Maximin Choice of Voting Rules for Committees
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Danilo Coelho∗,+ This version: July 12, 2003. ∗
Departament d’Economia i d’Hist` oria Econ`omica, Universitat Aut` onoma de Barcelona,
08193 Bellaterra, Espa˜ na. (email:
[email protected]) +
Diretoria de Estudos Sociais, Instituto de Pesquisa Econˆ omica Aplicada, Bras´ilia - DF,
70076-900, Brasil. (email:
[email protected])
Abstract. In the context of a probabilistic voting model, we investigate the consequences of choosing among voting rules according to the maximin criterion. We characterize the rules that satisfy the maximin criterion as a function of the distribution of voters’ probabilities to favor change. We prove that there are at most two maximin rules, at least one is Pareto efficient and often different to simple majority rule. If a committee is formed only by “conservative voters”, i.e. voters who are more likely to prefer the status quo than change, then the maximin criterion recommends voting rules that are not higher than the simple majority rule. If there are only “radical voters”, then this criterion recommends rules that are not lower than the 50% majority. Key words: maximin; voting; majority; committee. JEL classification: D71
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I am grateful to Salvador Barber`a, my adviser, for the various comments and suggestions that helped
improve this paper. I would also like to thank seminar audience at IPEA, at UAB Microworkshop and at 2003 Annual Meeting of Public Choice Society. Carmen Bevi´a, Mirko Cardinale, Joan Esteban, Wioletta Dziuda, Bernard Grofman, Matthew Jackson, Raul Lopez, Jordi Mass´ o, Hugh Mullan, Ana Pires do Prado and Arnold Urken provided helpful comments. Finally, I also acknowledge the financial support from Capes, Brazilian Ministry of Education and Spanish Ministry of Science and Technology (Project BEC2002-02130).
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1. Introduction The members of a committee, with conflicting interests, have to decide about the voting rule to be used in a series of dichotomous choices involving the rejection or the adoption of proposed changes from the status quo. There is uncertainty about the future proposals that will be voted on. However, they are able to form expectations about the proposals and about the behavior of other members. Based on these expectations, they form preferences over voting rules that best satisfy their own interests. The debate about which voting system should be adopted by the European Union Council of Ministers, with the arrival of new members, is an example of where this theoretical framework could be applied. In this context the voters might have different preferences over voting rules. We explore the idea that voters tend to agree on voting rules that generate patterns of outcomes that are considered reasonable by all of them. In other words, voting rules are selected according to principles such as fairness or Pareto efficiency. We analyse the endogenous choice of voting rules in a probabilistic voting model first proposed by Rae (1969). We consider the choice of voting rules according to the Rawls’s maximin criterion, with the suggestion that fairness considerations may recommend the choice of a rule that maximizes the expected utility of the worst off individual. In this model, a voting rule is characterized by the minimum number of votes needed to adopt a proposal. Each voter is characterized by a probability of being in favor of the status quo and the action each voter takes on any given proposal is completely independent of the action taken by others. Each voter casts only one vote and he gets utility 1 if his preferred alternative is chosen in the vote, and utility 0 otherwise. Given this setup, the voter’s expected utility over voting rules is the frequency, generated by the rule, with which his opinion about proposed changes of status quo coincides with the decisions taken by the committee. As part of our analysis of majoritarian rules that satisfies the maximin criterion, we are led to investigate who are the worst off voters in a committee. We show that there is an endogenous threshold rule such that for any rule lower than it the most conservative among the members of the committee is the worst off voter. And for rules higher than this threshold the reversal holds, i.e. the worst off voter is the most radical voter. After investigating the relationship between the distribution of well being across voters 2
and voting rules, we are able to prove that there are at most two voting rules that satisfy the maximin criterion, at least one is Pareto efficient and it is often different to simple majority rule. Indeed, if a committee is formed only by “conservative voters”, i.e. voters who are more likely prefer the status quo than change, then the maximin criterion recommends voting rules that are not higher than the simple majority rule. If there are only “radical voters”, then this criterion recommends rules that are not lower than the 50% majority. Early proponents of this model concentrated on the utilitarian perspective, which always recommends the choice of simple majority rule (Rae (1969) and Curtis (1972)). In a similar model, Guttman (1998) uses Harsanyi’s construction of veil of ignorance to justify the use of this criterion in the choice of voting rules. He assumes that each voter would participate in the choice of the voting rule with no knowledge whatsoever of whether his expected utility over voting rules will be that of voter 1, voter 2, etc. Each possibility is equally likely. Under this assumption, he proves that the optimal voting rule for any voter is the one that maximizes the sum of voters’ expected utilities (see Buchanan (1998), Tullock (1998) and Arrow (1998)), and this is simple majority rule in our context. By contrast, Badger (1972) and Barber`a and Jackson (2002) have discussed the choice of voting rules by means of a vote. Badger (1972) shows that the Condorcet Winner always exists proving that each voter’s preference over voting rules is single peaked. While Barber`a and Jackson (2002) claim that a voting rule is likely to persist in a group if it cannot be defeated by any other alternative rule if it itself is used to choose between the rules. They call this property self stability. They argue that the possibility of a lack of self-stable voting rules could be an explanation why most states’ Constitutions require super majorities in order to change the voting system used on day-to-day decisions. Other authors have different lines of argument to defend the choice of some voting rule over another.2 The rules used by a society to make constitutional choices are called meta rules by Brennan and Buchanan (1985). They argue that any rule is “ just ” if it is chosen by a society using an agreed meta rule. Moreover, they claim that the use of unanimity criterion to choose between rules is feasible because consensus about the choice of rules can be easily reached: “... The uncertainty introduced in any choice among rules 2
See Grofman(1979) and Esteban and Ray (2001).
