# Fair Allocation in Crowd-Sourced Systems

Mishal Assif P K†, William Kennedy‡, Iraj Saniee‡

(†ECE, University of Illinois, Urbana-Champaign
‡Math & Algorithms Group, Bell Labs, Nokia
Date: 00footnotetext:  Work done while author (${\dagger}$) was an intern at
Nokia Bell labs. )

###### Abstract

In this paper, we address the problem of fair sharing of the total value of a
crowd-sourced network system between major participants (founders) and minor
participants (crowd) using cooperative game theory. Shapley allocation is
regarded as a fair way for computing the shares of all participants in a
cooperative game when the values of all possible coalitions could be
quantified. We define a class of value functions for crowd-sourced systems
which capture the contributions of the founders and the crowd plausibly and
derive closed-form expressions for Shapley allocations to both. These value
functions are defined for different scenarios, such as presence of oligopolies
or geographic spread of the crowd, taking network effects, including
Metcalfe’s law, into account. A key result we obtain is that under quite
general conditions, the crowd participants are collectively owed a share
between $\frac{1}{2}$ to $\frac{2}{3}$ of the total value of the crowd-sourced
system. We close with an empirical analysis demonstrating consistency of our
results with the compensation offered to the crowd participants in some public
internet content sharing companies.

##  1 Introduction

Many existing and emerging online network systems and services are designed to
work semi-autonomously using cloud hardware and secure software created by a
small group of founders and enabled by mass participation of crowds. We refer
to (the small number of) the former as _major_ participants and (the large
number of) the latter as _minor_ participants. Without the hardware platform,
and the associated reliability and security provided by the software, there
would be no platform and service. But without participation of the crowd there
would be no data or content to enable the service. Recent examples of such
platforms include Waze[Waz] and Helium[Hel] and one can count Google and
Facebook as older and well-established instances of such services. Recently,
cryptographically strong decentralized mechanisms are also added for higher
levels of security in these systems [Hel]. There is a large body of literature
in the computer science community on the design and operation of such
decentralized services (see [Dea09] for an overview) and a smaller but growing
number of publications on reliable distributed mechanisms that automatically
account for and accumulate rewards for participants [Vos19].

What has not been discussed much in the said literature concerning crowd-
sourced network systems and services is the concept of fair allocation of the
service’s total value to the crowd participants who by constantly feeding data
and information to the system are key to its success. One observes a trend
towards rewarding crowd participants via automatically-counted tokens based on
such measures as the number of queries, messages or packets each participant
processes, independently of the total value that the network system and
service as a whole generates. This particularly obscures the fact that in many
such network systems crowd participants additionally provide critical local or
even private information free of charge purely as part of participation which
is monetized by the service provider, adding to the total value generated by
the service.

Lack of valued-based allocation to participants in large-scale crowd-sourced
systems is felt keenly by the general public after over two decades since the
emergence of these once-novel enterprises (see, for example, [Lan13]) but
interestingly, this topic has not been addressed adequately by the research
and networking communities; for some notable exceptions with respect to the
narrower Internet Service Provider settlements, see [MCLM10, MCL+10, CCH08,
MICM10].

Motivated by the interest in revisiting some of the structural aspects of
crowd-sourced systems by the Web 3.0 Foundation, see [Web], and more directly,
the new possibilities offered by the enterprise-sourced systems as part of the
Industry 4.0, this paper aims to help bridge this gap via a formal methodology
for fair allocation of the total value and a framework of what fair allocation
could potentially mean for crowd-sourced systems’ participants.

The paper is organized as follows. In section 2 we describe the role and
requirements of value functions that are necessary for computation of the
Shapley value in a cooperative games. Our aim is to make these functions as
simple as possible while capturing the key contributions of major and minor
participants in crowd-sourced systems. In section 3 we consider various models
for a single crowd-sourced system consisting of a major participant or
founder, and a large number of minor participants, or crowd. We derive closed-
formed expressions for Shapley allocations to both for a general class of
value functions under different regimes. A key result here is that under quite
general and plausible conditions, crowd participants collectively get a payoff
at least $\frac{1}{2}$ of the total value of the crowd-sourced system.

We next consider a broad extension of our methodology to oligopolies of
crowed-sourced network systems in section 4 whereby distinct collections of
single-founder crowd-sourced systems agree to cooperate via pair-wise
agreements. We show similar results hold in terms of fair share of
participants. Interestingly and in contrast to the single crowd-sourced
system, the ratio of the fair share of major to minor participants now
increases with the number of inter-crowd connections each major participant
contributes to. In section 5, we present a model of geographic crowd-source
systems, and obtain closed-form expressions for the Shapley allocation to each
community which continues to exhibit the $\frac{1}{2}$ to $\frac{2}{3}$
allocation to the crowd participants. We close with an empirical study of
crowd-sourced systems whose public financial statements help estimate the
actual revenue share of the crowd which we observe to be consistent with our
models’ predictions.

##  2 Fair Allocation in Cooperative Games

A formal derivation of fair allocation in cooperative games was first
introduced by Shapley [Sha51] and numerous extensions of it have been
considered since. A good introduction is found in [Rot88]. To summarize the
main concept, consider the setting where a group of participants $N$ are
cooperating towards a common goal and generate a value $\nu(N)$ as a result,
and this value needs to be distributed among all the participants in a fair
way. Suppose that there is a value function
$\nu:2^{N}\rightarrow\mathbb{R^{+}}$ that takes any subset $S$ of $N$ as input
and outputs the value that would have been generated by the coalition formed
by only the participants in $S$, as opposed to all of $N$. Now let us denote
by $\phi_{i}(\nu)$ the payoff that participant $i$ receives as part of the
grand coalition, i.e. when $S=N$.

_Fairness._ There are a few natural properties that a fair allocation scheme
must satisfy:

$\displaystyle 1.\ \sum_{i\in N}\phi_{i}(\nu)=\nu(N)$ (1) $\displaystyle 2.\
\nu(S\cup\\{i\\})=\nu(S\cup\\{j\\}))\ \text{for all
}S\implies\phi_{i}(\nu)=\phi_{j}(\nu)$ $\displaystyle 3.\
\nu(S\cup\\{i\\})=\nu(S)\ \text{for all }S\implies\phi_{i}(\nu)=0$
$\displaystyle 4.\
\phi_{i}(\nu_{1}+\nu_{2})=\phi_{i}(\nu_{1})+\phi_{i}(\nu_{2})$

Property (3) above _null player_ states that any player $i$ that adds no value
to any coalition $S$ does not receive any payoff in the grand coalition, while
property (2) _symmetry_ states that any two players $i$ and $j$ that add the
same value to every coalition $S$ should receive the same payoff. Participants
are rewarded only based on the value they add and no other factors. Property
(1) _Efficiency_ states that the total value generated $\nu(N)$ is completely
distributed among the participants and property (4) _linearity_ says that for
two independent games, the fair payoff to each participant must be the sum of
the fair payoffs in each independent game.

From these conditions Shapley derives the following expression for the unique
payoff $\phi_{i}(\nu)$ to each participant $i$:

$\displaystyle\phi_{i}(\nu)$ $\displaystyle=\sum_{S\subseteq
N-\\{i\\}}\frac{|S|!~{}(|N|-|S|-1)!}{|N|!}~{}(\nu(S\cup\\{i\\})-\nu(S)).$ (2)

Here the expression in brackets on the extreme right hand side is the marginal
value that agent $i$ adds to coalition $S$ and this marginal value is averaged
out for all possible $2^{|N|-1}$ coalitions. The payoff $\phi_{i}(\nu)$ in 2
is known as the _Shapley allocation_ to participant $i$.

##  3 A Single Crowd-Sourced System

We think of a crowd-sourced system ($CSS$ from here on) as a cooperative game
in which one entity (the founder) provides the main infrastructure, which
could be physical (hardware) or virtual (software), of a value-generating
service that can only succeed if a large number of spatio-temporal agents
participate to enable local match of supply and demand for the said
infrastructure service. In this setting, consider a collection of participants
denoted by $N=\\{g,u_{1},u_{2},...,u_{n}\\}$ where $g$ is the founder and
$u_{i},1\leq i\leq n$ are the crowd member identities. There is much symmetry
in this setting which we exploit fully in the sequel.

It will be assumed that the value $\nu(S)$ of any subset $S$ of $N$ is zero
unless it includes the founder, and in the case where the founder is in $S$
its value is equal to a function of the size of the crowd. This is a
restatement of the notion of a $CSS$ (crowd-sourced system) in that it gives
much credit to a founder for imagining and initiating a distributed enterprise
enabled by a large number of uncoordinated participants who constantly feed
the system. This form of the value function also makes it possible to derive
closed-form expressions for the Shapley payoff of all participants in various
settings, as we observe below. We use a power of the size of the crowd as its
value: a linear function corresponds to standard scaling, the quadratic
function has been widely discussed in networking literature as Metcalfe’s law
3.3, and higher, super quadratic, powers are included for benchmarking
purposes.

### Notation

$g$ | The unique major participant (founder)
---|---
$u_{i}$ |  $i$-th minor participant (crowd)
$n$ | Number of minor participants
$N$ | Set containing founder and
| crowd (grand coalition)
$S$ | A coalition of participants, a subset of $N$
$|S|$ | Cardinality of $S$
$\nu(S)$ | Value of the coalition $S$
$\phi_{u_{i}}(\nu)$ | Shapley payoff to participant $u_{i}$
$K_{g}$ | Cost incurred by founder per participant
$k_{u}$ | Cost incurred by each participant
$\rho$ | Constant unit in value functions

###  3.1 Coalition value based on revenue

#### Case 1 (identical participants)

The first situation we consider is one where all the potential participants
are identical. Any coalition that does not contain the major participant $g$
is treated as one with no value, while the value of a coalition including $g$
grows as a power of its size. The motivation for this model comes from
Metcalfe’s law which states that the value of any network grows proportional
to the square of its number of participants. This empirical law, originally
devised in the context of ethernet networks [Met13], seems to also hold
remarkably well in many modern situations such as cryptocurrency networks
[Pet18] and social networks [ZLX15]. The model can be formally written as
$N=\\{g,u_{1},...,u_{n}\\}$, $\nu:2^{N}\rightarrow\mathbb{R}$ and

$\nu(S):=\begin{cases}0\quad&\text{if}\,g\notin S\\\
\rho|S-\\{g\\}|^{k}\quad&\text{if}\,g\in S\\\ \end{cases}$ (3)

where $\rho$ is some constant. Metcalfe’s law is simply a special case of our
model with $k=2$.

Here we list a few key properties of $\nu(S)$ as defined above and explain why
we believe this simple form captures the key aspects of a crowd-sourced system
valuation. We observe that:

1) the founder $g$ plays a critical role in the definition of the value
function in that any subset which does not include it has value 0. It is not
hard to show that this value function $\nu$ defined in 3 is supermodular, and
the Shapley value’s derived in the sequel are stable, see the end of section
2.

2) for a any specific crowd participant, there is no special incentive to join
any of the numerous coalitions of size $S$, as the value function only depends
on the size of the coalition and not its constitution. This means the basic
assumption in (2) that all coalitions of size $S$ are equivalent is met.

