SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Since the expectation of Eq.(17) over  $\pmb{x} \in \mathcal{F}$  yields Eq.(16), we focus on the verification of Eq.(17). With out loss of generality, we also assume  $\sigma_{1} \geq \dots \geq \sigma_{i'} \geq \dots \geq \sigma_{i} \geq \dots \geq \sigma_{n}$ . After player-  $i$  changes her strategy from  $s_i$  to  $s_i'$ , the relevance ranking increases from  $i$  to  $i'$ , i.e.,  $\sigma_{1} \geq \dots \geq \sigma_{i' - 1} \geq \sigma_{i'}' \geq \sigma_{i'} \geq \dots \geq \sigma_{n}$ .
$$
\text {L H S} (1 7) = \int_ {\sigma_ {i ^ {\prime}}} ^ {\sigma_ {i} ^ {\prime}} f _ {i ^ {\prime}} (t) d t + \sum_ {k = i ^ {\prime} + 1} ^ {n} \int_ {\sigma_ {k - 1}} ^ {\sigma_ {k}} f _ {k} (t) d t, \tag {18}
$$
$$
\begin{array}{l} \text {R H S} (1 7) = \sum_ {k = 1} ^ {n} \int_ {0} ^ {\sigma_ {k}} f _ {k} (t) d t - \left(\sum_ {k = 1} ^ {i ^ {\prime} - 1} \int_ {0} ^ {\sigma_ {k}} f _ {k} (t) d t + \int_ {0} ^ {\sigma_ {i} ^ {\prime}} f _ {i ^ {\prime}} (t) d t + \sum_ {k = i ^ {\prime} + 1} ^ {n} \int_ {0} ^ {\sigma_ {k - 1}} f _ {k} (t) d t\right) \\ = \int_ {0} ^ {\sigma_ {i ^ {\prime}}} f _ {i ^ {\prime}} (t) d t + \sum_ {k = i ^ {\prime} + 1} ^ {n} \int_ {0} ^ {\sigma_ {k}} f _ {k} (t) d t - \int_ {0} ^ {\sigma_ {i} ^ {\prime}} f _ {i ^ {\prime}} (t) d t - \sum_ {k = i ^ {\prime} + 1} ^ {n} \int_ {0} ^ {\sigma_ {k - 1}} f _ {k} (t) d t \\ = \int_ {\sigma_ {i ^ {\prime}}} ^ {\sigma_ {i} ^ {\prime}} f _ {i ^ {\prime}} (t) d t + \sum_ {k = i ^ {\prime} + 1} ^ {n} \int_ {\sigma_ {k - 1}} ^ {\sigma_ {k}} f _ {k} (t) d t. \\ \end{array}
$$
Hence, Eq.(17) holds for any  $j$  which completes the proof.
For the second claim, we can verify that when  $f_{i} = r_{i}, \forall i$
$$
\begin{array}{l} P (\pmb {s}) = \mathbb {E} _ {\pmb {x} \in \mathcal {F}} \left[ \sum_ {i = 1} ^ {n} \int_ {0} ^ {\sigma_ {l (i)} (\pmb {x})} f _ {i} (t) d t \right] - \sum_ {i = 1} ^ {n} c _ {i} (\pmb {s} _ {i}) \\ = \mathbb {E} _ {\boldsymbol {x} \in \mathcal {F}} \left[ \sum_ {i = 1} ^ {n} r _ {i} \sigma_ {l (i)} (\boldsymbol {x}) \right] - \sum_ {i = 1} ^ {n} c _ {i} (\boldsymbol {s} _ {i}) \\ = \mathbb {E} _ {\boldsymbol {x} \in \mathcal {F}} [ W (\boldsymbol {s}; \boldsymbol {x}) ] - \sum_ {i = 1} ^ {n} c _ {i} (\boldsymbol {s} _ {i}) \\ = W (s). \\ \end{array}
$$
8.7 Proof of Corollary 1
Proof. We show that any TvN game instance  $\mathcal{G}$  with  $M = M[r_1, \dots, r_K, 0, \dots, 0] \in \mathrm{BRCM}$  possesses a unique NE  $s^*$  which maximizes  $W(s)$ . From Theorem 2 we know that under  $M$ ,  $\mathcal{G}$  is a potential game and its potential function  $P$  is identical to its welfare function  $W$ . Therefore, any PNE of  $\mathcal{G}$  belongs to  $Loc(W)$ . Next we show that all elements in  $Loc(W)$  yield the same value of  $W$ , thus any PNE of  $\mathcal{G}$  maximizes social welfare  $W$ .
