# The Bayesian Origins of Growth Rates
in Stochastic Environments

Jordan T. Kemp1 and Luís M. A. Bettencourt2,3 1Department of Physics,
University of Chicago, Chicago, Illinois 60637, USA  2Department of Ecology
and Evolution, University of Chicago, Chicago, Illinois 60637, USA  3Mansueto
Institute for Urban Innovation, University of Chicago, Chicago, Illinois
60637, USA

###### Abstract

Stochastic multiplicative dynamics characterize many complex natural phenomena
such as selection and mutation in evolving populations, and the generation and
distribution of wealth within social systems. Population heterogeneity in
stochastic growth rates has been shown to be the critical driver of diversity
dynamics and of the emergence of wealth inequality over long time scales.
However, we still lack a general statistical framework that systematically
explains the origins of these heterogeneities from the adaptation of agents to
their environment. In this paper, we derive population growth parameters
resulting from the interaction between agents and their knowable environment,
conditional on subjective signals each agent receives. We show that average
growth rates converge, under specific conditions, to their maximal value as
the mutual information between the agent’s signal and the environment, and
that sequential Bayesian inference is the optimal strategy for reaching this
maximum. It follows that when all agents access the same environment using the
same inference model, the learning process dynamically attenuates growth rate
disparities, reversing the long-term effects of heterogeneity on inequality.
Our approach lays the foundation for a unified general quantitative modeling
of social and biological phenomena such as the dynamical effects of
cooperation, and the effects of education on life history choices.

††preprint: APS/123-QED

Growth and inequality are fundamental properties of complex adaptive systems,
and are especially important in human societies where they drive issues of
prosperity and equity. Increased attention to these quantities and richer data
enables a new set of approaches based on the statistical dynamics of
populations of strategic, forward-looking agents. For example, we now have
general answers connecting growth and redistribution schemes to specific
standing levels of inequality [1, 2, 3]. However, more general questions about
how societies can promote long-term growth while controlling or mitigating
inequality remain rather open.

To deal with these issues, researchers have recently sought to better
understand the non-linear dynamics of wealth distributions by modeling the
generation and redistribution of incomes and costs among agents within model
societies [4, 5, 6, 7], and by identifying resulting long-term steady-state
wealth distributions [1, 8, 9, 2, 3] . In much of this work, agents
representing individuals or households (often with life cycles), grow or lose
wealth through a multiplicative (geometric) stochastic process. This modeling
choice is well supported empirically and introduces a number of key parameters
as an agent’s resources (or wealth), $r$, evolve exponentially with mean
growth rate (over time), $\gamma$, fluctuate with standard deviation
(volatility), $\sigma$ [1, 10, 11] and vary across individuals of a population
with standard deviation $\sigma_{\gamma}$ [12, 13].

These parameters determine the statistical dynamics of wealth in populations
and the emergence of inequality across short and long timescales. Particularly
important are the statistics of heterogeneous growth rates within the
population, which generally result in growing inequality over long times. In
such contexts, agents with higher average growth rates [12] amass larger and
larger amounts of relative wealth, thus reducing social mobility across the
population. This phenomenon has been well known to economists, who have
studied its emergence in models of elastic agent decision-making for goods
exchanges [14, 15, 16], and its aggregate impacts via heterogeneous growth
through firm innovation [17] and natural resource abundance [18]. Generally,
the sensitive trajectories of heterogeneous multiplicative growth, observed
both in multiplicative growth models and in empirical data, highlight the need
for theoretical developments that can explain the origins of growth rate
values, volatilities, and population variances. More broadly, there remain
analytical gaps in our understanding of how optimal agent decisions in
stochastic environments contribute to disparities in growth, and what
processes influence agent decisions over time and across levels of social
organization.

Recent developments in cognitive and ecological sciences provide some
additional insights into the dynamics of agent behavior, suggesting that
optimal decision-making and stochasticity can be understood in terms of a
formal treatment of information and learning in unknown, noisy environments.
In this vein, researchers seeking stochastic decision-making models to explore
child and adolescent development [19, 20, 21] have rethought the learning
process in terms of acquiring information through (active and passive)
interactions with a knowable external environment. Similarly, ecologists have
formulated natural selection, the process through which a genotype optimally
leverages its environment’s structure, to maximize population growth (fitness)
as a (Bayesian) learning process [22, 23, 24, 25, 26].

This connection between optimal intertemporal decisions, information, and
growth was originally developed as a mathematical formalism to optimize
betting and portfolio investment returns [27, 28]. However, it has now begun
to be explored as a basis for a general statistical mechanics of wealth growth
and inequality [26].

Here we bring these insights together to develop a statistical dynamics of
growth and inequality in a population of strategic agents, where the growth
rates result from investing and learning in a stochastic environment. In this
approach, agents invest in sequential, stochastic environmental events based
on signals they receive, and grow their wealth based on the quality of their
predicted allocations. Exploring this mechanism of information-driven growth
in the context of population dynamics yields a better understanding of wealth
growth and disparities. More broadly, this work adds a new dimension to the
study of wealth inequality that more fundamentally links disparities between
wealth, growth, and agent’s subjective characteristics, such as their present
knowledge and their singular experience of the environment.

Our approach treats both resources and information as dynamically coupled
quantities. To model information dynamics, we show that learning in the joint
space of environmental states and agents’ signals is developed optimally in
terms of Bayesian inference, translating a maximization of predictability of
environmental states into that of resource allocations and growth. We finish
by exploring the consequences of learning a shared environment on the
statistics of information and wealth, and discuss the consequences of these
results for the role of general education and training on population dynamics
and its potential to reverse long-term inequality [12].

## Theory and Modelling of Information-Based Growth

In this section, we derive a theory of growth rates in terms of informational
quantities. Here, information means an agent’s predictive knowledge of event
probabilities in a noisy environment. Agents seek to maximize the growth of
their resources over time by investing in a set of possible events in their
environment using their individual knowledge. This agent’s knowledge is
subjective in the sense that it is formed by the agent’s own observations and
is not shared or compared with other agents. This knowledge is improved over
time through an iterative process of (Bayesian) learning based on the agent’s
expectations (“beliefs”) and observed environmental outcomes. We illustrate
these dynamics using a multinomial model, for which we derive closed-form
expressions for the average resource growth rate and volatility in terms of
information-theoretic quantities. We will then show when this learning process
dynamically attenuates inequality in resource growth rates across populations.

Figure 1:  Agents earn resources from their environment based on the quality
of their information. A. During each time step, a. the agent’s private channel
outputs a signal $s\in S$ with probability $P(s)$. b. The agent observes $s$,
c. the agent consults their guess for the conditional outcome probability of
the environment, $X(E|s)$, and d. the agent makes proportional bets on all
outcomes $B(E|s)$. f,e. The true event $e\in E$ is revealed from the
environment with probability $P(e)$, and g. the agent receives a payout
proportional to the marginal probability of $e$. B. In a population
simulation, $N$ agents independently sample private signals and invest in
events sampled from the same environment.

### Growth from Information

We consider a population of $N$ agents, each initially assigned resources
$r_{i}$ that can be (re)invested into the set of outcomes of their environment
to generate returns. The agents have access to a private predictive signal
$s\in S$, which they use as a predictor to invest resources in events $e\in E$
generated by their environment. The set of signals and events are described by
the joint probability distribution, $P(E,S)$ with marginals $P(E)$ and $P(S)$.

At each time step, each agent observes its own signal $s$, and allocates its
resources on events, using a vector $B(E|s)$. As the event $e$ is revealed,
the agent is awarded returns, $w_{e}$ for the fraction of resources invested
in the correct outcome, $B(e|s)r_{i}$. After $n$ steps, the agent’s total
resources (wealth) is

$r_{n}=r_{i}\prod_{j=1}^{n}B(e_{j}|s_{j})w_{e_{j}}=r_{i}\prod_{s,e}\big{[}B(e|s)w_{e}\big{]}^{W_{s,e}},$
(1)

where $W_{s,e}$ is the number of occurrences (wins) of $s,e$. Note that
$\frac{W_{s,e}}{n}\rightarrow P(s,e)$ as $n\rightarrow\infty$ by the law of
large numbers. It follows that the average growth rate of resources over $n$
steps is

$\gamma_{i}\equiv\frac{1}{n}\log\frac{r_{n}}{r_{i}}\approx\sum_{e,s}P(s,e)\log[B(e|s)w_{e}].$
(2)

Kelly showed that the maximal growth rate as $n\rightarrow\infty$, obtained by
maximizing the previous expression with relation to $B(E|S)$, results in an
allocation mirroring the conditional probability, $B(E|S)=P(E|S)$. This
maximum growth rate is the mutual information, $\gamma_{max}=I(E,S)$ when the
odds are ”fair”, $w_{e}=1/P(e)$ [27].

