SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We will connect this concept of local smoothness with the linearization of the players' regrets. In particular, the utility of a player  $i \in [[n]]$  under a joint strategy  $\pmb{x}$  is defined as  $u_{i}(\pmb{x}) \coloneqq \sum_{e \in E} x_{i}[e] u_{e} \left( \sum_{i=1}^{n} x_{i}[e] \right)$ . Then,
$$
\frac {\partial u _ {i} (\boldsymbol {x})}{\partial \boldsymbol {x} _ {i} [ e ]} = u _ {e} \left(\sum_ {i = 1} ^ {n} \boldsymbol {x} _ {i} [ e ]\right) + \boldsymbol {x} _ {i} [ e ] u _ {e} ^ {\prime} \left(\sum_ {i = 1} ^ {n} \boldsymbol {x} _ {i} [ e ]\right).
$$
As a result, for any  $t\in \mathbb{N}$
$$
\begin{array}{l} \sum_ {i = 1} ^ {n} \langle \boldsymbol {x} _ {i} ^ {\star} - \boldsymbol {x} _ {i} ^ {(t)}, \nabla_ {\boldsymbol {x} _ {i}} u _ {i} (\boldsymbol {x} ^ {(t)}) \rangle = \sum_ {e \in E} u _ {e} (\boldsymbol {x} [ e ]) \boldsymbol {x} ^ {\star} [ e ] - \sum_ {e \in E} u _ {e} (\boldsymbol {x} [ e ]) \boldsymbol {x} [ e ] \\ + \sum_ {e \in E} \sum_ {i = 1} ^ {n} \boldsymbol {x} _ {i} [ e ] (\boldsymbol {x} _ {i} ^ {\star} [ e ] - \boldsymbol {x} _ {i} [ e ]) u _ {e} ^ {\prime} (\boldsymbol {x} [ e ]), \\ \end{array}
$$
with the understanding that  $\pmb{x}[e] := \sum_{i=1}^{n} \pmb{x}_i[e]$  and  $\pmb{x}^{\star}[e] := \sum_{i=1}^{n} \pmb{x}_i^{\star}[e]$  for any  $e \in E$ . Now let us define  $\mathrm{Reg}_{\mathcal{L},i}^{(T)}(\pmb{x}_i^{\star}) := \sum_{t=1}^{T} \langle \pmb{x}_i^{\star} - \pmb{x}_i^{(t)}, \nabla_{\pmb{x}_i} u_i(\pmb{x}) \rangle$ . Combining the last displayed equality with local smoothness (22), we get that
$$
\begin{array}{l} \sum_ {i = 1} ^ {n} \operatorname {R e g} _ {\mathcal {L}, i} ^ {(T)} \left(\boldsymbol {x} _ {i} ^ {\star}\right) \geq \lambda \sum_ {e \in E} \boldsymbol {x} ^ {\star} [ e ] u _ {e} \left(\boldsymbol {x} ^ {\star} [ e ]\right) - (\mu + 1) \sum_ {e \in E} \boldsymbol {x} [ e ] u _ {e} \left(\boldsymbol {x} [ e ]\right) \\ = \lambda \operatorname {S W} \left(\boldsymbol {x} ^ {\star}\right) - (\mu + 1) \operatorname {S W} (\boldsymbol {x}). \tag {23} \\ \end{array}
$$
Consequently, if we define a local smoothness bound,  $\rho_{\mathcal{L}} \coloneqq \frac{\lambda}{1 + \mu}$  with  $(\lambda, \mu)$  being subject to (22), Theorem 3.1 can be readily extended in the regime  $\rho_{\mathcal{L}} \to 1$  based on (23).
A.5 Considerations based on PoA
It is natural to ask if the conclusion of Theorem 3.1 can be relaxed to  $\mathrm{PoA} \to 1$ . Here, we point out that such an assumption does not suffice to obtain interesting guarantees. In particular, we next note Proposition 3.7, stated earlier in the main body, which is an immediate byproduct of the hardness of computing Nash equilibria in constant-sum games.
