{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T13:05:58Z","timestamp":1775480758258,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T00:00:00Z","timestamp":1766102400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003141","name":"Secretar\u00eda de Ciencia, Humanidades, Tecnolog\u00eda e Innovaci\u00f3n","doi-asserted-by":"crossref","award":["1562"],"award-info":[{"award-number":["1562"]}],"id":[{"id":"10.13039\/501100003141","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We introduce Causal Incentive Design (CID), a framework that applies causal inference to canonical single-stage principal\u2013agent problems (PAPs) characterized by bilateral private information. Within CID, the operating rules of PAPs are formalized using an additive-noise causal graphical model (CGM). Incentives are modeled as interventions on a function space variable, \u0393, which correspond to policy interventions in the principal\u2013follower causal relation. The causal inference target estimand V(\u0393) is defined as the expected value of the principal\u2019s utility variable under a specified policy intervention in the post-intervention distribution. In the context of additive-Gaussian independent noise, the estimand V(\u0393) decomposes into a two-layer expectation: (i) an inner Gaussian smoothing of the principal\u2019s utility regression; and (ii) an outer averaging over the conditional probability of the follower\u2019s action given the incentive policy. A Gauss\u2013Hermite quadrature method is employed to efficiently estimate the first layer, while a policy-local kernel reweighting approach is used for the second. For offline selection of a single incentive policy, a Functional Causal Bayesian Optimization (FCBO) algorithm is introduced. This algorithm models the objective functional \u03b3\u21a6V(\u03b3) using a functional Gaussian process surrogate defined on a Reproducing Kernel Hilbert Space (RKHS) domain and utilizes an Upper Confidence Bound (UCB) acquisition functional. Consequently, the policy value V(\u03b3) becomes an interventional query that can be answered using offline observational data under standard identifiability assumptions. High-probability cumulative-regret bounds are established in terms of differential information gain for the proposed FBO algorithm. Collectively, these elements constitute the central contributions of the CID framework, which integrates causal inference through identification and estimation with policy search in principal\u2013agent problems under private information. This approach establishes a causal decision-making pipeline that enables commitment to a high-performing incentive in a single-shot game, supported by regret guarantees. Provided that the data used for estimation is sufficient, the resulting offline pipeline is appropriate for scenarios where adaptive deployment is impractical or costly. Beyond the methodological contribution, this work introduces a novel application of causal graphical models and causal reasoning to incentive design and principal\u2013agent problems, which are central to economics and multi-agent systems.<\/jats:p>","DOI":"10.3390\/e28010004","type":"journal-article","created":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T14:27:16Z","timestamp":1766154436000},"page":"4","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Single-Stage Causal Incentive Design via Optimal Interventions"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8580-7361","authenticated-orcid":false,"given":"Sebasti\u00e1n","family":"Bejos","sequence":"first","affiliation":[{"name":"Computer Science Department, National Institute of Astrophysics, Optics and Electronics, Puebla 72840, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7618-8762","authenticated-orcid":false,"given":"Eduardo F.","family":"Morales","sequence":"additional","affiliation":[{"name":"Computer Science Department, National Institute of Astrophysics, Optics and Electronics, Puebla 72840, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3685-5567","authenticated-orcid":false,"given":"Luis Enrique","family":"Sucar","sequence":"additional","affiliation":[{"name":"Computer Science Department, National Institute of Astrophysics, Optics and Electronics, Puebla 72840, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3249-096X","authenticated-orcid":false,"given":"Enrique","family":"Munoz de Cote","sequence":"additional","affiliation":[{"name":"Mutable Tactics, London CB2 9PJ, UK"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1146\/annurev-control-053018-023634","article-title":"A perspective on incentive design: Challenges and opportunities","volume":"2","author":"Ratliff","year":"2009","journal-title":"Annu. Rev. Control Robot. Auton. Syst."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Pearl, J. (1994). A probabilistic calculus of actions. Uncertainty in Artificial Intelligence, Morgan Kaufmann.","DOI":"10.1016\/B978-1-55860-332-5.50062-6"},{"key":"ref_3","unstructured":"Correa, J., and Bareinboim, E. (2020, January 7\u201312). A Calculus for Stochastic Interventions: Causal Effect Identification and Surrogate Experiments. Proceedings of the AAAI Conference on Artificial Intelligence, New York, NY, USA."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Williams, C.K., and Rasmussen, C.E. (2006). Gaussian Processes for Machine Learning, The MIT Press.","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"ref_5","unstructured":"Snoek, J., Larochelle, H., and Adams, R.P. (2012). Practical bayesian optimization of machine learning algorithms. Adv. Neural Inf. Process. Syst., 25."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Vien, N.A., Zimmermann, H., and Toussaint, M. (2018, January 2\u20137). Bayesian Functional Optimization. Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, New Orleans, LA, USA.","DOI":"10.1609\/aaai.v32i1.11830"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3250","DOI":"10.1109\/TIT.2011.2182033","article-title":"Information-theoretic regret bounds for Gaussian process optimization in the bandit setting","volume":"58","author":"Srinivas","year":"2012","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_8","unstructured":"Chowdhury, S.R., and Gopalan, A. (2017, January 6\u201311). On kernelized multi-armed bandits. Proceedings of the International Conference on Machine Learning, PMLR, Sydney, Australia."