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Non-Holonomic Gradient Play: Leafwise Nash Equilibria, Stability, and Deception

Mahmoud Abdelgalil, Miroslav Krstic, Jorge I. Poveda

arXiv:2609.38051v1math.OCcs.SYeess.SY

TL;DR

The paper addresses multi-agent learning on manifolds with state-dependent actuation, in a setting where deception in Nash-seeking has been studied in flat spaces. It represents the dynamics as projected Riemannian gradients and analyzes their stability and geometric response to deception; exponentially stable equilibria persist locally as deception varies, with a first-order displacement characterization.

  • Problem

    Deception in Nash-seeking has been studied in repeated static games on flat spaces, while the paper considers multi-agent learning with state-dependent actuation on manifolds.

    Source: §I · p. 1, B. Asymmetric Information and Deception · p. 3

  • Method

    Under a bundle-splitting condition, the paper represents interaction dynamics as projected Riemannian gradients and analyzes stability and deception through intrinsic geometric tools.

    Source: §I · pp. 1–2

  • Results

    Exponentially stable equilibria and their stability persist over an open set of deception parameters, with an explicit first-order formula for equilibrium displacement.

    Source: §I · p. 2

  • Takeaways & Limitations

    A deceiver embedding victims’ exploration signals into its strategy tilts their effective actuation directions and generically shifts equilibria to locations of the deceiver’s choosing.

    Source: §VII · p. 16

Abstract

from arXiv · show

We study generalized learning dynamics in multi-agent systems whose joint state evolves on a manifold and whose agents act through state-dependent, potentially nonholonomic vector fields. Under a bundle-splitting condition, we show that these dynamics admit an intrinsic representation as projected Riemannian gradients, giving rise to a class of \emph{nonholonomic gradient play} dynamics. We characterize the local stability of its equilibria through an intrinsic linearization that explicitly captures the effects of the Riemannian connection and the nonholonomy of the actuation frame. The framework recovers classical gradient play on Euclidean spaces and manifolds as special cases while accommodating nonholonomic actuation. We then develop a geometric theory of deception under asymmetric information, whereby an agent exploits privileged knowledge of other agents' learning rules to manipulate the emerging equilibrium. We show that deception effectively \emph{tilts the Riemannian geometry} perceived by the oblivious agents, distorting their projected gradient directions. We establish persistence of exponentially stable equilibria over an open set of deception parameters and derive an explicit first-order characterization of the resulting equilibrium displacement. Analytical and numerical examples illustrate the framework and its implications for equilibrium manipulation in games.

I. INTRODUCTION

The paper formulates gradient play for games on manifolds where agents control state-dependent admissible directions, addressing the lack of intrinsic analysis for such nonholonomic interactions. It develops projected Riemannian-gradient dynamics and geometric equilibrium and stability concepts, then applies the framework to deception under asymmetric information.

I. INTRODUCTION

  • Existing manifold-game analyses use coordinate-dependent payoff derivatives and typically assume direct, independent strategy-coordinate actuation, leaving state-dependent noncommuting actuation without an intrinsic theory.The paper identifies these gaps in prior stability analyses of continuous-strategy games on manifolds.Source: §I · p. 1
  • Under bundle-splitting, the dynamics become per-agent projected Riemannian gradients, with coordinate-independent equilibria and stability linearization that captures actuation nonholonomy.The linearization includes Riemannian Hessian and connection terms; the latter encode nonholonomy.Source: §I · pp. 1–2
  • Under asymmetric information, a deceiver tilts oblivious agents’ effective actuation bundles and projected gradients; exponentially stable equilibria persist on an open parameter set with a first-order displacement formula.Earlier deception mechanisms cited in the paper rely on Euclidean product spaces with additive actuation.Source: §I · p. 2

II. PRELIMINARIES AND NOTATION

  • The preliminaries define Riemannian gradients through the metric and projected gradients as orthogonal projections onto a smooth subbundle.They also define the Riemannian Hessian using the metric’s Levi-Civita connection.Source: §II · p. 2

III. PROBLEM FORMULATION

  • The model has independent agents interacting through a smooth manifold, each influencing the environment state through its own control vector fields.Agents have individual payoff functions and seek Nash equilibria in the noncooperative setting.Source: A. Geometric Learning in Games · p. 2

A. Geometric Learning in Games

  • For control-affine actuation, payoff-gradient feedback along admissible fields yields nonholonomic gradient play under mild regularity, recovering standard gradient play in the constant, decoupled Euclidean case.The dynamics also arise as model-free approximations through Lie-bracket methods.Source: A. Geometric Learning in Games · p. 3

B. Asymmetric Information and Deception

The paper models asymmetric-information deception as a geometric tilt of oblivious agents’ actuation directions, within a framework that represents constrained learning on manifolds as projected Riemannian gradients. The resulting dynamics can shift equilibria, while their nominal form is recovered when deception parameters vanish.

