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arxiv: 2604.26746 · v1 · submitted 2026-04-29 · 🧮 math.OC

Induced Stackelberg Equilibrium Seeking via Iterative Tikhonov Regularization

Silvia Cianchi , Anibal Sanjab , Sergio Grammatico This is my paper

Pith reviewed 2026-05-07 13:00 UTC · model grok-4.3

classification 🧮 math.OC
keywords Stackelberg equilibriumTikhonov regularizationzeroth-order methodequilibrium selectionNash equilibriumgame theoryvariational inequalities
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The pith

A leader can select among multiple follower equilibria by adding a vanishing Tikhonov incentive term to their game, then converge to the resulting Stackelberg solution with a simple probing algorithm.

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

When followers can settle on any of several Nash equilibria, a Stackelberg leader faces an ill-posed problem because the outcome depends on an unspecified selection rule. The paper shows that the leader can resolve this by augmenting the followers' objectives with an extra regularization term whose strength shrinks to zero over time. This term acts as a vanishing incentive that picks one particular equilibrium while leaving the original set of equilibria unchanged in the limit. The leader then runs a zeroth-order method that repeatedly queries the followers' best responses and jointly updates its own decision variable together with the current strength of the incentive term.

Core claim

The leader augments the lower-level game with a vanishing Tikhonov regularization term that serves as an incentive for a specific equilibrium selection. A follower-agnostic zeroth-order algorithm then lets the leader converge to a solution of the induced Stackelberg problem by iteratively probing the followers and updating both its own action and the incentive parameter.

What carries the argument

The vanishing Tikhonov regularization term added to the followers' game, which induces a unique equilibrium selection while preserving the original equilibrium set in the limit and enabling joint zeroth-order updates.

Load-bearing premise

A vanishing Tikhonov term can be chosen so that it selects one equilibrium among several without changing the equilibrium set in the limit and without preventing the leader's updates from converging.

What would settle it

A simple two-player follower game with two distinct Nash equilibria where the leader's iterative updates either fail to converge or converge to an equilibrium different from the one the regularization was designed to select.

Figures

Figures reproduced from arXiv: 2604.26746 by Anibal Sanjab, Sergio Grammatico, Silvia Cianchi.

Figure 2
Figure 2. Figure 2: If a selection ϕ(x) is introduced, the follower reaction, x ∗ ϕ (y), is uniquely determined by y. Without accounting for this dependence, the ϕ-induced SE cannot be achieved. mechanism exists, it is beneficial for the leader to account for it, to achieve the corresponding ϕ-induced SE, which is, by definition, optimal for the leader. This motivates the adop￾tion (or enforcement by the leader) of optimal eq… view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of J0(yk, xk) throughout the iterations of Alg. 1 in log-log scale. βk = β¯(k + 1)−1 . In view at source ↗
read the original abstract

Existing methods for learning Stackelberg equilibria typically assume that the followers' (variational, generalized) Nash equilibrium is unique. However, in the presence of multiple equilibria, without a selection convention, the problem may become ill-posed, thus leading standard algorithms to potentially fail to converge. This paper addresses this issue by introducing an optimal selection at the lower-level game, hereby defining a Stackelberg game with induced equilibrium selection. To this end, we enable the leader to augment the followers' game with an additional vanishing term that acts as an incentive. We then propose a follower-agnostic zeroth-order method, whereby the leader converges to a solution of the resulting problem by iteratively probing the followers and jointly updating its decision variable and the incentive term.

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

3 major / 2 minor

Summary. The paper claims that when the lower-level (variational or generalized) Nash equilibrium set is non-unique, standard Stackelberg learning methods may fail to converge. It resolves this by letting the leader augment the followers' game with a vanishing Tikhonov regularization term that acts as an incentive, thereby inducing a specific equilibrium selection. A follower-agnostic zeroth-order algorithm is then proposed in which the leader iteratively probes the followers while jointly updating its own decision variable and the regularization parameter, with the claim that the iterates converge to a solution of the resulting induced Stackelberg problem.

Significance. If the regularization is shown to preserve the original equilibrium set while guaranteeing a continuous selection and if the joint zeroth-order updates are proven to converge under compatible rate schedules, the work would provide a practical, gradient-free method for handling ill-posed Stackelberg games without assuming uniqueness of the lower level. The follower-agnostic property and the explicit use of vanishing regularization for selection are potentially useful extensions of existing regularization ideas in game theory.

