Asymptotic values of zero sum repeated games : evolution equations in discrete and continuous time

We consider some discrete and continuous dynamics in a Banach space involving a non expansive operator J and a corresponding family of strictly contracting operators Φ(λ, x) := λJ( 1−λ λ x) for λ ∈]0, 1]. Our motivation comes from the study of two-player zero-sum repeated games, where the value of the n-stage game (resp. the value of the λdiscounted game) satisfies the relation vn = Φ( 1 n , vn−1) (resp. vλ = Φ(λ, vλ)) where J is the Shapley operator of the game. We study the evolution equation u′(t) = J(u(t))− u(t) as well as associated Eulerian schemes, establishing a new exponential formula and a Kobayashi-like inequality for such trajectories. We prove that the solution of the non-autonomous evolution equation u′(t) = Φ(λ(t), u(t))− u(t) has the same asymptotic behavior (even when it diverges) as the sequence vn (resp. as the family vλ) when λ(t) = 1/t (resp. when λ(t) converges slowly enough to 0). This is a short version, omitting most proofs, of the original article [30].