# Fast decisions reflect biases, slow decisions do not

Samantha Linn<EMAIL_ADDRESS>Department of Mathematics, University of
Utah, Salt Lake City, Utah, USA  Sean D. Lawley<EMAIL_ADDRESS>Department of Mathematics, University of Utah, Salt Lake City, Utah, USA
Bhargav R. Karamched<EMAIL_ADDRESS>Department of Mathematics, Florida
State University, Tallahassee, Florida 32306, USA  Institute of Molecular
Biophysics, Florida State University, Tallahassee, Florida 32306, USA  Program
in Neuroscience, Florida State University, Tallahassee, Florida 32306, USA
Zachary P. Kilpatrick<EMAIL_ADDRESS>Department of Applied
Mathematics, University of Colorado Boulder, Boulder, Colorado 80309, USA
Krešimir Josić<EMAIL_ADDRESS>Department of Mathematics, University
of Houston, Houston, Texas 77004, USA  Department of Biology and Biochemistry,
University of Houston, Houston, Texas 77004, USA

###### Abstract

Decisions are often made by heterogeneous groups of individuals, each with
distinct initial biases and access to information of different quality. We
show that in large groups of independent agents who accumulate evidence the
first to decide are those with the strongest initial biases. Their decisions
align with their initial bias, regardless of the underlying truth. In
contrast, agents who decide last make decisions as if they were initially
unbiased, and hence make better choices. We obtain asymptotic expressions in
the large population limit that quantify how agents’ initial inclinations
shape early decisions. Our analysis shows how bias, information quality, and
decision order interact in non-trivial ways to determine the reliability of
decisions in a group.

###### pacs:

Valid PACS appear here

Evidence accumulation models are used widely to describe how different
organisms integrate information to make choices Bogacz2006 . Experimental
evidence shows that these models capture the dynamics of the decision making
process of humans and other animals, including the tradeoff between speed and
accuracy Ratcliff1978theory ; chittka2003bees ; newsome1989neuronal ;
uchida2003speed ; swets1961decision . Such models can also be used to
understand how decisions are made in social groups, both when individuals
observe each other’s choices reina2023asynchrony ; karamched2020heterogeneity
; mann2018collective ; tump2022avoiding  and when they act independently
stickler2023impact .

The accumulation of evidence is often modeled using biased Brownian motion
with the quality of evidence determining the magnitude of drift and diffusion.
An agent is assumed to commit to a decision when the process crosses a
threshold. Most previous evidence accumulation models describe a single agent.
However, questions remain about how the order of choices in a group is related
to their accuracy zandbelt2014response . In a group of initially unbiased
individuals accumulating evidence of different quality, the fastest and most
accurate decisions are made by those accessing the highest quality information
reina2023asynchrony . Here we ask how the initial biases of individuals in a
group impact the order and accuracy of their choices. When is a decision
driven mainly by an agent’s initial bias as opposed to accumulated evidence?

We show that in large groups of agents starting with different initial biases,
early decisions tend to be made by agents with the most extreme
predispositions. The choices of these agents agree with their initial bias,
regardless of the quality of the evidence they have access to. On the other
hand, decisions of late deciders do not depend on their initial bias. Thus, in
large groups early decisions reflect only initial inclinations, regardless of
which choice is right. Late decisions reflect only accumulated evidence and
are more likely to be correct. These effects hold generically, but not in the
special case of initially unbiased agents reina2023asynchrony .

### Model description.

We first assume that each individual in a population of $N$ agents has to
decide between two choices (hypotheses), $H^{+}$ and $H^{-}$. They do so by
accumulating evidence and computing the conditional probabilities,
$P(H^{\pm}|\text{evidence})$, that one of the two hypotheses is correct. When
observations are independent and each provides weak evidence, the log
likelihood ratio, or _belief_ , of agent $i$ in the group,
$X_{i}=\log\left(P(H^{+}|\text{evidence}_{i})/P(H^{-}|\text{evidence}_{i})\right)$,
evolves approximately as a biased Brownian motion Bogacz2006 ; Gold02  (See
Fig. 1A),

$\textup{d}X_{i}=\mu_{i}\,\textup{d}t+\sqrt{2D_{i}}\,\textup{d}W_{i},$ (1)

where the drift, $\mu_{i},$ and diffusion coefficient, $D_{i},$ capture the
strength and noisiness of the evidence, respectively 111For an ideal observer
if only measurement noise is present then drift and diffusion are equal in the
continuum limit, $\mu_{i}=D_{i}=m_{i}$ Bogacz2006 ; veliz16 . We consider the
more general case where internal noise can increase $D_{i}>\mu_{i}$.. For all
agents the correct choice ($H\in H^{\pm}$) is given by the sign of the drift
(${\rm sign}[\mu_{i}]=\pm 1$). Eq. (1) is widely used and accurately captures
the dynamics of decisions in humans and animals, including variability in
response time and the impact of evidence quality and biases on choice
Ratcliff1978theory ; gold2007neural ; mulder2012 .

Agents start with an initial bias, $X_{i}(0)$, reflecting information or
assumptions they have about the prior probability of either hypothesis
mulder2012 . We denote by $y$ the initial data for a generic agent. Each agent
then accumulates evidence, and its beliefs evolve according to Eq. (1). Agent
$i$ makes a decision when its belief reaches one of two thresholds,
$-\theta<0<\theta,$ at _decision time_
$\tau_{i}:=\inf\\{t>0:X_{i}(t)\notin(-\theta,\theta)\\}$. This decision,
denoted by $d_{i}=H^{\pm}$, is determined by the sign of the threshold
reached, ${\rm sign}[X_{i}(\tau_{i})]$. If decision criteria differ between
agents an appropriate rescaling of $X_{i}(0)$, $\mu_{i}$, and $D_{i}$ allows
us to assume that all agents use the same thresholds Bogacz2006 .

### Agents with the most extreme initial biases decide first.

We show that in large groups agents whose initial biases are closest to one of
the thresholds make the earliest decisions. We first assume observers are
identical except for their initial biases, so that $\mu_{i}=\mu$ and $D_{i}=D$
in Eq. (1). We denote by $T_{i}$ the $i^{\rm th}$ decision time so that
$T_{1}\leq T_{2}\leq\cdots\leq T_{N}$, where $T_{i}=\tau_{n(i)}$ and $n(i)$ is
the index of the $i^{\rm th}$ agent to decide. Hence, the index of the first
decider is $n(1)$.

For simplicity, we assume that each agent starts with one of finitely many
initial beliefs, $\\{x_{0},x_{1},\ldots,x_{I-1}\\}$, sampled with probability
$q_{i}=P(X_{j}(0)=x_{i})$ for $i=0,...,I-1$. The distance of the initial
belief $x_{i}$ to the closest threshold is $L_{i}=\min\\{\theta-
x_{i},x_{i}+\theta\\}.$ Let $i=0$ be the index of the unique most extreme
initial belief held by an agent, so $L_{0}<L_{i}$ for $i\neq 0$.

Figure 1: Initial bias determines the choice of early deciders. (A) Evolution
of beliefs of $N=10^{4}$ agents who each have even odds of initially being
unbiased or biased ($P(X_{j}(0)=x_{i})=0.5$, $x_{i}=0,-0.5$). The first agent
(red) decides according to their initial bias, and makes the wrong decision at
$T_{1}\approx 0.01$. The last agent (blue) decides correctly at
$T_{10000}\approx 10$. (B) Probability that the agent with the largest initial
bias decides first as a function of population size, $N$. Solid curves were
determined by numerical quadrature (Eq. (S4)) with initial biases assigned
with uniform probability from values listed in the legend; black crosses
denote results of a stochastic simulation averaged over $10^{6}$ trials.
Inset: Log-log plot of the same results with dashed curves showing the
asymptotic results in Eq. (3). Throughout, agents use identical thresholds
$\pm\theta=\pm 1$, drift $\mu=1$, and diffusivity $D=1$.

For a fixed number of initial beliefs, $I$, the first agent to decide in a
large group is the one with the largest initial bias (Fig. 1A), in the sense
that

$\displaystyle P(X_{n(1)}(0)=x_{0})\to 1\quad\text{as }N\to\infty.$ (2)

More precisely, in the Supplemental Material (SM) we show that

$\displaystyle P(X_{n(1)}(0)=x_{i})\sim\eta_{i}(\ln
N)^{(\beta_{i}-1)/2}N^{1-\beta_{i}}$ (3)

as $N\to\infty$ for each $i\neq 0$, where

$\displaystyle\beta_{i}$ $\displaystyle=(L_{i}/L_{0})^{2}>1,$
$\displaystyle\eta_{i}$
$\displaystyle=\frac{q_{i}}{q_{0}^{\beta_{i}}}\exp\Big{(}\frac{\sqrt{\beta_{i}}}{2D}\big{(}\mu_{i}L_{0}-\mu_{0}L_{i}\big{)}\Big{)}\sqrt{\beta_{i}\pi^{\beta_{i}-1}}\Gamma(\beta_{i})>0,$

and $\mu_{i}=\pm\mu$ if $x_{i}\gtrless 0$. The same statement holds if $n(1)$
is replaced by $n(j)$ in Eq. (3), but with a change in the prefactor,
$\eta_{i}$ (See SM). Thus, the probability that the first decision is not made
by the agent with the most extreme initial belief decreases as a negative
power of the population size $N$ (Fig. 1B). The approximation given by Eq. (3)
is in excellent agreement with the true probabilities when $N\gtrapprox
10^{3}$ (See Fig 1B inset). Moreover, the probability that the agents with the
most extreme initial beliefs make the first decision is close to unity already
for $N\approx 100$ when initial beliefs are well separated and drift is not
too strong.

Figure 2: First decider accuracy is determined by its initial bias. (A) The
accuracy of the first decider as a function of population size, $N,$ for
different initial biases, $y,$ obtained by quadrature. Curves are ordered by
the proximity of the initial bias $y$ of the first decider to the correct
threshold $+\theta$. The drift, and hence the correct decision, are positive.
(B) Under the same assumptions a small deviation from an unbiased initial
belief strongly affects the probability of a correct first decision when $N$
is large. (C) Drift weakly affects the first decision in populations with
biased agents ($y=\theta/4$ here) when $N$ is large. See SM for decision
polarity formulas. (D) In large populations in which all agents have the same
initial bias, $y=\theta/2,$ but different diffusivities, early deciders (here
first and third) have the shortest diffusive timescale. X’s represent averages
of stochastic simulations over $10^{6}$ trials.

The choice of the fastest decider agrees with their initial bias: e.g., if
$\theta$ is the threshold closest to the most extreme initial belief, $x_{0}$,
then $P(X_{n(1)}(T_{1})=\theta)\to 1\text{ as }N\to\infty$ (See Fig. 2A,B).
Similar results hold when initial beliefs are drawn from a continuous
distribution (See SM and next section). Thus, although all agents behave
rationally, early decisions of biased agents tend to be less accurate
Karamched20 ; stickler2023impact .

In contrast, the probability that a single agent - or one chosen randomly
without regard to decision order - decides incorrectly can be made arbitrarily
small by increasing the drift or threshold Bogacz2006 . In large populations
with biased agents, drift and diffusion impact the probability of the first
decision only through the prefactor in Eq. (3), $\eta_{i}$, and thus decrease
in importance as population size diverges. If even a small proportion of a
large population holds an initial bias, early decisions are determined by the
most extreme bias (Fig. 2B) regardless of the drift (Fig. 2C). On the other
hand, if all deciders are initially unbiased ($X_{i}(0)=0$ for all $i$), the
probability the first decider makes a correct choice is
$(1+\exp(-\mu\theta/D))^{-1}$ Bogacz2006 .