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or institutions serves the salutary function of making potential agreement more rather than less likely....(Brennan and Buchanan (1985), pp. 29)”. The outline of this paper is as follows: In section 2, we describe the model. In section 3, we present some properties of individual preferences over voting rules, proved in Badger (1972), Barber`a and Jackson (2002) and Rae (1969), that help us to present our results. In section 4, we show our results about the properties of the voting rules that satisfy the maximin criteria. Finally, in section 5, we close with some final remarks. 2. The model Let’s represent the set of voters by N = {1, ..., n}. We shall assume that N is finite and n ≥ 3. The voters, with conflicting interests, have to decide about the voting rule to be used in a series of dichotomous choices involving the rejection or the adoption of proposed changes from the status quo. Each voter casts a vote in {yes, no}. Voting for “yes” is interpreted as being in favor of the proposed change from the status quo. Voting for “no” is interpreted as being against the change. A voting rule is characterized by a number s ∈ {1, ..., n}. The proposal for change is adopted if there are at least s voters in favor of it. The voters have expectations over future issues that will be voted on, but do not know their exact realization. The voters are simply characterized by a parameter pi ∈ (0, 1). This represents the probability that they will support change at the time of the vote. The realizations of voters’ support for the alternatives are independent. Badger (1972) offers a convincing justification for this assumption: “We shall also make the admittedly highly unrealistic assumption that the action each legislator takes on any given proposal is completely independent of the action taken by others. This eliminates the consideration of factional disputes, logrolling, and the entire gamut of political and historical dynamics which are basic to the evolution of any real legislative structure. But then we shall not attempt to analyse such structures. By eliminating “interactive” political dynamics entirely, we hope to get a much narrower yet somewhat clearer view of the relationship between an individual legislative will and optimal collective policy.( Badger (1972), pp.35 )”
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A voter gets utility 1 if his preferred alternative is chosen in the vote, and utility 0 otherwise. The committee is described by a set of voters N = {1, ..., n} and a vector p = (p1 , ..., pn ). For any k ∈ {1, ...., n − 1}, let Pi (k) denote the probability that exactly k individuals in N\ {i} support change. P Pi (k) = ×j∈B pj ×l∈B / (1 − pl ).
(1)
B⊂N \ {i}:|B|=k
Let Ui (s) be the expected utility of voter i if voting rule s is used. This is expressed
as follows. Ui (s) = pi
n−1 P
Pi (k) + (1 − pi )
k=s−1
s−1 P
Pi (k).3
(2)
k=0
For instance, Ui (s) can be interpreted as the frequency, generated by the rule s, with which voter i expects to support a proposal and have it adopted and to oppose a proposal and have it defeated. Example 1 Let C = (N = {1, 2, 3, 4, 5, 6, 7}, p1 = 0.9, p2 = 0.8, p3 = 0.7, p4 = 0.6, p5 = 0.5, p6 = 0.4, p7 = 0.1) be a representation of a committee. The expected utilities over voting rules of all the voters are illustrated in Figure 1.4 Expected Utility Over Voting Rules P=( 0.9, 0.8, 0.7, 0.6,0.5,0.4,0.1 )
1 0,9 0,8 0,7
p1=0.9 p2=0.8
0,6 U(s)
p3=0.7 0,5
p4=0.6 p5=0.5
0,4
p6=0.4 p7=0.1
0,3 0,2 0,1 0 s=1
s=2
s=3
s=4
s=5
s=6
s=7
Figure 1 3 4
Notice that expression (2) implies that: if pi = pj then Ui (s) = Uj (s) for any s ∈ {1, ..n}. The author has written a program in Matlab that computes the voters’ expected utilities over voting
rules. This program is available upon request.
