One stake one vote. Marc Fleurbaey. December Abstract. The weighted majority rule, which is widely applied much more than the


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1 One stake one vote Marc Fleurbaey December 2008 Abstract The weighted majority rule, which is widely applied much more than the simple majority rule has good properties when the weights reflect the stakes of the voters in the decision. These properties can be contrasted with classical problems of democratic theory which are associated with the one person, one vote principle and the simple majority rule: tyranny of the majority, intensity problem, and many voting paradoxes. These observations suggest that the egalitarian one person, one vote principle should be replaced in democratic theory by a more fundamental proportionality principle. A new measure of sensitivity of a voting rule to the population s preferences is proposed, which is useful in determining the optimal voting weights of representatives. Keywords: majority rule, weighted majority, voting game, Banzhaf measure, power, democratic theory. This paper has benefited from a seminar at the LSE, from advice and comments by G. Arrhenius, N. Baigent, C. Beisbart, L. Bovens, R. Bradley, S. Brams, R. Goodin, A. Laruelle, C. List, K. Spiekermann, A. Voorhoeve, and from discussions with H. Brighouse, J. Roemer. CNRSCERSES, University ParisDescartes, LSE and IDEP. 1
2 it would be helpful for the development of democratic theory if we could assume that some means exist for comparing intensities of preferences (R. Dahl, 1956, p. 99) 1 Introduction The egalitarian principle one person, one vote is a cornerstone of democratic theory. It is intimately linked to the simple majority rule, which, in one form or another, plays an important role in many democratic institutions. In modern political thought, the core of the notion of democracy is its etymological core rule by the people which translates most naturally as majority rule if there are divisions of opinion. (Hardin 1993, p. 158). Butdemocratictheoryisplaguedwithdifficulties, many of which stem from undesirable properties of the simple majority rule: in particular, majorities can by cyclic (intransitivity), tyrannical (oppression of minorities), and inefficient (a combination of separate majoritarian decisions can be bad for all). In this paper it is observed that the weighted majority rule, in which each voter has a certain number of votes, has some nice properties if the weights are well chosen, i.e., if they are proportional to the voters stakes in the decisions. These properties suggest that the classical difficulties of democratic theory can be alleviated in this way, and therefore this paper argues for substituting a proportionality principle, one stake, one vote, to the egalitarian one person, one vote principle. Moreover, as argued in this paper, the perspectives of applications of weighted procedures are much greater than usually thought, in particular because they are already widely applied in various ways. The possibility to designing the voting rule on the basis of interpersonally comparable 2
3 information is traditionally excluded from the literature on voting systems, which, as will be argued here, appears unduly restrictive compared to actual practice. In order to assess the actual or potential merits of weighted voting rules in practice, it is necessary to analyze the choice of weights in some detail. In this paper, Sections 26 study how to determine the optimal weights for the weighted majority rule, in various informational contexts and at various levels of decision, when the objective is to maximize the (expected) sum of individual indices measuring individual interests. Some of the results extend existing results about the simple majority rule, while others are more original. Specifically, Section 2 presents a basic result that links weighted majority to the maximization of the sum of individual indices. Section 3 makes use of this result to solve the constitutional question of choosing a voting rule when one knows the relative stakes of different individuals but not how they vote. This provides a direct extension, and new proof, of the classical RaeTaylor theorem that is limited to the case of equal intensities. Section 4 extends the analysis to the case when the stakes are not known with certainty and when several issues with different stakes will have to be treated with the same voting rule. Section 5 briefly explores the intriguing case of voters who do not vote in their best interest, preparing the stage for Section 6, which is about the allocation of power across jurisdictions whose representatives may be more or less faithful to their constituencies. Taking stock, Section 7 explains how such results highlight a particular interpretation of the underlying causes of the classical difficulties of democratic theory. Section 8 discusses the practical relevance and the possible applications of the proportionality principle. Section 9 concludes. This paper relates to various branches of the literature. First, there are many studies of the social optimality of various classical voting rules (e.g., Bordley 1983, Merrill 1984, Beisbart et al. 2005, Barbera and Jackson 2006, Beisbart and Bovens 2007, Bovens and 3
