Maximin Relative Improvement: Fair Learning as a Bargaining Problem

Jiwoo Han; Moulinath Banerjee; Yuekai Sun

REVIEW 2 major objections 6 minor 17 references

Maximin Relative Improvement: Fair Learning as a Bargaining Problem

T0 review · 2 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper proposes that fair learning across subpopulations should maximize worst-group relative improvement, and proves this criterion is exactly the Kalai-Smorodinsky bargaining solution, inheriting its axiomatic guarantees.

desk verdict Genuinely useful bargaining framing and a clean convergence result, but Proposition 4.2(4) misstates individual monotonicity and the stated version is false in the paper's own regression setup; fixable, not fatal. read the letter →

arxiv 2602.04155 v2 pith:NXVXCD6H submitted 2026-02-04 stat.ML cs.GTcs.LG

Jiwoo Han , Moulinath Banerjee , Yuekai Sun This is my paper · ORCID

classification stat.MLcs.GTcs.LG

keywords fairnessrelativeimprovementKalai-Smorodinskybargainingsolutioncooperativegametheoryrobustoptimizationgroupsubpopulationshiftconvergencerate

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 is trying to establish that fairness among subpopulations sharing one predictor is a bargaining problem, and that the right compromise is to maximize the worst group's relative improvement—the fraction of its own achievable risk reduction, measured from a baseline, that the shared model delivers. This criterion, they argue, is exactly the Kalai-Smorodinsky bargaining solution, so it inherits classical axioms: Pareto optimality, symmetry, scale invariance, and individual monotonicity (with a leximin refinement for more than two groups). If the argument holds, fairness no longer depends on comparing absolute losses across groups with different inherent predictability; instead, each group is guaranteed at least the same proportional share of its possible gain. The paper also provides finite-sample guarantees, showing the empirical maximin relative-improvement estimator converges at O(1/sqrt(n)) under uniform concentration and a positive improvement bound.

What carries the argument

The key object is the relative improvement ratio ρ_g(f) = (R_g(f0) − R_g(f)) / (R_g(f0) − R_g(f*_g)), which normalizes each group's risk reduction by its total achievable reduction. It carries the argument by mapping the learning problem into a cooperative bargaining game: groups are players, utilities are negative risks, the baseline predictor is the disagreement point, and group-optimal predictors form the ideal point. The Kalai-Smorodinsky solution—the Pareto-optimal point on the segment from disagreement to ideal—then coincides with maximizing the minimum relative improvement.

What would settle it

Compute the maximin relative-improvement solution in a two-group scalar linear regression with β1=1, β2=2, first with parameter interval [0,1], then with [0,2]. If the first group's risk increases under the expanded class, the individual-monotonicity axiom used in the characterization is false.

Watch

Extended reading notes

Core claim

The central discovery is that maximizing the minimum relative improvement ρ_g(f) = (R_g(f0) − R_g(f)) / (R_g(f0) − R_g(f*_g)) corresponds exactly to the Kalai-Smorodinsky bargaining solution, not merely as an analogy but as a mathematical equivalence. In the two-group case, the solution is the unique Pareto-optimal point with equal relative improvements; with more than two groups, the leximin refinement maintains uniqueness and Pareto optimality. Because the transformation from risks to relative improvements is affine, it preserves compactness, convexity, and Pareto structure of the feasible set. The paper argues this gives relative improvement a principled justification that absolute-scale

Load-bearing premise

The axiomatic story depends on the premise that expanding the function class can never make any group's relative improvement worse; if that monotonicity fails, the Kalai-Smorodinsky inheritance does not survive.

Editorial extensions

If this is right

If the baseline predictor belongs to the function class, the maximin relative-improvement solution guarantees no group is worse off than at baseline; minimax regret does not offer this guarantee.
Scale invariance means affine rescaling of each group's risk does not change the chosen model, so groups with different potential predictability can be compared fairly.
For two groups the solution is unique and Pareto-optimal; for more than two, the leximin refinement yields a unique risk vector and Pareto optimality.
Empirical implementation converges at O(1/sqrt(n)) under mild uniform-concentration and positive-improvement conditions, matching regret-based methods.
Existing robust fairness objectives—worst-group risk, worst-group explained variance, worst-group regret—can be reinterpreted as different bargaining solutions with weaker axiomatic profiles.

Reading between the lines

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

A testable consequence not pursued in the paper: on datasets where groups have very different achievable signal, maximin relative improvement should protect low-signal groups more than minimax regret; one could measure this by comparing each group's proportional gain under the two rules.
The characterization leans on an individual-monotonicity assumption about how the feasible set expands; a robustness check would be to see whether relative improvement solutions remain stable under non-nested function-class changes or misspecified baselines.
The bargaining view suggests a design principle for multi-objective learning beyond fairness: any collection of heterogeneous objectives could be balanced by normalizing each objective by its own achievable range.
A natural extension is algorithmic: the paper does not give efficient computation for the leximin refinement; designing scalable solvers for many groups would be the next step.