B. Asymmetric Information and Deception

  • Privileged knowledge of an oblivious agent’s feedback law lets a deceptive agent modify its inputs, biasing steady-state behavior without directly changing the other agent’s actions or measurements.The paper extends this study of deception from repeated games on flat spaces to equilibrium-seeking on smooth manifolds.Source: B. Asymmetric Information and Deception · pp. 3–4
  • Deception tilts the oblivious agent’s vector fields while preserving the nonholonomic gradient-play structure, generally shifting equilibria; zero deception parameters recover nominal dynamics.The tilt reshapes the effective directions of payoff dynamics.Source: B. Asymmetric Information and Deception · p. 4

A. Riemannian Pseudogradient Flow and its Equilibria

  • The examples establish the framework on an open region of a sphere and the interior of a simplex, where the agents’ vector fields span the tangent space and do not commute.The sphere example excludes the equator, while the simplex example excludes its boundary.Source: A. Riemannian Pseudogradient Flow and… · pp. 4–5
  • Under the bundle-splitting assumption, system (3) is a Riemannian pseudogradient flow on an open submanifold, with each agent’s payoff gradient projected onto its actuation distribution.The paper calls this non-holonomic gradient play; in Euclidean space with standard coordinate actuation, it reduces to classical gradient play.Source: A. Riemannian Pseudogradient Flow and… · p. 5
  • In the specified submanifold, the equilibrium condition is independent of agents’ gains, so equilibria are determined by payoffs and actuation vector fields.This independence applies under the conditions that simplify the equilibrium condition.Source: A. Riemannian Pseudogradient Flow and… · pp. 4–5

B. Stability via Intrinsic Jacobian Characterization

The intrinsic Jacobian supplies local stability tests for equilibria of nonholonomic gradient play. Its structure includes both payoff curvature and terms arising from the geometry of the agents’ actuation distributions.

1) Intrinsic Jacobian and Exponential Stability:

  • An equilibrium is isolated if its intrinsic Jacobian is an automorphism, and it is locally exponentially stable if and only if that Jacobian is Hurwitz.The Jacobian’s eigenvalues determine the equilibrium’s local properties.Source: 1) Intrinsic Jacobian and Exponential… · p. 5
  • For identical-interest games, the dynamics reduce to an intrinsic gradient flow, and local exponential stability is equivalent to a negative-definite Hessian at the equilibrium.This recovers the standard geometric optimization condition.Source: 1) Intrinsic Jacobian and Exponential… · p. 6
  • The Jacobian includes bundle-curvature contributions that can persist at equilibrium because each payoff need only be stationary along its agent’s distribution.The block decomposition uses the orthogonal splitting into actuation distributions.Source: 1) Intrinsic Jacobian and Exponential… · p. 6
  • A sufficient block condition makes the Jacobian an automorphism and Hurwitz, thereby ensuring an isolated, locally exponentially stable equilibrium.The condition uses positive constants for the agent-wise blocks.Source: 1) Intrinsic Jacobian and Exponential… · p. 6

2) Jacobian Structure of Nonholonomic Gradient Play:

The intrinsic Jacobian separates payoff-Hessian effects from connection terms that capture how the actuation frame varies. Player-wise involutivity simplifies its diagonal blocks, but does not generally give the manifold a product structure.