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 exist  $k' \in [n]$  such that  $e_{k'} \notin s_j$ . In this case,  $W$  strictly increases if  $s_j$  changes strategy 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, the welfare function writes
$$
W (\boldsymbol {s}) = (n + 1) \sum_ {i = 1} ^ {\min  (K, q)} r _ {i} + (n - q) r _ {1}. \tag {19}
$$
It is clear that the  $q$  that maximizes Eq.(19) satisfies  $1 \leq q \leq K$  and  $(n + 1)r_{q + 1} \leq r_1 \leq (n + 1)r_q$ , and all such  $q$  yields the same objective value of  $W$ . Therefore, we conclude that any PNE of  $\mathcal{G}$  attains the optimal social welfare  $W$ .
8.8 Content Creator Response Simulator
Algorithm 1 functions as follows: at each step, a random creator  $i$  selects a random improvement direction  $g_{i}$ . If creator  $i$  discovers that adjusting her strategy in this direction yields a higher utility,
Algorithm 1 (simStra) Simulate content creators' strategy evolving dynamic
Input: Time horizon  $T$ , learning rate  $\eta$ , utility function strategy set  $(u_{i}(\boldsymbol{s}),\mathcal{S}_{i})$  for each player, current mechanism  $M[\boldsymbol{f}]$  parameterized by  $\boldsymbol{f}$ .  
Initialization: Initial strategy profile  $\boldsymbol{s}^{(0)} = (\boldsymbol{s}_1^{(0)},\dots ,\boldsymbol{s}_n^{(0)})$ .  
for  $t = 0$  to  $T - 1$  do  
    Generate  $i\in [n]$  and  $\boldsymbol{g}_i\in \mathbb{S}^{d - 1}$  uniformly at random.  
    if  $u_{i}(\boldsymbol{s}_{i}^{(t)} + \eta \boldsymbol{g}_{i},\boldsymbol{s}_{-i}^{(t)})\geq u_{i}(\boldsymbol{s}^{(t)})$  then  
         $\boldsymbol{s}_{i}^{(t + \frac{1}{2})} = \boldsymbol{s}_{i}^{(t)} + \eta \boldsymbol{g}_{i}$ .  
    Find  $\boldsymbol{s}_{i}^{(t + 1)}$  as the projection of  $\boldsymbol{s}_{i}^{(t + \frac{1}{2})}$  in  $\mathcal{S}_i$ .  
else  
     $\boldsymbol{s}_{i}^{(t + 1)} = \boldsymbol{s}_{i}^{(t)}$   
Output:  $\boldsymbol{s}^{(T)}$ .
she updates her strategy along  $\pmb{g}_i$ ; otherwise, she retains her current strategy. This approach is designed to more closely mimic real-world scenarios where content creators may not have full access to their utility functions, but instead have to perceive them as black boxes. While they may aim to optimize their responses to the current incentive mechanism, identifying a new strategy that definitively increases their utilities can be challenging. Therefore, we model their strategy evolution as a trial-and-exploration process. We should note that the specifics of the simulator are not critical to our proposed solution: the optimizer can select any equilibrium-finding dynamic to replace our Algorithm 1, as long as it is believed to better represent creators' responses in reality.
8.9 Optimization Algorithm
Our proposed welfare optimization algorithm is presented in Algorithm 2, which is organized into  $L_{1}$  epochs. Each epoch starts with a random perturbation of the current mechanism  $M$  within the feasible polytope and conducts simulations of creators' responses for  $L_{2}$  steps using the simulator specified in Algorithm 1. Welfare is reassessed at the conclusion of each epoch, and the perturbation applied to  $M$  is adopted if it yields an increase in welfare.
Algorithm 2 Optimize  $W$  in BRCM
Input: Time horizon  $T = L_{1}L_{2}$  , learning rate  $\eta_1,\eta_2$ $(u_{i}(s),S_{i})$  for each creator.   
