SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\prod_ {i = 1} ^ {n} \mathcal {X} _ {i} \ni \boldsymbol {w} \mapsto \Pi_ {\boldsymbol {x} ^ {\prime}} (\eta F (\boldsymbol {w})) \tag {32}
$$
is  $(\eta L)$ -Lipschitz continuous. As a result, function (32) is a contraction mapping as long as  $\eta < \frac{1}{L}$ .
This follows directly from Lemma A.17:  $\| \Pi_{\pmb{x}'}(\eta F(\pmb{w})) - \Pi_{\pmb{x}'}(\eta F(\pmb{w}'))\|_2 \leq \eta \| F(\pmb{w}) - F(\pmb{w}')\|_2 \leq \eta L\|\pmb{w} - \pmb{w}'\|_2$ .
Proposition A.18 reassures us that fixed points of the contraction mapping (32) not only exist, but  $\epsilon$ -approximate fixed points can also be computed in a time proportional to  $\log(1/\epsilon)$  [85]. In this context, if  $\pmb{x}^{(0)} \in \prod_{i=1}^{n} \mathcal{X}_i$  is an arbitrary point, we consider the update rule defined for  $t \in \mathbb{N}$  via
$$
\boldsymbol {x} ^ {(t)} = \Pi_ {\boldsymbol {x} ^ {(t - 1)}} (\eta F (\boldsymbol {w} ^ {(t)})), \tag {33}
$$
where  $\pmb{w}^{(t)}\in \prod_{i = 1}^{n}\mathcal{X}_{i}$  is any point such that  $\| \pmb{w}^{(t)} - \Pi_{\pmb{x}^{(t - 1)}}(\eta F(\pmb{w}^{(t)}))\| _2\leq \epsilon^{(t)}$  . It is important to note that this sequence  $(\pmb{x}^{(t)})_{t\geq 1}$  is not uniquely defined, but with a slight abuse we refer to any sequence satisfying (33) as clairvoyant gradient descent (CGD). CGD satisfies the following remarkable regret bound.
Theorem A.19 ([85, 37]). For any  $T \in \mathbb{N}$ , the regret of player  $i \in [[n]]$  under CGD satisfies
$$
\operatorname {R e g} _ {i} ^ {(T)} \leq \frac {D _ {\mathcal {X} _ {i}} ^ {2}}{2 \eta} - \frac {1}{2 \eta} \sum_ {t = 1} ^ {T} \| \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {(t - 1)} \| _ {2} ^ {2} + D _ {\mathcal {X} _ {i}} L _ {i} \sum_ {t = 1} ^ {T} \epsilon^ {(t)}, \tag {34}
$$
where  $L_{i}\in \mathbb{R}_{>0}$  is the Lipschitz constant of  $\pmb{u}_i$  with respect to  $\| \cdot \| _2$ .
Proof. Given that  $\pmb{x}_i^{(t)}\coloneqq \Pi_{\pmb{x}_i^{(t - 1)}}(\eta \pmb {u}_i(\pmb{w}_{-i}^{(t)}))$ , the first-order optimality condition implies that for any  $\pmb{x}_i^\star \in \mathcal{X}_i$  and  $t\in [[T]]$
$$
\left\langle \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {\star}, \eta \boldsymbol {u} _ {i} \left(\boldsymbol {w} _ {- i} ^ {(t)}\right) - \left(\boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {(t - 1)}\right) \right\rangle \geq 0. \tag {35}
$$
We further have that
$$
\langle \pmb {x} _ {i} ^ {\star} - \pmb {x} _ {i} ^ {(t)}, \pmb {x} _ {i} ^ {(t)} - \pmb {x} _ {i} ^ {(t - 1)} \rangle = - \frac {1}{2} \| \pmb {x} _ {i} ^ {\star} - \pmb {x} _ {i} ^ {(t)} \| _ {2} ^ {2} + \frac {1}{2} \| \pmb {x} _ {i} ^ {\star} - \pmb {x} _ {i} ^ {(t - 1)} \| _ {2} ^ {2} - \frac {1}{2} \| \pmb {x} _ {i} ^ {(t)} - \pmb {x} _ {i} ^ {(t - 1)} \| _ {2} ^ {2},
$$
thereby implying through a telescopic summation that
$$
\sum_ {t = 1} ^ {T} \left\langle \boldsymbol {x} _ {i} ^ {\star} - \boldsymbol {x} _ {i} ^ {(t)}, \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {(t - 1)} \right\rangle \leq \frac {1}{2} D _ {\mathcal {X} _ {i}} ^ {2} - \frac {1}{2} \sum_ {t = 1} ^ {T} \| \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {(t - 1)} \| _ {2} ^ {2}.
