SlowGuess/NeurIPS_2023_OCR huggingface dataset:
[1] Kenneth J Arrow. A difficulty in the concept of social welfare. Journal of political economy, 58(4):328-346, 1950.
[2] Omer Ben-Porat, Gregory Goren, Itay Rosenberg, and Moshe Tennenholtz. From recommendation systems to facility location games. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 1772-1779, 2019.
[3] Omer Ben-Porat and Moshe Tennenholtz. Shapley facility location games. In International Conference on Web and Internet Economics, pages 58-73. Springer, 2017.
[4] Omer Ben-Porat and Moshe Tennenholtz. A game-theoretic approach to recommendation systems with strategic content providers. Advances in Neural Information Processing Systems, 31, 2018.
[5] Jesús Bobadilla, Fernando Ortega, Antonio Hernando, and Abraham Gutiérrez. Recommender systems survey. Knowledge-based systems, 46:109-132, 2013.
[6] Nicholas Carr. The shallows: What the Internet is doing to our brains. WW Norton & Company, 2020.
[7] Bruno De Borger and Kurt Van Dender. Prices, capacities and service levels in a congestible bertrand duopoly. Journal of Urban Economics, 60(2):264-283, 2006.
[8] Jicong Fan and Jieyu Cheng. Matrix completion by deep matrix factorization. Neural Networks, 98:34-41, 2018.
[9] Daniel Fleder and Kartik Hosanagar. Blockbuster culture's next rise or fall: The impact of recommender systems on sales diversity. Management science, 55(5):697-712, 2009.
[10] Angela Glotfelter. Algorithmic circulation: how content creators navigate the effects of algorithms on their work. Computers and composition, 54:102521, 2019.
[11] F Maxwell Harper and Joseph A Konstan. The movielens datasets: History and context. Acm transactions on interactive intelligent systems (tiis), 5(4):1-19, 2015.
[12] Thomas Hodgson. Spotify and the democratisation of music. Popular Music, 40(1):1-17, 2021.
[13] Jiri Hron, Karl Krauth, Michael I Jordan, Niki Kilbertus, and Sarah Dean. Modeling content creator incentives on algorithm-curated platforms. arXiv preprint arXiv:2206.13102, 2022.
[14] Meena Jagadeesan, Nikhil Garg, and Jacob Steinhardt. Supply-side equilibria in recommender systems. arXiv preprint arXiv:2206.13489, 2022.
[15] Meta. Meta is experimenting with new monetization options for creators, 2022. https://www.digitalinformationworld.com/2022/03/meta-is-experimenting-with-new.html.
[16] Dov Monderer and Lloyd S Shapley. Potential games. Games and economic behavior, 14(1):124-143, 1996.
[17] Kun Qian and Sanjay Jain. Digital content creation: An analysis of the impact of recommendation systems. Available at SSRN 4311562, 2022.
[18] Robert W Rosenthal. A class of games possessing pure-strategy nash equilibria. International Journal of Game Theory, 2:65-67, 1973.
[19] Savy. Will the new youtube algorithm impact your content?, 2019. https://savyagency.com/new-youtube-algorithm/.
[20] TikTok. What is the tiktok creator fund? here's how to join + start making money, 2022. https://www.backstage.com/magazine/article/tiktok-creator-fund-explained-how-to-join-75090/.
[21] Takashi Ui. Robust equilibria of potential games. *Econometrica*, 69(5):1373-1380, 2001.
[22] Hal R Varian and Christopher Harris. The vcg auction in theory and practice. American Economic Review, 104(5):442-445, 2014.
[23] Fan Yao, Chuanhao Li, Denis Nekipelov, Hongning Wang, and Haifeng Xu. How bad is top-k recommendation under competing content creators? In International Conference on Machine Learning. PMLR, 2023.
[24] Youtube. Youtube partner program overview & eligibility, 2023. https://support.google.com/youtube/answer/72851.
[25] Piercarlo Zanchettin. Differentiated duopoly with asymmetric costs. Journal of Economics & Management Strategy, 15(4):999-1015, 2006.
