ON THE JUSTICE OF DECISION RULES∗

JOSE APESTEGUIA†, MIGUEL A. BALLESTER‡, AND ROSA FERRER§

Abstract. Which decision rules are the most efficient? Which are the best in
terms of maximin, or maximax? We study these questions for the case of a group of
individuals faced with a choice from a set of alternatives. First, we show that the set
of optimal decision rules is well-defined, particularly simple, and well-known: the
class of scoring rules. Second, we provide the optimal decision rules for the three
different ideals of justice under consideration: utilitarianism (efficiency), maximin,
and maximax. We show that plurality, arguably the most widely used voting sys-
tem, is optimal in terms of maximax, while the best way to achieve maximin is by
means of negative voting, and the optimal utilitarian decision rule depends on the
culture of the society. We then provide the mapping between cultures and optimal
decision rules.

Keywords: Decision rules, Scoring Rules, Voting, Utilitarianism, Maximin, Maxi-
max.
JEL classification numbers: D00, D63, D71, D72.

1. Introduction

Should I implement a plurality voting rule if I aim to maximize efficiency? If I

am persuaded by the maximin ideal of justice, should I implement the Borda count?

What should I use if it is maximax that I wish to optimize? These are the type of

questions addressed in this paper. Our aim is to identify the best decision rules in

terms of these three ideals of justice.

In our setting, individuals’ preferences are cardinal and interpersonally comparable

utility random variables. The realization of the individual utility values depends on

the culture of the society, which is simply a probability distribution over the range of

utility intensities. We consider three prominent ideals of justice defined in cardinal

Date: September, 2008.
∗ We thank Toni Calvo, Gabor Lugosi, Larry Samuelson, and Rann Smorodinsky for valu-

able comments. Financial support by the Spanish Commission of Science and Technology CICYT
(SEJ2006-11510, SEJ2005-01481/ECON), the Barcelona GSE research network, and the Generalitat
de Catalunya is gratefully acknowledged.
† Universitat Pompeu Fabra. E-mail: jose.apesteguia@upf.edu.
‡ Universitat Autonoma de Barcelona. E-mail: miguelangel.ballester@uab.es.
§ Vanderbilt University. E-mail: rosa.ferrer@vanderbilt.edu.

1



2

terms: utilitarianism (or efficiency), maximin, and maximax. In short, utilitarian-

ism evaluates an alternative in terms of the average of the individual utility values.

The maximin principle disregards the utility values of the most favored individuals

to evaluate an alternative on the basis of the utility value of the worst-off agent. In

contrast with maximin, the maximax rule focuses on the best-off individuals. Maxi-

max as an ideal of justice may appear a mere formal curiosity. However, we will see

that, because of its close connection to plurality voting, the maximax principle plays

a more important role in democratic political institutions than might be expected.

In our model, n individuals must choose an alternative from a set of k ≥ 3 alter-
natives. Decision rules serve precisely this purpose. A decision rule transforms the

ordinal preferences of the individuals into a selection of alternatives. We impose no

restriction on the possible set of decision rules, other than assuming that it uses the

actual ordinal preferences of the individuals.

The question arises of what the optimal decision rules are in terms of a particular

cardinal ideal of justice. That is, we explore the connection between a given ideal

of justice and different decision rules, with the aim of identifying those that best

serve the pursued ideal of justice. There are different options for measuring decision

rule performance. A natural index can be obtained by evaluating decision rules in

terms of their expected attainable value for a given ideal of justice. Another index

emerges by judging decision rules on the basis of the probability of their selecting the

best alternative for a given theory of justice. Yet a third possibility is to measure

the probability of avoiding the worst alternatives. The three criteria approach the

same problem from different angles, and hence in principle complement each other.

It could be argued that the first index adopts a risk-neutral approach to measure the

success of decision rules, while the second and third indices could be understood as

representative of risk-loving and risk-averse positions, respectively.

Firstly, we show that a particularly prominent class of decision rules emerges as

optimal: the class of scoring rules. A scoring rule is a vector of fixed points that

individuals assign to the different alternatives. The plurality scoring rule, the negative

scoring rule and Borda’s scoring rule are especially salient. This result significantly

restricts the class of decision rules from which to seek the optimal ones. A further

point is that scoring rules are relatively easy to implement and widely used.

Secondly, our results indicate that the message for each ideal of justice is robust

across indices. That is, what matters is the nature of the ideal of justice, not the

particular way of measuring decision rule performance.

Thirdly, we provide the optimal decision rules for the three ideals of justice. In

utilitarian terms, we show that there is a mapping from the culture of the society to

the optimal decision rule. Moreover, we find the exact shape of the optimal decision

rule, conditional on the culture of the society. We then illustrate the result with a

number of examples. In particular, we show that for a range of cultures satisfying



3

a certain regularity condition, a prominent scoring rule emerges as optimal: Borda.

When the culture follows the normal distribution, the optimal decision rule for util-

itarianism is not Borda but a scoring rule where the differences in the values given

to consecutive alternatives is a symmetric and strictly convex function. That is, with

the normal distribution, the optimal scoring rule strongly discriminates between the

very best alternatives and also between the very worst alternatives. Alternatives in

between are treated as almost alike. We explore the connection between the optimal

utilitarian decision rule and other prominent cultures, such as the exponential or the

logistic distribution.

In contrast to utilitarianism, maximax offers a unique optimal decision rule for

every possible culture. It proves to be a remarkably important decision rule: plurality.

Although plurality is probably the most widely used voting system in the democratic

world, our results show that, instead of maximizing efficiency or the well-being of

the worst-off agents in a society, plurality maximizes the well-being of the best-off,

a property that may be considered of questionable merit. Finally, maximin is best

approached with the negative scoring rule. Given the link between plurality and

maximax, it is not surprising that negative, the symmetric scoring rule to plurality,

is the best decision rule in terms of maximin, which can be shown to be in a way

symmetric to maximax. Interestingly, the results for the three ideals of justice point

to the three best-known scoring rules: plurality, Borda, and negative.

We are certainly not the first to evaluate decision rules. There is a large and still

growing literature examining decision rules in terms of the properties they satisfy.

In other words, decision rules are judged on the basis of their capacity to meet cer-

tain desirable properties such as anonymity, independence of irrelevant alternatives,

strategy-proofness, consistency of the social preference ordering, Pareto-dominance,

path-independence, probability of selecting the Condorcet winner, etc.1

In this paper, we focus not on the specific properties that voting systems may or

may not satisfy, but on how well they perform in terms of an ideal of justice. If there

is agreement over which ideal of justice is to be pursued, it seems much more relevant

to evaluate voting systems on the basis of how well they serve that ideal, than it is to

calculate the probability of their electing the Condorcet winner, for example. In other

words, in this paper we shift the decision rule evaluation criterion from an ordinal to

a cardinal approach. The problem therefore reduces to inferring the relevant utility

intensities from ordinal rankings.

There is very little work on evaluating decision rules on the basis of some cardinal

theory of justice. Notable exceptions are the early simulation studies of Bordley

(1983) and Merrill (1984), and the theoretical work of Weber (1978). Bordley and

1Early studies are Brams and Fishburn (1978), Caplin and Nalebuff (1988), DeMeyer and Plott
(1970), and Nurmi (1983). See also Benoit and Kornhauser (2007), Dasgupta and Maskin (2008),
Gehrlein (1997), Levin and Nalebuff (1995), Myerson (2002), and Saari (1999).



4

Merrill use simulations to analyze the efficiency of different voting systems, including

plurality and Borda. Consistent with our results, they show that plurality is easily

outperformed in utilitarian terms by other decision rules. Weber (1978) studies the

performance of scoring rules for the case of utilitarianism and for cultures following

the uniform distribution. He shows that, asymptotically, Borda is the best scoring

rule in this case. This, again, is consistent with our results.

In another related strand of literature, there are papers that study how to select

a voting rule in a constitutional setting where there are two options, the status quo

and a second alternative, and individual preferences are uncertain. A voting rule is

characterized by the number of votes needed to accept the second alternative over

the status quo. The papers that comprise this literature examine issues such as

which voting rules maximize efficiency, which are self-stable, how to weight votes in

heterogeneous contexts, self-enforcement voting rules, etc. For examples in this vein,

see Rae (1969), Barbera and Jackson (2004, 2006), Maggi and Morelli (2006) and

papers cited therein.

