{"id":"7a2950a2-2033-4f0f-9b72-88744c43130f","arxiv_id":"2607.27128","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"IVFEs are poly-time computable offline yet not learnable by smooth no-swap-regret dynamics, so rationalizable learning must generate valuable information.","lead":"Information-value-free equilibria can be computed offline in polynomial time but cannot be learned online by standard smooth no-swap-regret algorithms. The result implies that rationalizable learning in games inevitably creates valuable implicit information, challenging antitrust rules that ban such exchange among pricing algorithms.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"Theorem 3.3 appears internally sound, but it is an existential statement over opponent-calibrated game sequences; in the canonical pricing/auction games motivating the policy claims, smooth no-swap-regret learners provably converge to Nash (⊂ IVFE), so the abstract's categorical \"cannot avoid\" over-","rationale":"The reader's verdict (CONDITIONAL, high confidence, low correctness risk) matches my own reading, so I recommend no adjustment. I verified the main proof components rather than taking them on faith: the LP feasibility argument in Theorem 2.3 is correct (existence via Nash, enumeration over best-fixed-action pairs); the contradiction in §B.2 is arithmetically sound (Lemma B.1 gives period-one payoff ≥ 6βT − o(T) via A = Ω(T) from the smoothness gap P−Q = Ω(T), Lemma B.2 caps it at (5/2)βT + o(T) via the clean algebraic identity G_M = −G_L − β in Eq. B.17); and Proposition 3.2's smoothness verifications use correct closed-form sums (Laplace-difference trick for FTPL, integral bounds for MWU). Minor blemishes (a trivially-satisfied \"if η ≤ A/2T\" guard, a stray \"2c/β\" typo for 2η/β) do not affect validity. Where I diverge from the reader is on what is most load-bearing. The reader's weakest_assumption is the smooth-learner scope condition. That is a real limitation, but it bounds the theorem's generality, not the interpretation of what is proven. I find the sharper soft spot one level up: even accepting that all deployed pricing algorithms are smooth no-swap-regret learners, the theorem only furnishes a per-opponent, per-horizon tailored hard game family, while results the paper itself cites establish that in first-price auctions and Bertrand games — the literal subject of the algorithmic-collusion policy debate — those same smooth learners converge to Nash, hence to IVFE, hence generate no valuable information. The abstract's \"cannot avoid\" phrasing and §4.1's regulatory incompatibility claim are therefore theorem-strength only as worst-case existential statements, and the paper's brief concession (\"one could argue that our worst-case construction might not reflect representative scenarios\") understates this, because the counterexamples are not hypothetical but published convergence theorems in the motivating domain. This supports keeping the verdict CONDITIONAL: accept the theory, condition the policy sentence. Hence agreement_with_reader = partial (same family of concern — reach of the impossibility — but a different, arguably sharper weakest link), and verdict_should_be = UNCHANGED.","tokens_in":29250,"tokens_out":8594,"duration_ms":187292,"concrete_test":"Run Blum–Mansour-reduced MWU (smooth and no-swap-regret per Prop 3.2) against itself in (a) the first-price auction instance of Deng et al. (2022, Thm: Nash convergence when ≥2 bidders tie for highest value) and (b) the Bertrand pricing game of Calvano et al. (2020)/Bichler et al. (2024), for T up to ~10^6, computing d_G(τ_T) per Definition 2.8. If d_G(τ_T) → 0 (as the published Nash-convergence results predict), then in the canonical algorithmic-collusion markets rationalizable learning provably avoids generating valuable information, and the abstract's \"cannot avoid\" sentence and the antitrust-incompatibility claim must be downgraded to worst-case statements. If d_G instead stays bounded away from 0 in these natural games, the policy claim gains real support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem's quantifiers are: for every no-swap-regret L and every smooth A, there EXISTS a game sequence {G_T} with liminf d_{G_T}(τ_T) ≥ c. The construction (§B.2) is calibrated to A itself: the payoffs depend on w̄ and β = A/8T, which are statistics of A's own weight sequence in the two-arm experiment with ε = ε(2T), and each horizon gets a different game. This is a legitimate diagonal/worst-case lower bound, and I found no correctness gap (the B.1/B.2 contradiction arithmetic, the B.6 accounting, and Prop 3.2's closed-form MWU/FTPL/OGA computations all check out). The load-bearing problem is the upgrade from this existential statement to the abstract's and §4.1's near-universal claim that valuable information exchange is \"unavoidable under rationalizable learning\" and hence that antitrust doctrine is incompatible with it. There is concrete countervailing evidence inside the paper's own frame: Prop 3.2 certifies MWU, FTPL, and OGA as smooth learners, and the cited collusion literature (Deng et al. 2022; Bichler et al. 2024; Ahunbay–Bichler 2025) proves exactly these mean-based/gradient learners converge to Nash equilibrium in first-price auctions and Bertrand competition. Since Nash ⊂ IVFE (Prop 2.2), in those games d_G(τ_T) → 0: rationalizable, smooth, no-swap-regret learning reaches information-value-free outcomes precisely in the markets the regulation debate is about. This is consistent with Theorem 3.3 (the hard games are simply elsewhere), but it shows the hard family is disjoint from the motivating application class. The policy incompatibility claim therefore needs an extra premise — that real algorithmic-pricing markets structurally resemble the adversarial G_T — which the paper does not supply and only half-acknowledges. This is distinct from the reader's flagged smoothness scope: it limits the conclusion's reach even granting that all pricing algorithms are smooth.","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper introduces information-value-free equilibrium (IVFE): a correlated equilibrium in which each player's equilibrium payoff is no higher than what some fixed action would yield against the opponents' equilibrium play (Definition 2.1). The authors show NE ⊊ IVFE ⊊ CE (Proposition 2.2) and that an IVFE always exists and is computable in polynomial time via finitely many LPs (Theorem 2.3). The main result (Theorem 3.3) is a lower bound: for any no-swap-regret learner L facing any \"smooth learner\" A (Definition 3.1, a gradual-response condition on two-arm experiments), there exists a sequence of games {G_T} such that empirical play stays a constant distance from every IVFE. Proposition 3.2 certifies MWU, FTPL, OGA, and their Blum–Mansour reductions as smooth learners, with closed-form weight-sum computations. Corollaries derive unilateral non-learnability against adversaries (3.4) and an informational obstruction to time-average Nash convergence (3.5). The authors interpret the lower bound information-economically (rationalizable learning generates valuable information) and draw policy conclusions for algorithmic-collusion regulation.","tokens_in":29674,"tokens_out":5919,"duration_ms":254225,"significance":"If the results hold — and on my reading the technical core is sound — the paper makes two genuine contributions. First, the offline/online separation is novel: to my knowledge IVFE is the first natural equilibrium concept that is poly-time computable from the normal form yet provably not learnable by a broad, explicitly verified class of uncoupled dynamics. This inverts the usual pattern (CE: easy both ways; Nash: hard both ways) and is of independent interest to learning-in-games. Second, the proof is fully constructive and checkable: the lower-bound game family is explicit (Table 2 with parameters w̄, β = A/8T), the smoothness verifications in §B.1 are closed-form, and the contradiction arithmetic in §B.2–B.7 is elementary and transparent. The paper is also unusually honest about its own scope: it states that guess-and-verify dynamics escape the result, flags the worst-case nature of the construction in §4.1, and lists the right open questions in §4.2. The policy connection to algorithmic-collusion regulation is timely, though, as detailed below, the strength of the stated policy conclusion currently exceeds what the theorem supports.","major_comments":[{"comment":"The headline claim — 'valuable and implicit information exchange is unavoidable under rationalizable learning, and traditional antitrust regulation against such exchange is therefore incompatible with rationalizable learning' (abstract; repeated in §1.1 and §4.1) — is a near-universal statement, but Theorem 3.3 is existential over opponent-calibrated game sequences: for each (L, A) the construction in §B.2 builds G_{2T} from statistics of A's own weight sequence (w̄, β = A/8T) in the two-arm experiment with ε = ε(2T), and each horizon gets a different game. This is a legitimate worst-case/diagonal lower bound, but it does not show that valuable information arises in any particular game of economic interest. Worse for the policy framing, there is countervailing evidence inside the paper's own frame: Proposition 3.2 certifies MWU, FTPL, and OGA as smooth, and the works the paper itself cit","section":"Abstract, §1, §4.1 vs. Theorem 3.3"},{"comment":"The entire impossibility rests on Definition 3.1, a bespoke condition (existence of a weakly decreasing weight subsequence of length T(1−h(T)) with an Ω(T) first-half/second-half mass gap in the two-arm experiment). The paper verifies it for three algorithms and their Blum–Mansour reductions, which is commendable, but two clarifications are needed for the result to be properly assessed. (a) Is the weakly-decreasing-subsequence requirement essential to the proof or an artifact of it? Several standard algorithms may satisfy the mass-gap property without admitting such a subsequence (e.g., algorithms with oscillatory but drifting weights); a sentence on what fails in §B.2 without monotonicity would help. (b) The class excludes exactly the dynamics known to reach Nash (Foster–Young, Hart–Mas-Colell, Germano–Lugosi), and the exclusion is doing real work: the theorem should be read as 'no smoo","section":"Definition 3.1 and §B.2"},{"comment":"The game family {G_T} varies with the horizon T and depends on the opponent algorithm A's realized behavior. Two consequences deserve explicit discussion. First, the result does not rule out convergence to IVFE in any fixed stage game; Corollary 3.4 gives a fixed game only for the unilateral/adversarial case. Whether a fixed-game version of Theorem 3.3 holds for smooth opponents is a natural strengthening (or refutation) target and should be stated as an open question, since a time-varying, opponent-tailored game family is a weaker notion of 'inherent' than the introduction suggests. Second, the construction requires knowing A's weight statistics (w̄, A, β); the lower bound is therefore