Discussion Paper No. 1059

THE WINNER-TAKE-ALL DILEMMA

Kazuya Kikuchi
Yukio Koriyama

Revised December 2019
June 2019

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

The Winner-Take-All Dilemma∗
Kazuya Kikuchi†

Yukio Koriyama‡

December 11, 2019

Abstract
This paper considers collective decision-making when individuals are partitioned into groups (e.g., states or parties) endowed withvoting weights. We study a game in which each group chooses aninternal rule that specifies the allocation of its weight to the alternatives as a function of its members’ preferences. We show that underquite general conditions, the game is a Prisoner’s Dilemma: while
the winner-take-all rule is a dominant strategy, the equilibrium isPareto dominated. We also show asymptotic Pareto dominance ofthe proportional rule. Our numerical computation for the US Electoral College verifies the sensibility of the asymptotic results.

JEL classification: C72, D70, D72∗

For their fruitful discussions and useful comments, we are thankful to Pierre Boyer,
Micael Castanheira, Quoc-Anh Do, Shuhei Kitamura, Michel Le Breton, Shintaro Miura,
Takeshi Murooka, Matı́as Núñez, Alessandro Riboni, and the seminar participants at
11th Pan Pacific Game Theory Conference, CREST, Hitotsubashi Summer Institute,
Institut Henri Poincaré, Institute of Social and Economic Research at Osaka University,
Osaka School of International Public Policy Lunch Seminar, Parisian Political Economy
Workshop, Philipps-University of Marburg, Transatlantic Theory Workshop, Univerisityof Montpelier, University of Tokyo, Waseda University and 2019 Autumn Meeting ofJapanese Economic Association. Financial support by Investissements d’Avenir, ANR11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047 and DynaMITE: Dynamic Matching and Interactions: Theory and Experiments, ANR-13-BSHS1-0010, Waseda University Grant for Special Research Projects (2018K-014), JSPS KAKENHI Grant Numbers
JP17K13706 and JP15H05728 is gratefully acknowledged.
†
Osaka University. E-mail: kazuya.kikuchi68@gmail.com.
‡
CREST, Ecole Polytechnique. E-mail: yukio.koriyama@polytechnique.edu.

1Keywords: Representative democracy, winner-take-all rule, proportional rule1

Introduction

A fundamental question about representative democracy is how social decisions should reflect the opinions of individuals belonging to distinct groups,
such as states or parties. Typically, each group has a voting weight, in theform of a number of representatives or a weighted vote assigned to a uniquerepresentative. The groups allocate the weights to decision alternatives,
and the one that receives the most weight becomes the social decision. In
such cases, the quality of social decision-making depends not only on theapportionment of weights among the groups, but also on the rules that allocate the groups’ weights to alternatives, based on the preferences of theirindividual members. The present paper is concerned with how the weight
allocation rules aﬀect individuals’ welfare.
Existing institutions use diﬀerent weight allocation rules. On the one
hand, the winner-take-all rule devotes all the weight of a group to thealternative preferred by the majority of its members. Most states in theUntied States use this rule to allocate presidential electoral votes. A council of national ministers, each with a weighted vote (e.g., the Council of
the European Union), is another example, provided the ministers can bethought of as representing their countries’ majority interests. Party discipline frequently observed in legislative voting may also be understood as
the winner-take-all rule used by parties.
On the other hand, the proportional rule allocates a group’s weight inproportion to the number of members who prefer the respective alternatives. In many parliamentary institutions at the national or international
level, each constituency (e.g., state or prefecture) elects a set of representatives whose composition more or less proportionally reflects its residents’ preferences. Alternatively, when the representatives are viewed as
standing for parties rather than states or prefectures, the proportional rule
corresponds to a party’s rule that allows its representatives to vote for oragainst proposals based on their own preferences, provided the composition of the party’s representatives proportionally reflects the opinions of all2

party members.
The weight allocation rules are often exogenously given to all groups,
but there are also cases where each group chooses its own rule. For instance,
in national parliaments, how the representatives are elected from the respective constituencies is stipulated by national law. By contrast, parties
often have control over how their representatives vote, by punishing those
who violate the party lines. As another example, the US Constitution stipulates that it is up to each state to decide the way in which the presidential
electoral votes are allocated (Article II, Section 1, Clause 2).
If groups are allowed to choose their rules, it is possible that each group
has an incentive to allocate the weight so as to increase the influence ofits members’ opinions on social decisions, at the cost of the other groups’
influence. It is not clear whether such an incentive at the group level iscompatible with desirable properties of the overall preference aggregation,such as eﬃciency. To address this issue, we need to model the choice ofrules as a non-cooperative game.
In this paper, we consider a model of social decision-making whereindividuals are partitioned into groups endowed with voting weights. The
society makes a binary decision through two stages: first, all individuals
vote; then each group allocates its weight to the alternatives, based on thenumber of votes they received from the group’s individual members. The
winner is the alternative with the most weight. A rule for a group is afunction that maps each possible vote result in the group to an allocationof its weight to the alternatives. Examples are the winner-take-all andproportional rules. A profile is a specification of rules for all groups. We
study the game in which the groups independently choose their rules, so as
to maximize their members’ expected welfare.
The main result of this paper is that the game is a n-player Prisoner’s
Dilemma (Theorem 1). On the one hand, the winner-take-all rule is adominant strategy, i.e., it is an optimal strategy for each group regardlessof the rules chosen by the other groups. On the other hand, if each group
has less than a half of the total weight, then the winner-take-all profile isPareto dominated, i.e., some other profile makes every group better oﬀ. In
brief, no group has an incentive to deviate from the winner-take-all rule, butevery group will be better oﬀ if all groups jointly move to another profile.
3

The dilemma structure exists for any number of groups (> 2) and with little
restriction on the joint distribution of preferences. Individuals’ preferences
may be biased, and also correlated within and across groups, or not, which
would be true when the groups are parties with diﬀerent but overlapping
political goals, or states that tend to support specific alternatives, e.g.,
blue, red or swing states in the US elections.
The observation that the winner-take-all rule is a dominant strategy
is consistent with the fact that it has been dominantly employed by thestates in the US Electoral College since 1830s in order to allocate presidential electoral votes,1 and also with the widely observed party disciplinein assemblies. Despite the various problems or limitations that have been
pointed out concerning the winner-take-all rule,2 it is still used prevalently.
Our conventional knowledge that direct majority voting by all individuals maximizes the utilitarian welfare of the society is not suﬃcient to see
whether every group is better oﬀ under the proportional profile than thewinner-take-all profile. We provide a counterexample later (Example 1):a small group may be strictly better oﬀ under the winner-take-all profile
than the proportional profile. Indeed, this is an oft-used argument by thesmall states in the US, on which their support for the winner-take-all ruleis based. The welfare criterion used in Theorem 1 (ii) is Pareto dominance,
which is obviously stronger than the utilitarian welfare evaluation: there
exists a profile under which every group is better oﬀ than the winner-takeall profile. Example 1 shows that it is not necessarily the proportional
profile. Then, what profile Pareto dominates the winner-take-all profile?
1

One of the recent attempts of reform by a state took place in 2004, when a ballot
initiative for an amendment to the state constitution was raised in Colorado. The suggested procedure is the proportional rule, in which the state electoral votes are allocatedproportionally to the state popular votes. The amendment did not pass, garnering only
34.1% approval.
2
Some of the major arguments against the winner-take-all rule are the following.
First, the winner of the election may be inconsistent with that of the popular votes(May (1948), Feix et al. (2004)). Such a discrepancy has happened five times in thehistory of the US presidential elections, including recently in 2000 and 2016. Second,it may cause reduced dimensionality: (i) the parties have an incentive to concentratecampaign resources only in the battleground states, and (ii) voters’ incentive to turnout or to invest in information may be small and/or uneven across states, since theprobability of each voter to be pivotal is so small under the winner-take-all rule, andeven smaller in the non-swing states. Although campaign resource allocation and voter
turnout are important issues, they are beyond the scope of this paper.

4

A full characterization of the Pareto set is provided in Lemma 1.
To further study welfare properties, we turn to an asymptotic and normative analysis of the model. We consider situations where the numberof groups is suﬃciently large, and the preferences are independent acrossgroups and distributed symmetrically with respect to the alternatives. In
this case, we show that the proportional profile Pareto dominates every
other symmetric profile (i.e., one in which all groups use the same rule),
including the winner-take-all one. The assumptions on the preference distribution abstract from the fact that in reality, some groups tend to prefer specific alternatives. Such an abstraction would be reasonable on the
grounds that normative judgment about rules should not favor particulargroups because of their characteristic preference biases. To see how manygroups are typically suﬃcient for the asymptotic result, we provide numerical computations in a model based on the US Electoral College, using thecurrent apportionment of electoral votes. The numerical comparisons indicate that the proportional profile does Pareto dominate the winner-take-all
profile in the model with fifty states and a federal district.
While the above result suggests that the proportional profile asymptotically performs well in terms of eﬃciency, it is silent about the equalityof individuals’ welfare. In fact, our model also provides some insight into
how rules aﬀect the distribution of welfare. We examine an asymmetricprofile called the congressional district profile. This profile is inspired by
the Congressional District Method currently used by Maine and Nebraska,
in which two electoral votes are allocated by the winner-take-all rule andthe remaining ones are awarded to the winner of each district-wide popular
vote.3 We show that the congressional district profile achieves a more equal
distribution of welfare than any symmetric profile by making individuals
in smaller groups better oﬀ.
A technical contribution of this paper is to develop an asymptotic
method for analyzing players’ expected welfare in weighted voting games.
3

The idea of allocating a part of the votes by the winner-take-all rule and allowing
the rest to be awarded to potentially distinct candidates can be seen as a compromise
between the winner-take-all and the proportional rules. Symbolically, the two votes
allocated by the winner-take-all rule is the same number as the Senators in each state,while the rest is equal to the number of the House representatives. The idea behindsuch a mixture is in line with the logic supporting bicameralism, which is supposed toprovide checks and balances between the states and the federal governance.

5One of the major challenges in analyzing such games is their discreteness.
By the nature of combinatorial problems, obtaining an analytical resultoften requires a large number of classifications by cases, which may include
prohibitively tedious and complex tasks in order to obtain general insights.
We overcome this diﬃculty by considering asymptotic properties of games
in which there are a suﬃciently large number of groups. This techniqueallows us to obtain an explicit formula that captures the asymptotic behavior of the probability of success for each individual, which holds for awide class of distributions of weights among groups (the correlation lemma:
Lemma 2).

