REVIEW 4 major objections 4 minor 1 cited by
Be Intentional About Fairness!: Fairness, Size, and Multiplicity in the Rashomon Set
T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read The largest possible set of equally accurate models has closed-form size, per-person flip odds, and a large fairness gap between random and best picks.
desk verdict The empirical message on intentional fairness is well supported and the algorithms are useful, but the headline asymptotic theorems are not actually proven as written: the derivation of the rate function is circular. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the flip-vector representation $\theta\in\{0,1\}^N$: every model is described by which records differ from the Bayes-optimal prediction, and membership in the Rashomon set becomes a single knapsack constraint $\theta\cdot W_N/N\le\varepsilon$, where $W_N$ holds the per-record weights $w_i=|2p_i-1|$. The same representation carries all five contributions: a 0-1 knapsack (for statistical parity) and a fractional knapsack (for FPR/TPR) find fair models in $O(N\log N)$; a Gibbs sampler over the constraint set samples models uniformly; and the integral equation $g(C)=\int_0^1 \frac{w f(w)}{1+\exp(Cw)}dw=\varepsilon$ pins down the asymptotic flip probabilities, from which the exponential growth base $B(\varepsilon)=\exp(\int_0^\varepsilon C(x)dx)$ and the full-tolerance-usage result follow.
What would settle it
On a synthetic dataset with known Bayes-optimal probabilities $p_i$ (so $w_i$ and the weight density $f(w)$ are exactly known), enumerate the full Rashomon set for moderate $N$ by checking all $2^N$ flip vectors, and compare the exact flip probability $q_{N,i}$, the exact size $|R_N(\varepsilon)|$, and the average error use to the asymptotic formulas $q(w)=1/(1+\exp(C(\varepsilon)w))$, $B(\varepsilon)^N$, and $\varepsilon$; a systematic gap that does not shrink as $N$ grows would falsify the large-sample claims.
Extended reading notes
Core claim
On its own terms, the paper's central claim is that the largest possible Rashomon set, the $2^N$ binary labelings of $N$ i.i.d. records within $\varepsilon$ of optimal expected accuracy, has a tractable large-sample geometry. As $N\to\infty$, the uniform distribution over the set gives each record $i$ a flip probability $q(w_i)=1/(1+\exp(C(\varepsilon)w_i))$, where $w_i=|2p_i-1|$ is the Bayes-optimal classifier's confidence and $C(\varepsilon)$ is the unique solution of $\int_0^1 \frac{w f(w)}{1+\exp(Cw)}dw=\varepsilon$. The set's cardinality converges to $|R_N(\varepsilon)|=B(\varepsilon)^N$ with $B(\varepsilon)=\exp(\int_0^\varepsilon C(x)dx)$, and the average model spends the entire tolerance, so $\mathrm{acc}(\theta_0)-\mathrm{acc}(R_N(\varepsilon))\to\varepsilon$. For fairness, the flip-vector representation turns optimization of statistical parity into an $O(N\log N)$ knapsack problem with an exact solution, and error-rate-balance optimization into a fractional knapsack with $O(1/N)$ suboptimality, starkly contrasting with the NP-hardness of searching over restricted model classes. Across three benchmark datasets, the gap between the fairest model and a uniformly random one is large, and disparities can often be eliminated with $\varepsilon$ below half a percent.
Load-bearing premise
The whole theory assumes the Rashomon set—the set of equally accurate models—is every possible assignment of predictions to the observed records, even assignments that treat identical people differently and that no training procedure could produce; if the models a company can actually build form a much smaller set, the computed flip chances, set sizes, and fairness gains are ceilings, not predictions.
Editorial extensions
If this is right
- Fairness within equal-accuracy sets is a design choice: a randomly chosen model sits far from the fairest one, so deploying any high-accuracy model without an explicit fairness search forgoes most of the achievable disparity reduction.
- Per-person flip probabilities are computable in closed form, so arbitrariness can be audited per individual and per group before deployment, without sampling any models.
- The Rashomon set grows as $B(\varepsilon)^N$, so each small increase in the allowed error tolerance multiplies the number of equally accurate alternatives; the same result shows the average model consumes the full tolerance, so a wider search comes with the certainty that the deployed model will sit at the edge of the accuracy budget.
- For statistical parity the fairest model can be found exactly in $O(N\log N)$ time, and for error-rate balance within $O(1/N)$ of optimal, giving practical less-discriminatory-alternative search algorithms for the largest possible Rashomon set.
Reading between the lines
- Because the theory characterizes the largest possible Rashomon set, its flip probabilities and fairness maxima are upper bounds for any realistic model class; for restricted classes (like the paper's linear baselines) the same quantities are likely smaller, so real LDA searches would need re-calibration rather than direct use of these formulas.
