Paired-Consistency: An Example-Based Model-Agnostic Approach to Fairness Regularization in Machine Learning

Elhanan Mishraky; Noa Haas; Shir Meir Lador; Yair Horesh; Yehezkel S. Resheff

REVIEW 5 major objections 5 minor 20 references

Paired-Consistency: An Example-Based Model-Agnostic Approach to Fairness Regularization in Machine Learning

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Pairing similar people lets a fairness regularizer reduce a model's reliance on protected attributes without explicit access to them.

desk verdict Paired-consistency is a clean, well-presented fairness regularizer for settings without protected attributes, but the experiments generate pairs with the protected attribute in hand, so the hard part of the claim is untested. read the letter →

arxiv 1908.02641 v2 pith:MNEHT2BF submitted 2019-08-07 cs.LG stat.ML

Yair Horesh , Noa Haas , Elhanan Mishraky , Yehezkel S. Resheff , Shir Meir Lador This is my paper · ORCID

classification cs.LGstat.ML

keywords pairedconsistencyfairnessregularizationindividualprotectedattributesmodel-agnosticdecisiontreeslogisticregressionCensusIncome

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper seeks to establish that fairness can be enforced without ever giving the model a protected attribute: instead, a fair domain expert supplies a small set of pairs of examples that differ on some protected construct but should receive the same prediction, and the training objective is augmented with a paired-consistency term that penalizes the model for treating the two members of a pair differently. This gives a model-agnostic regularization method that works for gradient-descent models (by adding the term to the loss) and for trees (by adding the fraction of pairs kept intact to the split criterion). On the Census Income dataset, with age as the protected construct, increasing the regularization weight lowers age's feature-importance rank from 5 to 10 while accuracy moves from 82.9% to 83.0%, and the fraction of pairs classified together reaches 100% at weight 1. A sympathetic reader should care because it offers a route to individual fairness in the common setting where protected attributes are absent, continuous, or too many to enumerate.

What carries the argument

The load-bearing object is the consistency set: $M$ pairs of examples supplied by a fair domain expert, each pair differing on a protected attribute or construct yet judged to warrant the same model response. Around this set the paper builds a paired-consistency score: for classification, the fraction $\frac{1}{M}\sum_{j=1}^{M} I[\hat{y}^{(j)}_1 = \hat{y}^{(j)}_2]$; for regression, $1 - \frac{1}{M\delta_{\max}}\sum_{j=1}^{M}(\hat{y}^{(j)}_1 - \hat{y}^{(j)}_2)^2$; and a confidence-weighted variant (Equation 3). This score is the mechanism that carries the argument: as a regularizer it penalizes the model exactly where fairness, as judged by the expert, requires identical treatment, and as a post-hoc metric it ranks models by their adherence to that requirement. The tree variant encodes the same idea locally by rewarding splits that keep both members of each pair in the same child node.

What would settle it

Generate consistency pairs from actual human experts on a dataset with a known protected attribute, train with paired-consistency regularization while excluding the protected attribute from the features, and compare demographic-parity and equal-opportunity gaps against an unregularized baseline; if the gaps do not shrink, or if experts' disagreements produce worse fairness than no regularization, the load-bearing expert assumption is refuted. A sharper check: on the same Census Income setup, measure whether the fairness improvement persists when the protected attribute is fully hidden and only correlated proxies remain.

Watch

Extended reading notes

Core claim

The central claim is that constraining a model to agree on a set of expert-judged pairs of examples is enough to suppress the model's use of protected information. The paper defines paired consistency as the fraction of consistency pairs on which the model assigns the same class (for classification) or a normalized squared-error closeness (for regression), and shows two ways to use it: as a regularizer, where the term is added to the loss or to a tree split criterion with a trade-off parameter, and as a post-hoc model-selection score, summarized by the PRC score, a weighted harmonic mean of precision, recall, and paired consistency. The Census Income experiments show that strengthening the regularizer monotonically reduces the importance rank of the protected variable age, that several hundred automatically generated pairs already capture most of the benefit, and that these fairness gains come with little or no accuracy loss in this dataset. The paper also compares the method against a prejudice-remover baseline and reports improvements on standard fairness metrics such as disparate impact, statistical parity difference, and equal opportunity difference.