1.1

Literature Review

The incentives for groups to use the winner-take-all rule have been studied
by several papers. Hummel (2011) and Beisbart and Bovens (2008) analyze
models of the US presidential elections. Gelman (2003) and Eguia (2011a,b)
give theoretical explanations as to why voters in an assembly form parties
or voting blocs to coordinate their votes. Their findings are coherent with
our observation that the winner-take-all rule is a dominant strategy.
Beisbart and Bovens (2008) and Gelman (2003) also contain comparisons of the winner-take-all and proportional profiles. Under the current
apportionment of electoral votes in the US, Beisbart and Bovens (2008) numerically compares these profiles, in terms of inequality indices on citizens’
voting power and the mean majority deficit, on the basis of a priori and aposteriori voting power measures. Gelman (2003) compares the case withcoalitions of equal sizes in which voters coordinate their votes to the casewithout such coordination. Our analysis is based on Pareto dominance between profiles, and provides results which hold under general distributionof groups’ weights or sizes. In that sense, Beisbart and Bovens’s positive
analysis is complementary to our normative analysis of properties of the
proportional profile.
De Mouzon et al. (2019) provides a welfare analysis of popular voteinterstate compacts, and shows that, for the regional compact, welfare of
the member states is single-peaked as a function of the number of the participating states, while it is monotonically decreasing for the non-member

6states. The second eﬀect dominates in terms of the social welfare, unless alarge majority (approximately more than 2/π ≃ 64%) of the states join thecompact, implying that a small- or middle-sized regional compact is welfare detrimental. For the national compact, the total welfare is increasing,as it turns out that even the non-members would mostly benefit from thecompact, implying that the social optimum is attained when a majority
joins the compact, i.e. the winner is determined by the national popular
vote. Their findings are coherent with ours: if the winner-take-all rule isapplied only to a subset of the groups, then the member states enjoy thebenefit at the expense of the welfare loss of the non-member states, andthe total welfare decreases. The social optimum is attained when the entire
nation uses the popular vote.
The winner-take-all rule has been a regular focus of the literature. The
history, objectives, problems, and reforms of the US Electoral College aresummarized, for example, in Edwards (2004) and Bugh (2010). One of themost scrutinized problems of the Electoral College is its reduced dimensionality. The incentive of the candidates to concentrate their campaign
resources in the swing and decisive states is modeled in Strömberg (2008),
which is coherent with the findings of the seminal paper in probabilistic
voting by Lindbeck and Weibull (1987). Strömberg (2008) also finds that
uneven resource allocation and unfavorable treatment of minority states
would be mitigated by implementing a national popular vote, which iscoherent with the classical findings by Brams and Davis (1974). Voters’
incentive to turn out is investigated by Kartal (2015), which finds that thewinner-take-all rule discourages turnout when the voting cost is heterogeneous.
Constitutional design of weighted voting is studied extensively in theliterature. Seminal contributions are found in the context of power measurement: Penrose (1946), Shapley and Shubik (1954), Banzhaf (1968)
and Rae (1946). Excellent summaries of theory and applications of power
measurement are given by, above all, Felsenthal and Machover (1998) and
Laruelle and Valenciano (2008). The tools and insights obtained in the
power measurement literature are often used in the apportionment problem: e.g., Barberà and Jackson (2006), Koriyama et al. (2013), and Kurz
et al. (2017).
7

2

The Model

Let us consider a society which consists of n disjoint groups: i ∈ {1, 2, · · · , n}.
The society makes a collective decision between two alternatives, denoted−1 and +1. For instance, the alternatives may represent presidential candidates in a two-party system, or the status quo and a proposal in a legislature. Each group i is endowed with a weight wi > 0.
Let Θi ∈ [−1, 1] be the random variable which represents the groupwide margin, i.e., the fraction of members of i preferring alternative +1
minus the fraction of those preferring −1.4 For instance, Θi = −0.2 meansthat 60% of members of group i prefer alternative −1 and the remaining
40% prefer +1. Since the model is concerned with the weight allocationby each group which aggregates the preferences of its members, it is mostappropriate to suppose that the groups’ aggregation rules are fixed prior
to the realization of the preferences, and hence of the group-wide margins.
The following is the assumption on the joint distribution of the group-wide
margins.
Assumption 1. The joint distribution of group-wide margins (Θi )ni=1 isabsolutely continuous and has full support [−1, 1]n .
Assumption 1 permits a wide variety of joint distributions of individuals’ preferences, in which intra- and inter-group correlations and biases are
possible. First, the assumption imposes no restriction on preference correlations within each group. Second, individuals’ preferences may also becorrelated across groups, since the group-wide margins (Θi )ni=1 can be correlated. This allows us to capture situations where, for instance, residents
of diﬀerent states or members of diﬀerent parties have common interest on
some issues. Third, preferences may be biased toward a particular alternative, since Θi can be asymmetrically distributed. For instance, blue (resp.
red) states in the US might be described as groups whose group-wide margins have a distribution biased to the left (resp. right). In contrast, swing
states might be described as groups whose distributions are concentrated
around zero.
4

Throughout the paper, we use capital Θi for representation of a random variable,
and small θi for the realization.

8

The society decides between the alternatives through two stages: (i)each individual votes for his preferred alternative; (ii) each group allocates
its weight to the two alternatives, based on the group-wide margin. The
winner is the alternative which receives a majority of the weight.
At the second stage, each group’s allocation of weight is determined asa function of the group-wide margin.
Definition 1. A rule for group i is defined as a Borel-measurable5 function:
ϕi : [−1, 1] → [−1, 1].
The value ϕi (θi ) is the fraction of the weight wi allocated to alternative+1 minus that allocated to −1, given that the group-wide margin is θi .
That is, the rule allocates wi ϕi (θi ) more weight to alternative +1 thanalternative −1. For example, if wi = 50 and ϕi (θi ) = −0.2, it means thatthe rule allocates 20 (resp. 30) units of weight to the alternative +1 (resp.
−1).
Let
Φ = {ϕi |Borel-measurable}
be the set of all admissible rules.
Examples of rules. Among all admissible rules, the following ones deserve
particular attention:(i) Winner-take-all rule: ϕWTA
(θi ) = sgn θi .
i(ii) Proportional rule: ϕPRi (θi ) = θi .
(θi ) + (1 − a)ϕPR(iii) Mixed rules: ϕai (θi ) = aϕWTAi
i (θi ), 0 ≤ a ≤ 1.
The winner-take-all rule devotes all the weight of a group to the winning
alternative in the group. The proportional rule allocates the weight inproportion to the vote shares of the respective alternatives in the group.
The mixed rule ϕa allocates the fixed ratio a of the weight by the winnertake-all rule and the remaining 1 − a part by the proportional rule.
5

Borel-measurability is needed to ensure that each ϕi (Θi ) is a well-defined randomvariable.

9A profile ϕ = (ϕi )ni=1 consists of rules specified for all groups. By symmetric profile, we mean that the same rule is used by all groups. For
instance, the above three rules naturally define the following symmetricprofiles: the winner-take-all profile ϕWTA = (ϕWTA)ni=1 , the proportional
i
n
a
a n
profile ϕPR = (ϕPRi )i=1 , and mixed profiles ϕ = (ϕi )i=1 , a ∈ [0, 1].
The winning alternative is the one which obtains more weight from thegroups. In the case of a tie, we assume that both alternatives are chosenwith equal probability. To define it formally, let
Sϕ =

n
∑

wi ϕi (Θi )i=1

be the diﬀerence between the total weight allocated for alternatives +1 and−1. The social decision Dϕ isDϕ =sgn Sif Sϕ ̸= 0ϕ

±1 with equal probabilities if S = 0.
ϕ(1)

The payoﬀ of each individual is 1 or −1, depending on whether she
prefers the social decision or not. We define group i’s (expected) payoﬀ as
the average expected payoﬀ of its members. Since the average payoﬀ ofgroup-i members is Θi or −Θi depending on whether the social decision is+1 or −1, the group’s expected payoﬀ is
πi (ϕ) = E(Θi Dϕ ).
Since each group chooses a rule as a function of the group-wide margin,
maximizing πi (ϕ) with respect to its own rule ϕi is equivalent to maximizing, for almost every θi ∈ [−1, 1], the conditional expected payoﬀ given thegroup-wide margin Θi = θi :
πi (ϕ|θi )= θi E(Dϕ |Θi = θi )= θi (P {Dϕ = +1|Θi = θi } − P {Dϕ = −1|Θi = θi })= θi (P{wi ϕi (θi ) + Sϕ−i > 0|Θi = θi } − P{wi ϕi (θi ) + Sϕ−i < 0|Θi = θi }),(2)10where Sϕ−i =∑6j̸=i wj ϕj (Θj ).

Remark 1. Our definition of group payoﬀs has the following interpretation
based on the members’ preferences. Let Mi be the set of individuals ingroup i, and Xim ∈ {−1, 1} be the preferred alternative of member m ∈ Mi
in group i. Let us here redefine Θi as a latent variable that parametrizes the
distribution of the random preferences in group i. Specifically, suppose Xim
are independently and identically distributed conditional on the realization(θi )ni=1 with the following probabilities for all i = 1, · · · , n and m ∈ Mi :{P {Xim = +1|Θ1 = θ1 , · · · , Θn = θn } = (1 + θi ) /2,
P {Xim = −1|Θ1 = θ1 , · · · , Θn = θn } = (1 − θi ) /2.

(3)

Then, as the group size becomes large (|Mi | → ∞), the Law of Large Num∑bers implies that the group-wide margin M1 i m∈Mi Xim indeed converges
to Θi almost surely, which is consistent with our original definition of Θias the group-wide margin. Moreover,P {Xim = Dϕ } = E [P {Xim = Dϕ |Θ}]= E [P {Xim = 1, Dϕ = 1|Θ} + P {Xim = −1, Dϕ = −1|Θ}]][
1 − Θi
1 + Θi+ P {Dϕ = −1|Θ}= E P {Dϕ = 1|Θ}2
2
1=
(1 + E [P {Dϕ = 1|Θ} Θi + P {Dϕ = −1|Θ} (−Θi )])2
1=(1 + E [Θi Dϕ ]) .
2
Therefore, πi (ϕ) = E (Θi Dϕ ) is an aﬃne transformation of the probability
that the preferred alternative of a member m in group i coincides withthe social decision (Xim = Dϕ ), which is called success in the literature of
voting power measurement (Laruelle and Valenciano (2008)). The objectiveof the group, formulated as the maximization of πi , is thus equivalent tomaximization of the probability of success.
Under the winner-take-all profile ϕWTA , πi is closely related to the classical voting power indices studied in the literature. If (Θi )ni=1 are independently, identically and symmetrically distributed (thus each group’s pre6

The last expression in (2) uses the assumption that ties are broken by tossing a fair
coin.

