Are Equivariant Equilibrium Approximators Beneficial?

Zhijian Duan¹, Yunxuan Ma¹, Xiaotie Deng^1,2
¹Center on Frontiers of Computing Studies, Peking University
²Center for Multi-Agent Research, Institute for AI, Peking University
{zjduan,charmingmyx,xiaotie}@pku.edu.cn

Abstract

Recently, remarkable progress has been made by approximating Nash equilibrium (NE), correlated equilibrium (CE), and coarse correlated equilibrium (CCE) through function approximation that trains a neural network to predict equilibria from game representations. Furthermore, equivariant architectures are widely adopted in designing such equilibrium approximators in normal-form games. In this paper, we theoretically characterize benefits and limitations of equivariant equilibrium approximators. For the benefits, we show that they enjoy better generalizability than general ones and can achieve better approximations when the payoff distribution is permutation-invariant. For the limitations, we discuss their drawbacks in terms of equilibrium selection and social welfare. Together, our results help to understand the role of equivariance in equilibrium approximators.

1 Introduction

The equivariant equilibrium property states that, given a Nash Equilibrium (NE) solution of a game, the permuted solution is also an NE for the game whose actions of representation are permuted in the same way. The same property also holds in correlated equilibrium (CE) and coarse correlated equilibrium (CCE), as well as the approximate solutions for all three solution concepts.

In this paper, we are interested in understanding the equivariant equilibrium property in designing neural networks that predict equilibria from game payoffs, following such recent approaches in designing equivariant equilibrium approximators (Feng et al., 2021; Marris et al., 2022) in normal-form games. Informally, such equivariant approximators keep the same permutation of the output strategies (represented as vectors or tensors) when the input game representations (e.g., the game payoff tensors) are permuted ¹¹1We will provide a formal definition of equivariance equilibrium approximators in Section 3. While equivariant approximators achieved empirical success, little work has theoretically discussed whether they are beneficial.

1.1 Our Contributions

We theoretically characterize benefits and limitations of equivariant NE, CE and CCE approximators. For the benefits, we first show that equivariant approximators enjoy better generalizability, where we evaluate the approximators using the maximum exploitability (Lockhart et al., 2019; Goktas and Greenwald, 2022) over all players. To get such a result, we derive the generalization bounds and the sample complexities of the NE, CE, and CCE approximators: The generalization bounds offer confidence intervals on the expected testing approximations based on the empirical training approximations; The sample complexities describe how many training samples the equilibrium approximators need to achieve desirable generalizability. The generalization bounds and sample complexities include the covering numbers (Shalev-Shwartz and Ben-David, 2014), which measure the representativeness of the approximators’ function classes. Afterward, we prove that the equivariant approximators have lower covering numbers than the general models, therefore have lower generalization bounds and sample complexities. We then show that the equivariant approximators can achieve better approximation when the payoff distribution is permutation-invariant.

As for the limitations, we find the equivariant approximators unable to find all the equilibria of some normal-form games. Such a result is caused by the limited representativeness of the equivariant approximators’ function class. Besides, we find that the equivariant NE approximator may lose social welfare. Specifically, in an example we constructed, while the maximum NE social welfare is large, the maximum social welfare of NEs that the equivariant NE approximators could find can be arbitrary close to zero. Such a negative result inspires us to balance generalizability and social welfare when we design the approximators’ architectures.

1.2 Further Related Work

Solving (approximate) NE, CE, and CCE for a single game are well studied (Fudenberg et al., 1998; Cesa-Bianchi and Lugosi, 2006). However, many similar games often need to be solved (harris2023metalearning) , both in practice and in some multi-agent learning algorithms (Marris et al., 2021; Liu et al., 2022). For instance, in repeated traffic routing games (Sessa et al., 2020), the payoffs of games depend on the capacity of the underlying network, which can vary with time and weather conditions. In repeated sponsored search auctions, advertisers value different keywords based on the current marketing environment (Nekipelov et al., 2015). In many multi-agent learning algorithms such as Nash Q-learning (Hu and Wellman, 2003), Correlated-Q learning (Greenwald et al., 2003), V-learning (Jin et al., 2022) and PSRO (Lanctot et al., 2017), an NE, CE or CCE of a normal-form game need to be solved in every update step.

In these settings, it is preferred to accelerate the speed of game solving by function approximation: Marris et al. (2022) introduces a neural equilibrium approximator to approximate CE and CCE for $n$ -player normal-form games; Feng et al. (2021) proposes a neural NE approximator in PSRO (Lanctot et al., 2017); Wu and Lisser (2022) designs a CNN-based NE approximator for zero-sum bimatrix games. Differentiable approximators have also been developed to learn QREs (Ling et al., 2018), NE in chance-constrained games (Wu and Lisser, 2023), and opponent’s strategy (Hartford et al., 2016).

Equivariance is an ideal property of the equilibrium approximator. We will discuss the literates of equivariant approximators after formally defining them in Section 3.

1.3 Organization

The rest of our paper is organized as follows: In Section 2 we introduce the preliminary of game theory and equilibrium approximators. In Section 3 we formally define the equivariance of equilibrium approximators. We present our theoretical analysis of benefits in Section 4 and limitations in Section 5. We conclude and point out the future work in Section 6.

2 Preliminary

In this section, we introduce the preliminary and notations of our paper. We also provide a brief introduction to equilibrium approximators.

2.1 Game Theory

Normal-Form Game

Let a normal-form game with joint payoff $u$ be $\Gamma_{u}=(n,\mathcal{A},u)$ , in which

•

$n\in\mathbb{N}_{\geq 2}$ is the number of players. Each player is represented by the index $i\in[n]$ .
•

$\mathcal{A}=\times_{i\in[n]}\mathcal{A}_{i}$ is the product action space of all players, where $\mathcal{A}_{i}=\{1,2,\dots,m_{i}\}$ . For player $i\in[n]$ , let $a_{i}\in\mathcal{A}_{i}$ be a specific action of $i$ (An action is also referred to as a pure strategy). A joint action $a=(a_{1},a_{2},\dots,a_{n})\in\mathcal{A}$ represents one play of the game in which the player $i$ takes action $a_{i}$ . The action space $\mathcal{A}$ is a Cartesian product that contains all possible joint actions. We have $|\mathcal{A}|=\prod_{i\in[n]}|\mathcal{A}_{i}|=\prod_{i\in[n]}m_{i}$ .
•

$u=(u_{i})_{i\in[n]}$ is the joint payoff or utility of the game. $u_{i}$ is an $n$ -dimensional tensor (or matrix if $n=2$ ) describing player $i$ ’s payoff on each joint action. In our paper, following previous literatures (Tsaknakis and Spirakis, 2007; Deligkas et al., 2022), we normalize all the elements of payoff into $[0,1]$ .

A joint (mixed) strategy is a distribution over $\mathcal{A}$ . Let $\sigma=\times_{i\in[n]}\sigma_{i}$ be a product strategy and $\pi\in\Delta\mathcal{A}$ be a joint (possibly correlated) strategy. Denote $\pi_{i}$ as the marginal strategy of player $i$ in $\pi$ . The expected utility of player $i$ under $\pi$ is

u_{i}(\pi)=\mathbb{E}_{a\sim\pi}[u_{i}(a)]=\sum_{a\in\mathcal{A}}\pi(a)u_{i}(a).

Besides, on behalf of player $i$ , the other players’ joint strategy is denoted as $\pi_{-i}$ , so as $a_{-i}$ and $\sigma_{-i}$ .

Nash Equilibrium (NE)

We say a product strategy $\sigma^{*}=(\sigma^{*}_{1},\sigma^{*}_{2},\dots,\sigma^{*}_{n})$ is a NE if each player’s strategy is the best response given the strategies of others, i.e.,

u_{i}(\sigma_{i},\sigma^{*}_{-i})\leq u_{i}(\sigma^{*}_{i},\sigma^{*}_{-i}),\leavevmode\nobreak\ \forall i\in[n],\sigma_{i}\in\Delta\mathcal{A}_{i}.

(NE)

Computing NE for even general $2$ -player or $3$ -player games is PPAD-hard (Chen et al., 2009; Daskalakis et al., 2009), which leads to research on approximate solutions. For arbitrary $\epsilon>0$ , we say a product strategy $\hat{\sigma}$ is an $\epsilon$ -approximate Nash equilibrium ( $\epsilon$ -NE) if no one can achieve more than $\epsilon$ utility gain by deviating from her current strategy. Formally,

u_{i}(\sigma_{i},\hat{\sigma}_{-i})\leq u_{i}(\hat{\sigma}_{i},\hat{\sigma}_{-i})+\epsilon,\leavevmode\nobreak\ \forall i\in[n],\sigma_{i}\in\Delta\mathcal{A}_{i}.

(

\epsilon

-NE)

The definition of $\epsilon$ -NE reflects the idea that players might not be willing to deviate from their strategies when the amount of utility they could gain by doing so is tiny (not more than $\epsilon$ ).

Coarse Correlated Equilibrium (CCE)

We say a joint (possibly correlated) strategy $\pi^{*}$ is a CCE if no player can receive a higher payoff by acting independently, i.e.,

u_{i}(\sigma_{i},\pi^{*}_{-i})\leq u_{i}(\pi^{*}),\leavevmode\nobreak\ \forall i\in[n],\sigma_{i}\in\Delta\mathcal{A}_{i},

(CCE)

and we say $\hat{\pi}$ is an $\epsilon$ -approximate coarse correlated equilibrium ( $\epsilon$ -CCE) for $\epsilon>0$ if

u_{i}(\sigma_{i},\hat{\pi}_{-i})\leq u_{i}(\hat{\pi})+\epsilon,\leavevmode\nobreak\ \forall i\in[n],\sigma_{i}\in\Delta\mathcal{A}_{i},

(

\epsilon

-CCE)

The difference between NE and CCE is that in an NE, players execute their strategy individually in a decentralized way, while in a CCE, players’ strategies are possibly correlated. A standard technique to correlate the strategy is sending each player a signal from a centralized controller (Shoham and Leyton-Brown, 2008).

Correlated Equilibrium (CE)

CE is similar to CCE, except that in a CE, each player can observe her recommended action before she acts. Thus, player $i$ deviates her strategy through strategy modification $\phi_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i}$ . $\phi_{i}$ maps actions in $\mathcal{A}_{i}$ to possibly different actions in $\mathcal{A}_{i}$ . Based on strategy modification, we say a joint (possibly correlated) strategy $\pi^{*}$ is a CE if

\sum_{a\in\mathcal{A}}\pi^{*}(a)u_{i}(\phi_{i}(a_{i}),a_{-i})\leq u_{i}(\pi^{*}),\leavevmode\nobreak\ \forall i,\forall\phi_{i},

(CE)

and a joint strategy $\hat{\pi}$ is an $\epsilon$ -approximate correlated equilibrium ( $\epsilon$ -CE) for $\epsilon>0$ if

\sum_{a\in\mathcal{A}}\hat{\pi}(a)u_{i}(\phi_{i}(a_{i}),a_{-i})\leq u_{i}(\hat{\pi})+\epsilon,\leavevmode\nobreak\ \forall i,\forall\phi_{i},

(

\epsilon

-CE)

Note that for a finite $n$ -player normal-form game, at least one NE, CE, and CCE must exist. This is because NE always exists (Nash et al., 1950) and NE $\subseteq$ CE $\subseteq$ CCE.

Equilibrium Approximation

To evaluate the quality of a joint strategy to approximate an equilibrium, we define approximation based on exploitability (Lockhart et al., 2019; Goktas and Greenwald, 2022).

Definition 2.1 (Exploitability and Approximation).

Given a joint strategy $\pi$ , the exploitability (or regret) $\mathcal{E}_{i}(\pi,u)$ of player $i$ is the maximum payoff gain of $i$ by deviating from her current strategy, i.e.,

\displaystyle\mathcal{E}_{i}(\pi,u)

\displaystyle\coloneqq\max_{\sigma^{\prime}_{i}}u_{i}(\sigma^{\prime}_{i},\pi_{-i})-u_{i}(\pi)=\max_{a^{\prime}_{i}}u_{i}(a^{\prime}_{i},\pi_{-i})-u_{i}(\pi)

and the exploitability under strategy modification $\mathcal{E}_{i}^{\mathrm{CE}}(\pi,u)$ of player $i$ is the maximum payoff gain of $i$ by deviating through strategy modification, i.e.,

\mathcal{E}_{i}^{\mathrm{CE}}(\pi,u)\coloneqq\max_{\phi_{i}}\sum_{a\in\mathcal{A}}\pi(a)u_{i}(\phi_{i}(a_{i}),a_{-i})-u_{i}(\pi).

The equilibrium approximation is defined as the maximum exploitability over all players ²²2 A similar metric of equilibrium approximation is called Nikaido-Isoda function (Nikaidô and Isoda, 1955) or NashConv (Lockhart et al., 2019), which is the sum of exploitability over all players, i.e., $\sum_{i\in[n]}\mathcal{E}_{i}(\pi,u)$ . , i.e.,

\mathcal{E}(\pi,u)\coloneqq\begin{cases}\max_{i\in[n]}\mathcal{E}_{i}(\pi,u)&,\text{for NE and CCE}\\ \max_{i\in[n]}\mathcal{E}^{\mathrm{CE}}_{i}(\pi,u)&,\text{for CE}\end{cases}

Based on approximation, we can restate the definition of solution concepts. A product strategy $\sigma$ is an NE of game $\Gamma_{u}$ if $\mathcal{E}(\sigma,u)=0$ and is an $\epsilon$ -NE if $\mathcal{E}(\sigma,u)\leq\epsilon$ . A joint strategy $\pi$ is a (C)CE of $\Gamma_{u}$ if $\mathcal{E}(\pi,u)=0$ and is an $\epsilon$ -(C)CE if $\mathcal{E}(\pi,u)\leq\epsilon$ .

2.2 Equilibrium Approximator

The equilibrium approximators, including NE, CE, and CCE approximators, aim to predict solution concepts from game representations. In our paper, we fix the number of players $n$ and action space $\mathcal{A}$ . We denote by $\mathcal{U}$ the set of all the possible game payoffs. The NE approximator $f^{\mathrm{NE}}:\mathcal{U}\to\times_{i\in[n]}\Delta\mathcal{A}_{i}$ maps a game payoff to a product strategy, where $f^{\mathrm{NE}}(u)_{i}\in\Delta\mathcal{A}_{i}$ is player $i$ ’s predicted strategy. We define $\mathcal{F}^{\mathrm{NE}}$ as the function class of the NE approximator. Similarly, we define (C)CE approximator as $f^{\mathrm{(C)CE}}:\mathcal{U}\to\Delta\mathcal{A}$ and (C)CE approximator class as $\mathcal{F}^{\mathrm{(C)CE}}$ .

Algorithm 1 Example: learning NE/CCE approximator via minibatch SGD

1: Input: Training set

S

2: Parameters: Number of iterations

T>0

, batch size

B>0

, learning rate

\eta>0

, initial parameters

w_{0}\in\mathbb{R}^{d}

of the approximator model.

3: for

t\leavevmode\nobreak\ =\leavevmode\nobreak\ 0

T

4: Receive minibatch

{S}_{t}\,=\,\{u^{(1)},\ldots,u^{(B)}\}\subset S

5: Compute the empirical average approximation of

S_{t}

L_{S_{t}}(f^{w_{t}})\leftarrow\frac{1}{B}\sum_{i=1}^{B}\mathcal{E}(f^{w_{t}}(u^{(i)}),u^{(i)})

7: Update model parameters:

w_{t+1}\leftarrow w_{t}-\eta\nabla_{w_{t}}L_{S_{t}}(f^{w_{t}})

9: end for

An equilibrium approximator can be learned through machine learning paradigms based on empirical data. For instance, Feng et al. (2021) learn the NE approximator using the game payoffs generated in the learning process of PSRO, and optimize the approximator by gradient descent in exploitability. Marris et al. (2022) learn the CE and CCE approximators using the games i.i.d. generated from a manually designed distribution, and optimize the approximators using maximum welfare minimum relative entropy loss. Such a loss balances the equilibrium approximation, the social welfare, and the relative entropy of the joint strategy. Additionally, another simple way to learn NE or CCE equilibrium approximator is to apply minibatch stochastic gradient descent (SGD) on approximation. Specifically, we denote $w\in\mathbb{R}^{d}$ as the $d$ -dimensional parameter of the approximator, such as the weights of the neural network. We can optimize $w$ by the standard minibatch SGD algorithm on approximation (See Algorithm 1).

3 Equivariant Equilibrium Approximator

In this section, we introduce the equivariance of the equilibrium approximators and the way how we apply orbit averaging (Elesedy and Zaidi, 2021) to construct equivariant approximators. Equivariant approximator has been developed in many literatures (Hartford et al., 2016; Feng et al., 2021; Marris et al., 2022; Wu and Lisser, 2022), which we will discuss latter.

We first define the permutation of a game. Let $\rho_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i}$ be a permutation function of player $i$ , which is a bijection from $\mathcal{A}_{i}$ to $\mathcal{A}_{i}$ itself. Define $\mathcal{G}_{i}\ni\rho_{i}$ as the class of permutation function for player $i$ , which forms an Abelian group.

Definition 3.1 (Permutation of a game).

For a normal-form game $\Gamma_{u}=(n,u,\mathcal{A})$ , we define the $\rho_{i}$ -permutation of payoff tensor $u$ as $\rho_{i}u=(\rho_{i}u_{j})_{j\in[n]}$ , in which

(\rho_{i}u_{j})(a_{i},a_{-i})=u_{j}(\rho_{i}^{-1}(a_{i}),a_{-i}),\leavevmode\nobreak\ \forall a\in\mathcal{A}.

We also define the $\rho_{i}$ -permutation of joint strategy $\pi$ as $\rho_{i}\pi$ , where

(\rho_{i}\pi)(a_{i},a_{-i})=\pi(\rho_{i}^{-1}(a_{i}),a_{-i}),\leavevmode\nobreak\ \forall a\in\mathcal{A},

and the $\rho_{i}$ -permutation of product strategy $\sigma$ as $\rho_{i}\sigma=(\rho_{i}\sigma_{j})_{j\in[n]}$ , where

\forall a_{j}\in\mathcal{A}_{j},\rho_{i}\sigma_{j}(a_{j})=\begin{cases}\sigma_{j}(a_{j})&,\text{if }j\neq i\\ \sigma_{i}(\rho_{i}^{-1}a_{i})&,\text{if }j=i\\ \end{cases}

Equivariant NE Approximator

Considering the relationship of $\rho_{i}$ -permuted game and $\rho_{i}$ -permuted product strategy, we have the following result for NE:

Lemma 3.2.

