# Three candidate election strategy

Dorje C. Brody1 and Tomooki Yuasa2 1 School of Mathematics and Physics,
University of Surrey, Guildford GU2 7XH, UK
2 Faculty of Economics and Business Administration, Graduate School of
Management, Tokyo Metropolitan University, Tokyo, 100-0005, Japan

###### Abstract

The probability of a given candidate winning a future election is worked out
in closed form as a function of (i) the current support rates for each
candidate, (ii) the relative positioning of the candidates within the
political spectrum, (iii) the time left to the election, and (iv) the rate at
which noisy information is revealed to the electorate from now to the election
day, when there are three or more candidates. It is shown, in particular, that
the optimal strategy for controlling information can be intricate and
nontrivial, in contrast to a two-candidate race. A surprising finding is that
for a candidate taking the centre ground in an electoral competition among a
polarised electorate, certain strategies are fatal in that the resulting
winning probability for that candidate vanishes identically.

## Introduction

This paper is concerned with informational strategies behind an election
involving three or more candidates. Suppose that there is an election coming
up next year, and that the three candidates have the support rates of, say,
38%, 26%, and 36%, according to the current opinion poll statistics. Do these
numbers represent the current likelihoods of the candidates winning the future
election? If not, what are they? Can we derive a formula for a given candidate
winning the election in the future? Because voter preferences change over time
in accordance with information revealed to them, such a formula ought to be
dependant on how information is managed between today and the future election
day. In this paper, an exact formula will be worked out for the probability of
winning a future election that depends explicitly on the model for the flow of
information.

With such a formula at hand, we are able to ask a range of strategic questions
and find quantitative answers. For example, for the candidate lagging behind
with only 26% support today, in which way should they reveal policy
information so as to maximise the realised probability of winning the future
election? How would that differ from the strategy for other candidates? Are
there advantages in the positioning of the candidate’s political party within
the political spectrum, for example, leaning further to the right or to the
left? Our purpose here is to provide a systematic framework to answer
questions of this kind, building on the previous work on two-candidate
election models Brody1 ; Brody2 ; Brody3 . In particular, we shall reveal some
surprising features that emerge from having more than two candidates in an
electoral competition.

It is worth remarking that there is a long history of building mathematical
models to analyse various aspects of electoral competitions (see, for example,
xxx ; Shepsle ; JH ; MO ; Coughlin ; FP ; FM ; Lloyd ; yyy ; YAKM ). The
models hitherto considered in the literature tend to be either deterministic,
or else probabilistic but static. While the importance of the role played by
information in electoral competitions is widely acknowledged, the models that
have been proposed in the literature omit the impact of noise such as rumours,
speculations, disinformation, and so on. In contrast, our formulation,
building on a successful application of the noisy information-based approach
to modelling the dynamics of financial assets BHM , takes on board the
revelation of noisy information, which in turn can be used to deduce the
statistics of the outcomes of future elections. In this way, a sensitivity
analysis can be pursued so as to determine the impact of different
informational strategies on the future statistics.

With these preliminaries, the present paper will be organised as follows. We
begin with a brief introduction to the information-based approach to election
modelling introduced in Brody1 . We then proceed by reviewing some of the key
findings of Brody2  in which the probability of winning a future election is
worked out in the case of a two-candidate electoral competition (or a “yes-no”
referendum) for benefits of readers less acquainted with the material. We
shall also indicate here how a large number of information sources can be
aggregated in the form of a single information process, and how individual
information source affects the overall information. We then examine the case
of an election involving three candidates, and work out explicit formulae for
the probabilities of six different outcomes (orderings of candidates) in a
future election. Our focus here will be a first-past-the-post electoral
system, although the fact that we have explicit formulae for all different
scenarios means that the approach can be used in other electoral systems such
as a proportional representation system. We then explore the impact of
candidate’s positioning within the political spectrum. In this context it will
be shown that among a polarised electorate the candidate with a centre-ground
position can find that the probability of winning a future election is
identically zero in a first-past-the-post electoral system, unless a
significant amount of reliable information is revealed to create higher
volatilities. We then discuss briefly how our model can be implemented in
practice in an electoral competition. We conclude with a discussion on how
closed-form formulae for the probabilities of the various different outcomes
of an election can be obtained in the present scheme when there are more than
three candidates.

## Information-based modelling of electoral competition

Let us begin by examining the role of information in modelling an electoral
competition. In a generic situation, voters will have a range of issues that
concern them in deciding which candidate to choose. The policy positions of
the candidates, if elected, on these issues, however, are known only partially
to the voters. Nevertheless, as time progresses, more information about the
candidates, or their views on different policy positions, are revealed, which
in turn will shift voter perceptions. We can model this dynamics by use of the
mathematical techniques of signal processing. Specifically, we model the
position of candidate $l$ on issue $k$ by a random variable $X^{k,l}$, taking
a range of values labelling different policy positions on this issue, where
the probability that $X^{k,l}$ taking a given value represents the voter
perception of the policy position of the candidate $l$ on this issue. Voters
will then acquire partial information about the values of these random
variables. For a fixed candidate and fixed issue, because the quantity of
interest to voters is the candidate’s position on that issue, represented by
the value of $X^{k,l}$, we can think of this random variable as the ‘signal’
in a communication channel, which is superimposed with noise that represents,
for instance, rumours, speculations, disinformation, and so on. The arrival of
information to voters can therefore be modelled in the form of a superposition
of signal and noise. The idea we evoke here is that we regard the environment
in which voters are living, in itself, as forming a communication channel,
where noisy information is transmitted through newspapers, radio and
television broadcasts, internet, word of mouth, and so on. Then based on this
partial information, voters will come up with best estimates for each of the
factors. Candidates are then ranked in accordance with these estimates,
reflecting the preferences of voters.

This concept has been formalised mathematically in Brody1  as a structural
approach to modelling electoral competition, with an emphasis to determine the
statistics of the impact of disinformation on democratic processes. For a
better understanding of the information-based formalism it will be useful to
explain the structural approach in more detail. The fundamental idea is to
first model the information-providing process $\\{\xi_{t}^{k,l}\\}$ associated
to the policy position $k$ of candidate $l$. For example, if the noise that
obscures the value of $X^{k,l}$ is modelled by an additive Gaussian noise
$\\{B_{t}^{k,l}\\}$, and if the rate at which information is revealed to the
electorates at time $t$ is given by $\sigma_{t}^{k,l}$, then the information
process takes the form of a ‘signal-plus-noise’ decomposition:

$\displaystyle\xi_{t}^{k,l}=X^{k,l}\int_{0}^{t}\sigma_{s}^{k,l}\,\mbox{$\rm
d$}s+B_{t}^{k,l}\,.$ (1)

If we let ${\cal F}_{t}$ denote the totality of information available to the
electorates at time $t$ generated by (1) for all $k,l$, then the best estimate
people will arrive at about the $k$-th policy position of candidate $l$ is
given by the conditional expectation
${\hat{X}}_{t}^{k,l}={\mathbb{E}}[X^{k,l}|{\cal F}_{t}]$.