11

ferred alternative is independently and equally distributed over {−1, +1},
called Impartial Culture), then πi corresponds to the Banzhaf-Penrose index (Banzhaf (1965), Penrose (1946)) and P {Xim = Dϕ } to the Rae index(Rae (1946)), up to a multiplication by the constant E [|Θi |]. If (Θi )ni=1
are perfectly correlated and symmetrically distributed (called Impartial
Anonymous Culture. See, for example, Le Breton et al. (2016)), then πicorresponds to the Shapley-Shubik index (Shapley and Shubik (1954)).
Remark 2. Our specification of group’s payoﬀ may sound at first as if
it excludes the case where individuals have diﬀerent preference intensities.
However, even for the cases in which each group is allowed to use not only
the ordinal, but also the cardinal information of its members’ preferences(as in Barberà and Jackson (2006), Beisbart et al. (2005), Beisbart andBovens (2008), Beisbart and Hartmann (2010)), our assumption comesalong with no loss of generality: it suﬃces to redefine Θi as the average
payoﬀ diﬀerence between alternatives +1 and −1 over group i’s members.

3

The Dilemma

We consider a non-cooperative game Γ in which each group chooses a ruleto allocate its weight to the alternatives. Each group’s objective is tomaximize the average expected payoﬀ of its members. Formally, the game Γis defined as follows. The set of players is the set of groups: {1, · · · , n}. The
strategy space for group i is the set of all rules: {all measurable functions
ϕi : [−1, 1] → [−1, 1]}. The payoﬀ of group i is its per capita expected
payoﬀ: πi (ϕ).
Two rules ϕi and ψi are called equivalent if ϕi (Θi ) = ψi (Θi ) almost
surely. Two profiles ϕ and ψ are called equivalent if Dϕ = Dψ almost
surely.
A rule (or strategy) ϕi for group i dominates another rule ψi if πi (ϕi , ϕ−i ) ≥πi (ψi , ϕ−i ) for any ϕ−i , with strict inequality for at least one ϕ−i . A rule ϕi
is a dominant strategy for group i if it dominates every rule not equivalent
to ϕi . A profile ϕ Pareto dominates another profile ψ if πi (ϕ) ≥ πi (ψ) for
all i, with strict inequality for at least one i. If ϕ is not Pareto dominatedby any profile, it is called Pareto eﬃcient.
12

To show the main theorem, we need the following assumption on the
allocation of weights among the groups.
Assumption 2. Each group has less than half the total weight: wi <∑n1
j=1 wj for all i = 1, · · · , n.
2Theorem 1. Under Assumptions 1 and 2, game Γ is a Prisoner’s Dilemma:(i) the winner-take-all rule ϕWTA
is a dominant strategy for each groupii;(ii) the winner-take-all profile ϕWTA is Pareto dominated.
We use the following lemma to prove the theorem. A generalized proportional profile refers to a profile in which ϕi (θi ) = λi θi , i = 1, · · · , n, forsome vector λ ∈ [0, 1]n \ {0}.
Lemma 1. (Characterization of the Pareto set) Under Assumption 1, a
profile ϕ is Pareto eﬃcient if and only if it is equivalent to some generalized
proportional profile.
Proof of Theorem 1. Part (i). By (2), if θi > 0 (resp. θi < 0), then the conditional expected payoﬀ πi (ϕ|θi ) is non-decreasing (resp. non-increasing)in ϕi (θi ) ∈ [−1, 1]. We thus have πi (ϕWTA, ϕ−i |θi ) ≥ πi (ϕi , ϕ−i |θi ) for anyi(ϕi , ϕ−i ) and θi . Therefore
πi (ϕWTA, ϕ−i ) ≥ πi (ϕi , ϕ−i )ifor any (ϕi , ϕ−i ). Now we show that for any subprofile ϕ−i in which eachϕj : [−1, 1] → [−1, 1] (j ̸= i) is onto (e.g., ϕPRj ), the strict inequality
πi (ϕWTA, ϕ−i ) > πi (ϕi , ϕ−i )i(4). To see this, note that
holds for any rule ϕi that is not equivalent to ϕWTAinfor such ϕ−i , the full-support assumption on (Θj )j=1 implies that the condi∑∑
tional distribution of Sϕ−i given Θi = θi has support [− j̸=i wj , j̸=i wj ].
∑Since wi < j̸=i wj by Assumption 2, formula (2) implies that if θi > 0(resp. θi < 0), then πi (ϕ|θi ) is strictly increasing (resp. decreasing) in, ϕ−i |θi ) > πi (ϕi , ϕ−i |θi ) holds at any θi forϕi (θi ) ∈ [−1, 1]. Thus πi (ϕWTAi13which ϕWTA
(θi ) ̸= ϕi (θi ). Since Θi has full support, this implies that (4)iholds for any ϕi that is not equivalent to ϕWTA.
iPart (ii). By the characterization of the Pareto set (Lemma 1), it
suﬃces to check that ϕWTA is not equivalent to any generalized proportional
profile. Suppose, on the contrary, that ϕWTA is equivalent to a generalized
proportional profile with coeﬃcients λ ∈ [0, 1]n \ {0}. Then, since (Θi )ni=1
has full support,
DϕWTA (θ) = sgnn∑wi λi θi at almost every θ ∈ [−1, 1]n .

(5)

i=1

Since no group dictates the social decision, the coeﬃcients λi are positive
for at least two groups. Without loss of generality, assume λ1 > 0 and
λ2 > 0. Now, fix θi for i ̸= 1, 2 so that they are suﬃciently small in
absolute value. Then, according to (5), for (almost any) suﬃciently small
ε > 0, DϕWTA (θ) = +1 if θ1 = 1 − ε and θ2 = −ε, while DϕWTA (θ) = −1 ifθ1 = ε and θ2 = −1 + ε. This contradicts the fact that DϕWTA (θ) depends
only on the signs of (θi )ni=1 .
Together with Lemma 1, Theorem 1 shows that while the dominant
strategy for each group is the winner-take-all rule, the dominant-strategy
equilibrium is Pareto dominated by some generalized proportional profile.
This typical Prisoner’s Dilemma situation suggests to us that a Pareto eﬃcient outcome is not expected to be achieved under decentralized decision
making, and a coordination device is necessary in order to attain a Pareto
improvement.
If Assumption 2 fails and some group has more than half the total
weight, the winner-take-all profile is Pareto eﬃcient.
Proposition 1. Under Assumption 1, if there exists a group i∗ such that∑wi∗ > 12 nj=1 wj , then the winner-take-all profile ϕWTA is Pareto eﬃcient.
Proof. Under ϕWTA , the social decision always coincides with group i∗ ’s majority preference. Thus ϕWTA is equivalent to the generalized proportional
profile with coeﬃcients λi∗ > 0 and λi = 0 for all i ̸= i∗ . The proposition
follows from the characterization of the Pareto set (Lemma 1).
14

Intuitively, the winner-take-all profile gives dictatorial power to thegroup with more than half the total weight, while any non-equivalent profile creates a positive probability of social decision against that group’s will.
Hence Pareto improvement is impossible. One might further speculate that
Pareto eﬃciency of the winner-take-all profile will still hold as long as there
are groups with suﬃciently large weights, even if no group’s weight exceeds
half the total weight. Theorem 1 disproves this possibility.
In contrast with the winner-take-all profile, the proportional profile isPareto eﬃcient, regardless of the allocation of weights across the groups.
Proposition 2. Under Assumption 1, the proportional profile ϕPR is Paretoeﬃcient.
Proof. This follows from the characterization of the Pareto set (Lemma
1).
However, the proportional profile does not necessarily Pareto dominate
the winner-take-all profile, even when Assumption 2 holds. This is illustrated by the following example.
Example 1. Let us consider three groups with weights (w1 , w2 , w3 ) =(49, 49, 2). The group-wide margins Θi are independent and uniformly distributed on [−1, 1]. On the one hand, under the winner-take-all profile
ϕWTA , all groups are perfectly symmetric, and a simple calculation showsthat the expected payoﬀ is πi (ϕWTA ) = 0.25 for all i = 1, 2, 3. On theother hand, under the proportional profile ϕPR , group 3 is extremely unlikely to aﬀect the social decision, and π3 (ϕPR ) is close to 0 (approximately
0.014). Group 3 is better oﬀ under ϕWTA than ϕPR , and so ϕPR does not
Pareto dominate ϕWTA . By what profile is ϕWTA Pareto dominated? The
characterization lemma provides an answer. Consider the generalized proportional profile ϕ̂ with coeﬃcients λi = 1/wi . Then, the distribution of
the weight assigned to the alternative is exactly the same across groups,
and thus πi (ϕ̂) is the same for all i. By Pareto eﬃciency of the generalized
proportional profile, πi (ϕ̂) > 0.25 for all i.
Remark 3. An interesting extension of the model would be to assume
that each group chooses a rule through voting by its members. Does this15

extension lead to an equilibrium diﬀerent from the winner-take-all profile?
The answer is relatively clear in the case of group-wide majority voting
between the winner-take-all rule and some other rule (e.g., the proportionalrule), as when a state in the US holds a referendum to switch from thecurrent winner-take-all rule to some proposed rule (see, e.g., Beisbart andBovens (2008)). In that case, a group’s choice of a rule depends on the priorjoint distribution of its members’ preferences, which we have not specified
so far. If the group is ex ante suﬃciently homogeneous,7 the choice bymajority voting will coincide with the choice that maximizes the per capita
expected payoﬀ, i.e., the winner-take-all rule. However, if the group is ex
ante suﬃciently heterogeneous, group-wide majority voting may select theother rule. The following example illustrates this point.
Suppose there are two groups with weights w1 = 4, w2 = 3.8 Eachgroup consists of two types of members, L and R. Type-L members are
more likely to prefer alternative −1 than +1, and type-R members are
more likely to prefer +1 than −1. The fraction of type-L members is 51%
in group 1, and 80% in group 2. In that sense, ex ante heterogeneity ishigh in group 1 and low in group 2.
To define the types more precisely, suppose there are four random variables (Θ1L , Θ1R , Θ2L , Θ2R ) in which Θit is the latent variable for the preferences of type-t members in group i, in the same sense as in Remark 1.9 The
latent variable ΘiL is uniformly distributed on [−1, 0], and ΘiR is uniformly
distributed on [0, 1]. In particular, each type-L (resp. type-R) member isalways more likely to prefer −1 than +1 (resp. +1 than −1). We also
7

Ex ante homogeneity of a group does not necessarily mean that most of its members
tend to support the same alternative. What it means is that the members’ preferences
follow similar probability laws. For instance, conditionally independent and identically
distributed preferences in Remark 1 provide an example of perfect ex ante homogeneity.
8
Here we use the case with n = 2 for ease of exposition. A similar example can beconstructed with a larger number of groups. We also note that while Assumption 2
excludes the case with n = 2, strategic dominance of the winner-take-all rule (i.e., part(i) of Theorem 1) does not depend on that assumption.
9
That is, conditional on the realization (θ1L , θ1R , θ2L , θ2R ), all individual members’
preferences are independent and identically distributed, where the alternative Xim preferred by a type-t member m of group i has the following conditional distribution:{
P {Xim = +1|Θ1L = θ1L , · · · , Θ2R = θ2R } = (1 + θit ) /2,
P {Xim = −1|Θ1L = θ1L , · · · , Θ2R = θ2R } = (1 − θit ) /2.

