# Information Synergy Maximizes the Growth Rate of Heterogeneous Groups

Jordan T. Kemp1, Adam G. Kline1, 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

Collective action and group formation are fundamental behaviors among both
organisms cooperating to maximize their fitness, and people forming
socioeconomic organizations. Researchers have extensively explored social
interaction structures via game theory and homophilic linkages, such as kin
selection and scalar stress, to understand emergent cooperation in complex
systems. However, we still lack a general theory capable of predicting how
agents benefit from heterogeneous preferences, joint information, or skill
complementarities in statistical environments. Here, we derive general
statistical dynamics for the origin of cooperation based on the management of
resources and pooled information. Specifically, we show how groups that
optimally combine complementary agent knowledge about resources in statistical
environments maximize their growth rate. We show that these advantages are
quantified by the information synergy embedded in the conditional probability
of environmental states given agents’ signals, such that groups with a greater
diversity of signals maximize their collective information. It follows that,
when constraints are placed on group formation, agents must intelligently
select with whom they cooperate to maximize the synergy available to their own
signal. Our results show how the general properties of information underlie
the optimal collective formation and dynamics of groups of heterogeneous
agents across social and biological phenomena.

††preprint: APS/123-QED

##  I Introduction

Collective behavior is a general feature of biological and social systems. It
mediates the survival and evolution of populations under resource constraints,
competition, or predation in natural systems [1] and the formation and
persistence of social organizations in human societies [2]. Much past work has
modeled collective dynamics using homogeneous interaction rules, common to all
agents, that are often also phenomenological. While these models have produced
diverse insights, they typically lack a theoretical foundation to explain how
specific social behavior emerges among individual agents with heterogeneous
information and behavior. Thus, there remain significant knowledge gaps in
most realistic situations, where agents with distinct but potentially
complementary traits act collectively to maximize their joint growth (fitness,
wealth) in knowable, noisy environments.

Some examples help illustrate the present situation. Game theorists and
ecologists have considered many different cooperative interaction schemes [3]
and explored evolutionary stable behavior [4], particularly on networks [5, 6,
7], where optimal behavior is identifiable under given interaction rules.
Elaborating these schemes by introducing higher order interactions has
broadened our understanding of more complex social networks [8, 9, 10], and
their dynamical phase-stability under varying interaction strengths [11].
Researchers have also studied, both theoretically and in the laboratory, how
memory of previous interactions influences agents’ preferences for future
encounters [12, 13, 14, 15], the spread of social crises across distance [16],
and the formation and scaling properties of social collectives [17, 18], such
as cities [19, 20].

In addition to interaction rules and associated payoffs, collective dynamics
is predicated on maximum principles, which specify agents’ preferences in view
of a goal and thus render their behavior intelligent (optimal). For example,
inclusive fitness theory, which assumes a reproductive benefit to cooperation
because of shared genes [21, 22] has been studied in mixing populations and
over networks [23] where it predicts population benefits to cooperation
through several forms of reciprocity [24]. More recently, researchers have
studied resource pooling in models of growth as a means to minimize
environmental uncertainty and associated loss of fitness among agents
experiencing independent fluctuations with shared statistics [25, 26]. Such
approaches remain limited by the association between collective behavior and
(genetic) homophily but they can help explain the existence of phase
transitions in cooperation networks [18, 11], and specify agents’ plausible
behavioral patterns [14], even if doubts remain about inclusive fitness’s
predictive power [27].

Generally, however, most current quantitative frameworks fail to address
collective dynamics when agents remain heterogeneous across skills, knowledge,
and behavior [28, 29]. Developing more general approaches to collective
behavior that include adaptation and learning along with heterogeneity, is a
crucial step towards understanding how agents self-organize in more complex
and dynamical environments, where specialization and the division of labor and
knowledge become key.

Adaptive behavior requires agents to acquire and process information over time
[30, 31] in response to their environments and to each other. In any realistic
situation, limited experience, specialization costs and physical limitations
of effort, energy and time, all prevent agents from perfecting their knowledge
of complex environments [32, 33]. A natural way to mitigate these individual
limitations is to pool knowledge across agents leading to the formation of
social organizations [34], and the division and coordination of labor in terms
of their behavior [35]. This is widely observed in human organizations, but
also in animal social behavior starting with the division of labor by age and
sex.

By working jointly to predict characteristics of their environment [29] and
gather resources, groups of agents can maximize their collective fitness even
when each individual has very limited knowledge. In a setting where there are
resource returns to successful prediction and behavior, information of the
state of a statistical environment determines the fitness of the population
[36, 37], though there are questions about how such benefits emerge
quantitatively [38]. Here we formalize the calculation of these social
benefits in terms of the properties of information and show how maximizing
knowledge complementarities (synergy) maximizes the long-term growth rate of
collectives. Specifically, we derive an expression for the additional payoff
to cooperative behavior in terms of the joint information synergy about the
agents’ dynamical environment.

These results lead us to introduce the principle of maximum synergy, which
maps the maximization of collective resource growth rates into optimal social
interaction structures. This work adds new dimensions to the study of
collective dynamics by connecting the structure of groups to that of
information in complex environments mediated by agents’ diverse subjective
characteristics, such as their present knowledge and their life histories.

##  II Theory of Collective Growth

We start by demonstrating how the benefits of collective action emerge from
pooling information in synergistic situations. Synergy means the combination
of behavior, knowledge, and skills that complement each other towards a goal.
This concept is necessary for creating effective organizations that embody
complex information [29], but it is often not sufficiently formalized in
common language, such as in discussions of innovation [39] or firm structure.

Here, we will refer to synergy as an explicit information theoretic quantity
that measures the additional predictive power that a group has upon pooling
its agent’s information together, relative to the knowledge of each individual
separately. This quantity has been introduced sometime ago in the context of
studying circuits in information processing systems [40, 41], and has provided
a framework for studying higher-order neuron interactions in the brain [42],
and causality and information in complex systems [43, 44]. As we will show,
synergy results formally from the conditional dependence between the
probability of predictive signals distributed in a population and events in a
shared environment. The gain in predictive power from agents pooling
information as collectives allows them to obtain additional resources from a
knowable environment beyond what agents alone can do, thus boosting their
relative fitness or productivity.