Load-bearing premise

The method assumes a fair domain expert can reliably identify pairs of examples that differ on protected attributes but ought to receive the same prediction; if those pair judgments are incomplete or biased, the regularizer will encode a flawed notion of fairness, and the paper's experiments use automatically generated pairs rather than real expert labels.

Editorial extensions

If this is right

With a fixed consistency set, raising the regularization weight monotonically lowers the protected feature's importance rank, reaching a plateau once all pairs are classified together (Table 2: rank 5 to 10 while accuracy rises from 82.9% to 83.0%).
A few hundred consistency pairs are enough to capture most of the fairness benefit, which matters because expert pair labeling is costly (Table 3 and Table 5).
The method applies to the two dominant model families: gradient-based training via a loss term and tree training via split-criterion augmentation.
Paired consistency can double as a model-selection metric when regularization is not feasible, via the PRC score combining precision, recall, and paired consistency.
Because the pair labels come from an expert and not from model internals, the method can be combined with other fairness techniques such as adversarial removal of protected information from learned representations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: if pair selection were made explicit as a distance or equivalence relation, paired consistency could be linked to Lipschitz-style individual fairness guarantees; the paper leaves that theoretical connection open.
Editorial inference: the near-flat accuracy-fairness trade-off seen on Census Income is likely not universal; on datasets where the protected attribute is strongly predictive and entangled with many features, the same regularization should produce a steeper accuracy cost, and that is a testable prediction.
Editorial inference: the choice of pairs determines which notion of fairness is optimized, so different pair-sampling strategies may collapse paired consistency onto demographic parity, equalized odds, or something new; the paper notes this as future work.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a look if you work on fairness regularization. The core idea is simple and genuinely useful: rather than needing protected attribute values, a fair expert supplies pairs that should get the same model response, and the model is regularized to be consistent on those pairs. The definition is clear, the tree and gradient-descent variants are sensible, and the paper is honest about its own limitations.

The new contribution is real relative to the consistency literature. Zemel et al. define consistency via k-NN on features; this redefines it via expert pairs and uses it as a loss term. That is a distinct step, and the model-agnostic framing is a nice practical touch.

What the paper does well: it reports a range of standard fairness metrics, not just its own; it shows the expected monotone drop in feature importance for age as the regularization weight increases; and it explicitly flags the reliance on a fair domain expert. The experiments are reproducible from public data, though no code is shipped.

The soft spot is exactly the one the stress-test note flags, and it is load-bearing: the consistency pairs are generated automatically using age, the very protected attribute the method is supposed to work without. The authors call it a toy example and say a real expert would be needed; that's honest, but it means the motivating scenario—no explicit protected attribute, or a non-measurable construct—is never actually exercised. No sensitivity analysis for pair noise, sampling bias, or expert disagreement is provided, so we don't know how robust the regularizer is to imperfect pairs.

One detail in the stress-test note is wrong: the 0.913 in Table 4 is the prejudice remover baseline, not the LR+pairs eta=0.4 result. The LR+pairs eta=0.4 score is 0.927, which matches Table 5's 3062-pair row. So the evaluation protocol is consistent.

Other weaknesses are minor for a proof-of-concept: a single dataset, no error bars or significance tests, and the paired-consistency metric doubles as loss and evaluation. The external fairness metrics mitigate the circularity, but the lack of variance estimates is still a limitation.

Who gets value: practitioners in industry who want a simple fairness knob and don't have protected attribute data; readers who want a clear statement of the expert-pair assumption and its costs. It deserves a serious referee: the idea is plausible, the presentation is clear, and the missing validation is fixable with a real-expert study or a hidden-construct simulation. My recommendation: send to peer review, and ask for exactly that experiment before acceptance.

Referee Report

5 major / 5 minor

Summary. The paper proposes paired-consistency, a model-agnostic fairness regularizer that uses a set of example pairs judged by a fair domain expert to warrant identical model responses despite differing on protected attributes. The authors define a paired-consistency score for classification and regression, show how to add it to gradient-descent and tree-split objectives, and report experiments on the Census Income dataset using automatically generated age-based pairs. They report that adding the regularizer reduces the feature-importance rank of age, leaves accuracy roughly unchanged or slightly improved, and improves a set of group and individual fairness metrics relative to baselines, including a prejudice-remover baseline.

