Regret Minimization with Adaptive Opponents in Repeated Games

Inference

Proof of Lemma K.1. Define δt(hk) = Vπ¯t(hk) − Vt(hk). Then, we have t # ∑ # αst ∑ πs(a|hk) Vπ¯s((h2:M,a)k+1) − Vs((h2:M,a)k+1) + βs δt(hk) ≤ s=1 a∈A t # ∑ # αst ∑ πs(a|hk) δs((h2:M,a)k+1) + βs = s=1 a∈A t ## 4 |A1|K3 √t # ∑ αstδsk+1 + ≤ s=1 where the last line is by our definition that δtk maxh∈HM δt(hk). Then

Hardware

Proof of Lemma K.1. Define δt(hk) = Vπ¯t(hk) − Vt(hk). Then, we have t # ∑ # αst ∑ πs(a|hk) Vπ¯s((h2:M,a)k+1) − Vs((h2:M,a)k+1) + βs δt(hk) ≤ s=1 a∈A t # ∑ # αst ∑ πs(a|hk) δs((h2:M,a)k+1) + βs = s=1 a∈A t ## 4 |A1|K3 √t # ∑ αstδsk+1 + ≤ s=1 where the last line is by our definition that δtk maxh∈HM δt(hk). Then