In a normal-form game $\Gamma_{u}=(n,u,\mathcal{A})$ , for arbitrary player $i\in[n]$ and any ( $\epsilon$ -)NE strategy $\sigma=(\sigma_{i},\sigma_{-i})$ , $\rho_{i}\sigma=(\rho_{i}\sigma_{i},\sigma_{-i})$ is also an ( $\epsilon$ -)NE for the $\rho_{i}$ -permuted game $\Gamma_{\rho_{i}u}$ .

3.2 tells the unimportance of action order with respect to NE approximation. Based on it, we can summarize two ideal properties for NE approximators, which are defined as follows:

Definition 3.3 (Player-Permutation-Equivariance, PPE).

We say an NE approximator $f^{\mathrm{NE}}$ satisfies player $i$ permutation-equivariant ( $i$ -PE) if for arbitrary permutation function $\rho_{i}\in\mathcal{G}_{i}$ we have

f^{\mathrm{NE}}(\rho_{i}u)_{i}=\rho_{i}f^{\mathrm{NE}}(u)_{i},

(

i

-PE)

Moreover, we say $f^{\mathrm{NE}}$ is player-permutation-equivariant (PPE) if $f^{\mathrm{NE}}$ satisfies $i$ -PE for all player $i\in[n]$ .

Definition 3.4 (Opponent-Permutation-Invariance, OPI).

We say an NE approximator $f^{\mathrm{NE}}$ is opponent $i$ permutation-invariant ( $i$ -PI) if for any other player $j\in[n]-\{i\}$ and arbitrary permutation function $\rho_{i}\in\mathcal{G}_{i}$ we have

f^{\mathrm{NE}}(\rho_{i}u)_{j}=f^{\mathrm{NE}}(u)_{j},\forall j\neq i

(

i

-PI)

and we say $f^{\mathrm{NE}}$ is opponent-permutation-invariant (OPI) if $f^{\mathrm{NE}}$ satisfies $i$ -PI for all player $i\in[n]$ .

Equivariant (C)CE approximator

Considering the relationship of $\rho_{i}$ -permuted game and $\rho_{i}$ -permuted joint strategy, we have a similar result for CE and CCE:

Lemma 3.5.

In a normal-form game $\Gamma_{u}=(n,u,\mathcal{A})$ , for an arbitrary player $i\in[n]$ and any ( $\varepsilon$ -)CE or ( $\epsilon$ -)CCE strategy $\pi$ , $\rho_{i}\pi$ is also an ( $\varepsilon$ -)CE or ( $\epsilon$ -)CCE for the $\rho_{i}$ -permuted game $\Gamma_{\rho_{i}u}$ .

Inspired by Lemma 3.5, we can also summarize an ideal property for CE and CCE approximators defined as follows.

Definition 3.6 (Permutation-Equivariance,PE).

We say an (C)CE approximator $f^{\text{(C)CE}}$ is player $i$ permutation-equivariant ( $i$ -PE) if for permutation function $\rho_{i}\in\mathcal{G}_{i}$ we have

f^{\mathrm{(C)CE}}(\rho_{i}u)=\rho_{i}f^{\mathrm{(C)CE}}(u),

and we say $f^{\text{(C)CE}}$ is permutation-equivariant (PE) if $f^{\text{(C)CE}}$ satisfies $i$ -PE for all player $i\in[n]$ .

Equivariant Approximators in Literature

For two-player games, Feng et al. (2021) propose an MLP-based NE approximator that satisfies both PPE and OPI for zero-sum games. Additionally, they also design a Conv $1$ d-based NE approximator that satisfies PPE only. Hartford et al. (2016) give a PPE approximator to predict players’ strategies. The traditional algorithms Tsaknakis and Spirakis (2007) and Deligkas et al. (2022), which approximate NE by optimization, are also PPE and OPI to payoff and the initial strategies. For $n$ -player general games, Marris et al. (2022) provide a permutation-equivariant approximator to approximate CE and CCE. Equivariant architectures are also adopted in optimal auction design (Rahme et al., 2021; Duan et al., 2022; Ivanov et al., 2022), and Qin et al. (2022) theoretically characterize the benefits of permutation-equivariant in auction mechanisms. We follow the rough idea of Qin et al. (2022) when we analyze the benefits of equivariant equilibrium approximators.

3.1 Orbit Averaging

Orbit averaging is a well-known method to enforce equivariance or invariance for a function (Schulz-Mirbach, 1994). It averages the inputs of a function over the orbit of a group (e.g., the permutation group in our paper).

Orbit Averaging for NE Approximator

For an NE approximator $f^{\mathrm{NE}}$ and any player $i\in[n]$ , we can construct a $i$ -PI or $i$ -PE NE approximator by averaging with respect to all the permutations of player $i$ . Specifically, we construct an $i$ -PI NE approximator by operator $\mathcal{O}_{i}$ with

(\mathcal{O}_{i}f^{\mathrm{NE}})(u)_{j}=\begin{cases}f^{\mathrm{NE}}(u)_{i}&,\text{if }j=i\\ \frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}f^{\mathrm{NE}}(\rho_{i}u)_{j}&,\text{otherwise}\end{cases}

and we construct an $i$ -PE NE approximator by operator $\mathcal{P}_{i}$ with:

(\mathcal{P}_{i}f^{\mathrm{NE}})(u)_{j}=\begin{cases}\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}f^{\mathrm{NE}}(\rho_{i}u)_{i}&,\text{if }j=i\\ f^{\mathrm{NE}}(u)_{j}&,\text{otherwise}\end{cases}

Lemma 3.7.

$\mathcal{O}_{i}f^{\mathrm{NE}}$ is $i$ -PI and $\mathcal{P}_{i}f^{\mathrm{NE}}$ is $i$ -PE. Specially, if $f^{\mathrm{NE}}$ is already $i$ -PI or $i$ -PE, then we have $\mathcal{O}_{i}f^{\mathrm{NE}}=f^{\mathrm{NE}}$ or $\mathcal{P}_{i}f^{\mathrm{NE}}=f^{\mathrm{NE}}$ , respectively.

To construct a PPE or OPI NE approximator, we composite the operators with respect to all players. Let $\mathcal{O}=\mathcal{O}_{1}\circ\mathcal{O}_{2}\circ\dots\circ\mathcal{O}_{n}$ and $\mathcal{P}=\mathcal{P}_{1}\circ\mathcal{P}_{2}\circ\dots\circ\mathcal{P}_{n}$ , we get the following corollary:

Lemma 3.8.

$\mathcal{O}f^{\mathrm{NE}}$ is OPI and $\mathcal{P}f^{\mathrm{NE}}$ is PPE. If $f^{\mathrm{NE}}$ is already OPI or PPE, we have $\mathcal{O}f^{\mathrm{NE}}=f^{\mathrm{NE}}$ or $\mathcal{P}f^{\mathrm{NE}}=f^{\mathrm{NE}}$ , respectively.

Furthermore, we can also compose $\mathcal{P}\circ\mathcal{O}$ to construct a NE approximator with both PPE and OPI.

Orbit Averaging for (C)CE Approximator

For CE or CCE approximator $f$ , we define $\mathcal{Q}_{i}$ -project for player $i\in[n]$ to construct an $i$ -PE approximator, which averages with respect to all the permutations of player $i$ .

(\mathcal{Q}_{i}f^{\text{(C)CE}})(u)=\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}f^{\text{(C)CE}}(\rho_{i}u)

Similarly, we define $\mathcal{Q}=\mathcal{Q}_{1}\circ\mathcal{Q}_{2}\circ\dots\circ\mathcal{Q}_{n}$ as the composite operator.

Lemma 3.9.

$\mathcal{Q}_{i}f^{\text{(C)CE}}$ is $i$ -PE and $\mathcal{Q}f^{\text{(C)CE}}$ is PE. Specifically, If $f^{\text{(C)CE}}$ is already $i$ -PE or PE, then we have $\mathcal{Q}_{i}f^{\text{(C)CE}}=f^{\text{(C)CE}}$ or $\mathcal{Q}f^{\text{(C)CE}}=f^{\text{(C)CE}}$ , respectively.

Combined with Lemma 3.7, Lemma 3.8 and Lemma 3.9, we can have the following corollary directly.

Corollary 3.10.

$\mathcal{O}^{2}=\mathcal{O},\mathcal{P}^{2}=\mathcal{P},\mathcal{Q}^{2}=\mathcal{Q}$ .

The benefit of using orbit averaging is shown in the following lemma:

Lemma 3.11.

Denote $\mathcal{X}$ as an idempotent operator, i.e. $\mathcal{X}^{2}=\mathcal{X}$ (e.g. $\mathcal{O},\mathcal{P}$ or $\mathcal{Q}$ ). For function class $\mathcal{F}$ of NE, CE or CCE approximator, let $\mathcal{F}_{\mathcal{X}}$ be any subset of $\mathcal{F}$ that is closed under $\mathcal{X}$ , then $\mathcal{X}\mathcal{F}_{\mathcal{X}}$ is the largest subset of $\mathcal{F}_{\mathcal{X}}$ that is invariant under $\mathcal{X}$ .

According to Lemma 3.8, Lemma 3.9 and Lemma 3.11, $\mathcal{O}\mathcal{F}^{\mathrm{NE}}$ (or $\mathcal{P}\mathcal{F}^{\mathrm{NE}}$ / $\mathcal{Q}\mathcal{F}^{\mathrm{(C)CE}}$ ) is the largest subset of $\mathcal{F}^{\mathrm{NE}}$ (or $\mathcal{F}^{\mathrm{NE}}$ / $\mathcal{F}^{\mathrm{(C)CE}}$ ) with the corresponding property (OPI, PPE, or PE) if $\mathcal{F}^{\mathrm{NE}}$ (or $\mathcal{F}^{\mathrm{NE}}$ / $\mathcal{F}^{\mathrm{(C)CE}}$ ) is closed operator under $\mathcal{O}$ (or $\mathcal{P}$ / $\mathcal{Q}$ ). The result tells that the orbit averaging operators, while enforcing the operated function to be equivariance or invariance, keep as large capacity of the function class as possible. Therefore, we believe that orbit averaging is an ideal approach to constructing equivariant or invariant functions.

4 Theoretical Analysis of Benefits

In this section, we theoretically analyze the benefits of equivariant approximators with respect to generalizability and approximation.

4.1 Benefits for Generalization

We first derive the generalization bound and sample complexity for general approximator classes, and then we show the benefits of equivariant approximators by applying orbit averaging to the approximators.

The representativeness of an approximator class is measured by the covering numbers (Shalev-Shwartz and Ben-David, 2014) under $\ell_{\infty}$ -distance, which are defined as follows:

Definition 4.1 ( $\ell_{\infty}$ -distance).

The $\ell_{\infty}$ -distance between two equilibrium approximators $f,g$ is:

\ell_{\infty}(f,g)=\max_{u\in\mathcal{U}}{\|f(u)-g(u)\|},

where we define the distance of two product strategies $\sigma$ and $\sigma^{\prime}$ as

{\|\sigma^{1}-\sigma^{2}\|}=\max_{i\in[n]}\sum_{a_{i}\in\mathcal{A}_{i}}|\sigma^{1}_{i}(a_{i})-\sigma^{2}_{i}(a_{i})|

and the distance of two joint strategy $\pi$ and $\pi^{\prime}$ as

{\|\pi^{1}-\pi^{2}\|}=\sum_{a\in\mathcal{A}}|\pi^{1}(a)-\pi^{2}(a)|

Definition 4.2 ( $r$ -covering number).

For $r>0$ , we say function class $\mathcal{F}_{r}$ $r$ -covers another function class $\mathcal{F}$ under $\ell_{\infty}$ -distance if for all function $f\in\mathcal{F}$ , there exists $f_{r}\in\mathcal{F}_{r}$ such that ${\|f-f_{r}\|}_{\infty}\leq r$ . The $r$ -covering number $\mathcal{N}_{\infty}(\mathcal{F},r)$ of $\mathcal{F}$ is the cardinality of the smallest function class $\mathcal{F}_{r}$ that $r$ -covers $\mathcal{F}$ under $\ell_{\infty}$ -distance.

Based on covering numbers, we provide the generalization bounds of NE, CE and CCE approximators. The bounds describe the difference between the expected testing approximation and empirical training approximation.

Theorem 4.3.

[Generalization bound] For function class $\mathcal{F}$ of NE, CE or CCE approximator, with probability at least $1-\delta$ over draw of the training set $S$ (with size $m$ ) from payoff distribution $\mathcal{D}$ , for all approximator $f\in\mathcal{F}$ we have

\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(f(u),u)]-\frac{1}{m}\sum_{u\in S}\mathcal{E}(f(u),u)\leq 2\cdot\inf_{r>0}\{\sqrt{\frac{2\ln\mathcal{N}_{\infty}(\mathcal{F},r)}{m}}+Lr\}+4\sqrt{\frac{2\ln(4/\delta)}{m}},

where $L=2n$ for NE approximator, and $L=2$ for CE and CCE approximators.

To get the theorem, we first show that all three equilibrium approximations are Lipschitz continuous with respect to strategies. Afterward, we derive the Rademacher complexity (Bartlett and Mendelson, 2002) of the expected approximation based on the Lipschitz continuity and covering numbers. See Section B.1 for the detailed proof.

We can see from Theorem 4.3 that, with a large enough training set, the generalization gaps of equilibrium approximators go to zero if the covering number $\mathcal{N}_{\infty}(\mathcal{F},r)$ is bounded. As a result, we can estimate the expected testing performance through the empirical training performance.

We can also derive the sample complexities of equilibrium approximators to achieve the desirable generalizability.

Theorem 4.4.

[Sample complexity] For $\epsilon,\delta\in(0,1)$ , function class $\mathcal{F}$ of NE, CE or CCE approximator and distribution $\mathcal{D}$ , with probability at least $1-\delta$ over draw of the training set $S$ with

m\geq\frac{9}{2\epsilon^{2}}\left(\ln\frac{2}{\delta}+\ln\mathcal{N}_{\infty}(\mathcal{F},\frac{\epsilon}{3L})\right)

games sampled from $\mathcal{D}$ , $\forall f\in\mathcal{F}$ we have

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(f(u),u)]\leq\frac{1}{m}\sum_{u\in S}\mathcal{E}(f(u),u)+\epsilon,

where $L=2n$ for NE approximators, and $L=2$ for CE and CCE approximators.

The proof is based on the Lipschitz continuity of approximation, uniform bound, and concentration inequality. See Section B.2 for details. Theorem 4.4 is also called the uniform convergence of function class $\mathcal{F}$ , which is a sufficient condition for agnostic PAC learnable (Shalev-Shwartz and Ben-David, 2014).

As for the benefits of equivariant approximators for generalizability, the following result indicates that the projected equilibrium approximators have smaller covering numbers.

Theorem 4.5.

The $\mathcal{O}$ -projected, $\mathcal{P}$ -projected and $\mathcal{Q}$ -projected approximator classes have smaller covering numbers, i.e., $\forall r>0$ we have

	$\displaystyle\mathcal{N}_{\infty}(\mathcal{O}\mathcal{F}^{\mathrm{NE}},r)$	$\displaystyle\leq\mathcal{N}_{\infty}(\mathcal{F}^{\mathrm{NE}},r),$
	$\displaystyle\mathcal{N}_{\infty}(\mathcal{P}\mathcal{F}^{\mathrm{NE}},r)$	$\displaystyle\leq\mathcal{N}_{\infty}(\mathcal{F}^{\mathrm{NE}},r),$
	$\displaystyle\mathcal{N}_{\infty}(\mathcal{Q}\mathcal{F}^{\mathrm{(C)CE}},r)$	$\displaystyle\leq\mathcal{N}_{\infty}(\mathcal{F}^{\mathrm{(C)CE}},r)$

The proof is done by showing all the operators are contraction mappings. See Section B.3 for details.

Both the generalization bounds in Theorem 4.3 and the sample complexities in Theorem 4.4 decrease with the decrease of covering numbers $\mathcal{N}_{\infty}(\mathcal{F},r)$ . Thus, we can see from Theorem 4.5 that both PPE and OPI can improve the generalizability of NE approximators, and PE can improve the generalizability of CE and CCE approximators.

4.2 Benefits for Approximation

We then show the benefits of equivariance for approximation when the payoff distribution is invariant under permutation. The permutation-invariant distribution holds when the action is anonymous or indifferent or when we pre-train the equilibrium approximators using a manually designed distribution (Marris et al., 2022).

(C)CE Approximator

The following theorem tells the benefit of permutation-equivariance in decreasing the exploitability of (C)CE approximators.

Theorem 4.6.

When the payoff distribution $\mathcal{D}$ is invariant under the permutation of payoffs, the $\mathcal{Q}$ -projected (C)CE approximator has a smaller expected equilibrium approximation. Formally, for all $f^{\mathrm{(C)CE}}\in\mathcal{F}^{\mathrm{(C)CE}}$ and permutation-invariant distribution $\mathcal{D}$ , we have

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(\mathcal{Q}f^{\mathrm{(C)CE}}(u),u)]\leq\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(f^{\mathrm{(C)CE}}(u),u)],

The proof is done by using the convexity of approximation. See Section B.4 for details. We can see from Theorem 4.6 that, when payoff distribution is invariant under permutation, it is beneficial to use equivariant architecture as the CE or CCE approximators.

NE Approximator

As for NE approximator, we have similar results.

Theorem 4.7.

For bimatrix constant-sum games, when the payoff distribution $\mathcal{D}$ is invariant under the permutation of payoffs, then the $\mathcal{X}$ -projected ( $\mathcal{X}\in\{\mathcal{O},\mathcal{P}\}$ ) NE approximator has a smaller expected exploitability. Formally, for all $f^{\mathrm{NE}}\in\mathcal{F}^{\mathrm{NE}}$ and permutation-invariant distribution $\mathcal{D}$ for bimatrix constant-sum games, we have

\mathbb{E}_{u\sim\mathcal{D}}[\sum_{i}\mathcal{E}_{i}((\mathcal{X}f^{\mathrm{NE}})(u),u)]\leq\mathbb{E}_{u\sim\mathcal{D}}[\sum_{i}\mathcal{E}_{i}(f^{\mathrm{NE}}(u),u)]

Theorem 4.8.

