{"id":"e08338c3-dd6a-4c8f-9ac0-c854cd566dca","arxiv_id":"2608.13315","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A Stackelberg equilibrium analysis shows an LLM provider's optimal default reasoning budget follows a three-regime rule, and defaults steer reasoning only when users value keeping them.","lead":"This paper models a provider that sells LLM reasoning tokens at a per-token price and sets a default reasoning budget; the user can keep it, customize it, or exit. It derives the resulting equilibrium price and default in closed form and shows defaults change behavior only when users value the convenience of not customizing.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Closed-form results hinge on the exponential accuracy model in Eq. (1); the experiments fit this form without error bars or model criticism, leaving the three-regime default rule empirically unsecured for real LLM services.","rationale":"The paper's mathematics is careful and self-consistent: Lemma 1's derivative calculation, Lemma 2's Lambert-W manipulation, and Proposition 1's three regimes check out, and Theorem 1's continuity argument via Berge's maximum theorem is sound. The weakest point is the bridge from the parametric model to real systems. The favorable properties of Q (strictly increasing, concave, saturating) are not empirically established: the experiments use three samples per point, report only point estimates of D, A, and b, and do not test the functional form. This matters because the qualitative conclusions--that the acceptance region is a compact interval, that a three-regime default rule exists, and that defaults have no independent allocative power when delta=0--are consequences of strict concavity and the exponential form. If a provider's actual accuracy-token curve is non-concave, the optimal default could occur at a discontinuity, and the one-dimensional price optimization would need a different algorithm. Since the reader already flags this as the weakest assumption and the fix (error bars, cross-validation, code release) is straightforward, the CONDITIONAL verdict is appropriate; no stronger penalty is warranted.","tokens_in":16376,"tokens_out":15678,"duration_ms":171792,"concrete_test":"Re-fit the accuracy model on the released or regenerated data with a flexible alternative, e.g., Q(r)=D+A(1-exp(-(br)^k)) with k free, using the same per-question triples; test k=1 against k>1 by likelihood-ratio or AIC comparison. Then bootstrap the fitted parameters by resampling questions and compute the implied customization shutoff price p_s=vAb-theta*c and the equilibrium default r_dagger(p*) of Section IV.C. If the flexible model significantly outperforms the exponential model, or if the bootstrap confidence interval for r_dagger(p*) spans a large fraction of the observed allocation range, the three-regime predictions are not empirically identified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theoretical core (Section III) is internally consistent: Lemma 1, Lemma 2, Proposition 1, and Theorem 1 follow from Q(r)=D+A(1-exp(-br)) in Eq. (1) combined with linear latency and token models. But this functional form is the load-bearing assumption: it yields strict concavity of the user's objective (Lemma 1), the compact-interval acceptance region (Lemma 2), and the projection-based three-regime default rule (Proposition 1). The experiments are the only evidence that real accuracy-budget curves have that shape: Table II reports fitted D, A, b without error bars, goodness-of-fit statistics, or cross-validation, and no code or data artifacts are released. With three sampled generations per question, each measured accuracy is a coarse Bernoulli average, so the exponential fit could easily be an artifact. If the true Q(r) is non-concave, staircase-like, or non-monotone at large budgets, Lemma 2's interval characterization and Proposition 1's unique maximizer can fail, and the constructive one-dimensional price optimization in Theorem 1 is no longer guaranteed. The theorem remains correct as a conditional result, but the paper's practical claim that a provider can compute the optimal default from this model is only as strong as this unvalidated fit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies pricing and default design in an LLM reasoning service. The provider commits to a per-token price p and a default reasoning-token allocation r_d; a representative user then keeps the default, customizes the allocation, or exits. Under an exponential accuracy-budget model Q(r)=D+A(1-exp(-br)) with linear latency and billed-token models, the paper derives the user's unique customized allocation r_c(p) (Lemma 1), characterizes the set of accepted defaults as a compact interval whose Lambert-W endpoints are given in closed form (Lemma 2), solves the provider's fixed-price default selection in three regimes (Proposition 1), and proves