### Heterogeneous population and continuous distribution of initial biases.

While we can obtain the most precise asymptotic results in the homogeneous
case, our conclusions extend to populations of agents with heterogeneous
distributions of initial biases, drifts, diffusivities, and thresholds. We
again assume that each agent again starts with one of finitely many initial
beliefs, $X_{i}(0)\in\\{x_{0},x_{1},\ldots,x_{I-1}\\}$ with drift and
diffusivity sampled from a finite set of fixed size. For each agent we define
the diffusive timescale,

$\displaystyle S_{i}=\frac{L_{i}^{2}}{4D_{i}}>0.$ (4)

By assumption, the timescales $S_{i}$ follow a discrete distribution
$\mathbb{P}(S=s_{i})>0$ with support on a finite set $0<s_{0}\leq s_{1}\leq
s_{2}\leq s_{3}\dots\leq s_{J},$ and $S_{n(j)}$ refers to the timescale of the
$j^{\rm th}$ agent to decide (See Fig. 2D). We denote by $s$ the diffusive
timescale of a generic agent.

In large populations, early deciders are those with the shortest diffusive
timescales. In particular, we show in the SM that for every $\varepsilon>0$
and _fixed_ $j\geq 1$,

$\displaystyle N^{1-s_{1}/s_{0}-\varepsilon}\ll\mathbb{P}(S_{n(j)}>s_{0})\ll
N^{1-s_{1}/s_{0}+\varepsilon}\quad\text{as }N\to\infty,$ (5)

where we use the notation $f\ll g$ to mean $\lim_{N\rightarrow\infty}f/g=0$.
We can thus conclude that
$N^{1-s_{1}/s_{0}-\varepsilon}=o\big{(}\mathbb{P}(S_{n(j)}>s_{0})\big{)}$ and
$\mathbb{P}(S_{n(j)}>s_{0})=o\big{(}N^{1-s_{1}/s_{0}+\varepsilon}\big{)}$ as
$N\to\infty$.

These results agree with our earlier conclusion: If all agents share the same
diffusivity, then the fastest deciders are the agents who start closest to
their decision thresholds. This is true regardless of the quality of the
evidence they receive. Diffusivity can reduce the effective distance to the
threshold according to Eq. (4). Thus, the fastest deciders are either those
with the most extreme initial biases or those with the noisiest integration
process, regardless of the drift, $\mu_{i}$. Indeed, how we model drift does
not impact these conclusions, and they hold even if we model the evolution of
beliefs as an Ornstein-Uhlenbeck process, as is frequently done in the
psychophysics literature busemeyer1993decision ; veliz16 .

Figure 3: Late deciders make choices as if they held no initial bias. (A) For
large $N$, decision accuracy monotonically increases with decision order. The
accuracy of late deciders approaches the accuracy of a single, initially
unbiased agent. Here, all agents have initial bias $\theta/3$, and on each
trial, $\mathbb{P}(H=H^{+})=0.5$. (B) In large groups even a large initial
bias has no impact on the decision of later agents. Here, initial biases are
sampled with uniform probability from $(-\theta,\theta)$.

### Late deciders make decisions as if initially unbiased.

We expect in large populations the inaccuracy of early deciders to be balanced
by higher accuracy of late deciders stickler2023impact . Thus, we next
determine the probability that the last agent to decide makes a correct
decision. In the SM we show that this probability has an intuitive form,

$\displaystyle\mathbb{P}(X_{n(N)}(T_{N})=\theta)\to\int_{-\theta}^{\theta}p_{\theta}(x)q(x)\,\textup{d}x\quad\text{as
}N\to\infty.$ (6)

Here $p_{\theta}(x)$ is the probability that a single agent with initial bias
$X(0)=x$ makes a correct decision, and $q(x)$ is the quasi-steady state
distribution 222This is the distribution of beliefs conditioned on the absence
of a decision after a long time. of beliefs evolving according to Eq. (1).
Thus the decision of the last decider is made as if they forget their actual
initial bias and instead sample an initial belief from the quasi-stationary
distribution, $q(x)$.

Eq. (6) is general and can be extended to arbitrary domains. When applied to
the drift-diffusion process with decision boundaries at $\pm\theta$ we show in
the SM that $\mathbb{P}(X_{n(N)}(T_{N})=\theta)\to(1+\exp(-\mu\theta/D))^{-1}$
as $N\to\infty$, which is the probability that a single, initially unbiased
decider makes a correct decision (See Fig. 3A) Bogacz2006 . Thus, the last
decider forgets their initial bias and makes decisions based only on the
accumulated evidence. The probability that an agent with a large initial bias
makes a late decision is small. But should this happen, the initial bias will
have little impact on their decision (See Fig. 3B).

Figure 4: Bias impacts multi-alternative and two-alternative decisions
similarly in large groups. Beliefs about three options evolve on an
equilateral triangle. Here, $\theta$ is the closest distance from the center
of the triangle (burgundy ring) to the boundary. The initial bias is the
distance from the triangle center to the initial belief, $X_{i}(0)$. As $N$
increases, the probability that the most biased agent chooses first grows.
Curves are computed by averaging $10^{6}$ stochastic simulations. Inset:
Sample trajectories from a trial with biases sampled with equal probability
from $\\{\theta/2,\theta/4,\theta/8\\}$. The first agent to decide (red) has
the largest initial bias. The belief of the last decider (blue) explores the
space before reaching a threshold.

### Extension to multiple alternatives.

We can extend these results to decisions between $k$ alternatives. Eq. (1)
again describes the evolution of beliefs, but now
$X_{i}(t),\mu_{i}\in\mathbb{R}^{k-1}$ and $W_{i}$ is a vector of independent
Wiener processes 333We interpret beliefs as log-likelihood ratios. Therefore,
with $k$ alternatives beliefs evolve in $k-1$ dimensions.. Each belief evolves
on a domain, $\Omega\subset\mathbb{R}^{k-1}$, with $k$ boundaries
mcmillen2006dynamics , each associated with one of the alternatives. Agent $i$
chooses alternative $j$ if its belief, $X_{i}(t)$, crosses the associated
boundary first. The boundaries that lead to the best decisions are difficult
to find analytically tajima2019optimal , but their exact shape is immaterial
for our result.

In the SM we show that Eq. (3) holds for general domains in arbitrary
dimensions (See Fig. 4). We therefore reach our earlier conclusions: In large
homogeneous populations, the agents holding the most extreme initial beliefs
make the first decisions, and their choices are consistent with their initial
biases. Our conclusions about the late decisions also carry over to agents
facing multiple choices: The natural extension of Eq. (6) holds with $q(x)$
the quasi-stationary distribution on $\Omega$. The last decider makes a choice
as if it sampled its initial belief from this quasi-stationary distribution.

### Discussion

Our decisions are often influenced by information we obtained previously and
predilections we develop. In drift-diffusion models, prior evidence and
initial inclinations are often represented by a shift in the initial state. We
have shown that initial biases determine early decisions and have a
diminishing impact on later decisions.

An agent unaware of the order of their decision would believe this decision
was made according to the evidence the agent accumulated and that the accuracy
of their choice is determined only by the decision threshold Bogacz2006 .
Though early decisions are not always necessarily less accurate
karamched2020heterogeneity , our work identifies a clear case in which hasty
choices tend to be the most unreliable. Our findings also suggest a means of
weighting choices of biased agents according to decision order in a large
group when formulating collective decisions marshall2017individual . However,
in social groups the exchange of social information between agents
Acemoglu2011 ; mossel2014opinion  or correlations in the evidence
stickler2023impact  will affect these results.

Ramping activity of individual neurons during decision making has been
observed across the brain shadlen1996 ; kiani2008bounded  (although see
latimer2015single ). Such dynamics may reflect the underlying evidence
accumulation process preceding a decision and is often modeled by a drift-
diffusion process. Decisions are thought to be triggered by the elevated
activity of sufficiently many choice-related neurons wang2012neural . Our
results suggest that in large neural populations decisions reflect the most
extreme initial neural states, rather than the accumulated evidence, if the
activity is uncorrelated. Since neural activity is often correlated
cohen2011measuring , the effect of such biases could be tempered.

While we have interpreted our results in the context of social decision
theory, they apply more generally to independently evolving drift-diffusion
processes on bounded domains linn2022extreme : In large populations early
threshold crossings reflect only the initial states, while late crossings are
independent of initial states and reflect the quasi-stationary distribution.
Hence, early crossings reflect initial biases providing fast reactions needed
for deadlined biophysical processes grebenkov2020single . If time allows,
quorum sensing processes that weight passages by order could be used
koriat2012two . Thus, our theory shows how initial biases can be used to
implement population level tradeoffs between speed and accuracy.

##  I Acknowledgements

ZPK and KJ were supported by NSF DMS-2207700. ZPK was supported by NIH BRAIN
1R01EB029847-01. SDL was supported by NSF DMS-2325258 and NSF DMS-1944574. KJ
was supported by NSF-DBI-1707400 and NIH RF1MH130416. SL was supported by NSF
Grant No. 2139322.

Supplemental Material

##  II Mathematical preliminaries

Suppose $\\{(\tau_{n},Z_{n})\\}_{n\geq 1}$ is an independent and identically
distributed (iid) sequence of realizations of the pair of (possibly
correlated) random variables $(\tau,Z)$. We have in mind that $\tau$ is the
decision time (or first passage time (FPT)) of some decider whose stochastic
evolution of beliefs is denoted by $\\{X(t)\\}_{t\geq 0}$ and $Z$ is a vector
containing information about this decider, such as their random initial
position, drift, diffusivity, and decision made. Define the cumulative
distribution function (CDF) of $\tau$,

$\displaystyle F(t):=\mathbb{P}(\tau\leq t).$

Further, for any event $E$ that is in the $\sigma$-algebra generated by $Z$,
define

$\displaystyle F_{E}(t):=\mathbb{P}(\tau\leq t\cap E).$

In words, $E$ is any event for which we can know whether or not it occurred by
knowing $Z$. For example, we are interested in events $E$ like
$E=\\{X(0)=\theta/2\\}$, $E=\\{X(0)\leq 0\\}$, $E=\\{X(\tau)=\theta\\}$, etc.

For a given $N\geq 1$, let $n(j)\in\\{1,\dots,N\\}$ denote the (random) index
of the $j$th fastest decider out of the first $N$ deciders to make a decision.
That is, suppose we order the first $N$ FPTs (or first decision times),

$\displaystyle T_{1,N}\leq T_{2,N}\leq\cdots\leq T_{N-1,N}\leq T_{N,N},$

where $T_{j,N}$ denotes the $j$th fastest FPT,

$\displaystyle
T_{j,N}:=\min\big{\\{}\\{\tau_{1},\dots,\tau_{N}\\}\backslash\cup_{i=1}^{j-1}\\{T_{i,N}\\}\big{\\}},\quad
j\in\\{1,\dots,N\\}.$ (7)

Then $n(j)$ is such that

$\displaystyle\tau_{n(j)}=T_{j,N}.$ (8)

In the examples of interest, the FPTs, $\tau,$ have continuous probability
distributions (i.e. $F(t)$ is a continuous function) so that the event
$\tau_{n^{*}}=\tau_{n^{\prime}}<\infty$ for $n^{*}\neq n^{\prime}$ has
probability zero so there is no ambiguity in Eq. (8).

