A POSSIBILITY THEOREM ON
INFORMATION AGGREGATION IN ELECTIONS
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
Abstract. We consider the problem of aggregation of private information common value
elections with general state and signal spaces, where the winning alternative is chosen via
majority rule. We show that, for a class of environments that is generic, there exists a
sequence of equilibria that e¢ ciently aggregates information in the limit. The result can be
interpreted in the following way: information aggregation is a generic property if the signal
space is rich enough compared to the state space. On the contrary, if the state space is
su¢ ciently rich compared to the signal space, preferences have to satisfy a strong property
in order for there to exist some strategy pro…le that aggregates information.
1. Introduction
In any election, decision relevant information is dispersed among the voters. It is necessary
to aggregate all the individuals’s privately held information to ascertain who the best candidate is. But since each voter is constrained to vote with his own private information, we have
the classic question of whether the electoral mechanism correctly aggregates information:
Does the electoral outcome coincide with the candidate the voters would have selected if all
the private information were commonly known? We analyze this question in a setting where
voters have the same underlying preference: given the aggregated information, all voters
would choose the same candidate.
This question was …rst posed and answered in the a¢ rmative by Condorcet (1786). If each
voter votes for the correct candidate with a probability p > 12 ; then individual, uncorrelated
errors do not matter in the aggregate and the majority is almost surely right in a large
electorate by the Law of Large Numbers. In the modern interpretation of the environment,
there are two states of the world, each corresponding to one candidate being the commonly
preferred choice and each voter’s private information is a noisy signal about the state. Recent
work (e.g., Ladha (1982), Austen-Smith and Banks (1996), Feddersen and Pesendorfer (1997),
Wit (1998), Myerson (1998), Duggan and Martinelli (2005)) has generalized this result, known
popularly as Condorcet Jury Theorem (CJT), to more general assumptions on information
structures, voting rules, voter rationality and so on. We o¤er a general characterization of
the common value environments in which information is aggregated e¢ ciently.
Date: November 1, 2016.
This paper is partly based on an earlier version with the same title, co-authored with Tim Feddersen and
Wolfgang Pesendorfer. We are deeply indebted to both of them for their inputs. We are also thankful to Hari
Govindan, Bard Harstad and Roee Teper for helpful suggestions. All errors that remain are our own.
1
2
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
The two-state formulation, however, substantially reduces the complexity of the information aggregation by equating a private signal to an aggregate assessment of which candidate
is better. Formally, the binary state assumption implies that there are exactly two possible
distributions of private signals in a large electorate. However, the electability of a candidate
depends on myriad factors like his policy positions on di¤erent issues, his past history, party
a¢ liation, the state of the economy, the geopolitical situation and so forth. Similarly, different voters are likely to hold information about di¤erent factors, and a private signal may
favor di¤erent candidates depending on the underlying situations arising as a combination of
these factors. We identity by the state variable the di¤erent possible situations that a¤ect
voter preference, and require the correct candidate to be elected for every possible state.
Given this yardstick, the objective of our paper is to examine how the property of information aggregation depends on the relationship between (common) preference and distribution
of information in the electorate. Our main message is that the substantive interpretation
of signals and states in a particular environment matters for the property of information
aggregation.
In the formal sense, we extend the CJT to general state and signal spaces. Our results can
be interpreted the following way: If there are at least as many signals as there are states, then
FIE holds generically. On the other hand, if the state space is in…nite and the signal space is
…nite, FIE is impossible except for very special situations. Thus, the property of information
aggregation depends on the complexity of the information structure. Aggregation failure in
our setting happens not due to perverse equilibrium inferences, but because the variation
in voter information is not su¢ ciently rich to be able to distinguish between the di¤erent
situations over which the preferences vary.
For an illustration of the idea, consider an incumbent A competing against a challenger B,
and suppose that each voter gets one of two signals: a or b about A: First assume that they
are competing on quality, and an a-signal is good news and b-signal is bad news about A’s
relative quality in the sense that as the quality of candidate A improves, each voter is more
likely to observe an a-signal and less likely to have signal b. In this case, we are in a setting
similar to the canonical CJT, and voting will aggregate information. Next, consider a di¤erent
setting where there is uncertainty about A’s policy position on the left-right dimension and
voters prefer to vote for A only if his policies are su¢ ciently moderate. Now suppose that
a-signals arise more relatively frequently for more leftist positions and b-signals arise more
frequently for more rightist positions. Since the signals only tell the voter about whether
the position is skewed to the left or to the right and not about how extreme the position is,
information aggregation is impossible.1 While the example seems a bit stylised, such problems
arise routinely in settings where more than one issue is involved. Consider a referendum on a
reform proposal against a status quo. The reform has uncertain consequences on two policy
dimensions, and voters prefer the proposal only if it is closer to their ideal policy compared
to the status quo on this two-dimensional policy space. By the same logic as above, if voters
1 We thank Timothy Feddersen for providing us this example.
INFORMATION AGGREGATION IN ELECTIONS
3
only obtain binary signals about the consequences of the reform on each issue dimension,
information aggregation fails generically.
In the formal model, there are several alternatives, a state and a signal space. Each voter
obtains a signal independently from a distribution over signals conditional on the state. The
voter votes for one of the alternatives, and the alternative getting most votes wins. Additionally, we assume that indi¤erence occurs with measure zero. We say that an environment
allows Full Information Equivalence (FIE) if there exists some feasible strategy pro…le which
induces the correct outcome ex-ante almost surely as the electorate size becomes large. Borrowing an insight from McLennan (1998), we show that as long as there exists a feasible
strategy pro…le that induces FIE, there is also an equilibrium pro…le that does the same.
Therefore, while aggregation is possible in (some) equilibrium in an environment that allows
FIE, aggregation is impossible in environments that violate FIE even if voters can commit
to any strategy pro…le.
While the set-up is very standard, we develop a novel way of looking at the primitives of
the problem. We consider preferences de…ned over the conditional distributions arising in the
states (rather than over the states themselves). Each state is thus mapped to a vector on the
simplex (the space of all probability distributions) over the signals, and each such vector is
associated with the corresponding ranking over alternatives. This re‡ects the intuitive idea
that voter preferences are de…ned over distributions of private information.
Our …rst result (Theorem 1) provides a necessary and su¢ cient condition for FIE in environments with two alternatives (say, a1 and a2 ). Denote the set of conditional distributions
arising in states where a1 (resp. a2 ) is preferred by A1 (resp. A2 ). An environment allows
FIE if and only if A1 and A2 are separable by a hyperplane on the simplex. This result
(and most others in the paper) follows from the observation that the expected vote share for
an alternative (given any strategy) is a linear function of the vectors on the simplex, which is
a direct implication of each voter having to vote only based on his own private information.
A reinterpretation of the above theorem is the following. An environment allows FIE if
and only if each signal can be identi…ed to be in favor of alternative a1 or a2 in the same
way in every state. The classi…cation is based on which side of the hyperplane the vertex
corresponding to a given signal lies. The strategies that aggregate information are sincere in
the following sense: signals than favor a1 are assigned a higher probability of voting for a1
than for a2 ; and vice versa for signals that favor a2 :
Theorem 1 has opposite implications for discrete and continous state spaces. First, suppose
that the state space is isomorphic to the simplex over signals, i.e., there are well de…ned
preferences for all probability distributions over signals. In this case, Theorem 1 says that
FIE holds if and only if A1 and A2 from a convex partition of the simplex. In other words,
all the (indi¤erent) vectors around which the ranking changes must lie on a hyperplane.
Therefore, if preferences were to be de…ned over a su¢ ciently rich space, information would
be aggregated only in special environments.
4
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
At the other extreme, suppose that there are only two states. If A1 and A2 are two
singletons, they can always be separated by a hyperplane. Therefore, in a two-state world
which has been the focus of much of the literature, information is trivially aggregated. If
there are three states and at least three signals, it is easy to check that we can separate any
two probability vectors on the simplex from the third one by a hyperplane, as long as the
three vectors are not collinear.
Theorem 2 presents a general condition on the information structure that is su¢ cient for
an environment (with two or more alternatives) to allow FIE. The condition simply requires
that the set of probability vectors arising in the di¤erent states be linearly independent. This
result implies that FIE holds generically in environments where the state space is discrete and
there are at least as many signals than states (corollary 1), or if the states have a natural order
(proposition 2). In this sense, FIE holds if the information structure is not very complex.
Notice also that this is a condition only on the information structure (mapping from states
to the simplex) and not on preferences (mapping from states to rankings over alternatives).
We already know that when the state space is rich in the sense that it is isomorphic to
the simplex, FIE holds for very special utility functions. Indeed, the linear independence
condition (Theorem 2) is violated for rich state spaces. We provide a separate su¢ cient
condition for FIE for this case. The environment in this situation is described by a partition
A of the simplex where each element Aj is the set of conditional probability distributions
for which aj is the most preferred alternative. Convexity of Aj for each alternative aj ; which
is already demanding, is necessary but not su¢ cient for FIE when there are more than three
alternatives. Theorem 3 says that an environment allows FIE if for every pair of alternatives
ai and aj ; the set of distributions for which ai is preferred over aj be separable from the set
for which aj is preferred.over ai :
Finally, we provide a “representation” theorem for environments that allow FIE. Proposition 1 says that (i) if the utility from each alternative is a linear function of the conditional
probability vectors arising in the possible states, then the environment allows FIE; and (ii)
conversely, for any environment that allows FIE, there exists another environment with the
same top-ranked alternative in every state) which allows FIE and admits a utility that is
linear in the probability vector. For rich state spaces, this means that the “marginal rate of
substitution” between any two signals must be independent of the proportion of any other
signal.
This result points to another source of aggregation failure: when the tradeo¤ between
two sides of the issue under contention is a¤ected by a third factor. Suppose a country is
voting on whether to remain in an open trade zone or to exit (a classic case would be the
British referendum in May 2016 on whether to continue in the Eurozone). A central tradeo¤
that drives voter preferences is that trade induces growth but leads to immigration as well,
leading to loss of jobs for the local population. However, the tradeo¤ between growth and
immigration depends on the extent to which immigrants contribute to the economy. Each
voter receives a signal about one of these three factors, and the frequency of a signal depends
INFORMATION AGGREGATION IN ELECTIONS
5
on the strength of the factor. A similar issue arises in case of minimum wage legislation:
the tradeo¤ of income and employment is a¤ected by other factors like in‡ation. In all these
cases, the information structure is too complex to guarantee that the correct outcome will
prevail in all possible situations.