existence of a Stackelberg equilibrium, with service offered iff the optimized service value is nonnegative (Theorem 1). It further shows that when the default convenience benefit delta=0, the acceptance region collapses to {r_c(p)}, so defaults have no independent allocative power. Experiments on Qwen3-8B and DeepSeek-R1-Distill-Llama-8B across five benchmarks fit the exponential accuracy model and illustrate equilibrium prices, defaults, and allocations under calibrated service parameters.","tokens_in":16640,"tokens_out":10055,"duration_ms":106043,"significance":"The theoretical core is sound and clean: the proofs of Lemma 1, Lemma 2, Proposition 1, and Theorem 1 are internally consistent, and the replication argument (customization outcomes can be reproduced by an accepted default) together with the Berge maximum-theorem continuity argument are well executed. The main qualitative insight—defaults matter only through a positive convenience benefit, and otherwise every service-providing outcome implements the user's customized allocation—is a useful and falsifiable statement for the design of configurable LLM services. The reduction of equilibrium computation to one-dimensional price optimization is a genuine convenience. The significance is currently limited by the empirical validation: the claim that experiments support the accuracy-token model rests on point fits without uncertainty quantification, so the practical applicability of the equilibrium rules is not yet established. If the empirical concerns are addressed, the paper would be a solid contribution to the emerging literature on test-time-compute economics.","major_comments":[{"comment":"The empirical support for Eq. (1) is not quantified. The fitted D, A, b in Table II are point estimates only; no standard errors, confidence intervals, R-squared, RMSE, residual plots, or cross-validation are reported. Since each empirical accuracy is computed from only three Bernoulli responses per question in Eq. (31), the noise in each point is substantial, and without uncertainty quantification or model criticism the statement that Fig. 2 supports the saturating accuracy-token model is not yet justified. This is a load-bearing gap because strict concavity of Q(r) is the foundation for the compact-interval acceptance region in Lemma 2 and for the three-regime default rule in Proposition 1. Please add inferential statistics for the fits and, if possible, release code/data or the per-allocation accuracy values used in the fitting.","section":"Section IV.A-B, Table II, Eq. (31)"},{"comment":"The closed-form equilibrium rules are conditional on the specific exponential form in Eq. (1). The paper does not test robustness to non-concave or non-monotone accuracy-budget curves, which are plausible given the discrete, step-like behavior of LLM reasoning traces. If the true Q(r) is staircase-like or has local plateaus, the compact-interval acceptance region in Lemma 2 and the projection-based three-regime rule in Eq. (25) can fail. I am not asking for a proof for general Q, but the paper should either provide evidence that the exponential form adequately captures the measured data, which connects to the previous comment, or explicitly limit the practical claims to the calibrated exponential model rather than presenting them as empirical validation of the equilibrium rule.","section":"Section III, Lemmas 1-2; Eq. (25)"}],"minor_comments":[{"comment":"The model name 'DeepSeek-R1-Distill-8B' in Table I is inconsistent with 'DeepSeek-R1-Distill-Llama-8B' used in the text and Fig. 2; please standardize.","section":"Table I and Fig. 2"},{"comment":"The fitting procedure is not described: please specify the loss function, the optimizer, the set of allocation points used, and the number of points per model-benchmark pair.","section":"Section IV.A"},{"comment":"The statement that accuracy 'generally increases' and 'exhibits diminishing returns' is not a quantitative validation; please support it with the fit statistics requested above or soften the wording.","section":"Section IV.B"},{"comment":"The tie-breaking robustness paragraph is informal; since Theorem 1 is stated conditional on the K > C > E rule, a precise statement of when the alternative tie-breaking gives the same value versus only a supremum would be helpful.","section":"Section III.D"},{"comment":"The notation \\hat{y}_{i,j}(\\ell) is used before being defined; please define it as the answer produced by the j-th sample for question i at allocation \\ell.","section":"Eq. (31)"},{"comment":"The discrete jump in p* and r_d* near b\\delta = 0.88 is explained in the text as a switch between co-optimal equilibria, but the caption should also state this so the plot is not misread as a discontinuity in the equilibrium value.","section":"Fig. 6"}],"recommendation":"major_revision","confidential_remarks":"The mathematical contribution