8.1 Additional Examples of User Utility Function
Fix any user  $\pmb{x} \in \mathcal{F}$ , let  $\sigma_{i} = \sigma(\pmb{s}_{i}; \pmb{x})$  and for simplicity of notations we assume  $\sigma_{1} \geq \dots \geq \sigma_{n}$ . As discussed in Section 3, if the platform presents the top-  $K$  ranked content in terms of their relevance quality, the user utility function has the following form:
$$
W (\boldsymbol {s}; \boldsymbol {x}) = \sum_ {k = 1} ^ {n} r _ {k} \sigma_ {k}, \tag {9}
$$
where  $\{r_k\}_{k=1}^n$  are the user's "attention" over the  $k$ -th ranked content such that  $r_k = 0, \forall k \geq K + 1$ . We emphasize that our user utility model given in Eq.(9) is compatible with various matching strategies and here we provide additional examples that incorporate a modified version of the top- $K$  approach, taking into account considerations of advertised content. For instance, considering a scenario where  $K = 5$  and the platform intends to promote the content originally ranked at position 6 to position 2 with probability  $p \in (0,1)$ . Consequently, the resulting utility function can be expressed as follows:
$$
\begin{array}{l} \tilde {W} _ {j} (s) = p (r _ {1} \sigma_ {1} + r _ {2} \sigma_ {6} + r _ {3} \sigma_ {2} + r _ {4} \sigma_ {3} + r _ {5} \sigma_ {4}) + (1 - p) (r _ {1} \sigma_ {1} + r _ {2} \sigma_ {2} + r _ {3} \sigma_ {3} + r _ {4} \sigma_ {4} + r _ {5} \sigma_ {5}) \\ = r _ {1} \sigma_ {1} + [ p r _ {3} + (1 - p) r _ {2} ] \sigma_ {2} + [ p r _ {4} + (1 - p) r _ {3} ] \sigma_ {3} + [ p r _ {5} + (1 - p) r _ {4} ] \sigma_ {4} + (1 - p) r _ {5} \sigma_ {5} + p r _ {2} \sigma_ {6} \\ \triangleq \sum_ {k = 1} ^ {n} \tilde {r} _ {k} \sigma_ {k}. \\ \end{array}
$$
This example shows that user utility function under any position-based perturbation of top-  $K$  ranking can be expressed in the form of Eq.(9), and in general the values of  $r_k, k > K$  can be non-zero.
8.2 Examples of  $\mathcal{M}^3$
In this section we formally justify that the two examples given in Section 4 belong to the class of  $\mathcal{M}^3$ .
1. When the creators' utilities are set to the total content exposure [2, 13, 14], we have  $M(\sigma_i; \sigma_{-i}) = \mathbb{I}[i \leq K] \frac{\exp(\beta^{-1} \sigma_i)}{\sum_{j=1}^{K} \exp(\beta^{-1} \sigma_j)}$ , with a temperature parameter  $\beta > 0$  controlling the spread of rewards. The validity of three merit-based properties are straightforward. In terms of monotonicity, we have  $\sum_{i=1}^{n} M(\sigma_i; \sigma_{-i}) = 1$  which is a constant and thus monotone.
2. When the creators' utilities are set to the total user engagement [23], we have  $M(\sigma_i; \sigma_{-i}) = \mathbb{I}[i \leq K] \frac{\exp(\beta^{-1} \sigma_i)}{\sum_{j=1}^{K} \exp(\beta^{-1} \sigma_j)} \pi(\sigma_1, \dots, \sigma_n)$ , where  $\pi(\sigma_1, \dots, \sigma_n) = \beta \log \left( \sum_{j=1}^{K} \exp(\beta^{-1} \sigma_j) \right)$ . The first two merit-based properties are obvious (Normality and Fairness). In terms of monotonicity, we have  $\sum_{i=1}^{n} M(\sigma_i; \sigma_{-i}) = \beta \log \left( \sum_{j=1}^{K} \exp(\beta^{-1} \sigma_j) \right)$  which is monotone in each  $\sigma_j$ . To verify negative externality, it suffices to show the function  $\frac{\log \left( \sum_{j=1}^{K} \exp(\beta^{-1} \sigma_j) \right)}{\sum_{j=1}^{K} \exp(\beta^{-1} \sigma_j)}$  is decreasing in  $\sigma_j, \forall j$ . Since  $\exp(x)$  is increasing in  $x$ , and function  $\frac{\log(t)}{t}$  is decreasing when  $t > e$ , we conclude that  $M$  satisfies negative externality when  $n \geq 3$ .
