Original: T EAMWORK MAKES VON N EUMANN WORK : M IN -M AX O PTIMIZATION IN T WO -T EAM Z ERO -S UM G AMES
Revised: T EAMWORK MAKES VON N EUMANN WORK : M IN -M AX O PTIMIZATION IN T WO -T EAM Z ERO -S UM G AMES

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Motivated by recent advances in both theoretical and applied aspects of multiplayer games, spanning from e-sports to multi-agent generative adversarial networks, we focus on min-max optimization in team zero-sum games.
Revised: Motivated by recent advances in both theoretical and applied aspects of multiplayer games, spanning from e-sports to multi-agent generative adversarial networks, we focus on min-max optimization in team zero-sum games.

Original: In this class of games, players are split into two teams with payoffs equal within the same team and of opposite sign across the opponent team.
Revised: In this class of games, players are split into two teams with payoffs equal within the same team and of opposite sign across the opponent team.

Original: Unlike the textbook twoplayer zero-sum games, ﬁnding a Nash equilibrium in our class can be shown to be CLS -hard, i.e., it is unlikely to have a polynomial-time algorithm for computing
Revised: Unlike the textbook twoplayer zero-sum games, ﬁnding a Nash equilibrium in our class can be shown to be CLS -hard, i.e., it is unlikely to have a polynomial-time algorithm for computing

Original: Nash equilibria.
Revised: Nash equilibria.

Original: Moreover, in this generalized framework, we establish that even asymptotic last iterate or time average convergence to a Nash Equilibrium is not possible using Gradient Descent Ascent (GDA), its optimistic variant, and extra gradient.
Revised: Moreover, in this generalized framework, we establish that even asymptotic last iterate or time average convergence to a Nash Equilibrium is not possible using Gradient Descent Ascent (GDA), its optimistic variant, and extra gradient.

Original: Speciﬁcally, we present a family of team games whose induced utility is non-multilinear with non-attractive per-se mixed Nash Equilibria, as strict saddle points of the underlying optimization landscape.
Revised: Speciﬁcally, we present a family of team games whose induced utility is non-multilinear with non-attractive per-se mixed Nash Equilibria, as strict saddle points of the underlying optimization landscape.

Original: Leveraging techniques from control theory, we complement these negative results by designing a modiﬁed GDA that converges locally to Nash equilibria.
Revised: Leveraging techniques from control theory, we complement these negative results by designing a modiﬁed GDA that converges locally to Nash equilibria.

Original: Finally, we discuss connections of our framework with AI architectures with team competition structures like multi-agent generative adversarial networks.
Revised: Finally, we discuss connections of our framework with AI architectures with team competition structures like multi-agent generative adversarial networks.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Team competition has played a central role in the development of Game Theory (Marschak, 1955; von Stengel & Koller, 1997; Bacharach, 1999; Gold, 2005), Economics (Marschak, 1955; Gottinger, 1974) and Evolutionary Biology (Nagylaki, 1993; Nowak et al., 2004), however, the behavior of the underlying dynamics within the teams are usually sidelined.
Revised: Team competition has played a central role in the development of Game Theory (Marschak, 1955; von Stengel & Koller, 1997; Bacharach, 1999; Gold, 2005), Economics (Marschak, 1955; Gottinger, 1974) and Evolutionary Biology (Nagylaki, 1993; Nowak et al., 2004), however, the behavior of the underlying dynamics within the teams are usually sidelined.

Original: Either for reasons of mathematical convenience or bigger picture understanding, “teams” in literature are typically modeled as if they were unitary actors, i.e., single individuals without unveiling the internal decision-making of the team members (see Kim et al.
Revised: Either for reasons of mathematical convenience or bigger picture understanding, “teams” in literature are typically modeled as if they were unitary actors, i.e., single individuals without unveiling the internal decision-making of the team members (see Kim et al.

Original: For instance, in the biology setting of weak selection model (Nagylaki, 1993; Chastain et al., 2014;
Revised: For instance, in the biology setting of weak selection model (Nagylaki, 1993; Chastain et al., 2014;

Original: Mehta et al., 2015) species are modeled to compete as teams, while at the crux of the matter the genes of each species are the actual players and their alleles are the actions in the survival game.
Revised: Mehta et al., 2015) species are modeled to compete as teams, while at the crux of the matter the genes of each species are the actual players and their alleles are the actions in the survival game.

Original: Similarly, it was the social-media collaboration of the Reddit retail trading crowd as a team that touches off the last year’s GameStop frenzy of short squeeze (Umar et al., 2021; Hasso et al., 2021) transforming the markets into a tug of war game against the team of Wall Street hedge funds.
Revised: Similarly, it was the social-media collaboration of the Reddit retail trading crowd as a team that touches off the last year’s GameStop frenzy of short squeeze (Umar et al., 2021; Hasso et al., 2021) transforming the markets into a tug of war game against the team of Wall Street hedge funds.

Original: Recently, these intrinsic details behind the competition among teams have attracted renewed interest in the Machine Learning community, motivated by the advent of multi-agent systems that are used for generative tasks or playing complex games like CTF (Jaderberg et al., 2019) or Starcraft (Vinyals et al., 2019).
Revised: Recently, these intrinsic details behind the competition among teams have attracted renewed interest in the Machine Learning community, motivated by the advent of multi-agent systems that are used for generative tasks or playing complex games like CTF (Jaderberg et al., 2019) or Starcraft (Vinyals et al., 2019).

Original: So as to win this kind of games, self-training AI systems have to develop both collaborative attributes (coordination within each team) as well as contesting ones (competition across the teams).
Revised: So as to win this kind of games, self-training AI systems have to develop both collaborative attributes (coordination within each team) as well as contesting ones (competition across the teams).

Original: Moreover, following the complementary thread of multi-agent generative adversarial network research, the creation of a pool by efﬁcient incumbent agents, either in generators (Arora et al., 2017; Hoang et al., 2017; 2018; Zhang et al., 2018; Tang, 2020), or discriminators (Hardy et al., 2019; Albuquerque et al., 2019) has been tested providing signiﬁcant statistical and computa- tional beneﬁts.
Revised: Moreover, following the complementary thread of multi-agent generative adversarial network research, the creation of a pool by efﬁcient incumbent agents, either in generators (Arora et al., 2017; Hoang et al., 2017; 2018; Zhang et al., 2018; Tang, 2020), or discriminators (Hardy et al., 2019; Albuquerque et al., 2019) has been tested providing signiﬁcant statistical and computa- tional beneﬁts.

Original: In this direction, researchers strive to harness the efﬁcacy of distributed processing, utilizing shallower networks that can learn all the while more diverse datasets 1 .
Revised: In this direction, researchers strive to harness the efﬁcacy of distributed processing, utilizing shallower networks that can learn all the while more diverse datasets 1 .

Original: In order to shed some light on this persistent strain of research, the main premise of the theoretical scaffolding developed in this paper is that
Revised: In order to shed some light on this persistent strain of research, the main premise of the theoretical scaffolding developed in this paper is that

Original: The “unitary
Revised: The “unitary

Original: two-players” min-max approach misses the critical component of the collective strategy making within each competing team
Revised: two-players” min-max approach misses the critical component of the collective strategy making within each competing team

Original: Our class of games.
Revised: Our class of games.

Original: In this regard, we turn our attention to Two-Team Zero-Sum games , proposed by Schulman & Vazirani (2019b), a quite general class of min-max optimization problems that include bilinear games as well as a wide range of non-convex non-concave games.
Revised: In this regard, we turn our attention to Two-Team Zero-Sum games , proposed by Schulman & Vazirani (2019b), a quite general class of min-max optimization problems that include bilinear games as well as a wide range of non-convex non-concave games.

Original: In this class, the players fall in two teams of size k 1 , k 2 and submit their own probabilistic strategy vector independently, akin to a general normal form multi-player game.
Revised: In this class, the players fall in two teams of size k 1 , k 2 and submit their own probabilistic strategy vector independently, akin to a general normal form multi-player game.

Original: Following the econometric common value assumption of Marschak (1955), what makes a group of players a team is that in any outcome the players of each team receive an identical payoff.
Revised: Following the econometric common value assumption of Marschak (1955), what makes a group of players a team is that in any outcome the players of each team receive an identical payoff.

Original: Thus, to build some intuition, it is easy to see that if perfect coordination existed within each team, the interaction between the teams is merely a zero-sum game between two “virtual” players.
Revised: Thus, to build some intuition, it is easy to see that if perfect coordination existed within each team, the interaction between the teams is merely a zero-sum game between two “virtual” players.

Original: To streamline our presentation here, we defer the more precise description of our model to Section 2.
Revised: To streamline our presentation here, we defer the more precise description of our model to Section 2.

Original: Challenges behind Two-Team Zero-Sum games.
Revised: Challenges behind Two-Team Zero-Sum games.

Original: In the archetypical case of two players, i.e., ( k 1 = k 2 = 1) , min-max strategies are typically thought of as the axiomatically correct predictions thanks to the seminal Von Neumann’s minmax theorem (Von Neumann, 1928).
Revised: In the archetypical case of two players, i.e., ( k 1 = k 2 = 1) , min-max strategies are typically thought of as the axiomatically correct predictions thanks to the seminal Von Neumann’s minmax theorem (Von Neumann, 1928).

Original: Unfortunately, min-max optimization for case k > 1 is a much more tenuous affair: Schulman & Vazirani (2019b) preclude the existence of unique value by presenting a family of team games where min max (cid:54) = max min together with bounds about this duality gap, which quantiﬁes exactly the effect of exchanging the order of strategy commitment either between the teams or the players thereof.
Revised: Unfortunately, min-max optimization for case k > 1 is a much more tenuous affair: Schulman & Vazirani (2019b) preclude the existence of unique value by presenting a family of team games where min max (cid:54) = max min together with bounds about this duality gap, which quantiﬁes exactly the effect of exchanging the order of strategy commitment either between the teams or the players thereof.

Original: If deﬁning the correct ﬁgure of merit for Team games is rife with frustration, what is even more demanding is understanding what kind of algorithms/dynamics are able to solve this problem when a game-theoretically meaningful solution exists: Firstly, computing local Nash Equilibria (NE) in general non-convex non-concave games is PPAD -complete
Revised: If deﬁning the correct ﬁgure of merit for Team games is rife with frustration, what is even more demanding is understanding what kind of algorithms/dynamics are able to solve this problem when a game-theoretically meaningful solution exists: Firstly, computing local Nash Equilibria (NE) in general non-convex non-concave games is PPAD -complete

Original: (Daskalakis et al., 2009; 2021).
Revised: (Daskalakis et al., 2009; 2021).

Original: Thus, all well-celebrated ﬁrst-order methods, like gradient descent-ascent (Lin et al., 2020; Daskalakis &
Revised: Thus, all well-celebrated ﬁrst-order methods, like gradient descent-ascent (Lin et al., 2020; Daskalakis &

Original: Panageas, 2019), its optimistic (Popov, 1980; Daskalakis & Panageas, 2018; Mertikopoulos et al., 2019) and extra gradient variant (Korpelevich, 1976) would require an exponential number of steps in the parameters of the problem to ﬁnd an appoximate NE under Nemirovsky-Yudin (Nemirovskij & Yudin, 1983) oracle optimization model.
Revised: Panageas, 2019), its optimistic (Popov, 1980; Daskalakis & Panageas, 2018; Mertikopoulos et al., 2019) and extra gradient variant (Korpelevich, 1976) would require an exponential number of steps in the parameters of the problem to ﬁnd an appoximate NE under Nemirovsky-Yudin (Nemirovskij & Yudin, 1983) oracle optimization model.

Original: Secondly, even if a regret notion could be deﬁned, noregret methodology is guarranteed to attract only to the set of coarse correlated equilibria (CCE) (Fudenberg, 1991; Hannan, 2016; Flokas et al., 2020; Giannou et al., 2021), a weaker notion that may be exclusively supported on strictly dominated strategies, even for simple symmetric two-player games (See also Viossat & Zapechelnyuk (2013)).
Revised: Secondly, even if a regret notion could be deﬁned, noregret methodology is guarranteed to attract only to the set of coarse correlated equilibria (CCE) (Fudenberg, 1991; Hannan, 2016; Flokas et al., 2020; Giannou et al., 2021), a weaker notion that may be exclusively supported on strictly dominated strategies, even for simple symmetric two-player games (See also Viossat & Zapechelnyuk (2013)).

Original: Whilst the aforementioned intractability failures for the general case of non-convex non-concave min-max problems provides signiﬁcant insights, they can not a fortiori answer the fundamental question, restricted in the model of Two-Team Zero-Sum Games:
Revised: Whilst the aforementioned intractability failures for the general case of non-convex non-concave min-max problems provides signiﬁcant insights, they can not a fortiori answer the fundamental question, restricted in the model of Two-Team Zero-Sum Games:

Original: Can we compute Nash equilibria in Two-Team Zero-Sum Games and ultimately are there ﬁrst-order methods that converge to them under tangible guarantees?
Revised: Can we compute Nash equilibria in Two-Team Zero-Sum Games and ultimately are there ﬁrst-order methods that converge to them under tangible guarantees?

Original: Our results.
Revised: Our results.

Original: To the best of our knowledge, the following contributions are the ﬁrst-of-its-kind type of results for the
Revised: To the best of our knowledge, the following contributions are the ﬁrst-of-its-kind type of results for the

Original: case of
Revised: case of

Original: Two-Team Zero-Sum games:
Revised: Two-Team Zero-Sum games:

Original: • For the case of the computational complexity of
Revised: • For the case of the computational complexity of

Original: approximate (possibly mixed) NE we establish a sweeping negative result proving that it is CLS -hard
Revised: approximate (possibly mixed) NE we establish a sweeping negative result proving that it is CLS -hard

Original: Theorem 3.1), i.e., is computationally harder than ﬁnding pure NE in a congestion game or ﬁnding approximate gradient descent ﬁxed points.
Revised: Theorem 3.1), i.e., is computationally harder than ﬁnding pure NE in a congestion game or ﬁnding approximate gradient descent ﬁxed points.

Original: • From an optimization perspective, we settle these questions with a resounding “no” for all the well-known discrete gradient ﬂow variations.
Revised: • From an optimization perspective, we settle these questions with a resounding “no” for all the well-known discrete gradient ﬂow variations.

Original: Speciﬁcally, we present a simple family of twoteam with two-players zero-sum games where Projected-GDA, Optimistic-GDA, and Extra Gradient fail even to stabilize around a mixed NE, when they are initialized nearby (Theorem 3.5).
Revised: Speciﬁcally, we present a simple family of twoteam with two-players zero-sum games where Projected-GDA, Optimistic-GDA, and Extra Gradient fail even to stabilize around a mixed NE, when they are initialized nearby (Theorem 3.5).

Original: Additionally, for the category GDA in the non-degenerate team games with unique mixed NE, one could acquire an even stronger result for any high-dimensional conﬁguration of actions and players.
Revised: Additionally, for the category GDA in the non-degenerate team games with unique mixed NE, one could acquire an even stronger result for any high-dimensional conﬁguration of actions and players.

Original: (Theorem 3.2) • In order to make some substantial headway under the burden of the above instability results, we shift our attention to adaptive control generalizations of the celebrated Washout ﬁlters–traditionally used for stabilizing the Dutch-roll motion of an aircraft during a ﬂight (Hassouneh et al., 2004; Grant & Reid, 1997).
Revised: (Theorem 3.2) • In order to make some substantial headway under the burden of the above instability results, we shift our attention to adaptive control generalizations of the celebrated Washout ﬁlters–traditionally used for stabilizing the Dutch-roll motion of an aircraft during a ﬂight (Hassouneh et al., 2004; Grant & Reid, 1997).

Original: Inspired by this framework
Revised: Inspired by this framework

Original: we propose the modiﬁed KPV-GDA 2 which consists of a tandem combination of GDA together with a stabilizing feedback introduces by Bazanella et al.
Revised: we propose the modiﬁed KPV-GDA 2 which consists of a tandem combination of GDA together with a stabilizing feedback introduces by Bazanella et al.