Our work speaks to a large body of literature on information aggregation in elections developed in the last three decades. The earlier statistical work (see Ladha (1992) and Berg
(1993) among others) assumed that voters voted for the alternative that their private signals
favored. Austen-Smith and Banks (1995) showed that such “sincere voting”is not necessarily
rational when voters condition their decision on a tie, which is when their vote matters. The
subsequent strand of game theoretic work (Feddersen and Pesendorfer (1997), Wit (1998),
Myerson (1998), Duggan and Martinelli (2005), Meirowitz and Acharya (2013), McMurray
(2016)) focussed on demonstrating that the equilibrium outcome of the voting game with large
populations approximated the full information outcome while making speci…c assumptions
on the informational and institutional environments (e.g., in…nite signal spaces, supermajority rules, abstention etc). Our approach has been to characterize the set of environments
where FIE is achievable in (some) equilibrium, rather than making speci…c assumptions on
the environment. Moreover, all these papers use a binary state space except for Feddersen
and Pesendorfer (1997) (diverse preferences and ordered state space) and McMurray (2016)
(common values and ordered state space). Thus, all except Feddersen and Pesendorfer (1997)
can be seen as instances of our result that aggregation is generically possible when the state
space is either discrete or ordered. We show that in order to consider the problem of information aggregation in a non-trivial way, one must consider move away from the two-state
formulation. Finally, we show in an extension that the characterization with majority rule
remains unchanged if we allow scoring rules, supermajority rules or abstention (Proposition
[[]]) in order to connect with this line of work.
McLennan (1998) shows that if in a given environment there is a type-symmetric voting
strategy pro…le that aggregates information, there is also a Nash equilibrium pro…le of strategies that aggregates information. We simply identify the environments where aggregation is
feasible in this sense, and adapt McLennan’s result to show that feasibility of FIE implies
FIE in equilibrium (Theorem 4). Therefore, our paper brings forward McLennan’s point that
under common values, aggregation problems arise due to failure of feasibility abd not due to
perverse equilibrium inferences.
There is another strand of literature that identi…es sources of aggregation failure in common
values environments. This includes unanimity rules (Feddersen and Pesendorfer (1998)),
alternative voter motivations (Razin (2003), Callander (2008)), cost of information acquisition
(Persico (2003), Martinelli (2005)), cost of voting (Krishna and Morgan (2012)), aggregate
uncertainty (Feddersen and Pesendorfer (1997)), diverse preferences (Bhattacharya 2013)
and so forth. Our work suggests that complexity of the information structure may itself be
a barrier to information aggregation.
6
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
Mandler (2013) shows that aggregation can break down in a common values model if the
same signal indicates opposite states in di¤erent situations. Bhattacharya (2013) presents a
condition called Weak Preference Monotonicity which says that aggregation can fail if the
same change in signal induces a randomly selected voter to vote for di¤erent alternatives
for di¤erent beliefs over states. While these papers have an analogous message, our paper
provides a stronger result in that we say that aggregation fails in all equilibria while they
only identify particular bad equilibria.
There is a literature on informational e¢ ciency of di¤erent scoring rules, in particular when
there are three alternatives. While Goertz and Maniquet (2011) and Bouton and Castanhiera
(2012) use diverse values, Ahn and Oliveros (forthcoming) have a common values model where
they show that the approval rule performs weakly better than all other scoring rules. We
show, conversely, that if a pure common values environment satis…es FIE for the approval
rule, it does so for other scoring rules too.
There is also a parallel literature on information aggregation in common value auctions
(Pesendorfer and Swinkels (1997), Siga (2016), and our Theorem 2 draws heavily from Siga
(2016), which also provides a characterization of environments where information is aggregated in the auction context.
In our …nal section we show that Theorem 1 can be extended to include the case where
voters have diverse preferences in addition to diverse information. However, in this case, we
cannot apply McLennan’s result to claim that whenever FIE is feasible, it is also achieved in
equilibrium. Therefore, we only have general conditions for feasibility of FIE when preferences
are diverse. Both Feddersen and Pesendorfer (1997) and Bhattacharya (2013) do satisfy the
feasibility conditions.
The rest of the paper is organized as follows. Section 2 lays out the model with common
voter preferences. Section 3 deals with the two-alternative case, where we provide a necessary
and su¢ cient condition under which an environment allows full information aggregation.
Section 4 provides su¢ cient conditions for FIE when there are multiple alternatives: We …rst
provide a su¢ cient condition for general environments, and then provide one for rich state
spaces where the previous condition fails. Section 5 shows that existence of a feasible strategy
pro…le that fully aggregates information implies the existence of an equilibrium sequence of
pro…les that does the same. Section 6 discusses three extensions: the …rst one shows that our
results do not change if we consider more general voting rules, the second one discusses the
genericity of information aggregation, and the other one derives the conditions for existence
of information-aggregating strategy pro…le in the case where voters may have preference
heterogeneity. Section 7 concludes.
INFORMATION AGGREGATION IN ELECTIONS
7
2. Model
In the model, there are n voters choosing among the set of k alternatives A = fa1 ; a2 ; :::; ak g:
Each voter casts a vote for one of these alternatives, and the alternative receiving the maximum number of votes wins the election. We shall consider abstention and other voting rules
later in section [[]]. Ties, if any, are broken randomly.
All voters have the same preference. The utility of a voter from an alternative depends on
an unobserved state variable 2 ; where
is a compact, separable metric space. Notice
that this formulation allows for both discrete and continuous state spaces. The utility of each
voter is given by a bounded and continuous function u :
A ! R. We denote by Aj the
set of states where alternative aj is weakly preferred to all other alternatives
Aj = f 2
: u( ; aj )
u( ; ai )g for all i
Voters do not observes the state : Instead, each voter receives a private signal x 2 X,
where X is a compact, separable, metric space. Pro…les of signals are denoted by xn 2 X n .
The information structure is captured by a probability measure on
X, as follows.
For a given ; each voter draws a signal x independently from the conditional distribution
( j ) 2 (X); where ( ) denotes the space of probability measures over “ ”, endowed with
the weak star topology. Likewise, given a signal x 2 X, he makes inferences about the true
state using the conditional ( jx) 2 ( ).
For in…nite state spaces we assume in addition that that 7! ( j ) is surjective and
strongly continuous: for each Borel measurable E
X, (Ej k ) ! (Ej ) as k ! . In
addition, we will assume absolute continuity, in the sense that there exists a Borel probability
measure 2 (X) such that ( j )
for every .
Let m 2 ( ) denote the marginal of
on . That is, m is the prior on . We
assume that (i) each alternative is the most preferred one with some positive prior probability,
i.e., m(Aj ) > 0 for all j = 1; 2; ::k; and (ii) indi¤erence occurs with zero probability, i.e.,
m(Ai \ Aj ) = 0 for all i 6= j: The second assumption e¤ectively implies that rankings are
strict when is discrete.
A tuple fu; A; ; X; g is de…ned as an environment. An environment in addition to an
electorate size n de…nes a game. In a game, a strategy for a voter speci…es a probability
of voting for each alternative given each signal. For reasons we explain later, we consider
only type-symmetric strategies where voters with the same signal use the same strategy.
Therefore, a strategy for a voter is a list of functions 1 ; 2 ; ::: k with j : X ! [0; 1]
P
satisfying j j (x) = 1 for all x 2 X: In short, is a behavioral strategy mapping X to
(A); where j (x) is understood to be the probability of voting for the alternative aj on
obtaining signal x. We denote a pure strategy s as a list s1 ; s2 ; ::sk with sj : X ! f0; 1g
likewise. Sometimes, with an abuse of terminology, we shall refer to or s as a pro…le of
strategies, with the understanding that every player uses the same or s; as the case may
be. While we use notation appropriate for …nite signal spaces, the same results go through
when we consider in…nite signal spaces (with summation replaced by integrals).
8
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
Given , the expected vote share of alternative aj at state
X
zj ( ) =
j (x) (xj )
is given by
x
By the appropriate version of the LLN, when the electorate grows without bounds and every
voter uses the strategy , zj ( ) approaches the realized proportion of votes for alternative
aj . Since we are interested only in large electorates, we call zj ( ) the vote share function
for alternative aj : The vote share function for each alternative has two properties which will
be very useful for our characterization: …rst, it is continuous in ; and second, it is linear in
( j ), i.e., in the vectors lying on the simplex (X):
Finally, while we consider symmetric strategies throughout the paper, it is without loss of
generality to do so. Consider any pro…le fbi gi=1;2;::n of asymmetric strategies, and suppose
P
that for each x 2 X; it produces an average probability bj (x) = n1 ni=1 bij (x) of voting for
alternative ak : The symmetric pro…le where each voter votes for alternative aj with probability
bj (x) for each x 2 X produces the same expected vote share as the asymmetric pro…le under
consideration. Since we are only interested in limit results as n grows, there is no loss in
considering only symmetric strategies.
Next, we de…ne the standard for information aggregation for a given strategy pro…le.
2.1. Full Information Equivalence. In a large electorate, the signal pro…le almost surely
reveals the state. Thus, if the signal pro…le were common knowledge, the most preferred
alternative would almost surely be elected. We say that information is aggregated by a
strategy pro…le if, under incomplete information, the most preferred alternative is guaranteed
to win with an arbitrarily high probability. We formalize this idea now.
Given any strategy pro…le , let zn denote the realized vector of proportion of votes for
alternatives a1 ; :::; ak . Observe that and induce a probability distribution over zn , since
the signal pro…le is drawn according to ( j ) and given the realized signal pro…le, the pro…le
of votes is drawn according to . Call such a probability distribution p on ([0; 1]k ). Let
Ljn [0; 1]k denote the set of vectors where the jth coordinate is not the unique highest. A
wrong outcome obtains if, in a state where aj is the most preferred alternative, it fails to
garner the unique maximum number of votes. Thus, the ex ante probability of a “wrong”
outcome obtaining is
XZ
Wn =
p (Ljn )m(d ):
j
Aj
We say that informaton is fully aggregated if Wn ! 0 as n ! 1. More formally, we say
that in an environment fu; A; ; X; g; the strategy achieves Full Information Equivalence
(FIE) if the ex-ante likelihood of error induced by converges to 0 as the number of voters
increases unboundedly.
Alternatively, for a given , let us de…ne the set of states where alternative aj is elected
by
Aj = f : zj ( ) > zi ( ); for i 6= jg:
INFORMATION AGGREGATION IN ELECTIONS
9
We say that achieves Full Information Equivalence (FIE) if the set of states where the
preferred alternative fails to win almost surely is of m-measure zero. In other words, FIE
obtains if for all j;
m(Aj n Aj ) = 0:
There is no guarantee that such a strategy will exist for every environment. In fact, if there
exists some strategy that aggregates information in a given environment, we say that the
environment allows FIE.
The next section discusses properties of the environments that allow FIE. Notice that
an environment allowing FIE is necessary but not su¢ cient for information aggregation in
equilibrium.