Since we have the sequence $\\{(\tau_{n},Z_{n})\\}_{n\geq 1}$, we denote
$E_{n}$ the event $E$ as it pertains to the $n$th element in the sequence
$\\{(\tau_{n},Z_{n})\\}_{n\geq 1}$. For example, if $E=\\{X(0)=\theta/2\\}$,
then $E_{n}=\\{X_{n}(0)=\theta/2\\}$. Similarly, $E_{n(j)}$ is the event $E$
as it pertains to $Z_{n(j)}$.

Throughout the Supplemental Material, we use the notation $\int
f(t)\,\textup{d}g(t)$ to denote the Riemann-Stieltjes integral of a function
$f$ with respect to a function $g$.

######  Proposition 1.

For any $j\in\\{1,2,\dots,N\\}$ (denoting an agent by the order $j$ of their
decision), we have that

$\displaystyle\mathbb{P}(E_{n(j)})=j{N\choose
j}\int_{0}^{\infty}[F(t)]^{j-1}[1-F(t)]^{N-j}\,\textup{d}F_{E}(t).$ (9)

In the case $j=1$ (i.e. the fastest decider), Proposition 1 implies

$\displaystyle\mathbb{P}(E_{n(1)})=N\int_{0}^{\infty}[1-F(t)]^{N-1}\,\textup{d}F_{E}(t).$
(10)

Since $1-F$ is a decreasing function, Eq. (10) implies that the short-time
behavior of $F$ and $F_{E}$ determine the large $N$ behavior of
$\mathbb{P}(E_{n(1)})$. More generally, Proposition 1 implies that the short-
time behavior of $F$ and $F_{E}$ determine the large $N$ behavior of
$\mathbb{P}(E_{n(j)})$ for $1\leq j\ll N$.

In the case $j=N$ (i.e. the slowest decider), Proposition 1 implies

$\displaystyle\mathbb{P}(E_{n(N)})=N\int_{0}^{\infty}[F(t)]^{N-1}\,\textup{d}F_{E}(t).$
(11)

Since $F$ is an increasing function, Eq. (11) implies that the large-time
behavior of $F$ and $F_{E}$ determine the large $N$ behavior of
$\mathbb{P}(E_{n(N)})$. More generally, Proposition 1 implies that the large-
time behavior of $F$ and $F_{E}$ determine the large $N$ behavior of
$\mathbb{P}(E_{n(N-j)})$ for $1\ll N-j$.

##  III Some integral asymptotics

The following proposition is useful for estimating the large $N$ behavior of
some integrals of the form in Eq. (9) and was proved in linn2022extreme  (See
Proposition 2 in linn2022extreme ). Throughout the Supplemental Material,
“$f\sim g$” denotes $f/g\to 1$ (e.g., as $N\to\infty$ or as $t\to 0$).

######  Proposition 2.

Assume $C_{+}>C>0$, $A>0$, and $p,q\in\mathbb{R}$. Then there exists a
$\delta_{0}>0$ so that for all $\delta\in(0,\delta_{0}]$, we have

$\displaystyle\int_{0}^{\delta}t^{q-2}e^{-C_{+}/t}\big{(}1-At^{p}e^{-C/t}\big{)}^{N-1}\,\textup{d}t\sim{{\eta}}(\ln
N)^{p{\beta}-q}N^{-{\beta}}\quad\text{as }N\to\infty,$

where

$\displaystyle{\beta}$
$\displaystyle=C_{+}/C>1,\quad{{\eta}}=C^{q-1}(AC^{p})^{-\beta}\Gamma(\beta)>0,$

and $\Gamma({\beta}):=\int_{0}^{\infty}z^{{\beta}-1}e^{-z}\,\textup{d}z$
denotes the gamma function.

The following result estimates integrals of the form in Eq. (9) for $1\leq
j\ll N$ assuming that $F(t)$ and $F_{+}(t)$ have short-time $t$ behavior that
is characteristic of diffusion.

######  Theorem 3.

Assume $F(t)$ and $F_{+}(t)$ are bounded, nondecreasing, continuous from the
right, and satisfy

$\displaystyle F(t)$ $\displaystyle\sim At^{p}e^{-C_{0}/t}\quad\text{as }t\to
0^{+},$ (12) $\displaystyle F_{+}(t)$ $\displaystyle\sim
Bt^{q}e^{-C_{+}/t}\quad\text{as }t\to 0^{+},$ (13)

where $C_{+}>C_{0}>0$, $A>0$, $B>0$, and $p,q\in\mathbb{R}$. Then for any
fixed integer $j\geq 1$, we have

$\displaystyle j{N\choose
j}\int_{0}^{\infty}[F(t)]^{j-1}[1-F(t)]^{N-j}\,\textup{d}F_{+}(t)\sim{{\eta(j)}}(\ln
N)^{p{\beta}-q}N^{1-{\beta}}\quad\text{as }N\to\infty,$

where

$\displaystyle{\beta}$
$\displaystyle:=C_{+}/C_{0}>1,\quad{{\eta(j)}}:=B(C_{0})^{q-p\beta}A^{-\beta}\Gamma(j)\Gamma(\beta+j)>0,$
(14)

and $\Gamma(x):=\int_{0}^{\infty}z^{x-1}e^{-z}\,\textup{d}z$ denotes the gamma
function.

Notice that the asymptotic behavior found in Theorem 3 as $N\to\infty$ is
independent of $j\geq 1$, except for the constant prefactor $\eta(j)$.
Further, this prefactor is an increasing function of $j$ and satisfies

$\displaystyle\eta(j)=\frac{(j-1)!\Gamma(\beta+j)}{\Gamma(\beta+1)}\eta(1),\quad
j\geq 1.$

The asymptotic behavior in Eq. (12)-(13) is typical for diffusion, but
computing the prefactors $A$ and $B$ and the powers $p$ and $q$ can be
challenging lawley2020dist . Indeed, these constants depend on the details of
the system (e.g., drift, space dimension, geometry of the domain, etc.).
However, the constants in the exponents $C_{0}$ and $C_{+}$ are more universal
and can be obtained in a very general mathematical setting Lawley20b . The
following result yields estimates on the fastest deciders when we only know
these constants, which is equivalent to knowing the short-time behavior of
$F_{+}(t)$ and $F(t)$ on a logarithmic scale.

######  Theorem 4.

Assume $F(t)$ and $F_{+}(t)$ are bounded, nondecreasing, continuous from the
right, and satisfy

$\displaystyle\lim_{t\to 0^{+}}t\ln F(t)$
$\displaystyle=-C_{0}<0,\quad\lim_{t\to 0^{+}}t\ln F_{+}(t)\leq-C_{+}<0,$ (15)

where $C_{+}>C_{0}>0$. Then for every $\varepsilon>0$,

$\displaystyle j{N\choose
j}\int_{0}^{\infty}[F(t)]^{j-1}[1-F(t)]^{N-j}\,\textup{d}F_{+}(t)=o(N^{1-{\beta}+\varepsilon})\quad\text{as
}N\to\infty,$ (16)

where

$\displaystyle{\beta}$ $\displaystyle:=C_{+}/C_{0}>1.$

If, in addition, we assume that

$\displaystyle\lim_{t\to 0^{+}}t\ln F_{+}(t)=-C_{+}<0,$ (17)

then for every $\varepsilon>0$,

$\displaystyle N^{1-\beta-\varepsilon}=o\bigg{(}j{N\choose
j}\int_{0}^{\infty}[F(t)]^{j-1}[1-F(t)]^{N-j}\,\textup{d}F_{+}(t)\bigg{)}\quad\text{as
}N\to\infty.$

The following result estimates integrals of the form in Eq. (9) for $1\ll
N-j\leq N$ assuming that $F(t)$ and $f_{i}(t)=F_{i}^{\prime}(t)$ have large-
time $t$ behavior that is characteristic of diffusion in a bounded domain.

######  Theorem 5.

Assume $F(t)\in[0,1)$ is continuous and nondecreasing and $f_{i}(t)$ is
continuous and bounded and

$\displaystyle F(t)$ $\displaystyle=1-ce^{-\lambda
t}+\textup{h.o.t.}\quad\text{as }t\to\infty,$ $\displaystyle f_{i}(t)$
$\displaystyle=\lambda c_{i}e^{-\lambda t}+\textup{h.o.t.}\quad\text{as
}t\to\infty,$

where $\lambda>0$, $c>0$, $c_{i}>0$. Then for any fixed $j\geq 0$, we have
that

$\displaystyle(N-j){N\choose
N-j}\int_{0}^{\infty}[F(t)]^{N-j-1}[1-F(t)]^{j}f_{i}(t)\,\textup{d}t\to\frac{c_{i}}{c}\quad\text{as
}N\to\infty.$

##  IV Proof of Eq. (3) in main text

We now apply Theorem 3 to obtain Eq. (3) in the main text. Suppose the belief
of each agent evolves independently according to the following stochastic
differential equation (SDE),

$\displaystyle\textup{d}X=\mu\,\textup{d}t+\sqrt{2D}\,\textup{d}W,$ (18)

where $\mu\in\mathbb{R}$ is a constant drift, $D>0$ is a constant diffusivity,
and $W=\\{W(t)\\}_{t\geq 0}$ is a standard Brownian motion. Define the FPT,

$\displaystyle\tau:=\inf\\{t>0:X(t)\notin(-\theta,\theta)\\},$

for some threshold $\theta>0$. Assume that the initial distribution
$\mathbb{P}(X(0)=x_{i})$ of each agent is a sum of Dirac masses at a finite
set of points $\\{x_{0},x_{1},\dots,x_{I-1}\\}$,

$\displaystyle\mathbb{P}(X(0)=x)=\begin{cases}q_{i}&\text{if }x=x_{i}\text{
for some }i\in\\{0,1,\dots,I-1\\},\\\ 0&\text{if
}x\notin\cup_{i=0}^{I-1}x_{i}.\end{cases},$

Letting $F_{i}(t)\equiv F_{X(0)=x_{i}}(t)=\mathbb{P}(\tau\leq t\cap
X(0)=x_{i})$, we have that linn2022extreme

$\displaystyle F_{i}(t)\sim q_{i}A_{i}t^{1/2}e^{-C_{i}/t}\quad\text{as }t\to
0^{+},$ (19)

where

$\displaystyle C_{i}=\frac{(L_{i})^{2}}{4D},$

and

$\displaystyle A_{i}=\begin{cases}\exp\big{(}\frac{-\mu
L_{i}}{2D}\big{)}\sqrt{\frac{4D}{\pi(L_{i})^{2}}}&\text{if }x_{i}<0\\\
\exp\big{(}\frac{\mu
L_{i}}{2D}\big{)}\sqrt{\frac{4D}{\pi(L_{i})^{2}}}&\text{if }x_{i}>0\\\
\big{[}\exp\big{(}\frac{-\mu L_{i}}{2D}\big{)}+\exp\big{(}\frac{\mu
L_{i}}{2D}\big{)}\big{]}\sqrt{\frac{4D}{\pi(L_{i})^{2}}}&\text{if
}x_{i}=0,\end{cases}$

where $L_{i}$ is the distance to the closest threshold from $x_{i}$,

$\displaystyle L_{i}=\min\\{\theta-x_{i},x_{i}+\theta\\}.$

Further, we assume ${0}\in\\{0,1,\dots,I-1\\}$ is the index of the unique
starting location closest to a threshold