3) the simple form of $\nu$ we assume, linear, quadratic or higher powers,
does not need a saturation effect as the more crowd participants, the larger
the total value of a coalition. This is in contrast with saturating systems
where (say) the first 100 crowd participants provide higher value than the
last 100.

4) we note that in the analysis in the following pages relatively small values
of $S$ give us the asymptotic results with respect to the exponent $k$ equals
1 and 2 corresponding to linear and quadratic value functions.

Our first result characterizes the Shapley value associated with each
participant as a fraction of the value of the total coalition. We now look at
a general computation that will be useful later as well.

######  Lemma 3.1.

Consider the coalition game with the set of agents $N=\\{g,u_{1},...,u_{n}\\}$
and value $\nu$ given by

$\nu(S):=\begin{cases}0\quad&\text{if}\,g\notin S\\\
f\left(|S|\right)\quad&\text{if}\,g\in S\\\ \end{cases}$

The Shapley values of the game satisfy

$\phi_{g}(\nu)=\frac{1}{n+1}\sum_{s=0}^{n}f(s),\quad\phi_{u_{i}}(\nu)=\frac{\nu(N)-\phi_{g}(\nu)}{n}.$

###### Proof.

See Appendix 8.1. ∎

######  Theorem 3.2.

Consider the coalition game with the set of agents $N=\\{g,u_{1},...,u_{n}\\}$
and value $\nu$ given by (3). The associated Shapley values satisfy

$\phi_{g}(\nu)=\frac{\rho
n^{k}}{k+1}+O(n^{k-1}),\quad\phi_{u_{i}}(\nu)=\frac{k\rho
n^{k-1}}{k+1}-O(n^{k-2})$ (4)

and

$\lim_{n\to\infty}\frac{\phi_{g}(\nu)}{\nu(N)}=\frac{1}{k+1},\quad\lim_{n\to\infty}\frac{\sum_{i=1}^{n}\phi_{u_{i}}(\nu)}{\nu(N)}=\frac{k}{k+1}.$
(5)

###### Proof.

Applying the results of Lemma 3.1 to the value (3), we get

$\displaystyle\phi_{g}(\nu)$
$\displaystyle=\frac{\rho}{n+1}\sum_{s=0}^{n}s^{k}=\frac{1}{n+1}\frac{\rho
n^{k+1}+O\left(n^{k}\right)}{k+1}$ $\displaystyle\approx\frac{\rho
n^{k}}{k+1}=\frac{1}{k+1}\nu(N)\quad\text{for large $n$,}$ $\displaystyle
n\phi_{u_{i}}(\nu)$ $\displaystyle=\nu(N)-\phi_{g}(\nu)\approx\frac{\rho
k}{k+1}\nu(N).$

∎

######  Remark 3.3.

_The results obtained in Theorem 3.2 tell us that when $k=1$, i.e. when the
power of a network grows proportional to the number of participants, the value
of the grand coalition is shared equally between the major participant and the
collection of minor participants. In other words,_
$\phi_{g}(\nu)\approx\frac{\nu(N)}{2}$ and
$\sum_{i=1}^{n}\phi_{u_{i}}(\nu)\approx\frac{\nu(N)}{2}$.

_In the popular case of Metcalfe’s law where $k=2$, the theorem says that the
major participant should get 1/3 of the value generated while 2/3 should be
distributed equally among the minor participants._

#### Case 2 (Non-identical participants)

The assumption that all participants are (nearly) identical might not be hold
in many settings. For example, when some participants contribute significantly
more to the value of the system than others due to their influence in a social
network. A slightly more refined model for the value would be

$\nu(S):=\begin{cases}0\quad&\text{if}\,g\notin S\\\ \rho\left(\sum_{i\in
S}W_{i}^{\alpha}\right)^{k}\quad&\text{if}\,g\in S\\\ \end{cases}$ (6)

where $W_{i}$ refers to some notion of amount of work done by participant $i$,
e.g., the number of messages routed by $i$. We will denote by
$f_{i}:=\frac{W_{i}^{\alpha}}{\sum_{j}W_{j}^{\alpha}}$ the share of total work
done by $i$.

######  Theorem 3.4.

Consider the coalition game with the set of agents $N=\\{g,u_{1},...,u_{n}\\}$
and value $\nu$ given by (6) where $k=2$. The associated Shapley values
satisfy

$\lim_{n\to\infty}\frac{\phi_{g}(\nu)}{\nu(N)}=\frac{1}{3}+\sum_{i}\frac{f_{i}^{2}}{6},\quad\lim_{n\to\infty}\frac{\phi_{u_{i}}(\nu)}{\nu(N)}=\frac{2}{3}f_{i}-\frac{f_{i}^{2}}{6}.$
(7)

###### Proof.

See Appendix 8.1. ∎

######  Remark 3.5.

_The results obtained in Theorem 3.2 under the identical participant model can
be recovered from the above theorem by assuming $W_{i}=1$. In fact, in the
case where the workload is more or less uniformly distributed and not too
concentrated, as formalized by the condition_

$\lim_{n\to\infty}\sum_{i}f_{i}^{2}=0,$

_the above theorem tells us that
$\frac{\phi_{g}(\nu)}{\nu(N)}\approx\frac{1}{3}$ and
$\frac{\phi_{u_{i}}(\nu)}{\nu(N)}\approx\frac{2}{3}f_{i}$, which is equivalent
to dividing the value between the founder and the crowd in the same way as in
the identical participant model, and then dividing the value among the crowd
proportional to the share of work done._

######  Remark 3.6.

_In theorem 3.4, we have treated the case $k=2$. However, analogous results
hold true for higher values of $k$ as well. It can be shown that_

$\displaystyle\lim_{n\to\infty}\frac{\phi_{g}(\nu)}{\nu(N)}$
$\displaystyle=\frac{1}{k+1}+O\left(\sum_{i}f_{i}^{2}\right)$
$\displaystyle\lim_{n\to\infty}\frac{\phi_{u_{i}}(\nu)}{\nu(N)}$
$\displaystyle=\frac{k}{k+1}f_{i}-O\left(f_{i}^{2}\right).$

###  3.2 Coalition value based on profit

A legitimate concern regarding the use of revenue as a proxy for value is that
the founder could be incurring significant costs for infrastructure which the
minor participants do not. In this case the resulting profit, revenue minus
cost, is a more natural proxy for value. A question which our Shapley
allocation derivation answers in the positive is the observable impact of the
large infrastructure cost incurred by the founder compared to the small cost
incurred by the crowd members.

We modify the model to take into account such costs. We assume that each minor
participant incurs a fixed constant cost $k_{u}$ and the major participant
pays a cost of $K_{g}$ per minor participant with $K_{g}>>K_{u}$. Formalizing
all this, we get a new value function $\nu:2^{N}\rightarrow\mathbb{R}$

$\nu(S):=\begin{cases}0\quad&\text{if}\,g\notin S\\\
\rho|S-\\{g\\}|^{k}-K_{g}|S-\\{g\\}|-\\\ \quad
k_{u}|S-\\{g\\}|\quad&\text{if}\,g\in S\\\ \end{cases}$ (8)

We compute the Shapley values in this case by appealing to Lemma 3.1 again.

######  Theorem 3.7.

Consider the coalition game with the set of agents $N=\\{g,u_{1},...,u_{n}\\}$
and value $\nu$ given by (8). The associated Shapley values satisfy

$\displaystyle\frac{\phi_{g}(\nu)}{\nu(N)}\approx\frac{\frac{\rho
n^{k}}{k+1}-K_{g}\frac{n}{2}-k_{u}\frac{n}{2}}{\rho n^{k}-K_{g}n-k_{u}n}$ (9)
$\displaystyle\frac{\sum_{i=1}^{n}\phi_{u_{i}}(\nu)}{\nu(N)}\approx\frac{\frac{\rho
kn^{k}}{k+1}-K_{g}\frac{n}{2}-k_{u}\frac{n}{2}}{\rho n^{k}-K_{g}n-k_{u}n}$

for large values of $n$.

###### Proof.

Using the results of Lemma 3.1, the exact same computation in the proof of
Theorem 3.2 delivers this result. ∎

######  Remark 3.8.

_Even though in the limit $n\to\infty$, the results of Theorem 3.7 collapse to
that of 3.2, the value of $K_{g}$ might be large enough that the contribution
of $K_{g}n$ cannot be ignored for the moderately large values of $n$ that
typically arise._

######  Remark 3.9.

_In this case, the ratio of the share of the founder to the share of the crowd
is_

$\frac{\phi_{g}(\nu)}{\sum_{i=1}^{n}\phi_{u_{i}}(\nu)}=\frac{\frac{\rho
n^{k}}{k+1}-K_{g}\frac{n}{2}-k_{u}\frac{n}{2}}{\frac{\rho
kn^{k}}{k+1}-K_{g}\frac{n}{2}-k_{u}\frac{n}{2}}.$

This shows that when $k>1$, as the cost of the system infrastructure $K_{g}$
increases, the share of the founder compared to the crowd of participants
decrease. This may seem at first counterintuitive or unfair, especially
compared to the case $k=1$ where the share is constant regardless of the cost
of the infrastructure. This phenomenon occurs because the revenue should be
divided according to $\frac{1}{k+1}$ and $\frac{k}{k+1}$ between the founder
and the crowd, whereas the total cost is split equally. Thus superlinear
scaling of revenue and linear scaling of cost give a decreasing proportional
share to the founder even though the absolute value of the founder share is
still increasing.

Another observation regarding (6) is that while the share of the founder
$\phi_{g}(\nu)$ is increasing in $k$, the founder’s profit decreases as a
percentage of the total profit. This again is a consequence of the revenue of
the network growing as a power of its size.

##  4 Oligopolies of Crowd-Sourced Systems

We now consider the case of multiple crowd-sourced systems where major
participants or founders are willing to cooperate, thus inter-working $CSS$s
for a higher total payoff to all. Let us assume that each major participant
$v$ has $n_{v}$ minor participants in its $CSS(v)$, crowd-sourced system
affiliated with $v$. The major participants now agree to cooperate for higher
payoff by inter-networking their disparate crowds. This is done through a
series of bilateral agreements between pairs of major players. The key
consequence of such cooperation between major participants $i$ and $j$ is
joint creation of a payoff which is proportional to $(n_{i}+n_{j})^{2}$ for
the Metcalfe value model we considered previously, in contrast to
$n_{i}^{2}+n_{j}^{2}$ for the case $i$ and $j$ do not cooperate. The question
we wish to answer is fair allocation to each $CSS$, and each major and minor
participants in each $CSS$ assuming global cooperation represented by a graph
$G$ as defined below.

