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arxiv: 2406.19607 · v2 · pith:QC2PLPTQnew · submitted 2024-06-28 · 🧮 math.OC

Closed-loop equilibria for Stackelberg games: a story about stochastic targets

Camilo Hern\'andez , Nicol\'as Hern\'andez Santib\'a\~nez , Emma Hubert , Dylan Possama\"i This is my paper

Pith reviewed 2026-05-23 23:51 UTC · model grok-4.3

classification 🧮 math.OC
keywords stochastic Stackelberg gamesclosed-loop strategiesbackward stochastic differential equationstarget constraintsHamilton-Jacobi-Bellman equationsstochastic control
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The pith

Stackelberg games under closed-loop strategies reduce to single-level stochastic control with target constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to turn any continuous-time stochastic Stackelberg differential game, where both leader and follower choose drift and volatility to maximize expected utilities, into a standard optimization problem when players use only the history of the output process. It treats the second-order BSDE that tracks the follower's continuation utility as an extra controlled state variable available to the leader. This converts the leader's problem into one with terminal target constraints that can be solved by adapting existing techniques for stochastic control with targets, yielding a system of Hamilton-Jacobi-Bellman equations whose solution gives the equilibrium strategies and value. The approach is illustrated on a simple example that compares closed-loop solutions to those under other information structures.

Core claim

By considering the second-order backward stochastic differential equation associated with the continuation utility of the follower as a controlled state variable for the leader, the latter's unconventional optimisation problem can be reformulated as a more standard stochastic control problem with target constraints. Thereafter the optimal strategies and equilibrium value are characterised through the solution of a system of Hamilton-Jacobi-Bellman equations.

What carries the argument

The second-order backward stochastic differential equation for the follower's continuation utility, adjoined as an additional controlled state variable for the leader.

If this is right

  • The bi-level Stackelberg problem becomes a single-level stochastic control problem with terminal constraints.
  • Equilibrium strategies and value functions are recovered from solutions to a system of Hamilton-Jacobi-Bellman equations.
  • The method applies to games where both players control drift and volatility of the same output process.
  • Numerical and theoretical comparisons with open-loop or other information structures become feasible via the resulting HJB system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reformulation may allow existing numerical solvers for target-constrained control to be applied directly to Stackelberg settings.
  • It suggests a route to closed-loop solutions in other hierarchical stochastic games where the follower's value process can be written as a BSDE.
  • If the HJB system admits unique solutions, the method would guarantee existence of closed-loop equilibria without requiring the leader to observe the noise.

Load-bearing premise

The follower's continuation-utility BSDE can be adjoined as a controlled state for the leader while preserving decisions based only on output history and without introducing explicit dependence on the unobservable driving noise.

What would settle it

An explicit closed-loop equilibrium strategy obtained from the HJB system that cannot be implemented using only the observed output path and requires direct knowledge of the driving noise.

Figures

Figures reproduced from arXiv: 2406.19607 by Camilo Hern\'andez, Dylan Possama\"i, Emma Hubert, Nicol\'as Hern\'andez Santib\'a\~nez.

Figure 1
Figure 1. Figure 1: Comparison of the value functions for various information concepts, with cF = cL = 1, and a◦ = 10. We first remark that for the four sets of parameters, we have the following inequalities for the leader’s value function, V AOL L = V AF L < V ACLM−K¯ L < V CL L < V ACL L = V FB L , and the converse inequalities for the follower’s value. Most of these inequalities were to be expected, as already mentioned in… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the value functions for various information concepts, with cF = 1, cL = 1.25, and a◦ = 10. Comparing [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the value functions for various information concepts, with cF = 1.25, cL = 1, and a◦ = 10. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the value functions for various information concepts, with cF = 1, cL = 1, and a◦ = 15. Finally, comparing [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
read the original abstract

We provide a general approach to reformulating any continuous-time stochastic Stackelberg differential game under closed-loop strategies as a single-level optimisation problem with target constraints. More precisely, we consider a Stackelberg game in which the leader and the follower can both control the drift and the volatility of a stochastic output process, in order to maximise their respective expected utility. The aim is to characterise the Stackelberg equilibrium when the players adopt 'closed-loop strategies', i.e. their decisions are based solely on the historical information of the output process, excluding especially any direct dependence on the underlying driving noise, often unobservable in real-world applications. We first show that, by considering the second-order backward stochastic differential equation associated with the continuation utility of the follower as a controlled state variable for the leader, the latter's unconventional optimisation problem can be reformulated as a more standard stochastic control problem with target constraints. Thereafter, adapting the methodology developed by Soner and Touzi (2002a) or Bouchard, Elie and Imbert (2010), the optimal strategies, as well as the corresponding value of the Stackelberg equilibrium, can be characterised through the solution of a well-specified system of Hamilton- Jacobi-Bellman equations. For a more comprehensive insight, we illustrate our approach through a simple example, facilitating both theoretical and numerical detailed comparisons with the solutions under different information structures studied in the literature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to reformulate any continuous-time stochastic Stackelberg differential game under closed-loop strategies (decisions based only on the history of the output process X) as a single-level stochastic control problem with target constraints. This is achieved by adjoining the second-order BSDE for the follower's continuation utility as an additional controlled state for the leader; the resulting problem is then solved via HJB equations adapting Soner-Touzi (2002) and Bouchard et al. (2010), with an illustrative example for comparison across information structures.

