{"id":"e0c91ec9-674d-402e-9247-3edf86c86484","arxiv_id":"2506.12677","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Swap rounding for budget-constrained designs is shown to be flawed as written: the pseudocode breaks marginal preservation and the variance decomposition omits indirect covariances.","lead":"This paper proposes using swap rounding, a combinatorial rounding method, to turn fractional treatment probabilities into exact binary assignments under a fixed budget, while preserving each unit's target probability. The promised benefit is unbiased, lower-variance estimates of treatment effects, which matters for trials with strict resource limits like vaccine allocation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 1's Case 2 uses the wrong update probability, so the stated swap-rounding procedure does not preserve marginals; the unbiasedness and variance-reduction theorems do not follow from the paper's own algorithm.","rationale":"The reader's weakest assumption is exactly the load-bearing point: Algorithm 1's Case 2 probability is inverted, so the martingale condition in Eq. (4) fails. This is an internal inconsistency, not a disagreement with the field's consensus: the pseudocode explicitly specifies a transition that does not preserve the conditional mean of p_i. Direct computation settles the issue. A secondary but independent gap is that Eq. (7) sums covariances only over the set S of directly swapped pairs, whereas a swap-rounding chain can induce nonzero covariance between units that are never directly paired; this further undermines the variance-reduction proof. Both problems are in the submitted manuscript, and neither is resolved by the appendix. Because the paper's central claims are unsupported by its own algorithm and derivations, I would keep the reader's REJECT verdict.","tokens_in":17236,"tokens_out":12475,"duration_ms":145703,"concrete_test":"Run one step of Algorithm 1, Case 2, with p_i=0.4 and p_j=0.8. The printed probability gives E[p_i'] = 0.75*1 + 0.25*(0.4+0.8-1) = 0.8, which is not 0.4. Recompute with the corrected probability (1-p_j)/(2-p_i-p_j) = 0.25, obtaining E[p_i'] = 0.25*1 + 0.75*0.2 = 0.4. Alternatively, simulate Algorithm 1 on a valid budget vector such as p=(0.4,0.8,0.8) with B=2 over many seeds and measure the empirical marginal for unit 1: it will converge to approximately 0.8 under the printed rule, not 0.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on Algorithm 1 preserving each unit's marginal treatment probability via the martingale condition E[p_i^{(t)} | p_i^{(t-1)}] = p_i^{(t-1)} in Eq. (4). In Case 2, when p_i + p_j > 1, Algorithm 1 rounds p_i up to 1 with probability (1-p_i)/(2-p_i-p_j). This is the wrong probability: preserving the marginal requires probability (1-p_j)/(2-p_i-p_j). For p_i=0.4, p_j=0.8, the printed rule gives E[p_i'] = 0.75*1 + 0.25*0.2 = 0.8, not 0.4. Thus Eq. (4) is false, Proposition 1 (unbiasedness) does not follow, and the variance arguments in Proposition 2 and Theorem 1 inherit the same failure. The same inverted ratio appears in Algorithm 2 in the appendix, so this is not a one-off display typo: the proofs require a property that the stated algorithm does not have. Using the corrected probability (1-p_j)/(2-p_i-p_j) would likely restore the marginal-preservation argument, but as submitted the algorithm and the theorems do not match.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes using swap rounding, in the sense of Chekuri and Vondrák, to convert a fractional assignment probability vector p (with sum B) into a binary treatment vector A with exactly B treated units. It claims that the procedure preserves each marginal probability Pr(A_i = 1) = p_i, induces negative correlations between assignments, and leads to an IPW estimator that is unbiased and has variance no larger than independent Bernoulli assignment. It further claims asymptotic normality, consistent variance estimation, an extension to general linear estimators, and a covariate-ordered variant, supported by synthetic and semi-synthetic experiments. The central theoretical results are Propositions 1 and 2 and Theorem 1.","tokens_in":17495,"tokens_out":5865,"duration_ms":79923,"significance":"If the main claims were correct, the paper would offer a practically valuable method for enforcing exact budget constraints without sacrificing unbiasedness and with improved precision. The connection between swap rounding and budget-constrained experimental design is a useful idea, and the authors provide code and a broad empirical comparison. However, the central algorithmic guarantee as stated is false: the printed update probabilities in Algorithm 1 do not preserve marginals, so the martingale property, unbiasedness, variance reduction, and confidence-interval results do not follow from the paper's own algorithm.","major_comments":[{"comment":"The Case 2 update probability is inverted. When p_i^{(t)} + p_j^{(t)} > 1, the algorithm rounds p_i up to 1 with probability (1-p_i^{(t)})/(2-p_i^{(t)}-p_j^{(t)}), but marginal