16

assume that the four latent variables are independent. Given any profile
ϕ = (ϕ1 , ϕ2 ), the (expected) payoﬀ for each type-t member of group i isπit (ϕ) = E(Θit Dϕ ).10
In other words, the probability of success for this member (i.e., the probability that the social decision will be his preferred alternative) is 1+π2it (ϕ) .
Let us assume that the only rules available for each group are thewinner-take-all rule ϕWTA
and the proportional rule ϕPRii . Each groupchooses a rule by majority voting, where each member votes for the rulethat gives him a higher payoﬀ. Since the type-L members are the majority
in both groups, group i’s majority preference over profiles is represented by
the payoﬀ function πiL (ϕ). Thus the situation can be formally represented
as the following 2×2 game played by groups 1 and 2, where group i’s payoﬀ
is the payoﬀ πiL (ϕ) for a type-L member11 (not the per capita expected
payoﬀ πi (ϕ)):
ϕWTA
1
ϕPR
1

ϕPR
ϕWTA
2
2
0.192, 0.020 0.192, 0.020
0.375, 0.479 0.386, 0.402

WTA
The equilibrium is (ϕPR). In contrast with Theorem 1, the pro1 , ϕ2
portional rule is the dominant strategy for group 1 (or more precisely, voting
for the proportional rule is the dominant strategy for each type-L member
in group 1). The intuitive reason is as follows. If group 1 uses the winnertake-all rule, then it will dictate the social decision, and hence the decision
will be alternative −1 or +1 with almost equal probabilities, since group 1
is almost evenly split into the two types. Alternatively, if group 1 adopts
the proportional rule, then the social decision will be more likely to be alternative −1 (i.e., the alternative which the majority of group 1’s members
are more likely to support), since there are now chances that the decisionreflects the will of group 2 in which 80% are of type L. This explains why
10

Assuming that the number of members of each type is suﬃciently large in eachgroup, the group-wide margins in groups 1 and 2 are Θ1 = 0.51Θ1L + 0.49Θ1R andΘ2 = 0.8Θ2L + 0.2Θ2R , respectively. The definition of the social decision Dϕ is then thesame as before (equation (1)).
11
We obtained the payoﬀs in the table by numerical computation.

17

the winner-take-all rule is the dominant strategy for group 1. For the relatively homogeneous group 2, on the other hand, the dominant strategy is
the winner-take-all rule.
This example highlights the importance of the internal decision procedure employed within each group. In order to fully explain the incentivesof the group, it would be interesting to build a full-fledge model which
includes a detailed description of internal heterogeneity of the preference
distributions, but it is beyond the scope of the current paper.

4

Asymptotic and Computational Results

4.1

Asymptotic Analysis

We saw above that the game is a Prisoner’s Dilemma. In this section, we
provide further insights on the welfare properties, by focusing on the following situations in which: (i) the number of groups is suﬃciently large, and(ii) the preferences of the members are distributed symmetrically. These
properties allows us to provide an asymptotic and normative analysis.
Often the diﬃculty of analysis arises from the discrete nature of the
problem. Since the social decision Dϕ is determined as a function of thesum of the weights allocated to the alternatives across the groups, computing the expected payoﬀs may require classification of a large numberof success configurations which increases exponentially as the number ofgroups increases, rendering the analysis prohibitively costly. We overcome
this diﬃculty by studying asymptotic properties. In order to check thesensibility of our analysis, we provide Monte Carlo simulation results laterin the section, using the example of the US Electoral College.
In order to study asymptotic properties, let us consider a sequence of
weights (wi )∞i=1 , exogenously given as a fixed parameter.
Assumption 3. The sequence of weights (wi )∞i=1 satisfies the following
properties.
(i) w1 , w2 , · · · are in a finite interval [w, w̄] for some 0 ≤ w < w̄.

18

(ii) As n → ∞, the statistical distribution Gn induced by (wi )ni=1 weakly
converges to a distribution G with support [w, w̄].12
Assumption 3 guarantees that for large n, the statistical distributionof weights Gn is suﬃciently close to some well-behaved distribution G, on
which our asymptotic analysis is based.
Additionally, we impose an impartiality assumption for our normative
analysis:Assumption 4. The variables (Θi )∞i=1 are drawn independently from acommon symmetric distribution F .
As in Felsenthal and Machover (1998), a normative analysis requires
impartiality, and a study of fundamental rules in the society, such as a constitution, should be free from any dependence on the ex post realizationof the group characteristics. Assumption 4 allows our normative analysis
to abstract away the distributional details. Of course, a normative analysis is best complemented by a positive analysis which takes into accountthe actual characteristics of the distributions (as in Beisbart and Bovens(2008)).
Following the symmetry of the preferences, our analysis also focuses on
symmetric profiles, in which all groups use the same rule: ϕi = ϕ for all i.
With a slight abuse of notation, we write ϕ both for a single rule ϕ and forthe symmetric profile (ϕ, ϕ, · · · ), which should not create any confusion aslong as we refer to symmetric profiles. As for the alternatives, it is natural
to consider that the label should not matter when the group-wide votemargin is translated into the weight allocation, given the symmetry of thepreferences.
Assumption 5. We assume that the rule is monotone and neutral, thatis, ϕ is a non-decreasing, odd function: ϕ(θi ) = −ϕ(−θi ).
Let πi (ϕ; n) denote the expected payoﬀ for group i(≤ n) under profile
ϕ when the set of groups is {1, · · · , n} and each group j’s weight is wj , the12

The statistical distribution function Gn induced by (wi )ni=1 is defined by Gn (x) =#{i ≤ n : wi ≤ x}/n for each x. Gn weakly converges to G if Gn (x) → G(x) at every
point x of continuity of G.

19jth component of the sequence of weights. The definition of πi (ϕ; n) is thesame as πi (ϕ) in the preceding sections; the new notation just clarifies its
dependence on the number of groups n.
The main welfare criterion employed in this section is the asymptotic
Pareto dominance.
Definition 2. For two symmetric profiles ϕ and ψ, we say that ϕ asymptotically Pareto dominates ψ if there exists N such that for all n > N and
all i = 1, · · · , n,πi (ϕ; n) > πi (ψ; n).

4.2

Pareto Dominance

The following is the main result in our asymptotic analysis.
Theorem 2. Under Assumptions 1-5, the proportional profile asymptotically Pareto dominates all other symmetric profiles. In particular, it asymptotically Pareto dominates the dominant-strategy equilibrium of the game,i.e., the symmetric winner-take-all profile.
We use the following lemma to prove Theorem 2. The proof of Lemma
2 is relegated to the Appendix. The proof of part (ii) uses a more general
result, Lemma 3, stated in the next subsection, whose proof also appears
in the Appendix.
Lemma 2. Under Assumptions 1-5, the following statements hold.
(i) For any symmetric profile ϕ,πi (ϕ; n)
{
}
∫ 1
∑
=2
θi P − wi ϕ(θi ) <wj ϕ(Θj ) ≤ wi ϕ(θi ) dF (θi ).
0j≤n, j̸=i(ii) For any symmetric profile ϕ, as n → ∞,√2πnπi (ϕ; n) → 2wi
13√

E[Θ2 ]
Corr [Θ, ϕ(Θ)], 132w dG(w)w∫ w̄

Since Θ and ϕ(Θ) are symmetrically distributed, the correlation is given by20

uniformly in wi ∈ [w, w̄], where Θ is a random variable having thesame distribution F as Θi . The limit depends on the profile ϕ only
through the factor Corr[Θ, ϕ(Θ)].

Proof of Theorem 2.
The heart of the proof is in the correlation result shown in part (ii) of
Lemma 2. It follows that if ϕ(Θ) is more correlated with Θ than ψ(Θ) is,then for each group i, there exists Ni such that if the number of groups (n)is greater than Ni , group i (≤ n) will be better oﬀ under ϕ than ψ.
Note that the convergence in part (ii) of Lemma 2 is uniform in wi ∈[w, w̄]. This implies that the convergence is uniform in i = 1, 2, · · · .14 Thus
there is N with the above property, without subscript i, which applies toall groups i = 1, 2, · · · . Therefore, if ϕ(Θ) is more correlated with Θ thanψ(Θ) is, then ϕ asymptotically Pareto dominates ψ.
Since the perfect correlation Corr[Θ, ϕPR (Θ)] = 1 is attained by the
proportional rule, Theorem 2 follows.
The above results show that the winner-take-all rule is characterized
by its strategic dominance, while the proportional rule is characterized by
its asymptotic Pareto dominance. The following proposition provides acomplete Pareto order among all the linear combinations of the two rules.
Remember that we defined the mixed rules in Section 2 above. For
0 ≤ a ≤ 1, a fraction a of the weight is assigned to the winner of thegroup-wide vote, while the rest, 1 − a, is distributed proportionally to eachalternative:ϕa (θi ) = aϕWTA (θi ) + (1 − a)ϕPR (θi ).
Proposition 3. Under Assumptions 1-4, mixed profile ϕa asymptotically′
Pareto dominates mixed profile ϕa for any 0 ≤ a < a′ ≤ 1. In particular,√
Corr [Θ, ϕ(Θ)] = E[Θϕ(Θ)]/ E[Θ2 ]E[ϕ(Θ)2 ] unless ϕ(Θ) is almost surely zero. If ϕ(Θ)
is almost surely zero, then the correlation is zero.
14
A more detailed explanation of this step is the following. By Lemma 2 (i),√√∫1∑2πnπi (ϕ; n)) asymptotically behaves asj≤n wj ϕ(Θj ) ≤∑ 2 2πn 0 θP{−wi ϕ(θ) <wi ϕ(θ)}dF (θ), where whether the sum j≤n wj ϕ(Θj ) includes the ith term or not is√immaterial in the limit. The estimate of 2πnπi (ϕ; n) therefore has the form fn (wi ),√∫1∑
where fn (x) := 2 2πn 0 θP{−xϕ(θ) < j≤n wj ϕ(Θj ) ≤ xϕ(θ)}dF (θ). Lemma 2 (ii)
implies that fn√(x) converges uniformly in x ∈ [w, w̄], which in turn implies that theconvergence of 2πnπi (ϕ; n) ≈ fn (wi ) is uniform in i = 1, 2, · · · .