a diagonalization against a known opponent, not against an unknown smooth learner. This is standard for such results but should be said plainly where the theorem is stated, since it bears on how the 'inco","section":"§B.2, Theorem 3.3 statement"}],"minor_comments":[{"comment":"Notation collisions impede checking: A denotes both the opponent algorithm (Theorem 3.3 statement) and the quantity Σ_{t: w_t > w̄}(w_t − w̄) in §B.2; η denotes both the MWU/FTPL learning rate in §B.1 and the distance threshold η = β² in §B.2; h(T) is used for the 2T-round run with a parenthetical acknowledging the ambiguity. Please rename at least the §B.2 instances.","section":"§B.2 and §B.1"},{"comment":"The IVFE definition and the computation result are stated for two-player games only, while the abstract and introduction speak of 'games' generally. Either state the two-player restriction up front or note that the LP-enumeration argument extends to n players with running time polynomial in the normal-form representation.","section":"Definition 2.1, Theorem 2.3"},{"comment":"The reduction 'assume d < η, then there exist infinitely many horizons with negative-BIH magnitude < 2ηT + O(TR(T)); take such a subsequence and re-index' is compressed. A sentence explaining why the liminf hypothesis yields this subsequence (and why re-indexing preserves the smoothness invocation with ε = ε(2T)) would make the contradiction argument self-contained.","section":"§B.2, Analysis paragraph"},{"comment":"The simplified adversarial-Nature sketch is helpful, but the mapping from its three claims to the actual Lemmas B.1–B.6 (which replace Nature's state switch with the smooth learner's weight drift) should be made explicit in one or two sentences; currently the reader must infer that the counterfactual row player of Claim B.5 plays the role of Nature continuing in state 1.","section":"§3, proof sketch"},{"comment":"The discussion of Blum et al. (2018) and Camara et al. (2020) would benefit from one sentence stating precisely what breaks in those settings if IVFE-style guarantees are demanded (the current text says the approach 'would not be successful' without specifying whether this is a corollary of Theorem 3.3 or of Corollary 3.4).","section":"References / §1.2"}],"recommendation":"major_revision","confidential_remarks":"The technical core appears correct and the construction is genuinely checkable; my concerns are about the gap between the existential theorem and the universal policy framing, not about correctness. I note a fairly heavy self-citation pattern to the authors' prior regulation work (Hartline et al. 2024, 2025; Hartline 2026), which is topically natural but means the policy section partly recapitulates the group's earlier position; the editor may wish to weigh whether the policy framing fits the journal's scope once re-scoped. I recommend revision rather than rejection because the framing problems are fixable without new technical results, though the required rewriting of the abstract and §4.1 is substantial enough that I would want to see the revision."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real contribution is clean: they define IVFE (CE plus non-negative external regret), show it is poly-time via finitely many LPs, and prove that no-swap-regret vs any smooth learner stays a constant distance from every IVFE on a constructed sequence of games. That offline/online gap is new relative to the Foster–Vohra / Hart–Mas-Colell canon and the usual Nash hardness story. The appendices check out—Prop 3.2’s closed forms for MWU/FTPL/OGA and the Blum–Mansour lifts are careful, and the B.1/B.2 contradiction arithmetic is tight.\n\nWhat the paper does well is make the information-economics reading precise (Table 1, Lemma 2.11) and keep the math self-contained. The smoothness definition is a legitimate scope condition; it honestly excludes guess-and-verify dynamics that can hit Nash.\n\nThe soft spot is not a proof gap. Theorem 3.3 is existential and diagonal: for every L and smooth A there exist games G_T (payoffs calibrated to A’s own weight sequence) on which liminf d ≥ c. That is fine as a worst-case lower bound. The abstract and §4.1 then upgrade it to “valuable information exchange is unavoidable under rationalizable learning” and therefore that bans on implicit communication are incompatible with learning. Inside the paper’s own citations that upgrade fails: the same smooth learners (MWU, OGA, etc.) are already known to converge to Nash ⊂ IVFE in first-price auctions and Bertrand competition—the exact markets the collusion debate cares about. So the hard family is disjoint from the motivating applications. The authors half-acknowledge this; they still write the policy sentence as if it were theorem-strength. That is the part that needs tightening before anyone treats the regulatory claim as settled.