When the payoff distribution $\mathcal{D}$ is invariant under the permutation of payoffs, and $f^{\mathrm{NE}}$ satisfies OPI, then the $\mathcal{P}$ -projected NE approximator has a smaller expected NE approximation. Formally, for all $f^{\mathrm{NE}}\in\mathcal{F}^{\mathrm{NE}}$ that is OPI and permutation-invariant distribution $\mathcal{D}$ , we have

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}((\mathcal{P}f^{\mathrm{NE}})(u),u)]\leq\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(f^{\mathrm{NE}}(u),u)].

Theorem 4.9.

For bimatrix games, when the payoff distribution $\mathcal{D}$ is invariant under the permutation of payoffs, and $f^{\mathrm{NE}}$ satisfies PPE, then the $\mathcal{O}$ -projected NE approximator has a smaller expected NE approximation. Formally, for all $f^{\mathrm{NE}}\in\mathcal{F}^{\mathrm{NE}}$ that is PPE and permutation-invariant distribution $\mathcal{D}$ of bimatrix games, we have

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}((\mathcal{O}f^{\mathrm{NE}})(u),u)]\leq\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(f^{\mathrm{NE}}(u),u)].

Theorem 4.8 and Theorem 4.9 tell that PPE and OPI approximators can achieve better approximation than ones with only PPE or OPI. Meanwhile, we can see from Theorem 4.7 that for bimatrix constant-sum games (such as zero-sum games), it can be preferred to introduce PPE or OPI to the architectures.

5 Theoretical Analysis of Limitations

As we discussed in Section 4, equivariant approximators enjoy better generalizability and better approximation sometimes. However, as we will show, they have some limitations regarding equilibrium selection and social welfare. Such limitations attribute to the limited representativeness caused by equivariance.

5.1 Equilibrium Selection

We first show that there may be equilibria points that equivariant approximators will never find. We illustrate such limitation in permutation-invariant games, which is defined as follows:

Definition 5.1 (Permutation- $\rho$ -Invariant Game).

We say a game $\Gamma_{u}$ is permutation- $\rho$ -invariant, where $\rho=\circ_{i\in[n]}\rho_{i}$ , if the payoff $u$ is permutation-invariant with respect to $\rho$ . That is, $\rho u=u$ .

Permutation- $\rho$ -invariance indicates that one cannot distinguish joint action $a$ from $\rho a$ using only the payoff $u$ . We’d like to provide an example to show more insight of permutation- $\rho$ -invariant games:

Example 5.2.

For a $2$ -player game $\Gamma_{u}=(2,u=(u_{1},u_{2}),\mathcal{A}=([m_{1}],[m_{2}]))$ , Let $\rho_{i}=(m_{i},m_{i}-1,\dots,1)$ and $\rho=\rho_{1}\circ\rho_{2}$ . If one of the following conditions holds, then $u$ is permutation- $\rho$ -invariant:

1.

$u_{1}$ and $u_{2}$ are symmetric and persymmetric (i.e., symmetric with respect to the northeast-to-southwest diagonal) squares.
2.

Both $u_{1}$ and $u_{2}$ are centrosymmetric, i.e., $u_{i}(x,y)=u_{i}(m_{1}+1-x,m_{2}+1-y)$ for $i\in\{1,2\},x\in[m_{1}]$ and $y\in[m_{2}]$ .

For permutation $\rho=(\circ_{i\in[n]}\rho_{i})$ and player $k\in[n]$ , we denote the set of non-fixed actions of player $k$ under $\rho_{k}$ as

V(\rho_{k})\coloneqq\{a_{k}|a_{k}\in\mathcal{A}_{k},\rho_{k}(a_{k})\neq a_{k}\}.

Based on $V(\rho_{k})$ , we find some equilibria points of permutation- $\rho$ -invariant games that any equivariant approximators will never find.

Theorem 5.3.

For a permutation- $\rho$ -invariant game $\Gamma_{u}$ . if there is a pure NE $a^{*}=(a^{*}_{i})_{i\in[n]}$ and at least one player $k\in[n]$ such that $a^{*}_{k}\in V(\rho_{k})$ , then $a^{*}$ will never be found by any NE approximator with both PPE and OPI. Besides, $a^{*}$ (as a pure CE or CCE) will also never be found by any CE or CCE approximator with PE.

We illustrate Theorem 5.3 by the following example:

Example 5.4.

Consider a bimatrix game with identity utility

u=\begin{bmatrix}\bm{1,1}&0,0\\ 0,0&\bm{1,1}\end{bmatrix}

There are two pure NE (bolded in the above matrix) and one mixed NE of $\sigma_{1}=(0.5,0.5)$ and $\sigma_{2}=(0.5,0.5)$ . Let $\rho_{i}$ be the unique permute function (except for identity function) of player $i\in[2]$ , and $\rho=\rho_{1}\circ\rho_{2}$ . The game is permutation- $\rho$ -invariant.

Case 1: Let $f$ be a permutation-equivariant CE or CCE approximator, and denote $\pi=f(u)$ . We have

\pi=f(u)\overset{(a)}{=}f(\rho u)\overset{(b)}{=}\rho f(u),

where $(a)$ holds by permutation- $\rho$ -invariance of $u$ , and $(b)$ holds by PE of $f$ . Thus, we have $\pi_{1,1}=\pi_{2,2}\in[0,\frac{1}{2}]$ and $\pi_{1,2}=\pi_{2,1}\in[0,\frac{1}{2}]$ . As a result, the two pure (C)CEs cannot be found.

Case 2: Let $f$ be a NE approximator that holds PPE and OPI. Denote $f(u)=(\sigma_{1},\sigma_{2})$ , where $\sigma_{1}=(p_{1},1-p_{1})$ and $\sigma_{2}=(p_{2},1-p_{2})$ . By PPE and OPI of $f$ , we have

\displaystyle f(u)_{1}

\displaystyle=(p_{1},1-p_{1})\overset{(a)}{=}f(\rho_{1}\rho_{2}u)_{1}\overset{(b)}{=}\rho_{1}f(\rho_{2}u)_{1}\overset{(c)}{=}\rho_{1}f(u)_{1}=(1-p_{1},p_{1}),

where $(a)$ holds by permutaion- $\rho$ -invariance of $u$ , $(b)$ holds by PPE of $f$ , and $(c)$ holds by OPI of $f$ . As a result, the only NE that $f$ could find is the mixed NE.

As we can see from the example and Theorem 5.3, the equivariance, while introducing inductive bias to the approximator architecture, is also a strong constraint. Such a constraint is why the equivariant approximators cannot find all the equilibria points.

5.2 Social Welfare

The social welfare of a joint strategy $\pi$ is defined as the sum of all players’ utilities, i.e.,

\mathrm{SW}(\pi,u)=\sum_{i\in[n]}u_{i}(\pi).

The equilibrium with higher social welfare is usually preferred (Marris et al., 2022).

To analyze the social welfare of equivariant approximators, we define the worst social welfare ratio as follows:

Definition 5.5.

For any $N,M\geq 2$ and two NE (or CE/CCE) approximator classes $\mathcal{F}_{1},\mathcal{F}_{2}$ that target on games with number of players $n\leq N$ and $|\mathcal{A}_{i}|\leq M$ , we define the worst social welfare ratio of $\mathcal{F}_{1}$ over $\mathcal{F}_{2}$ as:

\displaystyle\mathrm{SWR}_{\mathrm{N,M}}(\mathcal{F}_{1},\mathcal{F}_{2})\coloneqq\inf_{\mathcal{D}}\frac{\max_{f_{1}\in\mathcal{F}_{1}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f_{1}(u),u)}{\max_{f_{2}\in\mathcal{F}_{2}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f_{2}(u),u)}

$\mathrm{SWR}_{\mathrm{N,M}}(\mathcal{F}_{1},\mathcal{F}_{2})$ measures the relative representativeness of $\mathcal{F}_{1}$ over $\mathcal{F}_{2}$ in terms of social welfare. Based on that, we have the following result for equivariant CE and CCE approximator classes:

Theorem 5.6.

Given $N,M\geq 2$ , let $\mathcal{F}^{\mathrm{(C)CE}}_{\mathrm{PE}}$ be the function class (target on games with number of players $n\leq N$ and $|\mathcal{A}_{i}|\leq M$ ) of all the (C)CE approximators with PE. Denote by $\mathcal{F}^{\mathrm{(C)CE}}_{\mathrm{general}}$ the function class of all the (C)CE approximators. Then we have

\displaystyle\mathrm{SWR}_{\mathrm{N,M}}(\mathcal{F}^{\mathrm{(C)CE}}_{\mathrm{PE}},\mathcal{F}^{\mathrm{(C)CE}}_{\mathrm{general}})=1.

Theorem 5.6 tells that, while the permutation-equivariant (C)CE approximator class may not be able to find all the (C)CE in a game, it can keep the social welfare of the output solutions.

However, when considering equivariant NE approximators, we have the following negative result:

Theorem 5.7.

Given $N,M\geq 2$ , let ${\mathcal{F}}^{\mathrm{NE}}_{\mathrm{OPI}},{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{PPE}}$ and ${\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}}$ be the function classes (target on games with number of players $n\leq N$ and $|\mathcal{A}_{i}|\leq M$ ) of all the NE approximators with OPI, PPE and both. Denote the function class of all the NE approximators as $\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}}$ . Then we have

$\displaystyle\mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{NE}}_{\mathrm{OPI}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})$	$\displaystyle=\frac{1}{M^{N-1}},$	(1)
$\displaystyle\mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{NE}}_{\mathrm{PPE}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})$	$\displaystyle\leq\frac{1}{M},$	(2)
$\displaystyle\mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})$	$\displaystyle=\frac{1}{M^{N-1}}.$	(3)

Additionally, when $M\geq 3$ , denote by $\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}}$ the function class of all the NE oracles (functions that always output exact NE solutions of the input games) with both PPE and OPI, and by $\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{general}}$ the function class of all the NE oracles. Then we have

\mathrm{SWR}_{\mathrm{N,M}}(\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}},\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{general}})=0.

(4)

The proof is done by construction (See Section C.3 for details). As an illustration of Equation 4, consider a bimatrix game with the following payoff:

u=\begin{bmatrix}{1,1}&0,0&0,\frac{1}{2}+\varepsilon\\ 0,0&{1,1}&0,\frac{1}{2}+\varepsilon\\ \frac{1}{2}+\varepsilon,0&\frac{1}{2}+\varepsilon,0&\varepsilon,\varepsilon\end{bmatrix}

for $\epsilon\in(0,\frac{1}{2})$ . The maximum NE (the upper-left corner of $u$ ) social welfare is $2$ , which can be found by at least one NE oracle in $\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{general}}$ . However, the only NE (the lower-right corner of $u$ ) that the NE oracles in $\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}}$ could find only has a social welfare of $2\epsilon$ . As a result,

\mathrm{SWR}_{2,3}(\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}},\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{general}})\leq\frac{2\epsilon}{2}=\epsilon,

which goes to zero as $\epsilon\to 0$ . Recall that we always have $\mathrm{SWR}_{N,M}\geq 0$ , thus Equation 4 holds when $N=2$ and $M=3$ .

Theorem 5.7 tells that equivariant NE approximators may lose some social welfare while enjoying better generalizability. Such a result inspires us to balance generalizability and social welfare when designing the NE approximator architecture.

6 Conclusion and Future Work

In this paper, we theoretically analyze the benefits and limitations of equivariant equilibrium approximators, including player-permutation-equivariant (PPE) and opponent-permutation-invariant (OPI) NE approximator, and permutation-equivariant (PE) CE and CCE approximators. For the benefits, we first show that these equivariant approximators enjoy better generalizability. To get the result, we derive the generalization bounds and sample complexities based on covering numbers, and then we prove that the symmetric approximators have lower covering numbers. We then show that the equivariant approximators can decrease the exploitability when the payoff distribution is invariant under permutation. For the limitations, we find the equivariant approximators may fail to find some equilibria points due to their limited representativeness caused by equivariance. Besides, while equivariant (C)CE approximators can keep the social welfare, the equivariant NE approximators reach a small worst social welfare ratio comparing to the general approximators. Such a result indicates that equivariance may reduce social welfare; therefore, we’d better balance the generalizability and social welfare when we design the architectures of NE approximators.

As for future work, since in our paper we assume the training and testing payoff distribution are the same, an interesting topic is to study the benefits of equivariant approximators under the payoff distribution shift. Moreover, since we consider fixed and discrete action space, another interesting future direction is to analyze the benefits of equivariant approximators in varying or continuous action space.

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Appendix A Omitted Proofs in Section 3

A.1 Useful Lemma

We first introduce a lemma, which will be frequently used in the following proofs.

Lemma A.1.

$\forall i,j\in[n],\rho_{i}\in\mathcal{G}_{i}$ we have $(\rho_{i}u)_{j}(\sigma_{i},\sigma_{-i})=u_{j}(\rho_{i}^{-1}\sigma_{i},\sigma_{-i})$ and $(\rho_{i}u)_{j}(\pi)=u_{j}(\rho_{i}^{-1}\pi)$

Proof.

Define $\widehat{a}_{i}\coloneqq\rho_{i}^{-1}a_{i}$ . For product strategy $\sigma=(\sigma_{i})_{i\in[n]}$ ,

	$\displaystyle(\rho_{i}u)_{j}(\sigma_{i},\sigma_{-i})=$	$\displaystyle\sum_{a_{i}\in\mathcal{A}_{i}}\sum_{a_{-i}\in\mathcal{A}_{-i}}(\rho_{i}u)_{j}(a_{i},a_{-i})\cdot\sigma_{i}(a_{i})\cdot\sigma_{-i}(a_{-i})$
	$\displaystyle=$	$\displaystyle\sum_{a_{i}\in\mathcal{A}_{i}}\sum_{a_{-i}\in\mathcal{A}_{-i}}u_{j}(\rho_{i}^{-1}a_{i},a_{-i})\cdot\sigma_{i}(a_{i})\cdot\sigma_{-i}(a_{-i})$
	$\displaystyle=$	$\displaystyle\sum_{a_{i}\in\mathcal{A}_{i}}\sum_{a_{-i}\in\mathcal{A}_{-i}}u_{j}(\rho_{i}^{-1}a_{i},a_{-i})\cdot(\rho_{i}^{-1}\sigma_{i})(\rho_{i}^{-1}a_{i})\cdot\sigma_{-i}(a_{-i})$
	$\displaystyle=$	$\displaystyle\sum_{\widehat{a}_{i}\in\mathcal{A}_{i}}\sum_{a_{-i}\in\mathcal{A}_{-i}}u_{j}(\widehat{a}_{i},a_{-i})\cdot(\rho_{i}^{-1}\sigma_{i})(\widehat{a}_{i})\cdot\sigma_{-i}(a_{-i})$
	$\displaystyle=$	$\displaystyle u_{j}(\rho_{i}^{-1}\sigma_{i},\sigma_{-i})$

For joint strategy $\pi$ ,

	$\displaystyle(\rho_{i}u)_{j}(\pi)=$	$\displaystyle\sum_{a_{i}\in\mathcal{A}_{i}}\sum_{a_{-i}\in\mathcal{A}_{-i}}(\rho_{i}u_{j})(a_{i},a_{-i})\cdot\pi(a_{i},a_{-i})$
	$\displaystyle=$	$\displaystyle\sum_{a_{i}\in\mathcal{A}_{i}}\sum_{a_{-i}\in\mathcal{A}_{-i}}u_{j}(\rho_{i}^{-1}a_{i},a_{-i})\cdot\pi(a_{i},a_{-i})$
	$\displaystyle=$	$\displaystyle\sum_{a_{i}\in\mathcal{A}_{i}}\sum_{a_{-i}\in\mathcal{A}_{-i}}u_{j}(\rho_{i}^{-1}a_{i},a_{-i})\cdot(\rho_{i}^{-1}\pi)(\rho_{i}^{-1}a_{i},a_{-i})$
	$\displaystyle=$	$\displaystyle\sum_{\widehat{a}_{i}\in\mathcal{A}_{i}}\sum_{a_{-i}\in\mathcal{A}_{-i}}u_{j}(\widehat{a}_{i},a_{-i})\cdot(\rho_{i}^{-1}\pi)(\widehat{a}_{i},a_{-i})$
	$\displaystyle=$	$\displaystyle u_{j}(\rho_{i}^{-1}\pi)$

∎

A.2 Proof of Lemma 3.2

See 3.2

Proof.