It follows that collectives that seek to maximize their resources over long
times must combine the information from their agents’ individual models of the
world in a way that accesses the most synergy. Groups that do not know a
priori how to realize their synergies must adjust their collective knowledge
and interaction structure by observing outcomes of their environment in an
iterative learning process. After developing the general framework for group
formation and collective growth across group sizes, we demonstrate a model
environment that exhibits synergy using logic gates that take signals as
inputs, and output probabilistic events. We will demonstrate how synergy
scales with the number of unique signals in a collective, and how specific
combinations of signals affect the average growth of resources for the group.

###  II.1 Collective Growth in Synergistic Environments

We consider a population of $N$ agents, each with initial resources $r_{i}$
that can be (re)invested into the set of outcomes of their environment to
generate returns. Each agents has access to a private signal, $s_{j}\in
S_{j}$, which is used to predict the state of the environment and makes
resource allocations to possible outcomes $e\in E$. This signal may represent
a number of different processes such as sensory input, or a lead retrieved
from memory. With accurate parameterization of a model of the environment,
$P(E|S_{j})$, an agent’s optimal investment strategy leads to a resource
growth rate $\gamma=I(E;S_{j})$ [36]. Agents with better models (and better
statistical estimations) experience higher average growth rates given by the
information rate of the agent’s signal for the environment.

We now define our environment by a set of $l$ signals with unique statistics,
$\bm{S}\equiv\\{S_{1},\dots S_{l}\\}$ as $P(E|\bm{S})$, with marginals of
events $P(E)$ and signals $P(\bm{S})$. The joint information that $\bm{S}$ has
on $E$ is at least equal to $S_{j}$, that is $I(\bm{S};E)\geq I(S_{j};E)$.
Generally, this inequality is strict if the conditional information
$I(\bm{S}|E)>I(\bm{S})$ [40, 41]. We compute the total information by summing
over the mutual information between the signals independently, subtracted by
an interaction term across signals,

$I(E;\bm{S})=\sum_{j}I(E;S_{j})-R_{P}.$ (1)

The quantity, $R_{P}$, denoted the coefficient of redundancy, measures the
strength of this conditional dependence across larger sets of signal (two,
three, etc way). It is defined in App. V.1

$\begin{split}R_{P}=&\sum_{j>k=1}^{l}R(E;S_{j};S_{k})+\sum_{j>k>m=1}^{l}R(E;S_{j};S_{k};S_{m})\\\
&+\dots+R(E;S_{1};\dots;S_{l}).\end{split}$ (2)

Figure 1:  Groups of agents with different signals grow resources based on the
information between their signals and states of the environment. A. Groups,
denoted $g$, are composed of an arbitrary number of agents. Each agent belongs
to only one group and can observe and contribute one signal to the group. A
group contains $k_{g}$ unique signals. B. At each time step, (a) the groups’s
private channel outputs a signal $\bm{s}\in\bm{S}$ with probability
$P(\bm{s})$. (b) Each member of the group observes their signal $s_{j}$ and
(c) the group consults their collective belief for the conditional outcome
probability of the environment, $X(E|\bm{s})$. (d) The agents make
proportional resource allocations on all possible outcomes $B(E|\bm{s})$. (f
and e). The true event $e\in E$ is observed in the environment with
probability $P(e)$, and (g) the agents receive payouts proportional to the
marginal probability of $e$.

The coefficient of redundancy can have a positive or negative value,
indicating different conditional relationships between the signals and
environmental states. When $R_{P}>0$, there is information across signals
irrespective of environmental events. This means that signals are $redundant$,
and consequently there are diminished returns to pooling information as
$I(E;\bm{S})<\sum_{j}I(E;S_{j})$. Conversely, when $R(E;\bm{S})=0$, the
signals are statistically independent, and the benefits of pooling information
increase linearly with the information of each signal on the environment but
there is no synergy. Finally, when $R(E;\bm{S})<0$, there is conditional
dependence of the signals on the environment. This is called synergy and
corresponds to a superlinear benefit to pooling information in the number of
agents, above and beyond the information contributed from each signal
individually.

####  II.1.1 Group formation and collective decision making

We have now defined individual resource growth as a quantity of information
and discussed how information can be aggregated across signals to express
their synergy relative to states of the environment. Now we can explore how
agents with different signals can pool information together as coordinated
groups, and access the synergy in their environment through collective
decision-making.

Consider the undirected hypergraph $H=(A,G)$ of vertices, $A$, and hyperedges
$G$. We consider a discrete number of vertices,
$A=\\{a_{1},a_{2},\dots,a_{N}\\}$, where $a_{i}$ identifies agent $i$. The set
of hyperedges, $g\in G=\\{1,2,\dots\\}$, called groups, defines the number of
cooperating collectives. A hyperedge connects $1\leq N_{g}\leq N$ agents, and
we assume that agents can only belong to a single group. Therefore, by
construction, $\sum_{g}N_{g}=N$ and the sum over all nodes of every hyperedge
yields the number of agents in the population. There exist two extremes of
cooperation. First, when a single hyperedge spans every node, meaning all
agents pool information in one group. In the limit of no cooperation,
$N_{g}=1$ for all $g$, and no agents pool information. In this case, the
dynamics of the model are similar to previous work [36].

Let $\bm{S}_{g}$ be the set of unique signals held by the agents of a group
$g$ to be pooled, such that $\bm{S}_{g}\subseteq\bm{S}$. The number of
$cooperants$ is defined by the number of unique signals, $|\bm{S}_{g}|=k_{g}$,
and is bounded by $1\leq k_{g}\leq l$. When $k_{g}=l$ and the group has a
complete signal, the collective can make maximally informed decisions.
Conversely, when $k_{g}<l$, the signal is considered $incomplete$, and the
collective can only interpret and act on a subset of signals. As we will see,
the number of unique signals a collective can observe determines the amount of
information they can access.

Now that we have defined how agents organize into groups of various sizes, we
can discuss how agents pool their information to make collective decisions and
grow their resources in dynamic environments. At every time step, a collective
with access to all signal types observes a unique private signal
$\bm{s}=\\{s_{1},\dots,s_{l}\\}\in\bm{S}$. Each agent then allocates its
resources $r_{i}$ on events according to collective $g$’s allocation matrix
$B(E|\bm{s})$. As the event $e$ is observed, the agent is rewarded with
returns $w_{e}$ to the fraction of resources invested in $e$, $B(e|\bm{s})$.
In the limit of many sequential investments $n$, the average growth rate of
resources converges to

$\gamma=\frac{1}{n}\log\frac{r_{n}}{r_{i}}\approx\sum_{e,\bm{s}}P(e,\bm{s})\log\big{[}B(e|\bm{s})w_{e}\big{]},$
(3)

The optimal investment in the large $n$ limit is the conditional probability
of the event given the signals, $B(e|\bm{s})=P(e|\bm{s})$. When the rewards
are “fair”, and $w_{e}=1/P(e)$, the optimal growth rate is given by the mutual
information [45] defined in equation 1, $\gamma=I(E;\bm{S})$.

