The Distortion of Binomial Voting Defies Expectation

Summary

The authors extend the notion of a distortion of a voting rule to a distributional setting. That is, given a distribution over utilities (that the voters have i.i.d. on the candidates), the authors replace the classic distortion with a random variable that depends on the distribution. The expected value of this random variable is the expected distortion. The authors mostly seek rules that have good expected distortion, as compared to a voting rule with the best one possible. The two biggest issues with the paper are that the assumed model is highly unrealistic and, worse yet, that it does not seem to lead to any sort of robust conclusions. The assumption that each voter can have a arbitrary (and, possibly, correlated with other voters) preference order, but all the utilities come from the same distribution (i.e., there is a common distribution for each voter and each candidate, except that there is the hidden correlation that utilities that a voter generates for the candidates must respect their preference order) is highly artificial. Then, based on this highly artificial assumption, the authors claim that (under some further, arguably mild assumptions) some specific positional scoring rules (such as m/2-approval and the binomial rule from the title) perform particularly well (however, if I understand correctly, they do not perform _objectively_ well, but simply with respect to the best possible rule, which itself may or may not be very good---I missed if this is clarified in the paper). Now m/2-approval is well understood to be a poor rule because it can be very indecisivle (indeed, if all voters agree that the ranking of the candidates is a > b > c > ... then the rule would not realize that a is the best candidate). The binomial rule is certainly better, but---as argued by the authors---is not too far off m/2-approval. So, all in all, the authors make a highly questionable assumption and conclude that some rules that do not look too attractive on the outset, optimize some criterion. This is far too weak a conclusion for a paper that expects to be competitive for NeurIPS. On the positive side, the paper certainly includes high-quality mathematics. Still, I cannot really see it as more than an incremental addition to the distortion literature (I mean incrementality in terms of conceptual contribution, which---I guess---the authors wanted to stress; on the technical level the paper is far from being incremental).

Strengths

The mathematics behind the results is certainly appealing.

Weaknesses

The underlying assumption of drawing utilities IID is highly unnatural (especially when merged with the assumption that the utilities still have to respect submitted preference orders, which creates a weird form of correlation). The conclusions of the paper do not seem to have much practical meaning.

Questions

Q1: You write that "the metric assumption is arguably difficult to justify in most domains of interest". Could you provide the arguments that you have in mind? Q2: Does your research lead to high-level conclusions beyond "this rule performs well in our setting?". The only thing I could think of is that---perhaps---using some sort of family of distributions that either are IID as yours, or become highly correlated (between the candidates) perhaps one could show that the closer we are to the IID setting, the more natural it is to use positional scoring rules, whereas the more highly correlated are the utilities the better are Condorcet-consistent rules. However, this just a guess and I am not sure how such a result could be obtained. However, generally, resolving the argument between Borda and Condorcet based on the correlations between voters' utilities would certainly be something I would be far more willing to recommend for NeurIPS.

Rating

3: Reject: For instance, a paper with technical flaws, weak evaluation, inadequate reproducibility and incompletely addressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

3 good

Contribution

2 fair

Limitations

The authors admit that their assumption is highly unrealistic, but argue that others make similar assumptions. I understand it is tempting to say so, but given that their work does not seem to give truly valuable conclusions, I think the argument is too weak.

Summary

This paper studies the expected distortion of deterministic voting rules in a distributional setting proposed by Boutilier et al. in 2015. The setting assumes each voter has a random utility for each alternative drawn i.i.d. from a distribution $D$. Then for a given preference profile $\sigma$, the expected welfare of a voting rule is defined as the expectation of winner's total utility conditioned on utilities are consistent with the preference profile $\sigma$. The expected distortion of a voting rule $f$ for a fixed $\sigma$: $ddist(f, \sigma)$ is defined as the expected ratio between total utility of the winner determined by the voting rule and the total utility of the candidate with the highest utility. Note that although utilities are i.i.d. from a distribution, the preference profile can be arbitrary and considered in the worst case. The first contribution is the authors' demonstration that the majority voting rule results in optimal expected distortion in cases where only two candidates are present. This finding aligns with extensive literature suggesting that majority (or plurality) is the sole "reasonable" voting protocol when faced with a two-candidate scenario. The paper also explores the connections between expected social welfare and expected distortion. By some standard concentration bounds, the author prove that any voting rule that approximately maximizes expected social welfare also approximately achieves optimal distortion under mild conditions on the distribution. Finally, Boutilier et al. [2015] identifies the optimal voting rule in terms of expected welfare. However, this optimal voting rule requires the knowledge of the utility distribution. The main contribution of this paper is that they propose a class of positional voting rules called binomial voting that has good guarantees on expected welfare/distortion for a wide range of distributions. In particular (generalized) binomial voting require no (or limited) information about the distribution.

