# Approachability in Stackelberg Stochastic Games with Vector Costs

arXiv: https://arxiv.org/abs/1411.0728 · HTML: https://mlsanity.com/paper/1411.0728/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Nov 2014.

## Built on by (most cited first, 1 of 1)

- [An Online Multiobjective Policy Gradient for Long-run Average-reward Markov Decision Process](https://mlsanity.com/paper/2511.13034/) (2025)

## Abstract (arXiv)

The notion of approachability was introduced by Blackwell [1] in the context of vector-valued repeated games. The famous Blackwell's approachability theorem prescribes a strategy for approachability, i.e., for `steering' the average cost of a given agent towards a given target set, irrespective of the strategies of the other agents. In this paper, motivated by the multi-objective optimization/decision making problems in dynamically changing environments, we address the approachability problem in Stackelberg stochastic games with vector valued cost functions. We make two main contributions. Firstly, we give a simple and computationally tractable strategy for approachability for Stackelberg stochastic games along the lines of Blackwell's. Secondly, we give a reinforcement learning algorithm for learning the approachable strategy when the transition kernel is unknown. We also recover as a by-product Blackwell's necessary and sufficient condition for approachability for convex sets in this set up and thus a complete characterization. We also give sufficient conditions for non-convex sets.