### Additional Notation

$V$ | Set of vertices $v$, each representing a
---|---
| crowd-sourced system, $v\equiv CSS(v)$
$E$ | Set of edges, each edge denoting an agreement
| between its ends
$G$ | Oligopoly graph of $CSS$s and their bilateral
| agreements
$S$ | A subset of vertices $V$
$G(S)$ | Subgraph of $G$ induced by $S$

$v$ | A vertex/participant in $V$
---|---
$e$ | An edge/agreement in $E$
$N(v)$ | Set of neighbors of the vertex $v$ in graph $G$
$n_{v}$ | Number of participants(crowd) in $v$’s
| crowd-sourced system

###  4.1 Shapley value of an Oligopoly: Coarse-grain model

We now assume that major participants have formed a set of pairwise,
bilateral, agreements amongst themselves. We represent this by a connected
graph $G=(V,E)$, whose vertices $v\in V$ denote $CSS$s associated with major
participants and each edge in $E$ denotes a bilateral agreement. A natural
extension of the notion of Metcalfe value to a graph $G$ is given by

$\nu(G)=\sum_{v\in V}n_{v}^{2}+\sum_{(u,w)\in E}n_{u}n_{w}.$ (10)

Here $n_{v}^{2}$ is the Metcalfe law restricted to the system $v$ and each
edge $(u,w)$ adds an additional $n_{u}n_{w}$ to the value of $G$ due to inter-
networking between vertices $u$ and $w$.

We assume that the crowd-sourced systems associated with major participants
$v\in V$ form the set of agents of a cooperative game. The value of a subset
$S\subset V$ is the Metcalfe value (10) of the subgraph $G(S)$ induced by $S$.
This is the graph consisting of vertices $S$ and all edges of $G$ both ends of
which lie in $S$. This is implicitly an aggregate model that clubs together
all the minor participants of a $CSS$ with the major participant, modelling
the scenario where each minor participant has already been bound to its major
participant, perhaps by means of an agreement or due to some external
constraints such as geographic that limits the choice of the minor
participant. As a result, in the following $\phi_{u}(\nu)$ denotes the share
of the crowd-sourced system associated with major participant $u$, and not
just the share of the major participant $u$ in $CSS(u)$ by itself (which will
be discussed in Remark 4.4).

The following result characterizes the Shapley value of each $CSS$.

######  Theorem 4.1.

Consider a graph $G=(V,E)$ and a cooperative game with set of agents $N=V$ and
value function

$\nu(S)=\sum_{v\in S}n_{v}^{2}+\sum_{(u,w)\in S\times S\cap E}n_{u}n_{w}.$

The Shapley value of each vertex $u$ is given by

$\phi_{u}(\nu)(\equiv\phi_{CSS(u)}(\nu))=n_{u}^{2}+\sum_{w\in N(u)}n_{u}n_{w}$
(11)

###### Proof.

See Appendix 8.2. ∎

$n_{A}$$n_{B}$$n_{C}$$n_{D}$ Figure 1: Graph of four crowd-sourced systems
with four bilateral agreements.

For example, in the oligopoly shown in Figure 1, we can compute the Shapley
values as

$\displaystyle\phi_{A}(\nu)$ $\displaystyle=n_{A}(n_{A}+n_{B}+n_{C}),$
$\displaystyle\phi_{B}(\nu)$ $\displaystyle=n_{B}(n_{A}+n_{B}+n_{C}+n_{D}),$
$\displaystyle\phi_{C}(\nu)$ $\displaystyle=n_{C}(n_{A}+n_{B}+n_{C}),$
$\displaystyle\phi_{D}(\nu)$ $\displaystyle=n_{D}(n_{B}+n_{D}).$

######  Remark 4.2.

_In the special case where all major participants have equal number of minor
participants within their system, i.e. $n_{v}=n$ for all $v\in V$, we get_

$\phi_{u}(\nu)=n^{2}(1+\text{\emph{deg}}(u)).$ (12)

######  Remark 4.3.

_If we consider $V=\\{g,u_{1},...,u_{n}\\}$, that there is an edge between $g$
and $u_{i}$ for each $i=1,...,n$, and that $n_{u_{i}}=1$ and $n_{u_{g}}=1$, we
get a model that is very close to (3) with $k=2$. The only difference here is
that the coalitions that don’t contain $g$ have non-zero value that scales
linearly with its size. Unsurprisingly, we get_

$\displaystyle\phi_{g}(\nu)=1+\text{\emph{deg}}(g)=1+n,$
$\displaystyle\phi_{u_{i}}(\nu)=2\ \implies\nu(N)=3n+1,$
$\displaystyle\implies\lim_{n\to\infty}\frac{\phi_{g}(\nu)}{\nu(N)}=\frac{1}{3},\quad\lim_{n\to\infty}\frac{\sum_{i=1}^{n}\phi_{u_{i}}(\nu)}{\nu(N)}=\frac{2}{3}$

_exactly the same as in Theorem 3.2._

###  4.2 Shapley value of an Oligopoly: Fine-grain model

Theorem 4.1 and the Remarks 4.2 and 4.3 derive the Shapely value of each
crowd-sourced system $($CSS$)$ associated with each vertex in the oligopoly
game of graph $G$. What remains is to identify the Shapley fair share of each
participant in each $CSS$ associated with each major player $v\in G$. To do
this, we need to define a new value function on each $CSS$ which takes its
size and sizes of its neighboring $CSS$s explicitly into account.

To this end, we consider a more fine-grained model here. We consider the set
$U_{v}$ of minor participants in the $CSS$ associated with major participant
$v\in V$ as separate agents, making the total set of agents of the cooperative
game

$N=\cup_{v\in V}U_{v}\cup V.$

A subset $S\subset N$ can be partitioned as $S_{v}=S\cap U_{v}$ for each $v\in
V$ and $S_{V}=S\cap V$. The value of the coalition is computed based on two
simple principles: each $v\in S_{V}$ contributes $|S_{v}|^{2}$ and each
$e=(v,w)\in G(S_{V})$ contributes $|S_{v}||S_{w}|$ to the value of $S$. That
is,

$\xi(S)=\rho\left(\sum_{v\in S_{V}}|S_{v}|^{2}+\sum_{(u,w)\in S_{V}\times
S_{V}\cap E}|S_{u}||S_{w}|\right)$ (13)

This value function and agent set jointly capture the cooperative game
involving all $CSS$s and their minor and major participants. For $v\in V$, we
denote by $n_{v}:=|U_{v}|$ the size of its set of minor participants.

######  Theorem 4.4.

Consider a graph $G=(V,E)$ where each node $v\in V$ has an associated set
$U_{v}$, and a cooperative game with set of agents $N=\cup_{v\in V}U_{v}\cup
V$ and value function

$\xi(S)=\rho\left(\sum_{v\in S_{V}}|S_{v}|^{2}+\sum_{(u,w)\in S_{V}\times
S_{V}\cap E}|S_{u}||S_{w}|\right)$

where $S_{V}=S\cap V$ and $S_{v}=S\cap U_{v}$ for each $v\in V$. The Shapley
value of each major participant $v\in V$ is the sum of two terms, the intra
and inter crowd payoffs, given by

$\phi_{v}(\xi)=\rho\left(\frac{n_{v}^{2}}{3}-O(n_{v})\right)+\rho\left(\sum_{w\in
N(v)}\frac{n_{v}n_{w}}{2}\right)$ (14)

The Shapley value of each minor participant $u_{v}\in U_{v}$ is also the sum
of two terms

$\phi_{u_{v}}(\xi)=\rho\left(\frac{2n_{v}}{3}+O(1)\right)+\rho\left(\sum_{w\in
N(v)}\frac{n_{w}}{2}\right).$ (15)

###### Proof.

See Appendix 8.2. ∎

######  Remark 4.5.

_The Shapley share of the major participant $u$ in $CSS(u)$ (vertex $u\in G$
of the oligopoly graph $G$) can be written as_

$\displaystyle\frac{\phi_{v}(\xi)}{\phi_{v}(\xi)+\sum_{u_{v}\in
U_{v}}\phi_{u_{v}}(\xi)}$
$\displaystyle=\frac{\frac{n_{v}^{2}}{3}+O(n_{v})+\sum_{w\in
N(v)}\frac{n_{v}n_{w}}{2}}{n_{v}^{2}+\sum_{w\in N(v)}n_{v}n_{w}}$

_For large enough numbers of minor participants $n_{v}$, this ratio can be
simplified as_

$\displaystyle\frac{\phi_{v}(\xi)}{\phi_{v}(\xi)+\sum_{u_{v}\in
U_{v}}\phi_{u_{v}}(\xi)}=\frac{\frac{n_{v}}{3}+\sum_{w\in
N(v)}\frac{n_{w}}{2}}{n_{v}+\sum_{w\in N(v)}n_{w}}$
$\displaystyle=\frac{1}{3}\times\frac{n_{v}}{n_{v}+\sum_{w\in
N(v)}n_{w}}+\frac{1}{2}\times\frac{\sum_{w\in N(v)}n_{w}}{n_{v}+\sum_{w\in
N(v)}n_{w}}.$

_This ratio is greater than $\frac{1}{3}$ and less than $\frac{1}{2}$, moving
closer to $\frac{1}{2}$ as the number of minor participants in the
neighbouring $CSS$ grows larger than that in the $CSS$ corresponding to $v$._

##  5 Geographic Crowd-Sourced Systems

In this last section, we discuss the extension of the models we have
considered so far by considering $CSS$s which are geographically distributed.
More concretely, consider a collection of agents and for each agent the region
covered by it. For simplicity we refer to these regions as “disks", even
though these can be of arbitrary shape. Figure 2 shows five disks associated
with five agents.

$\displaystyle n_{1}$$\displaystyle n_{2}$$\displaystyle n_{12}$$\displaystyle
n_{123}$$\displaystyle n_{13}$$\displaystyle n_{3}$$\displaystyle
n_{5}$$\displaystyle n_{4}$$\displaystyle n_{45}$$\displaystyle n_{23}$15234
Figure 2: Geographic $CSS$ with five disks.

In this model there are $m$ agents $M=\\{1,2,...,m\\}$, and each agent $i$ has
a corresponding disk $D_{i}$. There are also users, set $U$, served by the
agents in $M$. A user $u$ may be served by an agent $i$ only if $u$ is in the
region $D_{i}$. Notice that a user in $D_{i}$ may also be in $D_{j}$ for one
or more $j\neq i$. We need notation for number of users in the intersection of
disks. For each subset of agents $S\subset M$, we will denote by $d_{S}$ the
number of users in the geographic area exclusively covered by all $D_{i}$ for
$i\in S$. For example, in Figure 2 $d_{\\{1,2\\}}$ is the number of users in
$D_{1}\cap D_{2}$ that don’t belong to any of the other disks, and
$d_{\\{1,2,3\\}}$ is the number of users in $D_{1}\cap D_{2}\cap D_{3}$, etc.
More formally, $d_{S}$ is the number of users in $\cap_{i\in S}D_{i}\cap_{j\in
M-S}D_{j}^{c}$.