4 Hartmann 2008). Although they all rely on specific assumptions about utilities, these studies tend to downplay the role of these informational assumptions and do not argue directly in favor of eliciting utility information in order to improve the voting rule. In these studies the weighted majority rule is considered only for representatives of unequally sized constituencies. Bordley (1985, 1986) came closer to the idea defended in this paper by arguing in favor of weighted majority rule for ordinary voters, with weights derived from specific utility assumptions. However, he also downplayed the role of utility assumptions, instead emphasizing the role of the observed distribution of votes in the definition of weights, focusing on correlations between individual utilities within factions. This paper proposes a more direct study of the idea of making use of utility information, highlighting cases in which weighted majority is the optimal rule. Another related branch of the literature is interested in letting individuals allocate a budget of voting weights or points between several issues and studies the advantages of such games over more rigid voting rules (Casella 2005, Casella et al. 2006, HortalaVallve 2007, Jackson and Sonnenschein 2007). Such complex voting games are not studied here but will be briefly discussed in Section 4. 2 Welfare maximum In this section we provide a simple extension of the wellknown fact that the simple majority rule maximizes the sum of utilities under the assumption of equal intensity across voters. This basic result is a key lemma for our analysis. The framework is as simple as possible and is borrowed from the theory of voting games and of power measurement. 1 There is a fixed population N = {1,..., n}. A simple 1 Excellent syntheses can be found in Mueller (2003) for the general literature on voting and in Felsenthal and Machover (1998) for voting games and power measurement. See also Laruelle and Valenciano 4
5 voting game is a set V of subsets of N called winning coalitions. 2 Concretely, one has to imagine that when a binary decision (yes or no) is on the table, one collects the votes, and if the set of yes voters is a winning coalition, the yes decision wins. 3 To shorten expressions in subscripts, let A i denote A\{i} and A+i denote A {i}. A coalition that is not winning (A / V) is called a losing coalition. The size of a coalition A is denoted #A. ThesimplemajorityruleisassociatedtothesimplevotinggameM such that A M if and only if #A >n/2. A weighted majority rule, with weights (w 1,..., w n ) R n, is associated to the simple voting game W such that A W if and only if P i A w i > P i N w i/2, i.e., the members of A have a majority of the total weights. 4 Obviously, the simple majority rule is a particular case of weighted majority rule, in which w i =1for all i N (one person, one vote). 5 Consider a given binary decision, yes or no. Each voter i N has a utility U + i with the favorable decision for him, and a utility U i with the unfavorable decision for him. Let i = U + i U i. The utility derived by voter i from a particular alternative, say yes, (2008). The paper is selfcontained in the sense that it does not assume prior knowledge of this theory from the reader. 2 In the literature, a simple voting game is required to satisfy restrictions: N V, / V, and if A V and A B, then B V. The class considered in this paper is larger. 3 This description, under the definition of a simple voting game, does not exclude the possibility that the set of no voters is also a winning coalition the procedure as described is biased in favor of the yes decision. A proper voting game would not have disjoint winning coalitions. Unless otherwise indicated, we do not restrict attention to proper voting games, but nothing of substance would be changed if we did. The weighted majority rule examined in this paper is a proper voting game. 4 The weights can be negative, in which case the weighted majority game is not a simple voting game in the usual sense because it may happen that A W, A B and B/ W. 5 A more satisfactory description of these rules would yield a neutral decision (i.e., tossing a coin) instead of a no decision when a vote is obtained. Again, apart from expositional complications nothing of substance would be changed. 5
