Deep Incentive Design with Differentiable Equilibrium Blocks

Vinzenz Thoma; Georgios Piliouras; Luke Marris

cs.GTMar 8, 2026

Deep Incentive Design with Differentiable Equilibrium Blocks

Authors: Vinzenz Thoma, Georgios Piliouras, Luke Marris

Organizations: Department of Computer Science, ETH Zurich & ETH AI Center · Google DeepMind

Abstract

Automated design of multi-agent interactions with desirable equilibrium outcomes is inherently difficult due to the computational hardness, non-uniqueness, and instability of the resulting equilibria. In this work, we propose the use of game-agnostic differentiable equilibrium blocks (DEBs) as modules in a novel, differentiable framework to address a wide variety of incentive design problems from economics and computer science. We call this framework deep incentive design (DID). To validate our approach, we examine three diverse, challenging incentive design tasks: contract design, machine scheduling, and inverse equilibrium problems. For each task, we train a single neural network using a unified pipeline and DEB. This architecture solves the full distribution of problem instances, parameterized by a context, handling all games across a wide range of scales (from two to sixteen actions per player).

Figures & tables

Panel of Figure 1 — Figure 1: Deep Incentive Design framework. Black (green) arrows show the forward (backward) computation.

Figure 2: A simple example of a contract design problem with action space $2{\times}2$ and 3 outcomes.

Table 1: Experimental results. We report results for our approach (DID) and with and without local polishing (+Local) to give an approximate ceiling on the optimal local performance missed by DID. We also evaluate at the equilibrium computed by the DEB and ECOS to measure discrepancies from the DEB.

Figure 3: $\sigma^{\omega}$ is plotted in cyan, the $0.01$ -ME-CCE in magenta. The red polytope corresponds to the $0.01$ -CCE set. The underlying game is between a row player with actions U p and D own and a column player with actions L eft and R ight. The four vertices represent the four deterministic joint actions. Figure was produced with a smaller network trained on $2{\times}2$ games.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Inverse equilibrium training pipeline.

Panel of Figure 5 — Figure 5: The architecture of the contract design mechanism generator. Blue boxes show the core generator and the green text shows the shapes.

Panel of Figure 6 — Figure 6: Training pipeline for the contract design task.

Panel of Figure 7 — Figure 7: Training pipeline for the machine scheduling problem.

Mechanism Design

Mechanism design studies rules for allocating resources and shaping incentives when participants hold private information or act strategically.

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DNQ: Deep Nash Q-Network for Partially Observable n-Player Games

Qintong Xie, Edward Koh, Xavier Cadet, Peter Chin

Nash Equilibrium Reinforcement Learning Imperfect-Information Games Game Theory Automated Bidding Repeated Games+3

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Many real-world competitive systems require multiple decision-makers to act simultaneously under shared constraints, limited information, and repeated interaction, as in auctions, resource allocation, and security competition. We study multi-turn simultaneous bidding as a controlled testbed for such problems and propose DNQ, a solver-in-the-loop equilibrium supervision framework for training bidding agents. DNQ alternates between trajectory collection, critic-based payoff estimation, equilibrium computation, and policy imitation. At each visited state, a shared critic predicts either pairwise payoff matrices or an exact N-player payoff tensor, an external solver computes equilibrium strategies, and the agents are trained by minimizing the KL divergence between their masked policies and the solver-derived equilibrium targets. We focus on a scalable pairwise formulation that greatly reduces equilibrium-solving cost and training time compared with the exact formulation, while the shared critic amortizes payoff learning across agents and states. Experiments compare the pairwise and exact variants using critic loss, policy entropy, bidding resource usage, and training cost, showing that the pairwise method scales to larger numbers of agents, whereas the exact method becomes computationally impractical as the joint game grows. These results illustrate the trade-off between strategic fidelity and scalability in repeated competitive environments.

Abstract

Mechanism Design

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