#  Information Aggregation with Delegation of Votes111We are thankful to
Alessandra Casella, Timothy Feddersen, Navin Kartik, Mark Dean, and Carlo
Prato and to seminar participants at Columbia University, TU Berlin and Indian
Statistical Institute, New Delhi, Delhi School of Economics, for their
excellent comments and suggestions. Dhillon and Kotsialou acknowledge funding
from EPSRC grant number EP/P031811/1. This work supersedes Dhillon et al.,
2021a  and Ravindran, (2021) which are subsumed into the current paper. The
authors’ emails are<EMAIL_ADDRESS>(AD<EMAIL_ADDRESS>(GK<EMAIL_ADDRESS>(DR<EMAIL_ADDRESS>(DX).

Amrita Dhillon
King’s College, London
Grammateia Kotsialou
London School of Economics
Dilip Ravindran
Humboldt University of Berlin
Dimitrios Xefteris
University of Cyprus

###### Abstract

Liquid democracy is a system that combines aspects of direct democracy and
representative democracy by allowing voters to either vote directly
themselves, or delegate their votes to others. In this paper we study the
information aggregation properties of liquid democracy in a setting with
heterogeneously informed truth-seeking voters—who want the election outcome to
match an underlying state of the world—and partisan voters. We establish that
liquid democracy admits equilibria which improve welfare and information
aggregation over direct and representative democracy when voters’ preferences
and information precisions are publicly or privately known. Liquid democracy
also admits equilibria which do worse than the other two systems. We discuss
features of efficient and inefficient equilibria and provide conditions under
which voters can more easily coordinate on the efficient equilibria in liquid
democracy than the other two systems.

Keywords: Liquid democracy, delegation, strategic voting, information
aggregation, Condorcet Jury theorem.

JEL Codes: D72

##  1 Introduction

Recent progress on distributed ledger technologies and, in particular, in
blockchain based aggregation processes has opened up the possibilities of new
and improved ways of voting222Brill, (2018) discusses emerging ideas of using
the internet for new interactive collective decision-making processes. (see
e.g. Dhillon et al., 2021b, ). Liquid Democracy, which was first suggested by
Miller, (1969) and discussed by Shubik, (1970),333Introducing the idea of
”Homo Politicus”: voters who are well informed about politics despite not
being directly involved in it. is one such electoral rule that combines
aspects of direct and representative democracy. Under liquid democracy, a
voter chooses whether to vote themselves or to delegate their vote to another
voter (potentially someone more knowledgeable on a particular issue). Voters
can choose to vote or delegate differently for every decision.

Several versions of liquid democracy have been proposed in the literature
(e.g. where only one round of delegation is allowed or where voters who
receive delegated votes may further delegate these votes444Delegated votes can
transitively ‘travel’ until a voter who casts is reached (usually known as a
‘guru’). This ‘travel’ route is decided by the delegation rule applied, which
is a function assigning the votes (of each voter who delegates) into a
delegation ‘path’ ultimately reaching a guru, see e.g. models with preferences
over delegates: Kotsialou and Riley,, 2020; Colley et al.,, 2020, 2021; Brill
et al.,, 2022. ), some of these versions have already been used in settings of
applied interest (e.g. by certain political parties in Europe, Argentina
and—more recently—also by mutual funds in the US).555Google Votes is an
implementation of a liquid democracy platform, and Pirate parties in Europe
have been using this for a while. Also, as mentioned in Kotsialou and Riley,
(2020) and Bloembergen et al., (2019), the Democracy Earth Foundation and the
EU Horizon 2020 project WeGov-Now use similar platforms. Mutual funds in the
US that adopted pass-through voting (e.g. BlackRock and Vanguard), give the
option to their investors to either let the fund’s management vote on their
account, or to vote directly themselves (see, e.g., Malenko and Malenko,,
2023). We emphasize that enhancing shareholders’ voice within organizations by
implementing more efficient and democratic management of their rights, such as
internal liquid democracy systems where the role of ‘gurus’ (i.e. their proxy
advisors) can be accurately and transparently tracked, can be supported by
distributed ledger technologies (see e.g. Kotsialou et al.,, 2018; Riley et
al.,, 2019; Dhillon et al., 2021b, ; Lafarre and Van der Elst,, 2021). Figure
1 illustrates the features of liquid democracy contrasted with direct and
representative democracy.

Representative Democracy |  | Direct Democracy |  | Liquid Democracy
---|---|---|---|---
|  |  |  |

Figure 1: Design differences in three types of democratic models:
representative, direct and liquid democracy. In representative democracy,
citizens are represented by elected officials who then vote on behalf of the
first. In direct democracy, citizens vote directly. In liquid democracy, every
citizen either votes directly or delegates to a representative of her choice.

Blum and Zuber, (2016) present two main normative arguments in favour of
liquid democracy. First, allowing for delegation to more informed people
allows for better decisions when there is an objectively “correct” decision.
Second, delegation allows greater equality in the following sense. Instead of
creating two types of citizens with unequal power to influence policy (the
representatives who vote on policy and ordinary voters who can only vote for
representatives), liquid democracy allows voters to either vote directly on
policy or, delegate their votes to other agents who can vote directly on
policy.

In this paper we focus on the first issue: We study the information
aggregation properties of delegation in elections with two candidates and two
states of nature, where truth-seeking and partisan voters co-exist. Truth-
seeking voters (or independents) have state-dependent preferences over
candidates, while partisan voters have fixed state-independent preferences
over candidates. Truth-seeking voters have some information regarding the
state of the world, but their information precision might be heterogeneous
(i.e. some voters might be better informed than others about the true state of
nature), and partisans are divided into two (potentially unequal) groups,
depending on which candidate they support. We investigate whether liquid
democracy improves upon direct and representative democracy in binary setups
in the sense of welfare (the probability that the ex-post welfare maximizing
alternative is chosen) and in the sense of information aggregation (the
probability that the alternative matching the state of nature is chosen).

A priori, it is not obvious if and when delegation improves outcomes. When the
preferences and information quality of each voter is private information, it
is possible for truth-seeking voters to unwittingly delegate to partisans or
to other truth-seeking voters who are worse informed. Such delegation can hurt
welfare and information aggregation. Even when the preferences and quality of
information of all voters are commonly known, the answer is not obvious. In
such settings, delegation introduces a tradeoff (Bloembergen et al.,, 2019;
Kahng et al.,, 2018; Halpern et al.,, 2021; Armstrong and Larson,, 2021):
while delegation from a poorly informed voter $i$ to a better informed voter
$j$ increases the electoral power of $j$ (which is desirable), it leaves $i$
without a vote to express her own private information and hence leaves
valuable information out of the aggregation process (which is undesirable).
Whether the net effect of delegation is positive or negative, depends on the
exact behavior employed by the voters (Kahng et al.,, 2018; Halpern et al.,,
2021). To our knowledge, the literature has not yet determined the net effects
of this tradeoff; we investigate how strategic voters resolve it in
equilibrium.

Our main contribution is to show that delegation can improve information
aggregation in finite collective decision making of any size, both when the
preferences and the information precisions of voters are known, and when they
are not.

When voter types are common knowledge, we show that liquid democracy admits a
better equilibrium—in terms of information aggregation and welfare—than direct
and representative democracy. In general, the best such equilibrium is hard to
characterize, but we provide some of its properties and characterize it in the
case of an electorate containing sufficiently many uninformed voters. In order
to tractably study how voters can use delegation to improve aggregate
information in other cases, we then focus on a class of committees in which
all voters are truth-seeking and are either nonexperts (who possess
information of homogenous quality) or experts (who are better informed than
nonexperts). Under some conditions, we characterize how voters solve the
tradeoff posed by delegation optimally amongst a natural class of
equilibria.666We look at _neutral_ equilibria in which voters adopt strategies
that are symmetric with respect to the state. This equilibrium selection is
natural in environments which are symmetric with respect to the state.
Efficient information aggregation requires some voters delegate to better
informed voters such that the number of votes held by non-delegators optimally
reflects the precision of their information. Hence, under the efficient
equilibrium, it is worth wasting some information (due to delegation) in order
to form a sub-electorate of decision makers who are ‘optimally weighted’
relative to their information qualities.

While collective decisions can perform strictly better with delegation than
without, as we show, liquid democracy also admits inefficient equilibria.
These inefficient equilibria can do worse than the best equilibria of direct
democracy and representative democracy, meaning the comparison between the
mechanisms is not unambiguous due to the plethora of equilibria. Indeed, it is
well known that multiplicity of (inefficient) equilibria is pervasive in
voting games, where there are many situations in which no voter is pivotal.
There can often be numerous Nash equilibria in which players use, for
instance, weakly dominated strategies. We show that delegation can exacerbate
this problem of multiplicity; for instance, additional inefficient equilibria
in liquid democracy arise when there is ‘too much’ delegation to a small set
of voters. The best equilibria in liquid democracy requires coordination
between voters on who delegates to whom; we show that mis-coordination can
create incentives for voters to delegate much more than what is efficient.
This concern of vote concentration in liquid democracy has been noted by Kahng
et al., (2018), by Campbell et al., (2022) who find subjects inefficiently
overdelegate in an experiment on liquid democracy, and in practice (the German
Pirate Party found that a few ‘celebrities’ were delegated large numbers of
votes when implementing liquid
democracy)777https://www.newscientist.com/article/mg23531424-500-bitcoin-tech-
to-put-political-power-in-the-hands-of-voters/25, but to our knowledge, it has
not been discussed as an equilibrium outcome.

We then identify a condition under which issues of mis-coordination in liquid
democracy may not be a huge concern. Eliminating weakly dominated strategies
is often used to rule out implausible Nash equilibria in voting games. Taking
this further in this context, Iterated Elimination of Weakly Dominated
Strategies (IEWDS) is a powerful tool for predicting outcomes (Dhillon and
Lockwood,, 2004)888The concept of IEWDS has the drawback that the order of
elimination may matter in reaching the dominance solvable outcome. However,
when voters have strict preferences over alternatives, as in our setting, Marx
and Swinkels, (2000) show that the order of elimination does not affect the
outcome. Finally, a common knowledge justification for IEWDS was provided by
Rajan, (1998).. We show that the game with delegation is dominance solvable as
long as at least one truth-seeking voter is sufficiently well informed, while
the game without delegation (i.e. direct democracy) is not. Importantly, the
outcome of IEWDS is the efficient one and requires only two rounds of iterated
elimination, i.e. it is cognitively easy. The IEWDS solution also only relies
on type-symmetric strategies; together this suggests that the presence of a
sufficiently well informed voter may make it easy for voters to coordinate on
the efficient equilibrium in liquid democracy. In contrast, reaching
efficiency under direct democracy may be much harder for voters as the game is
not dominance solvable even in the simplest settings and efficiency may
require voters of the same type to employ asymmetric strategies. In absence of
a coordination mechanism, electoral accidents may be more likely to occur
under direct democracy. Finally, it is worth noting that voter participation
is greater under the IEWDS solution with delegation than under the efficient
equilibrium without delegation. With delegation, all votes are cast either
directly or indirectly (i.e. via delegates). When delegation is not allowed,
substantial abstention takes place in every efficient equilibrium even when
voting is costless (Feddersen and Pesendorfer,, 1996).

What happens when the voters’ types are their private information? In this
case, vote delegation introduces additional trade-offs, leading to a
substantially more involved analysis. On the one hand, liquid democracy can
increase the vote-share of the efficient alternative if less informed voters
transfer their ballots to better informed truth-seeking voters; but it may
also harm the electoral prospects of the efficient alternative if uninformed
voters delegate their votes inadvertently to partisans. Indeed, since the
players types are unobservable, mistakes in vote transfers are unavoidable.
Again, we prove that the best undominated equilibrium in liquid democracy is
always at least as good as the best undominated equilibrium in direct and
representative democracy. This is a very strong result, since it holds for
every possible type of distribution and society size, and establishes that,
despite the additional trade-off, delegation is welfare improving.999In
settings composed only of truth-seeking voters, it is known that communication
before voting can also boost information aggregation (e.g. Gerardi and Yariv,,
2007). However, it is not clear that it would do so when there is also a group
of partisan voters—or, merely, a chance that such voters exist—in the
electorate.

Overall, our main findings combined strengthen the case for liquid democracy.
To our knowledge, this is the first paper to establish that rational truth-
seeking voters can always exploit delegation to improve their welfare, when
partisan voters also exist in the electorate. However, liquid democracy does
not guarantee that better outcomes will prevail, as the worst equilibrium of
liquid democracy is worse than the best of direct or representative democracy.
Despite that, liquid democracy can lead to efficient information aggregation
in some cases without the demanding cognitive requirements (e.g. with respect
to higher order beliefs) necessary for direct democracy to aggregate
information well (see, for instance, Feddersen and Pesendorfer,, 1996). This
indicates that liquid democracy deserves further academic exploration (e.g. by
means of laboratory and field experiments—see, also, Campbell et al.,, 2022).

The rest of the paper is organised as follows: Section 2 discusses the related
literature, Section 3 presents the definitions and notation we use, Section 4
deals with the case when information on voter types is common knowledge,
Section 5 discusses dominance solvability when types are common knowledge,
Section 6 analyses the case when voter types are private information, and we
conclude and discuss potential future avenues of research in Section 7.

