Title: Stochastic Principal-Agent Problems: Computing and Learning Optimal History-Dependent Policies
Venue: NeurIPS 2025
Accepted: yes
Keywords: stochastic games

Markov games

Stackelberg games

information design

mechanism design
Abstract: We study a stochastic principal-agent model. A principal and an agent interact in a stochastic environment, each privy to observations about the state not available to the other. The principal has the power of commitment, both to elicit information from the agent and to signal her own information. The players communicate with each other and then select actions independently. 
Both players are {\em far-sighted}, aiming to maximize their total payoffs over the entire time horizon. 
We consider both the computation and learning of the principal's optimal policy. 
The key challenge lies in enabling {\em history-dependent} policies, which are essential for achieving optimality in this model but difficult to cope with because of the exponential growth of possible histories as the size of the model increases; explicit representation of history-dependent policies is infeasible as a result.
To address this challenge, we develop algorithmic techniques based on the concept of {\em inducible value set}. The techniques yield an efficient algorithm that computes an $\epsilon$-approximate optimal policy in time polynomial in $1/\epsilon$. 
We also present an efficient learning algorithm for an episodic reinforcement learning setting with unknown transition probabilities. The algorithm achieves sublinear regret $\widetilde{\mathcal{O}}(T^{2/3})$ for both players over $T$ episodes.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work studies the problem of computing optimal history-dependent policies under a model of principal-agent style stochastic games. The problem is general-sum and the goal is to find a Stackelberg equilibrium among history-dependent strategies. This work makes a \"hindsight observability\" assumptions, where both players observe the full interaction history, including states, observations and actions at the end of each time step. Under this model, this work shows that it is possible to compute an $\\epsilon$-optimal policy with time complexity scaling polynomially with the problem parameters. The key idea underlying this result is the construct of \"inducible value sets\", enabling the use of a value-based approach through sets that can be estimated with dynamic programming to compute convex polytope approximations of the value sets. This work also presents results on learning in such principal-agent problems, making use of recent advancements in reward-free exploration in RL.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n* This work identifies conditions under which it is possible to computationally-efficiently compute solutions to principal-agent-style stochastic games with partial information, which are known to be computationally-intractable in general.\n* The importance of history-dependence in such principal-agent problems is well-motivated. The construct of inducible value sets effectively addresses the core challenge of the exponential growth of history-dependent policies.\n* This work addresses both computation and learning, attaining polynomial complexity in both cases\n\n**Weaknesses**\n* The \"hindsight observability\" assumption appears quite restrictive---assuming that the states, observations, and actions are observable to both agents---given the motivations of the model as capturing aspects of imperfect and imbalanced information. However, the authors provide some discussion of this assumption in the main text and the appendix.\n* The centralized learning assumption allows for the use of recent advancements in reward-free exploration in RL, but is perhaps not the best fit for the principal-agent style games in practice, where one would expect the two players to be learning in a decentralized manner with different observable information.\n* ~The paper's discussion of related work contained some confusing phrasing that warrants revision. For example, [3] is described as 'concurrent work' (lines 114-115), despite being published two years ago. Also, I would suggest avoiding the phrase \"while we discovered these techniques independently\" in lines 110-111, which refers to work dating back to at least 2009.~\n**Edit:** The authors have clarified this. Their preprint predates the mentioned works.\n\nQUESTIONS:\n* In line 141-142, you say that the *hindsight observability* assumption is necessary, as without it the model subsumes POMDPs. Can you clarify what you mean here? The claim that it is necessary seems to imply that this is the *only* structural condition that can be made on top of what was described to make planning tractable. Is this the case? If not, perhaps this sentence should be clarified in the paper.\n* You mention in the related work section that prior work [14] has shown optimal stationary strategies are inapproximable, leaving opening the tractability of optimal history-dependent policies. Is this work making the \"hindsight observability\" assumption? Wouldn't optimization over history-dependent policies be *harder*? Besides computational issues, a different motivation you discuss is the importance of history-dependence for optimality and capturing complex/interesting behaviors. Can you comment on the relationship between these two aspects?