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Example 1 has provided instances where the distribution of well being across voters in terms of expected utilities depends drastically on the voting rule. In this context, consensus over the choice of a voting rule to be adopted by the committee is very difficult. We explore the idea that voters tend to agree on voting rules that generate patterns of outcomes that are considered reasonable by all of them. We consider the possibility of choosing among rules according to the maximin criterion, with the suggestion that fairness considerations may recommend the choice of a rule that maximizes the expected utility of the worst off voter. Definition 1 A voting rule s ∈ {1, ..., n} satisfies the maximin criterion if
Min{U1 (s), ..., Un (s)} ≥ Min{U1 (s0 ), ..., Un (s0 )} for every s0 ∈ {1, ...., n}. We denote
by Smax i min the set of voting rules that satisfies the maximin criterion in a committee. A well known result proved by Rae (1969) and Curtis (1972) which can be found in any texbook of public choice theory tells us that simple majority rule maximizes the sum of voters’ expected utility (see for example Muller (1989) pp.100). In Example 1, Smax i min = {5} which is more demanding for change than simple majority rule. Next we provide two other examples where maximin choice of a rule is again different than the utilitarian choice. These two examples anticipate the results in Corollary 3 and Theorem 5. Example 2 (A committee formed only by “conservative voters”, i.e. pi ≤ 0.5 for any i ∈ N) Let C = (N = {1, ..., 9}, p1 = p2 = p3 = p4 = p5 = p6 = 0.2, p7 = p8 = p9 = 0.1) be a committee. Direct calculation using expression (1) and (2) leads to Smax i min = {4} .(see figure 2 below)
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N={1,2,3,4,5,6,7,8,9} and P=(0.2 ,0.2,0.2,0.2,0.2,0.2,0.1,0.1,0.1) expected utility U(s)
1,0 0,8 0,6 0,4 0,2
s=1
s=2
s=3
s=4
s=5
s=6
s=7
s=8
s=9
voters with p=0.2 : U(s) 0,391103 0,645907 0,776406 0,803652 0,802394 0,800485 0,800051 0,800003 0,800000 voter with p=0.1 : U(s) 0,291103 0,598991 0,807641 0,883466 0,898694 0,900075 0,900026 0,900002 0,900000
Figure 2 Example 3 (A committee formed only by “radical voters”, i.e. pi ≥ 0.5 for any i ∈ N) Let C = (N = {1, ..., 7}, p1 = p2 = p3 = 0.6, p4 = p5 = p6 = p7 = 0.95) be a committee. Direct calculation leads to Smax i min = {5} (see figure 3 below).
expected utility U (s )
N={1,2,3,4,5,6,7} and P=(0.95,0.95,0.95,0.95,0.6,0.6,0.6) 1,0 0,8 0,6 0,4 0,2
s=1
s=2
s=3
s=4
s=5
s=6
s=7
voters w ith p = 0.95 : U(s)
0,9500004
0,9500174
0,9500885
0,9447053
0,8679358
0,5963194
0,2259334
voters w ith p = 0.6 : U(s)
0,6000004
0,6000314
0,6009425
0,6131468
0,6796848
0,7302609
0,5759334
Figure 3
3. Individual preferences over voting rules Before presenting the characterization of the voting rules that satisfy the maximin criterion, we first discuss some properties of individual preferences over voting rules, as proved in Rae (1969), Badger (1972) and Barber`a and Jackson (2002). Definition 2 Ui is single peaked if there exists sb ∈ {1, ..., n} with Ui (b s) ≥ Ui (s) for all s ∈ {1, ..., n} such that Ui (s) > Ui (s − 1) for any sb > s > 1 and Ui (s − 1) > Ui (s) for any
n ≥ s > sb.
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The definition above implies that if Ui is single peaked then there are at most two voting rules that maximize voter i’s expected utility and these two voting rules are necessarily adjacent. Following the literature, the correct name for this property should be single-plateaued preferences. However, Badger (1972) and Barber`a and Jackson (2002) adopted the term single-peaked since, in this model, such indifference can only occur between two adjacent rules on top and happens non-generically (in p). The following theorem was proved in Badger (1972). Theorem 1 (Badger (1972)) For any committee, every voter’s preferences over voting rules are single-peaked. Let sbi denote the peak of voter i, i.e. the voting rule that maximizes the voter i’s
expected utility. In the case where a voter has twin-peaks, i.e. for example both s∗ and s∗ − 1 maximize Ui (·), the definition above selects the higher of the two peaks so in this
case sbi = s∗ . In Example 1, we have that sb1 = 2, sb2 = 3, sb3 = 3, sb4 = 4, sb5 = 5, sb6 = 5
and sb7 = 7.
Proposition 1 (Barber`a and Jackson (2002)) For any i, j ∈ N, sbj ≥ sbi whenever pi ≥ pj . In words, if voter i expects to support proposals more often than voter j, then voter
i’s preferred voting rule, sbi , cannot be larger than sbj . In Example 1, sb1 ≥ sb2 ≥ ... ≥ sb7 . Definition 3 The simple majority rule is defined as s =
(n+1) 2
if n is odd and
n 2
+ 1 if n
is even. Let us denote by smaj the simple majority rule. Theorem 2 (Barber`a and Jackson (2002)) There exist i, j ∈ N such that sbj ≥ smaj ≥ sbi .
The theorem above tells us that for any committee there exist at least one voter
that prefers a voting rule smaller or equal than the simple majority rule and there exist another voter that prefers a voting rule higher or equal than it. In Example 1, smaj = 4 and 2 = sb1 ≥ smaj ≥ sb7 = 7.
Take the voters characterized by the highest p (smallest p) in the committee and select
only one of them to call voter R (voter C ). Thus voter R (voter C ) is the voter that has
the highest probability to support (reject) a proposal of change at the time of the vote. Let sbR (b sC ) denote the peak of voter R (voter C ). Thus, in example 1, voter 1 with 8
p1 = 0.9 is the voter R since he has the highest p, while voter 7 with p7 = 0.1 is the voter C. Below we present two easy corollaries of Theorem 2 above. Corollary 1 For any committee, sbC ≥ smaj ≥ sbR .