21

the proportional profile asymptotically Pareto dominates any mixed profile
ϕa for 0 < a < 1, which in turn asymptotically Pareto dominates thewinner-take-all profile. In other words, all mixed profiles can be ordered byasymptotic Pareto dominance, from the proportional profile as the best, tothe winner-take-all profile as the worst.
Proof. In Appendix.
The winner-take-all rule is not only asymptotically Pareto ineﬃcient,but the worst among the symmetric mixed profiles. Is it worse than any
other symmetric profile? We provide an answer in Remark 4 below.
Remark 4. Theorem 2 leaves the natural question of whether the winnertake-all profile is the worst among all symmetric profiles, in terms of asymptotic Pareto dominance. The answer is negative. To see this, note first that,for the winner-take-all profile, the correlation in Lemma 2 is strictly posi√
tive: Corr[Θ, ϕWTA (Θ)] = E(|Θ|)/ E(Θ2 ) > 0. On the other hand, for thesymmetric profile ϕ0 in which the rule is defined by ϕ0 (θ) = 0 for almost
all θ, the correlation is obviously zero. This rule assigns exactly half of
the weight to each alternative, regardless of the group-wide vote. Thus the
profile ϕ0 is the worst among all symmetric profiles, as the social decision ismade by a coin toss almost surely, yielding expected payoﬀ 0 to all groups.
In the following, we exclude such a trivial profile from our consideration.

4.3

Congressional District Method

The analysis in the preceding subsection suggests that the proportional
profile is optimal in terms of Pareto eﬃciency. However, our model also
implies that this profile produces an unequal distribution of welfare; infact, this unequal nature pertains to all symmetric profiles. The Correlation
Lemma 2 (ii) shows that for these profiles, the expected payoﬀ for a group is
asymptotically proportional to its weight, providing high expected payoﬀs
to the members in a group with a large weight.
In this subsection, we examine whether such inequality can be alleviated without impairing eﬃciency by using an asymmetric profile, basedon the Congressional District Method, currently used in Maine and Nebraska. This profile allocates a fixed amount c of each group’s weight by22

the winner-take-all rule and the rest by the proportional rule:
wi ϕCD (θi , wi ) = cϕWTA (θi ) + (wi − c)ϕPR (θi ).
We consider the congressional district profile ϕCD in which the rule is usedby all groups. Note that this profile is not symmetric as it depends on wi ,but the way ϕCD depends on wi is the same for all groups. To ensure thatthe profile is well-defined, we impose that its parameter c is below the lower
bound of weights: c ∈ [0, w].
Theorem 3. Under Assumptions 1-5, let us consider the congressional
district profile with parameter c ≤ w. For any symmetric profile ϕ, there
exists w∗ ∈ [w, w̄] with the following property: for any ε > 0, there is N
such that for all n > N and i = 1, · · · , n,wi < w∗ − ε ⇒ πi (ϕCD ; n) > πi (ϕ; n),wi > w∗ + ε ⇒ πi (ϕCD ; n) < πi (ϕ; n).

The proof of Theorem 3 uses the following lemma, which shows that thecorrelation lemma holds for a class of profiles such that the weight allocation
rules have the following specific form of separability. Its proof and the Local
Limit Theorem used in the proof are relegated to the Appendix.
Assumption 6. Let ϕ = (ϕi )∞i=1 be a profile.
h1 , h2 , h3 such that

There exist functions

wi ϕi (θi , wi ) = h1 (wi )h2 (θi ) + h3 (wi ) sgn θi , for all iwhere (i) h1 is bounded, (ii) h2 is an odd function such that the supportof the distribution of h2 (Θi ) contains 0, and (iii) h3 is continuous but notconstant.15
It is straightforward to show that Assumption 6 is satisfied for any symmetric profile as well as the congressional district profile. For a symmetricprofile ϕ, let h1 (wi ) = wi , h2 (θi ) = ϕ(θi ) − r sgn θi , and h3 (wi ) = wi r
Under this form, ϕi (·, ·) is the same for all i so that we can omit subscript i whenever
there is no confusion.
15

23

where r > 0 is any positive number in the support of the distribution ofϕ(Θ).16 For the congressional district profile ϕCD , let h1 (wi ) = wi − c,h2 (θi ) = θi − sgn θi , and h3 (wi ) = wi .
Lemma 3. Under Assumptions 1-5, let ϕ be a profile which satisfies Assumption 6. Then, as n → ∞,√2πnπi (ϕ; n) → √∫2wi E[Θϕ(Θ, wi )]w̄ 2w E[ϕ(Θ, w)2 ]dG(w)w,

uniformly in wi ∈ [w, w̄], where Θ is a random variable having the same
distribution F as Θi .
Proof of Theorem 3. By Lemma 3, the expected payoﬀ for group i undera symmetric profile ϕ tends to a linear function of wi . Let Aϕ be thecoeﬃcient:
lim

n→∞

√

2πnπi (ϕ; n) = √2wi E[Θϕ(Θ)]∫ w̄
E[ϕ(Θ)2 ] w w2 dG(w)(6)=: Aϕ wi .
For the congressional district profile, remember the definition:
wj ϕCD (θj , wj ) = cϕWTA (θj ) + (wj − c) ϕPR (θj )= c sgn (θj ) + (wj − c) θj .
We claim that the limit function is aﬃne in wi :
lim

n→∞

√

2πnπi (ϕCD ; n) = Bwi + C.

(7)

To see that, let us apply Lemma 3 again:
[ CD
]
√
w
E
Θϕ
(Θ,
w
)
i
i
lim 2πnπi (ϕCD ; n) = 2 · √∫]w̄ 2 [ CDn→∞w E ϕ (Θ, w)2 dG(w)wcE [|Θ|] + (wi − c) E [Θ2 ]= 2 · √∫.
w̄ 2 [ CD2]w E ϕ (Θ, w) dG(w)w16This is possible since ϕ(Θ) is symmetrically distributed, and since we exclude thetrivial case in which ϕ(Θ) = 0 almost surely.

24

Since |θ| ≥ θ2 with a strict inequality for 0 < |θ| < 1, the full supportcondition for Θ implies E [|Θ|] > E [Θ2 ], which induces that the intercept
C is positive. The coeﬃcient of wi is:
B = √∫

2E [Θ2 ]w̄ 2 [ CD
2]
w
E
ϕ
(Θ,
w)
dG(w)
w

.

If Aϕ < B, combined with C > 0, the right-hand side of (7) is above thatof (6). Then, set w∗ = w̄. If Aϕ > B, again combined with C > 0, the twolimit functions (6) and (7) intersect only once at a positive value ŵ. Letw∗ = max {w, min{ŵ, w̄}}.
Since the convergences (6) and (7) are uniform in wi , for any ε > 0
there is N with the property stated in Theorem 3.
Theorem 3 implies that the congressional district profile makes themembers of groups with small weights better oﬀ, compared with any symmetric profile. If the weight is an increasing function of the group size, itmeans that the congressional district profile is favorable for the membersof small groups.
The intuitive reason why the congressional district profile is advantageous for small groups is as follows. Under this profile, the ratio of weights
cast by the winner-take-all rule (i.e. c/wi ) is higher for small groups thanlarge groups. The congressional district profile therefore resembles the situation where the rules used by the smaller groups are relatively close to
the winner-take-all rule, whereas those by the larger groups are close to
the proportional rule. The strategic dominance of the winner-take-all rulesuggests that this deviation is profitable for the small groups. We provide
a numerical result in the following subsection using an example of the US
Electoral College.
In addition to Theorem 3, we can also show that the congressional
district profile distributes payoﬀs more equally than any symmetric profile
does, in the sense of Lorenz dominance. A profile of per capita payoﬀs forthe groups, π = (π1 , · · · , πn ), is said to Lorenz dominate another profile
π ′ = (π1′ , · · · , πn′ ) if the share of payoﬀs acquired by any bottom fraction of25

groups is larger in the former profile than in the latter.17 Lorenz dominance,
whenever it occurs, agrees with equality comparisons by various inequality
indices including coeﬃcient of variation, Gini coeﬃcient, Atkinson index,
and Theil index (see Fields and Fei (1978) and Atkinson (1970)). To see
why the congressional district profile is more equal than any symmetricprofile, recall equations (6) and (7) in the proof of Theorem 3, which assertthat when the number of groups is large, the per capita payoﬀ for groupi is approximately Aϕ wi for the symmetric profile, and it is approximately
Bwi + C for the congressional district profile. The constant term C > 0
for the congressional district profile means equal additions to all groups’
payoﬀs, which result in a more equal distribution than when there is nosuch term. More precisely, we can prove the following statement. The
proof is relegated to the Appendix.
Theorem 4. Under Assumptions 1-5, let us consider the payoﬀ profile
(
) ( (
))n
under the congressional district profile: π ϕCD ; n = πi ϕCD ; n i=1 . Let
ϕ be any symmetric profile and π (ϕ; n) = (πi (ϕ; n))ni=1 the payoﬀ profile()
under ϕ. For suﬃciently large n, π ϕCD ; n Lorenz dominates π (ϕ; n).