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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

Colleague,

The paper is worth engaging with. The central mapping of maximin relative improvement to the Kalai–Smorodinsky solution is real, and the way the authors unify GDRO, maximin explained variance, and minimax regret as different bargaining solutions is a genuinely clarifying frame. The risk-set convexification result (Theorem 3.4) is also a substantive step—showing that no Pareto-optimal points live in the convex hull outside the original set means the bargaining problem is well-posed exactly where it needs to be. The convergence theorem (5.4) is stated carefully, with standard uniform-concentration assumptions, and the O(1/√n) rate is believable.

The soft spot is in Section 4.2. Proposition 4.2(4) labels a set-containment property as “Individual Monotonicity”: if F1 ⊆ F2 with the same baseline, then enlarging the class cannot raise any group’s risk. That is not the Kalai–Smorodinsky individual-monotonicity axiom. KS-IM only applies when the expansion changes the ideal point of the protected player while the other players’ ideal points and the disagreement point stay fixed. Under plain containment, the normalization term f*_g changes, so the relative-improvement space is not a fixed coordinate system. The paper’s own Appendix D.1 defines Strong Monotonicity with exactly this containment flavor and marks it as violated for KS in Table 1—so the text contradicts itself.

And the claimed property is false in the paper’s own linear-regression setup. Scalar θ, β1 = 1, β2 = 2, baseline f0 = 0, F1 = [0,1], F2 = [0,2]. For F1 the maximin RI solution is θ = 1, with both groups at their in-class optima. For F2 the solution is θ = 4/3, giving ρ1 = ρ2 = 8/9 and raising group 1’s risk from σ² to V/9 + σ². Enlarging the class increases a group’s risk. So Proposition 4.2(4) is not just mislabeled; it is false.

That said, this is a localized error. The equivalence to KS and the convergence theorem do not depend on this axiom. The paper can be repaired by adopting the standard conditional individual monotonicity and clarifying that the relative-improvement normalization depends on the function class. With that fix, the framework stands.

This paper is for people working on group fairness, robust optimization, or bargaining-theoretic foundations. It deserves a serious referee, and I would accept it with the expectation of a major revision on the axiomatic section. Don’t desk reject.

Referee Report

2 major / 6 minor

Summary. The paper proposes a group-fairness criterion, maximin relative improvement (Eq. 3): choose a predictor that maximizes the minimum, over groups, of the ratio of actual risk reduction to the group's achievable reduction from a baseline. The central claims are: (i) this criterion is exactly the Kalai-Smorodinsky bargaining solution and inherits its axiomatic properties; (ii) under compactness/convexity assumptions, the risk set can be convexified without changing Pareto-optimal outcomes; (iii) when the baseline is feasible, the solution never harms any group relative to the baseline; and (iv) the empirical maximizer attains an O(1/sqrt(n)) convergence rate under uniform concentration. For m>2 groups, a leximin refinement is introduced to recover uniqueness and Pareto optimality. The paper also positions existing methods (GDRO, maximin explained variance, minimax regret) as other bargaining solutions.

Significance. The identification of relative improvement with the Kalai-Smorodinsky solution is conceptually attractive and gives a principled way to compare fairness criteria through bargaining axioms. The risk-set convexification result (Theorem 3.4) and the no-harm guarantee (Theorem 4.1) are useful, and the finite-sample convergence bound (Theorem 5.4) is clean and matches minimax-regret rates. However, the axiomatic core is not currently sound: the stated individual monotonicity property is false as written, which undermines the advertised fairness justification and the uniqueness claim. The m>2 leximin discussion is also not matched by the estimation theory. These are fixable, but they require substantive revision rather than copyediting.

major comments (2)

[Section 4.2, Proposition 4.2(4); Table 1; Appendix D.1] Prop 4.2(4) defines IM as: if F1⊆F2 with the same f0, then R_g(f2,RI)≤R_g(f1,RI) for all g. This is the Strong Monotonicity axiom of App. D.1 (with risk for ρ), which Table 1 marks as violated for KS. The two differ because the normalization in Eq. (1) uses the in-class optimum f*_g; hence F1⊆F2 does not imply Ω(F1)⊆Ω(F2). The claim is false in the paper's own linear setup: scalar θ, β1=1, β2=2, f0=0, F1=[0,1], F2=[0,2]. For F1, fRI=1 gives ρ=(1,1), R1=σ1^2; for F2, fRI=4/3 gives ρ=(8/9,8/9), R1=σ1^2+1/9. Enlarging the class harms group 1, contradicting (4). Since this axiom supports the uniqueness characterization and the advertised IM guarantee, the axiomatic claims of §4.2 are unproven. Replace (4) with the true KS IM (expansion changes only the protected group's ideal point, other ideal points and d fixed) and revise the fairness discussion.
[Section 5 vs. Section 4.3] For m>2, the paper's axiomatic object is the leximin refinement (Prop 4.4), while Eq. (3) and the estimator in Section 5 are the maximin solution. The maximin solution is non-unique and may be Pareto-dominated for m>2, as the paper notes. Theorem 5.4 establishes convergence of the worst-group relative improvement value for a maximin estimator, but this does not imply convergence of the leximin selection or its risk vector. If the leximin solution is the proposed method, the paper must either analyze the empirical leximin estimator or explicitly restrict the convergence guarantee to the first stage of the refinement. As written, the m>2 theory and the m>2 estimation theory apply to different objects.