##  2 Related Literature

There is a vast literature on information aggregation in two candidate
elections starting with the seminal work on the Condorcet Jury theorem (1785),
which showed that, with two alternatives, two states of the world and common
values, if each individual voted for the correct alternative with
probability—same across subjects—strictly greater than half, then the
probability that a majority would choose the correct alternative $(1)$ is
higher than the probability that any one individual would, and $(2)$ converges
to one as the society grows large. The theorem assumed sincere voting and
follows from the law of large numbers. Austen-Smith and Banks, (1996) showed
that sincere voting was not always rational in such a setting. McLennan,
(1998) (for common value elections) and Feddersen and Pesendorfer, (1997)
(with heterogeneous in information voters and private information on voter
types) show that, for two candidate elections, there is always an equilibrium
that aggregates information efficiently, asymptotically as the size of the
electorate goes to infinity. They do not allow abstention. Feddersen and
Pesendorfer, (1996), allow for heterogeneous (in preference) voters who are
either fully informed or fully uninformed, private information and allow
abstention. In their setting where the size of the population is unknown, yet
large, and truth-seeking and partisan voters co-exist; they show that
elections aggregate information efficiently, i.e. the equilibrium outcome is
asymptotically the same as though information on the state were common
knowledge. In equilibrium, informed truth-seeking voters and partisan voters
vote for the alternative that they support, while uninformed truth-seeking
voters employ non-trivial behavior. More specifically, they are subject to a
“swing voters curse”: a substantial fraction of voters abstain strategically
to allow informed truth-seeking voters to be decisive, even when voting is
costless. For their main result, they do need large elections.101010As shown
by Shotts, (2006) and Meirowitz and Shotts, (2009) moderate voters need not
only abstain to avoid diluting the informativeness of the election, but also
to signal their private preferences to the politicians in settings of repeated
elections. In contrast, we show that delegation can improve on simple majority
voting with abstention in elections of any size, with and without private
information on voter types.

Also close to our paper is work in the weighted majority voting (WMV)
literature (Nitzan and Paroush,, 1982; Shapley and Grofman,, 1984; Ben-Yashar
and Danziger,, 2015). Papers in this literature often study voters making a
binary decision through majority voting and wanting this decision to match
some underlying state of the world. Typically, voters have heterogeneous
qualities of information about this underlying state, and this literature asks
(among other questions) what the optimal weight (or number of votes) a social
planner should _exogenously_ assign to each voter in order to maximize the
probability the election outcome is correct. The key difference between this
literature and liquid democracy, is that in liquid democracy voters obtain
different numbers of votes _endogenously_.111111Bouton et al. (2022) analyze a
model each voter is assigned a number of votes and decides instrumentally how
many of them she wants to use (i.e. she is allowed to partially abstain). As
we show in section 6, allowing agents to delegate voting power has distinct
effects from allowing partial abstention, and can lead in certain
circumstances to higher welfare. The results in certain common-interest
special cases of liquid democracy relate closely to those in the WMV
literature; we discuss the connection with Nitzan and Paroush, (1982), in
particular, in section 4.

Christoff and Grossi, (2017) and Kahng et al., (2018) also study information
aggregation with delegated voting, but do not employ an equilibrium approach.
Christoff and Grossi, (2017) focus on the aggregation of individual choices to
social choice and the unintended effects of delegation on the rationality
postulates satisfied by direct voting. Kahng et al., (2018) study the case
where voters have different levels of information, complete information and a
network setting. They show that when voters delegate only to more informed
voters who are within their local network, then delegation can lead to worse
outcomes than simple majority voting, due to the concentration of power. They
argue that if alternative behavioral assumptions are imposed then delegation
can lead to better outcomes. The key point of the paper is that too much
delegation to the same voters (given the network structure) risks losing out
on valuable information. Halpern et al., (2021), meanwhile, study a setting
similar to that of Kahng et al., (2018) and find sufficient conditions
satisfied by examples of delegation mechanisms under which LD must improve
upon DD in large committees. In both these papers, voters are not strategic
and behave as perscribed by a given mechanism. We complement their work by
showing that when behavior rules are not fixed but rather endogenously
determined by rational voters then delegation can lead to welfare improvements
in equilibrium but need not do so. Bloembergen et al., (2019), study
equilibrium behaviour in a network setting. Voters know which of two outcomes
they would prefer with a probability between 0.5 and 1 (voter accuracy). They
focus on the decision problem of voters, when voters have a choice of direct
voting, which incurs a cost (of getting information), and delegated voting,
when they know the accuracy type of the other voters and the probability that
they have a similar preference, but they do not know the true type. The
authors show existence of Nash equilibria and average accuracy achieved in a
network setting. In contrast, we study simultaneous delegation and costless
voting and we include abstention, complete and incomplete information
settings. While we focus on information aggregation, Green-Armytage, (2015)
also studies a non-strategic model of liquid democracy but focuses on
potential preference aggregation benefits of the system: unlike in many
implementations of representative democracy, voters can delegate to someone
who has close preferences to them, issue by issue. Recently, Armstrong and
Larson, (2021) in a pure common interest and complete information game, with
costly voting in a fully connected network, show theoretically that delegation
always reaches an equilibrium with weakly higher group accuracy at identifying
the "ground truth" (objectively correct) outcome. In their model, there is a
first stage of delegations in which every voter can observe the delegations
that have already taken place, and which ends only when no voter can improve
the probability of selecting the correct alternative by further delegation.
This is what ensures that only those delegations, which improve the chance of
a correct outcome, happen. In contrast, our model has partisan voters, has
costless voting, simultaneous delegation and voting, and we also handle the
incomplete information setting which might be more realistic.121212Notice
that, introducing a cost to voting makes it more attractive to delegate.
However, their experiments with machine learning classifiers show that neither
optimal delegations nor efficiently computable delegation strategies
significantly improve accuracy in small or realistically sized electorates,
respectively. Campbell et al., (2022) study a model of liquid democracy both
theoretically and experimentally. Their theoretical model is nested by ours
(modulo some technical details). Theoretically, they characterize an important
class of liquid democracy equilibria in specific types of common value
electorates, with no abstention; we discuss their theoretical results in
Section 6. Experimentally, they find that despite the theoretical possibility
of liquid democracy to improve outcomes, subjects systematically delegate too
much and do worse with liquid democracy than simple majority voting. While we
also discuss potential issues of overdelegation in liquid democracy (Section
4), in Section 5 we describe a setting where it is cognitively easier for
voters to coordinate when delegation is possible; such settings may improve
the odds for liquid democracy to perform well in experimental testing.

Beyond the specific issue of delegation, our work also relates to studies
which try to assess the information aggregation properties of different
electoral systems, most of which are conducted in an asymptotic setup (i.e.
they are relevant only for large elections, while most of our results pertain
to groups of any size). Bhattacharya, (2013) and Barelli et al., (2017) extend
the Condorcet’s Jury theorem to heterogeneous state-dependent preferences and
to general state and signal spaces respectively, and show that information
aggregation depends on the complexity of the preference and information
structure. Goertz and Maniquet, (2011), Bouton et al., (2016) and Ahn and
Oliveros, (2016) study the properties of approval voting and other scoring
rules, for any electorate size, when a divided majority occurs due to
disagreements among the truth-seeking voters regarding the most likely state
of the world. These papers focus on the case where the majority needs to
coordinate in order to eliminate the possibility of inefficient outcomes, and
show that approval voting performs better than plurality in settings where
truth-seeking voters are identical in terms of information precision. As far
as runoff systems with two (or more) voting rounds are concerned, Piketty,
(2000) observes that due to the existence of multiple voting rounds, the
majority group should have additional opportunities to coordinate and
aggregate their information. Martinelli, (2002) shows that efficient
aggregation of information is feasible in equilibrium under a two-round runoff
rule, in the setting of a divided majority with three alternatives. Tsakas and
Xefteris, (2021) extend this possibility result to more general settings.
Herrera et al., (2019) assess theoretically the information aggregation
properties of more “proportional” systems. That is, when a small change in the
vote-share distribution affects the outcome, even if it does not affect the
winner of the election. They find that in large societies, relatively
uninformed voters abstain, and information is aggregated efficiently. Finally,
McMurray, (2017) and Prato and Wolton, (2017) show that voting might be less
efficient in aggregating information, when the policy alternatives are
proposed by self-interested candidates. Our work contributes to this
literature by showing that, beyond the ballot space, the aggregation method,
and the process through which alternatives emerge, the possibility of
endogenous vote transfers can crucially affect the quality of electoral
outcomes, even in electorates with finite size.

##  3 A model of Liquid Democracy

Consider a set of voters $\mathcal{N}=\\{1,...,N\\}$ who vote to make a single
binary decision between two alternatives $C=\\{A,B\\}$. The alternative
receiving a simple majority of votes cast is the election outcome
$\mathcal{O}\in\\{A,B\\}$ (with ties broken uniformly at random). There is an
unknown state $\omega\in\\{a,b\\}$ and voters have common prior probability
$Pr(\omega|\omega=a)=\pi$, with $\pi\in(0,1)$.

Types of voters. At the start of the game, voter $i$’s type $t_{i}$ is
realized (we consider types being privately or publicly observed). After this,
$i$ receives a private signal $s_{i}\in\\{a,b\\}$ which has a distribution
depending on her type. Voters’ types consist of two components: (1) their
preferences over the election outcome under full information, and (2) the
quality of their private information about $\omega$.

A voter $i$ has preferences $p_{i}\in\\{A,B,I\\}$, where $A$ indicates
preference for alternative $A$, $B$ for alternative $B$, and $I$ for being
‘independent’. If $p_{i}=I$, then voter $i$ prefers that the election outcome
matches the state of the world and gets an ex-post payoff equal to 1. The ex-
post payoff of voter $i$ is denoted by

$u_{p_{i}}(\mathcal{O},\omega),$

where $\mathcal{O}$ is the election outcome, $\omega$ the state of the world,
and $p_{i}$ the type of voter $i$. If $p_{i}=I$, then the independent voter
$i$ prefers that the election outcome matches the state of the world and gets
an ex-post payoff equal to 1, that is $u_{I}(A,a)=1$ and $u_{I}(B,b)=1$.
Voters with $p_{i}\in\\{A,B\\}$ are ‘partisans’ and get an ex-post payoff
equal to 1 when the election outcome matches their preference regardless of
the state, i.e. $u_{A}(A,\omega)=1$ and $u_{B}(B,\omega)=1$.

A voter $i$ has information quality $q_{i}\in[0.5,1]$ drawn from a commonly
known distribution $Q_{i}$ (note that $q_{i}$ is independent of
$p_{i}$).131313It doesn’t matter that $q_{i}$ is independent of $p_{i}$. If
not, all we would care about is the distribution of $q_{i}$ conditional on
$p_{i}=I$. A higher $q_{i}$ means the voter is more likely to receive the
correct signal since $Pr(s_{i}=\omega|\omega)=q_{i}$, where $q_{i}$ is the
precision of voter $i$’s information.141414We assume that signal precision is
the same in both states. If $q_{i}=1$, the voter is perfectly informed and, if
$q_{i}=0.5$, the voter is then perfectly uninformed. Voter $i$’s information
precision $q_{i}$ is irrelevant when it comes to partisan voters who prefer
one alternative regardless of the state.

A voter $i$’s type is given by $t_{i}=(p_{i},q_{i})$, where $(p_{i},q_{i})\in
T=\\{A,B,I\\}\times[0.5,1]$. Then we denote a profile of all voters’ types as
$t=(t_{1},...,t_{N})\in T^{N}$.

Private signals. After types are realized (and observed privately or
publicly), each voter $i$ receives a private signal $s_{i}$ with information
precision $q_{i}$.

Strategies. We call a voter’s choice of what actions to take after her type is
realized an _interim strategy_. This type of strategy is useful when types are
publicly observed. A full contingent plan of what actions a voter takes as a
function of her realized type, which is useful when types are private, is an
_ex-ante_ strategy. We next describe these types of strategies.

Interim Strategies. When delegation is not allowed, a (pure) interim strategy
for a voter $i$ is a choice of action as a function of $i$’s signal
realization chosen after her type $t_{i}$ is realized. When delegation is not
allowed, an interim strategy for $i$ is given by the mapping
$\sigma_{i}:\\{a,b\\}\rightarrow\\{a,b,x\\}$. Function $\sigma_{i}$ maps $i$’s
signal realization $s_{i}$ to a decision on whether to cast her vote either in
favor of alternative $A$, alternative $B$, or to abstain (the choice denoted
by $x$). If $\sigma_{i}(a)=a$ and $\sigma_{i}(b)=b$, we say voter $i$ votes
_sincerely_ , while if $\sigma_{i}(a)=\sigma_{i}(b)=a$, we say $i$ votes
_unresponsively_ for $a$ (voting unresponsively for $b$ is defined similarly).
When delegation is allowed, an interim strategy for $i$ is a mapping
$\sigma_{i}:\\{a,b\\}\rightarrow\\{a,b,x,d_{j\neq i}\\}$, where the choice of
$d_{j\neq i}$ is a voter $j$ that $i$ chooses to delegate to. If $i$ delegates
to $j$ and $j$ does not delegate, then $i$’s vote is cast for whichever
alternative $j$ votes for (or neither if $j$ abstains). If $j$ delegates her
vote to another voter $k$, $i$’s vote is delegated to $k$ along with $j$’s
(that is, we allow for transitive delegation). We call a _delegation cycle_
any sequence of voters $i_{1},...,i_{K}$, with $K\geq 2$, such that each voter
$i_{k}$, with $1\leq k<K$, delegates to voter $i_{k+1}$ and $i_{K}$ delegates
to $i_{1}$. In the case of a delegation cycle, all votes held by the voters in
such a cycle are cast in abstention. 151515None of our results depend on
allowing for transitive delegation or nullifying all votes cast in a
delegation cycle. While changing either assumption may change the set of
equilibria of the game, they will not affect our main results concerning LD
having a better equilibrium than DD or RD, nor will they affect our results
characterizing best equilibria and best neutral equilibria in Section 4. We
assume that if a voter votes for an alternative (or abstains), all votes she
holds are cast for this alternative (or in abstention).161616Similar to the
previous footnote, none of our results would change if voters were allowed to
split the votes they hold amongst multiple alternatives. However, allowing for
this would require strategies to allow for complicated conditional behavior as
delegation and voting happen simultaneously.