\n\nLIMITATIONS:\nThe authors adequately addressed the limitations and potential negative societal impact of their work", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers a stochastic principal agent problem with history dependent policies. Once this generalized principal agent problem is introduced, the paper makes the following main contributions: \n\n1. It is shown that there exists an approximately optimal incentive compatible (IC) policy and the paper proposes a polynomial time procedure to compute the induced distribution for any given history of interaction and observation tuple in polynomial time (Theorem 5). From the technical viewpoint the procedure requires to compute polytope approximations of the so-called inducible value sets introduced by the paper using dynamic programming before one can unroll the optimal policy for a given history and observation tuple.\n\n2. In the episodic online learning setting where the transition model is unknown for the players, the paper proposes a centralized learning algorithm which guarantees a regret of order $T^{2/3}$ to both players with high probability, and which can be executed in polynomial time in the problem parameters (Theorem 10). The algorithm follows a explore-and-commit strategy which involves a first reward-free exploration step to learn an estimate of the transition kernel followed by a second step for computing an approximately optimal approximate IC policy using the estimated kernel. In particular, the learning algorithm requires relaxing the IC constraints compared to the result of the previous section.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths** \n\n- Solving repeated principal agent problems is an active topic of research nowadays, including in the ML community. The paper adds to this literature by considering a sequential stateful setting with far-sighted agents and history dependent policies beyond simpler stationary policies. In my opinion, this is an interesting addition although I am not an expert in principal-agent problems and I leave experts of the room provide a more accurate judgment in view of existing literature. Considering history dependent policies definitely brings some complexity to the problem which is balanced by the hindsight (full) observability assumption. The appendix and technical developments (which I did not go over in details) seem rather compact. \n\n- The paper is overall well-written, notations are precise and the paper is generally easy to follow. Although the mathematical objects involve some complexity, the paper did a good job introducing nice and rigorous notations which make the paper readable, especially for (recursive) induced value sets, dynamic programming procedures, summaries of the algorithms in frames and a precise introduction of the problem in section 2 including the communication protocol and the setting of interest with the required assumptions.   \n\n- The results are sound but I have not checked the proofs in the appendix.\n\n**Weaknesses**\n\n- One of the highlighted contributions is a **generalized** framework for principal-agent problems. With this regard, I would expect a clearer positioning of the paper w.r.t the literature. The paper discusses related work. Nevertheless, it is not immediately clear to me from the paper which principal-agent problems are particular cases of the setting of this paper in the literature, among the papers dealing with sequential **stateful** settings (MDPs) which are the most relevant to this paper (if there are any). I think that would clarify further the generality of the model and the positioning of the paper: \n\n-- More precisely, can you elaborate more precisely on the differences than in l. 95-108? The model seems different in several aspects via communication and the use of history dependent policies for instance as well as the setting itself (not contract design like in Wu et al. 24, Ivanov et al. 24 and others) … \n\n-- Can you also clarify the positioning w.r.t. Stackelberg (Markov) games? E.g. Bai et al. Neurips 2021 Sample-Eﬃcient Learning of Stackelberg Equilibria in General-Sum Games. There are also several more recent works about learning Stackelberg equilibria. \n\n- l. 317-323: About hindsight observability and reward-free learning: Are the sampling requirements aligned here? You reframe your problem as an MDP but do the MDP works you use rely on a generative model (which is a standard assumption in RL), i.e. the ability to query a next state from any given state action (not necessarily only the ones observed online during the interaction)? Is this a reasonable assumption in your principal-agent setting with hindsight observability? Is it reasonable to assume that one can query the next state from any state $\\theta$ (which combines actions of both players for instance) while hindsight observability seems to only give access to a history of interactions between the principal and the agent? \n\n- One limitation might be that the algorithm is centralized. I would imagine that players in a Principal-Agent setting would rather act independently. Who is computing the optimal policy exactly? The principal? Is it reasonable to assume the principal (or some central entity) has access to all information (in hindsight) for both players (actions for instance)? \n\n **Typos and suggestions**\n\n- l. 234: is it rather $\\tilde{\\omega}^A$ here for the IC constraint? \n- It might be useful to remind in Fig. 2 step 1 that $v= (v^P, v^A)$ and that $v^P \\in \\mathbb{R}$. \n- l. 340: I guess $\\delta$ should be $\\epsilon$ here?