SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \left\| \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i}\right) - \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i} ^ {\prime}\right) \right\| _ {2} \leq \sqrt {\left| \mathcal {A} _ {i} \right|} \| \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i}\right) - \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i} ^ {\prime}\right) \| _ {\infty} (13) \\ \leq \sqrt {\left| \mathcal {A} _ {i} \right|} \left\| \sum_ {\boldsymbol {a} _ {- i} \in \mathcal {A} _ {- i}} u _ {i} (\cdot , \boldsymbol {a} _ {- i}) \left(\prod_ {i ^ {\prime} \neq i} \boldsymbol {x} _ {i ^ {\prime}} [ a _ {i ^ {\prime}} ] - \prod_ {i ^ {\prime} \neq i} \boldsymbol {x} _ {i ^ {\prime}} [ a _ {i ^ {\prime}} ]\right) \right\| _ {1} (14) \\ \leq \sqrt {\left| \mathcal {A} _ {i} \right|} \left\| \sum_ {\boldsymbol {a} _ {- i} \in \mathcal {A} _ {- i}} \left(\prod_ {i ^ {\prime} \neq i} \boldsymbol {x} _ {i ^ {\prime}} \left[ a _ {i ^ {\prime}} \right] - \prod_ {i ^ {\prime} \neq i} \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime} \left[ a _ {i ^ {\prime}} \right]\right) \right\| _ {1} (15) \\ \leq \sqrt {\left| \mathcal {A} _ {i} \right|} \sum_ {i ^ {\prime} \neq i} \| \boldsymbol {x} _ {i ^ {\prime}} - \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime} \| _ {1} \leq \max  _ {1 \leq i \leq n} \left| \mathcal {A} _ {i} \right| \sum_ {i ^ {\prime} \neq i} \| \boldsymbol {x} _ {i ^ {\prime}} - \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime} \| _ {2}, (16) \\ \end{array}
$$
where (13) uses the equivalence between the  $\ell_2$  and the  $\ell_{\infty}$  norm; (14) follows from the definition of the (expected) utility:  $u_{i}(a_{i},\pmb{x}_{-i})\coloneqq \mathbb{E}_{\pmb{a}_{-i}\sim \pmb{x}_{-i}}[u_{i}(\pmb {a})] = \sum_{\pmb{a}_{-i}\in \mathcal{A}_{-i}}u_{i}(\pmb {a})\prod_{i^{\prime}\neq i}\pmb{x}_{i^{\prime}}[a_{i^{\prime}}]$ , for any  $a_i\in \mathcal{A}_i$ ; (15) uses the triangle inequality along with the assumption that  $|u_{i}(\pmb {a})|\leq 1$ ; and (16) follows from the well-known fact that the total variation distance between two product distributions can be upper bounded by the sum of the total variation distance of each individual component [54], as well as the equivalence between the  $\ell_1$  and the  $\ell_2$  norm. As a result, continuing from (16), we have
$$
\begin{array}{l} \| F (\boldsymbol {x}) - F \left(\boldsymbol {x} ^ {\prime}\right) \| _ {2} ^ {2} = \sum_ {i = 1} ^ {n} \| \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i}\right) - \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i} ^ {\prime}\right) \| _ {2} ^ {2} \leq \left(\max  _ {1 \leq i \leq n} | \mathcal {A} _ {i} |\right) ^ {2} \sum_ {i = 1} ^ {n} \left(\sum_ {i ^ {\prime} \neq i} \| \boldsymbol {x} _ {i ^ {\prime}} - \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime} \| _ {2}\right) ^ {2} \\ \leq n ^ {2} \left(\max  _ {1 \leq i \leq n} | \mathcal {A} _ {i} |\right) ^ {2} \| \boldsymbol {x} - \boldsymbol {x} ^ {\prime} \| _ {2} ^ {2}, \\ \end{array}
$$
where the last inequality used that, by Jensen's inequality,  $\left(\sum_{i' \neq i} \| \boldsymbol{x}_{i'} - \boldsymbol{x}_{i'}' \|_2\right)^2 \leq (n - 1) \sum_{i' \neq i} \| \boldsymbol{x}_{i'} - \boldsymbol{x}_{i'}' \|_2^2 \leq n \| \boldsymbol{x} - \boldsymbol{x}' \|_2^2$ . This concludes the proof.