Next, for a given voter $m$, we let $w_{m,k}$ denote the preference weight of
that voter for issue $k$. Then we can identify the ‘score’ $S_{m}^{l}(t)$
assigned at time $t$ by voter $m$ for candidate $l$. For example, in a linear
scoring system we have

$\displaystyle S_{m}^{l}(t)=\sum_{k}w_{m,k}\,{\hat{X}}_{t}^{k,l}.$ (2)

Then at time $T$ of the election, voter $m$ will choose the candidate with the
highest score at that time. Because a large number of the voter preferences
$\\{w_{m,k}\\}$ can be sampled from a distribution Brody1 , in a structural
approach it is possible to engage in a rather detailed issue-by-issue scenario
analysis to identify optimal informational strategies, as well as making
predictions of the statistics of a future election.

## Reduced-form approach to electoral competition

An alternative ‘reduced form’ approach has also been introduced in Brody1  and
was further developed in Brody2 . The idea of a reduced form approach is to
aggregate a broad range of issues into a single random variable $X$ that
labels different candidates. We remark that the terminologies of structural
versus reduced-form are derived from an analogous consideration in the
modelling of credit risk in financial markets. Here, for a given cash flow
associated to a given firm, one can attempt to go into a detailed structural
analysis of that firm in identifying the risk associated to that cash flow.
However, in most cases this is not feasible because relevant financial
structures – for instance other cash flows linked to that firm – are far too
complicated and often not even transparent. To remedy this issue, in credit
risk modelling an alternative reduced-form approach has been introduced as a
way for capturing the essence of credit risks without going into any of the
structural details so that the method can be applied universally and
systematically, in practical applications.

In contrast to credit risk modelling, for an electoral competition a
structural approach is entirely feasible. This is because the number of
significant issues that are of concern to a large number of electorates is
limited – perhaps a handful as opposed to thousands in the case of a typical
credit product – and likewise the number of candidates is also limited.
Nevertheless, the advantage of the reduced form formalism, which will be
explained in more detail now, is that just as in credit risk modelling, the
mathematical analysis can be simplified considerably without losing many of
the key features of the structural formalism. In the present paper we shall
therefore develop the theory underlying a three candidate race in the reduced-
form approach.

In a reduced form approach to modelling electoral competitions, a wide-ranging
information relevant to deciding which candidate to choose is aggregated in
the form of a single information process that represents abstractly the choice
of the candidates. More specifically, in an election with $N$ candidates we
let $X$ be the random variable taking the values $\\{x_{k}\\}$,
$k=1,2,\ldots,N$, that label different candidates, with the probabilities
$\\{p_{k}\\}$. These probabilities represent the current opinion poll
statistics. These statistics will change in time, in accordance with the
revelation of information related to the candidates. Recall that voters wish
to determine which candidate to vote for. Hence in a reduced form model the
random variable $X$ plays the role of the ‘signal’.

As an elementary model let us assume that information concerning the
candidates is revealed to the voters at a constant rate $\sigma$, and that
wide-ranging noise is modelled by a Brownian motion $\\{B_{t}\\}$. Then the
flow of information takes the familiar signal-plus-noise form

$\displaystyle\xi_{t}=\sigma Xt+B_{t}.$ (3)

In a more realistic scenario the information flow rate is time dependent, and
in this case the signal component is modified to
$X\int_{0}^{t}\sigma_{s}\,{\rm d}s$. However, for simplicity we shall consider
the special case in which $\sigma_{t}=\sigma$ is constant, with the remark
that all the results presented below can be extended to the time-dependent
case without losing analytical tractability.

It is worth remarking that in a real election there are many information
sources associated to the candidate choice $X$. We thus have the information
process $\xi_{t}^{j}=\sigma_{j}Xt+B_{t}^{j}$ generated by the $j$-th
information source. However, without loss of generality we can aggregate
different information sources in the form of a single information process (3).
To see this, let us first consider a pair of information sources:
$\xi_{t}^{1}=\sigma_{1}Xt+B_{t}^{1}$ and $\xi_{t}^{2}=\sigma_{2}Xt+B_{t}^{2}$.
Letting $\rho={\mathbb{E}}[B_{t}^{1}B_{t}^{2}]$ denote the correlation of the
two Brownian noise, it is an elementary fact that there exists a Brownian
motion ${\bar{B}}_{t}$, independent of $B_{t}^{1}$, such that we have
$B_{t}^{2}=\rho B_{t}^{1}+\sqrt{1-\rho^{2}}{\bar{B}}_{t}$. Defining

$\displaystyle{\bar{\sigma}}=\frac{\sigma_{2}-\rho\sigma_{1}}{\sqrt{1-\rho^{2}}}\quad{\rm
and}\quad{\bar{\xi}}_{t}={\bar{\sigma}}Xt+{\bar{B}}_{t}$ (4)

we deduce that the information generated jointly by $\xi_{t}^{1}$ and
$\xi_{t}^{2}$ is equivalent to that generated jointly by $\xi_{t}^{1}$ and
${\bar{\xi}}_{t}$. Next, let us define

$\displaystyle\sigma^{2}=\sigma_{1}^{2}+{\bar{\sigma}}^{2}\quad{\rm and}\quad
B_{t}=\frac{1}{\sigma}(\sigma_{1}B_{t}^{1}+{\bar{\sigma}}{\bar{B}}_{t}),$ (5)

and set

$\displaystyle\xi_{t}=\sigma Xt+B_{t}\quad{\rm
and}\quad\delta_{t}=\frac{\xi_{t}^{1}}{\sigma_{1}}-\frac{{\bar{\xi}}_{t}}{{\bar{\sigma}}}=\frac{B_{t}^{1}}{\sigma_{1}}-\frac{{\bar{B}}_{t}}{{\bar{\sigma}}}.$
(6)

Then a short calculation shows that the information generated jointly by
$\xi_{t}^{1}$ and $\xi_{t}^{2}$ is equivalent to that generated jointly by
$\xi_{t}$ and $\delta_{t}$, and that $\xi_{t}$ and $\delta_{t}$ are
independent. However, $\delta_{t}$ is independent of the random variable $X$.
It follows that
${\mathbb{E}}[X|\\{\xi_{t}^{1},\xi_{t}^{2}\\}]={\mathbb{E}}[X|\\{\xi_{t}\\}]$.
In other words, the aggregate of the two information sources can be
represented in the form of a single information process (3).