Definition 4 A committee is homogeneous if pi = pj for every i, j ∈ N. Corollary 2 (Rae 1969) If a committee is homogeneous then sbi = smaj for every i, j ∈ N. If in addition n is even then sbi = smaj is a twin peak.5
Corollary 1 above tell us that the optimal voting rule for the voter with the smallest p
(highest p) in a committee will be not smaller (not higher) than the simple majority rule. While the second corollary, proved by Rae (1969), says that the simple majority rule is unanimously the preferred voting rule if the committee is homogeneous.
4. Maximin results 4.1. Characterization of voting rules that satisfy the maximin criterion We are now ready to characterize the rules that satisfy maximin criterion as a function of the distribution of voters probabilities to favor changes. We begin this presentation analysing a homogeneous committee. Theorem 3 For any homogeneous committee, if n is odd then Smax i min = {smaj } otherwise Smax i min =
©n
ª maj , s . 2
In words, if a committee is homogeneous then the maximin choice of voting rules coincides with the utilitarian choice. Proof of Theorem 3.
Recall that expression (2) implies that: if pi = pj then
Ui (s) = Uj (s) for any s ∈ {1, ..., n}. Hence, in a homogeneous committee the voting rules that satisfy the maximin criterion are those that maximize the voters’ expected utilities. But then it follows by Corollary 2 (Rae 1969) that if a committee is homogeneous then 5
Schofield (1972) proved that the marginal advantage of simple majority over any other voting rule
does become vanishingly small as the size of the committee increases.
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sbi = smaj for every i, j ∈ N. If in addition n is even then sbi = smaj is a twin peak. Therefore the proof of Theorem 3 is established since sbi is the largest of the two peaks.
We now proceed, by showing who are the worst off voters in a heterogeneous committee
(Proposition 2 and Claim 1).6 Proposition 2 For any heterogeneous committee there is a sRC ∈ {2, .., n}, such that for all j ∈ N, Uj (s) ≥ UC (s) whenever s < sRC and Uj (s) ≥ UR (s) whenever s ≥ sRC . Proposition 2 says that there is a threshold rule such that for any rule lower than it the most conservative among the members of the committee (every i ∈ N such that pi = pC ) is the worst off voter. And for rules higher than this threshold the reversal holds, i.e. the worst off voter is the most radical voter (every i ∈ N such that pi = pR ). Figure 1 ilustrates Proposition 2. Observe that sRC = 5. Moreover, Min{U1 (s), U2 (s), ..., U7 (s)} = U7 (s) for any s < sRC and Min{U1 (s), U2 (s), ..., U7 (s)} = U1 (s) for any s ≥ sRC . For any k ∈ {0, ...., n − 2}, denote by PR, C (k) the probability that exactly k of the voters other than voter C and voter R support a proposal; that is, P PR, C (k) = ×j∈B pj ×l∈B / (1 − pl ) B⊂N \ {R,C}:|B|=k
After some algebraic manipulations of expression (2), it can be shown that the differ-
ence between UR (s) and UC (s) can be expressed as follows:7 s−2 P UR (s) − UC (s) = (pR − pC )(1 − 2 PR, C (k)) f or any s ∈ {2, ..., n} k=0
UR (1) − UC (1) = (pR − pC ).
The information above helps us, for a given committee, to identify among all possible voting rules which one is the voting rule that plays the role of sRC in Proposition 2 above.8 s−2 P For any s ∈ {1, ..., n}, let GR,C (s) ≡ PR, C (k) whenever s ∈ {2, ..., n} and let k=0
GR,C (s) ≡ 0 whenever s = 1. 6
A committte is heterogenous if it is not homogenous (see Definiton 4). The proof of this expression can be found in the Appendix, Lemma 5. 8 Recall the assumption that pi ∈ (0, 1) for any i ∈ N. Thus, PR, C (k) > 0 for any k ∈ {0, ..., n − 2} s−2 P and PR, C (k) is a strict increasing function on s. 7
k=0
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Claim 1 {sRC } = {s0 ∈ {2, ..., n}|GR,C (s0 ) ≥ 1/2 and GR,C (s) < 1/2 f or any s < s0 }. Recall that sRC is only defined for heterogeneous committees. Its existence is guaranteed because GR,C (n) = 1, GR,C (1) = 0 and GR,C (·) is a strict increasing function in s. Notice that in a heterogeneous committee only sRC and sRC − 1 can minimize the difference between UR (s) and UC (s). In order to prove and give a intuition of Proposition 2 and Claim 1, we need first to study the relationship between the distribution of well being across voters (in terms of expected utilities) and voting rules. This task is done in the next subsection. In subsection 4.2, we prove that for any heterogeneous committee, voters preferences over voting rules satisfy the p-monotonic single crossing property and Proposition 2 and Claim 1 are direct consequences of this property. Before it, let us proceed to show our characterization of voting rules that satisfy the maximin criterion. Definition 5 A voting rule s ∈ {1, ..., n} is Pareto efficient if there is no other voting rule s0 ∈ {1, ..., n} such that Ui (s0 ) ≥ Ui (s) for all i ∈ N and Uj (s0 ) > Uj (s) for some
j ∈ N. Claim 2 If s ∈ {1, ..., n} is Pareto efficient then sbR − 1 ≤ s ≤ sbC .