6 is denoted U i (yes). Suppose we are interested in maximizing social welfare, understood as the sum of individual utilities, P i N U i. This sum need not represent the classical utilitarian criterion because the numbers U i can be interpreted in many ways. The most convenient for our purposes is to understand them as measuring the social value of i s situation for the observer. This makes it possible for the welfare criterion to incorporate a priority for the worstoff as well as nonwelfarist considerations in the measurement of individual wellbeing. Any continuous separable social criterion, welfarist or nonwelfarist, is encompassed in our approach. The term utility is retained only for its simplicity and in order to make the results more easily comparable with those found in the literature. As announced, the following result extends the observation that the simple majority rule maximizes the sum of utilities in the case of equal intensity: the weighted majority rule maximizes the sum of utilities when voters have unequal intensities and their weights are proportional to the utility differences i. It is assumed that each voter votes for the option which is more favorable to him (this assumption will be relaxed in Section 5). The indifferent voters ( i =0) vote in an undetermined manner (they vote either yes or no, or can even abstain), this does not affecttheoutcomesincetheirvoteisnotcountedby the weighted majority rule in the case considered in the proposition. Proposition 1 If the weights w i are proportional to i, the weighted majority rule selects the decision which yields the greater sum of utilities. Proof. Suppose that yes yields a greater sum of utilities: P i N U i(yes) > P i N U i(no). Putting the yes voters on the left side of the inequality and the no voters on the right, one obtains (the yes and no voters may include some indifferent voters, but they can be 6
7 ignored in the formula because for them U i (yes) U i (no) =0): This also reads i N: U i (yes)>u i (no) U i (yes) U i (no) > i N: U i (yes)>u i (no) and by assumption is equivalent to i N: U i (yes)>u i (no) i > w i > i N: U i (yes)<u i (no) i N: U i (yes)<u i (no) i N: U i (yes)<u i (no) which implies that the weighted majority rule will select yes. U i (no) U i (yes). A symmetric argument applies if P i N U i(yes) < P i N U i(no). i w i, 3 Constitutionmaking According to the RaeTaylor theorem (Rae 1969, Taylor 1969), the simple majority rule minimizes the risk that a voter disagrees with the collective decision, over the family of qualified majority rules. 6 As a result, it maximizes the expected utility of a constitutionmaker who does not know in advance on which side of the issue he will be falling, or which voter he will be, on the assumption that all voters have an equal intensity of preferences over the issue. This section is about the fact that if the equal intensity assumption is relaxed, the weighted majority rule is singled out instead of the simple majority rule. A much simpler proof of the RaeTaylor theorem is a byproduct of this analysis. A word on the interpretation of uncertainty is in order here. Most of the literature seems to consider that the constitutionmaker is a real individual who either does not yet 6 Aqualified majority involves a threshold that differs from n/2. This theorem has been generalized by Straffin (1977) to all simple voting games. See also Curtis (1972), Dubey and Shapley (1979). 7
8 know what his interests will be, or decides behind a veil of ignorance which artificially renders him impartial. The former interpretation is not satisfactory because the situation of full uncertainty is never obtained in practice, as one always knows some key aspects of one s present and future preferences. The latter has been contested, because even if choices made behind a veil of ignorance are certainly impartial, they need not be fair in a more substantial sense. Behind a veil of ignorance, one may be willing to take a risk for oneself in order to obtain greater advantages for oneself in other states of nature translated into distributive judgments, this means that minorities can be sacrificed for the sake of greater pleasures of the rest of the population: it is dubious that the tradeoffs in the former context are relevant guidelines for tradeoffs in the latter. A different interpretation of uncertainty is proposed here. What we are after in constitutionmaking is a set of institutions that produce good outcomes on average over the possible cases that may occur. The formalism of uncertainty is adopted for convenience, but really we simply want to have an idea of the average performance of the considered rules. The results below will use the phrase on average while using the formalism of expected value. Consider a single binary decision again. Suppose first that uncertainty is limited to the decision (yes or no) that each voter will favor, whereas U + i,u i and therefore i is already known for every i N. Uncertainty about i is studied in the next section. Assuming that we know the voters intensities without knowing the direction of their preferences corresponds to a special but relevant case, as we often have indications about intensities by objective proxies (location, occupation, financial situation) whereas the direction of preferences may depend on beliefs or political views which are not observable. There is a given probability law over the 2 n possible coalitions of yes voters, with p A denoting the probability of coalition A N. 7 Once a simple voting game V has been 7 This general probabilistic description, which does not assume that voters are independent, is also found in Laruelle and Valenciano (2005). A similar kind of generalization can also be found in Curtis 8