Graphical games As a byproduct of the proof above, we next point out an important refinement of Lemma A.6 concerning graphical games. In particular, here we assume that the utility of each player  $i \in [[n]]$  only depends on the actions of players belonging to its neighborhood  $\mathcal{N}_i \subseteq [[n]] \setminus \{i\}$ .
We further assume that  $|\mathcal{N}_i| \leq \Delta$  for any player  $i \in [[n]]$ , where  $\Delta \in \mathbb{N}$  will be referred to as the degree of the graphical game. To conclude the definition, we also posit that each player  $i \in [[n]]$  can only affect the utilities of at most  $\Delta$  other players:  $|i' \in [[n]] : i \in \mathcal{N}_{i'}| \leq \Delta$ .
Lemma A.7 (Lipschitz constant in graphical games). For any  $n$ -player graphical game with degree  $\Delta \in \mathbb{N}$  and utilities bounded in  $[-1, 1]$ , the Lipschitz constant  $L_{n}$  of the game operator satisfies  $L_{n} \leq \Delta \max_{1 \leq i \leq n} |\mathcal{A}_{i}|$ .
$$
\begin{array}{l} \left\| \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i}\right) - \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i} ^ {\prime}\right) \right\| _ {2} \leq \sqrt {\left| \mathcal {A} _ {i} \right|} \left\| \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i}\right) - \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i} ^ {\prime}\right) \right\| _ {\infty} \\ \leq \sqrt {\left| \mathcal {A} _ {i} \right|} \left\| \sum_ {\prod_ {i ^ {\prime} \in \mathcal {N} _ {i}} \mathcal {A} _ {i ^ {\prime}}} \left(\prod_ {i ^ {\prime} \in \mathcal {N} _ {i}} \boldsymbol {x} _ {i ^ {\prime}} [ a _ {i ^ {\prime}} ] - \prod_ {i ^ {\prime} \in \mathcal {N} _ {i}} \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime} [ a _ {i ^ {\prime}} ]\right) \right\| _ {1} \\ \leq \sqrt {\left| \mathcal {A} _ {i} \right|} \sum_ {i ^ {\prime} \in \mathcal {N} _ {i}} \| \boldsymbol {x} _ {i ^ {\prime}} - \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime} \| _ {1} \leq \max  _ {1 \leq i \leq n} \left| \mathcal {A} _ {i} \right| \sum_ {i ^ {\prime} \in \mathcal {N} _ {i}} \| \boldsymbol {x} _ {i ^ {\prime}} - \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime} \| _ {2}, \\ \end{array}
$$
where the derivation above is similar to that in Lemma A.6. As a result,
$$
\begin{array}{l} \| F (\boldsymbol {x}) - F \left(\boldsymbol {x} ^ {\prime}\right) \| _ {2} ^ {2} = \sum_ {i = 1} ^ {n} \| \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i}\right) - \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i} ^ {\prime}\right) \| _ {2} ^ {2} \\ \leq \left(\max  _ {1 \leq i \leq n} | \mathcal {A} _ {i} |\right) ^ {2} \sum_ {i = 1} ^ {n} \left(\sum_ {i ^ {\prime} \in \mathcal {N} _ {i}} \| \boldsymbol {x} _ {i ^ {\prime}} - \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime} \| _ {2}\right) ^ {2} \\ \leq \Delta \left(\max _ {1 \leq i \leq n} | \mathcal {A} _ {i} |\right) ^ {2} \sum_ {i = 1} ^ {n} \sum_ {i ^ {\prime} \in \mathcal {N} _ {i}} \| \boldsymbol {x} _ {i ^ {\prime}} - \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime} \| _ {2} ^ {2} \\ = \Delta \left(\max _ {1 \leq i \leq n} | \mathcal {A} _ {i} |\right) ^ {2} \sum_ {i = 1} ^ {n} \| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {i} ^ {\prime} \| _ {2} ^ {2} | i ^ {\prime} \in [   [ n ]   ]: i \in \mathcal {N} _ {i ^ {\prime}} | \\ \leq \Delta^ {2} \left(\max  _ {1 \leq i \leq n} | \mathcal {A} _ {i} |\right) ^ {2} \| \boldsymbol {x} - \boldsymbol {x} ^ {\prime} \| _ {2} ^ {2}. \\ \end{array}
$$
Games with vanishing sensitivity We next focus on a different subclass of normal-form games; namely, games with small sensitivity (per Definition 3.5). Taking a step back, in graphical games every player can only be impacted by (and have an impact to) a small number of other players. Instead, here we consider games where a player's utility can be impacted by all other players, but only by a small amount.
