# Collaborative network formation games between competing channels in a streaming platform

Vineeth S. Varma

► To cite this version:

Vineeth S. Varma. Collaborative network formation games between competing channels in a streaming platform. Symposium on Nonlinear Control Systems, IFAC, Jul 2025, Reyjavik, Iceland. hal-05142538

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Submitted on 19 Sep 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Collaborative network formation games
between competing channels in a streaming
platform*

V.S. Varma*,**

* Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France.

** Automation Department, Technical University of Cluj-Napoca,
Memorandumului 28, 400114 Cluj-Napoca, Romania.

**Abstract:** We develop a compartmental model that describes the evolution of the popularity of various channels hosted on a digital platform. Free users on the platform pick a channel based on its popularity, and after interacting with it, may go back to the platform list or may get redirected/recommended to another channel. These channels could be YouTube channels or Twitch streamers. All these channels are competitive, as time spent by users on one channel implies less time spent on another. However, in practice, it has been observed that many channels support and collaborate with one another, i.e., redirect users to other channels. In this work, we first analyze the compartmental model and then try to explain this seemingly irrational behavior using game theory.

Keywords:

SIS epidemics, network formation game

# 1. INTRODUCTION

Live streaming and video hosting platforms that rely on user-generated content (UGC), like Twitch and YouTube, have become extremely popular in recent years for gaming, e-sports, reactions, and other user-generated content as presented in Pires and Simon (2015). With the expansion of the gaming, streaming, and e-sports industries, there is a growing body of literature examining the motivations and behaviors of consumers and participants. In this context, Watanabe et al. (2021) studies the “attention economy,” i.e., a theoretical approach that it is in fact the attention of consumers that becomes a scarce resource, for which organizations or content creators must compete. Various models have been used to capture how video or channel popularity evolves in time. Compartmental models are frequently used due to being easy to motivate and often being mathematically tractable Sachak-Patwa et al. (2018).

Compartmental models like Susceptible-Infected-Susceptible (SIS) or Susceptible-Infected-Recovered (SIR) have often been applied to model the spread of epidemics over networks. Many researchers have often used similar models to capture the spread of viral videos and trends over social media Liu et al. (2019). Song popularity over the internet has been studied using an epidemic model in Rosati et al. (2021). Consequently, the study of viral marketing and content has attracted attention from the optimization, economics, and game theory community to model the behavior of consumers, streaming services, and other content creators. For example, in De Silva et al. (2023), a Stackelberg game to describe the interaction of

streaming platforms is developed, where the user interest in each platform is assumed to evolve according to a bi-viral SIS model. In Thomes (2013), the economy of the music streaming service industry is analyzed. In Varma et al. (2017), an opinion dynamics model is used to model the interest of the population in two competing products that try to manipulate the market via social media.

Streamer behavior data has been studied by some researchers recently Zhao et al. (2019) in order to better understand their motivations and actions. As mentioned in Watanabe et al. (2021), the attention and time of the user pool are limited, and thus, all UGC creators hoping to generate revenue from these media platforms are in competition for this finite resource. Despite this competitive aspect, Youtubers, Twitch streamers, and content creators on various social media have been observed to collaborate and support one another Koch et al. (2018); Rosati et al. (2021). In this work, we aim to understand the dynamics of video or channel popularity and then use this to motivate the behavior of UGC creators. For this purpose, we use the concept of network formation games (NFG) introduced in Tardos and Wexler (2007) to define the behavior of UGC creators in forming or breaking collaborations with other UGC creators.

The main contributions of this paper are the following:

(1) Development of a compartmental model based on multi-Viral SIS to describe the evolution of UGC popularity.

(2) Equilibrium analysis of the proposed model.

(3) The introduction of a network formation game to describe the behavior of UGC creators.

<footnote>* This work was partially supported by ANR through the grant UMICrowd and the PNRR project DECIDE No. 760069.</footnote>
The paper is structured as follows: Section 2 presents the definitions of the measures that constitute the model. Characteristics of the equilibrium and asymptotic behavior are analyzed in Section 3. The resulting network formation game is described and some numerical examples are presented in Section 4. Finally, Section 5 concludes this work.