Significance. If the empirical claims held up, this would be a useful contribution: the method addresses an under-served setting (no explicit protected attribute), is model-agnostic, and the pair-based formulation lets a domain expert encode fairness without formalizing a similarity metric. The manuscript is honest about its limitations, including the reliance on a fair domain expert and the toy nature of the automated pairs. The method is described clearly enough to implement, and the inclusion of external fairness metrics such as equalized-odds-related differences is a strength. However, the evidence base is currently too thin to support the central claims: one dataset, no error bars or significance tests, and a key assumption, namely the availability of reliable expert pairs, is never exercised experimentally.

major comments (5)

[§3, Consistency pairs] Section 3 generates consistency pairs automatically by selecting individuals who are similar except for age, with an age gap of at least ten years, explicitly using the protected attribute that the method claims to avoid. This is an oracle construction: the setting the paper motivates in Section 2, in which no explicit protected attribute is available, is never tested. Since the reliability of a fair domain expert is a load-bearing assumption of the method, and the paper itself notes that "a fair domain expert will likely be necessary" in real use, the authors should either run a study with genuine expert-generated pairs or provide a sensitivity analysis with respect to pair noise, biased sampling, and expert disagreement; without this, the central advantage over methods requiring the protected attribute is unvalidated.
[§3, Tables 4 and 5] The same configuration (logistic regression with paired-consistency regularization, eta = 0.4, 3,062 pairs) reports a paired-consistency score of 0.913 in Table 4 but 0.927 in Table 5. Because the paired-consistency score is the paper's headline metric, this discrepancy needs to be explained (for example, by different train/test splits or different pair subsets); as written, the evaluation protocol is not pinned down and the reported numbers are not reproducible.
[§3, Tables 2-5] All experimental claims rest on single runs on a single dataset: no error bars, variance estimates, repeated trials, or significance tests are reported. In particular, the statement that accuracy is "virtually unaffected" (Section 3, discussion of Table 2) depends on differences of at most a few tenths of a percent; without any uncertainty quantification, the claim that fairness can be improved with little or no accuracy cost is not supported.
[§2, Eq. (1)-(2); §3, Tables 2-3] The paired-consistency score is used both as the training objective and as a primary evaluation metric, so the near-perfect or perfect "pairs intact" values for regularized models are partly by construction. The paper should therefore lead with the external fairness metrics, such as statistical parity difference, equal opportunity difference, average odds difference, and disparate impact, when arguing for fairness gains rather than presenting 100% pair agreement as standalone evidence; the external metrics in Table 4 are the stronger part of the evidence.
[§3, Table 4] Table 4 reports group-fairness metrics (statistical parity difference, disparate impact, equal opportunity difference, average odds difference) that are defined for a binary protected variable, but the manuscript never states how age was dichotomized to compute these metrics. Without this detail the numbers cannot be reproduced or compared across methods; this is a necessary protocol specification.

minor comments (5)

[§3, Census Income] The description "similar in all other aspects except for age" is not operationalized; please specify the feature subset, the distance or matching criterion, and the exact procedure used to generate the 3,062 pairs.
[Table 1; §3] There are typos: "diﬀerene" in Table 1 and "martial status" in Section 3; also "whereδmax" in Eq. (2) is missing a space and should be "where δ_max".
[§2, PRC score] The PRC score is defined in Section 2 but is never used in the experiments; either use it in the evaluation or remove it to avoid a dangling definition.
[Figure 3] The caption of Figure 3 does not explain the axes or the exact normalization; please describe what is plotted and how the fractional changes are computed.
[References] Reference [16] for the eli5 package is incomplete, and the bibliographic details for [15] and [6] are sparse; please complete these entries.

