__ Summary and Contributions__: This paper provides volume analyses of MWU and OMWU, using new techniques from dynamical systems. The new analyses provide new insights into the two learning dynamics, going beyond just zero-sum games, to coordination games. These volume analyses established two novel while negative properties of MWU for zero-sum games, and also OMWU for coordination games: extremism and unavoidability. This complementary property of the two settings also implies some kind of "no-free-lunch theorem" in learning for games.

__ Strengths__: The main strengths of the paper are the soundness of the theoretical claims, the novelty of the volume analyses for learning dynamics in games, and the significance of the results obtained via the new analyses. It is relevant to the community.

__ Weaknesses__: The only small weakness might be a bit lack of empirical studies. The current empirical study only concerns two simple examples (though illustrative enough). It might be better to include more simulation examples, e.g., for general-sum games, or how the different choices of stepsize may affect the results, etc.

__ Correctness__: The results seem mostly correct to me, though I did not check line-by-line. Empirical study is correct and clear.

__ Clarity__: The paper is in general well-written. The logic is clear, and the idea is easy to follow. I enjoy reading the draft.

__ Relation to Prior Work__: Mostly yes. It would be better to compare with the most relevant a prior work [9] with more details.

__ Reproducibility__: Yes

__ Additional Feedback__: Additional comments:
1. How may the results, particularly the volume analyses, be translated to the case with "diminishing stepsize"? What about other learning dynamics, e.g., those in [1][5][17]?
2. Typos: line 112, remove one "the"; the end of line 201, "following" -> "the following"; line 309 after system, need a "."; line 315, in "a" distributed manner; line 324, remove "the".
====================================
I have read the rebuttal, which has addressed my minor comments. Thanks for the clarifications.

__ Summary and Contributions__: The authors study continuous-time dynamical system analogues of multiplicative weights updates (MWU) and optimistic multiplicative weights updates (OMWU). They focus on zero-sum and coordination games, and use volume arguments to provide a theoretical analysis of the performance of MWU and OMWU in both of these settings. For zero-sum games, MWU exhibits exponential volume expansion and hence Lyapunov chaos with respect to the initial conditions of the learning dynamic, which gives theoretical evidence to numerical instability. On the other hand OMWU demonstrates volume contraction, which once again gives evidence towards the success of this dynamic.
The authors also provide new measures of instability in equilibrium learning dynamics which they call extremism and unavoidability. Extremism is behaviour whereby a dynamic repeatedly reaches points nearby pure equilibria, irrespective of the set of equilibria (for example the game can have a unique mixed NE), and unavoidability is a setting whereby undesirable strategy profiles (under mild constraints) cannot be avoided indefinitely.
With these definitions in hand, the authors demonstrate that for MWU in zero-sum games, the dynamic also suffers from extremism and unavoidability.
OMWU on the other hand is not ideal in all scenarios, and the authors demonstrate that performance is essentially reversed in the worst case for coordination games. They show that MWU exhibits volume contraction and OMWU exhibits exponential volume expansion (hence Lyapunov instability), as well as extremism and unavoidability.

__ Strengths__: These results seem to extend the existing literature on theoretical barriers to applying existing online learning algorithms to equilibrium computation. I am not as familiar with the techniques, but the work is relevant to the NeurIPS community, especially given increased interest in such decentralised equilibrium computation algorithms when applied to GAN training.
The authors have also addressed reviewer concerns in the rebuttal phase well. My score remains the same.

__ Weaknesses__: There could be more empirical results demonstrating the performance of MWU and OMWU side-by-side on given zero-sum and coordination games.
The authors have also addressed this concern in the rebuttal phase well. My score remains the same.

__ Correctness__: To my extent they look correct, though this is not my area of expertise.

__ Clarity__: The paper is clear, which is a strong point, given the complicated machinery used.

__ Relation to Prior Work__: This is done well.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: This paper studies the unstability of MWU and OMWU on zero-sum games and coordination games respectively through volume analysis for the dynamic processes. The unstability is indicated by two properties, extremism (visiting near pure strategies infinitely) and unavoidability (escape any set of inital points eventually). These two negative propeties are novel to define, and their proofs, especially the volume integrands, somewhat reflect inherent relationship between these two similar algorithms.

__ Strengths__: This paper provides new insights for two common game dynamic systems. The main technique, volume analysis, mainly base on the volume integrand results in [9], is novel and well-explained. The transformation from dynamics of OMWU to a backward finite-difference update for an ODE is quite interesting. As MWU and OMWU are basis for many online learning algorithms, it is good to know their flaws duiring the dynamic processes.

__ Weaknesses__: The main results, the two negative properties, do not undermine the practicalities of the algorithms at all, especially when they are used to find Nash equilibrium. MWU and OMWU are commobly applied mainly due to the convergence of their time average, while this paper is analyzing their day-to-day behaviors.

__ Correctness__: I checked most proofs and they are correct.

__ Clarity__: This paper is well written. The use of volume analysis is easy to follow, and the sketches of proofs are quite intuitive.

__ Relation to Prior Work__: It is clearly discussed how this work differs from previous contributions.

__ Reproducibility__: Yes

__ Additional Feedback__: Some phrases, "volume" in line 163, "diameter" in line 166, "game value" in line 265, are used before definitions, or never defined. \bar{C} in Theorem 3 and Lemma 4 should be defined as \inf_{(x,y)\in U}C(x,y) or \inf_{(x,y)\in V}C(x,y) (actually similar line 197, it may be better to use notation \bar{C}_S).
The discussion on RPS in Appendix C.1 is indeed comlicated. I wonder whether it is possible to explore a more general condition for the non-trivial matrix, including both conditions in Theorem 5 and RPS.
==========================
Regarding the author's response:
My minor concern has been addressed by the author's response.

__ Summary and Contributions__: The paper uses volume analysis to understand the behaviour of Multiplicative Weights Update (MWU) and its optimistic variant (OMWU) in zero-sum and coordination games. In particular, the paper:
1) Defines two properties, Extremism (where a system recurrently gets stuck near pure strategies) and Unavoidability (where a set of “bad” points cannot be avoided), and shows that MWU suffers from these properties in the case of zero-sum games.
2) Proves, on the other hand, that OMWU is Lyapunov-chaotic for coordination games, and gives an analysis (distinct from that done in earlier literature) showing that OMWU is stable on zero-sum games.

__ Strengths__: The negative result proved for OMWU in coordination games is novel and would be of interest to the NeurIPS community. The notions of Extremism and Unavoidability that the paper introduces are also sensible and useful. Of secondary importance, the paper was very well-written and could more generally be a useful introduction to the theoretical analysis of zero-sum and coordination games.

__ Weaknesses__: This was a strong paper with no major problems.

__ Correctness__: The motivation and theoretical analysis were logical and sound.

__ Clarity__: The paper was well-written and easy to follow throughout. There were no obvious notational errors or formatting problems.

__ Relation to Prior Work__: Papers published on similar topics were mentioned, and the differences between those and this paper were made clear.

__ Reproducibility__: Yes

__ Additional Feedback__: Just mentioning a few very minor typos:
L29 – most -> for most
L69 – family -> a family
L73 – phenomena -> phenomenon
L86 – mild -> a mild
L108 – thought -> thought of
L111 – condition -> conditions
L112 – game -> games
L245 – game -> games
L251 – boundary -> the boundary
L261 – same row -> the same row
# post-rebuttal edit
After reading the rebuttal I have decided to keep my original rating.