More generally, suppose that we have a series of information processes of the
form

$\displaystyle\left\\{\begin{array}[]{l}\xi^{1}_{t}=\sigma_{1}Xt+B^{1}_{t}\\\
\quad\,\,\,\vdots\\\ \xi^{n}_{t}=\sigma_{n}Xt+B^{n}_{t}\end{array}\right.$
(10)

available to the electorates about the choice of candidates, where the various
noise processes $\\{B_{t}^{i}\\}_{i=1,\ldots,n}$ in general may be mutually
correlated, with the correlation matrix $\rho_{ij}$. Then the aggregate of the
$n$ information processes (10) can be represented by means of a single
information process of the form (3), where

$\displaystyle\sigma^{2}=\frac{\sum^{n}_{i}\sigma_{i}^{2}\,\rho^{-1}_{ii}-2\sum_{i\neq
j}\sigma_{i}\,\sigma_{j}\,\rho_{ij}^{-1}}{\det(\rho)}$ (11)

represents the effective information flow rate and

$\displaystyle
B_{t}=\frac{1}{\sigma}\left(\sum^{n}_{i,j}\sigma_{i}\,\rho^{-1}_{ij}\,B_{t}^{i}\right)$
(12)

represents the effective noise. Here $\rho^{-1}_{ij}$ denotes the $ij$ element
of the inverse correlation matrix. This observation shows that although the
idea of representing a wide range of information flows in terms of a single
information process at first may seem restrictive, in fact it is quite
general.

Given our model (3) to represent the flow of information, the initial voter
preference for a candidate, embodied in the a priori probability
$p_{i}={\mathbb{P}}(X=x_{i})$, will change into the posterior preference
$\pi_{it}={\mathbb{P}}(X=x_{i}|\\{\xi_{s}\\}_{0\leq s\leq t})$, which is just
the conditional probability that candidate labelled by $x_{i}$ should be
chosen, given the information available up to time $t$. In the present example
in which the information flow rate is constant, the information providing
process $\\{\xi_{t}\\}$ is Markov, from which it follows that the conditional
probability simplifies into $\pi_{it}={\mathbb{P}}(X=x_{i}|\xi_{t})$. Then by
use of the Bayes formula

$\displaystyle{\mathbb{P}}(X=x_{i}|\xi_{t})$ $\displaystyle=$
$\displaystyle\frac{{\mathbb{P}}(X=x_{i})\rho(\xi_{t}|X=x_{i})}{\sum_{j}{\mathbb{P}}(X=x_{j})\rho(\xi_{t}|X=x_{j})},$
(13)

along with the fact that the conditional density function
$\rho(\xi_{t}|X=x_{i})$ for the random variable $\xi_{t}$ is Gaussian and is
given by

$\displaystyle\rho(\xi|X=x_{i})=\frac{1}{\sqrt{2\pi
t}}\exp\left(-\frac{(\xi-\sigma x_{i}t)^{2}}{2t}\right),$ (14)

we deduce at once that

$\displaystyle\pi_{it}=\frac{p_{i}\exp\left(\sigma
x_{i}\xi_{t}-\frac{1}{2}\sigma^{2}x_{i}^{2}t\right)}{\sum_{j}p_{j}\exp\left(\sigma
x_{j}\xi_{t}-\frac{1}{2}\sigma^{2}x_{j}^{2}t\right)}.$ (15)

Incidentally, this expression in the context of signal processing is known as
the Wonham filter wonham  associated to the detection of a random drift of a
Brownian motion; a problem that has also been explored more recently in
different contexts Buonaguidi ; Ekstrom .

## Two-candidate scenario

Given the expression (15) for the a posteriori probability that the $i$th
candidate should be chosen, we are able to ask a range of questions that link
informational strategies to the election outcome. To this end we determine
first the probability that the $i$th candidate wins a future election, given
(a) current support rate, and (b) how information is managed from now to the
election day, in the case of a two-candidate competition. In this case we may
let, without loss of generality, the random variable $X$ labelling the two
candidates be binary, taking the values $0$ and $1$. Let $p$ be the
probability that $X=0$ and $1-p$ be the probability that $X=1$. Then the a
posteriori probability that, say, candidate $0$ being chosen when an arbitrary
voter is sampled from the population is

$\displaystyle\pi_{0t}=\frac{p}{p+(1-p)\exp\left(\sigma\xi_{t}-\frac{1}{2}\sigma^{2}t\right)}.$
(16)

If the election were to take place in $T$ years time from today (today always
implies time $t=0$), then the probability that candidate zero should win the
election, subject to current poll and how information is revealed from today
to the election day, is therefore given by ${\mathbb{P}}(\pi_{0T}>1/2)$.

It is important to note that the realised winning probability
${\mathbb{P}}(\pi_{0T}>1/2)$ for candidate zero, as of today, can be very
different from the current support rate $p$. To understand this, imagine that
the election is to take place in a week, and that candidate zero currently has
$55\%$ support. Then unless something radical happens — such as a revelation
of a major scandal — it is likely that voter preferences will not change very
much in one week, hence candidate zero will receive nearly $55\%$ of the votes
to secure a victory. In other words, in such a scenario the realised
probability of candidate zero winning the future election is close to $100\%$,
even though the support rate remains only $55\%$. Putting the matter
differently, today’s poll statistics is not the predictor for the likelihood
of winning a future election, although it can be used to calculate the
likelihood.

One advantage of our approach is that we are able to derive an explicit
formula for the probability of a given candidate winning the election that
reflects this observation. The detailed derivation of the formula in the case
of a two-candidate election is provided in Brody2 , which we shall not
reproduce here. Instead, we mention some key steps that will be exploited in
the present context. Namely, that the denominator of the conditional
probability (15) can be used to change probability measure ${\mathbb{P}}$ into
a new measure ${\mathbb{Q}}$ such that under ${\mathbb{Q}}$ the information
process $\\{\xi_{t}\\}$ is a standard Brownian motion Brody2 . Specifically,
writing

$\displaystyle\Phi_{t}=p+(1-p)\exp\left(\sigma\xi_{t}-\mbox{$\textstyle\frac{1}{2}$}\sigma^{2}t\right)$
(17)

for the measure-change martingale, we have

$\displaystyle{\mathbb{P}}\left(\pi_{0T}>\mbox{$\textstyle\frac{1}{2}$}\right)={\mathbb{E}}^{\mathbb{P}}\left[{\mathds{1}}\left\\{\pi_{0T}>\mbox{$\textstyle\frac{1}{2}$}\right\\}\right]={\mathbb{E}}^{\mathbb{Q}}\left[\Phi_{T}\,{\mathds{1}}\left\\{\pi_{0T}>\mbox{$\textstyle\frac{1}{2}$}\right\\}\right].$
(18)

Now the condition that $\pi_{0T}>1/2$ is equivalent to the condition that

$\displaystyle\frac{\xi_{T}}{\sqrt{T}}<\frac{\log\left(\frac{p}{1-p}\right)+\frac{1}{2}\sigma^{2}T}{\sigma\sqrt{T}},$
(19)

but under ${\mathbb{Q}}$ the information process is a Brownian motion, and
hence $\xi_{T}/\sqrt{T}$ is a standard normal random variable. It then follows
at once that