For player $i$ , we have

	$\displaystyle\mathcal{E}_{i}(\rho_{i}\sigma,\rho_{i}u)=$	$\displaystyle\max_{a_{i}\in\mathcal{A}_{i}}\rho_{i}u_{i}(a_{i},\rho_{i}\sigma_{-i})-\rho_{i}u_{i}(\rho_{i}\sigma)=\max_{a_{i}\in\mathcal{A}_{i}}\rho_{i}u_{i}(a_{i},\sigma_{-i})-\rho_{i}u_{i}(\rho_{i}\sigma_{i},\sigma_{-i})$
	$\displaystyle=$	$\displaystyle\max_{a_{i}\in\mathcal{A}_{i}}u_{i}(\rho_{i}^{-1}a_{i},\sigma_{-i})-u_{i}(\rho_{i}^{-1}\rho_{i}\sigma_{i},\sigma_{-i})\overset{(a)}{=}\max_{a_{i}\in\mathcal{A}_{i}}u_{i}(a_{i},\sigma_{-i})-u_{i}(\sigma_{i},\sigma_{-i})=\mathcal{E}_{i}(\sigma,u),$

where $(a)$ holds since $\rho_{i}$ is a bijection on $\mathcal{A}_{i}$ . For player $j\neq i$ , we have

	$\displaystyle\mathcal{E}_{j}(\rho_{i}\sigma,\rho_{i}u)=$	$\displaystyle\max_{a_{j}\in\mathcal{A}}\rho_{i}u_{j}(a_{j},\rho_{i}\sigma_{-j})-\rho_{i}u_{j}(\rho_{i}\sigma)=\max_{a_{j}\in\mathcal{A}_{j}}u_{j}(a_{j},\rho_{i}^{-1}\rho_{i}\sigma_{-j})-u_{j}(\rho_{i}^{-1}\rho_{i}\sigma)$
	$\displaystyle=$	$\displaystyle\max_{a_{j}\in\mathcal{A}_{j}}u_{j}(a_{j},\sigma_{-j})-u_{j}(\sigma)=\mathcal{E}_{j}(\sigma,u)$

From above, we have $\mathcal{E}(\rho_{i}\sigma,\rho_{i}u)=\mathcal{E}(\sigma,u)$ , thus if $\sigma$ is a $\varepsilon$ -NE of $\Gamma_{u}$ , then $\rho_{i}\sigma$ must be a $\varepsilon$ -NE of $\Gamma_{\rho_{i}u}$ . ∎

A.3 Proof of Lemma 3.5

See 3.5

CCE

For player $i$ , we have

$\displaystyle\mathcal{E}_{i}(\rho_{i}\pi,\rho_{i}u)=$	$\displaystyle\max_{a_{i}\in\mathcal{A}_{i}}(\rho_{i}u_{i})(a_{i},(\rho_{i}\pi)_{-i})-(\rho_{i}u_{i})(\rho_{i}\pi_{i})$
$\displaystyle=$	$\displaystyle\max_{a_{i}\in\mathcal{A}_{i}}(\rho_{i}u_{i})(a_{i},(\rho_{i}\pi)_{-i})-u_{i}(\rho_{i}^{-1}\rho_{i}\pi_{i})$
$\displaystyle=$	$\displaystyle\max_{a_{i}\in\mathcal{A}_{i}}(\rho_{i}u_{i})(a_{i},(\rho_{i}\pi)_{-i})-u_{i}(\pi_{i})$
$\displaystyle=$	$\displaystyle\max_{a_{i}\in\mathcal{A}_{i}}\sum_{b\in\mathcal{A}}(\rho_{i}u_{i})(a_{i},b_{-i})\cdot(\rho_{i}\pi)(b)-u_{i}(\pi_{i})$
$\displaystyle=$	$\displaystyle\max_{a_{i}\in\mathcal{A}_{i}}\sum_{b_{i}\in\mathcal{A}_{i},b_{-i}\in\mathcal{A}_{-i}}u_{i}(\rho_{i}^{-1}a_{i},b_{-i})\cdot\pi(\rho_{i}^{-1}b_{i},b_{-i})-u_{i}(\pi_{i})$
$\displaystyle=$	$\displaystyle\max_{a_{i}\in\mathcal{A}_{i}}\sum_{b_{i}\in\mathcal{A}_{i},b_{-i}\in\mathcal{A}_{-i}}u_{i}(a_{i},b_{-i})\cdot\pi(b_{i},b_{-i})-u_{i}(\pi_{i})$	$\displaystyle,\text{$\rho_{i}$ is a bijection on $\mathcal{A}_{i}$}$
$\displaystyle=$	$\displaystyle\mathcal{E}_{i}(\pi,u)$

For player $j\neq i$ , we have

$\displaystyle\mathcal{E}_{j}(\rho_{i}\pi,\rho_{i}u)=$	$\displaystyle\max_{a_{j}\in\mathcal{A}_{j}}(\rho_{i}u_{j})(a_{j},(\rho_{i}\pi)_{-j})-(\rho_{i}u_{j})(\rho_{i}\pi_{j})$
$\displaystyle=$	$\displaystyle\max_{a_{j}\in\mathcal{A}_{j}}(\rho_{i}u_{j})(a_{j},(\rho_{i}\pi)_{-j})-u_{j}(\rho_{i}^{-1}\rho_{i}\pi_{j})$
$\displaystyle=$	$\displaystyle\max_{a_{j}\in\mathcal{A}_{j}}(\rho_{i}u_{j})(a_{j},(\rho_{i}\pi)_{-j})-u_{j}(\pi_{j})$
$\displaystyle=$	$\displaystyle\max_{a_{j}\in\mathcal{A}_{j}}\sum_{b\in\mathcal{A}}(\rho_{i}u_{j})(a_{j},b_{-j})\cdot(\rho_{i}\pi)(b)-u_{j}(\pi_{j})$
$\displaystyle=$	$\displaystyle\max_{a_{j}\in\mathcal{A}_{j}}\sum_{b_{i}\in\mathcal{A}_{i},b_{-i}\in\mathcal{A}_{-i}}u_{j}(a_{j},(b_{-j})_{-i},\rho_{i}^{-1}b_{i})\cdot\pi(\rho_{i}^{-1}b_{i},b_{-i})-u_{j}(\pi_{j})$
$\displaystyle=$	$\displaystyle\max_{a_{j}\in\mathcal{A}_{j}}\sum_{b_{i}\in\mathcal{A}_{i},b_{-i}\in\mathcal{A}_{-i}}u_{j}(a_{j},(b_{-j})_{-i},b_{i})\cdot\pi(b_{i},b_{-i})-u_{j}(\pi_{j})$	$\displaystyle,\text{$\rho_{i}$ is a bijection on $\mathcal{A}_{i}$}$
$\displaystyle=$	$\displaystyle\mathcal{E}_{j}(\pi,u)$

Thus, we have $\mathcal{E}(\rho_{i}\pi,\rho_{i}u)=\mathcal{E}(\pi,u)$ . Thus, if $\pi$ is a $\varepsilon$ -CCE of $\Gamma_{u}$ , then $\rho_{i}\pi$ must be a $\varepsilon$ -CCE of $\Gamma_{\rho_{i}u}$ .

CE

For player $j\neq i$ , we have

$\displaystyle\mathcal{E}_{j}^{\mathrm{CE}}(\rho_{i}\pi,\rho_{i}u)=$	$\displaystyle\max_{\phi_{j}:\mathcal{A}_{j}\to\mathcal{A}_{j}}\sum_{a\in\mathcal{A}}(\rho_{i}\pi)(a)\cdot(\rho_{i}u_{j})(\phi_{j}(a_{j}),a_{-j})-(\rho_{i}u_{j})(\rho_{i}\pi)$
$\displaystyle=$	$\displaystyle\max_{\phi_{j}:\mathcal{A}_{j}\to\mathcal{A}_{j}}\sum_{a\in\mathcal{A}}\pi(\rho_{i}^{-1}a_{i},a_{-i})\cdot u_{j}(\phi_{j}(a_{j}),a_{-i,j},\rho_{i}^{-1}a_{i})-u_{j}(\pi)$
$\displaystyle=$	$\displaystyle\max_{\phi_{j}:\mathcal{A}_{j}\to\mathcal{A}_{j}}\sum_{a\in\mathcal{A}}\pi(a_{i},a_{-i})\cdot u_{j}(\phi_{j}(a_{j}),a_{-i,j},a_{i})-u_{j}(\pi)$	$\displaystyle,\text{$\rho_{i}$ is a bijection on $\mathcal{A}_{i}$}$
$\displaystyle=$	$\displaystyle\mathcal{E}_{j}^{\mathrm{CE}}(\pi,u)$

For player $i$ , we define operator $\bar{\rho}_{i}$ as $(\bar{\rho}_{i}\phi_{i})(a_{i})=\rho_{i}^{-1}\phi_{i}(\rho_{i}a_{i})$ . We can verify that $\bar{\rho}_{i}$ is a bijection on $\{\phi_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i}\}$ , because $\bar{\cdot}$ is a homomorphism in the sense that $\overline{\rho_{i}^{1}}\circ\overline{\rho_{i}^{2}}=\overline{\rho_{i}^{2}\rho_{i}^{1}}$ and $\bar{\cdot}$ maps the identity mapping of $\mathcal{A}_{i}$ to the identity mapping of $\{\mathcal{A}_{i}\to\mathcal{A}_{i}\}$ . Specifically,

\displaystyle\overline{\rho_{i}^{1}}\circ\overline{\rho_{i}^{2}}\phi_{i}(a_{i})=(\rho_{i}^{1})^{-1}(\overline{\rho_{i}^{2}}\phi_{i})(\rho_{i}^{1}a_{i})=(\rho_{i}^{1})^{-1}(\rho_{i}^{2})^{-1}\phi_{i}(\rho_{i}^{2}\rho_{i}^{1}a_{i})=\overline{\rho_{i}^{2}\rho_{i}^{1}}\phi_{i}(a_{i}),

and

\displaystyle\overline{e_{i}}\phi_{i}(a_{i})=e_{i}^{-1}\phi_{i}(e_{i}a_{i})=\phi_{i}(a_{i}).

Based on $\bar{\rho}_{i}$ , we have

	$\displaystyle\mathcal{E}_{i}^{\mathrm{CE}}(\rho_{i}\pi,\rho_{i}u)$
$\displaystyle=$	$\displaystyle\max_{\phi_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i}}\sum_{a\in\mathcal{A}}(\rho_{i}\pi)(a)\cdot(\rho_{i}u_{i})(\phi_{i}(a_{i}),a_{-i})-u_{i}(\pi)$
$\displaystyle=$	$\displaystyle\max_{\phi_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i}}\sum_{a\in\mathcal{A}}\pi(\rho_{i}^{-1}a_{i},a_{-i})u_{i}(\rho_{i}^{-1}\phi_{i}(a_{i}),a_{-i})-u_{i}(\pi)$
$\displaystyle=$	$\displaystyle\max_{\phi_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i}}\sum_{a\in\mathcal{A}}\pi(\rho_{i}^{-1}a_{i},a_{-i})u_{i}(\rho_{i}^{-1}\phi_{i}(\rho_{i}(\rho_{i}^{-1}a_{i})),a_{-i})-u_{i}(\pi)$
$\displaystyle=$	$\displaystyle\max_{\phi_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i}}\sum_{a\in\mathcal{A}}\pi(a_{i},a_{-i})u_{i}(\rho_{i}^{-1}\phi_{i}(\rho_{i}a_{i}),a_{-i})-u_{i}(\pi)$	$\displaystyle,\text{$\rho_{i}$ is a bijection on $\mathcal{A}_{i}$}$
$\displaystyle=$	$\displaystyle\max_{\phi_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i}}\sum_{a\in\mathcal{A}}\pi(a_{i},a_{-i})u_{i}((\bar{\rho}_{i}\phi_{i})(a_{i}),a_{-i})-u_{i}(\pi)$
$\displaystyle=$	$\displaystyle\max_{\phi_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i}}\sum_{a\in\mathcal{A}}\pi(a_{i},a_{-i})u_{i}(\phi_{i}(a_{i}),a_{-i})-u_{i}(\pi)$	$\displaystyle,\text{$\bar{\rho}_{i}$ is a bijection on $\{\mathcal{A}_{i}\to\mathcal{A}_{i}\}$}$
$\displaystyle=$	$\displaystyle\mathcal{E}_{i}^{\mathrm{CE}}(\pi,u)$

Thus, we have $\mathcal{E}(\rho_{i}\pi,\rho_{i}u)=\mathcal{E}(\pi,u)$ , thus if $\pi$ is a $\varepsilon$ -CE of $\Gamma_{u}$ , then $\rho_{i}\pi$ must be a $\varepsilon$ -CE of $\Gamma_{\rho_{i}u}$ .

A.4 Proof of Lemma 3.7 to Lemma 3.9

See 3.7

Proof.

$\forall j\neq i,\rho_{0}\in\mathcal{G}_{i}$ , for operator $\mathcal{O}_{i}$ we have

\displaystyle(\mathcal{O}_{i}f^{\mathrm{NE}})(\rho_{0}u)_{j}=

\displaystyle\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}f^{\mathrm{NE}}(\rho_{i}\rho_{0}u)_{j}\overset{(a)}{=}\frac{1}{|\mathcal{A}_{i}|!}\sum_{\widehat{\rho}_{i}\in\mathcal{G}_{i}}f^{\mathrm{NE}}(\widehat{\rho}_{i}u)_{j}=(\mathcal{O}_{i}f^{\mathrm{NE}})(u)_{j}

where in $(a)$ we define $\widehat{\rho}_{i}=\rho_{i}\rho_{0}$ , and $(a)$ holds since $\rho_{0}$ is a bijection on $\mathcal{G}_{i}$ . As a result, $\mathcal{O}_{i}f^{\mathrm{NE}}$ is $i$ -PI.

For operator $\mathcal{P}_{i}$ we have

	$\displaystyle(\mathcal{P}_{i}f^{\mathrm{NE}})(\rho_{0}u)_{i}=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}f^{\mathrm{NE}}(\rho_{i}\rho_{0}u)_{j}=\rho_{0}\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{0}^{-1}\rho_{i}^{-1}f^{\mathrm{NE}}(\rho_{i}\rho_{0}u)_{j}$
	$\displaystyle=$	$\displaystyle\rho_{0}\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\widehat{\rho}_{i}\in\mathcal{G}_{i}}\widehat{\rho}_{i}^{-1}f^{\mathrm{NE}}(\widehat{\rho}_{i}u)_{j}=\rho_{0}(\mathcal{P}_{i}f^{\mathrm{NE}})(u)_{i},$

therefore $\mathcal{P}_{i}f^{\mathrm{NE}}$ is $i$ -PE.

If $f^{\mathrm{NE}}$ is already $i$ -PI, $\forall j\neq i$ we have

\displaystyle\mathcal{O}_{i}f^{\mathrm{NE}}(u)_{j}=

\displaystyle\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}f^{\mathrm{NE}}(\rho_{i}u)_{j}=\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}f^{\mathrm{NE}}(u)_{j}=f^{\mathrm{NE}}(u)_{j},

and $\mathcal{O}_{i}f^{\mathrm{NE}}(u)_{i}=f^{\mathrm{NE}}(u)_{i}$ according to definition of $\mathcal{O}_{i}$ . Therefore, $\mathcal{O}_{i}f^{\mathrm{NE}}=f^{\mathrm{NE}}$ for $i$ -PI $f^{\mathrm{NE}}$ .

If $f^{\mathrm{NE}}$ is already $i$ -PE, we have

\displaystyle\mathcal{P}_{i}f^{\mathrm{NE}}(u)_{i}=

\displaystyle\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}f^{\mathrm{NE}}(\rho_{i}u)_{i}=\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}\rho_{i}f^{\mathrm{NE}}(u)_{i}=\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}f^{\mathrm{NE}}(u)_{i}=f^{\mathrm{NE}}(u)_{i},

and $\forall j\neq i,\mathcal{P}_{i}f^{\mathrm{NE}}(u)_{j}=f^{\mathrm{NE}}(u)_{j}$ according to definition of $\mathcal{P}_{i}$ . Therefore, $\mathcal{P}_{i}f^{\mathrm{NE}}=f^{\mathrm{NE}}$ for $i$ -PE $f^{\mathrm{NE}}$ .

∎

See 3.8

Proof.

A direct inference from Lemma 3.7 ∎

See 3.9

Proof.

$\forall\rho_{0}\in\mathcal{G}_{i}$ , we have

	$\displaystyle(\mathcal{Q}_{i}f^{\mathrm{(C)CE}})(\rho_{0}u)=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}f^{\mathrm{(C)CE}}(\rho_{i}\rho_{0}u)=\rho_{0}\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{0}^{-1}\rho_{i}^{-1}f^{\mathrm{(C)CE}}(\rho_{i}\rho_{0}u)$
	$\displaystyle=$	$\displaystyle\rho_{0}\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\widehat{\rho}_{i}\in\mathcal{G}_{i}}\widehat{\rho}_{i}^{-1}f^{\mathrm{(C)CE}}(\widehat{\rho}_{i}u)=\rho_{0}(\mathcal{Q}_{i}f^{\mathrm{(C)CE}})(u)$

If $f^{\text{(C)CE}}$ is already $i$ -PE, we have

\displaystyle\mathcal{Q}_{i}f^{\text{(C)CE}}(u)=

\displaystyle\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}f^{\text{(C)CE}}(\rho_{i}u)=\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}\rho_{i}f^{\text{(C)CE}}(u)=\frac{1}{|\mathcal{A}_{i}|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}f^{\text{(C)CE}}(u)=f^{\text{(C)CE}}(u)

∎

A.5 Proof of Lemma 3.11

See 3.11

Proof.

We prove the three claims below.

1.

$\mathcal{X}\mathcal{F}_{\mathcal{X}}\subseteq\mathcal{F}_{\mathcal{X}}$ .
2.

$\mathcal{X}^{2}\mathcal{F}_{\mathcal{X}}=\mathcal{X}\mathcal{F}_{\mathcal{X}}$ .
3.

If $\mathcal{X}\mathcal{Y}=\mathcal{Y}\subseteq\mathcal{F}_{\mathcal{X}}$ , then $\mathcal{Y}\subseteq\mathcal{X}\mathcal{F}_{\mathcal{X}}$

The first claim holds because $\mathcal{F}_{\mathcal{X}}$ is closed under $\mathcal{X}$ , and the second claim holds because $\mathcal{X}$ is idempotent. For the third claim, from $\mathcal{Y}\subseteq\mathcal{F}_{\mathcal{X}}$ we know $\mathcal{X}\mathcal{Y}\subseteq\mathcal{X}\mathcal{F}_{\mathcal{X}}$ , then $\mathcal{Y}=\mathcal{X}\mathcal{Y}\subseteq\mathcal{X}\mathcal{F}_{\mathcal{X}}$ .

We immediately know $\mathcal{X}\mathcal{F}_{\mathcal{X}}$ is the largest subset of $\mathcal{F}_{\mathcal{X}}$ that is invariant under $\mathcal{X}$ . ∎

Appendix B Omitted Proofs in Section 4

B.1 Proof of Theorem 4.3

See 4.3

Some of the proof techniques come from Dütting et al. [2019] and Duan et al. [2021]. We first introduce some useful lemmas. Denote $\ell:\mathcal{F}\times\mathcal{U}\to\mathbb{R}$ as the loss function (such as $\ell(f,u)\coloneqq\mathcal{E}(f(u),u)$ ). We measure the capacity of the composite function class $\ell\circ\mathcal{F}$ using the empirical Rademacher complexity [Bartlett and Mendelson, 2002] on the training set $S$ , which is defined as:

\displaystyle\mathcal{R}_{S}(\ell\circ\mathcal{F})\coloneqq\frac{1}{m}\mathbb{E}_{\bm{x}\sim\{+1,-1\}^{m}}\Big{[}\sup_{f\in\mathcal{F}}\sum_{i=1}^{m}x_{i}\cdot\ell(f,u^{(i)})\Big{]},

where $\bm{x}$ is distributed i.i.d. according to uniform distribution in $\{+1,-1\}$ . We have

Lemma B.1 (Shalev-Shwartz and Ben-David [2014]).

Let $S$ be a training set of size $m$ drawn i.i.d. from distribution $\mathcal{D}$ over $\mathcal{U}$ . Then with probability at least $1-\delta$ over draw of $S$ from $\mathcal{D}$ , for all $f\in\mathcal{F}$ ,

\mathbb{E}_{u\sim\mathcal{D}}[\ell(f,u)]-\frac{1}{m}\sum_{u\in S}\ell(l,u)\leq 2\mathcal{R}_{S}(\ell\circ\mathcal{F})+4\sqrt{\frac{2\ln(4/\delta)}{m}}

Lemma B.2.