The typical collective may not have a complete signal, and instead may only
observe and interpret a subset of all unique signals $\bm{S}_{g}$. Their
optimal allocation, given by $P(E|\bm{S}_{g})$, then has mutual information
$I(E;\bm{S}_{g})\leq I(E;\bm{S})$, with equality only if the omitted signals
are completely redundant with present signals. Unless there are redundant
signals, an incomplete group is guaranteed to have suboptimal information and
growth rate.

Furthermore, agents may also not start out with perfect knowledge and must
invest using their best estimate of the true environmental probability,
$X(E|\bm{S}_{g})\neq P(E|\bm{S}_{g})$. In this case, the collective’s average
growth will be submaximal by the number of signals and lack of information on
signals, and is described by

$\gamma_{g}=I(E;\bm{S}_{g})-\textrm{E}_{\bm{s}_{g}}\big{(}D_{KL}\big{[}P(E|\bm{s}_{g})||X(E|\bm{s}_{g})\big{]}\big{)},$
(4)

where $\textrm{E}_{\bm{s}_{g}}$ is the expectation value over the states of
the group’s signals, and
$D_{KL}\big{[}P(E;\bm{s}_{g})||X(E;\bm{s}_{g})\big{]}=\sum_{e}P(e|\bm{s}_{g})\log(P(e|\bm{s}_{g})/X(e|\bm{s}_{g}))\geq
0$ is the Kullback-Leibler divergence, an information measure expressing how
similar the distributions are in their inputs. This result shows that
collectives with both a better model as reflected by the first term, a better
characterization of the model and its various synergies by the second, and a
more complete signal will experience higher growth rates. Furthermore,
$\gamma_{g}<\gamma$ unless $g$ is the full set of signals, so it is typically
valuable to add more signals to the group. This setup is illustrated in Figure
1.

###  II.2 Maximum synergy principle and optimal growth

These results introduce important considerations for how collective innovation
and growth determine strategies for group formation. In theories of
cooperation such as kin selection [46] and scalar stress [47], group formation
is advantaged by member relatedness and disadvantaged by unfamiliarity. This
is intuitive in many situations, as agents are more likely to cooperate when
they are more certain others will reciprocate [48], and cooperating with
similar agents minimizes this uncertainty. Equation 4 counters this intuition
by defining an explicit benefit to cooperating with dissimilar agents across
heterogeneous, complementary skills and information. Specifically, a group
with more synergistic signals, as defined through the conditional dependence
of their decisions on states of the environment, will experience higher
growth. So, even if there are additional coordination costs for more
heterogeneous agents, there is now a possiblity that cooperation will emerge
as there are also greater informational benefits, formalizing intuitive ideas
about the value of diversity [49].

The beneficial contribution of synergy to the growth rate of resources
provides an important input to models of random multiplicative growth, such as
those commonly used to study wealth dynamics and mathematical finance. In its
simplest form, the stochastic growth rate in such models is characterized by
its first two temporal moments. The average over time, $\eta$, and the
resource temporal standard deviation (volatility), $\sigma$, combine under Itô
integration to give actual growth rate $\gamma=\eta-\sigma^{2}/2$. Maximizing
this growth rate (as a positive quantity) entails maximizing $\eta$ and
minimizing $\sigma$, which at the individual agent level can be achieved by
(Bayesian) learning over time [36].

At the population level, it has been proposed that pooling resources in groups
would naturally emerge as a means to reduce $\sigma$, when growth rate
fluctuations are independent across agents, and thus maximize $\gamma$ [50,
25].

Our results introduce a different possibility of cooperation, through pooling
information in structured groups, that maximizes $\eta$ (and $\gamma$) through
synergy effects. Thus, to maximize $\gamma$, agents should pool information
with a most diverse set of collaborators possible to access the most mutual
synergy viz. the environment. This maximum synergy principle defines the
benefit of intelligent collective behavior in complex environments where there
are agent level limitations to knowing the environment fully and where
mechanisms of the division of labor and knowledge are favored. This principle
is general and applies across levels of cooperation, whether it be individuals
matching skills to form groups, or specialized groups organizing into more
complex collectives [38], all the way to large scale societies.

Generally, these two strategies– information synergy versus resource pooling
under independence– are distinct modes of cooperation over which groups can
maximize $\gamma$, as demonstrated in Figure 2.

Figure 2:  Complementary strategies for increasing the long-term growth rate
of resources from the environment in stochastic growth models. Pooling
resources can reduce the volatility through a hedging strategy while pooling
information creates synergy to increase average growth rates. The lines
represent contours of constant average growth rates $\gamma$.

As we will see later, the decision of who to cooperate with is not trivial, as
different combinations of signals may yield varying synergies. This means that
under constraints of group size, such as from cooperation costs per
connection, groups satisfying the maximum synergy principle must intelligently
select which signals and agents to integrate, and which to exclude as
redundant.

Furthermore, collectives may not a priori know the optimal allocation strategy
that leverages the synergy available to their signals, meaning that
intelligent collective behavior must itself be learned over time and by
exploring the best possible matchings. We will now develop the dynamics of how
a group can organize itself optimally so as to maximize its synergy.

####  II.2.1 Synergy maximization through Bayesian inference

Bayesian learning is the optimal strategy to incorporate new information from
observed events into the estimate of conditional probabilities, such as those
of environmental states given agents’ signals [36]. Agents can also learn the
synergy embedded in their environment in groups by collectively weighing their
conditional observations across their individual signals. A group wanting to
maximize their synergy must then update their conditional relationship through
a Bayesian inference process

$X_{n}(e|\bm{s})=AP(\bm{s}_{n}|e_{n})X(e_{n})=\bigg{[}\prod_{i=1}^{n}\frac{P(\bm{s}_{i}|e_{i})}{P(\bm{s}_{i})}\bigg{]}X(e),$
(5)

where the normalization $A=(\int de_{n}P(\bm{s}_{n}|e_{n})X(e_{n}))^{-1}$. We
take the prior probability, $X(e_{1})=X(e)$, because we are assuming that the
environment is stationary or at least slowly changing relative to groups’
learning rates.