$\displaystyle L_{{0}}=\min\\{L_{0},L_{1},\dots,L_{I-1}\\}<L_{i}\quad\text{if
}i\neq{0},$

then

$\displaystyle F(t)\sim F_{{0}}(t)\quad\text{as }t\to 0^{+}.$

We claim that

$\displaystyle\mathbb{P}(X_{n(1)}(0)=x_{{0}})\to 1\quad\text{as }N\to\infty,$
(20)

Thus, when $N$ is large the first decider out of many deciders is always the
one with the most extreme initial bias. Using the integral representation in
Proposition 1 and applying Theorem 3 yields

$\displaystyle\mathbb{P}(X_{n(1)}(0)=x_{i})\sim\eta_{i}(1)(\ln
N)^{(\beta_{i}-1)/2}N^{1-\beta_{i}}\quad\text{as }N\to\infty\quad\text{for
each }i\neq{0},$

where

$\displaystyle\beta_{i}$ $\displaystyle=(L_{i}/L_{{0}})^{2}>1,$

and

$\displaystyle\eta_{i}(1)$
$\displaystyle=\begin{cases}\displaystyle\frac{q_{i}}{q_{{0}}^{\beta_{i}}}\sqrt{\frac{\pi^{\beta_{i}-1}}{\beta_{i}}}\Gamma(\beta_{i}+1)\exp\big{(}\frac{\sqrt{\beta_{i}}}{2D}\big{(}\mu_{i}L_{{0}}-\mu_{{0}}L_{i}\big{)}\big{)}&\text{if
}x_{i}\neq 0,\\\
\displaystyle\frac{q_{i}}{q_{{0}}^{\beta_{i}}}\sqrt{\frac{\pi^{\beta_{i}-1}}{\beta_{i}}}\Gamma(\beta_{i}+1)\Big{[}\exp\big{(}\frac{\sqrt{\beta_{i}}}{2D}\big{(}\mu_{i}L_{{0}}-\mu_{{0}}L_{i}\big{)}\big{)}+\exp\big{(}\frac{\sqrt{\beta_{i}}}{2D}\big{(}-\mu_{i}L_{{0}}-\mu_{{0}}L_{i}\big{)}\big{)}\Big{]}&\text{if
}x_{i}=0,\end{cases}$

where $\mu_{i}=\pm\mu$ if $x_{i}\gtrless 0$.

##  V First decision agrees with initial bias

The analysis above shows that the first agent to decide in a large group has
the most extreme initial bias. We now show the intuitive result that this
first decider’s decision agrees with their initial bias. Without loss of
generality, assume that the most extreme initial bias is negative,
$x_{{0}}<0$. Letting $F_{+}(t)=\mathbb{P}(\tau\leq t\cap X(\tau)=+\theta)$, we
have

$\displaystyle F_{+}(t)$ $\displaystyle=\sum_{i}\mathbb{P}(\tau\leq t\cap
X(\tau)=+\theta\,|\,X(0)=x_{i})q_{i}$ $\displaystyle\sim\mathbb{P}(\tau\leq
t\cap X(\tau)=+\theta\,|\,X(0)=x_{i^{+}})q_{i^{+}}$ $\displaystyle\sim
q_{i^{+}}A_{i^{+}}t^{p}_{i^{+}}e^{-C_{i^{+}}/t}\quad\text{as }t\to 0^{+},$

where $i^{+}\in\\{1,\dots,I\\}$ is the index of the starting location closest
to $+\theta$. Using the integral representation in Proposition 1 and applying
Theorem 3 yields

$\displaystyle\mathbb{P}(X_{n(1)}(\tau)=+\theta)\sim\eta_{i^{+}}^{(1)}(\ln
N)^{(\beta_{i^{+}}-1)/2}N^{1-\beta_{i^{+}}}\quad\text{as }N\to\infty.$

##  VI Continuous initial belief distribution

In Section IV, we showed that the first of many deciders have the most extreme
initial beliefs in the case that the population has a discrete initial belief
distribution. We now generalize this calculation to the case that the deciders
have a continuous initial belief distribution. In particular, suppose that the
decider’s initial belief (position) has a smooth probability density $\nu(x)$
with support $(a,b)$ with $-\theta<a<b<\theta$. Suppose that

$\displaystyle\nu(x)$ $\displaystyle\sim(x-a)^{\alpha_{a}}\nu_{a}\quad\text{as
}x\to a^{+},$ $\displaystyle\nu(x)$
$\displaystyle\sim(b-x)^{\alpha_{b}}\nu_{b}\quad\text{as }x\to b^{-},$

where the coefficients are positive, $\nu_{a}>0$, $\nu_{b}>0$, and the powers
ensure that $\nu$ is integrable, $\alpha_{a}>-1$, $\alpha_{b}>-1$. In light of
(19), suppose that

$\displaystyle\mathbb{P}(\tau\leq t\,|\,X(0)=x)\sim
A(x)t^{p}e^{-C(x)/t}\quad\text{as }t\to 0^{+},\quad\text{uniformly for all
}x\in[a,b],$

where

$\displaystyle C(x)=(L(x))^{2}/(4D)>0,\quad L(x)=\min\\{\theta-x,\theta+x\\},$

and $A(x)>0$ for all $x\in[a,b]$.

It follows that

$\displaystyle F(t)=\mathbb{P}(\tau\leq t)$
$\displaystyle=\int_{a}^{b}\mathbb{P}(\tau\leq
t\,|\,X(0)=x)\nu(x)\,\textup{d}x$ $\displaystyle\sim
t^{p}\int_{a}^{b}A(x)\nu(x)e^{-C(x)/t}\,\textup{d}x\quad\text{as }t\to 0^{+}.$

We thus need to estimate the small time $t$ asymptotics of the integral

$\displaystyle I:=\int_{a}^{b}A(x)\nu(x)e^{-C(x)/t}\,\textup{d}x,$

which is an exercise in Laplace’s method bender2013 . If $b>0$, then for any
$\varepsilon\in(0,b)$, we have

$\displaystyle\int_{0}^{b}A(x)\nu(x)e^{-C(x)/t}\,\textup{d}x$
$\displaystyle\sim\int_{b-\varepsilon}^{b}A(x)\nu(x)e^{-C(x)/t}\,\textup{d}x$
$\displaystyle\sim
A(b)e^{-C(b)/t}\nu_{b}\Gamma(\alpha_{b}+1)t^{\alpha_{b}+1}\quad\text{as }t\to
0^{+}.$

Similarly, if $a<0$, then for any $\varepsilon\in(0,|a|)$, we have

$\displaystyle\int_{a}^{0}A(x)\nu(x)e^{-C(x)/t}\,\textup{d}x$
$\displaystyle\sim\int_{a}^{a+\varepsilon}A(x)\nu(x)e^{-C(x)/t}\,\textup{d}x$
$\displaystyle\sim
A(a)e^{-C(a)/t}\nu_{a}\Gamma(\alpha_{a}+1)t^{\alpha_{a}+1}\quad\text{as }t\to
0^{+}.$

Putting this together, we have that if $b>|a|$, then

$\displaystyle F(t)\sim
A(b)\nu_{b}\Gamma(\alpha_{b}+1)t^{p+\alpha_{b}+1}e^{-C(b)/t}\quad\text{as
}t\to 0^{+},$

and similarly if $|a|>b$ or $|a|=b$.

With these estimates, we can apply Theorem 3 to obtain estimates that the
fastest decider(s) have extreme initial beliefs. In particular, suppose we
want to estimate

$\displaystyle\mathbb{P}(a+\varepsilon<X_{n(1)}(0)<b-\varepsilon)\quad\text{for
some small $0<\varepsilon\ll 1$},$

which is the probability that the fastest decider does not have extreme
initial beliefs. If we define the event

$\displaystyle E=\\{a+\varepsilon<X(0)<b-\varepsilon\\},$

then using the notation of Section II, we have that

$\displaystyle F_{E}(t):=\mathbb{P}(\tau\leq t\cap E)$
$\displaystyle=\int_{a+\varepsilon}^{b-\varepsilon}\mathbb{P}(\tau\leq
t\,|\,X(0)=x)\nu(x)\,\textup{d}x$ $\displaystyle\sim
t^{p}\int_{a+\varepsilon}^{b-\varepsilon}A(x)\nu(x)e^{-C(x)/t}\,\textup{d}x\quad\text{as
}t\to 0^{+},$

which can be estimated as above using Laplace’s method bender2013 . In
particular, if $b>|a|$, then

$\displaystyle F_{E}(t)\sim
A(b-\varepsilon)\nu(b-\varepsilon)t^{p+1}e^{-C(b-\varepsilon)/t}\quad\text{as
}t\to 0^{+},$

assuming $\nu(b-\varepsilon)>0$, and similarly if $|a|>b$ or $|a|=b$. With
this short-time behavior of $F_{E}(t)$, we can then plug this into Theorem 3
to show that the first deciders have the most extreme initial beliefs.

##  VII Heterogeneous population with multiple alternatives

We next consider the generalized case where the beliefs of the agents in the
population evolve as processes with (possibly space-dependent) drift,
diffusion coefficient, initial position, and even domain (in their own
arbitrary space dimension $d\geq 1$). Suppose the belief of the $i$th decider
evolves according to the following $d$-dimensional SDE,

$\displaystyle\textup{d}X_{i}=\mu_{i}(X_{i})\,\textup{d}t+\sqrt{2D_{i}}\,\textup{d}W_{i},$
(21)

where $\mu_{i}:\mathbb{R}^{{d}}\to\mathbb{R}^{{d}}$ is a possibly space-
dependent drift, $D_{i}>0$ is the diffusion coefficient, and
$W(t)\in\mathbb{R}^{d}$ is a standard Brownian motion in $d$-dimensional
space.

Let $L>0$ denote an agent’s (random) shortest distance they must travel to hit
the closest target and let $D>0$ denote the agent’s diffusion coefficient.
Define the random timescale

$\displaystyle S=\frac{L^{2}}{4D}>0.$

Suppose that $S$ has a discrete distribution on a finite set

$\displaystyle 0<s_{0}<s_{1}<s_{2}<s_{3}\dots<s_{I},$

where

$\displaystyle\mathbb{P}(S=s_{i})=q_{i}>0,\quad\sum_{i=0}^{I}q_{i}=1.$

Since we have $N\geq 1$ iid agents indexed from $n=1$ to $n=N$, we let $S_{n}$
denote the value of $S$ for the $n$th agent and $S_{n(j)}$ the value of $S$
for the $j$th fastest to decide.

We have that Lawley20b

$\displaystyle\lim_{t\to 0^{+}}t\ln\mathbb{P}(\tau\leq t)$
$\displaystyle=-s_{0}<0,\quad\lim_{t\to 0^{+}}t\ln\mathbb{P}(\tau\leq t\cap
S=s_{i})=-s_{i}<0.$

Hence, Proposition 1 and Theorem 4 imply that for any fixed $j\geq 1$ and
$i\in\\{1,\dots,I\\}$ and any $\varepsilon>0$,

$\displaystyle N^{1-s_{i}/s_{0}-\varepsilon}\ll\mathbb{P}(S_{n(j)}=s_{i})\ll
N^{1-s_{i}/s_{0}^{\varepsilon}}\quad\text{as }N\to\infty,$ (22)

where we use the notation $f\ll g$ to mean $\lim f/g=0$. That is, in more
traditional notation,

$\displaystyle
N^{1-s_{i}/s_{0}-\varepsilon}=o\big{(}\mathbb{P}(S_{n(j)}=s_{i})\big{)}\quad\text{as
}N\to\infty,$
$\displaystyle\mathbb{P}(S_{n(j)}=s_{i})=o\big{(}N^{1-s_{i}/s_{0}+\varepsilon}\big{)}\quad\text{as
}N\to\infty.$

In the special case that the agents all move in one space dimension and the
drifts are spatially constant (but may differ between agents), we can get the
constant and logarithmic prefactors on the decay of
$\mathbb{P}(S_{n(j)}=s_{i})$ as $N\to\infty$.