Proof. Chen et al. [20] showed that computing a  $(1 / \mathrm{poly}(\max_i|\mathcal{A}_i|))$ -Nash equilibrium in a general-sum two-player game in normal form is PPAD-hard. As a result, the same applies in constant-sum 3-player games by suitably incorporating an additional player who has no strategic impact on the game. Further, in any constant-sum game  $\mathcal{G}$  it clearly holds that  $\mathrm{PoA}_{\mathcal{G}} = 1$ , concluding the proof.
Remark A.10. The positive result established in Theorem A.2 requires that  $\rho \approx 1$  under a bounded pair of smoothness parameters  $(\lambda, \mu)$ . One may wonder whether similar conclusions apply even
when the smoothness parameters are unbounded. In particular, the robust price of anarchy (rPoA) can be defined as the solution to the following linear program.
maximize  $\rho$
$$
\text {s u b j e c t} \quad \begin{array}{l} z \sum_ {i = 1} ^ {n} \left(u _ {i} \left(a _ {i} ^ {\star}, \boldsymbol {a} _ {- i}\right) - u _ {i} (\boldsymbol {a})\right) \geq \rho \mathrm {S W} \left(\boldsymbol {a} ^ {\star}\right) - \mathrm {S W} (\boldsymbol {a}), \boldsymbol {a} \in \prod_ {i = 1} ^ {n} \mathcal {A} _ {i}, \end{array} \tag {24}
$$
Above  $\pmb{a}^{\star} = (a_{1}^{\star},\dots ,a_{n}^{\star})\in \prod_{i = 1}^{n}\mathcal{A}_{i}$  is a welfare-maximizing action profile, which can be assumed to be unique for the purposes of our discussion here. In this context, can we extend the conclusion of Theorem A.2 under the assumption that  $\mathsf{rPoA}_{\mathcal{G}_n}\to 1$ ? In general, that is not possible. Indeed, in any constant-sum game one can make the LP (24) feasible by taking  $z = 0$  and  $\rho = 1$ ; that is,  $\mathsf{rPoA}_{\mathcal{G}_n} = 1$  for any constant-sum game  $\mathcal{G}$ . Thus, assuming merely that  $\mathsf{rPoA}_{\mathcal{G}}\to 1$  is not enough to obtain interesting guarantees for computing Nash equilibria (in accordance with Proposition 3.7). In other words, our underlying assumption that the smoothness parameters are bounded is necessary. To further explain this discrepancy, we note that a constant-sum game is generally not  $(1 + \mu ,\mu)$ -smooth (for a finite  $\mu > - 1$ ), for otherwise any constant-sum game would satisfy the Minty property (Proposition 3.3), which would in turn contradict Proposition 3.7. The case where  $z = 0$  in any optimal solution of (24) essentially corresponds to a pathological manifestation of smoothness in which the underlying parameters are unbounded. We stress again that throughout this paper, when we say that a game is  $(\lambda ,\mu)$ -smooth we, of course, posit that those smoothness parameters are finite. Further, when we say that  $\rho_{\mathcal{G}} = 1$  in a game  $\mathcal{G}$ , we accept that there are finite smoothness parameters associated with  $\rho_{\mathcal{G}}$ . With this convention, we reiterate that there are games in which  $\mathsf{rPoA}_{\mathcal{G}}\neq \rho_{\mathcal{G}}$ .
Next, we provide a concrete example based on Shapley's game [99] in which OGD (unconditionally) fails to converge to an  $\epsilon$ -Nash equilibrium, for a constant  $\epsilon > 0$ .