Original: [Equation]
Revised: [Equation]

Original: The main linchpin of KPV-GDA method is the Simon’s and Theil’s
Revised: The main linchpin of KPV-GDA method is the Simon’s and Theil’s

Original: (Simon, 1956; Theil, 1957) certainty equivalence principle , a widely used methodology in Control theory in developing applied dynamic rational expectations models.
Revised: (Simon, 1956; Theil, 1957) certainty equivalence principle , a widely used methodology in Control theory in developing applied dynamic rational expectations models.

Original: According
Revised: According

Original: to this principle, the feedback law is split into two optimization steps whereby K − step attracts quickly the state to the stress while
Revised: to this principle, the feedback law is split into two optimization steps whereby K − step attracts quickly the state to the stress while

Original: P
Revised: P

Original: − step converges slowly to the
Revised: − step converges slowly to the

Original: ﬁxed
Revised: ﬁxed

Original: points of
Revised: points of

Original: GDA.
Revised: GDA.

Original: Compared
Revised: Compared

Original: with the plethora of the proposed dynamics for min-max problems, the crucial advantage of the afore-described technique is that
Revised: with the plethora of the proposed dynamics for min-max problems, the crucial advantage of the afore-described technique is that

Original: does not introduce any extra ﬁxed points
Revised: does not introduce any extra ﬁxed points

Original: than
Revised: than

Original: GDA’s
Revised: GDA’s

Original: ones.
Revised: ones.

Original: In
Revised: In

Original: Section
Revised: Section

Original: 2.2, we provide some il lustrative
Revised: 2.2, we provide some il lustrative

Original: examples of KPV-GDA technique
Revised: examples of KPV-GDA technique

Original: , while in
Revised: , while in

Original: Theorem
Revised: Theorem

Original: 3.7 we prove the
Revised: 3.7 we prove the

Original: existence of such control feedback
Revised: existence of such control feedback

Original: for our
Revised: for our

Original: class of games.
Revised: class of games.

Original: • Finally, in Section 4 we provide a series of experiments in simple two-team zero-sum games showcasing both the messy behaviors of traditional methods like GDA,OGDA and the power of KPV-GDA
Revised: • Finally, in Section 4 we provide a series of experiments in simple two-team zero-sum games showcasing both the messy behaviors of traditional methods like GDA,OGDA and the power of KPV-GDA

Original: method in these optimization environments.
Revised: method in these optimization environments.

Original: Additionally, we show that multi-agent
Revised: Additionally, we show that multi-agent

Original: GAN architectures achieve better performance than the single-agent ones, in terms of network capacity, when they are trained in synthetic or real-world datasets like CIFAR10.
Revised: GAN architectures achieve better performance than the single-agent ones, in terms of network capacity, when they are trained in synthetic or real-world datasets like CIFAR10.

Original:  P RELIMINARIES 2. D EFINITIONS
Revised:  P RELIMINARIES 2. D EFINITIONS

Original: Our setting.
Revised: Our setting.

Original: Formally, a two-team game in normal form is deﬁned as a tuple Γ =
Revised: Formally, a two-team game in normal form is deﬁned as a tuple Γ =

Original: Γ( N , A , u ) consisting of ( i ) a ﬁnite set of players N , split into two teams A, B with k A and k B players correspondingly such that: N = N A ∪ N B =
Revised: Γ( N , A , u ) consisting of ( i ) a ﬁnite set of players N , split into two teams A, B with k A and k B players correspondingly such that: N = N A ∪ N B =

Original: { A 1 , · · · , A k A , B 1 , · · · , B k B } ; ( ii ) a ﬁnite set of actions (or pure strategies )
Revised: { A 1 , · · · , A k A , B 1 , · · · , B k B } ; ( ii ) a ﬁnite set of actions (or pure strategies )

Original: A i = { α 1 , . . .
Revised: A i = { α 1 , . . .

Original: , α n i } per player i ∈ N ; ( iii ) each team’s payoff function u A , u B : A → R , where A :
Revised: , α n i } per player i ∈ N ; ( iii ) each team’s payoff function u A , u B : A → R , where A :

Original: =
Revised: =

Original: (cid:81)
Revised: (cid:81)

Original: i A
Revised: i A

Original: i denotes the ensemble of all possible action proﬁles α
Revised: i denotes the ensemble of all possible action proﬁles α

Original: = ( α A 1 , . .
Revised: = ( α A 1 , . .

Original: α A k A , α B 1 , . . .
Revised: α A k A , α B 1 , . . .

Original: , α B k B ) while the individual utility of a player is identical to her teammates, i.e., u i =
Revised: , α B k B ) while the individual utility of a player is identical to her teammates, i.e., u i =

Original: u A & u j = u B ∀ ( i,
Revised: u A & u j = u B ∀ ( i,

Original: j ) ∈ N A ×N B .
Revised: j ) ∈ N A ×N B .

Original: In this general context, players could also adhere mixed strategies , i.e, probability distributions s k ∈ ∆( A k ) over the pure strategies α k ∈
Revised: In this general context, players could also adhere mixed strategies , i.e, probability distributions s k ∈ ∆( A k ) over the pure strategies α k ∈

Original: A k .
Revised: A k .

Original: Correspondingly, we deﬁne the product distributions x = s A 1 ⊗ · · · ⊗ s A kA , y =
Revised: Correspondingly, we deﬁne the product distributions x = s A 1 ⊗ · · · ⊗ s A kA , y =

Original: s B 1 ⊗ · · · ⊗ s B kB as the teams’ strategies.
Revised: s B 1 ⊗ · · · ⊗ s B kB as the teams’ strategies.

Original: Collectively, we will write X := (cid:81) i ∈N A X
Revised: Collectively, we will write X := (cid:81) i ∈N A X

Original: i = (cid:81) i ∈N A ∆( A i ) , Y :=
Revised: i = (cid:81) i ∈N A ∆( A i ) , Y :=

Original: (cid:81)
Revised: (cid:81)

Original: i ∈N A Y i =
Revised: i ∈N A Y i =

Original: (cid:81) i ∈N B ∆( A i ) the space of mixed strategy proﬁles of teams
Revised: (cid:81) i ∈N B ∆( A i ) the space of mixed strategy proﬁles of teams

Original: A, B .
Revised: A, B .

Original: Similarly with the bilinear two-player games, the teams’ utility functions can be expressed via the payoff- tensors
Revised: Similarly with the bilinear two-player games, the teams’ utility functions can be expressed via the payoff- tensors

Original: A , B ∈ R τ with τ = (cid:81) i ∈N |A
Revised: A , B ∈ R τ with τ = (cid:81) i ∈N |A

Original: i | and acquire the form 3 :
Revised: i | and acquire the form 3 :

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: shapes ( equivalent
Revised: shapes ( equivalent

Original: In terms of solutions, we focus on the per player Nash Equilibrium (NE), i.e., a state strategy proﬁle s ∗ =
Revised: In terms of solutions, we focus on the per player Nash Equilibrium (NE), i.e., a state strategy proﬁle s ∗ =

Original: ( x , y ) = (cid:16) ( s ∗ A 1 , . .
Revised: ( x , y ) = (cid:16) ( s ∗ A 1 , . .

Original: . , s ∗ A kA ) , ( s ∗ B 1 , . .
Revised: . , s ∗ A kA ) , ( s ∗ B 1 , . .

Original: , s ∗ B kB ) (cid:17) such that
Revised: , s ∗ B kB ) (cid:17) such that

Original: [Equation]
Revised: [Equation]

Original: The state strategy proﬁle s ∗ is called pure if every player of both teams chooses a single action; otherwise we say that it is mixed.
Revised: The state strategy proﬁle s ∗ is called pure if every player of both teams chooses a single action; otherwise we say that it is mixed.

Original: Finally, a two-team game is called two-team zero-sum if u A = − u B or equivalently
Revised: Finally, a two-team game is called two-team zero-sum if u A = − u B or equivalently

Original: A + B = O .
Revised: A + B = O .

Original: Remark 2.1 .
Revised: Remark 2.1 .

Original: A quite technical prerequisite for the rest of this work, we will assume that a succinct representation of the utility tensors of the game is available or equivalently that a payoff oracle provides efﬁciently both the value of the utility function and its derivatives for a speciﬁc input, which is consistent with the vast majority of the applications that are described in the literature (von
Revised: A quite technical prerequisite for the rest of this work, we will assume that a succinct representation of the utility tensors of the game is available or equivalently that a payoff oracle provides efﬁciently both the value of the utility function and its derivatives for a speciﬁc input, which is consistent with the vast majority of the applications that are described in the literature (von

Original: Stengel & Koller, 1997).
Revised: Stengel & Koller, 1997).

Original: A ﬁrst approach on computing Nash equilibria in Two-Team Zero-Sum games.
Revised: A ﬁrst approach on computing Nash equilibria in Two-Team Zero-Sum games.

Original: Given the existence of the duality-gap between the min max and max min , in lieu of the two-player zero-sum game, an equilibrium in our setting can not be computed via linear programming.
Revised: Given the existence of the duality-gap between the min max and max min , in lieu of the two-player zero-sum game, an equilibrium in our setting can not be computed via linear programming.

Original: For the goal of computing Nash equilibria in two-team zero-sum games, we have experimented with a selection of ﬁrst-order methods that have been utilized with varying success in the setting of the two-person zerosum case.
Revised: For the goal of computing Nash equilibria in two-team zero-sum games, we have experimented with a selection of ﬁrst-order methods that have been utilized with varying success in the setting of the two-person zerosum case.

Original: Namely, we analyze the following methods: i)
Revised: Namely, we analyze the following methods: i)

Original: Gradient Descent-Ascent ii)
Revised: Gradient Descent-Ascent ii)

Original: Optimistic
Revised: Optimistic

Original: Gradient Descent-Ascent iii)
Revised: Gradient Descent-Ascent iii)

Original: Extra Gradient Method iv) Optimistic Multiplicative Weights Update
Revised: Extra Gradient Method iv) Optimistic Multiplicative Weights Update

Original: Method We defer their precise deﬁnitions in Appendix B.
Revised: Method We defer their precise deﬁnitions in Appendix B.

Original: The below remark will play a key role in the sequel.
Revised: The below remark will play a key role in the sequel.

Original: Remark 2.2 .
Revised: Remark 2.2 .

Original: Any ﬁxed point of the aforementioned discrete-time dynamics on the utility function corresponds necessarily to the Nash equilibria of the game.
Revised: Any ﬁxed point of the aforementioned discrete-time dynamics on the utility function corresponds necessarily to the Nash equilibria of the game.

Original: Hence, an important testbed for the long-run behavior of GDA, OGDA, and EG methods is to examine whether these methods stabilize around their ﬁxed points, which effectively constitute the Nash equilibria of the game.
Revised: Hence, an important testbed for the long-run behavior of GDA, OGDA, and EG methods is to examine whether these methods stabilize around their ﬁxed points, which effectively constitute the Nash equilibria of the game.

Original: In Section 3.2, we show that in lack of pure Nash equilibria, all the above methods fail to stabilize on their ﬁxed points even for a simple class of (2 , 2) -players game, and as a consequence to the mixed Nash equilibria of the game.
Revised: In Section 3.2, we show that in lack of pure Nash equilibria, all the above methods fail to stabilize on their ﬁxed points even for a simple class of (2 , 2) -players game, and as a consequence to the mixed Nash equilibria of the game.

Original: The presence of these results showcases the need for a different approach that lies outside purely optimization-based ideas.
Revised: The presence of these results showcases the need for a different approach that lies outside purely optimization-based ideas.

Original: Inspired by the applications of washout ﬁlters to stabilize highly susceptible systems and their adaptive control generalizations, we design a new incarnation of GDA vaned by two matrices-control feedback.
Revised: Inspired by the applications of washout ﬁlters to stabilize highly susceptible systems and their adaptive control generalizations, we design a new incarnation of GDA vaned by two matrices-control feedback.

Original: Surprisingly, in contrast with the aforementioned traditional methods, our proposed technique accomplishes last-iterate stabilization on its ﬁxed point, i.e., the mixed Nash equilibria of the team game.
Revised: Surprisingly, in contrast with the aforementioned traditional methods, our proposed technique accomplishes last-iterate stabilization on its ﬁxed point, i.e., the mixed Nash equilibria of the team game.

Original: ( K, P ) -Vaned GDA Method.
Revised: ( K, P ) -Vaned GDA Method.

Original: After concatenating the vectors of the minimizing and the maximizing agents z ( k ) =
Revised: After concatenating the vectors of the minimizing and the maximizing agents z ( k ) =

Original: ( x ( k ) , y ( k ) )
Revised: ( x ( k ) , y ( k ) )

Original: we can write our method, for appropriate matrices K, P :
Revised: we can write our method, for appropriate matrices K, P :

Original: [Equation]
Revised: [Equation]

Original: Intuitively, the added variable θ ( k ) holds an estimate of the ﬁxed point, and through the feedback ηK ( z ( k ) − θ ( k ) )
Revised: Intuitively, the added variable θ ( k ) holds an estimate of the ﬁxed point, and through the feedback ηK ( z ( k ) − θ ( k ) )

Original: the vector z stabilizes around that estimate which slowly moves towards the real ﬁxed point of the plain GDA dynamic.
Revised: the vector z stabilizes around that estimate which slowly moves towards the real ﬁxed point of the plain GDA dynamic.

Original: It is crucial to note that no additional ﬁxed points are introduced to the system.
Revised: It is crucial to note that no additional ﬁxed points are introduced to the system.

Original: 2. T WO I LLUSTRATIVE E XAMPLES
Revised: 2. T WO I LLUSTRATIVE E XAMPLES

Original: Our ﬁrst example supports a double role: Firstly, it exempliﬁes how our two-team min-max competition can capture the formulation of multi-agent GANs’ architectures.
Revised: Our ﬁrst example supports a double role: Firstly, it exempliﬁes how our two-team min-max competition can capture the formulation of multi-agent GANs’ architectures.

Original: Secondly, it hints also at an early separation between the optimization methods, since as we will see GDA will not converge to the Nash Equilibrium/ground-truth distribution.
Revised: Secondly, it hints also at an early separation between the optimization methods, since as we will see GDA will not converge to the Nash Equilibrium/ground-truth distribution.

Original: Consider the case of M , a mixture of gaussian distribution with two components, C 1 ∼ N ( µ , I ) and
Revised: Consider the case of M , a mixture of gaussian distribution with two components, C 1 ∼ N ( µ , I ) and

Original: C 2 ∼ N ( − µ , I ) and mixture weights π 1 , π 2 to be positive such that π 1 + π 2 = 1 and π 1 , π 2 (cid:54) = 12 .
Revised: C 2 ∼ N ( − µ , I ) and mixture weights π 1 , π 2 to be positive such that π 1 + π 2 = 1 and π 1 , π 2 (cid:54) = 12 .

Original: To learn the distribution above, we utilize an instance of a Team -WGAN in which there exists a generating team of agents G p : R → R , G θ : R n → R n , and a discriminating team of agents
Revised: To learn the distribution above, we utilize an instance of a Team -WGAN in which there exists a generating team of agents G p : R → R , G θ : R n → R n , and a discriminating team of agents

Original: D v : R n → R , D w : R n → R , all described by the following equations:
Revised: D v : R n → R , D w : R n → R , all described by the following equations:

Original: [Equation]
Revised: [Equation]

Original: The generating agent G θ maps random noise z ∼ N (0 , I ) to samples while generating agent G p ( ζ ) , utilizing an independent source of randomness ζ ∼ N (0 , 1) , probabilistically controls the sign of the output of the generator G θ .
Revised: The generating agent G θ maps random noise z ∼ N (0 , I ) to samples while generating agent G p ( ζ ) , utilizing an independent source of randomness ζ ∼ N (0 , 1) , probabilistically controls the sign of the output of the generator G θ .

Original: The probability of ultimately generating a sample y =
Revised: The probability of ultimately generating a sample y =

Original: z + θ is equal to ζ + p , while the probability of the sample being y =
Revised: z + θ is equal to ζ + p , while the probability of the sample being y =

Original: −
Revised: −

Original: z − θ is equal to 1 − ( p + ζ ) .
Revised: z − θ is equal to 1 − ( p + ζ ) .

Original: On the other end, there stands the discriminating team of D v , D w .
Revised: On the other end, there stands the discriminating team of D v , D w .