Initialization: Unit basis  $\{e_i\}_{i = 1}^n$  in  $\mathbb{R}^n$  , initial strategy profile  $s^{(0)} = (s_1^{(0)},\dots ,s_n^{(0)})$  , initial parameter  $\pmb {f}^{(0)} = (f_1^{(0)},\dots ,f_n^{(0)})\in \mathcal{F}$  and mechanism  $M[\pmb{f}^{(0)}]$    
for  $t = 0$  to  $L_{1} - 1$  do Generate  $i\in [n]$  and  $\pmb {g}_i\in \{-\pmb {e}_i,\pmb {e}_i\}$  uniformly at random. Update  $\pmb {f}_i^{(t + \frac{1}{2})}$  as the projection of  $\pmb {f}_i^{(t)} + \eta_1\pmb {g}_i$  on  $\mathcal{F}$  Simulate  $s^{(t + 1)} = \mathrm{simStra}(s^{(t)};L_2,\eta_2,\{\pmb {u}_i,\pmb {S}_i\}_{i = 1}^n,M[\pmb {f}^{(t + \frac{1}{2})}])$  // Implemented by Algo. 1 if  $W(s^{(t + 1)}) > W(s^{(t)})$  then  $\pmb {f}^{(t + 1)} = \pmb {f}^{(t + \frac{1}{2})}$  else  $\pmb {f}^{(t + 1)} = \pmb {f}^{(t)}$
8.10 Additional Experiments
- Experiments using MovieLens-1m We use deep matrix factorization [8] to train user and movie embeddings predicting movie ratings from 1 to 5 and use them to construct the user population  $\mathcal{X}$  and creators' strategy set  $\{S_i\}$ . The dataset contains 6040 users and 3883 movies in total, and the embedding dimension is set to  $d = 32$ . To validate the quality of the trained representation, we first performed a 5-fold cross-validation and obtain an averaged RMSE  $= 0.739$  on the test sets, then train the user/item embeddings with the complete dataset.
To construct a more challenging environment for creators, we avoid using movies that are excessively popular and highly rated or users who are overly active and give high ratings to most movies. This
(b) User Util. Distr. on  $\mathcal{G}_1$
(c) Welfare on  $\mathcal{G}_2$
Figure 3: Social welfare curve and average user utilities under two different environments. Error bars represent 0.2 standard deviation range, and they are generated from 10 independent runs. Game instances are generated from MovieLens-1m dataset.
(d) User Util. Distrib. on  $\mathcal{G}_2$
ensures that the strategy of "producing popular content for the majority of active users" does not become a dominant strategy under any rewarding mechanism. Thus, we filtered out users and movies who have more than 500 predicted ratings higher than 4. After the filtering, we have a user population with size  $|\mathcal{X}| = 2550$  and movie set of size 1783. The user distribution are set to uniform distribution over  $\mathcal{X}$ , and the remaining movies become the action set  $\{S_i\}$  for each creator-  $i$  of  $n = 10$  creators. To normalize the relevance score to [0, 1], we set  $\sigma(\boldsymbol{s}, \boldsymbol{x}) = \mathrm{clip}(\langle \boldsymbol{s}, \boldsymbol{x} \rangle / 2.5 - 1, 0, 1)$ .  $\{r_i\}_{i=1}^n$  is set to  $\left\{\frac{1}{\log_2(2)}, \frac{1}{\log_2(3)}, \frac{1}{\log_2(4)}, \frac{1}{\log_2(5)}, \frac{1}{\log_2(6)}, 0, \dots, 0\right\}$ . We also consider two types of game instances, namely  $\mathcal{G}_1$  and  $\mathcal{G}_2$ , as we elaborated on in Section 6.1. Specifically, in  $\mathcal{G}_1$  creators' initial strategies are set to the most popular movie among all users (i.e., the movie that enjoys the highest average rating among  $\mathcal{X}$ ) and the cost functions are set to be zero. In  $\mathcal{G}_2$ , we set creators' cost functions to  $c_i = 10\|s_i - \bar{s}_i\|_2^2$  and let creator  $i$  start at the cost center  $\bar{s}_i$ .  $\{\bar{s}_i\}_{i=1}^n$  are sampled at random from all the movies.
For each baseline in Section 6.2, we plot the welfare curve over  $T = 500$  steps using Algorithm 1 and also the average user utility distribution at the end of simulations. The parameters of Algorithm 2 are set to  $L_{1} = 100$ ,  $L_{2} = 5$ ,  $\eta_{1} = 0.5$ ,  $\eta_{2} = 0.1$ ,  $f^{(0)} = (1,1,1,1,1,0,\dots ,0)$ . The results are presented in Figure 3.
The new results obtained reinforce the findings presented in Section 6. In both the  $\mathcal{G}_1$  and  $\mathcal{G}_2$  environments, the BRCM family continues to outperform  $\mathcal{M}^3$  overall. Specifically,  $\mathrm{BRCM}_{opt}$ ,  $\mathrm{BRCM}_1$ , and  $\mathrm{BRCM}^*$  consistently demonstrate strong performance in social welfare, highlighting the robustness of BRCM across different environments. When creators initially adopt the most popular strategy in  $\mathcal{G}_1$ ,  $M^3(0)$  does not yield any improvement since no creator would change their strategy in such a situation under  $M^3(0)$ . In the case of  $\mathcal{G}_2$ , the advantage of BRCM over  $\mathcal{M}^3$  diminishes slightly, which aligns with our observations from the synthetic dataset. The main reason is that the cost function discourages creators to deviate from their default strategies. Additionally, Figure 3b provides further evidence that the welfare gain achieved by BRCM arises from enhanced utility for a wider range of users.