Example A.11. We consider a 3-player game  $\mathcal{G}$  defined as follows. We let
$$
\mathbf {A} := \left[ \begin{array}{l l l} 1 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & 1 \end{array} \right], \mathbf {B} := \left[ \begin{array}{l l l} 1 & 2 & 1 \\ 1 & 1 & 2 \\ 2 & 1 & 1 \end{array} \right]. \tag {25}
$$
Then, we set  $u_{1}(\pmb{x}_{1},\pmb{x}_{2},\pmb{x}_{3}) \coloneqq \pmb{x}_{1}^{\top}\mathbf{A}\pmb{x}_{2}$ ,  $u_{2}(\pmb{x}_{1},\pmb{x}_{2},\pmb{x}_{3}) \coloneqq \pmb{x}_{1}^{\top}\mathbf{B}\pmb{x}_{2}$ , and  $u_{3}(\pmb{x}_{1},\pmb{x}_{2},\pmb{x}_{3}) \coloneqq 3 - \pmb{x}_{1}^{\top}\mathbf{A}\pmb{x}_{2} - \pmb{x}_{1}^{\top}\mathbf{B}\pmb{x}_{2}$ . Thus, for any joint strategy  $(\pmb{x}_{1},\pmb{x}_{2},\pmb{x}_{3})$  it holds that  $\mathrm{SW}(\pmb{x}_{1},\pmb{x}_{2},\pmb{x}_{3}) = 3$ , implying that  $\mathrm{PoA}_{\mathcal{G}} = 1$ . Further,  $\mathcal{G}$  is in normal form. Now, through a numerical simulation we draw the following conclusion: Although  $\mathrm{PoA}_{\mathcal{G}} = 1$ , for  $T \gg 10\mathrm{GD}$  with learning rate  $\eta \coloneqq 0.01$  and initialization  $(\hat{\pmb{x}}_1^{(1)},\hat{\pmb{x}}_2^{(1)},\cdot) \coloneqq ((0.5,0.25,0.25),(0.25,0.5,0.25),\cdot)$  satisfies  $\mathrm{NEGAP}(\pmb{x}^{(t)}) \geq 0.1875$  for any  $t \in [T]$ , where  $(\pmb{x}^{(t)})_{t\geq 1}$  is the sequence of iterates produced by  $0\mathrm{GD}$ . We note that here we do not consider the initialization from the uniform distribution simply because that happens to be the unique Nash equilibrium in Shapley's game; the conclusion above readily applies for any initialization by suitably modifying the underlying game. We also note that the specific value for the learning rate specified above is used for concreteness, and the conclusion is not tied to that specific value.
We next note another separation between the bound predicted by smoothness and PoA, which gives an additional reason why the smoothness framework is more suited as a criterion for determining tractability. Below, we make for simplicity the assumption that the game has a unique welfare-maximizing action profile, which can always be enforced by incorporating an arbitrarily small noise in the players' utilities.
Proposition A.12. Determining whether a game is  $(\lambda, \mu)$ -smooth can be done in polynomial time in explicitly represented normal-form games. In contrast, even in two-player games, determining PoA is NP-hard.
Proof. First of all, the welfare-maximizing action profile  $\pmb{a}^{\star} \in \prod_{i=1}^{n} \mathcal{A}_{i}$  can be trivially computed in polynomial time (in the size of the input) since the game is explicitly represented. For a legitimate  $(\lambda, \mu)$  pair, determining whether the game is  $(\lambda, \mu)$ -smooth can be phrased as a feasibility linear program, with a number of constraints equal to the number of possible joint action profiles, each corresponding to a separate constraint in (1); namely,  $\sum_{i=1}^{n} u_{i}(a_{i}^{\star}, \pmb{a}_{-i}) \geq \lambda \mathrm{OPT} - \mu \sum_{i=1}^{n} u_{i}(\pmb{a})$ . As a result, the number of constraints is polynomial in the description of the game (since it is assumed that the game is explicitly represented). Furthermore, one can optimize over the smoothness parameters by considering the LP (of polynomial size) given in (24), which determines the robust price
of anarchy (rPoA). In accordance with Theorem A.2 (see Remark A.10), one can also incorporate the constraint  $z \geq 1 / \mathrm{poly}(\mathcal{G})$ . Regarding the second claim, hardness of determining PoA even in two-player games follows directly from the reduction of Conitzer and Sandholm [27, Theorem 1]. In particular, an algorithm computing PoA would enable determining the satisfiability of a SAT formula.