4.4

Computational Results

The results in the previous subsection concern cases with a large number ofgroups. The question remains as to whether the conclusions obtained there
are also valid for a finite number of groups. In this section, we study thisquestion by numerically analyzing a model of the US presidential election.
There are 50 states and one federal district. The weight wi for state i isthe number of electoral votes currently assigned to it. This number equals
the state’s total number of seats in the Senate and House of Representatives. Thus, wi is two plus a number that is roughly proportional to thestate’s population. The first and second columns of Table 1 describe the
distribution of weights among the states.
Formally, for each x ∈ [−1, 1], let Hπ (x) be the total population share of thosegroups whose per capita welfare is not greater than x under the payoﬀ profile π. Then
Hπ is a distribution function. The Lorenz curve of Hπ is the graph of the function∫ Hπ−1 (p)∫1xdHπ (x)/ 0 xdHπ (x), 0 ≤ p ≤ 1, where we define Hπ−1 (p) = sup{x : Hπ (x) ≤0p}. A payoﬀ profile π Lorenz dominates another profile π ′ if the Lorenz curve of Hπ lies
above that of Hπ′ .
1726

We assume IAC* (Impartial Anonymous Culture*): the statewide popular vote margins Θi are independent and uniformly distributed on [−1, 1],
first introduced by May (1948) and studied thoroughly by, for example,
De Mouzon et al. (2019). For any profile ϕ, we can compute the per capita
payoﬀ for state i via the formula:∫ 150

πi (ϕ) = 0.5

−1

∫ 1
···

−1

θi 1A (θ1 , · · · , θ51 )dθ1 · · · dθ51

(8)

{
}
∑51
where A = (θ1 , · · · , θ51 )
w
ϕ
(θ
)
>
0
.18
j
j
j
j=1
We consider four distinct profiles: ϕWTA , ϕPR , ϕa with a = 102/538, andϕCD with coeﬃcient c = 2. As before, these are respectively the winnertake-all profile, the proportional profile, a mixed profile, and a congressional
district profile. The parameter c = 2 of the congressional district profile isthe number currently used in Maine and Nebraska, namely, it corresponds
to two seats assigned to each state in the Senate. The parameter a =102/538 of the mixed profile is chosen so that the proportion of electoral
votes allocated on the winner-take-all basis is the same for all states, andthe total number of electoral votes allocated in this way is the same as in
the congressional district profile.
We compute (8) under these four profiles by a Monte Carlo simulation with 1010 iterations. The results are summarized in Tables 1 and 2.
Table 1 shows the per capita payoﬀ (πi (ϕ)) under the respective profiles.
Table 2 shows the ratios of per capita payoﬀ between diﬀerent profiles(πi (ϕ)/πi (ψ)). If the ratio is below 1, state i prefers ψ to ϕ.
It follows from Lemma 2 (ii) that as the number n of states increases, the
(
)
(
)
(
)
ratios πi ϕWTA /πi ϕPR and πi (ϕa ) /πi ϕPR converge to the respective
correlations Corr[Θ, ϕWTA (Θ)] ≈ 0.866 and Corr[Θ, ϕa (Θ)] ≈ 0.989, wherethe values are computed for Θ uniformly distributed on [−1, 1]. Table
2 indicates that for the present example with 50 states plus DC, theseratios are indeed close to the respective correlations, which suggests thatconvergence of the π-ratios is fairly quick. In particular, as expected by
Theorem 2, the proportional profile Pareto dominates the winner-take-all
profile in the present case. As suggested by Proposition 3, all states prefer18

It is easy to check that under the uniform distribution assumption, (8) is equivalent
to the expression in Lemma 2 (i).

27the mixed profile ϕa to the winner-take-all profile, and the proportional
profile to ϕa .
The ratios πi (ϕCD )/πi (ϕPR ) in Table 2 are consistent with the resultin Theorem 3. Small states prefer the congressional district profile to the
proportional one.
In addition, the values of πi (ϕCD )/πi (ϕWTA ) in the table show that thewinner-take-all profile is Pareto dominated by the congressional district
profile, and the welfare improvement by switching to the congressional
district profile is greater for small states than for large states in terms of
the ratio.

28

Table 1: Estimated payoﬀs in the US presidential election, based on theapportionment in 2016, via Monte Carlo simulation with 1010 iterations.
The estimated standard errors are in the range between 3.9 and 4.1 × 10−6 .
electoral number π(ϕWTA ) π(ϕPR ) π(ϕa ) π(ϕCD )votesof states
380.01130.0133 0.0130 0.0167
45
0.01510.0177 0.0174 0.0209
53
0.01890.0221 0.0217 0.0251
66
0.0226
0.0266 0.0261 0.0293
7
3
0.02640.0310 0.0305 0.0335
82
0.03020.0354 0.0348 0.0377
93
0.03400.0399 0.0392 0.0419
10
40.03780.0443 0.0436 0.0461
114
0.0416
0.0488 0.0479 0.0503
12
1
0.04540.0532 0.0523 0.0545
13
1
0.04920.0577 0.0567 0.0587
14
1
0.0531
0.0622 0.0611 0.0630
151
0.05690.0666 0.0655 0.0672
162
0.06070.0711 0.0699 0.0715
18
1
0.06840.0801 0.0788 0.0800
202
0.07620.0891 0.0877 0.0885
29
2
0.11200.1303 0.1284 0.1275
38
1
0.14940.1729 0.1706 0.1677
55
1
0.2356
0.2614 0.2615 0.2507

29

Table 2: Ratios between payoﬀs.
electoral
votes
3
4
5
6
7
8
9
10
11
12
13
14
15
16
18
20
29
38
55

numberof states
8
5
3
6
3
2
3
4
4
1
1
1
1
2
1
2
2
1
1

π(ϕWTA )π(ϕPR )π(ϕa )π(ϕPR )π(ϕCD )π(ϕPR )π(ϕCD )
π(ϕWTA )

0.852
0.852
0.852
0.852
0.852
0.852
0.852
0.853
0.853
0.853
0.853
0.853
0.854
0.854
0.854
0.855
0.859
0.864
0.901

0.982
0.982
0.982
0.982
0.982
0.982
0.982
0.983
0.983
0.983
0.983
0.983
0.983
0.983
0.983
0.983
0.985
0.987
1.000

1.260
1.182
1.134
1.103
1.080
1.064
1.050
1.040
1.031
1.024
1.018
1.013
1.009
1.005
0.998
0.993
0.978
0.970
0.959

1.479
1.387
1.331
1.294
1.268
1.248
1.232
1.220
1.210
1.201
1.194
1.187
1.181
1.177
1.168
1.161
1.138
1.122
1.064

30

5

Concluding Remarks

This paper shows that the decentralized choice of the weight allocation rulein representative voting constitutes a Prisoner’s Dilemma: the winner-takeall rule is a dominant strategy for each group, whereas the Nash equilibrium
is Pareto dominated. We also show that the proportional rule Pareto dominates every other symmetric profile, when the number of the groups is
suﬃciently large. Each group has an incentive to put its entire weight on
the alternative supported by the majority of its members in order to reflect
their preferences in the social decision, although it fails to eﬃciently aggregate the preferences of all members in the society, if the winner-take-allrule is employed by all groups.
Our model may provide explanations for the phenomena that we observein existing collective decision making. In the United States Electoral College, the rule used by the states varied in early elections until it convergedby 1832 to the winner-take-all rule, which remains dominantly employed bynearly all states since then. In many parliamentary voting situations, we
often observe parties and/or factions forcing their members to align theirvotes in order to maximally reflect their preferences in the social decision,
although some members may disagree with the party’s alignment. The
voting outcome obtained by the winner-take-all rule may fail to eﬃciently
aggregate preferences, as observed in the discrepancy between the electoral
result and the national popular vote winner in the US presidential elections in 2000 and 2016. Party discipline or factional voting may also cause
welfare loss when each group pushes their votes maximally toward theirideological goals, failing to reflect all members’ preferences in the social
decision.
The Winner-Take-All Dilemma tells us that the society should call forsome device diﬀerent from each group’s unilateral eﬀort, in order to obtaina more socially preferable outcome. As we see in the failure of various
attempts to modify or abolish the winner-take-all rule, such as the ballot
initiative for an amendment to the State Constitution in Colorado in 2004,
each state has no incentive to unilaterally deviate from the equilibrium.
The National Popular Vote Interstate Compact is a well-suited example ofa coordination device (Koza et al. (2013)). As it comes into eﬀect only
31

when the number of electoral votes attains the majority, each state does
not suﬀer from the payoﬀ loss by unilateral (or coalitional) deviation until suﬃcient coordination is attained. The emergence of such an attempt
is coherent with the insights obtained in this paper that the game is a
Prisoner’s Dilemma, and a coordination device is necessary for a Pareto
improvement.
Our analysis is abstract in that we do not impose assumptions on thepreferences distribution based on the observed characteristics in the real
representative voting problems. Additionally, we impose an impartiality
assumption in our asymptotic analysis. Obviously, our normative analysis
would be best complemented by a positive analysis, which we leave for
future research.
We have assumed that social decisions are binary. There are situations
where this assumption may not fit. In the US presidential elections, thirdparty or independent candidates can, and do, have a non-negligible impact
on the election outcome. It is not clear how the presence of such candidates
alters the comparison of rules to allocate electoral votes. When the model
is applied to legislative voting, the assumption of binary decision might
be justified on the grounds that choices are ultimately made between thestatus quo and a proposal. However, such an argument abstracts away
the process that gives rise to the particular pair of alternatives (e.g., what
becomes the status quo, how much proposal power each party has, and soon). Cases with more than two alternatives require further investigation.

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35

Appendix“The Winner-Take-All Dilemma”
Kazuya Kikuchi and Yukio Koriyama
A1

Proof of Lemma 1

Preliminaries. In this proof, we denote by ϕλ the generalized proportional
profile with coeﬃcients λ ∈ [−1, 1]n \ {0}:ϕλi (θi ) = λi θi , i = 1, · · · , n,which should not be confused with the notation ϕa for mixed profiles.
We write π(ϕ) = (πi (ϕ))ni=1 for the vector of payoﬀs. Let Π be the setof all possible payoﬀ vectors in game Γ:Π = {π(ϕ) : ϕ is a profile}.
For any X ⊂ Rn , let Pareto (X) be the Pareto frontier of X, i.e., the setof points x ∈ X for which there exists no y ∈ X such that yi ≥ xi for all i,with strict inequality for at least one i. Let co X denote the closed convex
hull of X.
We will refer to the following maximization problem parametrized byvector q ∈ R+ \ {0} as Problem Mq :Problem Mq : max q · x.
x∈co Π

Note that the maximization is not directly with respect to profile ϕ.
Moreover, Π may be non-closed or non-convex. Thus, there may be asolution x ∈ co Π that is not the payoﬀ vector of any profile ϕ, although
we will later disprove this possibility.
We divide the proof of Lemma 1 into several claims. Claims 2.1-2.4
concern properties of the solutions to Problem Mq . Claim 2.5 describes
the relation between Problem Mq and Pareto eﬃciency. Finally, Claim 2.6
completes the proof.
q

Claim 2.1. A solution of Problem Mq is x = π(ϕλ ), where λqi = cqi /wi ,1

i = 1, · · · , n.19
Proof of Claim 2.1. Recall that the payoﬀ is given by
πi (ϕ) = E (Θi sgn Sϕ ) .