preservation requires probability (1-p_j^{(t)})/(2-p_i^{(t)}-p_j^{(t)}). For p = (0.4, 0.8), the printed rule gives E[p_i'] = 0.75*1 + 0.25*0.2 = 0.8, not 0.4. Consequently, Eq. (4) is false, the martingale argument in §4.1.2 fails, and Propositions 1 and 2, Theorem 1, and Theorem 2 do not follow for the algorithm as written. The same inverted ratio appears in Algorithm 2 in the appendix, so this is not an isolated typo; replacing the probability with (1-p_j^{(t)})/(2-p_i^{(t)}-p_j^{(t)}) would restore the intended identity, but that correction must be propagated through the proofs.","section":"§3.1, Algorithm 1"},{"comment":"The variance decomposition sums covariance terms only over S, the set of pairs that are directly swapped. The variance of a sum over all n units includes covariances for every pair of units, and Eq. (9) states a formula for ρ_{ij} for arbitrary i,j. Unless S is the set of all unordered pairs, Eq. (7) omits covariance terms, and the paper does not prove that cov(X_i, X_j) = 0 for pairs outside S. This affects Proposition 2 and the asymptotic variance used in Theorem 1, and it is inconsistent with the appendix's derivation in §10.8, which applies cov(X_i, X_j) to any pair.","section":"§4.1.3, Eq. (7)"},{"comment":"The claimed unbiasedness and consistency of the variance estimator σ̂ are not proved. The main text states the result without proof, and the appendix derives population variance quantities but does not derive the expectation of the plug-in estimator in Eqs. (12)–(14). In particular, replacing terms such as E[Y_i(1)^2] with A_i' Y_i^2 / (p_i^0)^2 and subtracting n τ_swap^2 requires a careful argument about the joint distribution of A' and the potential outcomes under swap rounding; no such argument is given. Since Proposition 3 underpins the confidence-interval claim in Theorem 1, this is a load-bearing gap.","section":"§4.1.6, Proposition 3"}],"minor_comments":[{"comment":"In the Introduction and references, 'V ondrák' should read 'Vondrák'.","section":"References"},{"comment":"The sentence 'We will also that Y (0) and Y (1) are always nonnegative' is missing the verb 'assume'.","section":"§3.2"},{"comment":"The sentence beginning 'Them, the IPW estimator' should read 'Then, the IPW estimator', and the last bullet in the assumption list ends with a stray comma.","section":"Theorem 1"},{"comment":"The synthetic experiment text reports that Covariate Based Swap Rounding performs best, while the Housing sub-experiment reports that regular swap rounding performs best; the presentation should reconcile these statements or state explicitly that the ranking varies by setting.","section":"§6.3.2 and §6.4.2"},{"comment":"The y-axis labels such as 'Empirical Variance of Uniform Distributed p-values' are awkward and should be reworded to describe the estimator variance rather than the variance of p-values.","section":"Figures"}],"recommendation":"reject","confidential_remarks":"The paper's central theoretical claim is invalid as written because Algorithm 1 fails the stated marginal-preservation property, and the variance decomposition in Eq. (7) omits covariance terms. A corrected resubmission might be viable, but the current manuscript would require substantial revision of the algorithm statements, proofs, and variance formulas before the results can be assessed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Khurram, Ed, Bryan's paper is a good idea with a load-bearing bug in the pseudocode and a deeper problem in the variance math. The core suggestion—apply swap rounding to make treatment assignments that respect an exact budget while keeping the marginal probabilities—is sensible and, as far as I can tell, new. If the rounding procedure actually did what they claim, the unbiasedness and variance-reduction results would be nice for practitioners.\n\nBut as written, the algorithm does not do what they claim. In Algorithm 1, Case 2, when p_i+p_j>1, they round p_i up with probability (1-p_i)/(2-p_i-p_j). The correct probability to keep E[p_i] fixed is (1-p_j)/(2-p_i-p_j). With their numbers, take p=(0.4,0.8): after one step the expected value of the first entry becomes 0.8, not 0.4. The same wrong ratio appears in Algorithm 2 in the appendix, so it's not a one-off typo. The martingale condition in Eq. (4) is false, and Proposition 1 falls.\n\nThe variance part has a separate issue. Eq. (7) sums covariances only over directly swapped pairs, but swap rounding induces covariance between any two units, not just pairs that meet head-to-head. The pairwise covariance formula in Eq. (9) is also too strong: for n=4, p=(0.5,0.5,0.5,0.5), B=2, the actual covariance between any two assignments is -1/12, not -1/4 as Eq. (9) claims. So the variance decomposition and Proposition 2 are unsupported even after fixing the Case 2 probability.\n\nOn the positive side: the covariate-ordered variant is a reasonable heuristic, and the experiments are broad (three synthetic setups, two semi-synthetic datasets). But because the printed algorithm distorts the marginals, the empirical claims rest on whatever the code actually did, and the paper doesn't verify that the implementation matches the intended rounding.