Finally, there is a growing literature addressing the question of the transmission

of utility intensities in collective decision problems (see Casella, 2005; Jackson and

Sonnenschein, 2007; and Hortala-Vallve, 2007). The innovation of these papers is

to consider a collective decision situation problem repeated over T times and endow

individuals with a maximum number of votes to allocate over the T problems. By so

doing, individuals may transmit utility intensities.

The remainder of the paper is organized as follows. Section 2 introduces in detail

the setting in which we will be working. Sections 3, 4, and 5 study the cases of

utilitarianism, maximax and maximin, respectively. Finally, Section 6 presents some

conclusions. All the proofs are given in Appendix A.

2. Environment

Let N be a finite set of individuals with cardinality n ≥ 2 and K the set of
alternatives with cardinality k ≥ 3. Typical elements of N are denoted by i and j
and those of K are denoted by l,h and q. We then say that a society is composed of

n individuals with preferences over k alternatives.

To address the question of finding the best ordinal decision rule to approach each

given ideal of justice, we first present the cardinal environment (utilities and ideals

of justice), then the ordinal setting (decision rules and ordinal preferences), and then

the one that connects the two worlds (indices evaluating the success of a decision rule

in terms of an ideal of justice).

2.1. Cardinal Preferences. Individual preferences over the set of alternatives are

cardinal, interpersonally comparable utility random variables in the [0, 1] interval. Uli
is the random variable representing the cardinal utility of individual i for alternative



5

l. We adopt the convention that subindices refer to individuals and superindices refer

to alternatives. Let culture C be the probability distribution from which the cardinal

individual utility values are drawn. We consider cultures with continuous probability

distributions on [0, 1]. Hence, the probability that Uli takes a particular value on [0, 1]

is zero, and then the probability of ties (i.e., of individual i being indifferent between

two alternatives) is also zero.2 Moreover, we assume that variables {Uli}i∈N,l∈K are
iid. We say that a culture is symmetric whenever its density function is symmetric.

Given the random variables U1i ,U
2
i , . . . ,U

k
i , the order statistics U

(1)
i ≤ U

(2)
i ≤ ···≤

U
(k)
i are also random variables, defined by sorting the realizations of U

(1)
i ≤ U

(2)
i ≤

··· ≤ U(k)i in increasing order of magnitude. U
(l)
i denotes the l-th order statistic of

individual i, representing the utility value for individual i of that alternative having l−
1 alternatives with lower utility values. Analogously, we will be using order statistics

given an alternative l. Consider the collection of random variables Ul1,U
l
2, . . . ,U

l
n, and

arrange them in order of magnitude by denoting Ul(1) ≤ U
l
(2) ≤ ··· ≤ U

l
(n). Here U

l
(i)

denotes the utility value of the i-th order statistic for alternative l, representing the

utility value of l for that individual who ranks l higher than exactly i−1 individuals.
A Social Welfare Function SWF is a mapping W from [0, 1]n×k to [0, 1]k, where

W l(Ul) ∈ [0, 1] denotes the social value of alternative l, given the realization of random
variables Ul = (Ul1, . . . ,U

l
n).

3 The three SWFs we consider here are utilitarianism,

maximin, and maximax. Utilitarianism evaluates an alternative by taking the average

of individual cardinal utilities. Formally, a SWF is utilitarian if W = WUT with

W lUT (U
l) =

∑n
i=1 U

l
i/n. The maximin principle evaluates an alternative on the basis

of the utility value of the worst-off agent, disregarding any other utility value. In

other words, a SWF is of the maximin type if W = WMN with W
l
MN (U

l) = Ul(1).

Consider also the maximax rule. In contrast to maximin, the maximax rule focuses

on the best-off individuals. That is, a SWF is of the maximax type if W = WMX
with W lMX(U

l) = Ul(n).

2.2. Ordinal Preferences. The following describes the ordinal rankings of the in-

dividuals over the set of alternatives. M is an n × k matrix with properties: (1)
mli ∈ {1, . . . ,k}, and (2) mli 6= mhi for all i ∈ N and for all l,h ∈ K, l 6= h. Entry
mli denotes the position of alternative l in the preferences of individual i, where the

higher mli is, the higher alternative l is ranked by individual i.
4 Mi denotes the i-

th row of matrix M. That is, Mi represents the ordinal preferences of individual i.

Ml denotes the l-th column of matrix M, representing the ordinal preferences of all

2Given our assumption that cultures are continuous probability distributions on [0, 1], all our
results naturally extend to other ranges such as (0, 1), (−∞,∞), or [0,∞).

3Throughout the paper we use the terms “social welfare function” and “ideal of justice”
indistinctly.

4Note that given the assumptions on continuous probability distributions, there are no ties, and
hence property (2) above follows.



6

individuals with respect to alternative l. The collection of all possible matrices M

is denoted by M. Finally, by l(h) we refer to the number of individuals that place
alternative l exactly above h− 1 alternatives. That is, l(h) = |{i ∈ N : mli = h}|.

A multi-valued decision rule f is a correspondence from M to K. Denote by F
the set of all decision rules f. That is, a decision rule takes ordinal information on

preferences as input to determine a set of alternatives as social choice. We assume

that f uses true information. That is, the particular matrix M over which f is applied

is assumed to be known.

Scoring rules are a particularly interesting class of decision rules. They are typically

simple to implement in practice and they encompass a number of widely-used decision

rules. Formally, a scoring rule S can be represented by a vector S ∈ [0, 1]k, with Sl
denoting the strength of an individual’s vote for whichever alternative she ranks

higher than exactly l − 1 alternatives. We normalize strength of vote by assigning a
value 1 to the best alternative, and a value 0 to the worst. That is, we set Sk = 1

and S1 = 0. Note that this normalization of the extreme values is without loss of

generality. We denote by S(M) ⊆ K the set of alternatives selected by scoring rule
S with preferences M. An alternative l is selected by scoring rule S when M, if l

gets the highest number of points across individuals when ordinal preferences are M.

The especially salient scoring rules are plurality, Borda, and negative. A scoring rule

is plurality if S = SPl with S
l
Pl = 0 for every l < k. It is negative if S = SNg with

SlNg = 1 for every l > 1, and it is Borda if S = SBd with S
l
Bd =

l−1
k−1 for every l.

2.3. Cardinal and Ordinal Preferences: Evaluating Decision Rules. We will

be using three different indices to evaluate the performance of alternatives in terms

of the ideals of justice under consideration.

The first, which we denote by the α-index, judges an alternative l on the basis of

the expected value that l provides in terms of an ideal of justice W. That is,

αlW (M) = E[W
l(Ul) | M].

Recall that W lUT (U
l) =

∑n
i=1 U

l
i

n
, W lMN (U

l) = Ul(1), and W
l
MX(U

l) = Ul(n). For the

case of utilitarianism and given ordinal preferences M, alternative l is better than

alternative q in terms of the α-index, if the expected average utility value of l is

larger than the expected average value of q. Therefore, the aim is to find decision

rules that select alternatives with the highest expected average utility value. Now

consider maximin. Then, for a given M, alternative l is better than alternative q in

terms of the α-index, if l provides a higher expected value of the lowest order statistic,

than q. Finally, l is better than q for maximax in terms of the α-index, if l provides a

higher expected value of the highest order statistic than q, given ordinal preferences

M.



7

The β-index, on the other hand, measures for a given SWF W, the probability of

an alternative l being optimal :

βlW (M) = P(W
l(Ul) ≥ Wh(Uh) for all h ∈ K \{l} | M).

Instead of looking to the expected value of an alternative, the β-index considers the

probability of that alternative providing the highest social welfare value, given ordinal

preferences M. Under utilitarianism, maximin, and maximax, we will be talking

about the probability of selecting those alternatives that give the highest average

utility value, the highest value among the first order statistics, and the highest value

among the highest order statistics, respectively.

Finally, for a given SWF W the γ-index evaluates alternatives in terms of the

probability of not choosing the worst alternative:

γlW (M) = 1 −P(W
l(Ul) < Wh(Uh) for all h ∈ K \{l} | M)

= P(W l(Ul) ≥ Wh(Uh) for some h ∈ K \{l} | M).

The three indices are different in nature. It could be argued that the α-index

adopts a risk neutral approach to measure the success of decision rules. Indices β and

γ, however, could be understood as adopting risk-loving and risk-averse positions,

respectively.