Circularity Check

1 steps flagged · score 6.0 of 10

Paired-consistency is used both as the training objective and as the headline evaluation metric, so its reported gains are partly by construction; external fairness metrics provide independent but partial support.

fitted input called prediction [Section 2 (paired-consistency score, Eqs. 1-2, and its embedding in the loss) and Tables 4-5 ('Paired consistency score' row)]
"The consistency score is embedded within the loss function as a fairness regularization term, to make the model consistency aware. This is done by adding the measure (eq. 2) to the objective, multiplied by a trade-off parameter... In addition (or alternatively for fairness-based model selection), the score is calculated post-hoc, and can be aggregated with other performance metrics."

The quantity in Eq. 2 (and its classification analogue Eq. 1) is exactly what the model is trained to optimize, and the same quantity is then reported as the 'Paired consistency score' evaluation row in Tables 4 and 5. Thus the improvement from baseline to Tree+pairs (0.976 to 1.000) and the LR+pairs scores in Table 5 are, by construction, consequences of minimizing the training loss rather than independent fairness measurements. The paper does provide independent evidence via external metrics (disparate impact, statistical parity difference, equal opportunity difference) that are not in the loss, so the circularity is partial, not total.

full rationale

The only reduction-by-construction I can exhibit is the training/evaluation metric overlap: paired-consistency (Eqs. 1-2) is added to the loss and then reused verbatim as the 'Paired consistency score' row in Tables 4-5. This makes the score gains for the regularized models internally driven rather than independently measured. The paper's central claim, however, does not rest solely on this score: it also reports reductions in age feature-importance rank (Table 2) and improvements in statistical parity difference, equal opportunity difference, and average odds difference (Table 4) relative to baselines, none of which are optimized directly. Those external metrics give the method independent content, so this is partial circularity, not a forced identity. The expert-pair assumption is a stated premise rather than a derived conclusion, and the use of automatically generated pairs with known age is a validation gap (the hidden-construct setting is not exercised), but that is a limitation in evidence, not circularity. The only author self-citation [18] is unrelated to the load-bearing argument. No uniqueness theorem or ansatz is smuggled in via citation. I therefore score 6: one evaluation metric reduces by construction, while external benchmarks provide independent support.

Assumptions & free parameters 2 free parameters · 2 assumptions · 0 invented entities

The central claim rests on the existence and reliability of a fair domain expert, the choice of pair-generation thresholds, and the regularization weight. The model's fairness behavior is defined operationally by the pairs; there are no invented physical entities.

free parameters (2)

Regularization trade-off parameter eta = 0.1, 0.5, 1, 10 (tree) and 0.4, 0.5 (logistic regression)
Controls the weight of the paired-consistency loss term relative to the prediction loss. Chosen by the authors per experiment, not derived from theory. Results depend on this value.
Pair generation thresholds = age gap >= 10 years; similarity on other features
The pairs that define fairness are generated by a heuristic with thresholds that are not justified or varied. This design choice determines what fairness means in the experimental demonstration.

assumptions (2)

domain assumption A fair domain expert exists who can generate pairs of examples that should receive similar model responses, with members differing only on protected attributes.
This is the core premise of the method, stated in Section 2. If the expert's pairs are incorrect or biased, the regularizer enforces the wrong notion of fairness.
ad hoc to paper The paired-consistency score is a valid operationalization of the desired fairness notion.
The paper defines fairness operationally as agreement on expert pairs. This is an assumption about what fairness means, and it is not derived from any axiomatic fairness principle.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Paired-Consistency: An Example-Based Model-Agnostic Approach to Fairness Regularization in Machine Learning." pith.science (2026). https://pith.science/paper/MNEHT2BF

@misc{pith2026190802641,
  author       = {Pith},
  title        = {Pith review of: Paired-Consistency: An Example-Based Model-Agnostic Approach to Fairness Regularization in Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MNEHT2BF}},
  note         = {Machine review of arXiv:1908.02641}
}

read the original abstract

As AI systems develop in complexity it is becoming increasingly hard to ensure non-discrimination on the basis of protected attributes such as gender, age, and race. Many recent methods have been developed for dealing with this issue as long as the protected attribute is explicitly available for the algorithm. We address the setting where this is not the case (with either no explicit protected attribute, or a large set of them). Instead, we assume the existence of a fair domain expert capable of generating an extension to the labeled dataset - a small set of example pairs, each having a different value on a subset of protected variables, but judged to warrant a similar model response. We define a performance metric - paired consistency. Paired consistency measures how close the output (assigned by a classifier or a regressor) is on these carefully selected pairs of examples for which fairness dictates identical decisions. In some cases consistency can be embedded within the loss function during optimization and serve as a fairness regularizer, and in others it is a tool for fair model selection. We demonstrate our method using the well studied Income Census dataset.

