SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Now, to be more concrete regarding the preconditions of Theorem 3.1, we first observe that the growth of the Lipschitz constant  $L_{n}$  depends on the normalization assumptions as well as the structure of the underlying game  $\mathcal{G}_n$ . In particular, let us assume—as is standard—that the range of the utility functions is independent of  $n$ , in which case  $\mathrm{OPT}_{\mathcal{G}_n} = O_n(n)$ . We then show that  $L_{n} = O_{n}(1)$  in each of the following cases: graphical games with bounded degree (Lemma A.7), games with  $O_{n}(1 / n)$  strategic sensitivity per Definition 3.5 (Lemma A.8), and polymatrix (general-sum) games even with unbounded neighborhoods (Lemma A.9); the first two of the aforementioned classes are the most commonly studied classes in the literature under the regime  $n \gg 1$ . For all of those classes, applying Theorem 3.1 yields an  $(o_{n}(1), o_{n}(1))$ -weak Nash equilibrium for any  $\epsilon_{n} \leq o_{n}(1)$ . More generally, the Lipschitz constant  $L_{n}$  can grow with  $n$  (Lemma A.6), in which case  $\epsilon_{n}$  must vanish with a faster rate for the conclusion of Theorem 3.1 to kick in. One important limitation of Theorem 3.1 is that, at least in a certain regime, it prescribes a strategy in which many players—albeit a vanishing fraction—may have a profitable deviation (in accordance with Definition 2.2); this is admittedly an inherent feature of our framework.
Remark 3.2. In a decentralized environment, one question that arises from Theorem 3.1 concerns the identification of a time index  $t^{\star} \in [[T]]$  that satisfies (2), which can be viewed as the stopping condition of the algorithm. We suggest two possible approaches. First, if we accept that players have access to a common source of randomness, then all players can sample the same index  $t^{\star}$  uniformly at random from the set  $[[T]]$ . As we point out in Remark A.4, this suffices to provide a guarantee with high probability with only a small degradation in the solution quality. The second approach, which does not rest any having a common source of randomness, involves a coordinator who can communicate with the players but possesses no information whatsoever about the underlying game. The coordinator sets a target solution quality parameterized by  $(\epsilon, \delta)$  (per Definition 2.2), and after each iteration  $t$  elicits from each player  $i \in [[n]]$  a single bit, encoding whether  $\mathbb{1}\{\mathrm{BRGAP}_i(\pmb{x}^{(t)}) > \epsilon\}$ . (We note that each player can indeed determine its best response gap with only its local information—namely,
the utility feedback.) The coordinator can then evaluate whether the fraction of the players with at most an  $\epsilon$  best response gap matches the desired accuracy. While the second approach makes for a less decentralized protocol, the communication overhead described above is arguably very limited.
The key precondition of Theorem 3.1 pertains the behavior of the smoothness parameters, to be discussed next after we first sketch the proof of Theorem 3.1, which extends a recent technique [1].
Proof sketch of Theorem 3.1 The proof is based on the fact that the sum of the players' regrets  $\sum_{i=1}^{n} \operatorname{Reg}_i^{(T)}$  cannot be too negative, which in turn follows from the assumption that  $\rho_n \geq 1 - \epsilon_n$ . The argument then proceeds by bounding the players' cumulative best response gap across the  $T$  iterations  $\sum_{t=1}^{T} \sum_{i=1}^{n} \left( \mathrm{BRGAP}_i(x^{(t)}) \right)^2$  as a function of  $\gamma_n$  and the time horizon  $T \in \mathbb{N}$ , ultimately leading to the conclusion of Theorem 3.1.