Proof of Claim 2. It follows by single peakedness and Corollary 1. Notice that sbC is Pareto inefficient if and only if sbC is a twin-peak and sbR 6= sbC .
Moreover sbR − 1 is Pareto efficient if and only if sbR is a twin-peak and the committee is
homogeneous.
Now we are ready to present our main theorem that provides a characterization of the
voting rules that satisfy the maximin criterion in heterogeneous committees. Theorem 4 For any heterogeneous committee, Smax min ⊆ {sR,C − 1, sR,C } whenever sR,C is Pareto efficient.
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Smax i min ⊆ {b sC − 1, sbC } whenever sR,C is Pareto inefficient and larger than smaj .9
Smax i min ⊆ {b sR − 1, sbR } whenever sR,C is Pareto inefficient and smaller than smaj .10
Note that, in this case, Smax i min = {b sC − 1, sbC } only if sbC is a twin-peak, otherwise Smax i min =
{b sC } . 10 Note that, in this case, Smax i min = {b sR − 1, sbR } only if sbR is a twin-peak, otherwise Smax i min = {b sR } .
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The proof of Theorem 4 is in the appendix. Figure 4 below helps us to understand Theorem 4. In this figure we describe again the committee of Example 1. Notice that the minimum voters’ expected utility under voting rule s = 5 is just slightly higher than the one under voting rule s = 4. If it were equal, it would be an example of a committee that has tow different voting rules that satisfies the maximin criterion. Figure 5 illustrates a committee in which sR,C is Pareto inefficient. While Figure 6 illustrates a committee in which sR,C is Pareto efficient.
Expected Utility Over Voting Rules P=( 0.9, 0.8, 0.7, 0.6,0.5,0.4,0.1 ) 1 0,9 0,8 0,7
U(s )
0,6
Voter R(p1=0.9)
0,5
Voter C(p7=0.1)
0,4
Min{U1(s),...,U7(s)}
0,3 0,2 0,1 0 s=1
s=2
s=3
s=4
s=5
s=6
s=7
Figure 4: sR,C = 5 and Smax i min = {sR,C } N={1,2,3,4} and p1>p2>p3>p4
12 10 voter 1
U(s)
8
voter 2
6
voter 3
4
voter 4
2 0 1
2
3
4
s
Figure 5: Smax i min = {ˆ sC }, sR,C = 4. Figure 6: Smax i min = {sR,C }, sR,C = 3. Next we present a direct corollary of Theorem 4. Corollary 3 For any committee, there are at most two voting rules that satisfy the maximin criterion and at least one is Pareto efficient.
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Proposition 3 For any committee, Min{U1 (s), U2 (s), ..., Un (s)} is single peaked.11 Proof of Proposition 3. It follows by single peakedness of Ui , Theorem 3, Proposition 2 and Theorem 4. Figures 4, 5 and 6 above give a good intuition why the function Min{U1 (s), U2 (s), ..., Un (s)} is single peaked. Theorem 5 For any committee and any s ∈ Smax i min we have that smaj − 1 ≤ s ≤ sbC whenever pi ≥
sbR − 1 ≤ s ≤ smaj whenever pi ≤
1 2 1 2
for every i ∈ N. for every i ∈ N.
The proof of Theorem 5 is in the appendix. This theorem states that if a committee
is formed only by “conservative voters”, i.e. pi ≤ 0.5 for any i ∈ N , then the maximin criterion recommends voting rules that are not higher than the simple majority rule and not lower than the optimal voting rule of the less conservative among these voters. If a committee is formed only by “radical voters”, then the maximin criterion recommends voting rules that are not lower than the fifty percent majority and not higher than the optimal voting rule of the less radical among these voters. The conditions of Theorem 5 are satisfied in the committees of Examples 2 and 3. In Example 2, all nine voters of the committee are characterized by p’s smaller than 0.5, Smax i min = {4} and smaj = 5. While
in Example 3 all seven voters have p’s higher than 0.5 and Smax i min = {5} and smaj = 4. 4.2. Distribution of well being across voters and voting rules We now present additional properties of the distribution of well being across voters
for different voting rules. At the end of this subsection the proofs of Proposition 2 and Claim 1 are provided. Even before we reach the end, it will be clear that Proposition 2 and Claim 1 presented above are a direct consequences of the properties presented below. Definition 6 A committee has preferences over voting rules that satisfy the strict single crossing property if for any pi > pj , there is a si,j ∈ {2, .., n} such that ·Ui (s) > Uj (s) f or any s < si,j .
·Ui (s) ≤ Uj (s) f or s = si,j . ·Ui (s) < Uj (s) f or any s > si,j . 11
See Definiton 2.