In reality, agents do not start out with perfect knowledge. In this case,
agents must invest using their present best estimate for the conditional
probability, $X(E|S)\neq P(E|S)$. Then, their resource growth rate will be
smaller than the maximum. This can still be written in terms of informational
quantities as the Kelly growth rate A.1,

$\gamma=I(E;S)-\mathrm{E}_{s}\big{(}D_{KL}\big{[}P(E|s)||X(E|s)\big{]}\big{)}.$
(3)

where $\textrm{E}_{s}$ is an expectation value over the states of the signal,
and
$D_{KL}\big{[}P(E|s)||X(E|s)\big{]}=\sum_{e}P(e|s)\log\frac{P(e|s)}{X(e|s)}$
is the Kullback-Leibler divergence, expressing how similar the two
distributions in its inputs are. This result shows that agents with better
information will experience greater resource growth rates, as long as they
invest optimally [29]. These compounding dynamics are illustrated in Fig. 1.

We will now illustrate these general results using a specific multinomial
model, which will allow us to further explore population dynamics.

#### Multinomial Choice Model of Growth Dynamics

Consider the space of signals, $S$ and events $E$, of equal size $l$ with
outcomes $s,e\in\mathbb{Z}$ and degenerate, multinomial conditional
probability

$P(e|s)=f(p,l)=\begin{cases}p&\text{if }s=e\\\ \frac{1-p}{l-1}&\text{if }s\neq
e,\end{cases}$ (4)

where $0<p<1$ is the binomial probability of guessing the correct outcome. For
simplicity, we assumed that the probability of a correct guess independent of
$l$. The distribution has uniform marginals, $P(e)=1/l$ and $P(s)=1/l$, for
all signals and events, such that $P(s|e)=P(e|s)$ via Bayes’ rule.

The mutual information is then $I(E;S)=\log l+p\log
p+(1-p)\log\frac{1-p}{l-1}$ (APP 16). For a binary choice, $l=2$, the first
term gives 1 bit as the entropy of the environment and the remaining is the
conditional entropy, expressing how well an agent could know the environment
given the signal. In the limit $p\rightarrow 1$, agents have perfect knowledge
of the marginal of $E$.

So far we considered that the agent has perfect knowledge of the joint
distribution of the signals and the environment. When this is not the case, we
can write a parametric expression of the agent’s ignorance in terms of an
estimated binomial probability $x\neq p$. The agent’s likelihood model of the
conditional probability is then $X(e|s)=f(x,l)$. The divergence term of Eqn. 3
becomes the divergence between $f(p,l)$ and $f(x,l)$ averaged over all
signals,
$\textrm{E}_{s}\big{[}D_{KL}\big{]}=p\log\frac{p}{x}+(1-p)\log\frac{1-p}{1-x}$.
Subtracting the mutual information by this term yields the agent’s growth rate
in parametric form under imperfect information as (APP A.2)

$\gamma=\log l+p\log x+(1-p)\log\frac{1-x}{l-1}.$ (5)

Figure 2:  The degenerate multinomial environmental structure permits the
parameterization of growth rates and volatilities, and can reproduce the
behavior of GBM wealth generation models. A. For $p=.7$, the computed growth
rate maximizes at $x=.7$, decreases as $x$ diverges from $p$, and scales with
the value of $l$. The parameter $l=2$ provides a realistic range of growth
rates. B. Monte Carlo simulations with $N=388$ homogeneous agents, all with
$\gamma=.03$ and $r_{0}=1$. The population resource statistics produce
lognormal exponential growth. The expected mean (red) and actual mean (orange)
overlap in value. The blue lines represent the 95% confidence interval. Inset:
The resource histogram is fit to a log-normal distribution of the same growth
and volatility parameters. C. Volatility minimizes at $x=l/2$ and increases
monotonically in either direction. Volatility increases more rapidly at higher
values of $l$. D. Over time, $\Delta_{\gamma}\rightarrow 0$ as agents’ growth
rates approach the Kelly growth rate. The average agent converges to within
15% the expected mean at $t\approx 80$.

This expression is plotted in Fig. 2A as a function of $x$ for various $l$
values and fixed $p$. We see that increasing the size of the event space, $l$,
reduces the probability of any individual outcome, making it harder to guess,
increasing the payouts and Kelly growth rate. The maximal growth rate is
obtained when $E_{e}[D_{KL}]\rightarrow 0$, when $x\rightarrow p$. Conversely,
$\gamma\rightarrow 0$ when $p\rightarrow 1/l$, indicating the signal and the
environment have become statistically independent.

Treating $\gamma$ as the expected growth rate, the volatility is calculated as
the second moment of the growth process. The volatility squared (variance) is
given as (APP A.3)

$\sigma^{2}={p(1-p)}\log^{2}\frac{x(l-1)}{1-x}.$ (6)

This expression is shown in Fig. 2C. The volatility vanishes in the limit
$x\rightarrow 1/l$, corresponding to when agents invest with equal probability
in all possible event types. A larger $l$ increases the magnitude of the
growth rate, but also the volatility. The volatility is highest when
$p\rightarrow 1/2$ and the environment is most uncertain, but the agents feel
sure of the outcomes when $x\rightarrow 0$ or $x\rightarrow 1$.

Kelly’s formulation describes the average growth rate of resources over a
large number of discrete investments [27]. To relate this situation to a
growth process in time, the agent makes $\omega$ bets per unit time such that
$\Delta t=1/\omega$ is the average interval of time between investments.
Resources at time $t$ are then the compound of all investments made in the
time interval $[t,t+\Delta t]$. In the continuous limit,
$\omega\rightarrow\infty$, $r_{n}\rightarrow r(t)$ and $\gamma$ describes the
average growth rate over long times. We consider $t\approx 10^{-2}yr$ so that
our results are comparable to previous work based on yearly growth rates of
the order of a few percent. Volatility is reduced
$\sigma_{t}=\sigma_{n}/\sqrt{\omega}$ as fluctuations are averaged out in each
time step (APP A3).

Fig. 2C demonstrates the two investment regimes for each value of $\gamma$;
where the growth rate maps to either high or low volatility depending on the
value of $x$. Using $x>p$ results in over-investment, denoted as aggressive
betting, as agents overestimate the dependence between their signal and the
environment. In this situation, the agent invests relatively more on diagonal
outcomes and experiences large gains or losses resulting in higher volatility.
With $x<p$, or in the conservative regime, agents underestimate $p$ and
distribute their wealth more equally across all outcomes, resulting in less
volatility. Agents can also experience $\gamma=0$ at two values of $x$. In the
trivial limit, as $x\rightarrow 1/l$, signals and agent investments become
statistically independent. The other trivial case can be solved for
numerically when $\gamma=0$.

The dynamics of this model closely resemble the well-known behavior of
geometric Brownian motion (GBM) with drift. Fig 2B shows the dynamics of a
population of agents with homogeneous parameters evolved using a Monte-Carlo
simulation. Over time, mean population resources grows with $\langle
r(t)\rangle=\frac{1}{N}\sum_{i}r_{i}(t)=\exp[\gamma t]$, in agreement with
[10]. We also demonstrate that the time-averaged growth rate of resources
converges to the Kelly growth rate over long times. Fig 2D shows the
asymptotic convergence of the normalized difference of averaged growth rate
for individual agents $\Delta_{G}=(\gamma-G)/\gamma\rightarrow 0$, where
$G=\frac{1}{t}\ln\frac{r(t)}{r(0)}$ (black) and population-averaged growth
rate, $\langle G\rangle=\frac{1}{N}\sum_{i}G_{i}$ (red).

So far we have considered $x$ as a static variable and explored the dynamics
of resources when $x\neq p$. We are now ready to consider $x$ as a dynamical
variable that converges to $p$ as a result of sequential (Bayesian) learning.

### Bayesian Dynamical Growth

More realistic agent trajectories are dynamical in the sense that investment
allocations become history-dependent and reflect the cumulative knowledge of
each agent’s past experience [10, 30]. In such a setting, agents can improve
their information by updating their model of the conditional relationship of
$S|E$ with each observation. In the absence of other processes, this learning
task is optimally achieved in terms of sequential Bayesian inference [31, 32]:

$X_{n}(e|s)=AP(s_{n}|e_{n})X(e_{n})=\left[\Pi_{i=1}^{n}\frac{P(s_{i}|e_{i})}{P(s_{i})}\right]X(e),$
(7)

where the normalization $A=\big{(}\int
de_{n}P(s_{n}|e_{n})X(e_{n})\big{)}^{-1}$, and where we take the prior
$X(e_{1})=X(e)$, as we are assuming that the environment is stationary.

Bayesian inference converges $X(E|S)\rightarrow P(E|S)$, maximizing the growth
rate in the long run. The agent’s interactions with the environment are thus
not only a way to gather resources in the short term, but also information
[26], as demonstrated in Fig. 3A. In minimizing information divergence, the
learning process maximizes resource growth over the long term. In the
following section, we describe a parametric Bayesian inference scheme applied
to the multinomial model of the previous section, via a Dirichlet prescription
of conjugate priors [33].

