Stackelberg Stopping Game

• Stackelberg stopping games are sequential stopping problems where the leader commits first and the follower best-responds using observed history.
• The framework distinguishes between precommitment optimality and time-consistent equilibria, highlighting issues like discontinuous best responses.
• Applications extend to secretary problems and cyber-deception, employing Bayesian filtering and dynamic programming for optimal decision-making.

Searching arXiv for relevant papers on Stackelberg stopping games and closely related formulations. A Stackelberg stopping game is a stopping problem with hierarchical timing and commitment: one player, the leader, commits to a stopping strategy first, and another player, the follower, best-responds after observing that commitment. In contrast to the classical Dynkin game, where players choose stopping times on equal footing, the Stackelberg formulation is intrinsically sequential and generally induces asymmetric information, endogenous best-response discontinuities, and time inconsistency. Recent arXiv work develops this idea in several settings: a secretary-problem-based leader–follower stopping game with history-dependent Bayesian inference (Ramsey, 2024), a general discrete-time Stackelberg variant of Dynkin games with precommitment, equilibrium, and entropy regularization (Zhang et al., 26 Jul 2025), and a cross-layer cyber-deception framework in which the leader solves a stopping/switching problem while anticipating equilibrium play in a lower-layer one-sided-information Markov game (Yang et al., 27 May 2025). A related but structurally different line studies Stackelberg differential games with a random exit time, where stopping enters through an exogenous exit rather than a strategic stopping control (Gou et al., 2021).

1. Formal definition and game-theoretic position

In its most general sense, a Stackelberg stopping game is a leader–follower stopping problem in which the leader chooses a stopping rule, or more generally a stopping/switching policy, anticipating the follower’s optimal response. The stopping component may refer to a literal stopping time, as in Dynkin-type models, or to a stopping/switching control over regimes, as in cyber deception (Yang et al., 27 May 2025). The essential structural feature is sequential commitment rather than simultaneous choice.

The discrete-time Dynkin-type formulation in "Stackelberg stopping games" (Zhang et al., 26 Jul 2025) makes the distinction explicit. On a finite or infinite horizon, the underlying process is a discrete-time Markov chain, and the leader’s stopping time $\tau$ and follower’s stopping time $\rho$ determine Dynkin-type nonzero-sum payoffs. In finite horizon, the payoffs are

$$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$

$$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$

with discounted infinite-horizon analogues $J_1(x,\tau,\rho)$ and $J_2(x,\tau,\rho)$ (Zhang et al., 26 Jul 2025). The leader’s value is defined after embedding the follower’s best response into her objective.

This formulation differs from a Nash stopping game in a specific way. In a classical Dynkin game, a Nash equilibrium requires mutual optimality under simultaneous strategic choice. In the Stackelberg version, the follower responds after observing the leader’s strategy, so the leader solves an optimal control problem over the space of stopping policies under an endogenous response map. The paper shows that precommitment strategy, time-consistent Stackelberg equilibrium strategy, and Dynkin Nash equilibrium are distinct concepts and may yield different initial actions (Zhang et al., 26 Jul 2025).

A closely related but more specialized model appears in the secretary setting of "A Stackelberg Game based on the Secretary Problem: Optimal Response is History Dependent" (Ramsey, 2024). There, Player 1 acts as leader by choosing according to the classical secretary protocol, and Player 2, observing the remaining objects in the original order, best-responds under incomplete rank information. This is a Stackelberg stopping game because the leader’s optimal threshold is fixed first, and the follower’s problem is an auxiliary optimal stopping problem conditioned on that commitment (Ramsey, 2024).

2. Information structure, commitment, and asymmetry

A defining feature of Stackelberg stopping games is that the follower typically conditions on the leader’s policy rather than merely on realized actions. This produces informational asymmetries that are absent, or less pronounced, in simultaneous stopping models.

In the secretary-based game (Ramsey, 2024), the leader observes a random permutation of $N$ distinct objects and uses the classical secretary strategy: reject the first $r$ objects and then accept the first record. The follower later observes all remaining objects in the same original order except the object selected by the leader. He observes relative ranks with respect to what he has seen, but cannot compare any candidate directly to the leader’s pick. This induces incomplete information on whether the current candidate is better than the leader’s chosen object. The follower’s information is therefore not exhausted by the current time index: it must incorporate inference about the hidden relative rank of the leader’s selected item (Ramsey, 2024).

The paper identifies two key inferences. Before the leader’s secretary threshold $n^*$, when the follower sees a record, he knows the leader has not yet accepted; after the leader has accepted, the follower’s records satisfy relative rank at most $2$, and the timing of post-threshold records reveals information about how many objects have appeared in total and about the probability that the current object dominates the leader’s choice (Ramsey, 2024). This makes the response genuinely history dependent.