If $|\ell(\cdot)|\leq c$ for constant $c>0$ and $\forall f,f^{\prime}\in\mathcal{F},|\ell(f,u)-\ell(f^{\prime},u)|\leq L{\|f-f^{\prime}\|}_{\infty}$ , then we have

\mathbb{E}_{u\sim\mathcal{D}}[\ell(f,u)]-\frac{1}{m}\sum_{u\in S}\ell(l,u)\leq 2\inf_{r>0}\left\{c\sqrt{\frac{2\ln\mathcal{N}_{\infty}(\mathcal{F},r)}{m}}+Lr\right\}+4\sqrt{\frac{2\ln(4/\delta)}{m}}

Proof.

For function class $\mathcal{F}$ , let $\mathcal{F}_{r}$ with $|\mathcal{F}_{r}|=\mathcal{N}_{\infty}(\mathcal{F},r)$ be the function class that $r$ -covers $\mathcal{F}$ for some $r>0$ . Similarly, $\forall f\in\mathcal{F}$ , denote $f_{r}\in\mathcal{F}_{r}$ be the function that $r$ -covers $f$ . We have

$\displaystyle\mathcal{R}_{S}(\ell\circ\mathcal{F})=$	$\displaystyle\frac{1}{m}\mathbb{E}_{\bm{x}}\Big{[}\sup_{f\in\mathcal{F}}\sum_{i=1}^{m}x_{i}\cdot\ell(f,u^{(i)})\Big{]}$		(5)
$\displaystyle=$	$\displaystyle\frac{1}{m}\mathbb{E}_{\bm{x}}\Big{[}\sup_{f\in\mathcal{F}}\sum_{i=1}^{m}x_{i}\cdot\big{(}\ell(f_{r},u^{(i)})+\ell(f,u^{(i)})-\ell(f_{r},u^{(i)})\big{)}\Big{]}$
$\displaystyle\leq$	$\displaystyle\frac{1}{m}\mathbb{E}_{\bm{x}}\Big{[}\sup_{f_{r}\in\mathcal{F}_{r}}\sum_{i=1}^{m}x_{i}\cdot\ell(f_{r},u^{(i)})\Big{]}+\frac{1}{m}\mathbb{E}_{\bm{x}}\Big{[}\sup_{f\in\mathcal{F}}\sum_{i=1}^{m}\|x_{i}\cdot Lr\|\Big{]}$	$\displaystyle,\|\ell(f,u)-\ell(f_{r},u)\|\leq L{\\|f-f_{r}\\|}_{\infty}=Lr$
$\displaystyle{\leq}$	$\displaystyle\sup_{f_{r}\in\mathcal{F}_{r}}\sqrt{\sum_{i=1}^{m}\ell^{2}(f_{r},u^{(i)})}\cdot\frac{\sqrt{2\ln\mathcal{N}_{\infty}(\mathcal{F},r)}}{m}+\frac{Lr}{m}\mathbb{E}_{\bm{x}}{\\|\bm{x}\\|}$	$\displaystyle,\text{the first term holds by Massart's lemma}$
$\displaystyle\leq$	$\displaystyle\sqrt{c^{2}m}\cdot\frac{\sqrt{2\ln\mathcal{N}_{\infty}(\mathcal{F},r)}}{m}+\frac{Lr}{m}\mathbb{E}_{\bm{x}}{\\|\bm{x}\\|}$
$\displaystyle\leq$	$\displaystyle c\sqrt{\frac{2\ln\mathcal{N}_{\infty}(\mathcal{F},r)}{m}}+Lr,$

Combining Lemma B.1 and Equation 5, we get

\mathbb{E}_{u\sim\mathcal{D}}[\ell(f,u)]-\frac{1}{m}\sum_{u\in S}\ell(l,u)\leq 2\inf_{r>0}\left\{c\sqrt{\frac{2\ln\mathcal{N}_{\infty}(\mathcal{F},r)}{m}}+Lr\right\}+4\sqrt{\frac{2\ln(4/\delta)}{m}}

∎

B.1.1 NE Approximator

Lemma B.3.

For arbitrary product mixed strategy $\sigma$ and $\sigma^{\prime}$ , we have

\displaystyle|\mathcal{E}(\sigma,u)-\mathcal{E}(\sigma^{\prime},u)|\leq 2n{\|\sigma-\sigma^{\prime}\|},

Proof.

$\forall\sigma,\sigma^{\prime}$ , we define $y_{-j}\coloneqq(\sigma_{1},\dots,\sigma_{j-1},\sigma^{\prime}_{j+1},\dots,\sigma^{\prime}_{n})$ . Then, $\forall i\in[n]$ we have

$\displaystyle\|u_{i}(\sigma)-u_{i}(\sigma^{\prime})\|=$	$\displaystyle\|u_{i}(\sigma_{1},\sigma_{2},\dots,\sigma_{n})-u_{i}(\sigma^{\prime},\sigma^{\prime}_{2},\dots,\sigma^{\prime}_{n})\|$
$\displaystyle=$	$\displaystyle\Big{\|}\sum_{j=1}^{n}\Big{(}u_{i}(\sigma_{1},\dots,\sigma_{j},\sigma^{\prime}_{j+1},\dots,\sigma^{\prime}_{n})-u_{i}(\sigma_{1},\dots,\sigma^{\prime}_{j},\sigma^{\prime}_{j+1},\dots,\sigma^{\prime}_{n})\Big{)}\Big{\|}$
$\displaystyle=$	$\displaystyle\Big{\|}\sum_{j=1}^{n}\Big{(}u_{i}(\sigma_{j},y_{-j})-u_{i}(\sigma^{\prime}_{j},y_{-j})\Big{)}\Big{\|}$
$\displaystyle=$	$\displaystyle\Big{\|}\sum_{j=1}^{n}\sum_{a_{j}}(\sigma_{j}(a_{j})-\sigma^{\prime}_{j}(a_{j}))\sum_{a_{-j}}u_{i}(a_{j},a_{-j})y_{-j}(a_{-j})\Big{\|}$
$\displaystyle\leq$	$\displaystyle\sum_{j=1}^{n}\sum_{a_{j}}\Big{\|}\sigma_{j}(a_{j})-\sigma^{\prime}_{j}(a_{j})\Big{\|}\sum_{a_{-j}}u_{i}(a_{j},a_{-j})y_{-j}(a_{-j})$
$\displaystyle{\leq}$	$\displaystyle\sum_{j=1}^{n}\sum_{a_{j}}\Big{\|}\sigma_{j}(a_{j})-\sigma^{\prime}_{j}(a_{j})\Big{\|}\sum_{a_{-j}}y_{-j}(a_{-j})$	$\displaystyle,u_{i}(\cdot)\in[0,1]$
$\displaystyle\leq$	$\displaystyle\sum_{j=1}^{n}\sum_{a_{j}\in A_{j}}\Big{\|}\sigma_{j}(a_{j})-\sigma^{\prime}_{j}(a_{j})\Big{\|}\leq n\max_{j\in[n]}\sum_{a_{j}\in A_{j}}\Big{\|}\sigma_{j}(a_{j})-\sigma^{\prime}_{j}(a_{j})\Big{\|}$
$\displaystyle=$	$\displaystyle n{\\|\sigma-\sigma^{\prime}\\|},$

Therefore, $\forall a_{i}\in A_{i}$ ,

	$\displaystyle u_{i}(a_{i},\sigma_{-i})-u_{i}(\sigma)=$	$\displaystyle u_{i}(a_{i},\sigma_{-i})-u_{i}(a_{i},\sigma^{\prime}_{-i})+u_{i}(a_{i},\sigma^{\prime}_{-i})-u_{i}(\sigma^{\prime})+u_{i}(\sigma^{\prime})-u_{i}(\sigma)$
	$\displaystyle\leq$	$\displaystyle n{\\|\sigma-\sigma^{\prime}\\|}+\mathcal{E}(\sigma^{\prime},u)+n{\\|\sigma-\sigma^{\prime}\\|}$
	$\displaystyle=$	$\displaystyle\mathcal{E}(\sigma^{\prime},u)+2n{\\|\sigma-\sigma^{\prime}\\|}.$

Based on that, we get

\displaystyle\mathcal{E}(\sigma,u)=\max_{i\in N,a_{i}\in A_{i}}[u_{i}(a_{i},\sigma_{-i})-u_{i}(\sigma)]\leq\mathcal{E}(\sigma^{\prime},u)+2n{\|\sigma-\sigma^{\prime}\|}

Similarly, we also have

\mathcal{E}(\sigma^{\prime},u)\leq\mathcal{E}(\sigma,u)+2n{\|\sigma-\sigma^{\prime}\|}

∎

Based on Lemma B.3, $\forall f,f^{\prime}\in\mathcal{F}^{\mathrm{NE}}$ , we have

\mathcal{E}(f(u),u)-\mathcal{E}(f^{\prime}(u),u)\leq 2{\|f(u)-f^{\prime}(u)\|}\leq 2{\|f-f^{\prime}\|}_{\infty}

Considering that $|\mathcal{E}(\cdot)|\leq 1$ , according to Lemma B.2, we have:

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(f^{\mathrm{NE}}(u),u)]-\frac{1}{m}\sum_{u\in S}\mathcal{E}(f^{\mathrm{NE}}(u),u)\leq 2\cdot\inf_{r>0}\Big{\{}\sqrt{\frac{2\ln\mathcal{N}_{\infty}(\mathcal{F}^{\mathrm{NE}},r)}{m}}+2nr\Big{\}}+4\sqrt{\frac{2\ln(4/\delta)}{m}}

B.1.2 CCE Approximator

Lemma B.4.

For arbitrary joint mixed strategy $\pi$ and $\pi^{\prime}$ , we have

|\mathcal{E}(\pi,u)-\mathcal{E}(\pi^{\prime},u)|\leq 2{\|\pi-\pi^{\prime}\|},

Proof.

$\forall\pi,\pi^{\prime},\forall i\in[n]$ we have

\displaystyle|u_{i}(\pi)-u_{i}(\pi^{\prime})|=\sum_{a\in\mathcal{A}}(\pi(a)-\pi^{\prime}(a))u_{i}(a)\overset{(a)}{\leq}\sum_{a\in\mathcal{A}}|\pi(a)-\pi^{\prime}(a)|={\|\pi-\pi^{\prime}\|}

(6)

where $(a)$ holds since $u_{i}(\cdot)\in[0,1]$ . Therefore, $\forall a_{i}\in A_{i}$ ,

	$\displaystyle u_{i}(a_{i},\pi_{-i})-u_{i}(\pi)=$	$\displaystyle u_{i}(a_{i},\pi_{-i})-u_{i}(a_{i},\pi^{\prime}_{-i})+u_{i}(a_{i},\pi^{\prime}_{-i})-u_{i}(\pi^{\prime})+u_{i}(\pi^{\prime})-u_{i}(\pi)$
	$\displaystyle\leq$	$\displaystyle{\\|\pi-\pi^{\prime}\\|}+\mathcal{E}(\pi^{\prime},u)+{\\|\pi-\pi^{\prime}\\|}$
	$\displaystyle=$	$\displaystyle\mathcal{E}(\pi^{\prime},u)+2{\\|\pi-\pi^{\prime}\\|}.$

Based on that, we get

\displaystyle\mathcal{E}(\pi,u)=\max_{i\in N,a_{i}\in A_{i}}[u_{i}(a_{i},\pi_{-i})-u_{i}(\pi)]\leq\mathcal{E}(\pi^{\prime},u)+2{\|\pi-\pi^{\prime}\|}

Similarly, we also have

\mathcal{E}(\pi^{\prime},u)\leq\mathcal{E}(\pi,u)+2{\|\pi-\pi^{\prime}\|}

∎

Based on Lemma B.4, $\forall f,f^{\prime}\in\mathcal{F}^{\mathrm{CCE}}$ , we have

\mathcal{E}(f(u),u)-\mathcal{E}(f^{\prime}(u),u)\leq 2{\|f(u)-f^{\prime}(u)\|}\leq 2{\|f-f^{\prime}\|}_{\infty}

Considering that $|\mathcal{E}(\cdot)|\leq 1$ , according to Lemma B.2, we have:

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(f^{\mathrm{CCE}}(u),u)]-\frac{1}{m}\sum_{u\in S}\mathcal{E}(f^{\mathrm{CCE}}(u),u)\leq 2\cdot\inf_{r>0}\Big{\{}\sqrt{\frac{2\ln\mathcal{N}_{\infty}(\mathcal{F}^{\mathrm{CCE}},r)}{m}}+2r\Big{\}}+4\sqrt{\frac{2\ln(4/\delta)}{m}}

B.1.3 CE Approximator

Lemma B.5.

For arbitrary joint mixed strategy $\pi$ and $\pi^{\prime}$ , we have

|\mathcal{E}^{\mathrm{CE}}(\pi,u)-\mathcal{E}^{\mathrm{CE}}(\pi^{\prime},u)|\leq 2{\|\pi-\pi^{\prime}\|},

Proof.

$\forall a_{i}\in A_{i},\forall\phi_{i}$ , we have

	$\displaystyle\sum_{a\in\mathcal{A}}\pi(a)u_{i}(\phi(a_{i}),a_{-i})-u_{i}(\pi)=$	$\displaystyle\sum_{a\in\mathcal{A}}\pi(a)u_{i}(\phi(a_{i}),a_{-i})-\sum_{a\in\mathcal{A}}\pi^{\prime}(a)u_{i}(\phi(a_{i}),a_{-i})$
		$\displaystyle\quad+\sum_{a\in\mathcal{A}}\pi^{\prime}(a)u_{i}(\phi(a_{i}),a_{-i})-u_{i}(\pi^{\prime})+u_{i}(\pi^{\prime})-u_{i}(\pi)$
	$\displaystyle\leq$	$\displaystyle{\\|\pi-\pi^{\prime}\\|}+\mathcal{E}^{\mathrm{CE}}(\pi^{\prime},u)+{\\|\pi-\pi^{\prime}\\|}$
	$\displaystyle=$	$\displaystyle\mathcal{E}^{\mathrm{CE}}(\pi^{\prime},u)+2{\\|\pi-\pi^{\prime}\\|}.$

Based on that, we get

\displaystyle\mathcal{E}^{\mathrm{CE}}(\pi,u)=\max_{i\in N}\max_{\phi_{i}}\sum_{a\in\mathcal{A}}\pi(a)u_{i}(\phi(a_{i}),a_{-i})-u_{i}(\pi)\leq\mathcal{E}^{\mathrm{CE}}(\pi^{\prime},u)+2{\|\pi-\pi^{\prime}\|}

Similarly, we also have

\mathcal{E}^{\mathrm{CE}}(\pi^{\prime},u)\leq\mathcal{E}^{\mathrm{CE}}(\pi,u)+2{\|\pi-\pi^{\prime}\|}

∎

Based on Lemma B.4, $\forall f,f^{\prime}\in\mathcal{F}^{\mathrm{CE}}$ , we have

\mathcal{E}^{\mathrm{CE}}(f(u),u)-\mathcal{E}^{\mathrm{CE}}(f^{\prime}(u),u)\leq 2{\|f(u)-f^{\prime}(u)\|}\leq 2{\|f-f^{\prime}\|}_{\infty}

Considering that $|\mathcal{E}(\cdot)|\leq 1$ , according to Lemma B.2, we have:

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}^{\mathrm{CE}}(f^{\mathrm{CE}}(u),u)]-\frac{1}{m}\sum_{u\in S}\mathcal{E}^{\mathrm{CE}}(f^{\mathrm{CE}}(u),u)\leq 2\cdot\inf_{r>0}\Big{\{}\sqrt{\frac{2\ln\mathcal{N}_{\infty}(\mathcal{F}^{\mathrm{CE}},r)}{m}}+2r\Big{\}}+4\sqrt{\frac{2\ln(4/\delta)}{m}}

B.2 Proof of Theorem 4.4

See 4.4

Proof.

For function class $\mathcal{F}$ of NE, CE or CCE approximators, according to Lemma B.3, Lemma B.4 and Lemma B.5, $\forall f,g\in\mathcal{F}$ we have

\mathcal{E}^{(\mathrm{CE})}(f(u),u)-\mathcal{E}^{(\mathrm{CE})}(g(u),u)\leq L{\|f(u)-g(u)\|}\leq L{\|f-g\|}_{\infty},

(7)

where $L=2n$ for NE approximators, and $L=2$ for CE and CCE approximators.

For simplicity, we denote $L_{S}(f)=\frac{1}{m}\sum_{u\in S}\mathcal{E}^{(\mathrm{CE})}(f(u),u)$ and $L_{\mathcal{D}}(f)=\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}^{(\mathrm{CE})}(f(u),u)]$ . let $\mathcal{F}_{r}$ with $|\mathcal{F}_{r}|=\mathcal{N}_{\infty}(\mathcal{F},r)$ be the function class that $r$ -covers $\mathcal{F}$ for some $r>0$ . $\forall\epsilon\in(0,1)$ , by setting $r=\frac{\epsilon}{3L}$ we have

		$\displaystyle\mathbb{P}_{S\sim\mathcal{D}^{m}}\Big{[}\exists f\in\mathcal{F},\big{\|}L_{S}(f)-L_{\mathcal{D}}(f)\big{\|}>\epsilon\Big{]}$
	$\displaystyle\leq$	$\displaystyle\mathbb{P}_{S\sim\mathcal{D}^{m}}\Big{[}\exists f\in\mathcal{F},\big{\|}L_{S}(f)-L_{S}({f}_{r})\big{\|}+\big{\|}L_{S}({f}_{r})-L_{\mathcal{D}}({f}_{r})\big{\|}+\big{\|}L_{\mathcal{D}}({f}_{r})-L_{\mathcal{D}}(f)\big{\|}>\epsilon\Big{]}$
	$\displaystyle\overset{(a)}{\leq}$	$\displaystyle\mathbb{P}_{S\sim\mathcal{D}^{m}}\Big{[}\exists f\in\mathcal{F},Lr+\big{\|}L_{S}({f}_{r})-L_{\mathcal{D}}({f}_{r})\big{\|}+Lr>\epsilon\Big{]}$
	$\displaystyle\leq$	$\displaystyle\mathbb{P}_{S\sim\mathcal{D}^{m}}\Big{[}\exists{f}_{r}\in{\mathcal{F}}_{r},\big{\|}L_{S}({f}_{r})-L_{\mathcal{D}}({f}_{r})\big{\|}>\epsilon-2Lr\Big{]}$
	$\displaystyle\overset{(b)}{\leq}$	$\displaystyle\mathcal{N}_{\infty}(\mathcal{F},r)\mathbb{P}_{S\sim\mathcal{D}^{m}}\Big{[}\big{\|}L_{S}({f})-L_{\mathcal{D}}({f})\big{\|}>\epsilon-2Lr\Big{]}$
	$\displaystyle\overset{(c)}{\leq}$	$\displaystyle 2\mathcal{N}_{\infty}(\mathcal{F},r)\exp(-2m(\epsilon-2Lr)^{2}),$
	$\displaystyle=$	$\displaystyle 2\mathcal{N}_{\infty}(\mathcal{F},\frac{\epsilon}{3L})\exp(-\frac{2}{9}m\epsilon^{2})$

where $(a)$ holds by Equation 7, $(b)$ holds by union bound, and $(c)$ holds by Hoeffding inequality. As a result, when $m\geq\frac{9}{2\epsilon^{2}}\left(\ln\frac{2}{\delta}+\ln\mathcal{N}_{\infty}(\mathcal{F},\frac{\epsilon}{3L})\right)$ , we have $\mathbb{P}_{S\sim\mathcal{D}^{m}}\Big{[}\exists f\in\mathcal{F},\Big{|}L_{S}(f)-L_{\mathcal{D}}(f)\Big{|}>\epsilon\Big{]}<\delta$ . ∎

B.3 Proof of Theorem 4.5

See 4.5

We first provide an auxiliary lemma.