Ex-ante Strategies. To describe a voter’s strategy before their type is
realized, we need them to choose a contingent plan for every possible type
realization. An ex-ante strategy for voter $i$ is a mapping
$\Sigma_{i}:T\times\\{a,b\\}\rightarrow\\{a,b,x\\}$ when delegation is not
allowed and is a mapping
$\Sigma_{i}:T\times\\{a,b\\}\rightarrow\\{a,b,x,d_{j\neq i}\\}$ when
delegation is allowed. Voter $i$ chooses an interim strategy for every
possible type they could be. For any ex-ante strategy of voter $i$,
$\Sigma_{i},$ and any possible realized type of $i$, $t_{i}$, let
$\Sigma_{i}(t_{i})$ be the interim strategy of voter $i$ with type $t_{i}$
under strategy $\Sigma_{i}$.

Timing. The timing of the game is as follows:

  1. 1.

Nature draws state $\omega$.

  2. 2.

Types: if there is complete information on types, the vector of types
$(t_{1},...,t_{N})$ is commonly observed. If there is incomplete information
on types, voter $i$’s type $t_{i}=(p_{i},q_{i})$ is drawn from $P_{i}\times
Q_{i}$ and is privately observed.

  3. 3.

Voters’ private signals are realized.

  4. 4.

Voters choose to vote/abstain/delegate. If there is complete information on
types, they do so following interim strategies; with incomplete information on
types they follow ex-ante strategies.

  5. 5.

The election outcome is determined.

  6. 6.

Payoffs are realized.

Strategy Profiles. An interim strategy profile
$\sigma=(\sigma_{1},...,\sigma_{N})$ is a vector of interim strategy profiles
for each voter. Similarly $\Sigma=(\Sigma_{1},...,\Sigma_{N})$ is an ex-ante
strategy profile. For any ex-ante strategy profile $\Sigma$ and profile of
realized types $t$, let
$\Sigma(t)=\left(\Sigma_{1}(t_{1}),...,\Sigma_{N}(t_{N})\right)$ be the
profile of interim strategies implied by $\Sigma$ at $t$.

Expected utilities and equilibrium. Given a publicly observed type profile
$t\in T^{N}$ and an interim strategy profile $\sigma$, let $U_{i}(\sigma,t)$
be voter $i$’s expected utility from voters following interim strategy profile
$\sigma$ and $U_{i}(\sigma_{i}^{\prime},\sigma_{-i},t)$ be the same expected
utility when voter $i$ deviates to interim strategy $\sigma_{i}^{\prime}$.

When types are privately observed, we are interested in ex-ante expected
payoffs. Fixing ex-ante strategy profile $\Sigma$, let

$\displaystyle EU_{i}(\Sigma)=\sum_{t\in
T^{N}}\left(U_{i}\left(\Sigma(t),t\right)\cdot Pr(t)\right)$ (1)

be $i$’s expected utility and $EU_{i}(\Sigma_{i}^{\prime},\Sigma_{-i})$ be the
same utility when $i$ deviates to $\Sigma_{i}^{\prime}$.

We look for Bayes Nash equilibria (BNE) of the game in interim strategies when
types are publicly observed and in ex-ante strategies when types are private
information. In both the publicly and privately observable types’ cases, we
will select for BNE in which weakly dominated strategies are not employed;
when we say ‘equilibrium’, henceforth this means BNE in undominated
strategies. As it is typical in voting games, there typically are many
equilibria; ruling out equilibria in dominated strategies alleviates this
issue.

Direct and Representative Democracy. Our main aim is to compare equilibrium
outcomes of liquid democracy, with equilibrium outcomes of direct (DD) and
representative democracy (RD). To this end, we need to provide a definition of
these two alternative systems, which formally differ from liquid democracy
(LD) only in terms of the involved interim strategies. In DD, an interim
strategy for voter $i$ is given by mapping
$\sigma_{i}:\\{a,b\\}\rightarrow\\{a,b,x\\}$.

RD differs from DD and LD in that there is a fixed set of representatives
$\mathcal{J}$. Set $\mathcal{J}$ can contain any subset of voters in
$\mathcal{N}$ and additionally contains two representatives $a^{*}$ and
$b^{*}$, where $a^{*}$ is a partisan representative, who is an advocate for
alternative $A$ and we assume always (mechanically) votes for $a$, and $b^{*}$
is an advocate for $B$ who always votes for alternative $B$. Representatives
$a^{*}$ and $b^{*}$ are hence not strategic and are not included in
$\mathcal{N}$. In RD, an interim strategy for $i\notin J$ is a mapping
$\sigma_{i}:\\{a,b\\}\rightarrow\\{x,d_{j\in J}\\}$ and for $i\in J$ is a
mapping $\sigma_{i}:\\{a,b\\}\rightarrow\\{a,b,x\\}$.171717Other than the
existence of $a^{*}$ and $b^{*}$, Representative Democracy (RD) differs from
Liquid Democracy (LD) in two ways: first the potential delegates are fixed
exogenously and second, if a voter delegates to a representative this
delegation applies for all issues/decisions faced by the committee. Liquid
Democracy meanwhile allows delegates to be different for different issues.
However in this paper we only focus on the first difference, as our model
concerns a single issue. The assumption that representatives $a^{*}$ and
$b^{*}$ always exist, vote for their preferred alternative, and are not
included in $\mathcal{N}$ is for technical convenience in comparing RD with DD
and LD in common interest committees.

In summary, under direct democracy DD, agents can assign votes either to $a$
or to $b$ (or abstain), while under representative democracy they can assign
votes to a fixed set of representatives.

Comparing mechanisms. There are multiple measures of interest in comparing
liquid democracy with representative and direct democracy. One natural measure
of the welfare attained under an equilibrium of a mechanism is the probability
it implements the ‘majoritarian outcome’ (i.e. the alternative that is
preferred ex post by most voters). Another important measure is the
probability the mechanism implements the ‘correct’ outcome: i.e. the outcome
that matches the state of the world. Implementing the correct outcome is not
only important to independent voters, but it also gives a measure of how well
the committee aggregates voters’ private information.

Note that if an equilibrium of a mechanism implements the correct alternative
with higher probability than another equilibrium of the same or of a different
mechanism, then it also implements the majoritarian outcome with higher
probability. This is so because for all type draws such that the truth-seeking
voters are not pivotal as a group (i.e. their choice cannot affect the outcome
when players use undominated strategies), then both equilibria lead to the
majoritarian outcome with certainty and to the correct outcome with the same
probability. Observe that in the remaining type draws the correct alternative
coincides with the majoritarian outcome. Therefore, since one equilibrium
implements the correct alternative with higher probability than the other
unconditionally, it must be the case that it implements the correct
alternative—and hence also the majoritarian outcome—with higher probability
conditional on the truth-seeking types being decisive. Since the majoritarian
outcome is implemented with higher probability under the former equilibrium
than the latter conditional on the group of truth-seeking voters being
decisive, and with equal probability conditional on truth-seeking voters not
being decisive, it leads to the implementation of the majoritarian alternative
with higher probability unconditionally. We call the equilibria that maximize
the probability of the majoritarian outcome as best equilibria.

In the next sections, we compare LD with DD and RD in terms of the best and
worst equilibria under the three voting sytems. We first analyse the complete
information setting and then the incomplete information setting.

##  4 Complete information about types

In this section, we assume each agent’s type is common-knowledge. We are
interested in the structure and information aggregation properties of
equilibria; of particular interest are equilibria which best aggregate
information (and hence implement the majoritarian outcome with the highest
probability). We compare the best equilibria in undominated strategies in LD
with both DD and RD. While we cannot characterize the set of equilibria, we
show the best equilibrium of LD does better than that of DD and RD, and
provide some examples of committees in which LD does strictly better. Below we
provide the intuition for our first result (Proposition 1).

Let $n_{I}$ be the number of independent voters and $n_{A}$ and $n_{B}$ be the
number of partisans for $A$ and $B$ respectively. Assume $n_{I}\geq 1$ or the
problem is uninteresting. The existence of an equilibrium in undominated
strategies is straightforward in LD, DD and RD when types are commonly known.
First, in LD and DD it is a weakly dominant strategy for every partisan $i$
with $p_{i}=A$ ($p_{i}=B$) to vote for $A$ ($B$) regardless of her signal.
This is simply because the partisan strictly prefers this strategy when
pivotal and is indifferent across all strategies otherwise; moreover, every
voter is pivotal with probability $>0$, given some strategy profile of her
opponents.181818 E.g. w.l.o.g consider an A partisan. If $N$ is even, let
$\frac{N-2}{2}$ voters vote for $A$ and $\frac{N}{2}$ vote for $B$. If N is
odd, then let $\frac{N}{2}$ vote for $A$ and $\frac{N}{2}$ vote for $B$. Given
this, in every equilibrium of LD and DD, each partisan votes unresponsively in
favor of her preferred alternative. Meanwhile in RD,
$a^{*},b^{*}\in\mathcal{J}$ always vote (mechanically) for their preferred
alternatives. Other partisans in $\mathcal{J}$ have weakly dominant strategies
of voting for their preferred alternative, and partisans not in $\mathcal{J}$
have a weakly dominant strategy of delegating to either $a^{*}$ or $b^{*}$.
Partisans will adopt these strategies in any equilibrium in undominated
strategies. Having pinned down the behavior of partisans in all three
mechanisms, in the case that $|n_{A}-n_{B}|>n_{I}$, in any equilibrium
independent voters are (collectively) never pivotal and hence any combination
of non-weakly dominated strategies they use constitutes an equilibrium. The
election outcome in equilibrium in this case is deterministic. The interesting
case is hence when $n_{I}\geq|n_{A}-n_{B}|$. Having fixed the equilibrium
behavior of partisans, from the perspective of independents, the game is
common-interest. As types are commonly known and strategy spaces are finite,
there is a profile of non-weakly dominated strategies that maximizes payoffs
for independents and, by McLennan (1996), this profile constitutes an
equilibrium for each of the LD, RD and DD systems. Note that every strategy
that is feasible in DD/RD is also feasible in LD and leads to weakly lower
welfare compared to the strategy profile that maximizes the probability of
implementing the correct alternative in LD—which constitutes an equilibrium.

###  4.1 Liquid Democracy

The best equilibrium for independents in LD is of interest beyond just proving
existence; it is also the equilibrium that best aggregates the information of
the committee. In general, we cannot pin down this strategy profile; the
following Proposition tells us that such an equilibrium in LD does better than
the corresponding equilibrium in RD/DD.

######  Proposition 1.

In liquid democracy with commonly known types, there is a best (in terms of
information aggregation) equilibrium $\sigma$ which does at least as well as
the best equilibrium in direct and representative democracy.

The proof follows from the logic at the end of the previous subsection. We
note that whenever the best equilibrium of liquid democracy requires use of
delegation, this best equilibrium will do strictly better than that of direct
democracy. The comparison between liquid and representative democracy is a bit
more nuanced. It can be possible to implement the best equilibrium of liquid
democracy under representative democracy if the set of representatives
$\mathcal{J}$ contains exactly the set of voters who are delegated to under
the best equilibrium of liquid democracy. If some voters who are delegated
votes under liquid democracy’s best equilibrium are not contained in
$\mathcal{J}$, then representative democracy will do strictly worse than
liquid democracy. Meanwhile, if the set of representatives $\mathcal{J}$ is
too large then representative democracy can also fail to do as well as liquid
democracy. For instance, if $\mathcal{J}$ contains all voters in the
committee, then representative democracy is equivalent to direct democracy.
Hence the welfare comparisons of liquid and representative democracy will
depend on the specific structure of the committee.

For some simple types of committees, we can pin down the best equilibrium in
LD. This will give some insight into the role delegation plays in information
aggregation. First, we introduce a result from the weighted majority voting
(WMV) literature, which helps us to characterize equilibria for special cases.

Connection to Weighted Majority Voting (WMV): The WMV literature assumes all
voters are independents and considers binary state models in which voters make
a binary decision seeking to match the state. Instead of allowing for
delegation before voting, voters are assigned weights (a number of votes which
can be any weakly positive real number) exogenously. Voters receive private
signals with varying precisions and cast all of their weight (or votes)
sincerely. The alternative $A$ or $B$ receiving the majority of votes wins and
voters prefer this outcome to match the state of the world.