#### Bayesian Inference in the Multinomial Model

In general, we define the agent’s likelihood of a sample of the signal, $s|e$,
as a categorical distribution with parameter vector
$\beta=\\{\bm{\beta}^{1},...,\bm{\beta}^{l}\\}\in\mathbb{R}^{l}$, with each
vector corresponding to an event and each component, $\beta^{e}_{s}$
corresponding to a signal, event pair. The probability mass function is given
by $P(s|e)=\prod_{s}(\beta_{s}^{e})^{s}$, with normalization
$\sum_{s}\beta_{s}^{e}=1$. The conjugate prior distribution of $E$ is given by
a Dirichlet with hyperprior vector $\bm{\alpha}\in\mathbb{R}^{l}$, and
distribution $P(e)=\alpha_{e}/A$, where magnitude
$A=\sum_{e}^{l}\alpha_{e}/l$. This scheme is illustrated in Fig. 3B.

We set $\alpha_{e}=1$ for all $e$ so that our prior is uniform. We ensure the
off-diagonal degenerate condition by setting $\beta^{e}_{s}=p_{e}$ for $s=e$,
and for off-diagonal events, $e\neq s$, $\beta^{e}_{s}=\frac{1-p_{e}}{l-1}$,
satisfying Eqn 14. Under this setup, the binomial parameter describing the
environment is given by the average along the diagonal,

$\begin{split}p&=\frac{1}{l}\sum_{s}^{l}\beta_{s}^{e},\hskip 14.22636pts=e.\\\
\end{split}$ (8)

We have thus far described the parameter configuration for an agent with
perfect information. An agent with imperfect information will have estimates
for the parameters, $\tilde{\bm{\alpha}}\neq\bm{\alpha}$ and
$\tilde{\beta}\neq\beta$, and posterior,
$X(E|S,\tilde{\beta},\tilde{\bm{\alpha}})\neq P(E|S,\beta,\bm{\alpha})$. With
each observation, the agent updates $X(E|S)$ via (APP B.1)

$X(e|s)\propto\frac{m_{(-s)}^{(-e)}/\omega
k+\tilde{\beta}_{s}^{e}}{M^{(-s)}/\omega k+1}(n_{(-e)}+\tilde{\alpha}_{e}),$
(9)

where $m^{(-e)}_{(-s)}$ and $n_{(-e)}$ are the cumulative number of
observations of $e,s$ pairs and $e$ excluding the current observation, and
$M^{(-s)}=\sum_{e}m^{(-e)}_{(-s)}$ is the total occurrences of $s$ excluding
the current. We also introduce an inference time, $k$, as a free parameter
that weighs the evidence versus the prior, with units $time/update$ such that
$t/k$ is unit-less. In the limit $k\rightarrow\infty$, the agent does not
update their prior with new evidence. In the opposite limit, $k\rightarrow 0$,
the agent ignores the prior and considers only the most recent evidence, and
this becomes a maximum likelihood model.

Figure 3:  Visualization of the learning process. A. In addition to earning
resources from the environment, the agent is now awarded information with each
investment. B. Plate notation for the latent Dirichlet inference process. The
agent is assigned prior parameters $\tilde{\mathbf{\alpha}},\tilde{\beta}$,
corresponding to their guess for the distributions of $E$ and $S$, which are
updated based on counts $M,n$ respectively.

During the inference process, the agent will break the degeneracy of their
posterior as they infer each $\beta^{e}_{s}$ individually. This is
inconsequential though, as $x(t)$ can still be computed similarly to equation
8 at any time. The degeneracy of $P(E|S)$ permits us to reduce the dynamics of
$X(E|S)$ to that of the diagonal probability $x(t)$, such that (APP B.2).

$x(t)=\frac{pt/kl+x_{0}}{1+t/kl},$ (10)

where $x_{0}$ is the agent’s initial binomial parameter. This equation is a
core result of this work, as the dynamics of the information stored in the
agent’s posterior determine the average dynamics of the growth rate, via the
functional $\gamma[x(t)]$. Over many observations, the agent refines their
guess, driving $X\rightarrow P$, minimizing their information divergence as
$D_{KL}(P||X)\rightarrow 0$. The agent thus maximizes the growth rate for
their signal over time with a power law $-1$ in terms of the unitless
inference parameter $\lambda\equiv t/kl$. For the remainder of this paper, we
will study the effects of this learning process on the population dynamics of
growth rates and wealth.

### Population effects of Bayesian Dynamics

Having defined the dynamics of single agents, we can now explore the dynamics
of growth rate statistics in a heterogeneous population. Growth rates can vary
because agents have different initial conditions, experience varying
stochastic histories, or have different inference models (likelihoods).

To better understand the effects of these sources of heterogeneity, we write
the population variance of growth rates in terms of information-theoretic
quantities. Where for convenience, $I_{i}\equiv I(E;S_{i})$ and
$D_{i}\equiv\textrm{E}_{s_{i}}\big{(}D_{KL}\big{[}P(E|s_{i})||X(E|s_{i})\big{]}\big{)}$,
the population variance is given as (APP B.3)

$\textrm{Var}_{N}[\gamma_{i}]=\textrm{Var}_{N}[I_{i}]+\textrm{Var}_{N}[D_{i}]-2\textrm{Covar}_{N}\big{[}I_{i}D_{i}\big{]}.$
(11)

The first two terms in this equation arise from different sources and have
different implications for inequality. The first term is independent of the
learning process and trajectory and depends only on the model of the
environment given the agent’s signal. The second term expresses variance in
the prior and learning trajectory, and eventually vanishes as agents learn
their environment fully. Thus, the population growth rate variance only
vanishes if every agent has a signal with the same statistics, and after every
agent has had time to learn their environment. The third term arises in
populations where the quality of signals correlates with agents’ information
on the signal, a notion particularly relevant in modeling environments with
high variability across signal types. For example, across the United States
towns of different sizes, growth rates and wealth positively covary with
inequality as cities are often both wealthier and more unequal wages than
smaller municipalities [34, 35]. In this work, we focus on the inference
process for agents with identically distributed signals. That is we
(implicitly) take $\textrm{Var}_{N}[I_{i}]=0$ and ignore any covariances.

For a population of agents in the multinomial environment with heterogeneous
information independently sampling the same signal, the initial variance in
growth rates is given by the variance in the initial binomial parameter,
$\sigma^{2}_{x}$. The dynamics of the variance in binomial parameter for a
population of size $N$ is (APP 5.1)

$\textrm{Var}_{N}\big{[}x_{i}(t)\big{]}\equiv\big{\langle}\big{[}x_{i}(t)-\langle
x(t)\rangle\big{]}^{2}\big{\rangle}=\frac{\sigma_{x}^{2}}{(1+t/kl)^{2}},$ (12)

Figure 4:  Monte Carlo simulations of a population undergoing the inference
process with parameters of mean growth rate, $\bar{\gamma}=.04$, and standard
deviation, $\sigma_{\gamma}=.641\bar{\gamma}$. A. The population and
theoretical means of $x$ converge to $p$, thus maximizing growth rate. The
parametric variance, $\sigma^{2}_{x}$, (blue) follows the theoretical
prediction (red). The linear behavior log-log plot demonstrates the power law
behavior of $\sigma^{2}_{x}$. B. The mean resources of three population types
are plotted with a shaded region providing $95\%$ CI bounds for single agent
trajectories. Heterogeneity broadens the range of possible wealth values,
while inference both increases mean growth while narrowing the shaded region
relative to no inference. Agent inference slows the increase in the Gini
coefficient introduced by heterogeneity, and reduces the coefficient of
variation.

where $\langle x(t)\rangle=\frac{1}{N}\sum_{i}x_{i}(t)$, and $\sigma_{x}$.
Assuming a population of entirely conservative (or aggressive) agents, such
that all growth rates map to a unique binomial parameter, we can approximate
the variance in growth rates,
$\sigma^{2}_{\gamma}(t)=\big{\langle}\big{(}\gamma[x_{i}(t)]-\gamma[\langle
x(t)\rangle]\big{)}^{2}\big{\rangle}$, by Taylor expanding the second moment
of the resource distribution. APP B.5 shows that variance decreases
asymptotically in polynomial $t^{-2}$ time. Figure 4A demonstrates that in a
population of agents sampled from a Gaussian distribution of growth rates and
resources learning their environment, $\Delta_{p,x}=p-x(t)\rightarrow 0$ as
$t\rightarrow\infty$, and their binomial parameters converge to the optimal.
On the population level, there is agreement between the empirical population
mean and theoretical mean trajectory, calculated by evolving $\langle
x(t)\rangle$ using Eqn 10. Similarly, the empirical population variance in $x$
matches the theoretical power law prediction given by Eqn 12.

