# Condorcet Markets

Stéphane Airiau  Nicholas Kees Dupuis  Davide Grossi
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The paper studies information markets about single events from an epistemic
social choice perspective. Within the classical Condorcet error model for
collective binary decisions, we establish equivalence results between
elections and markets, showing that the alternative that would be selected by
weighed majority voting (under specific weighting schemes) corresponds to the
alternative with highest price in the equilibrium of the market (under
specific assumptions on the market type). This makes it possible to implement
specific weighted majority elections, which are known to have superior truth-
tracking performance, through information markets and, crucially, without
needing to elicit voters’ competences.

##  1 Introduction

Information markets (also known as prediction markets) [1, 13, 5, 14] are
markets of all-or-nothing contracts (so-called Arrow securities) that pay one
unit of currency if a designated event occurs and nothing otherwise. Under the
view, inspired by [12], that markets are good aggregators of the information
dispersed among traders, proponents of information markets have argued that
equilibrium prices are accurate estimates of the probability of the designated
event. Much research—theoretical and empirical—has probed this interpretation
of prices in information markets, finding that equilibrium prices successfully
track the traders’ average belief about the event, under several models of
trader’s utilities [16, 18].

In this paper we address a closely related, but different question: if we are
to take a decision based on the information we extract from the equilibrium
price, how accurate would such a decision be? In other words, rather than
relating equilibrium prices to belief aggregation, we relate them directly to
the quality of the decision they would support. We frame the above question
within the standard binary choice framework of epistemic social choice,
stemming from the Condorcet jury theorem tradition [6, 11, 19] and the
maximum-likelihood estimation approach to voting [11, 7, 17, 10].

beliefsvotespricedecision

binarize

binarize

marketelection Figure 1: Elections and information markets commute.

##### Contribution

To the best of our knowledge, the above one is a novel perspective on
information markets. In particular, counter to the common assumption on
traders’ beliefs being subjective, we study information markets when traders’
beliefs are obtained by Bayesian update from a private independent signal with
known (to the trader) accuracy, just like in the classic jury theorems
setting. In other words, we study ‘jurors’ as if they were ‘traders’ who,
instead of relaying their vote to a central mechanism, trade in an information
market. In taking this perspective, we ask the above question by comparing the
decisions that would be taken based on the equilibrium price of an information
market, with the decisions that would be taken by specific weighted majority
elections, whose truth-tracking behavior is already well-understood [11].
Specifically, we aim at identifying correspondences between classes of markets
and of weighted majority elections which are equivalent from a decision-making
point of view. That is, they are such that the weighted majority when agents
vote according to the event they believe more likely, always coincides with
the event whose price is highest in equilibrium when agents trade in Arrow
securities according to their beliefs. Figure 1 depicts such relationship via
a commutative diagram. Such results open up the possibility of implementing
weighted majority voting with proven truth-tracking performance without
needing to know jurors’ competences, which may be hard to truthfully elicit or
estimate [3].

##### Paper outline

Section 2 introduces the standard binary truth-tracking framework and presents
our model of information markets. Section 3 presents results on equilibrium
prices in two of the three types of markets we consider (Naive and Kelly
markets) and Section 4 proves ‘Figure 1-type’ results for those markets.
Section 5 then shows how such results could be lifted even to the case of
majority voting where jurors are weighted perfectly according to their
competence. Section 6 outlines future research directions. Two examples
illustrating our framework and analysis are provided in Appendix A. All proofs
not directly included in the main text are available in Appendix B and C.

##  2 Preliminaries

###  2.1 Collective truth-tracking

We are concerned with a finite set of agents $N=\left\\{1,\ldots,n\right\\}$
who have to decide collectively on the correct state of the world
$x\in\\{A,B\\}$. A prior probability $P(x=A)=\pi=0.5$ is given, that the
correct state is $A$. Each agent $i$ observes a private independent signal
$y_{i}\in\left\\{A,B\right\\}$ that has quality $q_{i}\in(0.5,1)$, that is
$q_{i}=P(y=A\mid x=A)=P(y=B\mid x=B)$. Each $q_{i}$ represents the competence
or accuracy of $i$. We call each vector $\mathbf{q}=(q_{1},\ldots,q_{n})$ of
individual accuracies an accuracy or competence profile of the group. Having
observed her private signal, each agent then forms a posterior belief
$b_{i}=P(x=A\mid y=A)$ about state $x=A$ by Bayes rule. Observe that, by the
conditions we impose on competences and prior, if $b_{i}>0.5$ then, by Bayes
rule, $b_{i}=q_{i}$ (the belief in $A$ equals $q_{i}$), and $b_{i}=1-q_{i}$
(the belief in $A$ equals $1-q_{i}$) otherwise. This gives us, for all $i\in
N$:

$b_{i}=\mathbbm{1}(b_{i}>0.5)\cdot(2q_{i}-1)+(1-q_{i})$ (1)

where $\mathbbm{1}$ denotes the indicator function. Individual beliefs are
then collected in a belief profile
$\mathbf{b}=(b_{1},\ldots,b_{n})\in[0,1]^{n}$. Given an accuracy profile
$\mathbf{q}$, the set of possible belief profiles is denoted
$\mathcal{B}_{\mathbf{q}}=\\{\mathbf{b}\in[0,1]^{n}\mid
P(\mathbf{b}\mid\mathbf{q})>0\\}$. Observe that the size of this set equals
$2^{n}$: the number of all signal realizations.

Based on a profile $\mathbf{b}$ of individual beliefs, the group then takes a
decision by mapping the profile to $A$ or to $B$. In this process of
aggregation, agents may have different weights. These weights are collected in
a weight profile $\mathbf{w}=(w_{1},\ldots,w_{n})\in[0,1]^{n}$. We refer to
$\mathbf{1}=(1,\ldots,1)$ as the egalitarian weight profile in which all
agents have equal weight. So, assuming a given weight profile $\mathbf{w}$, we
call aggregator any function

$\mathcal{A}^{\mathbf{w}}:[0,1]^{n}\to 2^{\\{1,0\\}}\backslash\emptyset$ (2)

mapping belief profiles to alternatives, where $\left\\{1\right\\}$ denotes
$A$; $\left\\{0\right\\}$ denotes $B$; and $\left\\{1,0\right\\}$ denotes a
tie.

##### Types of aggregators

We will study two classes of mechanisms to implement aggregators. In the first
class, agents cast binary ballots based on their beliefs and these ballots are
submitted to a voting mechanism. The winning alternative is the outcome of the
aggregation process. In the second class, agents’ trade in special types of
securities, based on their beliefs. The equilibrium price of this securities
market is then used as a proxy for the group’s belief in the probability of
state $A$. In this case, it is the alternative favored by this collective
belief to be the outcome of the aggregation process.

Let us make the above notions more precise. First of all, a belief $b\in[0,1]$
is translated into binary opinions, or votes, for $A$ or $B$ via the following
binarization function:

$\displaystyle\widehat{b}$ $\displaystyle=\begin{cases}\\{1\\}&\mbox{if}\
b>0.5\\\ \\{0\\}&\mbox{if}\ b<0.5\\\ \\{0,1\\}&\mbox{otherwise}\end{cases}$
(3)

That is, agents are assumed to vote in accordance to their posterior belief
(this is sometimes referred to as sincere voting [2]). A binarized belief
profile $\widehat{\mathbf{b}}=(\widehat{b}_{1},\ldots,\widehat{b}_{n})$ is
therefore a binary vector and we will referred to such vectors also as voting
profiles and denote them by $\mathbf{v}=(v_{1},\ldots,v_{n})$.111As individual
beliefs cannot equal $0.5$, the reduction function always outputs a singleton
$\\{0\\}$ or $\\{1\\}$ on individual beliefs. This will not be the case,
however, for collective beliefs, which may be undecided ($\\{0,1\\}$).

Given a weight profile $\mathbf{w}$, a (belief) merger is a function
$F^{\mathbf{w}}:[0,1]^{n}\to[0,1]$ taking as input a belief profile and
outputting a group belief. A choice function is a function
$f^{\mathbf{w}}:\left\\{1,0\right\\}^{n}\to
2^{\left\\{0,1\right\\}}\backslash\emptyset$ taking as input a voting profile
and outputting a possibly tied choice between $1$, i.e., $A$, and $0$, i.e.,
$B$.

We will study aggregators of the type $f^{\mathbf{w}}\circ\widehat{\ }$
(voting) and $\widehat{\ }\circ F^{\mathbf{w}}$ (trading). A voting mechanism
is a choice function $f^{w}$ which, applied to a binarized belief profile
$\widehat{\mathbf{b}}$, yields a collective choice
$f^{\mathbf{w}}(\widehat{\mathbf{b}})$ (under the weight profile
$\mathbf{w}$). A market mechanism is a belief aggregation function $F$ that,
once applied to a belief profile $\mathbf{b}$, yields a collective belief
$F^{\mathbf{w}}(\mathbf{b})$ whose binarization
$\widehat{F^{\mathbf{w}}(\mathbf{b})}$ yields a collective choice (under the
weight profile $\mathbf{w}$).

##### Group accuracy

We will study aggregators from a truth-tracking perspective. The accuracy
$Q({\mathcal{A}^{\mathbf{w}}},\mathbf{q})$ of an aggregator
$\mathcal{A}^{\mathbf{w}}$ under the accuracy profile $\mathbf{q}$, is the
conditional probability that the outcome of the aggregator is $x$ if the state
of the world is $x$. The above describes an epistemic social choice setting
where the group is confronted with a maximum-likelihood estimation task in a
dichotomous choice situation (see [10]).

###  2.2 Voting and market mechanisms

We turn now to the description of the mechanisms we are concerned with.

