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There is widespread debate on whether to anonymize author identities in peer review. The key argument for anonymization is to mitigate bias, whereas arguments against anonymization posit various uses of author identities in the review process. The Innovations in Theoretical Computer Science (ITCS) 2023 conference adopted a middle ground by initially anonymizing the author identities from reviewers, revealing them after the reviewer had submitted their initial reviews, and allowing the reviewer to change their review subsequently. We present an analysis of the reviews pertaining to the identification and use of author identities. Our key findings are: (I) A majority of reviewers self-report not knowing and being unable to guess the authors' identities for the papers they were reviewing. (II) After the initial submission of reviews, 7.1% of reviews changed their overall merit score and 3.8% changed their self-reported reviewer expertise. (III) There is a very weak and statistically insignificant correlation of the rank of authors' affiliations with the change in overall merit; there is a weak but statistically significant correlation with respect to change in reviewer expertise. We also conducted an anonymous survey to obtain opinions from reviewers and authors. The main findings from the 200 survey responses are: (i) A vast majority of participants favor anonymizing author identities in some form. (ii) The "middle-ground" initiative of ITCS 2023 was appreciated. (iii) Detecting conflicts of interest is a challenge that needs to be addressed if author identities are anonymized. Overall, these findings support anonymization of author identities in some form (e.g., as was done in ITCS 2023), as long as there is a robust and efficient way to check conflicts of interest. [ABSTRACT FROM AUTHOR]