An important result (found in multiple papers starting with Nitzan and
Paroush, (1982)) is that there exist ‘optimal’ weights for each voter
$(w^{*}(1),...,w^{*}(N))\in\mathbb{R}_{++}^{N}$ such that the following holds:
Under these optimal weights and under sincere voting, the outcome of the
election is efficient – or first-best – in the sense that it is always the
same outcome as if a single decision maker (with the same preferences as the
voters) looked at all private signals and made the decision unilaterally. When
voters are assigned these optimal weights, the committee _efficiently
aggregates_ the private information of all committee members in the sense that
the election outcome is the first-best (absent asymmetric information) outcome
with probability equal to $1$. Nitzan and Paroush, (1982) show that for
$i\in\mathcal{N}$, the optimal weight is
$w^{*}(i)=\log(\frac{q_{i}}{1-q_{i}})$, where $q_{i}$ is the precision of
voter $i$.191919Note that if $q_{i}=1$ then the optimal weight is infinite.
This is because voter $i$ always knows the state and hence should optimally
dictate the election outcome. Under these weights, each voter’s power
optimally matches the quality of their information. If, for instance,
$w^{*}(i)=2w^{*}(j)$ for some $i,j$, this means that in terms of information
content, voter $i$’s private signal is worth two of voter $j$’s private
signal. If $s_{i}=a$ (voter $i$ receives information in favor of state $A$),
then it would take more than two independent signal realizations of precision
$q_{j}$ in favor of state $B$ to outweigh the information provided by $s_{i}$.
Importantly for some of our results, these weights are not unique; scaling the
weights of all voters by the same constant does not change voting outcomes.

In WMV, voters’ weights are exogenously assigned; this means some voters are
ex-ante more powerful than others, which may be arguably socially underisable
or hard to implement. In liquid democracy all voters are ex-ante equally
powerful (each is endowed with a single vote) but can use delegation to
endogenously determine weights. However, under liquid democracy the optimal
weights of Nitzan and Paroush, (1982) may not be attainable, as delegation
results in the delegator being left without a vote. Despite this, some of our
main results will show that it can be efficient for some voters to delegate to
enable other voters to endogenously attain these optimal weights.

While, the best equilibrium under LD is typically hard to characterize, we can
do so for an interesting class of committees.

Committees with uninformed voters and few experts. Consider a committee with
prior $\pi=\frac{1}{2}$ and $n_{A}$ $A$ partisans and $n_{B}$ $B$ partisans
(wlog assume $n_{A}\geq n_{B}$). Amongst independent voters, there are $n_{U}$
voters who are perfectly uninformed, that is $q_{i}=0.5$. There are $E$
independents who are ‘experts’; each such voter $i$ has precision $q_{i}>0.5$.
The ‘experts’ may be heterogeneous in their precisions.

Suppose $n_{U}$ is large: there is a large number of uninformed independents.
While these voters hold no information, they can aid other independents in
aggregating information. First, as $n_{A}-n_{B}$ extra votes are cast by
partisans for $A$ over $B$, if $n_{A}-n_{B}$ uninformed voters vote
unresponsively for $B$, they will entirely negate the partisans. There is no
information loss in doing this.

Given this behavior, weighted majority voting with the remaining uninformed
voters, each having a weight of $0$ and each expert $i$ having a weight of
$w^{*}(i)$, would deliver first-best outcomes. Attaining these outcomes only
depends on the ratio of votes held by any two experts reflecting their
relative information qualities: for any experts $i$ and $j$, the ratio of
votes held should be $\frac{w^{*}(i)}{w^{*}(j)}$. One can see that delegation
from uninformed voters (along with abstention from remaining uninformed
voters) can arbitrarily approximate optimal weights as $n_{U}$ gets large. As
the Nitzan and Paroush, (1982) weights are robust to certain small
perturbations, we can then attain first-best information aggregation with LD
for large enough (finite) $n_{U}$. Proposition 2 captures this.

######  Proposition 2.

Assume $\pi=\frac{1}{2},$ $n_{A}\geq n_{B}$, $n_{U}$ uninformed ($q_{i}=0.5$)
independent voters and $E$ independent experts ($q_{i}>0.5$). There exists
$n^{*}$ such that when $n_{U}>n^{*}$, under the best LD equilibrium $\sigma$:

  1. 1.

$n_{A}-n_{B}$ uninformed independents vote unresponsively for $B$.

  2. 2.

Each remaining uninformed independents either: (1) abstains at both signal
realizations, or (2) delegates to the same expert at both signal realizations.

  3. 3.

The election outcome under LD is the first-best outcome w.p. $1$.

  4. 4.

Let $v_{i}$ and $v_{j}$ be the number of votes held by experts $i$ and $j$,
respectively. Then, as $n_{U}\rightarrow\infty$, the ratio
$\frac{v_{i}}{v_{j}}$ converges to the ratio $\frac{w^{*}(i)}{w^{*}(j)}$
(experts approach optimal relative weights).

Importantly, this result does not qualitatively depend on the worst informed
voters being perfectly uninformed. In particular, whenever the result holds
and the committee attains first-best outcomes in equilibrium, there exists an
$\epsilon>0$ such that if all $n_{U}$ uninformed independents instead had
precisions in the range $[0.5,0.5+\epsilon]$, then the same equilibrium would
still deliver first-best outcomes. This is an implication of Nitzan and
Paroush, (1982)’s Corollary 1, which tells us that first-best solutions will
anyway ignore the information of poorly informed voters who do not hold enough
information to affect the best choice ex-interim.

While the logic of the Proposition is that with large enough number of
dispensable votes we can approximate Nitzan and Paroush, (1982) weights, the
result is not an asymptotic one, as we can attain first-best outcomes for
finite $n_{U}$. The following example demonstrates that in fact attaining
first-best outcomes do not require $n_{U}$ to be very large.

_Example._ There are four experts with
$q_{1}=0.8,q_{2}=0.7,q_{3}=0.65,q_{4}=0.6$. The log likelihood ratios are
respectively $1.3862,0.8473,0.618,0.4055.$ Rounding off to 2 decimals we have
$1.39,0.85,0.62,0.4.$ Dividing through by 0.4 we have the normalised weights
3.475, 2.125, 1.55 and 1, or 3.5, 2.1, 1.5 and 1. Therefore 1 and $i$ should
prevail over $j$ and $k,$ for any $i,j,k$ distinct from each other and from 1.
However 2,3,4 should prevail over 1. This remains true even with rounded down
weights. So we need $n_{U}\geq 3$ to achieve this result. With DD we cannot
achieve FB-at best 4 and 5 can abstain.

Proposition 2 highlights two important roles uninformed independent voters can
play in information aggregation: delegating to informed voters to enable them
to achieve optimal weights, and voting unresponsively to counter partisans in
the committee. As uninformed voters do not possess any private information,
these two types of strategies are costless, i.e. they waste no information.
This allows committees with enough uninformed independent voters to attain
first-best outcomes. In the next section, we study committees in which all
voters are informed (i.e. receive signals with $q_{i}>0.5$), which introduces
a tradeoff in delegation: a voter $i$ delegating to a better informed voter
can enable the better informed voter to have a weight closer to the optimal
weight, but this comes at the cost of $i$’s signal no longer being aggregated
into the election outcome. In order to elucidate how this tradeoff affects
information aggregation, we focus on pure common-interest committees.

###  4.2 Common-interest committees and neutral equilibria

In this section we consider environments which are symmetric with respect to
the state: there are no partisans ($n_{A}=n_{B}=0$), and the prior is
$\pi=1/2$.202020All results also apply if $\pi$ is close to $1/2$ or if the
number of $A$ and $B$ partisans is equal, or if there are sufficiently many
uninformed voters who can neutralise the partisans. In this symmetric
environment with no partisans, the best equilibrium is the one that best
aggregates voters’ information. Consider the best equilibrium of LD in the
setting of Proposition 2; according to Proposition 2, this best equilibrium is
neutral with respect to state in the sense that every voter’s strategy is
symmetric with respect to their signal realization. Formally, we say a
strategy profile $\sigma$ under LD is _neutral_ if under $\sigma$, every voter
$i$ employs a neutral strategy by either: voting sincerely, abstaining at both
signal realizations, or delegating to the same voter at both signal
realizations (if $\sigma_{i}(a)=d_{j}$ then $\sigma_{i}(b)=d_{j}$). For DD,
neutrality is defined similarly without allowing for delegation. In RD, a
neutral strategy for a representative $i\in\mathcal{J}$ is either voting
sincerely or abstaining at both signal realizations; a neutral strategy for a
nonrepresentative $i\not\in\mathcal{J}$ is to either abstain at both signal
realizations, delegate to a given independent partisan at both signal
realizations, or to delegate to $A$ partisan $a^{*}$ at signal realization
$s_{i}=a$ and to $B$ partisan $b^{*}$ at signal realization $s_{i}=b$ (this
last type of strategy is analogous to sincere voting under DD and LD).

Surprisingly, even in symmetric environments, it need not be the case that the
best equilibria of LD are neutral. We present an example in the Appendix in
which all best equilibria involve voters employing asymmetric strategies and
electing the correct alternative with higher probability in one state than the
other. These equilibria can rely on voters conditioning delegation on their
signal realization, which allows for correlating voting between multiple
voters.212121If voter $i$ delegates to $j$ only when her signal is $a$, then
the posterior probability of state A is higher conditional on $j$ casting an
additional vote. While delegation results in the loss of some private
information of the delegator, conditional delegation can be used to mitigate
this. However, these strategies require considerable coordination between
voters and characterizing the best equilibrium is difficult. Instead, we focus
on the best neutral equilibrium, which is a natural benchmark in symmetric
settings. We show that even when restricting to this natural class of
equilibria – hence shutting down conditional delegation – voters can still use
delegation in LD to do strictly better than under DD and RD.

Lemma 3 in the Appendix shows that, in committees with no partisans and a flat
prior, the best neutral strategy profile under all three mechanisms is an
equilibrium. This is because when the environment is symmetric, a voter always
has a neutral best response when all others use neutral strategies.222222In
fact, any neutral strategy profile at which there are no profitable deviations
using only neutral strategies will be an equilibrium. As strategy spaces are
finite, every symmetric game hence has a neutral equilibrium in LD, RD, and
DD. In the remainder of the section we analyze some natural classes of
committees. We demonstrate that the best neutral equilibrium of LD can do
strictly better than that of DD and RD and also discuss ‘bad’ neutral
equilibria under LD.

The single expert committee. Here we consider the simplest setting in which
liquid democracy may be useful in improving information aggregation. The
committee is composed of $N-1$ ‘nonexperts’ $1,...,N-1$ and a single ‘expert’
$e$. Assume $N$ is odd.232323Results are the same for $N$ even, but with $N$
odd ties are avoided which simplifies exposition. All committee members
$i\in\mathcal{N}$ are independents: $p_{i}=I$. The nonexperts $i=1,...,N-1$
receive private signals with precision $q_{i}=q\in(0.5,1)$ and the expert
receives information with superior precision $q_{e}=r\in(q,1)$. Additionally,
suppose that $\pi=\frac{1}{2}$ so the information setting is entirely
symmetric.

This is a common-interest setting where all voters are purely interested in
information aggregation. Intuitively, when the expert is sufficiently well
informed relative to the nonexperts ($r$ is high relative to $q$), there is an
incentive for the nonexperts to delegate to the expert. Delegation however
comes at a cost: if a voter $i$ delegates, she cannot express her own private
information in voting. We are interested in how voters solve this tradeoff in
delegation under the best neutral equilibrium and other neutral equilibria.

Let $w^{*}(e)=\frac{\log(\frac{r}{1-r})}{\log(\frac{q}{1-q})}$; $w^{*}(e)$ can
be thought of as the number of non-expert signals that collectively have
informational content equivalent to the expert’s private signal. By Nitzan and
Paroush, (1982), if the expert had $w^{*}(e)$ votes and the nonexperts had $1$
vote each, all voters voting sincerely would efficiently aggregate the
information of the committee. Proposition 3 characterizes the best neutral
equilibrium of the single-expert committee under LD as a function of
$w^{*}(e)$.

######  Proposition 3.

There is a best neutral equilibrium $\sigma$ of the single expert committee
characterized as follows:

  1. 1.

If $w^{*}(e)<2$: all voters vote sincerely under $\sigma$.

  2. 2.

If $w^{*}(e)\geq 2$: Exactly $\min\\{\lfloor
w^{*}(e)\rfloor-1,\frac{N-1}{2}\\}$ nonexperts delegate to the expert (i.e.
$\sigma_{i}(a)=\sigma_{i}(b)=d_{e}$). The remaining voters vote sincerely.

These strategies are independent of the number of nonexperts in the committee.
Whenever $w^{*}(e)\in(2,\frac{N+1}{2}]$, the best neutral equilibrium of LD
does strictly better than that of DD.

The proof is in the Appendix but we explain the intuition. Note that if
$w^{*}(e)<2$, then the best neutral equilibrium in liquid democracy, as per
the Proposition, requires no delegation and hence is replicable under direct
democracy. Meanwhile if $w^{*}(e)\geq\frac{N+1}{2}$, expert receives
$\frac{N-1}{2}$ delegated votes; as she votes sincerely, she then dictates the
election outcome. This outcome is also replicable under direct democracy by
all nonexperts abstaining. Hence, liquid democracy strictly improves on direct
democracy when the expert sufficiently well informed relative to nonexperts,
but not too well informed. A comparison between liquid and representantive
democracies will depend on the set of representatives $\mathcal{J}$, as
discussed at the beginning of Section 4. For instance, the best neutral
equilibrium of liquid democracy will do strictly better than that of
representative democracy whenever $w^{*}(e)>2$ and $e\not\in\mathcal{J}$ (as
the expert’s superior information cannot then be given additional weight via
delegation), or whenever $w^{*}(e)\in(2,\frac{N+1}{2}]$ and
$\mathcal{J}=\mathcal{N}$ (in which case representative and direct democracy
are equivalent).