\n\nThis is not a paper to desk-reject in my view—the idea deserves a serious referee and a careful resubmission. But it is not publishable as is. The authors need to correct the update probability, rewrite the variance proof with the full pairwise covariance matrix (or cite the known negative-association result and derive the actual covariance), and re-check the experiments. If they do that, the result could be a useful paper.\n\nI'd send it out for review, but expect the reviewers to come back with major revisions.","headline":"Worth a look for the idea, but the main algorithm as printed does not preserve marginals and the variance analysis is incomplete; the paper needs major repair.","tokens_in":17974,"tokens_out":7409,"would_cite":false,"duration_ms":80346,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62K05","62D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Swap rounding can fix the number of treated units while preserving each unit's planned probability and cutting estimator variance.","keywords":["swap rounding","budget-constrained experimental design","inverse probability weighting","treatment effect estimation","negative correlation","variance reduction","exact budget constraint","martingale"],"falsifier":"Run one documented swap step of Algorithm 1 on the pair $p = (0.4, 0.8)$ using the printed Case 2 probability: the probability of rounding $p_i$ up is $(1 - 0.4)/(2 - 0.4 - 0.8) = 0.75$, so the expected value of $p_i$ after the step is $0.75 \\cdot 1 + 0.25 \\cdot 0.2 = 0.8$, not the required $0.4$; observing this mismatch in simulation would settle that the marginal-preservation and martingale conditions fail for the algorithm as printed.","tokens_in":16997,"feed_emoji":"🎯","tokens_out":7666,"duration_ms":89208,"temperature":0.7,"pith_summary":"This paper argues that a combinatorial optimization technique called swap rounding can convert fractional treatment probabilities into binary treatment assignments that use exactly B treatments, preserve every unit's original probability of treatment, and induce negative correlations between assignments. If correct, this would remove a long-standing tension in budget-constrained experiments: an inverse-probability-weighted estimator based on the rounded assignments would remain unbiased and would have variance no larger than the same estimator under independent Bernoulli randomization. The authors also claim the same guarantees for any estimator linear in the treatment indicators, and they propose a covariate-ordered variant that pairs similar units to strengthen the variance reduction. A careful reader would care because public-health and social-policy trials often have a fixed number of treatments available and currently must choose between violating the budget or distorting the planned assignment probabilities.","feed_headline":"Swap rounding fixes treatment counts and shrinks variance","feed_subtitle":"Budget-constrained trials can keep target treatment probabilities while making exactly B assignments and reducing IPW variance.","key_machinery":"The load-bearing object is the swap rounding algorithm, which repeatedly chooses a pair of fractional probabilities, transfers probability mass between them in a randomized way, and continues until every entry is 0 or 1. The key identity is the pairwise covariance induced by the rounding: for two entries involved in a swap, Cov(A_i, A_j) is negative, equal to $-p_i p_j$ when $p_i + p_j \\leq 1$ and to $-(1-p_i)(1-p_j)$ when $p_i + p_j > 1$, and this negativity is what converts the coupling into variance reduction. The martingale argument runs through the intermediate fractional vectors $p^{(t)}$: if each step preserves conditional expectations of the entries, then the IPW estimator at the final rounded assignment has the same expectation as at the start, and its variance decomposition contains only nonpositive covariance terms.","core_discovery":"The central claim is that dependent randomized rounding via swap rounding gives a joint distribution over binary treatment vectors with three simultaneous properties: exact feasibility, because the sum of the assignments is always exactly B; marginal preservation, because Pr(A_i = 1) = p_i for every unit; and negative covariance between assignments. The paper shows that under these properties the IPW estimator for the average treatment effect is unbiased, has variance no larger than the IPW estimator under independent Bernoulli assignment, and can be paired with a consistent variance estimator and a martingale central limit theorem to produce asymptotically valid confidence intervals. The same argument is extended to any estimator linear in the treatment assignments with nonnegative coefficients, and a covariate-ordered swap variant is claimed to reduce variance further when outcomes vary smoothly with covariates.","pith_inferences":["The negative-correlation mechanism is broader than swap rounding