$$
Combining with (35),
$$
\sum_ {t = 1} ^ {T} \langle \boldsymbol {x} _ {i} ^ {\star} - \boldsymbol {x} _ {i} ^ {(t)}, \boldsymbol {u} _ {i} (\boldsymbol {w} _ {- i} ^ {(t)}) \rangle \leq \frac {1}{2 \eta} D _ {\mathcal {X} _ {i}} ^ {2} - \frac {1}{2 \eta} \sum_ {t = 1} ^ {T} \| \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {(t - 1)} \| _ {2} ^ {2}.
$$
The claim thus follows from the fact that
$$
\begin{array}{l} \sum_ {t = 1} ^ {T} \langle \boldsymbol {x} _ {i} ^ {\star} - \boldsymbol {x} _ {i} ^ {(t)}, \boldsymbol {u} _ {i} (\boldsymbol {w} _ {- i} ^ {(t)}) - \boldsymbol {u} _ {i} (\boldsymbol {x} _ {- i} ^ {(t)}) \rangle \geq - \sum_ {t = 1} ^ {T} \| \boldsymbol {x} _ {i} ^ {\star} - \boldsymbol {x} _ {i} ^ {(t)} \| _ {2} \| \boldsymbol {u} _ {i} (\boldsymbol {w} _ {- i} ^ {(t)}) - \boldsymbol {u} _ {i} (\boldsymbol {x} _ {- i} ^ {(t)}) \| _ {2} \\ \geq - D _ {\mathcal {X} _ {i}} L _ {i} \sum_ {t = 1} ^ {T} \epsilon^ {(t)}. \\ \end{array}
$$
Unlike prior work, here we will make crucial use of the negative term in (34). The important lemma below will enable us to express the regret bound (34) in terms of  $i$ 's best response gap; it is analogous to Lemma A.1 we stated earlier for OGD.
Lemma A.20. Fix any  $t \in \mathbb{N}$  and let  $\pmb{x}^{(t)} = \Pi_{\pmb{x}^{(t-1)}}(\eta F(\pmb{w}^{(t)}))$ , where  $\pmb{w}^{(t)} \in \prod_{i=1}^{n} \mathcal{X}_i$  is any point such that  $\| \pmb{w}^{(t)} - \Pi_{\pmb{x}^{(t-1)}}(\eta F(\pmb{w}^{(t)})) \|_2 \leq \epsilon^{(t)}$ . Then,
$$
\operatorname {B R G A P} _ {i} \left(\boldsymbol {x} ^ {(t)}\right) \leq D _ {\mathcal {X} _ {i}} \left\| \boldsymbol {u} _ {i} \left(\boldsymbol {w} _ {- i} ^ {(t)}\right) - \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i} ^ {(t)}\right) \right\| _ {2} + \frac {D _ {\mathcal {X} _ {i}}}{\eta} \left\| \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {(t - 1)} \right\| _ {2}.