## 2. PROBLEM FORMULATION AND PRELIMINARIES

We consider a framework in which $N$ UGC creators, henceforth referred to as channels or players, compete for viewers or user engagement in an online platform. We use $V := \{1, \dots, N\}$ to denote the set of channels. We will consider that their collaboration is fixed according to a pre-defined collaboration strategy and is given according to an undirected graph $G = (V, E)$. The neighborhood of a channel $i$ is denoted by $N_i$ and represents the set of channels that $i$ collaborates with according to the graph $G$ (i.e., $j \in N_i \Leftrightarrow (j, i) \in E$). Later on, in Section 4, this graph will be formed according to the collaboration strategy of each player (or channel).

### 2.1 User engagement with channels

We will use $S(t) \in [0, 1]$ to denote the fraction of time an average user spends at the platform interface browsing for UGC to engage with at time $t$ (averaged over a large pool of users). The platform offers a list of UGC, ranked according to its popularity for the user to select. We will use $I_n(t) \in [0, 1]$ to denote the fraction of time an average user spends engaging with channel $n$. The engagement with the channel $n$ will decay at rate $\delta_n$ and once done, a user will either go back to the platform interface (back to $S$) or follow the recommendation of $n$ to engage with the channel $j \in N_n$ with a uniform probability distribution. As these indicate the average user engagement, we have $S + \sum_{n \in V} I_n = 1$. For ease of exposition, unless required, we will remove the explicit dependence of $S$ or $I_n$ on $t$ and just write $S, I_n$.

We can thus write the popularity, i.e., average engagement of users with channel $n$, evolution with the following differential equations.

$$ 
\begin{aligned}
\dot{S} &= \sum_n \frac{\delta_n}{1 + |N_n|} I_n - I_n S \\
\dot{I}_n &= I_n S - \delta_n I_n + \sum_{j \in N_n} \frac{\delta_j}{|N_j| + 1} I_j, \quad \forall n \in V
\end{aligned}
\quad (1) 
$$

The term $I_n S$ indicates that users go from $S$ to $I_n$ with a rate proportional to the popularity of channel $n$ (and users have to be in $S$ to move from $S$ to $I_n$). The users then go away from $n$ at decay rate $\delta_n$, but at the same time new users may also visit $n$ due to collaborations.

**Remark 1.** Just like the classical SIS model, (1) also has a trivial equilibrium at $I_n = 0$ for all $n \in V$. However, as this is not interesting, our analysis will focus on the characterization of the other possible equilibria.

It is clear that the model (1) is equivalent to the multi-viral SIS model studied in Prakash et al. (2012); De Silva et al. (2023), among others, when $E = \emptyset$. Thus, when

no collaboration occurs, the resulting behavior is already well established in literature. The focus of this work is to first study the behavior of (1) for a fixed $E$ in Section 3. Specifically, we look to characterize the equilibrium points of this system of differential equations with nonlinear dynamics.

### 2.2 NFG preliminaries

The next step of this work is to model the behavior of these channels (as players in a game). NFGs have a long history in the social and economic sciences, and most NFGs in the economics literature use a bilateral process, i.e., edges only form between two agents with the consent of both parties, and any agent can break the agreement. As explained in Tardos and Wexler (2007), Nash equilibria are apparently ill-suited for modeling bilateral agreements and collaboration graphs. As Nash equilibrium is defined according to unilateral deviation, a graph is stable if we can ensure that no node wants to drop edges since a player cannot unilaterally add an edge. For example, the empty graph is always a Nash equilibrium in the bilateral game. Consequently, Jackson and Wolinsky (2003) suggested using the notion of pairwise stable equilibrium (PSE), i.e., a graph is considered to be an equilibrium if and only if no player wants to break a link or no two players want to create (or destroy and create) a new link.

We will define the NFG with the following three items.

(1) Set of players $V$.

(2) Collaboration strategy $\mathcal{E}$, where $(i, j) \in V$ implies both $i$ and $j$ agreed to collaborate with each other.

(3) $u_n : 2^{V \times V} \to \mathbb{R}_{\ge 0}$ for all $n \in V$, where $2^S$ denotes the power set of a set $S$. Thus, $u_n(\mathcal{E})$ denotes the utility obtained by player $n$ when the collaboration strategy $\mathcal{E}$ is chosen. For this work, we will set $u_n(\mathcal{E}) = I_n^*(\mathcal{E})$, where $I_n^*(\mathcal{E})$ is the value of $I_n(t)$ at the stable equilibrium of (1) when the graph $\mathcal{E}$ is used.

The stable equilibrium $I_n^*(\mathcal{E})$ will be shown to be unique in the next section. Throughout this section, we will use this notion of equilibrium defined as follows. First, we define a pairwise deviant graph.