(9)q · π(ϕ) = E [(q · Θ) (sgn Sϕ )] ≤ E(|q · Θ|).

(10)

Thus, for any profile ϕ,That is, q · x ≤ E(|q · Θ|) for any x ∈ Π. The linearity of the objectiveqfunction q·x implies that q·x ≤ E(|q·Θ|) for all x ∈ co Π. If ϕ = ϕλ , then Sϕq
has the same sign as q·Θ. Thus for x = π(ϕλ ), we have q·x = E(|q·Θ|).
Claim 2.2. Let v(q) := max q · x be the maximum value of Problem Mq .
x∈co Π
Thenv(q) = E(|q · Θ|).

Proof of Claim 2.2. This follows from the proof of Claim 2.1, in which we
showed that q · x ≤ E(|q · Θ|) for all x ∈ co Π and the upper bound isqattained by x = π(ϕλ ).
Claim 2.3. A profile ϕ satisfies q ·π(ϕ) = v(q) if and only if ϕ is equivalentq
to ϕλ .
Proof of Claim 2.3. Since q ̸= 0 and Θ is absolutely continuous, we haveq · Θ ̸= 0 almost surely. Thus, (10) holds with equality if and only if
sgn Sϕ = sgn (q · Θ) almost surely.
Since cq · Θ = Sϕλq , this holds if and only if ϕ is equivalent to ϕλ .
qqClaim 2.4. x = π(ϕλ ) is the unique solution of Problem Mq .
Proof of Claim 2.4. We use the absolute continuity of Θ to show that thevalue function v(q) = E(|q · Θ|) is diﬀerentiable, with gradient ∇v(q) =19

c > 0 is a constant such that cqi /wi ≤ 1 for all i.

2qπ(ϕλ ). Then the uniqueness follows by the Duality Theorem (Mas-Colell
et al. (1995, Proposition 3.F.1)).
To show that v(q) is diﬀerentiable, it suﬃces to show that as vector
ε ∈ Rn approaches 0,qv(q + ε) − v(q) − π(ϕλ ) · ε = o(∥ε∥).

(11)

Using (9) and Claim 2.2, we can rewrite the left-hand side of (11) as:qv(q + ε) − v(q) − π(ϕλ ) · ε= E [{(q + ε) · Θ} × sgn {(q + ε) · Θ}]− E [(q · Θ) × sgn (q · Θ)]− E [(ε · Θ) × sgn (q · Θ)]= E[{(q + ε) · Θ} × {sgn ((q + ε) · Θ) − sgn (q · Θ)}].
This expression has the following bound:()
q|v(q + ε) − v(q) − π(ϕλ ) · ε| ≤ 2E |(q + ε) · Θ| 1{sgn ((q+ε)·Θ)̸=sgn (q·Θ)} .
The expectation on the right-hand side is
∫

∫

+
Aq,ε

{(q + ε) · θ}h(θ)dθ −A−q,ε{(q + ε) · θ}h(θ)dθ,(12)

where h is the joint density of Θ andnA+q,ε = {θ ∈ [−1, 1] : (q + ε) · θ ≥ 0 ≥ q · θ},nA−q,ε = {θ ∈ [−1, 1] : (q + ε) · θ ≤ 0 ≤ q · θ}.

√
We show that for ε suﬃciently close to 0, (q + ε) · θ ≤ n∥ε∥ forall θ ∈ A+q,ε . To do this, we fix a suﬃciently small ε so that for eache ∈ {−1, 1}n (i.e., each vertex of the hypercube [−1, 1]n ), either both q · eand (q + ε) · e are non-negative or both are non-positive.20 Now, consider
the following linear-programming problem (Lq,ε ):For each vertex e ∈ {−1, 1}n there is δe > 0 such that if ∥ε∥ < δe then either bothq · e and (q + ε) · e are non-negative or both are non-positive. Thus, it suﬃces to choose
ε so that ∥ε∥ < min{δe : e ∈ {−1, 1}n }.
20

3Problem Lq,ε : max(q + ε) · θ.
+θ∈Aq,ε∗Let θ∗ be a solution of Problem Lq,ε that is a vertex of A+q,ε . Then θbelongs to at least one of the following sets:

Hq+ε = {θ : (q + ε) · θ = 0},
Hq = {θ : q · θ = 0},{−1, 1}n .
We claim that θ∗ ∈ Hq . First, we have θ∗ ∈/ Hq+ε , since otherwiseθ∗ minimizes the objective function (q + ε) · θ subject to θ ∈ A+q,ε , while+
the n-dimensional polytope Aq,ε contains points that attain larger valuesof the function. Now, suppose θ∗ ∈ {−1, 1}n \ Hq . The fact that θ∗ ∈cc∗{−1, 1}n ∩ A+q,ε ∩ Hq ∩ Hq+ε implies that θ is a vertex of the hypercube[−1, 1]n such that q · θ∗ < 0 < (q + ε) · θ∗ . This contradicts the fact that for
any vertex e of the hypercube, either both (q+ε)·e and q·e are non-negative
or both are non-positive. Therefore θ∗ ∈ Hq .
We have shown that q · θ∗ = 0. This implies that for any θ ∈ A+q,ε ,√(q + ε) · θ ≤ (q + ε) · θ∗ = ε · θ∗ ≤ ∥θ∗ ∥∥ε∥ ≤ n∥ε∥. It similarly follows√
that −(q + ε) · θ ≤ n∥ε∥ for any θ ∈ A−q,ε . Therefore, (12) is bounded by∫∫√n∥ε∥ A+q,ε ∪A−q,ε h(θ)dθ. Noting that the integral A+q,ε ∪A−q,ε h(θ)dθ vanishes
as ε → 0, we obtain (11).
Claim 2.5. Let x ∈ co Π. Then, x ∈ Pareto (co Π) if and only if there
exists q ∈ Rn+ \ {0} such that x is the unique solution of Problem Mq , i.e.,qx = π(ϕλ ).
Proof Claim 2.5. For any x ∈ Rn , let D(x) = {x + a : a ∈ Rn+ \ {0}} be the(convex) set of all points that dominate x. Note that x ∈ Pareto (co Π) if
and only if D(x)∩co Π = ∅. To prove Claim 2.5, suppose x ∈ Pareto (co Π).
Then there exists a hyperplane with some normal vector q ∈ Rn+ \ {0}that separates co Π and D(x).21 Clearly this hyperplane contains x, whichmeans that x is the solution of Problem Mq . Conversely, suppose x is
the unique solution of Problem Mq . Then the supporting hyperplane of
21

Here, separation is in the weak sense that the hyperplane may contain boundary
points of the two sets.

4co Π with normal vector q separates co Π and D(x). The uniqueness of thesolution implies that the hyperplane intersects co Π only at x. This implies
that D(x) ∩ co Π = ∅.
Claim 2.6. A profile ϕ satisfies π(ϕ) ∈ Pareto (Π) if and only if there
exists λ ∈ Rn+ \ {0} such that ϕ is equivalent to ϕλ . That is, Lemma 1
holds.
Proof of Claim 2.6. By Claim 2.5,qPareto (co Π) = Pareto (Π) = {π(ϕλ ) : q ∈ Rn+ \ {0}}.
q

By Claim 2.3, π(ϕ) belongs to this set if and only if ϕ is equivalent to ϕλfor some q ∈ Rn+ \ {0}. This condition is the same as saying that ϕ isequivalent to ϕλ for some λ ∈ Rn+ \ {0}.

A2Proof of Part (i) of Lemma 2

We prove the statement for group 1. Let π1 (ϕ; n|θ1 ) be the conditional
expected payoﬀ for group 1 given that the group-wide margin is Θ1 = θ1 ,which by (2) is:π1 (ϕ; n|θ1 ) = θ1 (P{w1 ϕ(θ1 ) + Sϕ−1 > 0} − P{w1 ϕ(θ1 ) + Sϕ−1 < 0}).
Since Sϕ−1 is symmetrically distributed, the second probability can be written as P{−w1 ϕ(θ1 ) + Sϕ−1 > 0}. Thus, for θ1 ∈ [0, 1], the above expression
equalsπ1 (ϕ; n|θ1 ) = θ1 P{−w1 ϕ(θ1 ) < Sϕ−1 ≤ w1 ϕ(θ1 )}.
By symmetry, twice the integral of this expression over θ1 ∈ [0, 1] (insteadof [−1, 1]) equals the unconditional expected payoﬀ π1 (ϕ; n), which provespart (i) of Lemma 2.

A3

Local Limit Theorem

We quote a version of the Local Limit Theorem shown in Mineka andSilverman (1970). We will use it in the proof of part (ii) of Lemma 2.

5

LLT. (Mineka and Silverman (1970, Theorem 1)) Let (Xi ) be a sequenceof independent random variables with mean 0 and variances 0 < σi2 < ∞.
∑Write Fi for the distribution of Xi . Write also Sn = ni=1 Xi and s2n =∑n2i=1 σi . Suppose the sequence (Xi ) satisfies the following conditions:(α) There exists x̄ > 0 and c > 0 such that for all i,1
σi2∫|x|<x̄

x2 dFi (x) > c.

(β) Define the setA(t, ε) = {x : |x| < x̄ and |xt−πm| > ε for all integer m with |m| < x̄}.
Then, for some bounded sequence (ai ) such that inf i P{|Xi − ai | <δ} > 0 for all δ > 0, and for any t ̸= 0, there exists ε > 0 such thatn
1 ∑P{Xi − ai ∈ A(t, ε)} → ∞.
log sn i=1(γ) (Lindeberg’s condition.) For any ε > 0,1 ∑s2n i=1n∫|x|/sn >ε

x2 dFi (x) → 0.

Under conditions (α)-(γ), if s2n → ∞, we have√
2πs2n P{Sn ∈ (a, b]} → b − a.22

22

(13)

The original conclusion of Theorem 1 in Mineka and Silverman (1970) is stated interms of the open interval (a, b). Applying the theorem to (a, b + c) and (b, b + c) andthen taking the diﬀerence gives the result for (a, b]. In addition, the original statement
allows for cases where s2n does not go to infinity, and also mentions uniform convergence.
These considerations are not necessary for our purpose, so we omit them.

