{"id":"9c3d7e7c-ab7b-4d66-a8f1-3ade19afa24f","arxiv_id":"2505.06691","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A distributed, event-triggered, model-free algorithm is proven to drive N-player quadratic games near the unique Nash equilibrium, with a guaranteed minimum time between communication events.","lead":"This paper gives competitors a recipe: probe, estimate, and broadcast only when something changes enough, using no model of the game. The authors prove that the players' actions settle near the balance point where nobody can improve alone, and that updates can never come infinitely fast.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The O(1/omega) averaging bridge (73) is load-bearing for Theorem 1, but Plotnikov's theorem requires a T-periodic state-only map and the event-triggered system (34)-(35) is not one; Appendix A concedes the event-time discontinuities are not periodic.","rationale":"Read in full and in good faith. The average-system Lyapunov argument in Section 5.A is internally coherent, and the decomposition of Ghat into a time-varying matrix H(t), a zero-mean disturbance Delta(t), and a quadratic term is standard for local ES analysis. The decisive issue is the transfer from the average system back to the event-triggered original system. Theorem 1's bound (50) and the Zeno lower bound (90) both add an O(1/omega) term to average-system estimates, and that term is justified only by (73)/(87), which invoke Plotnikov's theorem. The hypotheses of that theorem are not verified: the event-triggered vector field depends on the last broadcast values, so it is not a T-periodic function of the instantaneous state; the Appendix's statement that the solutions have no jumps does not address the missing T-periodicity. I therefore agree with the reader's identified weakest assumption. A second, independent defect is in the Zeno calculation: solving (85) with initial condition zero gives tilde_phi = sqrt(p/q)/(1-(||KH||/omega)t) - sqrt(p/q), not (86), whose denominator contains an extra q/p; consequently the displayed tau* in (90) is not derived. This does not prove the algorithm is wrong, but it confirms the proof is incomplete. The plausibility of the construction and the authors' prior results keep the appropriate verdict at CONDITIONAL; my stress test does not move it.","tokens_in":23236,"tokens_out":21032,"duration_ms":204763,"concrete_test":"Take N=2 and construct two admissible closed-loop trajectories that at the same tbar have identical instantaneous state (Ghat, thetatilde) but different event histories, e.g., player 1's last trigger at tbar=0 with held value Ghat_1(0)=1 versus tbar=0.1 with held value Ghat_1(0.1)=0.5, while current Ghat_1=0.8 in both. Evaluate the right-hand sides of (34)-(35) at that instant; if they differ, the vector field F in (37) is not a function of (tbar,X), so Plotnikov's Theorem 2 cannot be applied as stated. To check whether the conclusion nonetheless survives, simulate the full event-triggered loop and the average system (44)-(45) with identical high omega and compare max_{t in [0,L]} ||theta(t)-theta_av(t)|| against the claimed O(1/omega) bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem's central transfer is the O(1/omega) closeness (73), because it converts average-system exponential stability into the original-system bound (50) and also feeds the dwell-time bound via (87). Plotnikov's Theorem 2 in Appendix A requires the multivalued map X(tbar,x) in (A.1) to be T-periodic in tbar and Lipschitz in x. The closed loop (34)-(35) is not put in that form: the error e(tbar) in (27) contains Ghat_i(t_i_kappa), the value at the player's last event time, and t_i_kappa is generated by the state-dependent rule (31). Hence the right-hand side is a functional of the past trajectory, not a function of the instantaneous state (Ghat, thetatilde). Section 4.1 asserts that the system 'maintains its periodicity over time due to the periodic probing and demodulation signals,' but no argument shows the event schedule is periodic or that the triggering rule preserves T-periodicity of the inclusion. Appendix A explicitly concedes that 'the discontinuities induced by the increasing sequence of event times are not periodic' and then asserts the theorems apply; that assertion is the unproved premise. Without a constructed T-periodic, Lipschitz X(tbar,x), Eq. (73), the residual bound (50), and the Zeno bound (90) lack support.