- The $q(w)$ formula implies that flip chance is a function of confidence only, so any two individuals with the same distance from the decision boundary have the same arbitrariness risk; if confidence estimates are miscalibrated, flip probabilities will inherit the miscalibration.
- One could turn the flip-probability formula into a disclosure requirement for algorithmic systems: an individual could be told not just the prediction but the probability that the prediction would change across equally accurate models, making arbitrariness concrete per person.
- The full-tolerance result suggests a caution for LDA policies: if regulators require firms to search the Rashomon set, firms will exploit the full $\varepsilon$ cap, so policy should not only set the tolerance but also monitor the accuracy actually delivered by the deployed model.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the 'largest possible Rashomon set' for binary classification: all 2^N labelings of N i.i.d. records whose expected accuracy is within an error tolerance epsilon of the Bayes-optimal classifier. It contributes efficient algorithms for optimizing statistical parity and error-rate balance over this set, a Gibbs sampler for uniform sampling, and asymptotic formulas for individual flip probabilities, set size, and the average fraction of error tolerance used. These results are used to argue that intentionally searching for fair models within the Rashomon set yields much larger fairness gains than random sampling, and to draw policy conclusions about LDA searches. Experiments on German Credit, Adult, and Heritage Health support the qualitative findings.
Significance. If the theoretical results are correct, this is a significant contribution to the predictive-multiplicity literature: it provides the first general quantitative characterization of individual flip probabilities, exponential set size, and average error usage for the maximal Rashomon set, along with practical O(N log N) optimization algorithms and an efficient sampler. The paper is clearly written, is unusually candid about the limitations of the 'largest possible' model space, and includes extensive experiments with a robustness check. However, the three central asymptotic theorems rest on a circular derivation in Appendix C, and Algorithm 1 does not in fact solve the claimed exact optimization problem; both issues must be repaired before the results can be relied upon.
major comments (4)
- [Appendix C, Lemmas C.5-C.7, Corollary C.10, Theorems 5.1, 6.1, 6.2] The central asymptotic results are not independently proven. Lemma C.5 defines log B(epsilon) = lim_N log|R_N(epsilon)|/N, assumes this limit exists and is differentiable, and uses that assumption to prove pairwise independence of flip probabilities. Lemma C.6 then derives the logistic form q(w)=1/(1+exp(Cw)) from this pairwise independence, and Lemma C.7 uses that same logistic form to 'prove' the existence and formula for log B(epsilon). Corollary C.10 derives C(epsilon)=g^{-1}(epsilon) from Theorem C.9, which itself relies on Lemmas C.6 and C.7. Thus Theorems 5.1, 6.1, and 6.2 all depend on the very limit whose existence and differentiability they are supposed to establish. The formulas are plausible and match a standard large-deviations/Gibbs-conditioning calculation, but the paper does not supply that argument. A non-circular proof is required, for example by proving a large-deviation principle for the empirical mean of the weights under independent Bernoulli(1/2) flips and then applying the Gibbs conditioning principle to obtain the tilted product measure.
- [Section 4.1.1, Algorithm 1] Algorithm 1 does not find the exact fairest model for PPR disparity as claimed. The algorithm maximizes k_A/|A| + k_B/|B|, the total disparity reduction, without capping this reduction at the initial disparity id. If the maximum feasible reduction exceeds id, the resulting model overshoots and the absolute disparity |id - (k_A/|A| + k_B/|B|)| is not minimized; a smaller undershooting reduction can produce a strictly fairer model. Consequently the claim of exact optimality in Section 4.1.1 and the optimal-disparity curves in Figure 1 are not justified. The algorithm should minimize |id - reduction| (equivalently, stop flipping once disparity reaches zero) and the experiments should be rerun with the corrected objective.
- [Corollary C.11] The closed-form expression for g(C) is incorrect as written. The stated formula g(C) = ∫_0^1 w/(1+e^{Cw})dw = 12 Li_2(-e^{-C}) - 12 C log(e^{-C}+1) + pi^2/(12C^2) is dimensionally inconsistent and diverges as C->0, whereas the integral tends to 1/4. The correct expression is g(C) = [Li_2(-e^{-C}) - C log(1+e^{-C}) + pi^2/12]/C^2, or equivalently Li_2(-e^{-C})/C^2 - log(1+e^{-C})/C + pi^2/(12C^2). This error affects the claimed exact value for uniformly distributed weights in Corollary C.14 and the corresponding curve in Figure 3(left).