\n\nQUESTIONS:\n1. Clarification question about hindsight observability: Both players (and the agent in particular) observe the full history *at the end* of each time step. I find it a little confusing that the deviation plan (l. 176) for the agent takes the history as input (including the last principal’s observation and action?). Can you clarify this? \n\n2. Before committing to approximation, is there anything known or that can be said and exploited about the geometry of exact value sets beyond nonlinearity? Also can you make the point in l. 248 clearer? Ou argue about nonlinearity: What are the quadratic terms in the expressions? Also there is no max term in (3) which seems linear in v’. Nonlinearity is somehow clear from the max but less so from the rest.  \n\n3. What is exactly the output in step 3 of Fig. 2? You say $\\bar{\\pi}$ (which is a one-step policy as mentioned in l. 225), is it the last one computed in the loop? Is the output the ‘concatenation’ of the $h-1$ ones computed so that it’s a history dependent policy? It’s a little confusing. \n\n\n4. Is there a way to compute an optimal policy recursively without needing to compute first all the inducible value sets and then perform the unrolling? Say simultaneously using the inducible value sets and approximate the optimal policy for a given history and observation tuple if that makes sense. \n\n5. l. 341-348: I am a little confused by the comparator in this learning part. You resort to relaxation for IC constraints which seems reasonable, but then the final comparator in the regret is not adjusted for this, i.e. you still compare to V^* which is the optimal value of (2) where the lower level (agent’s problem) is not relaxed with $\\delta$-IC. \nCan you clarify why you can still achieve vanishing (average) regret w.r.t $V^*$ and the (true) best response for the agent? Do you reduce $\\delta$ in some way? Otherwise there should still be a bias in the final regret bound and a dependence on $\\delta$ I guess. \n\n6. Is there a way to provide a sharper characterization of the dependence on $\\epsilon$ for the problem, more than just polynomial? Is it related to the discretization? Can such a discretization be avoided? \n\n**Additional, more minor questions:**\n\n7. Optimal policy unrolling: This kind of procedure reminds me of what is sometimes called *local planning* in RL which can be done efficiently in polynomial time given a state for instance to compute the optimal policy at that state (see e.g. Szepesvari’s lecture notes online). Is this accurate?\n\n8. Is the order of regret a consequence of the reward free exploration procedure? Because the quality of the approximation (order in epsilon) is not really specified in Theorem 5 whereas the order is precisely given in Theorem 10. Is it optimal up to improving the reward free exploration step? Is there still room for improvement?\n\nLIMITATIONS:\nSee weaknesses section.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper shows that optimal and exactly IC solutions can be computed efficiently in two-player stochastic games with asymmetric information by extending a recent idea of using value polytopes.\n\nSTRENGTHS AND WEAKNESSES:\nComputation of Stackelberg equilibria in stochastic games is not my area of expertise, so I will focus on clarity of the contribution.\n\nOverall, I find the paper well written. The technique used in the paper, based on value polytopes, seems to have been used already in a few papers (as mentioned by the authors at the end of the related work section). However, if I understood correctly, the authors claim that they are able to use the idea of inducible values to guarantee exact incentive compatibility, unlike prior work. So, there seems to be a tangible improvement, though it is hard for me to quantify how technically challenging (or, conversely, incremental) this is. I would have appreciated more clarity regarding the technical obstacles that the authors had to overcome to establish this refined result. \n\nAnother point of comparison is the paper by Gan et al.. That paper is for turn-taking games, whereas the current paper seems to be more general, although somewhat related in terms of techniques (Gan et al. also keeps a sort of \"Pareto frontier\" via constraints if I understood correctly). Is this assessment accurate?\n\nOverall, I found the discussion in Section 3 quite terse. An example could have gone a long way. Furthermore, I am still slightly puzzled about how and why it is possible to compute the H-representation of the value polytope efficiently, given that usually converting from V- to H-polytopes (say, using double description techniques) cannot be tractable in general. Is there any intuition regarding this?\n\n\nMinor points:\n- Citations [2], [6], [17], and [40] are to an arXiv preprint. Were the corresponding papers published anywhere? If so, please update to use the published version, unless there is a specific reason not to.\n- Citations [3] and [22]: \"pomdp\" and \"mdp\" should be \"POMDP\" and \"MDP\", respectively.\n- Citation [34]: \"markov\" -> \"Markov\"\n- Citation [27]: the citation is missing a venue, journal, publisher, or similar.\n\nQUESTIONS:\nCould you please comment on my questions above? Specifically:\n\n1. Technical obstacles\n2. Assessment of the difference with Gan et al. and Bernasconi et al.\n3. Intuition for point 3 of figure 1.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
The paper studies a dynamic principal-agent problem modeled by POMDP, as the policy could depend on the history. It aims to find a policy for the principal so as to maximize its reward, assuming that the agent responds optimally. Under hindsight observability, the paper proposes an algorithm based on computing inducible value sets and optimal policies. Then the paper considers an online setting with episode RL.

