$First we would like to thank the reviewers for their useful and interesting comments.
Please find below our answers.

Reviewer 1:

Extending our approach to partial orders would indeed be very useful. Up to our knowledge, there are two main ways to extend Kemeny rank aggregation to partial orders:
1. Identify a partial order with the set of its linearly extended complete orders and define the distance between a permutation and a partial order as the average of the Kendall’s tau distances between the permutation and each of the linear extensions of the partial order.
2. See a partial order as a collection of pairwise comparisons and define the distance between a permutation and a partial order as the number of pairwise disagreements between the permutation and the collection.
We can adapt our approach to these two settings by extending the Kemeny embedding of a partial order in the following way:
1. The Kemeny embedding of a partial order is the mean embedding of all its linear extensions.
2. The Kemeny embedding of a partial order is the vector with coordinates: the sign of the pairwise comparison if it appears in the partial order and 0 otherwise.
In both cases, a Kemeny consensus minimizes the sum of its (extended) Kendall’s tau distances to partial orders. Our approach can then be applied in both cases with some slight modifications.

Reviewer 2:

- If we understand correctly, the proposed approach consists in: 
Compute a consensus sigma_S of the dataset for the Spearman’s Footrule distance (it seems to us however this is given by the Hungarian algorithm, while the Borda count gives the consensus under the squared Spearman’s rho distance)
Bound the distance between a permutation sigma and a Kemeny consensus sigma_K using d_K(sigma,sigma_K) <= d_K(sigma,sigma_S) + d_K(sigma_S,sigma_K)
Bound the two terms of the right-hand side using the Diaconis inequality
This would indeed be a very interesting approach. We do not see however how to perform the last step.
- It is a typo corrected by $sigma, sigma’ in Sn$

Reviewer 3:

- The case where the dataset is a composed of replicates of one single ranking is interesting but Theorem 1 still applies. Indeed, in this case the Kemeny consensus is equal to the ranking and the mean embedding of the dataset also. Let us consider that this ranking is the north pole of the sphere. Then for a permutation with Kendall tau distance to the ranking equal to m, the cosinus of the angle equal to 1 - 2m²/binom{n}{2} by the law of cosines. Theorem 1 thus would not be true if there existed a permutation with distance m >= k+1 and yet with 1 - 2m²/binom{n}{2} > sqrt(1 - (k+1)/binom{n}{2}). It is easy to see that this is not possible.
- In Table 1, we only showed one realization of each randomized voting rules (Quicksort and Pick-a-Perm). We will add an error bar taking into consideration the variance of the values returned from intrinsic algorithmic randomization.
- Yes indeed R denotes the norm of the embedded point of a single permutation.
- Intuitively speaking, conditioned on being a permutation within same distance, or equivalently having same angle, to Kemeny consensus, the worst-case tightness of the proposed bound occurs when the permutation is the one having the largest angle to phi(DN). As we deem a technical result on this aspect is not so straightforward, we devised a negative control experiment in the paper to mimic this worst-case scenario, namely Pick-a-Random where the permutation is chosen independent of DN.
- Yes indeed, PL consensus is the ranking returned by sorting the skill parameters of a PL model fitted to the data.
- We see that experiments involving only n<=5 is admittedly unavoidable due to computation burden for Kemeny consensuses but not entirely convincing when experimentally testing the tightness of the bound. As suggested by the reviewer, an interesting future work can be to analyze the bound under a probabilistic assumption, for instance establishing a relationship between the consensus of sampled permutations w.r.t. a Mallows model and the modal ranking, and study asymptotical behavior of the bound with large number of samples.