2) Jacobian Structure of Nonholonomic Gradient Play:

  • Each Jacobian block combines a payoff-Hessian term with a connection term that captures interaction between the payoff gradient and variations in the actuation frame.The connection contribution can remain at equilibrium because payoff stationarity holds only along the agent’s own actuation directions.Source: 2) Jacobian Structure of Nonholonomic… · p. 6
  • If inter-block connection coefficients vanish, off-diagonal blocks simplify; if intra-block coefficients vanish, diagonal blocks become pure Hessian blocks.For a holonomic frame, all connection terms vanish and the characterization reduces to the coordinate-based result in.Source: 2) Jacobian Structure of Nonholonomic… · p. 7

3) Player-wise Involutivity of Control Distributions:

  • Player-wise involutivity requires each agent’s vector-field Lie brackets to remain in its own distribution, preventing direct interference with other agents’ directions through those brackets.Under the stated assumptions, each distribution is integrable and has leaves, but those leaves need not form a product structure for the manifold.Source: 3) Player-wise Involutivity of Control… · p. 7
  • Under player-wise involutivity, each diagonal Jacobian block is the intrinsic Hessian of the player’s payoff restricted to its leaf, making that block symmetric.The Hessian is computed with the metric induced on the leaf.Source: 3) Player-wise Involutivity of Control… · p. 7
  • Deception can tilt oblivious agents’ distributions so they are no longer involutive, motivating the general geometric stability analysis.The simplified diagonal-block result therefore need not apply to deceptive dynamics.Source: 3) Player-wise Involutivity of Control… · p. 7

4) Intra-Agent Commutativity of Vector Fields:

Under cross-player commutativity, the actuation distributions admit a local product structure with a block-diagonal metric and no inter-agent connection components. The section relates the resulting dynamics’ equilibria to leafwise Nash equilibria and, under additional conditions, to local Nash equilibria in product coordinates.

4) Intra-Agent Commutativity of Vector Fields:

  • Cross-player commutativity yields a local product decomposition compatible with the actuation distributions, a block-diagonal metric, and no inter-agent components in the Levi-Civita connection.The intrinsic Jacobian consequently simplifies; its off-diagonal blocks are payoff-Hessian terms.Source: 4) Intra-Agent Commutativity of Vector… · pp. 7–8

C. Leafwise Nash Equilibrium: Regularity and Stability

  • An equilibrium satisfying the stated first- and second-order conditions is a strict local leafwise Nash equilibrium, defined by each player’s strict local payoff maximum on its own leaf.The figure contrasts that own-leaf maximum with a payoff that need not be stationary along another player’s leaf.Source: C. Leafwise Nash Equilibrium:… · p. 8, Figure 5 · p. 8
  • Negative diagonal blocks of the intrinsic Jacobian suffice for a strict local leafwise Nash equilibrium under Assumptions 1–3.The diagonal blocks represent the Hessians of players’ payoffs restricted to their leaves.Source: C. Leafwise Nash Equilibrium:… · p. 8
  • For two players, a strict local leafwise Nash equilibrium is equivalent to a strict local Nash equilibrium in local product coordinates without Assumption 4.For three or more players, the corresponding construction requires the sum of the other players’ distributions to be involutive, which may fail even when each distribution is involutive.Source: C. Leafwise Nash Equilibrium:… · p. 9

D. Examples (continued)

The examples illustrate stable equilibria for the nonholonomic gradient-play dynamics. The section then turns to information asymmetry, analyzing dynamics induced by a deceptive feedback law that lets an informed agent bias steady-state behavior through its own control inputs.

D. Examples (continued)

  • In Example 4, x⋆ = (0, 0, 1) is the only equilibrium, and its intrinsic Jacobian −I₂ is Hurwitz, making it isolated and locally exponentially stable.The negative diagonal entries also imply a strict local leafwise Nash equilibrium.Source: D. Examples (continued) · p. 9
  • Figure 6 illustrates the Example 1 payoffs in ξ-coordinates, with red dots marking the corresponding functions’ maximizers.The accompanying example identifies (0, 0, 1) as the only equilibrium in its domain.Source: Figure 6 · p. 9, D. Examples (continued) · p. 9
  • In Example 5, for α ∈ (1/6, 2/3), the equilibrium is non-degenerate, isolated, and locally exponentially stable.Its intrinsic Jacobian is Hurwitz, and its negative diagonal entries imply a strict local leafwise Nash equilibrium.Source: D. Examples (continued) · p. 9

V. INFORMATION ASYMMETRY AND DECEPTION

  • Under asymmetric information, a privileged agent can bias steady-state behavior by manipulating only its own control inputs, without interfering with oblivious agents’ actions or measurements.The section analyzes the interaction dynamics induced by the deceptive feedback law; Figure 7 illustrates Example 2 payoffs in ξ-coordinates, with red dots marking maximizers.Source: §V · p. 9, Figure 7 · p. 10

A. Regularity and Stability under Deception

Deception tilts the victim’s effective actuation bundle, changing its projected-gradient dynamics while preserving exponentially stable equilibria for sufficiently small parameter changes. The examples show that this manipulation can shift equilibria without necessarily sacrificing stability.