In this setting, we define the value functions $\nu_{lin}$ and $\nu_{Met}$ as
follows. For any $S\subset M$,

$\displaystyle\nu_{lin}(\\{i\\})$ $\displaystyle=\rho\sum_{i\in S\subset
M}\frac{d_{S}}{|S|},\quad\nu_{lin}(S)=\rho\sum_{i\in S}\nu_{lin}(\\{i\\}),$
(16) $\displaystyle\nu_{Met}(S)$
$\displaystyle=\left(\nu_{lin}(S)\right)^{2}/\rho.$

For example, in the configuration of Figure 2 we have

$\displaystyle\nu_{lin}(\\{1\\})$
$\displaystyle=\rho\left(d_{1}+\frac{d_{12}+d_{13}}{2}+\frac{d_{123}}{3}\right),$
$\displaystyle\nu_{lin}(\\{2,3\\})$
$\displaystyle=\rho\left(d_{2}+d_{3}+\frac{d_{12}+d_{13}}{2}+d_{23}+\frac{2d_{123}}{3}\right),$

etc. and

$\displaystyle\nu_{Met}(\\{1\\})$
$\displaystyle=\rho\left(d_{1}+\frac{d_{12}+d_{13}}{2}+\frac{d_{123}}{3}\right)^{2},$
$\displaystyle\nu_{Met}(\\{2,3\\})$
$\displaystyle=\rho\left(d_{2}+d_{3}+\frac{d_{12}+d_{13}}{2}+d_{23}+\frac{2d_{123}}{3}\right)^{2},$

etc.

With these definitions, the geographic $CSS$ model is equivalent to an
oligopoly of $CSS$s where the underlying graph $G$ is complete and
$n_{i}:=\nu_{lin}(\\{i\\})$. The Shapley value for each agent under the
Metcalfe model is therefore given by Theorem 4.1 while the linear model is an
additive cooperative game where Shapley value coincides with the value
function, namely

$\phi_{i}(\nu_{lin})=\nu_{lin}(i)=\rho n_{i},\quad\phi_{i}(\nu_{Met})=\rho
n_{i}\left(\sum_{i=1}^{m}n_{i}\right).$

###  5.1 Geographic Crowd-Sourced Systems with a Founder

As a conclusion to the discussion of geographic crowd-sourced systems, we
consider the contribution of a founder who establishes the infrastructure for
the collection of geographic crowd-sourced system. For example, the agent who
provides the software for connecting access points in a distributed wireless
network e.g. [Hel]. Just to set the notation, let

### Notation

$g$ | Main founder of the geographic $CSS$s
---|---
$M=\\{1,..,m\\}$ | Set of agents $i$, each representing disk $D_{i}$
$n_{i}$ | Number of users in disk $D_{i}$.

As in section 3.1, we define a macro value function $\xi$ which quantifies the
cooperative game involving the founder and any subset of agents
$M\cup\\{g\\}$.

$\xi_{lin}(S)=\begin{cases}\nu_{lin}(S-\\{g\\})&\text{if }g\in S\\\
0&\text{otherwise}\end{cases}$ (17)

and

$\xi_{Met}(S)=\begin{cases}\nu_{Met}(S-\\{g\\})&\text{if }g\in S\\\
0&\text{otherwise}.\end{cases}$ (18)

where $\nu_{lin}$ and $\nu_{Met}$ are defined in (16).

######  Theorem 5.1.

In a geographic crowd-sourced system with value function $\xi$ defined in
(17),(18), the Shapley allocation to the founder and agent $i$ for each $i\in
M$ is given by

$\phi_{g}(\xi_{lin})=\rho\frac{\sum_{i}n_{i}}{2},\quad\phi_{i}(\xi_{lin})=\rho\frac{n_{i}}{2}\\\
$ (19) $\displaystyle\phi_{g}(\xi_{Met})$
$\displaystyle=\rho\left(\frac{\left(\sum_{i}n_{i}\right)^{2}}{3}+\frac{\sum_{i}n_{i}^{2}}{6}\right)$
(20) $\displaystyle\phi_{i}(\xi_{Met})$
$\displaystyle=\rho\left(\frac{2(\sum_{i}n_{i})n_{i}}{3}-\frac{n_{i}^{2}}{6}\right).$

###### Proof.

Follows from theorem 3.4. ∎

As in Remark 3.5, notice that $\frac{\phi_{g}(\xi_{Met})}{\xi_{Met}(M)}$ and
$\frac{\sum_{i}\phi_{i}(\xi_{Met})}{\xi_{Met}(M)}$ asymptotically approach
$\frac{1}{3}$ and $\frac{2}{3}$ as the number of agents increases.

##  6 Empirical Analysis

Our main thesis in this paper is that many crowd-sourced systems, old or new,
do not compensate their crowd participants and when they do so, the share is a
relatively small fraction of their total revenue. Our goal is to determine
what a reasonable share would be if fair allocation existed. Unfortunately,
the information about sharing of the total value/revenue with crowd
participants, if and when this information exists, is hard to come by. Often
data or content from the crowd is collected as part of the service level
agreement signed by the crowd participants in exchange for a service, which
may be search, access to social media platform, use of navigational maps,
access to wireless spectrum and others services. In public records, it is
often hard to carve out compensation paid to crowd participants directly. In a
few instances, however, it is possible to estimate these payouts. Here, we
give 3 examples: Spotify, Pandora and YouTube:

  * •

Spotify compensates audio contributors an average of 0.003-0.005 cents per
stream which comes to about 2/3 of its total revenues in 2021 ($11.4B), see
[Ale22] and [Mul22].

  * •

Pandora pays an average of 0.00133 cents per stream which translates to 70% of
its total revenues to artists and publishers in 2021 (total revenue $3.64 B in
2021)) of which 57% goes to artists or record labels, ie 2/5 of the total
revenue, see [Kor20].

  * •

YouTube is our 3rd example and here it is possible to get more detailed
information from public records which we describe below.

YouTube, an Alphabet subsidiary, has for over a decade awarded its content
creators a large share of its advertising revenue. Since 2019, YouTube has
disclosed its payments to its crowd content creators. Table 1 below shows the
breakout of the total revenues of Alphabet and YouTube in 2017 through 2021.
Table 1 gives us the YouTube annual revenues for 5 recent years. But how much
of these were paid to content creators in each fiscal year? During August
2021, articles in Fortune [BS21], Barron’s and numerous other outlets reported
that over the previous three years Alphabet/YouTube had paid $30B to its
content creators. Adding the relevant figures from Table 1, this corresponds
to a fraction 30/(28.8/2+19.7+15.1+11.1/2) = 30/54.75 = 55% share of the total
revenues during 2nd half of 2018, 2019, 2020, and 1st half of 2021 to content
creators. A more recent article [Scr21] puts this ratio at 68%. Other figures
in the same range are also mentioned in various articles in the past 2 years.
Putting all of this together, we observe a range of 55%-68% of the total
YouTube revenues were paid out to content creators in each fiscal year, which
is in close agreement with the $\left[\frac{1}{2},\frac{2}{3}\right]$ range we
derived under the Shapley allocation using the simple value functions
discussed above.

Year | Revenue ($B) | Revenue ($B)
---|---|---
| Alphabet | YouTube
2017 | 110.8 | 8.1
2018 | 136.9 | 11.1
2019 | 161.8 | 15.1
2020 | 182.5 | 19.7
2021 | 257.6 | 28.8
Table 1: Annual revenues of Alphabet and YouTube, 2017-2021

Other video content crowd-sourced systems, such as TikTok, do not publish
corresponding data although it is believed they also compensate their content
creators a substantial share of their revenues. Meta/Facebook has also started
compensating its content
creators111https://www.facebook.com/business/learn/lessons/how-make-money-
facebook..

##  7 Conclusions and Future Work

In this paper, we defined value functions which succinctly capture the
interplay between a small number of infrastructure providers, founders, and a
large number of contributors, crowd participants, in cooperative value-
generating settings. We derived closed form expressions for fair allocation of
the total value of such crowed-sourced systems between the founders and the
crowd participants using Shapley allocation. We showed that Shapley allocation
gives the crowd participants one-half to two-thirds share of the total payoff
for the cases of linear and quadratic growth of revenue as a function of crowd
size. We further showed similar results hold in cases of oligopolies of such
network systems characterized by bilateral agreements between pairs of
independent crowd-sourced systems. The same holds true for geographic crowd-
sourced systems. These results are in contrast to the practice of many
existing crowd-sourced systems whereby participants are awarded tokens of
limited value compared to the global payoff of the full system but in close
agreement with some others which we documented. A natural next step would be
to consider integration of our proposed value-based allocation in the emerging
token-based reward systems which are being actively pursued by standards
bodies, including the Web3.0 foundation.

## References

  * [Ale22] Marco Alexis.  How much does spotify pay per stream?  https://twostorymelody.com/spotify-pay-per-stream/, 2022.
  * [BS21] Mark Bergen and Lucas Shaw.  Youtube has paid out $30 billion to creators as the competition for online content intensifies?  https://fortune.com/2021/08/23/youtube-30-billion-ad-sales-online-creators/, 2021\.
  * [CCH08] Yang Cheung, Dah Ming Chiu, and Jianwei Huang.  Can bilateral ISP peering lead to network-wide cooperative settlement.  In 2008 Proceedings of 17th International Conference on Computer Communications and Networks, pages 1–6. IEEE, 2008.
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  * [Hel] Helium, people-powered network.  http://helium.com.
  * [Kor20] Robert Kormoczi.  How much can an artist earn with pandora radio?  https://timesinternational.net/how-much-can-an-artist-earn-with-pandora-radio/, 2020\.
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##  8 Appendices

###  8.1 Proofs from Section 3

###### Proof of Lemma 3.1.

We can compute the Shapley value of $g$ as

$\phi_{g}(\nu)=\frac{1}{|N|!}\sum\limits_{S\subset
N-\\{g\\}}|S|!(|N|-|S|-1)!\nu(S\cup\\{g\\})$

since $\nu(S)=0$ as $g\notin S$. Since $\nu(S\cup\\{g\\})$ is just a function
of $|S|$, we can reduce the computation as

$\displaystyle\phi_{g}(\nu)$
$\displaystyle=\frac{1}{|N|!}\sum\limits_{S\subset
N-\\{g\\}}|S|!(|N|-|S|-1)!f(|S|)$
$\displaystyle=\frac{1}{(n+1)!}\sum_{s=0}^{n}{n\choose s}s!(n-s)!f(s)$
$\displaystyle=\frac{1}{n+1}\sum_{s=0}^{n}f(s).$

Since the participants are all identical, we derive the Shapley value as

$\displaystyle\phi_{u_{i}}(\nu)=\frac{\nu(N)-\phi_{g}(\nu)}{n}.$

∎

###### Proof of Theorem 3.4.

The Shapley value of participant $i$ can be computed as

$\displaystyle\phi_{u_{i}}(\nu)$ $\displaystyle=\sum_{S\subset
N-\\{u_{i}\\}}\frac{|S|!(|N|-|S|-1)!}{|N|!}\left(\nu(S\cup\\{u_{i}\\})-\nu(S)\right)$
$\displaystyle=\sum_{S\subset N-\\{u_{i}\\},g\in
S}\frac{|S|!(|N|-|S|-1)!}{|N|!}\left(\nu(S\cup\\{u_{i}\\})-\nu(S)\right).$