8.3 Examples of BRM
To get an better intuition of how BRM works, let us consider a special case  $M \in \mathrm{BRCM}$  such that  $f_{1} = \dots = f_{K} = 1$  and  $f_{k} = 0, k \geq K + 1$ . By the definition, any matching score sequence  $\sigma_{1} \geq \dots \geq \sigma_{n}$  will be mapped to a reward sequence of  $(\sigma_{1} - \sigma_{K + 1}, \dots, \sigma_{K} - \sigma_{K + 1}, 0, \dots, 0)$ . Consequently, the top-  $K$  ranked creators will experience a significant reduction in rewards if the  $(K + 1)$ -th ranked creator increases its matching score. This mechanism can deter an unnecessary concentration of creators on a specific strategy, as when the number of creators with high scores exceeds a certain threshold, even those ranked highly can receive a decreasing reward. This backward rewarding mechanism thus encourages diversity in content creation and mitigates the risk of oversaturation in any particular group of users.
Another notable special case within BRCM is  $M^{SM} = M[1, \frac{1}{2}, \dots, \frac{1}{n}]$ , which coincides with the Shapley mediator proposed in [4]. One key feature of  $M^{SM}$  is that for any sequence  $1 \geq \sigma_1 \geq \dots \geq \sigma_n \geq 0$ , it holds that  $\sum_{i=1}^{n} M^{SM}(\sigma_i; \sigma_{-i}) = \sigma_1 \leq 1$ . This implies that the platform can avoid providing explicit incentives and merely implement these rewards as matching probabilities. However, to do so, it must accommodate the possibility of not matching a user with any creators, corresponding to a probability of  $1 - \sigma_1$ . Furthermore, it does not support the top- $K$  ranking strategy.
A more comprehensive understanding about the construction of BRM can be obtained through the lens of congestion games. As pointed out by [16], every finite potential game is isomorphic to a congestion game. Furthermore, the definition of  $M$  as outlined in Eq.(5) can be interpreted as the utility that creator  $i$  acquires from the following congestion game:
1. The set of congestible elements are given by the continuum  $E = \mathcal{X} \times [0,1]$ , where each element  $(\pmb{x},t) \triangleq \pmb{e} \in E$  corresponds to a user  $\pmb{x}$  with satisfaction level  $t$ .
2. The  $n$  players are  $n$  content creators.
3. Each creator's pure action  $s_i \in S_i$  can be mapped to a subset of  $E$  in the following way: the action  $s_i$  determines the matching score  $\sigma(s_i; x)$  over each  $x \in \mathcal{X}$ , and then  $s_i$  is mapped to a subset  $\{(x, t) | x \in \mathcal{X}, t \in [0, \sigma(s_i; x)]\} \triangleq S_i \subseteq E$ .
4. For each element  $e$  and a vector of strategies  $(S_1, \dots, S_n)$ , the load of element  $e$  is defined as  $x_e = \# \{i : e \in S_i\}$ , i.e., the number of players who occupy  $e$ .
5. For each element  $e$ , there is a payoff function  $d_{e} : \mathbb{N} \to \mathbb{R}_{\geq 0}$  that only depends on the load of  $e$ .
6. For any joint strategy  $(S_{1},\dots ,S_{n})$ , the utility of player  $i$  is given by  $\sum_{e\in S_i}d_e(x_e)$ , i.e., the sum of reward he/she collects from all occupied elements. For each occupied element  $e$ , the reward is determined by its "congestion" level  $x_{e}$ , which is characterized by the payoff function  $d_{e}$ .