A. Regularity and Stability under Deception

  • The deceptive agent tilts the victim’s actuation bundle, biasing its projected gradients and generically shifting equilibrium; the tilted fields remain spanning for all deception parameters.The block-triangular structure of the tilt separates invertibility from the tilt magnitude.Source: A. Regularity and Stability under… · p. 10
  • The tilted dynamics can again be represented as projected Riemannian gradients, using a metric adapted to the tilted actuation distributions.The victim’s bundle is tilted along the deceiver’s bundle, while the deceiver’s own bundle remains unchanged.Source: A. Regularity and Stability under… · p. 10
  • An exponentially stable equilibrium persists as a smooth family of exponentially stable equilibria on an open neighborhood of zero deception parameters.To first order, displacement is weighted by the victim’s payoff sensitivities along the deceiver’s directions; if those sensitivities vanish, deception is ineffective to first order.Source: A. Regularity and Stability under… · p. 10

B. Deceptive Nash Equilibria

  • The deceptive Nash criteria distinguish leafwise equilibria from strict local Nash equilibria in local product coordinates through their assumptions and second-order conditions.The corollary states separate sufficient conditions for each notion.Source: B. Deceptive Nash Equilibria · pp. 10–11
  • In Example 6, the equilibrium varies with the deception parameter, yet remains a non-degenerate isolated exponentially stable equilibrium for every real parameter value.Its derivative at zero is (0, 1, 0), and the example identifies this shift with the victim’s payoff sensitivity along the deceiver’s direction.Source: B. Deceptive Nash Equilibria · p. 11
  • Example 7’s equilibrium shift is exactly linear in the deception parameter, with admissible values restricted to an explicit bounded open interval in the non-degenerate regimes.For the stated admissible parameters, the equilibrium remains locally exponentially stable.Source: B. Deceptive Nash Equilibria · pp. 11–12

VI. PROOFS

The proof starts from the agents’ actuation distributions and their pointwise decomposition of the tangent space.

VI. PROOFS

  • At each state, the tangent space decomposes as a direct sum of the agents’ actuation distributions.Each distribution is regular and generated by that agent’s vector fields.Source: 1) Proof of Theorem 1 · p. 12
  • The decomposition is pointwise, holding for every state in the domain.The claim is stated under Assumption 1.Source: 1) Proof of Theorem 1 · p. 12
  • Each agent’s distribution is generated by its corresponding family of actuation vector fields.These distributions are the summands in the tangent-space decomposition.Source: 1) Proof of Theorem 1 · p. 12

1) Proof of Theorem 1:

The proof constructs a Riemannian metric from the actuation splitting so that each agent’s gradient contribution projects onto that agent’s distribution.

1) Proof of Theorem 1:

  • The proof builds a Riemannian co-metric that is block-diagonal with respect to the splitting into actuation distributions.Smoothness of the vector fields and projections yields a Riemannian metric.Source: 1) Proof of Theorem 1 · p. 12
  • Under this metric, orthogonal projection onto an agent’s distribution extracts that component of the gradient.The proof uses the dual decomposition of covectors and the block structure of the metric.Source: 1) Proof of Theorem 1 · p. 12
  • The projected gradient for each agent equals the sum of its payoff derivatives along its control fields, weighted by those fields.Combining this identity with the metric construction gives the projected-gradient form of the dynamics.Source: 1) Proof of Theorem 1 · p. 12

2) Proof of Proposition 2:

The stability proof bounds the linearization’s spectrum, while the subsequent intrinsic linearization calculation separates Hessian and connection contributions.