####  2.2.1 Voting mechanisms

After observing their private signal, agents decide whether to vote for $A$ or
$B$ according to Equation (3). A weighted majority rule is then applied to
these votes to determine the group’s choice:

$\displaystyle M^{\mathbf{w}}(\mathbf{v})$
$\displaystyle=\begin{cases}\\{1\\}&\mbox{if}\ \sum_{i\in
N}w_{i}v_{i}>\frac{1}{2}\sum_{i\in N}w_{i}\\\ \\{0\\}&\mbox{if}\ \sum_{i\in
N}w_{i}v_{i}<\frac{1}{2}\sum_{i\in N}w_{i}\\\
\\{0,1\\}&\mbox{otherwise}\end{cases}$ (4)

We will be working in particular with three variants of Equation (4) defined
by three different weight profiles: the egalitarian weight profile
$\mathbf{1}$; the weight profile allocating to each agent $i$ a weight
proportional to $q_{i}-0.5$; the weight profile allocating to each agent $i$ a
weight proportional to $\log\frac{q_{i}}{1-q_{i}}$. The rule induced by the
egalitarian weight profile is the simple majority rule. We will see that the
second weight profile simulates decision-making according to the average
belief. The latter weight profile can be inferred from Bayes theorem and
induces the weighted majority rule which we refer to as perfect majority, and
which has been proven to optimize the truth-tracking ability of the group:

######  Theorem 1 ([11]).

For any accuracy profile $\mathbf{q}$, $Q(M^{\mathbf{w}},\mathbf{q})$ is
maximal if $\mathbf{w}$ is such that
$w_{i}\propto\ln\left(\frac{q_{i}}{1-q_{i}}\right)$ for all $i\in N$.

####  2.2.2 Markets

The market model we use is borrowed from [14, 5]. Two symmetric Arrow
securities are traded: securities of type $A$, which cost $p_{A}\in[0,1]$ and
pay $1$ unit of currency if $x=A$, and $0$ otherwise; securities of type $B$,
which cost $p_{B}\in[0,1]$ and pay $1$ unit if $x=B$ and $0$ otherwise. After
observing their private signal, agents decide what fraction of their endowment
to invest in which securities. We assume that all agents have the same
endowment consisting of $1$ unit of currency. We also assume that agents
invest in at most one of these securities, so if $s^{A}>0$ then $s^{B}=0$ and
vice versa. We call agents investing in $A$, $A$-traders and agents investing
in $B$, $B$-traders. In our setting, this assumption is without loss of
generality (see Proposition 1 in the appendix222We are indebted to Marcus
Pivato for bringing this issue to our attention.). When the true state of the
world is revealed, the market resolves and payouts based on the agents’
investments are distributed. We refer to tuples
$\mathbf{s}^{A}=\left(s^{A}_{1},\ldots,s^{A}_{n}\right)$ (respectively,
$\mathbf{s}^{B}=\left(s^{B}_{1},\ldots,s^{B}_{n}\right)$) as investment
profiles in $A$-securities (respectively, $B$-securities). We refer to a pair
$\mathbf{s}=(\mathbf{s}^{A},\mathbf{s}^{B})$ as an investment profile. We
proceed now to define the notions of price, utility and equilibrium.

##### Market mechanism

We assume that when the market opens all purchasing orders for each security
are executed by the the market operator, who therefore sells all requested
securities to agents when the market opens and pays the winning securities out
immediately when the market resolves, that is, when it is determined whether
$A$ or $B$ is the case. We further assume that the operator makes no profits
and incurs no losses. So, for every $A$-security sold at price $p^{A}$ a
$B$-security is sold at price $p^{B}=1-p^{A}$ and vice versa. In other words,
the price of the risk-less asset consisting of one of each security is
$p_{A}+p_{B}=1$. In this way the operator finances the payout of any bet by
the pay-in of the opposite bet.

Under the above assumptions, the market clears333A market is normally said to
clear when supply and demand match. In our model, supply and demand are
implicit in the following way: purchasing one $A$-security at price $p^{A}$
(i.e., reducing one’s endowment to $1-p^{A}$) is equivalent to selling one
$B$-security thereby $p^{B}=1-p^{A}$. when the total amount of individual
wealth invested in $A$-securities, divided by the price of $A$-securities
(demand of $A$-securities) matches the amount of individual wealth invested in
$B$-securities, divided by the price of $B$-securities (demand of
$B$-securities), that is:444It may be worth observing that by the above design
we are effectively treating the operator as an extra trader in the market, who
holds a risk-less asset consisting of $\frac{1}{p^{A}}\sum_{i\in N}s^{A}_{i}$
$A$-securities and $\frac{1}{1-p^{A}}\sum_{i\in N}s^{B}_{i}$ $B$-securities.
We are indebted to Marcus Pivato for this observation.

$\frac{1}{p^{A}}\sum_{i\in N}s^{A}_{i}=\frac{1}{1-p^{A}}\sum_{i\in
N}s^{B}_{i}.$ (5)

It follows that, given an investment profile $\mathbf{s}$, solving Equation
(5) for $p^{A}$, yields the clearing price $\frac{\sum_{i\in
N}s^{A}_{i}}{\sum_{i\in N}s^{A}_{i}+\sum_{i\in N}s^{B}_{i}}$ to which we refer
as $p(\mathbf{s})$. Note that the price is undefined if either $p_{A}=0$ or
$p_{A}=1$. We come back to this issue in Remark 2.

When the market resolves, each agent receives a different payout depending on
how much of each security she owns, how the market resolves, and how much of
her endowment is not invested. The payout, that is, the amount of wealth
obtained by an agent with a given strategy $s^{A}_{i}$ investing in $A$ under
a price $p^{A}$, is defined as follows:

$z(p^{A},s^{A}_{i})=\left\\{\begin{array}[]{ll}\frac{s^{A}_{i}}{p^{A}}&\quad
A\text{ is correct}\\\ 1-s^{A}_{i}&\quad\mbox{otherwise}\end{array}\right.$
(6)

The payout for an investment in $B$-securities is defined in the same manner.

######  Remark 1.

In what follows, to simplify notation, we will refer to the price of
$A$-securities as $p$ instead of $p^{A}$ and to the price of $B$-securities as
$1-p$ instead of $p^{B}$.

##### Utility

We study price $p$ by making assumptions on how much utility agents extract
from their payout at that price. We consider two types of utility functions:

Naive

Given a price $p\in[0,1]$, the naive utility function of a $A$-trader $i$ is
$u(p,s^{A}_{i})=z(p,s^{A}_{i})$ Similarly, for a $B$-trader, it is
$u(1-p,s^{B}_{i})=z(1-p,s^{B}_{i}).$ The expected utility for investment in
$A$-securities is then:

$U^{A}_{i}(p,s_{i})=\mathbb{E}[u(p,s^{A}_{i})]=b_{i}\left(\frac{s^{A}_{i}}{p}-s_{i}+1\right)+(1-b_{i})(1-s^{A}_{i}).$
(7)

The expected utility for investment in $B$-securities is, correspondingly,
$b_{i}(1-s^{B}_{i})+(1-b_{i})\left(\frac{s^{B}_{i}}{1-p}-s^{B}_{i}\right)$. We
will refer to markets under a naive utility assumption as Naive markets.

Kelly

Given a price $p\in[0,1]$ the Kelly [13] utility function of an $A$-trader $i$
is $u(p,s^{A}_{i})=\ln(z(p,s^{A}_{i}))$, and mutatis mutandis for $B$-traders.
The expected Kelly utility for an $A$-trader is therefore:

$U^{A}_{i}(p,s^{A}_{i})=\mathbb{E}[u(p,s^{A}_{i})]=b_{i}\ln\left(s^{A}_{i}\frac{1-p}{p}+1\right)+(1-b_{i})\ln(1-s^{A}_{i}).$
(8)

Correspondingly, the expected utility of investment $s^{B}_{i}$ for a
$B$-traders is
$b_{i}\ln(1-s^{B}_{i})+(1-b_{i})\ln\left(\frac{s^{B}_{i}}{1-p}-s^{B}_{i}\right)$.
We will refer to markets under such logarithmic utility assumption as Kelly
markets. Investing with a logarithmic utility function is known as Kelly
betting and is known to maximize bettor’s wealth over time [13]. Information
market traders with Kelly utilities have been studied, for instance, in [5].

##### Equilibria

For each of the above models of utility we will work with the notion of
equilibrium known as competitive equilibrium [16]. This equilibrium assumes
that agents optimize the choice of their investment strategy $s_{i}$ under the
balancing assumption of Equation (5), while not considering the effect of
their choice on the price (they behave as ‘price takers’).

######  Definition 1 (Competitive equilibrium).

Given a belief profile $\mathbf{b}$, an investment profile $\mathbf{s}$ is in
competitive equlibrium with respect to price $p$ if and only if:

  1. 1.

Equation (5) holds, that is, $p=p(\mathbf{s})$;

  2. 2.

for all $i\in N$, if $i$ is a $t$-trader in $\mathbf{s}$ then
$s^{t}_{i}\in\operatorname*{arg\,max}_{x\in[0,1]}U_{i}^{t}(p^{t},x)$, for
$t\in\left\\{A,B\right\\}$.

So, when the investment profile $\mathbf{s}$ is in equilibrium with respect to
price $p$, no agent would like to purchase more securities of any type when
the price of $A$-securities is $p$. If $\mathbf{s}$ is in equilibrium with
respect to $p(\mathbf{s})$ we say that $\mathbf{s}$ is an equilibrium. If
equilibria always exist and are all such with respect to one unique price,
then such price can be interpreted as the market’s belief that the state of
the world is $A$, given the agents’ underlying beliefs $\mathbf{b}$. We can
therefore view a market as a belief merger $F^{\mathbf{w}}:[0,1]^{n}\to[0,1]$
mapping belief profiles to the equilibrium price.

######  Remark 2 (Null price).

Under equation (5) a price $p=0$ (respectively, $p=1$) implies that there are
no $A$-traders (respectively, no $B$-traders). In such cases Equations (6),
(7) (Naive utility) and (8) (Kelly utility) would be formally undefined. Such
situations, however, cannot occur in equilibrium because as $p$ approaches $0$
(respectively, $1$), the utility for $s^{A}_{i}>0$ (respectively,
$s^{B}_{i}>0$) approaches $\infty$ under both utility models. No investment
profile can therefore be in equilibrium with respect to prices $p=0$ or $p=1$.

##  3 Equilibrium price in Naive and Kelly markets

In order to see markets as belief aggregators we need to show the above market
types always admit equilibria and, ideally, that equilibrium prices are
unique, thereby making the aggregator resolute. The present section is
concerned with these issues.

###  3.1  Equilibrium $p$ in Naive markets is the ($1-p$)-quantile belief

Let us start by observing that under naive utility agents maximize their
utility by investing all their wealth, unless their belief equals the price,
in which case any level of investment would yield the same utility to them.