The cyber-deception formulation (Yang et al., 27 May 2025) introduces a different asymmetry. The active deception mode $\rho$0 is known to the defender but hidden from the attacker, while the state $\rho$1 is commonly observed. The attacker maintains a belief $\rho$2 and updates it by Bayes’ rule from observed transitions and actions. Here the Stackelberg stopping structure lies in the upper-layer decision of when to stop the current mode and switch to a new deception mode, anticipating the lower-layer equilibrium response under one-sided information (Yang et al., 27 May 2025).

These examples show that Stackelberg stopping games need not be about stopping a single sample path in the narrow Dynkin sense. They may also be stopping/switching problems with belief-state dynamics, provided the leader’s stopping action is chosen in anticipation of a follower best response. A plausible implication is that the concept is best understood as a hierarchical stopping-control framework rather than a single canonical model class.

3. Equilibrium concepts: precommitment, best response, and time consistency

The equilibrium notion in a Stackelberg stopping game is subtler than in standard stopping games because the leader’s commitment may be optimal ex ante but not sequentially optimal from the perspective of future selves. This is the source of time inconsistency emphasized in (Zhang et al., 26 Jul 2025).

For pure strategies in finite horizon, the leader’s precommitment problem is to choose $\rho$3 maximizing

$\rho$4

where $\rho$5 is the follower’s best response, taken as the earliest maximizer (Zhang et al., 26 Jul 2025). A precommitment is optimal at the initial time but need not remain optimal after the game evolves. The paper’s deterministic two-period example shows precisely this phenomenon: under suitable inequalities, the leader’s precommitment at $\rho$6 is to continue to $\rho$7, while at $\rho$8 the optimal action becomes immediate stopping (Zhang et al., 26 Jul 2025).

To resolve this intertemporal inconsistency, the paper defines a time-consistent equilibrium strategy as a subgame-perfect policy. In finite horizon, a pure Markov stopping policy $\rho$9 is an equilibrium if, for each time-state pair, no one-step deviation followed by continuation with $$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$0 can improve the leader’s Stackelberg value. In the infinite-horizon randomized Markov setting, $$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$1 is an equilibrium if

$$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$2

where only the current-time action is deviated (Zhang et al., 26 Jul 2025).

The secretary-based model (Ramsey, 2024) uses a more specialized Stackelberg equilibrium concept. The leader uses the classical optimal secretary threshold, and the follower’s best response is characterized by dynamic programming and one-step look-ahead optimality. Because both players seek the globally best object and the leader has priority, the leader’s success probability is unaffected by the follower’s subsequent behavior, so the Stackelberg equilibrium preserves the classical secretary strategy for the leader (Ramsey, 2024).

The cyber-deception model (Yang et al., 27 May 2025) separates the equilibrium notions by layer. At the tactical layer, for each deception mode $$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$3, the players play a one-sided $$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$4-Perfect Bayesian Nash Equilibrium. At the strategic layer, the defender chooses stopping times $$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$5 for switching modes and maximizes the induced value function, anticipating those tactical equilibria. This yields a Stackelberg stopping equilibrium consisting of a leader stopping/switching policy and tactical-layer equilibrium strategies for each active mode (Yang et al., 27 May 2025).

4. History dependence and Bayesian filtering in the secretary-based model

The most explicit illustration of history dependence in a Stackelberg stopping game appears in (Ramsey, 2024). After the leader’s threshold $$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$6, the follower acts only when the current observed object is a record for him. The state of the problem is not summarized by time alone. Instead, the follower must track the conditional probability that the current candidate is better than the leader’s pick.

Let $$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$7 denote the time at which the follower sees the $$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$8-th record after $$F_1(\tau,\rho)=f_1(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+g_1(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+h_1(\tau,X_\tau)\mathbf{1}_{\{\rho=\tau\}},$$9, and let

$$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$0

where $$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$1 is the relevant history, represented sufficiently by the list of prior decision times $$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$2 (Ramsey, 2024). The base case is

$$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$3

and the update from decision time $$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$4 to next decision time $$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$5 is

$$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$6

The paper proves that $$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$7, that this updated belief increases in $$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$8 for fixed $$F_2(\tau,\rho)=f_2(\rho,X_\rho)\mathbf{1}_{\{\rho<\tau\}}+g_2(\tau,X_\tau)\mathbf{1}_{\{\tau<\rho\}}+h_2(\rho,X_\rho)\mathbf{1}_{\{\tau=\rho\}},$$9, and that $J_1(x,\tau,\rho)$0, with equality only in the worst-case history of immediate successive records (Ramsey, 2024).