$\displaystyle{\mathbb{P}}\left(\pi_{0T}>\mbox{$\textstyle\frac{1}{2}$}\right)=p\,N(d^{+})+(1-p)\,N(d^{-}),$
(20)

where

$\displaystyle
d^{\pm}=\frac{\log\left(\frac{p}{1-p}\right)\pm\frac{1}{2}\sigma^{2}T}{\sigma\sqrt{T}}$
(21)

and

$\displaystyle N(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}\mbox{$\rm
e$}^{-\frac{1}{2}z^{2}}\mbox{$\rm d$}z$ (22)

denotes the cumulative normal distribution function. It is a curious
coincidence that the winning probability of a candidate in a two-candidate
election is essentially the same as the option pricing formula of Black and
Scholes in financial modelling. The formula shows, for instance, that if
candidate zero has $55\%$ support rate today and if the election is to take
place in a week, then even if the information revelation rate is as large as,
say, $\sigma=1.2$, the winning probability will be about $89\%$; whereas if
the information revelation rate is reduced to, say, $\sigma=0.5$, then this
probability increases to $99.8\%$. In other words, the model reflects the
intuition described above. Putting it differently, formula (20) allows us to
interpolate between today’s and future’s statistics.

Figure 1:  Winning likelihood. The probability that candidate zero will win
the election in eighteen months ($T=1.5$), as a function of the current
support rate $p$ for the candidate. The realised likelihood of winning a
future election is always higher than today’s poll if $p>\frac{1}{2}$; and
conversely lower than the poll if $p<\frac{1}{2}$. How much the winning
probability deviates from the current poll depends on how much information is
revealed over the next eighteen months. Here, two examples are shown,
corresponding to the values $\sigma=0.2$ (in purple) and $\sigma=1.2$ (in
red).

We therefore see how an explicit formula (20) for a given candidate winning a
future election can be obtained in the case of a two-candidate competition.
The winning probability, more explicitly, depends on the following three
ingredients: (i) The current support rate $p$ for the candidate, (ii) the time
$T$ left to the election, and (iii) the rate $\sigma$ at which information is
revealed to the electorate from now until the election day. The only ‘control’
parameter at candidates’ disposal therefore is the information flow rate
$\sigma$. To gain a better intuition here, therefore, let us examine how the
winning probability depends on the current poll statistics for different
values of $\sigma$. In Figure 1 we plot the winning probability for candidate
zero as a function of the current support rate $p$ for two different values of
$\sigma$. It is evident that if very little information is revealed from today
to the election day, then the current support rate will not change
significantly so that the likelihood of winning the election is considerably
higher (lower) than the current poll if it is higher (lower) than $50\%$. What
this means is that in a two-candidate election, if a candidate is losing then
it is in their interest to release as much information as possible to generate
more volatility; whereas if the candidate is winning then it is in their
interest to conceal as much information as possible. This situation may be
empirically familiar to some election strategists. When there are three or
more candidates, however, there are some nontrivial situations that can arise,
as we shall discuss below.

## Three-candidate electoral competition

In a reduced-form approach, when there are three candidates we let the random
variable $X$ take the values $\\{x_{k}\\}$ $(k=1,2,3)$, labelling the these
candidates. Thus the event that $X=x_{3}$, say, means that the third candidate
is being selected as the ‘right’ choice, if the voters were to have access to
the information $\\{\xi_{t}\\}$ eternally. That is, in this scenario, which
occurs with probability $p_{3}$, the sample path of $\pi_{3t}$ will be such
that we have $\pi_{3t}\to 1$ as $t\to\infty$. Of course, the election will
take place earlier, so the vote share for the third candidate on the election
day will instead be $\pi_{3T}$, which may or may not be larger than the
support rates of the other two candidates. Hence even in the event in which
$X=x_{3}$, in general this has little to do with the likelihood of the third
candidate winning the election, unless the value of $\sigma$ is unusually
large.

As a matter of interpretation, to further clarify the meaning of the random
variable $X$, we remark that our model setup is such that the value of $X$
will only be revealed base on the information process $\\{\xi_{t}\\}$ over an
infinite time horizon, because

$\displaystyle\lim_{t\to\infty}\frac{\xi_{t}}{\sigma t}=X$ (23)

in a distributional sense. In other words, if hypothetically the electorates
were to live forever, then they will all learn which of the candidates they
should all be voting for. However, the election will take place considerably
sooner at time $T$, at which point there has not been sufficient information
generated by the electorates according to the information process (unless
$\sigma$ is very large). Thus the voter variability remains high, and the
election outcome remains uncertain. It follows that our model is only of use
until the election day, at which point it can be discarded, but this is all we
need in order to interpolate the statistics between today and the election
day.

Differently stated, in a typical election cycle the information-providing
process $\\{\xi_{t}\\}$ ceases to exist after the election day (or,
equivalently, the information flow rate parameter $\sigma$ goes to zero,
leaving behind nothing but noise), so the voters will never learn which
candidate would have been the ‘right’ candidate. Hence in reality none of the
a posteriori probabilities will actually converge to unity (except perhaps in
certain extreme circumstances). This is because no one will have access to
sufficient information to allow them to appropriately assign probabilities on
counterfactual events: What would have happened if that person were elected
rather than this person. Indeed, even in an event whereby a candidate who lost
the election remains engaged towards a subsequent election, it is not always
the case that much information about that candidate reaches the voters, as
exemplified by an 18 August 1996 Newsweek article titled “Forgotten, but not
gone” about the then presidential candidate Ross Perot.

In the case of a three-candidate race, there are six possible ordering
scenarios for the vote share. Hence in order to determine the probability that
the third candidate, say, wins the election, we need to work out the
probabilities for individual ordering. This follows because the current
probability for the third candidate to win the election is given by
${\mathbb{P}}(\pi_{3T}>\pi_{2T}>\pi_{1T})+{\mathbb{P}}(\pi_{3T}>\pi_{1T}>\pi_{2T})$;
and similarly for the other candidates to win the election. To this end we
note from (15) that for any $j\neq k$ the event $\pi_{kT}>\pi_{jT}$ holds true
if and only if

$\displaystyle p_{k}\exp\left(\sigma
x_{k}\,\xi_{T}-\mbox{$\textstyle\frac{1}{2}$}\sigma^{2}x_{k}^{2}T\right)>p_{j}\exp\left(\sigma
x_{j}\,\xi_{T}-\mbox{$\textstyle\frac{1}{2}$}\sigma^{2}x_{j}^{2}T\right).$
(24)

This condition can easily be solved for a condition on $\xi_{T}$, provided
that the ordering of the numbers $\\{x_{i}\\}$ are given. Without loss of
generality we may assume that $x_{3}>x_{2}>x_{1}$. Then from (24) it follows
that $\pi_{kT}>\pi_{jT}$ holds if and only if

$\displaystyle\xi_{T}>\frac{\log(p_{j}/p_{k})+\frac{1}{2}\sigma^{2}(x_{k}^{2}-x_{j}^{2})T}{\sigma(x_{k}-x_{j})},$
(25)

provided that $x_{k}>x_{j}$. Otherwise, the inequality is reversed.

