Partially Observable Stackelberg Game

• Partially observable Stackelberg game is a leader–follower model where the leader commits first and the follower reacts based on incomplete or noisy observations.
• The framework extends to stochastic differential, BSDE, and mean-field settings, using filtering, Bayesian inference, and Riccati equations for equilibrium analysis.
• Applications in finance, robotics, and cybersecurity underscore its significance in designing robust strategies under information asymmetry.

A partially observable Stackelberg game is a leader–follower model in which the strategic hierarchy of a Stackelberg game is combined with asymmetric or incomplete observation. In the literature, the term spans several formalizations: stochastic differential games in which admissible controls are adapted to different filtrations, partially observable stochastic games and POMDP-based formulations with hidden states or private types, security games in which only a coarse signal of the leader’s commitment is observed, and mean-field models in which public aggregate statistics coexist with privately observed individual states (Zheng et al., 2019, Conitzer, 2016, Zheng et al., 2021, Brero et al., 2022). The common structural feature is sequential rationality under informational asymmetry: the leader commits or moves first, the follower best-responds using only its available observations, and the leader optimizes with that observation-constrained response in mind.

1. Information structures and model families

The defining technical object is the information structure. In continuous-time stochastic formulations, partial observability is often expressed by nested filtrations. Zheng and Shi study a BSDE Stackelberg game with

$$\mathbb G^f=\{\mathcal G^f_t\}_{t\ge 0}\subseteq \mathbb G^l=\{\mathcal G^l_t\}_{t\ge 0}\subseteq \mathbb F,$$

so that the follower reacts under a smaller filtration than the leader (Zheng et al., 2019). A closely related asymmetric-information differential game uses

$$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$

with the follower observing a sub-$\sigma$-algebra of the leader’s information (Shi et al., 2015). By contrast, some partially observed LQ Stackelberg models reverse the direction of informational advantage: Zheng and Shi consider a formulation in which the leader’s filtration is contained in the follower’s, $\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$, and the followers know more than the leader (Zheng et al., 2020, Li et al., 2024).

Other formulations encode partial observability through observation channels rather than filtrations. In the security-game model of partial commitment, the defender’s pure action $a_D$ is mapped to a signal $o=\sigma(a_D)$, so the attacker observes only the signal induced by the action, not the action itself (Conitzer, 2016). In robust-commitment models, the follower does not observe the leader’s mixed commitment $\pi$ exactly; instead it receives a random signal $o$ drawn according to a known channel $P(o\mid \pi)$, such as $N$ i.i.d. samples from $$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$0 (Muthukumar et al., 2019). In one-sided zero-sum POSGs, the leader does not observe the true state and instead maintains a belief $$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$1 updated by Bayes’ rule, while the follower observes the state exactly (Zheng et al., 2021). In Stackelberg POMDPs for economic design, the hidden state includes the game state, the followers’ private types, and the internal state of the followers’ policy-interactive learning algorithm (Brero et al., 2022).

Model family	Partial observation object	Representative paper
Stochastic differential / BSDE game	Nested filtrations or noisy observations	(Zheng et al., 2019)
Security game with partial commitment	Signal partition $$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$2	(Conitzer, 2016)
Finite-observation commitment	Observation kernel $$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$3	(Muthukumar et al., 2019)
One-sided POSG	Belief state $$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$4 under Bayes update	(Zheng et al., 2021)
Stackelberg POMDP	Hidden state with private types and learning state	(Brero et al., 2022)
Mean-field Stackelberg game	Public mean field with private types or local filtrations	(Vasal et al., 2022)

This diversity shows that “partial observability” is not restricted to hidden physical state. It may refer to noisy state measurements, limited observation of the leader’s commitment, private types, public but information-revealing actions, overlapping observation channels, or clustered communication.

2. Equilibrium concepts and hierarchical solution structure

The sequential structure is the same across formulations: first the follower solves a best-response problem for a fixed leader strategy, then the leader solves an optimization problem anticipating that best response. In asymmetric-information differential games, the open-loop Stackelberg equilibrium $$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$5 is defined by the two nested optimizations

$$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$6

(Shi et al., 2015). In a partially observable stochastic game, the same idea is written in policy form: $$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$7 (Koh et al., 2020).