We now give intuition for the structure of the best neutral equilibrium of LD.
First note that if all nonexperts had a single vote and the expert held
$\lfloor w^{*}(e)\rfloor$ votes, all voters voting sincerely would produce
equivalent election outcomes to the case where the expert had exactly
$w^{*}(e)$ votes.242424This is simply because with an odd number of voters and
no abstentions under the strategy profile, all decisions are made by a margin
of at least $1$ vote; hence rounding the expert’s weight will not swing the
election outcome. In Case 1, when $w^{*}(e)<2$, $\lfloor w^{*}(e)\rfloor=1$
and so all voters voting sincerely will attain the first-best outcome and
hence must be the best neutral equilibrium. In this case, the best neutral
equilibria of LD and DD perform identically.

Case 2 is the more interesting case: The prescribed strategy profile allocates
$\min\\{\lfloor w^{*}\rfloor,\frac{N-1}{2}+1\\}$ votes to the expert. When
$w^{*}(e)\geq\frac{N-1}{2}+1$, the expert’s signal contains more information
than any $\frac{N-1}{2}+1$ other voters and this implies it is optimal for the
expert to dictate the election outcome. More interestingly, when $2\leq
w^{*}(e)<\frac{N-1}{2}+1$, then delegation to the expert such that she attains
a weight of $\lfloor w^{*}\rfloor$ is optimal. As all nondelegating nonexperts
are voting sincerely, this means the committee efficiently aggregates the
information of the expert and all $(N-1)-(\lfloor w^{*}\rfloor-1)$ nonexperts
holding votes. However, this comes at the cost of wasting the private
information of $(\lfloor w^{*}\rfloor-1)$ delegators, meaning all information
of the committee is not being efficiently aggregated. The implication is that
when the expert is sufficiently, but not too well, informed it is worth
wasting some voters’ information in order to efficiently aggregate the
information of the remaining voters. The best neutral equilibrium forms a
‘subcommittee’ of decision makers who endogenously attain optimal weights.

To understand why it is optimal to delegate exactly $\lfloor w^{*}\rfloor-1$
votes to the expert when $2\leq w^{*}(e)<\frac{N-1}{2}+1$, suppose the expert
is being delegated $k$ votes and consider the incentive of another nonexpert
$i$ to delegate to her. For simplicity, assume that all voters who are not
delegating vote sincerely (we show in the Appendix that this is the behavior
at the best neutral equilibrium). Voter $i$’s delegation decision will only
affect payoffs when: (1) $i$’s vote is pivotal, and (2) $s_{i}\neq s_{e}$ (if
$i$ and $e$ receive the same signal realization, as nondelegators are casting
their votes sincerely, delegation will not change how this vote is cast). When
both these conditions hold, it must be that $s_{i}$’s realization agrees with
that of $\frac{N-1}{2}$ other nonexperts; meanwhile $s_{e}\neq s_{i}$ and
$s_{e}$ agrees with $\frac{N-1}{2}-(k+1)$ nonexperts. Hence, in the event that
$i$ delegating to $e$ is pivotal, $i$’s private information agrees with that
of $k+1$ _excess_ nonexperts while disagreeing with the expert’s. Voter $i$
should delegate to the expert if and only if the expert is more likely to be
correct. This is true when the informational content of $s_{e}$ is worth that
of at least $k+2$ nonexpert signals; i.e. if and only if $k+2\leq w^{*}(e)\iff
k+1\leq\lfloor w^{*}\rfloor-1$.

There are a few important implications of this result. First, note that the
number of votes the expert should be delegated in the best neutral equilibrium
does not depend on the number of nonexperts in the committee. This is because
the expert’s optimal weight does not depend on this size of the committee, but
only on the quality of her information relative to that of a nonexpert.
Second, whenever $2\leq w^{*}(e)<\frac{N-1}{2}+1$, the best equilibrium
involves nonexperts employing asymmetric strategies: either some nonexperts
delegate while others vote sincerely. This means that there will be multiple
best equilibria as nonexperts are interchangeable. Coordinating on equilibria
in which voters of the same type use different strategies may be hard; we
discuss this in more detail in Section 5. The following example demonstrates
that while the best neutral equilibrium given in Proposition 2 highlights that
LD can do strictly better than DD, miscoordination in delegation can lead to
worse neutral equilibrium outcomes as well.

######  Example 1.

Consider a single expert committee with nonexperts $i=1,...,8$ (so $N=9$)
having precisions $q_{i}=0.6$ and the expert having precision $r=0.7$. There
is a (neutral) equilibrium in which:

  * •

Nonexperts $i=1,2$ delegate to the expert
($\sigma_{i}(a)=\sigma_{i}(b)=d_{e}$) at both signal realizations.

  * •

Nonexperts $i=3,5,7$ delegate to nonexpert $i+1$:
$\sigma_{i}(a)=\sigma_{i}(b)=d_{i+1}$ at both signal realizations.

  * •

Nonexperts $i=4,6,8$ vote sincerely.

When $q=0.6$ and $r=0.7$, $w^{*}(e)\in(2,3)$ and $\lfloor
w^{*}(e)\rfloor=\lfloor\frac{\log(0.7/0.3)}{\log(0.6/0.4)}\rfloor=2$. This
means the expert should optimally vote with twice the weight of nonexperts.
The best neutral equilibrium of Ls has one nonexpert delegating to the expert
and the rest voting sincerely. In the equilibrium above, there is
‘overdelegation’ to the expert, who ends up with $3$ votes. This
overdelegation would make the expert suboptimally too powerful relative to
nonexperts unless nonexperts delegate to one another in order to achieve the
correct balance of power amongst those casting votes. Notice that the strategy
profile given above leaves 3 nonexperts ($4,6,8$) each with $2$ votes. The
alternative the expert votes for will win the election unless all $3$
nonexperts receive information disagreeing with her; as $w^{*}(e)\in(2,3)$,
this means that this equilibrium first-best aggregates the information of the
expert and three nonexperts holding votes (the correct decision is always made
when only taking into account these four voters’ information). However,
obviously this equilibrium is inefficient as nonexperts signals are wasted due
to overdelegation.

The single expert committee demonstrates the role delegation can play in
allowing voters to optimally balance power relative to how informed voters
are. However even in such simple settings and selecting only for neutral
equilibria, mis-coordination in delegation can create incentives to waste
further information through delegation. These incentives to agglomerate votes
can result in inefficiency in equilibrium. Inefficiencies due to wasted
delegation stemming from overdelegation to a small number of experts have been
noted in the Computer Science literature on LD (Kahng et al., (2018),Halpern
et al., (2021)), when testing liquid democracy experimentally (Campbell et
al., (2022)), and in practice (see footnote 7) but to our knowledge have not
been shown to be possible in equilibrium when voters are strategic.

Committees with nonexperts and few experts. We now show that the intuition
from the best neutral LD equilibrium of the single expert committee extends to
another natural class of committees. Consider a committee in which all members
are independents and the prior is $\pi=\frac{1}{2}$. Suppose there are $n$
(even) voters who each have precision $q_{i}=q$; call these voters the
‘nonexperts’. The remaining $E=N-n$ (odd) voters are ‘experts’; each such
voter $i$ has precision $q_{i}>q$ and
$w^{*}(i)=\frac{\log(\frac{q_{i}}{1-q_{i}})}{\log(\frac{q}{1-q})}$.252525Assuming
$n$ even and $E$ odd is for convenience. This committee is an adaptation of
the extension of the single expert committee allowing for heterogeneity
amongst multiple experts. When the number of nonexperts is sufficiently large
(i.e. the number of experts is relatively few), the results from the single
expert committee extend intuitively.

######  Proposition 4.

There exists $\bar{n}\in\mathcal{N}$ s.t. for committee described above with
$n\geq\bar{n}$, there is a best neutral equilibrium of LD $\sigma$ in which:

  1. 1.

For each expert $i$, exactly $\lfloor w^{*}(i)\rfloor-1$ nonexperts delegate
to her (i.e. for $\lfloor w^{*}(i)\rfloor-1$ nonexperts $j$,
$\sigma_{j}(a)=\sigma_{j}(b)=d_{i}$).

  2. 2.

All nondelegators vote sincerely.

Moreover the best neutral equilibrium in LD is strictly better than the best
neutral equilibrium in DD whenever $w^{*}(i)>2$ for some expert $i$.

Proposition 4 shows that the logic of Proposition 3 extends to other settings.
As long as the number of worst informed voters is large enough, optimal
information aggregation—when restricted to neutral strategy profiles—in
common-interest committees requires delegation such that all nondelegators
attain (approximately) their Nitzan and Paroush, (1982) optimal weight. Again,
the committee forms a subcommittee of approximately optimally weighted voters
to make the decision. Here as in the single expert committee, the optimal
number of delegated votes each expert should receive does not depend on the
size of the committee. Also, the best neutral equilibria typically requires
nonexperts to employ asymmetric strategies (some delegate and some vote
sincerely) and hence there may be multiple such equilibria.

With multiple experts in the committee, we cannot directly extend the single
expert committee result – that experts attain their rounded optimal weights in
the best neutral equilibrium – without there being a sufficiently large number
of nonexperts. When the number of nonexperts is too small, the best neutral
equilibrium may not have this property. This stems from the fact that
efficient information aggregation requires every two voters to have relative
weights that correctly reflect their relative information qualities, but this
can be hard to achieve in small committees because delegation causes discrete
changes in voters’ weights. When the number of nonexperts is sufficiently
large, however, we can extend the logic of Proposition 3. Importantly,
Proposition 4 is not an asymptotic result; it characterizes the best neutral
equilibrium for finite (but large enough) $n$.

The proof is in the Appendix, but the logic is as follows. First, we show that
as the committee gets large, the best neutral equilibrium must have a bounded
number of voters delegating. This is because if the number of delegators grew
arbitrarily as the committee grew, the committee would be wasting an unbounded
amount of information (that of the delegators), which is inefficient. As the
number of nonexperts grows, the best neutral equilibrium must then require
that almost all of these nonexperts do not delegate. Then, voting nonexperts
dominate in the committee and what is most important for information
aggregation is that the experts are optimally weighted relative to them. Hence
the same incentives for nonexpert-expert delegation as in the single expert
committee prevail and we attain a similar result. Moreover, whenever some
expert has an optimal weight above $2$, the best neutral equilibrium of DD
does strictly worse as this expert cannot attain her optimal weight without
delegation. As with the single expert committee, the comparison with the best
neutral equilibrium of RD depends on the set of representatives, but LD will
do strictly better whenever, for instance, some expert $i$ with $w^{*}(i)>2$
is not a representative (preventing her from being delegated more votes), or
if all voters in the committee are representatives (rendering RD and DD
identical).

Numerous inefficient neutral equilibria will generally exist in these
committees under LD. As in the single expert committee, such equilibria can
for instance be generated by overdelegation to experts and delegation between
nonexperts to compensate.

###  4.3 Inefficiency in equilibrium

In the following we denote the number of independent, $A/B$ partisan and
uninformed independent voters as $n_{I},n_{A},n_{B}$ and $n_{U}$,
respectively. As discussed before, it is hard to characterize the best
equilibrium for general committees, let alone the full set of equilibria.
However, it is easy to see that bad equilibria always exist. For instance, in
LD, RD and DD, it is a BNE for all voters to vote unresponsively for option
$a$; as no voter is ever pivotal under this strategy profile, everyone is
totally indifferent across all strategies.262626Note that voting
unresponsively for $a$ may be weakly dominated for some voters. However, it is
not hard to come up with committees for which all independents voting for $a$
is an undominated equilibrium behavior and results in $A$ being the outcome of
the election w.p. $1$. This sort of equilibrium multiplicity is common in
voting games and the problem is exacerbated when delegation is allowed in LD.
For example, in any committee if we fix an independent voter $i$, whenever the
number of independent voters $n_{I}>|n_{A}-n_{B}|+1$, all independents
delegating to $i$ (and $i$ voting sincerely) can be supported in equilibrium.
Under such a strategy profile, $i$ dictates the election outcome and no
individual voter can change this. Considering again the single expert
committee, this means that it is an equilibrium for all voters to delegate to
any particular nonexpert $i$, who votes sincerely. Hence equilibrium can
support even the worst informed voter dictating the election outcome.
Importantly, both of the types of equilibria just discussed can result in the
committee failing to improve outcomes even as the committee (and hence the sum
of its’ members information) gets arbitrarily large. This means equilibria in
under all voting systems can fail to aggregate information asymptotically,
something that, for instance, the best equilibrium of DD is capable of doing
(e.g. sincere voting by all voters does so by the Condorcet Jury Theorem).
Further, the former equilibrium example (of everyone voting unresponsively for
one alternative) will typically be strictly worse than the best equilibria of
direct democracy and representative democracy, hence the welfare comparison
between liquid democracy and these other mechanisms is not clear-cut.