Indices and ideals of justice are intimately linked. There is a natural link between

utilitarianism and the α-index, between maximax and the β-index, and between max-

imin and the γ-index. It is in fact the case that these natural links translate into ana-

lytical tractability. We will see that utilitarianism is best approached analytically by

the α-index, while the β-index is poorly suited to the formal study of utilitarianism.

With respect to maximax, in contrast, great analytical progress can be made with

the β-index, but the tractability of maximax with regard to α becomes complex. For

intuition, consider for instance the α-index of an alternative l in utilitarian terms.

The value of the α-index is the average of the expected value of the different order

statistics that make up alternative l. In terms of maximax (and similarly for max-

imin), one has to obtain the expected value of the highest order statistic of l. This

is a challenge. If there is an individual who ranks l as her best alternative, then this

individual is a good candidate to represent the highest order statistic of l. However,

if for example there are many individuals who ranks l as their second best alterna-

tive, then it is likely that the highest order statistic of l is among the latter group of

individuals. Thus, there is a complex dependent structure that forces consideration

of the interrelation between the random utility variables of all the individuals in the

society for all the alternatives.

An XW -optimal decision rule, with XW = {αW ,βW ,γW}, is a decision rule f that
always selects alternatives with the largest X-index for ideal of justice W. That is,



8

f is XW -optimal decision rule if for all M ∈M:

l ∈ f(M) ⇒ XlW (M) ≥ X
h
W (M) for all h ∈ K.

It is straightforward to see that, given an XW -optimal decision rule f, any other

decision rule in the family δ(f) = {g ∈F : ∅ 6= g(M) ⊆ f(M) for all M ∈M} is also
an XW -optimal decision rule. In order to characterize the set of XW -optimal decision

rules, it is sufficient to identify the maximum XW -optimal f. We will say that f is

the maximum XW -optimal decision rule if whenever f and g are XW -optimal, then

it must be that g ∈ δ(f).

3. Utilitarianism

We first show that for every n×k society and every culture, the optimal decision
rule to maximize the utilitarian expected value is a scoring rule. This is good news.

It implies that if the interest is to maximize the expected value of utilitarianism, it

is advisable to implement a scoring rule, which is a relatively simple decision rule.

Furthermore, we provide the exact form of the optimal scoring rule, conditional on

the culture under consideration. More specifically, a culture determines the expected

value of the order statistics, which in turn determines the optimal value of the scoring

rule. Knowledge of the culture of the society provides information on the expected

value of the order statistics, which can be used to design the optimal scoring rule. In

particular, we show that for a range of cultures with symmetric density functions, a

prominent scoring rule emerges as the maximum optimal: Borda. The intuition goes

as follows. The jumps in strength that Borda assigns to consecutive order statistics

are constant. Hence, cultures in which the expected value of the order statistics has

this property are the most closely linked to Borda. Also, when the culture follows the

normal distribution, we show that the optimal decision rule is a scoring rule where the

differences in the values given to consecutive alternatives is a symmetric convex func-

tion. That is, under the normal distribution, it is optimal to discriminate between the

very best alternatives and also between the very worst alternatives, and less impor-

tant to discriminate between the intermediate alternatives. Interestingly, when the

culture of the society follows a Cauchy distribution then the pattern becomes even

more pronounced. In this case, the optimal scoring rule tends to discriminate be-

tween the very best alternative and the very worst alternative, treating all remaining

alternatives in almost the same manner.

We then turn to the analysis of the β- and γ-indices. The β-index (as well as

the γ) is less well suited than the α-index to the analytical study of utilitarianism.

The source of difficulty lies in the dependency across alternatives. To determine the

probability of an alternative being the utilitarian alternative, it is necessary to take

the whole society into account. This dependency drastically limits the analytical

tractability of utilitarianism in terms of the β-index. For this reason, we begin with



9

the formal study of a small society, a 2×3 society, for a complete range of cultures with
symmetric density functions, and then we run a set of simulations to approach larger

societies. Our analytical results for small societies and the computational analysis for

larger societies provide analogous results as those obtained in the formal study of the

α-index. This is reassuring since it indicates that there is a great deal of consistency

across indices. What matters is the social welfare function, not the particular way of

measuring success. Finally, our analysis of the γ-index exploits a symmetry relation

with the β-index, allowing us to immediately reach the same conclusions as those

reached with the β-index.

We organize the study of utilitarianism by first giving the general results obtained

with the α-index, and we then turn to the study of the β- and γ-indices.

3.1. The αUT -index: Evaluating Decision Rules in terms of Expected Value.

Theorem 3.1 below provides the main result for utilitarianism. It first identifies a scor-

ing rule as the maximum optimal decision rule, and then it characterizes the exact

shape of the scoring rule. Importantly, it does so for any n × k society and for any
culture.

Theorem 3.1. For n×k societies and for every culture, the maximum αUT -optimal
rule is a scoring rule with S∗l =

E[U(l)]−E[U(1)]
E[U(k)]−E[U(1)] , l ∈{1, . . . ,k}.

Theorem 3.1 shows that there is a mapping from the culture of the society to the

optimal decision rule. A culture determines the first moment of the order statistics,

and this in turn characterizes the optimal decision rule. This raises the question of

what the optimal decision rules of especially prominent cultures are like. The follow-

ing corollary represents a first illustration.

Corollary 3.2. The maximum αUT -optimal rule is Borda:

• For n×k societies and for every culture such that E[U(l+1)]−E[U(l)] = c, with
1 ≤ l ≤ n− 1.
• In particular for the uniform distribution.
• For n× 3 societies and for every culture with a symmetric density function.

Corollary 3.2 points to a prominent decision rule, Borda, as the optimal rule for a

wide range of cultures with density functions satisfying a regularity condition. But

there are many other relevant cultures for which to obtain the optimal scoring rule.

Consider the normal distribution, for example. The expected value of the order sta-

tistics from the normal distribution are tabularized (see, e.g., Teichroew, 1956). From



10

inspection of these values, it immediately emerges that the αUT -optimal decision rule

in this case is a scoring rule where S∗l+1 −S∗l is a symmetric and convex function. A
more extreme version of the above emerges when considering the Cauchy distribution

(see, e.g., Barnett (1966) for the expected values of the Cauchy order statistics). In

the latter case, the optimal scoring rule tends to discriminate only between the very

best alternative, the very worst alternative and all the rest.

We can further apply our Theorem 3.1 to other classes of probability distributions

via known results in the literature of order statistics. By way of illustration consider

the following class of probability distributions F (see Kamps, 1991, 1992; see also

Balakrishnan and Sultan 1998):

d

dt
F−1(t) =

1

d
tp(1 − t)q−p−1,

t ∈ (0, 1), where p and q are integers and d > 0. The values of parameters p and q
determine the probability distribution. For example, if p = q = 0 we get the expo-

nential distribution, if p = q = −1 we get the logistic distribution, if p > −1 and
q = p + 1 we get the power function, etc. We can then establish the following result

for this class of probability distributions.

Theorem 3.3. For n × k societies and for the class of cultures F , the maximum
αUT -optimal rule is a scoring rule with

S∗l =

∑l
h=2 E[U

(h)] − E[U(h−1)]∑k
h=2 E[U(h)] − E[U(h−1)]

,

l ∈{1, . . . ,k}, with E[U(h)] − E[U(h−1)] = (
k

h−1)
(h+p)(k+qh+p)

.

The introduction of the values of the parameters into the equations of Theorem

3.3 immediately gives the optimal scoring rule. By way of illustration, let k = 3.

It is immediate that when there are three alternatives, the optimal scoring rule is

well-defined by the value assigned to the middle alternative.

Corollary 3.4. For n × 3 societies and for the class of cultures F , the maximum
αUT -optimal rule is a scoring rule with S

∗2 = 1−p+q
3+q

, where p and q are integers.

The above corollary provides the optimal scoring rule contingent on the values of

parameters p and q, and hence contingent on the culture of the society.



11

3.2. The βUT -index: Evaluating Decision Rules in terms of the Probability

of Selecting an Optimal Alternative. Due to the analytical complexity of the

study of utilitarianism through the lens of the βUT -index, we begin with the formal

study of a small society, a 2×3 society, for a complete range of cultures with symmetric
density functions, and we then run a set of simulations to approach larger societies.