To better understand the constructed congestion game and the utility definition given in Eq.(5), we can consider each element in  $E$  (i.e., a user with a particular satisfaction level) as an atomic "resource". Each production strategy adopted by an individual creator can be thought of as occupying a subset of these resources. Given a fixed strategy profile, the load of  $e = (x,t)$  is determined by the number of creators who achieve a matching score exceeding  $t$  for user  $x$ , thereby linking the ranking of each creator in the matching score sequence for  $x$ . Consequently, we can reformulate the utility for a creator who is ranked in the  $i$ -th position for user  $x$  as
$$
\begin{array}{l} \sum_ {\boldsymbol {e} \in S _ {i}} d _ {\boldsymbol {e}} \left(x _ {\boldsymbol {e}}\right) = \sum_ {t \in [ 0, \sigma \left(\boldsymbol {s} _ {i}; \boldsymbol {x}\right) ]} d _ {t} \left(x _ {\boldsymbol {e}}\right) = \sum_ {k = i} ^ {n} \sum_ {t \in [ \sigma \left(\boldsymbol {s} _ {k + 1}; \boldsymbol {x}\right), \sigma \left(\boldsymbol {s} _ {k}, \boldsymbol {x}\right) ]} d _ {t} \left(x _ {\boldsymbol {e}}\right) \\ = \sum_ {k = i} ^ {n} \sum_ {t \in [ \sigma (\boldsymbol {s} _ {k + 1}; \boldsymbol {x}), \sigma (\boldsymbol {s} _ {k}; \boldsymbol {x}) ]} d _ {t} (k) \tag {10} \\ \triangleq \sum_ {k = i} ^ {n} \int_ {\sigma_ {k + 1}} ^ {\sigma_ {k}} f _ {k} (t) d t. \\ \end{array}
$$
Eq.(10) holds because for any resource  $e = (x, t)$  such that  $t \in [\sigma(s_{k+1}; x), \sigma(s_k; x])$ , the load of  $e$  is exactly given by  $k$ . As a result, by letting  $f_k(t) = d_t(k)$ , we recover the utility function defined in Eq.(5), where the value of function  $f_i(t)$  at  $t = t_0$  indicates the atomic reward for each creator if his/her strategy covers "resource"  $(x, t_0)$ , given that there are exactly  $i$  creators occupy  $(x, t_0)$ . This relationship also rationalizes why it is natural to assume that  $f_1 \geq \dots \geq f_n$ : as an increase in competition for the same resource from multiple creators should correspondingly reduce the return that can be accrued from that resource.
8.4 Proof of Theorem 1
Before showing the proof, we define the following notion of local maximizer:
Definition 4. We say  $s = (s_1, \dots, s_n)$  is a local maximizer of  $W(s)$  if for any  $i \in [n]$  and any  $s_i' \in S_i$ ,
$$
W \left(\boldsymbol {s} _ {1}, \dots , \boldsymbol {s} _ {i}, \dots , \boldsymbol {s} _ {n}\right) \geq W \left(\boldsymbol {s} _ {1}, \dots , \boldsymbol {s} _ {i} ^ {\prime}, \dots , \boldsymbol {s} _ {n}\right).
$$
The set of all the local maximizers of  $W$  is denoted by  $\operatorname{Loc}(W)$ .
According to the definition, for any join strategy profile  $s \in \text{Loc}(W)$ , no creator can unilaterally change his/her strategy to increase the value of function  $W$ . And clearly we have  $\arg \max_{s \in S} W(s) \in \text{Loc}(W)$ . To simplify notation we define  $\pi(\sigma_1, \dots, \sigma_n) = \sum_{i=1}^{n} M(\sigma_i; \sigma_{-i})$ . Now we are ready to present the proof of Theorem 1. To avoid complex notations, with a slight abuse of notation we use  $M(\sigma_1, \sigma_2, \dots, \sigma_n)$  to denote  $M(\sigma_1; \{\sigma_2, \dots, \sigma_n\})$  in the following proof.