Past literature has focused on undominated strategies as a selection criterion
to deal with equilibrium multiplicity (Feddersen and Pesendorfer,, 1996,
1997). We note that, even when making this selection, typical committees still
admit inefficient equilibria of the types mentioned in the previous paragraph
under all three mechanisms. However, in the next section we show that, for
certain types of committees, applying _iterated_ deletion of weakly dominated
strategies can be very powerful under LD.

##  5 Dominance solvability with complete information

While we have shown that when there is complete information about types,
liquid democracy admits better equilibrium than mechanisms without delegation,
LD also has numerous bad equilibria. Such equilibria can occur due to
overdelegation or mis-coordination between voters in delegation. As the best
equilibrium typically involves agents of the same type employing asymmetric
strategies (e.g. some delegating and some not delegating), miscoordination is
a concern. In Section 5.1 we discuss one way in which this concern is
mitigated: Proposition 5 implies that when some independent member of the
committee is fully informed, then the game with delegation is dominance
solvable (DS) and the DS solution is the best equilibrium. In such situations,
it may be reasonable to expect voters to coordinate in playing the best
equilibrium. Meanwhile, the game without delegation is not necessarily DS
suggesting that, in certain settings, liquid democracy may make it easier to
coordinate on the efficient equilibrium.

We have already argued that it is weakly dominant for partisan voters (or
partisan representatives in RD) to always vote in favor of their preferred
alternative. Assume that $n_{I}\geq|n_{A}-n_{B}|$. Suppose that some
independent member of the committee $e\in\mathcal{N}$ has precision $q_{e}=1$;
that is, at least one member of the committee learns the state with certainty.
It is straightforward to see that it is weakly dominant for $e$ to vote
sincerely; conditional on her pivotality, sincere voting is uniquely optimal
(as she knows $\omega$ and wants the election outcome to match it) and
conditional on her not being pivotal, she is indifferent across all
strategies.

Below we provide an example showing that the game without delegation (direct
democracy) is not DS while the game with delegation (liquid democracy) is.

######  Example 2.

Suppose there are 3 $A$-partisans, 1 $B$-partisan, 1 expert independent
($q_{e}=1$) and 4 non expert independents ($q_{i}<1$). It is easy to see that
the partisans and expert voters have dominant strategies: an $A$ ($B$ partisan
, expert independent) is pivotal if excluding her, the remaining votes are
divided equally between $a$ and $b$. In all other cases they are indifferent
between their top ranked candidate and others. Thus, assume that partisan and
expert independents are voting their dominant strategies and consider the
reduced game and the strategies of non-experts.

The efficient equilibrium is chosen when non expert independents neutralise
the partisans - so 2 vote $b$ unresponsively and two abstain unresponsively,
ending up with a tie (excluding the expert). This ensures that the expert
decides the election. However this strategy requires a lot of coordination
among the non expert independent voters. Moreover, in this example, each of
the non experts is pivotal so the strategies of voting $b$ unresponsively and
voting $x$ unresponsively are a unique best response. The strategy of voting
$a$ unresponsively is a unique best response if 3 non experts vote $b$
unresponsively while one votes $a$ unresponsively. (This requires at least
$n_{e}+|n_{A}-n_{B}|+1$ non experts.) This ensures that even if the partisans
and expert voters have weakly dominant strategies, the reduced game does not
allow further elimination of dominated strategies and is not dominance
solvable.

With delegation, each of the non experts can delegate to the expert and
achieve the efficient outcome. Moreover, delegation weakly dominates all other
strategies: consider the profile where 3 of the non experts vote $b$
unresponsively, then there is a tie in state $a$ excluding one non expert.
Delegation ensures the correct outcome in both states while voting $b$
unresponsively delivers $b$ in state $a$. Voting $x$ unresponsively delivers a
tie in state $a$. Thus delegation is strictly better than these two
strategies. Consider the profile where 1 of the non experts votes $b$
unresponsively while 2 abstain, then the remaining non expert is pivotal in
state $b$. Delegation achieves the correct outcome in both states, while
voting $a$ unresponsively delivers $a$ in state $b$. Therefore delegation is a
weakly dominant strategy and the game is dominance solvable.

Let $n_{e}$ be the number of perfectly informed independent voters. Applying
iterated deletion of weakly dominated strategies, assume all partisans and
$n_{e}$ perfectly informed independents vote as prescribed and consider the
behavior of independents $i$ with $q_{i}<1$. We can show that as long as these
nonperfectly informed independent voters are pivotal with positive probability
in both states, the game is DS because it is iteratively weakly dominant for
each independents with $q_{i}<1$ to delegate to some voter $e$ with $q_{e}=1$.
This is simply because delegating to a perfectly informed voter can never hurt
(as the perfectly informed voter always makes the correct voting decision) and
will be strictly better for some strategies of others’. As
$n_{I}\geq|n_{A}-n_{B}|$ and $n_{e}\geq 1$, under the DS solution the election
outcome is determined by the perfectly informed independents and hence will
always match the state of nature.272727Strictly speaking, when
$n_{I}=|n_{A}-n_{B}|$, the election outcome will be correct in one state and a
tie —and hence correct w.p. $0.5$ —in the other state. This will still be the
best equilibrium. Note that if voters are able to understand that perfectly
informed voters have an undominated strategy, the DS solution should be easy
to coordinate on. The solution is also ‘type-symmetric’ in a sense that all
nonperfectly informed independents can delegate to any perfectly informed
voter. All perfectly informed voters adopt the same strategy as do all
partisan voters of each type. One can argue that reaching this DS solution has
low cognitive cost. After presenting Proposition 5, we argue that the same
argument goes through as long as some independent voter in the electorate is
sufficiently well informed, even if no one is perfectly informed.

In contrast, the game without delegation is not DS (Proposition 6). As a
result, reaching the efficient pure strategy Nash equilibria (PSNE) is not a
straightforward task: it requires a lot of coordination among the nonperfectly
informed independent voters as efficient equilibria are in type-asymmetric
strategies. Moreover, other types of equilibria that require less coordination
(type-symmetric PSNE or mixed strategy Nash equilibria) exist, but they
frequently lead to inefficient outcomes.

Now we extend this example to the case of RD. The results are easy to show
when the set of delegates includes the expert, in addition to one of each
partisan. As before, each of the delegates is pivotal when votes are equally
split among the two other voters. Therefore voting sincerely is a dominant
strategy in the second stage.

The remaining voters are 2 $A$ partisans, and 4 non expert independents.
Consider the reduced game where delegates use their dominant strategies. In
the first stage, therefore it is easy to show that voting for delegate $A$ is
a dominant strategy for the $A$ partisans and voting for the expert is a
dominant strategy for the non experts.282828Since there are 6 voters, we can
create a profile where a tie is created in favour of the favoured delegate. .
In this case, RD reproduces what LD does endogenously and the outcomes are the
same.

Consider a different set of delegates which includes a non-expert independent
voter. The expert voter has a dominant strategy of voting for partisan
delegate $A$ when the state is $a$ and delegate $B$ when the state is $b$. The
non expert delegate must coordinate with the non expert voters. To achieve the
correct outcome for sure, two of the non expert delegates must vote to
neutralise the $A$ partisans, therefore two of the nonexperts vote for $B$
unresponsively while two nonexperts (including the delegate) abstain. Moreover
each of these strategies (voting $b$ unresponsively or abstaining
unresponsively) is a unique best response for the nonexperts. Since nonexperts
are identical, this implies that these strategies cannot be eliminated for any
of them. Suppose, instead, in the reduced game, two of the nonexperts vote for
partisan $B$, one of them votes for $A$ and the remaining one abstains. In
this case, the expert decides the election but each of the strategies of the
nonexperts in a unique best response, therefore the game is not DS. Overall,
therefore, there exists a committee where the best RD equilibrium is
equivalent to the best equilibrium in the LD game and the game is DS but the
expert is chosen exogenously to be in the committee, and there exists a
committee where the best RD equilibria are equivalent to the best DD
equilibria, and the game is not DS. In this case, in order to achieve full
information equivalence, the non-experts must have asymmetric strategies.

In Section 4.2, we showed that the best equilibria in liquid democracy can
involve asymmetric voting/delegating amongst voters of the same type. In the
LD cases discussed, one voter’s decision to delegate may affect another’s
incentive to do so. For example, in the single expert committee, nonexpert $i$
may want to delegate to the expert if and only if sufficiently few other
nonexperts have delegated. The argument presented in this section shows that
issues of coordination on the best equilibria are less of an issue when a
voter’s delegation incentives do not depend on others’ behavior: when some
voter is sufficiently well informed, others delegating to her is good
regardless of what other voters are doing and hence the best equilibrium can
be easy to coordinate on.

In summary, there are two reasons why we might want the game to be DS. First,
because, as we see in the example, this property rules out many other
equilibria which are inefficient and implausible in the game with delegation.
Second, without this property, reaching the efficient equilibria requires a
fine balancing act, where independent voters must use type-asymmetric
strategies. We have seen in the previous section how coordination mistakes can
lead to far from efficient informational outcomes. When the game is DS,
however, liquid democracy allows for easier coordination on the best
equilibrium in an unambiguous sense.

###  5.1 With delegation

Proposition 5 below shows that the election game with delegation is dominance
solvable, and characterizes the DS outcome. Recall that $n_{U}$ is the number
of independents who have type $(I,q_{i})$, where $q_{i}<1$.

######  Proposition 5.

Suppose that $n_{e}\geq 1$, $n_{U}\geq n_{e}+|n_{A}-n_{B}|+1$ and
$max(n_{e}+n_{A},n_{e}+n_{B})\leq\frac{N}{2}$. Then the election game with
delegation is dominance solvable. The DS outcome is the efficient outcome.

The proof of this proposition is in the Appendix.

The dominant strategies for $A$, $B$ and independent and perfectly informed
voters are $\sigma_{i}(a)=\sigma_{i}(b)=a$, $\sigma_{i}(a)=\sigma_{i}(b)=b$,
and $\sigma_{i}(a)=a,\sigma_{i}(b)=b$ respectively, while the strategy for an
independent voter $i$ who is not perfectly informed in the DS equilibrium is
$\sigma_{i}(a)=\sigma_{i}(b)=d_{j}$ for some perfectly informed $j$. Given the
conditions on numbers of each type, this profile ensures that alternative $a$
wins in state $\omega=a$ and alternative $b$ wins in state $\omega=b$. The
inequality $n_{I}\geq|n_{A}-n_{B}|$ ensures independents are pivotal with
positive probability which ensures sincere voting is a weakly dominant
strategy for perfectly informed voters ($n_{e}\geq 1$). Finally,
$max(n_{e}+n_{A},n_{e}+n_{B})\leq\frac{N}{2}$ is needed to guarantee each
nonperfectly informed independent $i$ is pivotal under some profile in each
state. Given this, $i$ delegating to a perfectly informed voter is weakly
dominant as it is strictly preferable to all other strategies when $i$ is
pivotal (and weakly preferable otherwise).

While the assumption that some voter in the committee is perfectly informed
($q_{i}=1$) is a strong one, this is not a necessary condition for Proposition
5 to hold. It would be sufficient, for instance, for there to be some
independent member of the committee $e$ whose private signal is more
informative than all other independents’ private signals combined. In other
words,

$w^{*}(e)>\sum_{\\{i\in\mathcal{N}|i\neq e,p_{i}=I\\}}w^{*}(i).$

Again, it is weakly dominant for partisans to vote for their preferred
outcome. Then, if $n_{I}\geq|n_{A}-n_{B}|$, it is a weakly dominant strategy
for $e$ to vote sincerely. This is because, any updating she would do when
conditioning on her own pivotality is based on the information held by other
independents (as partisans’ behavior is deterministic). If her own signal is
relatively informative enough, this updating will be outweighed by the
updating from her private signal. Note that under this condition, a best
equilibrium (in terms of information aggregation) is $e$ dictating. Again, an
independent voter $i$, such as $i\neq e$ cannot be harmed by delegating to $e$
and strictly benefits when pivotal with positive probability. Note that in the
case of RD, the game is DS by similar arguments iff at least one sufficiently
well informed expert is in $\cal J.$ But the latter is not guaranteed, so RD
could be equivalent to DD or to LD depending on the independent experts in the
set $J$.

Next, we move to the game without delegation and show that it is not DS.

###  5.2 Without delegation

######  Proposition 6.

Suppose that $n_{e}\geq 1$, $n_{I}\geq|n_{A}-n_{B}|$,
$max(n_{e}+n_{A},n_{e}+n_{B})\leq\frac{N}{2}$, and that delegation is not
allowed. Then:

  1. 1.

The game is not dominance solvable.

  2. 2.

There exist multiple efficient equilibria in asymmetric strategies for
nonperfectly informed voters where the outcome is $a$ in state $\omega=a$ and
$b$ in state $\omega=b$. Assume $(n_{I}-n_{e})\geq n_{I}+|n_{A}-n_{B}|+1$. If
$n_{e}>|n_{A}-n_{B}|,$ there exists an efficient equilibrium where all
nonperfectly informed independent voters abstain. If $n_{e}<|n_{A}-n_{B}|$
then there exists an inefficient equilibrium where all nonperfectly informed
independent voters abstain and the correct outcome is not chosen in at least
one state. There also exist inefficient symmetric PSNE where the outcome is
$i\in\\{a,b\\}$ regardless of the state.

The proof of this proposition can also be found in the Appendix.