The result in Eq. (22) says that in a large population if all the agents have
the same diffusion coefficient, then the fastest deciders started closest to
their decision thresholds (targets). If we allow the diffusion coefficients to
vary between agents, then (22) implies that the fastest deciders started close
to their decision thresholds and/or they had big diffusion coefficients.

##  VIII Slowest deciders

Suppose the beliefs of the iid agents diffuse in some $d$-dimensional spatial
domain $U\subset\mathbb{R}^{d}$ and can be absorbed at one of $m\geq 2$
targets $V_{0},\dots,V_{m-1}$ and let $\kappa\in\\{0,\dots,m-1\\}$ indicate
which target the decider eventually hits. Here, we will think of the $m$
targets as parts of the $d-1$ dimensional boundary of the domain, and assume
that hitting one of the targets triggers a decision. Following pavliotis2014 ;
madrid2020comp , suppose the beliefs of the deciders evolve as stochastic
process $\\{X(t)\\}_{t\geq 0}$ that diffuse according to the SDE

$\displaystyle\textup{d}X(t)=-\nabla
V(X(t))\,\textup{d}t+\sqrt{2D}\,\textup{d}W(t),$ (23)

with reflecting boundary conditions. In Eq. (23), the drift term is the
gradient of a given potential, $V(x)$, and the noise term depends on the
diffusion coefficient $D>0$ and a standard $d$-dimensional Brownian motion
(Wiener process) $\\{W(t)\\}_{t\geq 0}$. The survival probability conditioned
on the initial position,

$\displaystyle\mathbf{S}(x,t):=\mathbb{P}(\tau>t\,|\,X(0)=x),$

satisfies the backward Kolmogorov (also called backward Fokker-Planck)
equation,

$\displaystyle\begin{split}\tfrac{\partial}{\partial
t}\mathbf{S}&=\mathcal{L}\mathbf{S},\quad x\in{U},\\\ \mathbf{S}&=0,\quad
x\in\textup{targets},\\\
\tfrac{\partial}{\partial\textbf{n}}\mathbf{S}&=0,\quad x\in\textup{reflecting
boundary (if there is one)},\\\ S&=1,\quad t=0.\end{split}$ (24)

In Eq. (24), the differential operator $\mathcal{L}$ is the generator (i.e.
the backward operator) of Eq. (23),

$\displaystyle\mathcal{L}=-\nabla V(x)\cdot\nabla+D\Delta,$

and $\frac{\partial}{\partial\textbf{n}}$ is the derivative with respect to
the inward unit normal $\textbf{n}:\partial{U}\to\mathbb{R}^{d}$.

Using the following weight function of Boltzmann form ,

$\displaystyle\rho(x):=\frac{e^{-V(x)/D}}{\int_{{U}}e^{-V(y)/D}\,\textup{d}y},$
(25)

one can check that the differential operator $\mathcal{L}$ is formally self-
adjoint on the weighted space of square integrable functions (see, for
example, Ref. pavliotis2014 ),

$\displaystyle
L_{\rho}^{2}({U}):=\Big{\\{}f:\int_{{U}}|f(x)|^{2}\rho(x)\,\textup{d}x<\infty\Big{\\}},$

using the boundary conditions in (24) and the following weighted inner
product,

$\displaystyle(f,g)_{\rho}:=(f,g\rho)=\int_{{U}}f(x)g(x)\rho(x)\,\textup{d}x,$

where $(f,g)=\int_{U}f(x)g(x)\,\textup{d}x$ denotes the standard $L^{2}$-inner
product (i.e. with no weight function). Expanding the solution to (24) yields,

$\displaystyle\mathbf{S}(x,t)=\sum_{n\geq
1}(u_{n},1)_{\rho}e^{-\lambda_{n}t}u_{n}(x)=\sum_{n\geq
1}(u_{n},\rho)e^{-\lambda_{n}t}u_{n}(x),$ (26)

where

$\displaystyle 0<\lambda_{1}<\lambda_{2}\leq\dots,$ (27)

denote the (necessarily positive) eigenvalues of $-\mathcal{L}$. The
corresponding with eigenfunctions $\\{u_{n}(x)\\}_{n\geq 1}$ satisfy the
following time-independent equation,

$\displaystyle\begin{split}-\mathcal{L}u_{n}&=\lambda_{n}u_{n},\quad
x\in{U},\end{split}$ (28)

and identical boundary conditions as $\mathbf{S}$. Further, the eigenfunctions
are orthogonal and are taken to be orthonormal, which means that

$\displaystyle(u_{n},u_{m})_{\rho}=\delta_{nm}\in\\{0,1\\},$ (29)

where $\delta_{nm}$ denotes the Kronecker delta function (i.e. $\delta_{nn}=1$
and $\delta_{mn}=0$ if $n\neq m$).

If the initial distribution of an agent has probability measure $\mu_{0}$,

$\displaystyle\mathbb{P}(X(0)\in
B)=\mu_{0}(B)=\int_{B}1\,\textup{d}\mu_{0}(x),\quad B\subset{U},$ (30)

then the FPT $\tau$ has survival probability given by

$\displaystyle S(t)$
$\displaystyle:=\mathbb{P}(\tau>t\,|\,X(0)=_{\textup{d}}\mu_{0})=\int_{{U}}\mathbf{S}(x,t)\,\textup{d}\mu_{0}(x),$

where the condition $X(0)=_{\textup{d}}\mu_{0}$ in the conditional probability
merely denotes that $X(0)$ has initial distribution given by $\mu_{0}$. Hence,
we obtain the following representation for the survival probability,

$\displaystyle S(t)=\sum_{n\geq 1}A_{n}e^{-\lambda_{n}t}=\sum_{n\geq
1}(u_{n},\rho)(u_{n},\textup{d}\mu_{0})e^{-\lambda_{n}t},$ (31)

where the coefficients are given by the following integrals,

$\displaystyle
A_{n}:=(u_{n},1)_{\rho}\int_{{U}}u_{n}(x)\,\textup{d}\mu_{0}(x),\quad n\geq
1.$ (32)

We have that the FPT $\tau$ to one of the targets has CDF

$\displaystyle F(t)=\mathbb{P}(\tau\leq t)$
$\displaystyle=1-\mathbb{P}(\tau>t)$ $\displaystyle=1-\sum_{k\geq
1}(u_{k},\rho)(u_{k},\textup{d}\mu_{0})e^{-\lambda_{k}t},$

If

$\displaystyle p_{i}(x)=\mathbb{P}(\kappa=i|X(0)=x),$

then

$\displaystyle F_{i}(t):=\mathbb{P}(\tau\leq t\cap\kappa=i)$
$\displaystyle=\mathbb{P}(\kappa=i)-\mathbb{P}(\tau>t\cap\kappa=i)$
$\displaystyle=\mathbb{P}(\kappa=i)-\sum_{k\geq
1}(u_{k},p_{i}\rho)(u_{k},\textup{d}\mu_{0})e^{-\lambda_{k}t},$

and therefore

$\displaystyle f_{i}(t):=F_{i}^{\prime}(t)=\sum_{k\geq
1}\lambda_{k}(u_{k},p_{i}\rho)(u_{k},\textup{d}\mu_{0})e^{-\lambda_{k}t}$

Applying Proposition 1 and Theorem 5 yields

$\displaystyle\mathbb{P}(\kappa_{n(N-j)}=i)\to\frac{(u_{1},p_{i}\rho)}{(u_{1},\rho)}=\frac{(u_{1}\rho,p_{i})}{(u_{1},\rho)}\quad\text{as
}N\to\infty.$

Now, the solution to the forward Fokker-Planck equation is given by

$\displaystyle p(x,t)=\mathbb{P}(X(t)=\textup{d}x\,|\,\tau>t)=\sum_{k\geq
1}e^{-\lambda_{k}t}(u_{k},\textup{d}\mu_{0})\rho(x)u_{k}(x).$

Hence, $u_{1}(x)\rho(x)/(u_{1},\rho)$ is the quasi-stationary distribution
(QSD), $q(x),$ defined by

$\displaystyle q(x)=\lim_{t\to\infty}\mathbb{P}(X(t)=\textup{d}x\,|\,\tau>t)$
$\displaystyle=\lim_{t\to\infty}\frac{\mathbb{P}(X(t)=\textup{d}x\cap\tau>t)}{\mathbb{P}(\tau>t)}$
$\displaystyle=\lim_{t\to\infty}\frac{\sum_{k\geq
1}e^{-\lambda_{k}t}(u_{k},\textup{d}\mu_{0})\rho(x)u_{k}(x)}{\sum_{k\geq
1}(u_{k},\rho)(u_{k},\textup{d}\mu_{0})e^{-\lambda_{k}t}}$
$\displaystyle=\lim_{t\to\infty}\frac{e^{-\lambda_{1}t}(u_{1},\textup{d}\mu_{0})\rho(x)u_{1}(x)}{(u_{1},\rho)(u_{1},\textup{d}\mu_{0})e^{-\lambda_{1}t}}$
$\displaystyle=\frac{\rho(x)u_{1}(x)}{(u_{1},\rho)}.$

Summarizing, we have shown that

$\displaystyle\mathbb{P}(\kappa_{n(N-j)}=i)\to\int_{U}p_{i}(x)q(x)\,\textup{d}x\quad\text{as
}N\to\infty.$ (33)

### The case of drift-diffusion processes in one dimension.

For the one-dimensional example in which all the beliefs of all the agents
evolve according to (18), we can compute the QSD, and find that

$\displaystyle
q(x)=\frac{\left(\pi^{2}D^{2}+\theta^{2}\mu^{2}\right)\cos(\frac{\pi
x}{2\theta})e^{\frac{\mu(\theta+x)}{2D}}}{2\pi
D^{2}\theta(e^{\frac{\theta\mu}{D}}+1)}.$

Further, it is straightforward to show that the probability that a decider
reaches $+\theta$ before $-\theta$ conditioned on the initial belief
$x\in[-\theta,\theta]$ is

$\displaystyle
p_{1}(x):=\mathbb{P}(X(\tau)=+\theta)=\frac{1}{2}\Big{(}\coth\Big{(}\frac{\theta\mu}{D}\Big{)}-1\Big{)}e^{\frac{\mu(\theta-x)}{D}}\left(e^{\frac{\mu(\theta+x)}{D}}-1\right)$

Therefore, applying (33) and explicitly computing the integral yields

$\displaystyle\mathbb{P}(\kappa_{n(N-j)}=1)\to\int_{-\theta}^{\theta}p_{1}(x)q(x)\,\textup{d}x=\frac{1}{1+e^{-\frac{\theta\mu}{D}}}=p_{1}(0)\quad\text{as
}N\to\infty.$

Hence, the slowest deciders out of $N\gg 1$ deciders make a decision as if
they were initially unbiased (i.e. as if $X(0)=0$).