9 picked, one can compute the expected utility EU i of each voter. Let P(e) denote the probability of event e. One has: EU i = U + i P(i agrees with the decision)+u i P(i disagrees with the decision) = i P(i agrees with the decision)+u i. (1) In particular, P(i agrees with the decision) = P(i votes yes and yes wins) +P (i votes no and no wins) = A V: i A p A + A/ V: i/ A p A. (2) We seek a rule that maximizes the sum of utilities on average, i.e., that maximizes E P i N U i. Theansweristhesameasintheprevioussection,forasimplereason:whatever the profile of preferences of the voters, the weighted majority rule selects the alternative which yields the greater sum of utilities. In addition, it is shown that this is essentially the only rule that maximizes the sum of utilities on average: other acceptable rules may differ only on the decision made for ties. Proposition 2 If the weights w i are proportional to i, theweightedmajorityrulemaximizes the sum of utilities on average, over the whole set of simple voting games. Conversely, if p A > 0 for all A N, any simple voting game maximizing the sum of utilities on average is such that every coalition A for which P i A i > P i N i/2 is a winning coalition, and no coalition A for which P i A i < P i N i/2 is a winning coalition. (1972), who restricts attention to symmetric games, seeks to minimize the expected number of disappointed voters, and assumes a general probability distribution over the possible numbers (not the coalitions) of yes voters. 9
10 Proof. By Prop. 1, we know that for each coalition of yes voters, the weighted majority rule selects the option with greater P i N U i. Thereforenosimplevotinggamecanyield agreatere P i N U i. For the converse, let V be any simple voting game. If P i A i > P i N i/2 and A is the coalition of yes voters, then yes is the decision yielding greater total utility. Since p A > 0, one should retain A as a winning coalition for V. Symmetrically, if P i A i < P i N i/2 and A is the coalition of yes voters, then no is the decision yielding greater total utility. One should not retain A as a winning coalition for V. Failure to satisfy any of these two properties reduces total expected utility if p A > 0 for all A N. Note that the argument in the proof applies for any probability law defined over the coalitions of yes voters, which is much more general than the standard formulations of the RaeTaylor theorem in which either equally probable coalitions or independence between voters is assumed. And this argument provides a much simpler way to prove the Rae Taylor theorem than the standard proofs which involve a computation of the probability P(i agrees with the decision) (as in Straffin 1977) or P(i disagrees with the decision) (as in Taylor 1969). In fact, it is also easy to prove the result directly by looking at the probabilities but with very little algebra. Maximizing the expected sum of utilities is equivalent to maximizing the sum of expected utilities, which itself is equivalent, by (1), to maximizing P i N ip(i agrees with the decision). One computes: i P(i agrees with the decision) = i p A + p A i N i N A V: A/ V: i A i/ A = Ã! p A i + Ã! p A i. A V: i A A/ V: i/ A One sees that moving a losing coalition A into V changes the total sum by Ã p A i! i, i A i/ A 10
11 which is beneficial if and only if P i A i > P i/ A i. One should then put into V exactly the coalitions satisfying this condition. This is precisely what W does. Theargumentinthefirst proof above is in fact more powerful than this second one because the former also implies that the weighted majority rule maximizes Ef P i N U i for any arbitrary increasing function f. This means that the weighted majority rule is not only good in terms of average performance, but in every aspect of the distribution of performance. Let us now restrict attention to the uniform law (p A =1/2 n for all A), i.e., the case in which every voter has an equal chance of voting yes or no and votes independently of the others. This particular case is central in the literature on voting power and will enable us to relate the above result to some wellknown results about the Banzhaf measure of voting power. The Banzhaf measure can be defined as the probability that i is critical, i.e. that yes wins if i votes yes and no wins if i votes no. The probability is usually computed under the assumption that all coalitions of yes voters are equally likely. This makes sense if the goal is to measure the structural power of a voter in the game form rather than his effective influence given the likely preference profile of the population. Therefore the Banzhaf measure is simply the number η i of winning coalitions in which i is critical divided by the total number of possible coalitions to which i may belong, 2 n 1. Following standard conventions, let β 0 i denote the Banzhaf measure of power for voter i in a given simple voting game: β 0 i = η i /2 n 1. The sum of these numbers over the population, P i N β0 i, is commonly used as a measure of the sensitivity of the simple voting game to the preferences of the population. A result closely related to the RaeTaylor theorem is that the simple majority rule maximizes sensitivity over the whole set of simple voting games. The relation between the two results comes from the following fact, which states a simple link between the probability for a voter of being critical with the probability of 11