One important question is whether smoothness can be identified in polynomial time even in succinctly represented games [79]. Indeed, the obvious algorithm for identifying smoothness described above requires a number of constraints that scales exponentially with the number of players, which is especially problematic in the regime of large games we focus on in Section 3.
We need to clarify, however, that knowing the smoothness parameters is certainly not a prerequisite for applying our approach. In detail, one can first of all apply Theorem 3.1 in classes of games where there are available analytical bounds for  $\rho_{n}$  as a function of  $n$ , obviating the need to determine whether  $\rho_{n}$  is close to 1. Besides this point, and more importantly, one can always execute the algorithm prescribed by Theorem 3.1—which does not require any knowledge regarding the smoothness parameters—and then efficiently evaluate the solution quality (per Definition 2.2). If the desired accuracy has been reached, then this is precisely the initial goal; otherwise, we can safely assume that the preconditions of Theorem 3.1 are not met.
A.6 Smoothness does not suffice for convergence
Next, we provide for completeness an example of a smooth game where OGD fails to converge to Nash equilibria. This shows that, as expected, it is not merely enough to know that  $\rho \neq 0$  to obtain interesting guarantees.
Example A.13. This example is based on a bimatrix game in normal form described with the payoff matrices
$$
\mathbf {A} = \left[ \begin{array}{l l l l} 0. 2 & 0. 8 & 0. 9 & 0. 3 \\ 0. 2 & 0. 8 & 0. 2 & 0. 3 \\ 0. 9 & 0. 2 & 0. 4 & 0. 4 \\ 0. 6 & 0. 9 & 0. 3 & 0. 1 \end{array} \right], \mathbf {B} = \left[ \begin{array}{l l l l} 0. 4 & 0. 2 & 0 & 0. 1 \\ 0. 5 & 0 & 0. 2 & 0. 8 \\ 0. 7 & 0. 8 & 0 & 0. 4 \\ 0 & 0 & 0. 1 & 0. 4 \end{array} \right]. \tag {26}
$$
We first claim that this bimatrix game  $\mathcal{G}$  satisfies  $\rho_{\mathcal{G}} \geq 0.125$ . Indeed, we first see that the welfare-maximizing profile of (26) reads  $(\pmb{x}_1^{\star}, \pmb{x}_2^{\star}) = ((0,0,1,0), (1,0,0,0))$ . We also claim that for any pair of actions  $a_1 \in \mathcal{A}_1$  and  $a_2 \in \mathcal{A}_2$  it holds that
$$
\sum_ {i = 1} ^ {2} u _ {i} \left(\boldsymbol {x} _ {i} ^ {\star}, a _ {- i}\right) \geq 0. 1 2 5 \cdot \operatorname {O P T} _ {\mathcal {G}},
$$
where  $\mathrm{OPT}_{\mathcal{G}} = 1.6$ . As a result, it follows that for any  $\pmb{x}_1 \in \Delta(\mathcal{A}_1)$  and  $\pmb{x}_2 \in \Delta(\mathcal{A}_2)$  it holds that  $\mathbb{E}_{a_1 \sim \pmb{x}_1, a_2 \sim \pmb{x}_2}[\sum_{i=1}^{2} u_i(\pmb{x}_i^\star, a_{-i})] \geq 0.125 \cdot \mathrm{OPT}_{\mathcal{G}}$ , in turn implying that  $\sum_{i=1}^{2} u_i(\pmb{x}_i^\star, \pmb{x}_{-i}) \geq 0.125 \cdot \mathrm{OPT}_{\mathcal{G}}$ . This means that  $\mathcal{G}$  is (0.125, 0)-smooth. Furthermore, through a numerical simulation we draw the following conclusion: for  $T \gg 1$ , OGD with learning rate  $\eta := 0.01$  satisfies  $\mathrm{NEGAP}(\pmb{x}_1^{(t)}, \pmb{x}_2^{(t)}) \geq 0.046$  for any  $t \in [T]$ , where  $(\pmb{x}_1^{(t)}, \pmb{x}_2^{(t)})_{t \geq 1}$  is the sequence of iterates produced by OGD. Once again, it is worth noting that this conclusion is not tied to the specific choice of learning rate we chose above, which is only used for concreteness.