##  IX Proofs

###### Proof of Proposition 1.

Since $\\{(\tau_{n},Z_{n})\\}_{n\geq 1}$ are identically distributed, we have
that

$\displaystyle\begin{split}\mathbb{P}(A_{n(j)})&=\sum_{\textup{distinct
indices}\atop
n_{1},\dots,n_{N}\in\\{1,\dots,N\\}}\mathbb{P}(\max\\{\tau_{n_{1}},\dots,\tau_{n_{j-1}}\\}<\tau_{n_{j}}<\min\\{\tau_{n_{j+1}},\dots,\tau_{n_{N}}\\}\cap
A_{n_{j}})\\\ &=j{N\choose
j}\mathbb{P}(\max\\{\tau_{1},\dots,\tau_{j-1}\\}<\tau_{j}<\min\\{\tau_{j+1},\dots,\tau_{N}\\}\cap
A_{j}),\end{split}$ (34)

where the coefficient comes from noting that the number of terms in the sum is
obtained by choosing the $j$ fastest FPTs out of $N$ and then choosing which
of those $j$ will be the $j$th fastest. Define

$\displaystyle\tau_{j}^{(A_{j})}=\begin{cases}\tau_{j}&\text{if $A_{j}$
occurs},\\\ +\infty&\text{if $A_{j}$ does not occur},\end{cases}$

so that if $j<N$,

$\displaystyle\mathbb{P}(\max\\{\tau_{1},\dots,\tau_{j-1}\\}<\tau_{j}<\min\\{\tau_{j+1},\dots,\tau_{N}\\}\cap
A_{j})$
$\displaystyle\quad=\mathbb{P}(\max\\{\tau_{1},\dots,\tau_{j-1}\\}<\tau_{j}^{(A_{j})}<\min\\{\tau_{j+1},\dots,\tau_{N}\\})$

To handle the case $j=N$, we can simply replace $+\infty$ by $-\infty$ in the
definition of $\tau_{j}^{(A_{j})}$.

Since $\\{\tau_{n}\\}_{n\geq 1}$ are iid, we have that

$\displaystyle\mathbb{P}(\max\\{\tau_{1},\dots,\tau_{j-1}\\}<t)=\mathbb{P}(\max\\{\tau_{1},\dots,\tau_{j-1}\\}\leq
t)=[F(t)]^{j-1},$

where we have used that $F(t)$ is continuous. Similarly,

$\displaystyle\mathbb{P}(\min\\{\tau_{j+1},\dots,\tau_{N}\\}>t)=[1-F(t)]^{N-j},$

Using that $\\{\tau_{n}\\}_{n\geq 1}$ are independent, we have

$\displaystyle G(t):$
$\displaystyle=\mathbb{P}(\max\\{\tau_{1},\dots,\tau_{j-1}\\}<t<\min\\{\tau_{j+1},\dots,\tau_{N}\\})$
$\displaystyle=\mathbb{P}(\max\\{\tau_{1},\dots,\tau_{j-1}\\}<t)\mathbb{P}(t<\min\\{\tau_{j+1},\dots,\tau_{N}\\})$
$\displaystyle=[F(t)]^{j-1}[1-F(t)]^{N-j}.$

Combining the above finally yields

$\displaystyle\mathbb{P}(A_{n(j)})$ $\displaystyle=j{N\choose
j}\mathbb{P}(\max\\{\tau_{1},\dots,\tau_{j-1}\\}<\tau_{j}^{(A_{j})}<\min\\{\tau_{j+1},\dots,\tau_{N}\\})$
$\displaystyle=j{N\choose j}\mathbb{E}[G(\tau_{j}^{(A_{j})})]$
$\displaystyle=j{N\choose
j}\int_{0}^{\infty}[F(t)]^{j-1}[1-F(t)]^{N-j}\,\textup{d}F_{E}(t),$

which completes the proof. ∎

The proof of Theorem 3 is similar to the proof of Theorem 3 in linn2022extreme
.

###### Proof of Theorem 3.

Define the integral from $t=a$ to $t=b$,

$\displaystyle
I_{a,b}:=\int_{a}^{b}[F(t)]^{j-1}[1-F(t)]^{N-j}\,\textup{d}F_{+}(t).$

Let $\varepsilon\in(0,1)$. By the assumptions in Eq. (12)-(13), there exists a
$\delta>0$ so that

$\displaystyle A_{-\varepsilon}t^{p}e^{-C_{0}/t}$ $\displaystyle\leq F(t)\leq
A_{+\varepsilon}t^{p}e^{-C_{0}/t}\quad\text{for all }t\in(0,\delta),$ (35)
$\displaystyle B_{-\varepsilon}t^{q}e^{-C_{+}/t}$ $\displaystyle\leq
F_{+}(t)\leq B_{+\varepsilon}t^{q}e^{-C_{+}/t}\quad\text{for all
}t\in(0,\delta),$ (36)

where $A_{\pm\varepsilon}:=A(1\pm\varepsilon)$ and
$B_{\pm\varepsilon}:=B(1\pm\varepsilon)$. Using Eq. (35) and integrating by
parts yields

$\displaystyle\begin{split}&I_{0,\delta}\leq\int_{0}^{\delta}(A_{+\varepsilon}t^{p}e^{-C_{0}/t})^{j-1}(1-A_{-\varepsilon}t^{p}e^{-C_{0}/t})^{N-j}\,\textup{d}F_{+}(t)\\\
&=(A_{+\varepsilon}\delta^{p}e^{-C_{0}/\delta})^{j-1}(1-A_{-\varepsilon}\delta^{p}e^{-C_{0}/\delta})^{N-j}F_{+}(\delta)\\\
&+(N-j)\int_{0}^{\delta}(A_{+\varepsilon}t^{p}e^{-C_{0}/t})^{j-1}(pt^{-1}+C_{0}t^{-2})A_{-\varepsilon}t^{p}e^{-C_{0}/t}\big{(}1-A_{-\varepsilon}t^{p}e^{-C_{0}/t}\big{)}^{N-j-1}F_{+}(t)\,\textup{d}t\\\
&-(j-1)\int_{0}^{\delta}(A_{+\varepsilon}t^{p}e^{-C_{0}/t})^{j-1}(pt^{-1}+C_{0}t^{-2})\big{(}1-A_{-\varepsilon}t^{p}e^{-C_{0}/t}\big{)}^{N-j}F_{+}(t)\,\textup{d}t\\\
&\leq(A_{+\varepsilon}\delta^{p}e^{-C_{0}/\delta})^{j-1}(1-A_{-\varepsilon}\delta^{p}e^{-C_{0}/\delta})^{N-j}F_{+}(\delta)\\\
&+(N-j)\int_{0}^{\delta}(A_{+\varepsilon}t^{p}e^{-C_{0}/t})^{j}(pt^{-1}+C_{0}t^{-2})\big{(}1-A_{-\varepsilon}t^{p}e^{-C_{0}/t}\big{)}^{N-j-1}B_{+\varepsilon}t^{q}e^{-C_{+}/t}\,\textup{d}t\\\
&-(j-1)\int_{0}^{\delta}(A_{+\varepsilon}t^{p}e^{-C_{0}/t})^{j-1}(pt^{-1}+C_{0}t^{-2})\big{(}1-A_{-\varepsilon}t^{p}e^{-C_{0}/t}\big{)}^{N-j}B_{-\varepsilon}t^{q}e^{-C_{+}/t}\,\textup{d}t,\end{split}$
(37)

where we have used Eq. (36) in the final inequality. The first term in the
righthand side of Eq. (37) vanishes exponentially fast as $N\to\infty$. Using
Proposition 2 to find the large $N$ behavior of the second two terms in the
righthand side of Eq. (37) and the fact that $I_{\delta,\infty}$ vanishes
exponentially fast as $N\to\infty$ yields

$\displaystyle\limsup_{N\to\infty}\frac{j{N\choose
j}I_{0,\infty}}{{{\eta_{j}}}(\ln
N)^{p{\beta}-q}N^{1-{\beta}}}\leq\frac{(1+\varepsilon)}{(1-\varepsilon)^{\beta}}.$

The analogous argument yields the lower bound

$\displaystyle\liminf_{N\to\infty}\frac{j{N\choose
j}I_{0,\infty}}{{{\eta_{j}}}(\ln
N)^{p{\beta}-q}N^{1-{\beta}}}\geq\frac{(1-\varepsilon)}{(1+\varepsilon)^{\beta}}.$

Since $\varepsilon\in(0,1)$ is arbitrary, the proof is complete. ∎

###### Proof of Theorem 4.

Define the integral from $t=a$ to $t=b$,

$\displaystyle
I_{a,b}:=\int_{a}^{b}[F(t)]^{j-1}[1-F(t)]^{N-j}\,\textup{d}F_{+}(t).$

By Eq. (15), there exists a $\delta>0$ so that

$\displaystyle e^{-(C_{0}+\varepsilon)/t}\leq F(t)$ $\displaystyle\leq
e^{-(C_{0}-\varepsilon)/t}\quad\text{for all }t\in(0,\delta),$ (38)
$\displaystyle F_{+}(t)$ $\displaystyle\leq
e^{-(C_{+}-\varepsilon)/t}\quad\text{for all }t\in(0,\delta).$ (39)

Using Eq. (38) and integrating by parts yields

$\displaystyle\begin{split}I_{0,\delta}&\leq\int_{0}^{\delta}e^{-(j-1)(C_{0}-\varepsilon)/t}\big{(}1-e^{-(C_{0}+\varepsilon)/t}\big{)}^{N-j}\,\textup{d}F_{+}(t)\\\
&=F_{+}(\delta)e^{-(j-1)(C_{0}-\varepsilon)/\delta}\big{(}1-e^{-(C_{0}+\varepsilon)/\delta}\big{)}^{N-j}\\\
&\quad+(N-j)(C_{0}+\varepsilon)\int_{0}^{\delta}e^{-(j-1)(C_{0}-\varepsilon)/t}t^{-2}e^{-(C_{0}+\varepsilon)/t}\big{(}1-e^{-(C_{0}+\varepsilon)/t}\big{)}^{N-j-1}F_{+}(t)\,\textup{d}t\\\
&\quad-\int_{0}^{\delta}(j-1)(C_{0}-\varepsilon)t^{-2}e^{-(j-1)(C_{0}-\varepsilon)/t}\big{(}1-e^{-(C_{0}+\varepsilon)/t}\big{)}^{N-j}F_{+}(t)\,\textup{d}t.\end{split}$
(40)

The first term in the righthand side of Eq. (40) vanishes exponentially fast
as $N\to\infty$. To handle the second term in the righthand side of Eq. (40),
note that Eq. (39) implies that

$\displaystyle\begin{split}&\int_{0}^{\delta}e^{-(j-1)(C_{0}-\varepsilon)/t}t^{-2}e^{-(C_{0}+\varepsilon)/t}\big{(}1-e^{-(C_{0}+\varepsilon)/t}\big{)}^{N-j-1}F_{+}(t)\,\textup{d}t\\\
&\quad\leq\int_{0}^{\delta}e^{-(j-1)(C_{0}-\varepsilon)/t}t^{-2}e^{-(C_{0}+\varepsilon)/t}\big{(}1-e^{-(C_{0}+\varepsilon)/t}\big{)}^{N-j-1}e^{-(C_{+}-\varepsilon)/t}\,\textup{d}t.\end{split}$
(41)

Since the third term in the righthand side of Eq. (40) is nonpositive,
applying Proposition 2 to Eq. (41) and using Eq. (40) and the fact that
$I_{\delta,\infty}$ vanishes exponentially fast as $N\to\infty$ completes the
proof of Eq. (16).