Lemma A.8 (Lipschitz constant in games with vanishing sensitivity). For any  $n$ -player normal-form game with sensitivity  $\epsilon_{n} \in \mathbb{R}_{>0}$ , the Lipschitz constant  $L_{n}$  of the game operator satisfies  $L_{n} \leq \epsilon_{n} n \max_{1 \leq i \leq n} |\mathcal{A}_{i}|$ .
Proof. Let  $i \in \llbracket n\rrbracket$ . For  $x_{1}, x_{1}^{\prime} \in \Delta(\mathcal{A}_{1})$  and  $a_{-1} \in \mathcal{A}_{-1}$  (restricting on Player 1 here is without any loss, and only made for the sake of simplicity in the notation), it follows that
$$
\begin{array}{l} u _ {i} \left(\boldsymbol {x} _ {1}, \boldsymbol {a} _ {- 1}\right) - u _ {i} \left(\boldsymbol {x} _ {1} ^ {\prime}, \boldsymbol {a} _ {- 1}\right) = \sum_ {a _ {1} \in \mathcal {A} _ {1}} \boldsymbol {x} _ {1} \left[ a _ {1} \right] u _ {i} \left(a _ {1}, \boldsymbol {a} _ {- 1}\right) - \sum_ {a _ {1} \in \mathcal {A} _ {1}} \boldsymbol {x} _ {1} ^ {\prime} \left[ a _ {1} \right] u _ {i} \left(a _ {1}, \boldsymbol {a} _ {- 1}\right) \\ = \sum_ {a _ {1} \in \mathcal {A} _ {1} \backslash \{a _ {1} ^ {\prime} \}} \left(\boldsymbol {x} _ {1} \left[ a _ {1} \right] - \boldsymbol {x} _ {1} ^ {\prime} \left[ a _ {1} \right]\right) u _ {i} \left(a _ {1}, \cdot\right) + \left(\boldsymbol {x} _ {1} \left[ a _ {1} ^ {\prime} \right] - \boldsymbol {x} _ {1} ^ {\prime} \left[ a _ {1} ^ {\prime} \right]\right) u _ {i} \left(a _ {1} ^ {\prime}, \cdot\right) \\ = \sum_ {a _ {1} \in \mathcal {A} _ {1} \backslash \left\{a _ {1} ^ {\prime} \right\}} \left(\boldsymbol {x} _ {1} \left[ a _ {1} \right] - \boldsymbol {x} _ {1} ^ {\prime} \left[ a _ {1} \right]\right) \left(u _ {i} \left(a _ {1}, \boldsymbol {a} _ {- 1}\right) - u _ {i} \left(a _ {1} ^ {\prime}, \boldsymbol {a} _ {- 1}\right)\right), \tag {17} \\ \end{array}
$$
for some  $a_1' \in \mathcal{A}_1$ , where we used that  $(\pmb{x}_1[a_1'] - \pmb{x}_1'[a_1']) = \sum_{a_1 \in \mathcal{A}_1 \setminus \{a_1'\}} (\pmb{x}_1'[a_1] - \pmb{x}_1[a_1])$  since  $\pmb{x}_1, \pmb{x}_1' \in \Delta(\mathcal{A}_1)$ . Continuing from (17), we have
$$