3. Feasibility of Information Aggregation: Two Alternatives
In this section, we discuss the special case of two alternatives, i.e., k = 2. Starting with
the case of binary alternatives has several advantages. First, this case is of great interest
in itself. Aside from empirical relevance, almost the entire literature on CJT concentrates
on the binary alternatives case. As a basis of comparison with the rest of the literature,
it is useful to study this case separately. Second, with two alternatives, we obtain a sharp
characterization of the environments that allow FIE by deriving a condition that is both
necessary and su¢ cient. This condition is very useful in understanding the issues that drive
FIE. Finally, the binary case serves as a starting point for understanding the more general
case of multiple alternatives.
To demonstrate the condition that determines whether an environment allows FIE or not,
we start with two of examples. In the …rst example, we show that when a third state is added
to a very standard Condorcet Jury setup, we may fail to get information aggregation.
For this section, we have A = fa1 ; a2 g; and strategy = ( 1 ( ); 2 ( )); where j (x) is the
probability of voting for alternative aj on obtaining signal x 2 X: Given the strategy, we
write the di¤erence in expected vote share between a1 and a2 as
z12 ( )
z1 ( ) z2 ( )
X
=
[ 1 (x)
2 (x)] (xj )
x
X
12 (x)
(xj )
x
where 12 (x) = 1 (x)
2 (x) for all x 2 X:
In order for FIE to obtain, the function z12 ( ) should be positive in states belonging to A1
(where a1 is preferred) and negative in states belonging to A2 (where a2 is preferred). While
most of the CJT concentrates on two states, the next example shows that aggregation can
break down if there are three states.
Example 1. Suppose A = fa1 ; a2 g and = fL; M; Rg; with each state having positive prior
probability: Assume a1 is preferred in L and R while a2 is preferred in M . Also, X = fx; yg;
10
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
and for some p > 21 ;
1
Pr(xjL) = p; Pr(xjM ) = ; and Pr(xjR) = 1
2
This environment does not allow FIE.
p
Proof. In order for a1 to win almost surely in both L and R; we require the strategy ( ) to
satisfy the following two conditions respectively
z12 (L) = p
12 (x)
z12 (R) = (1
Taken together, we must have 12 (x)+
almost surely in state M; which is
z12 (M ) =
p)
12 (y)
1
2
+ (1
12 (x)
p)
12 (y)
>0
+p
12 (y)
>0
> 0; which violates the condition for a2 winning
12 (x)
+
1
2
12 (y)
<0
The problem with information aggregation in this example is the following: In order for
a1 to win in state L where x is the more frequent signal, voters with signal x should vote
for a1 with su¢ ciently high probability. Similarly, in order for a1 to win in state R where
y is the more frequent signal, voters with signal y should vote for a1 with su¢ ciently high
probability. But then, for both the signals, voters vote for a1 more frequently than for a2 :
As a consequence, in state M where both these signals occur with moderate probability, the
vote share for a1 is already high and a2 cannot win.
The next example develops similar ideas in a richer state space.
Example 2. Suppose A = fa1 ; a2 g and the state is distributed uniformly over [0; 1]: The
signal space X = fx; yg; and Pr(xj ) = : Consider two di¤ erent preference environments. In
the …rst case, suppose all voters prefer a1 if > t and a2 for < t; for some t 2 (0; 1): In the
second case, for some 0 < t1 < t2 < 1; a1 is preferred whenever 2 (t1 ; t2 ) and a2 is preferred
when < t1 or > t2 : The …rst environment allows FIE but the second environment does
not.
Proof. For any strategy
=(
1(
);
2(
z12 ( ) =
)), the vote share di¤erence function is given by
12 (x)
+ (1
)
12 (y)
Notice that this function is continuous and linear in :
For to satisfy FIE in the …rst environment, we must have z12 ( ) > 0 for > t and
z1 ( ) < 0 for < t: By continuity, we must have z12 (t) = 0: Any that satis…es (i) z12 (t) = 0
and (ii) z12 ( ) is an increasing function leads to FIE. It is easy to check that we can always
…nd some with these properties.
For FIE in the second environment, we must have z12 ( ) > 0 for 2 (t1 ; t2 ) and z12 ( ) < 0
for 2 [0; t1 ) [ (t2 ; 1]: However, since z12 ( ) is linear in for every strategy ; there is no
symmetric strategy pro…le that achieves FIE in this case.
INFORMATION AGGREGATION IN ELECTIONS
11
Example 2 has the following real world implication. Suppose the voters have only one of
two kinds of information. If signals are about relative quality of the candidates, one signal
is interpreted as positive information about a1 and the other about a2 . However, if signals
are about whether a candidate leans to the left or right, and voter preferences are depend on
whether the candidate is moderate or extreme, signals cannot be classi…ed as each favoring
one candidate. Aggregation fails in this second case.
The main idea underlying the characterization theorem is contained in example 2. In this
example, FIE depends on the convexity of the set of states for which a given alternative is
preferred: in the …rst environment both the sets A1 and A2 are convex, while in the second
environment the set A2 is non-convex. The example, however, is special since the state space
is isopmorphic over the simplex over signals. Therefore, convexity of the sets A1 and A2
are incidental to the example: in general, the condition for an environment allowing FIE
is convexity of set of the probability distributions arising in the states for which the same
alternative is preferred. Theorem 1 formalizes the idea.
3.1. De…nitions: Pivotal states and hyperplanes. We say that a state 2 is pivotal
if, for each " > 0, m(A1 \ B" ( )) and m(A2 \ B" ( )) are positive, where B" ( ) is the " open
ball around . Let M piv
denote the set of pivotal states. By continuity of the utility
function, at all pivotal states, the voters must be indi¤erent between a1 and a2 : Given our
assumption that indi¤erence occurs only with zero measure, pivotal states occur only when
the state space is continuous.
A hyperplane in (X) is denoted by
X
H = f 2 (X) :
h(x) (x) = 0g;
x2X
for a given measurable function h : X ! R. Given a hyperplane H, we use
X
H + = f 2 (X) :
h(x) (x) > 0g and
x2X
H
= f 2
(X) :
X
h(x) (x) < 0g
x2X
to denote the two associated half-spaces. We shall denote the vector h as the normal to the
hyperplane H:
Theorem 1. Suppose A = fa1 ; a2 g. An environment (u; A; ; X; ) allows FIE if and only
if there exists a hyperplane H in (X) such that ( j ) 2 H + for 2 A1 , ( j ) 2 H for
2 A2 , and, if M piv 6= ;, ( j ) 2 H for 2 M piv .
We provide the proof in the appendix. The intuition for necessity follows example 2. For
FIE, it must be the case that, that z12 ( ) has to be positive in the A1 -states and negative in
the A2 -states. Since z12 ( ) is a linear functional of the conditional probability vectors, the
respective images of the set of A1 -states and the A2 -states on the simplex must be convex
(or, contained in disjoint convex hulls). In particular, for any pivotal state, we must have
12
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
z12 ( ) = 0 by continuity. For su¢ ciency, suppose there is a hyperplane (h; 0) on the simplex
that separates the distributions arising in A1 -states from those in the A2 -states. Any strategy
pro…le with 12 ( ) proportional to the normal h will deliver FIE.
It must be noted that while we cannot uniquely identify the strategies that aggregate
information, all strategy pro…les that aggregate information are characterized by a speci…c
property: in pivotal states, they lead to a vote share exactly equal to the threshold necessary
for the alternative A to win. In section [[]], we show that in environments that allow FIE, there
also exists an equilibrium strategy pro…le with the same properties in the limit. McMurray
(2014) …nds a result with a similar ‡avor and interprets it as the endogenous emergence of a
single dimension of political con‡ict in a multidimensional world.
3.2. Discussion of Theorem 1. A concise way to describe the theorem is as follows. Denote
by Aj the image of Aj on the simplex, i.e.,
Aj = f ( j ) 2
:
2 Aj g
The theorem says that the environment allows FIE if and only if A1 and A2 are separated
by a hyperplane. Figure 1 illustrates an environment with discrete states that allows FIE.
Figure 1 here
[[Theorem 1 o¤ers a clear interpretation of signals in environments (with two alternatives)
that allow FIE. In any such environment, the set of signals X can be subdivided into two
classes X1 and X2 in the sense that those in X1 (resp. X2 ) have their corresponding vertices
in the halfspace of the simplex where a1 (resp. a2 ) is preferred. Signals in X1 favor a1 and
those in X2 favor a2 in the following sense: for any signal in X1 ; the strategy that achieves
FIE attaches higher probability to a1 than to a2 ( 12 (x) > 0 if x 2 X1 ) and for any signal
in X2 ; the strategy attaches higher probability to a2 : In case of discrete state spaces, this
classi…cation may be endogenous. Thus, voting is endogenously sincere if FIE holds.]]
Theorem 1 can be seen as a positive or a negative result for the genericity of information
aggregation depending on the class of information structures that one considers. To see the
idea, …rst consider a continuous state space so that there are pivotal states. The theorem
says that the images of all pivotal states must lie on a hyperplane on the simplex. Thus, FIE
imposes strong restrictions on the utility functions when the state space is continuous. A
particular case of importance is when preferences are de…ned over all probability distributions
over signals, where the hyperplane condition is satis…ed only in special environments. One
must remember that in an environment that does not allow FIE, information cannot be
aggregated even if voters can co-ordinate on a strategy pro…le.
Example 3 simply extends example 2 from a unidimensional to a multidimensional simplex
over signals.
INFORMATION AGGREGATION IN ELECTIONS
13
Example 3. Suppose that X = f 1; 0; 1g; and denote a generic vector in (X) by
=
( 1 ; 0 ; 1 ); with x being the probability of signal x: Assume, as in example 2, that preferences are de…ned directly on (X); and let the function v12 : (X) ! R denote the utility
di¤ erence between the two alternatives a1 and a2 : Suppose that the signal x = 1 is positive
private information about a1 (or negative information about a2 ) and the signal x = 1 denote
the opposite. It is reasonable given these interpretations that v12 ( ) > 0 when 1 = 1 and
v12 ( ) < 0 when 1 = 1: It is also reasonable that along the path of indi¤ erence, more
x = 1 signals would compensate for more x = 1 signals, i.e., dd 1
> 0: However, the
1
v( )=0
content of Theorem 1 is that environments that allow FIE must satisfy
some
d
d
1
1
v( )=0
=
for
> 0:
For a real world interpretation, consider an electorate voting on a referendum, and signals
provide information about the proposal. One can think of x = 1 and x = 1 being information on two sides of a tradeo¤ that the proposal entails. For example, in case of a vote over
whether to remain in a common politico-economic union or not (e.g. the “Brexit” vote in
May 2016). Here, x = 1 is a signal that says that that staying will be good for growth and
x = 1 is a signal that says there will be loss of local jobs due to migration. The stronger the
likely growth e¤ect is, the more x = 1 signals are received, and the larger the likely job loss
is, the more x = 1 signals are received. Now, think of x = 0 being a signal on a third factor,
say, the extent to which migrants make a net contribution to the local economy. Theorem 1
says that the proportion of x = 0 signals should not a¤ect the rate at whic 1 (the proportion
of signals about growth) is traded o¤ against 2 (the proportion of signals about job loss).