In societies with n = 2 and k = 3 there are 5 different scoring rules, since every

S2 ∈ (0, 1/2) represents the same scoring rule, and similarly every S2 ∈ (1/2, 1) also
represents the same scoring rule.5 Hence, the set of different scoring rules when n = 2

and k = 3 comprises plurality with S2Pl = 0, Borda with S
2
Bd = 1/2, negative with

S2Ng = 1, an explosion rule with any value S
2
Ex ∈ (1/2, 1), and an implosion rule with

any value S2Im ∈ (0, 1/2).
We study a collection of cultures with symmetric density functions. Take the para-

bolic function f(x) = ax2 +bx+c, with x ∈ [0, 1]. Now, in order for f(x) to represent
a symmetric density function it must be the case that a = −b = −(1−c), with c tak-
ing values in [0, 3]. Note that the c = 1 case corresponds to the uniform distribution.

Values of c ∈ [0, 1) correspond to strictly concave density functions. That is, lower
values of c represent higher levels of consensus on the evaluation of an alternative.

Finally, values of c ∈ (1, 3] correspond to strictly convex density functions. Hence,
higher values of c represent higher levels of extremism.

Theorem 3.5. For 2 × 3 societies and for any culture with a symmetric parabolic
function:

(1) The maximum βUT -optimal rule is the implosion scoring rule.

(2) The intersection between the alternatives selected by Borda and those selected

by the maximum βUT -optimal rule is always non-empty.

The analysis of the 2 × 3 case shows that any scoring rule with S2 ∈ (0, 1/2) is
βUT -optimal. Moreover, it also shows that although Borda is not βUT -optimal, it

comes close to being so. For every type of society M, the alternatives selected by

Borda and implosion coincide, except for one (which in the proof of Theorem 3.5

we call societies of type 4) where Borda randomizes between the 3 alternatives while

implosion randomizes between 2 alternatives.

We extend the study of the βUT -optimal decision rule by using computational

methods. We generate random cardinal societies from three different probability

distributions: the uniform distribution on the interval [0, 1], the standard normal

distribution on the interval (−∞,∞), and the exponential distribution on the interval
[0,∞). We study societies of 2, 3, 5, 10 and 100 individuals with preferences over 3, 4, 5
and 7 alternatives. The technical details are contained in Appendix B.

5We say that two scoring rules S and S′ are the same if S(M) = S′(M) for all M.



12

The main conclusion of the computational analysis is very clear. The results across

probability distributions and sizes of societies are fully in line with the analytical

results we presented for the case of the αUT -index. That is, for each of the three

probability distributions and sizes of societies scrutinized, the scoring rule providing

the highest βUT -value is the corresponding one as stated in Theorem 3.1. In particular,

for the uniform distribution the highest βUT -value is given by Borda; for the normal

distribution it is given by a scoring rule where S∗l+1 −S∗l is a symmetric and convex
function; and finally for the exponential distribution, the highest βUT -value is given

by a scoring rule where S∗l+1 −S∗l grows exponentially.6

3.3. The γUT -index: Evaluating Decision Rules in terms of the Probabil-

ity of Not Choosing the Worst Alternative. It is not difficult to appreciate

that for utilitarianism and symmetry of the culture C, the γ-index is symmetric to

the β-index in the following sense: take a society {Uli}i∈N,l∈K, and let l be an al-
ternative giving the highest utilitarian value. Write the following symmetric society

{U ′li }i∈N,l∈K with U
′l
i = 1 −Ulσ(i) where σ(i) = n− i + 1. It follows that alternative l

in society {U ′li }i∈N,l∈K is one of the worst alternatives in utilitarian terms. Then, the
analytical and simulation results that we obtained for the β-index and for symmetric

cultures can be extrapolated to the case of the γ-index. To complete the picture, we

run an extra set of simulations for the exponential distribution, an asymmetric distri-

bution, that points to, once again, that the particular index to measure performance

is not essential (see Appendix B). We may conclude, therefore, that at least for the

cases computationally scrutinized, the optimal decision rules in terms of γUT are well

captured in Theorem 3.1.

4. Maximax

We start the analysis of maximax by showing formally that for any n×k society
and for any culture, the maximum optimal decision rule in terms of the probability

of selecting the maximax alternatives is a scoring rule. In fact, it is a particularly

simple and widely used scoring rule: plurality. The intuition is very simple. Maxi-

max evaluates an alternative l in terms of that individual that most values alternative

l. It is very likely that such an individual ranks l as her most preferred alternative.

Consequently, those alternatives that are considered by the highest number of indi-

viduals as best are the most likely maximax alternatives. Plurality selects precisely

these alternatives. The analytical and computational study of the two other indices

reinforce the connection between plurality and maximax.

6For the sake of comparison, Appendix B reports the βU T -values of the optimal scoring rules as
predicted by Theorem 3.1, together with the three prominent scoring rules–plurality, Borda, and
negative–for each of the societies and cultures under scrutiny.



13

We start the analysis of the maximax case by first giving the general results reached

with the β-index and then we turn to the study of the α and γ-indices.

4.1. The βMX-index: Evaluating Decision Rules in terms of the Probability

of Selecting an Optimal Alternative. Theorem 4.1 below shows that something

as simple as the plurality decision rule offers the highest probability of selecting the

maximax alternative. Theorem 4.1 does not assume any particular probability distri-

bution over the individual utility values, or any society size.

Theorem 4.1. For n×k societies and for any culture, the maximum βMX-optimal
rule is the plurality scoring rule.

Corollary 4.2 reinforces the role of plurality in the maximax ideal of justice. It

shows that, as the society grows, plurality is the unique optimal scoring rule.

Corollary 4.2. For any given k and for any culture, when n tends to infinity plural-

ity is the only scoring rule in the set of βMX-optimal scoring rules.

Theorem 4.1 and Corollary 4.2 give a very clear message: if the interest is in

maximax, then the decision rule that should be implemented is as simple as plurality,

irrespective of the culture and size of the society.

4.2. The αMX-index: Evaluating Decision Rules in terms of Expected Value.

We start the analysis of maximax in terms of the α-index by studying the contribu-

tion made to the maximax expected value of an alternative l by adding one extra

best order statistic, as opposed to the value added by a second best order statistic,

when the culture follows the uniform distribution. We explore the extreme case where

everybody ranks alternative l as the best. It is in this case that the extra expected

maximax value of adding one best alternative is intuitively the smallest. We show

that, even in this case, the contribution made to the expected maximax value by

adding one extra best order statistic is of a higher order of magnitude than that

made by adding a second best order statistic. Then, plurality is the clear candidate

to be the optimal maximax decision rule in terms of the α-index, since it gives weight

only to the best random variables.

Consider the n×k society M such that mli = k for every i ∈{1, . . . ,n}. That is, all
n individuals in society M rank alternative l as the best. Consider also the following

two (n + 1)×k societies, M ′ and M, such that m′li = k for every i ∈{1, . . . ,n,n + 1},
and mli = k for every i ∈ {1, . . . ,n,n + 1} \ {j} and mlj = k − 1. In M ′, all n + 1



14

individuals rank l as the best alternative, while in M, there are n individuals who

rank l as best, and one that ranks l as second best.

Now, taking the n × k society M as the baseline, we ask what is the marginal
contribution to the l-th maximax expected value of adding one extra individual who

ranks l as best (society M ′). Also, we wonder about the marginal contribution of

adding one extra individual who ranks l as second-best (society M). Denote these

marginal contributions by ∆
(k)
n and ∆

(k−1)
n , where ∆

(k)
n = E[Ul(n+1)|M

′l] − E[Ul(n)|M
l]

and ∆
(k−1)
n = E[Ul(n+1)|M

l
] − E[Ul(n)|M

l]. The ratio ∆
(k)
n

∆
(k−1)
n

represents the ratio of the

marginal contributions of an extra order statistic in the top positions k and k − 1.
Lemma 4.3 below shows that, for the uniform distribution, when l(k) = n, the ad-

ditional expected value provided by one extra order statistic in the same position k

is considerably higher than that provided by one extra order statistic in position k−1.

Lemma 4.3. For n×k societies and for the uniform distribution, ∆
(k)
n

∆
(k−1)
n

=
k(n+1)
k−1 .

Intuitively the situation where everybody ranks l as best is the worst-case scenario

for the expected marginal contribution of adding one extra individual who rank l the

best alternative. Even in this case, Lemma 4.3 shows that the order of magnitude in

the expected maximax value of an extra best random variable is considerably higher

than that of an extra second best random variable. This clearly points to plurality

as the candidate for the optimal maximax decision rule. Plurality gives weight only

to best random variables, and hence selects alternatives with the highest number of

best random variables.