is solid and the paper is within scope for cs.GT. My main concern is the gap between the conditional theoretical results and the empirical support claimed in the abstract; the experiments need uncertainty quantification and model criticism before the practical claims can stand. I do not see a correctness issue in the proofs. The paper could become acceptable after a major revision that strengthens the empirical section and moderates the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before you read it. The new result is a clean closed-form characterization of a provider's optimal price and default reasoning allocation in a single-user LLM service, with the default treated as an endogenous strategic variable. The soft spot is the empirical section: it fits a saturating exponential accuracy model without error bars or code and treats the fit as support. The theory is the contribution; the experiments are illustrative at best.\n\nThe math is genuinely good. Lemma 1 gives the user's unique customized allocation, Lemma 2 characterizes the acceptance region through Lambert-W branches, Proposition 1 gives the three-regime fixed-price default rule, and Theorem 1 proves equilibrium existence and reduces the provider's problem to a one-dimensional price search. I checked the derivations carefully; they are internally consistent and don't smuggle in the conclusions. The continuity argument via Berge's maximum theorem is clean. As a conditional result, the equilibrium analysis holds.\n\nNow the soft spots, in proportion to how soft they are. The exponential accuracy model Q(r)=D+A(1-exp(-br)) is load-bearing: it gives strict concavity of the user's objective, the interval acceptance region, and the projection-based three-regime default. The experiments fit this form to accuracy measurements that are coarse Bernoulli averages over three samples per question, with no error bars, goodness-of-fit statistics, or cross-validation. The fit could easily be an artifact. Without released code or data, the empirical support is weak. Also, the finding that defaults have allocative power only when delta>0 is largely by construction, since delta is defined as the convenience benefit of keeping the default. That is a modeling choice, not a discovered phenomenon, and the paper should say so more plainly.\n\nThese are fixable. The theoretical core stands on its own, and the paper is transparent about its assumptions and the representative-user simplification. The right audience is applied game theorists and mechanism designers in the LLM-pricing subfield, who would get a usable closed-form toolkit. I would send this to a serious referee rather than desk-reject it. My recommendation: conditional acceptance or accept-with-revisions, with the empirical claims recast as calibration for the model rather than validation.","headline":"Solid Stackelberg analysis of default reasoning budgets; the theory is clean, the empirics are thin, but the paper deserves serious review.","tokens_in":17177,"tokens_out":2129,"would_cite":true,"duration_ms":23972,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A65","91B24"],"pacs":[],"model":"deepseek-v4-flash","headline":"A default reasoning allocation only changes what an LLM user gets when that user values keeping the default; otherwise the implemented allocation is the user's own customized optimum.","keywords":["LLM reasoning services","token pricing","Stackelberg game","default effects","reasoning-token allocation","test-time scaling","Lambert W function"],"falsifier":"Measure accuracy at many fine-grained token budgets on a current reasoning model and check whether the accuracy-versus-budget curve is concave everywhere. If any dataset shows a flat region followed by a jump, the derived unique customized allocation and compact acceptance interval fail, and the optimal default would sit at the jump rather than at the Lambert $W$ boundary.","tokens_in":16134,"feed_emoji":"🧠","tokens_out":10399,"duration_ms":94158,"temperature":0.7,"pith_summary":"This paper asks how a provider of an LLM reasoning service should jointly set two levers: the price per token and the default amount of reasoning tokens a user receives. It models the interaction as a leader-follower game in which the provider commits to an offer and a representative user then keeps the default, customizes the reasoning allocation, or exits. The key claim is that at any price the defaults a user will accept form either an empty set or a single closed range of budgets, and the provider's optimal default inside that range follows a three-regime rule. This reduces the equilibrium computation to a one-dimensional price search and proves that defaults change the implemented reasoning allocation only when users attach a convenience benefit to keeping the default. A provider that trusts this model can compute its optimal price and default from measured