This seems to be a strong restriction on preferences.
The next example is an application of Theorem 1 to a very standard case of spatial model
of political competition between two alternatives.
Example 4. We continue the metaphor of a policy proposal (alternative a1 ) being voted
on against a status quo (alternative a2 ). There is a policy space Y = [0; 1]2 ; in which both
alternatives are located. Voter utility for policy y is given by u(jy y j); u0 < 0: Thus, y 2 Y
is the voter ideal policy and voters prefer policies closer to y than those further from it. The
status quo is known to be located at ySQ 6= y on the policy space. On the other hand, there is
uncertainty about the location of the proposed policy: we denote the location of the proposed
alternative on the policy space by = ( 1 ; 2 ) 2 [0; 1]2 . In this setting, the voters prefer a1
(resp. a2 ) in a given state if j
y j is less (greater) than jySQ y j. The pivotal states
are given by
(1)
I=f :j
y j = jySQ -y jg
which is the circumference of a circle (or part thereof ). This is entirely a property of the
utility function. Now assume that the signal x = (x1 ; x2 ) is also two dimensional, and
suppose that xi 2 f0; 1g; with Pr(xi = 1j i ) = i : Thus, x1 provides information on 1 and
x2 on 2 independently of each other. Alternatively, the state i can simply be thought of as
14
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
the proportion of 1-signals in dimension i: There is no strategy pro…le for which FIE can be
obtained in this setting.
The proof is provided in the appendix.
Both examples 3 and 4 feature rich (continuous) state spaces but …nite number of signals.
In these examples, the voters have very limited private information and votes conditioned
on such information fails to produce the correct outcome in every state. Therefore, if voters
have limited richness in private information but their preferences have a rich variation across
di¤erent circumstances (states), then even with two alternatives it can be impossible to
guarantee information aggregation despite common preferences.
The next pair of examples presents the polar opposite case: the state space is discrete
(and there are at least as many signals as states). In this case, we shall see that irrespective
of the particular utility function, FIE obtains except for special circumstances, even when
there are more than two alternatives in question. We shall develop the result more generally
in the next section (Corollary 1), the examples in this section simply illustrate the idea. As
the examples make it clear, the crucial di¤erence between continuous-state and discrete-state
environments is that in the former, FIE requires a strategy to produces equal vote shares for
the two alternatives at all pivotal states but there is no such strict requirement in the latter.
In this pair of examples, there are r states { 1 ; 2 ; ::: r g occuring with positive probability
and s signals. The model assumptions imply that in each state, the ranking over alternatives
is strict. Each state t leads to a distinct probability distribution ( j t ) denoted as t :
Example 5. FIE obtains whenever r = 2 and
1
6=
2:
Example 6. FIE obtains if r = s = 3 and the conditionals arising in the three states
f 1 ; 2 ; 3 g are linearly independent.
We skip the proof as these are special cases of Corollary 1. However, for these cases, the
statements can simply be veri…ed by inspecting the …gures. Figure 3 demonstrates the case
of r = 2: it is always possible to pass a hyperplane separating 1 and 2 irrespective of their
location on the simplex. Notice that this example is of independent interest given that there
is a large literature that looks speci…cally at this case.
Similarly, Figure 3a illustrates the case of r = 3; s = 3 when the conditional vectors satisfy
linear independence. It is easy to see from the …gure that one can always separate any two
vectors from the third by a suitable hyperplane. Figure 3b demonstrates the necessity of
linear independence. In this case, if a1 is preferred in 1 and 3 while a2 is preferred in 2 :
It is easy to see from the …gure that one cannot …nd a hyperplane that separates f 1 ; 3 g
from 2 :
Fig 2 here Fig 3a here Fig 3b here
INFORMATION AGGREGATION IN ELECTIONS
15
4. Multiple alternatives: Feasibility
In this section, we attempt to characterize the conditions for an environment to allow FIE.
The two-alternative case is special in that it it allows for a necessary and su¢ cient condition
for an environment allowing FIE. In general however, such conditions are hard to obtain. We
provide two sets of results. The …rst set of results provides a simple su¢ cient condition for
FIE in general informational environments. However, the …rst set of conditions do not apply
to the case of rich state spaces (in the sense that the state space is isomorphic to the simplex
over signals). We develop a separate, second set of conditions for rich state spaces.
Recall that Aj is the set of states where alternative aj is (weakly) the most preferred,
and Aj is its image on the simplex over signals. We denote the partition of the state space
fAj gkj=1 by A and, correspondingly, fAj gkj=1 by A :
While the rest of the paper analyzes su¢ cient conditions for di¤erent environments, the
next remark identi…es the necessary condition
Remark 1. A necessary condition for an environment to allow FIE is that, for each for each
i 6= j; Ai and Aj are separable by a hyperplane on the simplex.
We skip the proof as it is immediate, and follows the same logic as Theorem 1.
4.1. Su¢ ciency condition for General Information Environments. Our next task is
to …nd a su¢ cient condition for FIE in environments with multiple alternatives. At this
point, it would be important to recall that the class of admissible information structures F
comprises of joint distributions over
X that satisfy both strong continuity and absolute
continuity. Notice that when the state space is discrete, these assumptions are trivially
satis…ed. For continuous ; let f ( j ) denote the density of ( j ): We use the notation for a
…nite signal space with the understanding that all results go through if the signal space were
continuous.
Example 6 shows that for three states, FIE is always satis…ed if the set of conditionals
is linearly independent. We now show that this is an instance of a more general result: as
long as the information structure satis…ed a form of independence that is even weaker than
linear independence, the environment allows FIE. This property is a¢ ne independence which
is de…ned as follows.
De…nition 1 (A¢ ne Independence). Let X denote the zero measure on X. We say that
2 F satis…es a¢ ne independence if there does not exist a nonzero, [ 1; 1]-valued, Borel
signed measure on the Borel sigma algebra of such that
Z
( j ) (d ) = X :
When the state space is discrete, i.e., = f 1 ; 2 ; ::: r g and the conditional ( j t ) in the
generic state t is denoted by t ; a¢ ne independence boils down to the following de…nition:
There does not exist r scalars f 1 ; 2 ; ::: r g with each scalar t 2 [ 1; 1] and not all zero such
16
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
that
r
X
t t
=0
t=1
The next theorem says that a¢ ne independence is su¢ cient for an environment to allow
FIE. Notice that this is a property of the information structure: as long as the joint distribution F on
X satis…es a¢ ne independence, FIE obtains irrespective of the preference
mapping from states to rankings over alternatives. We make heavy use of the techniques
developed in the auction context by Siga (2016) in order to prove the theorem.
Theorem 2. If the information structure
environment allows FIE.
2 F satis…es a¢ ne independence, then the
We develop the proof in two steps. First, we present a property which we call PS. An
environment is said to have this property if we can …nd a set of parallel hyperplanes that
separate each Ai and Aj (for i 6= j): Lemma 1 shows that this if an environment satis…es
this property, we can construct a strategy that satis…es FIE. Lemma 2 shows that a¢ ne
independence implies Property PS.
De…nition 2. We say that Property PS holds if there exists h : X ! [ 1; 1] and 0 < c1 <
P
c2 <
< ck 1 such that for all j < k, Aj f : x h(x) (xj ) 2 (cj 1 ; cj 11 )g.
Lemma 1. If property PS holds, then there exists a strategy that achieves FIE.
Proof. In the appendix
The intuition for the proof is the following. The property PS means that each Al and Aj
are separated by a hyperplane Hij on the simplex, and all the hyperplanes have a common
normal, which is given by the vector h(x): We show that we can de…ne a strategy function
such that ij (x) = h(x) c1i for all x; and all i 6= j: This strategy function achieves the
required separation and delivers FIE. This Lemma is based on Theorem 2 in Siga (2016).
2 F satis…es a¢ ne independence then property PS holds.
Lemma 2. If
Proof. In the Appendix
In the proof, we show that the information structure satis…es a¢ ne independence, we
can always slice the simplex by parallel hyperplanes such that each slice contains the set
Aj corresponding to a given alternative aj using Alaoglu’s theorem (Aliprantis and Border
(2006), Theorem 6.21). Lemma 1 and Lemma 2 together prove the statement of Theorem 2.2
2 Here is an example to show that the condition for su¢ ciency cannot be relaxed further to convex in-
dependence from a¢ ne independence. Suppose r = s = 4 and k = 2. The conditionals
2
=
( 12 ; 0; 0; 12 ),
and A2 = f 3 ;
3
4 g.
=
( 31 ; 13 ; 13 ; 0),
and
4
3
5
= (0; 13 ; 13 ; 13 ),
= (0; 0; 0; 1) satisfy convex independence; assume that A1 = f 1 ;
It is clear that the sets f
fact, the vector ( 51 ; 51 ; 15 ; 25 ) is equal to
1
1 +
2
5
1;
2
2g
and f
2g
3 ; 4 g cannot be separated by a hyperplane. In
3
+ 25 4 , so it lies in the convex hull of each
5 3
and also to
of these two sets; thus we cannot separate these two convex sets.
INFORMATION AGGREGATION IN ELECTIONS
17
Theorem 2 allows a simple corollary for the case when both the state and signal space are
…nite.
Corollary 1. Suppose there are r states f 1 ; :::: r g; k alternatives and s signals with s
r k: The environment allows FIE if the conditional vectors are a¢ nely independent, i.e.,
P
there exists no scalars j 2 [ 1; 1]; j = 1; 2; ::r (except all j = 0) such that j j ( j j ) = 0:
The above corollary generalizes examples 5 and 6. The general lesson with …nite state
and signal spaces is that FIE obtains except for very special cases as long as the number
of signals is more than the number of states, i.e., the signal space is richer than the state
space. Corollary 1 is important because a large number of papers the literature on the topic
have concentrated on exactly this set-up. The point that this corollary makes is that if it is
known that only certain probability distributions are possible but it is uncertain which one
has realized, then information aggregation is generically possible. We later show that FIE is
generic if the state space is either (i) …nite or (ii) continuous, compact and ordered. For now,
we postpone the formal analysis of genericity of FIE (to section [[]]) and turn to the case of
rich state spaces.
4.2. Rich State Spaces. There is one class of important information structures that violate
the su¢ cient conditions in Theorem 2. This is the case where the state space is isomorphic
to the signal space. We have seen that in this case, Theorem 1 puts strong restrictions on
the set of preferences that allow FIE (ref examples 3 and 4). This case is also of independent
theoretical interest as it does not restrict the set of signal distributions over which voter
preferences are de…ned. In this section we develop su¢ cient conditions for FIE when the
state space is isomorphic to the simplex over signals, i.e., when preferences are de…ned over
the entire simplex. Formally, this is equivalent to the following assumption which we make
for the remainder of this section.