Bayesian inference converges $X(E|\bm{S})\rightarrow P(E|\bm{S})$ over time,
decreasing the information divergence, and maximizing synergy and average
growth. For groups with incomplete signals, the information acquired through
learning is still bounded by what is available in the incomplete signal space.

We have thus far defined collective growth in terms of information synergy,
and shown how agents can learn as a collective to increase their growth rate
over time. We will now illustrate these general results using a model based on
logic circuits.

##  III Modeling Synergy with Logic Circuits

Logic circuits have been used extensively as models for synergistic
interactions [38, 40, 41]. This is because their outputs are predicted by
combinations of inputs, much like events are predicted by combinations of
signals. Among other logic circuits (like AND or OR), the XOR gate is unique
in that information between inputs and outputs only exists as synergy across
all inputs [51]; no individual input has mutual information with the output.

In the following, we will show how modifying the XOR gate relaxes this
condition, such that information exists for any input and scales on average
with the number of cooperating signals. Similar to [36], while this model will
be used to study synergy in a simplified setting, the theory is defined for
general dynamical environments.

###  III.1 The Uniform XOR Gate

Consider the space of statistically independent binary signals $s_{j}\in 0,1$,
such that a sample set $\bm{s}$ has uniform probability $P(\bm{s})=2^{-l}$. We
assign each input $\bm{s}$ a binary event, $e\in 0,1$, using the generalized
XOR rule, $e=M_{2}(\bm{s})\equiv\big{[}\sum_{j=1}^{l}s_{j}\big{]}(\textrm{mod
2})$ with binomial probability $p_{\bm{s}}$. From the sets of sampled signals,
$\bm{s}$, and binomial coefficients $\bm{p}=\\{p_{\bm{s}}\\}$, we can define
this generalized XOR circuit as a joint distribution on signals and events as

$\begin{split}P(E,\bm{S}|\bm{p})\equiv&f(\bm{p},l)=\frac{1}{2^{l}}\prod_{\bm{s}}(p_{\bm{s}})^{M_{2}(\bm{s})}(1-p_{\bm{s}})^{1-M_{2}(\bm{s})}.\end{split}$
(6)

This distribution is called the uniform XOR (UXOR). It performs a unique, $l$
dimensional XOR gate on each input $\bm{s}$ with probability $p_{\bm{s}}$. In
the case where $p_{\bm{s}}=1$ for all input permutations, this circuit behaves
deterministically like an XOR gate, and the complete group has 1 bit of
information. In the limit of $p_{\bm{s}}=.5$, this no longer models a logic
gate as the output is uncorrelated to the inputs. The truth table of this
circuit is shown in Figure 3A for an environment with two signals.

####  III.1.1 Information scaling in the UXOR environment

With this explicit choice of distribution, we can explore quantities of
information that will define a group’s growth process. For simplicity, we
choose a uniform prior for the distribution of $\bm{p}$, but in principle any
prior distribution is admissible. The information available in the environment
measures the maximum average growth rate a group with a complete signal can
experience. When averaged over all configurations of $\bm{p}$, the information
is given by $I(E;\bm{S})=\log 2-1/2\approx.28$ bits (App. V.3).

For groups with incomplete signals (when $k_{g}<l$), we compute the
information by marginalizing equation 6 over the $\lambda_{g}=l-k_{g}$ signals
unavailable to the group. The procedure for marginalization is defined in App.
V.3, but in general, marginalization of one signal halves the size of the
parameter space $\bm{p}$ that describes the distribution. The average
information for an incomplete signal is approximately (App. V.4)

$I(E;\bm{S}_{g}|\bm{p})\approx 2^{-\lambda}\left(\log 2-\frac{1}{2}\right).$
(7)

Average information increases exponentially, $\sim 2^{k}$, as more signals are
included. The mutual information of the complete signal is independent of the
number of signals, so the information of a single signal must converge to zero
in the limit of large $l$.

The exponential scaling of the information with the number of cooperants is
demonstrated in Figure 3B, as lines on a logarithmic scale for environments of
increasing $l$. The curves are computed by Monte Carlo sampling circuits for
$l$ signals by measuring the information after $\lambda=l-k$ marginalizations.

####  III.1.2 Growth and group learning

Until now we have explored the mean behavior of this environment subject to a
uniform prior. In general, collectives do not have perfect information on a
single prior. In this case, their inaccurate guess for the set of binomial
coefficients is parameterized by $\bm{x}_{g}\equiv\\{x_{\bm{s}_{g}}\\}$,
indexed by the signals available to the group $\bm{s}_{g}\in\bm{S}_{g}$, and
the collective’s likelihood model becomes
$X(e|\bm{s}_{g})=f(\bm{x}_{g},k_{g})$. The information divergence term of
equation 4 becomes the divergence between $f(\bm{x}_{g},k_{g})$ and
$f(\bm{p}_{g},k_{g})$, where $\bm{p}$ has been projected into the subspace
spanned by $\bm{S}_{g}$, averaged over all signals
$\textrm{E}_{\bm{s}_{g}}[D_{KL}]=\langle
p_{\bm{s}_{g}}\log(p_{\bm{s}_{g}}/x_{\bm{s}_{g}})+(1-p_{\bm{s}_{g}})\log[(1-p_{\bm{s}_{g}}/(1-x_{\bm{s}_{g}})]\rangle$
here angle brackets denote sample averages over the binomial values.
Subtracting the mutual information by this term yields the growth rate under
imperfect, incomplete group information.