\n\nFor theory people who care about regret refinements and learnability separations this is worth reading. For the antitrust audience the math is solid but the leap is not. I would send it to referees; they will force the quantifiers and the policy language into alignment. Engage the theory result; quote the policy claim only with the existential caveat attached.","headline":"Solid offline/online separation for a new CE refinement, but the antitrust punchline overreaches the existential lower bound.","tokens_in":30885,"tokens_out":542,"would_cite":true,"duration_ms":10207,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Rationalizable learning cannot reach equilibria that generate no valuable information, even though those equilibria are easy to compute offline.","keywords":["information-value-free equilibrium","algorithmic collusion","no-swap-regret learning","smooth learners","correlated equilibrium","valuable information","antitrust regulation","online learning in games"],"falsifier":"Exhibit a smooth no-swap-regret algorithm pair whose empirical play on the paper's constructed family of games approaches an information-value-free equilibrium (distance tending to zero), or show that a standard pricing algorithm fails the smooth-learner definition yet still satisfies no swap regret.","tokens_in":30522,"feed_emoji":"⚖️","tokens_out":791,"duration_ms":15889,"temperature":0.7,"pith_summary":"The paper introduces information-value-free equilibria: correlated equilibria in which no player is strictly better off than by playing some fixed action, so the correlation carries no valuable private information. These equilibria always exist and can be found offline by a simple linear program. Yet the paper proves they cannot be learned online. When one player uses any no-swap-regret algorithm and the other uses any smooth learner (algorithms that gradually abandon a slightly worse action), the empirical history of play stays a fixed positive distance from every information-value-free equilibrium. The same barrier rules out time-average convergence to Nash. The authors read the result as showing that rationalizable learning must generate valuable information, so antitrust rules that try to ban all implicit information exchange among pricing algorithms are incompatible with competitive learning itself.","feed_headline":"Learning can't reach no-value equilibria, even if computers can","feed_subtitle":"Rationalizable algorithms must create valuable information, clashing with bans on implicit collusion","key_machinery":"The smooth-learner property (Definition 3.1): when two arms differ by a vanishingly small payoff gap, the algorithm still shifts probability mass only gradually, admitting a long decreasing weight subsequence whose first-half versus second-half gap is linear in T. This property is used to construct stage games in which any no-swap-regret column player is forced into persistently valuable correlation.","core_discovery":"Although an information-value-free equilibrium can be computed in polynomial time from the normal-form game, no pair consisting of a no-swap-regret learner and a smooth learner can make the empirical distribution of play converge to one: the distance remains bounded below by a positive constant that depends only on the smoothness of the opponent.","pith_inferences":["Regulators seeking competitive prices may need outcome- or audit-based tests rather than communication bans, because any algorithm that fully exploits its own payoff history will create usable correlation.","The same smooth-learner barrier may obstruct other refinements that simultaneously demand no swap regret and non-negative external regret.","If practical pricing algorithms turn out to be non-smooth, the informational lower bound leaves open the possibility that they could still learn information-value-free play."],"forward_implications":["Rationalizable learning of correlated equilibria necessarily generates valuable information.","Blanket bans on implicit information exchange among pricing algorithms are incompatible with competitive learning.","Time-average convergence to Nash is informationally impossible for the same broad class of smooth learners.","Offline tractability of an equilibrium concept need not imply online learnability under uncoupled dynamics."],"fun_headline_variants":["Computable no-value equilibria remain unreachable for learners","Offline poly-time finds value-free equilibria; learners cannot converge","No-swap-regret plus smooth learning stays bounded from value-free equilibria","Value-free equilibria: efficiently computable yet unlearnable online","Rationalizable learners cannot reach information-value-free equilibria"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The impossibility holds only when the opponent is a smooth learner that cannot abruptly abandon a slightly worse action; algorithms that jump discontinuously fall outside the result.","fun_headline_variants_meta":{"raw":{"variants":["Computable no-value equilibria remain unreachable for learners","Offline poly-time finds value-free equilibria; learners cannot converge","No-swap-regret plus smooth learning stays bounded from value-free equilibria","Value-free equilibria: efficiently computable yet unlearnable online","Rationalizable learners cannot reach information-value-free equilibria"]},"model":"grok-4.5","effort":"low","cost_usd":0.005205,"raw_usage":{"total_tokens":1424,"prompt_tokens":726,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":52048000,"prompt_tokens_details":{"text_tokens":726,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":608,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":726,"tokens_out":90,"duration_ms":9878,"temperature":1.0,"reasoning_tokens":608,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T11:33:55.172740+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a smooth no-swap-regret algorithm pair whose empirical play on the paper's constructed family of games approaches an information-value-free equilibrium (distance tending to zero), or show that a standard pricing algorithm fails the smooth-learner definition yet still satisfies no swap regret.","supporting_citations":[],"review_version":1}