Assumption M: M = f 2
(X) :
= ( j ); 2
g has full support over
(X):
We can now write the utility function de…ned over the states as
: (X) A ! R
such that, ( ( j ); a) := u( ; a), for all 2 , and all a 2 A. Given the isomorphism, we
will refer to the prior over (X) as m; with some abuse of notation. Similarly, we shall
write za ( ) to mean za ( ) where (j ) = . We also assume that the signal space is …nite
X = fx1 ; x2 ; :::; xs g for this section only. Notice also that when the state space is rich, not
only does A describe a partition of the state space, its image A also describes a partition
of the simplex.
An adaptation of remark 1 tells us that the necessary condition for FIE for rich state spaces
is that Aj is convex for all j: As argued earlier, this condition seems rather strong and will
be satis…ed only in special cases. This result stands in stark contrast to the genericity of FIE
for …nite state spaces (see corollary 1).
The next example shows that a convex partitional structure is not su¢ cient for FIE.
18
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
Example 7. Suppose there are three signals and the simplex over the signals is represented
by the right angled tiangle ABC, as shown in Figure 4 below. There is a smaller right angled
triangle DEF inside ABC; with side EF parallel to BC: The line EF intersects AC at G.
The most favored alternatives for di¤ erent probability vectors in the triangle are as follows:
a1 for the trapezium ADEB; a2 for the trapezium ADF G; a3 for the trapezium GEBC; and
a4 for the triangle DEF: While each Aj is convex, FIE is not achievable in this environment.
To see why, suppose strategy achieves FIE. Now, it must be the case that along all points on
AD (and the entire line along AD on the simplex); z1 ( ) = z2 ( ): Similarly, for all points on
the line along BE; z3 ( ) = z1 ( ): By linearity of vote shares in ; if BE and AD intersect
at H; then at = H; z1 ( ) = z2 ( ) = z3 ( ) Similarly, at F which is the intersection of
DF and EF , we must have z4 ( ) = z2 ( ) = z3 ( ): By linearity, z23 ( ) = 0 must trace a line
on the simplex, but we already know two points on this line: H and F: Therefore, z23 ( ) = 0
must be represented by the line along F H: However, for FIE we need z23 ( ) = 0 to coincide
with the line through GF; which is impossible.
Fig 4 here
There are actually two reasons why the convex partitional structure may not be su¢ cient
for FIE. Suppose, for each pair (i; j); the sets Ai and Aj are separated by the hyperplane
with norm hij : By the logic of pivotal states for Theorem 1, we need to …nd a strategy
which induces zij ( ) = 0 along the hyperplane that separates Ai and Aj : In other words,
ij must be proportional to hij for all pairs (i; j): Now, when there are k alternatives, there
are potentially k C2 such hyperplanes hij : On the other hand, a choice of a strategy function
= f 1 ; 2 ; ::: k g is essentially a choice of of k vectors (each with s elements). Therefore, we
have k unknowns with which to satisfy potentially k C2 equations, and that cannot be done
unless we impose extra conditions. Notice that with k = 2, we had one hyperplane and we
could choose two vectors 1 and 2 to “match” it. Therefore, with k = 2; convexity of Aj
was also su¢ cient for FIE. However, this problem is no longer trivial if k > 3 when there
are potentially more hyperplanes than alternatives. The second reason that complicates the
problem is that for any strategy ; the hyperplanes described by zij ( ) = 0 for di¤erent pairs
(i; j) are not necessarily linearly independent. This is already evident from example 7.
In order to develop further conditions that guarantee FIE, we need to extend the domain
of preferences from the simplex, which is
= f 2 Rs :
e = 1;
0g to the space
s
=f 2R :
e = 1g; which is the space of s-vectors that sum to 1:
Assumption E: There exists u
^ :
restriction of u
^ to (X) A.
A ! R such that
:
(X)
A ! R is the
The next assumption is the key property that guarantees su¢ ciency.
Assumption H: For all ai ; aj 2 A, there exists a hyperplane Hij de…ned by the normal
vector hij 2 Rs such that such that Hij+ = f 2 : u
^( ; ai ) u
^( ; aj )g and Hij =
f 2 :u
^( ; ai ) u
^( ; aj )g:
INFORMATION AGGREGATION IN ELECTIONS
19
Given assumption H, we can represent the preferences by the set of normals fhij gi;j :
Assumption H strengthens convexity of Aj in the following sense. Convexity of Aj imposes conditions only on the most preferred alternative in each state. Assumption H requires
that for each pair of alternatives ai and aj , the simplex be separated into two convex sets one where ai is preferred and the other where aj is preferred. For any ; the set of states
for which any two alternatives receive equal vote shares is characterized by a hyperplane,
irrespective of the vote shares received by the other alternatives for these states. Assumption
H imposes a similar structure on the preferences, requiring that the states where the voter
is indi¤erent between any two alternatives lie on a hyperplane, irrespective of whether these
alternatives are top-ranked or not in these indi¤erent states.
The next Lemma shows that assumptions E and H impose a particular linear dependence
on the set of hyperplanes fhij g through transitivity.
Lemma 3. Suppose assumptions M, E and H hold.
ai ; aj ; al 2 A; there exist positive constants ij ; jl ; and
(2)
ij hij
+
jl hjl
=
Then, for any three alternatives
il such that
il hil :
Proof. In the appendix
In words, for every three alternatives, the three hyperplanes that describe indi¤erence between each pair are either parallel to each other or have a common intersection. In absence of
assumption E, the hyperplanes could intersect pairwise outside the simplex. Thus, assumption E simply imposes a restriction on the preferences over distributions within the simplex,
i.e., on preferences described by v( ).
Fig 5 here
Figure 5 makes the argument graphically by contradiction. Suppose there are three alternatives fa1 ; a2 ; a3 g and fH12 ; H23 ; H13 g describe indi¤erence between the respective pair of
alternatives. Suppose also that they do not have a common intersection: as depicted in the
…gure, A; B and C are the points of pairwise intersection. It is easy to see that we have a
cycle of strict preferences in the triangle ABC.
[[Note that when there are exactly three alternatives, convecity of Aj directly implies
Lemma 3.]] (Do we need E? Proof?)
Now, we are ready to state and prove the main theorem.
Theorem 3. If Assumptions M, E and H hold, then there exists a strategy that achieves
FIE.
Proof. In the appendix
We have argued earlier that the challenge is to choose k strategy vectors to satisfy k C2
linear equations (“match” k C2 hyperplanes). The proof consists in showing that Lemma 3
imposes su¢ cient dependence among these k C2 equations so that we can guarantee a solution.
20
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
The main message of the section on rich state spaces is the following. For two or three
alternatives, necessary and su¢ cient for information aggregation is convexity of the set of
states for which a given alternative is top-ranked: For k > 3; the su¢ cient condition for FIE
is that this convexity property must hold for every pairwise comparison of alternatives. This
property has the ‡avor of Independence of Irrelevant Alternatives in Social Choice Theory.
[[Is this too much?]]We stress again that these conditions are satis…ed only by special utility
functions.
4.2.1. Linear utility representation. Before concluding this section, we present the following “representation theorem” for environments that allow FIE. This is a representation for
utilities de…ned over probability distributions. More formally, de…ne the set of possible probability distributions as M = f 2 (X) : = ( j ); 2 g; and denote the utility function
over vectors on the simplex as : M A ! R such that, ( ( j ); a) := u( ; a); for all a 2 A
and 2 : While this characterization is sharp when the state space is rich in the sense that
M = (X); the rich state assumption is not necessary for the characterization result to hold.
We shall denote by A( ) the partition of the state space induced by the utility function .
We will continue to assume that the state space is X = fx1 ; x2 ; :::; xs g:
Our result is the following. If the utility from each alternative is a linear function of
the probability (or population proportion) of each signal, then the environment allows FIE.
Conversely, for every environment that allows FIE, there exists a utility function linear in
the probabilities which induces the same top-ranked alternative for each state.
Proposition 1. If there exists a 2 Rs such that ( ; a) = a
, for all a 2 A and
all 2 M
(X); then there exists a strategy that achieves FIE. Conversely, given some
v : M A ! R, suppose there exists a strategy that achieves FIE. Then, there exists ~( ; a) =
~a , with ~a 2 Rs , such that A(v) = A(~) for all j:
Proof. We can assume that a 2 (X) without loss of generality because a positive a¢ ne
transformation of va always lies in (X). Choose the strategy a = a with the interpretation
that a (x` ) is the `-th co-ordinate of va :Then, we have za ( ) = a
= ( ; a) for all
2 (X), which guarantees FIE. For the converse, simply let ~a := a for every alternative
a.
This characterization is also portrayed in example 3 and the discussion following it. An
interpretation of the proposition (for a rich state space) is that the marginal change in utility
from an alternative with respect to the proportion of any signal is independent of the proportion of the other signals. Alternatively, along the locus of indi¤erence of any two alternatives,
the rate at which the change in the proportion of one signal compensates for the change in
proportion of another signal must be constant. In this sense, the tradeo¤ between any two
signals should be una¤ected by a third signal.
IWhen the state space is rich, for every pair of alternatives ai and aj , the characterization
allows us to develop a classi…cation of signals in terms of which of the two alternatives is
INFORMATION AGGREGATION IN ELECTIONS
21
favored. Notice that this characterization depicts an environment that satis…es assumptions
E and H. Also, the characterization It is unique to the extent that, for all states lying
on the hyperplane Hij that describes indi¤erence between ai and aj the vectors vai and vaj
satisfy
(vai
vaj )
= 0:
Denote the r-th co-ordinate of the vector va by var : Now, de…ne a partition fXi ; Xj ; X g of
the signal space X in the following way: Xi = fxr 2 X if vari > varj g; Xj = fxr 2 X if
vari > varj g and Xj = fxr 2 X if vari = varj g: Signals in Xi favor ai and those in Xj favor aj
in the sense that higher proportion of any signals in Xi at the expense of any signal in Xj
raises the utility di¤erence between ai and aj : Moreover, since the strategy must have ij (xr )
in proportional to vai vaj ; it must be the case that ij (x) > 0 if x 2 Xi and ij (x) < 0 if
x 2 Xj : In this sense, voting is sincere when FIE is achieved.
5. Equilibrium analysis
From the previous sections, it is clear that certain environments allow full information
equivalence in the sense that there exist strategies that achieve FIE. However, it is not clear
whether, even in such environments, voters have an incentive to use such strategies. In order
to check whether voters …nd it in their interest to use such strategies, we consider voting as
a game played in such environments. A game is de…ned as an environment fu; A; ; X; g
along with a number of players n: We …x an environment and consider a sequence of games
by letting the number of voters grow. Following the logic in McLennan (1998), we show that
under common preferences, any environment that allows FIE also has a sequence of Nash
equilibrium pro…les that achieves FIE.
First, we de…ne the game derived from the environment fu; A; ; X; g along with a number
of players n more formally. We provide the analysis for the case where is a continuous
random variable, the case where is discrete is exactly analogous.