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Definition 6 tells us that if a committee has preferences that satisfy the strict single crossing property, then for any pair of voters, with different expectations, there is a threshold majority size si,j ∈ {2, ..., n} such that, for any majority size higher or equal to it, the one that rejects proposals more often has a higher utility than the other, and for any majority size below it, the reverse holds. In other words, the expected utility curves of any pair of voters that have different probabilities to support proposals cross only once. Theorem 6 Every committee has preferences over voting rules that satisfy the strict single crossing property. In Example 1, s1,7 = 5 . Notice that U1 (s) > U7 (s) for any s ∈ {1, 2, 3, 4} and U1 (s) < U7 (s) for any s ∈ {5, 6, 7}. Definition 7 For any k ∈ {2, ..., n} and i, j ∈ N, let Gi,j (k) be the probability that there are no more than k − 2 voters, other than i and j that support change.12 .
Claim 3 For any pi > pj , {si,j } = {s0 ∈ {1, ..., n}|Gi,j (s0 ) ≥ 1/2 and Gi,j (s) < 1/2 f or any s < s0 }.13 The intuition of Claim 3 is that Gi,j (s) >
1 2
means that under the voting rule s if
voter i or voter j do not support a proposal the probability of the proposal be rejected is higher than fifty percent. Thus voter i and j are decisive under voting rule s. Moreover, since pi > pj , the probability that voter i supports change and voter j does not is higher than the reverse since pi (1 − pj ) > (1 − pi )pj . It means that voter j is more decisive that voter i under voting rule s. Therefore, under this rule, voter j will have a higher expected utility than voter i. Not only for s, but for any s0 ≥ s since Gi,j (s) is a strictly increasing function. When Gi,j (s) < 12 , the situation is the reverse and then voter i has a higher expected utility than voter j. Definition 8 Every committee has preferences over voting rules that satisfy p-monotonic strict single crossing property if for any i, j, x ∈ N si,j ≤ si,x ≤ sj,x whenever pi > pj > px 12 13
Let Gi,j (1) ≡ 0. Note that for any pi > pj , {si,j } 6= {1} since Gi,j (1) = 0.
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Theorem 7 Every committee has preferences over voting rules that satisfy p-monotonic strict single crossing property. In Example 1, as you can check in Figure 1, we have that s1,3 ≤ s1,7 ≤ s3,7 since s1,3 = 4 and s1,7 = s3,7 = 5. Theorem 7 has the following useful corollary. Corollary 4 For any i, x, j ∈ N with pi ≥ pj ≥ px , Min{Ui (s), Uj (s), Ux (s)} = Min{Ui (s), Ux (s)}.
We now have all that we need to prove Proposition 2 and Claim 1. Proof of Proposition 2 and Claim 1.
Proposition 2 follows by Theorem 6 and
Theorem 7. Notice that Theorem 7 implies that sR,i ≤ sR,C ≤ si,C for any i ∈ N and pR > pi > pC . Claim 1 is a particular case of Claim 3.
5. Concluding remarks In contrast with Barber`a and Jackson (2002) and Badger (1972), who concentrate on the voting rules that might be chosen if voters vote on rules, we take a normative point of view and investigate the choice of voting rules according to the Rawls’s maximin criterion. Specifically, we hope to expand awareness, of choice subject to criteria, by complementing the utilitarian view (which leads to the choice of simple majority as proved in Rae (1969) and Curtis (1972)), with the suggestion that fairness considerations may recommend the choice of a rule that maximizes the expected utility of the worst off individual. As part of our comparison of different rules, we are led to study their implications on the distribution of well being across voters. From this investigation, we have discovered new properties of the model regarding voters’ preferences over voting rules. The main property which we have proved is that, for any pair of voters, with different probability of being in favor of the status quo, there is a threshold majority size such that, for any majority size higher than it, the one with the highest probability to reject changes of status quo has a higher utility. And for any majority size below this threshold, the reverse holds. Moreover, this threshold changes depending on the pair of voters under analysis, but in a particular way such that there a is threshold rule, such that for any rule lower than 15
it the most conservative among the members of the committee (every i ∈ N such that pi = Min{p1 , ..., pn }) is the worst off voter. And for rules higher than this threshold the reversal holds, i.e. the worst off voter is the most radical voter (every i ∈ N such that pi = Max{p1 , ..., pn }). This last result is important because it restricts the set of rules which are candidates to satisfy the maximin criterion. We proved that the function Min{U1 (s), U2 (s), ..., Un (s)} is single peaked (single-plateaued), there are at most two voting rules that maximizes this function, at least one is Pareto efficient and it is often different from the simple majority. Indeed, if a committee is formed only by “conservative voters”, i.e. pi ≤ 0.5 for any i ∈ N, then the maximin criterion recommends voting rules that are between the optimal voting rule of the less conservative among these voters and simple majority rule, i.e. sbR − 1 ≤ s ≤ smaj . If a committee is formed only by “radical voters”, i.e. pi ≥ 0.5 for
any i ∈ N, then the maximin criterion recommends voting rules that are between fifty percent majority and the optimal voting rule of the less radical among these voters, i.e. smaj − 1 ≤ s ≤ sbC .
