters vote as a bloc. We may therefore label the representatives elected by the ith group of voters the ith representative in the voting body. Thus, a (representational) voting body v(d,w) can be described by a vector of decision rules d - (dJ), j - t, ... ,k, such that t ~ dj > 0.5, and a vector of voting weights ( vote distribution") w - (Wi)' 1 - t, ... ,n, such that IWi .. t and Wi ~ O. n is the cardinality of the set of representatives N, and Wi represents the share of seats which are occupied by the representatives who were elected by the ith group. 428

Given that 0.5 < dj :ii 1 for all j .. 1, ... ,k, v(d,w) describes a proper simple game: S is either a winning or a losing coalition. In what follows, the groups of homogeneous representatives, i.e., dements of N, will also be called players. If we assume that the ith player is prepared to collude with the other players in N on unbiased terms so that all (winning coalitions) are of equal probability, the a priori voting power of i can be expressed by the (nonnormalized) Banzhaf index:

(1)

bi -

number of swings for player i

2n

I

Player i causes a swing to coalition S if S is a winning coalition and S\{i} is a losing coalition. We get the normalized Banzhaf index: b~

1

By this measure, the (a priori) voting power of the dements in N is expressed by the vector b - (b), i - 1, ... ,n. The Banzhaf index (as introduced by Banzhaf, 1965) weights all coalitions with an equal number of swings equally. As pointed out by Brams and Affuso (1976), it is equivalent to the Coleman index. It therefore does not show the rather peculiar results which Coleman (1972) attained from the examination of the Shapley - Shubik index. Neverthdess, there are good reasons to use the Shapley - Shubik index, the Deegan - Packd index (Packel and Deegan, 1982), or the public good index (Holler, 1982b, 1984aj and Holler and Packel, 1983» for the analysis of the a priori voting power in situations of strategic interdependence and sharing of private or public goods. Since, however, we will not specify the quality of the issue voted on, the Banzhaf index seems to be a suitable measure of the voting power for the following analysis.

4. Solutions to SPP

Having defined a measure of voting power, we can give the following definition: Definition 1: Strict proportional power (SPP) is given for a voting body 429

v(d,w), if for every i in N, bi(d) - wi and thus b - w. The examples given in Section 2 (as well as those given below) illustrate that, for an arbitrarily chosen decision rule dj , we cannot expect the power distribution b to be identical (or very close) to the seat distribution w. In fact, because of the discontinuity of the power measures related with the continuum of decision rules 0.5 < d. ::i 1, in many cases there is no (pure) decision rule dj available which guaiantees b(d} - w. By the application of adequate randomization procedures, however, the set of power measures becomes continuous. In principle, SPP becomes feasible for any given vote distribution w if we do not distinguish between the power measure and its expected (or the average) value. If we conceive power as the capacity of a player, alone or in combination with others, to bring about or preclude certain outcomes (see Miller, 1982), this capacity implies a potential to act. Thus, it can be identified with expectations as expressed by power indices. Given that the rules of probability calculation apply for these measures, the expected value of a power index can be interpreted as a power index as well. Moreover, the expected value of a power index is fully characterized by the properties of the original index if the rules of probability calculation apply to it. This defines the new category of power indices which so far has not been explicitly considered in the literature: the expected. value of power indices, or in short, the expected. power indices (see Holler, 1986a, for further discussion). Dubey and Shapley (1979) prove that SPP is attained if power is measured by the Banzhaf index as defined in (2) and d is taken to be a random variable, uniformly distributed over the interval (0.5,1]. A second candidate for an adequate randomization procedure is based on the random choice of a dictator, Le., a -randomized dictator- rule. Given player i's voting weight Wi' we can bring about bi - Wi by relating to i a chance (or a relative frequency) of qi to form the winning coalition S - {i} which implies b i - 1 for i in S and b i = 0 for all i (in N) not in S.·4 Given qi - Wi' qibi'- Wi can be assured. There might be political as well as practical reasons (a) for restricting the potential application of decision rules to a smaller and discrete set or (b) to avoid a (random) dictator. The corresponding discrete randomized. decision rule (discrete RD rule) can be represented by a mixture of voting games:

DeJinition 2: The set of games v(dj,w), dj in d together with an associated probability vector q = (ql' .... 'qn)' Iqj - 1 and 1 > qj > 0, is a mixture of voting games, if the constituent game v(dj,w) is played with probability (or frequency) qj' j - 1, ... ,k, k ~ 2. The probability interpretation of qj applies to the (discrete) RD rule. Given wand the corresponding RD rule (d,q), the expected. power index 430

of player i is computed by:

(3)

bi - l:q.b.(d.), j - 1, ... ,k J

~

1

J

The values b i constitute a vector b. SPP can be achieved if, for a given weight distribution w, we can fmd a RD rule (d,q), such that b(d,q) .. w. In general, there is more than one RD rule available which assures SPP as will be seen in the following example.