Figures

Figures reproduced from arXiv: 1908.02641 by the authors.

**Figure 2.** Top-6 feature importance in the basic tree model for [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Trade-off of Logistic Regression loss and paired-consistency loss compo [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 16 canonical work pages

[1]

arXiv preprint arXiv:1810.01943 (2018)

Bellamy, R.K., Dey, K., Hind, M., Hoﬀman, S.C., Houde, S., Kannan, K., Lohia, P., Martino, J., Mehta, S., Mojsilovic, A., et al.: Ai fairness 360: An extensible toolkit for detecting, understanding, and mitigating unwanted algorithmic bias. arXiv preprint arXiv:1810.01943 (2018)

arXiv 2018
[2]

Br¨ auer, J., Call, J., Tomasello, M.: Are apes really inequity averse? Proceedings of the Royal Society B: Biological Sciences 273(1605), 3123–3128 (2006)

work page 2006
[3]

Nature 425(6955), 297 (2003)

Brosnan, S.F., De Waal, F.B.: Monkeys reject unequal pay. Nature 425(6955), 297 (2003)

work page 2003
[4]

In: 2009 IEEE International Conference on Data Mining Workshops

Calders, T., Kamiran, F., Pechenizkiy, M.: Building classiﬁers with independency constraints. In: 2009 IEEE International Conference on Data Mining Workshops. pp. 13–18. IEEE (2009)

work page 2009
[5]

Science Advances 4, eaao5580 (Jan 2018)

Dressel, J., Farid, H.: The accuracy, fairness, and limits of predicting recidivism. Science Advances 4, eaao5580 (Jan 2018). https://doi.org/10.1126/sciadv.aao5580 Paired-Consistency 15

doi:10.1126/sciadv.aao5580 2018
[6]

uci.edu/ml

Dua, D., Graﬀ, C.: UCI machine learning repository (2017), http://archive.ics. uci.edu/ml

work page 2017
[7]

arXiv preprint arXiv:1808.06640 (2018)

Elazar, Y., Goldberg, Y.: Adversarial removal of demographic attributes from text data. arXiv preprint arXiv:1808.06640 (2018)

arXiv 2018
[8]

The quarterly journal of economics 114(3), 817–868 (1999)

Fehr, E., Schmidt, K.M.: A theory of fairness, competition, and cooperation. The quarterly journal of economics 114(3), 817–868 (1999)

work page 1999

Show all 20 references

[9]

Games and Economic behavior 6(3), 347–369 (1994)

Forsythe, R., Horowitz, J.L., Savin, N.E., Sefton, M.: Fairness in simple bargaining experiments. Games and Economic behavior 6(3), 347–369 (1994)

1994
[10]

CoRR abs/1903.03862 (2019), http://arxiv.org/abs/1903.03862

Gonen, H., Goldberg, Y.: Lipstick on a pig: Debiasing methods cover up sys- tematic gender biases in word embeddings but do not remove them. CoRR abs/1903.03862 (2019), http://arxiv.org/abs/1903.03862

2019 arXiv
[11]

In: Advances in neural information processing systems

Hardt, M., Price, E., Srebro, N., et al.: Equality of opportunity in supervised learning. In: Advances in neural information processing systems. pp. 3315–3323 (2016)

2016
[12]

Angwin, J

J. Angwin, J. Larson, S.M.L.K.: Machine bias: Theres software used across the country to predict future criminals. And its biased against blacks. propublica (May 2016), www.propublica.org/article/ machine-bias-risk-assessments-in-criminal-sentencing

2016
[13]

Journal of business pp

Kahneman, D., Knetsch, J.L., Thaler, R.H.: Fairness and the assumptions of eco- nomics. Journal of business pp. S285–S300 (1986)