**Definition 1.** (Pairwise deviation). We say that $\mathcal{E}'$ is a pairwise deviant graph of $\mathcal{E}$ by pair $i, j$ if and only if

(1) $\mathcal{E}' \setminus \{(i, j)\} \subseteq \mathcal{E}$,

(2) $\forall (l, m) \in \mathcal{E}$ such that $l, m \notin \{i, j\}$, we have $(l, m) \in \mathcal{E}'$.

This means that for any given graph, if a pair $i, j$ deviates from their strategy, i.e., they can break any of their links but can only create a link with each other, a pairwise deviant graph is formed.

**Definition 2.** (Pairwise stable equilibrium). A strategy $\mathcal{E}$ is said to be a PSE of the NFG if and only if any pairwise deviant graph $\mathcal{E}'$ for any pair $i, j$ satisfies $u_i(\mathcal{E}) \ge u_i(\mathcal{E}')$ and $u_j(\mathcal{E}) \ge u_j(\mathcal{E}')$.

It is important to note that in this work, we focus on static strategies (graphs). In practice, the dynamic nature of (1) coupled with the fact that a PSE might not exist could result in a dynamic MFG, i.e., the players may change their collaboration strategies in time. However, as this is
a preliminary work, we focus on the static game, and the dynamic game is left for future work.

### 3. ANALYSIS

Let us consider that the collaboration graph comprises K disconnected sub-graphs that are locally connected, which we call teams. If the entire graph is connected, we end up with a single team or $K = 1$.

**Definition 3.** (Team). We say that channels i, j belong to team $\mathcal{T}_k$ if there exists a path in the collaboration graph $\mathcal{G}$ from i to j. We have the union of all teams to be $\mathcal{V}$ and the intersection of any two teams to the empty set.

We also define

$$T_k := \sum_{n \in \mathcal{T}_k} I_n \qquad (2)$$

for the engagement of users with some UGC of channels in team k.

**Theorem 1.** The set of equilibria for the dynamics (1) is such that if $T_k^* > 0$ for some $k$, then $T_m^* = 0$ for all $m \neq k$ for almost all values of $\delta$.

**Proof:** First, we will write the equilibrium $I_n^*$ for each player as $I_n^* = \alpha_n T_m^*$ where $n \in \mathcal{T}_m$ and $\alpha_n \in [0, 1]$. This is always possible as $T_m$ is the sum of all $I_n$ for $n \in \mathcal{T}_m$. As each $I_n^* \in [0, 1]$, we have that $T_m^* = 0$ implies $I_n^* = 0$ for any $n \in \mathcal{T}_m$.

At equilibrium, we must have

$$0 = I_n S - \delta_n I_n + \sum_{j \in \mathcal{N}_n} \frac{\delta_j}{|\mathcal{N}_j| + 1} I_j \qquad (3)$$

By definition of a team, if $j, i$ are connected, they belong to the same team and we can thus write $I_j = \alpha_j T_m$ and $I_n = \alpha_n T_m$. This leads to:

$$\dot{T}_m = \alpha_n T_m S - \delta_n \alpha_n T_m + \sum_{j \in \mathcal{N}_n} \frac{\delta_j}{|\mathcal{N}_j| + 1} \alpha_j T_m \qquad (4)$$

which has a trivial solution for $\dot{T}_m = 0$ at $T_m = 0$. If $T_m > 0$, then we have

$$0 = \alpha_n S - \delta_n \alpha_n + \sum_{j \in \mathcal{N}_n} \frac{\delta_j}{|\mathcal{N}_j| + 1} \alpha_j \qquad (5)$$

which we can sum over all $n \in \mathcal{T}_m$ to obtain

$$0 = \sum_{n \in \mathcal{T}_m} \alpha_n S - \frac{\delta_n \alpha_n}{|\mathcal{N}_n| + 1} \qquad (6)$$

Therefore, at equilibrium if $T_m > 0$, as $\sum_n \alpha_n = 1$, we must have

$$S = \sum_{n \in \mathcal{T}_m} \frac{\delta_n \alpha_n}{|\mathcal{N}_n| + 1} \qquad (7)$$

which should hold for any team m such that $T_m > 0$. Substituting this back, we get

$$0 = \alpha_n \sum_{j \in \mathcal{T}_m} \frac{\delta_j \alpha_j}{|\mathcal{N}_j| + 1} - \delta_n \alpha_n + \sum_{j \in \mathcal{N}_n} \frac{\delta_j}{|\mathcal{N}_j| + 1} \alpha_j, \qquad (8)$$

which must hold for all $n \in \mathcal{T}_m$. This means we have $|\mathcal{T}_m|$ variables and $|\mathcal{T}_m| - 1$ equations (one is redundant due to the process of obtaining $S$). However, we also require $\sum_{n \in \mathcal{T}_m} \alpha_n = 1$, which implies that the values of each $\alpha_n$

are determined automatically (and are unique) according to the graph and $\delta$.