Original: Discriminators, D v ( y ) , D w ( y ) map any given sample y to a scalar value accounting for the realness or fakeness of it – negative meaning fake, positive meaning real.
Revised: Discriminators, D v ( y ) , D w ( y ) map any given sample y to a scalar value accounting for the realness or fakeness of it – negative meaning fake, positive meaning real.

Original: The discriminators are disparate in the way they measure the realness of samples as seen in their deﬁnitions.
Revised: The discriminators are disparate in the way they measure the realness of samples as seen in their deﬁnitions.

Original: We follow the formalism of the Wasserstein GAN to form the optimization objective:
Revised: We follow the formalism of the Wasserstein GAN to form the optimization objective:

Original: [Equation]
Revised: [Equation]

Original: Equation equation 2.4 yields the simpler form:
Revised: Equation equation 2.4 yields the simpler form:

Original: [Equation]
Revised: [Equation]

Original: It is easy to check that Nash equilibria of Equation equation 2.4 must satisfy:
Revised: It is easy to check that Nash equilibria of Equation equation 2.4 must satisfy:

Original: [Equation]
Revised: [Equation]

Original: Figure 1 demonstrates both GDA’s failure and OGDA, EG
Revised: Figure 1 demonstrates both GDA’s failure and OGDA, EG

Original: [Equation]
Revised: [Equation]

Original: and our KPV-GDA method’s success to converge to the above Nash equilibria and simultaneously to discover the groud truth mixture.
Revised: and our KPV-GDA method’s success to converge to the above Nash equilibria and simultaneously to discover the groud truth mixture.

Original: 2.2. M ULTIPLAYER M ATCHING P ENNIES
Revised: 2.2. M ULTIPLAYER M ATCHING P ENNIES

Original: Interestingly enough, there are non-trivial instances of two-team competition settings that even Optimistic GDA and EG Method fail to converge.
Revised: Interestingly enough, there are non-trivial instances of two-team competition settings that even Optimistic GDA and EG Method fail to converge.

Original: Such is the case for a team version of the well-known game of matching pennies.
Revised: Such is the case for a team version of the well-known game of matching pennies.

Original: The game can be shortly described as such: “ coordinate with your teammates to play a game of matching pennies against the opposing team, coordinate not and pay a penalty ”.
Revised: The game can be shortly described as such: “ coordinate with your teammates to play a game of matching pennies against the opposing team, coordinate not and pay a penalty ”.

Original: The penalty is set to 12 .
Revised: The penalty is set to 12 .

Original: For the interested reader, we defer to appendix C.2 the precise description of the game in a contracted tensor/table.
Revised: For the interested reader, we defer to appendix C.2 the precise description of the game in a contracted tensor/table.

Original: Since every player has simply two actions, their probability vector can be represented by a single variable in [0 , 1] .
Revised: Since every player has simply two actions, their probability vector can be represented by a single variable in [0 , 1] .

Original: Considering the minimizing team x its players are x 1 , x 2 , while the players of the maximizing team y are y 1 , y 2 .
Revised: Considering the minimizing team x its players are x 1 , x 2 , while the players of the maximizing team y are y 1 , y 2 .

Original: The multiplayer of matching pennies is described by the utility function:
Revised: The multiplayer of matching pennies is described by the utility function:

Original: [Equation]
Revised: [Equation]

Original: As we can see in Figures 2 and 3, multiplayer matching pennies game consists an excellent benchmark where all traditional gradient ﬂow discretizations fail under perfect competition setting.
Revised: As we can see in Figures 2 and 3, multiplayer matching pennies game consists an excellent benchmark where all traditional gradient ﬂow discretizations fail under perfect competition setting.

Original: Interestingly, we are not aware of a similar example in min-max literature and it has been our starting point for seeking new optimization techniques inspired by Control theory.
Revised: Interestingly, we are not aware of a similar example in min-max literature and it has been our starting point for seeking new optimization techniques inspired by Control theory.

Original: Indeed, KPV-GDA variation with ( K, P ) =
Revised: Indeed, KPV-GDA variation with ( K, P ) =

Original: ( − 1 .
Revised: ( − 1 .

Original: 1 I , 0 .
Revised: 1 I , 0 .

Original: 3 I ) achieves to converge to the unique mixed Nash Equilibrium of the game.
Revised: 3 I ) achieves to converge to the unique mixed Nash Equilibrium of the game.

Original: In the following sections, we provide theorems that explained formally this long-run behavior of the examined dynamics.
Revised: In the following sections, we provide theorems that explained formally this long-run behavior of the examined dynamics.

Original:  O UR MAIN RESULTS 3. O N THE COMPLEXITY OF T WO -T EAM Z ERO -S UM G AMES
Revised:  O UR MAIN RESULTS 3. O N THE COMPLEXITY OF T WO -T EAM Z ERO -S UM G AMES

Original: We start this section by showing that computing a Nash equilibrium in two-team zero-sum games is computationally hard and thus getting a polynomial-time algorithm that computes a Nash equilibrium is unlikely.
Revised: We start this section by showing that computing a Nash equilibrium in two-team zero-sum games is computationally hard and thus getting a polynomial-time algorithm that computes a Nash equilibrium is unlikely.

Original: Theorem 3.1 ( CLS -hard) .
Revised: Theorem 3.1 ( CLS -hard) .

Original: Computing a Nash equilibrium in two-team zero-sum games is CLS -hard.
Revised: Computing a Nash equilibrium in two-team zero-sum games is CLS -hard.

Original: The main idea of the proof of Theorem 3.1 relies on a reduction of approximating Nash equilibria in congestion games, which has been shown to be complete for the interesting class of CLS , which contains the problem of continuous optimization.
Revised: The main idea of the proof of Theorem 3.1 relies on a reduction of approximating Nash equilibria in congestion games, which has been shown to be complete for the interesting class of CLS , which contains the problem of continuous optimization.

Original: For concision, we defer the proof of the above theorem to the paper’s supplement.
Revised: For concision, we defer the proof of the above theorem to the paper’s supplement.

Original: 3. F IRST - ORDER METHODS FAIL TO STABILIZE
Revised: 3. F IRST - ORDER METHODS FAIL TO STABILIZE

Original: The negative computational complexity result we proved for two-team zero-sum games (Theorem 3.1) does not conclude the prospect of having algorithms (learning dynamics, ﬁrst-order methods) that converge to Nash equilibria and thus can approximate them well enough.
Revised: The negative computational complexity result we proved for two-team zero-sum games (Theorem 3.1) does not conclude the prospect of having algorithms (learning dynamics, ﬁrst-order methods) that converge to Nash equilibria and thus can approximate them well enough.

Original: Unfortunately, we can even prove negative results about convergence to Nash equilibria in two-team zero-sum games of well-established methods broadly used in classic two-player zero-sum games.
Revised: Unfortunately, we can even prove negative results about convergence to Nash equilibria in two-team zero-sum games of well-established methods broadly used in classic two-player zero-sum games.

Original: In this section, we are going to construct a family of two-team zero-sum games with the property that GDA, OGDA, EG, and OMWU fail to stabilize to Nash equilibria.
Revised: In this section, we are going to construct a family of two-team zero-sum games with the property that GDA, OGDA, EG, and OMWU fail to stabilize to Nash equilibria.

Original: This result indicates how challenging and rich the setting of team zero-sum games can be and why provable guarantees about convergence have not been established yet.
Revised: This result indicates how challenging and rich the setting of team zero-sum games can be and why provable guarantees about convergence have not been established yet.

Original: Before deﬁning the family of two-team zero-sum games, we prove an important theorem which states that GDA does not stabilize around mixed Nash equilibria.
Revised: Before deﬁning the family of two-team zero-sum games, we prove an important theorem which states that GDA does not stabilize around mixed Nash equilibria.

Original: This fact is a stepping stone in constructing the family of team
Revised: This fact is a stepping stone in constructing the family of team

Original: -zero sum games later.
Revised: -zero sum games later.

Original: We present the proof of all of the below statements in detail in the paper’s appendix.
Revised: We present the proof of all of the below statements in detail in the paper’s appendix.

Original: Weakly-stable Nash equilibrium (Kleinberg et al., 2009; Mehta et al., 2015).
Revised: Weakly-stable Nash equilibrium (Kleinberg et al., 2009; Mehta et al., 2015).

Original: Consider the set of Nash equilibria with the property that if any single randomizing agent of one team is forced to play any strategy in her current support with probability one, all other agents of the same team must remain indifferent between the strategies in their support.
Revised: Consider the set of Nash equilibria with the property that if any single randomizing agent of one team is forced to play any strategy in her current support with probability one, all other agents of the same team must remain indifferent between the strategies in their support.

Original: This type of Nash equilibria is called weakly-stable.
Revised: This type of Nash equilibria is called weakly-stable.

Original: Note that trivially pure Nash equilibria are weakly-stable.
Revised: Note that trivially pure Nash equilibria are weakly-stable.

Original: It has been shown that mixed Nash equilibria are not weakly-stable in generic games 5 (Kleinberg et al., 2009).
Revised: It has been shown that mixed Nash equilibria are not weakly-stable in generic games 5 (Kleinberg et al., 2009).

Original: We can show that Nash equilibria that are not weakly-stable Nash are actually unstable for GDA moreover, through standard dynamical systems machinery, that the set of initial conditions that converges to
Revised: We can show that Nash equilibria that are not weakly-stable Nash are actually unstable for GDA moreover, through standard dynamical systems machinery, that the set of initial conditions that converges to

Original: Nash equilibria that are not weakly-stable should be of measure zero.
Revised: Nash equilibria that are not weakly-stable should be of measure zero.

Original: Formally, we prove that:
Revised: Formally, we prove that:

Original: Theorem 3.2 (Non weakly-stable Nash are unstable) .
Revised: Theorem 3.2 (Non weakly-stable Nash are unstable) .

Original: Consider a two-team zero-sum game with utility function of Team B ( y vector) being U ( x , y ) and Team A ( x vector) being − U ( x , y ) .
Revised: Consider a two-team zero-sum game with utility function of Team B ( y vector) being U ( x , y ) and Team A ( x vector) being − U ( x , y ) .

Original: Moreover, assume that ( x ∗ , y ∗ ) is a Nash equilibrium of full support that is not weakly-stable.
Revised: Moreover, assume that ( x ∗ , y ∗ ) is a Nash equilibrium of full support that is not weakly-stable.

Original: It follows that the set of initial conditions so that GDA converges to ( x ∗ , y ∗ ) is of measure zero for step size
Revised: It follows that the set of initial conditions so that GDA converges to ( x ∗ , y ∗ ) is of measure zero for step size

Original: η < 1 L where L is the Lipschitz constant of ∇ U .
Revised: η < 1 L where L is the Lipschitz constant of ∇ U .

Original: 3. G ENERALIZED M ATCHING P ENNIES (GMP)
Revised: 3. G ENERALIZED M ATCHING P ENNIES (GMP)

Original: Inspired by Theorem 3.2, in this section we construct a family of team zero-sum games so that GDA,
Revised: Inspired by Theorem 3.2, in this section we construct a family of team zero-sum games so that GDA,

Original: OGDA, and EG methods fail to converge (if the initialization is a random point in the simplex, the probability of convergence of the aforementioned methods is zero).
Revised: OGDA, and EG methods fail to converge (if the initialization is a random point in the simplex, the probability of convergence of the aforementioned methods is zero).

Original: The intuition is to construct a family of games, each of which has only mixed Nash equilibria (that are not weakly-stable), i.e., the constructed games should lack pure Nash equilibria; using Theorem 3.2, it would immediately imply our claim for GDA.
Revised: The intuition is to construct a family of games, each of which has only mixed Nash equilibria (that are not weakly-stable), i.e., the constructed games should lack pure Nash equilibria; using Theorem 3.2, it would immediately imply our claim for GDA.

Original: It turns out that OGDA and EG also fail to converge for the same family.
Revised: It turns out that OGDA and EG also fail to converge for the same family.

Original: Deﬁnition of GMP.
Revised: Deﬁnition of GMP.

Original: Consider a setting with two teams (Team A , Team B ), each of which has n = 2 players.
Revised: Consider a setting with two teams (Team A , Team B ), each of which has n = 2 players.

Original: Inspired by the standard matching pennies game and the game deﬁned in Schulman & Vazirani (2019a), we allow each agent i to have two strategies/actions that is S
Revised: Inspired by the standard matching pennies game and the game deﬁned in Schulman & Vazirani (2019a), we allow each agent i to have two strategies/actions that is S

Original: =
Revised: =

Original: { H, T } for both teams with 2 4 possible strategy proﬁles.
Revised: { H, T } for both teams with 2 4 possible strategy proﬁles.

Original: In case all the members of a Team choose the same strategy say H or T then the Team “agrees” to play H or T (otherwise the Team “does not agree”).
Revised: In case all the members of a Team choose the same strategy say H or T then the Team “agrees” to play H or T (otherwise the Team “does not agree”).

Original: Thus, in the case both teams “agree”, the payoff of each team is actually the payoffs for the twoplayer matching pennies.
Revised: Thus, in the case both teams “agree”, the payoff of each team is actually the payoffs for the twoplayer matching pennies.

Original: If one team “agrees” and the other does not, the team that “agrees” gets payoff ω ∈ (0 , 1) and the other team gets penalty ω .
Revised: If one team “agrees” and the other does not, the team that “agrees” gets payoff ω ∈ (0 , 1) and the other team gets penalty ω .

Original: If both teams fail to “agree”, both teams get payoff zero.
Revised: If both teams fail to “agree”, both teams get payoff zero.

Original: Let x i with i ∈ { 1 , 2 } be the probability that agent i of Team A chooses H and 1 − x i the probability that she chooses T .
Revised: Let x i with i ∈ { 1 , 2 } be the probability that agent i of Team A chooses H and 1 − x i the probability that she chooses T .

Original: We also denote x the vector of probabilities for Team
Revised: We also denote x the vector of probabilities for Team

Original: A .
Revised: A .

Original: Similarly, we denote y i for i ∈ { 1 , 2 } be the probability that agent i of Team B chooses H and 1 − y i the probability that she chooses T and y the probability vector.
Revised: Similarly, we denote y i for i ∈ { 1 , 2 } be the probability that agent i of Team B chooses H and 1 − y i the probability that she chooses T and y the probability vector.

Original: The ﬁrst fact about the game that we deﬁned is that for ω ∈ (0 , 1) , there is only one Nash equilibrium ( x ∗ , y ∗ ) , which is the uniform, i.e., x ∗ 1 = x ∗ 2 =
Revised: The ﬁrst fact about the game that we deﬁned is that for ω ∈ (0 , 1) , there is only one Nash equilibrium ( x ∗ , y ∗ ) , which is the uniform, i.e., x ∗ 1 = x ∗ 2 =

Original: y ∗ 

Revised: y ∗

Original: =
Revised: =

Original: y ∗ 2 = 12 for all agents i.
Revised: y ∗ 2 = 12 for all agents i.

Original: Lemma 3.3 (GMP has a unique Nash) .
Revised: Lemma 3.3 (GMP has a unique Nash) .

Original: The Generalized Matching Pennies game exhibits a unique
Revised: The Generalized Matching Pennies game exhibits a unique

Original: Nash equilibrium which is ( x ∗ , y ∗ ) =
Revised: Nash equilibrium which is ( x ∗ , y ∗ ) =

Original: Remark 3.4 .
Revised: Remark 3.4 .

Original: The fact that the game we deﬁned has a unique Nash equilibrium that is in the interior of [0 , 1] 4 is really crucial for our negative convergence results later in the section as we will show that it is not weakly-stable Nash equilibrium and the negative result about GDA will be a corollary due to Theorem 3.2.
Revised: The fact that the game we deﬁned has a unique Nash equilibrium that is in the interior of [0 , 1] 4 is really crucial for our negative convergence results later in the section as we will show that it is not weakly-stable Nash equilibrium and the negative result about GDA will be a corollary due to Theorem 3.2.

Original: Please also note that if ω = 1 then there are more Nash equilibria, in particular the ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) are also Nash equilibria (which are pure).
Revised: Please also note that if ω = 1 then there are more Nash equilibria, in particular the ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) are also Nash equilibria (which are pure).