2) Proof of Proposition 2:

  • The proof uses a block Gershgorin argument to exclude eigenvalues with nonnegative real part, establishing that the linearization is Hurwitz.Similarity preserves the spectrum between the transformed operator and the original linearization.Source: 2) Proof of Proposition 2 · p. 13

3) Proof of Theorem 2:

  • The linearization is evaluated on arbitrary vector fields, then its components are extracted in the orthonormal actuation frame.The calculation applies the product rule and the covariant derivative of a one-form.Source: 3) Proof of Theorem 2 · p. 13
  • At equilibrium, stationarity removes a term in the linearization, and the Riemannian Hessian identity yields its intrinsic expression.The resulting formula includes both Hessian and connection contributions.Source: 3) Proof of Theorem 2 · p. 13

4) Proof of Proposition 3:

The proof rewrites payoff derivatives using the Riemannian Hessian and connection, then uses the assumptions on the actuation distributions to eliminate cross-player connection terms.

4) Proof of Proposition 3:

  • At equilibrium, the second derivative of a payoff along an agent’s control fields reduces to its Riemannian Hessian term after second-fundamental-form contributions cancel.The equilibrium condition makes the derivative along the projected connection term equal the contribution from the second fundamental form, which cancels in the Hessian identity.Source: 4) Proof of Proposition 3 · p. 14

5) Proof of Proposition 4:

  • The proof uses a global orthonormal frame to make the metric coefficients constant, so the Lie-derivative terms in the Koszul formula vanish.This leaves only bracket terms in the displayed connection expression.Source: 5) Proof of Proposition 4 · p. 14
  • Under Assumption 4, Lie brackets between distinct agents’ fields vanish, while Assumption 3 keeps same-agent brackets in their own orthogonal distribution, eliminating the cross-player connection terms.The proof checks the cases where the frame index belongs to neither player, the first player, or the second player.Source: 5) Proof of Proposition 4 · p. 14

6) Proof of Proposition 5:

The proofs establish strict local leafwise optimality from negative definite payoff Hessians, then show when this condition coincides with ordinary local Nash equilibrium in product coordinates.

6) Proof of Proposition 5:

  • A negative definite Riemannian Hessian at a stationary point makes each player’s payoff strictly smaller nearby on that player’s leaf, yielding a strict local leafwise Nash equilibrium.The proof applies the second-order sufficient condition on each integral manifold and obtains a neighborhood where unilateral deviations along the leaf lower payoff.Source: 6) Proof of Proposition 5 · p. 14

7) Proof of Proposition 6:

  • When each actuation distribution is involutive and distinct distributions commute, local product coordinates identify each player’s unilateral slice with its leaf, making strict local Nash and leafwise Nash equivalent.Under these assumptions, the Frobenius theorem supplies the local product diffeomorphism.Source: 7) Proof of Proposition 6 · p. 14
  • The proof of Proposition 6 verifies Proposition 5’s second-order condition by relating the nominal Jacobian’s diagonal block to the payoff Hessian on each player’s leaf.The stated Hessian condition therefore implies strict local leafwise Nash equilibrium, and hence strict local Nash equilibrium in the local product coordinates.Source: 7) Proof of Proposition 6 · pp. 14–15

8) Proof of Proposition 7:

The proofs show that the tilted actuation fields remain a basis, and that a nominal exponentially stable equilibrium persists as a locally exponentially stable equilibrium for nearby deception parameters. The conclusion connects deception to tilting the victims’ effective actuation directions and shifting equilibria.

8) Proof of Proposition 7:

  • The tilted fields remain a basis because their transformation matrix is block triangular with determinant one, so the tilted distributions retain a direct-sum decomposition.This establishes that the tilted bundles are well-defined across the state space.Source: 8) Proof of Proposition 7 · p. 15

9) Proof of Theorem 3:

  • The implicit function theorem gives a smooth, locally unique equilibrium branch from the nominal equilibrium for deception parameters near zero.The proof casts the deceptive equilibrium conditions as a smooth map whose state derivative at the nominal equilibrium is invertible.Source: 9) Proof of Theorem 3 · p. 15
  • The equilibria on that branch remain locally exponentially stable for deception parameters in an open neighborhood of zero.The deceptive Jacobian varies continuously and remains Hurwitz near its nominal Hurwitz value.Source: 9) Proof of Theorem 3 · p. 15

VII. CONCLUSION

  • Embedding victims’ exploration signals in the deceiver’s strategy tilts their effective actuation directions and generically shifts equilibria to locations of the deceiver’s choosing.The paper interprets this manipulation geometrically as a tilt of the oblivious agents’ perceived Riemannian geometry.Source: §VII · pp. 15–16
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