A.7 Bayesian mechanisms
In this subsection, we consider the standard independent private value model of Bayesian mechanisms. In particular, each player  $i \in [[n]]$  has a type  $v_i$  drawn from a distribution  $\mathcal{F}_i$  over a finite set of types  $\mathcal{V}_i$ ; without any loss, we may assume that  $\mathcal{F}_i$  is the uniform distribution over  $\mathcal{V}_i$ . It is further assumed that players' types are pairwise independent. After each player  $i \in [[n]]$  draws a type  $v_i \sim \mathcal{F}_i$ ,  $i$  selects an action  $a_i(v_i) \in \mathcal{A}_i$ , which is a function of its type  $v_i$ . Now consider a fixed profile of types  $\mathbf{v} \in \mathcal{V} := \mathcal{V}_1 \times \mathcal{V}_2 \dots \mathcal{V}_n$ . The (expected) utility of Player  $i$  under a joint strategy profile  $\mathbf{x}(\mathbf{v})$  is denoted by  $u_i(\mathbf{x}; v_i) := \mathbb{E}_{\mathbf{a} \sim \mathbf{x}}[u_i(\mathbf{a}; v_i)]$ . There is also a principal agent who does not take an action in the game, and whose utility under  $\mathbf{x}$  is given by  $R(\mathbf{x}) := \mathbb{E}_{\mathbf{a} \sim \mathbf{x}}[R(\mathbf{a})]$ . Accordingly, the social welfare is defined as  $\mathrm{SW}(\mathbf{x}, \mathbf{v}) := \sum_{i=1}^{n} u_i(\mathbf{x}; v_i) + R(\mathbf{x})$ , while  $\mathrm{OPT}_{\mathcal{M}}(\mathbf{v})$  represents the optimal social welfare of mechanism  $\mathcal{M}$  as a function of the joint type  $\mathbf{v} \in \mathcal{V}$ . We will make the assumption that the utility functions assign solely nonnegative values.
Smooth mechanisms Analogously to the notion of a smooth game (Definition 2.1), Syrgkanis and Tardos [102] introduced the notion of a smooth mechanism, formally recalled below.
Definition A.14 (Smooth mechanism [102]). A mechanism  $\mathcal{M}$  is  $(\lambda, \mu)$ -smooth, where  $\lambda, \mu \geq 0$ , if there exists an strategy profile  $\pmb{x}^{\star}(\pmb{v}) \in \prod_{i=1}^{n} \Delta(\mathcal{A}_i)$ , for every type profile  $\pmb{v} \in \mathcal{V}$ , such that for any action profile  $\pmb{a} \in \mathcal{A}$  and type profile  $\pmb{v} = (v_1, \dots, v_n) \in \mathcal{V}$ ,
$$
\sum_ {i = 1} ^ {n} u _ {i} \left(\boldsymbol {x} _ {i} ^ {\star} (\boldsymbol {v}), \boldsymbol {a} _ {- i}; v _ {i}\right) \geq \lambda \operatorname {O P T} _ {\mathcal {M}} (\boldsymbol {v}) - \mu R (\boldsymbol {a}). \tag {27}
$$
As it turns out, many important mechanisms satisfy the above definition under various parameters  $(\lambda ,\mu)$  [102]. For a  $(\lambda ,\mu)$ -smooth mechanism  $\mathcal{M}$ , we define  $\rho_{\mathcal{M}}\coloneqq \frac{\lambda}{\max\{1,\mu\}}$
Definition A.15. Let  $\pmb{x}_i: \mathcal{V}_i \to \Delta(\mathcal{A}_i)$  be the strategy of each player  $i \in [[n]]$ . A joint strategy profile  $\pmb{x}$  is a Bayes-Nash equilibrium (BNE) if for any player  $i \in [[n]]$ , any type  $v_i \in \mathcal{V}_i$  and deviation  $a_i' \in \mathcal{A}_i$ ,
$$
\mathbb {E} _ {\boldsymbol {v} _ {- i} \sim \mathcal {F} _ {- i}} [ u _ {i} (\boldsymbol {x} (\boldsymbol {v}); v _ {i}) ] \geq \mathbb {E} _ {\boldsymbol {v} _ {- i} \sim \mathcal {F} _ {- i}} [ u _ {i} \left(a _ {i} ^ {\prime}, \boldsymbol {x} _ {- i} (\boldsymbol {v} _ {- i}); v _ {i}\right) ]. \tag {28}
$$
This definition coincides with the standard notion of Nash equilibrium we saw in Definition 2.2 when each distribution  $\mathcal{F}_i$  is a point mass. We also note that an  $\epsilon$ -BNE incorporates an  $\epsilon \geq 0$  additive slackness in (28).