Original: The following Theorem is the main (negative) result of this section.
Revised: The following Theorem is the main (negative) result of this section.

Original: Theorem 3.5 (GDA, OGDA, EG, and OMWU fail) .
Revised: Theorem 3.5 (GDA, OGDA, EG, and OMWU fail) .

Original: Consider GMP game with ω ∈ (0 , 1) .
Revised: Consider GMP game with ω ∈ (0 , 1) .

Original: Assume that η GDA < 14 , η OGDA < min( ω, 18 ) , η EG < min( 12  ω 2 )) , and η OMWU < ω 2 (bound on the stepsizefor
Revised: Assume that η GDA < 14 , η OGDA < min( ω, 18 ) , η EG < min( 12 , ω 2 )) , and η OMWU < ω 2 (bound on the stepsize

Original: It holds that the set of initial conditions so that GDA, OGDA, EG, OMWU converge (stabilize to any point) is of measure zero.

Original: Remark 3.6 (Average iterate also fails) .
Revised: Remark 3.6 (Average iterate also fails) .

Original: One might ask what happens when we consider average iterates instead of the last iterate.
Revised: One might ask what happens when we consider average iterates instead of the last iterate.

Original: It is well-known fact Syrgkanis et al.
Revised: It is well-known fact Syrgkanis et al.

Original: (2015) that the average iterate of no-regret algorithms converges to coarse correlated equilibria (CCE) so we expect that the average iterate stabilizes.
Revised: (2015) that the average iterate of no-regret algorithms converges to coarse correlated equilibria (CCE) so we expect that the average iterate stabilizes.

Original: Nevertheless, CCE might not be Nash equilibria.
Revised: Nevertheless, CCE might not be Nash equilibria.

Original: Indeed we can construct examples in which the average iterate of GDA, OGDA, and EG experimentally fail to stabilize to
Revised: Indeed we can construct examples in which the average iterate of GDA, OGDA, and EG experimentally fail to stabilize to

Original: Nash equilibria.
Revised: Nash equilibria.

Original: In particular, we consider a slight modiﬁcation of GMP; players and strategies are the same but the payoff matrix has changed and can be found below (table on the right):
Revised: In particular, we consider a slight modiﬁcation of GMP; players and strategies are the same but the payoff matrix has changed and can be found below (table on the right):

Original: HH
Revised: HH

Original: HT/TH
Revised: HT/TH

Original: TT
Revised: TT

Original: HH 1 , −ω, − ω − 1 ,
Revised: HH 1 , −ω, − ω − 1 ,

Original: HT/TH − ω, ω 0 ,− ω, ω
Revised: HT/TH − ω, ω 0 ,− ω, ω

Original: TT − 1 ,ω, − ω 1 , −
Revised: TT − 1 ,ω, − ω 1 , −

Original: HH
Revised: HH

Original: HT/TH
Revised: HT/TH

Original: TT
Revised: TT

Original: HH 2 , −12 , −− 2 ,
Revised: HH 2 , −12 , −− 2 ,

Original: HT/TH − 12 ,0 ,− 12 ,
Revised: HT/TH − 12 ,0 ,− 12 ,

Original: TT − 1 ,12 , −1 , −
Revised: TT − 1 ,12 , −1 , −

Original: Figure 4 illustrates that the average iterates of GDA, OGDA, and EG stabilize to points that are not
Revised: Figure 4 illustrates that the average iterates of GDA, OGDA, and EG stabilize to points that are not

Original: Nash equilibria.
Revised: Nash equilibria.

Original: Note that since our method (see next subsection) converges locally, the average iterate should converge locally to a Nash equilibrium.
Revised: Note that since our method (see next subsection) converges locally, the average iterate should converge locally to a Nash equilibrium.

Original: 3. W ASH - OUT F ILTERS & A DAPTIVE C ONTROL
Revised: 3. W ASH - OUT F ILTERS & A DAPTIVE C ONTROL

Original: The aforementioned results indicate that to answer the tantalizing question of ﬁnding NE in twoteam zero-sum games, our machinery should be broadened outside the limits of textbook optimization arsenal.
Revised: The aforementioned results indicate that to answer the tantalizing question of ﬁnding NE in twoteam zero-sum games, our machinery should be broadened outside the limits of textbook optimization arsenal.

Original: The mainstay of this effort and our positive result is KPV-GDA method deﬁned in (2.2), inspired by the adaptive control toolbox and washout ﬁlters.
Revised: The mainstay of this effort and our positive result is KPV-GDA method deﬁned in (2.2), inspired by the adaptive control toolbox and washout ﬁlters.

Original: Our main statement shows that
Revised: Our main statement shows that

Original: KPV-GDA stabilizes around any Nash equilibrium for appropriate choices of matrices K, P .
Revised: KPV-GDA stabilizes around any Nash equilibrium for appropriate choices of matrices K, P .

Original: The formal theorem is given below:
Revised: The formal theorem is given below:

Original: Theorem 3.7 (KPV-GDA stabilizes) .
Revised: Theorem 3.7 (KPV-GDA stabilizes) .

Original: Consider a team zero-sum game so that the utility of Team
Revised: Consider a team zero-sum game so that the utility of Team

Original: B is U ( x , y ) and hence the utility of Team A is − U ( x , y ) and a Nash equilibrium ( x ∗ , y ∗ ) of the game.
Revised: B is U ( x , y ) and hence the utility of Team A is − U ( x , y ) and a Nash equilibrium ( x ∗ , y ∗ ) of the game.

Original: Moreover we assume
Revised: Moreover we assume

Original: [Equation]
Revised: [Equation]

Original: For any ﬁxed step size η > 0 , we can always ﬁnd matrices K, P so that KPV-GDA method deﬁned in (2.2) converges locally to ( x ∗ , y ∗ ) .
Revised: For any ﬁxed step size η > 0 , we can always ﬁnd matrices K, P so that KPV-GDA method deﬁned in (2.2) converges locally to ( x ∗ , y ∗ ) .

Original: The latter statement concerns the existence of matrices K, P .
Revised: The latter statement concerns the existence of matrices K, P .

Original: Below, we prove that there also exists a sufﬁcient condition under which a simple parametrization is possible while convergence is guaranteed.
Revised: Below, we provide a sufﬁcient condition under which a simple parametrization of K, P (provably) guarantees convergence.

Original: Theorem 3.8.
Revised: Theorem 3.8.

Original: Consider a two-team zero-sum game so that the utility of Team B is U ( x , y ) and hence the utility of Team A is − U ( x , y ) and a Nash equilibrium ( x ∗ , y ∗ ) of the game.
Revised: Consider a two-team zero-sum game so that the utility of Team B is U ( x , y ) and hence the utility of Team A is − U ( x , y ) and a Nash equilibrium ( x ∗ , y ∗ ) of the game.

Original: Moreover let
Revised: Moreover let

Original: [Equation]
Revised: [Equation]

Original: and E be the set of eigenvalues ρ of H with real part positive, that is E
Revised: and E be the set of eigenvalues ρ of H with real part positive, that is E

Original: = { H (cid:48)
Revised: = { H (cid:48)

Original: s eigenvalues ρ :
Revised: s eigenvalues ρ :

Original: Re ( ρ ) > 0 } .
Revised: Re ( ρ ) > 0 } .

Original: We assume that H is invertible and moreover
Revised: We assume that H is invertible and moreover

Original: [Equation]
Revised: [Equation]

Original: We set K = k · I , P = p · I .
Revised: We set K = k · I , P = p · I .

Original: There exist small enough step size η > 0 and scalar p > 0 and for any k ∈ ( − β, − α ) so that KPV-GDA method deﬁned in (2.2) with chosen K, P converges locally to ( x ∗ , y ∗ ) .
Revised: There exist small enough step size η > 0 and scalar p > 0 and for any k ∈ ( − β, − α ) so that KPV-GDA method deﬁned in (2.2) with chosen K, P converges locally to ( x ∗ , y ∗ ) .

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: In this section, we perform a series of numerical experiments to validate our theoretical ﬁndings.
Revised: In this section, we perform a series of numerical experiments to validate our theoretical ﬁndings.

Original: Our experiment setting includes a 2-D Gaussian Mixture Model with 8 modes.
Revised: Our experiment setting includes a 2-D Gaussian Mixture Model with 8 modes.

Original: Our architecture includes 8 “shallow” generators and discriminators with 2 layers of 2-16-2 ReLUs activations, compared with a giant single-agent GAN with 4 layers of 2-128-256-1024-2 activations.
Revised: Our architecture includes 8 “shallow” generators and discriminators with 2 layers of 2-16-2 ReLUs activations, compared with a giant single-agent GAN with 4 layers of 2-128-256-1024-2 activations.

Original: Interestingly, the giant one fails in a double sense; It demonstrates both mode-collapsing and mode-drop phenomena without stabilizing.
Revised: Interestingly, the giant one fails in a double sense; It demonstrates both mode-collapsing and mode-drop phenomena without stabilizing.

Original: On the other hand, our architecture with a small number of neurons achieves to ﬁt the data well.
Revised: On the other hand, our architecture with a small number of neurons achieves to ﬁt the data well.

Original: We defer the discussion of the experiments with multi-generators multi-discriminator architectures for CIFAR-10 again to the paper’s supplement.
Revised: We defer the discussion of the experiments with multi-generators multi-discriminator architectures for CIFAR-10 again to the paper’s supplement.

Original: [Figure]
Revised: [Figure]

Original:  C ONCLUSIONS AND O PEN P ROBLEMS
Revised:  C ONCLUSIONS AND O PEN P ROBLEMS

Original: In this paper, we have presented a number of negative results about the problem of ﬁnding a Nash equilibrium in team zero-sum games and moreover about the inability of commonly used methods for min-max optimization such as GDA, OGDA, and EG to stabilize.
Revised: In this paper, we have presented a number of negative results about the problem of ﬁnding a Nash equilibrium in team zero-sum games and moreover about the inability of commonly used methods for min-max optimization such as GDA, OGDA, and EG to stabilize.

Original: We also presented a method (called KPV-GDA) that manages to stabilize around Nash equilibria.
Revised: We also presented a method (called KPV-GDA) that manages to stabilize around Nash equilibria.

Original: Given these results, a number of interesting open questions emerge.
Revised: Given these results, a number of interesting open questions emerge.

Original: Open Questions.
Revised: Open Questions.

Original: One question for future consideration is the global convergence and the rates of convergence of KPV-GDA method.
Revised: One question for future consideration is the global convergence and the rates of convergence of KPV-GDA method.

Original: We believe that the KPV-GDA converges globally for an appropriate choice of matrices K, P .
Revised: We believe that the KPV-GDA converges globally for an appropriate choice of matrices K, P .

Original: One other possible direction is to ﬁnd a systematic way to get the matrices K, P.
Revised: One other possible direction is to ﬁnd a systematic way to get the matrices K, P.

Original: R EPRODUCIBILITY S TATEMENT
Revised: R EPRODUCIBILITY S TATEMENT

Original: In our submission folder, we provide all the necessary additional technical materials and complete proofs of the main draft’s statements in the appendix section.
Revised: In our submission folder, we provide all the necessary additional technical materials and complete proofs of the main draft’s statements in the appendix section.

Original: We also uploaded the code of our experiments (Python/PyTorch/Tensorﬂow).
Revised: We also uploaded the code of our experiments (Python/PyTorch/Tensorﬂow).

Original: A A PPENDIX A. F URTHER R ELATED W ORK
Revised: A A PPENDIX A. F URTHER R ELATED W ORK

Original: The properties of team games have given rise to a vast corpus of literature which we cannot hope to review here; for an appetizer, we refer the reader to Kim et al.
Revised: The properties of team games have given rise to a vast corpus of literature which we cannot hope to review here; for an appetizer, we refer the reader to Kim et al.

Original: ; Gold (2005) and reference therein.
Revised: ; Gold (2005) and reference therein.

Original: On the other hand, the convergence of ﬁrst-order methods to NE, like GDA, OGDA, or
Revised: On the other hand, the convergence of ﬁrst-order methods to NE, like GDA, OGDA, or

Original: Extragradient in team games (even ﬁnite ones) is nowhere near as well understood.
Revised: Extragradient in team games (even ﬁnite ones) is nowhere near as well understood.

Original: A notable exception to this is the case of team-maxmin equilibria (TME), a notion introduced by von Stengel & Koller (1997).
Revised: A notable exception to this is the case of team-maxmin equilibria (TME), a notion introduced by von Stengel & Koller (1997).

Original: In this category of two-team zero-sum games, a team of n − players plays defensively against a single adversarial player.
Revised: In this category of two-team zero-sum games, a team of n − players plays defensively against a single adversarial player.

Original: TMEs provide crucial information about the robustness since they give the team players the highest payoff they employ uncoordinated strategies.
Revised: TMEs provide crucial information about the robustness since they give the team players the highest payoff they employ uncoordinated strategies.

Original: Unfortunately, TMEs correspond to the solution of a non–linear non–convex program.
Revised: Unfortunately, TMEs correspond to the solution of a non–linear non–convex program.

Original: (Basilico et al., 2017b) provided heuristics with iterative LP and support enumeration methods with proved approximation guarantees.
Revised: (Basilico et al., 2017b) provided heuristics with iterative LP and support enumeration methods with proved approximation guarantees.

Original: For the case of extensive or large-sized normal form games, Basilico et al. (2017a); Celli & Gatti (2018) introduced a relaxed notion of team-maxmin equilibrium with coordination device (TMECor).
Revised: For the case of extensive or large-sized normal form games, Basilico et al. (2017a); Celli & Gatti (2018) introduced a relaxed notion of team-maxmin equilibrium with coordination device (TMECor).

Original: In TMECor solutions, the team of players shares their strategies exante 6 against the adversary.
Revised: In TMECor solutions, the team of players shares their strategies exante 6 against the adversary.

Original: While TMECor shares some properties of NEs in zero-sum two-player games (e.g., exchangeability), the proposed algorithms by (Celli & Gatti, 2018) leverages tools like the Mixed-Integer Linear Program (MILP) that involves a large number of integer variables.
Revised: While TMECor shares some properties of NEs in zero-sum two-player games (e.g., exchangeability), the proposed algorithms by (Celli & Gatti, 2018) leverages tools like the Mixed-Integer Linear Program (MILP) that involves a large number of integer variables.

Original: To tackle heuristically these hurdles, it (Zhang et al., 2020; Zhang & An, 2020) had been proposed modiﬁed versions of incremental strategy generation (ISG) algorithm
Revised: To tackle heuristically these hurdles, it (Zhang et al., 2020; Zhang & An, 2020) had been proposed modiﬁed versions of incremental strategy generation (ISG) algorithm

Original: (See (McMahan et al., 2003)) for large-sized games.
Revised: (See (McMahan et al., 2003)) for large-sized games.

Original: Describing the history and the literature surrounding multi-agent GANs would take us too far aﬁeld, so we do not attempt it.
Revised: Describing the history and the literature surrounding multi-agent GANs would take us too far aﬁeld, so we do not attempt it.

Original: Scratching only the surface, we mention that: a) In MGAN (Hoang et al., 2018) and MAD-GAN (Li et al., 2019), we have a mixture of generative models with asymptotically inﬁnite capacity networks.
Revised: Scratching only the surface, we mention that: a) In MGAN (Hoang et al., 2018) and MAD-GAN (Li et al., 2019), we have a mixture of generative models with asymptotically inﬁnite capacity networks.

Original: b) In the seminal case of MIX+GAN (Arora et al., 2017), an extra regularization term is typically added to discourage the weights of a mixture of generators being too far away from uniform.
Revised: b) In the seminal case of MIX+GAN (Arora et al., 2017), an extra regularization term is typically added to discourage the weights of a mixture of generators being too far away from uniform.

Original: c)
Revised: c)

Original: In Stackelberg GAN, (Zhang et al., 2018) exploit the leader-followers relation of discriminator vs generators to achieve smaller minimax gap and better Frech ´ et metrics.
Revised: In Stackelberg GAN, (Zhang et al., 2018) exploit the leader-followers relation of discriminator vs generators to achieve smaller minimax gap and better Frech ´ et metrics.