As $S$ can not take multiple values, this requires that either $T_k = 0$ for all $k \neq m$ or

$$\sum_{n \in \mathcal{T}_k} \frac{\delta_n \alpha_n}{|\mathcal{N}_n| + 1} = \sum_{j \in \mathcal{T}_m} \frac{\delta_j \alpha_j}{|\mathcal{N}_j| + 1} \qquad (9)$$

which can not happen unless the values of $\delta$ are such that (9) hold, which is a measure-zero set in general. ■

This theorem implies a behavior similar to the result in Prakash et al. (2012) which states that in a competitive viral SIS model, at most one virus survives at equilibrium (except for a measure zero set of parameters). In the framework of (1) with collaborative elements, we get that rather than the winning virus taking all, we will have the winning team taking all and the other teams having 0 share of user engagement at equilibrium.

**Proposition 1.** If the teams are locally complete (all agents connected to all other team members), then for all $n \in \mathcal{T}_k$

$$I_n^* = \frac{S_k^*(1 - S_k^*)}{\delta_n + \delta_n |\mathcal{T}_k|^{-1} - S_k^*} \qquad (10)$$

where $S_k^* \in (0, 1)$ is a solution to

$$\sum_{n \in \mathcal{T}_k} \frac{S_k^*}{\delta_n + \delta_n |\mathcal{T}_k|^{-1} - S_k^*} = 1 \qquad (11)$$

For all $n \notin \mathcal{T}_k$, $I_n^* = 0$. If no $S_k^* \in (0, 1)$ solves it for all teams, then $I_n = 0$ for all $n \in \mathcal{T}_k$ is the only equilibrium.

**Proof:** At equilibrium, the following must hold

$$0 = \alpha_n S |\mathcal{T}_k| - \delta_n \alpha_n |\mathcal{T}_k| + \sum_{k \neq n} \delta_k \alpha_k \qquad (12)$$

and

$$0 = S |\mathcal{T}_k| - \sum \delta_n \alpha_n \qquad (13)$$

i.e., $\sum_{j \neq n} \delta_j \alpha_j = S |\mathcal{T}_k| - \alpha_n \delta_n$. Substituting back, we have

$$\delta_n \alpha_n |\mathcal{T}_k| = \alpha_n S |\mathcal{T}_k| + S |\mathcal{T}_k| - \alpha_n \delta_n \qquad (14)$$

Thus, we have

$$\alpha_n = \frac{S}{\delta_n + \delta_n |\mathcal{T}_k|^{-1} - S} \qquad (15)$$

Recall that $I_n^* = \alpha_n T_m^*$. Taking the sum of $\alpha_n$ in (15) we get the main result on $S_k^*$ and we use $\alpha_n$ from (15) to get the main result. This concludes our proof. ■

As a special case, we can consider a symmetric decay rate, i.e., if $\delta_n = \delta$. In this case, we have a tractable solution for the equilibrium with

$$S = \delta(1 + |\mathcal{T}_k|^{-1})/(1 + |\mathcal{T}_k|) = \delta |\mathcal{T}_k|^{-1}. \qquad (16)$$

We can also expect that the stable equilibrium for this system is that $T_m > 0$ for the team m with the smallest $S$ satisfying (11) by applying the same proof as in Prakash et al. (2012) for the winner takes all model. We study the local vector fields to prove this.

**Proposition 2.** The stable equilibrium with $I_n > 0$ is given by the $n \in \mathcal{T}_m$ such that $m = \arg \min_{k \in \{1, \dots, K\}} S_k^*$ if $S_m^* < 1$.

**Proof:** Looking at any point $S, I$ in a neighborhood of the equilibrium with $S_m^* < S < S_k^*$, we notice by studying
(4) that $\dot{T}_m > 0$ and $\dot{T}_k < 0$ for any $k \neq m$. This implies that the vector field points towards this equilibrium. When $S < S_m^*$ we have $\dot{T}_k < 0$ for all teams, which means that in all cases, the vector field around the equilibrium is directed towards it. On the other hand, we also observe that for any other equilibrium $\dot{T}_m > 0$ for any $S_m^* < S < S_k^*$ implies that all the other equilibria are unstable. Please refer to Prakash et al. (2012) for a more detailed proof for a similar system.

