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Asynchronous Replanning in Two Population Linear Quadratic Mean Field Games: Information Requirements and Stability

Yuxin Jin, Wang Yao, Xiao Zhang

arXiv:2609.11424v2math.OCcs.SYeess.SY

TL;DR

Asynchronous continuation replanning requires a state–plan target that local aggregate observations may reveal even when the hidden initial input is not fully recoverable. The paper analyzes repeated responses, constructs a local causal implementation, and establishes stability and finite-prefix robustness results, with bounds limited to the stated settings.

  • Problem

    Replanning responses depend on the current aggregate state and opponent’s remaining plan, while a population observes only its own aggregate component and may not know the hidden initial belief.

    Source: §1 · pp. 1–2

  • Method

    The paper characterizes target recovery by a kernel-inclusion condition and studies asynchronous frozen-opponent responses using zero-extended plans and a limiting response cycle.

    Source: §1 · p. 2

  • Results

    For pre-terminal accumulation, spectral stability yields bounded physical execution, a finite state left limit, and strong plan convergence to the restarted equilibrium; on fixed finite prefixes, revision records match with probability tending to one and error is O(N_*^-1).

    Source: §1 · pp. 2–3

  • Takeaways & Limitations

    Local observations need determine the response-relevant state–plan target, not the entire hidden input, to initialize the continuation protocol.

    Source: §1 · p. 2

    Finite-population robustness is not uniform through Zeno accumulation and makes no ε-Nash claim for the revised finite-player process.

    Source: §4 · p. 20

Abstract

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We study asynchronous continuation replanning in a two-population linear--quadratic mean field game with deterministic open-loop controls and heterogeneous, possibly erroneous initial information. At exogenous opportunities, one population recomputes its continuation response from the physical aggregate state reached, with the opponent's active plan frozen. The state--plan target needed to initialize replanning is recoverable from local aggregate observations if and only if a kernel-inclusion condition holds; full recovery of the hidden initial input is unnecessary. For revisions accumulating before the terminal time, spectral stability of the limiting two-response cycle yields bounded physical execution and a finite state left limit. The zero-extended continuation plans then converge strongly to the mean field equilibrium restarted from that state. A scalar counterexample shows that well-posed continuation problems and a convergent physical state can coexist with divergent remaining plans. We also construct an exact causal implementation from local aggregate observations and establish finite-population robustness on every fixed finite opportunity prefix. Revision records match with probability tending to one, and the mean-square implementation error truncated to the matching event is \(O(N_*^{-1})\), where \(N_*\) is the smaller population size.

1. Introduction.

The introduction frames asynchronous continuation replanning around recovering the state–plan target from local observations and controlling repeated responses when revision times accumulate. It presents stability, implementation, and finite-population results, while distinguishing physical-state regularity from convergence of remaining plans.

1. Introduction.

  • A bounded initializer recovers the continuation state–plan target exactly when indistinguishable hidden inputs yield the same target; full recovery of the hidden input is unnecessary.Full observability is sufficient but not necessary, and the target initializes the entire protocol.Source: §1 · p. 2
  • For pre-terminal accumulation, spectral stability of the limiting response cycle bounds physical execution, gives a finite state left limit, and yields strong convergence to the restarted MFG equilibrium.The restarted equilibrium is determined by the state actually reached, which need not lie on the original complete-information equilibrium trajectory.Source: §1 · p. 2
  • Well-posed continuation problems and square-integrable execution can coexist with divergent continuation-plan norms, so regular physical execution alone does not ensure plan convergence.The scalar counterexample has genuine scheduled updates and a finite state left limit.Source: §1 · pp. 2–3
  • The protocol can be reproduced exactly on every finite chronological opportunity prefix using local aggregate observations, each population’s own plan, and the public revision record.The implementation starts from the recovered state–plan pair and regenerates the opponent plan after announced revisions.Source: §1 · p. 3
  • On each fixed finite opportunity prefix, revision records match with probability tending to one and the matching-event-truncated mean-square implementation error is O(N_*^-1).The estimate is not uniform through Zeno accumulation and makes no ε-Nash claim for the revised finite-player process.Source: §1 · p. 3
  • The paper distinguishes its physical-state-based, frozen-opponent continuation response from updates using history averages, current distributions, or short receding horizons.It situates repeated replanning alongside work on heterogeneous initial information [19] and related asynchronous adjustment approaches,,.Source: §1 · p. 3

2. Model and continuation response maps.

This section sets up a deterministic two-population LQ mean field game and defines initial plans, local observations, and population-wise continuation responses. Later responses use the physical boundary state and a frozen opponent plan, under local solvability conditions.