To proceed, let us introduce the notation

$\displaystyle
z_{kj}=\frac{\log(p_{j}/p_{k})+\frac{1}{2}\sigma^{2}(x_{k}^{2}-x_{j}^{2})T}{\sigma(x_{k}-x_{j})}.$
(26)

Then a short calculation shows, on account of the symmetry property
$z_{kj}=z_{jk}$, that

$\displaystyle\pi_{3T}<\pi_{2T}<\pi_{1T}\quad$
$\displaystyle\Leftrightarrow\quad\xi_{T}<\min\\{z_{12},z_{23}\\},$
$\displaystyle\pi_{2T}<\pi_{3T}<\pi_{1T}\quad$
$\displaystyle\Leftrightarrow\quad
z_{23}<\xi_{T}<z_{31}\qquad(\text{if}~{}z_{23}<z_{31}),$
$\displaystyle\pi_{3T}<\pi_{1T}<\pi_{2T}\quad$
$\displaystyle\Leftrightarrow\quad
z_{12}<\xi_{T}<z_{31}\qquad(\text{if}~{}z_{12}<z_{31}),$
$\displaystyle\pi_{1T}<\pi_{3T}<\pi_{2T}\quad$
$\displaystyle\Leftrightarrow\quad
z_{31}<\xi_{T}<z_{23}\qquad(\text{if}~{}z_{31}<z_{23}),$
$\displaystyle\pi_{2T}<\pi_{1T}<\pi_{3T}\quad$
$\displaystyle\Leftrightarrow\quad
z_{31}<\xi_{T}<z_{12}\qquad(\text{if}~{}z_{31}<z_{12}),$
$\displaystyle\pi_{1T}<\pi_{2T}<\pi_{3T}\quad$
$\displaystyle\Leftrightarrow\quad\max\\{z_{12},z_{23}\\}<\xi_{T}.$

Note here that, for example, the event $\pi_{2T}<\pi_{3T}<\pi_{1T}$ cannot be
realised if $z_{23}>z_{31}$, and similarly for other three intermediate cases.
With these conditions at hand, let us note that the probability
${\mathbb{P}}(a<\xi_{T}<b)$ for any $a<b$ can be worked out by changing the
probability measure. Specifically, we use the common denominator

$\displaystyle\Phi_{t}=\sum_{j=1}^{3}p_{j}\exp\left(\sigma
x_{j}\xi_{t}-\mbox{$\textstyle\frac{1}{2}$}\sigma^{2}x_{j}^{2}t\right)$ (27)

to effect a measure change ${\mathbb{P}}\to{\mathbb{Q}}$ so that under
${\mathbb{Q}}$ the information process $\\{\xi_{t}\\}$ is a standard Brownian
motion. Then we have

$\displaystyle{\mathbb{P}}\left(a<\xi_{T}<b\right)$ $\displaystyle=$
$\displaystyle{\mathbb{E}}^{{\mathbb{Q}}}\left[\Phi_{T}{\mathds{1}}\\{a<\xi_{T}<b\\}\right]$
(28) $\displaystyle=$
$\displaystyle{\mathbb{E}}^{{\mathbb{Q}}}\left[\sum_{j=1}^{3}p_{j}\exp\left(\sigma
x_{j}\xi_{T}-\mbox{$\textstyle\frac{1}{2}$}\sigma^{2}x_{j}^{2}T\right){\mathds{1}}\\{a<\xi_{T}<b\\}\right]$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{3}p_{j}\left[N\left(\frac{b-\sigma
x_{j}T}{\sqrt[]{T}}\right)-N\left(\frac{a-\sigma
x_{j}T}{\sqrt[]{T}}\right)\right].$

Figure 2:  Winning probabilities as functions of $(p_{1},p_{2})$. The
probabilities of winning a future election to take place in one year time
($T=1$), when the information flow rate is set at $\sigma=1$, are plotted here
for the parameter choice $(x_{1},x_{2},x_{3})=(1,2,3)$. The forms of the
probabilities for candidate 1 (left panel, in red) and candidate 3 (right
panel, in blue) are entirely symmetric. However, the behaviour of the
probability for candidate 2 (centre panel, in purple) is slightly different in
that there is a region in the parameter space $(p_{1},p_{2})$ of the current
support rates for which the probability of candidate 2 winning is identically
zero. We will have more to say about this in the next section.

With these results at hand we are able to work out the probabilities for the
six possible outcomes. For concreteness let us write them down explicitly
here. They are

$\displaystyle{\mathbb{P}}(\pi_{3T}<\pi_{2T}<\pi_{1T})=\sum_{j=1}^{3}p_{j}\left[N\left(\frac{z_{12}-\sigma
x_{j}T}{\sqrt[]{T}}\right){\mathds{1}}\\{z_{12}<z_{23}\\}+N\left(\frac{z_{23}-\sigma
x_{j}T}{\sqrt[]{T}}\right){\mathds{1}}\\{z_{12}\geq z_{23}\\}\right]$ (29)

and

$\displaystyle{\mathbb{P}}(\pi_{2T}<\pi_{3T}<\pi_{1T})=\sum_{j=1}^{3}p_{j}\left[N\left(\frac{z_{31}-\sigma
x_{j}T}{\sqrt[]{T}}\right)-N\left(\frac{z_{23}-\sigma
x_{j}T}{\sqrt[]{T}}\right)\right]{\mathds{1}}\\{z_{23}<z_{31}\\},$ (30)

together determine the probability of the first candidate winning,

$\displaystyle{\mathbb{P}}(\pi_{3T}<\pi_{1T}<\pi_{2T})=\sum_{j=1}^{3}p_{j}\left[N\left(\frac{z_{31}-\sigma
x_{j}T}{\sqrt[]{T}}\right)-N\left(\frac{z_{12}-\sigma
x_{j}T}{\sqrt[]{T}}\right)\right]{\mathds{1}}\\{z_{12}<z_{31}\\}$ (31)

and

$\displaystyle{\mathbb{P}}(\pi_{1T}<\pi_{3T}<\pi_{2T})=\sum_{j=1}^{3}p_{j}\left[N\left(\frac{z_{23}-\sigma
x_{j}T}{\sqrt[]{T}}\right)-N\left(\frac{z_{31}-\sigma
x_{j}T}{\sqrt[]{T}}\right)\right]{\mathds{1}}\\{z_{31}<z_{23}\\},$ (32)

together determine the probability of the second candidate winning, and

$\displaystyle{\mathbb{P}}(\pi_{2T}<\pi_{1T}<\pi_{3T})=\sum_{j=1}^{3}p_{j}\left[N\left(\frac{z_{12}-\sigma
x_{j}T}{\sqrt[]{T}}\right)-N\left(\frac{z_{31}-\sigma
x_{j}T}{\sqrt[]{T}}\right)\right]{\mathds{1}}\\{z_{31}<z_{12}\\}$ (33)

and

$\displaystyle{\mathbb{P}}(\pi_{1T}<\pi_{2T}<\pi_{3T})=\sum_{j=1}^{3}p_{i}\left[N\left(-\frac{z_{23}-\sigma
x_{j}T}{\sqrt[]{T}}\right){\mathds{1}}\\{z_{12}<z_{23}\\}+N\left(-\frac{z_{12}-\sigma
x_{j}T}{\sqrt[]{T}}\right){\mathds{1}}\\{z_{12}\geq z_{23}\\}\right],$ (34)

together determine the probability of the third candidate winning. In this way
we obtain an explicit formula for each of the candidate winning the future
election, as functions of (i) the current support rates $\\{p_{j}\\}$ for the
candidates, (ii) the time $T$ left to the election, (iii) the rate $\sigma$ at
which information is revealed to the electorate, and (iv) the choice of the
candidate labels $\\{x_{j}\\}$.

