# Minimax representation of nonexpansive functions and application to zero-sum recursive games

arXiv: https://arxiv.org/abs/1605.04518 · HTML: https://mlsanity.com/paper/1605.04518/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2016.

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

- [Relu and softplus neural nets as zero-sum turn-based games](https://mlsanity.com/paper/2512.20582/) (2025)

## Abstract (arXiv)

We show that a real-valued function on a topological vector space is positively homogeneous of degree one and nonexpansive with respect to a weak Minkowski norm if and only if it can be written as a minimax of linear forms that are nonexpansive with respect to the same norm. We derive a representation of monotone, additively and positively homogeneous functions on L^∞ spaces and on ℝn, which extend results of Kolokoltsov, Rubinov, Singer, and others. We apply this representation to nonconvex risk measures and to zero-sum games. We derive in particular results of representation and polyhedral approximation for the class of Shapley operators arising from games without instantaneous payments (Everett's recursive games).