The follower’s immediate reward from accepting a current candidate with belief $J_1(x,\tau,\rho)$1 is

$J_1(x,\tau,\rho)$2

where

$J_1(x,\tau,\rho)$3

The optimal rule after $J_1(x,\tau,\rho)$4 is one-step look-ahead: accept the current candidate if and only if

$J_1(x,\tau,\rho)$5

where

$J_1(x,\tau,\rho)$6

$J_1(x,\tau,\rho)$7

Since $J_1(x,\tau,\rho)$8 is decreasing in $J_1(x,\tau,\rho)$9 while the belief $J_2(x,\tau,\rho)$0 increases from one decision point to the next, one-step look-ahead is optimal for $J_2(x,\tau,\rho)$1 (Ramsey, 2024).

This result is conceptually important because the follower’s optimal response depends jointly on elapsed time and on a filtered belief state derived from the entire sequence of prior record times. The paper explicitly links this to Robbins’ problem, where optimal stopping also depends on the full observation history rather than a low-dimensional Markovian sufficient statistic (Ramsey, 2024). This suggests that Stackelberg stopping games can generate endogenous non-Markovianity even when the underlying arrival process is simple.

5. Dynamic programming, value characterization, and computational structure

Dynamic programming remains central, but the state space and Bellman structure depend strongly on the specific Stackelberg stopping model.

In the secretary-based game (Ramsey, 2024), the leader’s baseline problem is the classical secretary dynamic program. Writing $J_2(x,\tau,\rho)$2 for the success probability after $J_2(x,\tau,\rho)$3 rejections,

$J_2(x,\tau,\rho)$4

$J_2(x,\tau,\rho)$5

The optimal threshold $J_2(x,\tau,\rho)$6, denoted $J_2(x,\tau,\rho)$7 in the paper, is the smallest integer satisfying

$J_2(x,\tau,\rho)$8

The follower’s Bellman equations are more involved because they incorporate incomplete information and history dependence. For $J_2(x,\tau,\rho)$9 and $N$0,

$N$1

with the expectation taken over the next decision time under a mixture distribution induced by the unknown relative rank of the current candidate (Ramsey, 2024). Additional recursions govern the pre-threshold phase, including a preemption region where the follower may accept before the leader’s threshold.

In the general Dynkin-type model (Zhang et al., 26 Jul 2025), the follower’s and leader’s infinite-horizon values satisfy Bellman recursions. For a path-dependent randomized policy $N$2,

$N$3

$N$4

with

$N$5

and

$N$6

A major technical contribution of the paper is to reparametrize the leader’s continuation problem in the follower-utility space. For each state $N$7, the feasible set of follower continuation values is an interval $N$8, and the leader’s constrained continuation value $N$9 satisfies a Bellman equation over admissible follower-value/stop-probability pairs (Zhang et al., 26 Jul 2025). This converts an apparently path-dependent leader control problem into a dynamic program on an auxiliary state variable $r$0.

The cyber-deception paper (Yang et al., 27 May 2025) introduces a two-layer dynamic programming architecture. Tactical-layer equilibria are computed offline for each deception mode $r$1, and then embedded in a strategic-layer Bellman recursion on the extended state

$r$2

where $r$3 is the remaining switch budget and $r$4 is the set of already-used modes (Yang et al., 27 May 2025). The strategic value satisfies

$r$5

and for $r$6,

$r$7

with Bellman operators defined under the tactical-layer equilibrium policies and Bayesian belief update (Yang et al., 27 May 2025). This is structurally a stopping/switching dynamic program whose transition law depends on equilibrium play in a hidden-mode Markov game.

6. Existence, nonexistence, regularization, and asymptotic bounds

A central issue in Stackelberg stopping games is that the follower’s best response may be discontinuous in the leader’s strategy. This can destroy attainment of the leader’s supremum and can even preclude equilibrium existence.

The finite-horizon example in (Zhang et al., 26 Jul 2025) shows that randomized precommitment can dominate pure strategies but still fail to attain the supremum because the leader’s continuation value has a jump discontinuity induced by a switch in the follower’s best response. In the infinite-horizon setting, the paper constructs a three-state counterexample showing that a randomized Markov equilibrium may not exist in general, even under war-of-attrition-type ordered payoffs (Zhang et al., 26 Jul 2025). This sharply contrasts with classical Dynkin games, where randomized Markov Nash equilibria exist under such conditions.

To address this, the paper introduces an entropy-regularized Stackelberg stopping game (Zhang et al., 26 Jul 2025). The follower’s objective is augmented by a Shannon entropy term with parameter $r$8, producing logistic best responses:

$r$9

$n^*$0

The regularized continuation mapping becomes continuous and contractive, so the regularized equilibrium map admits a fixed point by Kakutani’s theorem. The paper proves that for each $n^*$1 there exists a regular randomized equilibrium, and that such equilibria provide $n^*$2-approximations to the original game as $n^*$3 (Zhang et al., 26 Jul 2025).