Partial observation changes the equilibrium concept whenever observation affects incentives beyond simple state uncertainty. In security games with limited observability, Conitzer introduces the Partially Observable Stackelberg Equilibrium (POSE) for uncorrelated commitment and the correlated variant SESLO, where the attacker conditions on the signal class rather than the exact defender action (Conitzer, 2016). In finite-horizon OTZ-POSGs with public actions, Wang and colleagues use an $$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$8-Stackelberg equilibrium because public run-time actions can leak private information and allow the opponent to gain in the future; they show that if the follower sacrifices $$\mathcal G_{1,t}\subset \mathcal G_{2,t}\subset \mathcal F_t,$$9 at each of the $\sigma$0 stages, the resulting pair achieves an overall $\sigma$1-Stackelberg equilibrium (Zheng et al., 2021). In large-population settings, the equilibrium notion becomes a Stackelberg–Nash hybrid: the followers form a Nash response system conditional on the leader’s policy, and the resulting decentralized strategies are shown to be $\sigma$2-Stackelberg-Nash equilibria with $\sigma$3 in the mean-field limit (Si et al., 2024, Si et al., 20 Mar 2025).

A recurring consequence of partial observability is that the follower’s best response depends on filtered estimates, beliefs, or coarse signals rather than on the true state. This suggests that equilibrium analysis must couple strategic anticipation with filtering or Bayesian inference, rather than treating information as an exogenous side condition.

3. Continuous-time stochastic and BSDE formulations

In continuous time, partially observable Stackelberg games are typically formulated as controlled SDEs, BSDEs, or FBSDEs. Zheng and Shi study a Stackelberg game of BSDEs under partial information with state equation

$\sigma$4

and cost functionals

$\sigma$5

$\sigma$6

(Zheng et al., 2019). For the follower, the analysis yields a partial-information stochastic maximum principle based on the Hamiltonian $\sigma$7 and an adjoint BSDE; optimality requires

$\sigma$8

For the leader, once the follower’s optimal response is substituted, the problem becomes an optimal control problem for an FBSDE, with a mean-field-type adjoint system and a corresponding maximum principle (Zheng et al., 2019).

The LQ specialization exposes the core mechanism of partial observability. In the BSDE setting, the follower’s optimal control has the state-estimate feedback form

$\sigma$9

obtained through two Riccati equations and a Kalman-type filter for $\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$0. The leader’s feedback law is then expressed through four high-dimensional Riccati equations: $\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$1 (Zheng et al., 2019). In the nested-observation LQG model, the follower’s control is

$\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$2

while the leader’s control is

$\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$3

the resulting solution requires two coupled matrix Riccati ODEs and two layers of Kalman–Bucy filtering, one under the follower’s observation filtration and one under the leader’s finer filtration (Li et al., 2022).

Several later models retain the same pattern but with different stochastic structure. Under asymmetric noisy observations, the follower solves a partial-observation control problem after a change of measure, and the leader solves a conditional mean-field FBSDE once the follower’s best response is substituted (Zheng et al., 2020). In partially observed linear Stackelberg games with Poisson jumps and mean–variance criteria, orthogonal decomposition is used to break the circular dependence between controls and observation filtrations, and observable state-feedback Stackelberg equilibria are obtained via Riccati equations (Lin et al., 24 Jan 2026). Across these formulations, state decomposition, backward separation, stochastic maximum principles, completion of squares, and filtering are the standard analytic tools.

4. Computation, dynamic programming, and learning-based methods

The computational picture depends strongly on the information model. In finite-horizon one-sided zero-sum POSGs with public actions, the one-stage Stackelberg equilibrium can be converted into a linear-fractional programming problem and therefore solved by linear programming. For multiple stages, Wang and colleagues propose a space-partition method and show that the leader’s value function is piece-wise linear while the follower’s value function is piece-wise constant (Zheng et al., 2021). The key difficulty is information leakage: public actions reveal private information and alter continuation values.