Proof. We start by showing that any TvN game instance with  $M \in \mathcal{M}^3$  possesses a unique NE at  $s^* = (e_1, \dots, e_1)$ . It suffices to show that:
1. For any joint strategy profile  $(s_1, \dots, s_n)$  in which there are  $k < n$  creators occupy  $e_1$ , there exists a creator who can receive a strict utility gain if she changes her strategy to  $e_1$ .
2. At  $s^* = (e_1, \dots, e_1)$ , any player would suffer a utility loss when changing her strategy.
For the first claim, suppose there are  $k$  players in  $s$  who play  $e_1$  and let  $i$  be any player who does not play  $e_1$ . In addition, there are  $t \leq n - k$  players who play the same strategy as  $s_i$ . By the definition of  $M_3$ , we have
$$
\begin{array}{l} u _ {i} (\boldsymbol {s} _ {i}; \boldsymbol {s} _ {- i}) = 1 \cdot M (\underbrace {1 , \cdots , 1} _ {t}, \underbrace {0 , \cdots , 0} _ {n - t}) + (n + 1) \cdot M (\underbrace {0 , \cdots , 0} _ {n - k}, \underbrace {1 , \cdots , 1} _ {k}) \\ = 1 \cdot \frac {1}{t} \cdot \pi (\underbrace {1 , \cdots , 1} _ {t}, \underbrace {0 , \cdots , 0} _ {n - t}) + (n + 1) \cdot 0 \\ = \frac {1}{t} \cdot \pi (\underbrace {1 , \cdots , 1} _ {t}, \underbrace {0 , \cdots , 0} _ {n - t}). \tag {11} \\ \end{array}
$$
If player-  $i$  changes her strategy from  $\pmb{s}_i$  to  $s_i^{\prime} = e_1$ , the new utility would be
$$
\begin{array}{l} u _ {i} \left(\boldsymbol {s} _ {i} ^ {\prime}; \boldsymbol {s} _ {- i}\right) = (n + 1) \cdot M (\underbrace {1 , \cdots , 1} _ {k + 1}, \underbrace {0 , \cdots , 0} _ {n - k - 1}) + \sum_ {j \neq i} 1 \cdot M (0, \dots) \\ = (n + 1) \cdot \frac {1}{k + 1} \cdot \pi (\underbrace {1 , \cdots , 1} _ {k + 1}, \underbrace {0 , \cdots , 0} _ {n - k - 1}) + 0 \\ = \frac {n + 1}{k + 1} \cdot \pi \left(\underbrace {1 , \cdots , 1} _ {k + 1}, \underbrace {0 , \cdots , 0} _ {n - k - 1}\right), \tag {12} \\ \end{array}
$$
From Eq.(11) and Eq.(12),  $u_{i}(s_{i}'; s_{-i}) > u_{i}(s_{i}; s_{-i})$  holds if and only if
$$
\frac {1}{t} \cdot \pi (\underbrace {1 , \cdots , 1} _ {t}, \underbrace {0 , \cdots , 0} _ {n - t}) <   \frac {n + 1}{k + 1} \cdot \pi (\underbrace {1 , \cdots , 1} _ {k + 1}, \underbrace {0 , \cdots , 0} _ {n - k - 1}). \tag {13}
$$
And a sufficient condition for Eq.(13) to hold is
$$
m = 2 n > n - 1 + \max  _ {0 \leq k \leq n - 1} \left\{\frac {k + 1}{t} \cdot \frac {\pi (\overbrace {1 , \cdots , 1} ^ {t} , \overbrace {0 , \cdots , 0} ^ {n - t})}{\pi (\underbrace {1 , \cdots , 1} _ {k + 1} , \underbrace {0 , \cdots , 0} _ {n - k - 1})} \right\}. \tag {14}
$$
Denote  $\tilde{\pi}_k = \pi(\underbrace{1, \cdots, 1}_{k}, \underbrace{0, \cdots, 0}_{n-k})$ . By the monotonicity of  $\pi$ , we have  $\tilde{\pi}_n \geq \dots \geq \tilde{\pi}_1 = M(1, 0, \dots, 0) > 0$ . Therefore, the RHS of Eq.(14) is a finite number. Moreover, when  $t \leq k + 1$ , we have
$$
\frac {k + 1}{t} \cdot \frac {\tilde {\pi} _ {t}}{\tilde {\pi} _ {k + 1}} \leq \frac {k + 1}{t} \cdot \frac {\tilde {\pi} _ {k + 1}}{\tilde {\pi} _ {k + 1}} \leq \frac {n - 1 + 1}{1} = n,
$$
and when  $t > k + 1$ , based on the negative externality principle of merit-based rewarding mechanism we have
$$
\frac {k + 1}{t} \cdot \frac {\tilde {\pi} _ {t}}{\tilde {\pi} _ {k + 1}} = \frac {M (\overbrace {1 , \cdots , 1} ^ {t} , \overbrace {0 , \cdots , 0} ^ {n - t})}{M (\underbrace {1 , \cdots , 1} _ {k + 1} , \underbrace {0 , \cdots , 0} _ {n - k - 1})} \leq 1.
