Finite Dimensional Approximations of Incentive Functions for Causal Incentive Design Models

Sebastián Bejos, Eduardo F. Morales, Enrique Munoz de Cote, Luis Enrique Sucar
Proceedings of the 13th International Conference on Probabilistic Graphical Models, PMLR 343:18-35, 2026.

Abstract

Single-stage principal-agent problems with bilateral private information can be represented as additive-noise Causal Incentive Design Models (CIDMs), where an incentive policy $\gamma$ acts as a functional intervention on a function-valued variable $\Gamma$. This research extends the Causal Incentive Design framework to address the computational challenge of densely sampling CIDMs when the intervention space $\Gamma$ is a Reproducing Kernel Hilbert Space (RKHS). This challenge arises because RKHS feature maps are typically infinite-dimensional, while CIDM simulation requires that each incentive policy be instantiated as an explicit callable function and evaluated within a structural equation involving the follower’s best-response optimization. To overcome this, quasi-Monte Carlo (QMC) and MC Random Fourier Features (RFF) frameworks are used to construct finite-dimensional approximations of RKHS-valued incentive functions and to densely sample CIDMs. The developed samplers are evaluated on five distinct CIDM instantiations. The experiments compare MC-RFF and QMC-RFF in terms of the approximation error of the RKHS kernel and the subsequent estimation of the causal policy-value target $V(\gamma)$ using Functional Kernel Ridge Regression and a Functional Multi-Layer Perceptron, benchmarked against an analytical Gaussian-smoothed baseline.

Cite this Paper

BibTeX
@InProceedings{pmlr-v343-bejos26a, title = {Finite Dimensional Approximations of Incentive Functions for Causal Incentive Design Models}, author = {Bejos, Sebasti\'an and Morales, Eduardo F. and {Munoz de Cote}, Enrique and Sucar, Luis Enrique}, booktitle = {Proceedings of the 13th International Conference on Probabilistic Graphical Models}, pages = {18--35}, year = {2026}, editor = {Camps-Valls, Gustau and Leonelli, Manuele and Varando, Gherardo}, volume = {343}, series = {Proceedings of Machine Learning Research}, month = {09--11 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v343/main/assets/bejos26a/bejos26a.pdf}, url = {https://proceedings.mlr.press/v343/bejos26a.html}, abstract = {Single-stage principal-agent problems with bilateral private information can be represented as additive-noise Causal Incentive Design Models (CIDMs), where an incentive policy $\gamma$ acts as a functional intervention on a function-valued variable $\Gamma$. This research extends the Causal Incentive Design framework to address the computational challenge of densely sampling CIDMs when the intervention space $\Gamma$ is a Reproducing Kernel Hilbert Space (RKHS). This challenge arises because RKHS feature maps are typically infinite-dimensional, while CIDM simulation requires that each incentive policy be instantiated as an explicit callable function and evaluated within a structural equation involving the follower’s best-response optimization. To overcome this, quasi-Monte Carlo (QMC) and MC Random Fourier Features (RFF) frameworks are used to construct finite-dimensional approximations of RKHS-valued incentive functions and to densely sample CIDMs. The developed samplers are evaluated on five distinct CIDM instantiations. The experiments compare MC-RFF and QMC-RFF in terms of the approximation error of the RKHS kernel and the subsequent estimation of the causal policy-value target $V(\gamma)$ using Functional Kernel Ridge Regression and a Functional Multi-Layer Perceptron, benchmarked against an analytical Gaussian-smoothed baseline.} }
Endnote
%0 Conference Paper %T Finite Dimensional Approximations of Incentive Functions for Causal Incentive Design Models %A Sebastián Bejos %A Eduardo F. Morales %A Enrique Munoz de Cote %A Luis Enrique Sucar %B Proceedings of the 13th International Conference on Probabilistic Graphical Models %C Proceedings of Machine Learning Research %D 2026 %E Gustau Camps-Valls %E Manuele Leonelli %E Gherardo Varando %F pmlr-v343-bejos26a %I PMLR %P 18--35 %U https://proceedings.mlr.press/v343/bejos26a.html %V 343 %X Single-stage principal-agent problems with bilateral private information can be represented as additive-noise Causal Incentive Design Models (CIDMs), where an incentive policy $\gamma$ acts as a functional intervention on a function-valued variable $\Gamma$. This research extends the Causal Incentive Design framework to address the computational challenge of densely sampling CIDMs when the intervention space $\Gamma$ is a Reproducing Kernel Hilbert Space (RKHS). This challenge arises because RKHS feature maps are typically infinite-dimensional, while CIDM simulation requires that each incentive policy be instantiated as an explicit callable function and evaluated within a structural equation involving the follower’s best-response optimization. To overcome this, quasi-Monte Carlo (QMC) and MC Random Fourier Features (RFF) frameworks are used to construct finite-dimensional approximations of RKHS-valued incentive functions and to densely sample CIDMs. The developed samplers are evaluated on five distinct CIDM instantiations. The experiments compare MC-RFF and QMC-RFF in terms of the approximation error of the RKHS kernel and the subsequent estimation of the causal policy-value target $V(\gamma)$ using Functional Kernel Ridge Regression and a Functional Multi-Layer Perceptron, benchmarked against an analytical Gaussian-smoothed baseline.
APA
Bejos, S., Morales, E.F., Munoz de Cote, E. & Sucar, L.E.. (2026). Finite Dimensional Approximations of Incentive Functions for Causal Incentive Design Models. Proceedings of the 13th International Conference on Probabilistic Graphical Models, in Proceedings of Machine Learning Research 343:18-35 Available from https://proceedings.mlr.press/v343/bejos26a.html.

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