6. References Arrow, K.J. (1998) The external costs of voting rules: A note on Guttman, Buchanan
and Tullock. European Journal of Political Economy 14: 219-222. Badger, W.W. (1972) Political individualism, positional preferences and optimal decisionrules. In: R.G. Niemi, H.F. Weisberg (eds.) Probability methods for collective decision making. Merril Publishing. Columbus Ohio. Barbera, S. and Jackson, M. (2002) Choosing how to choose: Self-stable majority rules. mimeo. Brennan, G. and Buchanan, M. (1985) The reason of rules: Constitutional political economy. Cambridge University Press, Cambridge, MA. Buchanan, J. (1998) Agreement and efficiency: Response to Guttman. European Journal of Political Economy 14: 209-213. Curtis, R. (1972) Decision rules collective values in constitutional choice. In: R.G. Niemi, H.F. Weisberg (eds.) Probability methods for collective decision making. Merril Publishing. Columbus Ohio.
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Esteban, J. and Ray, D. Social decision rules are not immune to conflicts. Economics of Governance 2(2001) 1: 59-67. Grofman, B. (1979) A preliminary model of jury decision making as a function of jury size, effective jury decision rule and mean juror judgmental competence. In Tullock, G. (ed.) Frontiers of economics. Blacksburg, VA: Center for Study of Public Choice. Guttman, J.M. (1998) Unanimity and majority rules: The calculus of consent reconsidered. European Journal of Political Economy 14: 189-207. Muller, D.C. (1989) Public choice II. Cambridge University Press, Cambridge, MA. Rae, D. (1969) Decision rules and individual values in constitutional choice. American Political Science Review 63: 40-56. Schofield, N. J. (1972) Is majority rule special? In: R.G. Niemi, H.F. Weisberg (eds.) Probability methods for collective decision making. Merril Publishing. Columbus Ohio. Tullock, G. (1998) Reply to Guttman. European Journal of Political Economy 14: 215-218.
Appendix To prove Theorem 4 we need the following lemmas: Lemma 1 a) If sR,C is Pareto inefficient and higher than smaj then sR,C ≥ sbC . b) If sR,C is Pareto inefficient and smaller than smaj then sR,C < sbR .
Proof of Lemma 1. It follows by Claim 2 and Corollary 1. Recall that sbC is Pareto
inefficient if and only if sbC is a twin-peak and sbR 6= sbC .
Lemma 2 If sbR ≤ sR,C ≤ sbC then Smax i min ⊆ {sR,C − 1, sR,C }
Proof of Lemma 2. Let sbR ≤ sR,C ≤ sbC . Take any x ∈ Smax i min .
Suppose by contradiction that x > sR,C . Note that by Proposition 2 and single peakedness, we have that
Min{U1 (x), ..., Un (x)} = UR (x) < UR (sR,C ) = = Min{U1 (sR,C ), ..., Un (sR,C )} It contradicts the maximin criterion so x ≤ sR,C Take any x ∈ Smax i min . Suppose by contradiction that x < sR,C − 1. Note that by Proposition 2 and single peakedness, we have that 17
Min{U1 (x), ..., Un (x)} = UC (x) < UC (sR,C − 1) = = Min{U1 (sR,C − 1), ..., Un (sR,C − 1)} It contradicts the maximin criterion so x ≥ sR,C − 1. Therefore we can conclude that x ∈ {sR,C − 1, sR,C } but then Smax i min ⊆ {sR,C − 1, sR,C }. Lemma 3 a) If sbC < sR,C then Smax i min ⊆ {b sC − 1, sbC } b) If sR,C < sbR then Smax i min ⊆ {b sR − 1, sbR }
Proof of Lemma 3. Let sbC < sR,C . Take any x ∈ Smax i min .
First suppose by contradiction that x < sbC − 1. But then it implies that x < sR,C .
Note that by Proposition 2 and single peakedness, x < sR,C implies that: Min{U1 (x), ..., Un (x)} = UC (x) < UC (b sC − 1) = = Min{U1 (b sC − 1), ..., Un (b sC − 1)}.
It contradicts the maximin criterion so x ≥ sbC − 1.
Now suppose that x > sbC . First note that it implies by Proposition 1 that x > sbC ≥ sbR .
Thus by single peakedness: UR (x) < UR (b sC ) and UC (x) < UC (b sC ). It leads a contradiction since by Proposition 2 and single peakedness, we have that
Min{U1 (x), ..., Un (x)} = Min{UR (x), UC (x)} < Min{UR (b sC ), UC (b sC )} It contradicts the maximin criterion. Thus x ≤ sbC .
Therefore Smax i min ⊆ {b sC − 1, sbC }. Notice that sbC − 1 ∈ Smax i min only if sbC is a twin
peak.
The proof of the part b) of the lemma is very similar from the part a so it is omitted. Proof of Theorem 4. Claim 2 and Lemma 2 imply that Smax min ⊆ {sR,C − 1, sR,C } whenever sR,C is Pareto efficient. Lemma 1a and Lemma 3a imply that Smax i min ⊆ {b sC − 1, sbC } whenever sR,C is Pareto
inefficient and larger than smaj .