When $|T_m| = 1$, we trivially have $S = 1 - \delta_n$ and $n \in T_m$. Next, we look at the special case of $|T_m| = 2$ with $i, j \in T_m$. In this case, we have

$$ \frac{S_m^*}{1.5\delta_i - S_m^*} + \frac{S_m^*}{1.5\delta_j - S_m^*} = 1 \quad (17) $$

which is a quadratic equation with the smaller root

$$ S_m^* = \frac{\delta_i + \delta_j - \sqrt{\delta_i^2 + \delta_j^2 - \delta_i \delta_j}}{2} \quad (18) $$

If $\delta_i = \delta_j$, we have $S_m^* = 0.5\delta_i$, which means that $I_j^* = I_i^* = 1 - 0.5\delta_i$. This can also be a good approximation when $|\delta_i - \delta_j| \ll \delta_i$.

## 4. NETWORK FORMATION GAME

From the previous section, we know that if $\mathcal{E}$ is such that there are several teams, only one team will have a non-zero utility. However, due to the fact that the sum of all $I_n^*$ is bounded by 1, a large team also implies a smaller share per player. Since the number of graphs is always finite for a finite number of players (even if large), we can explicitly list all the possible strategies for an NFG with a small number of players.

### 4.1 Two-player game

There are only two possible strategies in this game: will both players collaborate or not? Collaboration can occur only if this benefits both players. When no collaboration occurs, if $\delta_1 < 1$ and $\delta_1 < \delta_2$ we have

$$ u_1(\emptyset) = I_1^* = 1 - \delta_1, \quad (19) $$

and $u_1(\emptyset)$ if $\delta_1 \ge 1$. We always have $u_2(\emptyset) = 0$. If $\delta_2 > \delta_1$, the result in vice versa happens. If $\delta_1 = \delta_2$, we set $u_1(\emptyset) = u_2(\emptyset) = 1 - 0.5\delta_1$ as the fair equilibrium (considering the SIS system admits multiple equilibria).

Let us consider the case where $\delta_2 = \delta_1 + \epsilon$ where $\epsilon \ll \delta_1$. Now, we can use Theorem 2 and (18) to approximate $S_1^* = 0.5\delta_1$ and

$$ u_1(\{(1, 2)\}) = 0.5 - 0.25\delta_1 = u_2(\{(1, 2)\}). \quad (20) $$

This means that for any $\delta \in (2/3, 2)$, the two players can collaborate to improve both their utilities. If $\delta \ge 2$, the utilities are 0 for both players regardless of their strategy. For general $\delta_1 < \delta_2$, we can compute

$$ u_1(\{(1, 2)\}) = \frac{S_1^*(1 - S_1^*)}{1.5\delta_1 - S_1^*}, \quad (21) $$

and collaboration will occur only if $\frac{S_1^*(1-S_1^*)}{1.5\delta_1-S_1^*} > 1 - \delta_1$, with $S_1^*$ according to (18). When $\delta_1 = \delta_2$, collaboration or not

<figure><img src="image_2.png" /><figcaption>Fig. 1. PSE for various values of $\delta_1$ and $\delta_2$. The region in black has $\emptyset$ as the PSE while the white region has $\{(1, 2)\}$ as the PSE. When $\delta_1 = \delta_2$ both strategies are PSE.</figcaption></figure>

does not change the utility, and therefore all strategies are trivially PSE. In Figure 1, we mark the region of $\delta_1$ and $\delta_2$ according to which strategy becomes the PSE for the two-player game. As expected, when $\delta_1$ is close to $\delta_2$ (but not equal), the collaboration is a PSE only when $\delta_1 > 2/3$.

While the case of $\delta_1 \approx \delta_2$ is as expected, what is surprising is that when there is a large asymmetry, the collaboration is actually beneficial to both parties. In practice, this situation would correspond to an extremely popular channel collaborating with an unknown channel to help both channels improve their popularity. If it collaborated with another popular channel, it would lose out by the collaboration due to an equal share of the limited user pool. When there is a large asymmetry, the popular channel is able to attract a larger user pool while grabbing a majority of the team's users. As an example, we take $\delta_1 = 0.2$ and $\delta_2 = 2$ and observe that $u_1(\{(1, 2)\}) = 0.81$ while $u_1(\emptyset) = 0.8$, resulting in collaboration being the PSE.