Original: For the case of multiple-discriminators : d) In GMAN (Durugkar et al., 2016), an increased number of agents are used to achieve higher quality images, while e) MD-GAN (Hardy et al., 2019) is designed to generalize over different datasets and multiple tasks that are spread on multiple workers.
Revised: For the case of multiple-discriminators : d) In GMAN (Durugkar et al., 2016), an increased number of agents are used to achieve higher quality images, while e) MD-GAN (Hardy et al., 2019) is designed to generalize over different datasets and multiple tasks that are spread on multiple workers.

Original: Interestingly, in the continuous regime, gradient ﬂow has been recently studied under the perspective of dynamical systems.
Revised: Interestingly, in the continuous regime, gradient ﬂow has been recently studied under the perspective of dynamical systems.

Original: In the setting of Hidden Games , two competitive non-convex operators join in a two-player min-max game.
Revised: In the setting of Hidden Games , two competitive non-convex operators join in a two-player min-max game.

Original: In Piliouras & Schulman (2018); Vlatakis-Gkaragkounis et al.
Revised: In Piliouras & Schulman (2018); Vlatakis-Gkaragkounis et al.

Original: (2019), the authors exploit the bi-linear form of the hidden game to prove antithetical to convergence, like spurious ﬁxed points, cycles, and generalized high-dimensional recurrence phenomena.
Revised: (2019), the authors exploit the bi-linear form of the hidden game to prove antithetical to convergence, like spurious ﬁxed points, cycles, and generalized high-dimensional recurrence phenomena.

Original: Additionally, in the case of general convex-concave settings, Flokas et al.
Revised: Additionally, in the case of general convex-concave settings, Flokas et al.

Original: (2021) proved that local stability can be achieved around the von-Neumann equilibria while a global convergent result can be described for strictly convex hidden games.
Revised: (2021) proved that local stability can be achieved around the von-Neumann equilibria while a global convergent result can be described for strictly convex hidden games.

Original: A crucial comparison of the hidden model with the team games as described in our setting is the capacity & independence of the player’s choice.
Revised: A crucial comparison of the hidden model with the team games as described in our setting is the capacity & independence of the player’s choice.

Original: While the main contribution of this work is to highlight the inherent difﬁculties that the above heuristic architectures are cursed, our theoretical ﬁndings in team competition hint also at new optimization schemes like KPV-GDA whose performance is illustrated in Section 4 with delicate network architectures that contain even multiple generators and discriminators simultaneously.
Revised: While the main contribution of this work is to highlight the inherent difﬁculties that the above heuristic architectures are cursed, our theoretical ﬁndings in team competition hint also at new optimization schemes like KPV-GDA whose performance is illustrated in Section 4 with delicate network architectures that contain even multiple generators and discriminators simultaneously.

Original: A. D ISCUSSION ABOUT DIFFERENT SOLUTION CONCEPTS
Revised: A. D ISCUSSION ABOUT DIFFERENT SOLUTION CONCEPTS

Original: In this subsection, we would like to stress some differences between the different equilibrium notions ( TMECor, TME, and per-player NE ) explaining the game-theoretic rationalism of our choice.
Revised: In this subsection, we would like to stress some differences between the different equilibrium notions ( TMECor, TME, and per-player NE ) explaining the game-theoretic rationalism of our choice.

Original: Initially, it is easy to see that TMECor similarly with the Coarse correlated equilibria can achieve better social welfare but as in the case of congestion games, found in literature, their price of anarchy can become signiﬁcantly worse Roughgarden (2009).
Revised: Initially, it is easy to see that TMECor similarly with the Coarse correlated equilibria can achieve better social welfare but as in the case of congestion games, found in literature, their price of anarchy can become signiﬁcantly worse Roughgarden (2009).

Original: In order to give a better result, TMECor requests ana priori knowledge of the game.
Revised: In order to give a better result, TMECor requests ana priori knowledge of the game.

Original: For example, as Zhang et al.
Revised: For example, as Zhang et al.

Original: (2020) mentioned:
Revised: (2020) mentioned:

Original: For example, in multiplayer poker games, a team may play against an adversary player, but they cannot communicate and discuss their strategy during the game due to the rule.
Revised: For example, in multiplayer poker games, a team may play against an adversary player, but they cannot communicate and discuss their strategy during the game due to the rule.

Original: However in many nowadays challenging AI models, the agents/players of each team do not necessarily have knowledge of the actual game.
Revised: However in many nowadays challenging AI models, the agents/players of each team do not necessarily have knowledge of the actual game.

Original: Thus, the team players can only decide in advance only about their own dynamics.
Revised: Thus, the team players can only decide in advance only about their own dynamics.

Original: Thus, the main conceptual reason that we focus on NE per player in comparison with the aforementioned TME relaxations is to study the interplay between the individual and team incentives in competitive tasks.
Revised: Thus, the main conceptual reason that we focus on NE per player in comparison with the aforementioned TME relaxations is to study the interplay between the individual and team incentives in competitive tasks.

Original: It is also a way of reasoning about games in which players do not necessarily need to know in which team they fall into in advance.
Revised: It is also a way of reasoning about games in which players do not necessarily need to know in which team they fall into in advance.

Original: Problems of this kind are prevalent in modern ML applications like strategic conﬂict resolution Leibo et al.
Revised: Problems of this kind are prevalent in modern ML applications like strategic conﬂict resolution Leibo et al.

Original: (2017), coordination between autonomous vehicles Cao et al.
Revised: (2017), coordination between autonomous vehicles Cao et al.

Original: (2012) , or collaboration of agents in defensive escort teams Sheikh et al.
Revised: (2012) , or collaboration of agents in defensive escort teams Sheikh et al.

Original: In this kind of games each agent is simultaneously working towards maximizing its own payoff (local reward) as well as the collective success of the team (global reward) .
Revised: In this kind of games each agent is simultaneously working towards maximizing its own payoff (local reward) as well as the collective success of the team (global reward) .

Original: As it is stated again in Zhang et al.
Revised: As it is stated again in Zhang et al.

Original: (2020), Celli and Gatti (2018) show that the “ex ante” coordination can be modeled using a coordination device, assuming that the adversary does not observe any signal from the device.
Revised: (2020), Celli and Gatti (2018) show that the “ex ante” coordination can be modeled using a coordination device, assuming that the adversary does not observe any signal from the device.

Original: The team members agree on a planned strategy (e.g., a mixed strategy) in the planning phase, and then, just before the game starts, the coordination device randomly picks a pure joint strategy (from the planned strategy) for the team members to act upon.
Revised: The team members agree on a planned strategy (e.g., a mixed strategy) in the planning phase, and then, just before the game starts, the coordination device randomly picks a pure joint strategy (from the planned strategy) for the team members to act upon.

Original: Therefore, this TMECor interpretation would critically correspond to a high-level two-metaplayer zero-sum game, where the metaplayer is the team as a whole entity with every individual player in perfect coordination with its teammates.
Revised: Therefore, this TMECor interpretation would critically correspond to a high-level two-metaplayer zero-sum game, where the metaplayer is the team as a whole entity with every individual player in perfect coordination with its teammates.

Original: Finally, an additional reason, the notion of “per player Nash equilibria” can be seen as a smoother ﬁgure of merit for the performance of a defensive team against an adversary.
Revised: Finally, an additional reason, the notion of “per player Nash equilibria” can be seen as a smoother ﬁgure of merit for the performance of a defensive team against an adversary.

Original: Notably, in the vast majority of the aforementioned literature, a single superpowerful adversary has been used.
Revised: Notably, in the vast majority of the aforementioned literature, a single superpowerful adversary has been used.

Original: However, using a single-agent adversary can describe a universally fair model.
Revised: However, using a single-agent adversary can describe a universally fair model.

Original: By contrast, the notion of per player NE expresses the model hurdles of both simultaneous team intra-collaboration and intercompetition.
Revised: By contrast, the notion of per player NE expresses the model hurdles of both simultaneous team intra-collaboration and intercompetition.

Original: B G AME D YNAMICS
Revised: B G AME D YNAMICS

Original: The following algorithms are ubiquitous in the literature of min max optimization and have known variable success under different settings.
Revised: The following algorithms are ubiquitous in the literature of min max optimization and have known variable success under different settings.

Original: Gradient descent-ascent (GDA) is the prototypical method upon which Optimistic Gradient Descent (OGDA), Extra Gradient (EG), and our proposed method
Revised: Gradient descent-ascent (GDA) is the prototypical method upon which Optimistic Gradient Descent (OGDA), Extra Gradient (EG), and our proposed method

Original: K, P -vaned Gradient Descent-Ascent (KPV-GDA) are built.
Revised: K, P -vaned Gradient Descent-Ascent (KPV-GDA) are built.

Original: Optimistic Multiplicative Weights Update Method is the optimistic variant of the Multiplicative Weights Update Method.
Revised: Optimistic Multiplicative Weights Update Method is the optimistic variant of the Multiplicative Weights Update Method.

Original: B. G RADIENT D ESCENT -A SCENT
Revised: B. G RADIENT D ESCENT -A SCENT

Original: [Equation]
Revised: [Equation]

Original: B. O PTIMISTIC G RADIENT D ESCENT -A SCENT
Revised: B. O PTIMISTIC G RADIENT D ESCENT -A SCENT

Original: [Equation]
Revised: [Equation]

Original: B. E XTRA G RADIENT M ETHOD  
Revised: B. E XTRA G RADIENT M ETHOD  

Original: [Equation]
Revised: [Equation]

Original: k ) , y ( k ) ) (cid:111) , x ( k +1)
Revised: k ) , y ( k ) ) (cid:111) , x ( k +1)

Original: i =
Revised: i =

Original: Π X
Revised: Π X

Original: i (cid:110) x ( k ) i − η ∇ x i f ( x ( k + 12 ) , y ( k + 12 ) )
Revised: i (cid:110) x ( k ) i − η ∇ x i f ( x ( k + 12 ) , y ( k + 12 ) )

Original: (cid:111)
Revised: (cid:111)

Original: y ( k + 12 ) j
Revised: y ( k + 12 ) j

Original: = Π Y j (cid:110)
Revised: = Π Y j (cid:110)

Original: y ( k ) j + η ∇ y j f ( x ( k ) , y ( k ) )
Revised: y ( k ) j + η ∇ y j f ( x ( k ) , y ( k ) )

Original: (cid:111) , y ( k +1)
Revised: (cid:111) , y ( k +1)

Original: j = Π Y j (cid:110)
Revised: j = Π Y j (cid:110)

Original: y
Revised: y

Original: ( k ) j
Revised: ( k ) j

Original: + η ∇ y j f ( x ( k + 1 2 ) , y ( k + 1 2 ) ) (cid:111)
Revised: + η ∇ y j f ( x ( k + 1 2 ) , y ( k + 1 2 ) ) (cid:111)

Original: B. O PTIMISTIC M ULTIPLICATIVE W EIGHTS U PDATE M ETHOD
Revised: B. O PTIMISTIC M ULTIPLICATIVE W EIGHTS U PDATE M ETHOD

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation] [Table] 
Revised: [Equation] [Table] [Equation]

Original: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: Notations x ( k ) i ( or y ( k ) j ) stand for the strategy vector of the i -th minimizing (or j -th maximizing) agent at time-step k .
Revised: Notations x ( k ) i ( or y ( k ) j ) stand for the strategy vector of the i -th minimizing (or j -th maximizing) agent at time-step k .

Original: The step size is denoted as η .
Revised: The step size is denoted as η .

Original: Operators Π X
Revised: Operators Π X

Original: i , Π Y j , Π Z are the projection operators to the corresponding simplices.
Revised: i , Π Y j , Π Z are the projection operators to the corresponding simplices.

Original: The projection operator is important since the context of operation is that of constrained optimization and remaining inside the set of feasible solutions is not guaranteed at every step except for the case of the OMWU method.
Revised: The projection operator is important since the context of operation is that of constrained optimization and remaining inside the set of feasible solutions is not guaranteed at every step except for the case of the OMWU method.

Original: C D ERIVATION OF THE M IN -M AX O BJECTIVE IN EQUATION 2. 
Revised: C D ERIVATION OF THE M IN -M AX O BJECTIVE IN EQUATION 2.

Original: By the deﬁnition of Variance, we get that : Var (cid:2)
Revised: By the deﬁnition of Variance, we get that : Var (cid:2)

Original: x
Revised: x

Original: i (cid:3) =
Revised: i (cid:3) =

Original: E [ x 2 i ] − (cid:0) E [ x i ] (cid:1) 2 ⇔ E
Revised: E [ x 2 i ] − (cid:0) E [ x i ] (cid:1) 2 ⇔ E

Original: [ x 2 i ] =
Revised: [ x 2 i ] =

Original: Var (cid:2) x
Revised: Var (cid:2) x

Original: i (cid:3) + (cid:0) E
Revised: i (cid:3) + (cid:0) E

Original: [ x i ] (cid:1) 2 .
Revised: [ x i ] (cid:1) 2 .

Original: More precisely for any x ∼ N ( µ, I ) , we get E [ x 2 i ] = 1 + µ 2 i .
Revised: More precisely for any x ∼ N ( µ, I ) , we get E [ x 2 i ] = 1 + µ 2 i .

Original: Additionally, after calculations, we can get that:
Revised: Additionally, after calculations, we can get that:

Original: [Equation]
Revised: [Equation]

Original: Moreover, it is easy to check that for a mixture D ( µ 1 , σ 1 I , µ 2 , σ 2 I , π 1 , π 2 ) of two multidimensional normal distributions N ( µ 1 , σ 1 I ) , N ( µ 1 , σ 1 I ) with corresponding weights π 1 , π 2 :
Revised: Moreover, it is easy to check that for a mixture D ( µ 1 , σ 1 I , µ 2 , σ 2 I , π 1 , π 2 ) of two multidimensional normal distributions N ( µ 1 , σ 1 I ) , N ( µ 1 , σ 1 I ) with corresponding weights π 1 , π 2 :

Original: Var x ∼D [ x i ] =
Revised: Var x ∼D [ x i ] =

Original: π 1 σ 21 + π 2 σ 

Revised: π 1 σ 21 + π 2 σ

Original: + π 1 µ 21 ,i + π 2 µ 22 ,i − ( π 1 µ 1 ,i + π 2 µ 2 ,i )
Revised: + π 1 µ 21 ,i + π 2 µ 22 ,i − ( π 1 µ 1 ,i + π 2 µ 2 ,i )

Original: Speciﬁcally for the case of µ 1 = µ = − µ 2 , this equivalent with:
Revised: Speciﬁcally for the case of µ 1 = µ = − µ 2 , this equivalent with:

Original: [Equation]
Revised: [Equation]

Original: Hence:
Revised: Hence:

Original: [Equation]
Revised: [Equation]

Original: So we can apply the following manipulations in the objective (The notation [ x 2 i ] stands for the vector that has as entries the values of the vector x squared):
Revised: So we can apply the following manipulations in the objective (The notation [ x 2 i ] stands for the vector that has as entries the values of the vector x squared):

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: C.
Revised: C.

Original: [Equation]
Revised: [Equation]

Original: P ROOF OF T HEOREM 3.
Revised: P ROOF OF T HEOREM 3.

Original: We will reduce the problem of ﬁnding a Nash equilibrium in congestion games to the problem of ﬁnding a Nash equilibrium in two-team zero-sum games.
Revised: We will reduce the problem of ﬁnding a Nash equilibrium in congestion games to the problem of ﬁnding a Nash equilibrium in two-team zero-sum games.

Original: The result then will follow since computing Nash equilibria in congestion games is CLS-hard (Babichenko & Rubinstein (2021)).
Revised: The result then will follow since computing Nash equilibria in congestion games is CLS-hard (Babichenko & Rubinstein (2021)).

Original: As a recent result (Fearnley et al., 2021) shows CLS is equal to the intersection to PLS and PPAD, two important classes of total problems.
Revised: As a recent result (Fearnley et al., 2021) shows CLS is equal to the intersection to PLS and PPAD, two important classes of total problems.