The marginal utility of participant i can be written as

$\displaystyle\nu(S\cup\\{u_{i}\\})-\nu(S)=W_{i}^{2\alpha}+\sum_{j\in
S}2W_{i}^{\alpha}W_{j}^{\alpha},$

which gives

$\displaystyle\phi_{u_{i}}(\nu)=$ $\displaystyle\sum_{S\subset
N-\\{u_{i}\\},g\in
S}\frac{|S|!(|N|-|S|-1)!}{|N|!}\left(W_{i}^{2\alpha}+\sum_{j\in
S}2W_{i}^{\alpha}W_{j}^{\alpha}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{(n+1)!}\sum_{s=1}^{n}s!(n-s)!{n-1\choose
s-1}W_{i}^{2\alpha}\ +$ $\displaystyle\sum_{S\subset N-\\{u_{i}\\},g\in
S}\frac{|S|!(|N|-|S|-1)!}{|N|!}\left(\sum_{j\in
S}2W_{i}^{\alpha}W_{j}^{\alpha}\right).$

The first sum can be simplified directly and the second sum maybe rewritten
using Fubini’s theorem as follows

$\displaystyle\phi_{u_{i}}(\nu)$
$\displaystyle=W_{i}^{2\alpha}\sum_{s=1}^{n}\frac{s}{n(n+1)}+\sum_{j\neq
i}\frac{2W_{i}^{\alpha}W_{j}^{\alpha}}{(n+1)!}\sum\limits_{\\{g,u_{j}\\}\subset
S\subset N-\\{u_{i}\\}}|S|!(|N|-|S|-1)!$
$\displaystyle=\frac{W_{i}^{2\alpha}}{2}+\sum_{j\neq
i}2W_{i}^{\alpha}W_{j}^{\alpha}\sum_{s=2}^{n}\frac{s!(n-s)!}{(n+1)!}{n-2\choose
s-2}$ $\displaystyle=\frac{W_{i}^{2\alpha}}{2}+\sum_{j\neq
i}2W_{i}^{\alpha}W_{j}^{\alpha}\sum_{s=2}^{n}\frac{s(s-1)}{(n+1)(n-1)n}$
$\displaystyle=\frac{W_{i}^{2\alpha}}{2}+\sum_{j\neq
i}\left(\frac{2}{3}+O(n^{-1})\right)W_{i}^{\alpha}W_{j}^{\alpha}.$

Dividing by the value of the grand coalition $\nu(N)$, we get

$\displaystyle\frac{\phi_{u_{i}}(\nu)}{\nu(N)}$
$\displaystyle=\frac{1}{2}\frac{W_{i}^{2\alpha}}{\left(\sum_{j}W_{j}^{\alpha}\right)^{2}}+\frac{2}{3}\sum_{j\neq
i}\frac{W_{i}^{\alpha}W_{j}^{\alpha}}{\left(\sum_{k}W_{k}^{\alpha}\right)^{2}}+O(n^{-1})\sum_{j\neq
i}\frac{W_{i}^{\alpha}W_{j}^{\alpha}}{\left(\sum_{k}W_{k}^{\alpha}\right)^{2}}$
$\displaystyle=\frac{f_{i}^{2}}{2}+\frac{2}{3}f_{i}\sum_{j\neq
i}f_{j}+O(n^{-1})=\frac{f_{i}^{2}}{2}+\frac{2}{3}f_{i}(1-f_{i})+O(n^{-1})$
$\displaystyle=\frac{2}{3}f_{i}-\frac{f_{i}^{2}}{6}+O(n^{-1}).$

It follows that

$\displaystyle\lim_{n\to\infty}\frac{\phi_{u_{i}}(\nu)}{\nu(N)}$
$\displaystyle=\frac{2}{3}f_{i}-\frac{f_{i}^{2}}{6},$
$\displaystyle\lim_{n\to\infty}\frac{\phi_{g}(\nu)}{\nu(N)}$
$\displaystyle=1-\frac{2}{3}\sum_{i}f_{i}+\sum_{i}\frac{f_{i}^{2}}{6}=\frac{1}{3}+\sum_{i}\frac{f_{i}^{2}}{6}.$

∎

###  8.2 Proofs from Section 4

###### Proof of Theorem 4.1.

The marginal utility of $u$ to coalition $S$ is

$\displaystyle\nu(S\cup\\{u\\})-\nu(S)=$ $\displaystyle
n_{u}^{2}+\sum_{(u,w)\in E,w\in S}n_{u}n_{w}+\sum_{(w,u)\in E,w\in
S}n_{u}n_{w}$ $\displaystyle=$ $\displaystyle n_{u}^{2}+\sum_{(u,w)\in E,w\in
S}2n_{u}n_{w}.$

We can then compute

$\displaystyle\phi_{u}(\nu)=$ $\displaystyle\sum_{S\subset
V-\\{u\\}}\frac{|S|!(|V|-|S|-1)!}{|V|!}\left(\nu(S\cup\\{u\\})-\nu(S)\right)$
$\displaystyle=$ $\displaystyle\sum_{S\subset
V-\\{u\\}}\frac{|S|!(|V|-|S|-1)!}{|V|!}n_{u}^{2}+$
$\displaystyle\sum_{S\subset
V-\\{u\\}}\frac{|S|!(|V|-|S|-1)!}{|V|!}\sum_{(u,w)\in E,w\in S}2n_{u}n_{w}$
$\displaystyle=$ $\displaystyle n_{u}^{2}+\sum_{S\subset
V-\\{u\\}}\frac{|S|!(|V|-|S|-1)!}{|V|!}\sum_{(u,w)\in E,w\in S}2n_{u}n_{w}.$

The second term above can be reduced using Fubini’s theorem as we did before
to get

$\displaystyle\phi_{u}(\nu)$ $\displaystyle=n_{u}^{2}+\sum_{(u,w)\in
E}\sum_{w\in S\subset V-\\{u\\}}\frac{|S|!(|V|-|S|-1)!}{|V|!}2n_{u}n_{w}$
$\displaystyle=n_{u}^{2}+\sum_{(u,w)\in E}2n_{u}n_{w}\sum_{w\in S\subset
V-\\{u\\}}\frac{|S|!(|V|-|S|-1)!}{|V|!}$
$\displaystyle=n_{u}^{2}+\sum_{(u,w)\in
E}2n_{u}n_{w}\sum_{s=1}^{|V|-1}\frac{s!(|V|-s-1)!}{|V|!}{|V|-2\choose s-1}$
$\displaystyle=n_{u}^{2}+\sum_{(u,w)\in
E}2n_{u}n_{w}\sum_{s=1}^{|V|-1}\frac{s}{|V|(|V|-1)}$
$\displaystyle=n_{u}^{2}+\sum_{w\in N(u)}n_{u}n_{w}.$

∎

###### Proof of Theorem 4.4.

The value function $\xi$ can be split as

$\xi(S)=\sum_{v\in V}\xi_{u}(S)+\sum_{(v,w)\in E}\xi_{u,w}(S)$ (21)

where

$\xi_{v}(S)=\begin{cases}|S_{v}|^{2}&\text{if }v\in S\\\
0&\text{otherwise},\end{cases}$

and

$\xi_{v,w}(S)=\begin{cases}2|S_{v}||S_{w}|&\text{if }v,w\in S\\\
0&\text{otherwise}.\end{cases}$

Each of these value functions can be considered as the value function of a
cooperative game in its own right. By Linearity of Shapley value,

$\phi_{a}(\xi)=\sum_{v\in V}\phi_{a}(\xi_{v})+\sum_{(v,w)\in
E}\phi_{a}(\xi_{v,w})$ (22)

for any player $a\in N$.

Notice that for any $v\in V$, if $a\notin\\{v\\}\cup U_{v}$ then

$\xi_{v}(S\cup\\{a\\})-\xi_{v}(S)=0,\quad\text{for all $S\subset N$},$

which means $a$ is a null player in the game $\xi_{v}$. This implies
$\phi_{a}(\xi_{v})=0$ by the null player axiom of Shapley value. The game
$\xi_{v}$ on the remaining set $\\{v\\}\cup U_{v}$ is just the single major
player game 3, and the result of Theorem 3.2 gives us

$\phi_{v}(\xi_{v})=\frac{n_{v}^{2}}{3}+O(n_{v}),\quad\phi_{u_{v}}(\xi_{v})=\frac{2n_{v}}{3}-O(1).$

Similarly, for any $a\notin\\{v,w\\}\cup U_{v}\cup U_{w}$

$\xi_{v,w}(S\cup\\{a\\})-\xi_{v,w}(S)=0,\quad\text{for all $S\subset N$},$

which implies $\phi_{a}(\xi_{v,w})=0$ by the null axiom of Shapley value.

All that remains to be computed is the Shapley value of players $\\{v,w\\}\cup
U_{v}\cup U_{w}$ with the value function $\xi_{v,w}$. Since we have shown that
all other players in this game are dummy, we can find the Shapley value of
players $\\{v,w\\}\cup U_{v}\cup U_{w}$ as their Shapley value in the
simplified game whose set of players is $\tilde{N}=\\{v,w\\}\cup U_{v}\cup
U_{w}$ and value function $\xi_{v,w}$. To do this, we again split the value
function $\xi_{v,w}$ as

$\xi_{v,w}(S)=\sum_{u_{v}\in U_{v}}\sum_{u_{w}\in
U_{w}}\xi_{v,w}^{u_{v},u_{w}}(S)$

where

$\xi_{v,w}^{u_{v},u_{w}}(S)=\begin{cases}2&\text{if
}S=\\{v,w,u_{v},u_{w}\\}\\\ 0&\text{otherwise},\end{cases}$

and compute the Shapley value of the players for each of these value functions
separately. Observe that any player $\bar{u_{v}}\in U_{v}-\\{u_{v}\\}$ or
$\bar{u_{w}}\in U_{w}-\\{u_{w}\\}$ is a null player of the value function
$\xi_{v,w}^{u_{v},u_{w}}$, which means

$\phi_{a}(\xi_{v,w}^{u_{v},u_{w}})=0,\quad\text{if }U_{v}\ni a\neq u_{v}\text{
or }U_{w}\ni a\neq u_{w}.$

This means we can again reduce the set of players to $\\{v,w,u_{v},u_{w}\\}$
and value function to $\xi_{v,w}^{u_{v},u_{w}}$, and notice that this value
function is symmetric with respect to all four players $\\{v,w,u_{v},u_{w}\\}$
which implies

$\phi_{a}(\xi_{v,w}^{u_{v},u_{w}})=\begin{cases}\frac{1}{2}&\text{if
}a\in\\{v,w,u_{v},u_{w}\\}\\\ 0&\text{otherwise}.\end{cases}$

We can now compute the original Shapley values as

$\displaystyle\phi_{a}(\xi_{v,w})=\sum_{u_{v}\in U_{v}}\sum_{u_{w}\in
U_{w}}\phi_{a}(\xi_{v,w}^{u_{v},u_{w}}).$