$\gamma_{g}=\big{\langle}p_{\bm{s}_{g}}\log
x_{\bm{s}_{g}}+(1-p_{\bm{s}_{g}})\log\big{(}1-x_{\bm{s}_{g}}\big{)}\big{\rangle}+\log
2,$ (8)

We have so far described growth rate dynamics under a stationary $\bm{x}_{g}$.
To illustrate growth dynamics under group learning, we turn to the Latent
Dirichlet Allocation (LDA) model. Through a categorical description of pairs
of events and signals, agents experience average dynamics to $\bm{x}_{g}$ in
the limit of high sampling rate $\omega=n/t\gg 1$

$\bm{x}_{g}(t)=\frac{\bm{p}_{g}t/2\kappa+\bm{x}_{g}}{1+t/2\kappa},$ (9)

where $\kappa$ defines the Bayesian update time. The details of LDA are given
in Ref. [36] and provide parametric

Figure 3: The UXOR model provides an environment for exploring synergy across
groups of arbitrary size. A. The UXOR circuit, demonstrated by the modified
XOR symbol, and its truth table for $l=2$. B. The information of a circuit of
size $l$ scales exponentially in cooperating signals, $k$.

dynamics that converge to full information as a power law in time, in
stationary environments.

Figure 4: Groups learning an $l=4$ environment using more unique signals
acquire more resources and information, but combinations of signals have
unique amounts of information. A. Temporal resource trajectories, grouped by
number of unique signals in the corresponding group show that growth increases
with the number of signals. B. Groups with more signals can gather more
information from the environment. There is high variability when $k_{g}<l$, as
different combinations of signals access different amounts of information. C.
Top For $k_{g}=2,3$, the synergy benefits of a parameter configuration are
given by the difference between the information when averaged (small dot) and
pooled (large dot). Bottom Parameter values exist where no signal combinations
hold synergy (left) and synergy is equivalent across signal combinations
(right).

To study the dynamics of resources in the UXOR environment, we simulated agent
investments in a Monte Carlo sampled environment. We randomly assigned
$N=2000$ agents signals in an $l=4$ environment, then randomly assigned them
to groups sized $l\leq N_{g}\leq 11$. This results in an ensemble of groups
with cooperants $1\leq k_{g}\leq 4$. We reveal Bernoulli-sampled signals to
the groups, whose agents make collective decisions on which events to allocate
resources. For each group, we track the resources of a representative agent,
informed by the group, investing their individual resources through time.

Figure 4 illustrates the results of this simulation. In subfigures A and B,
the Monte Carlo simulated means are shown as solid lines, with 95% Confidence
Interval (CI) shaded regions. Theoretical means are computed from the initial
population configuration using equation 9, plotted as dashed lines, with hash-
filled uncertainty regions. The more unique signals a group can access
increases, the more they can learn, and the more resources they acquire over
time. A high signal-to-noise ratio when $k_{g}=1,2$ causes growth rates lower
than the theoretical mean, and cumulatively fewer resources over time.

####  III.1.3 Constrained intelligent group formation

For the groups with $k_{g}<4$ (incomplete signals) there is significantly
higher variance in both information and resources compared to $k_{g}=4$. This
is attributed to differences in synergy between groups with different
combinations of signals of order $k$. This illustrates a general feature of
the maximal synergy principal; that signal combinations with higher
conditional dependence on the environment will have higher synergy and
experience higher growth rates than other combinations. Figure 4 demonstrates
the synergy effects across different combinations of signals. For each group
of size $k$, the left, smaller dot indicates the amount of information each
signal has averaged over the signals present. The right, larger dot indicates
the total information the combination of signals has when pooled. The
difference between the two dots gives the amount of synergy. We see, for
example, that even though signals 0 and 3 have less information than signal 2,
both signals have higher synergy effects when pooled with 1 individually, as
indicated by their crossover with the 1, 2 line. For a group aggregator, not
only does this mean that signal choice is nontrivial, but also that individual
information is not generally a good indicator of synergy benefits that can be
realized when pooled. This result reinforces the complexity that fulfilling
the maximum synergy principle entails, as one must understand signal
complementarities for a given model of the environment to all orders, a likely
costly process.

As demonstrated by the bottom plots in Figure 4C, through a smart selection of
$\bm{p}$, we can also design special environments such as where either no
synergy is present, or where there are uniform benefits of synergy across
combinations of signals. The procedure for constructing environments with
specific synergy profiles will developed in future work.

##  IV Discussion

In this paper, we developed a novel mechanism of cooperation among
heterogeneous agents that use shared information to grow resources in noisy
environments. We derived the benefits of cooperation in terms of synergy
gained by pooling information across agents’ unique signals. This motivates
the principle of maximum synergy, whereby a group’s aggregate growth is
optimized when that group maximizes the synergy of its members relative to a
statistical environment. We proposed this principle as a complementary avenue
to cooperation compared to the reduction of volatility through resource
pooling in multiplicative growth models. We then showed that a group with no a
priori knowledge of its potential synergy can learn it through Bayesian
inference. We illustrated these principles using a model of a high-dimensional
probabilistic logic gate and showed that, on average, group synergy scales
superlinearly (exponentially) with the number of unique signals in the group.
We also illustrated the challenge faced by groups under constraints to size to
pick not just unique signals but also admit new group members as additional
signals that maximize their potential collective synergy.

These results formalize several insights into the causes and benefits of
cooperation. First, the properties of information allow us to consider how the
limits to human effort and ability motivate group formation. Specialization
through learning or adaptation is costly in terms of time and resources,
motivating a division of labor to fully learn and maximize productivity across
disparate but synergistic agents [34].

Second, these results motivate analyses of how information and resource
pooling strategies affect different levels of selection within an
organizational hierarchy. Effective resource pooling relies on uncorrelated
fluctuations across participants, which is not possible when agents are making
coordinated decisions across correlated signals. We therefore expect
information and resource pooling strategies to create tradeoffs in group
formation, and apply to different environmental features and levels of
selection.

Groups lacking informational complementarities (because they are homogeneous)
operating in variable environments should pool resources. This may apply to
people in insurance pools, or independent economic sectors within a common
population, such as a city or nation. Conversely, groups in complex
environments made up of agents with complementary knowledge, such as within a
firm or innovation ecology, should engage in information pooling and skills
specialization to maximize their production whenever the variability of the
environment and costs of cooperation are sufficiently low.

Parsing out these modes of cooperation becomes more important when considering
how groups respond to changing environmental or social conditions. As new
environmental conditions emerge, such as new industries, the distribution of
synergy across different group configurations will also change, selecting for
different group compositions and skills combinations. This has the interesting
implication that new knowledge (science, technology, institutional change)
should be disruptive of established social and economic structures because it
enables new synergies. This also has implications for natural ecosystems [52]
where changing environmental conditions and variability, such as via climate
change, may alters their structures.