In security games with partial commitment, computation separates sharply by whether correlation is allowed. POSE is NP-hard in general; Conitzer states that even if every SIS has size $\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$4, deciding if the defender can guarantee positive utility is NP-hard, and no polynomial-time approximation is possible unless $\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$5. By contrast, SESLO is solvable in polynomial time by a linear program, and a practical method for POSE is enumeration of attacker-response functions $\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$6, yielding $\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$7 linear programs (Conitzer, 2016). Robust-commitment analysis sharpens the computational issue further: when the leader’s mixed commitment is observed only through finite samples, the vanilla Stackelberg commitment $\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$8 may be non-robust in non-zero-sum games. Muthukumar and Sahai construct an observation-robust commitment by shrinking $\mathcal F_t^{Y^2}\subset \mathcal F_t^{Y^1}$9 into the interior of its best-response region with

$a_D$0

and show

$a_D$1

with the particular rate $a_D$2 when $a_D$3 (Muthukumar et al., 2019).

Learning-based formulations treat the leader’s problem itself as a partially observable sequential decision problem. In the Stackelberg POMDP framework, the leader’s interaction with adaptive followers is embedded into a POMDP whose hidden state includes the underlying game state, follower types, and the state of the followers’ learning algorithm; the optimal leader strategy in the original Stackelberg problem is proved to coincide with the optimal policy in the Stackelberg POMDP under a restricted policy class (Brero et al., 2022). The proposed solution uses centralized training with decentralized execution. In SLiCC, the stage game is approximated by learned payoff tables: a leader network $a_D$4 and follower network $a_D$5 produce approximate payoff matrices, and a pure-strategy Stackelberg equilibrium is computed by enumeration in $a_D$6 (Koh et al., 2020). In the reported bi-robot cooperative transport study, SLiCC converges in $a_D$7 episodes to a higher combined reward than centralized Q-learning, remains stably above it with approximately $a_D$8–$a_D$9 higher utility, reaches success ratio $o=\sigma(a_D)$0, and attains the same utility level approximately $o=\sigma(a_D)$1 faster (Koh et al., 2020).

Distributed algorithms also appear in networked multi-leader multi-follower games under clustered information. Chen and Yi propose a distributed seeking algorithm based on implicit gradient estimation and network consensus, and prove convergence under diminishing and constant step sizes under strict and strong monotonicity conditions, respectively (Chen et al., 2024). This indicates that, beyond centralized bilevel optimization, partial-information Stackelberg computation can be organized through local gradient surrogates and communication protocols.

5. Mean-field, multi-leader, and multi-follower extensions

A major development is the extension from a single leader and a single follower to population models. In the discrete-time Stackelberg mean field game with a single leader, the leader and followers observe types privately, the public state is the mean-field distribution $o=\sigma(a_D)$2, and the solution is characterized by a master-equation-style backward recursion over public beliefs and mean-field states (Vasal et al., 2022). In the multiple-leader extension, leaders, major followers, and infinitely many minor followers all have private types, and a corresponding master equation characterizes Stackelberg mean field equilibria with multiple leaders (Vasal, 2022).

Continuous-time mean-field models incorporate partial information by local observations, common noise, or hidden states. In the linear-quadratic mean-field Stackelberg game with partial information and common noise, followers use $o=\sigma(a_D)$3-adapted controls, the leader uses $o=\sigma(a_D)$4-adapted controls, and the resulting decentralized strategies are shown to form an $o=\sigma(a_D)$5-Stackelberg–Nash equilibrium with

$o=\sigma(a_D)$6

(Si et al., 2024). The 2025 partially observed mean-field Stackelberg stochastic differential game generalizes this by allowing both state average terms and state expectation terms in the dynamics and by deriving open-loop adapted decentralized strategies and feedback decentralized strategies through state decomposition and backward separation principle, again with $o=\sigma(a_D)$7 (Si et al., 20 Mar 2025).