\begin{array}{l} |u_{i}(\boldsymbol{x}_{1},\boldsymbol{a}_{-1}) - u_{i}(\boldsymbol{x}_{1}^{\prime},\boldsymbol{a}_{-1})|\leq \sum_{a_{1}\in \mathcal{A}_{1}\setminus \{a_{1}^{\prime}\}}|\boldsymbol{x}_{1}[a_{1}] - \boldsymbol{x}_{1}^{\prime}[a_{1} ]||u_{i}(a_{1},\boldsymbol{a}_{-1}) - u_{i}(a_{1}^{\prime},\boldsymbol{a}_{-1})| \\ \leq \epsilon_ {n} \sum_ {a _ {1} \in \mathcal {A} _ {1} \backslash \{a _ {1} ^ {\prime} \}} | \boldsymbol {x} _ {1} [ a _ {1} ] - \boldsymbol {x} _ {1} ^ {\prime} [ a _ {1} ] | \leq \epsilon_ {n} \| \boldsymbol {x} _ {1} - \boldsymbol {x} _ {1} ^ {\prime} \| _ {1}, \\ \end{array}
$$
where  $\epsilon_{n}$  is the sensitivity of the game. Similar reasoning yields that  $|u_{i}(\pmb {x}) - u_{i}(\pmb{x}_{1}^{\prime},\pmb{x}_{-1})|\leq$ $\epsilon_{n}\| \pmb{x}_{1} - \pmb{x}_{1}^{\prime}\|_{1}$ , for any  $\pmb{x}_{-1}\in \prod_{i = 2}^{n}\Delta (\mathcal{A}_{i})$ . As a result, we have
$$
\begin{array}{l} \left\| \boldsymbol {u} _ {1} (\boldsymbol {x} _ {- 1}) - \boldsymbol {u} _ {1} \left(\boldsymbol {x} _ {- 1} ^ {\prime}\right) \right\| _ {\infty} \leq \left| u _ {1} \left(\cdot , \boldsymbol {x} _ {2}, \boldsymbol {x} _ {3}, \dots , \boldsymbol {x} _ {n}\right) - u _ {1} \left(\cdot , \boldsymbol {x} _ {2} ^ {\prime}, \boldsymbol {x} _ {3}, \dots , \boldsymbol {x} _ {n}\right) \right| \\ + \left| u _ {1} \left(\cdot , \boldsymbol {x} _ {2} ^ {\prime}, \boldsymbol {x} _ {3}, \dots , \boldsymbol {x} _ {n}\right) - u _ {1} \left(\cdot , \boldsymbol {x} _ {2} ^ {\prime}, \boldsymbol {x} _ {3} ^ {\prime}, \dots , \boldsymbol {x} _ {n}\right) \right| \\ + \dots \\ + \left| u _ {1} \left(\cdot , \boldsymbol {x} _ {2} ^ {\prime}, \boldsymbol {x} _ {3} ^ {\prime}, \dots , \boldsymbol {x} _ {n - 1} ^ {\prime} \boldsymbol {x} _ {n}\right) - u _ {1} \left(\cdot , \boldsymbol {x} _ {2} ^ {\prime}, \boldsymbol {x} _ {3} ^ {\prime}, \dots , \boldsymbol {x} _ {n} ^ {\prime}\right) \right| \\ \leq \epsilon_ {n} \sum_ {i \neq 1} \| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {i} ^ {\prime} \| _ {1}. \\ \end{array}
$$
By symmetry, we have shown that  $\| \pmb{u}_i(\pmb{x}_{-i}) - \pmb{u}_i(\pmb{x}_{-i}')\|_{\infty} \leq \epsilon_n \sum_{i' \neq i} \| \pmb{x}_{i'} - \pmb{x}_{i'}'\|_1$ , and the rest of the argument is identical to that of Lemma A.6.