Third, the framework developed here describes a general approach to describing
interaction dynamics in many fields. The conditional probabilities
$P(e|\bm{s})$ capture the general structure of information between populations
and their environment. Through synergy, that information becomes encoded in
how groups form and are structured, and which sets of coordinated behaviors
produce beneficial or detrimental behaviors across agents. By tracing over
states of $E$, averaging over (stationary) environments, we can produce a set
of rules for (average) rewards associated with agents’ perceptions and
actions. This shows how general conditional probabilities of choices and
behaviors in given environments may underlie particular games and other
phenomenological agent interaction rules [53]. Furthermore, because
conditional distributions are general and multi-dimensional they also provide
natural models of higher order interactions expressing large groups’ synergy,
such as reciprocal cooperation and the emergence of culture as shared
knowledge and behavior [24]. In summary, the formal properties of information,
made explicit over group structures and time, provide the theoretical basis
for a broad class of agent interaction models found throughout the social and
ecological sciences. This includes the formation of complex societies made up
of diverse cooperating agents in situations where large scale synergy becomes
possible.

This work is supported by the Mansueto Institute for Urban Innovation and the
Department of Physics at the University of Chicago and by a Na- tional Science
Foundation Graduate Research Fellowship (Grant No. DGE 1746045 to JTK), and by
the National Science Foundation, through the Center for the Physics of
Biological Function (PHY-1734030), as well as the National Institutes of
Health BRAIN initiaitive (R01EB026943) to AGK.

##  V Appendix

###  V.1 Information Aggregation

Consider a target statistical variable $E$ (environment), that we wish to
predict using $l$ other variables (signals) $\bm{S}=\\{S_{1},\dots,S_{l}\\}$.
The mutual information between each signal $S_{i}$ separately and $E$ is given
by [54]

$I(E;S_{i})=H(E)-H(E|S_{i})=-\frac{\Delta H(E)}{\Delta S_{i}}.$ (10)

where $H(E)$ is the Shannon entropy of $E$, and the variation measures the
difference in entropy of the event when conditioned on the signal. From the
rules of information aggregation, this expression generalizes to information
across every added signal [55]. The mutual information between the event and
the set of several signals is given by

$\begin{split}I(E;\bm{S})=&-\sum_{i=1}^{l}\frac{\Delta H(E)}{\Delta
S_{i}}-\sum_{i>j=1}^{l}\frac{\Delta^{2}H(E)}{\Delta S_{i}\Delta
S_{j}}-\dots-\frac{\Delta^{l}H(E)}{\Delta S_{1}\dots\Delta S_{l}}.\end{split}$
(11)

The first term of this expansion is just a sum over the mutual information of
each individual signal and the environment. The goal of this section is to
show that the inclusion of each new signal introduces a coefficient of
redundancy of progressively higher order. The first term is

$\begin{split}\frac{\Delta^{2}H(E)}{\Delta S_{i}\Delta
S_{j}}=&H(S_{i})-H(S_{i}|E)+H(S_{j})-H(S_{j}|E)-H(S_{i},S_{j})+H(S_{i},S_{j}|E)\\\
=&H(S_{i})-H(S_{i}|E)-H(S_{i}|S_{2})+H(S_{i}|S_{j},E)\\\
=&I(S_{i};S_{j})-I(S_{i};S_{j}|E)\equiv R(E;S_{i};S_{j}),\end{split}$ (12)

where we used the identity $H(A,B)=H(A|B)+H(B)$. We denote $R$ as the
coefficient of redundancy, which measures the difference in mutual information
between the variables, $I(\bm{S})\equiv I(S_{1};\dots,S_{k})$, and the mutual
information of the variables conditioned on $E$, $I(\bm{S}|E)$. When
$I(S_{i};S_{j})<I(S_{i};S_{j}|R)$, the signals contain less mutual information
in the absence of the event (we gain information by considering the event),
and $R(S_{i};S_{j};X)<0$. In this case agents experience a positive benefit
from pooling information, which we call synergy.

To demonstrate this effect to higher orders in goups of signals, we perform a
similar calculation for a three-signal interaction.

$\begin{split}\frac{\Delta^{3}H(E)}{\Delta S_{i}\Delta S_{j}\Delta
S_{k}}=&H(E)-H(E|\\{S_{i},S_{j},S_{k}\\})\\\
=&H(S_{i})+H(S_{j})+H(S_{k})-H(S_{i}|E)-H(S_{j}|E)-H(S_{k}|E)-H(S_{i},S_{j})-H(S_{j},S_{k})\\\
&-H(S_{k},S_{i})+H(S_{i},S_{j}|E)+H(S_{j},S_{k}|E)+H(S_{k},S_{i}|E)+H(S_{i},S_{j},S_{k})-H(S_{i},S_{j},S_{k}|E)\\\
=&H(S_{i},S_{j},S_{k})-H(S_{i}|S_{j})-H(S_{j}|S_{k})-H(S_{k}|S_{i})+H(S_{i}|S_{j},E)+H(S_{j}|S_{k},E)\\\
&+H(S_{k}|S_{i},E)-H(S_{i},S_{j},S_{k}|E)\\\
=&I(S_{i};S_{j};S_{k})-I(S_{i};S_{j};S_{k}|E)\equiv
R(E;S_{i},S_{j},S_{k}).\end{split}$ (13)

We see that an analogous redundancy coefficient arises in three dimensions.
This can generally be retrieved for arbitrary number of dimensions through a
similar iterative procedure. We refer to the sum of these moments collectively
as the redundancy of the joint distribution, denoted $R_{P}$ [55],

$\begin{split}R_{P}\equiv&-\sum_{i>j=1}^{l}\frac{\Delta^{2}H(E)}{\Delta
S_{i}\Delta S_{j}}-\dots-\frac{\Delta^{l}H(E)}{\Delta S_{1}\dots\Delta
S_{l}}\\\
=&\sum_{i>j=0}^{l}\big{[}I(S_{i};S_{j})-I(S_{i};S_{j}|E)\big{]}+\dots+I(S_{1};\dots;S_{l})-I(S_{1},\dots,S_{l}|E)\end{split}.$
(14)

Note that redundancies of lower order than cardinality of the signal space
must be computed over every combination of signals. For example, when $l=3$,
there are three second order redundancy terms.

This expansion generally defines the benefits to cooperation over increasingly
higher orders of cooperation (number of signals). This expression can be used
to compute the relative strengths of the various orders of interaction for any
set of signals and environmental variables, given their conditional
distributions.