5. The Three - Player Case of SPP

In order to analyse the properties and conditions of SPP for the case of three players (N - 1,2,3), we introduce an ordered simplex denoted by

The simplex S(3) can be described by the triangle ABC in Figure 1. C (1,0,0)

(0,1.0)

(0,0,1)

The coordinates Xl, X 2, and x) are represented by the distances measured on the vertical to the corresponding legs of the triangle CFG. Because of the properties l:w.1 = 1 and w·1 > 0 and l:b.1 - 1 and b.1 ~ 0, both the distributions of the voting weights w and the normalized Banzhaf indices b can be expressed as points in S(3). E.g., as each vertex and leg of CFG correspond to a player, any point in S(3) represents a vote 431

distribution so that the ratio defined by the Euclidean distance from the representing point to the player i's leg of the triangle CFG, divided by the height of the triangle, is the percentage of votes assigned to player i. Since the height is standardized by IXi = 1, the percentage follows immediately. The ranking of the elements of N is in accordance with the size of w·1 . and/or b i, only. It can be shown (see Holler, 1985) that in the case of three players the set of (non - randomized but normalized.) Banzhaf power indices is given by (5) ~ .. {b 1 ,b 2 ,b',b 4 : b l _ (3/5,115,115), b 2 .. (112,112,0), b' = (113,113,113), b 4 = (1,0,0), for all w, IWi - 1 and Wi > 0, and all djE d}. For a given vote distribution w, each element in pj corresponds to a different decision rule d.. However, this does not imply that each d. corresponds to a differentJ hi.. In fact, the set p'J is fully represented byJ four points in S(3), only, i.e., A, B, C, and D (see Figure 1). From simplex algebra we know that for any vote distribution w represented as a point in the triangle ABC, SPP can be assured by a (discrete) RD rule randomizing on the power indices depicted by the points A, B, and C. However, we can further specify the choice of the RD rule by taking into account the following properties (which have been defined in Holler, 1985): SSP minimal: Given that the RD rules (d,q) and (d',q') are chosen so that b(d,q) - b(d' ,q') .. w, then composition (d,q) is SPP minimal, if its dimension k is equal to or smaller than the dimension k' of every (d' ,q') satisfying SSP. SPP dictatorial: A RD rule (d,q) is SPP dictatorial, if it assures SPP for a given w, and, for some dj contained in d, assigns the power index bi(d.) = 1 to one of the players. sp~ vetoer: A RD rule (d,q) is a SPP vetoer, if it assures SPP, and, for {i}1 and some dj contained in d, given w, dj > Iw t , for all t = 1, ... all i E N. If we apply these properties to the three - player case represented by the simplex S(3) and Figure 1, we first notice that triangle BCE (and thus point E) represents the vote distributions w for which b 4 is feasible for an adequate choice of d.. For any vote distribution w in the interior of ABE, we can therefore select a RD rule which is SPP dictatorial. On the other hand, for proper simple games (i.e., .5 < d.J. ~ 1) the power index b 4 cannot be achieved by the choice of some bl for a vote distribution w which is represented by a point in the interior of the triangle ABE. Thus, dictatorial RD rules cannot be applied to achieve SPP for a given vote distribution w in ABE.