accuracy, latency, and token-cost curves rather than by heuristic tuning.","feed_headline":"LLM default budgets only steer users who value keeping them","feed_subtitle":"Without a convenience benefit, the default cannot move the implemented reasoning allocation; pricing alone decides.","key_machinery":"The argument is carried by the acceptance region $D(p)$, the set of default reasoning budgets the user will keep at a given price, defined by the default's utility clearing both the customization and participation thresholds. Strict concavity of the user's utility in the reasoning budget, coming from the exponential accuracy model $Q(r)=D+A(1-e^{-br})$ together with linear latency and billed-token models, makes $D(p)$ either empty or a compact interval, with endpoints given in closed form by the two real branches of the Lambert $W$ function. Inside that interval the provider's optimal default is fixed by a three-regime rule: take the largest accepted default when the net marginal revenue per token $\\gamma(p)=p-\\rho-\\beta c$ is nonnegative, take the smallest when $\\gamma(p)\\leq -\\alpha Ab$, and otherwise project the unconstrained optimum $\\frac{1}{b}\\log\\left(-\\frac{\\alpha Ab}{\\gamma(p)}\\right)$ onto the interval. This collapses the provider's two-dimensional choice into a one-dimensional continuous maximization over prices, and the sign of the optimized value decides whether service is offered.","core_discovery":"On the paper's own terms, the discovery is that default design and token pricing in an LLM reasoning service are jointly tractable: the user's optimal customized reasoning allocation has a closed form, the acceptance region $D(p)$ is either empty or a compact interval whose endpoints are expressible through the Lambert $W$ function, and the provider's best default at a fixed price is described by a three-regime rule based on the net marginal revenue per token. A Stackelberg equilibrium exists, and the provider serves the user exactly when the optimized service value is nonnegative. The paper further establishes that the default has independent allocative power only when the convenience benefit $\\delta$ is positive; at $\\delta=0$ the acceptance region collapses to the singleton $\\{r_c(p)\\}$, so whatever default is shown, the implemented allocation is the user's customized optimum. With $\\delta>0$ the provider can deliberately offer a default above the user's customized allocation and capture the convenience margin, and the gap between the two measures the extra reasoning the default induces.","pith_inferences":["If real accuracy-token curves are non-concave with plateaus or jumps, the compact-interval acceptance region and the three-regime rule should fail in a specific way: the optimal default would sit at the edge of a plateau rather than at a Lambert $W$ boundary, which is directly testable with fine-grained budget sweeps.","The convenience benefit $\\delta$ is empirically identifiable: in a user study that presents the same allocation once as a default and once as a preselected customization value, the rate at which users keep the default encodes $\\delta$, and the model predicts how that rate varies with the allocation's distance from the user's optimum.","Because the provider extracts the convenience margin, the framework supplies a quantitative accounting for how much extra reasoning a default induces, which could inform welfare discussions about whether defaults steer users away from their independently chosen levels.","The same leader-follower structure should transfer to other metered reasoning resources, such as search steps or ensemble size, whenever quality is a saturating function of the resource and users face a keep-or-customize choice."],"forward_implications":["A provider can compute the equilibrium price and default by maximizing a single continuous function of price, then serving only if that maximum is nonnegative.","When the convenience benefit is zero, the default is allocationally inert: every accepted default implements the user's customized reasoning budget, although pricing still determines whether service is offered.","With a positive convenience benefit, the provider can place the default above the user's customized budget and the gap measures the extra reasoning the default induces.","The set of feasible prices is a compact interval that shrinks as price rises, closing at a unique maximum price where the default coincides with the customized allocation.","The provider's equilibrium payoff is nondecreasing in the convenience benefit, and the equilibrium price can switch between co-optimal values without any discontinuity in the payoff."],"supporting_citations":[{"why":"Supplies the test-time scaling finding that extra inference compute improves reasoning, motivating the accuracy-token model.","marker":"[1]"},{"why":"Establishes inference scaling laws, the empirical basis