- [Section 3 vs. Appendix C] The definition of a model is internally inconsistent. Section 3 defines a predictive model as a mapping from input features x_i to decisions, which implies deterministic functions of x, while Appendix C states that the classifier can be probabilistic and that two records with identical x_i may receive different predictions. Since the entire theoretical framework counts arbitrary labelings of the N records as distinct models, the paper should define a model as an assignment of labels to the N records (an element of {0,1}^N) rather than as a function of x. This would also make the 'largest possible Rashomon set' assumption explicit from the outset.
minor comments (4)
- [Section 5.1, Assumption (2)] The assumption that the weight distribution f(w) is continuous and positive on [0,1] is stated as 'reasonable' for the three datasets, but no diagnostic is provided. A histogram of the estimated weights, or a sensitivity analysis showing that the conclusions are robust to discretization, would make the application of the large-sample theorems to finite datasets more credible.
- [Appendix B] The Gibbs sampler's uniform stationary distribution is argued heuristically, but no mixing-time bound or convergence diagnostics (e.g., trace plots, Gelman-Rubin statistic) are reported. The 950 samples used in the experiments may therefore be correlated; adding diagnostics or a rigorous ergodicity statement for this down-closed state space would strengthen the empirical claims.
- [Sections 4.4 and 6.4] Several policy takeaways are phrased for companies performing LDA searches without repeating the caveat that the quantitative results apply to the largest possible Rashomon set of all 2^N labelings, not to restricted model classes. The paper should state in the abstract and conclusion that the computed set sizes and flip probabilities are upper bounds for practical model classes, as the linear-model experiments in Section 6.4 already demonstrate.
- [General typography] There are several typographical issues, including 'q_N.j' in the proof of Lemma C.5, an apparent stray '0 0' in Section 5.2, and inconsistent use of 'FPR' vs. 'FPR disparity' in the appendix. These should be cleaned up before final submission.
Circularity Check
Central asymptotic formulas (Theorems 5.1, 6.1, 6.2) are derived circularly: Lemma C.5 assumes the existence and differentiability of the Rashomon-set growth rate log B(ε) that Lemma C.7 then purports to prove.
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self definitional
[Appendix C, Lemma C.5 (proof)]
"Next, we define logB(ε) = lim_{N→∞} log|R_N(ε)|/N, and note that, since R_N(ε) has minimum size 1 (for ε = 0) and maximum size 2^N (for large ε), B(ε)∈[1,2] for all 0≤ε≤1. We can also write logB(ε) = lim_{N→∞} log|R_{N,−i,−j}(ε)|/(N−2)."
Lemma C.5 assumes, as a definition, the existence of the asymptotic rate function log B(ε) = lim_N log|R_N(ε)|/N and then differentiates it with respect to ε. That rate function is exactly the quantity that Theorem 6.1 and Appendix C Theorem C.13 claim to derive. Pairwise independence of flip probabilities is then proved from this assumed rate function, and Lemma C.7 uses the resulting flip-probability form q_i = 1/(1+exp(Cw_i)) to 'prove' the same rate function and B(ε)=exp(∫_0^ε C(x)dx). Thus Theorems 5.1, 6.1, and 6.2 form a closed loop: the growth rate is assumed to derive the flip probabilities, and the flip probabilities are then used to derive the growth rate. No independent large-deviations or Gibbs-conditioning argument is supplied.
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self definitional
[Appendix C, Lemma C.7 (proof)]
"And from Lemma C.6, we know that q_i = 1/(1+exp(C(ε)w_i)). Setting these quantities equal to each other... By the definition of derivative... Equivalently, we can write: lim_{N→∞} (1/N) d log|R_N(ε)|/dε |_ε = C(ε). Integrating both sides with respect to ε: lim_{N→∞} (1/N) log|R_N(ε)| = ∫_0^ε C(x)dx + constant."
Lemma C.7 does not independently prove the asymptotic log-size limit; it takes the limit log|R_N(ε)|/N as already given, differentiates it, and integrates. The C(ε) in the integrand was introduced in Lemma C.6 as the coefficient of the flip-probability formula, which itself was obtained only after Lemma C.5 assumed the existence and differentiability of log B. Therefore the identity B(ε)=exp(∫_0^ε C(x)dx) is a rearrangement of the assumed rate function, not a derivation. Theorem C.13 then cites Lemma C.7 and Corollary C.10, so the headline set-size formula and the full-tolerance result inherit the circularity. The paper presents these as five main theoretical contributions, but the quantitative asymptotic claims are not supported by an independent proof as written.
1 more flagged steps
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self definitional
[Appendix C, Theorem C.9 and Corollary C.10]
"This implies that for large N, the number of flip vectors (models) with total error in the interval [ε−δ, ε] dominates the Rashomon set... Combining with the initial inequality, we have: ε−δ ≤ ∫_0^1 wq(w)f(w)dw ≤ ε for all δ>0, and thus ∫_0^1 wq(w)f(w)dw = ε."