$$
Therefore, the RHS of Eq.(14) is strictly less than  $2n - 1$ .
For the second claim, we have
$$
u _ {i} (\pmb {s} _ {i} ^ {*}; \pmb {s} _ {- i} ^ {*}) = \frac {n + 1}{n} \tilde {\pi} _ {n},
$$
and if player-  $i$  changes her strategy from  $\pmb{s}_i^* = \pmb{e}_1$  to any  $\pmb {s}_i^\prime = \pmb {e}_j,j\neq 1$  , her new utility becomes
$$
u _ {i} (\pmb {s} _ {i} ^ {\prime}; \pmb {s} _ {- i} ^ {*}) = \tilde {\pi} _ {1} \leq \tilde {\pi} _ {n} <   \frac {n + 1}{n} \tilde {\pi} _ {n} = u _ {i} (\pmb {s} _ {i} ^ {*}; \pmb {s} _ {- i} ^ {*}).
$$
Therefore, we conclude that  $s^* = (e_1, \dots, e_1)$  is the unique NE of  $\mathcal{G}$ .
Next we estimate the welfare loss of  $s^*$  under any sequence  $\{r_i\}_{i=1}^K$ . First of all, note that for any  $s = (s_1, \dots, s_n) \in \text{Loc}(W)$  and any  $2 \leq k \leq n$ , if there exists  $i \neq j$  such that  $s_i = s_j = e_k$ , then there must be  $k' \in [n]$  such that  $e_{k'} \notin s_j$ . In this case,  $W$  strictly increases if  $s_j$  changes to  $e_{k'}$ . Therefore, for any  $2 \leq k \leq n$ , the number of elements in  $s$  that equal to  $e_k$  is either 0 or 1. Let the number of elements in  $s$  that equal to  $e_1$  be  $q$ . By definition,
$$
W (\boldsymbol {s}) = (n + 1) \sum_ {i = 1} ^ {\min  (K, q)} r _ {i} + (n - q) r _ {1}, \tag {15}
$$
$$
W (\boldsymbol {s} ^ {*}) = (n + 1) \sum_ {i = 1} ^ {K} r _ {i}.