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a distributed event-triggered extremum-seeking scheme for N-player noncooperative games with unknown quadratic payoff functions. Each player injects sinusoidal dither, demodulates its own payoff to estimate its pseudo-gradient, and updates/broadcasts the pseudo-gradient estimate only when a local static triggering condition is violated. The main result (Theorem 1) claims local exponential stability of an averaged event-triggered system and, for the original system, convergence to a neighborhood of the Nash equilibrium of radius O(a + 1/ω), together with a uniform positive lower bound on inter-event times that excludes Zeno behavior. The proof combines time scaling, a Lyapunov function for the average system, Plotnikov's averaging theorem for discontinuous right-hand sides, and a comparison-based dwell-time estimate. A four-player oligopoly simulation illustrates the proposed scheme.","tokens_in":23420,"tokens_out":10720,"duration_ms":114025,"significance":"If the averaging step were rigorously justified, the paper would make a useful contribution: it addresses a genuinely open combination of model-free Nash equilibrium seeking with event-triggered communication, provides quantitative residual bounds, and gives an explicit Zeno-avoidance guarantee. The decentralized, per-player triggering design and the independent pseudo-gradient estimates are attractive features. The Lyapunov analysis of the average system is internally coherent, and the simulation supports the qualitative claims. However, the central transfer from the average system to the original closed loop currently rests on an application of Plotnikov's averaging theorem whose hypotheses are not verified; this gap is load-bearing for both the convergence bound (50) and the dwell-time bound (90).","major_comments":[{"comment":"Plotnikov's averaging theorem (Appendix A, Theorems 2 and 3) is applied to a closed-loop system that does not satisfy the theorem's hypotheses. The theorem requires the multivalued map X(tbar, x) to be T-periodic in tbar and Lipschitz in x. In the closed loop (34)–(35), the error e(tbar) defined in (27) contains Ghat_i(t_i_kappa), the value at the player's last event time, and t_i_kappa is generated by the state-dependent rule (31). Hence the right-hand side is a functional of the past trajectory rather than a function of the instantaneous state, and there is no reason for the event schedule to be T-periodic. Appendix A explicitly concedes that \"the discontinuities induced by the increasing sequence of event times are not periodic,\" but then asserts that the theorems nevertheless apply. No argument or construction of a T-periodic, Lipschitz map X(tbar,x) is given. Consequently, the O(1/ω) closeness estimate (73), the residual bound (50), and the Zeno lower bound (90) are not established as they stand.","section":"Section 4.1 and Appendix A, Eqs. (34)–(37), (73)"},{"comment":"The average system (44)–(46) is not obtained from the averaging operation (38)–(39). The averaging integral freezes the state and averages the explicitly time-periodic terms H(tbar), dH(tbar)/dtbar, Delta(tbar), and dDelta(tbar)/dtbar, but the event-triggered error e(tbar) is not a T-periodic function of tbar and is not averaged by the computation in (40)–(43). Replacing e(tbar) with e_av(tbar) defined through the \"average\" event-triggering rule (48) constructs a different switched system. No theorem is presented that connects this constructed average system to the original switched system (34)–(35); the stability of (44)–(45) therefore does not, by itself, imply the claimed behavior of the original system.","section":"Section 4.2, Eqs. (38)–(46)"},{"comment":"The Zeno-avoidance proof for the original system inherits the averaging gap. Equation (87) asserts |phi(t) - phi_av(t)| <= O(1/ω) by invoking the same Theorem 2 of [64], but phi(t) is built from the original event-triggered error and pseudo-gradient signals, whose relation to the average variables is precisely what is not established. Even if trajectory closeness (73) were available, the closeness of the ratios |e(t)|/|Ghat(t)| would require a separate argument near points where Ghat(t) vanishes. Thus the lower bound tau* in (90) is unsupported, and with it the claim that Zeno behavior is avoided for the original system.","section":"Section 5.B, Eqs. (87)–(90)"}],"minor_comments":[{"comment":"The matrix order in (44) appears inconsistent: from (34) and the relation G_av = H theta_av, the average dynamics should read dG_av/dtbar = (1/ω) H K G_av + (1/ω) H K e_av, not (1/ω) K H G_av + (1/ω) K H e_av. Since H K and K H are similar, the stability conclusion is unaffected, but the text, the Lyapunov equation in (52), and the norm bound in (56) mix the two orderings and should be made consistent.","section":"Eq. (44)"},{"comment":"In the bound for ||e_av(tbar)||, the summation index is written as j while the terms are |e_av_i(tbar)|; the index should be i.","section":"Eq. (54)"},{"comment":"The parameter list sets sigma1 = 0.65 and then repeatedly lists sigma1 = 0.75; the second occurrence should presumably be sigma3 = 0.75.","section":"Section 6, simulation parameters"},{"comment":"The third payoff function is labeled J2(t) in (93); it should be J3(t).","section":"Eqs. (91)–(94)"},{"comment":"The text refers to \"Fig. 7.1\" when describing the closed-loop block diagram; the correct reference appears to be Fig. A.1.","section":"Appendix A"},{"comment":"The bound in (82) should be (1 + ||e_av||/||G_av||)^2 rather than 1 + (||e_av||/||G_av||)^2; the subsequent inequality (83) implicitly uses the squared sum, so this appears to be a typo rather than a substantive error.","section":"Eq. (82)"}],"recommendation":"major_revision","confidential_remarks":"The central gap is confined to the averaging bridge: the paper needs either a valid argument showing that the closed-loop inclusion satisfies Plotnikov's hypotheses, a reformulation of the system for which a suitable averaging theorem applies, or a direct Lyapunov/perturbation analysis that avoids the unjustified averaging step. The Lyapunov core and the overall design idea are plausible, and the issue appears fixable in revision, so I would not reject on the basis of this gap alone."