Connection with the Minty property While we have stated Theorem 3.1 in the regime  $n \gg 1$ , under the premise that  $\rho_{n}$  is sufficiently close to 1 (as a function of  $n$ ), its conclusion is in fact interesting beyond that regime. Indeed, an important observation is that any two-player constant-sum game  $\mathcal{G} \coloneqq (\mathbf{A}, \mathbf{B})$ , with  $\langle \pmb{x}_1, \mathbf{A}\pmb{x}_2 \rangle + \langle \pmb{x}_1, \mathbf{B}\pmb{x}_2 \rangle = V$  for any  $(\pmb{x}_1, \pmb{x}_2) \in \mathcal{X}_1 \times \mathcal{X}_2$ , satisfies  $\rho_{\mathcal{G}} = 1$ . This is indeed a consequence of Von Neumann's minimax theorem:  $\exists (\pmb{x}_1^\star, \pmb{x}_2^\star) \in \mathcal{X}_1 \times \mathcal{X}_2$  such that  $u_1(\pmb{x}_1^\star, \pmb{x}_2) + u_2(\pmb{x}_2^\star, \pmb{x}_1) - V = \langle \pmb{x}_1^\star, \mathbf{A}\pmb{x}_2 \rangle - \langle \pmb{x}_1, \mathbf{A}\pmb{x}_2^\star \rangle \geq 0$  for any  $(\pmb{x}_1, \pmb{x}_2) \in \mathcal{X}_1 \times \mathcal{X}_2$ , in turn implying that  $u_1(\pmb{x}_1^\star, \pmb{x}_2) + u_2(\pmb{x}_2^\star, \pmb{x}_1) \geq (1 + \mu)\mathrm{OPT}_{\mathcal{G}} - \mu \mathrm{SW}(\pmb{x}) = V$ ; that is, any two-player constant-sum game is  $(1 + \mu, \mu)$ -smooth, thereby making the conclusion of Theorem 3.1 readily applicable (by taking  $\rho \geq 1 - \epsilon^2$  for any  $\epsilon > 0$ ; see the explicit statement of Theorem A.2). This captures and unifies earlier iteration complexity bounds under OGDP and the extra-gradient method [15, 48, 46].
Proposition 3.3. Any two-player constant-sum game  $\mathcal{G}$  is  $(1 + \mu, \mu)$ -smooth for  $\mu > -1$ , implying that  $\rho_{\mathcal{G}} = 1$ . Thus,  $O(1 / \epsilon^2)$  iterations of OGD suffice to obtain an  $\epsilon$ -Nash equilibrium, for any  $\epsilon > 0$ .
More broadly, there is a surprisingly overlooked but immediate connection between smooth games (per Definition 2.1) and the Minty property, a well-known condition in the literature on variational inequalities (VIs) [36, 66, 71]. More precisely, the Minty property postulates the existence of a strategy profile  $\pmb{x}^{\star} \in \prod_{i=1}^{n} \mathcal{X}_i$  such that  $\langle \pmb{x}^{\star} - \pmb{x}, F(\pmb{x}) \rangle \geq 0$  for any  $\pmb{x} \in \prod_{i=1}^{n} \mathcal{X}_i$ , where we recall that  $F$  is the operator of the game. (We caution that the last inequality is typically stated with the opposite sign since the operator  $F$  is defined oppositely.) The following connection is thus immediate from the fact that  $\langle \pmb{x}^{\star}, F(\pmb{x}) \rangle = \sum_{i=1}^{n} \langle \pmb{x}_i^{\star}, \pmb{u}_i(\pmb{x}_{-i}) \rangle = \sum_{i=1}^{n} u_i(\pmb{x}_i^{\star}, \pmb{x}_{-i})$  and  $\langle \pmb{x}, F(\pmb{x}) \rangle = \sum_{i=1}^{n} \langle \pmb{x}_i, \pmb{u}_i(\pmb{x}_{-i}) \rangle = \mathrm{SW}(\pmb{x})$  (by multilinearity).
Observation 3.4. For any (multi-player) constant-sum game  $\mathcal{G}$ , the Minty property is equivalent to  $\mathcal{G}$  being  $(1 + \mu, \mu)$ -smooth for some  $\mu > -1$ .
Indeed, the Minty property implies that  $\sum_{i=1}^{n} u_i(\pmb{x}_i^\star, \pmb{x}_{-i}) \geq V = (1 + \mu)\mathrm{OPT}_{\mathcal{G}} - \mu \mathrm{SW}(\pmb{x})$ , and the converse direction is also immediate. In fact, for (multi-player) zero-sum games, the Minty property is equivalent to  $\mathcal{G}$  satisfying (1) under some pair  $(\lambda, \mu) \in \mathbb{R}^2$ . We stress that even if  $\rho_{\mathcal{G}} = 1$ , traditional no-regret learning algorithms such as online mirror descent do not generally enjoy iterate convergence to Nash equilibria [70], which stands in stark contrast to the behavior of OGDP (Proposition 3.3). In light of Observation 3.4, Theorem 3.1 should also be viewed as part of an ongoing effort to establish sufficient conditions of tractability that are more permissive than the Minty property (e.g., [33, 12, 14, 84, 11]). The criterion we furnish herein, based on the smoothness framework, has the important benefit of enjoying a natural economic interpretation, as well as having been extensively studied in the literature. Indeed, we will leverage insights from prior work to obtain several interesting extensions in the remainder of this section.
