Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: The paper suggests using a representation similar to PCA to highlight important tradeoffs associated with different winning conditions in arbitrary games in a way that lends itself to 2D visualization.
strength_and_weaknesses: I really enjoyed the paper. To use the same wording as the authors, my research mostly focuses on the "policy problem", not on the "problem problem", so my point of view is that of a non-expert in the specific research niche of the paper. I found the writing style engaging, never boring, very approachable, and exceptionally persuasive.

For those reasons, assuming that none of the reviewers that are expert on the topic have issues with the positioning (e.g., all relevant literature has been correctly and fairly represented) and claims of the paper, I recommend acceptance of the paper.

That being said, I think that the authors should revise the paper to improve certain aspects:
- throughout the paper, a majority of the citations should have been contained in parentheses and instead are just part of the text; this is rather distracting. Please make sure you're using the \citep and \citet commands adequately.
- many plots have axes whose tick labels are straight up unreadable.
- many (all?) plots are raster images, rather than vectorial graphic. Please consider re-exporting all your plots to be vectorial.
- the bibliography section seems to be using the wrong style file. This is the only paper I'm reviewing that has underlined venues in the bibliography. Please align the bibliographic style to the rules of the conference.
- some of the bibliographic entries have wrong capitalizations: for example, I believe Go and Blotto should be spelt with capitals, to name two examples 
clarity,_quality,_novelty_and_reproducibility: The writing is high-quality, though the typesetting could definitely be improved (see above for detailed comments). There are no obvious reproducibility concerns. 
summary_of_the_review: Overall, I appreciated the high-quality writing and persuasiveness of the paper. I don't have any particular concern with the paper, or reason to believe the proposed method has obvious shortcomings that would prevent it from being useful beyond the games considered in the paper. The typesetting quality needs to be improved for the final version.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This paper introduces a method of decomposing games in low dimensional feature spaces called PTA. This method allows some characterization of game structure and allows for a general technique for visualizing data arising from competitive tasks or pairwise choice tasks. The relationships between embedding coordinates represent performance relations and elucidate the principal trade-offs responsible for cyclic competition in each game.

strength_and_weaknesses: Good visualization and coherent writing. I particularly liked the game balancing concept introduced and how we can visualize it through disk games. Similar insights are valuable for game understanding, beyond the utility function. It would be even more interesting to depart from the consideration of the reconstruction of the matrix Fc with a given accuracy and focus more on how to maximally utilize the information given by the first k disc games and capture elements of the game without the need of visualization. Higher dimensions wouldn’t allow visualization of the disc games.

Questions: 
Given a reconstruction accuracy delta, how many disc games are needed to extract the useful information presented by the experiments? (such as the attack matrix). How does one argue about the redundancy of disc games in higher dimensions when visualization and understanding by humans is not possible?
clarity,_quality,_novelty_and_reproducibility: The paper is written in a clear way. It seems that it has conceptual merits.
summary_of_the_review: The visualization aspect of disc games is insightful. However, the results are mostly empirical. Formulating the PCA analogy for game structures allows the reconstruction of the matrix Fc with minimal disc game information. However, as the authors state, the difficulty in many games is creating the cyclic component Fc altogether. It would be interesting to answer how someone can optimally use the first k linear combinations of disc games without necessarily requiring visual feedback to understand.
review_title: null

final_score: rating: 3: reject, not good enough
review_content: summary_of_the_paper: The authors study the overall structure of games. They propose the Principle Trade-off Analysis (PTA) a decomposition of two-player zero-sum game. Similar to Principal Component Analysis PTA represents a game as the weighted sum of pairs of orthogonal plane. Where each plane is associated with a strategic trade-off from a simple cyclic game. Furthermore truncation to the first planes provides a meaningful reduction of the game. The authors illustrate the usefulness of the method on two games Blotto and Pokemon. In particular in  Blotto allows to recover specific symmetries whereas in Pokemon PTA recovers clusters that naturally correspond to Pokemon
types and correctly identifies a strategic trade-off in the Pokemon generation type.
strength_and_weaknesses: Strengths:

- Understanding the game structure rather than focusing on the equilibrium computation.

- PTA reduction of a game in simple cyclic games.


Weaknesses:

- Lack of clarity.
clarity,_quality,_novelty_and_reproducibility: See Summary Of The Review
summary_of_the_review: #Review
Understanding the overall structure of a game rather that focusing only on computing equilibrium is a interesting research direction. And in particular the provided decomposition method is a valuable contribution. But the paper lacks of clarity in order to fully present the PTA method. In particular I find the presentation a bit too verbose and it could be improved with supporting precise quantitative statement and definition of the introduced quantities. It would be also useful to illustrate the different visual properties on a very simple example before jumping to the Blotto and Pokemon games. Finally the literature section could be improve by explaining what are the other methods of reductions, compare them to PTA in the experimental section.


#Specific comments:

- P2, related work: It is not very clear at this point what are the other methods and what are the differences between them and PTA.