$$
Proof. By the first-order optimality condition, it follows that for any  $\pmb{x}_i^\star \in \mathcal{X}_i$
$$
\left\langle \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {\star}, \eta \boldsymbol {u} _ {i} (\boldsymbol {w} _ {- i} ^ {(t)}) - (\boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {(t - 1)}) \right\rangle \geq 0,
$$
$$
\left\langle \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {\star}, \boldsymbol {u} _ {i} \left(\boldsymbol {w} _ {- i} ^ {(t)}\right) \right\rangle \geq - \frac {1}{\eta} \left\langle \boldsymbol {x} _ {i} ^ {\star} - \boldsymbol {x} _ {i} ^ {(t)}, \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {(t - 1)} \right\rangle . \tag {36}
$$
$$
\begin{array}{l} \langle \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {\star}, \boldsymbol {u} _ {i} (\boldsymbol {w} _ {- i} ^ {(t)}) \rangle = \langle \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {\star}, \boldsymbol {u} _ {i} (\boldsymbol {x} _ {- i} ^ {(t)}) \rangle + \langle \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {\star}, \boldsymbol {u} _ {i} (\boldsymbol {w} _ {- i} ^ {(t)}) - \boldsymbol {u} _ {i} (\boldsymbol {x} _ {- i} ^ {(t)}) \rangle \\ \leq \langle \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {\star}, \boldsymbol {u} _ {i} (\boldsymbol {x} _ {- i} ^ {(t)}) \rangle + \| \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {\star} \| _ {2} \| \boldsymbol {u} _ {i} (\boldsymbol {w} _ {- i} ^ {(t)}) - \boldsymbol {u} _ {i} (\boldsymbol {x} _ {- i} ^ {(t)}) \| _ {2} \\ \leq \left\langle \boldsymbol {x} _ {i} ^ {(t)} - \boldsymbol {x} _ {i} ^ {\star}, \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i} ^ {(t)}\right) \right\rangle + D _ {\chi_ {i}} \| \boldsymbol {u} _ {i} \left(\boldsymbol {w} _ {- i} ^ {(t)}\right) - \boldsymbol {u} _ {i} \left(\boldsymbol {x} _ {- i} ^ {(t)}\right) \| _ {2}. \tag {37} \\ \end{array}
$$
Combining (36) and (37) yields that
$$
\langle \pmb {x} _ {i} ^ {(t)}, \pmb {u} _ {i} (\pmb {x} _ {- i} ^ {(t)}) \rangle - \max _ {\pmb {x} _ {i} ^ {t} \in \mathcal {X} _ {i}} \langle \pmb {x} _ {i} ^ {\star}, \pmb {u} _ {i} (\pmb {x} _ {- i} ^ {(t)}) \rangle \geq - D _ {\mathcal {X} _ {i}} \| \pmb {u} _ {i} (\pmb {w} _ {- i} ^ {(t)}) - \pmb {u} _ {i} (\pmb {x} _ {- i} ^ {(t)}) \| _ {2} - \frac {D _ {\mathcal {X} _ {i}}}{\eta} \| \pmb {x} _ {i} ^ {(t)} - \pmb {x} _ {i} ^ {(t - 1)} \| _ {2},
$$
Corollary A.21. For any  $T \in \mathbb{N}$ , the regret of player  $i \in [[n]]$  under CGD can be bounded as
$$
\mathsf {R e g} _ {i} ^ {(T)} \leq \frac {D _ {\mathcal {X} _ {i}} ^ {2}}{2 \eta} - \frac {\eta}{4 D _ {\mathcal {X} _ {i}} ^ {2}} \sum_ {t = 1} ^ {T} \left(\mathsf {B R G A P} _ {i} (\boldsymbol {x} ^ {(t)})\right) ^ {2} + L _ {i} D _ {\mathcal {X} _ {i}} \sum_ {t = 1} ^ {T} \epsilon^ {(t)} + \frac {1}{2} \eta \sum_ {t = 1} ^ {T} \| \boldsymbol {u} _ {i} (\boldsymbol {w} _ {- i} ^ {(t)}) - \boldsymbol {u} _ {i} (\boldsymbol {x} _ {- i} ^ {(t)}) \| _ {2} ^ {2}.