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read arXiv:2505.06691. The new thing here is real within its subfield: N-player quadratic games, model-free extremum seeking, distributed event-triggered broadcasts, no sharing of payoff or action info. That combination hasn't been done before, and the authors' own prior work on event-triggered ES for static maps plus a duopoly conference version does not preempt it. The average-system part (Section 5.A) is clean. The candidate Lyapunov function, the triggering inequality |e| <= sigma|G|, the exponential decay rate with sigma — that all hangs together. The dwell-time derivation for the average system is also standard, and the Zeno bound is at least the right shape.\n\nBut there is a load-bearing gap at the bridge between the average system and the original system. The paper invokes Plotnikov's averaging theorem, which requires a T-periodic, Lipschitz multivalued map X(t,x) in t. The closed loop (34)-(35) has event times generated by the state-dependent rule (31). Those are not periodic in general, and the paper never proves they are. Appendix A concedes 'the discontinuities induced by the increasing sequence of event times are not periodic' and then simply asserts the theorems apply. That assertion is doing all the work for Eq. (73), the O(1/omega) closeness that transfers exponential stability to the original system and also feeds the dwell-time bound. Without it, Theorem 1's residual bound and the Zeno lower bound are unsupported. The stress-test note is right on this.\n\nThere is also a smaller mechanical error in the Zeno bound. The claimed solution (86) does not solve the comparison ODE (85); the denominators don't match. So the explicit tau* in (90) is not justified as written, even for the average system.\n\nOn the positive side, the paper is not circular. The averaging computation (40)-(43) is independent, and the trigger condition is a premise, not a conclusion. The simulations show the expected behavior, though there's no code, and there's a typo in the sigma parameters (sigma1 twice). Minor stuff.\n\nWho is this for? People working on extremum seeking and event-triggered control, especially the Krstic group lineage. It would be a useful conference/journal paper once the averaging bridge is fixed. As is, it deserves a serious referee — the idea is good, and the gap is identifiable rather than fatal. My recommendation: send to peer review, but the referee should ask for a proper justification of the averaging theorem, or a workaround (e.g., a self-triggered or periodic-event formulation, or a two-timescale argument that doesn't need periodicity of events). If that can't be done, the theorem is not established.","headline":"First model-free event-triggered Nash seeking for N-player games, but the averaging bridge to the original system is not actually shown.","tokens_in":24077,"tokens_out":3886,"would_cite":false,"duration_ms":33297,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A10","93C57","93D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"N players with unknown quadratic payoffs can be driven to the unique Nash equilibrium by distributed event-triggered pseudo-gradient feedback, up to a residual of order $\\mathcal{O}(a + 1/\\omega)$, with a guaranteed minimum time between…","keywords":["Nash equilibrium seeking","event-triggered control","extremum seeking","noncooperative games","pseudo-gradient estimation","averaging for discontinuous systems","Zeno behavior avoidance","model-free optimization"],"falsifier":"Simulate a two-player version with rational probing frequencies and a nonzero initial error, and record the event times modulo the common period $T$ over several periods. If the pattern of event times differs from one period to the next, the vector field in (37) is not $T$-periodic, and the averaging theorem cannot supply the $\\mathcal{O}(1/\\omega)$ closeness estimate on which the residual bound (50) rests; one could also check directly whether the measured inter-event intervals stay above the claimed lower bound $\\tau^*$ as the state approaches equilibrium.","tokens_in":22873,"feed_emoji":"🎯","tokens_out":14055,"duration_ms":121982,"temperature":0.7,"pith_summary":"The paper proposes the first model-free, distributed event-triggered scheme for Nash equilibrium seeking in $N$-player noncooperative games whose payoff functions are unknown