Strengths

- The distributional model considered in the paper is reasonable. Instead of resorting to questionable random ranking assumptions such as impartial culture, this paper only makes distributional assumptions for utilities but not preferences. - The new definition expected distortion under this model is well motivated and the authors provide diverse results under various assumptions on the distribution. - The binomial scoring rule proposed in the paper is novel. It is a nice result that a single voting rule achieves good expected welfare/distortion for a wide range of distributions. - This paper is very well-written.

Weaknesses

- Though preferences are considered in the worst case, the assumption that all utilities are i.i.d. seems a bit strong. - The results for $m=2$ and connections between expected welfare and expected distortion are not very surprising and techniques involved are fairly straightforward.

Questions

- Do authors know any results if utilities are still independent but from different distributions for different candidates?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

4 excellent

Contribution

3 good

Limitations

The authors addressed all limitations.

Summary

This paper studies the distortion of voting rules, which is a measure of how well a voting rule performs with respect to optimal social welfare while having access to limited information about preferences. In particular, the focus is on expected distortion where the underlying utility vectors are drawn from an arbitrary prior distribution and the goal is to design a voting rule that doesn't use any information about such prior distribution, similar in spirit to the design of prior-independent auction mechanisms. The main approach is to first design an approximate expected welfare maximizing rule (EWMR) and then derive conditions under which EWMR approximates expected distortion maximizing rule (EDMR). When there are only two alternatives ($m=2$), it turns out that the majority is rule is both an EDMR and EWMR. However, this is not true for $m>2$ and the authors consider scoring rule based methods. The authors show that approving the top half is $1/3$-approximate EWMR for all symmetric distributions. For asymmetric distributions, the authors propose a new scoring called the Binomial voting rule which is $\nu/2$-EDMR for any distribution with median $\nu$. Furthermore, it is also shown that the voting rule can be adapted to incorporate partial knowledge about the underlying distribution e.g. various quantiles.

Strengths

1. I think the notion of prior independent voting rules is quite interesting as the voting rules don't need to depend on the underlying distribution. Furthermore, the results show that prior independent voting rules can achieve constant expected distortion. 2. The connection between EWMR and EDMR voting rules is non-trivial and might turn out to be an interesting approach to designing voting rules that aim to maximize welfare. 3. Finally, the Binomial voting rule is quite interesting and I also liked the result showing that the voting rule can be adapted to incorporate additional information regarding the prior distribution.

Weaknesses

1. The main weakness of the paper is that the utilities of the voters for the $m$ items are assumed to be i.i.d. from the prior distribution. In practice, the utilities of different alternatives are often correlated and this assumption excludes distribution like Gaussian with general covariance matrix. However, the authors have mentioned the limitations in the paper. 2. I am also a bit concerned about the guarantees provided by prior-independent voting rules. Repeated auctions are quite frequent in online platforms and providing bound on performance in expectation makes sense. On the other hand, voting rules are used less frequently -- political elections happen every couple of years, and people's preferences change from one iteration to the next. In such a situation, providing instance dependent bound on distortion is more practical.