By contrast, the secretary-based game (Ramsey, 2024) does not present a nonexistence pathology. Instead, the challenge is exact characterization of the follower’s history-dependent value. The paper develops lower bounds by restricting memory and upper bounds by information relaxation. In asymptotic continuous time, the leader’s threshold is $n^*$4 with value $n^*$5, while the follower’s value satisfies

$n^*$6

and remembering more than three prior records yields only minimal gains (Ramsey, 2024). The paper reports lower bounds $n^*$7 of $n^*$8, $n^*$9, $2$0, and $2$1 for $2$2, with an upper bound gap below $2$3 at $2$4 (Ramsey, 2024).

These results indicate two different mathematical obstacles. In Dynkin-type Stackelberg stopping games, discontinuous best responses create equilibrium-selection and existence issues. In the secretary-based model, equilibrium exists in operational form, but exact value characterization is obstructed by history dependence and incomplete rank information. The distinction is substantive rather than merely technical.

7. Applications, variants, and relation to adjacent models

The current literature shows that Stackelberg stopping games arise in at least three distinct ways.

First, they appear as hierarchical variants of classical stopping games. The discrete-time Dynkin-type model of Zhang and Zhou (Zhang et al., 26 Jul 2025) is the cleanest abstract formulation. Its contribution is to show that Stackelberg sequencing alone generates time inconsistency, separates precommitment from equilibrium, and invalidates some existence intuitions imported from Nash Dynkin games.

Second, they arise as stopping problems with incomplete information induced by a leader’s prior action. The secretary-based formulation (Ramsey, 2024) is notable because the leader’s optimal policy remains the classical secretary threshold, while the follower’s response becomes history dependent through Bayesian updating of the probability that the current record beats the leader’s pick. This creates an auxiliary stopping problem whose structure is closer to partially observed optimal stopping than to standard secretary theory.

Third, they arise as stopping/switching control layered above a dynamic game. In the cyber-deception framework (Yang et al., 27 May 2025), the defender’s upper-layer problem is to choose when to stop the current deception mode and switch to another one under resource constraints, while the attacker best-responds in a lower-layer one-sided-information Markov game. The paper interprets this as a Stackelberg stopping game because the leader’s stopping/switching decisions are taken in anticipation of the follower’s equilibrium adaptation. Numerical experiments on states $2$5, modes $2$6, and horizons $2$7 show that switching improves the defender’s expected value and that the proposed dynamic-programming policy can reduce critical-asset compromise probabilities relative to baselines, with reported reductions of $2$8–$2$9 in representative scenarios (Yang et al., 27 May 2025).

A related but conceptually separate model is the linear-quadratic mean-field stochastic Stackelberg differential game with random exit time (Gou et al., 2021). There the leader’s strategy stops at an exogenous random exit time $\rho$00, and the analysis uses progressive enlargement of filtration, two-stage decomposition, the stochastic maximum principle, and verification. Because the exit time is not chosen strategically, this is not a Dynkin-type Stackelberg stopping game. However, it belongs to the broader family of Stackelberg games with stopping structure, and it shows how stopping interacts with mean-field terms, jump-adjoint equations, and backward induction in continuous time (Gou et al., 2021).

A common misconception is to treat all Stackelberg stopping games as simple Dynkin games with priority. The recent literature suggests a broader taxonomy. Some models are genuine leader–follower stopping-time games on a stochastic process (Zhang et al., 26 Jul 2025); some are secretary-style problems where the follower stops under partial rank information generated by the leader’s prior stop (Ramsey, 2024); others are stopping/switching control problems over modes or regimes embedded in stochastic games (Yang et al., 27 May 2025). Another misconception is that the leader’s commitment necessarily changes the leader’s baseline optimal strategy. In the secretary model, it does not: the leader still uses the classical optimal secretary threshold because her success probability is unaffected by the follower’s subsequent behavior (Ramsey, 2024). In the Dynkin-type model, by contrast, commitment structure fundamentally alters both the leader’s optimization problem and the equilibrium notion (Zhang et al., 26 Jul 2025).

Taken together, these papers establish Stackelberg stopping games as a distinct research area at the intersection of optimal stopping, stochastic games, information design, and dynamic commitment. The most persistent technical themes are endogenous informational asymmetry, discontinuous best responses, history-dependent sufficient statistics, and the tension between precommitment optimality and time-consistent equilibrium (Ramsey, 2024, Zhang et al., 26 Jul 2025, Yang et al., 27 May 2025).