,IN -

432

Next, let us take a vote distribution w which is located in the interior of triangle BCD and vary the decision rule dj from .5 to 1. We thereby get the Banzhaf indices which correspond to the points A, B, C, and D. For an appropriate mixture (in accordance with Definition 2) of A, B, and C we can get SPP with respect to the given w. The corresponding RD rule is SPP dictatorial as well as a SPP vetoer. However, we can also use a mixture of voting games which relates to the points B, C, and D to achieve SPP. The corresponding RD rule is not a SPP vetoer, however, it is SPP dictatorial. In fact, SPP has to be dictatorial for all vote distributions located in the interior of the triangle BCD. If we take a point w in the interior of the triangle ABE, we can achieve SPP by the application of a RD rule which is SPP dictatorial (implying point C) and a SPP vetoer (implying A). Instead of relying on C we could choose a RD rule which implies D but not C, and thus is not SPP dictatorial. We can, however, not avoid the RD rule to be a SPP vetoer. As already noticed, for vote distributions w allocated in the interior of triangle ABE dictatorial RD rules are not available to achieve SPP. All RD rules which are used to assure SPP for interior points of this triangle are SPP vetoers. The logic of simplex algebra follows that a RD rule (d,q) is SPP minimal for vote distributions w in the interior of the triangles ABD and BDC, respectively, if it assures SPP and its dimension k is equal to the number of players n. Of course, this only applies if any two of the related Banzhaf index vectors of b(d) are different. For boundary points of the considered triangles (which are not corner points) SPP can be achieved by a minimal RD rule based on two pure power measures. Trivially, a vote distribution which is represented by a corner point of the triangles relates to a SPP minimal RD rule constisting of one pure power measure, only. The corresponding qj is equal to 1.

6. Interpretation

RD rules allow for two rather different interpretations: The frequency interpretation implies that each decision rule dj in d (d 1 , ... ,dk) will be realized qj times, whereas the probability interpretation implies that each decision rUle dj has a chance of probability 'lj to be put into reality. The latter interpretation is consistent with the fact that only one d. will be singled out by some random mechanism, while the f1l"st inte;pretation implies that we will see all dj (which are ascribed by the chosen RD rule (d,q) with q. > 0) applied "in the course of . time". The two interpretations lead to quite different problems when it comes to the application of RD rules. Risk averse voters will prefer a RD rule :0

~

433

which concurs with the frequency interpretation. In addition, the intertemporal alternation of the decision rule invites logrolling solutions (Bernholz, 1973). These may (a) contribute to overcome shortcomings of non - random majority rules which are known as Condorcet cycles (the voting paradox) and Borda effects (see Berg, 1985), and (b) lead to joint efficient solutions. On the other hand, we may expect resistance to the application and implementation of RD rules in collective decision making. Some of these problems are discussed in Berg and Holler (1986) and Holler (1986b). The conditions of paradox - proofness which we analysed in the preceding sections with respect to proper simple games can be generalized inasmuch as we can associate to every effectivity function 5 a simple game, made up of its winning coalitions (see Moulin, 1983, p. 159). From this we can deduce sufficient conditions for paradox - proofness of social choice functions (and social choice correspondences, respectively) such as the veto function, the Borda scoring correspondence, and the Copeland and Kramer scores (see Moulin, 1983, for an overview). On this, however, I will elaborate in future work.

Notes

·1 would like to thank Sven Berg for previous collaboration on ideas prescnted in this paper. Thanks also to GiRle Dc Mcur, Jan - Erik Lane, Hannu Nurmi, and Bjorn Erik Rasch for helpful comments. Sections 2, 3 and 5 are derived from part of Berg and Holler (1986). Earlier results were published in Holler (1985). The article was written while the author participated in a research project on "The Efficiency and Stability of Multipany Systems", fmanced by the Deutsche Forschungsgemeinschaft (DFG). 1

The Shapley - Shubik index has been introduced by Shapley and Shubik (1954) as an application of the Shapley value (Shapley, 1953) to the weighted majority games. The Shapley value expresscs the players' expected values of games.

Hereby, a game is

considered a sct of rules. 2 Regardless

of the choscn electoral rule, it is possible that a party increases its vote share

and nevenheless its voting weight decreases. In this paper, however, the apponionment paradox should not be dealt with. , A proof of Theorem 1 and Lemma 1 can be based on Moulin's (1983) treatment of the proponional veto core (Ch. 6) and its generalization via the effectivity function approach (Ch. 7). • Note that the decision rule applied here does not imply a comparison of a number of votes to a specific majority rule dj as assumed in the preceding and subsequent discussion. , An effectivity function is a binary relation which says for each coalition S, sublet of N, and each subset B of the set of outcomes A whether or not S can force the fmal outcome within B (or, equivalently, whether S can veto all clements in A\B) (sce Moulin, 1983, pp.

434

155 - 157).

For proper simple games, a winning coalition S is defmed by its power to

force the fuW decision within every subset B of A.

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