Attainability in Repeated Games with Vector Payoffs

Dario Bauso (1) , Ehud Lehrer (2) , Eilon Solan (2) , Xavier Venel (3)

1 DICGIM - Dipartimento di Ingegneria Chimica, Gestionale, Informatica, Meccanica [Palermo]
2 TAU - School of Mathematical Sciences [Tel Aviv]
3 CES - Centre d'économie de la Sorbonne

We introduce the concept of attainable sets of payoffs in two-player repeated games with vector payoffs. A set of payoff vectors is called attainable by a player if there is a finite horizon T such that the player can guarantee that after time T the distance between the set and the cumulative payoff is arbitrarily small, regardless of the strategy Player 2 is using. We provide a necessary and sufficient condition for the attainability of a convex set, using the concept of B-sets. We then particularize the condition to the case in which the set is a singleton, and provide some equivalent conditions. We finally characterize when all vectors are attainable.

Keywords: Attainability; continuous time; repeated games; vector-payoffs; dynamic games; approachability

Domains: Mathematics [math]; Optimization and Control [math.OC]

Dario Bauso, Ehud Lehrer, Eilon Solan, Xavier Venel. Attainability in Repeated Games with Vector Payoffs. Mathematics of Operations Research, 2015, 40 (3), pp.739-755. ⟨10.1287/moor.2014.0693⟩. ⟨hal-01302514⟩