Original: PPAD captures diverse problems in combinatorics and (non- )cooperative game theory, like the ε -approximation of a mixed Nash Equilibrium in a graphical game or the computation of market equilibria.
Revised: PPAD captures diverse problems in combinatorics and (non- )cooperative game theory, like the ε -approximation of a mixed Nash Equilibrium in a graphical game or the computation of market equilibria.

Original: PLS, for “Polynomial Local Search”, captures problems of ﬁnding a local minimum of an objective function f, in contexts where any candidate solution x has a local neighbourhood within which we can readily check for the existence of some other point having a lower value of f. Many diverse local optimization problems have been shown complete for PLS, attesting to its importance.
Revised: PLS, for “Polynomial Local Search”, captures problems of ﬁnding a local minimum of an objective function f, in contexts where any candidate solution x has a local neighbourhood within which we can readily check for the existence of some other point having a lower value of f. Many diverse local optimization problems have been shown complete for PLS, attesting to its importance.

Original: Examples include searching for a local optimum of the TSP according to the Lin-Kernighan heuristic (Papadimitriou, 1992), and ﬁnding pure Nash equilibria in many-player congestion games (Fabrikant et al., 2004).
Revised: Examples include searching for a local optimum of the TSP according to the Lin-Kernighan heuristic (Papadimitriou, 1992), and ﬁnding pure Nash equilibria in many-player congestion games (Fabrikant et al., 2004).

Original: The complexity class CLS (“Continuous Local Search”) was introduced by Daskalakis and Papadimitriou (Daskalakis & Papadimitriou, 2011) to classify various important problems that lie in both PPAD and PLS.
Revised: The complexity class CLS (“Continuous Local Search”) was introduced by Daskalakis and Papadimitriou (Daskalakis & Papadimitriou, 2011) to classify various important problems that lie in both PPAD and PLS.

Original: CLS is seen as a strong candidate for capturing the complexity of some of those important problems, like the general versions of Banach’s ﬁxed point theorem, computation of KKT points, computation of gradient descent ﬁxed points etc.
Revised: CLS is seen as a strong candidate for capturing the complexity of some of those important problems, like the general versions of Banach’s ﬁxed point theorem, computation of KKT points, computation of gradient descent ﬁxed points etc.

Original: In our reduction, a congestion game is deﬁned by the tuple ( N ; E ; ( S i ) i ∈ N ; ( c e ) e ∈ E ) where N is the set of agents , E is a set of resources (also known as edges or facilities ), and each player i has a set S i of subsets of E .
Revised: In our reduction, a congestion game is deﬁned by the tuple ( N ; E ; ( S i ) i ∈ N ; ( c e ) e ∈ E ) where N is the set of agents , E is a set of resources (also known as edges or facilities ), and each player i has a set S i of subsets of E .

Original: Each strategy s i ∈ S
Revised: Each strategy s i ∈ S

Original: i is a set of edges (a path ), and c e is a cost (negative utility) function associated with facility e .
Revised: i is a set of edges (a path ), and c e is a cost (negative utility) function associated with facility e .

Original: For a strategy proﬁle s =
Revised: For a strategy proﬁle s =

Original: ( s 1 , s 2 , . .
Revised: ( s 1 , s 2 , . .

Original: , s N ) , the cost of player i is given by c i
Revised: , s N ) , the cost of player i is given by c i

Original: [Equation]
Revised: [Equation]

Original: s
Revised: s

Original: [Equation]
Revised: [Equation]

Original: )) , where (cid:96) e ( s ) is the number of players using e in s (the load of edge e ).
Revised: )) , where (cid:96) e ( s ) is the number of players using e in s (the load of edge e ).

Original: It is a well-known result Rosenthal (1973) that congestion games exhibit a potential function Φ( s ) , that
Revised: It is a well-known result Rosenthal (1973) that congestion games exhibit a potential function Φ( s ) , that

Original: [Equation]
Revised: [Equation]

Original: with the property that if any agent i changes her strategy to s (cid:48)
Revised: with the property that if any agent i changes her strategy to s (cid:48)

Original: i it holds that Φ( s (cid:48)
Revised: i it holds that Φ( s (cid:48)

Original: i , s − i ) − Φ( s i , s − i )
Revised: i , s − i ) − Φ( s i , s − i )

Original: =
Revised: =

Original: c i ( s (cid:48)
Revised: c i ( s (cid:48)

Original: i , s − i )
Revised: i , s − i )

Original: − c
Revised: − c

Original: i
Revised: i

Original: s i , s − i ) .
Revised: s i , s − i ) .

Original: Reduction.
Revised: Reduction.

Original: Consider a congestion game ( N ; E ; ( S i ) i ∈ N ; ( c e ) e ∈ E ) with n = | N | players and potential function Φ .
Revised: Consider a congestion game ( N ; E ; ( S i ) i ∈ N ; ( c e ) e ∈ E ) with n = | N | players and potential function Φ .

Original: We deﬁne a team zero-sum game as follows:
Revised: We deﬁne a team zero-sum game as follows:

Original: Team A has n players, in which each agent i chooses strategies from S i .
Revised: Team A has n players, in which each agent i chooses strategies from S i .

Original: Team B has n players, with each agent j having only one possible choice (singleton set of actions) call it d , i.e., these are dummy players.
Revised: Team B has n players, with each agent j having only one possible choice (singleton set of actions) call it d , i.e., these are dummy players.

Original: If players from
Revised: If players from

Original: Team A choose strategy proﬁle
Revised: Team A choose strategy proﬁle

Original: s (Team B has only one choice) then they get utility u A ( s , d )
Revised: s (Team B has only one choice) then they get utility u A ( s , d )

Original: = − Φ( s ) .
Revised: = − Φ( s ) .

Original: The utility members of Team B get is u B ( s , d ) =
Revised: The utility members of Team B get is u B ( s , d ) =

Original: Φ( s ) .
Revised: Φ( s ) .

Original: Let x ∗ ≡ ( x ∗ 1 , ..., x ∗ n ) and ( d, ..., d ) be a (possibly mixed)
Revised: Let x ∗ ≡ ( x ∗ 1 , ..., x ∗ n ) and ( d, ..., d ) be a (possibly mixed)

Original: Nash equilibrium in the team zero-sum game we deﬁned.
Revised: Nash equilibrium in the team zero-sum game we deﬁned.

Original: We shall show that ( x ∗ 1 , ..., x ∗ n ) is a Nash equilibrium of the original congestion game and the reduction will be complete.
Revised: We shall show that ( x ∗ 1 , ..., x ∗ n ) is a Nash equilibrium of the original congestion game and the reduction will be complete.

Original: Aiming for contradiction, suppose ( x ∗ 1 , ..., x ∗ n ) is not a
Revised: Aiming for contradiction, suppose ( x ∗ 1 , ..., x ∗ n ) is not a

Original: Nash equilibrium of the original congestion game.
Revised: Nash equilibrium of the original congestion game.

Original: Then there exists an agent i that can deviate from strategy x ∗ i to ˜ x
Revised: Then there exists an agent i that can deviate from strategy x ∗ i to ˜ x

Original: i and decrease her expected cost.
Revised: i and decrease her expected cost.

Original: Hence we have that
Revised: Hence we have that

Original: [Equation]
Revised: [Equation]

Original: potential
Revised: potential

Original: [Equation]
Revised: [Equation]

Original: We conclude that E s ∼ x ∗ [ u A ( s , d )]
Revised: We conclude that E s ∼ x ∗ [ u A ( s , d )]

Original: [Equation]
Revised: [Equation]

Original: E s ∼ (˜ x ∗ i , x ∗− i )
Revised: E s ∼ (˜ x ∗ i , x ∗− i )

Original: [ u A ( s , d )] which is a contradiction since ( x ∗ 1 , ..., x ∗ n ) is a Nash equilibrium for the team zero-sum game hence if player i deviates, her payoff (i.e., the payoff of her Team) should not increase.
Revised: [ u A ( s , d )] which is a contradiction since ( x ∗ 1 , ..., x ∗ n ) is a Nash equilibrium for the team zero-sum game hence if player i deviates, her payoff (i.e., the payoff of her Team) should not increase.

Original: C. M ULTIPLAYER M ATCHING P ENNIES
Revised: C. M ULTIPLAYER M ATCHING P ENNIES

Original: Below, we present the exact deﬁnition of the (2x2) two-team of two-players matching pennies that we discuss in Sec
Revised: Below, we present the exact deﬁnition of the (2x2) two-team of two-players matching pennies that we discuss in Sec

Original: Team B
Revised: Team B

Original: HH
Revised: HH

Original: HT/TH
Revised: HT/TH

Original: TT
Revised: TT

Original: Team A
Revised: Team A

Original: HH − 1 ,− 1 / 2 , 1 /1 , −
Revised: HH − 1 ,− 1 / 2 , 1 /1 , −

Original: HT/TH 1 / 2 , − 1 /0 ,1 / 2 , − 1 /
Revised: HT/TH 1 / 2 , − 1 /0 ,1 / 2 , − 1 /

Original: TT 1 , −− 1 / 2 , 1 /− 1 ,
Revised: TT 1 , −− 1 / 2 , 1 /− 1 ,

Original: C. P ROOF OF T HEOREM 3. 
Revised: C. P ROOF OF T HEOREM 3.

Original: Since ( x ∗ , y ∗ ) is not weakly-stable, there exist players i, j from the same team (say B without loss of generality) and strategies k, l, l (cid:48) so that if i is forced to play k , then j ’s best response is l and that gives larger payoff than another strategy l (cid:48) in her support.
Revised: Since ( x ∗ , y ∗ ) is not weakly-stable, there exist players i, j from the same team (say B without loss of generality) and strategies k, l, l (cid:48) so that if i is forced to play k , then j ’s best response is l and that gives larger payoff than another strategy l (cid:48) in her support.

Original: Formally it holds that (by multi-linearity of U ) ( payoff if i, j choose k, l ) ∂ 2 U ( x ∗ , y ∗ )
Revised: Formally it holds that (by multi-linearity of U ) ( payoff if i, j choose k, l ) ∂ 2 U ( x ∗ , y ∗ )

Original: [Equation]
Revised: [Equation]

Original: ( payoff if i, j choose k, l (cid:48) ) , (C.1) and also ∂U ( x ∗ , y ∗ ) ∂y jl = ∂U
Revised: ( payoff if i, j choose k, l (cid:48) ) , (C.1) and also ∂U ( x ∗ , y ∗ ) ∂y jl = ∂U

Original: ( x ∗ , y ∗ )
Revised: ( x ∗ , y ∗ )

Original: ∂y jl (cid:48)
Revised: ∂y jl (cid:48)

Original: We shall show that ∇ 2 yy U ( x ∗ , y ∗ ) has a strictly positive eigenvalue tangent in the product of simplices (note that if we were working with A, we would show that ∇ 2 xx U ( x ∗ , y ∗ ) has a strictly negative eigenvalue).
Revised: We shall show that ∇ 2 yy U ( x ∗ , y ∗ ) has a strictly positive eigenvalue tangent in the product of simplices (note that if we were working with A, we would show that ∇ 2 xx U ( x ∗ , y ∗ ) has a strictly negative eigenvalue).

Original: Consider a vector of size (cid:80) i | S i | , (where | S i | is the cardinality of the strategy space of agent i in Team B ) which has
Revised: Consider a vector of size (cid:80) i | S i | , (where | S i | is the cardinality of the strategy space of agent i in Team B ) which has

Original: By multilinearity of U it follows that
Revised: By multilinearity of U it follows that

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: **) for all agents j and strategies s, s (cid:48) and the same is true for x variables (team A).
Revised: **) for all agents j and strategies s, s (cid:48) and the same is true for x variables (team A).

Original: We conclude that
Revised: We conclude that

Original: [Equation]
Revised: [Equation]

Original: Therefore ∇ 2 yy U ( x ∗ , y ∗ ) has a positive eigenvalue and as a result
Revised: Therefore ∇ 2 yy U ( x ∗ , y ∗ ) has a positive eigenvalue and as a result

Original: [Equation]
Revised: [Equation]

Original: must have a positive and a negative eigenvalue (since the trace is zero).
Revised: must have a positive and a negative eigenvalue (since the trace is zero).

Original: We consider the Jacobian of the GDA dynamics at ( x ∗ , y ∗ ) .
Revised: We consider the Jacobian of the GDA dynamics at ( x ∗ , y ∗ ) .

Original: The corresponding matrix is the following:
Revised: The corresponding matrix is the following:

Original: [Equation]
Revised: [Equation]

Original: We will show that J GDA has an eigenvalue (possible complex) with an absolute value greater than one.
Revised: We will show that J GDA has an eigenvalue (possible complex) with an absolute value greater than one.

Original: It sufﬁces to show that J GDA − I has an eigenvalue with positive real part (because then J GDA would have an eigenvalue with real part greater than 1 and hence magnitude greater than one).
Revised: It sufﬁces to show that J GDA − I has an eigenvalue with positive real part (because then J GDA would have an eigenvalue with real part greater than 1 and hence magnitude greater than one).

Original: Due to (**), we get that J GDA −
Revised: Due to (**), we get that J GDA −

Original: I has trace zero.
Revised: I has trace zero.

Original: To reach contradiction suppose that no eigenvalue of
Revised: To reach contradiction suppose that no eigenvalue of

Original: J GDA − I has positive real part, then all eigenvalues of J GDA − I should be imaginary or zero.
Revised: J GDA − I has positive real part, then all eigenvalues of J GDA − I should be imaginary or zero.

Original: But an imaginary eigenvalue (that is not zero) also results in an eigenvalue with magnitude greater than one for J GDA , therefore all eigenvalues of J GDA − I should be zero.
Revised: But an imaginary eigenvalue (that is not zero) also results in an eigenvalue with magnitude greater than one for J GDA , therefore all eigenvalues of J GDA − I should be zero.

Original: We use Ky Fan inequalities which states that the sequence (in decreasing order) of the eigenvalues of 12 ( H + H (cid:62) ) majorizes the real part of the sequence of the eigenvalues of H (see Moslehian (2011), page 4) for any matrix H .
Revised: We use Ky Fan inequalities which states that the sequence (in decreasing order) of the eigenvalues of 12 ( H + H (cid:62) ) majorizes the real part of the sequence of the eigenvalues of H (see Moslehian (2011), page 4) for any matrix H .

Original: We choose H = 1 η · ( J GDA − I ) .
Revised: We choose H = 1 η · ( J GDA − I ) .

Original: Since R has both a negative and a positive eigenvalue, we get that H has an eigenvalue with negative real part.
Revised: Since R has both a negative and a positive eigenvalue, we get that H has an eigenvalue with negative real part.

Original: The claim follows since H has trace zero, thus it should have an eigenvalue with positive real part as well.
Revised: The claim follows since H has trace zero, thus it should have an eigenvalue with positive real part as well.

Original: We conclude that J GDA has an eigenvalue with an absolute value greater than one.
Revised: We conclude that J GDA has an eigenvalue with an absolute value greater than one.

Original: Using Theorem 2.2 in Daskalakis & Panageas (2018), it occurs that the set of initial conditions so that GDA converge to ( x ∗ , y ∗ ) is of measure zero (for the particular choice of the step size).
Revised: Using Theorem 2.2 in Daskalakis & Panageas (2018), it occurs that the set of initial conditions so that GDA converge to ( x ∗ , y ∗ ) is of measure zero (for the particular choice of the step size).

Original: D P ROOF OF L EMMA 3. 
Revised: D P ROOF OF L EMMA 3.