This result shows that learning a shared environment reduces growth rate
variance on the same time scale as the dynamical effects introduced by growth
rate variance [12]. It suggests that fast learning (by a sufficiently low $k$)
equalizes information access, and is a suitable mechanism for reversing the
long-term effects of heterogeneous growth on inequality. We demonstrate this
by comparing the statistics of resources across Monte Carlo-simulated
populations; first with homogeneous initial conditions, then with
heterogeneous initial conditions with and without inference. To measure the
increase in inequality, we track the Gini coefficient, denoted $G_{ini}$, a
value that scales from 0, for uniformly distributed resources, to 1, for
maximally unequal resources. In a distribution that is lognormal in the
continuous limit such as in these simulations,
$G_{ini}(t)\approx\textrm{Erf}[\sigma_{r}^{2}(t)]$. We furthermore measure the
relative increase in variance to resources using the coefficient of variation
$c_{v}=\sigma_{r}/\langle r\rangle$, to assess whether the increase in
inequality outpaces the overall resource growth. More on this analysis is
given in [12].

The resource time series in Figure 4B demonstrates that growth rate
heterogeneity dramatically broadens the wealth distribution, in agreement with
[12]. Accordingly, heterogeneity increases $G_{ini}$ and $c_{v}$ as compared
to a homogeneous population. The introduction of learning increases the
average growth rate in a heterogeneous population, as demonstrated by the
higher mean wealth, while reducing the variance in resources. The former slows
the rapid increase $G_{ini}$, while the combination of both reduces $c_{v}$ to
levels comparable to the homogeneous trajectory, confirming that learning
reverses the effects of heterogeneity on inequality.

While this simplified model does not capture the nuanced effects of
educational systems in real societies, the connection between convergent
learning in a population and growth is general and provides a sound
theoretical basis for the observed benefit of education on national growth,
human capital, and inequality reduction [36, 37, 38].

## Discussion

We developed a statistical theory for the origin of resource growth rates in
populations of learning agents experiencing a correlated stochastic
environment. We showed that an agent’s growth rate is, in the long time limit,
the quantity of mutual information they possess about their environment, and
that learning through Bayesian inference provides a natural (optimal)
mechanism for increasing agent’s growth rates, managing volatility, and
reducing disparities across populations over time. We demonstrated that with
relatively simple modeling assumptions, this theory produces similar behaviors
as GBM models widely used in studies of wealth dynamics and inequality. The
present treatment answers an important open question on how to mechanistically
control variances in growth rates across a society while maximizing learning
and growth, and generally enriches the typical modeling schema of wealth
dynamics by incorporating agents’ subjective choices in a structured
statistical environment. Beyond these results, this work provides a modular
foundation for incorporating information and strategic subjective agent
behavior in statistical mechanics, bridging a gap between physics and computer
science, and biological and social science.

There are a number of important developments that this type of theory suggests
and that will be necessary to model realistic social systems. First, learning
is not uniform across populations or time, varying across the life course,
with some agents being able to dedicate more time and effort to it. This issue
can be modeled by making inference rates dynamic and heterogeneous, for
example, through coupling to agents’ socioeconomic status (SES) or age.
Importantly, SES has been shown to correlate negatively with the presence of
stressors that inhibit the cognitive ability of people to learn [39, 40, 41],
and positively with educational outcomes [42, 43, 44]. Coupling learning rates
to SES would alter the population learning trajectory and potentially
attenuate its effectiveness in reducing information and wealth inequality.
Moreover, our analysis has assumed that each agent samples identically
distributed signals. In reality, people across different societies, cities, or
even neighborhoods have access to different signals, with implications for
what they can learn and for resulting social equity. Future studies of the
origins of inequality and social equity should consider these structural
complexities from the general point of view of access to information and
learning.

Second, from the point of view of maximizing resources, there are familiar
trade-offs between learning and investing. These can be modeled in terms of
the inference process, which can be divided into passive experiential
learning, resembling the “learning by doing” featured above, and,
additionally, emulating formal, institutional education wherein agents
sacrifice short-term gains in income to more rapidly increase their knowledge
and learning rates. These considerations define agent trade-offs between
actively exploring and passively exploiting the environment, an important
topic in experimental neuroscience and machine learning [45, 46]. Furthermore,
while information is a non-rival good that can be made available to a society
with minimal cost or degradation, the generation and dissemination of
information through teaching is a costly process that can produce additional
non-trivial dynamics. Incorporating the social costs of education through
mechanisms of finite learning resources could help explore trade-offs in
investing in human capital over various timescales [47, 48] and determining
when they are worth it in inter-temporal settings.

Thirdly, tracking individual agent dynamics under constraints of finite
(varying) lifespans can determine the effects of generational wealth transfers
on inequality, and provide insight into life-course strategies [49] and issues
of valuing the future. Thus, an extended framework can help us explore the
scope of education under the discounting of delayed resources by longevity and
lived volatility [50]; including the implications of costs and expected
earnings with or without an education over time. Lastly, agents in this model
experience the same environment and learn the same information, whereas actual
communities specialize in different, complementary skills that minimize
knowledge redundancy. These complementarities and exchanges are known commonly
in the social and ecological sciences in terms of the division of labor and
knowledge in societies. How agents decide which information to learn and what
profession to choose based on their environments begets different growth rates
across a population, altering emerging inequality and influencing how social
groups cooperate or compete across community or institutional social levels
[51]. Cooperation among agents with synergistic information in a stochastic
environment has been shown to produce non-linear additive effects on aggregate
information, suggesting that cooperative agents would experience larger growth
rates when coordinated, compared to the sum of agents acting independently
[52, 53]. Studying this connection between social behavior and growth from the
point of view of information and learning will provide insights into the
circumstances when cooperative and altruistic behavior becomes favored from
the point of view of both shared resources and information.

We thank Arvind Murugan, Marc Berman, and Adam Kline for their discussions and
comments on the manuscript. This work is supported by the Mansueto Institute
for Urban Innovation and the Department of Physics at the University of
Chicago and by a National Science Foundation Graduate Research Fellowship
(Grant No. DGE 1746045 to JTK).

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##  Appendix A Information quantities

###  A.1 Kelly growth rate

Applying $P(e|s)/P(e|s)$ to the log of Eqn. 2 yields

$\begin{split}\gamma&=\sum_{e,s}P(e,s)\log\bigg{[}w_{e}P(e|s)\frac{X(e|s)}{P(e|s)}\bigg{]}\\\
&=\sum_{e,s}P(e,s)\log\frac{P(e|s)}{P(e)}-P(s)P(e|s)\log\frac{P(e|s)}{X(e|s)}\\\
&=I(E;S)-\mathrm{E}_{s}\big{(}D_{KL}\big{[}P(E|s)||X(E|s)\big{]}\big{)},\end{split}$
(13)

where E denotes an expectation value over all sample outcome states.

###  A.2 Simplified growth model

Consider a conditional probability that is degenerate off-diagonal,

$P(e|s)=f(p,l)=\begin{cases}p&\text{if }s=e\\\ \frac{1-p}{l-1}&\text{if }s\neq
e,\end{cases}$ (14)

The “correct” outcome corresponding to the sampled event occurs with
conditional probability $0<p\leq 1$, and all other ”incorrect” guesses occur
with some uniform probability normalized to

$\sum_{e}^{l-1}P(e|s)=1-p;\hskip 10.00002pts\neq e.$ (15)

We describe the agent’s posterior of all bettors with the same form, with the
”correct” bet binomial coefficient $x$. Thus, calculating the growth rate
becomes an expectation calculation over the set of received tips, summing over
diagonal and off-diagonal components separately.