###  V.2 Kelly Growth rate

Consider an environment with events conditionally dependent on signals
characterized by a joint distribution $P(E,\bm{S})$ for event $E$ and $l$
signals $S$. Consider a cooperative Kelly investment scheme whereby each
participant, agent $i$, witnesses signal $s_{i}\in S_{i}$, and informs the
collective how to invest their shared resources $r$. The mechanics of pooling
resources and collectively investing will be discussed below. Kelly’s
formalism can be adapted by expanding the environmental probability to contain
$l$ signals, $P(E,S)\rightarrow P(E,\bm{S})$, as can the betting matrix
$X(E|S)\rightarrow X(E|\bm{S})$, where $\bm{S}=\\{S_{1},\dots,S_{l}\\}$. When
odds are fair, the Kelly growth rate is given by the returns to each
investment, averaged over the probability of that signal, event pair

$G=\sum_{e,\bm{s}}^{E,\bm{S}}p(e,\bm{s})\log\frac{x(e|\bm{s})}{p(e)}.$ (15)

We expand this equation by inserting $p(e,\bm{s})$ into the numerator and
denominator of the log

$G=\sum_{e,\bm{s}}^{E,\bm{s}}p(e,\bm{s})\bigg{[}\log\frac{p(e,\bm{s})}{p(\bm{s})p(e)}-\log\frac{p(e|\bm{s})}{x(e|\bm{s})}\bigg{]}.$
(16)

These two terms can be simply expressed as
$G=I(E;\bm{S})-\textrm{E}_{\bm{s}}\big{[}D_{KL}\big{(}P(E|\bm{s})||X(E|\bm{s})\big{)}\big{]}$,
similar to previous work, but we can decomposes this equation in terms of
redundant information across the signals using equations 11 and 14.

###  V.3 Information for UXOR circuits

Here we compute $I(E;\bm{S})$ for the UXOR logic circuit. This represents the
information that a group of agents with $l$ distinct signals have about the
output $E$ of the probabilistic gate, averaged over all configurations of the
gate for a uniformly distributed prior. Because the signals $s_{i}$ are
independent Bernoulli trials with probability $1/2$,

$I(E;\bm{S})=\sum_{e,\bm{s}}P(e,\bm{s})\log\frac{P(e,\bm{s})}{P(e)P(\bm{s})}=\frac{1}{2^{l}}\sum_{e,\bm{s}}P(e|\bm{s})\log\frac{P(e|\bm{s})}{P(e)}\,.$

Then, using the fact that the output $E$ is also a Bernoulli variable,
$P(e=1|\bm{s})=1-P(e=0|\bm{s})$, and

$I_{l}=I(E;\bm{S})=\frac{1}{2^{l}}\sum_{\bm{s}}g(P(e=0|\bm{s}))-g(P(e=0))\,,$
(17)

where $g$ is a function representing application of the UXOR gate and is
defined over binomial parameters $x$ as

$g(x)=x\log x-(1-x)\log(1-x)\,.$

The number of terms in (17) grows exponentially with $l$ and quickly becomes
large. When it is sufficiently large, the sum can be approximated by an
average. In particular, for uniformly distributed $P(e=0|\bm{s})$,

$\frac{1}{2^{l}}\sum_{\bm{s}}g(P(e=0|\bm{s}))\approx\langle g(x)\rangle_{x\sim
U(0,1)}.$

This expectation can be analytically evaluated. With this, (17) gives

$I_{l}=\langle g(x)\rangle_{x\sim U(0,1)}-g(1/2)=\log 2-\frac{1}{2}\,.$

###  V.4 Mutual Information for incomplete signal sets

Here, we demonstrate that the information the collective has about the gate
output scales exponentially with respect to the number of cooperants, $k$, as
is depicted in Figure 3. Following the previous section, the introduction of
the function $g(x)=x\log x+(1-x)\log(1-x)$ simplifies the expression for
mutual information

$I_{k}=I(E;\bm{S})=\frac{1}{2^{k}}\sum_{\bm{s}}g(P(e=0|\bm{s}))-g(P(e=0))\,.$
(18)

As before, this sum can be interpreted as an average over the uniform
distribution when the number of terms is large. Here, the removal of parts of
the signal changes the distribution of parameters, so the measure that
approximates this sum also changes. We call this new measure $P_{k}$.
Furthermore, whereas in the main text, the subscript of $\bm{S}_{g}$ denoted
the signals of group $g$, here the subscript of $\bm{S}_{k}$ will denote the
signal set of cardinality $k$ to be marginalized. With this new notation, the
first term in (18) may be written approximately as:

$\frac{1}{2^{k}}\sum_{\bm{s}_{k}}g(P(e=0|\bm{s}_{k}))\approx\langle
g(x)\rangle_{x\sim P_{k}}.$

To compute $P_{k}$ for $k<l$, we need to calculate how the probability of $E$
conditional on the remaining signals $\bm{s}_{k}$ changes under the removal of
the $k^{\text{th}}$ signal. For this model,

$P(e|\bm{s}_{k-1})=\frac{1}{2}\left(P(e|\bm{s}_{k-1},s_{k}=0)+P(e|\bm{s}_{k-1},s_{k}=1)\right).$
(19)

By iterating this sum, we reduce the number of parameters required to describe
$P_{k}(k)$, which in the main text are given by the set of binomial parameters
$\bm{p}$. Additionally, the distribution $P_{k}(x)$ becomes increasingly
narrow, centered around 1/2, which is the mean of all probabilities
$P(e|\bm{s})$. Parameterizing $P_{k}(x)$ by its moments allows us to directly
compute the mutual information. The moment expansion of the distribution is
given by

$I_{k}\approx\langle
g(x)\rangle_{P_{k}}-f\left(\frac{1}{2}\right)=\sum_{a=0}^{\infty}\frac{1}{a!}f^{(a)}(x_{0})\langle(x-x_{0})^{a}\rangle_{P_{k}}-f\left(\frac{1}{2}\right).$

Using standard arguments, which we provide in the following section, these
moments approximately scale like