Polymatrix games A careful examination of the proof of Lemma A.8 reveals that its conclusion in fact applies under a more relaxed condition compared to what imposed by Definition 3.5; namely, we can define
$$
\epsilon := \max  _ {1 \leq i \leq n} \max  _ {\boldsymbol {a} \in \mathcal {A}} \max  _ {i ^ {\prime} \neq i} \max  _ {a _ {i ^ {\prime}} ^ {\prime} \in \mathcal {A} _ {i ^ {\prime}}} | u _ {i} (\boldsymbol {a}) - u _ {i} \left(a _ {i ^ {\prime}} ^ {\prime}, \boldsymbol {a} _ {- i ^ {\prime}}\right) |. \tag {18}
$$
In words, when considering the utility of a player  $i \in [[n]]$ , we only bound deviations by players besides  $i$ . It is easy to see that Lemma A.8 in fact applies even if  $\epsilon \in \mathbb{R}_{>0}$  is defined as in (18). This observation enables capturing other interesting classes of games under the premise that  $L_{n} = O_{n}(1)$ , such as polymatrix games. Specifically, a polymatrix game is defined with respect to an underlying directed graph  $G = ([n]], E)$ , so that each node of  $G$  is uniquely associated with the corresponding player. For every edge  $(i, i') \in E$  there is a matrix  $\mathbf{A}_{i,i'} \in \mathbb{R}^{\mathcal{A}_i \times \mathcal{A}_{i'}}$  so that the utility of Player  $i$  is defined as
$$
u _ {i} (\boldsymbol {x}) := \frac {1}{n} \sum_ {i ^ {\prime} \in \mathcal {N} _ {i}} \boldsymbol {x} _ {i} ^ {\top} \mathbf {A} _ {i, i ^ {\prime}} \boldsymbol {x} _ {i ^ {\prime}}, \tag {19}
$$
where  $\mathcal{N}_i\coloneqq \{i'\in [[n]]:(i,i')\in E\}$ . Unlike the class of graphical games we saw earlier, here we do not restrict the size of the neighborhoods. For this reason, we have normalized each player's utility by a  $1 / n$  factor in (19), for otherwise the utilities are not guaranteed to be bounded (independent of the number of players  $n$ ). It is then easy to see that polymatrix games are subject to Lemma A.8 under  $\epsilon$  defined in (18), which here satisfies  $\epsilon = O(1 / n)$ . Below, we provide a simpler and sharper argument compared to Lemma A.8.
Lemma A.9. For any  $n$ -player polymatrix game, the Lipschitz constant  $L_{n}$  of the game operator satisfies  $L_{n} \leq \max_{(i,i^{\prime}) \in E} \| \mathbf{A}_{i,i^{\prime}} \|_{2}$ , where  $\| \cdot \|_{2}$  here denotes the spectral norm.