2.1. Baseline two-population LQ MFG.

  • The baseline model uses deterministic open-loop controls and a representative-state system with constant deterministic coefficients and a unique complete-information equilibrium under the stated assumptions.The equilibrium is the unique tuple satisfying representative-agent optimality and consistency with the aggregate mean.Source: §2 · p. 4, §2.1 · p. 4

2.2. Initial plans and local observations.

  • Each population knows its own initial mean but may use an erroneous estimate of the opponent’s mean to form its initial plan; execution follows the active plan until replacement.The hidden input is the opponent’s plan-generating belief, distinct from the physical initial state, while observations are local aggregate paths.Source: §2.2 · p. 5

2.3. Population-wise continuation response maps.

  • A continuation response on [τ, T] takes the physical boundary state and frozen opponent plan as inputs, without anticipating future revisions.The response is defined by representative-agent optimality followed by population consistency; continuation domains vary with τ.Source: §2.2 · p. 5, §2.3 · p. 5
  • Under the local frozen-response no-conjugate condition, each continuation problem is uniquely solvable and its response has a bounded affine representation.The stated bound scales with T − τ; neither no-conjugate condition asserts stability of repeated responses.Source: §2.3 · p. 6

3. Response-relevant information.

This section characterizes exactly which information local observations must provide to initialize replanning. The required state–plan target can be recoverable even when the full hidden input is not.

3.1. Response-relevant target and factorization.

  • Full observability is sufficient but not necessary for recovering the target, which initializes the entire continuation protocol and need not be minimal for one isolated response.The exact criterion is that indistinguishable inputs have the same target.Source: §3.1 · pp. 7–8
  • A bounded protocol initializer exists exactly when observationally indistinguishable hidden inputs produce the same state–plan target.The canonical initializer uses a Moore–Penrose inverse, whose norm is the smallest observation-to-target amplification among bounded factors.Source: §3.1 · p. 7

3.2. Recoverability without full observability.

  • Both populations’ state–plan targets admit bounded initializers in the example despite singular full observability Gramians.The derivative calculation identifies the observation kernels; a remaining hidden coordinate changes neither the actual opponent contribution nor its plan.Source: §3.2 · p. 8

4. Asynchronous continuation replanning and stability.

This section formalizes asynchronous continuation replanning from the physical state and the opponent’s frozen plan, examining stable response cycles, convergence to a restarted equilibrium, and the distinction between physical-state and plan convergence. It also describes a local-observation implementation that agrees with the ideal process on every finite opportunity prefix.

4. Asynchronous continuation replanning and stability.

  • The ideal recursion replans from the actual aggregate state using the opponent’s active plan, while executed portions remain fixed between opportunities.Opportunities are exogenous, and only the actor receives the current signal.Source: §4 · p. 9

4.1. Persistence and alternation of genuine revisions.

  • After a population adopts its response or passes with its current response, it keeps passing at later own opportunities until the opponent changes plans, so genuine revisions alternate.If both populations pass at their first opportunities, no revision occurs; otherwise the first revision is at its actor’s first opportunity.Source: §4.1 · p. 9

4.2. Mutual responses and stability of response cycles.

  • At each boundary, the mutual frozen-response equations have the unique continuation MFG equilibrium plan pair, and spectral radius below one is the stated response-cycle stability criterion.Source: §4.2 · p. 10

4.3. State limits at pre-terminal accumulation.

  • If executed controls are integrable up to a pre-terminal accumulation, the physical state has a finite left limit, even without convergence of the full plans.Square-integrable executed controls suffice because they are integrable on a finite interval.Source: §4.3 · p. 11

4.4. Convergence to the restarted continuation MFG equilibrium.

  • Under spectral stability and locally finite opportunities, executions stay bounded, the physical state has a finite left limit, and zero-extended plans converge strongly to the equilibrium restarted from that state.The restart uses the physical state actually reached, which need not lie on the original equilibrium trajectory.Source: §4.4 · pp. 12, 14

4.5. A counterexample: convergent state and divergent plans.

  • A well-posed scalar two-population model has a pre-terminal Zeno sequence with convergent execution but divergent remaining plans that do not approach the restarted equilibrium.Every scheduled update in the example is genuine.Source: §4.5 · p. 14