Figure 3:  Dynamical behaviours of the poll statistics $\\{\pi_{it}\\}$ and
the corresponding winning probabilities. Sample paths for the support rates
$(\pi_{1t},\pi_{2t},\pi_{3t})$ for the three candidates are shown on the left
panels. The corresponding winning probability processes for each candidate are
shown on the right panels. The parameters are chosen as
$(x_{1},x_{2},x_{3})=(1,2,3)$ for the values of the random variable $X$,
$(p_{1},p_{2},p_{3})=(0.38,0.26,0.36)$ for the current support level so that
the electorates are slightly polarised, and $T=1$ year for the time left to
the election day. The top two panels correspond to the value $\sigma=0.25$ for
the information flow rate. In this case, the probability for the second
candidate to win the election is identically zero. For a comparison, the
corresponding results for the choice $\sigma=1$ are plotted in the bottom two
panels, in which the second candidate narrowly secures a victory.

In Figure 2 we sketch the behaviours of the winning probabilities for the
three candidates as functions of the current support rates $(p_{1},p_{2})$ for
the first two candidates. One distinguishing feature here, as compared to the
results for two candidate scenario, is the dependence on the information flow
rate $\sigma$. In the two candidate case, for a given level $(p,1-p)$ of
current support, the winning probabilities are either increasing or decreasing
in $\sigma$. That is, if the candidate is leading the poll, then it is best
not to reveal information, and conversely for the other candidate. In the
three candidate case, depending on the level $(p_{1},p_{2},1-p_{1}-p_{2})$ of
current support, the winning probabilities can lack monotonicity. That is,
there are values of $(p_{1},p_{2},1-p_{1}-p_{2})$ for which a candidate will
benefit from, say, increasing the information flow rate slightly to enhance
the probability of winning the future election, but if it is increased too
much, then this will result in decreasing the probability again. It follows
that in a three candidate race, the optimal strategy of controlling
information can be quite nontrivial for certain values of the current support
rates $(p_{1},p_{2},1-p_{1}-p_{2})$.

Although we have worked out here the initial (time $t=0$) probability of a
given candidate winning the election, it is straightforward to work out the
corresponding conditional probabilities, such as
${\mathbb{P}}(\pi_{1T}<\pi_{2T}<\pi_{3T}|\xi_{t})$ and so on, that depend on
how information has been unravelled up to time $t$. Then we are able to
simulate not just the support rates $\\{\pi_{it}\\}$ but also the realised
winning probabilities, as illustrated in Figure 3.

## Representing political spectrum

It is important to emphasise the fact that while in a two-candidate electoral
competition the choice of the labelling numbers $\\{x_{0},x_{1}\\}$ can be
made arbitrarily, this is no longer the case if there are more than two
candidates. The reason can be explained as follows.

We note first that from a signal detection perspective, whether the signal is
an unknown random variable $X$ or a (known) constant addition of an unknown
$X$, there is no difference in the inference. This follows from the fact that
the filtration generated by $\sigma Xt+B_{t}$ is identical to that generated
by $\sigma(X+c)t+B_{t}$, where $c$ is a known constant. Hence the only
quantities of relevance in the choice of $X$ are the gaps
$\omega_{ij}=x_{j}-x_{i}$. Now in the binary case there is only one such gap
$\omega=x_{1}-x_{0}$, but scaling the gap according to
$\omega\to\lambda\omega$ is entirely equivalent to scaling the information
flow rate $\sigma\to\lambda\sigma$. Putting it differently, the scaling
$\omega\to\lambda\omega$ can be compensated by the scaling
$\sigma\to\lambda^{-1}\sigma$ so that $\omega$ can be chosen arbitrarily by
regarding $\sigma$ as the variable parameter.

In the case of an election with three candidates, there are three gaps
$\omega_{12}$, $\omega_{23}$, and $\omega_{31}$, with one constraint
$\omega_{12}+\omega_{23}+\omega_{31}=0$. Hence there are two independent
scaling parameters, which cannot be simultaneously absorbed by scaling
$\sigma$. It follows that the probability of a given candidate winning the
election is dependent on the choice of the gaps $\\{\omega_{ij}\\}$.
Alternatively stated, there is a natural ordering (i.e. spectrum) encoded in
the random variable $X$ representing the candidates. In particular, the three
candidates cannot be placed on an equal footing, for, while it is possible to
set $\omega_{12}=\omega_{23}$, it is not possible to set
$\omega_{12}=\omega_{23}=\omega_{31}$.

We can take advantage of this feature of the model by observing that there is
a long established notion of a ‘political spectrum’ in an electoral process,
and we can encode this information naturally in the choices of the gaps
$\\{\omega_{ij}\\}$. Thus, for example, if candidate $1$ is on the left,
candidate $2$ is moderately on the right, and candidate $3$ is further on the
right, then we can let $\omega_{12}>\omega_{23}$ to capture this composition;
and similarly for other situations. Realising this, we see that the choice of
the gaps $\\{\omega_{ij}\\}$ is not up to the modeller, but it is up to the
candidates in terms of where they place themselves in the political spectrum.

With this in mind, we find that there are circumstances in which taking the
political centre ground leads to a disadvantage. This follows from the
observation that the probability for candidate $2$ to win the election is
identically zero if $z_{12}>z_{31}>z_{23}$, while such a constraint does not
exist for the candidate to the left or to the right. Note however that this
does not mean that taking the centre ground is always disadvantageous — it
merely shows that in certain situations, having popular competitors to both
the left and the right can be fatal. In particular, such a trap for candidate
$2$ can be created among a politically polarised set of voters so that
$p_{2}<p_{1},p_{3}$ holds while at the same time $p_{1}\sim p_{3}$.

Figure 4:  Winning probabilities as functions of $\sigma$. The probability of
winning an election in one year time ($T=1$), as a function of the information
flow rate $\sigma$, is shown for the three candidates, labelled according to
$x_{1}=1$, $x_{2}=2$, and $x_{3}=3$. The current poll statistics are taken to
be $p_{1}=0.38$ for the first candidate on the left (red), $p_{2}=0.26$ for
the second candidate taking the centre ground (purple), and $p_{3}=0.36$ for
the third candidate on the right (blue).