The mutual information separates into a term of only $l=1/P(e)$, an on-
diagonal, and off-diagonal term

$\begin{split}I(E;S)&=\sum_{e,s}^{l}P(e,s)\big{[}\log l+\log P(e|s)\big{]}\\\
&=\log l+p\log p+(1-p)\log\frac{1-p}{l-1}\\\ &=H(E)-H(E|S),\end{split}$ (16)

with the entropy of the outcome given by $H(E)=\log l$ and the reduction in
entropy by the signal given by $H(E|S)=-p\log p-(1-p)\log\frac{1-p}{l-1}$. The
information maximizes as $p\rightarrow 1$ and increases with $l$. The
information vanishes at $p\rightarrow 1/l$. The divergence is

$\begin{split}\textrm{E}_{s}\big{[}D_{KL}(P||X)\big{]}&=\sum_{e,s}P(e,s)\log\frac{P(e|s)}{X(e|s)}\\\
&=p\log\frac{p}{x}+(1-p)\log\frac{1-p}{1-x},\end{split}$ (17)

which is always nonnegative and vanishes when $x\rightarrow p$. We can write
the growth rate as the difference between these two terms

$\gamma=\textrm{E}\big{[}\log lf(x,l)\big{]}=\log l+p\log
x+(1-p)\log\frac{1-x}{l-1}.$ (18)

###  A.3 Variance of growth model

The volatility can be calculated as the second moment of growth. Standard
deviation is computed with the equation

$\sigma=\sqrt{E\big{[}\log\big{(}lf(x,l)\big{)}^{2}]-E\big{[}\log
lf(x,l)\big{]}^{2}}.$ (19)

$\textrm{E}\big{[}\log lf(x,l)\big{]}$ is simply $\gamma$, and the second term
is

$\begin{split}\textrm{E}\big{[}\log lf(x,l)\big{]}^{2}&=\bigg{(}\log l+p\log
x+(1-p)\log\frac{1-x}{l-1}\bigg{)}^{2}\\\
&=\log^{2}l+p^{2}\log^{2}x+(1-p)^{2}\log^{2}\frac{1-x}{l-1}\\\ &+2p\log l\log
x+2(1-p)\log l\log\frac{1-x}{l-1}\\\ &+2p(1-p)\log
x\log\frac{1-x}{l-1}.\end{split}$ (20)

The first term expands to

$\begin{split}\textrm{E}\big{[}\log\big{(}lf(x,l)\big{)}^{2}\big{]}&=\textrm{E}_{e,s}\big{[}(\log
P(e|s)+\log l\big{)}^{2}\big{]}\\\
&=\log^{2}l+p\log^{2}x+(1-p)\log^{2}\frac{1-x}{l-1}\\\ &+2p\log l\log
x+2(1-p)\log l\log\frac{1-x}{l-1}.\end{split}$ (21)

Combining these two quantities yields the volatility, where
$(1-p)-(1-p)^{2}=p(1-p)$,

$\begin{split}\sigma_{n}&=\sqrt{p(1-p)\bigg{[}\log^{2}x+\log^{2}\frac{1-x}{l-1}-2\log
x\log\frac{1-x}{l-1}\bigg{]}}\\\
&=\sqrt{p(1-p)}\log\frac{x(l-1)}{1-x}.\end{split}$ (22)

The variance of investment clusters of size $1/\omega$ scales as

$\sigma_{t}^{2}=\frac{1}{\gamma}\sigma_{n}^{2},$ (23)

where the subscript $t$ denotes the temporal variance.

##  Appendix B Latent Dirichlet Allocation

###  B.1 Defining the model

In this section, we derive the Latent Dirichlet Allocation (LDA) mode for the
degenerate multinomial environment. The Bayesian update equation is given by

$\begin{split}X(e|s)&\propto\frac{\big{(}m_{(-s)}^{(-e)}+\tilde{\beta}_{s}^{e}\big{)}}{(M^{(-s)}+\tilde{B}^{e})}(n_{(-e)}+\tilde{\alpha}_{e}),\end{split}$
(24)

for $m^{(-e)}_{(-s)}$ occurrences of $s$ conditional on $e$ excluding the
current, $n_{(-e)}$ occurrences of $e$ excluding the current in a batch of $n$
trials. We set $\alpha_{e}=1$, as every event is equally likely. For $s=e$,
$\beta_{es}=x$, and for $s\neq e$, $\beta_{se}=\frac{(l-1)}{1-x}$ to impose
degenerate off-diagonal conditions on $s|e$. We introduce
$B_{s}=\sum_{e}\beta^{e}_{s}$, whereby symmetry, $B_{s}\equiv B=1$, and we
count over the diagonals, $n_{e=s}$, and off diagonals, $n_{e\neq s}$.
Therefore the diagonal environmental posterior is

$P(e|s)\propto\frac{\big{(}m_{(-s=e)}^{(-e)}+x_{e}\big{)}}{\big{(}M^{(-s)}+1\big{)}}(n_{(-e)}+1),$
(25)

and the off-diagonal is

$P(e|s)\propto\frac{\big{(}m_{(-s\neq
e)}^{(-e)}+\frac{1-x_{e}}{l-1}\big{)}}{\big{(}M^{(-s)}+1\big{)}}(n_{(-e)}+1).$
(26)

###  B.2 Asymptotic, temporal behavior

We introduce the temporal behavior, with two constants. We multiply the number
of observations by the observation rate $\omega$, with units $samples/time$
and the inference rate $k$, with unit $time/update$. The inference rate counts
the number of samples per Bayesian update, and the observation rate counts the
updates per unit time. We multiply through by $k$ so that it becomes a
magnitude on the hyperprior, leaving

$\begin{split}P(e|s)&\propto\frac{(m_{(-s)}^{(-e)}/\omega+\tilde{\beta}_{s}^{e}k)}{(M^{(-s)}/\omega+1k)}(n_{(-e)}/\omega+\tilde{1}/k).\end{split}$
(27)

Over many observations, the law of large numbers argues that each outcome
count converges to the environmental posterior with some noise, $\xi_{i}$ as

$\begin{split}M^{(-s)}/\omega&\rightarrow P(s)Nt+\xi_{s}\\\
n_{(-e)}/\omega&\rightarrow P(e)Nt+\xi_{e}\\\
m_{(-s)}^{(-e)}/\omega&\rightarrow P(s|e)Nt+\xi_{s|e},\end{split}$ (28)

where the $\xi^{\prime}$s are fluctuation terms representing deviations from
the mean. Over many i.i.d observations of events, $\xi\rightarrow 0$. The
marginal terms converge to uniform over all states and become unity, and the
agent posterior converges to the dynamical distribution (APP 5)

$X(e,\lambda|s)=\frac{P(s|e)\lambda+X(s|e)}{1+\lambda},$ (29)

where we have converted to the time domain $t=N/\omega$, and substituted the
unitless inference sample size $\lambda=t/kl$. Over long times, the
distribution converges to the environmental posterior by

$\begin{split}X(e,\lambda|s)&\propto\frac{P(s|e)\lambda+X(s|e)}{P(s)\lambda+1}\big{(}P(e)\lambda+\alpha_{e}\big{)}\\\
\\\
&\rightarrow\bigg{(}P(s|e)+\frac{X(s|e,0)}{\lambda}\bigg{)}\frac{P(e)}{P(s)}=P(e|s),\end{split}$
(30)

yielding power law time-averaged behavior. At early times, as $t\rightarrow 0$
the posterior is proportional to the agent’s initial agent posterior,
$X(E|S)$, and converges to $P(E|S)$ as $kl\ll t\rightarrow\infty$. If agents
are initialized with the same diagonal posterior value such that
$X(s|e)=X(e^{\prime},s^{\prime})$ for all $e=e,s^{\prime}=s^{\prime}$, we can
assume that the diagonals of an agent uniformly converge to $p$ in time such
that $X(s,t|e)\equiv x(t)$ for all $s=e$,

###  B.3 Growth rate variance

The mean growth rate is computed, where for brevity, the expected divergence
for agent $i$ with signals $s_{i}\in S_{i}$ is given as
$\mathrm{E}_{s_{i}}\big{(}D_{KL}\big{[}P(E|s_{i})||X(E|s_{i}\big{]}\big{)}\equiv
D_{i}$, and the mutual information between individual signals and the
environment, $I(E;S_{i})\equiv I_{i}$

$\begin{split}\langle\gamma\rangle&=\frac{1}{N}\sum_{i}I(E|S_{i})-\textrm{E}_{s_{i}}\big{[}D_{i}\big{]}\\\
&=\big{\langle}I\big{\rangle}-\langle D\rangle,\end{split}$ (31)

where angle brackets denote population arithmetic means. The variance in
growth rates is calculated

$\begin{split}\textrm{Var}_{N}[\gamma_{i}]&=\big{\langle}(\gamma-\langle\gamma\rangle)^{2}\big{\rangle},\\\
&=\big{\langle}\gamma^{2}+\langle\gamma\rangle^{2}-2\gamma\langle\gamma\rangle\big{\rangle}\\\
&=\langle I^{2}\rangle-\langle I\rangle^{2}+\langle D^{2}\rangle-\langle
D\rangle^{2}\\\ &-2\big{(}\langle ID\rangle-\langle I\rangle\langle
D\rangle\big{)}\\\
&=\textrm{Var}_{N}[I_{i}]+\textrm{Var}_{N}[D_{i}]-2\textrm{Covar}_{N}\big{[}I_{i}D_{i}\big{]}.\end{split}$
(32)

When all agents are exposed to the same environment, the first and third terms
vanish, leaving

$\textrm{Var}_{N}[\gamma_{i}]=\textrm{Var}_{N}\bigg{[}\textrm{E}_{s_{i}}\bigg{(}D_{KL}\big{[}P(S|s_{i})||X(E|s_{i})\big{]}\bigg{)}\bigg{]}.$
(33)