In the context of n individuals and 3 alternatives, and for the uniform distribu-

tion, Proposition 4.4 below first provides the exact formula for the computation of

the maximax expected value of any alternative l. Then, it shows that only under

extreme circumstances does plurality prove not to be the αMX-optimal decision rule.

Proposition 4.4. For n× 3 societies and for the uniform distribution,
(1) E[Ul(n) | M] =

∑l(2)
a=0

∑l(1)
b=0

∑3b
c=0

(
l(2)

a

)(
l(1)

b

)(
3b
c

)
(−1)a+b+c3l(2)−a2a 3l

(3)+2l(2)+a+c
3l(3)+2l(2)+a+c+1

(2) Let h be an αMX-optimal alternative and let p ∈ SPl(M)\{h}. Then, M must
satisfy that either

(a) h(3) = p(3) and h(2) ≥ p(2), or
(b) h(3) = p(3) − 1 and h(2) > n−1

2

Proposition 4.4 shows that plurality is very difficult to beat in terms of its capacity

to optimize the maximax expected value. The plurality alternative can be surpassed



15

by another alternative in terms of αMX if about the same people rank it as the best

alternative and (many) more people rank it as second-best. Clearly, these are low-

probability events, and hence plurality is very likely to be the αMX-optimal decision

rule.

In accordance with the above, the computational studies for the uniform, normal,

and exponential distributions show that, for every single size of society scrutinized,

plurality or a scoring rule in the neighborhood of plurality are always the αMX-optimal

decision rule among the set of scoring rules evaluated (see Appendix B for details).

4.3. The γMX-index: Evaluating Decision Rules in terms of the Probability

of Not Choosing the Worst Alternative. We start with the characterization of

a small society.

Theorem 4.5. For 2 × 3 societies and for any culture with a symmetric parabolic
function:

(1) Implosion is a γMX-optimal decision rule.

(2) The intersection between the alternatives selected by plurality and those selected

by the maximum γMX-optimal decision rule is always non-empty.

Note that implosion is not the maximum γMX-optimal decision rule since, in soci-

eties where the two individuals share the same ranking over the alternatives, implosion

selects the single best alternative, instead of the two better alternatives.

However, it is not difficult to see that, for general n×k societies, plurality is the
natural candidate to select γMX-optimal alternatives. Below, in Proposition 4.6, we

show that, for any n × k society and for any culture, whenever there is a set of
alternatives Pareto-dominating some other alternatives, any decision rule selecting

this set of Pareto-dominating alternatives is optimal in the sense of maximizing the

probability of avoiding the selection of worst maximax alternatives. Clearly, plurality

alternatives tend to be Pareto-dominating, and hence plurality voting is the most

obvious candidate to be the optimal decision rule in terms of the γMX-index.

We write Ml > Mh whenever mli > m
h
i for every i. That is, every individ-

ual i ranks alternative l higher than alternative h. This is equivalent to Pareto-

dominance. Then, consider the following set P(M) = {l ∈ K such that there exists
h ∈ K with Ml > Mh}.

Proposition 4.6. For n×k societies and for any culture, whenever the set P(M) is
non-empty, P(M) is the maximum γMX-optimal rule.



16

Finally, the simulations also generally select plurality as the best decision rule,

across society sizes and cultures (see Appendix B for a discussion).

5. Maximin

For the analysis of the maximin case we exploit a symmetry between maximin

and maximax. This allows us to directly extrapolate results that we obtained from

the maximax case to the maximin case. In particular, the results point to negative,

the symmetric scoring rule of plurality, as the obvious candidate to be the optimal

decision rule for maximin in terms of the three indices of consistency, α, β, and γ.

Denote by 1 an n×k matrix with all entries equal to 1. Then, consider the follow-
ing Theorem.

Theorem 5.1. For n×k societies, symmetric cultures, every M ∈M, and l ∈ K:
(1) αlMN (M) = 1 −α

l
MX((k + 1)1 −M),

(2) βlMN (M) = 1 −γ
l
MX((k + 1)1 −M), and

(3) γlMN (M) = 1 −β
l
MX((k + 1)1 −M).

Theorem 5.1 allows the extrapolation of all analytical and simulations results ob-

tained in the previous section for maximax that involve symmetric cultures to the

case of maximin. This means that negative is the decision rule best suited to optimize

maximin. For asymmetric cultures, Theorem 5.1 does not apply. To test the robust-

ness of our theoretical findings when the culture is asymmetric, we estimate the value

of the three indices for the case of the exponential distribution. Table 3 in Appen-

dix B reports the results. Once more, it can be appreciated that negative generally

emerges as the optimal scoring rule in the computational analysis (see Appendix B

for a more detailed account of the results of the simulations).

6. Conclusions

This paper explores the relation between ideals of justice and decision rules. Whereas

ideals of justice are typically presented in cardinal terms, decision rules are primarily

constructed on the basis of ordinal information. We study the cardinal consequences

of using ordinal-based decision rules.

We have shown that the optimal choice of decision rules depends on the criterion

of justice that one wishes to follow. Among our specific findings we emphasize that

our results identify a particularly prominent set of decision rules as optimal: the

set of scoring rules. Moreover, the three most prominent scoring rules, plurality,

Borda, and negative, are optimal for the three different ideals of justice. Maximax

is best approached by plurality for any possible culture, maximin is best approached

by negative, and, for a range of cultures, utilitarianism is best approached by Borda.



17

Moreover, in the latter case of utilitarianism we provide the mapping between cultures

and optimal decision rules.

Appendix A. Proofs

Proof of Theorem 3.1. Recall that αlUT (M) = E[
∑

i U
l
i

n
| M]. The independence

of the random variables Uli across individuals allows us to write α
l
UT (M) =

∑
i E[U

l
i |

Mi]/n. In addition, since, for every individual i, all the random variables are identi-

cally distributed, we can write αlUT (M) =
∑

i E[U
l
i | Mli ]/n. Finally, since Ul1, . . . ,Uln

are also identically distributed, we can write the last expression in terms of the num-

ber of order statistics. First, note that, since U
(h)
i and U

(h)
j are identically distributed,

we may simply write U(h). Then, αlUT (M) =
∑

h l
(h)E[U(h)]
n

.

The value of the difference between two alternatives l and q is αlUT (M)−α
q
UT (M) =∑

h l
(h)E[U(h)]
n

−
∑

h q
(h)E[U(h)]
n

=
∑

h(l
(h)−q(h))E[U(h)]

n
. Note that

∑
h l

(h) =
∑

h q
(h) = n, and

therefore
∑

h E[U
(1)](l(h)−q(h))
n

= 0. We then write αlUT (M)−α
q
UT (M) =

∑
h(l

(h)−q(h))E[U(h)]
n

−∑
h E[U

(1)](l(h)−q(h))
n

and hence αlUT (M) − α
q
UT (M) =

∑
h(l

(h)−q(h))(E[U(h)]−E[U(1)])
n

. Note

that E[U(k)] − E[U(1)] > 0. Then, multiplying and dividing by E[U(k)] − E[U(1)], we
have

αlUT (M) −α
q
UT (M) =

E[U(k)] − E[U(1)]
n

∑
h(l

(h) − q(h))(E[U(h)] − E[U(1)])
E[U(k)] − E[U(1)]

=
E[U(k)] − E[U(1)]

n
(
∑
h

[l(h) − q(h)]S∗h).

Since E[U(k)] − E[U(1)] > 0, it follows that, for every M, S∗ gives exactly the same
ranking over the alternatives as the one given by utilitarianism. Hence, the two claims

of the theorem follow. �

Proof of Corollary 3.2. The first two points in the corollary follow directly from

Theorem 3.1. To see the third claim, consider that a well known fact in order sta-

tistics theory is that
∑

l E[U
(l)] = kE[Uh] for any alternative h. Consequently, since

the symmetry of the density function implies that E[Uh] = E[U(2)] = 1
2
, it follows

immediately that E[U(l+1)] − E[U(l)] = c with 1 ≤ l ≤ 2. Then, from Theorem 3.1,
Borda is the αUT -optimal decision rule. �

Proof of Theorem 3.5. We report the proof for the uniform distribution. The cor-

responding proof for any other parabolic function is analogous, and hence is omitted.

We organize the proof by types of societies in ordinal terms.

Type 1: Perfect correlation. This is the case when M1 = M2. Let, w.l.o.g.,

M1 = (3, 2, 1). Utilitarianism always selects alternative 1. Hence, βUT (M) = (1, 0, 0).