######  Lemma 1.

For any $\mathbf{q}$ and $\mathbf{b}\in\mathcal{B}_{\mathbf{q}}$, and
$p\in[0,1]$ we have that, for any $i\in N$:

$\displaystyle\operatorname*{arg\,max}_{x\in[0,1]}U_{i}^{A}(p,x)=\begin{cases}\\{1\\}&\mbox{if}\
p<b_{i}\\\ \\{0\\}&\mbox{if}\ p>b_{i}\\\ [0,1]&\mbox{otherwise}\end{cases};$
$\displaystyle\operatorname*{arg\,max}_{x\in[0,1]}U_{i}^{B}(p,x)=\begin{cases}\\{1\\}&\mbox{if}\
(1-p)<(1-b_{i})\\\ \\{0\\}&\mbox{if}\ (1-p)>(1-b_{i})\\\
[0,1]&\mbox{otherwise}\end{cases}$

###### Proof.

We reason for $A$. The argument for $B$ is symmetric. Observe first of all
that Equation (7) can be rewritten as
$U^{A}_{i}(p,s^{A}_{i})=\frac{b_{i}}{p}(s^{A}_{i}(1-p)+p)+(1-b_{i})(1-s^{A}_{i}).$
So, the utility for strategy $s^{A}_{i}=1$ is $\frac{b_{i}}{p}$ and for
$s^{A}_{i}=0$ is $1$. If $\frac{b_{i}}{p}>1$,
$U^{A}_{i}(p,s_{i})\in[1,\frac{b_{i}}{p}]$ and so $s^{A}_{1}=1$ maximizes
Equation (7). By our assumptions, we therefore also have $s_{i}^{B}=0$. If
$\frac{b_{i}}{p}<1$, instead, $U^{A}_{i}(p,s_{i})\in[\frac{b_{i}}{p},1]$ and
$s^{A}_{1}=0$ maximizes Equation (7). The agent then takes the opposite side
of the bet and maximizes $U^{B}_{i}(p,s^{B}_{i})$ by setting $s_{i}^{B}=1$.
Finally, if $\frac{b_{i}}{p}=1$, all investment strategies yield utility of
$1$. ∎

So, if $\mathbf{s}$ is in competitive equilbrium with respect to price
$p(\mathbf{s})$ in a Naive market, then for each agent $i$: $s^{A}_{i}=1$ if
$b_{i}>p(\mathbf{s})$, $s^{A}_{i}=0$ if $b_{i}<p(\mathbf{s})$, and
$s_{i}\in[0,1]$ if $b_{i}=p(\mathbf{s})$, and correspondingly for $s_{i}^{B}$.

1

input : A belief profile $\mathbf{b}=(b_{1},\ldots,b_{n})$ ordered from
highest to lowest beliefs

output : An investment profile $\mathbf{s}=(\mathbf{s}^{A},\mathbf{s}^{B})$

2 $\mathbf{s}^{A}\leftarrow(0,\ldots,0)$ ;

3 /* We start by assuming no agent invests in $A$ */

4 for  _$1\leq i <n$_ do

5 if  _ $b_{i}\geq\frac{i}{n}\geq b_{i+1}$ _ then

6
$\mathbf{s}^{A}\leftarrow(\underbrace{1,\ldots,1}_{i~{}\mathit{times}},0,\ldots,0)$
and
$\mathbf{s}^{B}\leftarrow(\underbrace{0,\ldots,0}_{i~{}\mathit{times}},1,\ldots,1)$
;

7 return _$(\mathbf{s}^{A},\mathbf{s}^{B})$_ and exit ;

8 /* profile with price $\frac{i}{n}$ */

9

10 end if

11

12 end for

13for  _$1\leq i\leq n$_ do

14 if  _ $\frac{i-1}{n}<b_{i}<\frac{i}{n}$ _ then

15 $x\leftarrow$ solve $\frac{1}{b_{i}}((i-1)+x)=\frac{1}{1-b_{i}}(n-i)$ ;

16 /* partial $A$ investment */

17 if  _ $x\geq 0$ _ then

18 $s^{A}_{i}\leftarrow x$ ;

19
$\mathbf{s}^{A}\leftarrow(\underbrace{1,\ldots,1}_{i-1~{}\mathit{times}},s^{A}_{i},0,\ldots,0)$
and
$\mathbf{s}^{B}\leftarrow(\underbrace{0,\ldots,0}_{i-1~{}\mathit{times}},0,1,\ldots,1)$
;

20 return _$(\mathbf{s}^{A},\mathbf{s}^{B})$_ and exit

21 else

22 $x\leftarrow$ solve $\frac{1}{b_{i}}(i-1)=\frac{1}{1-b_{i}}((n-i)+x)$ ;

23 /* partial $B$ investment */

24 $s^{B}_{i}\leftarrow x$ ;

25
$\mathbf{s}^{B}\leftarrow(\underbrace{0,\ldots,0}_{i-1~{}\mathit{times}},s^{B}_{i},1,\ldots,1)$
and
$\mathbf{s}^{A}\leftarrow(\underbrace{1,\ldots,1}_{i-1~{}\mathit{times}},0,0,\ldots,0)$
;

26 return _$(\mathbf{s}^{A},\mathbf{s}^{B})$_ and exit ;

27 /* profile with price $b_{i}$ */

28

29 end if

30

31 end if

32

33 end for

Algorithm 1 Competitive equilibria in Naive markets

Let us denote by $\mathit{NC}(\mathbf{b})$ the set of investment profiles
$\mathbf{s}$ in competitive equilibrium (under naive utilities). We show now
that such equilibria always exist and are unique.

######  Lemma 2.

For any $\mathbf{q}$ and $\mathbf{b}\in\mathcal{B}_{\mathbf{q}}$,
$|NC(\mathbf{b})|\geq 1$.

###### Proof.

We prove the claim by construction via Algorithm 1, showing that the algorithm
outputs an investment profile which is in competitive equilibrium. The
algorithm consists of two routines: lines 1-7, lines 8-21. We first show that
the conditions of the loops of the two routines are such that an output is
always obtained. The two routines compare entries in two vectors: the $n$-long
vector of beliefs $(b_{1},\ldots,b_{n})$, assumed to be ordered by decreasing
values (thus, stronger beliefs first); the $n+1$-long vector
$(0,\frac{1}{n},\frac{2}{n},\ldots,\frac{n}{n})$, ordered therefore by
increasing values. The two vectors define two functions from
$\\{0,\ldots,n\\}$ to $[0,1]$ (we postulate $b_{0}=1$). Because the first
function is non-increasing, and the second one is increasing and its image
contains both $0$ and $1$, there exists $i\in\\{0,\ldots n\\}$ such that the
two segments $[b_{i+1},b_{i}]$ and $[\frac{i}{n},\frac{i+1}{n}]$ have a non-
empty intersection. There are four ways in which the two segments can overlap
giving rise to two exhaustive cases: $\frac{i}{n}$ lies in $[b_{i+1},b_{i}]$,
in which case the condition of the first routine applies; or $b_{i+1}$ lies in
$[\frac{i}{n},\frac{i+1}{n}]$, in which case the condition of the second loop
applies.

It remains to be shown that the outputs of the two routines are equilibria.
The output of the first routine is an investment profile
$\mathbf{s}=(\mathbf{s}^{A},\mathbf{s}^{B})$ where $i$ agents fully invest in
$A$ and $n-i$ agents fully invest in $B$, yielding a price
$p(\mathbf{s})=\frac{i}{n}\in[b_{i},b_{i+1}]$. By Lemma 1 such a profile is an
equilibrium. The output of the second routine is an investment profile
$\mathbf{s}$ where $i-1$ agents fully invest in $A$, $n-i$ agents fully invest
in $B$ and agent $i$, whose belief equals the price, invests partially in
either $A$ or $B$ in order for the market to clear (Equation (5)). Again by
Lemma 1 we conclude that the profile is in equilibrium with respect to
$b_{i}$. ∎

Observe that the price constructed by Algorithm 1 is either equal to the
belief of the $i$-th agent (ordered from stronger to weaker beliefs) or falls
in the interval between the belief of the $i$-th and the $i+1$-th agents.

######  Lemma 3.

For any $\mathbf{q}$ and $\mathbf{b}\in\mathcal{B}_{\mathbf{q}}$,
$|NC(\mathbf{b})|\leq 1$.

###### Proof.

Assume towards a contradiction there exist $\mathbf{s}\neq\mathbf{t}\in
NC(\mathbf{b})$. It follows that $p(\mathbf{s})\neq p(\mathbf{t})$. Assume
w.l.o.g. that $p(\mathbf{s})<p(\mathbf{t})$. By Equation 5 and the definition
of competitive equilibrium, it follows that $\sum_{i\in
N}s^{A}_{i}\leq\sum_{i\in N}t^{A}_{i}$ (larger $A$-investment in
$\mathbf{t}$). By Lemma 1 it follows that there are more agents $i$ such that
$b_{i}>p(\mathbf{t})$ rather than $b_{i}>p(\mathbf{s})$, and therefore that
$p(\mathbf{t})<p(\mathbf{s})$. Contradiction. ∎

We can thus conclude that the equilibrium price in Naive markets is unique.

######  Theorem 2.

For any $\mathbf{q}$ and $\mathbf{b}\in\mathcal{B}_{\mathbf{q}}$,
$NC(\mathbf{b})$ is a singleton.

We will refer to such equilibrium profile as $\mathbf{s}_{NC}(\mathbf{b})$ and
to its equilibrium price as $p_{NC}(\mathbf{b})$. An interesting consequence
of the above results is that such equilibrium price behaves like a quantile of
$\mathbf{b}$, splitting the belief profile into segments roughly proportional
to the price.

######  Corollary 1.

Fix $\mathbf{q}$. For any belief profile
$\mathbf{b}\in\mathcal{B}_{\mathbf{q}}$ there are $n\cdot p(\mathbf{s})$
agents $i$ such that $b_{i}\geq p_{NC}(\mathbf{b})$ and there are
$n\cdot(1-p(\mathbf{s}))$ agents $i$ such that $b_{i}\leq p_{NC}(\mathbf{b})$.

The equilibrium price $p_{NC}(\mathbf{b})$ corresponds to the
$(1-p_{NC}(\mathbf{b}))$-quantile of $\mathbf{b}$.555A similar observation,
but for a continuum of players ($N=[0,1]$) and for subjective beliefs is
reported in [15].