Questions

1. If the assumption of symmetric distribution is violated, can you obtain $\alpha$-EWMR for some distribution independent constant $\alpha$? I understand that Theorem 5.2 provides a negative result but the distribution in the theorem needs to depend on $n$, and $m$. Is it still the case for asymmetric distributions independent of $n$, and $m$? 2. Since the distortion of the binomial voting rule depends on the median of the prior distribution, a natural question is why median and why not some other statistic? In particular, if the median of the prior distribution is very small, should the distortion of any voting rule be small as well? Note that, mean can be large in this case as $E[X]$ can be as large as $\textrm{Median}(X) + \sqrt{\textrm{Var}(X)}$.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

The authors have addressed the limitations of the paper.

Summary

The paper studies voting rules in the context of expected distortion and expected welfare. In more detail, one of the trending topics in social choice theory is the design of voting rules with low distortion. The underlying assumption for this is that voters have utilities over the alternatives, but only report ordinal preferences. Then, the distortion quantifies the loss of social welfare (=sum of the voters utilities) caused by not knowing the cardinal utilities. To this end, one typically investigates the utility profile (and induced preference profile), where the ratio between the social welfare of the optimal alternative and the alternative chosen by the voting rule is maximal. This is clearly a worst-case measure and allows, without additional assumptions, only for rather negative results. The paper at hand therefore analyzes expected distortion: given a preference profile, the authors assume that, for each voter, the utilities for the alternative are drawn i.i.d from some distribution, and then try to find alternatives that have in expectation a high social welfare (and therefore a good distortion). Clearly, if we have knowledge about the distribution, it is theoretically possible to compute the alternative that maximizes the expected welfare. The authors thus rely on a model where the prior is not known (and we are thus not in the bayesian setting). In particular, while there is an underlying distribution, it is not known to the mechanism designer. In this setting, the authors then show that for 2 alternatives, the majority rule is the rule that optimizes the expected social welfare and the expected distortion for every underlying distribution. For more than 2 alternatives, the authors furthermore design a new rule called binomial voting (which is a scoring rule relying on the binomial coefficient for the score vector) which always chooses an alternative that has an expected welfare of at least v/2 times the expected welfare of the optimal alternative (where v is the median of the underlying, unknown distribution of utilities). For deriving this result, the authors establish a link between the maximization of expected social welfare and expected distortion. Finally, the authors also discuss several variants of their main result and the limitations of their approach.

Strengths

The paper is well-written and, despite its technical complexity, I found it easy to follow the main ideas of the paper. Furthermore, the paper opens an interesting new direction for going beyond the worst-case distortion in voting by studying the expected distortion. Since the worst-case distortion is known to be prohibitive and the new model seems reasonable, I find the approach quite attractive. Moreover, the results are interesting as they give first, clearly non-trivial bounds for this new setting. Finally, since distortion in voting is a topic that has appeared in NeurIPS before.

Weaknesses

On the negative side, it should be mentioned that many of the main claims of the paper cannot be verified based on the paper itself (as all proofs are deferred to the appendix). (I also did not read the appendix and can therefore not vouch for the correctness of the given results; the proof of Theorem 3.1, which is the only one that is (almost completely) presented in the paper seems correct). Furthermore, I find the comparison to the literature somewhat lacking. For instance, Ebadian et al. (Optimized Distortion and Proportional Fairness in Voting, 2022) suggest several voting rules with close to optimal worst-case distortion and it seems interesting or even necessary to reason why these rules are not good in the given setting. In particular, the harmonic rule by Boutilier et al. (Optimal social choice functions: A utilitarian view, 2015) is also a scoring rule and it thus seems like an interesting question whether it also has a good expected distortion. However, I have to acknowledge that the paper is already quite dense and that the discussion of these results might be too much. Feedback: 1) I am not a fan of the pun in the title. 2) I think it should be better motivated why the authors use the distribution to draw utilities. This assumption is crucial for the results, not the only way to define expected distortion, and I am not entirely sure about how convincing this model is. In particular, the assumption implies that the preference intensities of voters are (in average) similar, which may not capture reality. 3) Instead of Section 5.2 (which is certainly interesting, but rather dense and technical), one could think of introducing more detailed proof sketches or to discuss the expected distortion of known rules. In principle, I feel that this space is just not used in the most effective way (but the authors may disagree with me here, this is clearly subjective)