Proof. By definition of the utility functions in (19), we have that for any player  $i \in [[n]]$  and  $\pmb{x}, \pmb{x}' \in \prod_{i=1}^{n} \Delta(\mathcal{A}_i)$ ,
$$
\begin{array}{l} \left\| \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i}\right) - \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i} ^ {\prime}\right) \right\| _ {2} \leq \frac {1}{n} \left\| \sum_ {i ^ {\prime} \in \mathcal {N} _ {i}} \mathbf {A} _ {i, i ^ {\prime}} \left(\boldsymbol {x} _ {i ^ {\prime}} - \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime}\right) \right\| _ {2} \\ \leq \frac {1}{n} \sum_ {i ^ {\prime} \in \mathcal {N} _ {i}} \| \mathbf {A} _ {i, i ^ {\prime}} \left(\boldsymbol {x} _ {i ^ {\prime}} - \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime}\right) \| _ {2} \\ \leq \frac {1}{n} \sum_ {i ^ {\prime} \in \mathcal {N} _ {i}} \| \mathbf {A} _ {i, i ^ {\prime}} \| _ {2} \| \boldsymbol {x} _ {i ^ {\prime}} - \boldsymbol {x} _ {i ^ {\prime}} ^ {\prime} \| _ {2} \\ \leq \frac {\max _ {(i , i ^ {\prime}) \in E} \| \mathbf {A} _ {i , i ^ {\prime}} \| _ {2}}{n} \sum_ {i ^ {\prime} \in \mathcal {N} _ {i}} \| \pmb {x} _ {i ^ {\prime}} - \pmb {x} _ {i ^ {\prime}} ^ {\prime} \| _ {2}, \\ \end{array}
$$
A.2 Games with vanishing sensitivity
We next establish an important implication of our framework concerning the class of games with vanishing strategic sensitivity (per Definition 3.5); the statement of the theorem is recalled below.
Proof. Let  $i \in \mathbb{[}n\mathbb{]}$ . Following the proof of Lemma A.8, the definition of sensitivity implies that  $|u_i(\pmb{x}_i', \pmb{x}_{-i}) - u_i(\pmb{x})| \leq \epsilon_n \| \pmb{x}_i' - \pmb{x}_i \|_1 \leq 2\epsilon_n$ , for any  $\pmb{x} \in \prod_{i=1}^n \Delta(\mathcal{A}_i)$  and  $\pmb{x}_i' \in \Delta(\mathcal{A}_i)$ . The proof now follows that of Theorem A.2. In particular, we can lower bound the sum of the players' regrets as follows.
$$
\begin{array}{l} \sum_ {i = 1} ^ {n} \operatorname {R e g} _ {i} ^ {(T)} = \sum_ {i = 1} ^ {n} \left(\max  _ {\boldsymbol {x} _ {i} ^ {\star} \in \Delta (\mathcal {A} _ {i})} \left\{\sum_ {t = 1} ^ {T} u _ {i} (\boldsymbol {x} _ {i} ^ {\star}, \boldsymbol {x} _ {- i} ^ {(t)}) \right\} - \sum_ {t = 1} ^ {T} u _ {i} (\boldsymbol {x} ^ {(t)})\right) \\ \geq - \sum_ {i = 1} ^ {n} \sum_ {t = 1} ^ {T} | u _ {i} (\boldsymbol {x} _ {i} ^ {\prime}, \boldsymbol {x} _ {- i} ^ {(t)}) - u _ {i} (\boldsymbol {x} ^ {(t)}) | \geq - 2 T n \epsilon_ {n}. \\ \end{array}
$$
As a result, following the proof of Theorem A.2, we conclude that for learning rate  $\eta_{n} := \frac{1}{4L_{n}}$  there exists a time  $t^{\star} \in [[T]]$  such that
$$
\sum_ {i = 1} ^ {n} \left(\mathrm {B R G A P} _ {i} (\pmb {x} ^ {(t ^ {\star})})\right) ^ {2} \leq 4 \left(\frac {\max _ {1 \leq i \leq n} D _ {\mathcal {X} _ {i}} ^ {2}}{\eta_ {n} ^ {2}} + \max _ {1 \leq i \leq n} B _ {i} ^ {2}\right) \left(\frac {2 D _ {\mathcal {X}} ^ {2}}{T} + 4 \eta_ {n} n \epsilon_ {n}\right),
$$
Thus, setting
$$
T := \frac {D _ {\mathcal {X}} ^ {2}}{2 \eta_ {n} n \epsilon_ {n}} = \frac {2 D _ {\mathcal {X}} ^ {2} L _ {n}}{n \epsilon_ {n}} \leq 2 D _ {\mathcal {X}} ^ {2} \left(\max _ {1 \leq i \leq n} | \mathcal {A} _ {i} |\right) = O _ {n} (n),
$$
by Lemma A.8, yields that
$$
\sum_ {i = 1} ^ {n} \left(\mathrm {B R G A P} _ {i} \left(\boldsymbol {x} ^ {(t ^ {*})}\right)\right) ^ {2} \leq O _ {n} \left(L _ {n} n \epsilon_ {n}\right) = O _ {n} \left(n ^ {2} \epsilon_ {n} ^ {2}\right),
$$
where the asymptotic notation above applies in the regime  $L_{n}\geq \Omega_{n}(1)$ . The statement thus follows.