### 4.2 Three player example

In this section, we show an example of an NFG with just three players. For certain values of $\delta$, we are able to explicitly find the equilibrium strategy of the NFG. We list all the possible team formations and the relevant payoffs for the case of $\delta = (0.3, 0.5, 0.55)$.

(1) $\mathcal{E} = \emptyset$ gives $(0.7, 0, 0)$. This case is also plotted in Figure 2.

(2) $\mathcal{E} = \{(2, 3)\}$ gives $(0, 0.3954, 0.3428)$, and this is plotted in Figure 3.

(3) $\mathcal{E} = \{(1, 2)\}$ gives $(0.5558, 0.2622, 0)$

(4) Any collaboration with 1 gives 2, 3 a smaller payoff.

Neither player 2 nor 3 will break or create new links when $\mathcal{E} = \{(2, 3)\}$. Therefore, $\mathcal{E} = \{(2, 3)\}$ is a PSE of the NFG.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Simulation of (1) for $\mathcal{E} = \emptyset$, i.e., when no collaboration occurs. Player 1 has the highest user retention (lowest $\delta_n$) and thus captures the entire user attention asymptotically.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Simulation of (1) for $\mathcal{E} = \{(2,3)\}$, i.e., when players 2 and 3 collaborate. In this case, players 2 and 3 are able to beat player 1 despite player 1 having the smallest $\delta$.</figcaption></figure>

### 4.3 Four player example

Consider the MFG when there are four players with $\delta = (0.3, 0.32, 0.34, 0.36)$. We list out some of the interesting cases. The graphs where the utilities are inferior are dominated by the listed strategies are not listed.

(1) $\mathcal{E} = \{(1, 2), (3, 4)\}$ gives (0.443, 0.4, 0, 0).

(2) $\mathcal{E} = \{(1, 3), (2, 4)\}$ gives (0.46, 0, 38, 0).

(3) $\mathcal{E} = \{(2, 3)\}$ gives (0, 0, 43, 0.4, 0).

(4) $\mathcal{E} = \{(1, 4)\}$ gives (0.47, 0, 0, 0.36).

(5) $\mathcal{E} = \{(3, 4)\}$ gives (0, 0, 43, 0.4).

In this example, we observe that every single graph strategy is unstable, as some pair can deviate to improve their utility. For example, the graph $\mathcal{E} = \{(1, 2), (3, 4)\}$ will have 1, 4 deviate to form the graph $\mathcal{E} = \{(1, 4)\}$. The graph $\mathcal{E} = \{(1, 4)\}$ will have 3, 4 deviate to form the graph $\mathcal{E} = \{(3, 4)\}$ which will in turn have 1, 2 deviate to form the graph $\mathcal{E} = \{(1, 2), (3, 4)\}$. Consequently, no pure PSE exists in this NFG, as for any possible graph, a pairwise deviation can improve the utility of some pair.

**Discussion:** So far, we have presented some models motivated by practical examples and a game theoretical analysis of the resulting system. What is interesting is that when there are a sufficiently large number of players, no pure PSE may exist. In game theory this would mean that all players will deploy a mixed strategy (as in rock-paper-scissors, for example). In the context of channels streaming content, a reasonable interpretation is that teams are not

eternal. That is, players may form teams and then break to form new teams over time.

Indeed, one of the limitations of this work is that we focused on a static NFG. In future work, we will explore a dynamic game in which $\mathcal{E}$ can vary in time. The player utility will be the expected user engagement over a long period of time, accounting for the fact that the network can evolve in time. In this case, $I_n$ may never actually converge to an equilibrium and may instead exhibit oscillations or limit cycles.

## 5. CONCLUSION

In this paper, we have introduced and analyzed a compartmental model to describe the evolution of UGC popularity when some of the channels establish collaborative links with each other. We characterize the equilibria of this dynamics based on the network of channels. The resulting equilibrium is used to define the utility of channels that are the players in an NFG. We demonstrate that there are certain parameters for which the NGF has no pairwise stable equilibrium. This work can be seen as a first step towards a more realistic model of streamer interactions. Despite the dynamic nature of the user engagement model, we studied a static game, which means that the collaboration strategy was fixed once and for all. A dynamic game formulation and a more thorough stability analysis of the equilibria are planned for future work.

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