Questions

1) Can the authors say something about the expected distortion of known rules? This seems necessary to get some intuition about how strong the results are. 2) Similarly, can the authors say more about upper bounds on expected distortion? Currently, Theorem 5.2 rules out the existence of voting rules with constant expected distortion, but can we, e.g., hope to find a voting rule which is v-EDMR (where v is the median of the distribution) rather than v/2-EDMR? 3) The importance of the results hinges on some implicit assumption. For instance, the main result of the authors is only appealing if the median of the underlying (but unknown) distribution is not too small. Can the authors comment on whether there is some evidence that this might be the case?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

4 excellent

Contribution

4 excellent

Limitations

N/A

Summary

This work introduces the notion of the expected distortion of voting rules. The setup involves n voters and m alternatives. Each voter $i$ has a utility $u_{ij}$ for alternative j, which induces a ranking $\sigma_i$ of the alternatives, where $j$ is ranked higher than $k$ in $\sigma_i$ only if $u_{ij} \geq u_{ik}$. A deterministic voting rule $f$ only has access to the ordinal preferences profiles $\boldsymbol{\sigma}$ instead of the actual underlying utilities $\mathbf{u}$. In the standard worst-case analysis, the distortion of a voting rule is defined as the social welfare incurred by the information loss, i.e.,$$\sup_{\boldsymbol{\sigma}} \sup_{\mathbf{u}: \mathbf{u} \triangleright \boldsymbol{\sigma}} \frac{\max_{j \in A} sw(j,\mathbf{u})}{sw(f(\boldsymbol{\sigma},\mathbf{u}))}.$$ This work considers a different setting. Given a preference profile $\boldsymbol{\sigma}$, a utility vector consistent with the preference profile $\mathbf{u}$ is drawn from some distribution $D$ as follows: Each voter $i$ draws $m$ utilities i.i.d from $D$, and these $m$ utilities are assigned from highest to lowest to the $m$ alternatives according to the order of $\sigma_i$. The paper then defines the distributional distortion of a voting rule $f$ for a given preference $\boldsymbol{\sigma}$ as a random variable $$ddis(f,\boldsymbol{\sigma}) = \frac{sw(f(\boldsymbol{\sigma}), \mathbf{u})}{\max_{j \in A} sw(j,\mathbf{u})}.$$ Subsequently, the expected distortion of $f$ for a given $\boldsymbol{\sigma}$ is the expectation of this ratio, taken over the random process described above. Similarly, it defines distributional social welfare $dsw(f, \sigma) = sw(f(\boldsymbol{\sigma}), \mathbf{u})$ and expected social welfare. Unlike the worst-case distortion, the benchmarks the paper considers is agnostic to the actual underlying utility profile, but rather the best voting rules that can know only to the distribution $D$ the utilities are drawn from. The best such rules are termed EDMR (expected-distortion-maximizing-rule) and EWMR (expected-welfare-maximizing-rule) for the expected distortion and expected social welfare objective, respectively. Regarding the results, for the two-alternative case, the paper shows that the Majority rule is both EDMR and EWMR. For multiple agents, by restricting attention to distributions that are independent of $n$ and $m$, and for sufficiently large $n$ or sufficiently large $m$, the expected distortion of an EWMR is a $1-\epsilon$ approximation of the expected distribution of an EDMR. The paper then attempts to approximate the expected welfare which in terms to approximate the expected distortion of EDMR, under the stated conditions. To achieve this, they first present a negative result, demonstrating that it is impossible for a single voting rule to achieve any constant $\alpha$ approximation of the expected social welfare for all distributions supported on $[0,1]$. To overcome this negative result, the paper shows that for symmetric distributions, a scoring rule that assigns a score of 1 to alternatives ranking in the highest half position and 0 to others, and outputs the alternative with the highest score, achieves an expected welfare 1/3 of the welfare of EWMR. IT also show that no voting rule can achieve a $\sqrt{1/3}$ approximation for this setting. Another way to overcome the aforementioned negative result is by assigning a score of $\sum_{\ell = k}^{m} {m\choose\ell}$ to an alternative ranked $k$. The paper demonstrates that this smoother version of the top-half score rule is a $v/2$ approximation to the EWMR, where $v$ is the largest median.

