Online Learning in Periodic Zero-Sum Games Supplementary MaterialAppendix Organization and Contents

The organization and contents of this appendix is as follows. Following Appendix A, proofs for the theoretical results in the paper are presented in the order that they appeared. Specifically, Appendix B contains the proofs for the results presented in Section 4.1 on This includes the proofs of Proposition 1, Lemma 1, Lemma 2, and Theorem 1. Appendix C, we provide the proof of Proposition 2 from Section 4.2 on the time-average behavior of This appendix includes the proofs of Proposition 1, Lemma 1, Lemma 2, and Theorem 1. B.1 Proof of Proposition 1 This will immediately allow us to conclude the system is not Poincaré recurrent by definition. Given the previous intermediate results, Theorem 1 follows from the arguments presented in Section 3.3. This appendix includes the proofs of Lemma 3, Lemma 4, and Theorem 2. D.1 Proof of Lemma 3 Then, we show that the divergence of this vector field is zero, from which we conclude the dynamics are volume preserving by Liouville's theorem.