6

A4

Proof of Lemma 3

Preliminaries. We prove the lemma for group 1. In the proof, we use thenotation of LLT. Let
Xi := wi ϕ(Θi , wi ), i = 1, 2, · · · ,∑and Sn := ni=1 Xi . Then Xi has mean 0 and variance σi2 := wi2 E[ϕ(Θ, wi )2 ],∑and so the partial sum of variances is s2n := ni=1 wi2 E[ϕ(Θ, wi )2 ], where Θ
represents a random variable that has the same distribution F as Θi .
Define the event{}n∑Ωn (θ1 , w1 ) = −w1 ϕ(θ1 , w1 ) <Xi ≤ w1 ϕ(θ1 , w1 ) .
i=2

We divide the proof into several claims. Claims 5.1-5.3 show that thesequence (Xi ) defined above satisfies the conditions of the Local Limit
Theorem (LLT) in Section A4. Claim 5.4 applies LLT to complete theproof of Lemma 3.
2

Claim 5.1. snn →∫ w̄
w

w2 E[ϕ(Θ, w)2 ]dG(w).

Proof of Claim 5.1. This holds since sequence (σi2 ) is bounded and the statistical distribution Gn induced by (wi )ni=1 converges weakly to G.
Claim 5.2. Conditions (α) and (γ) in LLT hold.
Proof of Claim 5.2. This immediately follows from the fact that sequence(Xi ) is bounded and s2n → ∞. In particular, it is enough to define x̄ to be
any finite number greater than w̄.
Claim 5.3. Condition (β) in LLT holds.
Proof of Claim 5.3. Recall that ϕ has the formwi ϕ(θi , wi ) = h1 (wi )h2 (θi ) + h3 (wi ) sgn θi .
Let ai = h3 (wi ). We first check that the sequence (ai ) satisfies therequirements in condition (β). First, (ai ) is bounded since h3 is bounded.

7

Now, for any i and any δ > 0,
P{|Xi − ai | < δ} ≥ P{|Xi − ai | < δ and Θi > 0}
≥ P {|wi ϕ(Θi , wi ) − h3 (wi ) sgn Θi | < δ and Θi > 0}
= P{|h1 (wi )h2 (Θi )| < δ and Θi > 0}.
Letting h̄1 > 0 be an upper bound of |h1 | and Θ a random variable distributed as Θi , the last expression has the following lower bound independent of i:
P{|h2 (Θ)| < δ/h̄1 and Θ > 0} > 0,
which is positive by the assumptions on h2 and on the distribution of Θ.
Next we check the limit condition in (β). Recall that A(t, ε) is the unionof intervals()
πm + ε π(m + 1) − ε,, m = 0, ±1, ±2, · · · ,|t||t|restricted to (−x̄, x̄), where we can choose x̄ to be any number greater thanw̄. To prove the limit condition in (β), it therefore suﬃces to verify thatone such interval contains Xi −ai with probability bounded away from zero,for all groups i in some suﬃciently large subset of groups. To do this, notethat if Θi < 0, then Xi − ai = h1 (wi )h2 (Θi ) − 2h3 (wi ). The assumptions on
h2 and on the distribution of Θ imply that for any η > 0, there exists a setOη ⊂ [−1, 0] with P{Θ ∈ Oη } > 0 such that if Θ ∈ Oη then |h2 (Θ)| ≤ η.
Therefore,Θi ∈ Oη ⇒ Xi − ai ∈ Twi ,η ,where
Twi ,η := [−2h3 (wi ) − ηh1 (wi ), −2h3 (wi ) + ηh1 (wi )].
Since h1 is bounded, we can make Twi ,η an arbitrarily small interval around−2h3 (wi ) by letting η > 0 suﬃciently small. Moreover, since h3 is continuous and not a constant, we can find a suﬃciently small interval [v, v̄] ⊂[w, w̄] with v < v̄ such that if wi ∈ [v, v̄], then −2h3 (wi ) is between, andbounded away from, πmand π(m+1)
for some integer m. Fix such an interval|t||t|[v, v̄] and define
I := {i : wi ∈ [v, v̄]}.
8

Then, for suﬃciently small η > 0 and ε > 0, we have Twi ,η ⊂ A(t, ε) for alli ∈ I. Fixing such η > 0 and ε > 0, it follows thatΘi ∈ Oη and i ∈ I ⇒ Xi − ai ∈ A(t, ε).
This implies that
P{Xi − ai ∈ A(t, ε)} ≥ P{Θ ∈ Oη } =: p > 0 for all i ∈ I,
and hencen
1 ∑n♯{i ∈ I : i ≤ n}
P{Xi − ai ∈ A(t, ε)} ≥·· p.
log sn i=1
log snn

As n → ∞, the first factor on the right-hand side tends to ∞ since sn has√
an asymptotic order of n. The second factor tends to G(v̄) − G(v) > 0,
which is positive since G has full support on [w, w̄]. Therefore the left-handside tends to ∞.
Claim 5.4. As n → ∞, uniformly in w1 ∈ [w, w̄],∫ 1
2√
2w1 E[Θϕ(Θ, w1 )]θ1 2πnP{Ωn (θ1 , w1 )}dF (θ1 ) → √∫.
w̄ 22 ]dG(w)0w
E[ϕ(Θ,w)
w

By part (i) of Lemma 2,23 the left-hand side of (14) istherefore Lemma 3 holds.

√

(14)

2πnπi (ϕ; n), andProof of Claim 5.4. By Claims 5.2 and 5.3, we may apply LLT to obtain√
2πs2n P{Ωn (θ1 , w1 )} → 2w1 ϕ(θ1 , w1 ).
By Claim 5.1, this means that√
2πnθ1 P{Ωn (θ1 , w1 )} → √∫2w1 θ1 ϕ(θ1 , w1 )w̄ 2
w E[ϕ(Θ, w)2 ]dG(w)
w

.

(15)

Letting θ1 = 1 maximizes the left-hand side of (15) with the maximum
23

It is easy to check that part (i) of Lemma 2 holds for rules ϕ(·, wi ) that depend on
weight wi as well.

9√value 2πnP{Ωn (1, w1 )}. This maximum value itself converges to a finite√
limit. Hence the expression 2πnθ1 P{Ωn (θ1 , w1 )} is uniformly bounded
for all n and θ1 ∈ [0, 1]. By the Bounded Convergence Theorem,
∫ 1
2

θ1
0

√

∫1
2w1 0 θ1 ϕ(θ1 , w1 )dF (θ1 ).
2πnP{Ωn (θ1 , w1 )}dF (θ1 ) → 2 · √∫w̄ 22w E[ϕ(Θ, w) ]dG(w)w

Since F is symmetric and ϕ is odd, this limit is exactly the one in (14).
To check the uniform convergence, note that for each n, the integral on the left-hand side of (14) is non-decreasing in w1 , since eventΩn (θ1 , w1 ) weakly expands as w1 increases.24 We have shown that thisintegral converges pointwise to a limit that is proportional to the factor
w1 E[Θϕ(Θ, w1 )], which is continuous in w1 .25 Therefore, the convergencein (14) is uniform in w1 ∈ [w, w̄].26

A5

Proof of Part (ii) of Lemma 2

This follows immediately from Lemma 3, by noting that if ϕ is a symmetricprofile, each group’s rule can be written as ϕ(θj , wj ) = ϕ(θj ).

A6

Proof of Proposition 2

By part (ii) of Lemma 2, we must show that Corr [Θ, ϕa (Θ)] is decreasing in a ∈ [0, 1]. By simple calculation,
E(Θ2 ) · Corr [Θ, ϕa (Θ)]2 =aE(|Θ|) + (1 − a)E(Θ2 ).
a2 + 2a(1 − a)E(|Θ|) + (1 − a)2 E(Θ2 )Let θ1 ∈ [0, 1]. If ϕ is a symmetric profile, i.e. if ϕ(θ1 , w1 ) = ϕ(θ1 ), then w1 ϕ(θ1 ) is
non-decreasing in w1 . If ϕ = ϕCD , then w1 ϕCD (θ1 , w1 ) = c sgn(θ1 ) + (w1 − c)θ1 , which
is non-decreasing in w1 again. Thus event Ωn (θ1 , w1 ) weakly expands as w1 increases.
25
If ϕ is a symmetric profile, this factor is linear in wi . If ϕ = ϕCD , the factor equalscE(|Θ|) + (wi − c)E(Θ2 ), which is aﬃne in wi .
26
It is known that if (fn ) is a sequence of non-decreasing functions on a fixed finite
interval and fn converges pointwise to a continuous function, then the convergence isuniform. See Buchanan and Hildebrandt (1908).
24

10

The derivative of this expression with respect to a has the same sign as{}()
a2 + 2a(1 − a)E(|Θ|) + (1 − a)2 E(Θ2 )
(
)2 {
}
d
2
2
2
(a
+
2a(1
−
a)E(|Θ|)
+
(1
−
a)
E(Θ
))
− aE(|Θ|) + (1 − a)E(Θ2 )dad(aE(|Θ|) + (1 − a)E(Θ2 ))2da= a(aE(|Θ|) + (1 − a)E(Θ2 ))(E(|Θ|)2 − E(Θ2 )).
This is negative for any a ∈ (0, 1], since E(|Θ|)2 ≤ E(Θ2 ) in general, and thefull-support assumption implies that this holds with strict inequality.

A7

Proof of Theorem 4

Clearly, Lorenz dominance is invariant to linear transformations of payoﬀs. Thus, it suﬃces to prove that for large enough n, the payoﬀ pro√√file 2πnπ(ϕCD ; n) Lorenz dominates the payoﬀ profile 2πnπ(ϕ; n). Byequations (6) and (7) in the proof of Theorem 3, as n → ∞ these amounts
converge to Bwi + C and Aϕ wi , respectively. A result by Moyes (1994,
Proposition 2.3) implies that if f and g are continuous, nondecreasing,and positive-valued functions such that f (wi )/g(wi ) is decreasing in wi ,then the distribution of f (wi ) Lorenz dominates that of g(wi ). The ratio(Bwi +C)/(Aϕ wi ) is decreasing in wi , and so the claimed Lorenz dominanceholds in the limit as n → ∞. Recalling that the convergences are uniform,
the dominance holds for suﬃciently large n.

11