4.5. A counterexample: convergent state and divergent plans

  • The counterexample bounds physical boundary states by 0.1|b_k−1| while continuation amplitudes grow by factors between 3 and 5.1.The shrinking execution intervals allow the executed controls to remain square-integrable even as the plans diverge and the physical state has a finite left limit.Source: §4.5 · pp. 15–16
  • Well-posed continuation problems and a unique mutual frozen-response equilibrium coexist with continuation plans that diverge despite a finite physical-state left limit.The counterexample has square-integrable executed controls, but its continuation plans do not converge to the equilibrium restarted from the limiting state.Source: §4.5 · pp. 15–16
  • Local aggregate observations, each population’s own plan, and public revision events let populations maintain the state and opponent-plan inputs needed to match the ideal process on every finite opportunity prefix.Source: §4.5 · p. 16

5. Local implementation from aggregate observations.

The section specifies how populations initialize and update continuation plans from local aggregate information, then establishes agreement with ideal replanning on finite opportunity prefixes. It also addresses empirical-aggregate fluctuations with a vanishing tolerance.

5.1. Available local information and initialization.

  • Under initial recoverability, the initializer recovers the state–plan pair needed to start replanning from locally available information.The section distinguishes recovery of this target from recovery of the hidden initial input.Source: §5.1 · p. 16

5.2. Causal propagation and event updates.

  • Between revisions, a causal integral observer estimates the physical aggregate state while the stored opponent plan remains fixed and its unexecuted restriction shortens.The observer uses the locally observed aggregate and a known bounded measurable gain; a zero gain is allowed.Source: §5.2 · pp. 16–17
  • At an own opportunity, the population compares its candidate response with its active plan, passing on equality or adopting and announcing a genuine revision otherwise.After an opponent revision, it regenerates and replaces the stored opponent continuation plan.Source: §5.2 · p. 17

5.3. Exact local implementation.

  • On every finite chronological opportunity prefix, the local implementation matches ideal replanning in decisions, revision records, active plans, and physical aggregate path.Under Theorem 4.6, the implementation also inherits bounded execution, a state left limit, and strong plan convergence before the accumulation time.Source: §5.3 · pp. 17–18
  • Because empirical aggregates fluctuate around their mean-field values, the finite-population implementation uses a vanishing tolerance and analyzes record agreement and mean-square error on fixed finite prefixes.Exact equality of candidate and active plans is not a robust pass criterion under empirical fluctuations.Source: §6 · p. 18

6. Finite-population robustness.

On every fixed finite opportunity prefix, the analysis establishes robustness of the local implementation under finite-population sampling assumptions. It controls reconstruction and plan errors and bounds the chance of revision-record disagreement.

6. Finite-population robustness

  • Local observations and the population’s own plan support empirical residual reconstruction, state propagation, and opponent-plan regeneration after revisions.The observer uses the empirical residual while retaining the known base trajectory and law means.Source: §6 · p. 18
  • On each fixed finite prefix, revision records agree with high probability and mean-square implementation error is controlled on the matching event.The theorem assumes finite-population sampling conditions and a fixed exogenous opportunity grid.Source: §6 · pp. 18–19
  • Initialization errors are controlled by the aggregate fluctuation estimate, while active-plan errors initially vanish.The proof uses sampling bounds, BDG, and Gronwall on the interval before the first opportunity.Source: §6 · p. 19

2. Inter-event propagation.

Between opportunities, the proof propagates state and plan errors from one post-opportunity boundary to the next. Its bounds depend on preceding errors and aggregate martingale fluctuations.

2. Inter-event propagation

  • State suprema on an inter-event interval are bounded by preceding state errors, L2 plan errors, and the aggregate martingale supremum.The errors on the interval are controlled by post-opportunity errors at its left boundary; restricting plans cannot increase their L2 norm.Source: §2 · p. 19
  • The inter-event estimate is applied after multiplying by the event that the preceding opportunity records match, without requiring conditional martingale properties.The pathwise bounds and the stated estimate therefore support error control on matching prefixes.Source: §2 · p. 19
  • A pathwise dependence estimate applies to random inputs during propagation.The passage introduces the estimate as the next step in controlling inter-event errors.Source: §3 · p. 19

3. Response and regeneration.

At a common revision, the acting population takes its candidate response and the other population regenerates the acting population’s plan. The resulting bounds close an induction over the finite sequence of updates.