When $a=v$ or $a=w$,

$\displaystyle\phi_{a}(\xi_{v,w})=\sum_{u_{v}\in U_{v}}\sum_{u_{w}\in
U_{w}}\frac{1}{2}=\frac{|U_{v}||U_{w}|}{2}=\frac{n_{v}n_{w}}{2}.$

When $a\in U_{v}$,

$\displaystyle\phi_{a}(\xi_{v,w})=\sum_{u_{v}\in U_{v}}\sum_{u_{w}\in
U_{w}}\frac{1}{2}\delta_{u_{v},a}=\frac{|U_{w}|}{2}=\frac{n_{w}}{2}$

and when $a\in U_{w}$,

$\displaystyle\phi_{a}(\xi_{v,w})=\sum_{u_{v}\in U_{v}}\sum_{u_{w}\in
U_{w}}\frac{1}{2}\delta_{u_{w},a}=\frac{|U_{v}|}{2}=\frac{n_{v}}{2}.$

Putting everything together in (22), we get the results of equation (11) and
(15). ∎


∎

<EMAIL_ADDRESS><EMAIL_ADDRESS>
11institutetext:  Department of Physics, Duke University, Durham, NC 27708,
USA 22institutetext:  Department of Physics and Astronomy, Stony Brook
University, Stony Brook, New York 11794-3800, USA 33institutetext:  Institute
of Theoretical Physics, Jagiellonian University, S. Łojasiewicza 11, 30-348
Kraków, Poland

# Geometrical Scaling of Direct Photons in
Relativistic Heavy Ion and d+Au Collisions

Vladimir Khachatryan e1,addr1,addr2
Michał Praszałowicz e2,addr3

(Received: date / Accepted: date)

###### Abstract

In this paper, we show that multiplicity spectra of direct photons in A+A and
d+Au collisions at different centrality classes and different energies exhibit
geometrical scaling, i.e., they depend on a specific combination of number of
participants $N_{\rm part}$, collisions energy $W$, and transverse momentum
$p_{T}$ – called saturation scale – rather than on all these three variables
separately. In particular, the dependence on the geometry of collisions
encoded in the dependence on $N_{\rm part}$ is in agreement with the
expectations based on the Color Glass Condensate theory.

††journal: Eur. Phys. J. C

##  1  Introduction

Direct photons possibly originate from the entire space-time evolution of the
matter produced in relativistic heavy ion collisions (HIC): from the pre-
equilibrium initial stage, hot fireball of the Quark-Gluon Plasma (QGP), as
well as the late hadronic phase. These photons are excellent probes for
studying the properties and dynamics of the produced matter, and by definition
do not originate from hadronic decays. Since they do not interact strongly
while passing through the QGP environment, the information that they carry is
not washed out by final state interactions. In p+p and p(d)+Au collisions,
they are basically produced only from the initial state wave function.
Therefore, one may expect qualitative differences in the photon transverse
momentum ($p_{T}$) spectra (invariant yields) with the change of the sizes of
colliding systems/species. Indeed, the thermal photon $p_{T}$-spectra in HIC
are enhanced with respect to a p+p reference at small $p_{T}$, and show large
anisotropy (elliptic flow) Adare:2015lcd  – a phenomenon dubbed thermal photon
puzzle. The photon puzzle is by now less surprising after the collaborations
at LHC have measured elliptic flow of charged particles in p+p
Sirunyan:2017uyl ; Aaboud:2018syf ; Acharya:2019vdf , suggesting a possibility
of initial state correlations not associated with hydrodynamical evolution.

In this paper, we show that the direct photon $p_{T}$-spectra exhibit
geometrical scaling (GS), which is a property of initial state dynamics
associated with over-occupation of gluons at small Bjorken $x$, and even more
so in large nuclei. GS has been first observed in the inclusive deep inelastic
electron-proton scattering (DIS) Stasto:2000er , where the reduced cross
section, essentially $F_{2}(x,Q^{2})/Q^{2}$, which is in principle a function
of two variables $x$ and $Q^{2}$, depends only on the scaling variable
$\tau^{2}=Q^{2}/Q^{2}_{\rm sat}(x)$ for small values of $x$. In this regard,
the saturation scale defined as

$Q^{2}_{\rm sat}(x)={\bar{Q}}^{2}\left(x/x_{0}\right)^{-\lambda},$ (1)

is proportional to the gluon density in the proton. In the case of DIS,
$x_{0}$ and ${\bar{Q}}=Q_{0}$ are fixed parameters of the order of $10^{-3}$
and 1 GeV/c, respectively. The exponent $\lambda\approx 0.33$ is a
nonperturbative dynamical quantity following from the properties of the (non-
linear) QCD evolution BK ; Mueller:2002zm ; Munier:2003vc , with its numerical
value fixed from the data (for review see Ref. McLerran:2010ub ;
Praszalowicz:2018vfw ). GS has also been observed in the charged particle
spectra in hadronic collisions McLerran:2010ex ; McLerran:2010wm ;
Praszalowicz:2015dta . In HIC, $Q^{2}_{\rm sat}$ is enhanced by the overlap of
interacting nucleons in the colliding nuclei (see Sec. 2).

Physically, the saturation scale corresponds to the gluon density in colliding
systems. However, the photons are produced from quarks (through $q\bar{q}$
annihilation and QCD Compton scattering), and therefore they do not probe
directly the over-occupied gluonic cloud. To this end, one can employ the
Color Glass Condensate (CGC) theory to describe the initial hadronic wave
function, which evolves into an intermediate state called Glasma, being kind
of a strongly interacting and not thermalized QGP McLerran:2010ub . The quarks
are produced in the thermalization process, and – if there are no other mass
scales around – their distribution should exhibit geometrical scaling111The
detailed description of the photon production mechanism in Glasma is beyond
the scope of this paper; we refer the reader to Refs. Chiu:2012ij ;
McLerran:2015mda ; Khachatryan:2018ori  for details. in terms of scaling
variable $\tau$:

$\tau=p_{T}/Q_{\rm sat}(x),~{}~{}{\rm where}~{}~{}x=e^{\pm
y}\,p_{T}/\sqrt{s_{NN}}.$ (2)

Consequently, the direct photon $p_{T}$-spectra should scale as well. Indeed,
the analysis of Ref. Klein-Bosing:2014uaa  showed the emergence of GS in
direct photon production, where the authors assumed the functional form of the
photon spectra to be $p_{T}^{-n}$ (see also Refs. Chiu:2012ij ;
McLerran:2015mda ; Khachatryan:2018ori ). In this paper we extend the analysis
of Refs. Klein-Bosing:2014uaa ; Khachatryan:2019uqn ; Praszalowicz:2018vfw
including later experimental data and employing a model-independent method to
search for GS, rather than assuming an approximate functional shape of the
scaling function. Preliminary results have been already presented by one of us
in Ref. Praszalowicz:2018vfw .

Although the GS scenario has been postulated in Ref. Stasto:2000er  following
the idea of the saturation exemplified by the GBW model GolecBiernat:1998js
for deep inelastic $e-p$ scattering, it in fact arises also in the kinematical
regions beyond saturation Iancu:2002tr ; Kwiecinski:2002ep ; Caola:2008xr .
Indeed, Iancu, Itacura and McLerran have shown in Ref. Iancu:2002tr  that the
BFKL evolution preserves GS beyond saturation when starting from the boundary
conditions in the saturation region. Kwieciński and Staśto have in turn
studied the DGLAP evolution in Ref. Kwiecinski:2002ep , showing that GS is
stable for sufficiently large values of the parameter $\lambda$ at low $x$ in
the fixed coupling case. In the running coupling case, GS has been found to be
only approximately preserved at small $x$. These studies have been further
extended by Caola and Forte in Ref. Caola:2008xr , where they have shown that
an approximate GS is in fact a general property of the DGLAP evolution with
general boundary conditions. The saturation scale emerges from a saddle point
of the DGLAP kernel being a generic property of perturbative evolution. As
such, it also generates GS for the BFKL evolution. For low $Q^{2}$ and small
$x$, perturbative solutions violate GS, therefore its emergence provides
genuine evidence for parton saturation.

Even though GS may not be a sufficient argument in favor of saturation, also
in the case of hadronic collisions, it is hard to imagine that the scaling
behaviour of direct photon $p_{T}$-spectra that we observe in this paper
(following from geometrical characteristics of the saturation scale) can be
explained without invoking the saturation mechanism.

Our paper is organized as follows. In Sec. 2, we discuss the geometry of HIC,
d+Au and p+p collisions222Note that throughout the paper we also use the
expressions large systems (for HIC) and small systems (for d+Au and p+p
collisions). and introduce pertinent scaling variables. In Sec. 3, we analyze
available direct photon data concentrating on two aspects of GS: geometry and
energy dependence. We summarize our findings in Sec. 4.

##  2  Saturation and scaling variables

The GS physics emerges when particles are produced in a kinematical region,
where the only relevant intrinsic scale is the saturation momentum of a
radiation source. Then, by dimensional analysis, the particle multiplicity is
proportional to the overlap transverse area $S_{T}$ and a universal function
of the scaling variable (2):

$\frac{1}{S_{T}}\frac{dN_{\gamma}}{\,2\pi p_{T}\,d\eta dp_{T}}=\,F(\tau).$ (3)

The energy dependence of $Q_{\rm sat}$ is given by Eq. (1).

As far as the value of $\bar{Q}^{2}$ is concerned, we have to distinguish two
geometrical setups corresponding to symmetric and asymmetric collisions
Kharzeev:2000ph ; Kharzeev:2002ei . A photon with transverse momentum $p_{T}$
in the mid-rapidity region resolves, by uncertainty principle, partons of the
characteristic transverse size $s_{T}\sim\pi/p_{T}^{2}$. For symmetric A+A
collisions, the interaction region approximately corresponds to a sphere of
radius $R$, as shown in Fig. 1. The number of all active sources of radiation,
_i.e._ , the number of participants $N_{\rm{part}}$, is proportional to $\pi
R^{3}$. Therefore,

$R=N_{\rm{part}}^{1/3}/Q_{0},~{}~{}S_{T}=\pi N_{\rm{part}}^{2/3}/Q_{0}^{2}.$
(4)

Here $S_{T}$ is centrality-dependent transverse overlap of the colliding
nuclei. When the number of active partons reaches a critical value
$N_{\rm{cr}}$, where the whole transverse area is covered by resolved partons

$N_{\rm{cr}}=\frac{S_{T}}{s_{T}}=N_{\rm{part}}^{2/3}\,\frac{p_{T}^{2}}{Q_{0}^{2}},$
(5)

then the number of partons cannot grow any more and the system reaches the
saturation, $N_{\rm{part}}\approx N_{\rm{cr}}$. Consequently, the average
transverse momentum, $\left\langle p_{T}^{2}\right\rangle$ (related to the
saturation scale) when this happens, calculated from (5) is equal to

${\bar{Q}}^{2}\equiv\left\langle
p_{T}^{2}\right\rangle=\frac{N_{\rm{cr}}}{N_{\rm{part}}^{2/3}}\,Q_{0}^{2}=N_{\rm{part}}^{1/3}Q_{0}^{2}.$
(6)

This means, getting back to Eqs.(1) and (2), that for mid-rapidity photons we
will have

$Q_{\rm{sat}}=N_{\rm{part}}^{1/6}Q_{0}\left(\frac{p_{T}}{x_{0}\sqrt{s_{NN}}}\right)^{-\lambda/2}.$
(7)

Introducing the variable $W=\sqrt{s_{{NN}}}\times x_{0}=\sqrt{s_{{NN}}}\times
10^{-3}$, we will further have

$\tau=\frac{p_{T}}{Q_{\rm
sat}}=\frac{1}{N_{\rm{part}}^{1/6}}\frac{p_{T}}{Q_{0}}\left(\frac{p_{T}}{W}\right)^{\lambda/2},$
(8)

where $p_{T}$ is given in GeV and $W$ in TeV.