Overall, we observe that the efficient outcome is more "likely" (in the sense
of DS of the game) to be reached in the game with delegation than in the game
without delegation. The reason is that it is much easier for voters to
coordinate their strategies in the game with delegation as the DS equilibrium
strategies are type-symmetric. In contrast, if the game is not DS, the
efficient equilibrium requires high level of coordination as there are
multiple such equilibria (any permutation of an equilibrium strategy profile
across nonperfectly informed independent voters is an equilibrium). Moreover
there are also multiple inefficient equilibria even if we restrict ourselves
to PSNE. All outcomes are possible in this game.

Unlike much of the literature on information aggregation (e.g. Feddersen and
Pesendorfer,, 1996), we do not need to assume large elections for this result.
With delegation, when types are known e.g. in committee elections, delegation
ensures that the efficient outcome is reached. When delegation of votes is not
allowed it is "likely" that inefficient outcomes prevail. Of course the more
plausible case is of incomplete information on types. Section 6 describes
this.

##  6 Incomplete information about types

In this section we consider the type of each agent as her private information.
This is an important generalisation as preferences and information precision
cannot be publicly observed in reality unless a truth revelation mechanism is
in place. We still allow each agent to be characterised by a different
likelihood of being assigned each specific type, which makes the model more
realistic compared to the standard assumption that all agents draw their type
from a common distribution.

A first challenge that we have to face is with respect to equilibrium
existence. Indeed, if the space of types was finite, existence of a (Bayesian
Nash) equilibrium would follow trivially from standard equilibrium existence
arguments for finite games. Since we want to be more general and permit
atomless distributions, we have to provide independent arguments that an
equilibrium exists. Throughout this section, we focus on pure undominated
equilibria in threshold strategies. That is, we do not constrain the strategy
set of the players, but only the set of equilibria that we are interested in.
Focusing on undominated strategies imposes that partisans do not delegate and
simply vote for their preferred party. Threshold strategies imply that if two
common value types of an agent characterized by different precisions take the
same action, then no common value type of this agent with an intermediate
precision takes another action. Whenever we refer to an equilibrium from now
on, we implicitly assume that it is one characterized by the features
mentioned.

The second challenge we encounter relates to the welfare effects of liquid
democracy vis-a-vis direct and representative democracy. Since the environment
is much more complex than before, one cannot provide a similarly detailed
description of the set of equilibria in the general case. What we can provide
though is a comparative analysis of the best equilibria of all systems in
terms of information aggregation. This is compatible with similar studies that
also provide welfare comparisons based on the best equilibria (given some
welfare benchmark) of alternative collective choice mechanisms (see, for
instance, Ahn and Oliveros, (2016)).

To establish our first result and to be able to proceed with the one regarding
welfare, we work in the following manner. First, we define an auxiliary game
that is identical to the one that we study except for a twist: in this
auxiliary game, even types $A$ and $B$ have the same ex post utility function
as the common value voters. However, the action space of these types is now
constrained to only one action: type $A$ voters are only allowed to vote for
$A$, and type $B$ voters are only allowed to vote for $B$. It is
straightforward that each equilibrium of the auxiliary game with liquid,
direct or representative democracy corresponds to a unique undominated
equilibrium of the game we are interested in with liquid, direct or
representative democracy, and vice versa. Therefore, we formally analyze the
auxiliary game, to understand whether the game of interest admits an
equilibrium and how the best equilibria of different kinds of democracy
compare.292929 Formally, a threshold strategy in the auxiliary game is an ex-
ante strategy and is characterized by a collection of thresholds $t_{i}$
$t_{i}\in[0.5,1]^{n+1}$ for each player and signal realization, and a strict
ordering $r_{i}$ of the $n+2$ available actions (i.e. the actions $a$, $b$,
$x$, $d_{j\neq i}$) for each player and signal realization. When a player has
a type from $0.5$ up to the first threshold, she employs the action at the top
of the ordering, when her type is larger than the first threshold and at least
as large as the second threshold she employs the action that ranks second in
the ordering, and so on. A threshold strategy in the standard game is
identical to the one of the auxiliary game as far as independent types are
concerned, and employs the weakly dominant action for the partisan types.

Second, we observe that for any ex ante strategy profile, each player has a
best response in threshold strategies. That is, the profile of threshold
strategies that maximizes the ex ante common expected utility of the agents
must be an equilibrium of the auxiliary game (McLennan, (1998)), and it is
also the best equilibrium in terms of information aggregation. Hence, our
equilibrium existence proof amounts to establishing that such a maximizer
profile of threshold strategies exists. We achieve this by arguing that the ex
ante common expected utility of the players in the auxiliary game is
continuous in threshold strategies, and that the set of threshold strategies
is compact.

######  Lemma 1.

When types are the agents’ private information, an equilibrium always exists
under liquid democracy.

###### Proof.

As explained above, we focus on the auxiliary game and proceed in distinct
steps. First we show that for every strategy profile, there always exists a
best response that is a threshold strategy. Then, we prove that a utilitarian
planner who is constrained to choose a threshold strategy for every player,
can always solve her problem (i.e. a maximizer strategy profile exists among
the ones involving only threshold strategies). And finally we combine
McLennan, (1998), along with our first two steps, and show that an equilibrium
in threshold strategies always exists.

Notice that the interim expected utility of player $i$ in the auxiliary game
when she is of type $(I,q_{i})$, receives signal $a$, takes action
$y\in\\{a,b,x,d_{j\neq i}\\},$ and the other players are expected to use the
ex-ante strategy profile $\sum_{-i}$, which is given by

$\displaystyle
q_{i}\cdot\Pr(A\text{wins}|\omega=a,y,\sum_{-i})+(1-q_{i})\cdot\Pr(B\text{
wins}|\omega=b,y,\sum_{-i}).$

This is a linear function of $q_{i}\in[0.5,1]$. Therefore, if we also take the
expected utility of this player for any of the other action $y^{\prime}$, the
two of them—unless they fully coincide—can intersect at most one $q_{i}$. That
is, one can always find a threshold value within $[0.5,1]$ such that using $y$
is optimal if we are below (above) the threshold, and $y^{\prime}$ is optimal
if we are above (below) the threshold. This trivially generalizes to multiple
actions, and hence for any ex-ante strategy profile $\sum_{-i}$ of the other
players, there always exists a best response for player $i$ that is a
threshold strategy (even if $\sum_{-i}$ does not involve threshold
strategies).

Now we move to the problem of a social planner who can fix the ex-ante
strategy of all the players (she is allowed to use only threshold strategies),
and wishes to maximize the sum of the players’ utilities. Since all the
players have the same ex-post utility, the planner essentially maximizes the
probability that the outcome is correct (i.e. it matches the state of the
world). Given that the set of threshold strategies is compact, and the
probability the correct outcome wins is continuous in the threshold strategy
employed for each player and signal realization (because of the fact that
precisions are drawn from an atomless distribution), it follows that the
planners problem has a solution. That is, there exists a profile of threshold
strategies that maximizes the ex-ante probability of election of the
alternative that matches the state of the world.

By McLennan, (1998), we know that the strategy profile that solves the
planner’s problem is an equilibrium of the game in which players are
constrained to use only threshold strategies. While we do not constrain the
player to use threshold strategies, we have already established that among a
player’s best responses there is always a threshold strategy. Hence, the
auxiliary game—and thereafter the game we are interested in—always admits a
pure undominated equilibrium in threshold strategies.303030Notice that an ex-
ante BNE in threshold strategies is trivially an interim BNE of the game as
well. ∎

Given the foundation laid above, and the arguments used to establish existence
of an equilibrium under liquid democracy, the main welfare results follow
directly313131Notice that the above proof can be adapted to also establish
equilibrium existence under direct and representative democracy in this
environment of incomplete information. Indeed, the only thing we need to
change is the interim action set so that our argument corresponds to each
specific form of democracy, and the rest follows..

######  Theorem 1.

In liquid democracy, the best equilibrium (in terms of information
aggregation) is at least as good as the best equilibrium in direct and
representative democracy.

In the auxiliary common-value game under direct/representative democracy, the
threshold strategy profile that maximizes the probability of the correct
alternative winning the election is the best equilibrium in terms of
information aggregation. This strategy profile is also feasible in the
auxiliary game under liquid democracy, and leads to weakly lower welfare
compared to the threshold strategy profile that maximizes the probability of
implementing the correct alternative—which constitutes an equilibrium. Since
each equilibrium of an auxiliary game corresponds to a unique equilibrium of
the corresponding game we are interested in, the result described in Theorem 1
holds.

To show that the best equilibrium in liquid democracy can be strictly better
than the best equilibrium in direct and representative democracy, we consider
the following example.

An example: common-interest committee with experts and nonexperts and
incomplete information on types. Consider the following modification of the
committee with nonexperts and multiple experts introduced in Section 4. As
before, all $N$ committee members are independents, the prior is
$\pi=\frac{1}{2}$, and there are $E$ (odd) experts with precisions $q_{e}=r$.
Suppose that the $N-E$ (even) other voters, nonexperts $i=1,...,N-E$, each
have precision drawn iid from a continuous distribution with support on
$[0.5,\bar{q}]$ for some $\bar{q}\leq r$. While all voters are certain of the
quality of the experts’ information, and are certain this information is
superior to that of other voters, the precision of each nonexpert’s
information is random and privately learned. This setup is studied in Campbell
et al., (2022); they focus on type-symmetric equilibria in which: (1) non-
experts do not delegate to one another and all adopt the same ex-ante
strategy, (2) all non-delegating voters vote sincerely. They show that such an
equilibrium exists that strictly improves payoffs (and information
aggregation) relative to simple majority voting. This equilibrium takes the
form of some cutoff $\tilde{q}\in[0.5,\bar{q}]$ such that a voter $i$
delegates to an expert (picking one uniformly at random) if and only if
$q_{i}<\tilde{q}$ and votes sincerely otherwise.323232Campbell et al., (2022)
does not allow for abstention, but the equilibria they find are still
equilibria when allowing for abstention. This type of equilibrium is very
intuitive: voters delegate their vote if and only if the information they
receive is of low enough quality. Interestingly, this equilibrium maximizes
payoffs amongst all sincere voting semi-symmetric equilibria: the equilibrium
cutoff $\tilde{q}$ balances the costs and benefits of delegation optimally.
For example (from Campbell et al., (2022)) when $N=3$, $K=1$, $r=0.7$, and
$F(q)$ is uniform on $[0.5,0.7],$ $\tilde{q}=0.572$ i.e. a nonexpert delegates
if and only if their precision is below $0.572.$ This cutoff stems from an
indifference condition for a nonexpert with exactly the cutoff precision. A
nonexpert $i$’s decision to delegate their vote is conditioned on the event
that their vote is pivotal, i.e. (i) $i$’s signal disagrees with the expert
(ii) so does the other nonexpert’s signal (iii) the other non expert is not
delegating. In this case $\tilde{q}$ solves

$0.7\times(1-\tilde{q})\times[1-\mu(\tilde{q})]=0.3\times\tilde{q}\times\mu(\tilde{q}),$

where $\mu(\tilde{q})$ is the expected precision of the other non expert given
her precision is above $\tilde{q}$. I.e. indifference at $\tilde{q}$ requires
that the probability that two non experts are correct and the expert is wrong
equals the probability that the expert is correct and both non experts are
wrong.

Liquid democracy with partisans. As it has been recently shown, in settings
like this one, one can improve upon simple majority voting by introducing
cumulative voting, according to which each agent is allocated a number of
votes and she is allowed to distribute them anyway she likes between the two
alternatives (see Bouton et al.,, 2021). Liquid democracy, like cumulative
voting, allows agents to better correlate their voting influence with their
information precision, leading also to better outcomes than simple majority
voting. But the underlying mechanisms behind these welfare improvements are
not equivalent. As the following example shows, liquid democracy can aggregate
information better in some cases than any voting rule that assigns the same
voting power to each agent, independently of whether one is allowed to abstain
partially on some of her votes or not.

Consider that there are three players, and that the priors are even as in the
previous example. The first player is a partisan but it is not known which
party she supports (she is a partisan of each alternative with equal
probability), the second and the third player are informed (i.e. their signal
is always correct) independent voters with probability $1/2-x$, uninformed
independent voters with probability $1/2-x$, and partisans of each party with
probability $x$, for $x\in[0,1/2)$. Without delegation everybody votes
sincerely (and the uninformed are indifferent between voting or abstaining),
but the outcome is not always optimal (i.e. when 2 players are independent but
only 1 is informed, the correct outcome does not win for sure). With
delegation, though, the following equilibrium exists: All types except
uninformed independent (henceforth, uninformed) ones vote sincerely, and the
uninformed type of player two (three) delegates her vote to player three
(two). Hence, whenever there are two independent voters and at least one of
them is informed, the correct outcome wins for sure (which is strictly better
compared to no delegation), and the correct outcome wins with probability
$1/2$ otherwise (which is the same as with no delegation).

The reason why this happens is because the uninformed type of player two
understands that increasing the voting power of player three, can only affect
her expected utility if player three is informed. In that case, by delegating
her vote to player three she can guarantee that the correct outcome will
prevail, even when player one is a partisan supporting the incorrect
alternative. When player three is uninformed or partisan, then, from the point
of view of player two, the probability that the correct outcome wins is $1/2$
independently of her action. Therefore, with liquid democracy full information
equivalence obtains in this case. Notice that in direct democracy _under any
anonymous voting rule_ , there is no way that the correct outcome wins with
certainty when player two is informed, player three is uninformed and player
one is a partisan of the incorrect alternative, establishing our claim that
vote-delegation is not equivalent to any anonymous voting system which
precludes transfers of votes from one player to another.