Lemma B.6.

For function class $\mathcal{F}$ and orbit averaging operator $\mathcal{X}$ , if $\forall f,g\in\mathcal{F},\ell_{\infty}(\mathcal{X}f,\mathcal{X}g)\leq\ell_{\infty}(f,g)$ , then $\mathcal{N}_{\infty}(\mathcal{X}\mathcal{F},r)\leq\mathcal{N}_{\infty}(\mathcal{F},r)$ for any $r>0$ .

Proof.

$\forall r>0$ , Denote $\mathcal{F}_{r}$ as the smallest $r$ -covering set that covers $\mathcal{F}$ with size $\mathcal{N}_{\infty}(\mathcal{F},r)$ . $\forall f\in\mathcal{F}$ , let $f_{r}\in\mathcal{F}_{r}$ be the function that $r$ -covers $f$ . We have $\ell_{\infty}(\mathcal{X}f_{r},\mathcal{X}f)\leq\ell_{\infty}(f_{r},f)\leq r$ . Therefore, $\mathcal{X}\mathcal{F}_{r}$ is a $r$ -covering set of $\mathcal{X}\mathcal{F}$ , and we have $\mathcal{N}_{\infty}(\mathcal{X}\mathcal{F},r)\leq|\mathcal{X}\mathcal{F}_{r}|\leq|\mathcal{F}_{r}|=\mathcal{N}_{\infty}$ . ∎

Proof of Theorem 4.5.

For player $i\in[n]$ and $\forall f^{\mathrm{NE}},g^{\mathrm{NE}}\in\mathcal{F}^{\mathrm{NE}}$ , assuming $\mathcal{U}$ is closed under any $\rho_{i}\in\mathcal{G}_{i}$ . For $\mathcal{O}_{i}$ ,

	$\displaystyle l_{\infty}(\mathcal{O}_{i}f^{\mathrm{NE}},\mathcal{O}_{i}g^{\mathrm{NE}})=$	$\displaystyle\max_{u\in\mathcal{U}}\\|\mathcal{O}_{i}f^{\mathrm{NE}}(u)-\mathcal{O}_{i}g^{\mathrm{NE}}(u)\\|$
	$\displaystyle=$	$\displaystyle\max_{j\in[n]}\max_{u\in\mathcal{U}}\\|(\mathcal{O}_{i}f^{\mathrm{NE}})(u)_{j}-(\mathcal{O}_{i}g^{\mathrm{NE}})(u)_{j}\\|$
	$\displaystyle=$	$\displaystyle\max\Big{\{}\max_{u\in\mathcal{U}}\\|f^{\mathrm{NE}}(u)_{i}-g^{\mathrm{NE}}(u)_{i}\\|,\leavevmode\nobreak\ \max_{j\neq i}\max_{u\in\mathcal{U}}\\|\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}(f^{\mathrm{NE}}(\rho_{i}u)_{j}-g^{\mathrm{NE}}(\rho_{i}u)_{j})\\|\Big{\}}$
	$\displaystyle\leq$	$\displaystyle\max\Big{\{}\max_{u\in\mathcal{U}}\\|f^{\mathrm{NE}}(u)_{i}-g^{\mathrm{NE}}(u)_{i}\\|,\leavevmode\nobreak\ \max_{j\neq i}\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\max_{u\in\mathcal{U}}\\|f^{\mathrm{NE}}(\rho_{i}u)_{j}-g^{\mathrm{NE}}(\rho_{i}u)_{j}\\|\Big{\}}$
	$\displaystyle=$	$\displaystyle\max\Big{\{}\max_{u\in\mathcal{U}}\\|f^{\mathrm{NE}}(u)_{i}-g^{\mathrm{NE}}(u)_{i}\\|,\leavevmode\nobreak\ \max_{j\neq i}\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\max_{u\in\mathcal{U}}\\|f^{\mathrm{NE}}(u)_{j}-g^{\mathrm{NE}}(u)_{j}\\|\Big{\}}$
	$\displaystyle=$	$\displaystyle\max\Big{\{}\max_{u\in\mathcal{U}}\\|f^{\mathrm{NE}}(u)_{i}-g^{\mathrm{NE}}(u)_{i}\\|,\leavevmode\nobreak\ \max_{j\neq i}\max_{u}\\|f^{\mathrm{NE}}(u)_{j}-g^{\mathrm{NE}}(u)_{j}\\|\Big{\}}$
	$\displaystyle=$	$\displaystyle l_{\infty}(f^{\mathrm{NE}},g^{\mathrm{NE}})$

Since $\mathcal{O}=\mathcal{O}_{1}\circ\cdots\circ\mathcal{O}_{n}$ , we have

\ell_{\infty}(\mathcal{O}f^{\mathrm{NE}},\mathcal{O}g^{\mathrm{NE}})\leq\ell_{\infty}(f^{\mathrm{NE}},g^{\mathrm{NE}}).

(8)

For $\mathcal{P}_{i}$ ,

	$\displaystyle l_{\infty}(\mathcal{P}_{i}f^{\mathrm{NE}},\mathcal{P}_{i}g^{\mathrm{NE}})=$	$\displaystyle\max_{u\in\mathcal{U}}\max_{j\in[n]}\\|(\mathcal{P}_{i}f^{\mathrm{NE}})(u)_{j}-(\mathcal{P}_{i}g^{\mathrm{NE}})(u)_{j}\\|$
	$\displaystyle=$	$\displaystyle\max\Big{\{}\max_{j\neq i}\max_{u}\\|f^{\mathrm{NE}}(u)_{j}-g^{\mathrm{NE}}(u)_{j}\\|,\leavevmode\nobreak\ \max_{u}\\|\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}(f^{\mathrm{NE}}(\rho_{i}u)_{i}-g^{\mathrm{NE}}(\rho_{i}u)_{i})\\|\Big{\}}$
	$\displaystyle=$	$\displaystyle\max\Big{\{}\max_{j\neq i}\max_{u}\\|f^{\mathrm{NE}}(u)_{j}-g^{\mathrm{NE}}(u)_{j}\\|,\leavevmode\nobreak\ \max_{u}\\|\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}(f^{\mathrm{NE}}(\rho_{i}u)_{i}-g^{\mathrm{NE}}(\rho_{i}u)_{i})\\|\Big{\}}$
	$\displaystyle\leq$	$\displaystyle\max\Big{\{}\max_{j\neq i}\max_{u}\\|f^{\mathrm{NE}}(u)_{j}-g^{\mathrm{NE}}(u)_{j}\\|,\leavevmode\nobreak\ \frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\max_{u}\\|f^{\mathrm{NE}}(\rho_{i}u)_{i}-g^{\mathrm{NE}}(\rho_{i}u)_{i}\\|\Big{\}}$
	$\displaystyle=$	$\displaystyle\max\Big{\{}\max_{j\neq i}\max_{u}\\|f^{\mathrm{NE}}(u)_{j}-g^{\mathrm{NE}}(u)_{j}\\|,\leavevmode\nobreak\ \frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\max_{u}\\|f^{\mathrm{NE}}(u)_{i}-g^{\mathrm{NE}}(u)_{i}\\|\Big{\}}$
	$\displaystyle=$	$\displaystyle\max\Big{\{}\max_{j\neq i}\max_{u}\\|f^{\mathrm{NE}}(u)_{j}-g^{\mathrm{NE}}(u)_{j}\\|,\leavevmode\nobreak\ \max_{u}\\|f^{\mathrm{NE}}(u)_{i}-g^{\mathrm{NE}}(u)_{i}\\|\Big{\}}$
	$\displaystyle=$	$\displaystyle l_{\infty}(f^{\mathrm{NE}},g^{\mathrm{NE}})$

Since $\mathcal{P}=\mathcal{P}_{1}\circ\cdots\circ\mathcal{P}_{n}$ , we have

\ell_{\infty}(\mathcal{P}f^{\mathrm{NE}},\mathcal{P}g^{\mathrm{NE}})\leq\ell_{\infty}(f^{\mathrm{NE}},g^{\mathrm{NE}}).

(9)

For CE or CCE approximator $f^{\mathrm{(C)CE}}\in\mathcal{F}^{\mathrm{(C)CE}}$ and $\mathcal{Q}_{i}$ , we have

	$\displaystyle l_{\infty}(\mathcal{Q}_{i}f^{\mathrm{(C)CE}},\mathcal{Q}_{i}g^{\mathrm{(C)CE}})=$	$\displaystyle\max_{u\in\mathcal{U}}\\|(\mathcal{Q}_{i}f^{\mathrm{(C)CE}})(u)-(\mathcal{Q}_{i}g^{\mathrm{(C)CE}})(u)\\|$
	$\displaystyle=$	$\displaystyle\max_{u}\\|\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}(f^{\mathrm{(C)CE}}(\rho_{i}u)-g^{\mathrm{(C)CE}}(\rho_{i}u))\\|$
	$\displaystyle\leq$	$\displaystyle\max_{u}\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\\|\rho_{i}^{-1}(f^{\mathrm{(C)CE}}(\rho_{i}u)-g^{\mathrm{(C)CE}}(\rho_{i}u))\\|$
	$\displaystyle\leq$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\max_{u}\\|\rho_{i}^{-1}(f^{\mathrm{(C)CE}}(\rho_{i}u)-g^{\mathrm{(C)CE}}(\rho_{i}u))\\|$
	$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\max_{u}\\|f^{\mathrm{(C)CE}}(\rho_{i}u)-g^{\mathrm{(C)CE}}(\rho_{i}u)\\|$
	$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\max_{u}\\|f^{\mathrm{(C)CE}}(u)-g^{\mathrm{(C)CE}}(u)\\|$
	$\displaystyle=$	$\displaystyle l_{\infty}(f^{\mathrm{(C)CE}},g^{\mathrm{(C)CE}})$

Since $\mathcal{Q}=\mathcal{Q}_{1}\circ\cdots\circ\mathcal{Q}_{n}$ , we have

\ell_{\infty}(\mathcal{Q}f^{\mathrm{(C)CE}},\mathcal{Q}g^{\mathrm{(C)CE}})\leq\ell_{\infty}(f^{\mathrm{(C)CE}},g^{\mathrm{(C)CE}}).

(10)

Combing Lemma B.6, Equation 8, Equation 9 and Equation 10, we finish the proof. ∎

B.4 Proof of Theorem 4.6

See 4.6

We first prove a lemma about the property of $\mathcal{E}_{i}(\pi,u)$ and $\mathcal{E}_{i}^{\mathrm{CE}}(\pi,u)$ .

Lemma B.7.

$\mathcal{E}_{i}(\pi,u)$ and $\mathcal{E}_{i}^{\mathrm{CE}}(\pi,u)$ are convex on $\pi$ , i.e.

\displaystyle p\mathcal{E}_{i}^{\text{(CE)}}(\pi_{1},u)+(1-p)\mathcal{E}_{i}^{\text{(CE)}}(\pi_{2},u)\geq\mathcal{E}_{i}^{\text{(CE)}}(p\pi_{1}+(1-p)\pi_{2},u),\quad\forall p\in[0,1]

Proof.

We recall the definition $\mathcal{E}_{i}(\pi,u)=\max_{a_{i}\in\mathcal{A}_{i}}u_{i}(a_{i},\pi_{-i})-u_{i}(\pi)$ for CCE approximator and $\mathcal{E}_{i}^{\mathrm{CE}}(\pi,u)=\max_{\phi_{i}\in\mathcal{A}_{i}\to\mathcal{A}_{i}}\sum_{a}\pi(a)u_{i}(\phi_{i}(a_{i}),a_{-i})-u_{i}(\pi)$ for CE approximator. $u_{i}(a_{i},\pi_{-i})$ is linear on $\pi$ . Given $\phi$ , $\sum_{a}\pi(a)u_{i}(\phi_{i}(a_{i}),a_{-i})$ is also linear on $\pi$ . Moreover, the maximum operator on a set of linear functions will induce a convex function.

∎

Proof of Theorem 4.6.

For $f\in\mathcal{F}^{\mathrm{(C)CE}}$ and $\forall i,j\in[n]$ ,

$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}^{\text{(CE)}}(\mathcal{Q}_{j}f(u),u)]=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}^{\text{(CE)}}(\frac{1}{\|\mathcal{A}_{j}\|!}\sum_{\rho_{j}\in\mathcal{G}_{j}}\rho_{j}^{-1}f(\rho_{j}u),u)]$	$\displaystyle,\text{by definition}$
$\displaystyle\leq$	$\displaystyle\frac{1}{\|\mathcal{A}_{j}\|!}\sum_{\rho_{j}\in\mathcal{G}_{j}}\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}^{\text{(CE)}}(\rho_{j}^{-1}f(\rho_{j}u),u)]$	$\displaystyle,\text{by convexity}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{j}\|!}\sum_{\rho_{j}\in\mathcal{G}_{j}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{i}^{\text{(CE)}}(\rho_{j}^{-1}f(v),\rho_{j}^{-1}v)]$	$\displaystyle,\text{let $v=\rho_{j}u$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{j}\|!}\sum_{\rho_{j}\in\mathcal{G}_{j}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{i}^{\text{(CE)}}(f(v),v)]$	$\displaystyle,\text{invariance of $\mathcal{E}_{i}^{\text{(CE)}}(\pi,u)$ under $\rho_{j}^{-1}\in\mathcal{G}_{j}$}$
$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}^{\text{(CE)}}(f(u),u)]$

Since $\mathcal{Q}=\circ_{i}\mathcal{Q}_{i}$ and $\mathcal{E}=\max_{i}\mathcal{E}_{i}$ , we have

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(\mathcal{Q}f(u),u)]\leq\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(f(u),u)]

∎

B.5 Proof of Theorem 4.7

See 4.7

Proof.

We only prove for the $\mathcal{P}$ -projected case; the proof for $\mathcal{O}$ -projected case is similar and therefore omitted.

Recall

\displaystyle\mathcal{E}_{i}(\sigma,u)=\max_{a_{i}\in\mathcal{A}_{i}}u_{i}(a_{i},\sigma_{-i})-u_{i}(\sigma)

Denote $u_{1}(\sigma)+u_{2}(\sigma)\equiv c$ , then

\displaystyle\sum_{i}\mathcal{E}_{i}(\sigma,u)=\max_{a_{1}\in\mathcal{A}_{1},a_{2}\in\mathcal{A}_{2}}u_{1}(a_{1},\sigma_{2})+u_{2}(a_{2},\sigma_{1})-c

Then we have

	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\sum_{i}\mathcal{E}_{i}((\mathcal{P}f^{\mathrm{NE}})(u),u)]=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{1},a_{2}}u_{1}(a_{1},(\mathcal{P}f^{\mathrm{NE}})(u)_{2})+u_{2}(a_{2},(\mathcal{P}f^{\mathrm{NE}})(u)_{1})-c]$
	$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{1}}u_{1}(a_{1},(\mathcal{P}f^{\mathrm{NE}})(u)_{2})]+\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{2}}u_{2}(a_{2},(\mathcal{P}f^{\mathrm{NE}})(u)_{1})]-c$

For the first term,

	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{1}}u_{1}(a_{1},(\mathcal{P}f^{\mathrm{NE}})(u)_{2})]=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{1}}u_{1}(a_{1},\frac{1}{\|\mathcal{A}_{2}\|!}\sum_{\rho_{2}\in\mathcal{G}_{2}}\rho_{2}^{-1}f^{\mathrm{NE}}(\rho_{2}u)_{2})]$
	$\displaystyle\leq$	$\displaystyle\frac{1}{\|\mathcal{A}_{2}\|!}\sum_{\rho_{2}\in\mathcal{G}_{2}}\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{1}}u_{1}(a_{1},\rho_{2}^{-1}f^{\mathrm{NE}}(\rho_{2}u)_{2})]$
	$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{2}\|!}\sum_{\rho_{2}\in\mathcal{G}_{2}}\mathbb{E}_{v\sim\mathcal{D}}[\max_{a_{1}}(\rho_{2}^{-1}v)_{1}(a_{1},\rho_{2}^{-1}f^{\mathrm{NE}}(v)_{2})]$
	$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{2}\|!}\sum_{\rho_{2}\in\mathcal{G}_{2}}\mathbb{E}_{v\sim\mathcal{D}}[\max_{a_{1}}v_{1}(a_{1},f^{\mathrm{NE}}(v)_{2})]$
	$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{1}}u_{1}(a_{1},f^{\mathrm{NE}}(u)_{2})]$

Similarly, for the second term,

\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{2}}u_{2}(a_{2},(\mathcal{P}f^{\mathrm{NE}})(u)_{1})]\leq\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{2}}u_{2}(a_{2},f^{\mathrm{NE}}(u)_{1})]

Above all,

	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\sum_{i}\mathcal{E}_{i}((\mathcal{P}f^{\mathrm{NE}})(u),u)]=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{1}}u_{1}(a_{1},(\mathcal{P}f^{\mathrm{NE}})(u)_{2})]+\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{2}}u_{2}(a_{2},(\mathcal{P}f^{\mathrm{NE}})(u)_{1})]-c$
	$\displaystyle\leq$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{1}}u_{1}(a_{1},f^{\mathrm{NE}}(u)_{2})]+\mathbb{E}_{u\sim\mathcal{D}}[\max_{a_{2}}u_{2}(a_{2},f^{\mathrm{NE}}(u)_{1})]-c$
	$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\sum_{i}\mathcal{E}_{i}(f^{\mathrm{NE}}(u),u)]$

∎

B.6 Proof of Theorem 4.8

See 4.8

We first introduce a useful lemma. It is about the property of $\mathcal{E}_{i}(\sigma,u)$

Lemma B.8.