Strengths

The paper is well organized. The high-level idea of moving beyond worst-case analysis for distortion is novel and of great interest. The observation of the connection between the optimal expected welfare and optimal expected distortion under different distributional assumptions is interesting. The distributional-independent adaptation of the scoring rule of Boutilier et al [2015] is intuitive (in a good way) and clever.

Weaknesses

I find the benchmark to be relatively weak, and the positive results are somewhat marginal. Additionally, there are some correctness issues in the proofs. To expand on the first point regarding the benchmarks, let us recall that the standard worst-case distortion measures the greatest difference between the social welfare of a voting rule that only has access to the ordinal preference and the optimal social welfare of the utilities that are consistent with the preference. This work relaxes the measure in two ways: First, the ratio is now measured in expectation, which aligns well with the idea of the paper. Second, the benchmarks are voting rules that are also agnostic to the underlying utilities but can have knowledge of the distributions from which the values are drawn (EDMR and EWMR). However, it is not clear to me why the second relaxation is needed or interesting, and no motivation is provided regarding the choice of such benchmarks. In fact, without the second relaxation, we can define the expected distortion of a voting rule for $D$ as follows: $$\inf_{\boldsymbol{\sigma}} \mathbb{E}\left[\frac{sw(f(\boldsymbol{\sigma},\mathbf{u}))}{\max_{j \in A} sw(j,\mathbf{u})}\right].$$ Results regarding the above evaluation seem to better capture the ''average-case analysis'' the paper refers to. To this end, it is unclear how well the EDMR (possibly distribution-dependent) performs with respect to the above benchmarks, which is also interesting and arguably a more natural question to tackle first before distribution-independent voting rules. Regarding the second point, please note that the positive results in section 5 are all with respect to the expected social welfare considered by Boutilier et al. [2015]. The results presented in this work are stronger since they are distribution-independent, but they are weaker since the distributions are assumed to be bounded (supported on [0,1]). As far as I know, no such restriction is needed for Boutilier et al.'s results to hold. To translate these expected social welfare results into expected distortion results, one needs to further restrict distributions to the ones that are independent with respect to both $n$ and $m$ and with sufficiently large $n$ or $m$, which weakens the positive results. Additionally, none of the bounds in section 5 are tight. There are a few incorrect arguments in the proofs, the most important one I spotted being in the proof of Lemma F1. The first sentence states that by linearity of expectation, $\mathbb{E}[dsw(j, \sigma)] = \sum^n_{i = 1} \mathbb{E}[u_{ij}] = n\mu$ for all $j$, where $\mu$ is the mean of distribution $D$. The second equality seems to be incorrect. Consider a simple example with 2 alternatives and 2 agents with $D$ being Uniform$[0,1]$, and alternative $1$ is ranked second by any $i$. Then $u_{i1}$ will be the first-order statistic out of two draws for both agents, making $\mathbb{E}[u_{ij}] = 1/3$ instead of $=\mu = 1/2$. This, in turn, makes the proof of F1 invalid. With that said, I believe the statement of Lemma F1 is correct. A possible alternative proof is as follows: since $\sum_{j} dsw(j,\sigma) = \sum_{j}\sum_{i} u_{ij} = mn\mu$, and the maximum $dsw$ is always weakly higher than the average $dsw = n\mu$. Similarly, in the proof of theorem 4.2, the claim that the social welfare of any alternative $j$ can be represented as $dsw(j,\sigma) = \sum^n_{i=1} u_{ij}$, where $u_{ij}$ are $n$ i.i.d. random variables, is also incorrect since $j$ could be ranked in different positions for different voters $i$, and $u_{ij}$s are therefore only independent, not identical. This issue is again not major, as Hoeffding’s inequality only requires the variables to be independent and bounded. However, these false claims hinder my confidence in the rest of the proofs that I did not check to the same level of detail.