18

Type 2: Only best-correlation. Both individuals agree on the best alternative, but

disagree on the others. Let, w.l.o.g., M1 = (3, 2, 1) and M2 = (3, 1, 2). Utilitarianism

selects alternative 1, and then βUT (M) = (1, 0, 0).

Type 3: Only worst-correlation. Both individuals coincide in the worst alternative,

but disagree on the others. Let, w.l.o.g., M1 = (3, 2, 1) and M2 = (2, 3, 1). It is

straightforward to see that βUT (M) = (1/2, 1/2, 0).

Type 4: Only middle-correlation. Both individuals agree on the middle alternative,

but disagree on the others.7 Let, w.l.o.g., M1 = (3, 2, 1) and M2 = (1, 2, 3). To

compute the selection of utilitarianism we first need to derive the joint probability

distribution associated to the continuous random variables U
(1)
i , U

(2)
i and U

(3)
i .

The joint probability density function is

f(u
(1)
i ,u

(2)
i ,u

(3)
i ) =

{
6 if 1 ≥ u(3)i ≥ u

(2)
i ≥ u

(1)
i ≥ 0

0 otherwise.

Hence, the density function for the two agents is

f(u
(1)
i ,u

(2)
i ,u

(3)
i ,u

(1)
j ,u

(2)
j ,u

(3)
j ) =




36 if 1 ≥ u(3)i ≥ u
(2)
i ≥ u

(1)
i ≥ 0, and

1 ≥ u(3)j ≥ u
(2)
j ≥ u

(1)
j ≥ 0

0 otherwise.

We can now calculate the probability of utilitarianism resulting in the selection of

alternative 2. Alternative 2 will be the utilitarian winner whenever the sum of U
(2)
i

and U
(2)
j is greater than U

(1)
i + U

(3)
j and U

(3)
i + U

(1)
j . Equivalently,

(A.1)∫
R

∫
R

∫
R

∫
R

∫
Ha

∫
Hb

f(u
(1)
i ,u

(2)
i ,u

(3)
i ,u

(1)
j ,u

(2)
j ,u

(3)
j )du

(3)
i du

(3)
j du

(2)
i du

(2)
j du

(1)
i du

(1)
j ,

where Ha and Hb are the sets Ha = {t : t ≤ u
(2)
i + u

(2)
j − u

(1)
i } and Hb = {t : t ≤

u
(2)
i + u

(2)
j −u

(1)
j }. Then, (A.1) can be expressed as

(A.2)∫ 1
0

∫ 1
0

∫ 1
u
(1)
j

∫ 1
u
(1)
i

∫ min{1,u(2)i +u(2)j −u(1)i }
u
(2)
j

∫ min{1,u(2)i +u(2)j −u(1)j }
u
(2)
i

36du
(3)
i du

(3)
j du

(2)
i du

(2)
j du

(1)
i du

(1)
j .

It can be shown that the above is .3 (for any parabolic distribution, the value of

the integral is strictly lower than 1/3). Clearly, the utilitarian choice is, with equal

probability, either alternative 1 or 3. Then, βUT (M) = (.35, .3, .35) (for the remaining

parabolic functions, β1UT (M) = β
3
UT (M) > β

2
UT (M).

7This type of societies has attracted a good deal of attention. Interestingly, Börgers and Postl
(2007) study whether for this class of societies there are incentive-compatible rules which elicit
utilities and implement efficient decisions.



19

Type 5. No correlation. Individuals do not agree on the ranking of any alternative.

Let, w.l.o.g., M1 = (3, 2, 1) and M1 = (2, 1, 3). Hence, alternative 1 dominates

alternative 2, and consequently the utilitarian choice will never be alternative 2. We

now show that alternative 1 has a higher probability of being the utilitarian choice

than alternative 3. For any society where alternative 3 is better in utilitarian terms,

a society can be injectively constructed such that: (i) it is of type 5, and (ii) the

utilitarian choice is alternative 1. Simply write u
′1
1 = u

3
2, u

′2
1 = u

1
2, u

′3
1 = u

2
2, u

′1
2 = u

2
1,

u
′2
2 = u

3
1, and u

′3
2 = u

1
1.

8 Clearly, the constructed society is of type 5, and to see that

the utilitarian choice is alternative 1, notice that u
′1
1 + u

′1
2 = u

3
2 + u

2
1 > u

3
2 + u

3
1 >

u11 + u
1
2 > u

1
1 + u

2
2 = u

′3
2 + u

′3
1 . Then, β

1
UT (M) > β

3
UT (M) > β

2
UT (M) = 0.

Taking into consideration all five cases together, implosion is the βUT -optimal gen-

erator. This proves the first part of the theorem. The second part follows immediately

from the above analysis.�

Proof of Theorem 4.1. First note that only alternatives ranked best by some

agents can provide the highest maximax value. In fact, due to the i.i.d. nature of

utility realizations across individuals, for all M ∈M and for all h,l ∈ K

βlMX(M) ≥ β
h
MX(M) ⇔ l

(k) ≥ h(k).

It is immediate that every decision rule f in δ(SPl) is a βMX-optimal decision rule

since it selects precisely that subset of the alternatives that are ranked best by the

largest number of individuals. Furthermore, for every f 6∈ δ(SPl) there exists an
M ∈ M and an alternative h ∈ K such that h ∈ f(M) \ SPl(M). Consequently,
there is an l ∈ K such that l(k) > h(k) and hence f is not a βMX-optimal decision
rule.�

Proof of Corollary 4.2. We prove the corollary by showing that, for any given K,

when n tends to infinity, plurality is the only scoring rule in δ(SPl). Take a scoring

rule f different from plurality. We now find a real number nf such that for every

positive integer n with n > nf , there exists a profile M ∈M with f(M) 6⊆ SPl(M).
Given that f is not plurality, Sk−1 > 0. Take nf = 2(

1
Sk−1
− 1), and n > nf . If

n is odd, consider the following matrix M: Mi = (k,k − 1,k − 2, . . . , 2, 1) for all
1 ≤ i ≤ n+1

2
, and Mj = (1,k,k − 1, . . . , 3, 2) for all n+32 ≤ j ≤ n. Plurality selects

alternative 1. Scoring rule f assigns to alternative 1 a total score of n+1
2

, while the

total score of alternative 2 is n−1
2

+ n+1
2
Sk−1. We now show that n−1

2
+ n+1

2
Sk−1 > n+1

2
,

or equivalently n+1
2
Sk−1 > 1. Our original assumption that n > nf = 2(

1
Sk−1

− 1)
guarantees the latter inequality.

If n is even, consider the matrix M with Mi = (k,k − 1,k − 2, . . . , 2, 1) for all
1 ≤ i ≤ n

2
, and Mj = (1,k,k − 1, . . . , 3, 2) for all n+22 ≤ j ≤ n − 1 and Mn =

8Note, that under i.i.d., this argument is distribution free.



20

(1,k − 1,k − 2, . . . , 2,k) . An argument analogous to the one above shows that
plurality selects alternative 1 while f selects alternative 2.

Then, we have shown that, for n > nf , decision rule f is not a βMX-optimal scoring

rule, which proves the result.�

Proof of Lemma 4.3. The distribution function of a uniform k order statistic is

F (k)(t) = tk and of a (k−1) order statistic is F (k−1)(t) = tk+ktk−1(1−t). Now, the dis-
tribution function of Ul(n) when M

l such that mli = k for every i ∈{1, . . . ,n}, is F(t) =
tnk. Then, E[Ul(n) | M] =

∫ 1
0
t
dF(t)
dt
dt = nk

nk+1
. Analogously, E[Ul(n+1) | M

′] =
(n+1)k

(n+1)k+1
,

where m
′l
i = k for every i ∈ {1, . . . ,n,n + 1}. Finally, the distribution function of

Ul(n+1) when M
l

with mli = k for every i ∈{1, . . . ,n,n + 1}\{j} and mlj = k − 1, is
F(t) = tnk(tk+ktk−1(1−t)). Then, E[Ul(n+1) | M] =

∫ 1
0
t
dF(t)
dt
dt =

k(n2k+2nk+k−1)
(nk+k)(nk+k+1)

. Con-

sequently, E[Ul(n+1)|M
′l] − E[Ul(n)|M

l] = k
(nk+1)(nk+k+1)

and ∆
(k−1)
n = E[Ul(n+1)|M

l
] −

E[Ul(n)|M
l] =

k(k−1)
(nk+1)(nk+k)(nk+k+1)

. Then, the result follows.�

Proof of Proposition 4.4. We start by proving point (1). For every individual

i, the distribution functions of her best, medium and worst random variables are

F (3)(t) = t3, F (2)(t) = t3 + 3t2(1 − t), and F (1)(t) = 1 − (1 − t)3, respectively. The
distribution function of the maximum of l(3), l(2) and n−l(3)−l(2) = l(1) best, medium
and worst random variables is

F(t) = (t3)l
(3)

(t3 + 3t2(1 − t))l
(2)

(1 − (1 − t)3)l
(1)

.