Original: Firstly, we start with the min-max formulation of GMP game:
Revised: Firstly, we start with the min-max formulation of GMP game:

Original: [Equation]
Revised: [Equation]

Original: , 1] 2 − x 1 x 2 y 1 y 2 − (1 − x 1 )(1 − x 2 )(1 − y 1 )(1 − y 2 ) + x 1 x 2 (1 − y 1 )(1 − y 2 )+
Revised: , 1] 2 − x 1 x 2 y 1 y 2 − (1 − x 1 )(1 − x 2 )(1 − y 1 )(1 − y 2 ) + x 1 x 2 (1 − y 1 )(1 − y 2 )+

Original: + (1 − x 1 )(1 − x 2 )
Revised: + (1 − x 1 )(1 − x 2 )

Original: y 1 y 2 + ω (1 − x 1 x 2 − (1 − x 1 )(1 − x 2 ))
Revised: y 1 y 2 + ω (1 − x 1 x 2 − (1 − x 1 )(1 − x 2 ))

Original: ( y 1 y 2 + (1 − y 1 )(1 − y 2 ))
Revised: ( y 1 y 2 + (1 − y 1 )(1 − y 2 ))

Original: − − ω (1 − y 1 y 2 − (1 − y 1 )(1 − y 2 ))
Revised: − − ω (1 − y 1 y 2 − (1 − y 1 )(1 − y 2 ))

Original: ( x 1 x 2 + (1 − x 1 )(1 − x 2 )) or after simpliﬁcation, it is equivalent with
Revised: ( x 1 x 2 + (1 − x 1 )(1 − x 2 )) or after simpliﬁcation, it is equivalent with

Original: [Equation]
Revised: [Equation]

Original: Remark D.1 .
Revised: Remark D.1 .

Original: Please note that the objective in the min-max is multi-linear and the degree of each variable in every summand is at most one (total degree is 2 ).
Revised: Please note that the objective in the min-max is multi-linear and the degree of each variable in every summand is at most one (total degree is 2 ).

Original: Moreover note that due to the non convexity-concavity of the function above, the max-min is not equal to the min-max.
Revised: Moreover note that due to the non convexity-concavity of the function above, the max-min is not equal to the min-max.

Original: Let ( x ∗ 1 , x ∗ 2 , y ∗ 1 , y ∗ 2 ) be a Nash equilibrium.
Revised: Let ( x ∗ 1 , x ∗ 2 , y ∗ 1 , y ∗ 2 ) be a Nash equilibrium.

Original: Assuming x ∗ 1 , x ∗ 2 , y ∗ 1 , y ∗ 2 ∈ (0 , 1) from (D.1) and ﬁrstorder conditions we get the system of equations
Revised: Assuming x ∗ 1 , x ∗ 2 , y ∗ 1 , y ∗ 2 ∈ (0 , 1) from (D.1) and ﬁrstorder conditions we get the system of equations

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Combining the ﬁrst two equations, we have x ∗ 1 = x ∗ 2 and combining the last two it follows y ∗ 

Revised: Combining the ﬁrst two equations, we have x ∗ 1 = x ∗ 2 and combining the last two it follows y ∗

Original: = y ∗ 2 .
Revised: = y ∗ 2 .

Original: Dividing equation one by ω and subtracting three, it follows that 1 ω − 2 y ∗ 1 ω + ω − 2 ωy ∗ 1 = 0 .
Revised: Dividing equation one by ω and subtracting three, it follows that 1 ω − 2 y ∗ 1 ω + ω − 2 ωy ∗ 1 = 0 .

Original: Hence we conclude that y ∗ 1 = 12 .
Revised: Hence we conclude that y ∗ 1 = 12 .

Original: As a result y ∗ 2 = 12 and substituting in ﬁrst equation x ∗ 1 = x ∗ 2 = 12 .
Revised: As a result y ∗ 2 = 12 and substituting in ﬁrst equation x ∗ 1 = x ∗ 2 = 12 .

Original: • Assume now that x ∗ 1 = 0 and x ∗ 2 , y ∗ 1 , y ∗ 2 ∈ (0 , 1) .
Revised: • Assume now that x ∗ 1 = 0 and x ∗ 2 , y ∗ 1 , y ∗ 2 ∈ (0 , 1) .

Original: Following the same idea, now only equations 2, 3, 4 hold and instead of the ﬁrst we have the constraint ω +1 − y ∗ 1 − y ∗ 2 − 2 ωx ∗ 2 ≥ 0 .
Revised: Following the same idea, now only equations 2, 3, 4 hold and instead of the ﬁrst we have the constraint ω +1 − y ∗ 1 − y ∗ 2 − 2 ωx ∗ 2 ≥ 0 .

Original: From 3, 4 we conclude that y ∗ 1 =
Revised: From 3, 4 we conclude that y ∗ 1 =

Original: y ∗ 2 and using 2 it holds that y ∗ 1 =
Revised: y ∗ 2 and using 2 it holds that y ∗ 1 =

Original: y ∗ 2 = 1+ ω 2 .
Revised: y ∗ 2 = 1+ ω 2 .

Original: Using 3 follows that 1 − ω − x ∗ 2 + ω ( ω + 1) = 0 .
Revised: Using 3 follows that 1 − ω − x ∗ 2 + ω ( ω + 1) = 0 .

Original: Thus x ∗ 2 = 1 + ω 2 > 1 (this is not possible because x ∗ 2 ∈ [0 , 1] ).
Revised: Thus x ∗ 2 = 1 + ω 2 > 1 (this is not possible because x ∗ 2 ∈ [0 , 1] ).

Original: • Consider the case
Revised: • Consider the case

Original: that x ∗ 1 = 1 and x ∗ 2 , y ∗ 1 , y ∗ 2 ∈ (0 , 1) .
Revised: that x ∗ 1 = 1 and x ∗ 2 , y ∗ 1 , y ∗ 2 ∈ (0 , 1) .

Original: Only equations 2, 3, 4 hold and instead of the ﬁrst we have the constraint ω
Revised: Only equations 2, 3, 4 hold and instead of the ﬁrst we have the constraint ω

Original: + 1 − y ∗ 1 − y ∗ 2 − 2 ωx ∗ 2 ≤ 0 .
Revised: + 1 − y ∗ 1 − y ∗ 2 − 2 ωx ∗ 2 ≤ 0 .

Original: From 3, 4 we conclude that y ∗ 1 =
Revised: From 3, 4 we conclude that y ∗ 1 =

Original: y ∗ 2 and using 2 it holds that y ∗ 1 =
Revised: y ∗ 2 and using 2 it holds that y ∗ 1 =

Original: y ∗ 2 = 1 − ω 2 .
Revised: y ∗ 2 = 1 − ω 2 .

Original: Using 3 follows that − ω − x ∗ 2 + ω (1 − ω ) = 0 .
Revised: Using 3 follows that − ω − x ∗ 2 + ω (1 − ω ) = 0 .

Original: Thus x ∗ 2 = − ω 2 < 

Revised: Thus x ∗ 2 = − ω 2 <

Original: (this is not possible because x ∗ 2 ∈ [0 , 1] ).
Revised: (this is not possible because x ∗ 2 ∈ [0 , 1] ).

Original: By symmetry the same happens when x ∗ 2 = 0 or x ∗ 2 = 1 and x ∗ 1 , y ∗ 1 , y ∗ 2 ∈ (0 , 1) .
Revised: By symmetry the same happens when x ∗ 2 = 0 or x ∗ 2 = 1 and x ∗ 1 , y ∗ 1 , y ∗ 2 ∈ (0 , 1) .

Original: • Case x ∗ 1 = x ∗ 2 = 0 and y ∗ 1 , y ∗ 2 ∈ (0 , 1) .
Revised: • Case x ∗ 1 = x ∗ 2 = 0 and y ∗ 1 , y ∗ 2 ∈ (0 , 1) .

Original: Using 3, 4 we get y ∗ 

Revised: Using 3, 4 we get y ∗

Original: = y ∗ 2 = ω − 1 2 ω < 

Revised: = y ∗ 2 = ω − 1 2 ω <

Original: (this is not possible).
Revised: (this is not possible).

Original: • Case x ∗ 1 = x ∗ 2 = 1 and y ∗ 1 , y ∗ 2 ∈ (0 , 1) .
Revised: • Case x ∗ 1 = x ∗ 2 = 1 and y ∗ 1 , y ∗ 2 ∈ (0 , 1) .

Original: Using 3, 4 we get y ∗ 

Revised: Using 3, 4 we get y ∗

Original: = y ∗ 2 = ω
Revised: = y ∗ 2 = ω

Original: +12 ω > 1 (this is not possible).
Revised: +12 ω > 1 (this is not possible).

Original: • Case x ∗ 1 = 0 and x ∗ 2 = 

Revised: • Case x ∗ 1 = 0 and x ∗ 2 =

Original: and y ∗ 1 , y ∗ 2 ∈ (0 , 1) .
Revised: and y ∗ 1 , y ∗ 2 ∈ (0 , 1) .

Original: Using 3, 4 we get y ∗ 

Revised: Using 3, 4 we get y ∗

Original: = y ∗ 2 = 12 .
Revised: = y ∗ 2 = 12 .

Original: Moreover one becomes ω − 2 ωx ∗ 2 ≥ 0 and two ω − 2 ωx ∗ 1 ≤ 0 , that is x ∗ 1 ≥ 12 and x ∗ 2 ≤(contradiction).
Revised: Moreover one becomes ω − 2 ωx ∗ 2 ≥ 0 and two ω − 2 ωx ∗ 1 ≤ 0 , that is x ∗ 1 ≥ 12 and x ∗ 2 ≤(contradiction).

Original: The case x ∗ 1 = 1 and x ∗ 2 = 0 and y ∗ 1 , y ∗ 2 ∈ (0 , 1) is symmetric.
Revised: The case x ∗ 1 = 1 and x ∗ 2 = 0 and y ∗ 1 , y ∗ 2 ∈ (0 , 1) is symmetric.

Original: Similarly one can consider the case where the y team plays pure and x ∗ 1 , x ∗ 2 ∈ (0 , 1) .
Revised: Similarly one can consider the case where the y team plays pure and x ∗ 1 , x ∗ 2 ∈ (0 , 1) .

Original: One can also check that all possible pure strategy proﬁles are not Nash equilibria.
Revised: One can also check that all possible pure strategy proﬁles are not Nash equilibria.

Original: E P ROOF OF T HEOREM 3. 
Revised: E P ROOF OF T HEOREM 3.

Original: We split the proof in 3 parts.
Revised: We split the proof in 3 parts.

Original: Before we start the proof, note that the Hessian of U (E.4) has inﬁnity norm less than 4 (since ω ∈ (0 , 1) ), so U has gradient Lipschitz with L ≤ 4 .
Revised: Before we start the proof, note that the Hessian of U (E.4) has inﬁnity norm less than 4 (since ω ∈ (0 , 1) ), so U has gradient Lipschitz with L ≤ 4 .

Original: Thus for the rest of the proof for GDA, we choose η OGDA < 14 .
Revised: Thus for the rest of the proof for GDA, we choose η OGDA < 14 .

Original: GDA.
Revised: GDA.

Original: For GDA the proof will be straightforward.
Revised: For GDA the proof will be straightforward.

Original: We will show that ( x ∗ 1 , x ∗ 2 , y ∗ 1 , y ∗ 2 ) is a weakly Nash equilibrium.
Revised: We will show that ( x ∗ 1 , x ∗ 2 , y ∗ 1 , y ∗ 2 ) is a weakly Nash equilibrium.

Original: Then the claim about GDA will follow because of Lemma 3.3, Theorem, 3.2 and Remark 2.2.
Revised: Then the claim about GDA will follow because of Lemma 3.3, Theorem, 3.2 and Remark 2.2.

Original: Assume that player x 1 ﬁxes his strategy to x 1 = 0 .
Revised: Assume that player x 1 ﬁxes his strategy to x 1 = 0 .

Original: and y 1 , y 2 keep their strategy ( 12 , 12 ) .
Revised: and y 1 , y 2 keep their strategy ( 12 , 12 ) .

Original: We shall show that x 2 is not indifferent in his support and would like to change his mixed strategy x 2 = 12 to pure.
Revised: We shall show that x 2 is not indifferent in his support and would like to change his mixed strategy x 2 = 12 to pure.

Original: When x 1 = 0 and y 1 =
Revised: When x 1 = 0 and y 1 =

Original: y 2 = 12 the payoff of Team A ( x variables) becomes − ωx 2 − 1 + ω 2 .
Revised: y 2 = 12 the payoff of Team A ( x variables) becomes − ωx 2 − 1 + ω 2 .

Original: Since ω ∈ (0 , 1) , x 2 prefers to play x 2 = 0 (instead of 12 she had).
Revised: Since ω ∈ (0 , 1) , x 2 prefers to play x 2 = 0 (instead of 12 she had).

Original: We conclude that ( 12 , 12 , 12 , 12 ) is not a weakly-stable Nash equilibrium.
Revised: We conclude that ( 12 , 12 , 12 , 12 ) is not a weakly-stable Nash equilibrium.

Original: OGDA.
Revised: OGDA.

Original: The Jacobian of the update rule of OGDA dynamics (use the same machinery of
Revised: The Jacobian of the update rule of OGDA dynamics (use the same machinery of

Original: Section 3 in Daskalakis & Panageas (2018)) is the following:
Revised: Section 3 in Daskalakis & Panageas (2018)) is the following:

Original: [Equation]
Revised: [Equation]

Original: where U is the payoff of Team B (max) and − U is the payoff of team A .
Revised: where U is the payoff of Team B (max) and − U is the payoff of team A .

Original: Substituting for MPG payoff at Nash equilibrium, we have
Revised: Substituting for MPG payoff at Nash equilibrium, we have

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: −
Revised: −

Original: [Equation]
Revised: [Equation]

Original: −
Revised: −

Original: [Equation]
Revised: [Equation]

Original: The corresponding Jacobian matrix becomes:
Revised: The corresponding Jacobian matrix becomes:

Original: J OGDA =  
Revised: J OGDA =  

Original: I 0 

Revised: I 0

Original: I 

Revised: I

Original: I 0 
 I 0 
Revised: I 0 
 I 0 

Original: [Equation]
Revised: [Equation]

Original: OGDA   4 ω   − 2 ω −−4 ω   − 2 ω
Revised: OGDA   4 ω   − 2 ω −−4 ω   − 2 ω

Original: −−− 2 − 4 ω   − 2 ω − 2 − 2 4 ω   − 2 ω                                   , (E.2)
Revised: −−− 2 − 4 ω   − 2 ω − 2 − 2 4 ω   − 2 ω                                   , (E.2)

Original: It turns out that matrix (E.2) has characteristic polynomial (help of Mathematica) π
Revised: It turns out that matrix (E.2) has characteristic polynomial (help of Mathematica) π

Original: [Equation]
Revised: [Equation]

Original: E.
Revised: E.

Original: [Equation]
Revised: [Equation]

Original: A root of π ( λ ) is 12 (cid:16)
Revised: A root of π ( λ ) is 12 (cid:16)

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: (cid:17) , which is in absolute value greater than one for 0 < η OGDA ≤ ω
Revised: (cid:17) , which is in absolute value greater than one for 0 < η OGDA ≤ ω

Original: It is also easy to see the Hessian of
Revised: It is also easy to see the Hessian of

Original: U , that is:
Revised: U , that is:

Original: [Equation]
Revised: [Equation]

Original: is invertible for ω ∈ (0 , 1) (this is true as the eigenvalues are − 2 ω, 2 ω, − 2 √ 1 + ω 2 , 2 √ 1 + ω 2 , non of which is zero).
Revised: is invertible for ω ∈ (0 , 1) (this is true as the eigenvalues are − 2 ω, 2 ω, − 2 √ 1 + ω 2 , 2 √ 1 + ω 2 , non of which is zero).

Original: From Theorem 3.2 in Daskalakis & Panageas (2018) follows that the initial conditions so that OGDA converges to the Nash equilibrium is of measure zero for step size η OGDA < 12 L ≤ 18 (where L is the Lipschitz constant of ∇ U ).
Revised: From Theorem 3.2 in Daskalakis & Panageas (2018) follows that the initial conditions so that OGDA converges to the Nash equilibrium is of measure zero for step size η OGDA < 12 L ≤ 18 (where L is the Lipschitz constant of ∇ U ).

Original: We choose η OGDA ≤ min( 18 , ω ) and the claim follows.
Revised: We choose η OGDA ≤ min( 18 , ω ) and the claim follows.

Original: EG.
Revised: EG.