###  B.4 Binomial parameter variance

The binomial variance can be computed exactly as

$\begin{split}\textrm{Var}_{N}\big{[}x_{i}(\lambda)\big{]}&=\frac{1}{N}\sum_{i}^{N}\bigg{[}\frac{p\lambda+x_{i}}{1+\lambda}\bigg{]}^{2}-\bigg{[}\frac{p\lambda+\langle
x\rangle}{1+\lambda}\bigg{]}^{2}\\\
&=\frac{1}{N}\sum_{i}^{N}\bigg{[}2\frac{x_{i}p\lambda+x_{i}^{2}}{(1+\lambda)^{2}}-2\frac{\langle
x\rangle p\lambda-\langle x\rangle^{2}}{(1+\lambda)^{2}}\bigg{]}\\\
&=\frac{\langle x^{2}\rangle-\langle
x\rangle^{2}}{(1+\lambda)^{2}}=\frac{\sigma_{x}^{2}}{(1+\lambda)^{2}}.\end{split}$
(34)

###  B.5 Multinomial growth rate variance

The variance of a function, $\gamma(x)$, of a random variable, $x$, is given
generally by the Taylor expansion of that function [54]. It is written as

$\begin{split}\textrm{Var}_{N}\big{(}\gamma[x_{i}(\lambda)])&=\gamma^{\prime}\big{[}\big{\langle}x(\lambda)\big{\rangle}\big{]}\textrm{Var}_{N}\big{[}x_{i}(\lambda)\big{]}\\\
&-\frac{\gamma^{\prime\prime}\big{[}\big{\langle}x(\lambda)\big{\rangle}\big{]}^{2}}{4}\textrm{Var}_{N}^{2}\big{[}x_{i}(\lambda)\big{]}+\bar{T}^{3},\end{split}$
(35)

where primes denote differentiation with respect to $x$, and $\bar{T}^{3}$ are
higher order terms that are only relevant at small times. The first and
second-order derivatives of $\gamma$ are given by

$\begin{split}\gamma^{\prime}(x)&=\frac{p}{x}-\frac{1-p}{1-x}\\\
\gamma^{\prime\prime}(x)&=-\bigg{[}\frac{p}{x^{2}}+\frac{1-p}{(1-x)^{2}}\bigg{]},\end{split}$
(36)

and the variance term is given by

$\textrm{Var}_{N}\big{[}x_{i}(\lambda)\big{]}=\frac{\sigma_{x}^{2}}{(1+\lambda)^{2}}.$
(37)

The growth rate variance after small times is given by

$\begin{split}\textrm{Var}_{N}\big{(}\gamma[x_{i}(\lambda)]\big{)}&=\bigg{[}\frac{p}{\bar{x}}-\frac{1-p}{1-\bar{x}}\bigg{]}\frac{\sigma_{x}^{2}}{(1+\lambda)^{2}}\\\
&+\bigg{[}\frac{p}{\bar{x}^{2}}+\frac{1-p}{(1-\bar{x})^{2}}\bigg{]}\bigg{[}\frac{\sigma_{x}^{2}}{(1+\lambda)^{2}}\bigg{]}^{2},\end{split}$
(38)

were for brevity, $\bar{x}\equiv\big{\langle}x(\lambda)\big{\rangle}$.


# Towards a New Interface for Music Listening:
A User Experience Study on YouTube

###### Abstract

In light of the enduring success of music streaming services, it is noteworthy
that an increasing number of users are positively gravitating toward YouTube
as their preferred platform for listening to music. YouTube differs from
typical music streaming services in that they provide a diverse range of
music-related videos as well as soundtracks. However, despite the increasing
popularity of using YouTube as a platform for music consumption, there is
still a lack of comprehensive research on this phenomenon. As independent
researchers unaffiliated with YouTube, we conducted semi-structured interviews
with 27 users who listen to music through YouTube more than three times a week
to investigate its usability and interface satisfaction. Our qualitative
analysis found that YouTube has five main meanings for users as a music
streaming service: 1) exploring musical diversity, 2) sharing unique
playlists, 3) providing visual satisfaction, 4) facilitating user interaction,
and 5) allowing free and easy access. We also propose wireframes of a video
streaming service for better audio-visual music listening in two stages:
search and listening. By these wireframes, we offer practical solutions to
enhance user satisfaction with YouTube for music listening. These findings
have wider implications beyond YouTube and could inform enhancements in other
music streaming services as well.

##  1 Introduction

In recent years, the music streaming industry has witnessed a significant
surge in popularity, with market leaders such as Spotify, Apple Music, and
Amazon Music dominating the market[1]. Alongside this trend, YouTube has
solidified its position as a prominent platform for diverse video content,
including documentaries, daily vlogs, entertainment shows, and more. As users
flocked to YouTube for various types of content, the platform naturally became
a hub for music-related videos as well. Users now have easy access to a wide
range of music video content on YouTube, contributing to the growing trend of
consuming music through video formats[2].

Indeed, YouTube delivers a distinctive multi-sensory experience by showcasing
a vast variety of music-related videos such as music videos, live
performances, curated playlists with visual artworks, and cover performances,
enabling users to enjoy music through a fusion of visual and auditory
elements. Despite Spotify’s global prominence based on subscribers, YouTube
has seen an increasing number of users turning to its platform for music
consumption[1, 3]. This trend is evident in regions like South Korea[4, 5] and
Latin America[6], where YouTube dominates as a preferred music platform.

Given YouTube’s current dominance in music consumption, there’s a need for a
more comprehensive investigation into this behavior and patterns. Earlier
studies have explored YouTube’s role as a streaming service[7], compared its
usability with Spotify[3], and analyzed music consumption behavior on
YouTube[8]. However, the elements contributing to YouTube’s rise as a primary
music platform and the actual levels of user satisfaction are still not fully
understood, indicating a need for further user-focused research.

Thus, this study aims to conduct in-depth interviews with music consumers on
YouTube, examining their behavior, comparing the advantages and disadvantages
of using YouTube as a music consumption tool with other music streaming
services, and reevaluating YouTube’s standing as a tool for music consumption.
Additionally, we propose a new interface design that enhances the usability of
music-related searches and listening. This research was conducted
independently by our team, with no financial backing or data provided by
YouTube or any associated organization. With our study, we aim to contribute
to the ongoing conversation on YouTube’s role as a music platform and offer
insights into developing an innovative interface that elevates the user’s
music listening experience.

##  2 Related Work

In the field of music information retrieval (MIR), research on music streaming
services includes studies on improving recommendation algorithms [9, 10, 11],
understanding user behavior and patterns of use [12, 13, 14, 15, 16], and
studying user experiences and interfaces [17, 18, 19, 20, 21]. These studies
aimed to enhance overall user satisfaction and engagement with music streaming
services by providing personalized recommendations, improving the user
interface, and identifying the factors that influenced user behaviors and
preferences.

Compared to other music streaming services, research on music consumption
through YouTube has only recently gained attention due to the platform’s
relatively late recognition as a music consumption platform. Early studies on
YouTube’s music videos have revealed that music is the most consumed content
category on YouTube, and researchers have classified the types of YouTube’s
music content while analyzing their differences[7]. Furthermore, [3] reported
that YouTube is used as frequently as Spotify and is perceived as superior to
Spotify in terms of its shareability and accessibility.

As YouTube’s influence in music consumption grows, recent research has
examined three types of online music practices according to the role YouTube
plays: default, soundtracking, and complementary platforms[8]. Authors report
that one of the main results is that YouTube’s music videos are listened to,
rather than watched. However, the significance of visual elements in music
listening can differ based on the genre or content. Additionally, it is worth
mentioning that the participants in the study reported only occasional use of
YouTube for music, which may limit the generalizability of the findings to
other contexts, such as frequent YouTube users.

Therefore, this study aims to examine the usage behavior of users who use
YouTube more than three times a week in everyday situations, report on the
characteristics of the subject group, and classify the content used. In
addition, we draw out advantages and disadvantages through usability tests to
newly consider the role of YouTube as a music-listening tool. Moreover, the
study proposes interface improvement measures to fill the research gap on "how
to improve the music listening environment through YouTube." Considering the
diverse range of devices used to access YouTube, including mobile devices,
PCs, tablets, and TVs, we primarily focus on the mobile device, taking into
account its widespread usage among participants.

##  3 Methods

###  3.1 Participant

We recruited 27 Seoul National University students (12 males, 15 females) aged
18 or older (mean=23.40, sd=3.13). Our recruitment focused on participants who
listen to music on YouTube at least three times a week while excluding those
who rely solely on YouTube Music without using YouTube. This approach allowed
us to concentrate on the distinct characteristics of consuming music through
videos on YouTube, which encompass both visual elements and audio.
Participants were compensated with a cash payment of KRW 10,000. Ethics
approval was obtained from the Institutional Review Board of SNU.