1986
[14]

In: 2011 IEEE 11th International Conference on Data Mining Workshops

Kamishima, T., Akaho, S., Sakuma, J.: Fairness-aware learning through regular- ization approach. In: 2011 IEEE 11th International Conference on Data Mining Workshops. pp. 643–650. IEEE (2011)

2011
[15]

Kohavi, R.: Scaling up the accuracy of naive-bayes classiﬁers: A decision-tree hy- brid. In: Kdd. vol. 96, pp. 202–207. Citeseer (1996)

1996
[16]

Mikhail Korobov, K.L.: (2016), https://pypi.org/project/eli5/

2016
[17]

In: Pro- ceedings of the 14th ACM SIGKDD international conference on Knowledge dis- covery and data mining

Pedreshi, D., Ruggieri, S., Turini, F.: Discrimination-aware data mining. In: Pro- ceedings of the 14th ACM SIGKDD international conference on Knowledge dis- covery and data mining. pp. 560–568. ACM (2008)

2008
[18]

arXiv preprint arXiv:1807.03521 (2018)

Resheﬀ, Y.S., Elazar, Y., Shahar, M., Shalom, O.S.: Privacy-adversarial user rep- resentations in recommender systems. arXiv preprint arXiv:1807.03521 (2018)

2018 arXiv
[19]

In: International Conference on Machine Learning

Zemel, R., Wu, Y., Swersky, K., Pitassi, T., Dwork, C.: Learning fair representa- tions. In: International Conference on Machine Learning. pp. 325–333 (2013)

2013
[20]

In: Proceedings of the 2018 AAAI/ACM Conference on AI, Ethics, and Society

Zhang, B.H., Lemoine, B., Mitchell, M.: Mitigating unwanted biases with adver- sarial learning. In: Proceedings of the 2018 AAAI/ACM Conference on AI, Ethics, and Society. pp. 335–340. AIES ’18, ACM, New York, NY, USA (2018)

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

arXiv preprint arXiv:1810.01943 (2018)

Bellamy, R.K., Dey, K., Hind, M., Hoﬀman, S.C., Houde, S., Kannan, K., Lohia, P., Martino, J., Mehta, S., Mojsilovic, A., et al.: Ai fairness 360: An extensible toolkit for detecting, understanding, and mitigating unwanted algorithmic bias. arXiv preprint arXiv:1810.01943 (2018)

arXiv 2018

[2] [2]

Br¨ auer, J., Call, J., Tomasello, M.: Are apes really inequity averse? Proceedings of the Royal Society B: Biological Sciences 273(1605), 3123–3128 (2006)

work page 2006

[3] [3]

Nature 425(6955), 297 (2003)

Brosnan, S.F., De Waal, F.B.: Monkeys reject unequal pay. Nature 425(6955), 297 (2003)

work page 2003

[4] [4]

In: 2009 IEEE International Conference on Data Mining Workshops

Calders, T., Kamiran, F., Pechenizkiy, M.: Building classiﬁers with independency constraints. In: 2009 IEEE International Conference on Data Mining Workshops. pp. 13–18. IEEE (2009)

work page 2009

[5] [5]

Science Advances 4, eaao5580 (Jan 2018)

Dressel, J., Farid, H.: The accuracy, fairness, and limits of predicting recidivism. Science Advances 4, eaao5580 (Jan 2018). https://doi.org/10.1126/sciadv.aao5580 Paired-Consistency 15

doi:10.1126/sciadv.aao5580 2018

[6] [6]

uci.edu/ml

Dua, D., Graﬀ, C.: UCI machine learning repository (2017), http://archive.ics. uci.edu/ml

work page 2017

[7] [7]

arXiv preprint arXiv:1808.06640 (2018)

Elazar, Y., Goldberg, Y.: Adversarial removal of demographic attributes from text data. arXiv preprint arXiv:1808.06640 (2018)

arXiv 2018

[8] [8]

The quarterly journal of economics 114(3), 817–868 (1999)

Fehr, E., Schmidt, K.M.: A theory of fairness, competition, and cooperation. The quarterly journal of economics 114(3), 817–868 (1999)

work page 1999