itself: any randomized assignment scheme that preserves marginals and induces nonpositive pairwise covariances would dominate independent Bernoulli assignment in variance, so the paper's argument points toward a general design principle for dependent randomization.","The covariate-ordered variant is one instance of a larger family of pair-selection rules; choosing pairs to minimize within-pair differences in the effective weight M_i could be optimized inside the same swap framework, with direct connections to rerandomization and matched-pair designs.","A natural empirical test beyond the paper's experiments is to repeatedly re-randomize a real trial dataset with known propensity scores, comparing the empirical variance and confidence-interval coverage of the swap-rounded estimator against independent Bernoulli draws.","The paper's guarantees depend only on the marginal-preservation and nonpositive-covariance structure, which suggests the same rounding approach could be combined with stratification or rerandomization to satisfy resource limits and covariate-balance goals simultaneously."],"forward_implications":["An experimenter can fix the number of treated units at B while keeping each unit's planned treatment probability exactly as designed, so the hard budget constraint no longer forces a trade-off against unbiasedness.","Because assignments are negatively correlated, the IPW estimate of the average treatment effect has variance no larger than independent Bernoulli assignment, meaning the exact-budget requirement does not cost precision.","Any estimator linear in the treatment indicators, not just IPW, inherits the same unbiasedness and variance-reduction guarantee under the stated conditions.","Pairing swaps between covariate-similar units can strengthen the variance reduction when outcomes and effective weights vary smoothly with covariates.","A martingale central limit theorem, together with the proposed variance estimator, gives confidence intervals for the average treatment effect that are asymptotically valid under the superpopulation assumption."],"supporting_citations":[{"why":"Supplies the swap rounding algorithm that the paper adapts from matroid polytopes to budget-constrained treatment assignment.","marker":"[Chekuri and Vondrák, 2009]"},{"why":"Defines conditional Poisson sampling, the classical exact-sample-size method whose distortion of probabilities the paper aims to avoid.","marker":"[Hájek, 1964]"},{"why":"Provides comparisons of conditional Poisson and related sampling designs that motivate the need for dependent rounding with preserved marginals.","marker":"[Grafström, 2005]"},{"why":"Introduces rerandomization, a baseline the paper contrasts with direct constraint enforcement and covariate-based pairing.","marker":"[Morgan and Rubin, 2012]"},{"why":"Motivates pairing similar units in causal inference, which underlies the covariate-ordered swap rounding variant.","marker":"[Stuart, 2010]"}],"fun_headline_variants":["Swap rounding keeps treatment counts exact and cuts variance","Exact B assignments with lower IPW variance via swap rounding","Dependent rounding leads to sharper estimates under fixed budgets","Negative correlation among assignments reduces estimator variance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every swap step leaves each unit's expected probability unchanged; if a printed update probability violates this marginal-preservation condition, the claimed unbiasedness and variance reduction do not follow from the algorithm as written.","fun_headline_variants_meta":{"raw":{"variants":["Swap rounding keeps treatment counts exact and cuts variance","Exact B assignments with lower IPW variance via swap rounding","Dependent rounding leads to sharper estimates under fixed budgets","Negative correlation among assignments reduces estimator variance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000219,"raw_usage":{"total_tokens":1353,"prompt_tokens":763,"completion_tokens":590,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":379,"completion_tokens_details":{"reasoning_tokens":529}},"tokens_in":379,"tokens_out":590,"duration_ms":6784,"temperature":1.0,"reasoning_tokens":529,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:44:57.775189+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run one documented swap step of Algorithm 1 on the pair $p = (0.4, 0.8)$ using the printed Case 2 probability: the probability of rounding $p_i$ up is $(1 - 0.4)/(2 - 0.4 - 0.8) = 0.75$, so the expected value of $p_i$ after the step is $0.75 \\cdot 1 + 0.25 \\cdot 0.2 = 0.8$, not the required $0.4$; observing this mismatch in simulation would settle that the marginal-preservation and martingale conditions fail for the algorithm as printed.","supporting_citations":[{"cited_title":"Dependent Randomized Rounding for Matroid Polytopes and Applications","cited_arxiv_id":"0909.4348","evidence_quote":"Supplies the swap rounding algorithm that the paper adapts from matroid polytopes to budget-constrained treatment assignment."}],"review_version":1}