More specifically, a calculation shows that the condition
$z_{12}>z_{31}>z_{23}$ can be translated into a bound on the information flow
rate $\sigma$ as follows:

$\displaystyle\sigma^{2}<\frac{2}{\omega_{12}\omega_{23}\omega_{31}T}\,\min\left\\{\omega_{12}\log\frac{p_{1}}{p_{3}}-\omega_{13}\log\frac{p_{1}}{p_{2}}\,,\,\omega_{32}\log\frac{p_{1}}{p_{3}}-\omega_{13}\log\frac{p_{3}}{p_{2}}\right\\}.$
(35)

That is, provided that the inequality (35) holds, the probability of candidate
$2$ winning the election is identically zero. This situation is illustrated in
Figure 4. A closer inspection shows that if $p_{2}$ is small, then the bound
on $\sigma$ can be large. It follows that among a politically polarised
electorate, the only way in which a candidate holding the centre ground has
any chance of winning the election is to ensure that a lot of reliable
information is revealed so as to increase the volatility of the poll
statistics $\\{\pi_{it}\\}$.

We now ask a related question on positioning within a political spectrum. For
this purpose we shall take the convention that if $x_{j}<x_{k}$ then candidate
$j$ is placed politically to the left of candidate $k$. The question that we
ask here more specifically is whether the winning probability can be enhanced
by leaning further to the left or to the right. The answer will be dependent
on the various parameter values, but let us consider the politically polarised
case as shown in Figure 4. In this case, if we keep the value of $x_{2}$
unchanged but increase $x_{3}$ and simultaneously decrease $x_{1}$, then we
find that the probability of winning the election for the candidate on the
left decreases for all values of $\sigma$. However, for the candidate on the
right, the situation is a little more complex. When the election process is
overshadowed by noise (i.e. small $\sigma$ values), the winning probability of
the candidate on the right can be enhanced considerably by leaning further to
the right; whereas if the election process is not dominated by noise, then the
winning probability decreases by leaning further to the right. Hence in this
scenario there is no advantage for the candidate on the left to lean further
to the left, but the candidate on the right has the advantage of turning more
extreme, provided that the noise level is sufficiently high. If however the
candidate misjudges the level of noise, then this strategy will backfire. Some
examples illustrating this feature are illustrated in Figure 5, which shows,
for example, that if the candidate on the right leans further to the right,
while the candidate on the left leans slightly to the right, then there is a
significant benefit to the candidate on the left, provided that the level of
noise is not overwhelming.

Figure 5:  Gains in winning probabilities as functions of $\sigma$. If the
political positioning $(x_{1},x_{2},x_{3})=(1,2,3)$ considered in Figure 4 is
shifted, how would that affect the winning probabilities? Here, the difference
of the resulting winning probabilities to the one in Figure 4 is shown for
three different cases: $(x_{1},x_{2},x_{3})=(0.1,2,3.9)$ (left panel),
$(x_{1},x_{2},x_{3})=(1,2,3.9)$ (central panel), and
$(x_{1},x_{2},x_{3})=(1.5,2,3.9)$ (right panel). Other parameters are kept
unchanged ($p_{1}=0.38$, $p_{2}=0.26$, $p_{3}=0.36$, and $T=1$). If the
difference is negative, then clearly the shift is disadvantageous. The result
shows that among a polarised electorate, if the candidate on the left of the
political spectrum leans further to the left and the candidate on the right
leans further to the right, then this is generally disadvantageous for both.
However, if the competition is dominated by noise (small $\sigma$ values),
then the candidate on the right can benefit by leaning further to the right.

## Calibration and implementation

As regards the practical implementation of the model, let us examine the model
parameters that can be calibrated, and those that can be controlled. To this
end, the current support rates $\\{p_{j}\\}$ can be fixed from today’s poll,
whereas $T$ is fixed by the date of the election. The variables $\\{x_{j}\\}$
that the random variable $X$ can take can then be fixed, up to an overall
scale that can be absorbed into $\sigma$, by the relative positioning of the
candidates within the political spectrum. A candidate, in particular, will
have the choice for their own value of the number $x_{k}$, but will not have
control over the positioning of other candidates.

Finally, as for the value of the information flow rate $\sigma$, its value
cannot be controlled by any individual, but its current value can be estimated
by studying the time series for the poll statistics. This follows from the
fact that the volatility of the support rates $\\{\pi_{it}\\}$ is given by
$\sigma$. Hence a historic estimate can be used to fix the value of $\sigma$.
Alternatively, from the odds of election betting it is possible to work out
the implied volatility, which can be used to fix the value of $\sigma$.

Having fixed all the parameters, the model can be used to interpolate the
statistics from today to the election day. If the resulting probability of a
given candidate winning the election appears undesirable, then that candidate
will have a small room to manoeuvre so as to increase the success probability.
First, the candidate can adjust their positioning within the political
spectrum. Of course other candidates may also adjust their positions as a
consequence of this, but if everything else being the same, our formula shows
in which way the candidate should position themselves within the political
spectrum.

The other variable that a candidate can adjust is the information flow rate.
While no individual can fix the value of $\sigma$, the result of (11) shows in
which way an individual flow rate $\sigma_{k}$ affects the overall value of
$\sigma$. In particular, $\sigma$ is a monotonic function of each
$\sigma_{k}$, so increasing the value of any one of $\sigma_{k}$ will increase
the overall value of $\sigma$, and similarly decreasing the value of
$\sigma_{k}$ will decrease $\sigma$. Again, other candidates, or other
information source such as the press, may adjust their information revelation
rate as a result to counterbalance the impact. This, however, is just a fact
about a democratic process – no one candidate can control its outcome.
Nevertheless, our framework offers an immediately implementable procedure for
guiding the candidates to identify which informational strategy will increase
their chances of success, if everything else remained the same.

## Discussion

We have examined in some detail the probability of a candidate winning a
future election and how it is affected by control variables such as the level
of information revelation, or noise, and the positioning of the candidates
within the political spectrum, in the case of an electoral competition
involving three candidates. It should be evident that a closed form expression
for a given candidate winning a future election can be obtained when there are
more than three candidates. For example, if there are four candidates, then
there are 24 different ways in which the support rates for the candidates on
the election day can be ordered, e.g., $\pi_{2T}<\pi_{3T}<\pi_{1T}<\pi_{4T}$
and so on. Each one of these will give rise to a bound on the random variable
$\xi_{T}$ in the form of $\xi_{T}\in{\cal D}_{2314}$ for some domain ${\cal
D}_{2314}$ on the real line. (The analogue of these domains in the case of a
three-candidate electoral competition would be ${\cal
D}_{231}=[z_{23},z_{31}]$, and so on.) The probability of this event being
realised is therefore given by

$\displaystyle{\mathbb{P}}\left(\pi_{2T}<\pi_{3T}<\pi_{1T}<\pi_{4T}\right)={\mathbb{E}}^{{\mathbb{Q}}}\left[\sum_{j=1}^{4}p_{j}\exp\left(\sigma
x_{j}\xi_{T}-\mbox{$\textstyle\frac{1}{2}$}\sigma^{2}x_{j}^{2}T\right){\mathds{1}}\\{\xi_{T}\in{\cal
D}_{2314}\\}\right],$ (36)

but because $\xi_{T}$ under ${\mathbb{Q}}$ is Gaussian with mean zero and
variance $T$, this expectation can easily be worked out. By repeating the
procedure for the five other domains ${\cal D}_{2134}$, ${\cal D}_{3124}$,
${\cal D}_{3214}$, ${\cal D}_{1234}$, and ${\cal D}_{1324}$, and adding the
results, we obtain the probability of the fourth candidate winning the
election. Evidently, an analogous calculation can be performed for each of the
other candidates to work out their success probabilities.