###  3.2 The average belief is the equilibrium price in Kelly markets

The two following lemmas are known results, which we restate here for
completeness.

######  Lemma 4 ([13]).

For any $b_{i}\in[0,1]$ and $p\in[0,1]$:

$\displaystyle\operatorname*{arg\,max}_{x\in[0,1]}U_{i}^{A}(p,x)=\begin{cases}\frac{b_{i}-p}{1-p}&\mbox{if}\
p<b_{i}\\\ 0&\mbox{otherwise}\end{cases};$
$\displaystyle\operatorname*{arg\,max}_{x\in[0,1]}U_{i}^{B}(p,x)=\begin{cases}\frac{p-b_{i}}{p}&\mbox{if}\
(1-p)<(1-b_{i})\\\ 0&\mbox{otherwise}\end{cases}$

So, a strategy profile $\mathbf{s}$ is in Kelly competitive equilibrium with
respect to price $p(\mathbf{s})$ whenever Equation (5) is satisfied together
with the ‘Kelly conditions’ of Lemma 4. Unlike in the case of Naive markets it
is easy to see that such equilibrium is unique. So, for a given belief profile
$\mathbf{b}$ let us denote by $\mathbf{s}_{\mathit{KC}}(\mathbf{b})$ such
competitive equilibrium and by $p_{\mathit{KC}}(\mathbf{b})$ the price at such
equilibrium. We then have:

######  Lemma 5 ([5]).

For any $\mathbf{q}$ and $\mathbf{b}\in\mathcal{B}_{\mathbf{q}}$,
$p_{\mathit{KC}}(\mathbf{b})=\frac{1}{|N|}\sum_{i\in N}b_{i}.$

##  4 Truth-Tracking via Equilibrium Prices

In this section we show how competitive equilibria in Naive and Kelly markets
correspond to election by simple majority and, respectively, by a majority in
which agents carry weight proportional to their competence minus $0.5$.

###  4.1 Simple majority and Naive markets

Simple majority is implemented in competitive equilibrium by a Naive market:

######  Theorem 3.

For any $\mathbf{q}$ and $\mathbf{b}\in\mathcal{B}_{\mathbf{q}}$:
$M^{\mathbf{1}}(\widehat{\mathbf{b}})=\widehat{p_{NC}(\mathbf{b})}.$

###### Proof.

The claim follows from the observation that, by Corollary 1,
$p_{NC}(\mathbf{b})>0.5$ if and only if there exists a majority of traders
whose beliefs are higher than the price. From which we conclude that
$\hat{\mathbf{b}}$ determines a voting profile with a majority of votes for
$A$. ∎

$\mathbf{b}$$\widehat{\mathbf{b}}$$p_{\mathit{NC}}(\mathbf{b})$$M^{\mathbf{1}}(\widehat{\mathbf{b}})$$\widehat{}$$\widehat{}$$\mathit{NC}$$M^{\mathbf{1}}$

Put otherwise, the theorem tells us that the outcome of simple majority always
consists of the security that the $(1-p)$-quantile belief (where $p$ is the
equilibrium price) would invest in equilibrium when the market is naive. So we
can treat $\mathit{NC}$ as a belief aggregator $[0,1]^{n}\to[0,1]$ mapping
belief profiles to prices induced by competitive equilibria. In other words,
for any belief profile $\mathbf{b}$ induced by independent individual
competences in $(0.5,1]$, the diagram on the right commutes.

######  Remark 3.

It is worth observing that, by Theorem 3, known extensions of the Condorcet
Jury Theorem with heterogeneous competences [11] directly apply to Naive
markets in competitive equilibrium. In particular with $N\to\infty$ the
probability that $p_{\mathit{NC}}(\mathbf{b})$ is correct approaches $1$ for
any $\mathbf{b}$ induced by a given competence profile.

###  4.2 Weighted majority and Kelly markets

A weighted majority rule with weights for each $i$ proportional to $q_{i}-0.5$
is implemented in competitive equilibrium by Kelly markets. Intuitively, such
markets then implement a majority election where individuals’ weights are
proportional to how much better the individual is compared to an unbiased
coin.

######  Theorem 4.

For any $\mathbf{q}$, and $\mathbf{b}\in\mathcal{B}_{\mathbf{q}}$:
$M^{\mathbf{w}}(\widehat{\mathbf{b}})=\widehat{p_{\mathit{KC}}(\mathbf{b})},$
where $\mathbf{w}$ is s.t. for all $i\in N$, $w_{i}\propto 2q_{i}-1$.

$\mathbf{b}$$\widehat{\mathbf{b}}$$p_{\mathit{KC}}(\mathbf{b})$$M^{\mathbf{w}}(\widehat{\mathbf{b}})$$\widehat{}$$\widehat{}$$\mathit{KC}$$M^{\mathbf{w}}$

Intuitively, by implementing a weighted average of the beliefs, the
competitive equilibrium price for Kelly utilities behaves like a weighted
majority where agents’ weights are a linear function of their individual
competence ($2q_{i}-1$). So, for any belief profile $\mathbf{b}$ induced by a
competence profile $\mathbf{q}$ and weights $w_{i}=2q_{i}-1$, the diagram on
the right commutes.

##  5 Markets for Perfect Elections

In this section we show how, by introducing a specific tax scheme, we can
modify Kelly markets in such a way as to make their equilibrium price
implement a perfect weighted majority, that is, a majority in which the weight
of each individual is proportional to the natural logarithm of their
competence ratio. The intuition of our approach is the following: Theorem 4
has shown that Kelly markets correspond to elections where individuals are
weighted proportionally to their competence in excess of $0.5$; in order to
bring such weights closer to the ideal values of Theorem 1 we need therefore
to allow more competent agents to exert substantially more influence on the
equilibrium price; we do so by designing a tax scheme which achieves such
effect asymptotically in one parameter of the scheme.

###  5.1 Taxing payouts

We modify Equation (8) by building in the effects of a tax scheme $T$ on
utility as follows:

$U^{A}_{i}(p,s_{i})=b_{i}\ln
T\left(s_{i}\frac{1-p}{p}+1\right)+(1-b_{i})\ln(1-s_{i})$ (9)

where

$T(x)=\frac{1-e^{-kx\frac{p}{1-p}}}{k\frac{p}{1-p}}$ (10)

with $k\in\mathbb{R}^{>0}$. Observe that as parameter $k$ approaches $0$,
$T(x)$ approaches $x$, that is, null taxation is approached.

The best way to gain an intuition of the working of function $T$ is to observe
its effects on the agent’s optimal investment strategy supposing the price
$0.5$. For $p=0.5$ the optimal strategy of a Kelly trader is $2b_{i}-1$ (Lemma
4). Function $T$ makes that strategy asymptotically proportional to
$\ln\left(\frac{b_{i}}{1-b_{i}}\right)$ (Figure 2) as $k$ grows.

Figure 2: Left: returns after taxation by $T$ as a function of investment
(Equation (9)). Right: investment strategy (red) approximating
$\ln\left(\frac{b_{i}}{1-b_{i}}\right)\frac{1}{k}$ (blue) as $k$ grows when
price equals $0.5$. Functions plotted for $k\in\\{0.1,0.2,1,2,10,20\\}$.

We call markets under the utility in Equation (9) taxed markets and denote
their equilibrium prize by $p_{\mathit{TC}}(\mathbf{b})$, for any belief
profile $\mathbf{b}$.

###  5.2 Equilibria in taxed Kelly markets

Like for Naive and Kelly markets, we first determine the optimal strategy of
the traders. We do that for $A$-traders, as the lemma for $B$-traders is
symmetric.

######  Lemma 6.

For any $i\in N$, if $b_{i}>p$, then as $k\to\infty$,

$\operatorname*{arg\,max}_{x\in[0,1]}U_{i}^{A}(p,x)\propto\ln\left(\frac{1-p}{p}\cdot\frac{b_{i}}{1-b_{i}}\right).$

###### Proof.

We start from $i$’s utility, given by Equation (9). By setting
$\frac{dU_{i}^{A}}{ds_{i}}=0$ (first order condition) we obtain:

$\frac{bT^{\prime}(s\frac{1-p}{p})\frac{1-p}{p}}{1+T(s\frac{1-p}{p})}=\frac{1-b_{i}}{1-s_{i}}$
(11)

If we plug Equation (10) into Equation (11), we obtain:

$\frac{be^{-ks_{i}}\frac{1-p}{p}}{1+\frac{1-e^{-ks_{i}}}{k\frac{p}{1-p}}}=\frac{1-b_{i}}{1-s_{i}}$
(12)

and therefore

$\frac{kbe^{-ks_{i}}}{k\frac{p}{1-p}+1-e^{-ks_{i}}}=\frac{1-b}{1-s_{i}}.$ (13)

As $k$ approaches infinity, $s_{i}$ approaches zero. For this reason we
rescale strategies by $k$ and consider a value $y=sk$. This allows us to
understand the form to which strategies tend to as they approach zero. We thus
obtain:

$\frac{kbe^{-y}}{k\frac{p}{1-p}+1-e^{-y}}=\frac{1-b}{1-\frac{y}{k}}.$ (14)

As $k$ approaches infinity this approaches:

$\frac{be^{-y}}{\frac{p}{1-p}}=(1-b)$ (15)

which can be rewritten in turn as

$y=\ln\left(\frac{1-p}{p}\frac{b}{1-b}\right)$ (16)

from which we conclude $s_{i}=\frac{1}{k}\log(\frac{1-p}{p}\frac{b}{1-b})$ as
desired. ∎

As $k$ tends to infinity, the optimal investment strategy will tend to $0$ for
all agents. However, it will do so in such a way that as $k$ grows, the
optimal investment strategy tends to be proportional to
$\ln(\frac{1-p}{p}\cdot\frac{b_{i}}{1-b_{i}})$ as desired.