12 his agreeing with the decision. 8 Lemma 1 Under the uniform probability assumption, P(i is critical) =2P(i agrees with the decision) 1. Proof. Let ω denote the number of winning coalitions and ω i the number of winning coalitions containing i. The number of nonwinning coalitions not containing i equals 2 n 1 (ω ω i ) i.e., the number of coalitions not containing i minus the number of winning coalitions not containing i. From equation (2) one computes P(i agrees with the decision) = 1 2 n ωi +2 n 1 (ω ω i ). The number of coalitions in which i is critical, η i, equals the number of winning coalitions containing i minus the number of winning coalitions in which i is not critical. The latter number equals the number of winning coalitions not containing i (adding i to any such coalition gives a winning coalition in which i is not critical). Therefore one has: P(i is critical) = 1 2 n 1 [ω i (ω ω i )]. Simple algebraic manipulation yields the desired relation between the two probabilities. The result about majority rule and sensitivity P i N β0 i can therefore be extended: the weighted majority rule maximizes P i N iβ 0 i and is almost characterized by this property. This result is not particularly interesting in itself because P i N iβ 0 i is not an immediately intuitive objective, but it will be extended and explained in the comment that follows. 9 8 It is attributed to Penrose (1946) by Felsenthal and Machover (1998). See also Brams and Lake (1978), Straffin (1978), Dubey and Shapley (1979) and Laruelle et al. (2006) for further developments. 9 This result is also mentioned and proved in Beisbart and Bovens (2007, Theorem 1). 12
13 Proposition 3 Under the uniform probability assumption, if the weights w i are proportional to i, the weighted majority rule maximizes P i N iβ 0 i overthewholesetofsimple voting games. Conversely, any simple voting game maximizing P i N iβ 0 i is such that every coalition A for which P i A i > P i N i/2 is a winning coalition, and no coalition A for which P i A i < P i N i/2 is a winning coalition. Proof. This is an immediate corollary of Prop. 2, in view of Lemma 1. Here is another, direct, proof. Let 1[e] denote the function which equals 1 if e is true and 0 otherwise. In the proof of Lemma 1 it is shown that η i =2ω i ω. Onetherefore has i η i = 2 i ω i i ω i N i N i N = 2 i 1[i A] i 1 i N A V: i N A V = Ã 2 i! i. A V i A i N In order to maximize this sum, one would ideally retain A if 2 P i A i P i N i > 0 and exclude A if 2 P i A i P i N i < 0, as done by W. The validity of Proposition 3 cannot be extended much beyond the case of uniform probability. It is possible to generalize the Banzhaf measure to any given probability law over the formation of coalitions 10 but the above result does not typically hold for this generalized measure of power. It is, however, possible to extend Proposition 3 to the 10 As proposed in Laruelle and Valenciano (2005). 13
14 general case of probability p if sensitivity is redefined as σ = i N i [P(i agrees) P(i disagrees)] = i i N = A V: A V: i A p A Ã i A p A + A/ V: i/ A i i/ A p A A V: i/ A! i A/ V: p A A/ V: i A p A Ã i A p A i i/ A i!. Indeed, EU i = U + i P(i agrees with the decision)+u i P(i disagrees with the decision) = i P(i agrees with the decision)+u i = U + i i P(i disagrees with the decision), so that 2EU i = U + i + U i + i [P(i agrees) P(i disagrees)]. From this equation it is clear that Proposition 3 holds in the general case when sensitivity is redefined as suggested. In the uniform probability case, from Lemma 1 one has P(i agrees) P(i disagrees) =P(i is critical), implying that the two definitions of sensitivity are then identical. 11 But in the general case it appears that the new notion σ is superior. Indeed, what is important, if one wants the decision to be sensitive to the voters preferences, is not so much that they should have a maximal probability of being critical, but that they should have a maximal expected 11 Laruelle et al. (2006), who argue that agreeing with the decision is more important than being critical, show that this equality holds for all voters and all voting games only inthecaseofuniform probability. Curtis (1972) and Brams and Lake (1978) study the average value of P(i agrees) over all voters, which is linearly related to the measure of sensitivity proposed here. 14