One of the nontrivial features that emerges when the number of candidates is
greater than two is that there is a disadvantage for candidates positioning in
the middle of the political spectrum, in a situation where the voters do not
have strong preferences on centre grounds. An analogous property is seen also
in the structural approach. The mathematical reason underlying this feature is
as follows. If we label the $N$ candidates such that
$x_{1}<x_{2}<\cdots<x_{N}$, then for each fixed $T$ and $\sigma$ we find that
$\pi_{1T}(\xi_{T})$, viewed as a function of $\xi_{T}$, is monotonically
decreasing in $\xi_{T}$ without bound in the range $[0,1]$ and
$\pi_{NT}(\xi_{T})$ is monotonically increasing in $\xi_{T}$ without bound in
the range $[0,1]$. However, for any $k\neq 1,N$, the function
$\pi_{kT}(\xi_{T})$, which gives the support rate for the $k$th candidate on
the election day, is unimodal and has a maximum value at $\xi_{T}=\xi_{k}^{*}$
that is strictly less than one, where $\xi_{k}^{*}$ is the unique solution to
the equation

$\displaystyle\sum_{j=1}^{N}(x_{j}-x_{k})p_{j}\exp\left(\sigma
x_{j}\xi_{k}^{*}-\mbox{$\textstyle\frac{1}{2}$}\sigma^{2}x_{j}^{2}T\right)=0.$
(37)

If at least one of the variables $p_{k}$, $\sigma$, or $T$ is large, then the
upper bound on $\pi_{kT}$ will be close to one, so there is little concern for
the candidate, but otherwise, the upper bound can be smaller than $1/N$. In
the latter case, whatever information is to be circulated, the probability of
the $k$th candidate winning a future election is identically zero. When there
are many candidates, the threshold value $1/N$ is small, but if there are only
three or four candidates then this effect is highly nontrivial and should not
be ignored. Indeed, as we have seen in the case of a three candidate race in
Figure 4, there is a wide range of values for the information flow rate
$\sigma$ for which there is no chance for the second candidate to win the
election. In Figure 6 we show the maximum attainable support rates for each
candidate in the case of an election with five candidates. The central panel
in Figure 6 shows that while the initial support rates $p_{2}$ and $p_{4}$ for
the centre left and centre right candidates are very close, the existence of a
far-right candidate with negligible current support level implies that the
maximum attainable value of $\pi_{4T}$ is close to $1$, whereas the existence
of a far-left candidate with a moderate current support level implies that the
maximum attainable value of $\pi_{2T}$ is considerably lower than that of
$\pi_{4T}$.

Figure 6:  Maximum attainable support rates $\pi_{kT}(\xi_{k}^{*})$ for the
five candidates. For a range of values for the information flow rate $\sigma$,
the maximum values of $\\{\pi_{kT}\\}$ on the election day are shown by the
dots, interpolated by lines to make the comparison easy. The current support
rates $\\{p_{k}\\}$ are given by the bottom values (in purple). In the left
panel the five candidates are all assumed to have an equal support rate of
20%, whereas they are chosen at random in the central and right panels. The
results shows how the maximum attainable support rates for different
candidates vary rather dramatically, depending on the existence of candidates
having different political leanings and their associated current support
rates. The parameters are chosen to be
$(x_{1},x_{2},x_{3},x_{4},x_{5})=(1,2,3,4,5)$ for the positioning of the
candidates and $T=1$ year for time left to the election.

Another nontrivial feature that emerges when there are more than two
candidates is the subtle dependence of the winning probability on the
positioning of the candidates within the political spectrum. We have merely
uncovered for illustration a small number of features shown in Figure 5 and
Figure 6, but a detailed sensitivity study of the winning probabilities on the
spectrum $\\{x_{j}\\}$ is entirely feasible on account of the fact that we
have closed-form expressions for these probabilities.

We conclude by remarking how our model might be extended. Throughout the paper
we have assumed that the information revelation rate $\sigma$ is constant, but
in reality this is never the case. The analysis presented here nevertheless
extends naturally to the case in which $\\{\sigma_{t}\\}$ has a deterministic
time dependency. Specifically, in this case the candidate support rates take
the form

$\displaystyle\pi_{it}=\frac{p_{i}\exp\left(x_{i}\int_{0}^{t}\sigma_{s}\mbox{$\rm
d$}\xi_{s}-\frac{1}{2}x_{i}^{2}\int_{0}^{t}\sigma_{s}^{2}\mbox{$\rm
d$}s\right)}{\sum_{j}p_{j}\exp\left(x_{j}\int_{0}^{t}\sigma_{s}\mbox{$\rm
d$}\xi_{s}-\frac{1}{2}x_{j}^{2}\int_{0}^{t}\sigma_{s}^{2}\mbox{$\rm
d$}s\right)}.$ (38)

Then the relevant random variable $\xi_{T}$ is replaced with
$\int_{0}^{T}\sigma_{s}\mbox{$\rm d$}\xi_{s}$, which remains Gaussian under
${\mathbb{Q}}$, so the various probabilities can still be worked out while
taking into account a time-varying informational strategy $\\{\sigma_{t}\\}$.
In this way, impacts of a range of time-dependent informational strategies can
be investigated. Of course, the information process $\\{\xi_{t}\\}$ is meant
to represent the aggregate of the various information sources, and so is the
variable $\sigma$ (see Brody4  for how the aggregated information flow rate is
related to that of the individual information source); whether it is time
dependent or not. Thus no one candidate has the access to control the overall
value of $\sigma$. Nevertheless, the idea is that each candidate can influence
the value of $\sigma$, which in turn will modify the likelihoods of the
candidates winning the election.

Acknowledgements. The authors thank Bernhard Meister for stimulating
discussion. DCB acknowledges support from the EPSRC (EP/X019926) and the John
Templeton Foundation (62210). The opinions expressed in this publication are
those of the authors and do not necessarily reflect the views of the John
Templeton Foundation. TY is supported by JSPS KAKENHI (22K13965).

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