###  3.2 Study Design

Phase |  Requirements
---|---
A Verbal Interview |  Asking about participants’ music listening habits and preferences along with the motivation to use YouTube.
Usability Test |  Comparison of YouTube and other music streaming services and feedback on the interface of YouTube for searching and listening to music.
UI proposal |  Propose YouTube interface design for music listening freely, and explain yourself.
Table 1: Three steps of semi-structured interview

Informed by previous studies’ methodologies and the specific needs of our
research, we designed our interview in two stages: a preliminary
questionnaire[22, 23], followed by a semi-structured interview[22, 24, 23]
that includes a brief ice-breaking session[25]. The preliminary questionnaire
collects demographic data and music consumption habits of the participants,
such as their academic majors, relationship with music, frequency and duration
of YouTube use for music, and specific contexts of YouTube music consumption
(excluding YouTube Music). Additionally, we also sought information regarding
their subscription to YouTube Premium or usage of YouTube Music.

Following an ice-breaking session, the semi-structured interview proceeded
with three main segments (Table 1). First, we explored participants’ regular
music consumption habits, such as frequency, platform preference, and content
preferences. Second, participants were asked to demonstrate the process of
searching and listening to music on YouTube, which allowed for a natural
exploration of the platform’s advantages and disadvantages in comparison to
other music streaming services. Third, participants utilized empty interface
templates on iPad to design a new interface for music searching and listening,
enabling them to customize the screen ratio, functions, buttons, and more.
Each interview, lasting roughly 30-40 minutes, was recorded and transcribed
using NAVER Clova Note, with participant consent.

###  3.3 Analysis

We identified the overarching themes and trends of the participants’ responses
and organized the data accordingly. The data were categorized into the
following topics: primary streaming service, weekly listening time, music
listening type, preferred music genres or content on YouTube, situations
YouTube is used for music listening, reasons for using YouTube as a music
consumption tool, music search methods on YouTube, criteria for video
selection, advantages and disadvantages of YouTube compared to other services,
and a summary of interface proposal sessions.

We generated a list of keywords for the qualitative analysis, which includes
the advantages and disadvantages of using YouTube for music listening and user
interface proposals from interviewees. To validate our classifications and
identify commonalities, we repeated the process of analysis and consensus-
building three times among the researchers similar to the analysis process in
[22, 26, 27]. Grounded theory[28] and content analysis[29] were also used as a
guide throughout the process of keyword generation. We also referred to
previous qualitative studies in the field of MIR[15, 26, 24, 27, 8] to guide
our data analysis, as well as to ensure consistency in our reporting and
citation practices. Finally, we thoroughly reviewed the keyword lists to
extract the main findings of how YouTube is used as a music consumption tool
by the participants based on the method of theme analysis[30].

To better understand the participants’ interface design proposals, we compared
the proposals from the participants and reviewed the summary of the interface
designing sessions. From this process, we synthesized useful design
implications and arrived at wireframe designs for the music search and
listening screens.

##  4 Result

###  4.1 Behavior and Characteristics of Music Consumption on YouTube

As the current study investigates interview data from a sample of 27 users, it
is important to take into account the unique characteristics of this group.
Therefore, information concerning the participants’ music consumption
behaviors and preferences was gathered through preliminary surveys and
interviews. The results showed that participants typically used YouTube about
five times per week (mean = 4.89, sd = 1.93), for a total of approximately
five hours (mean = 5.35, sd = 3.76), to listen to music while engaging in
various activities, such as studying, relaxing, commuting, and exercising. No
one specialized in music. The majority of participants used the free version
of YouTube and did not subscribe to YouTube Premium. Additionally, some
participants supplemented their music listening with other platforms such as
YouTube Music, Melon, Spotify, and Genie.

Participants enjoyed a diverse range of music genres on YouTube. The top five
genres mentioned most frequently were OST (original soundtrack of movies or
dramas, 13 times), pop (12 times), K-pop (11 times), classical music (7
times), and indie music (7 times). Other genres mentioned in order of
frequency include J-pop, ballads, old-fashioned music (mid-20th-century Korean
pop and ballads), jazz, rock, band music (with live instrumentation and
elements of rock, pop, and indie), new age, hip-hop, EDM, and R&B. Music
content can be broadly divided into three categories: 1) Official music
content such as music videos, 2) Live music content such as performances,
concerts, festivals, and 3) User-generated content such as playlists and cover
videos. In terms of frequency of mention, the order was 3-2-1 (27 times, 25
times, 8 times) respectively.

###  4.2 Advantages of using YouTube for music listening

Alongside our anticipation that YouTube serves as an audiovisual music
listening tool, we found that YouTube possesses various strengths compared to
other streaming services (Table 2). Musical diversity was the most frequently
mentioned category, with two main points: the availability of non-official
music in addition to official releases, and the diversity of playlist content
compared to other streaming services.

With streaming services, I can only listen to official releases, but with
YouTube, I can listen to not only official releases but covers and other user-
generated content. (P11)

Unlike other services, YouTube’s diverse playlists prevent repetitive
listening by offering a wide range of songs within similar genres. (P26)

Category |  Keyword |  Freq |  Total
---|---|---|---
Musical Diversity |  official soundtrack + $\alpha$ |  14 |  23
|  playlist |  9 |
Convenience |  familiarity |  4 |  15
|  accessibility |  4 |
|  subscription fee |  4 |
|  Customizing |  3 |
User Interaction |  recommendation |  10 |  13
|  comments |  3 |
Visual Contents |  thumbnail |  6 |  10
|  video |  4 |
etc. |  etc. |  1 |  1
Table 2: Pros. keywords of usability test

Also, convenience was mentioned as an advantage, with familiarity,
accessibility, no subscription fee, and user customization.

I use it because I’m used to it. I’ve used Melon and YouTube Music before, but
I settled with YouTube because it was more convenient. (P12)

Since YouTube is free, there’s no need to pay for other services. (P8)

As for user interaction, most users mentioned the recommendation algorithm
itself and the ability to view other users’ opinions through comments.

The recommendation algorithm is good. I often find great new songs through it.
(P17)

It’s good to be able to see other people’s opinions and sympathize by reading
comments. (P4)

Lastly, the visual content of thumbnails and videos was mentioned as an
advantage.

I can use both sight and sound when listening to music with videos. (P4)

When I play playlists with thumbnails, like at a housewarming party, it adds
to the atmosphere, and it’s good for interior purposes too. (P6)

While we initially expected the inclusion of visual elements to be a
significant advantage of YouTube, the participants’ usage patterns proved more
diverse. Some appreciated the visual components, while others turned to
YouTube strictly for audio during activities like work or sleep (P2, P5, P22,
P23, P24). These observations align with prior research[8], showing the varied
ways users utilize YouTube for music. Although some mentioned listening to
audio with the screen off (P7, P23, P24), we excluded this aspect from our
analysis as it’s a feature exclusive to YouTube Premium subscribers.

###  4.3 Disadvantages of using YouTube for music listening

The inconveniences and disadvantages of listening to music on YouTube were
categorized into seven major themes (Table 3). The most frequently mentioned
inconvenience was related to user interaction, with many complaints about the
inconvenience of filtering the desired information while exploring recommended
videos and comments.

Category |  Keyword |  Freq |  Total
---|---|---|---
User Interaction |  comments |  12 |  21
|  recommendation |  9 |
Manipulation |  button / tap |  10 |  17
|  display ratio |  7 |
Playlist |  playlist contents |  4 |  10
|  making playlist |  4 |
|  mix playlist |  2 |
Section Search |  timestamp |  6 |  9
|  playback bar |  3 |
Lack of Info. |  song information |  5 |  8
|  log information |  3 |
Underutilization |  replay |  5 |  8
|  volume control |  3 |
Contents Quality |  sound quality |  5 |  6
|  video quality |  1 |
etc. |  (video) data size |  4 |  6
|  etc. |  2 |
Table 3: Cons. keywords of usability test

In other music streaming services, genre separation is clearly done, but
YouTube recommends based on the videos you watch, so there is a tendency to
lean towards a specific genre. (P26)

When watching music videos and reading comments, it’s hard to find South
Korean users’ reactions when most comments are in foreign languages. (P11)

The second most frequently mentioned disadvantage was related to screen
manipulation, such as fixed thumbnails, the ratio of videos, and accidental
button presses.

It would be nice if I could reduce the screen ratio. I want to watch the small
screen when exercising or doing other things. (P8)

There are cases where I accidentally press the Shorts button and the music
stops. (P22)

Regarding playlists, users complained about not having timestamps for
individual songs, the content of playlists made by others, the process of
creating playlists themselves, and the mixes provided by YouTube.

It’s inconvenient to switch to another song if there is no timestamp in the
playlist. (P18)

Since playlists are made by others, there are few cases where all songs suit
my taste, and there are mediocre songs in between. (P18)

It’s inconvenient to save songs one by one in my library. It feels slow every
time I press the save button, and it is a hassle to press the button several
times to save. (P27)

Some users mentioned the lack of information about album or song information
and lyrics, as well as the lack of log information about previously watched
videos as a disadvantage.