Of course, liquid democracy—as modelled here—does not always perform better
than any kind of voting system. For instance, when all voters are common
value, but their information precision is their private information, then
direct democracy with cumulative voting can do better than liquid
democracy.333333Consider the following simple version of cumulative voting:
each player has one vote and can cast any fraction of it to any of the two
alternatives. Under this formulation, there is always an equilibrium that
achieves full-information equivalence (Bouton et al.,, 2021). Indeed, if we
have three players and two of them are independent with information precision
$q^{\prime}$, while the third is an independent either with precision
$q^{\prime}$ or with precision one, then with cumulative voting full-
information equivalence obtains in equilibrium, while with liquid democracy it
does not.

These examples imply that the optimal way to go is by merging the two
interventions: i.e. allow agents to transfer votes both between alternatives
and voters. Indeed, one can trivially adapt the arguments supporting Theorem
2, to the case in which each agent is endowed not with a unique, but with
several votes instead, and establish that a hybrid system of liquid democracy
with cumulative voting supports an equilibrium that aggregates information at
least as well as the best equilibrium of the standard version of liquid
democracy analyzed here, and of direct democracy with cumulative voting.

##  7 Conclusion

We consider a model of liquid democracy focused on studying the information
aggregation properties of this growingly popular voting system. We show that
liquid democracy admits an equilibrium that outperforms, in terms of
information aggregation and welfare, direct and representative democracies.
Such an equilibrium exists even there are partisan voters who are not
interested in information aggregation, and even when voters’ types are not
commonly known. When types are commonly known, we unpack what drives a voter’s
incentive to delegate and how voters optimally solve the tradeoff presented by
delegation. We discuss when delegation is needed to strictly improve outcomes
over other voting mechanisms, but also show that miscoordination in delegation
can lead to bad outcomes. Finally, we argue that despite the presence of bad
equilibria, in some settings liquid democracy may make it easier for voters to
coordinate on efficient equilibria than other voting systems.

Finally, one might wonder whether the specifics of the mechanism in place
matter greatly in settings where communication is allowed. To be sure, in
common value settings communication is known to improve welfare and to
diminish differences between voting systems (see, for instance, Gerardi and
Yariv,, 2007). In settings with partisan voters, though, communication is not
enough to guarantee an efficient outcome. Consider that after types and
signals are drawn voters are asked to submit a report (e.g. a binary message),
and then the distribution of reports is presented publicly. If the frequency
of a given report affects monotonically the beliefs of the agents regarding
the likelihood that state $a$ is correct, then such a report would also be
chosen in equilibrium by partisan voters who support alternative $A$, making
the distribution of reports less informative.

Perhaps more importantly, even if there exists an efficient equilibrium with
communication, it is never unique (i.e. an inefficient babbling equilibrium
always exists) and it is never the case that the efficient outcome is
implemented with iterated elimination of dominated strategies. Unlike
delegation, where agents with high information precision have a weakly
dominant strategy to vote for the alternative they think is most likely
correct, when agents are asked to first communicate and then vote, sending a
given message is optimal only conditional on how one believes that it will be
perceived by others. That is, a perfectly informed common value voter, whose
type is known to the other voters, will find it optimal to send message $x$ in
the communication stage only if the other agents interpret $x$ in a certain
way, and might not find it optimal otherwise. Therefore, while communication
is a valid means to improve welfare, it does not seem to eliminate all the
advantages of delegation, like the possibility of implementing the efficient
outcome in certain settings without imposing demanding assumptions on beliefs.

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##  Appendix A Proofs

###  A.1 Proofs of Section 4.

We first prove a Lemma.

######  Lemma 2.

For any independent voter $i$ in LD or DD, and any independent
$i\in\mathcal{J}$ in RD, the following strategies are weakly dominated:

  * (1)

$\sigma_{i}(a)=b$ and $\sigma_{i}(b)\neq b$

  * (2)

$\sigma_{i}(a)\neq a$ and $\sigma_{i}(b)=a$.

###### Proof.

We first prove the result under LD and then show that the same applies for DD
and RD. We prove (1) and by symmetry (2) holds as well. Let
payoff$(s_{i},y,\sigma_{-i})$, $s_{i}\in\\{a,b\\}$, $y\in\\{a,b,x,d_{j\neq
i}\\}$ be $i$’s payoff from taking action $y$ when her signal is $s_{i}$ and
others are playing strategy profile $\sigma_{-i}$. $i$’s payoff from using
strategy $\sigma_{i}(a)=b,\sigma_{i}(b)=y^{\prime}\neq b$ is:

$\displaystyle
Pr(s_{i}=a)payoff(a,b,\sigma_{-i})+Pr(s_{i}=b)payoff(b,y^{\prime},\sigma_{-i})$

As $i$ is an independent, we have:

$\displaystyle payoff(a,b,\sigma_{-i})\geq payoff(a,y^{\prime},\sigma_{-i})$
$\displaystyle\iff
q_{i}(Pr(\mathcal{O}=A|\omega=A,\sigma_{-i},y=b)-Pr(\mathcal{O}=A|\omega=A,\sigma_{-i},y=y^{\prime}))$
$\displaystyle+(1-q_{i})(Pr(\mathcal{O}=B|\omega=B,\sigma_{-i},y=b)-Pr(\mathcal{O}=B|\omega=B,\sigma_{-i},y=y^{\prime}))\geq
0$
$\displaystyle\iff|Pr(\mathcal{O}=B|\omega=B,\sigma_{-i},y=b)-Pr(\mathcal{O}=B|\omega=B,\sigma_{-i},y=y^{\prime})|$
$\displaystyle\geq\frac{q_{i}}{1-q_{i}}|Pr(\mathcal{O}=A|\omega=A,\sigma_{-i},y=b)-Pr(\mathcal{O}=A|\omega=A,\sigma_{-i},y=y^{\prime})|$

where the last step comes from the fact that the first term on the LHS of the
second equation is weakly negative; voting for $b$ when the state is $A$
results in a weakly lower probability of $A$ winning than taking any other
action (this is because other player’s actions are independent from $i$’s
conditional on the state; note that this statement is true even if the other
action we considering for $i$ is delegation to some $j$ and creating a cycle:
then all the votes in the cycle would be abstained if $i$ delegates to $j$ but
would be cast for $b$ if $i$ votes for $b$). Similarly, the second term on the
LHS is weakly positive.

Meanwhile, similarly:

$\displaystyle payoff(b,y^{\prime},\sigma_{-i})\geq payoff(b,b,\sigma_{-i})$
$\displaystyle\iff
q_{i}(Pr(\mathcal{O}=B|\omega=B,\sigma_{-i},y=y^{\prime})-Pr(\mathcal{O}=B|\omega=B,\sigma_{-i},y=b))$
$\displaystyle+(1-q_{i})(Pr(\mathcal{O}=A|\omega=A,\sigma_{-i},y=y^{\prime})-Pr(\mathcal{O}=A|\omega=A,\sigma_{-i},y=b))\geq
0$
$\displaystyle\iff|Pr(\mathcal{O}=A|\omega=A,\sigma_{-i},y=y^{\prime})-Pr(\mathcal{O}=A|\omega=A,\sigma_{-i},y=b)|$
$\displaystyle\geq\frac{q_{i}}{1-q_{i}}|Pr(\mathcal{O}=B|\omega=B,\sigma_{-i},y=y^{\prime})-Pr(\mathcal{O}=B|\omega=B,\sigma_{-i},y=b)|$

As $q_{i}>0.5$, both of these two expressions cannot hold and so the strategy
must be dominated by either $\sigma_{i}(a)=\sigma_{i}(b)=b$ or
$\sigma_{i}(a)=\sigma_{i}(b)=y^{\prime}$.

The same argument applies directly to DD and independents in $\mathcal{J}$ in
RD, but without delegation in the strategy space.

∎

Recall Proposition 2. We provide a proof below.

Proposition 2. Assume $\pi=\frac{1}{2},$ $n_{A}\geq n_{B}$, $n_{U}$ uninformed
($q_{i}=0.5$) independent voters and $E$ independent experts ($q_{i}>0.5$).
There exists $n^{*}$ such that when $n_{U}>n^{*}$, under the best LD
equilibrium $\sigma$:

  1. 1.

$n_{A}-n_{B}$ uninformed independents vote unresponsively for $B$.

  2. 2.

Each remaining uninformed independents either: (1) abstains at both signal
realizations, or (2) delegates to the same expert at both signal realizations.

  3. 3.

The election outcome under LD is the first-best outcome w.p. $1$.

  4. 4.

Let $v_{i},v_{j}$ be the number of votes held by experts $i$ and $j$. As
$n_{U}\rightarrow\infty$, $v_{i}/v_{j}\rightarrow w^{*}(i)/w^{*}(j)$ (experts
approach optimal weights).

Before going on to the proof, we define some useful notation. First, let
$W^{*}(\mathcal{J})=\sum_{i\in\mathcal{J}}w^{*}(i)$ be the summed Nitzan and
Paroush, (1982) optimal weight of $\mathcal{J}\subset\mathcal{N}$.

In the case of $\pi=\frac{1}{2}$, we define the following. Let $\mathcal{C}$
represent the set of all partitions of all independents into two subsets or
coalitions. Here we represent profiles of signal realizations by the
coalitions, $(C,C^{\prime})\in\mathcal{C}$, they induce; $(C,C^{\prime})$
denotes the event where all voters $i,j\in C$ receive signal realization
$s_{i}=s_{j}$ ($=s_{C}$) and all voters in the complementary set,
$C^{\prime}$, get signal realization $s_{C^{\prime}}\neq s_{C}$. Note that
each $(C,C^{\prime})\in\mathcal{C}$ occurs with positive probability. Each
$(C,C^{\prime})$ also corresponds to two different signal realization
profiles: one where voters in $C$ get signal $a$ and $C^{\prime}$ get signal
$b$ and vice versa; because $\pi=1/2$ and there are no partisans, these are
informationally equivalent in that both events contain the same strength of
evidence in favor of $\omega=s_{C}$ versus $\omega=s_{C^{\prime}}$. We use the
term ‘signal realization’ to denote both a profile $(s_{1},...,s_{I+1})$ and
the coalitions $(C,C^{\prime})$.

Proof of Proposition 2.

###### Proof.

The best equilibrium implements the correct outcome with the highest
probability. Hence any equilibrium which first-best aggregates the information
of all experts must be the best equilibrium (uninformed independents have no
information). It remains to show that such an equilibrium exists for $n_{U}$
large enough.

Assume $n_{U}\geq n_{A}-n_{B}$. For every expert $i\in E$ let
$w^{*k}(i)=kw^{*}(i)$ for $k>0$. Suppose $n_{A}-n_{B}$ uninformed voters vote
unresponsively for $B$. Fix $k>0$ and suppose that the remaining uninformed
voters abstained and all experts $i$ had exactly weight $w^{*k}(i)$. As
optimal weights can be scaled without changing election outcomes, this would
first-best aggregate the information of experts. Consider any signal
realizations partitioning the set of experts into subsets $C$ and $C^{\prime}$
with disagreeing signals and, wlog, $W^{*}(C)\geq W^{*}(C^{\prime})$ (note the
prior is flat, so it does not matter which signal each group received).
Whenever $W^{*}(C)>W^{*}(C^{\prime})$, the decision will be made by a margin
of $k(W^{*}(C)-W^{*}(C^{\prime}))$ votes. For large enough $k$ this quantity
is greater than $E$ and hence the same decision would be implemented if we
rounded down each $w^{*k}(i)$ to an integer (as rounding each $w^{*k}(i)$ to
an integer will affect vote tallies by at most $E$). Whenever
$W^{*}(C)=W^{*}(C^{\prime})$, voters are indifferent between options under the
first-best, and so rounding weights will leave payoffs first-best.

Fix some such large $k$ and let the rounded weights be $\lfloor
w^{*k}(i)\rfloor$. Then for large enough $n_{U}$, the same outcomes can be
replicated by an LD strategy profile where experts vote sincerely and amongst
uninformed voters: $n_{A}-n_{B}$ vote unresponsively for $B$, $\lfloor
w^{*k}(i)\rfloor-1$ delegate at both signals to each expert $i\in E$, and the
remaining always abstain.

For the final statement, as $k\rightarrow\infty$, $\lfloor
w^{*k}(i)\rfloor/\lfloor w^{*k}(j)\rfloor\rightarrow w^{*}(i)/w^{*}(j)$. ∎

####  A.1.1 Example of Best Non-Neutral Equilibria

Example 1 Suppose $\pi=0.5$ and consider a committee with: (1) one ‘expert’
$e$ with precision $0.8$, (2) ‘nonexperts’ $1,...,4$ with precisions $0.6$.
All voters are independents.

The Nitzan and Paroush (1982) weight of the expert is $w^{*}\in(3,4)$ and that
of the nonexperts is $1$. The best neutral equilibrium is as follows:

  * •

Expert votes sincerely: $\sigma_{e}(a)=a$, $\sigma_{e}(b)=b$.

  * •

(At least) two experts delegate to the expert so she attains her rounded down
Nitzan and Paroush (1982) weight ($\sigma_{i}(a)=\sigma_{i}(b)=d_{e}$ for at