It will be necessary to distinguish …nite electorates, so let us use the notation xn =
(x1 ; :::; xn ), sn = (s1 ; :::; sn ), and n = ( 1 ; :::; n ) for pro…les of signals and strategies in a …nite electorate f1; :::; ng. Given on
X, construct on
X n as d = dm ( ni=1 d ( j ))
and let x be the marginal of on X n . Denote by ( jxn ) the conditional of on given a
pro…le xn .3 For a given xn , let
Z
un (a; xn ) =
u(a; ) (d jxn );
for a 2 A. Given a pro…le of pure strategies sn , let u(sn (xn ); xn ) denote the utility at pro…le
xn at the outcome induced by sn (xn ). This outcome is simply the alternative ak obtaining
the maximum number of votes under sn (xn ); with ties broken by a coin ‡ip. The multilinear
3 The construction is presented in general form so that it also covers the case of uncountable X that we will
deal later.
22
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
extension at a pro…le of behavioral strategies is denoted u( n (xn ); xn ). Finally, the ex ante
expected utility for a given pro…le n is
X
un ( n ) =
u( n (xn ); xn ) x (xn ):
xn 2X n
The Bayesian game Gn played by the n voters is the game where each voter has the same
P
space of behavioral strategies, [[ = f k : X ! [0; 1]; k = 1; 2; ::K;
k k (x) = 1 for all
x 2 Xg (should we also index it with voter i?]] endowed with the narrow topology that
makes it a compact, convex LCTVS, and the payo¤s are the ones we just derived.
Suppose that n is a maximizer of un ( n ): The existence of such a maximizer follows from
compactness of the domain and continuity of u on n .4 Following McLennan (1998), n is
a Bayesian Nash equilibrium of the game Gn : It is straightforward to restrict to pro…les of
symmetric strategies and ensure existence of a symmetric BNE. The next theorem tells us
that the sequence n achieves FIE as long as the environment fu; A; ; X; g allows FIE.
Theorem 4. If the environment (u; A; ; X; ) allows FIE, there exists a sequence
n
equilibria of the game Gn that achieves FIE., i.e., Wn ! 0:
Proof. Observe that u(
n
of Nash
n)
can be written as
Z X
X
n
u(a; )
'a (xn ) (xn j )m(d )
xn 2X n
a
where a 2 A [ D, “D" standing for a draw (and the associated coin ‡ip to decide the
n
winner)[[bad notation??]], and 'a (xn ) is the probability that a is the outcome of the election
at the pro…le xn . For each size n of electorate, consider a symmetric pro…le of strategies n =
( ; :::; ), so that 1 = ( ; ; :::). For each 2 , the proportion of votes for ak converges
P
n
n
n
to zk ( ) ( j )-almost surely as n ! 1. Hence 6=
xn 2X n 'a (x ) (x j ) converges, so
Lebesgue Dominated Convergence implies that u( 1 ) = limn!1 u( n ) is well de…ned.
Observe that if the symmetric pro…le ^ 1 achieves FIE, then u(^ 1 ) is the maximum attainable value: for m-almost every 2 Ak , A wins. States in Aj \ Ak are irrelevant for
the evaluation above because they are of m-measure zero. So, given that u( n ) is linear in
u(a; ), the claim is veri…ed.
For each …nite electorate f1; :::; ng, choose n as a maximizer of u( n ). We know such
pro…le is an equilibrium of the corresponding game Gn . We also know that u(^ 1 ) is the
maximum feasible value of the ex ante utility. Hence
u(^ 1 )
u(
1
) = lim u(
n
n
)
lim u(^ n ) = u(^ 1 );
n
n
establishing the result. In fact, if Wn were not to converge to zero, then we would have to
1
have, say, m(Ak nAk ) > 0 for some k. That is, a set of positive measure in Ak where an
alternative aj 6= ak wins under 1 , whereas we know that no such set exists for ^ 1 . But
4 By construction, for each given , the probability measure on pro…les is the product
integral of the utility function is a continuous function of the pro…le
ex ante utility, which must then be a continuous function of
n
.
n
; integrating out
n
i=1
( j ); hence the
then recovers the
INFORMATION AGGREGATION IN ELECTIONS
then u(^ 1 ) > u(
loose notation]]
1 ),
23
contradicting what we just established. [[Please check this part for
6. Extensions and Discussion
6.1. Genericity of Full Inforamtion Equivalence. In this section, we formalize the idea
that FIE is a generic property of …nite state spaces as well as continuous but ordered and
compact state spaces. In order to do so, we …rst need to develop a notion of genericity
appropriate for the class of information structures F. [[A one-sentence description of the
genericity standard to be inserted here]] Proposition 2 shows that within this class, FIE
is generic if the state-space is isomorphic to the unit interval. By similar reasoning, FIE is
generic for …nite state spaces too. It must be mentioned here that all of the existing literature
on information aggregation in large elections has concentrated on subcases of these two kinds
of state spcaces. This includes [[insert examples here]] which have concentrated on binary
state spaces. Feddersen and Pesendorfer (1997), [[McMurray (201x, 201y)]] considers a state
space that is a unit interval, albeit with limited preference variation. Finally, [[McMurray
(201z)]] considers a state space that is a circle, but shows that it generically boils down to the
unit interval. This result also tells us that the failure of FIE in example 2 was a non-generic
case.
Let M( ) be the closed unit ball of the space of Borel signed measures on and M(X)
the closed unit ball of the space of Borel signed measures on X. By Alaoglu’s theorem
(Aliprantis and Border (2006), Theorem 6.21) M( ) and M(X) are weak* compact. As the
space of real-valued continuous functions C( ) is separable, choose a countable dense subset
fg1 ; g2 ; :::g of C( ). By Theorem 6.30 of Aliprantis and Border (2006) the metric
Z
Z
1
X
1
( ; )=
g
(
)
(d
)
gn ( ) (d ) for ; 2 M( )
n
2n
n=1
metrizes the weak* topology on M( ). Given our focus on information structures such that
( j n ) ! ( j ) strongly whenever n ! , we will consider a stronger topology on (X).
Let
m(p; q) =
sup jp(E)
E2B(X)
q(E)j for p; q 2 M(X);
where B(X) denotes the Borel sets of X, and endow C( ; (X)) with the metric
d( ; 0 ) = max m( ( j ); 0 ( j )):
2
R
For ease of notation, for 2 M( ), let ( ) denote
( j ) (d ), and note that ( ) 2
R
RR
f (x) (dxj ) (d ). Let
and X denote the zero
M(X). Also note that f d ( ) =
measures in M( ) and M(X).
Let F ac denote the set of 2 F that are absolutely continuous, and let
F = f 2 F ac : m( ( );
X)
= 0 for
2 M( ) ) ( ;
) = 0g:
24
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
We now show that F is a generic set in F when is one-dimensional. The extension for the
…nite-dimensional case would follow on the same lines and is omitted. We do not know if the
result is true for a general compact metric .
Proposition 2. When
= [0; 1], the set F is a residual set in F.
6.2. Scoring rules. TBA
6.3. Diverse Preferences. So far we have assumed that all voters have the same preferences. In this section we extend our results to a case where the voters in the electorate may
have di¤erent preferences. We maintain the assumption that all voters are ex ante identical,
and draw their information and preferences from some distribution conditional on the state.
To do so, we retain the elements of the set-up in the main section and assume in addition
that the private signal x is also payo¤ relevant. Thus, the private draw of an individual serves
two functions: it is a view about the outcomes and it provides information about how others
view the outcomes. We may think of xi = (si ; ti ); where si is the common value component
and ti is the private value component of the preference. Notice that this is a general setting
that can encompass many di¤erent environments. In particular, it admits the environments
studied in Feddersen and Pesendorfer (1997) with continuous state space and Bhattacharya
(2013) with just two states.
Consider, therefore, that voters preferences are given by u :
X fA; Bg ! R. [[We
deal only with the case that X is at most countable.]]
We now normalize u( ; x) = 1 if u( ; x; A) > u( ; x; B), u( ; x) = 0 if u( ; A) < u( ; x; B),
u( ; x) = 21 if u( ; x; A) = u( ; x; B). This normalization is innocuous for the feasibility
result.
Since there is no commonly preferred candidate under diverse preferences, we have to be
careful while de…ning the standard for information aggregation. We say that a strategy pro…le
aggregates information if, for a large electorate, the outcome is the same as the outcome when
the state is commonly known. Notice that in a large electorate with …nite signals, knowing
the state can be interpreted as knowing the entire pro…le of private signals.
By the SLLN, asymptotically the proportion of voters that prefer A to B, given a state ,
is
X
u( ) =
u( ; x) (xj ):
x2X
In a large electorate, A would get a vote share very close to u( ) if the state were known to
be : Therefore, under full information A wins depending on whether u( ) is greater or less
than the threshold q: Now …x q 2 (0; 1) and rede…ne the sets A, B, and I as
Aq = f 2
: u( ) > qg
Bq = f 2
: u( ) < qg
Iq = f 2
: u( ) = q:g
INFORMATION AGGREGATION IN ELECTIONS
25
In states Aq (resp.Bq ); alternative A (resp. B) wins under full information and large electorates. We then say that an environment (u; ; X; ; q) allows FIE if there exists a strategy
such that
m(Aq nA ) = m(Bq nB ) = 0:
It is simple to verify now that the argument in the proof of Theorem 1 follows through
line-by-line, so FIE is again characterized by the hyperplane condition. More formally, we
have the following result,
Proposition 3. An environment (u; ; X; ; q) and diverse preference allows FIE if and only
if there exists a hyperplane H in (X) such that ( j ) 2 H + for 2 Aq , ( j ) 2 H for
2 Bq , and, if M piv 6= ;, ( j ) 2 H for 2 Mqpiv .
It is also simple to verify that the arguments for Corollaries ?? and ?? remain valid.
Corollaries ?? and ?? on the other hand, do not go through any longer: changing q changes
the sets Aq , Bq , and Iq , so the arguments given above do not show that FIE is obtained for
the new level q^.
Notice that since Feddersen and Pesendorfer (1997) result already tells us that information
is aggregated in equilibrium, the existence of FIE strategies is trivial in their setting. More
interestingly, while Bhattacharya (2013) concentrates on showing that, for any consequential
rule, there exists an equilibrium that fails to aggregate information, it can be checked that
in Bhattacharya’s two-state setting, there always exists some feasible strategy that achieves
FIE. It would therefore be very interesting to examine the conditions under which, in a general setting with diverse preferences, there exists some equilibrium sequence that aggregates
information.
However, the proof of Theorem 4 explicitly utilizes the common value setting, and therefore
does not automatically generalize to an environment with diverse preferences. In particular,
we do not know yet whether, given preference diversity in the electorate, the existence of a
feasible strategy pro…le guaranteeing FIE also implies that FIE is achieved in equilibrium. Our
e¤orts are currently focussed on analyzing conditions under which the hyperplane result also
implies that information is aggregated in equilibrium in presence of preference heterogeneity.