15 utility, which depends on the probability of agreeing, or equivalently, on the gap between the probability of agreeing and the probability of disagreeing. Whether they agree by being critical or not is not so relevant. This notion of sensitivity will turn out to be relevant in the analysis of representative democracy in Section 7. 4 Multiple issues and uncertainty about stakes In the previous section, we have considered a single binary decision and assumed that the vector of stakes ( 1,..., n ) is known. If one extends the veil of ignorance and assumes that many binary decisions k = 1,...,m will be taken in a sequence, with additively separable utilities over the various decisions (i.e., U i = U 1 i +...+U m i ) and a priori unknown ( k 1,..., k n) for each specific decision k, the results are untouched if one assumes that the simple voting game can be adapted to each decision k and that ( k 1,..., k n) is disclosed in due time before each decision. In this case taking the weighted majority rule with weights proportional to ( k 1,..., k n) for each decision k is optimal. More interesting is the case when the same simple voting game has to be kept throughout the whole sequence of future decisions. One must be precise about the probabilistic setting. To begin with, let us assume that for each issue k, the value of the pair (Ui k,ui k+ ) is drawn from a probability law that may be specific toi and k, independently of the values for other issues. The same probability law p defines the probability of coalitions of yes voters over every issue (the probability of i voting yes or no over some issue is assumed to be independent of the values of Ui k,ui k+ and of his vote on other issues). Let E U denote expectation with respect to possible values of utilities, and E A 15
16 denote expectation with respect to possible coalitions over all issues. One computes m E U E A U i = E U E A Ui k = k=1 m k=1 E U k i P(i agrees with decision k)+ui k. Because the same simple voting game is operating for all k and the same probability law governs the formation of coalitions over all issues, P(i agrees with decision k) is a constant P(i agrees with decision), so the formula simplifies into m m E U E A U i = EU k i P(i agrees with decision)+ E U Ui k. k=1 Similarity with the formulae relative to Proposition 2 implies that the weighted majority rule with weights proportional to P m P k=1 EU k i for i =1,..., n maximizes E i N U i.in other words, voter i s weight should be proportional to the expected value of the sum of hisstakesoverallissues. k=1 Proposition 4 If the weights w i are proportional to P m k=1 EU k i, the weighted majority rule maximizes the sum of utilities on average, over the whole set of simple voting games. Conversely, if p A > 0 for all A N, any simple voting game maximizing the sum of utilities on average is such that every coalition A for which P P m P P i A k=1 EU k i > m i N k=1 EU k i /2 is a winning coalition, and no coalition A for which P P m P P i A k=1 EU k i < m i N k=1 EU k i /2 is a winning coalition. This result covers the following two subcases: Subcase 1: The k i are known but the same simple voting game must be retained for all k in this subcase the average value of k i over all values of k defines a suitable weigthing scheme. Subcase 2: There is only one issue but uncertainty prevails about i in this subcase theexpectedvalueof i defines a suitable weighting scheme. 16
17 A situation that is likely to occur in many applications is that a clear separation will be observed between two subpopulations. In one subpopulation, the individuals will have high and roughly equal values of average expected stake, whereas this value is very low for the members of the other subpopulation. Such a situation may be observed in particular for the issues affecting a local population (e.g., inhabitants of a town or any particular geographical district), the outsiders being largely unaffected on average by such issues. In this case a reasonable application of the above result is to restrict voting rights on such local issues to the local population, and grant them equal weights. A generalization to the case when each issue k has a specific probability distribution p k over coalitions is possible but one no longer obtains an ordinary weighted majority rule. One then computes m E U k i P(i agrees with decision k) = k=1 m E U k i k=1 = A V m k=1 p k A A V: i A p k A + A/ V: i/ A i A E U k i + A/ V p k A m p k A k=1 i/ A E U k i Moving a losing coalition A into V changes this expression by the amount Ã m p k A E U k i! E U k i = m p k AE U k i m p k AE U k i. k=1 i A i/ A i A k=1 i/ A k=1 We should therefore make A a winning coalition if this term is positive, and a losing coalition if it is negative. This yields a voting game in which the weight of a voter does not depend on the issue itself but depends on the coalition of yes voters 12 for the current issue: when this coalition is A, voter i s weight is P m k=1 pk A E U k i. This means that voter i s weight is greater, other things equal, when A is more likely to be the coalition of yes voters over issues for which his expected stakes are greater Voting rules with weights depending on the realized votes are mentioned in particular in Bordley (1986) and Barbera and Jackson (2006). 13 One may raise the second best question of what the optimal weights would be for weighted majority 17
