# Relu and softplus neural nets as zero-sum turn-based games

arXiv: https://arxiv.org/abs/2512.20582 · HTML: https://mlsanity.com/paper/2512.20582/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 22 references that are in the map.
- First posted on arXiv 23 Dec 2025.

## Builds on (most cited first, 6 of 6)

- [A Theory of Regularized Markov Decision Processes](https://mlsanity.com/paper/1901.11275/) (2019)
- [Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verification](https://mlsanity.com/paper/2103.06624/) (2021)
- [Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums](https://mlsanity.com/paper/2104.08135/) (2021)
- [A Tropical Approach to Neural Networks with Piecewise Linear Activations](https://mlsanity.com/paper/1805.08749/) (2018)
- [Revisiting Tropical Polynomial Division: Theory, Algorithms and Application to Neural Networks](https://mlsanity.com/paper/2306.15157/) (2023)
- [Minimax representation of nonexpansive functions and application to zero-sum recursive games](https://mlsanity.com/paper/1605.04518/) (2016)

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

- [Playing the network backward: A Game Theoretic Attribution Framework](https://mlsanity.com/paper/2605.06212/) (2026)

## Abstract (arXiv)

We show that the output of a ReLU neural network can be interpreted as the value of a zero-sum, turn-based, stopping game, which we call the ReLU net game. The game runs in the direction opposite to that of the network, and the input of the network serves as the terminal reward of the game. In fact, evaluating the network is the same as running the Shapley-Bellman backward recursion for the value of the game. Using the expression of the value of the game as an expected total payoff with respect to the path measure induced by the transition probabilities and a pair of optimal policies, we derive a discrete Feynman-Kac-type path-integral formula for the network output. This game-theoretic representation can be used to derive bounds on the output from bounds on the input, leveraging the monotonicity of Shapley operators, and to verify robustness properties using policies as certificates. Moreover, training the neural network becomes an inverse game problem: given pairs of terminal rewards and corresponding values, one seeks transition probabilities and rewards of a game that reproduces them. Finally, we show that a similar approach applies to neural networks with Softplus activation functions, where the ReLU net game is replaced by its entropic regularization.