Attainability in Repeated Games with Vector Payoffs

TL;DR

This paper studies the concept of attainable payoff vectors in two-player repeated games with vector payoffs, characterizing conditions under which specific vectors or all vectors are attainable by one player.

Contribution

It introduces the notion of attainable sets in vector payoff games and provides characterizations for when particular vectors are attainable.

Findings

Characterization of attainable vectors in repeated games.

Conditions under which a specific vector is attainable.

Analysis of when all vectors are attainable.

Abstract

We introduce the concept of attainable sets of payoffs in two-player repeated games with vector payoffs. A set of payoff vectors is called {\em attainable} if player 1 can ensure that there is a finite horizon $T$ such that after time $T$ the distance between the set and the cumulative payoff is arbitrarily small, regardless of what strategy player 2 is using. This paper focuses on the case where the attainable set consists of one payoff vector. In this case the vector is called an attainable vector. We study properties of the set of attainable vectors, and characterize when a specific vector is attainable and when every vector is attainable.