$\mathcal{E}_{i}(\sigma,u)$ is

Linear on $\sigma_{i}$ , i.e.

p\mathcal{E}_{i}((\sigma_{i}^{1},\sigma_{-i}),u)+(1-p)\mathcal{E}_{i}((\sigma_{i}^{2},\sigma_{-i}),u)=\mathcal{E}_{i}((p\sigma_{i}^{1}+(1-p)\sigma_{i}^{2},\sigma_{-i}),u),\leavevmode\nobreak\ \forall p\in[0,1]

Convex on $\sigma_{j}$ , i.e.

p\mathcal{E}_{i}((\sigma_{j}^{1},\sigma_{-j}),u)+(1-p)\mathcal{E}_{i}((\sigma_{j}^{2},\sigma_{-j}),u)\geq\mathcal{E}_{i}((p\sigma_{j}^{1}+(1-p)\sigma_{j}^{2},\sigma_{-j}),u),\leavevmode\nobreak\ \forall p\in[0,1],j\neq i

Proof.

We recall the definition $\mathcal{E}_{i}(\sigma,u)=\max_{a_{i}\in\mathcal{A}_{i}}u_{i}(a_{i},\sigma_{-i})-u_{i}(\sigma)$ . Notice that $u_{i}(\sigma)$ is linear on $\sigma_{k}$ for all $k\in[n]$ , thus both $u_{i}(a_{i},\sigma_{-i})$ and $u_{i}(\sigma)$ are linear on $\sigma_{k}$ for any $k\in[n]$ . Moreover, the maximum operator on a set of linear functions will induce a convex function.

∎

Proof of Theorem 4.8.

We prove the theorem in two steps.

Step 1

First, we show that

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}((\mathcal{P}_{i}f^{\mathrm{NE}})(u),u)]=\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}(f^{\mathrm{NE}}(u),u)],\quad\forall f^{\mathrm{NE}}\in\mathcal{F}^{\mathrm{NE}}

By definition,

	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}(\mathcal{P}_{i}f^{\mathrm{NE}}(u),u)]$
$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}((\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}f(\rho_{i}u)_{i},f(u)_{-i}),u)]$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}((\rho_{i}^{-1}f(\rho_{i}u)_{i},f(u)_{-i}),u)]$	$\displaystyle,\text{by linearity of $\mathcal{E}_{i}(\sigma,u)$ on $\sigma_{i}$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{i}((\rho_{i}^{-1}f(v)_{i},f(\rho_{i}^{-1}v)_{-i}),\rho_{i}^{-1}v)]$	$\displaystyle,\text{let $v=\rho_{i}u$ and use the invariance of $\mathcal{D}$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{i}((\rho_{i}^{-1}f(v)_{i},f(v)_{-i}),\rho_{i}^{-1}v)]$	$\displaystyle,\text{OPI of $f$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}((f(u)_{i},f(u)_{-i}),u)]$	$\displaystyle,\text{invariance of $\mathcal{E}_{i}(\sigma,u)$ under $\rho_{i}^{-1}\in\mathcal{G}_{i}$}$
$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}(f^{\mathrm{NE}}(u),u)]$

Step 2

Then we show that

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}((\mathcal{P}_{i}f^{\mathrm{NE}})(u),u)]\leq\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}(f^{\mathrm{NE}}(u),u)],\quad\forall f^{\mathrm{NE}}\in\mathcal{F}^{\mathrm{NE}},j\neq i

	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}((\mathcal{P}_{i}f^{\mathrm{NE}})(u),u)]$
$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}((\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}f(\rho_{i}u)_{i},f(u)_{-i}),u)]$
$\displaystyle\leq$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}((\rho_{i}^{-1}f(\rho_{i}u)_{i},f(u)_{-i}),u)]$	$\displaystyle,\text{by convexity of $\mathcal{E}_{j}(\sigma,u)$ on $\sigma_{i}$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{j}((\rho_{i}^{-1}f(v)_{i},f(\rho_{i}^{-1}v)_{-i}),\rho_{i}^{-1}v)]$	$\displaystyle,\text{let $v=\rho_{i}u$ and use the invariance of $\mathcal{D}$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{j}((\rho_{i}^{-1}f(v)_{i},f(v)_{-i}),\rho_{i}^{-1}v)]$	$\displaystyle,\text{OPI of $f$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}((f(u)_{i},f(u)_{-i}),u)]$	$\displaystyle,\text{invariance of $\mathcal{E}_{j}(\sigma,u)$ under $\rho_{i}^{-1}\in\mathcal{G}_{i}$}$
$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}(f^{\mathrm{NE}}(u),u)]$

Since $\mathcal{P}=\circ_{i}\mathcal{P}_{i}$ and $\mathcal{E}=\max_{i}\mathcal{E}_{i}$ , we have

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}((\mathcal{P}f^{\mathrm{NE}})(u),u)]\leq\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(f^{\mathrm{NE}}(u),u)]

∎

B.7 Proof of Theorem 4.9

See 4.9

Proof.

We prove the theorem in two steps, similar to the proof of Theorem 4.8.

Step 1

First we show that for player $i\in\{1,2\}$ , let $\{j\}=\{1,2\}\backslash\{i\}$ ,

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}((\mathcal{O}_{i}f^{\mathrm{NE}})(u),u)]\leq\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}(f^{\mathrm{NE}}(u),u)]

This is because

$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}((\mathcal{O}_{i}f^{\mathrm{NE}})(u),u)]=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}((f^{\mathrm{NE}}(u)_{i},\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}f^{\mathrm{NE}}(\rho_{i}u)_{j}),u)]$
$\displaystyle\leq$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}((f^{\mathrm{NE}}(u)_{i},f^{\mathrm{NE}}(\rho_{i}u)_{j}),u)]$	$\displaystyle,\text{by convexity of $\mathcal{E}_{i}$ on $\sigma_{j}$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{i}((f^{\mathrm{NE}}(\rho_{i}^{-1}v)_{i},f^{\mathrm{NE}}(v)_{j}),\rho_{i}^{-1}v)]$	$\displaystyle,\text{let $v=\rho_{i}u$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{i}((\rho_{i}^{-1}f^{\mathrm{NE}}(v)_{i},f^{\mathrm{NE}}(v)_{j}),\rho_{i}^{-1}v)]$	$\displaystyle,\text{by PPE of $f^{\mathrm{NE}}$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{i}((f^{\mathrm{NE}}(v)_{i},f^{\mathrm{NE}}(v)_{j}),v)]$	$\displaystyle,\text{invariance of $\mathcal{E}_{i}(\sigma,u)$ under $\rho_{i}^{-1}\in\mathcal{G}$}$
$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{i}((f^{\mathrm{NE}})(u),u)]$

Step 2

Then we show that if $j\neq i$ and $\{i,j\}=\{1,2\}$

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}((\mathcal{O}_{i}f^{\mathrm{NE}})(u),u)]=\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}(f^{\mathrm{NE}}(u),u)]

This is because

$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}((\mathcal{O}_{i}f^{\mathrm{NE}})(u),u)]=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}((f^{\mathrm{NE}}(u)_{i},\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}f^{\mathrm{NE}}(\rho_{i}u)_{j}),u)]$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}((f^{\mathrm{NE}}(u)_{i},f^{\mathrm{NE}}(\rho_{i}u)_{j}),u)]$	$\displaystyle,\text{by linearity of $\mathcal{E}_{j}$ on $\sigma_{j}$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{j}((f^{\mathrm{NE}}(\rho_{i}^{-1}v)_{i},f^{\mathrm{NE}}(v)_{j}),\rho_{i}^{-1}v)]$	$\displaystyle,\text{let $v=\rho_{i}u$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{j}((\rho_{i}^{-1}f^{\mathrm{NE}}(v)_{i},f^{\mathrm{NE}}(v)_{j}),\rho_{i}^{-1}v)]$	$\displaystyle,\text{by PPE of $f^{\mathrm{NE}}$}$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}[\mathcal{E}_{j}((f^{\mathrm{NE}}(v)_{i},f^{\mathrm{NE}}(v)_{j}),v)]$	$\displaystyle,\text{invariance of $\mathcal{E}_{j}(\sigma,u)$ under $\rho_{i}^{-1}\in\mathcal{G}_{i}$}$
$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}_{j}(f^{\mathrm{NE}}(u),u)]$

Since $\mathcal{O}=\circ_{i}\mathcal{O}_{i}$ and $\mathcal{E}=\max_{i}\mathcal{E}_{i}$ , we have

\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(\mathcal{O}f^{\mathrm{NE}}(u),u)]\leq\mathbb{E}_{u\sim\mathcal{D}}[\mathcal{E}(f^{\mathrm{NE}}(u),u)]

∎

Appendix C Omitted Proofs in Section 5

C.1 Proof of Theorem 5.3

See 5.3

Proof.

Let $f$ be a PPE and OPI NE approximator. Denote $f(u)=(\sigma_{i})_{i\in[n]}$ . For player $k$ that $a^{*}_{k}\in V(\rho_{k})$ , we get

\sigma_{k}=f(u)_{k}\overset{(a)}{=}f(\rho u)_{k}\overset{(b)}{=}f(\rho_{k}u)_{k}\overset{(c)}{=}\rho_{k}f(u)_{k}=\rho_{k}\sigma_{k},

(11)

where $(a)$ holds since $u$ is permutable w.r.t. $\rho$ , $(b)$ holds by OPI of $f$ , and $(c)$ holds by PPE of $f$ . If $a^{*}$ can be found by $f$ , we will have $1=\sigma_{k}(a^{*}_{k})\overset{(d)}{=}\rho_{k}\sigma_{k}(a^{*}_{k})=\sigma_{k}(\rho_{k}^{-1}(a^{*}_{k}))$ , where $(d)$ holds by Equation 11. However, such result leads to a contradiction, because $a^{*}_{k}\neq\rho_{k}^{-1}(a_{k})$ but $\sigma_{k}(a^{*}_{k})=\sigma(\rho_{k}^{-1}(a^{*}_{k}))=1$ .

Let $f$ be a PE (C)CE approximator. Denote $f(u)=\pi$ , we have

\pi=f(u)\overset{(a)}{=}f(\rho u)\overset{(b)}{=}\rho f(u)=\rho\pi

(12)

where $(a)$ holds since $u$ is permutable w.r.t. $\rho$ , $(b)$ holds by PE of $f$ . If $a^{*}$ can be found by $f$ , we will have $1=\pi(a^{*})\overset{(c)}{=}\rho\pi(a^{*})=\pi(\rho^{-1}a^{*})=\pi(\rho_{1}^{-1}a_{1}^{*},\cdots,\rho_{n}^{-1}a_{n}^{*})$ , where $(c)$ holds by Equation 12. However, from $a_{k}^{*}\in V(\rho_{k})$ we know $\rho_{k}^{-1}(a_{k}^{*})\neq a_{k}^{*}$ , then $\rho^{-1}a^{*}\neq a^{*}$ , but $\pi(a^{*})=\pi(\rho^{-1}a^{*})=1$ . ∎

C.2 Proof of Theorem 5.6

See 5.6

Proof.

Assume $f\in\mathcal{F}^{\text{(C)CE}}_{\mathrm{general}}$ is an (C)CE approximator that always finds the strategy that maximizes the social welfare. Afterward, we construct another $f_{0}$ that satisfies PE and always finds the strategy that maximizes social welfare. $f_{0}$ is constructed by orbit averaging:

\displaystyle f_{0}(u)=\mathcal{Q}f(u),

thus $f_{0}$ is PE.

Denote $\mathcal{D}$ as an arbitrary payoff distribution of $u$ such that $\mathcal{D}$ is invariant under permutation and the cardinality of its support is finite. We have

$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(\mathcal{Q}_{i}f(u),u)=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}f(\rho_{i}u),u)$
$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}\sum_{i=1}^{n}u_{i}(\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\rho_{i}^{-1}f(\rho_{i}u))$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{u\sim\mathcal{D}}\sum_{i=1}^{n}u_{i}(\rho_{i}^{-1}f(\rho_{i}u))$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}\sum_{i=1}^{n}(\rho_{i}^{-1}v)_{i}(\rho_{i}^{-1}f(v))$	$\displaystyle,\text{let }v=\rho_{i}u$
$\displaystyle=$	$\displaystyle\frac{1}{\|\mathcal{A}_{i}\|!}\sum_{\rho_{i}\in\mathcal{G}_{i}}\mathbb{E}_{v\sim\mathcal{D}}\sum_{i=1}^{n}v_{i}(f(v))$
$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}\sum_{i=1}^{n}u_{i}(f(u))$
$\displaystyle=$	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)$

Due to that $\mathcal{Q}=\mathcal{Q}_{1}\circ\cdots\circ\mathcal{Q}_{n}$ , we have

\displaystyle\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f_{0}(u),u)=\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)

Due to the arbitrariness of $\mathcal{D}$ , we know that $f_{0}$ maximizes the social welfare w.r.t. any $u$ .

From above, we immediately know

\displaystyle\mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{(C)CE}}_{\mathrm{PE}},\mathcal{F}^{\mathrm{(C)CE}}_{\mathrm{general}})=1

∎

C.3 Proof of Theorem 5.7

See 5.7

C.3.1 Proof of Equation 1 and Equation 3

We first prove the theorem with respect to $\mathcal{F}^{\mathrm{NE}}_{\mathrm{OPI}}$ and $\mathcal{F}^{\mathrm{NE}}_{\mathrm{both}}$

Step 1

On the one part, we prove

\displaystyle\left.\begin{aligned} \mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{NE}}_{\mathrm{OPI}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})\\ \mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})\end{aligned}\right\}\leq\frac{1}{M^{N-1}}

We prove this by construction.

Consider a game with $N$ player and $\mathcal{A}_{i}=[M]$ for $i\in[N]$ . $\forall a\in\mathcal{A},i\in[N]$ , define the payoff $\bar{u}$ as follows:

\bar{u}_{i}(a)=\begin{cases}1&,\text{if }a_{1}=a_{2}=\dots=a_{N}\\ 0&,\text{otherwise}\end{cases}

Define $U=\{u^{\prime}|u^{\prime}=\circ_{i}\rho_{i}\bar{u},\rho_{i}\in\mathcal{G}_{i}\}$ and $\mathcal{D}$ as a uniform distribution on $U$ . Easy to certify that $\mathcal{D}$ is a permutation-invariant distribution.

Let $\tilde{f}\in\tilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{general}}$ be the NE oracle that $\tilde{f}(\bar{u})_{i}=1$ and for any $u^{\prime}=\circ_{i}\rho_{i}\bar{u}\in U$ , $\tilde{f}(u^{\prime})_{i}=\rho_{i}(1)$ . Intuitively, the oracle will choose the action that will provide all players with revenue $1$ , leading to a social welfare of $N$ . Since each player has got her maximum possible utility, we have

\max_{f\in\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)=\max_{\tilde{f}\in\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{general}}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(\tilde{f}(u),u)=N.

(13)

For any $j_{1},j_{2}\in[M]$ and $j_{1}<j_{2}$ , let $\rho^{(j_{1},j_{2})}_{i}=(1,\dots,j_{2},\dots,j_{1},\dots,M)$ for all player $i\in[N]$ be the swap permutation that swaps actions $j_{1}$ and $j_{2}$ and keeps other actions still. Then $\circ_{i\neq j}\rho_{i}^{(j_{1},j_{2})}\bar{u}=\rho_{j}^{(j_{1},j_{2})}\bar{u}$ for player $j$ . For $f\in\mathcal{F}^{\mathrm{NE}}_{\mathrm{OPI}}$ , we have $f(\bar{u})_{j}=f(\circ_{i\neq j}\rho_{i}^{(j_{1},j_{2})}\bar{u})_{j}=f(\rho_{j}^{(j_{1},j_{2})}\bar{u})_{j}$ for arbitrary swap permutation $\rho_{j}^{(j_{1},j_{2})}$ . Since any permutation can be achieved by composition of swap permutations, we have $\forall\rho_{j}\in\mathcal{G}_{j}$ , $f(\bar{u})_{j}=f(\rho_{j}\bar{u})_{j}$ . Based on that, and by OPI of $f$ , $\forall\rho=\circ_{i\in[N]}\rho_{i}$ we have $f(\bar{u})_{j}=f(\rho\bar{u})_{j}$ , i.e. $f$ is a constant function on $U$ . Without loss of generality, we denote $f(u)\equiv\sigma$ for all $u\in U$ . Then

\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)=\frac{1}{|U|}\sum_{u^{\prime}\in U}\mathrm{SW}(\sigma,u^{\prime})=\frac{1}{(M!)^{N-1}}\mathrm{SW}(\sigma,\sum_{u^{\prime}\in U}u^{\prime}).

Additionally, we have $(\sum_{u^{\prime}\in U}u^{\prime})(a)=((M-1)!)^{N-1}$ for any $a\in\mathcal{A}$ . Based on that, we have

\max_{f\in\mathcal{F}^{\mathrm{NE}}_{\mathrm{OPI}}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)=\frac{1}{(M!)^{N-1}}\cdot N((M-1)!)^{N-1}=\frac{N}{M^{N-1}}.

(14)

Combining Equation 13 and Equation 14, we have

\mathrm{SWR}_{\mathrm{N,M}}(\mathcal{F}^{\mathrm{NE}}_{\mathrm{OPI}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})\leq\frac{1}{M^{N-1}}.