quadratics. Each player injects a small sinusoidal perturbation into its own action, demodulates its own payoff to form a pseudo-gradient estimate, and broadcasts that estimate only when the error between the current estimate and the last broadcast crosses a state-dependent threshold. The authors claim that for sufficiently large probing frequency $\\omega$ the average closed-loop system is locally exponentially stable at the unique Nash equilibrium, while the true system converges to a residual ball of radius $\\mathcal{O}(a + 1/\\omega)$ and never exhibits Zeno behavior. If true, this closes the gap between extremum-seeking Nash equilibrium seeking, which needs continuous communication, and event-triggered control, which makes communication aperiodic to save bandwidth.","feed_headline":"Event-triggered probing finds Nash equilibria in unknown games","feed_subtitle":"Players broadcast occasionally and still converge to the unique Nash equilibrium, with no payoff models and no Zeno","key_machinery":"The machinery has three parts. The pseudo-gradient estimate $\\hat{G}_i(t) = (2/a_i)\\sin(\\omega_i t)\\,J_i(\\theta(t))$ uses sinusoidal demodulation to expose the average gradient $H\\tilde{\\theta}(t)$ while all other terms have zero mean; this is what lets players work without knowing their payoff functions. The static event-triggering rule $\\sigma_i|\\hat{G}_i(t)| - |e_i(t)| < 0$ decides when the zero-order hold refreshes the broadcast $\\hat{G}_i(t_i^\\kappa)$, generating the piecewise-constant tuning law. The proof runs the closed loop in the scaled time $\\bar{t} = \\omega t$ and invokes the averaging theorem for differential inclusions with discontinuous right-hand sides, together with a Lyapunov function for the average system and a comparison argument for the dwell time, to obtain exponential convergence and the positive bound $\\tau^*$ on inter-event intervals.","core_discovery":"Under strict diagonal dominance of the game matrix $H$ and a frequency-separation condition on the probing signals, the event-triggered tuning law $u_i(t) = K_i \\hat{G}_i(t_i^\\kappa)$ with trigger times set by $\\sigma_i|\\hat{G}_i(t)| - |e_i(t)| < 0$ drives the action estimate $\\hat{\\theta}(t)$ to the Nash equilibrium $\\theta^*$ within the error bound $\\|\\theta(t)-\\theta^*\\| \\le M_\\theta e^{-mt}\\|\\theta(0)-\\theta^*\\| + \\mathcal{O}(a + 1/\\omega)$. The average system obtained from the averaging theorem for discontinuous right-hand sides has $\\hat{G}_{\\mathrm{av}} = 0$ locally exponentially stable, and a lower bound $\\tau^*$ on inter-execution times rules out infinitely many updates in any finite interval. The residual term reflects the persistent sinusoidal dither and the finite probing frequency, so the convergence guarantee is practical rather than asymptotic.","pith_inferences":["As an extension not analyzed in the paper, time-varying dither amplitudes $a_i(t)$ decaying to zero near equilibrium could in principle remove the $\\mathcal{O}(a)$ part of the residual and yield asymptotic instead of practical convergence; the proof as written treats constant amplitudes only.","The analysis assumes a strictly diagonally dominant, hence unique, Nash equilibrium. For games with merely local diagonal dominance or multiple equilibria, a modified triggering condition or a projection mechanism would be needed; the paper does not address those cases.","Because each player's trigger threshold $\\sigma_i$ is chosen independently, heterogeneous players can trade bandwidth against accuracy: a larger $\\sigma_i$ reduces broadcasts but enlarges that player's contribution to the residual set.","The $\\mathcal{O}(1/\\omega)$ closeness between true and average trajectories rests on the periodicity premise of the averaging theorem; a direct numerical check of whether event-time patterns repeat with period $T$ would show whether that premise holds in the original system."],"forward_implications":["Any $N$-player game with strictly diagonally dominant quadratic payoffs can be solved online using only each player's own payoff measurement and its own trigger logic; no payoff models and no inter-player communication of actions are needed.","Raising the probing frequency $\\omega$ and lowering the dither amplitudes $a$ shrinks the guaranteed residual ball $\\mathcal{O}(a + 1/\\omega)$, at the cost of a slower effective convergence rate and more demanding probing signals.","The positive dwell time $\\tau^*$ means the event-triggered law can be implemented on digital hardware that samples faster than $\\tau^*$, so the scheme does not rely on infinitely fast switching.","The local stability of the average system plus the input-to-state stability bound with respect to the measurement error $e$ suggests the same static-triggering structure tolerates small measurement noise without losing convergence to a slightly larger residual set.","The paper claims this is the first combination of extremum seeking and event-triggered communication for noncooperative games, opening the design to networked