Lemma 1b and Lemma 3b imply that Smax i min ⊆ {b sR − 1, sbR } whenever sR,C is Pareto
inefficient and smaller than smaj . Therefore the Theorem 4 is proved.
In the proof of Theorem 5, we need the following lemma: Lemma 4 a) sR,C ≥ smaj whenever pi ≥ 0.5 for every i ∈ N b) sR,C ≤ smaj whenever pi ≤ 0.5 for every i ∈ N. 18
Proof of Lemma 4. Let for any k ∈ {2, ..., n}, let GR,C (k) be the probability that there are no more than k − 2 voters, other than voter R and voter C that support change and GR,C (1) = 0. For any k ∈ {0, ..., n − 2}, denote by PR, C (k) the probability that exactly k of the voters other than voter C and voter R support a proposal. s−2 P GR,C (s) = PR, C (k) for any s ∈ {2, ..., n} and GR,C (1) = 0 k=0
It can be easily verified that:
c) For any k ∈ {0, ...., n − 2}, PR, C (k) ≤ PR, C (n − 2 − k) whenever pi ≥ 0.5 for every i∈N d) For any k ∈ {0, ...., n − 2}, PR, C (k) ≥ PR, C (n − 2 − k) whenever pi ≤ 0.5 for every i∈N But then since
n−2 P
PR, C (k) = 1, the information (c) and (d) imply that:
k=0
e)
smaj P−2
PR, C (k) ≥
1 2
whenever pi ≤ 0.5 for every i ∈ N
PR, C (k)
Gi,j (s) < Gi,j (n) = 1 for any s ∈ {2, ..., n − 1} 20
These two properties hold because for any k ∈ {1, ..., n}, Pi,j (k) > 0 (This later argument follows by the assumption of independence of the realization of voters’ support). ·Ui (s) > Uj (s) whenever Gi,j (s) < ·Ui (s) = Uj (s) whenever Gi,j (s) = ·Ui (s) < Uj (s) whenever Gi,j (s) >
1 2 1 2 1 2
Therefore Theorem 6 and Claim 3 is established. We need the following lemma to the Proof of Theorem 7 Lemma 6 For any pi > pj > px . ·Gi,j (s) ≥ Gi,x (s) ·Gi,j (s) ≥ Gj,x (s) ·Gi,x (s) ≥ Gj,x (s) Proof of Lemma 6. Gi,j (s) − Gi,x (s) = and Gi,j (0) − Gi,j (0) = 0
s−2 P
Pi,j (k) − Pi,x (k)
k=0
for any s ∈ {2, ...., n}
For any k ∈ {0, ...., n − 3} and i, j, x ∈ N, Pi, j,x (k) is the probability that exactly k of the voters other than i, j and x support the change. First note that: Pi,j (k) = [px Pi,j,x (k − 1) + (1 − px )Pi,j,x (k)]
for any k ∈ {2, ...., n − 3}
Pi,x (k) = [pj Pi,j,x (k − 1) + (1 − pj )Pi,j,x (k)]
for any k ∈ {2, ...., n − 3}
Pi,j (0) = (1 − px )Pi,j,x (0)
(12)
Pi,x (0) = (1 − pj )Pi,j,x (0)
(13)
Pi,j (n − 2) = px Pi,j,x (n − 3)
(14)
Pi,x (n − 2) = pj Pi,j,x (n − 3)
(15)
(10) (11)
It follows from (10) to (15) that Pi,j (k) − Pi,x (k) = (pj − px )Pi,j,x (k) − (pj − px )Pi,j,x (k − 1) for any k ∈ {2, ...., n − 3} (16) Pi,j (0) − Pi,x (0) = (pj − px )Pi,j,x (0)
(17)
Taking (16) and summing up over s we have that: s−2 P Pi,j (k) − Pi,x (k) = (pj − px )(Pi,j,x (s − 2) − Pi,j,x (0)) for any s ∈ {3, ...., n − 1} (18)
k=1
Using (17) and (18) and the definition of Gi,j (·), we have that :
Gi,j (s) − Gi,x (s) = (pj − px )Pi,j,x (s − 2) for any s ∈ {2, ...., n − 1}. 21
Hence we have that For any i, j, x ∈ N, we have that: Gi,j (s) − Gi,x (s) = (pj − px )Pi,j,x (s − 2) for any s ∈ {2, ..., n − 1}, Gi,j (1) − Gi,x (1) = 0 and Gi,j (n) − Gi,x (n) = 0 Therefore we have established Lemma 6. Proof of Theorem 7. Suppose pi > pj > px . It implies by Lemma 6 that ·Gi,j (s) ≥ Gi,x (s)
(19)
·Gi,j (s) ≥ Gj,x (s)
(20)
·Gi,x (s) ≥ Gj,x (s)
(21)
Using the information in Claim 3: Inequality (19) implies that si,j ≤ si,x , while inequality (20) implies si,j ≤ sj,x and inequality (21) implies si,x ≤ sj,x . Therefore si,j ≤ si,x ≤ sj,x and the proof of Theorem 7 is established.
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