# Structure in the Value Function of Two-Player Zero-Sum Games of Incomplete Information

arXiv: https://arxiv.org/abs/1606.06888 · HTML: https://mlsanity.com/paper/1606.06888/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2016.

## Built on by (most cited first, 6 of 6)

- [HSVI can solve zero-sum Partially Observable Stochastic Games](https://mlsanity.com/paper/2210.14640/) (2022)
- [Abstracting Imperfect Information Away from Two-Player Zero-Sum Games](https://mlsanity.com/paper/2301.09159/) (2023)
- [Partially Observable Stochastic Games with Neural Perception Mechanisms](https://mlsanity.com/paper/2310.11566/) (2023)
- [HSVI for zs-POSGs using Concavity, Convexity and Lipschitz Properties](https://mlsanity.com/paper/2110.14529/) (2021)
- [ε-Optimally Solving Zero-Sum POSGs](https://mlsanity.com/paper/2406.00054/) (2024)
- [The Curious Case of Exploding DecPOMDPs: Containing the Fire through Policy Counting](https://mlsanity.com/paper/2608.17749/) (2026)

## Abstract (arXiv)

Zero-sum stochastic games provide a rich model for competitive decision making. However, under general forms of state uncertainty as considered in the Partially Observable Stochastic Game (POSG), such decision making problems are still not very well understood. This paper makes a contribution to the theory of zero-sum POSGs by characterizing structure in their value function. In particular, we introduce a new formulation of the value function for zs-POSGs as a function of the "plan-time sufficient statistics" (roughly speaking the information distribution in the POSG), which has the potential to enable generalization over such information distributions. We further delineate this generalization capability by proving a structural result on the shape of value function: it exhibits concavity and convexity with respect to appropriately chosen marginals of the statistic space. This result is a key pre-cursor for developing solution methods that may be able to exploit such structure. Finally, we show how these results allow us to reduce a zs-POSG to a "centralized" model with shared observations, thereby transferring results for the latter, narrower class, to games with individual (private) observations.