Population interpretation of Bayesian games In the agent-form representation of a Bayesian game [53], it is assumed that there are  $n$  finite subpopulations of players, each corresponding to a player  $i \in \mathbb{I}[n]$ . Each player belonging to population  $i$  corresponds to a type  $v_{i} \in \mathcal{V}_{i}$ , which is distinct in each population and across populations. In this induced population game, nature first draws one player from each population, and then each player  $v_{i}$  selects an action  $a_{i}(v_{i})$ ; the game is then played under those selected actions. In symbols, the utility of Player  $v_{i}$  from population  $i$  reads
$$
u _ {i, v _ {i}} ^ {\mathrm {A G}} (\boldsymbol {a}) := \mathbb {E} _ {\boldsymbol {v} \sim \mathcal {F}} [ u _ {i} (\boldsymbol {a} (\boldsymbol {v}); v _ {i}) \mathbb {1} \left\{v _ {i} \right\} ], \tag {29}
$$
where by  $\{v_{i}\}$  above we denote the event that type  $v_{i}$  is selected by nature among the population corresponding to Player  $i$ . The importance of the population interpretation of the Bayesian game is that it induces a mechanism of complete information, which will be denoted by  $\mathcal{M}^{\mathrm{AG}} = \mathcal{M}^{\mathrm{AG}}(\mathcal{M})$ . The following characterization highlights an important connection between  $\mathcal{M}^{\mathrm{AG}}$  and  $\mathcal{M}$ .
Theorem A.16 ([53]). If a mechanism  $\mathcal{M}$  is  $(\lambda, \mu)$ -smooth, then the complete information mechanism  $\mathcal{M}^{AG} = \mathcal{M}^{AG}(\mathcal{M})$  is also  $(\lambda, \mu)$ -smooth.
We now proceed with the proof of Theorem 3.8. To keep the exposition self-contained, we will not make explicit use of Theorem A.16.