Figure 1: (Color online) Overview of the collision geometry. Left panel: the
side view of large and small system collisions showing relation of the number
of participants to the collision geometry. Right panel: the front view of d+Au
and p+p collisions. Active transverse area (shown in colors) in d+Au is
entirely due to the average size of the smaller nucleus. In p+p, since
collisions are mostly _peripheral_ , active transverse area is close to twice
that of the proton.

For asymmetric systems, the overlapping region is not spherical but rather
one-dimensional (see Fig. 1). Therefore, we should have

$S_{T}\sim n_{\rm{part}}^{2/3}/Q_{0}^{2},$ (9)

where $n_{\rm{part}}$ is a number of participants in a small system.

In this paper, we consider only d+Au asymmetric collisions, such that we take
$n_{\rm part}=1.62$ following Adare:2013nff . In order to estimate the length
$l$ shown in Fig. 1 we take $l=\sqrt[3]{A}=\sqrt[3]{197}=5.82$, following Ref.
Klein-Bosing:2014uaa . Effectively, this means that for the minimum bias d+Au
collisions, the number of participants in the Gold nucleus is $N_{\rm
part}[{\rm Au}]=A^{1/3}$, rather than the one determined in Ref. Adare:2013nff
, where $N_{\rm part}[{\rm Au}]=7\pm 0.4$. Meanwhile, note that the volume of
the cylinder shown in Fig. 1 (i.e., the total number of participants) is equal
to

$V_{\rm cyl.}=\sqrt[3]{197}\times 1.62^{2/3}=8.03,$

being quite close to $N_{\rm part}=8.6\pm 0.4$ in minimum bias d+Au, as shown
in Adare:2013nff .

In view of the above-mentioned discussion, one can write that

$\bar{Q}^{2}=Q_{0}^{2}\;{N_{\rm part}}/{n_{\rm{part}}^{2/3}}.$ (10)

Hence the saturation scale of a large nucleus probed by a small one is equal
to:

$Q_{\rm{sat}}[{\rm
large}]=Q_{0}\left(\frac{N_{\rm{part}}}{n_{\rm{part}}^{2/3}}\right)^{1/2}\left(\frac{p_{T}}{W}\right)^{-\lambda/2}.$
(11)

For a small nucleus nothing changes with respect to the symmetric collisions,
and

$Q_{\rm{sat}}\left[\rm{small}\right]=Q_{0}\,n_{\rm{part}}^{1/6}\left(\frac{p_{T}}{W}\right)^{-\lambda/2}.$
(12)

It has been argued in Ref. Dumitru:2001ux  that in an asymmetric case the
effective saturation scale takes the following form:

$\displaystyle Q_{\rm sat}^{\rm eff}$ $\displaystyle=$
$\displaystyle\sqrt{Q_{\rm{sat}}[{\rm large}]\,Q_{\rm{sat}}[{\rm{small}}]}$
(13) $\displaystyle=$
$\displaystyle\left(\frac{N_{\rm{part}}^{3}}{n_{\rm{part}}}\right)^{1/12}Q_{0}\left(\frac{p_{T}}{W}\right)^{-\lambda/2}.$

In order to test the geometrical assumptions described hitherto, we will
release the relation $S_{T}\sim N_{\rm part}^{2/3}$ in what follows, treating
the power of $N_{\rm part}$ as a free parameter $\delta$:

$S_{T}\sim N_{\rm part}^{\delta},$ (14)

which leads to the scaling variables

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\frac{1}{N_{\rm{part}}^{\delta/4}}\frac{p_{T}}{Q_{0}}\left(\frac{p_{T}}{W}\right)^{\lambda/2},$
(15)

and

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\frac{1}{\left(N_{\rm{part}}^{3}/n_{\rm{part}}\right)^{\delta/8}}\frac{p_{T}}{Q_{0}}\left(\frac{p_{T}}{W}\right)^{\lambda/2},$
(16)

for large-large and small-large collision systems, respectively. For p+p
collisions there is no $N_{\rm part}$ dependence. In this way we are able to
test two distinct aspects of GS: the energy dependence by varying parameter
$\lambda$, and dependence on geometry by varying $\delta$. Note that in the
limit when the small nucleus size increases, $n_{\mathrm{part}}\rightarrow
N_{\rm{part}}$, the scaling variable (16) reduces to (15).

The exact coefficient linking $S_{T}$ and the number of participants may be of
importance for small nuclei, i.e., for $n_{\rm part}$ (9). We expect it to
vary at most between 1 and $2^{2/3}$. It is therefore of no numerical
importance as far as the scaling variable $\tau$ is concerned, due to the
power-like suppression with $\delta/8$ in (16). As a matter of fact, because
of rather small exponent in the definition of $\tau$ in (16), rescaling
$n_{\rm part}$ by a factor of 2 has rather negligible effect: $2^{1/12}=1.06$.

$W$ [GeV] | System | Centrality | $N_{\rm{part}}$ | Experiment | References
---|---|---|---|---|---
200 | Au+Au | c1 | 0–20 % | 277.5 | PHENIX | Adare:2008ab
|  | c2 | 20–40 % | 135.6 |  |
|  | mb | 0–92 % | 106.3 |  |
200 | Au+Au | c1 | 0–20 % | 277.5 | PHENIX | Adare:2014fwh
|  | c2 | 20–40 % | 135.6 |  | Afanasiev:2012dg
|  | c3 | 40–60 % | 56.0 |  |
|  | c4 | 60–92 % | 12.5 |  |
62.4 | Au+Au | mb | 0–86 % | 114.5 | PHENIX | Adare:2018wgc
200 | Cu+Cu | c1 | 0-40 % | 66.4 | PHENIX | Adare:2018jsz
|  | mb | 0–94 % | 34.6 |  |
2760 | Pb+Pb | c1 | 0–20 % | 308.0 | ALICE | Adam:2015lda
|  | c2 | 20–40 % | 157.0 |  |
|  | c3 | 40–80 % | 45.7 |  |
200 | d+Au |  |  | $-$ | PHENIX | Adare:2012vn
| p+p |  |  | $-$ |  |
Table 1: The data sets for direct photon $p_{T}$-spectra used in this paper.
For the small system $N_{\rm part}$ see the text.

On the contrary, the effect on $F(\tau)$ in (3) due to the linear factor of
$1/S_{T}$ is numerically important. It is instructive to have some intuition
concerning the possible relation between $S_{T}$ and $n_{\rm part}$. Let us go
back again to Fig. 1, where we schematically show the active transverse
geometry in d+Au and p+p collisions. In d+Au collisions, the average $N_{\rm
part}$ of the deuteron related to the transverse size $S_{T}$ in (3) has been
determined to be $1.62$ Adare:2013nff , due to a large asymmetry of the
deuteron. The actual transverse size may be larger due to the corona effect.
Here we shall neglect the corona effect in the d+Au case333The corona effect
has been taken into account in Ref. Klein-Bosing:2014uaa , where the authors
used $S_{T}\sim(2\,n_{\rm part})^{2/3}=3.2^{2/3}$.. On the other hand, in p+p
collisions, due to rather a small probability of “central” collisions, the
transverse size as shown in Fig. 1, is larger than the size of one proton, so
that in this case we simply assume $S_{T}\sim 2^{2/3}$. We discuss the
sensitivity of our results to these choices in the next section.

##  3  Geometrical scaling of direct photon $p_{T}$-spectra

In our analysis, we have used direct photon data collected in Table 1
Adare:2008ab ; Adare:2014fwh ; Afanasiev:2012dg ; Adare:2018wgc ;
Adare:2018jsz ; Adam:2015lda ; Adare:2012vn . These data sets cover rather
large range of centrality classes, different beam energies ranging from 62.4
GeV to 2760 GeV, and different colliding systems/species. This variety gives
us a unique opportunity to study the two distinct features of GS: (i) the
centrality dependence encoded in the exponent $\delta$, (ii) the energy
dependence encoded in the exponent $\lambda$, both shown in Eqs. (15) and
(16). In the first case we compare the data with different $N_{\rm part}$ but
at the same energy; initially carrying it out for the large systems only, and
then also for d+Au, trying to assess the best value of $\delta$. In the second
case we take the data at different energies to study the GS dependence on
$\lambda$.

Direct photon $p_{T}$-spectra in Au+Au at 200 GeV have also been measured by
the STAR Collaboration at RHIC STAR:2016use  several years ago, but those data
are not included in our analysis here444We would expect GS holding for the
STAR data as well, in spite of the systematic difference between the direct
photon invariant yields measured by STAR and PHENIX in Au+Au collisions..
There are also recent results on direct photon $p_{T}$-spectra in p+p at 2760
GeV and 8000 GeV beam energies from the ALICE Collaboration at LHC, where
upper bounds are given Acharya:2018dqe .

###  3.1  Centrality scaling

In Fig. 2, we plot different centrality-binned PHENIX 200 GeV Au+Au and 2760
GeV ALICE Pb+Pb direct photon invariant yield data, scaled by $1/N_{\rm
part}^{\delta}$ in terms of the scaling variable (15) for three choices of
$\delta$. We see that the data at lower $\delta=1/3$ initially form a wide
spread band (upper red points), then come closer to each other overlapping at
$\delta\approx 2/3$ (middle green points), to disperse again at larger
$\delta=1$ (lower blue points). The overlap occurs both for PHENIX and ALICE
data at $\delta\approx 2/3$, which is expected on the basis of the GS
arguments brought up in Sec. 2.

In order to see this with a better resolution, we plot ratios (in Fig. 3) of
the most central scaled spectra to those in the other centrality classes at
different values of $\delta$ (see also Appendix). These ratios should all be
equal to unity if GS is present, and in fact this happens to a good accuracy
at $\delta\approx 2/3$.

Figure 2: (Color online) Direct photon $p_{T}$-spectra scaled according to Eq.
(3) with $S_{T}$ and $\tau$ given by Eqs. (14) and (15) respectively, plotted
– from top to bottom – at $\delta=1/3$ (red points), 2/3 (green points) and 1
(blue points). The left panel corresponds to plotting of the PHENIX Au+Au data
at 200 GeV, the right panel to Pb+Pb ALICE data at 2760 GeV (see Table 1). The
exponent $\lambda=0.2$ does not play any role here since we compare data at
the same energies.