The polynomial decomposition of the latter expression gives

F(t) =
l(2)∑
a=0

l(1)∑
b=0

3b∑
c=0

(
l(2)

a

)(
l(1)

b

)(
3b

c

)
(−1)a+b+c3l

(2)−a2at3l
(3)+2l(2)+a+c.

The density function is

f(t) =
l(2)∑
a=0

l(1)∑
b=0

3b∑
c=0

(
l(2)

a

)(
l(1)

b

)(
3b

c

)
(−1)a+b+c3l

(2)−a2a(3l(3)+2l(2)+a+c)t3l
(3)+2l(2)+a+c−1.

Finally, the expected value is

E[Ul(n)|M] =
l(2)∑
a=0

l(1)∑
b=0

3b∑
c=0

(
l(2)

a

)(
l(1)

b

)(
3b

c

)
(−1)a+b+c3l

(2)−a2a
3l(3) + 2l(2) + a + c

3l(3) + 2l(2) + a + c + 1
.

We start the proof of part (2) of the theorem by stating the following lemma.

Lemma A.1. Let {xa}a∈T∪{b}, b /∈ T 6= ∅ be a set of independent random variables.
Then E[max{xa}a∈T∪{b}]−E[max{xa}a∈T ] ≤ E[max{xa}a∈S∪{b}]−E[max{xa}a∈S] for
all S ⊆ T .



21

The lemma is straightforward, and hence we omit its proof. Since p ∈ SPl(M) it
must be that p(3) ≥ h(3). Let h(3) < p(3) − 1. Vector (x,y,z) represents the case
where x = p(3), y = p(2), and z = p(1). Then, it is immediate that E[Up

(n)
| M] ≥

E[(p(3), 0, 0)].9 Since h(3) < p(3)−1 it is immediate that E[Uh(n) | M] ≤ E[(p
(3)−2,n−

p(3) + 2, 0)]. By Lemma A.1 E[(p(3) − 2,n− p(3) + 2, 0)] ≤ E[(p(3) − 2, 0, 0)] + (n−
p(3) + 2)∆

(2)

p(3)−2. We now prove that E[(p
(3), 0, 0)] > E[(p(3) − 2,n− p(3) + 2, 0)]. By

showing the latter, we are excluding the possibility of h being an optimal alternative

whenever h(3) < p(3) − 1.
Let us assume by way of contradiction that E[(p(3), 0, 0)] ≤ E[(p(3) − 2,n− p(3) +

2, 0)]. Note E[(p(3), 0, 0)] = E[(p(3)−2, 0, 0)] + ∆(3)
p(3)−2 + ∆

(3)

p(3)−1. Then it must be that

∆
(3)

p(3)−2 + ∆
(3)

p(3)−1 ≤ (n− p
(3) + 2)∆

(2)

p(3)−2. The latter leads to n ≥
(3p(3)−3)(3p(3)−2)

(3p(3)+1)
+

p(3) − 2. Now, since p is a plurality alternative, n ≤ 3p(3) − 2. Then it must be that
3p(3)−2 ≥ (3p

(3)−3)(3p(3)−2)
(3p(3)+1)

+p(3)−2. This can only be the case if p(3) ≤ 5. For all 2 ≤
p(3) ≤ 5, and for all p(3) ≤ n ≤ 3p(3)−2 the direct evaluation of the expression in point
(1) of the proposition shows that E[(p(3), 0, 0)] > E[(p(3) −2,n−p(3) + 2, 0)], hence, a
contradiction arises, and then it must be that E[(p(3), 0, 0)] > E[(p(3)−2,n−p(3)+2, 0)]
for all n and all p defined as in the proposition, as desired.

The latter shows that it can only be either h(3) = p(3) or h(3) = p(3) − 1. In
the first case it is immediate that, for h to be an optimal alternative, then it can

only be that h(2) ≥ p(2). For the second case, note that E[Uh(n) | M] = E[(p
(3) −

1, h(2),n−p(3) + 1−h(2))]. Now, by Lemma A.1 E[(p(3)−1, h(2),n−p(3) + 1−h(2))] ≤
E[(p(3)−1, 0, 0)]+h(2)∆(2)

p(3)−1 +(n−p
(3) +1−h(2))∆(1)

p(3)−1. Now, since p is the plurality

alternative, it must be that n ≤ 3p(3)−1. Hence, (n−p(3) +1−h(2)) ≤ 2p(3)−h(2). For

all p(3) ≥ 1, 2p(3)−h(2) ≤ 3p(3)−1. It can be checked that
∆

(2)

p(3)−1

∆
(1)

p(3)−1

= 3p(3)−1. Hence,

(n− p(3) + 1 − h(2))∆(1)
p(3)−1 ≤ ∆

(2)

p(3)−1. Then, E[(p
(3) − 1, h(2),n− p(3) + 1 − h(2))] ≤

E[(p(3) −1, 0, 0)] + (h(2) + 1)∆(2)
p(3)−1. For alternative p, E[U

p
(n)
| M] ≥ E[(p(3), 0, 0)] =

E[(p(3)−1, 0, 0)] + ∆(3)
p(3)−1. Hence, for alternative h to be an αMX-optimal alternative,

it must be that (h(2) + 1)∆
(2)

p(3)−1 ≥ ∆
(3)

p(3)−1. The latter implies that h
(2) ≥ 3p

(3)

2
− 1

and hence h(2) ≥ n−1
2

, as desired.�

Proof of Proposition 4.6. We start by noticing that, for every l ∈ P(M), index
γlMX = 1. This follows trivially from the fact that, for every l ∈ P(M), there is an
h ∈ K (possibly a different h for every l) such that Ml > Mh and hence Ul(n) > U

h
(n).

We now show that for every l ∈ K \P(M), index γlMX < 1. Suppose, w.l.o.g. that
l = 1. Since 1 6∈ P(M) for every h > 1 there is an agent ih such that m1ih < m

h
ih

.

9Note that vector (p(3), 0, 0) may not be feasible in M.



22

Denote all these agents by I = ∪h∈K\{1}{ih}. P(U1(n) < U
h
(n) for all h > 1 | M) ≥

P(U1(n) = U
1
j for any j ∈ I | M) ·P(U2(n) = U

2
i2
| M) · . . . ·P(Uk(n) = U

k
ik
| M). Clearly,

the latter is strictly above 0, and the result follows.�

Proof of Theorem 4.5. Take the structure of the proof of Theorem 3.5. Here,

too, we provide the proof for the uniform distribution. The values of γMX(M) for

any other parabolic function are exactly the ones provided below.

Then, the values of γMX(M) for the different types of societies are:

Type 1: It is straightforward that γMX(M) = (1, 1, 0).

Type 2: It is straightforward that γMX(M) = (1, 1/2, 1/2).

Type 3: It is straightforward that γMX(M) = (1, 1, 0).

Type 4: It is easy to appreciate that alternative 2 can never be the maximax

alternative, and hence γMX(M) = (1, 0, 1).

Type 5: It is straightforward that 1 = γ1MX(M) > γ
3
MX(M) > γ

2
MX(M).

Finally, taking into consideration all five cases together, explosion is an optimal

decision rule. The second part of the Theorem follows trivially from the above anal-

ysis. �

Proof of Theorem 5.1. (1) Recall that we denote the distribution function of

the l-th order statistic by F (l)(t). First note that P(Ul(1) ≤ t|M) = 1 − P(U
l
(1) >

t|M) = 1 − (1 −F (k)(t))l(k) · . . . · (1 −F (1)(t))l(1) . Given the symmetry of the culture,
it follows that, for every l ∈ K, F (l)(t) = 1 − F (k−l+1)(1 − t). Therefore, P(Ul(1) ≤
t|M) = 1 −F (1)(1 − t)l(k) · . . . ·F (k)(1 − t)l(1) . The latter expression is 1 −P(Ul(n) ≤
t|(k + 1)1 − M). It follows that E[Ul(1) | M] = E[U

l
(n) | (k + 1)1 − M], and hence

αlMN (M) = 1 −α
l
MX((k + 1)1 −M).