So, as $k$ grows large, a strategy profile $\mathbf{s}$ is in competitive
equilibrium in a taxed market with respect to price $p(\mathbf{s})$ whenever
Equation (5) is satisfied together with the condition identified by Lemma 6.
We denote by $\mathbf{s}_{\mathit{TC}}(\mathbf{b})$ such competitive
equilibrium and by $p_{\mathit{TC}}(\mathbf{b})$ the price at such
equilibrium. We then have:

######  Lemma 7.

For any $\mathbf{q}$ and $\mathbf{b}\in\mathcal{B}_{\mathbf{q}}$, as
$k\to\infty$,

$\ln\left(\frac{p_{\mathit{TC}}(\mathbf{b})}{1-p_{\mathit{TC}}(\mathbf{b})}\right)\propto\sum_{i}^{n}\ln\left(\frac{b_{i}}{1-b_{i}}\right).$

###### Proof.

To lighten notation we write $p$ for $p_{\mathit{TC}}(\mathbf{b})$. From the
equilibrium condition (Equation (5)) and Lemma 6 we have that

$\displaystyle\frac{1}{p}\sum_{i\in N^{A}}\ln\frac{b_{i}}{1-b_{i}}$
$\displaystyle=\frac{1}{1-p}\sum_{i\in N^{B}}\frac{1-b_{i}}{b_{i}}$ (17)

where $N^{A}=\left\\{i\in N\mid b_{i}>p\right\\}$ and $N^{B}=\left\\{i\in
N\mid b_{i}<p\right\\}$. From the above we obtain:

$0=\sum_{i}^{N}\ln\left(\frac{1-p}{p}\frac{b_{i}}{1-b_{i}}\right)$ (18)

which rewrites to

$\ln\left(\frac{p}{1-p}\right)=\frac{1}{N}\sum_{i}^{N}\ln\left(\frac{b_{i}}{1-b_{i}}\right)$
(19)

as desired. ∎

That is, the equilibrium price ratio between $A$ and $B$ securities in a taxed
market tends to be proportional, in logarithmic scale, to the average belief
ratio.

######  Theorem 5.

For any $\mathbf{q}$, $\mathbf{b}\in\mathcal{B}_{\mathbf{q}}$ and
$k\to\infty$:

$M^{\mathbf{w}}(\widehat{\mathbf{b}})=\widehat{p_{\mathit{TC}}(\mathbf{b})}$

where that $\mathbf{w}$ is s.t. for all $i\in N$
$w_{i}\propto\ln\frac{q_{i}}{1-q_{1}}$.

$\mathbf{b}$$\widehat{\mathbf{b}}$$p_{\mathit{TC}}(\mathbf{b})$$M^{\mathbf{w}}(\widehat{\mathbf{b}})$$\widehat{}$$\widehat{}$$\mathit{TC}$$M^{\mathbf{w}}$

This last result shows that elections that are perfect from a truth-tracking
perspective (Theorem 1) can be implemented increasingly faithfully by Kelly
markets once the taxation scheme $T$ is applied and the taxation parameter $k$
in Equation (10) grows larger. So, for any belief profile $\mathbf{b}$ induced
by a competence profile $\mathbf{q}$ and weights
$w_{i}=\frac{q_{i}}{1-q_{i}}$, the diagram on the right commutes as $k$ tends
to infinity and, therefore, taxation grows.

##  6 Conclusions and Outlook

Our paper is, to the best of our knowledge, the first one establishing a
formal link between voting and information markets from an epistemic social
choice perspective. The link consists specifically of correspondence results
between weighted majority voting on the one hand, and information markets
under three types of utility on the other. Such results open up the
possibility to implement weighted majority voting with strong epistemic
guarantees even without having access to individual competences because such
information becomes indirectly available in the market via the equilibrium
price. Notice that, in particular, while it may be difficult to elicit
truthful weights from agents, investment strategies are subject to the natural
incentive of maximizing investment returns. Whether this can prove
advantageous also in practice, for instance in the setting of classification
markets [4] or voting-based ensembles [8], should definitely be object of
future research.

The study we presented is subject to at least three main limitations. First,
our analysis inherits all assumptions built into standard jury theorems, in
particular: jurors’ independence; homogeneous priors; equivalence of type-$1$
and type-$2$ errors in jurors’ competences. Future research should try to lift
our correspondence to more general settings relaxing the above assumptions
(see [9] for a recent overview). Secondly, our study limited itself to one-
shot interactions. However, markets and specifically Kelly betting make most
sense in a context of iterated decisions. Extending our results to the
iterated setting, along the lines followed for instance in [5], is also a
natural avenue for future research. Thirdly, our market model makes use of the
notion of competitive equilibrium. Although such notion of equilibrium is
standard in information markets, it responds to the intuition individuals
operate in a large group and, therefore, behave as price takers. We consider
it interesting to study, at least experimentally, how different notions of
equilibrium that do not make such assumption (e.g., Nash equilibrium) would
behave within our framework.

##### Acknowledgments

This research was (partially) funded by the Hybrid Intelligence Center, a
10-year programme funded by the Dutch Ministry of Education, Culture and
Science through the Netherlands Organisation for Scientific Research,
https://hybrid-intelligence-centre.nl, grant number 024.004.022. Davide Grossi
wishes to also thank Université Paris Dauphine and the Netherlands Institute
for Advanced Studies (NIAS), where parts of this research were completed. The
authors also wish to thank the anonymous reviewers of COMSOC’23 for several
helpful suggestions.

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##  Appendix A Examples

Assume $N=\left\\{1,\ldots,5\right\\}$ with competence profile
$\mathbf{q}=\left(0.9,0.7,0.6,0.6,0.6\right)$. Assume further that only the
first and last agent receive signal $A$ while the rest receives signal $B$.
This gives us the following belief profile by Bayesian update:
$\mathbf{b}=\left(0.9,0.3,0.4,0.4,0.6\right)$.

The above beliefs result in the voting profile
$\mathbf{v}=\left(1,0,0,0,1\right)$, from which we obtain:

  * •

$M^{\mathbf{1}}(\mathbf{v})=\left\\{0\right\\}$, that is, standard majority
selects $B$

  * •

$M^{\mathbf{w}}(\mathbf{v})=\left\\{1\right\\}$ where
$\mathbf{w}=\left(0.8,0.4,0.2,0.2,0.2\right)$ (weight profile given by
$2q_{i}-1$) as $0.8+0.2-(0.4+0.2+0.2)>0$, that is, the sum of weights of the
first and last agents are larger then the sum of weights of the remaining
agents.

  * •

$M^{\mathbf{w}}(\mathbf{v})=\left\\{1\right\\}$ where
$\mathbf{w}=\left(\ln\frac{0.9}{0.1},\ln\frac{0.7}{0.3},\ln\frac{0.6}{0.4},\ln\frac{0.6}{0.4},\ln\frac{0.6}{0.4}\right)$
(optimal weights) as

$\ln\frac{0.9}{0.1}+\ln\frac{0.6}{0.4}-\left(\ln\frac{0.7}{0.3}+2\cdot\ln\frac{0.6}{0.4}\right)$
(20)

is positive.

We move now to the choices made by the markets based on equilibrium prices. We
have that: by Algorithm 1, $p_{\mathit{NC}}(\mathbf{b})=\frac{2}{5}$ (Naive
market equilibrium) where the two agents who received the $A$ signal invest
all their endowment in $A$-securities, and the remaining agents invest all
their endowment in $B$-securities; $p_{\mathit{KC}}(\mathbf{b})=\frac{2.6}{5}$
(Kelly market equilibrium) corresponding to the mean belief in $\mathbf{b}$.
So, a Naive market given the above beliefs selects $B$ while the Kelly market
selects $A$ by a very small margin. As to the taxed markets for perfect
elections, our results do not give us a closed expression for
$p_{\mathit{TC}}(\mathbf{b})$ but rather determine whether the price favors
$A$\- or $B$-securities based on the logarithm of the ratio between the two
prices, which is proportional to the logarithm of the weighed support for $A$
and for $B$ when the taxation parameter $k$ tends to infinity (Theorem 5). In
this example, we thus have that
$\ln\frac{p_{\mathit{TC}}(\mathbf{b})}{1-p_{\mathit{TC}}(\mathbf{b})}$ is
proportional to Equation (21) and therefore points to security $A$.

Assume $N=\left\\{1,\ldots,4\right\\}$ with competence profile
$\mathbf{q}=\left(0.8,0.6,0.6,0.6\right)$. Assume further that only the first
agent receives signal $A$ while the rest receives signal $B$. This gives us
the following belief profile by Bayesian update:
$\mathbf{b}=\left(0.8,0.4,0.4,0.4\right)$.

The above beliefs result in the voting profile
$\mathbf{v}=\left(1,0,0,0\right)$, from which we obtain:

  * •

$M^{\mathbf{1}}(\mathbf{v})=\left\\{0\right\\}$, that is, standard majority
selects $B$

  * •

$M^{\mathbf{w}}(\mathbf{v})=\left\\{0,1\right\\}$ where
$\mathbf{w}=\left(0.6,0.2,0.2,0.2\right)$ (weight profile given by $2q_{i}-1$)
as $0.6-(0.2+0.2+0.2)=0$. That is, we have a split weighted majority.

  * •

$M^{\mathbf{w}}(\mathbf{v})=\left\\{1\right\\}$ where
$\mathbf{w}=\left(\ln\frac{0.8}{0.2},\ln\frac{0.6}{0.4},\ln\frac{0.6}{0.4},\ln\frac{0.6}{0.4}\right)$
(optimal weights) as

$\ln\frac{0.8}{0.2}-3\cdot\ln\frac{0.6}{0.4}$ (21)

is positive.

As to equilibrium prices, by applying Algorithm 1, we have that also in this
case $p_{\mathit{NC}}(\mathbf{b})=\frac{2}{5}$ (Naive market equilibrium).
This price equals the posterior beliefs of the three agents that receive
signal $B$. By the algorithm, the agent receiving signal $A$ invests all its
wealth in $A$, one of the agents receiving signal $B$ invests $\frac{1}{3}$ of
their wealth in $A$ (to guarantee market clearing at that price, line 10 of
Algorithm 1), and the remaining agents invest all their endowment in
$B$-securities. The equilibrium price in Kelly markets is in this case $0.5$
(mean belief). So, a Naive market given the above beliefs selects $B$ while
the Kelly market remains undecided. In taxed markets, we have that
$\ln\frac{p_{\mathit{TC}}(\mathbf{b})}{1-p_{\mathit{TC}}(\mathbf{b})}$ (ratio
of prices in logarithmic scale) is proportional to Equation (21) and therefore
points to security $A$.