- P3, Section 3.1: Can you detail the Helmholtz-Hodge decomposition and precise what is 'least squares ratings that evaluate the average performance of each agent'.

- P3, Section 3.1: Often we do not have access to the exact advantage F but rather samples, how does you  model can be adapted to this case?

- P3 Section 4: 'Each pair of consecutive columns' of which matrix?

- P3, Section 4: define SVD.

- P3, Section 4: It'll be easier to follow if you define precisely what are F_t and F_c.

- P3, (2): Ok but you already used the notations Q and U for the decomposition of F when you said you replaced A by F.

- P3 below (2):  Can you define what is the optimal rank 2r approximation for F_c.

- P4, Section 4: Actually you never defined properly what is PTA.

- P4, above (4): Add a comma for the lower index of F_{c,i,j}.

- P4, last paragraphs: Can you  details these properties quantitatively?

- P5, last paragraph: How do you get discrete allocation from the sample of a Dirichlet?

- P6, Blotto Example 1: At this point it is hard to understand Table 1, especially since some notations used in it are still not introduced.

- P6: What is exactly a trade-off? Is it the same thing as a disc-game?

- P6, exchange symmetry: I'm not sure I follow this part. Why such subtlety is not  treated in the previous section presenting PTA? And since generic games should not exhibit such strong symmetries why choosing this game as first example?

- P7 Second paragraph: At this point it is not completely clear for me what you plot in Figure 1. Can you detail precisely which quantity in function of what is plotted.

- P8, Section 5.2: What are the agents in this case?
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: The authors introduce principal trade-off analysis, which is in some sense an analogue of PCA for zero-sum games. They propose an algorithm for computing PTA, and run it on several games of interest, producing informative visualizations.
strength_and_weaknesses: The concept of applying a PCA-like analysis to zero-sum games is very intriguing and, to my knowledge, novel. The method presented is simple yet powerful, and creates very interesting analyses of games. I think this is a good contribution to the literature, and as such I vote to accept.

One potential avenue of improvement is to include explicitly-worked decompositions of simple zero-sum games. For example, what is the decomposition of generalized RPS (i.e., 2k+1 actions of each player; action i beats action j if (i-j) mod (2k+1) <= k) or generalized matching pennies (normal-form game, n actions per player, payoff = identity matrix)? (of course, properly symmetrized.) That may give some intuition for what the method is doing--for example, we know exactly what the equilibria should be in both cases (uniform random).

I have only a few minor questions:

1. Why decompose $F_c$ instead of directly decomposing $F$ itself? That is, why ever explicitly separate out the transitive component? The decomposition could also be used on $F$ itself, and the transitive component would naturally appear as a component of the decomposition if it is prominent enough. This is what happened in the Pokemon example, and it produced a nice-looking plot of the transitive component. It would be interesting, in my opinion, to see that component explicitly in the Blotto example.

1. Does this method have any implications for *solving* games? For example, if a straightforward method exists to solve games with low rank, one could apply that method to a low-rank approximation of a game that is otherwise too complex to be solved, and hopefully recover a good strategy in that game as well. 

1. How dependent are the embeddings $\vec y_k(i)$ on the particular choice of (low-dimensional) embedding of the strategy space? For example, in the example Blotto game, one could hypothetically simply represent each pure strategy $i$ as a basis vector $\vec e_i \in \mathbb{R}^{2926}$. Would that result in different visualizations? How different?

Post-response: Thank you for the detailed response. I thought before that this was a strong paper, and I still do. I maintain my score.

clarity,_quality,_novelty_and_reproducibility: The paper is very well written, and the idea is very novel. I include more details in the main review.

Nitpick: use \citep to get parenthetical citations.
summary_of_the_review: A solid paper, introducing a new and seemingly very powerful method of analyzing two-player zero-sum games. I vote to accept.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: This paper looks at 0-sum game that are skew-symmetric (which is *not* compulsory, unlike the authors say). Using the Schor decomposition, it is possible to "replace" the original payoff matrix by a sum of $k$ "simpler" ones (as in PCA, hence the terminology).

Then this idea is illustrated on two different games, a variant of Blotto and another game related to the Pokemon franchise.

Honestly, the pictures are beautiful. But that's it. There is not much gain of intuition in any of those games, and the choice to focus on those two is quite disputable.

If there is a real interest of this approach, then it should be demonstrated properly (and not just verbosely as one reviewer mentions) on more than two (maybe) well-chosen games. For instance, how does the equilibria of the sub- game relate to the equilibria of the original one ? I wanted to ask how the value of the game is impacted, but since the construction works for skew-symmetric game only, this is irrelevant. Can it be generalized, somehow, to a larger class of games ?

I have the feeling that there might be something behind this idea, but it has not been investigating enough to be certain of it.




justification_for_why_not_higher_score: There are absolutely no evidence that this approach  can be generalized, apart from 2 use-case quite specific.
Decision: Reject