$$
In particular, if  $\eta \coloneqq \frac{1}{2L}$  and  $\epsilon^{(t)}\leq \frac{D_{x_i}}{t^2}$  for any  $t\in [[T]]$
$$
\operatorname {R e g} _ {i} ^ {(T)} \leq 3 L D _ {\mathcal {X} _ {i}} ^ {2} - \frac {1}{8 L D _ {\mathcal {X} _ {i}} ^ {2}} \sum_ {t = 1} ^ {T} \left(\operatorname {B R G A P} _ {i} (\boldsymbol {x} ^ {(t)})\right) ^ {2} + \frac {1}{2} \eta \sum_ {t = 1} ^ {T} \| \boldsymbol {u} _ {i} (\boldsymbol {w} _ {- i} ^ {(t)}) - \boldsymbol {u} _ {i} (\boldsymbol {x} _ {- i} ^ {(t)}) \| _ {2} ^ {2}.
$$
Proof. By Lemma A.20, it follows that for any  $t \in [[T]]$ ,
$$
\frac {1}{2 \eta} \| \pmb {x} _ {i} ^ {(t)} - \pmb {x} _ {i} ^ {(t - 1)} \| _ {2} ^ {2} \geq \frac {\eta}{4 D _ {\mathcal {X} _ {i}} ^ {2}} \left(\mathrm {B R G A P} _ {i} (\pmb {x} ^ {(t)})\right) ^ {2} - \frac {1}{2} \eta \| \pmb {u} _ {i} (\pmb {w} _ {- i} ^ {(t)}) - \pmb {u} _ {i} (\pmb {x} _ {- i} ^ {(t)}) \| _ {2} ^ {2}.
$$
Summing over all  $t \in [[T]]$  and combining with Theorem A.19 implies the statement since  $L_{i}D_{\mathcal{X}_{i}}\sum_{t = 1}^{T}\epsilon^{(t)} \leq L_{i}D_{\mathcal{X}_{i}}^{2}\sum_{t = 1}^{T}\frac{1}{t^{2}} \leq 2LD_{\mathcal{X}_{i}}^{2}$ .
We are now ready to prove Theorem 4.1, which is recalled below.
1. the average correlated distribution of play is a  $\frac{4LD_{\chi}^2}{T} - CCE$ ;
2. there is a time  $t^{\star}\in [[T]]$  such that
$$
\operatorname {S W} \left(\boldsymbol {x} ^ {\left(t ^ {\star}\right)}\right) \geq \sup  _ {\epsilon \geq \epsilon_ {0}} \min  \left\{\rho_ {\mathcal {G}} (\lambda , \mu) \cdot \mathrm {O P T} _ {\mathcal {G}} + \frac {\epsilon^ {2}}{1 6 (\mu + 1) L D _ {\chi} ^ {2}}, \mathrm {P o A} _ {\mathcal {G}} ^ {\epsilon} \cdot \mathrm {O P T} _ {\mathcal {G}} \right\}. \tag {3}
$$
Proof. First, by Corollary A.21,
$$
\begin{array}{l} \sum_ {i = 1} ^ {n} \operatorname {R e g} _ {i} ^ {(T)} \leq 3 L \sum_ {i = 1} ^ {n} D _ {\mathcal {X} _ {i}} ^ {2} - \frac {1}{8 L D _ {\mathcal {X}} ^ {2}} \sum_ {i = 1} ^ {n} \sum_ {t = 1} ^ {T} \left(\operatorname {B R G A P} _ {i} (\boldsymbol {x} ^ {(t)})\right) ^ {2} + \frac {1}{2} \eta \sum_ {t = 1} ^ {T} \sum_ {i = 1} ^ {n} \| \boldsymbol {u} _ {i} (\boldsymbol {w} _ {- i} ^ {(t)}) - \boldsymbol {u} _ {i} (\boldsymbol {x} _ {- i} ^ {(t)}) \| _ {2} ^ {2} \\ = 3 L D _ {\mathcal {X}} ^ {2} - \frac {1}{8 L D _ {\mathcal {X}} ^ {2}} \sum_ {i = 1} ^ {n} \sum_ {t = 1} ^ {T} \left(\mathrm {B R G A P} _ {i} (\boldsymbol {x} ^ {(t)})\right) ^ {2} + \frac {1}{2} \eta \sum_ {t = 1} ^ {T} \| F (\boldsymbol {w} ^ {(t)}) - F (\boldsymbol {x} ^ {(t)}) \| _ {2} ^ {2} \\ \leq 3 L D _ {\mathcal {X}} ^ {2} - \frac {1}{8 L D _ {\mathcal {X}} ^ {2}} \sum_ {i = 1} ^ {n} \sum_ {t = 1} ^ {T} \left(\mathrm {B R G A P} _ {i} (\boldsymbol {x} ^ {(t)})\right) ^ {2} + \frac {1}{2} \eta L ^ {2} \sum_ {t = 1} ^ {T} \| \boldsymbol {w} ^ {(t)} - \boldsymbol {x} ^ {(t)} \| _ {2} ^ {2} \\ \leq 3 L D _ {\mathcal {X}} ^ {2} - \frac {1}{8 L D _ {\mathcal {X}} ^ {2}} \sum_ {i = 1} ^ {n} \sum_ {t = 1} ^ {T} \left(\mathrm {B R G A P} _ {i} (\boldsymbol {x} ^ {(t)})\right) ^ {2} + \frac {1}{2} \eta L ^ {2} \sum_ {t = 1} ^ {T} (\epsilon^ {(t)}) ^ {2} \\ \leq 4 L D _ {\mathcal {X}} ^ {2} - \frac {1}{8 L D _ {\mathcal {X}} ^ {2}} \sum_ {i = 1} ^ {n} \sum_ {t = 1} ^ {T} \left(\mathrm {B R G A P} _ {i} \left(\boldsymbol {x} ^ {(t)}\right)\right) ^ {2}, \tag {38} \\ \end{array}
$$
where we used the fact that  $\sum_{t=1}^{T} (\epsilon^{(t)})^2 \leq D_{\mathcal{X}}^2 \sum_{t=1}^{T} \frac{1}{t^4} \leq 2D_{\mathcal{X}}^2$  and  $\eta = \frac{1}{2L}$ . As a result, Item 1 follows directly from (38) by invoking the well-known fact that the CCE gap is bounded by  $\max_{1 \leq i \leq n} \operatorname{Reg}_i^{(T)}$ .
For Item 2, let us fix any  $\epsilon \geq \epsilon_0$ . Suppose that at every time  $t \in [[T]]$  it holds that  $\mathrm{BRGAP}_i(\pmb{x}^{(t)}) > \epsilon$  for some player  $i \in [[n]]$ . Then, by (38) we conclude that
$$
\sum_ {i = 1} ^ {n} \operatorname {R e g} _ {i} ^ {(T)} \leq 4 L D _ {\mathcal {X}} ^ {2} - \frac {1}{8 L D _ {\mathcal {X}} ^ {2}} \epsilon^ {2} T \leq - \frac {1}{1 6 L D _ {\mathcal {X}} ^ {2}} \epsilon^ {2} T,
$$
since  $T \geq \frac{64L^2D_{\mathcal{X}}^4}{\epsilon_0^2} \geq \frac{64L^2D_{\mathcal{X}}^4}{\epsilon^2}$ . As a result, by  $(\lambda, \mu)$ -smoothness, it follows that
$$
\sum_ {i = 1} ^ {n} \operatorname {R e g} _ {i} ^ {(T)} \geq \lambda \mathrm {O P T} _ {\mathcal {G}} T - (1 + \mu) \sum_ {t = 1} ^ {T} \operatorname {S W} (\boldsymbol {x} ^ {(t)}),
$$
$$
\frac {1}{T} \sum_ {t = 1} ^ {T} \operatorname {S W} \left(\boldsymbol {x} ^ {(t)}\right) \geq \rho_ {\mathcal {G}} (\lambda , \mu) \cdot \operatorname {O P T} _ {\mathcal {G}} - \frac {1}{T} \sum_ {i = 1} ^ {n} \operatorname {R e g} _ {i} ^ {(T)} \geq \rho_ {\mathcal {G}} (\lambda , \mu) \cdot \operatorname {O P T} _ {\mathcal {G}} + \frac {\epsilon^ {2}}{1 6 (\mu + 1) L D _ {\mathcal {X}} ^ {2}}.