3. Response and regeneration

  • At a common revision, the actor adopts its candidate and the non-actor regenerates the actor’s plan.The regenerated-plan bound uses the response-dependence estimate.Source: §3 · p. 20
  • A common pass only restricts the existing plans, while the revision update requires bounds for candidate and regenerated plans.The response estimate also bounds the non-actor’s regenerated plan.Source: §3 · p. 20
  • Pathwise error bounds close the induction on matching events across a fixed finite number of intervals and updates.Taking the maximum over finitely many intervals and updates changes only the constant.Source: §3 · p. 20

4. Decision consistency.

Decision consistency is controlled by bounding first-mismatch probabilities at individual opportunities and summing over a fixed prefix. The robustness constant is not asserted to remain uniform as the number of opportunities grows.

4. Decision consistency

  • Chebyshev bounds the probability of a first decision mismatch at an opportunity using the implementation-error bound and the decision margin.The supplied passage gives bounds involving the population size and the squared margin.Source: §4 · p. 20
  • Summing the individual mismatch bounds over the fixed number of opportunities yields the stated prefix estimate.The argument takes the tolerance no greater than one.Source: §4 · p. 20
  • The robustness constant is not claimed uniform as the opportunity count grows, so the result does not extend to finite-population approximation through Zeno accumulation.The result concerns implementation robustness and makes no ε-Nash claim for the revised finite-player process.Source: §4 · p. 20
  • The consistency analysis concerns successive decisions, while the broader replanning problem also requires control of the full remaining plans.The paper distinguishes decision/physical execution behavior from convergence of continuation plans.Source: §7 · p. 20

7. Conclusion.

The conclusion presents exact causal implementation from local aggregate observations and finite-population robustness on fixed finite opportunity prefixes. It identifies the limits of those guarantees and two directions for further work.

7. Conclusion.

  • Revision records match with probability tending to one, and the mean-square implementation error on the matching event is O(N_*^-1).N_* is the smaller population size.Source: §7 · p. 20
  • The robustness estimates are limited to fixed finite opportunity prefixes; they are not uniform through Zeno accumulation and make no ε-Nash claim.Source: §7 · p. 20
  • The conclusion leaves open robustness to noisy or sampled observations and conditions for convergence at the spectral boundary r(QZ) = 1.Source: §7 · p. 20

Appendix A. Baseline Open-Loop LQ–MFG Derivation.

The appendix establishes unique deterministic open-loop optimization and equilibrium, and characterizes continuation boundary solvability through a global Riccati chart. It also introduces actor-first coordinates for the local frozen-response derivation.

Appendix A. Baseline Open-Loop LQ–MFG Derivation.

  • For every continuation boundary, the global problem has a unique solution exactly when the chart is invertible; singularity yields a zero-state homogeneous solution and unattainable inhomogeneous data.Source: Appendix A · p. 21
  • The complete-information deterministic open-loop mean-field equilibrium is unique under the stated theorems.Source: Appendix A · pp. 21–22
  • For fixed deterministic aggregates, the representative-agent problem has a unique optimizer, characterized through its adjoint along the optimal mean state.Source: Appendix A · p. 22

Appendix B. Local Frozen-Response Derivation.

  • The local frozen-response derivation uses actor-first coordinates that order the acting population's aggregate before the opponent's.Source: Appendix B · p. 22

B.1. Frozen boundary system and local Riccati chart.

The appendices characterize frozen continuation responses, their regularity and operator continuity as the replanning boundary moves, and calculations for a divergent-plan counterexample. They also establish compactness and short-horizon spectral stability under the stated bounds.

B.1. Frozen boundary system and local Riccati chart.

  • A frozen response is representative-agent optimal and population-consistent under the stated control characterization; its boundary problem is uniquely solvable when the local chart is invertible.Source: §B.1 · p. 23

B.2. Affine response formulas.

  • Continuation responses have uniformly bounded C1 representatives across the stated continuation intervals.Source: §B.2 · p. 24

B.3. Compactness and a coefficient-level gain bound.

  • The response operator is Hilbert–Schmidt, hence compact, and sufficiently short remaining horizons are spectrally stable.Source: §B.3 · p. 25

Appendix C. Continuity of the response at a moving boundary.

  • Zero-extended response maps vary continuously in their stated operator norms as the continuation boundary moves, including at the terminal time.Source: Appendix C · pp. 25–26

Appendix D. Calculations for the divergent-plan counterexample.

  • The constant-plan fixed point is unique, and the cycle gain tends to -4 as τ approaches 1.Source: Appendix D · p. 26
  • In the counterexample calculations, scheduled updates remain genuine and |b_k| > 3^k|b_0|, while the executed-control estimate holds.Source: Appendix D · p. 27
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