settings where communication bandwidth is the scarce resource."],"supporting_citations":[{"why":"Supplies the continuous-time Nash equilibrium seeking baseline, the sinusoidal probing structure, and the strict diagonal dominance assumption that the event-triggered design extends.","marker":"[24]"},{"why":"Supplies the averaging theorem for differential inclusions with discontinuous right-hand sides that justifies the $\\mathcal{O}(1/\\omega)$ closeness between true and average trajectories.","marker":"[64]"},{"why":"Introduces event-triggered extremum seeking for single systems, the design pattern that this paper extends to multi-player games.","marker":"[68]"},{"why":"Provides the event-triggered and periodic event-triggered extremum seeking results whose input-to-state stability and dwell-time techniques are adapted to the $N$-player setting.","marker":"[69]"},{"why":"Supplies the multi-sinusoidal probing-frequency separation condition and multivariable extremum seeking machinery used to estimate the pseudo-gradients.","marker":"[26]"},{"why":"Provides the ratio function and comparison argument used to lower-bound inter-execution intervals and exclude Zeno behavior.","marker":"[27]"},{"why":"Provides the comparison lemma, eigenvalue inequalities, and Lyapunov stability tools used for the average system.","marker":"[39]"},{"why":"Supplies the static event-triggering condition format $\\sigma|\\text{state}| - |\\text{error}| < 0$ used in Definitions 1 and 2.","marker":"[31]"}],"fun_headline_variants":["Event-triggered probing finds Nash equilibria without payoff models","Zeno-free event-triggered Nash seeking for unknown games","Distributed event-triggered Nash equilibrium seeking in games","Practical convergence to Nash via event-triggered estimates","First model-free event-triggered solution for noncooperative games"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is that the event-triggered closed-loop system is $T$-periodic in time, with the same period $T$ as the probing signals, so that the averaging theorem for discontinuous right-hand sides applies; the event times themselves are state-dependent and are never proved to be periodic.","fun_headline_variants_meta":{"raw":{"variants":["Event-triggered probing finds Nash equilibria without payoff models","Zeno-free event-triggered Nash seeking for unknown games","Distributed event-triggered Nash equilibrium seeking in games","Practical convergence to Nash via event-triggered estimates","First model-free event-triggered solution for noncooperative games"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000765,"raw_usage":{"total_tokens":3377,"prompt_tokens":914,"completion_tokens":2463,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":2374}},"tokens_in":530,"tokens_out":2463,"duration_ms":16927,"temperature":1.0,"reasoning_tokens":2374,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:39:14.920949+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a two-player version with rational probing frequencies and a nonzero initial error, and record the event times modulo the common period $T$ over several periods. If the pattern of event times differs from one period to the next, the vector field in (37) is not $T$-periodic, and the averaging theorem cannot supply the $\\mathcal{O}(1/\\omega)$ closeness estimate on which the residual bound (50) rests; one could also check directly whether the measured inter-event intervals stay above the claimed lower bound $\\tau^*$ as the state approaches equilibrium.","supporting_citations":[{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the continuous-time Nash equilibrium seeking baseline, the sinusoidal probing structure, and the strict diagonal dominance assumption that the event-triggered design extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the averaging theorem for differential inclusions with discontinuous right-hand sides that justifies the $\\mathcal{O}(1/\\omega)$ closeness between true and average trajectories."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the event-triggered and periodic event-triggered extremum seeking results whose input-to-state stability and dwell-time techniques are adapted to the $N$-player setting."},{"cited_title":"Ghaffari, M","cited_arxiv_id":null,"evidence_quote":"Supplies the multi-sinusoidal probing-frequency separation condition and multivariable extremum seeking machinery used to estimate the pseudo-gradients."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Provides the ratio function and comparison argument used to lower-bound inter-execution intervals and exclude Zeno behavior."},{"cited_title":"PrenticeHall,UpperSaddle River, New Jersey, 2002","cited_arxiv_id":null,"evidence_quote":"Provides the comparison lemma, eigenvalue inequalities, and Lyapunov stability tools used for the average system."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the static event-triggering condition format $\\sigma|\\text{state}| - |\\text{error}| < 0$ used in Definitions 1 and 2."}],"review_version":1}