Games with vanishing sensitivity Returning to the regime  $n \gg 1$ , why should we expect  $\rho_n \to 1$ ? Indeed, if anything large games are more general than games with a small number of players since one can always incorporate "dummy" players into the game. Yet, the point is that large games oftentimes exhibit a structure that leads to more efficient outcomes. For example, one immediate implication of our framework relates to games with a vanishing (strategic) sensitivity (see, e.g., [57, 3, 72, 31, 45]). There are various ways of defining sensitivity; here, we adopt the following standard definition.
Definition 3.5. The strategic sensitivity  $\epsilon \in \mathbb{R}_{>0}$  of an  $n$ -player game in normal form is defined as
$$
\epsilon := \max  _ {1 \leq i \leq n} \max  _ {\boldsymbol {a} \in \mathcal {A}} \max  _ {1 \leq i ^ {\prime} \leq n} \max  _ {a _ {i ^ {\prime}} ^ {\prime} \in \mathcal {A} _ {i ^ {\prime}}} | u _ {i} (a _ {i ^ {\prime}} ^ {\prime}, \boldsymbol {a} _ {- i}) - u _ {i} (\boldsymbol {a}) |.
$$
In words, a unilateral deviation can only impact a player's utility by an additive  $\epsilon$ . Now, as long as the sensitivity decays fast enough, a proof analogous to that of Theorem 3.1 implies the following.
Theorem 3.6. Consider an  $n$ -player game  $\mathcal{G}_n$  with sensitivity  $\epsilon_n \in \mathbb{R}_{>0}$ . Then,  $T = O_n(n)$  iterations of OGDP suffice to obtain a  $\left(\frac{1}{\sqrt{\delta}} O_n(\epsilon_n \sqrt{n}), \delta\right)$ -weak Nash equilibrium, for  $\delta \in (0,1)$ .
In particular, Theorem 3.6 yields an  $(o_n(1), o_n(1))$ -weak Nash equilibrium as long as  $\epsilon_n = o_n(\sqrt{n})$ . Further, in the canonical regime where  $\epsilon_n = O_n(1/n)$ , Theorem 3.6 circumscribes the best response gap for all but a constant number of players (by taking  $\delta = O_n(1/n)$ ). There are many natural settings where we should expect results such as Theorem 3.6 to be applicable [57, 58, 69]. We find it conceptually compelling that our framework can provide in a unifying way equilibrium guarantees for two seemingly disparate classes of games, namely two-player zero-sum games and games with vanishing strategic sensitivity.
In a similar vein, Feldman et al. [39] showed that  $\rho_{n} \to 1$  with a rate of  $1 / \sqrt{n}$  in a general auction design problem (see also [23]) under the relatively mild assumption that each player participates in the market with some constant probability (aka. probabilistic demand), thereby bypassing known barriers regarding the inefficiency of equilibria in general combinatorial domains. Their proof is based on the fact that each bidder's impact on the prices—under a simultaneous uniform-price auction format—becomes asymptotically negligible, in the spirit of Definition 3.5 introduced above. The difficulty that arises in that setting is that the natural representation of the utility functions violates our multilinearity assumption (postulated in Section 2). Instead, one would have to resort to some form of discretization before attempting to apply Theorem 3.1, which could be computationally prohibitive; the other prerequisite is that  $L_{n} = o_{n}(\sqrt{n})$ , for which our approach in Lemma A.8 in conjunction with the insights of Feldman et al. [39] could be useful. Understanding whether our techniques can be applied in the combinatorial auction setting of Feldman et al. [39] is left as a challenging open question.
Efficiency of equilibria does not suffice for tractability A natural question arising from Theorem 3.1 is whether a similar statement applies under the assumption that  $\mathsf{PoA} \to 1$ , that is, assuming that all Nash equilibria of  $\mathcal{G}$  are (approximately) fully efficient. This is clearly a weaker assumption, but it is unfortunately not sufficient to yield any non-trivial guarantees even in normal-form games:
Proposition 3.7. Even under the promise that  $\mathsf{PoA}_{\mathcal{G}} = 1$ , computing a  $(1 / \mathrm{poly}(\mathcal{G}))$ -Nash equilibrium in normal-form games in polynomial time is impossible when  $n \geq 3$ , unless  $\mathsf{PPAD} \subseteq \mathsf{P}$ .