Significance. If the adjoining step is rigorously justified, the approach would extend standard stochastic control methods (with target constraints) to closed-loop Stackelberg games where the driving noise is unobservable, providing a general framework beyond open-loop or full-information cases and enabling numerical comparisons in the example.

major comments (2)
  1. [Reformulation of the leader's problem] The central reformulation (abstract and the step adjoining the follower's 2BSDE) treats the 2BSDE solution as a controlled state while claiming to preserve the closed-loop filtration generated by X alone. No verification is supplied that this state process remains adapted to the observable sigma-field of X (without explicit dependence on the unobservable W), which is load-bearing for the single-level problem to be a valid closed-loop Stackelberg equilibrium; the cited external methods do not automatically guarantee this under the paper's information structure.
  2. [HJB characterization step] The subsequent HJB characterization (abstract) is asserted to yield the optimal strategies and equilibrium value, but the manuscript supplies no proof sketch, verification theorem, or error analysis for the reformulated control problem with the adjoined state; this leaves the soundness of the equilibrium characterization unassessed beyond the high-level claim.
minor comments (2)
  1. The abstract refers to 'a well-specified system of Hamilton-Jacobi-Bellman equations' without indicating its dimension or the form of the target constraints in the general case; a brief outline would improve readability.
  2. Notation for the output process, controls, and filtrations could be introduced more explicitly at the start of the main text to aid comparison with the cited literature on different information structures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding the rigor of the reformulation. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Reformulation of the leader's problem] The central reformulation (abstract and the step adjoining the follower's 2BSDE) treats the 2BSDE solution as a controlled state while claiming to preserve the closed-loop filtration generated by X alone. No verification is supplied that this state process remains adapted to the observable sigma-field of X (without explicit dependence on the unobservable W), which is load-bearing for the single-level problem to be a valid closed-loop Stackelberg equilibrium; the cited external methods do not automatically guarantee this under the paper's information structure.

    Authors: We agree that an explicit verification of the adaptation to the filtration generated by X is necessary to confirm that the reformulated problem indeed corresponds to a closed-loop equilibrium. In the manuscript, the 2BSDE is constructed with coefficients that depend on the closed-loop controls, which are functions of the history of X, ensuring by construction that its solution is adapted to the observable filtration. However, we acknowledge that this is not stated explicitly as a separate result. In the revision, we will add a short lemma verifying that the solution to the follower's 2BSDE remains adapted to the sigma-field generated by X, without direct dependence on the unobservable Brownian motion W. This will strengthen the justification for adjoining it as a controlled state. revision: yes

  2. Referee: [HJB characterization step] The subsequent HJB characterization (abstract) is asserted to yield the optimal strategies and equilibrium value, but the manuscript supplies no proof sketch, verification theorem, or error analysis for the reformulated control problem with the adjoined state; this leaves the soundness of the equilibrium characterization unassessed beyond the high-level claim.

    Authors: The HJB system is obtained by direct application of the methodology in Soner and Touzi (2002) and Bouchard et al. (2010) to the reformulated stochastic control problem with target constraints and the additional controlled state from the 2BSDE. While the manuscript relies on these references for the characterization, we concur that including a brief outline of how the verification theorem applies in this augmented setting would be beneficial. We will add a short proof sketch in the revised version, detailing the key steps of the verification argument and noting any modifications required due to the presence of the adjoined state and the target constraints. revision: yes

Circularity Check

0 steps flagged

No circularity; reformulation adapts external Soner-Touzi and Bouchard methodologies without self-reduction

full rationale

The paper's core step adjoins the follower's 2BSDE as a controlled state and then applies the cited external frameworks of Soner-Touzi (2002) and Bouchard-Elie-Imbert (2010) to obtain an HJB system. These citations are independent, externally published, and not self-citations by the present authors. No equation or claim reduces by construction to a fitted parameter or prior result defined inside this manuscript; the closed-loop information structure is preserved by the problem setup rather than by redefinition. The derivation therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are identifiable from the provided text.

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