Figure 3: (Color online) Another illustration of GS of direct photon
$p_{T}$-spectra as ratios of the scaled data in the most central to the scaled
data in the other centrality classes but at the same energies. Left panels:
ALICE; the red triangles correspond to $R_{c_{1}/c_{2}}$ and the green ones to
$R_{c_{1}/c_{3}}$ for Pb+Pb collisions. Right panel: PHENIX; the red squares
correspond to $R_{c_{1}/c_{2}}$, the green circles to $R_{c_{1}/c_{3}}$, and
the blue diamonds to $R_{c_{1}/c_{4}}$ for Au+Au collisions. The orange
triangles are for Cu+Cu collisions. The upper plots are obtained at
$\delta=1$, middle plots at $\delta=2/3$, and lower plots at $\delta=1/3$.

###  3.2  Centrality scaling for small systems

After there are new direct photon invariant yield data on small systems, for
example, in p+p, p+Au, d+Au, 3He+Au collisions at RHIC, as well as in p+p,
p+Pb collisions at LHC, then our analysis of direct photon GS in small systems
will certainly be more complete. For now we have at our disposal only two data
sets from PHENIX, both at 200 GeV (see Table 1). The p+p data do not scale, so
the d+Au collisions render a more interesting case. Here, following the
discussion of Sec. 2, we use the scaling variable $\tau$ given by Eq. (16),
and also the transverse size entering the scaling function $F(\tau)$ in Eq.
(3) that is given by Eq. (9). This follows from the fact that in asymmetric
collisions we have in fact two-saturation scales, and the effective saturation
scale is taken as a square root of the two (see Eq. (13)). We need to have
therefore two different number of participants, for which – as discussed after
Eq. (9) – we take $n_{\rm part}=1.6$ as the number of participants in
deuteron, and $N_{\rm part}=\sqrt[3]{197}\simeq 5.8$ as the number of
participants in the Gold nucleus. For comparison, we also use the one-
saturation scale formulae (4) and (15) for $S_{T}$ and $\tau$, respectively,
given $N_{\rm part}=8.6$. In both cases, we consider that the exponent
$\delta$ is fixed to 2/3.

Thus, in Fig. 4 we show the ratios of scaled $p_{T}$-spectra in Au+Au
centrality $c_{1}$ to d+A, with the two-saturation scales (green circles) and
the one-saturation scale (blue diamonds). Also, the ratio of two Au+Au
centralities $c_{1}/c_{2}$ (red squares) is also given for comparison.
Besides, we include the ratio of scaled $p_{T}$-spectra in Au+Au $c_{1}$ to
p+p (magenta triangles). At large values of $\tau$, the d+A ratio behaves very
much the same as the p+p ratio, irrespective whether we use the one- or two-
saturation scales formulae. Interestingly enough, a difference appears at
small $\tau$. The d+Au ratio with the two-saturation scales follows
approximately the $c_{1}/c_{2}$ ratio of Au+Au collisions. The one-saturation
scale formula still shows a sign of GS, albeit within the limits of
uncertainties.

We have also investigated the corona effect in d+A collisions discussed in
Sec. 2, and we have established that it violates GS at small values of $\tau$.

Figure 4: (Color online) Similar ratios as in Fig. 3 but for small collision
systems. We show the d+Au scaled yields (ratio of Au+Au $c_{1}$ over d+Au),
with the green circles describing two-saturation scales, and the blue diamonds
describing one-saturation scale (see the text). The red squares correspond to
the Au+Au ratio of $c_{1}/c_{2}$ centrality classes, and the magenta triangles
correspond to Au+Au $c_{1}$ scaled yield divided by the scaled yield in p+p.

###  3.3  Energy scaling

After having fixed $\delta=2/3$, we can now study the GS dependence on the
parameter $\lambda$. To this end, we choose one centrality class at different
collision energies, namely the most central Au+Au collisions at 200 GeV
(designated as $e_{1}$), minimum bias Au+Au collisions at 62.4 GeV
($e_{2}$)555We have not used the Au+Au data a 39 GeV from Adare:2018wgc , as
they have rather large uncertainties and the collisions energy is not that
different from 62.4 GeV., and the most central Pb+Pb collisions at 2760 GeV
($e_{3}$). We form two ratios, $e_{1}/e_{2}$ and $e_{1}/e_{3}$, by plotting
them for different values of $\lambda$. The result is shown in Fig. 5. We see
that the best scaling is achieved for rather low value of $\lambda$, somewhere
between 0.1 and 0.2. Certainly, the canonical value666Recall that GS in DIS
Stasto:2000er  and in the cross-sections in p+p collisions
Praszalowicz:2015dta  is achieved at $\lambda\sim 0.33$. of $\sim 0.33$ is
excluded. $\lambda=0.2$ is already used in Fig. 2 and Fig. 3. Note that this
value is not so far from the one used in the multiplicity scaling of the
charged-particle $p_{T}$-spectra in p+p collisions (i.e. $\lambda=0.22-0.24$)
Praszalowicz:2015dta .

Figure 5: (Color online) Illustration of GS of direct photon $p_{T}$-spectra
in large collision systems at different energies. The figures show ratios of
the scaled yields plotted in terms of the scaling variable $\tau$ at three
different energies and for four different values of $\lambda$ (here $\delta$
is fixed to 2/3): $e_{1}$ at Au+Au at 200 GeV divided by $e_{2}$ at Au+Au at
62.4 GeV (red squares), and $e_{1}$ at Au+Au at 200 GeV divided by $e_{3}$ at
Pb+Pb at 2760 GeV (blue diamonds). See text for more details.

##  4  Summary and Conclusions

In this paper, we have studied the geometrical scaling of direct photon
$p_{T}$-spectra (invariant yields) in relativistic heavy ion and d+Au
collisions. We have followed the initial study of Ref. Klein-Bosing:2014uaa
with, however, newer data and using a method, which does not rely on any
specific parametrization of photon yields. If GS is present than all $N_{\rm
part}^{\delta}$-scaled spectra should be aligned if plotted in terms of the
scaling variable $\tau$. If so, their ratios should be equal to unity (within
experimental uncertainties) for some extended interval of the $\tau$ variable,
regardless of the centrality class, collision energy, and the colliding nuclei
type.

The forms of the scaling variables $\tau$ in (15) and (16), and the scaling
function in $F(\tau)$ (3) are dictated by the Color Glass Condensate theory
McLerran:2010ub . The fact that we indeed observe GS in HIC, where the QGP is
produced, means that the information encoded in the initial wave function of
colliding systems is transferred to direct photons. This is a remarkable
situation, because GS follows from the scaling properties of the initial wave
function of gluons, whereas photons are produced from quarks.

GS has two distinct features: it depends on the collision geometry and on the
energy behavior of the saturation scale. We have encoded these two
dependencies in the parameter $\delta$ related to the geometry, and the
parameter $\lambda$ related to the collision energy. While these parameters
are in principle fixed to $\delta\simeq 2/3$ and $\lambda\simeq 0.33$ from
simple geometrical arguments and the DIS studies respectively, we have allowed
for their variation in order to asses the best values at which GS in HIC is
achieved. We have found that the best value for the scaling with centrality is
$\delta=2/3$, whereas for the scaling with energy we need to take $\lambda$
close to 0.2. The reason why $\lambda$ is substantially lower than its
expected value $\sim 0.33$ might be based on some scaling violations discussed
by us in Ref. Khachatryan:2019uqn , and based also on possible other
contributions in direct photon production that are not directly related to the
initial state physics.

Note that our analysis is not anchored on a detailed $\chi^{2}$ study. One
faces here three problems: (i) the accuracy of the alignment of different
$p_{T}$-spectra, (ii) how close this alignment is to unity (for the ratios of
spectra), and (iii) how large is the $\tau$ interval. A good example where the
ratios of spectra do align but below unity is the Au+Au $c_{1}/c_{2}$ ratio,
and the Au+Au $c_{1}$ to d+Au ratio at small $\tau\sim 1.5$ shown in Fig. 4.
We see that both ratios are close to each other, but below one. One should,
however, remember that in the small $\tau$ region (i.e., small $p_{T}$), mass
scales other than the saturation scale come into play (particle masses,
nonperturbative effects related to $\Lambda_{\rm QCD}$) that spoil dimensional
arguments on which GS is based. Further studies with possibly upcoming data on
small-large collision systems will be of importance to establish GS for
asymmetric collisions at low-$\tau$ region.

To summarize: we have established GS for direct photon invariant yield data in
HIC and observed a GS sign in d+Au collisions. Our analysis of the p+p data is
rather inconclusive, due to the large uncertainties of the corresponding ratio
shown in Fig. 4. However, as seen in Fig. 6 (right panel), the yield from p+p
overall has smaller slope than the yields from HIC. Our final conclusion is
best demonstrated in Fig. 6, where we show all data sets – non-scaled ones
(left panel) and scaled ones (right panel).

Figure 6: (Color online) Direct photon $p_{T}$-spectra of different colliding
systems at various energies and centrality classes, all scaled according to
Eq. (3) with $S_{T}$ and $\tau$ given by Eqs. (14) and (15) respectively. The
left figure is obtained at $\delta=0$ and $\lambda=0$, i.e., shows the
original unscaled $p_{T}$-spectra. The right figure is obtained using
$\delta=2/3$ and $\lambda=0.2$. The legends show all the references of the
data used. The QM2018 PHENIX fit to p+p data is taken from Khachatryan:2018evz
.

## Acknowledgements

We are grateful to Axel Drees and Larry McLerran for many important and
fruitful discussions. MP thanks the Institute for Nuclear Theory at the
University of Washington for its kind hospitality and stimulating research
environment. Research of MP was supported in part by the INT’s U.S. Department
of Energy grant No. DE-FG02- 00ER41132.

## Appendix

The data points describing direct photon invariant yield in different
centrality classes are measured at the same $p_{T}$ bins. However, when we
represent them in terms of the scaling variables, the values of $\tau$ of
those points are different at each centrality, as they depend on $N_{\rm
part}$. Therefore, if we wish to calculate the ratios of the $p_{T}$-spectra
in different centrality classes, we need to interpolate the data points of at
least one centrality to the $\tau$ values, where we have data from another
centrality. For example, we have selected the PHENIX Au+Au 200 GeV
$c_{1}=0-20\%$ data set as the reference spectrum, which we divide by other
data sets shown in Table 1. All this data has been published in Refs.
Adare:2014fwh ; Afanasiev:2012dg  for low and high $p_{T}$ ranges,
respectively. They have one overlapping point at $p_{T}=4.25$ GeV, for which
we take the data point from Ref. Afanasiev:2012dg , as it has substantially
smaller error than that in Adare:2014fwh .

The interpolation is carried out in a simplistic way. For each data point at
respective $\tau$ value, we find two adjacent data points, such that
$\tau_{1}<\tau<\tau_{2}$, and we fit the power law function $A/\tau^{n}$ to
these two points, both for the yield and for the total uncertainty. We do not
take into account any additional error coming from this procedure, as for
rather densely distributed points such an error is much smaller than the
experimental uncertainties.

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