Blackwell's Approachability with Time-Dependent Outcome Functions and Dot Products. Application to the Big Match

Blackwell's approachability is a very general sequential decision framework where a Decision Maker obtains vector-valued outcomes, and aims at the convergence of the average outcome to a given "target" set. Blackwell gave a sufficient condition for the decision maker having a strategy guaranteeing such a convergence against an adversarial environment, as well as what we now call the Blackwell's algorithm, which then ensures convergence. Blackwell's approachability has since been applied to numerous problems, in online learning and game theory, in particular. We extend this framework by allowing the outcome function and the dot product to be time-dependent. We establish a general guarantee for the natural extension to this framework of Blackwell's algorithm. In the case where the target set is an orthant, we present a family of time-dependent dot products which yields different convergence speeds for each coordinate of the average outcome. We apply this framework to the Big Match (one of the most important toy examples of stochastic games) where an $\epsilon$-uniformly optimal strategy for Player I is given by Blackwell's algorithm in a well-chosen auxiliary approachability problem.

Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture $\operatorname{maxmin}=\operatorname{lim}v_n$

Mertens [In Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986) (1987) 1528-1577 Amer. Math. Soc.] proposed two general conjectures about repeated games: the first one is that, in any two-person zero-sum repeated game, the asymptotic value exists, and the second one is that, when Player 1 is more informed than Player 2, in the long run Player 1 is able to guarantee the asymptotic value. We disprove these two long-standing conjectures by providing an example of a zero-sum repeated game with public signals and perfect observation of the actions, where the value of the $\lambda$-discounted game does not converge when $\lambda$ goes to 0. The aforementioned example involves seven states, two actions and two signals for each player. Remarkably, players observe the payoffs, and play in turn.