(2) Given M, βlMN (M) is the probability that alternative l is the best maximin

alternative. That is, βlMN (M) = P(U
l
(1) > U

h
(1) for all h ∈ K \ {l} | M). Equiv-

alently, βlMN (M) = 1 − P(There exists h ∈ K \ {l} such that U
l
(1) < U

h
(1) | M).

For every symmetric culture we then have that βlMN (M) = 1 − P(There exists h ∈
K \{l} such that Ul(n) > U

h
(n) | (k + 1)1 −M). The latter expression is equivalent to

1 −γlMX((k + 1)1 −M).
(3) Given M, γlMN (M) is the probability that alternative l is not the worst max-

imin alternative. That is, γlMN (M) = 1 − P(U
l
(1) < U

h
(1) for all h ∈ K \ {l} | M).

Equivalently, γlMN (M) = P(There exists h ∈ K \ {l} such that U
l
(1) > U

h
(1) | M).

For every symmetric culture we then have that γlMN (M) = P(There exists h ∈
K \ {l} such that Ul(n) < U

h
(n) | (k + 1)1 − M). The latter expression is equiva-

lent to 1−P(Ul(n) > U
h
(n) for all h ∈ K\{l} | (k+1)1−M) = 1−β

l
MX((k+1)1−M).�



23

Appendix B. Computational Analysis10

In this section we provide the technical details of the computational analysis, and

report some further results.

We study societies of size n×k, with n×k ∈{2, 3, 5, 10, 100}×{3, 4, 5, 7}, and three
different cultures. First, we consider societies where the utility values of the individu-

als are drawn independently from a uniform distribution on the interval [0, 1]. Second,

we consider the standard normal distribution in the interval (−∞,∞). Finally, to
study societies with cultures having asymmetric density functions, we study the case

where the utility values are drawn independently from an exponential distribution

with λ = 1.

For every single size of society and culture, we independently draw 3, 841, 600 car-

dinal societies. Then, for every combination of size of society and culture, we check

the performance of a set of scoring rules across all the 3, 841, 600 simulated cardinal

societies. The set of scoring rules is defined as follows. We randomly generate 47

scoring rules by independently drawing values from a uniform distribution on the in-

terval [0, 1]. Apart from this 47 randomly-generated scoring rules, our analysis always

includes the three most prominent ones: plurality, Borda, and negative. Finally, for

the case of the analysis of utilitarianism, we also include the optimal scoring rules as

predicted in Theorem 3.1.

Since the β and γ indices measure probabilities of success, we use interval estimation

methods for binomial distributions. Our sample size provides a confidence interval of

length .001 around the value of the corresponding index,11 assuming that the variance

of the value of the index is the highest possible for a binomial distribution (i.e., .25).

This is a conservative method to compare significant differences across scoring rules.12

For the α−index, we estimate both the value of the index and its variance in order
to construct confidence intervals that depend on the estimated variance.13

The simulations complement the theoretical results obtained in the text by studying

the following cases:

• Utilitarianism: βUT for the uniform, normal and exponential distributions,
and γUT for the exponential distribution.

10We are grateful to the Advanced Computing Center for Research and Education at Vanderbilt
University for providing the necessary computing resources.

11To determine that the difference between two values is significantly different than zero, this
method provides a confidence interval of length .001 × 21/2 for such difference.

12The use of this method might provide unnecessarily wide intervals; however, it allows us to find
significant differences across scoring rules without having to estimate their covariance.

13Given our sample size, with the exponential distribution and the normal distribution the length
of these intervals varies between .002 and .004 for the case of maximax and between .0002 and .002
for the case of maximin. With the uniform distribution, the length of the interval varies between
.00001 and .0005. Note that the differences across distributions are due to their different ranges.



24

• Maximax: αMX for the uniform (with k > 3), normal and exponential distri-
butions, and γMX for the exponential distribution.

• Maximin: βMN for the uniform, normal and exponential distributions, αMN
for the exponential distribution, and γMN for the exponential distribution.

Table 1 reports the βUT and γUT values of the optimal scoring rules according to

Theorem 3.1. Moreover, for the sake of comparison, the table also includes the βUT
and γUT -values that the three most prominent scoring rules, plurality, Borda and neg-

ative, attain in the simulations. Accordingly, Tables 2 and 3 report the corresponding

values attained by plurality, Borda and negative.

As shown in Tables 1, 2 and 3, for each of the three ideals of justice, each probability

distributions, and each performance index, the scoring rule that provides the highest

value is generally the one predicted in the theoretical results presented in the text.

This means that the scoring rule determined by Theorem 3.1 is the best in the case

of utilitarianism, plurality is the best in the case of maximax, and negative is the

best in the case of maximin, across probability distributions. There are only some

exceptions for small society sizes, in which the frequency of ties in the number of

points sometimes causes distortions.

More specifically, for utilitarianism the scoring rule predicted in Theorem 3.1 pro-

vides a higher value than not only plurality and negative (and Borda, when Borda is

not the S∗) but also than the best randomly-generated scoring rule. In addition, we

find that S∗ cannot be rejected as the best-performing scoring rule.14

For maximax, plurality has a significantly higher expected value and a significantly

higher probability of not selecting the worst alternative than Borda and negative, as

predicted. There are only a few exceptions for small societies, wherein the frequency

of ties causes distortions. In addition, the best randomly-generated scoring rule is

in a neighborhood of plurality. Moreover, as the size of the society increases (i.e., as

the probability of ties decreases) the values of plurality and of the best randomly-

generated scoring rule converge. In particular, for societies of 100 individuals we find

that plurality is significantly better than the best randomly-generated scoring rule.

For maximin, when the number of individuals is larger than the number of alter-

natives, the results are analogous to the case of maximax. That is, negative has a

significantly higher expected value, a significantly higher probability of selecting the

best alternative, and a significantly higher probability of not selecting the worst alter-

native than Borda and plurality. Furthermore, the best randomly-generated scoring

rule is in a neighborhood around negative. In contrast, notice that for societies with n

below or around k, it is likely than more that one alternative is not the worst for any of

the individuals. In such cases, negative chooses randomly between these alternatives,

which is not necessarily optimal and is the cause of some distortions. As the size of

14We find a few exceptions in small societies due to the frequency of ties.



25

the society increases, this effect vanishes. In particular, for societies of 100 individuals

we find that negative is significantly better than the best randomly-generated scoring

rule.



26

T
a
b
l
e

1
.

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ti

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ta

ri
a
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is

m
:

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in
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fr
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s†

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∗

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2
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8
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8
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1

2
×

5
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4
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2
1
4
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4
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1

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0
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9
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3
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2

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3
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3

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2
×

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4
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3
×

3
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3
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6
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3
×

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5
3
3
2

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9
2
8
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4
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1
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6

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4
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7
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3
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5
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4
7
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3
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3
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6
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5
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3
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7
9
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4
6
6
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3
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1

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8
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5
×

4
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4
5
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0
3
3
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4
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5

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1
5
9
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2
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2
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4
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8
2
2
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6
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5
×

5
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4
1
8
6

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4
3
3
8

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4
1
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6

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1
9
2
9

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2
5
3
9

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5
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1
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1
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4
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8
2
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2

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1
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3
2
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9
4
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8
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8
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9
4
4
5

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1
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5
×

7
.7

4
5
2
3

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5
0
5
6

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4
5
2
3

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5
8
4
6

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2
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8
1
1
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1
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5
6
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8

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2
6
7
8

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2
8
1
3

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7
1
0
2

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1
1
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7

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9
9
4
5

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9
1
4
9

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9
9
4
7

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4
2
0
2

1
0
×

3
.7

5
8
3
6

.6
8
5
9
3

.7
5
8
3
6

.6
4
7
4
9

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4
4
1
6

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7
7
4
1

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4
4
1
6

.6
3
7
9
5

.6
8
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2

†
S

P
l,
S

B
d
,
S

N
g

d
en

ot
e

p
lu

ra
li

ty
,

B
or

d
a,

an
d

n
eg

at
iv

e,
re

sp
ec

ti
ve

ly
.



29

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30

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