$$
Since  $q$  maximizes the RHS of Eq.(15), we have  $1 \leq q \leq K$  and  $(n + 1)r_{q + 1} \leq r_1 \leq (n + 1)r_q$ . Therefore,
$$
\begin{array}{l} \frac {\operatorname* {m a x} _ {\boldsymbol {s} \in \mathcal {S}} W (\boldsymbol {s})}{W (\boldsymbol {s} ^ {*})} \geq \frac {\operatorname* {m i n} _ {\boldsymbol {s} \in L o c (W)} W (\boldsymbol {s})}{W (\boldsymbol {s} ^ {*})} \\ \geq \frac {(n + 1) \sum_ {i = 1} ^ {\min  (K , q)} r _ {i} + (n - q) r _ {1}}{(n + 1) \sum_ {i = 1} ^ {K} r _ {i}} \\ \geq \frac {(n + 1) \sum_ {i = 1} ^ {q} r _ {i} + (n - q) r _ {1}}{(n + 1) \sum_ {i = 1} ^ {q} r _ {i} + (K - q) r _ {1}} \\ = 1 + \frac {(n - K) r _ {1}}{(n + 1) \sum_ {i = 1} ^ {q} r _ {i} + (K - q) r _ {1}} \\ \geq 1 + \frac {(n - K) r _ {1}}{[ (n + 1) q + (K - q) ] r _ {1}} \\ \rightarrow 1 - \frac {1 + 1 / q}{1 + n q / K} + \frac {1}{q}, n \rightarrow \infty . \\ \end{array}
$$
Since  $1 \leq q \leq K$ , we conclude that  $\frac{\max_{s \in S} W(s)}{W(s^*)} > 1 - O\left(\frac{1}{n}\right) + \frac{1}{K}$  when  $n$  is sufficiently large. And therefore we conclude that
$$
\frac {W (\boldsymbol {s} ^ {*})}{\operatorname* {m a x} _ {\boldsymbol {s} \in \mathcal {S}} W (\boldsymbol {s})} \leq \frac {K}{K + 1} + O \left(\frac {1}{n}\right).
$$
8.5 Proof of Proposition 1
Proof. To prove that any  $M \in \mathrm{BRM}$  is merit-based, we need to verify the following by definition:
1.  $M(0; \sigma_{-i}) = \int_0^0 f_n(t) dt = 0, M(1; \{0, \dots, 0\}) = \int_0^1 f_1(t) dt > 0.$
2.  $M(\sigma_{i}; \sigma_{-i}) - M(\sigma_{j}; \sigma_{-j}) = \sum_{k = j}^{i - 1} \int_{\sigma_{k + 1}}^{\sigma_{k}} f_{k}(t) dt \geq 0$ .
3. for any  $\{\sigma_j\}_{j = 1}^n,\{\sigma_j'\}_{j = 1}^n$  such that  $\sigma_{-i}\preceq \sigma_{-i}'$ , we can transform  $\{\sigma_j\}_{j = 1}^n$  to  $\{\sigma_j'\}_{j = 1}^n$  by taking finite steps of the following operations: 1. increase a certain value of  $\sigma_{j},j\neq i$  to  $\tilde{\sigma}_j$  and it does not change the order of the current sequence; 2. increase a certain value of  $\sigma_{j},j\neq i$  to  $\tilde{\sigma}_j$ , and  $\sigma_{i}$ 's ranking position decreases after this change. We will show that after each operation the value of  $M(\sigma_i,\cdot)$  under the perturbed sequence does not increase.
Let the perturbed sequence be  $\tilde{\sigma}$ . For the first type of operation, if  $j < i$ , we have  $M(\sigma_i; \tilde{\sigma}_{-i}) = M(\sigma_i; \sigma_{-i})$ . If  $j > i$ , we have
$$
\begin{array}{l} M \left(\sigma_ {i}; \tilde {\sigma} _ {- i}\right) - M \left(\sigma_ {i}; \sigma_ {- i}\right) = \int_ {\tilde {\sigma} _ {j}} ^ {\sigma_ {j - 1}} f _ {j - 1} (t) d t + \int_ {\sigma_ {j + 1}} ^ {\tilde {\sigma} _ {j}} f _ {j} (t) d t - \int_ {\sigma_ {j}} ^ {\sigma_ {j - 1}} f _ {j - 1} (t) d t - \int_ {\sigma_ {j + 1}} ^ {\sigma_ {j}} f _ {j} (t) d t \\ = \int_ {\sigma_ {j}} ^ {\tilde {\sigma} _ {j}} \left(f _ {j} - f _ {j - 1}\right) (t) d t \leq 0. \\ \end{array}