Multi-agent extensions also include overlapping information structures and several hierarchical layers. The overlapping-information LQ Stackelberg game with two leaders and two followers assigns followers the filtration generated by $o=\sigma(a_D)$8 and leaders the filtration generated by $o=\sigma(a_D)$9, so the information sets are asymmetric but share an overlapping component (Si et al., 2024). The solution uses two levels of partial-information Nash analysis and a conditional mean-field FBSDE for the leaders. In the multiple-follower partially observed Stackelberg game for formation control, none of the agents observes complete information and the followers know more than the leader; orthogonal decomposition and forward-backward LQ decoupling yield state-feedback strategies for one leader and multiple followers (Li et al., 2024).

These extensions show that partial observability is compatible with both mean-field and networked hierarchies. A plausible implication is that filtering and belief compression become even more central as the number of players grows, because the public state typically collapses to low-dimensional sufficient statistics such as mean fields, conditional expectations, or clustered local summaries.

6. Applications, robustness questions, and conceptual boundaries

The application range is broad. In finance, the BSDE Stackelberg framework is applied to a two-player pension fund management problem (Zheng et al., 2019), and asymmetric-information differential games are explicitly motivated by finance, economics, and management engineering applications such as newsvendor problems, cooperative advertising, and pricing problems (Shi et al., 2015). In a later partially observed LQ model, Zheng and Shi study a dynamic advertising problem with asymmetric information and analyze the relation between optimal control, state estimate, and model parameters (Zheng et al., 2020). In control and robotics, SLiCC is demonstrated on bi-robot cooperative object transportation (Koh et al., 2020), and the multi-follower partially observed Stackelberg differential game is specialized to stochastic multi-agent formation control (Li et al., 2024). In cybersecurity, turn-based OTZ-POSGs are proposed for attacker–defender interaction with public actions (Zheng et al., 2021). In economic design, the Stackelberg POMDP framework is used for indirect mechanism design with turn-taking and limited communication by agents (Brero et al., 2022). Mean-field applications include vaccine-subsidy epidemic control and technology adoption (Vasal et al., 2022). Deception-oriented variants address secure wireless communication, insider-assisted false data injection, and sequential hypothesis testing in adversarial partially observable systems (Xin et al., 3 Apr 2026, Zhou et al., 3 Sep 2025).

The literature also clarifies several conceptual boundaries. First, more observability is not uniformly beneficial in partially observable Stackelberg settings. Conitzer proves that the impact of additional observability can be highly non-monotonic: for any $\pi$0 and any $\pi$1, there exist games in which full observability yields defender utility at least $\pi$2, while any coarser signal partition yields utility at most $\pi$3; there are also constructions in which moving from one SIS to more than one SIS raises the defender’s utility from $\pi$4 to at least $\pi$5 (Conitzer, 2016). Second, exact Stackelberg commitment is not automatically robust when followers only partially observe the commitment. Muthukumar and Sahai formally show that, when the game is not zero-sum and the vanilla Stackelberg commitment is mixed, it is not robust to observational uncertainty; in one $\pi$6 non-zero-sum toy example, $\pi$7 for all finite $\pi$8, whereas in a zero-sum toy example Stackelberg is robust and even strictly better by an exponentially small term (Muthukumar et al., 2019). Third, public actions can create an endogenous information channel. In OTZ-POSGs, the public run-time action reveals certain private information to the opponent, which is precisely why the finite-horizon analysis is organized around $\pi$9-Stackelberg equilibrium and space partition (Zheng et al., 2021).

A final boundary concerns the relation between hierarchy and simultaneity. In a three-party deception game with insider and attackers, Xin, Xu, and Hong introduce Deception Stackelberg equilibria and Hyper Nash equilibria, and derive necessary and sufficient conditions under which the defender’s utility remains invariant when the hierarchical structure degenerates into a simultaneous-move scenario (Xin et al., 3 Apr 2026). This suggests that partial observability can be used not only to model informational limits, but also to analyze when the sequential advantage of commitment survives under misperception, deception, or information leakage.