Thank you for your comments and follow-up questions. We respond to these questions below:
**Question**: *Given algorithms with input predictions and output predictions, respectively, each having an approximation ratio depending on the respective prediction error defined, how can we compare these two algorithms and their performance?*
**Our response**: Indeed, this is an excellent, deep and quite general question, that could be explored in future work. Even comparing algorithms with *different* input predictions can be challenging and interesting, and comparison may vary according to the respected prediction error. We emphasize that it is not our intention (and beyond the scope of our work) to do such a general comparison so we only do, as this reviewer noted, whenever there is existed related work, i.e. for the facility location problem and for the scheduling problem.
## Further questions:
**Our response to Question 1:** Indeed, the reviewer is right. As we mention in lines 58-59 and 69-70, one can see our model as a restriction of models that have input predictions, which makes the design of mechanisms much more challenging comparing to those models that have input predictions. The scheduling problem is an excellent example for this discrepancy. The ScaledGreedy mechanism [9] that uses the input as a prediction can achieve a constant consistency together with a linear robustness (see Remark 2, lines 363-365, and Related Work, lines 529-531). Our theorem (Theorem 5) shows a separation result. It essentially stated that VCG-based mechanisms, which is a wide class of known mechanisms, when equipped with output predictions and when requiring constant consistency, are restricted to quadratic robustness which is more limited than mechanisms with input predictions.
**Our response to Question 2:** Those lemmas concern tight results for the proposed mechanisms. However, we also provide more general lower bounds for classes of mechanisms (that satisfy some natural properties), e.g. for all VCG-based in scheduling problem (Theorem 5) and for general mechanisms for the house allocation problem (Theorem 8). We remark that providing negative results for all mechanisms, even in the standard mechanism design setting (without any sort of prediction) may be notoriously hard, and only few such results exist (see e.g. [16] for scheduling, and [19, 36] for combinatorial auctions).
**Our response to Question 3**: Yes, there exist such instances. The value of the two expressions in the inequality above depend on the value of $\hat{\rho}$, $\eta$ as a function of the output advice $\hat{a}$. For example, taking into account Lemma 5, it is possible that $\hat{\rho} = 1$, $\eta = 0$ and $\lambda$ is close to 1. The comparison of the two upper bounds is inconclusive, and this is the reason that we conduceted experiments (section B.5) in order to contribute to the better understanding of the relation between the two errors.
**Our response to Question 4**: Actually, the reason is that Maximal in Range (MIR) compute the *optimal* allocation w.r.t social welfare among a subset of allocations (the range). One can enhance this subset of allocations by plugging-in the predicted output $a^*$. Then by definition of MIR, the MIR mechanism will output $a^*$ if this happens to be the best among the subset of allocations selected. Notice that this is a *deterministic* not a randomized mechanism. Studying randomized mechanisms with predictions is an interesting future direction. However, any randomized mechanisms that outputs the prediction with constant probability cannot have the same robustness guarantee (in expectation) with the MIR mechanism we use.
**Our response to Question 5**: Yes, this is exactly what we do in the facility location problem. Take the egalitarian cost for instance. The bound provided in [2] for the Minimum Bounding Box mechanism is tight. However, we manage to provide an even more refined upper bound in Theorem 1. This bound is tight (see Lemma 1), in the sense that there is no upper bound $\hat{\rho}-\epsilon$ *strictly* lower than the one we provide. As mentioned in Remark 1, this upper bound is tight whenever the output advice is inside the minimum bounding box.
Concerning the minor comments, again thank you for catching this typo, and we will consider further defining MIR mechanisms, even though they are generally described in lines 972-974.