If the main results survive, then publish a correction and give credit to the authors of the Arxiv note appropriately. If they don't, retract the paper. Simple as that.
Kasy and Sautmann Econometrica 2021 proven wrong

The main result in K&S has three parts. As the arxiv note shows, the proofs of the first two parts are incorrect, and the third part of the theorem can be disproved by a counterexample. The arxiv note than moves on to provide a version of the K&S result that is correct and provide a proof. The credit thus needs to go to the authors of the arxiv note. At a minimum, the arxiv note should be published in Econometrica (after being vetted by a referee process, of course). Ideally, the K&S paper would be retracted.

"Imbens accepts what he wants without listening to referees."
So trueAccording to the job description, the Editor may consult referees, but he is certainly not required to do what (some junior angry) folks tell the editor what he should do (ususally junior academics are extremely beligerent on each other papers). If Editors listened to the referees, then no paper would ever be published in Econometrica. Or in the better scenarios, the 300 page papers would be going 10 rounds.

"Imbens accepts what he wants without listening to referees."
So trueAccording to the job description, the Editor may consult referees, but he is certainly not required to do what (some junior angry) folks tell the editor what he should do (ususally junior academics are extremely beligerent on each other papers). If Editors listened to the referees, then no paper would ever be published in Econometrica. Or in the better scenarios, the 300 page papers would be going 10 rounds.
Relax G;ido.
G;ido is now a banned word on EJMR!

The main result in K&S has three parts. As the arxiv note shows, the proofs of the first two parts are incorrect, and the third part of the theorem can be disproved by a counterexample. The arxiv note than moves on to provide a version of the K&S result that is correct and provide a proof. The credit thus needs to go to the authors of the arxiv note. At a minimum, the arxiv note should be published in Econometrica (after being vetted by a referee process, of course). Ideally, the K&S paper would be retracted.
Let's stipulate that, if the first KS paper had been correct, it would have been EMA worthy. (I have no way of assessing whether this is correct, just for the sake of argument.)
If the differences between the results of the corrected paper and the original are small enough that it doesn't knock the paper out of EMAworthiness, then it seems like the right thing to do would be for EMA to retract the first KS paper and publish a corrected version by KS and the arxiv note authors. They all get credit for the total contribution and the original doesn't stick around to get misused.
If the results now fall below EMA, but are still worthwhile, again retract the original EMA but publish the corrected version in QE or TE, with KS + arxiv as coauthors.

The main result in K&S has three parts. As the arxiv note shows, the proofs of the first two parts are incorrect, and the third part of the theorem can be disproved by a counterexample. The arxiv note than moves on to provide a version of the K&S result that is correct and provide a proof. The credit thus needs to go to the authors of the arxiv note. At a minimum, the arxiv note should be published in Econometrica (after being vetted by a referee process, of course). Ideally, the K&S paper would be retracted.
Let's stipulate that, if the first KS paper had been correct, it would have been EMA worthy. (I have no way of assessing whether this is correct, just for the sake of argument.)
If the differences between the results of the corrected paper and the original are small enough that it doesn't knock the paper out of EMAworthiness, then it seems like the right thing to do would be for EMA to retract the first KS paper and publish a corrected version by KS and the arxiv note authors. They all get credit for the total contribution and the original doesn't stick around to get misused.
If the results now fall below EMA, but are still worthwhile, again retract the original EMA but publish the corrected version in QE or TE, with KS + arxiv as coauthors.This is a way forward, but assumes that both groups are willing to cooperate, something that can not be enforced.

"Imbens accepts what he wants without listening to referees."
So trueAccording to the job description, the Editor may consult referees, but he is certainly not required to do what (some junior angry) folks tell the editor what he should do (ususally junior academics are extremely beligerent on each other papers). If Editors listened to the referees, then no paper would ever be published in Econometrica. Or in the better scenarios, the 300 page papers would be going 10 rounds.
Relax G;ido.
G;ido is now a banned word on EJMR!I am not G;ido, but I don't envy him in his position. More exactly, I envy his Nobel Prize, but I don't envy being the editor for the Econometrica. Number 1, you make 97% of people angry by rejecting their papers, and, Number 2, well that gets very heavy, so let me explain that. You have to deal with both type 1 and type 2 errors. Type 1 error happens when you falsely reject a null hypothesis that the submission is EMAworthy. Type 2 error happens when you accept a non EMAworth paper. The stringent control of type 1 error makes a lot of people angry. Type 2 error also makes a lot of people angry, particularly here on. Therefore their is no happy ending in any of this job.

https://maxkasy.github.io/home/files/papers/correction_adaptiveexperimentspolicy.pdf
It looks like only part 3 of Theorem 1 needs a correction. The other parts are mostly okay. The computer science guys make a big deal out of other parts being wrong (mostly that they are confused by the point that a.s. convergence implies convergence in probability).
By the way, Sannikov paper in Econometrica was also wrong  See https://onlinelibrary.wiley.com/doi/10.3982/ECTA7441, but Econometrica published a correction and people moved on. Big deal.
The CS guys writing is very condescending and "talking down" to economists. Kind of j; erks.

https://maxkasy.github.io/home/files/papers/correction_adaptiveexperimentspolicy.pdf
It looks like only part 3 of Theorem 1 needs a correction. The other parts are mostly okay. The computer science guys make a big deal out of other parts being wrong (mostly that they are confused by the point that a.s. convergence implies convergence in probability).
By the way, Sannikov paper in Econometrica was also wrong  See https://onlinelibrary.wiley.com/doi/10.3982/ECTA7441, but Econometrica published a correction and people moved on. Big deal.
The CS guys writing is very condescending and "talking down" to economists. Kind of j; erks.Triggered, Max?

Even the question is just a restated thing from computer science.
Wt; f, the basic questions com from classical stats, and even key results were not obtained by "computer scientists". For example, Russo paper was published in Operations Research. This computer science thing is big b's. Read the critique  Aromi et al don't even understand that a.s convergence implies convergence in probability. They don't even understand basic stuff like that from prob theory, because they are too arrogant to take a basic probability course from the math department, and not some "sexy probability" from computer science.
This critique is honestly is big, pompous b.s. Part (iii) is wrong, Max hecked up. People just move. Stop looking up to "Cs" as some kind of gods.

Even the question is just a restated thing from computer science.
Wt; f, the basic questions com from classical stats, and even key results were not obtained by "computer scientists". For example, Russo paper was published in Operations Research. This computer science thing is big b's. Read the critique  Aromi et al don't even understand that a.s convergence implies convergence in probability. They don't even understand basic stuff like that from prob theory, because they are too arrogant to take a basic probability course from the math department, and not some "sexy probability" from computer science.
This critique is honestly is big, pompous b.s. Part (iii) is wrong, Max hecked up. People just move. Stop looking up to "Cs" as some kind of gods.wow, Max unhinged.

https://maxkasy.github.io/home/files/papers/correction_adaptiveexperimentspolicy.pdf
It looks like only part 3 of Theorem 1 needs a correction. The other parts are mostly okay. The computer science guys make a big deal out of other parts being wrong (mostly that they are confused by the point that a.s. convergence implies convergence in probability).
By the way, Sannikov paper in Econometrica was also wrong  See https://onlinelibrary.wiley.com/doi/10.3982/ECTA7441, but Econometrica published a correction and people moved on. Big deal.
The CS guys writing is very condescending and "talking down" to economists. Kind of j; erks.In the linked ECTA, the original result still holds. I think it is a big deal, but not big enough to retract the original article. The author of the correction deserves much more than that correction, though.