7. Conclusion
The existing literature on information aggregation in large elections has largely focussed
on speci…c preference and information environments. In this paper, we consider general environments in order to analyze conditions under which information is aggregated. Preferences
depend on the state of the world, and each state of the world is synonymous with a probability
distribution over private signals. Therefore, preferences are simply mappings from allowable
probability distributions over private information to rankings over the two alternatives A and
B. In a large electorate, the frequency distribution over signals is approximately the same as
the probability distribution. Thus, our question is whether the election achieves the outcome
26
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
that would obtain if the entire pro…le of private signals were publicly known. If an environment permits a feasible strategy pro…le that can induce the full information outcome with
a high probability in almost all states, we say that the environment allows Full Information
Equivalence (FIE). Moreover, we are interested in whether such a strategy pro…le is incentive
compatible, i.e., it constitutes a Nash equilibrium in the underlying game.
We study both environments with and without preference heterogeneity. In both cases,
we …nd that an environment allows FIE if and only if a hyperplane on the simplex over
signals separates the probability distributions arising in states where A is preferred from
those arising in states where B is preferred. Therefore, if the information environment is
su¢ ciently complex, there is no strategy pro…le that aggregates information. We like to
stress here that the failure of FIE has nothing to do with equilibrium assessments over the
states based on the criterion of one’s vote being pivotal in deciding the election.
We obtain sharper positive results in the common preference case where all voters would
have agreed on their rankings if they had known the pro…le of signals. In this case, voting
aggregates information alone (and not preference). We …nd that as long as an environment
allows FIE, there is a sequence of equilibria that achieves FIE. We must mention here that
there may be other equilibrium sequences that do not aggregate information - but ours is only
a possibility result. As implications of this result, we provide several examples of common
preference environments where information will be aggregated. We show that if there are only
a …nite number of states and signals, FIE holds in equilibrium for any preference mapping
from states to alternatives as long as the signal space is su¢ ciently rich compared to the space
of states. As a special case, we show that whenever there are two states, FIE is generically
achieved in equilibrium. We also show that in the common preference environment, the
voting rule does not matter for information aggregation: as long as FIE is achieved under
some voting rule, FIE is achieved under every other non-unanimous voting rule.
Feasibility of FIE does not automatically imply FIE in equilibrium when voting has the
burden of aggregating information and preferences simultaneously, i.e., in the diverse preference case. However, we conjecture that under some conditions, feasibility of FIE indeed
implies the existence of an equilibrium sequence that also achieves FIE in the case with
diverse preferences. Our current research e¤orts are focussed on unveiling these conditions.
8. References
(1) Acharya, Avidit. “Equilibrium False Consciousness." mimeo, Stanford University
(2014).
(2) Aliprantis, C. D., and K. C. Border. “In…nite Dimensional Analysis", Springer-Verlag.
(2006).
(3) Aubin, Jean-Pierre, and Hélène Frankowska. “Set-Valued Analysis", Birkhäuser
(2009).
(4) Austen-Smith, David, and Je¤rey S. Banks. “Information aggregation, rationality,
and the Condorcet jury theorem." American Political Science Review (1996): 34-45.
INFORMATION AGGREGATION IN ELECTIONS
27
(5) Berg, Sven. “Condorcet’s jury theorem, dependency among jurors." Social Choice
and Welfare 10, no. 1 (1993): 87-95.
(6) Bhattacharya, Sourav. “Preference Monotonicity and Information Aggregation in
Elections." Econometrica 81, no. 3 (2013): 1229-1247
(7) Cremer, Jaques, and Richard P. McLean, “Full Extraction of Surplus in Bayesian and
Dominant Strategy Auctions”, Econometrica 56, no. 6, (1988), 1247-1257.
(8) Duggan, John, and César Martinelli. “A Bayesian model of voting in juries." Games
and Economic Behavior 37, no. 2 (2001): 259-294.
(9) Feddersen, Timothy, and Wolfgang Pesendorfer. “Voting behavior and information
aggregation in elections with private information." Econometrica: Journal of the
Econometric Society (1997): 1029-1058.
(10) Ladha, Krishna K. “The Condorcet jury theorem, free speech, and correlated votes."
American Journal of Political Science (1992): 617-634.
(11) Mandler, Michael. “The fragility of information aggregation in large elections."
Games and Economic Behavior 74, no. 1 (2012): 257-268.
(12) McLennan, Andrew. “Consequences of the Condorcet jury theorem for bene…cial information aggregation by rational agents." American Political Science Review (1998):
413-418.
(13) McMurray, Joseph: “Why the Political World is Flat: An Endogenous Left-Right
Spectrum in Multidimensional Political Con‡ict”, mimeo, Brigham Young University
(14) Milgrom, Paul R. “Good news and bad news: Representation theorems and applications." The Bell Journal of Economics (1981): 380-391.
(15) Myerson, Roger B. “Extended Poisson games and the Condorcet jury theorem."
Games and Economic Behavior 25, no. 1 (1998): 111-131.
(16) Siga, Lucas. “Complex Signals and Information Aggregation in Large Markets"
mimeo, New York University (2013)
(17) Wit, Jörgen. “Rational choice and the Condorcet jury theorem." Games and Economic Behavior 22, no. 2 (1998): 364-376.
(18) To be completed
9. Appendix
9.1. Proof for Example 4. TBA
9.2. Proof of Lemma 1.
28
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
Proof. Let f^ k gK
k=1 be such that, for every x, we have
^ 1 (x)
^ 2 (x) = h(x)
c1 1
^ 2 (x)
^ 3 (x) = h(x)
c2 1
::: = :::
^K
1 (x)
^ K (x) = h(x)
cK1
1
Clearly, the ^ ’s are bounded, so we can choose su¢ ciently large so that ^ k (x) +
0 for all
P
k and all x. By the same reason, we can choose " > 0 su¢ ciently small such that k (^ k (x) +
P
)" 1 for every x. Let R(x) = 1
k (^ k (x) + )", and set k (x) = (^ k (x) + )" + R(x)=K,
P
so that k k (x) = 1 for every x.
Then, = ( 1 ; :::; K ) is a well de…ned strategy. To show that achieves FIE, consider
2 such that alternative j is best, that is, 2 Aj . The mass of votes for alternative k is
R
R
k (x) (dxj ). We want to show that ( j (x)
k (x)) (dxj ) > 0 for all k 6= j. First note
that
Z
Z
(^ j (x) ^ k (x)) (dxj ) > 0 ) (^ j (x) +
^ k (x)
)" (dxj ) > 0
)
Z
[(^ j (x) + )" + R(x)
Analogously
Z
(^ j (x)
[(^ k (x) + ) + R(x)]] (dxj ) > 0 )
^ k (x)) (dxj ) < 0 )
Consider k > j. By Property PS,
R
ck > cj for all k > j, we have (h(x)
Z
(^ j (x)
R
Z
(
j (x)
Z
(
k (x))
j (x)
k (x))
(dxj ) > 0:
(dxj ) < 0:
R
h(x) (dxj ) > cj 1 , so (h(x) cj 1 ) (dxj ) > 0. As
ck 1 ) (dxj ) > 0 for all k > j. Then
^ k (x)) (dxj ) =
k Z
X
(h(x)
ci 1 ) (dxj ) > 0
i=j
and hence
Z
(
j (x)
k (x))
(dxj ) > 0:
R
R
Consider k < j. By Property PS, h(x) (dxj ) < cj 11 . So (h(x) cj 11 ) (dxj ) < 0. As
R
ck < cj for all k < j, we have (h(x) ck 1 1 ) (dxj ) < 0 for all k < j. Again, this means that
Z
and hence
(^ k (x)
^ j (x)) (dxj ) =
j 1Z
X
(h(x)
ci 1 ) (dxj ) < 0
i=k
Z
(
k (x)
j (x))
(dxj ) < 0:
INFORMATION AGGREGATION IN ELECTIONS
29
9.3. Proof of Lemma 2.
m+1 , (ii) m !
Proof. Consider a sequence of …nite subsets m of such that (i) m
in Hausdor¤ sense, and (iii) the densities f ( j ) are a¢ nely independent, for all 2 m . We
can do this because satis…es a¢ ne independence. In fact, if for any …nite set f 1 ; :::; L g the
P
densities f ( j ` ), ` = 1; :::; L were not a¢ nely independent, we would have ` f (xj ` ) ` = 0
for every x with some of the weights ` being non-zero. But then, setting to be the signedR
P
measure ` ` ` , where ` is the point mass at ` , we would have f (xj ` ) (d ) = 0 for
R
every x, which implies that
( j ) (d ) = X , which in turn implies that all weights `
must be zero (by a¢ ne independence of ).
Fix a list 0 < c1 < c2 <
< cK 1 . We want to show that property PS holds. Let
1
c^i = ci + ", for " > 0 smaller than the di¤erence between any two ci and cj . By a¢ ne
independence, for each m there is hm 2 L1 (X) (in fact, we can choose hm to have range in
[ 1; 1]) such that
Z
hm (x)f (xj )dx = c^i ; for all
2 Am
i ;
m\A .
where Am
i
i =
By Alaoglu’s theorem (Aliprantis and Border (2006), Theorem 6.21), the so constructed
sequence hm has a weak -convergent subsequence, so let h be its limit. By construction, for
each 2 Am
i
Z
h(x)f (xj )dx = c^i :
R
As Am
h(x)f (xj )dx = c^i . Indeed, there
i converges to Ai , for each such we must also have
m
m
m
m
must exist a sequence
with
2 Ai such that
! ; by strong continuity of , f ( j m )
R
R
norm-converges to f ( j ), so h(x)f (xj m )dx ! h(x)f (xj )dx. Property PS is therefore
veri…ed.
9.4. Proof of Lemma 3.
Proof. Let I = Hij \ Hjk . Suppose …rst that I is non-empty. If
2 I then u
^( ; ai ) =
u
^( ; aj ) = u
^( ; ak ); hence 2 Hik . By assumption H, I is a vector space of dimension K 2.
It’s orthogonal complement therefore has dimension 2. In this orthogonal complement, hij
and hjk lie on a two dimensional linear space, they are independent (because their corresponding hyperplanes intersect), and therefore, they span the space. Since hik also lies in
this space it must be true that there exists constants ij ; jk such that hik = ij hij + jk hjk .
Suppose instead I is empty. This is true whenever Hij and Hjk are parallel. It is clear that
there is always constants satisfying equation (2). Therefore, we establish that whether I is
empty or not, there will always exist constants satisfying equation (2).
Next we show by contradiction that the constants need to be positive.