If Eq. (17) holds, then there exists a $\delta>0$ so that

$\displaystyle\begin{split}e^{-(C_{0}+\varepsilon)/t}\leq F(t)&\leq
e^{-(C_{0}-\varepsilon)/t}\quad\text{for all }t\in(0,\delta),\\\
e^{-(C_{+}+\varepsilon)/t}\leq F_{+}(t)&\leq
e^{-(C_{+}-\varepsilon)/t}\quad\text{for all }t\in(0,\delta).\end{split}$ (42)

Using Eq. (42) and integrating by parts yields

$\displaystyle\begin{split}I_{0,\delta}&\geq\int_{0}^{\delta}e^{-(j-1)(C_{0}+\varepsilon)/t}\big{(}1-e^{-(C_{0}-\varepsilon)/t}\big{)}^{N-j}\,\textup{d}F_{+}(t)\\\
&=F_{+}(\delta)e^{-(j-1)(C_{0}+\varepsilon)/\delta}\big{(}1-e^{-(C_{0}-\varepsilon)/\delta}\big{)}^{N-j}\\\
&\quad+(N-j)(C_{0}-\varepsilon)\int_{0}^{\delta}e^{-(j-1)(C_{0}+\varepsilon)/t}t^{-2}e^{-(C_{0}-\varepsilon)/t}\big{(}1-e^{-(C_{0}-\varepsilon)/t}\big{)}^{N-j-1}F_{+}(t)\,\textup{d}t\\\
&\quad-\int_{0}^{\delta}(j-1)(C_{0}+\varepsilon)t^{-2}e^{-(j-1)(C_{0}+\varepsilon)/t}\big{(}1-e^{-(C_{0}-\varepsilon)/t}\big{)}^{N-j}F_{+}(t)\,\textup{d}t\\\
&\geq
F_{+}(\delta)e^{-(j-1)(C_{0}+\varepsilon)/\delta}\big{(}1-e^{-(C_{0}-\varepsilon)/\delta}\big{)}^{N-j}\\\
&\quad+(N-j)(C_{0}-\varepsilon)\int_{0}^{\delta}e^{-(j-1)(C_{0}+\varepsilon)/t}t^{-2}e^{-(C_{0}-\varepsilon)/t}\big{(}1-e^{-(C_{0}-\varepsilon)/t}\big{)}^{N-j-1}e^{-(C_{+}+\varepsilon)/t}\,\textup{d}t\\\
&\quad-\int_{0}^{\delta}(j-1)(C_{0}+\varepsilon)t^{-2}e^{-(j-1)(C_{0}+\varepsilon)/t}\big{(}1-e^{-(C_{0}-\varepsilon)/t}\big{)}^{N-j}e^{-(C_{+}-\varepsilon)/t}\,\textup{d}t.\end{split}$
(43)

The first term in the righthand side of Eq. (43) vanishes exponentially as
$N\to\infty$. Using Proposition 2 to estimate the second two terms in the
righthand side of Eq. (43) completes the proof. ∎

######  Lemma 6.

For fixed $j\in\\{0,1,\dots\\}$, $c>0$, $\lambda>0$, and $\delta>0$, we have
that

$\displaystyle(N-j){N\choose N-j}\int_{1/\delta}^{\infty}\big{[}1-ce^{-\lambda
t}]^{N-j-1}e^{-(j+1)\lambda t}\,\textup{d}t\to\frac{1}{\lambda
c^{j+1}}\quad\text{as }N\to\infty.$

###### Proof of Lemma 6.

Changing variables

$\displaystyle u=1-ce^{-\lambda t},\quad\textup{d}u=\lambda ce^{-\lambda
t}\,\textup{d}t$

yields

$\displaystyle\int_{1/\varepsilon}^{\infty}\big{[}1-ce^{-\lambda
t}]^{N-j-1}e^{-(j+1)\lambda t}\,\textup{d}t$ $\displaystyle=\frac{1}{\lambda
c^{j+1}}\int_{1-ce^{-\lambda t}}^{1}u^{N-j-1}(1-u)^{j}\,\textup{d}u$
$\displaystyle=\frac{1}{\lambda
c^{j+1}}\bigg{[}\frac{(N-j-1)!j!}{N!}-\int_{0}^{1-ce^{-\lambda
t}}u^{N-j-1}(1-u)^{j}\,\textup{d}u\bigg{]},$ (44)

where we have used that
$\int_{0}^{1}u^{a-1}(1-u)^{b-1}\,\textup{d}u=\Gamma(a)\Gamma(b)/\Gamma(a+b)$.
Since the integral in Eq. (44) vanishes exponentially fast, the proof is
complete. ∎

###### Proof of Theorem 5.

Let $\varepsilon\in(0,1)$. By assumption, there exists $\delta>0$ so that

$\displaystyle 1-(1+\varepsilon)ce^{-\lambda t}\leq F(t)\leq
1-(1-\varepsilon)ce^{-\lambda t}\quad\text{for all }t\geq 1/\delta,$
$\displaystyle\lambda(1-\varepsilon)c_{i}e^{-\lambda t}\leq
f_{i}(t)\leq\lambda(1+\varepsilon)c_{i}e^{-\lambda t}\quad\text{for all }t\geq
1/\delta.$

Defining the integral from $t=a$ to $t=b$,

$\displaystyle
I_{a,b}:=\int_{a}^{b}[F(t)]^{N-j-1}[1-F(t)]^{j}f_{i}(t)\,\textup{d}t,$

we therefore have that

$\displaystyle(1-\varepsilon)^{j+1}\lambda
c_{i}c^{j}\int_{1/\delta}^{\infty}\big{[}1-(1+\varepsilon)ce^{-\lambda
t}]^{N-j-1}e^{-(j+1)\lambda t}\,\textup{d}t\leq I_{1/\delta,\infty}$
$\displaystyle\quad\leq(1+\varepsilon)^{j+1}\lambda
c_{i}c^{j}\int_{1/\delta}^{\infty}\big{[}1-(1-\varepsilon)ce^{-\lambda
t}]^{N-j-1}e^{-(j+1)\lambda t}\,\textup{d}t.$

Since $I_{0,1/\delta}$ vanishes exponentially fast as $N\to\infty$, Lemma 6
implies that

$\displaystyle\Big{(}\frac{1-\varepsilon}{1+\varepsilon}\Big{)}^{j+1}\frac{c_{i}}{c}$
$\displaystyle\leq\liminf_{N\to\infty}(N-j){N\choose N-j}I_{0,\infty}$
$\displaystyle\leq\limsup_{N\to\infty}(N-j){N\choose
N-j}I_{0,\infty}\leq\Big{(}\frac{1+\varepsilon}{1-\varepsilon}\Big{)}^{j+1}\frac{c_{i}}{c}.$

Since $\varepsilon\in(0,1)$ is arbitrary, the proof is complete. ∎

##  X Numerical solutions

Numerical solutions were computed via trapezoidal quadrature on Eq. (9) in
Proposition 1. In each set of dynamics, we rescaled the drift-diffusion
process on $[-\theta,\theta]$ to the interval $[0,\ell]$. The probability
density function for hitting the left boundary in this system is navarro2009

$\displaystyle
f_{0}(t):=\frac{\textup{d}}{\textup{d}t}F_{0}(t)=\textup{exp}\Big{(}-\frac{\mu
x_{0}}{2D}-\frac{\mu^{2}t}{4D}\Big{)}\frac{D}{\ell^{2}}\phi\Big{(}\frac{Dt}{\ell^{2}},\frac{x_{0}}{\ell}\Big{)},$
(45)

where

$\displaystyle\phi(s,w):=\begin{cases}\sum_{k=1}^{\infty}\textup{exp}(-k^{2}\pi^{2}s)2k\pi\sin(k\pi
w),\\\ (4\pi
s^{3})^{-1/2}\sum_{k=-\infty}^{\infty}(w+2k)\textup{exp}\Big{(}-\frac{(w+2k)^{2}}{4s}\Big{)}.\end{cases}$
(46)

The expressions in Eq. (46) are equivalent but have distinct utility: the top
expansion converges quickly for large $s$ while the bottom expansion converges
quickly for small $s$. Hence, we utilize both expressions to more accurately
compute probabilities associated with slow and fast deciders, respectively.

Integrating Eq. (45) yields

$\displaystyle
F_{0}(t)=\int_{0}^{t}f_{0}(t^{\prime})\,\textup{d}t^{\prime}=\textup{exp}\Big{(}-\frac{\mu
x_{0}}{2D}\Big{)}\Phi\Big{(}\frac{Dt}{\ell^{2}},\frac{x_{0}}{\ell}\Big{)}$

with long- and short-time expansions of $\Phi(s,w)$ given by

$\displaystyle\Phi(s,w)$
$\displaystyle=\int_{0}^{s}\phi(s^{\prime},w)\,\textup{d}s^{\prime}$
$\displaystyle=\begin{cases}\sum_{k=1}^{\infty}\Big{(}1-\textup{exp}[-(b+k^{2}\pi^{2})s]\Big{)}\frac{2k\pi}{b+k^{2}\pi^{2}}\sin(k\pi
w),\\\
\sum_{k=-\infty}^{\infty}\frac{\textup{sgn}(2k+w)}{2}\Big{(}e^{-\sqrt{\frac{b}{D}}|2k+w|}\textup{erfc}\big{(}\frac{|2k+w|}{\sqrt{4s}}-\sqrt{bs}\big{)}+e^{\sqrt{\frac{b}{D}}|2k+w|}\textup{erfc}\big{(}\frac{|2k+w|}{\sqrt{4s}}+\sqrt{bs}\big{)}\Big{)}\end{cases}$

where $b=(\mu\ell/2D)^{2}$. By symmetry one can determine the corresponding
probability density and cumulative distribution functions for hitting the
right boundary. Altogether, we acquire long- and short-time expressions for
the cumulative distribution function of an agent making a decision,

$\displaystyle F(t):=F_{0}(t)+F_{1}(t).$

Where numerical solutions are illustrated, we use the short-time expressions
of $\phi$ and $\Phi$ for $10^{-10}\leq t\leq 1$ and the complementary long-
time expressions for $1<t\leq 100$, discretizing each time interval into
$10^{3}$ log-spaced points. We consider $10^{3}$ terms in each series
expansion. Moreover, we take $\ell=1$ and unless otherwise stated $D=1$.
Finally, where more than one but finitely many initial beliefs are considered,
we scale the probability functions according to the corresponding initial
distribution as outlined in Section IV.

Specific details of figures with numerical solutions are as follows: In Fig 1B
we illustrate in color Eq. (10) where $F_{E}=F$ as defined above with
$X_{n(1)}(0)=y$. The black curve, which contains the remaining mass of the
total probability, is computed as the sum of the colored curves subtracted
from one. In Fig 2A-C we illustrate the probability that the first decider
chooses the decision at $X(T_{1})=\theta$ conditioned on having a particular
initial bias. Hence, by definition of conditional probability, the numerical
solutions are produced from quadrature on ratios of Eq. (10) with
$F_{E}=F_{1}$ in the numerator and $F_{E}=F$ in the denominator with
$X_{n(1)}=y$. The inset of Fig 2C is one minus the outset. In Fig 2D we
illustrate Eq. (9) where $F_{E}=F$ and $S_{n(j)}=s$. In Fig 3B we illustrate
the probability that the last decider chooses the decision at
$X(T_{N})=\theta$ conditioned on having a particular initial bias. Similar to
Fig 2B, the numerical solutions are produced from quadrature on ratios of Eq.
(11) with $F_{E}=F_{1}$ in the numerator and $F_{E}=F$ in the denominator with
$X_{n(N)}(0)=y$.