This stands in contrast to the class of  $(1 + \mu, \mu)$ -smooth games, where a fully polynomial-time approximation scheme (FPTAS) is implied by Theorem 3.1—assuming access to a utility and a projection oracle, both of which are available in, for example, most succinct normal-form games. Proposition 3.7 is a straightforward consequence of the fact that Nash equilibria are hard to compute even in constant-sum 3-player games [20]. Furthermore, in Example A.11 we identify a specific 3-player game in which OGD fails (unconditionally) to converge to  $\epsilon$ -Nash equilibria for a constant value of  $\epsilon > 0$ , even though  $\mathsf{PoA} = 1$ . Our example is based on a variant of Shapley's game [99].
Another notable advantage of smoothness as a criterion of convergence is that, at least when the game is represented explicitly, it is easy to compute; this is in contrast to PoA, whose identification even in two-player games is NP-hard (Proposition A.12).
Extensions It turns out that smoothness per Definition 2.1 can be further sharpened using a primal-dual framework [75]. Such refined guarantees can also be translated into our setting (in the context of Theorem 3.1), as we elaborate on in Appendix A.3. The upshot is that the modification of Nadav and Roughgarden [75] necessitates analyzing the weighted sum of the players' regrets  $\sum_{i=1}^{n} z_i \mathrm{Reg}_i^{(T)}$  under a dual set of variables  $\{z_i\}_{i=1}^n$ . Another interesting extension worth noting relies on the local smoothness framework [92, 78], as we explain in Appendix A.4; the key observation is that local smoothness per Nguyen [78] can be associated with a linearized notion of regret, at which point the analysis of Theorem 3.1 readily carries over. Finally, we also expand our scope to Bayesian mechanisms, as we expound in the upcoming subsection.
Before we proceed, it is important to point out that, unsurprisingly, smoothness is not merely enough to guarantee convergence of OGD; see Example A.13. It is instead crucial to additionally ensure that  $\rho \approx 1$  in order to obtain interesting guarantees for the behavior of algorithms such as OGD.
3.1 Bayesian mechanisms
Next, we leverage the connection between smoothness and convergence to Nash equilibria to extend the scope of our framework to Bayesian mechanisms. In particular, analogously to Definition 2.1, Syrgkanis and Tardos [102] have introduced the notion of a smooth mechanism (Definition A.14); detailed background on Bayesian mechanisms and smoothness in that realm is provided in Appendix A.7. In this context, we leverage a reduction of Hartline et al. [53] from an incomplete-information to a complete-information mechanism to arrive at the following theorem. (The standard notion of a Bayes-Nash equilibrium is analogous to Definition 2.2, and is recalled in Definition A.15 of Appendix A.7.)
Theorem 3.8. Consider a Bayesian mechanism  $\mathcal{M}$  such that  $\rho_{\mathcal{M}} = 1$ . Then, for any  $\epsilon > 0$ ,  $T = O(1 / \epsilon^2)$  iterations of OGD suffice to obtain an  $\epsilon$ -Bayes-Nash equilibrium of  $\mathcal{M}$ .
In particular, OGD above is executed on the so-called agent-form representation of  $\mathcal{M}$  (Appendix A.7). Analogously to Theorem 3.1, the above theorem can also be extended in the large  $n \gg 1$  under the assumption that  $\rho_n \to 1$ . It is worth noting that Theorem 3.1 already can be applied to certain games of incomplete information (such as imperfect-information extensive-form games), but Theorem 3.8 additionally makes a connection with the literature on smoothness in mechanism design, which facilitates characterizing the smoothness parameters.
4 Improved welfare for no-regret dynamics
Roughgarden's seminal work [90] established that no-regret learning algorithms always attain asymptotically at least rPoA fraction of the optimal social welfare (on average). This guarantee is satisfactory for many classes of games where rPoA is close to 1 (emphatically those studied earlier in Section 3), but smoothness is certainly not a universal phenomenon: there are simple games in which the smoothness framework only provides vacuous guarantees; one such example is Shapley's game, discussed in Appendix A.10. As a result, one important question arising is whether it is possible to improve the efficiency bound predicted by smoothness when specific learning algorithms are in place, while at the same time still guaranteeing convergence to the set of coarse correlated equilibria (CCE). We stress that optimizing over the set of CCE is typically NP-hard in succinct games [79, 6], making this question interesting also from a complexity-theoretic standpoint.