Original: The Jacobian of EG dynamics computed at the Nash equilibrium (ﬁxed point) is given below (made use of chain rule):
Revised: The Jacobian of EG dynamics computed at the Nash equilibrium (ﬁxed point) is given below (made use of chain rule):

Original: [Equation]
Revised: [Equation]

Original: We substitute with the values and we get
Revised: We substitute with the values and we get

Original: J EG
Revised: J EG

Original: [Equation]
Revised: [Equation]

Original:   + η 2 EG   4 ω 2 −−4 ω 4 ω −4 ω 2 −4 ω 4 ω − 4 ω − 4 ω − 2 + 4 ω−− 4 ω − 4 ω
Revised:   + η 2 EG   4 ω 2 −−4 ω 4 ω −4 ω 2 −4 ω 4 ω − 4 ω − 4 ω − 2 + 4 ω−− 4 ω − 4 ω

Original: −− 2 + 4 ω  .
Revised: −− 2 + 4 ω  .

Original: (E.6)
Revised: (E.6)

Original: The eigenvalues of J EG are 1 + η EG ( − 2 ω
Revised: The eigenvalues of J EG are 1 + η EG ( − 2 ω

Original: + 4 η EG ω 2 ) , 1 + η EG ( − 2 ω + 4 η EG ω 2 ) , 1 + η EG (cid:16) − 4 η EG + 2 ω + 4 η EG ω 2 − 2 (cid:112) − 1 − 8 η EG ω − 16 η 2 EG ω 2 (cid:17) , and 1 + η EG (cid:16) − 4 η EG + 2 ω + 4 η EG ω 2 + 2 (cid:112) − 1 − 8 η EG ω − 16 η 2 EG ω 2 (cid:17) .
Revised: + 4 η EG ω 2 ) , 1 + η EG ( − 2 ω + 4 η EG ω 2 ) , 1 + η EG (cid:16) − 4 η EG + 2 ω + 4 η EG ω 2 − 2 (cid:112) − 1 − 8 η EG ω − 16 η 2 EG ω 2 (cid:17) , and 1 + η EG (cid:16) − 4 η EG + 2 ω + 4 η EG ω 2 + 2 (cid:112) − 1 − 8 η EG ω − 16 η 2 EG ω 2 (cid:17) .

Original: If − 4 η EG + 2 ω + 4 η EG ω 2 >then J EG will have an eigenvalue with absolute value greater than one.
Revised: If − 4 η EG + 2 ω + 4 η EG ω 2 >then J EG will have an eigenvalue with absolute value greater than one.

Original: A sufﬁcient condition for that is η EG ≤ ω 2 .
Revised: A sufﬁcient condition for that is η EG ≤ ω 2 .

Original: Finally, for the same choice of stepsizes we have that J EG is invertible for all ( x , y ) , hence the EG dynamics is a local diffeomorphism.
Revised: Finally, for the same choice of stepsizes we have that J EG is invertible for all ( x , y ) , hence the EG dynamics is a local diffeomorphism.

Original: From Theorem 2 in arxiv version of Lee et al.
Revised: From Theorem 2 in arxiv version of Lee et al.

Original: (2019) the claim for EG method follows.
Revised: (2019) the claim for EG method follows.

Original: OMWU.
Revised: OMWU.

Original: The Jacobian of the update rule of OMWU dynamics (use the same machinery of Section 3 in Daskalakis & Panageas (2019)) is the following:
Revised: The Jacobian of the update rule of OMWU dynamics (use the same machinery of Section 3 in Daskalakis & Panageas (2019)) is the following:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original:  
Revised:  

Original: [Equation]
Revised: [Equation]

Original: E.
Revised: E.

Original: [Equation]
Revised: [Equation]

Original: It turns out that matrix (E.7) has characteristic polynomial (help of Mathematica)
Revised: It turns out that matrix (E.7) has characteristic polynomial (help of Mathematica)

Original: [Equation]
Revised: [Equation]

Original: One root of the polynomial π ( λ ) above is 12 (1 + 2 ηj + ηω +
Revised: One root of the polynomial π ( λ ) above is 12 (1 + 2 ηj + ηω +

Original: [Equation]
Revised: [Equation]

Original: 1 − 4 η 2 + 4 η 2 ωj + η 2 ω 2 ) .
Revised: 1 − 4 η 2 + 4 η 2 ωj + η 2 ω 2 ) .

Original: As in the analysis of OGDA, it turns out that for 0 < η < ω the aforementioned root has absolute value greater than one.
Revised: As in the analysis of OGDA, it turns out that for 0 < η < ω the aforementioned root has absolute value greater than one.

Original: Therefore we conclude that the Nash equilibrium where each agent plays ( 12 , 12 ) is repelling (and thus the fact that the set of initial conditions so that OMWU converges to that particular ﬁxed point is of measure zero is derived from standard arguments from Lee et al.
Revised: Therefore we conclude that the Nash equilibrium where each agent plays ( 12 , 12 ) is repelling (and thus the fact that the set of initial conditions so that OMWU converges to that particular ﬁxed point is of measure zero is derived from standard arguments from Lee et al.

Original: To conclude the proof we need to exclude that OMWU will stabilize in other ﬁxed points.
Revised: To conclude the proof we need to exclude that OMWU will stabilize in other ﬁxed points.

Original: Note that OMWU can stabilize to points that are not Nash equilibria (e.g., all pure strategy proﬁles are ﬁxed points).
Revised: Note that OMWU can stabilize to points that are not Nash equilibria (e.g., all pure strategy proﬁles are ﬁxed points).

Original: To exclude such stabilization, the trick is that if OMWU stabilizes to a point, it should be (approximate coarse correlated equilibrium where the approximation depends on the size of the stepsize).
Revised: To exclude such stabilization, the trick is that if OMWU stabilizes to a point, it should be (approximate coarse correlated equilibrium where the approximation depends on the size of the stepsize).

Original: However, none of the pure strategy proﬁles is a ω/ 2 - approximate coarse correlated equilibrium, so OMWU does not stabilize.
Revised: However, none of the pure strategy proﬁles is a ω/ 2 - approximate coarse correlated equilibrium, so OMWU does not stabilize.

Original: Hence if we choose 0 < η < ω/ 2 , OMWU does not stabilize.
Revised: Hence if we choose 0 < η < ω/ 2 , OMWU does not stabilize.

Original: F P ROOF OF T HEOREM 3. 
Revised: F P ROOF OF T HEOREM 3.

Original: We ﬁrst compute the Jacobian of KPV-GDA dynamics (2.2) where we have eliminated one variable per agent (to avoid using the projection operator).
Revised: We ﬁrst compute the Jacobian of KPV-GDA dynamics (2.2) where we have eliminated one variable per agent (to avoid using the projection operator).

Original: The Jacobian has the following form:
Revised: The Jacobian has the following form:

Original: [Equation]
Revised: [Equation]

Original: It sufﬁces to show that the spectral radius of J KPV-GDA is less than one for step size η small enough.
Revised: It sufﬁces to show that the spectral radius of J KPV-GDA is less than one for step size η small enough.

Original: We ﬁrst show the claim below
Revised: We ﬁrst show the claim below

Original: Claim F.1.
Revised: Claim F.1.

Original: Let J
Revised: Let J

Original: = I + ηM where I is the identity matrix.
Revised: = I + ηM where I is the identity matrix.

Original: Suppose that M has all of its eigenvalues with a real part that is negative.
Revised: Suppose that M has all of its eigenvalues with a real part that is negative.

Original: Then, there exists an interval (0 , η 0 ) so that if η ∈ (0 , η 0 ) then J has all of its eigenvalues with an absolute value less than one.
Revised: Then, there exists an interval (0 , η 0 ) so that if η ∈ (0 , η 0 ) then J has all of its eigenvalues with an absolute value less than one.

Original: Proof.
Revised: Proof.

Original: Let a i
Revised: Let a i

Original: + b i · j be an eigenvalue of M such that a i < 0 .
Revised: + b i · j be an eigenvalue of M such that a i < 0 .

Original: Assume that η < − 2 a i a 2 i + b 2 i , then the corresponding eigenvalue of J is 1 + η ( a i + b i · j ) , the magnitude of which is (1 + ηa i ) 2 + η 2 b 2 i = 1 + 

Revised: Assume that η < − 2 a i a 2 i + b 2 i , then the corresponding eigenvalue of J is 1 + η ( a i + b i · j ) , the magnitude of which is (1 + ηa i ) 2 + η 2 b 2 i = 1 +

Original: ηa
Revised: ηa

Original: i + η 2 ( a 2 i + b 2 i ) < 1 since − 2 a i > η ( a 2 i + b 2 i ) (by assumption).
Revised: i + η 2 ( a 2 i + b 2 i ) < 1 since − 2 a i > η ( a 2 i + b 2 i ) (by assumption).

Original: Hence we can choose η 0 to be min i − 2 a i a 2 i + b 2 i > 0 .
Revised: Hence we can choose η 0 to be min i − 2 a i a 2 i + b 2 i > 0 .

Original: Using Claim F.1, we conclude that as long as the matrix M below has eigenvalues with real part negative then ( x ∗ , y ∗ ) is attracting:
Revised: Using Claim F.1, we conclude that as long as the matrix M below has eigenvalues with real part negative then ( x ∗ , y ∗ ) is attracting:

Original: [Equation]
Revised: [Equation]

Original: By setting M 11 , M 12 , M 21 , M 22 as the corresponding block matrices we compute the characteristic polynomial of M .
Revised: By setting M 11 , M 12 , M 21 , M 22 as the corresponding block matrices we compute the characteristic polynomial of M .

Original: It holds
Revised: It holds

Original: [Equation]
Revised: [Equation]

Original: det( M − λ I ) has roots at λ = − p and when
Revised: det( M − λ I ) has roots at λ = − p and when

Original: [Equation]
Revised: [Equation]

Original: Therefore, if M has all eigenvalues with real part negative, we must have p > 0 .
Revised: Therefore, if M has all eigenvalues with real part negative, we must have p > 0 .

Original: Let ρ be an eigenvalue of
Revised: Let ρ be an eigenvalue of

Original: [Equation]
Revised: [Equation]

Original: It must hold that
Revised: It must hold that

Original: [Equation]
Revised: [Equation]

Original: We need to investigate under what assumptions λ will have real part negative.
Revised: We need to investigate under what assumptions λ will have real part negative.

Original: We expand (F.6) and we get
Revised: We expand (F.6) and we get

Original: [Equation]
Revised: [Equation]

Original: We solve Equation (F.7) and we get
Revised: We solve Equation (F.7) and we get

Original: [Equation]
Revised: [Equation]

Original: We provide sufﬁcient conditions so that Re ( λ 1 , 2 ) < 0 .
Revised: We provide sufﬁcient conditions so that Re ( λ 1 , 2 ) < 0 .

Original: We consider the following cases:
Revised: We consider the following cases:

Original: • ρ is real and negative.
Revised: • ρ is real and negative.

Original: In this case observe that for p = 

Revised: In this case observe that for p =

Original: we have that λ 1 , 

Revised: we have that λ 1 ,

Original: =
Revised: =

Original: k
Revised: k

Original: + ρ and 0 .
Revised: + ρ and 0 .

Original: Note that for k < 0 , the ﬁrst eigenvalue has real part negative and the second eigenvalue is zero.
Revised: Note that for k < 0 , the ﬁrst eigenvalue has real part negative and the second eigenvalue is zero.

Original: We compute the ﬁrst derivative at p = 

Revised: We compute the ﬁrst derivative at p =

Original: and this gives 12 (cid:16) − 1 + − k
Revised: and this gives 12 (cid:16) − 1 + − k

Original: + ρ | k + ρ |
Revised: + ρ | k + ρ |

Original: (cid:17)
Revised: (cid:17)

Original: =
Revised: =

Original: ρ − k − ρ <
Revised: ρ − k − ρ <

Original: Therefore the λ 1 , 2 as a function of p is strictly decreasing at zero.
Revised: Therefore the λ 1 , 2 as a function of p is strictly decreasing at zero.

Original: We conclude that for p sufﬁciently small and positive, both eigenvalues of M will be real and negative.
Revised: We conclude that for p sufﬁciently small and positive, both eigenvalues of M will be real and negative.

Original: Hence if an eigenvalue of H is real and negative then for p sufﬁciently small positive and k <both eigenvalues of M will be negative.
Revised: Hence if an eigenvalue of H is real and negative then for p sufﬁciently small positive and k <both eigenvalues of M will be negative.

Original: • ρ is complex with | Im ( ρ ) | (cid:54) = 0 .
Revised: • ρ is complex with | Im ( ρ ) | (cid:54) = 0 .

Original: In this case observe that for p = 

Revised: In this case observe that for p =

Original: we have that λ 1 , 

Revised: we have that λ 1 ,

Original: =
Revised: =

Original: k
Revised: k

Original: + ρ and 0 .
Revised: + ρ and 0 .

Original: If we choose min(0 , − Re ( ρ ))
Revised: If we choose min(0 , − Re ( ρ ))

Original: > k , then the ﬁrst eigenvalue will have real part negative.
Revised: > k , then the ﬁrst eigenvalue will have real part negative.

Original: Now using the same idea as for the ﬁrst case, we compute the ﬁrst derivative of the real part of λ 1 , 2 as a function of p and we get
Revised: Now using the same idea as for the ﬁrst case, we compute the ﬁrst derivative of the real part of λ 1 , 2 as a function of p and we get

Original: [Equation]
Revised: [Equation]

Original: The expression above is negative as long as − Re ( ρ ) 2 − Im ( ρ ) 2 − k
Revised: The expression above is negative as long as − Re ( ρ ) 2 − Im ( ρ ) 2 − k

Original: Re ( ρ ) < 0 .
Revised: Re ( ρ ) < 0 .

Original: The equation is trivially satisﬁed when Re ( ρ ) < 0 since k is chosen to be negative.
Revised: The equation is trivially satisﬁed when Re ( ρ ) < 0 since k is chosen to be negative.

Original: Assume Re ( ρ ) >(if it is zero then the above is trivially true since the imaginary part is non-zero).
Revised: Assume Re ( ρ ) >(if it is zero then the above is trivially true since the imaginary part is non-zero).

Original: It occurs that the inequality above is true when k > − | ρ | 2 Re ( ρ ) .
Revised: It occurs that the inequality above is true when k > − | ρ | 2 Re ( ρ ) .

Original: Since k is chosen to be smaller than min(0 , − Re ( ρ )) , it sufﬁces to show that
Revised: Since k is chosen to be smaller than min(0 , − Re ( ρ )) , it sufﬁces to show that

Original: [Equation]
Revised: [Equation]

Original: assumptions
Revised: assumptions

Original: [Equation]
Revised: [Equation]

Original: The above is true as long as Im ( ρ ) (cid:54) = 0 .
Revised: The above is true as long as Im ( ρ ) (cid:54) = 0 .

Original: Let E be the set of eigenvalues ρ of H with real part positive (and non-zero imaginary part by assumption).
Revised: Let E be the set of eigenvalues ρ of H with real part positive (and non-zero imaginary part by assumption).

Original: There exists a choice for η, k, p , in which η, p > 0 and sufﬁciently small and k is negative and chosen to be
Revised: There exists a choice for η, k, p , in which η, p > 0 and sufﬁciently small and k is negative and chosen to be

Original: [Equation]
Revised: [Equation]

Original: G CIFAR- 
Revised: G CIFAR-

Original: CIFAR-10 is a well-established testbed for various GAN architectures.
Revised: CIFAR-10 is a well-established testbed for various GAN architectures.

Original: It contains 10 balanced classes of images of different objects.
Revised: It contains 10 balanced classes of images of different objects.

Original: Just to further illustrate the merits of multi-agent GANs, we offer a selection of images generated by WGAN-GP Arjovsky et al.
Revised: Just to further illustrate the merits of multi-agent GANs, we offer a selection of images generated by WGAN-GP Arjovsky et al.

Original: (2017) and the MGAN Hoang et al.
Revised: (2017) and the MGAN Hoang et al.

Original: In the pictures provided by the WGAN-GP we see that across iterations, the generated samples tend to cover only certain classes of the dataset while it takes longer for the samples to become realistic.
Revised: In the pictures provided by the WGAN-GP we see that across iterations, the generated samples tend to cover only certain classes of the dataset while it takes longer for the samples to become realistic.

Original: On the other hand, the MGAN architecture from an early stage provides with diverse samples that tend to be more realistic from early on.
Revised: On the other hand, the MGAN architecture from an early stage provides with diverse samples that tend to be more realistic from early on.