$$
As a result, there is a time  $t^{\star} \in [[T]]$  such that  $\mathrm{SW}(\pmb{x}^{(t^{\star})}) \geq \rho_{\mathcal{G}}(\lambda, \mu) \cdot \mathrm{OPT}_{\mathcal{G}} + \frac{\epsilon^2}{16(\mu + 1)LD_{\mathcal{X}}^2}$ . In the contrary case, if there is a time  $t^{\star} \in [[T]]$  such that  $\mathrm{BRGAP}_i(\pmb{x}^{(t^{\star})}) \leq \epsilon$  for any player  $i \in [[n]]$ , it follows that  $\mathrm{SW}(\pmb{x}^{(t^{\star})}) \geq \mathrm{PoA}_{\mathcal{G}}^{\epsilon} \cdot \mathrm{OPT}_{\mathcal{G}}$  (by definition). This concludes the proof.
In particular, we note that Corollary 4.3 stated earlier in the main body is an immediate consequence of Theorem 4.1 under Condition 4.2 since  $\mathsf{rPoA}_{\mathcal{G}} \geq \rho_{\mathcal{G}}$ . It is also worth noting that for the special case of normal-form games, one can state Theorem 4.1 so that the first term in the right-hand side of (3) reads  $\rho_{\mathcal{G}}(\lambda, \mu) \cdot \mathrm{OPT}_{\mathcal{G}} + \frac{\epsilon^2}{32(\mu + 1)L}$ ; thus, for a broad class of games (see Lemmas A.7 to A.9), the improvement over the smoothness bound is an absolute constant when  $\epsilon$  and  $\mu$  are also bounded by absolute constants. Relatedly, we should note that Theorem 6.2 applies in the regime where  $\mu$  is polynomially bounded (we are not aware of any application of smoothness where this is not the case). Remark A.22. It is direct to see that Item 2 of Theorem 4.1 can be refined so that there is a set  $S \subseteq [[T]]$ , with  $|S| \geq (1 - \gamma)T$ , so that
$$
\frac {1}{| S |} \sum_ {t \in S} \operatorname {S W} \left(\boldsymbol {x} ^ {(t)}\right) \geq \sup  _ {\epsilon \geq \epsilon_ {0}} \min  \left\{\rho_ {\mathcal {G}} \cdot \operatorname {O P T} _ {\mathcal {G}} + \frac {\gamma \epsilon^ {2}}{1 6 (\mu + 1) L D _ {\mathcal {X}} ^ {2}}, \mathrm {P o A} _ {\mathcal {G}} ^ {\epsilon} \cdot \operatorname {O P T} _ {\mathcal {G}} \right\}.
$$
Remark A.23. Theorem 4.1 can also be refined using the primal-dual framework of Nadav and Roughgarden [75] discussed in Appendix A.3, with the caveat that the variables  $\{z_i\}_{i=1}^n$  need to
again have a bounded pairwise ratio. It is interesting to note, however, that using CGD enables making non-trivial conclusions even when a subset of the variables  $\{z_i\}_{i=1}^n$  are zero in an optimal solution. In particular, when the quantity  $\sum_{i=1}^n z_i \operatorname{Reg}_i^{(T)}$  is nonnegative, we can readily draw the non-trivial conclusion that all players in the set  $\{i \in [n] : z_i \neq 0\}$  are eventually best responding (by virtue of Corollary A.21). It is not at all clear if such conclusions apply to OGD as well. In other words, using CGD one can essentially replace  $\mathsf{PoA}_{\mathcal{G}}^\epsilon$  in Theorem 4.1 by the price of anarchy with respect to a solution concept in which a particular subset of players are best responding. Shedding light to this type of guarantee necessitates understanding the implications of having some of the variables  $\{z_i\}_{i=1}^n$  being zero in an optimal solution of the LP (21). Relatedly, even if we constraint  $z_1 = z_2 = \dots = z_n = z$ , can we characterize the games for which  $z \approx 0$ ? We explained earlier in Remark A.10 that such pathologies (typically) occur in constant-sum games, for which questions concerning social welfare are trivial.