We note that the conclusion of Theorem 3.6 also applies under the weaker notion of sensitivity defined in (18), if we additionally assume that there exists  $\pmb{x}^{\star} \in \prod_{i=1}^{n} \Delta(\mathcal{A}_i)$  such that for any  $\pmb{x} \in \prod_{i=1}^{n} \Delta(\mathcal{A}_i)$ ,
$$
\sum_ {i = 1} ^ {n} u _ {i} \left(\boldsymbol {x} _ {i} ^ {\star}, \boldsymbol {x} _ {- i}\right) - \sum_ {i = 1} ^ {n} u _ {i} (\boldsymbol {x}) \geq - n \epsilon_ {n}. \tag {20}
$$
Condition (20) can be met even in games with large sensitivity (Observation 3.4), which is why we chose to state Theorem 3.6 under a stronger but more interpretable condition.
Next, we point out two interesting extensions of our approach based on developments following the original smoothness framework of Roughgarden [90].
A.3 Refined smoothness
Throughout this paper we have relied on the original smoothness framework [90] to derive much of our results. Nevertheless, it is worth noting that there are certain extensions documented in the literature that can make our predictions sharper. To be more precise, focusing on normal-form games, Nadav and Roughgarden [75] noted that the original smoothness framework is in fact able to provide efficiency bounds for a set of equilibria even more permissive than CCE, which they refer to as average CCE with respect to a welfare-maximizing strategy  $\pmb{x}^{\star} \in \prod_{i=1}^{n} \Delta(\mathcal{A}_i)$  (ACCE*). In particular, the latter relaxation only requires a guarantee for the average regret over the players,  $\frac{1}{n} \sum_{i=1}^{n} \operatorname{Reg}_i^{(T)}(\pmb{x}_i^{\star})$ , while CCE instead requires a guarantee for the maximum regret  $\max_{1 \leq i \leq n} \operatorname{Reg}_i^{(T)}$ . Based on this observation, Nadav and Roughgarden [75] developed a primal-dual framework in order to provide refined guarantees (beyond the original smoothness framework) for CCE*, which boils down to solving the following linear program.
$$
\begin{array}{l} \text {m a x i m i z e} \\ \text {s u b j e c t t o} \end{array} \begin{array}{l} \rho \\ \sum_ {i = 1} ^ {n} z _ {i} \left(u _ {i} \left(a _ {i} ^ {\star}, \boldsymbol {a} _ {- i}\right) - u _ {i} (\boldsymbol {a})\right) \geq \rho \mathrm {S W} \left(\boldsymbol {a} ^ {\star}\right) - \mathrm {S W} (\boldsymbol {a}), \boldsymbol {a} \in \prod_ {i = 1} ^ {n} \mathcal {A} _ {i}, \end{array} \tag {21}
$$
Here, it is assumed that the underlying game is in normal form, with  $\prod_{i=1}^{n}\mathcal{A}_{i}$  being the set of joint action profiles, and  $a^{\star} \in \prod_{i=1}^{n}\mathcal{A}_{i}$  a welfare-maximizing joint action. (The formulation of Nadav and Roughgarden [75] is based on cost-minimization games, but of course their LP can be cast directly for payoff-maximization games in the form of (21).) The above LP is a simple transformation of the fractional-linear program corresponding to optimizing  $\rho \coloneqq \frac{\lambda}{1 + \mu}$  subject to the smoothness constraints of Definition 2.1, but with the additional flexibility of optimizing over a vector  $z \in \mathbb{R}_{\geq 0}^{n}$ . In particular, when restricting  $z_{1} = z_{2} = \dots = z_{n}$ , this exactly recovers the LP for computing the robust price of anarchy—given in (24). Nevertheless, Nadav and Roughgarden [75] pointed out that the additional flexibility of (21) can have an arbitrarily large impact on the predicted efficiency [75, Remark 1].