Due to $\mathcal{F}^{\mathrm{NE}}_{\mathrm{both}}\subseteq\mathcal{F}^{\mathrm{NE}}_{\mathrm{OPI}}$ , we immediately know

\displaystyle\mathrm{SWR}_{\mathrm{N,M}}(\mathcal{F}^{\mathrm{NE}}_{\mathrm{both}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})\leq\frac{1}{M^{N-1}}

Step 2

On the other part, we prove

\displaystyle\left.\begin{aligned} \mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{NE}}_{\mathrm{OPI}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})\\ \mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})\end{aligned}\right\}\geq 1/M^{N-1}

Define the maximum possible utility (MPU) for player $i$ with respect to utility $u_{i}$ and action $a_{i}$ as

\mathrm{MPU}(u_{i},a_{i})\coloneqq\max_{a_{-i}\in\mathcal{A}_{-i}}u_{i}(a_{i},a_{-i})

(15)

Define the set of maximum possible utility best response for player $i$ w.r.t. $u_{i}$ as

\mathcal{B}_{i}(u_{i})\coloneqq\{a_{i}\in\mathcal{A}_{i}:\mathrm{MPU}(u_{i},a_{i})=\max_{a^{\prime}_{i}\in\mathcal{A}_{i}}\mathrm{MPU}(u_{i},a^{\prime}_{i})\}

We first conduct some simplification to the target.

\displaystyle\mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})=\inf_{\mathcal{D}}\frac{\max_{f\in\mathcal{F}^{\mathrm{NE}}_{\mathrm{both}}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)}{\max_{f\in{\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}}}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)}\geq\inf_{\mathcal{D}}\frac{\max_{f\in\mathcal{F}^{\mathrm{NE}}_{\mathrm{both}}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)}{\mathbb{E}_{u\sim\mathcal{D}}\max_{\sigma}\mathrm{SW}(\sigma,u)}

Then we constrain $u$ to be a cooperation game. For a normal form game $\Gamma_{u}$ , we define $\tilde{u}=(\tilde{u}_{i})_{i\in[n]}$ in which $\tilde{u}_{i}=\frac{1}{n}\sum_{i=1}^{n}u_{i}$ . Then we have $\mathrm{SW}(\sigma,u)=\mathrm{SW}(\sigma,\tilde{u})$ , which means that constraining $u$ to be a cooperation game will induce the same social welfare. Then

\displaystyle\inf_{\mathcal{D}}\frac{\max_{f\in\mathcal{F}^{\mathrm{NE}}_{\mathrm{both}}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)}{\mathbb{E}_{u\sim\mathcal{D}}\max_{\sigma}\mathrm{SW}(\sigma,u)}=\inf_{\mathcal{D}}\frac{\max_{f\in\mathcal{F}^{\mathrm{NE}}_{\mathrm{both}}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),\tilde{u})}{\mathbb{E}_{u\sim\mathcal{D}}\max_{\sigma}\mathrm{SW}(\sigma,\tilde{u})}

Denote $f_{0}$ be the approximator that always outputs uniform strategy on $\mathcal{B}_{i}(\tilde{u}_{i})$ for player $i$ . It’s obvious that $f_{0}$ is both OPI and PPE because the operations from $u$ to $f_{0}(u)$ are all permutation-equivariant. Then,

\displaystyle\inf_{\mathcal{D}}\frac{\max_{f\in\mathcal{F}^{\mathrm{NE}}_{\mathrm{both}}}\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),\tilde{u})}{\mathbb{E}_{u\sim\mathcal{D}}\max_{\sigma}\mathrm{SW}(\sigma,\tilde{u})}\geq\inf_{\mathcal{D}}\frac{\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f_{0}(u),\tilde{u})}{\mathbb{E}_{u\sim\mathcal{D}}\max_{\sigma}\mathrm{SW}(\sigma,\tilde{u})}

Ignore the infimum and the expectation operator, consider $\frac{\mathrm{SW}(f_{0}(u),\tilde{u})}{\max_{\sigma}\mathrm{SW}(\sigma,\tilde{u})}$ for arbitrary $\tilde{u}$ , denote $b$ be the maximum element appeared in $\tilde{u}$ , then the denominator equals $Nb$ . But for the numerator, for player $i$ , no matter what action $a_{i}\in\mathcal{B}_{i}(\tilde{u}_{i})$ she chooses, she always has probability at least $\prod_{j\neq i}\frac{1}{|\mathcal{B}_{j}|}\geq\frac{1}{M^{N-1}}$ to achieve revenue $b$ , therefore inducing $\mathrm{SW}(f_{0}(u),\tilde{u})\geq\frac{Nb}{M^{N-1}}$ .

Then, $\frac{\mathrm{SW}(f_{0}(u),\tilde{u})}{\max_{\sigma}\mathrm{SW}(\sigma,\tilde{u})}\geq\frac{1}{M^{N-1}}$ , so as $\inf_{\mathcal{D}}\frac{\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f_{0}(u),\tilde{u})}{\mathbb{E}_{u\sim\mathcal{D}}\max_{\sigma}\mathrm{SW}(\sigma,\tilde{u})}$ , $\mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}})$ and $\mathrm{SWR}_{\mathrm{N,M}}({\mathcal{F}}^{\mathrm{NE}}_{\mathrm{OPI}})$ .

Above all,

\displaystyle\left.\begin{aligned} \mathrm{SWR}_{\mathrm{N,M}}(\mathcal{F}^{\mathrm{NE}}_{\mathrm{OPI}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})\\ \mathrm{SWR}_{\mathrm{N,M}}(\mathcal{F}^{\mathrm{NE}}_{\mathrm{both}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})\end{aligned}\right\}=\frac{1}{M^{N-1}}

C.3.2 Proof of Equation 2

We next prove the theorem with respect to $\mathcal{F}^{\mathrm{NE}}_{\mathrm{PPE}}$ that

\displaystyle\mathrm{SWR}_{\mathrm{N,M}}(\mathcal{F}^{\mathrm{NE}}_{\mathrm{PPE}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})\leq\frac{1}{M}

Consider a bimatrix game and $\mathcal{A}_{i}=[M]$ for $i\in[2]$ . $\forall a\in\mathcal{A},i\in[2]$ , define the payoff $\bar{u}$ as follows:

\bar{u}_{i}(a)=\begin{cases}1&,\text{if }a_{1}=a_{2}\\ 0&,\text{otherwise}\end{cases}

Define $U\coloneqq\{u^{\prime}|u^{\prime}=\rho_{1}\rho_{2}\bar{u},\rho_{i}\in\mathcal{G}_{i}\}$ and $\mathcal{D}$ as a uniform distribution on $U$ . Easy to certify that $U=\{u^{\prime}|u^{\prime}=\rho_{1}\bar{u},\rho_{1}\in\mathcal{G}_{1}\}=\{u^{\prime}|u^{\prime}=\rho_{2}\bar{u},\rho_{2}\in\mathcal{G}_{2}\}$ and $\mathcal{D}$ is a permutation-invariant distribution.

For a permutation $\varrho$ on $[M]$ , let $P_{\varrho}\in\{0,1\}^{M\times M}$ be the corresponding permutation matrix. Denote $\mathcal{P}$ as the set of all permutation matrice. As a result, $\forall u\in U,\forall\rho_{1}\in\mathcal{G}_{1},\rho_{1}u=(P_{\rho_{1}}{u}_{1},P_{\rho_{1}}{u}_{2})=:P_{\rho_{1}}u$ and $\forall\rho_{2}\in\mathcal{G}_{2},\rho_{2}u=({u}_{1}P_{\rho_{2}}^{T},{u}_{2}P_{\rho_{2}}^{T})=:uP_{\rho_{2}}^{T}$ . Specially, we have $P_{\varrho}\bar{u}P_{\varrho}^{T}=\bar{u}$ . For $f\in\mathcal{F}^{\mathrm{NE}}_{\mathrm{PPE}}$ , Denote $f(\bar{u})=\sigma=(\sigma_{1},\sigma_{2})$ . For permutation $\varrho$ in $[M]$ and payoff $u^{\prime}=P_{\varrho}\bar{u}=\bar{u}(P_{\varrho}^{T})^{-1}$ , by PPE of $f$ , we have $f(u^{\prime})_{1}=f(P_{\varrho}\bar{u})_{1}=P_{\varrho}\sigma_{1}=\varrho\sigma_{1},$ and $f(u^{\prime})_{2}=f(\bar{u}(P_{\varrho}^{T})^{-1})_{2}=(P_{\varrho})^{-1}\sigma_{2}=\varrho^{-1}\sigma_{2}.$ Then we have

\mathrm{SW}(f(u^{\prime}),u^{\prime})=\sum_{i}(P_{\varrho}\bar{u})_{i}(\varrho\sigma_{1},\varrho^{-1}\sigma_{2})=\sum_{i}\bar{u}_{i}(\sigma_{1},\varrho^{-1}\sigma_{2})=\sum_{i}(\bar{u}P_{\varrho}^{T})_{i}(\sigma_{1},\sigma_{2})=\mathrm{SW}(f(\bar{u}),\bar{u}P_{\varrho}^{T})

Therefore

	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)$	$\displaystyle=\frac{1}{\|U\|}\sum_{u^{\prime}\in U}\mathrm{SW}(f(u^{\prime}),u^{\prime})$
		$\displaystyle=\frac{1}{\|U\|}\sum_{P_{\varrho}\in\mathcal{P}}\mathrm{SW}(f(\bar{u}),\bar{u}P_{\varrho}^{T})$
		$\displaystyle=\frac{1}{\|U\|}\sum_{u=\bar{u}(P_{\varrho}^{T})\in U}\mathrm{SW}(f(\bar{u}),u)$
		$\displaystyle=\frac{1}{\|U\|}\mathrm{SW}(\sigma,\sum_{u^{\prime}\in U}u^{\prime}).$

Since $|U|=\frac{1}{M!}$ and $\sum_{u^{\prime}\in U}u^{\prime}$ is a tensor with all elements equal to $(M-1)!$ . Thus $\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)=\frac{2}{M}$ and

\displaystyle\mathrm{SWR}_{\mathrm{N,M}}(\mathcal{F}^{\mathrm{NE}}_{\mathrm{PPE}},\mathcal{F}^{\mathrm{NE}}_{\mathrm{general}})\leq\frac{1}{M}

C.3.3 Proof of Equation 4

Consider a $3\times 3$ game as follows, where $\epsilon\in(0,\frac{1}{2})$ :

u=\begin{bmatrix}\bm{1,1}&0,0&0,\frac{1}{2}+\varepsilon\\ 0,0&\bm{1,1}&0,\frac{1}{2}+\varepsilon\\ \frac{1}{2}+\varepsilon,0&\frac{1}{2}+\varepsilon,0&\varepsilon,\varepsilon\end{bmatrix}

It is obvious that $\max_{\sigma^{*}\subseteq\text{NE}(\Gamma_{u})}\text{SW}(\sigma^{*},u)=2$ , and the corresponding strategy has been bolded. However, for NE oracles with both PPE and OPI, it can only output a unique NE with a pure strategy that induces utility $(\varepsilon,\varepsilon)$ .

Let $\rho_{1}=\rho_{2}=(2,1,3)$ , we have $\rho_{1}\rho_{2}u=u$ . From the analysis above we know if $f^{\mathrm{NE}}\in\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}}$ and $f^{\mathrm{NE}}(u)=(\sigma_{1},\sigma_{2})$ , then $\sigma_{1}(1)=\sigma_{1}(2)$ , $\sigma_{2}(1)=\sigma_{2}(2)$ . We integrate the first two actions of player $1$ and player $2$ into a new action that will choose randomly between the first two actions, then we form the utility matrix below:

\overline{u}=\begin{bmatrix}\frac{1}{2},\frac{1}{2}&0,\frac{1}{2}+\varepsilon\\ \frac{1}{2}+\varepsilon,0&\bm{\varepsilon,\varepsilon}\end{bmatrix}

There is a unique NE in this Prisoner’s Dilemma, which has been bolded. The game $\overline{u}$ is the same with the game $u$ under the assumption that $\sigma_{1}(1)=\sigma_{1}(2)$ and $\sigma_{2}(1)=\sigma_{2}(2)$ in $u$ . Then $\max_{f\in\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}}}\text{SW}(f(u),u)=2\varepsilon$ . Since $\varepsilon$ can be arbitrarily small, we have $\mathrm{SWR}_{2,3}(\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}},\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{general}})=0$ . As a result, we have $\mathrm{SWR}_{N,M}(\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{both}},\widetilde{\mathcal{F}}^{\mathrm{NE}}_{\mathrm{general}})=0$ for all $N\geq 2$ and $M\geq 3$ .

$\displaystyle\|u_{i}(\sigma)-u_{i}(\sigma^{\prime})\|=$	$\displaystyle\|u_{i}(\sigma_{1},\sigma_{2},\dots,\sigma_{n})-u_{i}(\sigma^{\prime},\sigma^{\prime}_{2},\dots,\sigma^{\prime}_{n})\|$
$\displaystyle=$	$\displaystyle\Big{\|}\sum_{j=1}^{n}\Big{(}u_{i}(\sigma_{1},\dots,\sigma_{j},\sigma^{\prime}_{j+1},\dots,\sigma^{\prime}_{n})-u_{i}(\sigma_{1},\dots,\sigma^{\prime}_{j},\sigma^{\prime}_{j+1},\dots,\sigma^{\prime}_{n})\Big{)}\Big{\|}$
$\displaystyle=$	$\displaystyle\Big{\|}\sum_{j=1}^{n}\Big{(}u_{i}(\sigma_{j},y_{-j})-u_{i}(\sigma^{\prime}_{j},y_{-j})\Big{)}\Big{\|}$
$\displaystyle=$	$\displaystyle\Big{\|}\sum_{j=1}^{n}\sum_{a_{j}}(\sigma_{j}(a_{j})-\sigma^{\prime}_{j}(a_{j}))\sum_{a_{-j}}u_{i}(a_{j},a_{-j})y_{-j}(a_{-j})\Big{\|}$
$\displaystyle\leq$	$\displaystyle\sum_{j=1}^{n}\sum_{a_{j}}\Big{\|}\sigma_{j}(a_{j})-\sigma^{\prime}_{j}(a_{j})\Big{\|}\sum_{a_{-j}}u_{i}(a_{j},a_{-j})y_{-j}(a_{-j})$
$\displaystyle{\leq}$	$\displaystyle\sum_{j=1}^{n}\sum_{a_{j}}\Big{\|}\sigma_{j}(a_{j})-\sigma^{\prime}_{j}(a_{j})\Big{\|}\sum_{a_{-j}}y_{-j}(a_{-j})$	$\displaystyle,u_{i}(\cdot)\in[0,1]$
$\displaystyle\leq$	$\displaystyle\sum_{j=1}^{n}\sum_{a_{j}\in A_{j}}\Big{\|}\sigma_{j}(a_{j})-\sigma^{\prime}_{j}(a_{j})\Big{\|}\leq n\max_{j\in[n]}\sum_{a_{j}\in A_{j}}\Big{\|}\sigma_{j}(a_{j})-\sigma^{\prime}_{j}(a_{j})\Big{\|}$
$\displaystyle=$	$\displaystyle n{\\|\sigma-\sigma^{\prime}\\|},$

	$\displaystyle\mathbb{E}_{u\sim\mathcal{D}}\mathrm{SW}(f(u),u)$	$\displaystyle=\frac{1}{\|U\|}\sum_{u^{\prime}\in U}\mathrm{SW}(f(u^{\prime}),u^{\prime})$
		$\displaystyle=\frac{1}{\|U\|}\sum_{P_{\varrho}\in\mathcal{P}}\mathrm{SW}(f(\bar{u}),\bar{u}P_{\varrho}^{T})$
		$\displaystyle=\frac{1}{\|U\|}\sum_{u=\bar{u}(P_{\varrho}^{T})\in U}\mathrm{SW}(f(\bar{u}),u)$
		$\displaystyle=\frac{1}{\|U\|}\mathrm{SW}(\sigma,\sum_{u^{\prime}\in U}u^{\prime}).$

Are Equivariant Equilibrium Approximators Beneficial?

Abstract

1 Introduction

1.1 Our Contributions

1.2 Further Related Work

1.3 Organization

2 Preliminary

2.1 Game Theory

Normal-Form Game

Nash Equilibrium (NE)

Coarse Correlated Equilibrium (CCE)

Correlated Equilibrium (CE)

Equilibrium Approximation

Definition 2.1 (Exploitability and Approximation).

2.2 Equilibrium Approximator

3 Equivariant Equilibrium Approximator

Definition 3.1 (Permutation of a game).

Equivariant NE Approximator

Lemma 3.2.

Definition 3.3 (Player-Permutation-Equivariance, PPE).

Definition 3.4 (Opponent-Permutation-Invariance, OPI).

Equivariant (C)CE approximator

Lemma 3.5.

Definition 3.6 (Permutation-Equivariance,PE).

Equivariant Approximators in Literature

3.1 Orbit Averaging

Orbit Averaging for NE Approximator

Lemma 3.7.

Lemma 3.8.

Orbit Averaging for (C)CE Approximator

Lemma 3.9.

Corollary 3.10.

Lemma 3.11.

4 Theoretical Analysis of Benefits

4.1 Benefits for Generalization

Definition 4.1 (ℓ∞\ell_{\infty}-distance).

Definition 4.2 (rr-covering number).

Theorem 4.3.

Theorem 4.4.

Theorem 4.5.

4.2 Benefits for Approximation

(C)CE Approximator

Theorem 4.6.

NE Approximator

Theorem 4.7.

Theorem 4.8.

Theorem 4.9.

5 Theoretical Analysis of Limitations

5.1 Equilibrium Selection

Definition 5.1 (Permutation-ρ\rho-Invariant Game).

Example 5.2.

Theorem 5.3.

Example 5.4.

5.2 Social Welfare

Definition 5.5.

Theorem 5.6.

Theorem 5.7.

6 Conclusion and Future Work

References

Appendix A Omitted Proofs in Section 3

A.1 Useful Lemma

Lemma A.1.

Proof.

A.2 Proof of Lemma 3.2

Proof.

A.3 Proof of Lemma 3.5

CCE

CE

A.4 Proof of Lemma 3.7 to Lemma 3.9

Proof.

Proof.

Proof.

A.5 Proof of Lemma 3.11

Proof.

Appendix B Omitted Proofs in Section 4

B.1 Proof of Theorem 4.3

Lemma B.1 (Shalev-Shwartz and Ben-David [2014]).

Lemma B.2.

Proof.

B.1.1 NE Approximator

Definition 4.1 ( $\ell_{\infty}$ -distance).

Definition 4.2 ( $r$ -covering number).

Definition 5.1 (Permutation- $\rho$ -Invariant Game).