Appendix for " Unifying Behavioral and Response Diversity for Open-ended Learning in Zero-sum Games " Table of Contents 1 A.1 Proof of Theorem 1

A.1 Proof of Theorem 1 To prove Theorem 1, we need the help of the following Lemma Lemma 1. See Proposition 7.1 in [3]. Now we can prove our Theorem 1. Proof. For games with only one step (normal-form games, functional-form games), there is only one fixed state. Therefore, the distribution of state-action is equivalent to the distribution of the action. A.2 Proof of Theorem 2 Let us restate our Theorem 2 Theorem 2. For a given empirical payoff matrix A R A.3 Proof of Theorem 3 Now let us first restate the propositions.