##  Appendix B Omitted proofs

###### Proof of Lemma 4 [13].

We reason for $A$. The argument for $B$ is symmetric. First we compute the
first derivative of $U^{A}_{i}(p,x)$:

$\displaystyle\frac{d}{dx}U^{A}_{i}(p,x)$
$\displaystyle=\frac{d}{dx}\left(b_{i}\ln(x\frac{1-p}{p}+1)+(1-b_{i})\ln(1-x)\right)$
$\displaystyle=b_{i}\cdot\frac{\frac{d}{dx}(x\frac{1-p}{p}+1)}{x\frac{1-p}{p}+1}+(1-b_{i})\cdot\frac{\frac{d}{dx}(1-x)}{1-x}$
$\displaystyle=b_{i}\cdot\frac{\frac{1-p}{p}}{x\frac{1-p}{p}+1}+(1-b_{i})\cdot\frac{-1}{1-x}$
$\displaystyle=\frac{b_{i}(1-p)}{x+p(1-x)}-\frac{1-b_{i}}{1-x}$

The first-order condition is
$\frac{b_{i}(1-p)}{x+p(1-x)}-\frac{1-b_{i}}{1-x}=0$, that is,
$\frac{b_{i}-p}{1-b}=\frac{x}{1-x}$. Solving for $x$ we obtain
$\frac{b_{i}-p}{1-b_{i}}$. We need then to check the second-order condition.
The second derivative is
$-\frac{1-b_{i}}{(1-x)^{2}}-\frac{b_{i}(1-p)^{2}}{(p(1-x)+x)^{2}}$, which is
negative at every point and therefore, as desired, also at
$\frac{b_{i}-p}{1-b_{i}}$. Now observe that $\frac{b_{i}-p}{1-b_{i}}$ is
positive whenever $b_{i}>p$ and equals $0$ when $b_{i}=p$. When $b_{i}<p$ then
the expression is negative, and the trader takes the opposite side of the bet,
investing in $B$. ∎

###### Proof of Lemma 5 [5].

Let us denote $p_{\mathit{KC}}(\mathbf{b})$ by $p$. By Lemma 4, Equation (5)
becomes:

$\displaystyle\frac{1}{p}\sum_{i\in N^{A}}\frac{b_{i}-p}{1-p}$
$\displaystyle=\frac{1}{1-p}\sum_{i\in N^{B}}\frac{p-b_{i}}{p}$ (22)

where $N^{A}=\left\\{i\in N\mid b_{i}>p\right\\}$ and $N^{B}=\left\\{i\in
N\mid b_{i}<p\right\\}$. Notice we can assume $N=N^{A}\cup N^{B}$ as any agent
with belief equal to the price does not invest in either $A$\- or
$B$-securities in Kelly markets (by Lemma 4). From Equation (22), by some
basic algebra, we obtain the following series of equations

$\displaystyle\frac{1-p}{p}\sum_{i\in N^{A}}\frac{b_{i}-p}{1-p}$
$\displaystyle=\sum_{i\in N^{B}}\frac{p-b_{i}}{p}$ $\displaystyle\sum_{i\in
N^{A}}b_{i}-p$ $\displaystyle=\sum_{i\in N^{B}}p-b_{i}$ $\displaystyle np$
$\displaystyle=\sum_{i\in N}b_{i}$ $\displaystyle p$
$\displaystyle=\frac{1}{n}\sum_{i\in N}b_{i}$

thereby proving the claim. ∎

###### Proof of Theorem 4.

By the assumed weight profile $\mathbf{v}$ and Equation (4) we can rewrite
$M^{\mathbf{v}}(\widehat{\mathbf{b}})$ as $\widehat{R(\mathbf{b})}$ where:

$\displaystyle R(\mathbf{b})$ $\displaystyle=\frac{\sum_{i\in
N}\mathbbm{1}(b_{i}>0.5)(2q_{i}-1)}{\sum_{i\in N}(2q_{i}-1)}$
$\displaystyle=\frac{\sum_{i\in N}(2q_{i}-1)}{n}\cdot\left(\frac{\sum_{i\in
N}\mathbbm{1}(b_{i}>0.5}{(}2q_{i}-1)){\sum_{i\in
N}(2q_{i}-1)}-\frac{1}{2}\right)+\frac{1}{2}$ $\displaystyle=\frac{\sum_{i\in
N}(2q_{i}-1)}{n}\cdot\left(\frac{2\sum_{i\in
N}\mathbbm{1}(b_{i}>0.5)(2q_{i}-1)-\sum_{i\in N}(2q_{i}-1)}{2\sum_{i\in
N}(2q_{i}-1)}\right)+\frac{1}{2}$ $\displaystyle=\frac{\sum_{i\in
N}\mathbbm{1}(b_{i}>0.5)(2q_{i}-1)-\sum_{i\in
N}(q_{i}-\frac{1}{2})}{n}+\frac{1}{2}$ $\displaystyle=\frac{\sum_{i\in
N}\mathbbm{1}(b_{i}>0.5)(2q_{i}-1)-\sum_{i\in
N}(q_{i}-\frac{1}{2})+\frac{n}{2}}{n}$ $\displaystyle=\frac{\sum_{i\in
N}\mathbbm{1}(b_{i}>0.5)(2q_{i}-1)-\sum_{i\in
N}q_{i}+\frac{n}{2}+\frac{n}{2}}{n}$ $\displaystyle=\frac{\sum_{i\in
N}\mathbbm{1}(b_{i}>0.5)(2q_{i}-1)+\sum_{i\in N}(1-q_{i})}{n}$
$\displaystyle=\frac{\sum_{i\in
N}\mathbbm{1}(b_{i}>0.5)(2q_{i}-1)+(1-q_{i})}{n}\ \ \ \ \ \ \ ~{}\mbox{(recall
Equation \eqref{eq:belief})}$ $\displaystyle=\frac{\sum_{i\in N}b_{i}}{n}.$

From this and Lemma 5 we obtain
$M^{\mathbf{v}}(\widehat{\mathbf{b}})=\widehat{\frac{\sum_{i\in
N}b_{i}}{n}}=p_{\mathit{KC}}(\mathbf{b})$ as desired. ∎

###### Proof of Theorem 5.

First of all, observe that:
$\widehat{p_{\mathit{TC}}(\mathbf{b})}=\left\\{1\right\\}$ iff
$\ln\left(\frac{p_{\mathit{TC}}(\mathbf{b})}{1-p_{\mathit{TC}}(\mathbf{b})}\right)>0$;
$\widehat{p_{\mathit{TC}}(\mathbf{b})}=\left\\{0,1\right\\}$ iff
$\ln\left(\frac{p_{\mathit{TC}}(\mathbf{b})}{1-p_{\mathit{TC}}(\mathbf{b})}\right)=0$;
and $\widehat{p_{\mathit{TC}}(\mathbf{b})}=\left\\{0\right\\}$ iff
$\ln\left(\frac{p_{\mathit{TC}}(\mathbf{b})}{1-p_{\mathit{TC}}(\mathbf{b})}\right)<0$.
Then, by Lemma 7, Equation (1) and some algebra we obtain the following
relations:

$\displaystyle\ln\left(\frac{p_{\mathit{TC}}(\mathbf{b})}{1-p_{\mathit{TC}}(\mathbf{b})}\right)$
$\displaystyle\propto\sum_{i}^{n}\ln\left(\frac{b_{i}}{1-b_{i}}\right)$
$\displaystyle=\sum_{i}^{n}\mathbbm{1}(b_{i}>0.5)\cdot\ln\left(\frac{q_{i}}{1-q_{i}}\right)$
$\displaystyle=\sum_{i:b_{i}>0.5}\ln\left(\frac{q_{i}}{1-q_{i}}\right)+\sum_{i:b_{i}<0.5}\ln\left(\frac{1-q_{i}}{q_{i}}\right)$
$\displaystyle=\sum_{i:b_{i}>0.5}\ln\left(\frac{q_{i}}{1-q_{i}}\right)-\sum_{i:b_{i}<0.5}\ln\left(\frac{q_{i}}{1-q_{i}}\right)$

The last expression is: positive whenever weighted voting with optimal weights
returns $\left\\{1\right\\}$; negative whenever it returns
$\left\\{0\right\\}$; and $0$ whenever it returns $\left\\{0,1\right\\}$
(Equation (4)). ∎

##  Appendix C Supplementary results

Let us define the investment strategy of a trader $i$ by the pair
$s_{i}=(s_{i}^{A},s_{i}^{B})$ where $s_{i}^{A}\in[0,1]$ is the amount of
endowment invested in $A$-securities (similarly for $s_{i}^{B}$). Let then
$c_{i}=w_{i}-(s_{i}^{A}+s_{i}^{B})$, that is, the unspent endowment of $i$
given strategy $(s_{i}^{A},s_{i}^{B})$. The assumption that traders invest
their full endowment amounts to $s_{i}^{A}+s_{i}^{B}=1$. We can therefore
simply refer to $s^{A}_{i}$ as $s$ and to $s^{B}_{i}$ as $1-s$.

In Naive markets, it is a corollary of Lemma 1 that, in equilibrium, any
strategy in which traders invest all their endowment is equivalent to a
strategy investing in at most one security.

For Kelly markets we can prove that the same observation holds. Under a full
investment assumption Equation (8) needs to be modified to:

$U_{i}(p,s)=b_{i}\ln\left(\frac{s}{p}\right)+(1-b_{i})\ln\left(\frac{1-s}{1-p}\right).$
(23)

which can be further generalized to

$U_{i}(p,s)=b_{i}\ln\left(\frac{s}{p}+c\right)+(1-b_{i})\ln\left(\frac{1-s}{1-p}+c\right).$
(24)

when $i$ can invest any amount of endowment in $A$\- or $B$-securities and
still keep unspent endowment $c$.

######  Proposition 1.

In Kelly markets, for all $i\in N$, $p\in[0,1]$ and $s_{i}$ such that
$s_{i}^{A}+s_{i}^{B}=1$ (whole endowment is spent) there exists
$t_{i}=(t^{A}_{i},0)$ or $t_{i}=(0,t_{i}^{B})$ which yield the same utility as
$s_{i}$, and vice versa.