Case 1. Suppose ij < 0, and jk < 0. There are two possible scenarios: either there
exists such that
hij > 0 and
hjk > 0, or there exists such that
hij < 0 and
hjk < 0. Otherwise, one of the alternatives is never the best choice. We will focus on
the former and note that the analogous argument holds inverting the inequalities. Thus,
30
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
u
^( ; ai ) > u
^( ; aj ) and u
^( ; aj ) > u
^( ; ak ). By transitivity, u
^( ; ai ) > u
^( ; ak ). On the other
hand, ( ij hij + jk hjk ) < 0. Thus hik < 0, and then, u
^( ; ai ) < u
^( ; ak ), a contradiction.
Case 2. Suppose ij > 0, and jk < 0. Consider such that
hik = 0 and
hij 6= 0.
hik = 0 implies
( ij hij + jk hjk ) = 0. Then ij hij =
hjk . There are two
jk
possibilities: (i)
hij > 0. This implies
hjk > 0. Then u
^( ; ai ) > u
^( ; aj ) > u
^( ; ak ). (ii)
hij < 0. This implies
hjk < 0. Then u
^( ; ai ) < u
^( ; aj ) < u
^( ; ak ). In either case this
is a contradiction since
hik = 0 implies that u
^( ; ai ) = u
^( ; ak ).
Case 3. Suppose ij < 0, and jk > 0. This is symmetric to the case 2.
Finally, let ik = 1 and the result follows.
9.5. Proof of Theorem 3.
Proof. Consider the system,
(3)
ij hij
+
jk hjk
=
ik hik
for all i; j; k
Lemma [[3]] guarantees that the equations is well de…ned. The number of equations in the
system (3) is given by K C3 . Notice that hij are parameters of the equation given by the
preferences. We will show that the system in (3) has a non trivial solution for the variables
’s. Furthermore, if the solution is non trivial, all ’s are strictly positive. To show this last
assertion, suppose instead that there exists some ij = 0, then hjk = c:hik , for some constant
c. [[However, Hjk = Hik 6= Hij .]] This is not possible because the …rst equality implies that
there exist some such that u
^( ; ai ) = u
^( ; aj ) = u
^( ; ak ) but the second inequality implies
that u
^( ; ai ) 6= u
^( ; aj ) for all .
The number of variables ’s is K C2 . For K < 6, K C2 > K C3 and therefore the system
has a non trivial solution. However, for K
6, there are more equations than unknowns.
We need to show that there are su¢ ciently many linearly dependent equations so that the
system has a solution.
Consider the subsystem of equations in which we …x an alternative, that without loss of
generality we will call alternative 1, and we combine with all the other possible combinations
of the remaining two alternatives. This is the set of equations containing all equations in which
alternative 1 is present. The number of equations in this subsystem is given by (n 1) C2 < n C2
and contains all ’s, and therefore it has a non trivial solution. It only remains to show that
any equation in the system given by (3) can be generated using this subsystem. For simplicity
of exposition, and without loss of generality, consider an equation with alternatives (2; 3; 4):
(4)
23 h23
+
34 h34
=
24 h24
This equation will not be contained in our subsystem because alternative 1 is not present.
Consider the following three equations from our subsystem:
(5)
12 h12
+
23 h23
=
13 h13
(6)
12 h12
+
24 h24
=
14 h14
(7)
13 h13
+
34 h24
=
14 h14
INFORMATION AGGREGATION IN ELECTIONS
31
We do the following operation: equations (5) minus equation (6) plus equation (7) and we
note that this is equal to equation (4). Since the choice of alternatives is without loss of
generality we convince ourselves that our subsystem generates the full system.
Let = fh1j gj>1 . For all h1j 2 let 1 , and j be such that 1
j = 1j h1j . There are
n variables ’s and n 1 equations so this system has a solution.
Finally, for all i, choose i = ( i + ) , where is a su¢ ciently large constant such that
P
P
0, and = (e i ( i + )) 1 . Then, i 0, and i i = e, and therefore constituting
i+
a well de…ned symmetric mixed strategy pro…le.
To show that this strategy aggregates information we need to show that if alternative ai
is the best, then the strategy selects alternative ai over aj , for any aj 2 A.
Consider the simplest case where alternative 1 is the best alternative. By construction, for
all j 6= 1, u( ; a1 ) > u( ; aj ) ()
h1j > 0 ()
( 1
( 1
j ) > 0 ()
j ) > 0.
Then, alternative 1 obtains more votes than alternative j. The same relationship holds with
weak inequality and equality.
Consider the case where alternative ai 6= 1 is the best alternative. Then, u( ; ai ) >
u( ; aj ) ()
hij > 0 ()
( 1j h1j
( 1
1i h1i ) > 0 ()
j
1 + i) =
( i
( i
j ) ()
j ) > 0.
Therefore, the strategy chooses the right candidate always in the limit and this concludes
the proof.
9.6. Proof of Proposition 2.
Proof. We shall show that, for each n > 0, the set Fn given by
1
Fn = f 2 F ac : m( ( ); X ) = 0 for 2 M( ) ) ( ; ) < g
n
T
is open and dense, and that F = n Fn , establishing that F is a residual set.
T
Starting with the latter, if 2 n Fn , then ( ; ) < n1 for every n whenever m( ( ); X ) =
0, which then means that ( ; ) = 0, or that 2 F ; conversely, if 2 F and m( ( ); X ) =
T
0, then ( ; )) < n1 for every n, so 2 n Fn .
Move now to showing that Fn is open, or equivalently that its complement is closed. Take
a sequence k 2 F ac nFn such that k ! . Hence for each k there is a signed measure k
1
with k ( k ) = X and ( k ; )
be the
n . Passing to a subsequence if necessary, let
weak limit of k . For each E 2 B(X) and k such that d( k ; ) < ",
Z
Z
Z
(Ej ) k (d ) ";
(Ej ) k (d ) + "):
k ( k )(E) =
k (Ej ) k (d ) 2 (
R
(Ej ) k (d ) !
(Ej ) (d ), we have established that k ( k ) ! ( ). Since k ( k ) =
1
)
2 F ac nFn as we wanted to
X for all k, we have ( ) = X as well, and ( ;
n , so
show.
Lastly, let’s establish that Fn is dense in F for each n 2 N. Let us …x 2 F, " > 0,
and n 2 N. We shall construct ^ 2 Fn with dC ( ; ^) < ". As
is compact, the mapping
7! ( j ) is uniformly continuous. So, we can …nd > 0 such that m( ( j ); ( j 0 )) < 2"
As
R
32
PAULO BARELLI, SOURAV BHATTACHARYA AND LUCAS SIGA
whenever and 0 are -close. For each k = 1; 2; ::: and m k, let 0 = 0 < 1 < 1 < 2 <
< Nk 1 < Nk 1 < Nk = 1 where s
s < km , for s = 1; :::; Nk .
s 1 < k and s
Then the family f[ 0 ; 1 ); ( 1 ; 2 );
; ( Nk 1 ; Nk ]g is a …nite open covering of . For each
s = 1; :::; Nk , pick vsk;m 2 (X), absolutely continuous with respect to a Borel measure ,
k
such that the associated densities (zsk;m )N
s=1 are linearly independent and
"
m(vsk;m ; ( j^s )) < , for s = 1; :::; Nk ;
2
Nk
be a continuous
where ^s is the midpoint of the interval ( s ;
), with
0. Let ( k;m
s )
s 1
s=1
0
partition of unity subordinated to the covering and put
X
k;m
k;m
^k;m (Ej ) =
s ( )vs (E); for every Borel E:
s
Observe that ^k;m (Ej ` ) ! ^k;m (Ej ) for each Borel E, whenever ` ! , so ^k;m 2 F ac .
For each 2 , let s0 index the intervals that contain (there are at most two such
intervals). Then m( ( j ); ( j^s0 )) < 2" , so
m(vs0 ; ( j ))
m(vs0 ; ( j^s0 )) + m( ( j^s0 ); ( j )) < ":
Hence
sup j^k;m (Ej ); (Ej )j = sup j
E
E
X
s
or
k;m
(
X
k;m k;m
s (vs (E)
(Ej ))j
s
) sup j(vsk;m (E)
E
(Ej ))j < ";
m(^k;m ( j ); ( j )) < ":
Because this is true for every , we have established that d(^k;m ; ) < ". Observe that this is
true for all k = 1; 2; ::: and m k.
Let k;m 2 M( ) be such that m( k;m (^k;m ); X ) = 0. This means that
Z
X
k;m
k;m
k;m
zsk;m k;m
=
0
where
=
(d ):
s
s
s ( )
s
By linear independence, we must have k;m
= 0 for all s. Also, we have
s
Z
Z
Z
X
X
k;m
k;m
k;m
k;m
k;m
(
)
(d
)
=
0=
(
)
(d
)
=
(d ) =
s
s
s
k;m
([0; 1]):
s
Let 'k;m ( ) = k;m ([0; ]) denote the distribution function associated with k;m , and note
that 'k;m (1) = 0. Let us approximate 'k;m by a step function '
^ k;m de…ned as
'
^ k;m ( ) = 'k;m ( s ) when
2 [ s;
s+1 );
for s = 0; :::; Nk 1.
For …xed k, as we increase m the overlaps of the intervals shrink to a singleton. Let us
keep the points s …xed and have s # s for every s as m ! 1. As such, k;m
converges
s
k;m
to the characteristic function of the limiting interval, ( s 1 ; s ]. Because s = 0, we have
k;m ((
k;m ([0;
s ]) ! 0 for every
s 1 ; s ]) ! 0 for every s = 1; :::; Nk as m ! 1. But then
INFORMATION AGGREGATION IN ELECTIONS
33
s, so 'k;m ( s ) ! 0 as m ! 1, and a fortiori the step function '
^ k;m converges to the zero
function.
Let
be a limit point of the sequence k;m as k ! 1 (recall that m
k, so m necessarily goes to in…nity as well; by abuse of notation, use k; m to index the corresponding
subsequence.) Observe that limk!1 '
^ k;m = limk!1 'k;m , which is the distribution function
associated with , and, by the above argument, it is identically equal to zero. By Proposition
3 in Högnäs (1977),
must be the zero measure
.
For each 2 F, the set f 2 M( ) : ( ) = X g is weak* compact. In fact, for each
R
Borel E, the function 7! (Ej ) is continuous, so the integral
(Ej ) (d ) is continuous
k;m
in , and the set above is weak* closed in M( ). For each
, consider a …nite sub
k;m
1
cover of f 2 M( ) : (^ ) = X g composed of balls of radius less than 2n
. Pick k
(and consequently m) large enough so that all centers of the …nitely many balls that cover
1
f 2 M( ) : (^k;m ) = X g are within 2n
distance of the zero measure
. It then follows
k;m
k;m
that the constructed ^
belongs to Fn , because we have d (
; ) < n1 for any k;m
with m( k;m (^k;m ); X ) = 0, and the proof is complete.
Department of Economics, University of Rochester, Email:[email protected].
Department of Economics, Royal Holloway University of London, Email:[email protected].
Department of Economics, New York University Abu Dhabi, Email: [email protected].