It’s hard to find album or song information, and it’s frustrating not knowing
the information of the concert I am watching. (P7)

When I use the autoplay function, it is hard to find which song I thought I
liked. (P17)

Despite the existence of autoplay and volume control features on YouTube, user
complaints arose from a lack of information about these functions. Some users
viewed them as drawbacks, unaware of their existence or location.
Specifically, enabling autoplay requires navigating to the settings, while
fine-tuning volume necessitates physical device button use. This complexity
may have heightened user frustration and dissatisfaction.

I wish there is a autoplay button. (P12)

I want to make minute adjustments, but even if I increase the volume level by
just one, the volume suddenly becomes too loud. (P16)

The quality of the content is related to the audio or video quality. Some
responses showed low reliability in audio quality when used for music
listening.

There are cases where the sound quality is poor in content uploaded by
individual users. (P12)

Aside from that, there were four mentions of concerns about mobile data usage
due to large video data size (P2, P9, P11, P25), one mention of discomfort
with provocative titles (P23) and experiencing an error when randomly playing
saved videos (P14). There were nine mentions related to ads or background
playback (P2, P5, P6, P8, P9, P12, P17, P19, P25), but these were excluded
from the analysis since they can be resolved with a YouTube premium
subscription.

###  4.4 User Feedback for Interface Improvements

We analyzed users’ explanations and drawings of the searching and listening
screens, categorizing their demands for interface improvement into three
categories: addition, modification, and deletion. These categories, along with
relevant quotes, provide specific descriptions of users’ interface improvement
suggestions.

The first is to request the addition of new information or functions that are
currently absent on YouTube, such as a new button or tab, new sorting and
filtering criteria, or more information about contents and songs.

It would be great if there was a detailed search button under the search bar,
where you could search by year, album, composer, etc. (P7)

It would be nice if I can choose options between "all videos recommended" and
"music-related recommendations" in the recommendation section. (P9)

The second is to modify the existing functions or configuration of YouTube to
increase operability and efficiency when searching and listening to music,
such as changing the ratio of various spaces on the interface or changing the
positions of existing buttons and information.

It would be great if the thumbnail (album cover) could be smaller, and the
title, artist, etc. could be displayed next to it. (P12)

It would be nice to adjust the ratio of the comment box and recommended videos
so that you can view them together. (P4)

Lastly, there were cases where demands were made to remove things from the
existing YouTube interface that are not directly related to music searching or
listening.

We don’t need the buttons for uploading videos on the bottom menu bar. It
would be great if we could freely configure this menu bar. (P15)

If we could hide the buttons we don’t use often and press the detailed button
to show them, it would be neat. (P19)

##  5 Findings and Dicsussion

(a) Searching

(b) Listening

Figure 1: A Wireframe of youtube UI for music listening

###  5.1 Role of YouTube as a music streaming service

Users desire an improved interface for YouTube to maximize its potential as a
music consumption tool. We have identified five key roles that YouTube plays
in music listening, and based on this, we propose design implications to
enhance the user experience.

####  5.1.1 Exploring musical diversity

YouTube offers users a wide variety of musical genres, artists, and songs to
discover and explore, including rare or unreleased music not found on other
streaming services. Users can also enjoy various versions of the same song
through covers or live performances by different artists. Design Implication:
To improve search efficiency, music content should be categorized by genre,
artist, and mood, and album information such as lyrics should be provided to
reduce the need to search for information on other platforms.

####  5.1.2 Sharing unique playlists

YouTube creators can create and share playlists, simplifying the search
process and enabling them to select playlists based on keywords like mood or
activity (e.g. warm spring day, driving playlist). Design Implication:
Playlists should provide song and timeline information, and allow users to
switch to the next song with a button. Allowing users to customize songs
within the playlist, such as adding or removing them, and saving these
changes, would enhance the playlist’s functionality.

####  5.1.3 Providing visual satisfaction

By offering sensory satisfaction beyond just videos, YouTube’s visual content
enhances the music listening experience. Users appreciate being able to
observe the musicians’ expressions, gestures, and style, and sometimes even
watch music videos solely for visual gratification like repetitive animations
or thumbnail images paired with the music. Design Implication: The screen size
and ratio of the video should be customizable based on the content’s
characteristics and users’ listening environment. For example, users would
like the option to decrease the video screen size in public settings or
enlarge it to focus on a particular idol member or musician’s finger
movements.

####  5.1.4 Facilitating user interaction

YouTube’s likes, dislikes, subscriptions, and comments features enable users
to interact with the platform and foster a sense of community, resulting in a
more engaging music listening experience. Additionally, the recommendation
algorithm lets users explore new content and see how others react to music,
which is a key motivation for users to use YouTube. Design Implication: Users
should be able to sort and filter recommendations and comments based on
various criteria, such as timeline, keyword, lyrics or the most frequently
mentioned, to expand YouTube’s social function. Pinning specific comments that
users like or refer to frequently could also reduce search time.

####  5.1.5 Allowing free and easy access

YouTube’s accessibility, cost-effectiveness, and cross-device compatibility
make it a convenient option for users to listen to music in various
situations. This versatility has led some users to cease subscribing to other
streaming services. Primarily, the appeal lies in the free access to a diverse
library of music videos, live performances, covers, and user-generated
content, resonating with users disinclined to pay for music subscriptions.
Design Implication: While device-specific interfaces are important, consistent
usability is crucial to prevent user confusion or inconvenience.

It is crucial to acknowledge that while some of the proposed features (e.g.,
album and artist filters, lyrics, smaller screen mode) have already been
implemented in the YouTube Music App, users still rely on YouTube to access a
diverse range of music videos that are not available on the YouTube Music App.
Therefore, our design implications hold the potential to differentiate YouTube
from YouTube Music by catering to the experience of video streaming alongside
music consumption.

###  5.2 UI for Audio-Visual Music Streaming Platform

Taking into account the role of YouTube as a music streaming service, the
needs of its users, and the interface designs of typical music streaming
services, we have developed an ideal wireframe for an audio-visual music
listening platform. It consists of two stages: (a) searching and (b) listening
screens (Figure 1).

(a) Searching To display diverse music content tailored to users’ interests,
we added 1) advanced search functionality to the top keyword search bar,
allowing users to filter by era, genre, artist, and other details.
Additionally, we added 2) a button to easily add multiple videos to a user’s
playlist, and 3) reduced the thumbnail size to show more videos on one screen.
Next to the thumbnail, we included 4) information about the songs in the
video, and if the video is a playlist, we added 5) a timeline and information
about the included songs. Finally, we made 6) the bottom menu buttons
customizable, allowing users to remove buttons when they feel unnecessary and
create their own menu.

(b) Listening While maintaining the current structure of the interface, we
adjusted the layout and added new features to enhance the music listening
experience. 1) Adding a toggle button that allows users to exchange between
video watching and music listening. Users can use their fingers to 2) zoom in
or out of the video to adjust its size. Previously, users had to click the
video to access playback and skip buttons, but we located 3) the playback bar
and related functions at the bottom of the video. We also made the 4) repeat
button more visible. We added 5) a toggle button to expand or shorten album
information or lyrics, and made 6) comments expandable in a similar manner,
with a function for users to pin comments they want to keep visible. We added
7)a filtered recommendation feature to suggest reduced-size videos based on
specific user-selected filters. This allows for easier exploration of related
content through horizontal scrolling.

The findings of this research hold potential for application across a variety
of streaming services. Features such as advanced search functions,
customizable menus, and enhanced playlist capabilities can improve user
engagement and satisfaction. Effective presentation of music-related
information enriches the listening experience, while additional
functionalities such as video zoom or comment pinning foster a personalized
user experience. These findings can significantly benefit YouTube, as well as
aid other music streaming platforms like Spotify, Apple Music, and Amazon
Music, and video streaming services including music videos, like Bilibili and
Vimeo, in optimizing their interfaces according to their unique
characteristics and users’ needs.

##  6 Conclusion

This study explored the music listening behaviors of YouTube users and
analyzed the advantages and disadvantages of YouTube as a music streaming
service. We proposed new interface wireframes to improve usability and re-
examined YouTube’s role as a tool for music listening. Undoubtedly, there are
constraints in actualizing the proposed interface fully on YouTube.
Nevertheless, some suggestions on improving visual satisfaction, comment
exploration, and toggle button to exchange the interface between video
watching and music listening mode could be considered in designing the overall
interface of video streaming platforms.

Our study has limitations depending on the small sample size of Korean users,
and it is essential to consider several important factors. Firstly, our
interface design primarily focused on mobile environments, which may limit its
direct applicability to other devices like PCs and TVs. Secondly, the
relatively narrow age range and educational levels of our participants may
affect the generalizability of our findings. Thirdly, the absence of
comparative studies on similar video platforms and services hinders our
understanding of YouTube’s performance as a video streaming service. However,
these limitations present opportunities for future research to explore and
address the diverse needs of users across different devices, demographics, and
video services. Overall, our study provides valuable insights and paves the
way for further advancements in user-centered design for music streaming
services.

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