###### Proof.

First observe that the first-order condition for Equation (23) gives us the
optimal investment strategy, which is $b_{i}$. Such optimal strategy will give
$i$ $\frac{b_{i}}{p}$ $A$-securities and $\frac{1-b_{i}}{1-p}$ $B$-securities.
Left to right Suppose that $b_{i}>p$. It follows that
$\frac{b_{i}}{p}>\frac{1-b_{i}}{1-p}$, that is, $i$ purchases more
$A$-securities than $B$-securities. Were $i$ not be compelled to invest all
her endowment, she would invest according to Equation (24). Now it is easy to
see that if $i$ invests $b_{i}$ in $A$-securities, $0$ in $B$-securities, and
keeps $1-b_{i}$ as cash, she obtains the same payoff by Equation (24) as she
would obtain by Equation (23) when she invests $1-b_{i}$ in $B$-securities.
There exists therefore $t_{i}=(t^{A}_{i},0)$ yielding the same utility. The
same reasoning applies, symmetrically, for $b_{i}<p$. Right to left. Given an
investment strategy $t_{i}$ investing in only one security, a utility-
equivalent full-investment strategy can be constructed using the same
reasoning used in the previous case via the utility for unrestricted
investments of Equation (24). ∎

Stéphane Airiau
University Paris-Dauphine
France

Nicholas Kees Dupuis

Davide Grossi
University of Groningen and University of Amsterdam
The Netherlands


# Mpemba effect for a Brownian particle trapped in a single well potential

Apurba Biswas<EMAIL_ADDRESS>The Institute of Mathematical Sciences,
C.I.T. Campus, Taramani, Chennai 600113, India  Homi Bhabha National
Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India  R.
Rajesh<EMAIL_ADDRESS>The Institute of Mathematical Sciences, C.I.T.
Campus, Taramani, Chennai 600113, India  Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

Mpemba effect refers to the counterintuitive phenomenon of a hotter system
equilibrating faster than a colder system when both are quenched to the same
low temperature. For a Brownian particle trapped in a piece-wise linear single
well potential that is devoid of any other metastable minima we show the
existence of Mpemba effect for a wide range of parameters through an exact
solution. This result challenges the prevalent explanation of the Mpemba
effect that requires the energy landscape to be rugged with multiple minima.
We also demonstrate the existence of inverse and strong Mpemba effects.

Mpemba effect describes an anomalous relaxation phenomenon wherein a system
that is initially hotter equilibrates faster than a system that is initially
cooler, when both systems are quenched to the same low temperature [1].
Recently, there has been considerable experimental and theoretical interest in
the Mpemba effect. First observed in water [1, 2, 3, 4, 5, 6, 7, 8, 9], it has
now been experimentally observed in a wide range of physical systems such as
magnetic alloys [10], polylactides [11], clathrate hydrates [12], colloidal
systems [13, 14, 15], etc. Theoretical studies, focusing on model systems,
have shown the existence of Mpemba effect in spin systems [16, 17, 18, 19, 20,
21], Markovian systems with few states [22, 23], particles diffusing in a
potential [24, 25, 26, 27, 28], active systems [29], spin glasses [30],
molecular gases in contact with a thermal reservoir [31, 32, 33, 34], quantum
systems [35, 36, 37], systems with phase transitions [38, 19, 39, 40], and
granular systems [41, 42, 43, 44, 45, 46, 47, 48].

Several system-specific reasons have been proposed to explain the Mpemba
effect. For example, the different reasons proposed for Mpemba effect in water
include evaporation [2], convection [3], dissolved gases [4], supercooling
[5], hydrogen bonding [6, 7, 8] and non-equipartition of energy [9]. For
clathrate hydrates [12] interplay between evaporation and properties of the
hydrogen bonds has been suggested while Mpemba effect in magnetic alloys [10]
has been attributed to the kinetic arrest of non-equilibrium phase during the
relaxation. However, a general understanding of the origin of Mpemba effect is
lacking. Recently, insights obtained from the theoretical studies, in
particular the analytically tractable models with only few degrees of freedom,
suggest that the main driver of the Mpemba effect is the ruggedness or the
presence of multiple minima in the energy landscape. In particular, it has
been suggested that a metastable minimum, in addition to the global minimum,
tends to trap a system at lower temperature more effectively than a system at
higher temperature, resulting in a faster relaxation of the hotter system [22,
16, 17, 24, 29, 49]. This viewpoint was further supported in a recent
experiment on a single Brownian particle diffusing in an asymmetric double
well harmonic potential with linear slopes near the boundaries of the domain,
where Mpemba effect was clearly demonstrated [13]. However, the necessity of a
metastable minimum for the Mpemba effect has been questioned in recent works
[24, 49]. For a particular choice of a piece-wise constant potential, it was
shown that the Mpemba effect is observed when the metastable state has neutral
equilibrium [24], while its existence was shown for the particular case of a
double well potential when the metastability is just lifted [49].

In this paper, we solve exactly for the relaxation dynamics of a Brownian
particle in a piece-wise linear single well potential. By obtaining the phase
diagram for different combinations of the parameters defining the potential,
we show that the Mpemba effect is observable for a wide choice of potentials,
conclusively showing that the origin of the Mpemba effect does not require the
energy landscape to have metastable minima, in addition to the global minimum.
We also demonstrate the existence of inverse Mpemba effect for systems that
are heated, as well as the strong Mpemba effect when the colder system cools
exponentially faster. In addition, we show numerically the presence of the
Mpemba effect for an overdamped particle in a single well asymmetric harmonic
potential that can be realized in the experimental set up similar to that of
Ref. [13].

We consider a Brownian particle in one dimension trapped in a single well
potential $\tilde{U}(\tilde{x})$ that is infinite outside a domain of length
$L$. The thermal environment is characterized by noise $\eta$ that has the
characteristics $\langle\eta(\tilde{t})\rangle=0$ and
$\langle\eta(\tilde{t})\eta(\tilde{t}^{\prime})\rangle=2\gamma
k_{B}\tilde{T}_{b}\delta(\tilde{t}-\tilde{t}^{\prime})$, where $\gamma$ is the
damping coefficient, $\tilde{t}$ is time, $\tilde{T}_{b}$ is the temperature
of the thermal bath and $k_{B}$ is the Boltzmann’s constant. The motion of the
particle is described by the Langevin equation

$\gamma\frac{d\tilde{x}}{d\tilde{t}}=-\frac{d\tilde{U}}{d\tilde{x}}+\eta(\tilde{t}).$
(1)

We consider dimensionless variables $x=(2\pi/L)\tilde{x}$,
$T=\tilde{T}/\tilde{T}_{b}$, $U=\tilde{U}/(k_{B}\tilde{T}_{b})$ and
$t=(4\pi^{2}k_{B}\tilde{T}_{b}/\gamma L^{2})\tilde{t}$. The corresponding
Fokker-Planck (FP) equation for the evolution of the probability density
$p(x,t)$ in the non-dimensionalised variables is given by [50, 51]

$\frac{\partial p(x,t)}{\partial t}=\frac{\partial}{\partial
x}\Big{[}\frac{dU(x)}{dx}p(x,t)\Big{]}+\frac{\partial^{2}p(x,t)}{\partial
x^{2}}=\mathcal{L}_{FP}p(x,t),$ (2)

where $\mathcal{L}_{FP}$ is the FP operator:

$\mathcal{L}_{FP}=\partial_{x}U^{\prime}+\partial^{2}_{x}.$ (3)

With the aim to demonstrate and characterize Mpemba effect in the absence of
metastable states, we consider a single well potential which is piece-wise
linear, as shown in Fig. 1, and is given by

$U(x)=\begin{cases}U_{\ell}+k_{1}(x+\frac{2\pi}{1+\alpha}),&\frac{-2\pi}{1+\alpha}<x<0\\\
k_{2}x,&0<x<\frac{2\pi\alpha}{1+\alpha},\end{cases}$ (4)

where $k_{1}$, $k_{2}$ are the slopes and $U_{\ell}$, $U_{r}$ are the values
of the potential at the boundaries. The minimum of the potential, set equal to
zero, is fixed at $x=0$. The parameter $\alpha$ is the ratio of the length of
the right domain to that to the left domain. The piece-wise linearity makes
the problem analytically tractable.

Figure 1: Schematic diagram of the piecewise linear single well potential. The
parameters $k_{1}$ and $k_{2}$ refer to the slopes, $\alpha$ denotes the ratio
of width of right domain to the left domain, and $U_{\ell}$ and $U_{r}$ are
the values of the potential at the boundaries.

To solve the FP equation in Eq. (2), we closely follow the known methods in
Refs. [50, 51] and the solution in Ref. [49]. We first transform the
associated FP operator $\mathcal{L}_{FP}$ to an self-adjoint operator
$\mathcal{L}$:

$\mathcal{L}=e^{U(x)/2}\mathcal{L}_{FP}e^{-U(x)/2}=\frac{\partial^{2}}{\partial
x^{2}}-\frac{1}{4}\Big{(}\frac{dU}{dx}\Big{)}^{2}+\frac{1}{2}\frac{d^{2}U}{dx^{2}}.$
(5)

The problem then reduces to solving the eigenvalue equation

$\mathcal{L}\psi_{n}=-|\lambda_{n}|\psi_{n},$ (6)

where $\psi_{n}$ are the eigenfunctions of the operator $\mathcal{L}$
corresponding to the eigenvalue $\lambda_{n}$. Note that both the operators
$\mathcal{L}$ and $\mathcal{L}_{FP}$ have the same eigenvalue $\lambda_{n}$
but their respective eigenfunctions $\psi_{n}(x)$ and $\phi_{n}(x)$ are
related by $\psi_{n}(x)=\exp\left(U(x)/2\right)\phi_{n}(x)$. The details of
the solution of the Eq. (6) for the eigenvalues and eigenfunctions can be
found in Supplemental Information [52].