Proof. Under the assumption that  $\rho_{\mathcal{M}} = 1$ , it follows that there exists a pair  $(\lambda, \mu) \in \mathbb{R}_{\geq 0}^{2}$  such that  $\mathcal{M}$  is  $(\lambda, \mu)$ -smooth with  $\lambda = \max\{1, \mu\}$ . Further, by definition it holds that  $\mathrm{OPT}_{\mathcal{M}}(\pmb{v}) \geq \sum_{i=1}^{n} u_i(\pmb{a}; v_i) + R(\pmb{a})$ , for any action profile  $\pmb{a} \in \mathcal{A}$ . Combining with (27), we have that there exists a strategy profile  $\pmb{x}^{\star}(\pmb{v}) \in \prod_{i=1}^{n} \Delta(\mathcal{A}_i)$  such that for every type profile  $\pmb{v} \in \mathcal{V}$  and action profile  $\pmb{a} \in \mathcal{A}$ ,
$$
\sum_ {i = 1} ^ {n} u _ {i} \left(\boldsymbol {x} _ {i} ^ {\star} (\boldsymbol {v}), \boldsymbol {a} _ {- i}; v _ {i}\right) \geq \lambda \sum_ {i = 1} ^ {n} u _ {i} (\boldsymbol {a}; v _ {i}) + \lambda R (\boldsymbol {a}) - \mu R (\boldsymbol {a}) \geq \sum_ {i = 1} ^ {n} u _ {i} (\boldsymbol {a}; v _ {i}),
$$
since  $\lambda = \max \{1,\mu \}$  and utilities are nonnegative. As a result, for any  $\pmb {x}(\pmb {v})\in \prod_{i = 1}^{n}\Delta (\mathcal{A}_{i})$
$$
\sum_ {i = 1} ^ {n} \mathbb {E} _ {\boldsymbol {v} \sim \mathcal {F}} [ u _ {i} (\boldsymbol {x} _ {i} ^ {\star} (\boldsymbol {v}), \boldsymbol {x} _ {- i} (\boldsymbol {v} _ {- i}); v _ {i}) ] \geq \sum_ {i = 1} ^ {n} \mathbb {E} _ {\boldsymbol {v} \sim \mathcal {F}} [ u _ {i} (\boldsymbol {x} (\boldsymbol {v}); v _ {i}) ]. \tag {30}
$$
Further, by definition of the agent-form utilities (29) and the law of total expectation, (30) can be equivalently cast as
$$
\sum_ {i = 1} ^ {n} \sum_ {v _ {i} \in \mathcal {V} _ {i}} u _ {i, v _ {i}} ^ {\mathbf {A G}} \left(\boldsymbol {x} _ {i, v _ {i}} ^ {\star}, \boldsymbol {x} _ {- (i, v _ {i})}\right) \geq \sum_ {i = 1} ^ {n} \sum_ {v _ {i} \in \mathcal {V} _ {i}} u _ {i, v _ {i}} ^ {\mathbf {A G}} (\boldsymbol {x}),
$$
where we used the notation  $\boldsymbol{x}_{i,v_i} \coloneqq \boldsymbol{x}_i(v_i)$ . This implies that the sum of the players' regrets in the agent-form representation is nonnegative. The proof of Theorem 3.8 then follows from the correspondence between Nash equilibria in the agent-form representation of  $\mathcal{M}$  and Bayes-Nash equilibria in the original incomplete-information mechanism  $\mathcal{M}$ .
A.8 PoA vs robust PoA
In this subsection, we provide some additional justification for the condition  $rPoA \neq PoA$  required in Corollary 4.3. In particular, we conduct experiments on a set of random normal-form games. Some illustrative results for 10 random games are demonstrated in Figure 1. Overall, we observe that not only  $rPoA \neq PoA$ , but in fact the gap between the two quantities is typically substantial. This discrepancy is expected since, as we have already stressed,  $rPoA$  quantifies the worst-case welfare over a (typically) much broader set of equilibria.
Figure 1: PoA versus rPoA in random normal-form games.
A.9 Proof of Theorem 4.1
In this subsection, we provide the proof of Theorem 4.1. For completeness, we also include the proofs of certain results known from prior work [37, 85, 77].
To this end, we first recall that the prox operator associated with the (squared) Euclidean regularizer  $\frac{1}{2} \| \cdot \|_2^2$  is defined as
$$
\Pi_ {\boldsymbol {x} _ {i}} \left(\boldsymbol {u} _ {i}\right) := \arg \max  _ {\boldsymbol {x} _ {i} ^ {\prime} \in \mathcal {X} _ {i}} \left\{\left\langle \boldsymbol {x} _ {i} ^ {\prime}, \boldsymbol {u} _ {i} \right\rangle - \frac {1}{2} \| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {i} ^ {\prime} \| _ {2} ^ {2} \right\}, \tag {31}
$$
under some utility vector  $\pmb{u}_i\in \mathbb{R}^{d_i}$ . Accordingly, we let  $\Pi_{\pmb{x}}(\pmb {u})\coloneqq (\Pi_{\pmb{x}_1}(\pmb {u}_1),\dots ,\Pi_{\pmb{x}_n}(\pmb {u}_n))$ , where  $\pmb {u}\coloneqq (u_1,\ldots ,u_n)$ . With this definition in mind, we note the following property of the prox operator.