for the saturating accuracy function $Q(r)$.","marker":"[2]"},{"why":"Provides budget-forcing decoding, the mechanism that lets the provider enforce a binding reasoning-token allocation.","marker":"[3]"},{"why":"Gives the status-quo bias evidence that underlies the convenience benefit $\\delta$.","marker":"[5]"},{"why":"Shows behavioral inertia in plan participation, supporting the modeling of a default-specific convenience benefit.","marker":"[6]"},{"why":"Compares optimal defaults with active choices, grounding the three-action keep/customize/exit structure.","marker":"[7]"},{"why":"Closest prior work pricing LLM services as a Stackelberg game; the paper contrasts its single-provider default design with this setting.","marker":"[32]"},{"why":"Models competition over test-time compute and motivates the paper's focus on a default that prior games lack.","marker":"[33]"},{"why":"Supplies one of the two open-weight reasoning models used in the experiments.","marker":"[34]"},{"why":"Supplies the second open-weight reasoning model used in the experiments.","marker":"[35]"}],"fun_headline_variants":["Default reasoning budgets matter only if users avoid customizing","LLM default reasoning only shifts users who value convenience","When convenience is zero, default budget leaves allocation unchanged","Default design only steers LLM reasoning if users pay for convenience","The power of default budgets hinges on customization cost"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole closed-form analysis assumes that each extra reasoning token buys a smaller accuracy gain than the one before, smoothly and without plateaus or sudden jumps; if real reasoning models have accuracy curves with flat stretches or jumps, the unique customized allocation, the interval-shaped acceptance region, and the three-regime default rule need not hold.","fun_headline_variants_meta":{"raw":{"variants":["Default reasoning budgets matter only if users avoid customizing","LLM default reasoning only shifts users who value convenience","When convenience is zero, default budget leaves allocation unchanged","Default design only steers LLM reasoning if users pay for convenience","The power of default budgets hinges on customization cost"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000383,"raw_usage":{"total_tokens":2018,"prompt_tokens":925,"completion_tokens":1093,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":541,"completion_tokens_details":{"reasoning_tokens":1015}},"tokens_in":541,"tokens_out":1093,"duration_ms":10413,"temperature":1.0,"reasoning_tokens":1015,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:50:54.464559+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure accuracy at many fine-grained token budgets on a current reasoning model and check whether the accuracy-versus-budget curve is concave everywhere. If any dataset shows a flat region followed by a jump, the derived unique customized allocation and compact acceptance interval fail, and the optimal default would sit at the jump rather than at the Lambert $W$ boundary.","supporting_citations":[{"cited_title":"Inference scaling laws: An empirical analysis of compute-optimal inference for LLM problem- solving,","cited_arxiv_id":null,"evidence_quote":"Establishes inference scaling laws, the empirical basis for the saturating accuracy function $Q(r)$."},{"cited_title":"s1: Simple test-time scaling,","cited_arxiv_id":null,"evidence_quote":"Provides budget-forcing decoding, the mechanism that lets the provider enforce a binding reasoning-token allocation."},{"cited_title":"Status quo bias in decision making,","cited_arxiv_id":null,"evidence_quote":"Gives the status-quo bias evidence that underlies the convenience benefit $\\delta$."},{"cited_title":"The power of suggestion: Inertia in 401(k) participation and savings behavior,","cited_arxiv_id":null,"evidence_quote":"Shows behavioral inertia in plan participation, supporting the modeling of a default-specific convenience benefit."},{"cited_title":"Optimal defaults and active decisions,","cited_arxiv_id":null,"evidence_quote":"Compares optimal defaults with active choices, grounding the three-action keep/customize/exit structure."},{"cited_title":"Pricing online LLM services with data- calibrated Stackelberg routing game,","cited_arxiv_id":null,"evidence_quote":"Closest prior work pricing LLM services as a Stackelberg game; the paper contrasts its single-provider default design with this setting."},{"cited_title":"Test-Time Compute Games","cited_arxiv_id":"2601.21839","evidence_quote":"Models competition over test-time compute and motivates the paper's focus on a default that prior games lack."},{"cited_title":"Deepseek-r1 incentivizes reasoning in LLMs through reinforcement learning,","cited_arxiv_id":null,"evidence_quote":"Supplies the second open-weight reasoning model used in the experiments."}],"review_version":1}