Given the eigenspectrum, the probability distribution function $p(x,t)$ is
given by

$p(x,t)=\frac{e^{-U(x)}}{\mathcal{Z}(1)}+\sum_{n\geq
2}a_{n}e^{\frac{-U(x)}{2}}\psi_{n}(x)e^{-|\lambda_{n}|t},$ (7)

where the coefficients are fixed by the initial distribution
$p(x^{\prime},0)$: $a_{n}=\int
dx^{\prime}~{}p(x^{\prime},0)~{}e^{\frac{U(x^{\prime})}{2}}~{}\psi^{*}_{n}(x^{\prime})$.
The first term in the right hand side of Eq. (7) corresponds to the eigenvalue
$\lambda_{1}=0$ and describes the final equilibrium Boltzmann distribution
with partition function $\mathcal{Z}(1)=\int e^{-U(x)}dx$ at the bath
temperature $T_{b}=1$. Since the eigenvalues $\lambda_{n}$ of the FP operator
follow the order: $\lambda_{1}=0>\lambda_{2}>\lambda_{3}\ldots$, we can
approximate for $p(x,t)$ at large times as

$p(x,t)\simeq\frac{e^{-U(x)}}{\mathcal{Z}(1)}+a_{2}e^{\frac{-U(x)}{2}}\psi_{2}(x)e^{-|\lambda_{2}|t},~{}t\gg\frac{1}{|\lambda_{3}|}.$
(8)

Figure 2: Illustration of the Mpemba effect and the inverse Mpemba effect for
particular choices of the potential $U(x)$. (a) Two choices of the potential
corresponding to $U_{\ell}=4.0$, (red) and $U_{\ell}=6.0$, (blue) keeping
$U_{r}=10.0$, $\alpha=1.0$ fixed. (b) For $U(x)$ in (a) with smaller (larger)
value of $U_{\ell}$, $|a_{2}(T)|$ is non-monotonic (monotonic) for $T>1$,
demonstrating presence (absence) of the Mpemba effect. (c) Two choices of the
potential corresponding to $U_{\ell}=0.2$, (red) and $U_{\ell}=2.0$ (blue)
keeping $U_{r}=8.0$, $\alpha=1.0$ fixed. (d) For $U(x)$ in (c) with smaller
(larger) value of $U_{\ell}$, $|a_{2}(T)|$ is non-monotonic (monotonic) for
$T<1$, demonstrating presence (absence) of the inverse Mpemba effect. Figure
3: The $U_{\ell}$–$U_{r}$ phase diagram showing regions where Mpemba effect is
present (green) and absent (grey) for (a) $\alpha=1.0$, (b) $\alpha=1.2$, and
(c) $\alpha=1.4$. (d) The $U_{\ell}$–$U_{r}$ phase diagram showing regions
where strong Mpemba effect is present (yellow) and absent (grey) for
$\alpha=1.2$.

We will now use Eq. (8), that describes the relaxation dynamics at late times,
to quantitatively define Mpemba effect for this system. Consider two systems,
$P$ and $Q$ initially in equilibrium at temperatures $T_{h}$ and $T_{c}$
respectively, where $T_{h}>T_{c}$. Let $\pi(T)$ denote the corresponding
equilibrium probability distribution at temperature $T$. Both the systems are
then simultaneously quenched to a common bath temperature $T_{b}=1$ where
$T_{h}>T_{c}>1$. If $P$ equilibrates faster than $Q$, the Mpemba effect is
said to exist. Faster equilibration is quantified in terms of a distance from
equilibrium function $D[p(t),\pi(1)]$ which measures the distance between the
instantaneous distribution $p(x,t)$ from the final equilibrium Boltzmann
distribution, $\pi(1)$. It has been argued [22] that the existence of Mpemba
effect is independent of the choice of $D[p(t),\pi(1)]$ provided that the
distance measure obeys certain properties. These are (i) If $T_{h}>T_{c}>1$,
then the distance function should satisfy
$D[\pi(T_{h}),\pi(1)]>D[\pi(T_{c}),\pi(1)]$, i.e., higher the temperature
larger the distance, (ii) $D[p(t),\pi(1)]$ should be a non-increasing function
of time, and (iii) $D[p(t),\pi(1)]$ should be a convex function of $p(t)$.

A convenient choice of such a distance measure is the total variation distance
defined as $D[p(t),\pi(1)]\equiv L_{1}(t)=\int dx|p(x,t)-\pi(x,1)|$. When
$t=0$, since $T_{h}>T_{c}$, initially $L^{h}_{1}>L^{c}_{1}$. For Mpemba effect
to exist, we require that $L^{h}_{1}<L^{c}_{1}$ at late times. In Eq. (8),
since $\lambda_{2}$ is independent of initial condition, the above condition
for Mpemba effect reduces to $|a^{c}_{2}|>|a^{h}_{2}|$ [22, 17]. Equivalently,
if $|a_{2}(T)|$ is non-monotonic with temperature, then we would always be
able to make a choice of $T_{h}$ and $T_{c}$ such that
$|a^{c}_{2}|>|a^{h}_{2}|$.

The methodology we employ is as follows. Given a potential, we solve for the
eigenspectrum, which in turn allows us to compute the time evolution of the
probability distribution of the particle during equilibration using Eq. (7).
We then determine $|a_{2}(T)|$ for $T>1$ and check for non-monotonicity. A
similar analysis can be done for the inverse Mpemba effect in which the
systems are quenched to a temperature that is higher than the initial
temperatures. To check for presence of inverse Mpemba effect, we check whether
$|a_{2}(T)|$ for $T<1$ is non-monotonic.

We first show the existence of Mpemba effect and inverse Mpemba effect for a
single well potential in order to demonstrate that the origin of the effect
does not require more than one energy minima. For this purpose, we make
specific choices of the single well potential, as shown in Fig. 2. For the two
instances of $U(x)$, shown in Fig. 2(a), $|a_{2}(T)|$ for $T>1$ is non-
monotonic for one and monotonic for the other, showing the presence and
absence of Mpemba effect depending on the choice of potential. Similar
construction is possible for the inverse Mpemba effect. For the two instances
of $U(x)$, shown in Fig. 2(b), $|a_{2}(T)|$ for $T<1$ is non-monotonic for one
and monotonic for the other, showing the presence and absence of inverse
Mpemba effect depending on the choice of potential. Beyond showing its
existence, we will not discuss the inverse Mpemba effect further.

To get an insight of what potentials allow for the Mpemba effect, we now
construct phase diagrams demarcating regions that show Mpemba effect from
regions that do not. The potential $U(x)$ is characterized by three
parameters: $U_{\ell}$, $U_{r}$, and $\alpha$. Asymmetry in the potential can
be introduced through $U_{\ell}\neq U_{r}$ and/or $\alpha\neq 1$. The fully
symmetric potential is not expected to show Mpemba effect [24, 28]. To explore
the effect of all three parameters, we determine the phase diagram in the
$U_{\ell}$–$U_{r}$ plane for fixed $\alpha$. The phase diagrams for
$\alpha=1.0$, $1.2$, and $1.4$ are shown in Fig. 3(a)–(c). First, we observe
that, for all three values of $\alpha$, the fraction of the parameter space
that shows Mpemba effect is not negligible. This implies that, for the single
well potential, Mpemba effect can be observed for generic choices of
potentials. Second, we observe that when $U_{\ell}=U_{r}$, Mpemba effect does
not exist even if asymmetry is introduced through $\alpha\neq 1$. Thus
asymmetry through only different domain widths is not sufficient. Third, when
$\alpha$ is increased, thereby increasing asymmetry in the potential, the area
of the region showing Mpemba effect decreases. This is contrary to expectation
based on the experiment on colloids [13] where it was observed that asymmetry
in domain widths in double well potentials enhanced Mpemba effect. To further
demonstrate that increasing $\alpha$ may not be beneficial, we analyzed the
particular case of $U_{\ell}=2.0$ and $U_{r}=8.0$ and find that Mpemba effect
is present for $\alpha$ only in the domain that is approximately $(0.3,1.1)$.

We now discuss the existence of strong Mpemba effect for single well
potentials. Strong Mpemba effect refers to the case when the hotter system
cools exponentially faster than the colder system. In terms of Eq. (8), this
can be achieved only if the coefficient $a_{2}(T)$ is zero. When $a_{2}=0$,
the slowest relaxation mode does not contribute. For $\alpha=1.2$, we identify
those potentials for which $a_{2}$ is zero for some temperature. Then, if we
choose $T_{c}$ to be the same temperature, we would observe strong Mpemba
effect. Figure 3(d) shows the phase space region in the $U_{\ell}-U_{r}$ plane
where the strong Mpemba effect is present/absent for $\alpha=1.2$. Thus, even
for the existence of strong Mpemba effect, a rugged energy landscape is not
necessary.

Finally, we discuss the connection to experiments. In the experiment of Kumar
and Bechhoefer [13], Mpemba effect was demonstrated for a colloidal particle
trapped in an asymmetric double well harmonic potential with linear behavior
near the edges. We now show that the piece-wise linear single well potential
considered in this paper, when modified to have a harmonic minimum and linear
behavior near the edges, continues to exhibit Mpemba effect. Consider the
particular realization of the single well potential, shown in Fig. 4(a), which
has a harmonic minimum. The precise details of the potential are given in
[52]. However, for such potentials, it is no longer possible to analytically
solve for the eigenspectrum. Instead, we can solve the eigenspectrum
numerically, and thus obtain the coefficient $a_{2}$. For the particular
choice of potential, $|a_{2}(T)|$ is clearly non-monotonic [see Fig. 4(b)],
showing the existence of the Mpemba effect in single well harmonic potentials.
Thus, we expect that if the experiment in Ref. [13] is repeated with a single
well potential, Mpemba effect will be observed.

Figure 4: Numerical results for the illustration of the Mpemba effect for a
single well harmonic potential. (a) Shape of the potential with harmonic well
(red region)and linear slopes near the boundaries (blue regions). (b) The
variation of $|a_{2}(T)|$ with temperature $T$, obtained by a numerical
solution, is non-monotonic, showing the presence of the Mpemba effect.

In summary, we solved exactly the relaxation dynamics of a Brownian particle
in an asymmetric single well potential that is piecewise linear. By
identifying the regions in parameter space that exhibits the Mpemba effect, we
not only show that the presence of the Mpemba effect does not require the
energy landscape to have multiple minima, but also that the Mpemba effect can
be realized for generic choices of potentials. We also demonstrated that both
the inverse and strong Mpemba effects can also be realized for single well
potentials. We also showed numerically that single well harmonic potentials,
as opposed to piece-wise linear single well potentials, continue to exhibit
Mpemba effect, opening up the possibility of experimental realization along
the lines of the experiment in Ref [13].

We thank S. Vemparala for a critical reading of the manuscript.

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