Lemma A.17 ([37, 77]). The prox operator  $\Pi_{\pmb{x}}(\cdot)$  is 1-Lipschitz continuous with respect to  $\| \cdot \| _2$  for any  $\pmb {x}\in \prod_{i = 1}^{n}\mathcal{X}_{i}$ .
Proof. Let  $i \in [[n]]$ . For any  $\pmb{u}_i, \pmb{u}_i' \in \mathbb{R}^{d_i}$  and  $\pmb{x}_i \in \mathcal{X}_i$
$$
\| \Pi_ {\boldsymbol {x} _ {i}} (\boldsymbol {u} _ {i}) - \Pi_ {\boldsymbol {x} _ {i}} (\boldsymbol {u} _ {i} ^ {\prime}) \| _ {2} = \| \mathcal {P} _ {\mathcal {X} _ {i}} (\boldsymbol {x} _ {i} + \boldsymbol {u} _ {i}) - \mathcal {P} _ {\mathcal {X} _ {i}} (\boldsymbol {x} _ {i} + \boldsymbol {u} _ {i} ^ {\prime}) \| _ {2} \leq \| \boldsymbol {u} _ {i} - \boldsymbol {u} _ {i} ^ {\prime} \| _ {2},
$$
where we used the fact that the projection operator is non-expansive with respect to  $\| \cdot \|_2$ , and that
$$
\begin{array}{l} \Pi_ {\boldsymbol {x} _ {i}} (\boldsymbol {u} _ {i}) = \arg \max  _ {\boldsymbol {x} _ {i} ^ {\prime} \in \mathcal {X} _ {i}} \left\{\langle \boldsymbol {x} _ {i} ^ {\prime}, \boldsymbol {u} _ {i} \rangle - \frac {1}{2} \| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {i} ^ {\prime} \| _ {2} ^ {2} \right\} = \arg \max  _ {\boldsymbol {x} _ {i} ^ {\prime} \in \mathcal {X} _ {i}} \left\{- \frac {1}{2} \| \boldsymbol {x} _ {i} ^ {\prime} \| _ {2} ^ {2} + \langle \boldsymbol {x} _ {i} ^ {\prime}, \boldsymbol {u} _ {i} + \boldsymbol {x} _ {i} \rangle \right\} \\ = \arg \min  _ {\boldsymbol {x} _ {i} ^ {\prime} \in \mathcal {X} _ {i}} \left\{\left\| \boldsymbol {x} _ {i} ^ {\prime} - \left(\boldsymbol {u} _ {i} + \boldsymbol {x} _ {i}\right) \right\| _ {2} ^ {2} \right\} \\ = \mathcal {P} _ {\boldsymbol {\chi} _ {i}} (\boldsymbol {x} _ {i} + \boldsymbol {u} _ {i}). \\ \end{array}
$$
This implies that  $\| \Pi_{\pmb{x}}(\pmb{u}) - \Pi_{\pmb{x}}(\pmb{u}^{\prime})\|_{2}\leq \| \pmb{u} - \pmb{u}^{\prime}\|$ , where  $\pmb {u} = (u_{1},\dots ,u_{n})$  and  $\pmb {u}' = (u_1',\ldots ,u_n')$
Using this lemma, a key observation for the analysis of clairvoyant mirror descent is the following contraction property [85, 37].
Proposition A.18 ([85, 37]). Suppose that  $F$  is  $L$ -Lipschitz continuous. For any  $\pmb{x}^{\prime}\in \prod_{i = 1}^{n}\mathcal{X}_{i}$  the function