STRENGTHS AND WEAKNESSES:
The idea to overcome NP hardness via hindsight observability and computing the optimal policy in polynomial time is quite surprising. 

The major weakness is that the writing makes it very hard to comprehend the main idea and verify the claims made in the paper. 

A few questions related to motivation: why it is important to consider history-dependent policy instead of a Markov policy? 

Despite the fact that the paper seems to achieve a poly time algorithm for the very hard POMDP problem, the lack of experiments in the main context makes it hard to judge how significant the improvement it achieves.

QUESTIONS:
Motivation: Why is it necessary to consider POMDP instead of a normal MDP?

Since the policy is history-dependent, there could be a scenario tree getting involved in the algorithm, but how the authors address the curse of dimension remains unclear. 

Contribution in Line 62, it should be stated what the key motivation is for addressing history dependence. 

The notation system is very hard to understand. For example, there are multiple notations at the beginning of Sec 2 and a lot of new notations in Sec 2.1. In addition, the same notation is widely abused with several meanings and several different input set. 

Thm 3.7, what is the definition of  an $\epsilon$-optimal IC policy?

LIMITATIONS:
NA
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper analyzes a sequential principal-agent game in which, at every round, the principal observes her private signal, elicits (truthfully or not) the agent’s private signal, and then sends non-binding action recommendation for the agent.  Crucially, it imposes hindsight observability: once the round ends, the entire log—hidden state, both private signals, recommendation, chosen actions, and pay-offs—becomes common knowledge.  \n\nWith the transition dynamics known, the authors show that an approximate incentive-compatible commitment policy can be computed in polynomial time.  The key device is the inducible value set: a two-dimensional convex polytope that lists every pair of (principal payoff, agent payoff) that remains achievable from a given state-action pair while keeping the agent willing to comply.  A backward dynamic program updates these polytopes, yielding both the optimal value and a constructive policy.\n\nWhen transition dynamics are unknown, the method is extended with reward-free exploration. After collecting enough trajectories, a modified version of the algorithm yields an approximate incentive complatible policy with regret growing as T^2/3.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- The paper is well written: definitions, lemmas and algorithms are laid out cleanly, and intuition is provided alongside formal results.\n- The paper introduces hindsight observability, a theoretically valuable assumption that renders the model tractable yet still incentive-rich.\n- The introduction of inducible value sets, enabling a polynomial-time dynamic-programming solution while embedding incentive constraints, is a solid technical contribution.\n\nWeaknesses\n- Despite clean, hindsight observability means that every private signal and action becoming public at the end of each round, which significantly limits practical reach.\n- As the author also acknowledges, the method does not scale well to multiple followers, further restricting its scope.\n- The paper provides no empirical validation, so the practical runtime and sample-complexity costs remain untested.\n\nQUESTIONS:\n- In settings where only part of the interaction log can be revealed, do you see a principled way to adapt inducible value sets so that tractability is preserved, perhaps with approximation guarantees?\n- Do you intend to implement the algorithm on synthetic or real-world benchmarks to test runtime and sample complexity? If such experiments are not planned, what do you see as the primary bottlenecks for implementation?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}