The sharper definition of smoothness introduced by Nadav and Roughgarden [75] can be leveraged in our framework as follows. We can define a generalized robust price of anarchy as the solution of the LP (21). If this quantity approaches to 1 with a sufficiently fast rate (in terms of the number of players), we can extend Theorem A.2 by analyzing the weighted sum of the players' regrets  $\sum_{i=1}^{n} z_i \mathrm{Reg}_i^{(T)}$ . It is easy to see that the argument of Theorem 3.1 readily carries over under the condition that the ratio  $z_i / z_{i'}$  is bounded for any  $i, i' \in [[n]]$ , as well as the assumption that each  $z_i$  is bounded away from 0. In contrast, if there exists a pair  $i, i' \in [[n]]$  for which the ratio  $z_i / z_{i'}$  is unbounded, our current techniques do not appear to be applicable; this is related to the well-known difficulty of deriving so-called RVU bounds for the maximum of the players' regret, instead of their sum [103]. In other words, our current techniques can sharpen Theorem 3.1 by considering solutions of (21) under the additional (linear) constraint that the variables  $\{z_i\}_{i=1}^n$  have bounded pairwise ratio. Given that the more general framework of Nadav and Roughgarden [75] leads to improved bounds, we expect that this modification should have applications in our setting as well. In particular, we point out that the 2-player example of Nadav and Roughgarden [75] that separates  $\mathrm{ACCE}^*$  from  $\mathrm{CCE}^*$  indeed satisfies  $z_1 / z_2 \approx 2.3$  [75, Proposition 2], so the separation manifests itself even when the pairwise ratio is bounded by an absolute constant.
We finally refer to the works of Nguyen [78] and Kulkarni and Mirrokni [62] for a different primal-dual take on smoothness.
A.4 Local smoothness
Another interesting extension of our techniques can be obtained using the framework of local smoothness, first introduced by Roughgarden and Schoppmann [92] and recently refined by Nguyen [78] in the context of splittable congestion games. In what follows, we follow the treatment of Nguyen
[78] as it fits our framework. In this context, Nguyen [78] introduced the notion of a  $(\lambda, \mu)$ -dual-smooth differentiable utility function  $u: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$  with the property that for every vectors  $\pmb{z} = (z_1, \dots, z_n) \in \mathbb{R}_{\geq 0}^n$  and  $\pmb{z}' = (z_1', \dots, z_n') \in \mathbb{R}_{\geq 0}^n$ ,
$$
z ^ {\prime} u (z) + \sum_ {i = 1} ^ {n} z _ {i} \left(z _ {i} ^ {\prime} - z _ {i}\right) u ^ {\prime} (z) \geq \lambda z ^ {\prime} u \left(z ^ {\prime}\right) - \mu z u (z), \tag {22}
$$
where  $z \coloneqq \sum_{i=1}^{n} z_i$  and  $z' \coloneqq \sum_{i=1}^{n} z_i'$ . We also note that  $u'$  above denotes the derivative of  $u$ . (Again, (22) has been translated to utility-maximization games compared to the original formulation of Nguyen [78].) Furthermore, a splittable congestion game is called  $(\lambda, \mu)$ -dual-smooth [78] if for every resource  $e \in E$  the utility function  $u_e : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$  is  $(\lambda, \mu)$ -dual-smooth in the sense of (22). The importance of Nguyen's extension is that it can be applied to coarse correlated equilibria, as opposed to the original definition of Roughgarden and Schoppmann [92] that was applicable to correlated equilibria; this distinction is incidentally important for our framework.