$m_{k-1}^{(a)}=\left\langle\left(x-\frac{1}{2}\right)^{a}\right\rangle_{P_{k-1}}\approx\frac{m_{k}^{(a)}}{2^{a/2}}\,,\qquad
m_{l}^{(a)}=\frac{1}{(a+1)2^{a}}\,,$ (20)

which is related to the onset of central limit theorem behavior. This provides
us with a heuristic explanation for why $I_{k}$ scales as $1/2^{\lambda}$.
After only a few cooperants are removed, higher order terms in the expansion
(with order denoted by $a$) quickly die away, leaving only the second-order
term

$I_{k}\approx\sum_{n=0}^{\infty}\frac{f^{(a)}(1/2)}{a!}m_{k}^{(a)}\to
2m_{k}^{(2)}+O((m_{k}^{(2)})^{2})\,.$

Once this occurs, we can see plainly that between each marginalization the
mutual information reduces by half,

$\frac{I_{k}}{I_{k+1}}\to\frac{1}{2}.$

While this explanation gives approximately the correct scaling behavior, it
does not admit a good estimate of $I_{k}$ near full cooperation, since there
the higher-order terms in the expansion are not small. To explicitly include
these terms, we need all derivatives of $f$, evaluated at $x=1/2$

$f^{(a)}(1/2)=\frac{(-1)^{a}2^{a}a!}{a(a-1)}.$

Inserting these derivatives and the approximation (20) for moments of $P_{k}$
gives an approximation of $I_{k}$ as a series. Then, evaluating this series
analytically yields a closed form expression.

$\displaystyle I_{k}$
$\displaystyle=\sum_{a=0}\frac{f^{(a)}(1/2)}{a!}m_{k}^{(a)}-g(1/2)\approx\sum_{a=1}^{\infty}\frac{(2a+1)!}{(2a-1)!}\frac{1}{2^{\lambda
a}}$
$\displaystyle=\frac{1}{2}\left[(2^{-\lambda/2}+2^{\lambda/2})\,\text{arctanh}(2^{-\lambda/2})+\log(1-2^{-\lambda})-1\right]\,,\qquad\lambda>0$

This expression gives a good approximation for $I_{k}$ in the small $\lambda$
regime and also captures the scaling in the intermediate regime. For
$\lambda\to\infty$, an estimate of the asymptotic behavior is given by setting
$z=2^{-\lambda/2}$ and Taylor expanding around $0$.

$\displaystyle\frac{1}{2}\left[(z+z^{-1})\,\text{arctanh}(z)+\log(1-z^{2})-1\right]$
$\displaystyle\to\frac{1}{2}\left[(z+z^{-1})(z+\frac{1}{3}z^{3}+O(z^{4}))-z^{2}+O(z^{4})-1\right]$
$\displaystyle=\frac{1}{6}z^{2}+O(z^{4})$

Because $z=2^{-\lambda/2}$, the quadratic leading term agrees with the
observed exponential scaling $I_{k}\propto 2^{-\lambda}$. Although the scaling
prediction is correct, there is not a regime where this high-$\lambda$
expression consistently estimates $I_{k}$ in its actual value. The essential
reason is that as $k$ decreases, the number of terms in the sum over remaining
signal states decreases, and the approximation of that sum as an average over
$P_{k}(x)$ begins to break down. This manifests as an error in the overall
scale of the estimate, but not in the exponential dependence on $k$.

Instead, using the fact that $I_{l}$ is known to a very good approximation and
further that information is approximately exponential in $k$, a good estimate
for $I_{k}$ at large to intermediate $k$ is given by

$I_{k}=2^{-\lambda}I_{l}=2^{-\lambda}\left(\log 2-\frac{1}{2}\right)\,.$

This is the quantity quoted in (7) in the main text.

####  V.4.1 Moment scaling with $k$

The following argument justifies (20) and is standard. We produce it here for
completeness. Upon moving from $k$ to $k-1$ cooperants, the new conditional
distribution is given by (19). This means that $P_{k-1}$ is given by a
convolution of $P_{k}$ with itself.

$P_{k-1}(x)=\int_{\mathbb{R}^{2}}dy\,dz\,\delta\left(x-\frac{1}{2}(y+z)\right)P_{k}(y)P_{k}(z)=2\int_{\mathbb{R}}dy\,P_{k}(2x-y)P_{k}(y)$

The characteristic function of a distribution over a continuous variable is
its Fourier transform. Since $P_{k-1}$ is a convolution, its characteristic
function is the product of characteristic function of $P_{k}$ with itself. A
slightly more convenient object to work with is therefore the logarithm of the
characteristic function:

$\varphi_{k}(z)=\log\int_{\mathbb{R}}dxP_{k}(x)e^{izx}$

The sum rule above gives a recursion relation for $\varphi_{k}$

$\varphi_{k-1}(z)=2\varphi_{k}(z/2)+\log 2\,.$

Now, the cumulants of $P_{k}$ can be calculated from $\varphi_{k}$

$c^{(a)}_{k}=\frac{1}{i^{a}}\frac{d^{a}}{dz^{a}}\varphi_{k}(z)\bigg{|}_{z=0}\,,$

Meaning there are also recursion relations for the cumulants:

$c^{(a)}_{k-1}=2^{1-a}c^{(a)}_{k}$ (21)

This leads directly to the central limit theorem, since when the second
cumulant is rescaled to remain constant with respect to $n$, all higher-order
cumulants scale to zero. Here, by using the fact that the $n^{\text{th}}$
moment $m^{(a)}_{k}$ can be expressed in terms of cumulants $c^{(b)}_{k}$ for
$b\leq a$, we can see that the second-order cumulant dominates all of these
expressions once $k$ is sufficiently small. For example, the fourth moment
quickly scales like $(m^{(2)})^{2}$ as $k$ is decreased from $l$ because the
second cumulant begins much larger than the fourth cumulant and remains
dominant.

$\displaystyle c^{(2)}_{k}$
$\displaystyle=2^{-\lambda}c^{(2)}_{l}=\frac{1}{12\cdot 2^{\lambda}}$
$\displaystyle c^{(4)}_{k}$
$\displaystyle=2^{-3\lambda}c^{(4)}_{l}=\frac{-1}{120\cdot 2^{3\lambda}}$
$\displaystyle m^{(4)}_{k}$
$\displaystyle=c^{(4)}_{k}+3(c^{(2)}_{k})^{2}\approx 3(c^{(2)}_{k})^{2}\sim
2^{-2\lambda}$

Hence, due to (21), $m^{(4)}_{k-1}\approx m^{(4)}_{k}/4$, which agrees with
(20).

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