major comments (3)
  1. [Problem formulation / Definition of induced Stackelberg equilibrium] The central construction requires that, for each fixed positive regularization strength, the perturbed lower-level game admits a unique equilibrium that is a continuous selection from the original set; the manuscript must supply an explicit argument (with a cited theorem or proposition) showing that this selection property holds uniformly with respect to the leader's decision variable.
  2. [Convergence analysis / Main theorem on zeroth-order updates] The convergence analysis of the joint zeroth-order iteration must control the relative rates between the probing step-size schedule and the decay of the regularization parameter; without such control the limit point may fail to lie in the original equilibrium set. A concrete rate condition (e.g., relating the two sequences) is needed in the main convergence theorem.
  3. [Algorithm description / Zeroth-order probing step] The claim that the method is follower-agnostic rests on the assumption that the leader can obtain only zeroth-order information about the lower-level equilibrium map; the error bounds or bias introduced by this probing must be shown to vanish at a rate compatible with the regularization schedule, otherwise the overall convergence guarantee is compromised.
minor comments (2)
  1. [Notation and preliminaries] Clarify the precise definition of the Tikhonov term (e.g., whether it is added to the variational inequality or to the objective) and ensure consistent notation between the regularization parameter and the step-size sequences.
  2. [Numerical experiments] The numerical section should include at least one example in which the lower-level game has a continuum of equilibria, so that the selection effect of the vanishing term can be visually verified.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the insightful comments and the recommendation for major revision. We believe the suggested clarifications will strengthen the paper. We address each major comment below, indicating the changes we will make in the revised manuscript.

read point-by-point responses

  1. Referee: [Problem formulation / Definition of induced Stackelberg equilibrium] The central construction requires that, for each fixed positive regularization strength, the perturbed lower-level game admits a unique equilibrium that is a continuous selection from the original set; the manuscript must supply an explicit argument (with a cited theorem or proposition) showing that this selection property holds uniformly with respect to the leader's decision variable.

    Authors: We agree that the manuscript would benefit from an explicit statement of this property. In the revised version, we will insert a dedicated proposition immediately after the definition of the induced Stackelberg equilibrium. This proposition will cite a standard result on Tikhonov regularization for monotone variational inequalities (e.g., Theorem 2.1 in Facchinei and Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, or a more recent game-theoretic reference) to prove that, under the maintained assumptions of strong monotonicity of the regularized operator, the lower-level game has a unique equilibrium for every fixed leader decision and positive regularization parameter. Moreover, we will show that the map from the leader's decision to this unique equilibrium is continuous, and that the selection is uniform over compact sets of leader decisions. This addresses the uniformity requirement. revision: yes

  2. Referee: [Convergence analysis / Main theorem on zeroth-order updates] The convergence analysis of the joint zeroth-order iteration must control the relative rates between the probing step-size schedule and the decay of the regularization parameter; without such control the limit point may fail to lie in the original equilibrium set. A concrete rate condition (e.g., relating the two sequences) is needed in the main convergence theorem.

    Authors: The referee correctly identifies a gap in the explicit rate conditions. While the proof of Theorem 4.2 implicitly relies on the regularization parameter decaying slower than the probing steps, this is not stated. In the revision, we will add to the statement of the main convergence theorem (Theorem 4.2) the explicit condition that the regularization sequence ε_k satisfies ε_k → 0 with ε_k / α_k → 0, where α_k is the step-size for the leader's update, and the probing accuracy δ_k = o(ε_k). We will then adjust the proof to verify that these rates ensure the limit point satisfies the original (unregularized) lower-level equilibrium condition. revision: yes

  3. Referee: [Algorithm description / Zeroth-order probing step] The claim that the method is follower-agnostic rests on the assumption that the leader can obtain only zeroth-order information about the lower-level equilibrium map; the error bounds or bias introduced by this probing must be shown to vanish at a rate compatible with the regularization schedule, otherwise the overall convergence guarantee is compromised.

    Authors: We will enhance the analysis of the zeroth-order probing in Section 3.2. Specifically, we will derive an explicit bound on the probing error, showing that the bias in the estimated lower-level equilibrium is bounded by O(δ_k + ε_k), where δ_k is the finite-difference step for probing. We will then impose the rate condition δ_k = o(ε_k) in the theorem and prove that this ensures the error vanishes sufficiently fast for the overall convergence to the induced Stackelberg solution. This will be incorporated into the revised convergence proof. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies established regularization to new selection problem without reduction to inputs

full rationale

The paper defines an induced Stackelberg problem by augmenting the lower-level game with a vanishing Tikhonov term chosen by the leader, then proposes a joint zeroth-order update scheme. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The central construction relies on external regularization concepts applied to equilibrium selection without the result being equivalent to its inputs by construction. This is the expected non-circular outcome for a paper introducing a method on top of known tools.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the domain assumption that Tikhonov regularization can select among multiple equilibria without fundamentally altering the game, plus a tunable vanishing schedule for the added term.

free parameters (1)
  • vanishing regularization schedule
    The rate and manner in which the added incentive term goes to zero is a free design choice that must be chosen to induce selection while preserving convergence.
axioms (1)
  • domain assumption The lower-level followers' game is a variational or generalized Nash equilibrium problem whose equilibria can be selected by adding a vanishing regularization term.
    Invoked to make the Stackelberg problem well-posed when multiple equilibria exist.

pith-pipeline@v0.9.0 · 5424 in / 1331 out tokens · 99172 ms · 2026-05-07T13:00:28.324455+00:00 · methodology

discussion (0)

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Reference graph

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