In this section, we show that it is indeed possible to obtain improved efficiency bounds under a generic condition, while at the same time guaranteeing the no-regret property for each player. A key ingredient in our improvement is the use of clairvoyant mirror descent, an algorithm recently introduced by Piliouras et al. [85]. More precisely, we will instantiate that algorithm with (squared) Euclidean regularization, which can be defined as follows. Let  $\Pi_{\boldsymbol{x}_i}(\boldsymbol{u}_i) \coloneqq \arg \max_{\boldsymbol{x}_i' \in \mathcal{X}_i} \left\{\langle \boldsymbol{x}_i', \boldsymbol{u}_i \rangle - \frac{1}{2} \| \boldsymbol{x}_i - \boldsymbol{x}_i' \|_2^2 \right\}$  be the induced prox operator, where  $\boldsymbol{x}_i \in \mathcal{X}_i$  and  $\boldsymbol{u}_i \in \mathbb{R}^{d_i}$ . Clairvoyant gradient descent (henceforth CGD) at time  $t \in \mathbb{N}$  outputs  $\boldsymbol{x}^{(t)} = \Pi_{\boldsymbol{x}^{(t-1)}} (\eta F(\boldsymbol{w}^{(t)})) \coloneqq (\Pi_{\boldsymbol{x}_1^{(t-1)}} (\eta \boldsymbol{u}_1(\boldsymbol{w}_{-1}^{(t)})), \ldots, \Pi_{\boldsymbol{x}_n^{(t-1)}} (\eta \boldsymbol{u}_n(\boldsymbol{w}_{-n}^{(t)})))$ , where  $\boldsymbol{w}^{(t)}$  is any  $\epsilon^{(t)}$ -approximate fixed point of the map  $\prod_{i=1}^n \mathcal{X}_i \ni \boldsymbol{w} \mapsto \Pi_{\boldsymbol{x}^{(t-1)}} (\eta F(\boldsymbol{w}))$ , and  $\boldsymbol{x}^{(0)} \in \mathcal{X}$  is an arbitrary initialization. It turns out that for  $\eta < 1/L$ , this map is a contraction [37, 85, 19], thereby making approximate fixed points easy to compute. Furthermore, there is also an uncoupled implementation of CGD [85], making the algorithm compelling from a decentralized standpoint as well, but we will not dwell on this issue here. We are now ready to state the main result of this section.
Theorem 4.1. Suppose that all players are updating their strategies using CGD with  $\epsilon^{(t)}\leq \frac{\min_iDx_i}{t^2}$  and learning rate  $\eta = \frac{1}{2L}$  in a  $(\lambda ,\mu)$ -smooth game  $\mathcal{G}$ , where  $L$  is the Lipschitz-continuity parameter of  $F$ . Then, for any  $\epsilon_0 > 0$  and  $T\geq \frac{64L^2D_x^4}{\epsilon_0^2}$  iterations,
1. the average correlated distribution of play is a  $\frac{4LD_X^2}{T} - CCE$ ;
2. there is a time  $t^{\star}\in [[T]]$  such that
$$
\operatorname {S W} \left(\boldsymbol {x} ^ {(t ^ {\star})}\right) \geq \sup  _ {\epsilon \geq \epsilon_ {0}} \min  \left\{\rho_ {\mathcal {G}} (\lambda , \mu) \cdot \mathrm {O P T} _ {\mathcal {G}} + \frac {\epsilon^ {2}}{1 6 (\mu + 1) L D _ {\chi} ^ {2}}, \mathrm {P o A} _ {\mathcal {G}} ^ {\epsilon} \cdot \mathrm {O P T} _ {\mathcal {G}} \right\}. \tag {3}
$$
This is the first result that establishes simultaneously these properties under a computationally efficient algorithm, improving a recent work [1] (see also [42, 67] for related results) that failed to
guarantee convergence to CCE. In particular, that earlier work was analyzing OGD, and as it turns out, under a time-invariant learning rate  $\eta$  it is not even known whether OGD ensures sublinear per-player regret, let alone constant (as in Corollary 4.3). The basic ingredient to this improvement is a new property of CGD, which we explain below. Before we sketch the proof, we note that Item 2 above can be readily strengthened so that the improvement holds for the average welfare of most of the strategies, not just a single one (Remark A.22).