Questions

Is there any results or intuition regarding how bad is the worst expected distortion for EDMR for different $D$? For example,for the special case of two alternatives, what's the worst expected distortion for majority for different $D$? It seems like all the utilities i.i.d drawn from the same distribution is need for a lost of the concentration bounds to go through. Does the results still hold if voters are drawing the values for candidates i.i.d from $D_i$? Regrading the assumption that D is supported on [0,1], it is true that for the results to go through one only need the utilities to be bounded? Or the maximum of the distribution indeed affects the guarantees?

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

4 excellent

Contribution

3 good

Limitations

The assumptions are distributions for the positive results to hold are rather restricted. The results are not tight.

Summary

This paper studies the concept of expected distortion for voting rules, where distortion measures the worst-case ratio between the maximum social welfare and the rule's welfare. Expected distortion considers the expectation over consistent utility profiles drawn from an underlying i.i.d. distribution. The paper shows majority is optimal for two alternatives. For more alternatives, expected welfare maximization approximates expected distortion maximization for large electorates or alternatives. A novel voting rule called binomial voting is proposed and shown to approximate expected welfare maximization in a distribution-independent manner. Its guarantee depends on the median of the distribution.

Strengths

Provides an interesting perspective by analyzing distortion in an average-case Bayesian setting rather than worst-case. Establishes asymptotic equivalence between expected distortion and welfare maximization. Binomial voting is intuitive and has strong guarantees dependent on the median. Approaches expected welfare maximization. Solid theoretical analysis with proofs of key hardness and approximation results. Does a good job exploring the space.

Weaknesses

Assumption of i.i.d. utilities may be unrealistic for some domains like politics. Correlated utilities more reflective of reality. The paper oversells itself. In the abstract it works for “all distributions”; in the intro, i.i.d. distributions; and then in the main body consistent utility profiles drawn where each value is drawn from some underlying i.i.d. distribution. I think I am in the minority on this, but I don’t love distortion as something to optimize. It assumes truthful voting, which is very dubious in positional voting schemes (like is suggested in the paper).

Questions

I got lost in the proof of Theorem 3.1. In particular, could not see why a coupling argument was necessary (or sufficient). I am not suggesting that the proof is wrong, rather (likely) unclear in its current condensed form (it could just be a me thing, but I am not a newbie to complex proofs). Could you help me out here?

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

4 excellent

Contribution

3 good

Limitations

NA

Thanks for your response

Dear authors, thank you for your rebuttal and answering my questions. I have read the rebuttal and other reviews as well. Overall, I think this work presents an interesting take on distortion, and would vote to accept the paper.

Update

Thanks for the response, I have no further questions.

Response to rebuttal

I certainly see your contribution as something that adds possibly useful results to the distortion literature, but I do not see it as valuable enough to be accepted for NeurIPS. In particular, I am not convinced that your results are useful beyond the distortion literature. If, indeed, you showed a rule that were appealing under a number of criteria _and_ additionally was good with respect to your distortion notion (and, better yet, the distortion view would help you in selecting among several such rules), I would be far more convinced. Q1: "We note that you are asking us to justify that something is hard to justify :-) " <-- No, I am asking you to be responsible for your words. If you write that something "arguably holds" you need to be able to give the arguments. "Weird form of correlation" <-- I see your point of view and I understand that it is better than looking at impartial culture. That said, I think it still involves a form of correlation that is hard to justify. I guess that if you want to convince readers like myself in the future (or in the revised version of the paper) then you would have to make your view as to why this unrealistic assumption makes sense (the argument that "we know it is not really realistic, but is a clear improvement over status quo and this is the best we can do for now" would be good for me, for most venues, but is not sufficient for NeurIPS in my view). All in all, your papers that two rules are good according to a criterion you invented. Should we use these rules? Should we recommend them? Why is your result important beyond the realm of distortion literature?

Thank you!

We would like to thank the authors for replying to our questions. We have no further comments or questions at this point.

Thanks

Thank you for the thoughtful rebuttal.