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Laffont, J.J., and Martimort, D. (2009). The Theory of Incentives: The Principal-Agent Model, Princeton University Press.","DOI":"10.2307\/j.ctv7h0rwr"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1109\/TAC.1979.1101999","article-title":"Closed-Loop Stackelberg Strategies with Applications in the Optimal Control of Multilevel Systems","volume":"24","author":"Basar","year":"1979","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Dempe, S., and Zemkoho, A. (2020). Bilevel Optimization: Advances and Next Challenges, Springer International Publishing.","DOI":"10.1007\/978-3-030-52119-6"},{"key":"ref_12","unstructured":"Yang, J., Wang, E., Trivedi, R., Zhao, T., and Zha, H. (2022, January 9\u201313). Adaptive Incentive Design with Multi-Agent Meta-Gradient Reinforcement Learning. Proceedings of the 21st International Conference on Autonomous Agents and Multiagent Systems, Virtual."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1613\/jair.4940","article-title":"Adaptive Contract Design for Crowdsourcing Markets: Bandit Algorithms for Repeated Principal\u2013Agent Problems","volume":"55","author":"Ho","year":"2016","journal-title":"J. Artif. Intell. Res."},{"key":"ref_14","unstructured":"Fiez, T., Sekar, S., Zheng, L., and Ratliff, L.J. (2018, January 6\u201310). Combinatorial Bandits for Incentivizing Agents with Dynamic Preferences. Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence, Monterey, CA, USA."},{"key":"ref_15","unstructured":"Guresti, B., Vanlioglu, A., and Ure, N.K. (June, January 29). IQ-Flow: Mechanism Design for Inducing Cooperative Behavior to Self-Interested Agents in Sequential Social Dilemmas. Proceedings of the 22nd International Conference on Autonomous Agents and Multiagent Systems, London, UK."},{"key":"ref_16","unstructured":"Mguni, D., Jennings, J., Macua, S.V., Sison, E., Ceppi, S., and Cote, E.M.D. (2019, January 13\u201317). Coordinating the Crowd: Inducing Desirable Equilibria in Non-Cooperative Systems. Proceedings of the 18th International Conference on Autonomous Agents and Multiagent Systems, Montreal, QC, Canada."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"104071","DOI":"10.1016\/j.trd.2024.104071","article-title":"Robustness of bilayer railway-aviation transportation network considering discrete cross-layer traffic flow assignment","volume":"127","author":"Jiang","year":"2024","journal-title":"Transp. Res. Part D Transp. Environ."},{"key":"ref_18","first-page":"1181","article-title":"Causal Bandits: Learning Good Interventions via Causal Inference","volume":"29","author":"Lattimore","year":"2016","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_19","first-page":"1342","article-title":"Bandits with Unobserved Confounders: A Causal Approach","volume":"28","author":"Bareinboim","year":"2015","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_20","first-page":"6276","article-title":"Structural Causal Bandits: Where to Intervene?","volume":"31","author":"Lee","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_21","unstructured":"Lee, S., and Bareinboim, E. (February, January 27). Structural Causal Bandits with Non-Manipulable Variables. Proceedings of the Thirty-Third AAAI Conference on Artificial Intelligence, Honolulu, HI, USA."},{"key":"ref_22","unstructured":"Buesing, L., Weber, T., Zwols, Y., Racaniere, S., Guez, A., Lespiau, J.B., and Heess, N. (2019). Woulda, Coulda, Shoulda: Counterfactually- Guided Policy Search. arXiv."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Madumal, P., Miller, T., Sonenberg, L., and Vetere, F. (2020, January 7\u201312). Explainable Reinforcement Learning Through a Causal Lens. Proceedings of the AAAI Conference on Artificial Intelligence, New York, NY, USA.","DOI":"10.1609\/aaai.v34i03.5631"},{"key":"ref_24","unstructured":"Aglietti, V., Lu, X., Paleyes, A., and Gonz\u00e1lez, J. (2020, January 26\u201328). Causal Bayesian Optimization. Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics, Online."},{"key":"ref_25","unstructured":"Gultchin, L., Virginia, A., Alexis, B., and Silvia, C. (August, January 31). Functional Causal Bayesian Optimization. Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence, Pittsburgh, PA, USA."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Pearl, J. (2009). Causality: Models, Reasoning, and Inference, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511803161"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0273-0979-01-00923-5","article-title":"On the mathematical foundations of learning","volume":"39","author":"Cucker","year":"2002","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_28","unstructured":"Thomas, M.T.C.A.J., and Joy, A.T. (2006). Elements of Information Theory, Wiley-Interscience."},{"key":"ref_29","first-page":"493","article-title":"Regression with input-dependent noise: A Gaussian process treatment","volume":"10","author":"Goldberg","year":"1997","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_30","unstructured":"Bejos, S., Sucar, L.E., and Morales, E.F. (2024, January 11\u201313). Estimating Bounds on Causal Effects Considering Unmeasured Common Causes. Proceedings of the International Conference on Probabilistic Graphical Models (PMLR), Nijmegen, The Netherlands."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Zhang, T. (2023). Mathematical Analysis of Machine Learning Algorithms, Cambridge University Press.","DOI":"10.1017\/9781009093057"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1090\/S0002-9947-1950-0051437-7","article-title":"Theory of reproducing kernels","volume":"68","author":"Aronszajn","year":"1950","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_33","unstructured":"Rudin, W. (1976). Principles of Mathematical Analysis, McGraw-Hill. [3rd ed.]. International Series in Pure and Applied Mathematics."},{"key":"ref_34","first-page":"1","article-title":"On the equivalence between kernel quadrature rules and random feature expansions","volume":"18","author":"Bach","year":"2017","journal-title":"J. Mach. Learn. Res."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,25]],"date-time":"2025-12-25T05:13:54Z","timestamp":1766639634000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,19]]},"references-count":34,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["e28010004"],"URL":"https:\/\/doi.org\/10.3390\/e28010004","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,19]]}}}