##  XI Agent-Based stochastic simulations

a. One-dimensional drift diffusion equation. To test the analytical solutions,
we solved Eq. (1) in the main text using the Euler-Maruyama method, which
describes the evidence accumulation process preceding binary decisions. In
this approximation scheme, the true solution to the stochastic differential
equation is approximated by a Markov chain $Y$ constructed by setting
$Y_{0}=X(0)$ and updating $Y$ according to the iterative scheme

$Y_{n+1}=Y_{n}+\mu\Delta t+\sqrt{2D}\Delta W$

where $Y_{n}\equiv Y(n\Delta t)$ is the value of the Markov chain after the
$n$th update, and the random variables $\Delta W$ are independent and
identically distributed Gaussian random variables with mean 0 and variance
$\Delta t$. The equations were integrated until the value of $Y_{n}$ exceeded
$\pm\theta$.

The temporal discretization, $\Delta t,$ is user-defined. As $N$ grows, the
time to first decision decays slowly. Thus, for large $N$, $\Delta t$ must be
taken to be sufficiently small for accurate representation of decision
dynamics. For simulations here, we chose $\Delta t=10^{-3}$ for $1\leq N\leq
1000$. For $N>1000$, we chose $\Delta t=N^{-1}$.

b. Two-dimensional drift diffusion equation. Decisions between three choices
require a drift-diffusion model evolving on a planar domain
mcmillen2006dynamics . Updating the discrete-time approximation of Eq. (21)
for each observer (dropping the $i$ subscript) using Euler-Maruyama provides
the following iterative scheme

$\displaystyle Y_{n+1}^{1}$ $\displaystyle=Y_{n}^{1}+\mu^{1}\Delta
t+\sqrt{2D}\Delta W^{1},$ $\displaystyle Y_{n+1}^{2}$
$\displaystyle=Y_{n}^{2}+\mu^{2}\Delta t+\sqrt{2D}\Delta W^{2},$

where $Y_{n}^{j}=Y^{j}(n\Delta t)$ is the value of the belief after the $n$th
update, the random variables $\Delta W^{j}$ are Gaussian random variables with
mean 0 and variance $\sigma^{2}$. Equations are integrated until the vector
$\left(\begin{array}[]{c}Y_{n}^{1}\\\ Y_{n}^{2}\end{array}\right)$ departs the
triangular domain

$\displaystyle\\{(Y^{1},Y^{2})|Y^{2}<h\ \&\ Y^{2}>-2h(3Y^{1}+1)\ \&\
Y^{2}>2h(3Y^{1}-1)\\}$

where $h=1/2/\sqrt{3}$. Choices of each agent are determined by whether the
agent crosses the $Y^{2}=h$ or $Y^{2}=-2h(3Y^{1}+1)$ or $Y^{2}=2h(3Y^{1}-1)$
boundary. For simulations again we use $\Delta t=10^{-3}$ for $1\leq N\leq
1000$ and $\Delta t=N^{-1}$ for $N>1000$.

For the 2D case in an equilateral triangle, the threshold $\theta$ is taken to
be equal to the length of the apothem—defined as a line from the center of a
regular polygon at right angles to any of its sides. Hence, an unbiased agent
begins at the centre of the equilateral triangle. We prescribe initial data
for biased agents to be anywhere along an apothem except the centre of the
triangle.

## References

  * [1] Daron Acemoglu, Munther A Dahleh, Ilan Lobel, and Asuman Ozdaglar.  Bayesian learning in social networks.  The Review of Economic Studies, 78(4):1201–1236, 2011.
  * [2] CM Bender and SA Orszag.  Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory.  Springer Science & Business Media, 2013.
  * [3] Rafal Bogacz, Eric Brown, Jeff Moehlis, Philip Holmes, and Jonathan D. Cohen.  The physics of optimal decision making: A formal analysis of models of performance in two-alternative forced-choice tasks.  Psychological Review, 113(4):700–765, 2006.
  * [4] Jerome R Busemeyer and James T Townsend.  Decision field theory: a dynamic-cognitive approach to decision making in an uncertain environment.  Psychological review, 100(3):432, 1993.
  * [5] Lars Chittka, Adrian G Dyer, Fiola Bock, and Anna Dornhaus.  Bees trade off foraging speed for accuracy.  Nature, 424(6947):388–388, 2003.
  * [6] Marlene R Cohen and Adam Kohn.  Measuring and interpreting neuronal correlations.  Nature neuroscience, 14(7):811–819, 2011.
  * [7] Joshua I Gold and Michael N Shadlen.  Banburismus and the brain: decoding the relationship between sensory stimuli, decisions, and reward.  Neuron, 36(2):299–308, 2002.
  * [8] Joshua I Gold and Michael N Shadlen.  The neural basis of decision making.  Annu. Rev. Neurosci., 30:535–574, 2007.
  * [9] Denis S Grebenkov, Ralf Metzler, and Gleb Oshanin.  From single-particle stochastic kinetics to macroscopic reaction rates: fastest first-passage time of n random walkers.  New Journal of Physics, 22(10):103004, 2020.
  * [10] Bhargav Karamched, Megan Stickler, William Ott, Benjamin Lindner, Zachary P Kilpatrick, and Krešimir Josić.  Heterogeneity improves speed and accuracy in social networks.  Physical review letters, 125(21):218302, 2020.
  * [11] Bhargav Karamched, Simon Stolarczyk, Zachary P Kilpatrick, and Krešimir Josić.  Bayesian evidence accumulation on social networks.  SIAM Journal on Applied Dynamical Systems, 19(3):1884–1919, 2020\.
  * [12] Roozbeh Kiani, Timothy D Hanks, and Michael N Shadlen.  Bounded integration in parietal cortex underlies decisions even when viewing duration is dictated by the environment.  Journal of Neuroscience, 28(12):3017–3029, 2008.
  * [13] Asher Koriat.  When are two heads better than one and why?  Science, 336(6079):360–362, 2012.
  * [14] Kenneth W Latimer, Jacob L Yates, Miriam LR Meister, Alexander C Huk, and Jonathan W Pillow.  Single-trial spike trains in parietal cortex reveal discrete steps during decision-making.  Science, 349(6244):184–187, 2015.
  * [15] S D Lawley.  Distribution of extreme first passage times of diffusion.  Journal of Mathematical Biology, 2020.
  * [16] S. D. Lawley.  Universal formula for extreme first passage statistics of diffusion.  Physical Review E, (1):012413, 2020.
  * [17] Samantha Linn and Sean D Lawley.  Extreme hitting probabilities for diffusion.  Journal of Physics A: Mathematical and Theoretical, 55(34):345002, 2022.
  * [18] Jacob B Madrid and Sean D Lawley.  Competition between slow and fast regimes for extreme first passage times of diffusion.  Journal of Physics A: Mathematical and Theoretical, 53(33):335002, 2020.
  * [19] Richard P Mann.  Collective decision making by rational individuals.  Proceedings of the National Academy of Sciences, 115(44):E10387–E10396, 2018.
  * [20] James AR Marshall, Gavin Brown, and Andrew N Radford.  Individual confidence-weighting and group decision-making.  Trends in ecology & evolution, 32(9):636–645, 2017.
  * [21] Tyler McMillen and Philip Holmes.  The dynamics of choice among multiple alternatives.  Journal of Mathematical Psychology, 50(1):30–57, 2006.
  * [22] Elchanan Mossel and Omer Tamuz.  Opinion exchange dynamics.  arXiv preprint arXiv:1401.4770, 2014.
  * [23] Martijn J Mulder, Eric-Jan Wagenmakers, Roger Ratcliff, Wouter Boekel, and Birte U Forstmann.  Bias in the brain: a diffusion model analysis of prior probability and potential payoff.  Journal of Neuroscience, 32(7):2335–2343, 2012.
  * [24] Daniel J Navarro and Ian G Fuss.  Fast and accurate calculations for first-passage times in wiener diffusion models.  Journal of Mathematical Psychology, 53(4):222–230, 2009.
  * [25] William T Newsome, Kenneth H Britten, and J Anthony Movshon.  Neuronal correlates of a perceptual decision.  Nature, 341(6237):52–54, 1989.
  * [26] For an ideal observer if only measurement noise is present then drift and diffusion are equal in the continuum limit, $\mu_{i}=D_{i}=m_{i}$ [3, 38]. We consider the more general case where internal noise can increase $D_{i}>\mu_{i}$.
  * [27] This is the distribution of beliefs conditioned on the absence of a decision after a long time.
  * [28] We interpret beliefs as log-likelihood ratios. Therefore, with $k$ alternatives beliefs evolve in $k-1$ dimensions.
  * [29] G A Pavliotis.  Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations, volume 60.  Springer, 2014.
  * [30] Roger Ratcliff.  A theory of memory retrieval.  Psychological review, 85(2):59, 1978.
  * [31] Andreagiovanni Reina, Thomas Bose, Vaibhav Srivastava, and James AR Marshall.  Asynchrony rescues statistically optimal group decisions from information cascades through emergent leaders.  Royal Society Open Science, 10(3):230175, 2023.
  * [32] Michael N Shadlen and William T Newsome.  Motion perception: seeing and deciding.  Proceedings of the national academy of sciences, 93(2):628–633, 1996.
  * [33] Megan Stickler, William Ott, Zachary P Kilpatrick, Krešimir Josić, and Bhargav R Karamched.  Impact of correlated information on pioneering decisions.  Physical Review Research, 5(3):033020, 2023.
  * [34] John A Swets, Wilson P Tanner Jr, and Theodore G Birdsall.  Decision processes in perception.  Psychological review, 68(5):301, 1961.
  * [35] Satohiro Tajima, Jan Drugowitsch, Nisheet Patel, and Alexandre Pouget.  Optimal policy for multi-alternative decisions.  Nature neuroscience, 22(9):1503–1511, 2019.
  * [36] Alan N Tump, Max Wolf, Pawel Romanczuk, and Ralf HJM Kurvers.  Avoiding costly mistakes in groups: the evolution of error management in collective decision making.  PLoS Computational Biology, 18(8):e1010442, 2022.
  * [37] Naoshige Uchida and Zachary F Mainen.  Speed and accuracy of olfactory discrimination in the rat.  Nature neuroscience, 6(11):1224–1229, 2003.
  * [38] A. Veliz-Cuba, Z. P. Kilpatrick, and K. Josić.  Stochastic models of evidence accumulation in changing environments.  SIAM Review, 58(2):264–289, 2016.
  * [39] Xiao-Jing Wang.  Neural dynamics and circuit mechanisms of decision-making.  Current opinion in neurobiology, 22(6):1039–1046, 2012.
  * [40] Bram Zandbelt, Braden A Purcell, Thomas J Palmeri, Gordon D Logan, and Jeffrey D Schall.  Response times from ensembles of accumulators.  Proceedings of the National Academy of Sciences, 111(7):2848–2853, 2014.