A.10 Beyond smoothness
In this subsection, we discuss an example in which the welfare predicted by the smoothness framework is far from the welfare obtained by learning algorithms such as OGD. We then suggest a natural direction that enables obtaining sharper predictions; we have already seen refined guarantees beyond the standard smoothness framework in Appendices A.3 and A.4, but here we explore a different direction.
Our example is again based on Shapley's game, a bimatrix game in normal form defined with the matrices
$$
\mathbf {A} = \left[ \begin{array}{l l l} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right], \mathbf {B} = \left[ \begin{array}{l l l} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right]. \tag {39}
$$
Claim A.24. Let  $\mathcal{G}$  be the bimatrix game defined in (39). Then,  $\mathrm{rPoA}_{\mathcal{G}} = 0$
Proof. Let  $(\pmb{x}_1^\star, \pmb{x}_2^\star) \coloneqq ((1,0,0), (0,1,0))$  be a welfare-maximizing joint action. By symmetry, the following argument will apply to any welfare-maximizing joint action. Consider  $(\pmb{x}_1, \pmb{x}_2) = ((0,1,0), (0,1,0))$ . Then, we have that  $u_1(\pmb{x}_1^\star, \pmb{x}_2) + u_2(\pmb{x}_2^\star, \pmb{x}_1) = (\pmb{x}_1^\star)^\top \mathbf{A}\pmb{x}_2 + \pmb{x}_1^\top \mathbf{B}\pmb{x}_2^\star = 0 + 0$ . As a result, the smoothness constraint corresponding to  $(\pmb{x}_1, \pmb{x}_2)$  necessitates that  $0 \geq \lambda$ , which is incompatible with the constraint that  $\lambda > 0$  (Definition 2.1).
In spite of the fact that  $\mathsf{rPoA}_{\mathcal{G}} = 0$ , we see in Figure 2 (left) that OGD approaches close to the optimal social welfare 1. This can be explained by the fact that certain joint actions—in particular, those in the diagonal—are played with small probability under OGD (right-side of Figure 2). This motivates introducing a more refined notion of smoothness in which the smoothness constraints are only enforced on the joint actions that are visited with a non-negligible probability.
Figure 2: The behavior of OG in the bimatix game (39) with  $\eta \coloneqq 0.01$ . On the left, we plot the average social welfare of the dynamics. On the right, we plot the probability of playing a joint action in the diagonal.
One class of games in which one can easily make such refinements concerns games solvable under iterated elimination of strictly dominated actions. As a concrete example, let
$$
\mathbf {A} = \left[ \begin{array}{l l} 0 & 0 \\ 1 & 1 \end{array} \right], \mathbf {B} = \left[ \begin{array}{l l} 1 & 0 \\ 0 & 1 \end{array} \right]. \tag {40}
$$
Observation A.25. Let  $\mathcal{G}$  be the game defined in (40). Then,  $\mathsf{rPoA}_{\mathcal{G}} = \frac{1}{2}$ . Furthermore, if  $\mathcal{G}'$  is the game<sup>4</sup> resulting from iterative removal of strictly dominated actions, then  $\mathsf{rPoA}_{\mathcal{G}'} = 1$ .
4Under elimination of strictly dominated actions, this is always uniquely defined.