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Many peer-review processes involve reviewers submitting their independent reviews, followed by a discussion between the reviewers of each paper. A common question among policymakers is whether the reviewers of a paper should be anonymous to each other during the discussion. We shed light on this question by conducting a randomized controlled trial at the Conference on Uncertainty in Artificial Intelligence (UAI) 2022 conference where reviewer discussions were conducted over a typed forum. We randomly split the reviewers and papers into two conditions–one with anonymous discussions and the other with non-anonymous discussions. We also conduct an anonymous survey of all reviewers to understand their experience and opinions. We compare the two conditions in terms of the amount of discussion, influence of seniority on the final decisions, politeness, reviewers’ self-reported experiences and preferences. Overall, this experiment finds small, significant differences favoring the anonymous discussion setup based on the evaluation criteria considered in this work.