18 When the distribution of utilities is not independent on the distribution of coalitions of yes voters, one obtains a similar result in which voter i s weight is P m k=1 pk AE U/A k i, where E U/A denotes the expected value conditional on A being the coalition of yes voters. Proposition 5 Inthefullygeneralcaseinwhicheachissuehasaspecific probability distribution p k and uncertain utilities for an issue may depend on the realized coalition, the following voting game maximizes the sum of utilities on average, over the whole set of simple voting games: a coalition is winning if and only if wi A > wi A /2, i A i N where wi A = P m k=1 pk AE U/A k i. Conversely, any simple voting game maximizing the sum of utilities on average is such that every coalition A for which P i A wa i > P i N wa i /2 is a winning coalition, and no coalition A for which P i A wa i < P i N wa i /2 is a winning coalition. Contrary to Proposition 2, Propositions 4 and 5 cannot be generalized to the maximization of Ef P i N U P i because there is no voting game that maximizes i N U i for every configuration of utilities. The situation is even bleak in some configurations of utilities because the constraint to have the same game over all issues can generate bad decisions. Consider a threeagent, twoissue situation in which the k i areasinthefollowing table (a positive number means the agent votes yes, and vice versa). 1 i 2 i Ann +31 Bob Chris 11 rule in this context. Simulations done by the author suggest that, although weights proportional to P m k=1 E U k i are no longer optimal in general, they are typically close to optimal weights. This issue is left for future research. 18
19 In this configuration of utilities, the optimal decision (maximizing the sum of utilities) is yes for both issues. But if the probability distribution of coalitions is the same for both issues, it is optimal to use a weighted majority rule, and by symmetry Ann and Bob must get the same weight. The consequence is that they neutralize each other over every issue and no wins in both decisions. It is possible to obtain a yes decision over one issue by giving unequal weights to Ann and Bob (in violation of impartiality), but it is impossible to obtain a yes decision over the two issues. This kind of example highlights the importance of defining specific weightsforeach issue separately. Interestingly, in some cases the specific weightscan bemadetorelyon voters private information, as shown in the literature on more complex voting games in which voters can allocate points between issues (e.g., Casella 2005, Jackson and Sonnenschein2007). IfAnnandBobwereassignedfourpointsandChristwopoints,Annand Bob could secure a yes vote on their most important issue by putting at least three of their pointsonit.toachievethisoutcomethedesigneroftherulemustknowthatann sand Bob sstakesareonaveragegreaterthanchris sbutneednotknowwhichissueismore important for either of them. More generally, granting a number of points proportional P to E m U k=1 k i to each i and letting i allocate this quantity across issues may be an interesting way of improving upon the simple voting games considered in this paper. 5 Erratic voters Now let us introduce the complication that some voters may vote against their interests as measured by U i. This may be due to a discrepancy between U i and their preferences (e.g., if U i is a nonwelfarist measure), or to an informational problem that gives them mistaken beliefs about their interests. We first consider the simple setting of Section 2 and assume that N is partitioned 19
20 into N p (the prointerest voters who always vote in their interest) and N c (the contrainterest voters who always vote against their interest). Now, maximizing the sum of utilities requires giving the latter a negative weight. This result has no practical interest, if only because negative weights make the voting rule highly manipulable. But it is useful as a theoretical clarification and as a preparation for the next section. Proposition 6 If the weights w i are proportional to i for i N p and to i for i N c, the weighted majority rule selects the decision which yields the greater sum of utilities. Proof. Suppose P i N U i(yes) > P i N U i(no). Putting the yes voters on the left side of theinequalityandthenovotersontheright,asintheproofofprop.1,nowyields i N p : U i (yes)>u i (no) i i N c : U i (yes)<u i (no) i > i N p : U i (yes)<u i (no) i i N c : U i (yes)>u i (no) i. If we turn to the framework of Section 3, we may assume that there is a probability law such that p A,B is the probability that A is the coalition of voters whose interest is yes and B is the coalition of yes voters. An agent votes in his interest ( pro ) if i A B or i N \ (A B). We will say that voter i is steady when the following conditional probability is the same for all B N : P(i is pro B vote yes) = A N p A,B i A A N i/ A A N p A,B A N p A,B p A,B if i B if i/ B. The next result defines optimal weights which are jointly proportional to the stakes and to an index of reliability of the voter. This index, equal to 2P(i pro) 1, goes from 1 for a voter who consistently votes in his interest to 1 foravoterwhoalwaysvotes against his interest. 14 It is zero for a voter who has an equal chance of voting pro or 14 There is some similarity between this result and the Condorcet jury theorem where badly informed members of the jury should have a negative weight (BenYashar and Nitzan 1997). 20
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