Theorem C.9 (the source of Theorem 6.2) proves that the full error tolerance is used by comparing |R_N(ε)| with |R_N(ε−δ)| using Lemma C.7's formula log B(ε)=∫_0^ε C(x)dx. Since Lemma C.7 rests on the assumed rate function, this proof is not independent. Corollary C.10 then obtains C(ε)=g^{-1}(ε) by combining Theorem C.9 with Lemma C.6's functional form, so the final value of C is determined by the same circular chain rather than by a self-contained argument. Consequently, the paper's central predictions—individual flip probabilities, Rashomon set size, and full error-tolerance usage—all reduce, by the paper's own equations, to the existence and differentiability of log B that Lemma C.5 assumed at the start.
full rationale
The fairness-optimization results (Section 4 and Appendices A-B) and the experiments comparing uniform sampling, linear-model sampling, and optimal fairness searches are self-contained and do not depend on the asymptotic formulas; those parts are not circular. The paper's self-citations (e.g., [3]) are used for legal motivation and policy context, not to supply the central derivation, so they are not load-bearing. However, the central asymptotic claims—Theorems 5.1, 6.1, and 6.2—are not independently proven as written. Lemma C.5 assumes the existence and differentiability of the rate function log B(ε) that Lemma C.7 claims to establish, and Corollary C.10's C(ε)=g^{-1}(ε) is obtained from Theorem C.9, which in turn relies on Lemma C.7. The formulas are plausible and match a standard Gibbs/large-deviations calculation, but the paper does not provide that calculation; instead it proves the rate function from the flip-probability form that was derived from the assumed rate function. Hence the quantitative predictions reduce, by the paper's own equations, to the assumed growth-rate limit. I therefore assign a circularity score of 7: the central derivation is substantially circular, although the experimental and algorithmic contributions retain independent content.
Assumptions & free parameters
free parameters (1)
- Bayes-optimal probability estimates p_hat_i =
Not listed per record; logistic regression (main) and XGBoost (Appendix H), 5-fold cross-validation
assumptions (5)
- domain assumption Data records are i.i.d. from distribution D, and the weight distribution W has a continuous positive density f(w) on [0,1].
- domain assumption Error tolerance eps is less than half the average weight.
- ad hoc to paper Every binary labeling of the N records is a valid model, including labelings that are not functions of x (probabilistic for identical x).
- standard math Existence and differentiability of log B(eps), and interchange of limits in Lemma C.5, are treated as standard.
- domain assumption Estimated Bayes-optimal probabilities p_hat_i are accurate proxies for true p_i.
Cite this review
Pith. "Pith review of Be Intentional About Fairness!: Fairness, Size, and Multiplicity in the Rashomon Set." pith.science (2026). https://pith.science/paper/RAXLDHMT
@misc{pith2026250115634,
author = {Pith},
title = {Pith review of: Be Intentional About Fairness!: Fairness, Size, and Multiplicity in the Rashomon Set},
year = {2026},
howpublished = {\url{https://pith.science/paper/RAXLDHMT}},
note = {Machine review of arXiv:2501.15634}
}
read the original abstract
When selecting a model from a set of equally performant models, how much unfairness can you really reduce? Is it important to be intentional about fairness when choosing among this set, or is arbitrarily choosing among the set of ''good'' models good enough? Recent work has highlighted that the phenomenon of model multiplicity-where multiple models with nearly identical predictive accuracy exist for the same task-has both positive and negative implications for fairness, from strengthening the enforcement of civil rights law in AI systems to showcasing arbitrariness in AI decision-making. Despite the enormous implications of model multiplicity, there is little work that explores the properties of sets of equally accurate models, or Rashomon sets, in general. In this paper, we present five main theoretical and methodological contributions which help us to understand the relatively unexplored properties of the Rashomon set, in particular with regards to fairness. Our contributions include methods for efficiently sampling models from this set and techniques for identifying the fairest models according to key fairness metrics such as statistical parity. We also derive the probability that an individual's prediction will be flipped within the Rashomon set, as well as expressions for the set's size and the distribution of error tolerance used across models. These results lead to policy-relevant takeaways, such as the importance of intentionally looking for fair models within the Rashomon set, and understanding which individuals or groups may be more susceptible to arbitrary decisions.
Figures
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Forward citations
Cited by 1 Pith paper
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Fairness and Sparsity within Rashomon sets: Enumeration-Free Exploration and Characterization
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for FPR and TPR respectively. For FPR, assume without loss of generality that subgroup𝐴 has higher FPR, (1−𝑃𝐴)·𝐹𝐴 || 1−𝑃𝐴|| 1 > (1−𝑃𝐵)·𝐹𝐵 || 1−𝑃𝐵|| 1 . Then flipping a prediction in group𝐴 from 1 to 0, or flipping a prediction in group𝐵 from 0 to 1, reduces the disparity by 1−...
Reviewed August 10, 2026 · model on record in the stance chip above.
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