Proof sketch of Theorem 4.1 The key step in the proof is showing (in Corollary A.21) that CGD satisfies the remarkable per-player regret bound  $\mathrm{Reg}_i^{(T)}\leq \alpha -\gamma \sum_{t = 1}^T\left(\mathrm{BRGAP}_i(\pmb{x}^{(t)})\right)^2$  where  $\alpha >0$  depends on the approximation error of the fixed points of CGD—and can be made time-invariant with only an  $O(\log T)$  per-iteration overhead—and  $\gamma >0$ . To do this, we crucially rely on a certain property of the Euclidean regularizer (Lemma A.20), which we use in conjunction with the analysis of Farina et al. [37] who extended the original argument of Piliouras et al. [85] beyond entropic regularization.
It is worth noting that the above per-player regret bound (Corollary A.21) implies that a player with nonnegative regret will be almost always approximately best responding, a rather singular occurrence in the context of learning in games; this has interesting implications and goes well-beyond what is currently known for OGD. In particular, it is an open question whether Theorem 4.1 holds under OGD.
Next, we shall describe a concrete implication of Theorem 4.1 under a generic condition. To do so, let us denote by  $\mathsf{PoA}_{\mathcal{G}}^{\epsilon}$  the price of anarchy in  $\mathcal{G}$  with respect to the worst-case  $\epsilon$ -Nash equilibrium (so that  $\mathsf{PoA}_{\mathcal{G}}^{0} \equiv \mathsf{PoA}_{\mathcal{G}}$ ).
Condition 4.2. Consider a game  $\mathcal{G}$  and some game-dependent parameter  $C = C(\mathcal{G}) > 0$ . There exists an  $\epsilon_0 > 0$  such that  $\mathsf{P o A}_{\mathcal{G}}^{\epsilon_0} > \mathsf{r P o A}_{\mathcal{G}} + \epsilon_0^2 C$ .
Naturally, it is always the case that  $\mathsf{PoA}_{\mathcal{G}} \geq \mathsf{rPoA}_{\mathcal{G}}$ . Further,  $\mathsf{rPoA}_{\mathcal{G}}$  is in general strictly smaller since it measures the worst-case welfare over a larger set than  $\mathsf{PoA}_{\mathcal{G}}$  (even broader than outcomes of no-regret learning); Figure 1 in Appendix A.8 further corroborates this premise in a sequence of random normal-form games. Now assuming that  $\mathsf{PoA}_{\mathcal{G}} > \mathsf{rPoA}_{\mathcal{G}}$ , Condition 4.2 is met if  $\lim_{\epsilon \to 0} \mathsf{PoA}_{\mathcal{G}}^{\epsilon} = \mathsf{PoA}_{\mathcal{G}}$ , a mild continuity condition (see, for example, the discussion by Roughgarden [89]).
Corollary 4.3. Consider a  $(\lambda, \mu)$ -smooth game  $\mathcal{G}$  that satisfies Condition 4.2 under some  $\epsilon_0 > 0$ . Then, CGD after  $T \geq \frac{64L^2D_x^4}{\epsilon_0^2}$  iterations and  $\eta = \frac{1}{2L}$  satisfies the following:
1. the average correlated distribution of play is an  $O\left(\frac{1}{T}\right)$ -CCE;
2. there is a time  $t^{\star}\in \llbracket T\rrbracket$  and  $C^\prime (\mathcal{G}) > 0$  such that  $\mathrm{SW}(\pmb{x}^{(t^{\star})})\geq (\rho_{\mathcal{G}}(\lambda ,\mu) + \epsilon_0^2 C^\prime (\mathcal{G}))\mathrm{OPT}_{\mathcal{G}}$
A fundamental question that arises from Theorem 4.1 is whether there exists a computationally efficient algorithm that determines a CCE with social welfare at least a PoA fraction of the optimal welfare. In games where  $\mathsf{PoA} = \mathsf{rPoA}$  this is clearly possible; in contrast, while Theorem 4.1 improves over the smoothness bound, it does not always guarantee welfare up to PoA. This is a central question in light of the intractability of Nash equilibria [30, 20], which has indeed served as a primary critique to the literature quantifying the price of anarchy of Nash equilibria [90].
Another promising avenue to improving the welfare predicted by the smoothness framework revolves around eliminating certain strategy profiles by arguing that they are reached with negligible probability. For example, in Appendix A.10 we identify an example where iteratively eliminating strictly dominated actions can improve the predictive power of the smoothness framework.
5 Further related work