$$
For the second type of operation, with out loss of generality let's assume  $\sigma_{i + 1}$  has increased to  $\tilde{\sigma}_{i + 1}$  such that  $\sigma_{i} \leq \tilde{\sigma}_{i + 1} \leq \sigma_{i - 1}$ . In this case we have
$$
\begin{array}{l} M \left(\sigma_ {i}; \tilde {\sigma} _ {- i}\right) - M \left(\sigma_ {i}; \sigma_ {- i}\right) = \int_ {\sigma_ {i + 2}} ^ {\sigma_ {i}} f _ {i + 1} (t) d t - \int_ {\sigma_ {i + 1}} ^ {\sigma_ {i}} f _ {i} (t) d t - \int_ {\sigma_ {i + 2}} ^ {\sigma_ {i + 1}} f _ {i + 1} (t) d t \\ = \int_ {\sigma_ {i + 1}} ^ {\sigma_ {i}} \left(f _ {i + 1} - f _ {i}\right) (t) d t \leq 0. \\ \end{array}
$$
Therefore,  $M$  is merit-based. On the other hand, there exist instances in BRM that are not monotone. For example, if we let  $f_{1}(t) = 1$  and  $f_{k}(t) = 0, \forall k \geq 2$ . Then we have
$$
M (1, 0, 0, \dots , 0) = \int_ {0} ^ {1} f _ {1} (t) d t > 0,
$$
$$
M (1, 1, 0, \dots , 0) = \int_ {0} ^ {0} f _ {1} (t) d t + \int_ {0} ^ {1} f _ {2} (t) d t = 0.
$$
As a result,  $\pi (1,0,0,\dots ,0) > 0 = \pi (1,1,0,\dots ,0)$ , which violates monotonicity.
8.6 Proof of Theorem 2
Proof. For the first claim, consider the potential function of the following form:
$$
P (\boldsymbol {s}) = \mathbb {E} _ {\boldsymbol {x} \in \mathcal {F}} \left[ \sum_ {i = 1} ^ {n} \int_ {0} ^ {\sigma_ {l (i)} (\boldsymbol {x})} f _ {i} (t) d t \right] - \sum_ {i = 1} ^ {n} c _ {i} (\boldsymbol {s} _ {i}),
$$
where  $\sigma_{i}(\pmb{x}) = \sigma(\pmb{s}_{i};\pmb{x})$  and  $\{l(i)\}_{i=1}^{n}$  is a permutation such that  $\sigma_{l(1)}(\pmb{x}) \geq \sigma_{l(2)}(\pmb{x}) \geq \dots \geq \sigma_{l(n)}(\pmb{x})$ .
By the definition of potential games, we need to verify that for any set of functions  $\{f_i\}$  and a strategy pair  $s_i, s_i' \in S_i$  for player- $i$ , it holds that
$$
u _ {i} \left(\boldsymbol {s} _ {i} ^ {\prime}, \boldsymbol {s} _ {- i}\right) - u _ {i} \left(\boldsymbol {s} _ {i}, \boldsymbol {s} _ {- i}\right) = P \left(\boldsymbol {s} _ {i} ^ {\prime}, \boldsymbol {s} _ {- i}\right) - P \left(\boldsymbol {s} _ {i}, \boldsymbol {s} _ {- i}\right). \tag {16}
$$
For any user  $\pmb{x} \in \mathcal{F}$ , let  $\sigma_{i} = \sigma(\pmb{s}_{i}; \pmb{x}_{j})$ ,  $\sigma_{i}' = \sigma(\pmb{s}_{i}'; \pmb{x})$ ,  $\forall i \in [n]$ . It suffices to show that
$$
M \left(\sigma_ {i}; \sigma_ {- i}\right) - M \left(\sigma_ {i} ^ {\prime}; \sigma_ {- i}\right) = \sum_ {i = 1} ^ {n} \int_ {0} ^ {\sigma_ {l (i)}} f _ {i} (t) d t - \sum_ {i = 1} ^ {n} \int_ {0} ^ {\sigma_ {l (i)} ^ {\prime}} f _ {i} (t) d t. \tag {17}
$$