minor comments (6)

[Notation, Eq. (3) and Prop. 4.4] The symbol fRI is used both for the maximin solution (Eq. 3) and for the leximin solution (Prop 4.4). Please disambiguate, e.g., fRI^max and fRI^lex.
[Prop. 4.4(4)] The IIA axiom states 'same group-optimal predictor f*_g' but the proof invokes 'the same ideal point.' The statement should require equal ideal points explicitly; a shared predictor function is not the same condition unless it is also optimal in both classes.
[Theorem B.1(1)] The claim that the diagonal ρ1=ρ2 intersects the Pareto frontier at exactly one point is too strong if the frontier has a flat segment along the diagonal. The maximin selection is then the upper endpoint, so the uniqueness conclusion survives, but the wording should be corrected.
[Table 1] The symbols □ and ◦ in the table are not explained in the caption; please define them (or add a table note).
[Section C, Logistic Regression] With Y=1{β_g^T X+ϵ>0} and Gaussian ϵ, the conditional mean is a probit, not σ(β_g^T x). The statement that the group-optimal predictor is σ(β_g^T x) is therefore not exact for logistic loss over the stated class; this should be rephrased or the generative model adjusted.
[Lemma 5.2(B)] The assertion that (P̂_g−P_g)Z is σ/√n_g-sub-Gaussian is stated without proof; a brief justification (e.g., one-dimensional sub-Gaussian concentration for a fixed Z) would help the reader.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the KS identification is a transparent mapping to external bargaining theorems; only same-author citation is background.

full rationale

The derivation chain is not circular. Equation (3) defines relative improvement as a risk-reduction ratio; Equation (10) is the Kalai–Smorodinsky ratio in utility coordinates. The paper explicitly says “Our optimization in Equation (3) corresponds exactly to the Kalai–Smorodinsky (KS) bargaining solution” and then adapts the external characterization theorems of Kalai & Smorodinsky (1975) and Imai (1983). These are independent published results, not the authors' own prior claims, and the paper supplies the needed compactness/convexity conditions via Theorem 3.4 rather than assuming the conclusion. The finite-sample result, Theorem 5.4, follows from the uniform concentration assumption (Assumption 5.1) and the positive-improvement bound (Assumption 5.3) through a standard bound on |ρ̂_g(f) − ρ_g(f)|; no fitted parameter is renamed as a prediction. The only same-author citation is Maity et al. (2021), used as background related work and not load-bearing. The apparent weakness in Proposition 4.2(4) — the stated individual monotonicity uses plain set containment F1⊆F2, which is not the KS individual-monotonicity axiom and can fail even in the paper's own linear-regression setting — is a correctness/monotonicity defect rather than circularity: it does not make a predicted quantity equal to its input by construction. Thus the circularity score is at most 2, reflecting only the minor non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numeric free parameters are fitted. The main assumptions are the classical bargaining theorems, the convexity/compactness conditions on the loss and model class, feasibility of the baseline, uniform concentration, and positive improvement. The m>2 axiomatic story additionally assumes comprehensive closure is harmless. The paper introduces no new entities such as particles, forces, or latent variables.

assumptions (7)

standard math Kalai-Smorodinsky and Imai axiomatic characterizations are correct and apply under the paper's risk-set geometry.
Proposition 4.2 and 4.4 import uniqueness and axioms from Kalai-Smorodinsky (1975) and Imai (1983); the paper does not reprove them.
domain assumption Assumption 3.2/3.3: convex continuous loss, integrable envelope, compact convex Θ (or weakly compact convex F).
Needed for a compact, convexifiable risk set and for Pareto-preserving convexification; excludes nonconvex model classes such as neural networks.
domain assumption Baseline predictor f0 belongs to the function class F for the no-harm guarantee.
Theorem 4.1 uses feasibility of f0; if f0∉F, maximin RI can put a group below baseline.
domain assumption Assumption 5.1: uniform concentration of empirical group risks with rate r_ng(t).
Underlies Theorem 5.4; satisfied under Lipschitz loss, entropy condition, and bounded or sub-Gaussian loss (Lemma 5.2).
domain assumption Assumption 5.3: Δ = min_g (R_g(f0)-R_g(f*_g)) > 0.
Keeps relative-improvement denominators bounded away from zero; the convergence rate scales as 1/Δ.
domain assumption R_g(f0) > R_g(f*_g) for every group g.
Stated in Section 2 to ensure the relative-improvement transformation is well-defined.
standard math For m>2, d-comprehensive closure preserves the leximin solution.
Theorem 4.3 and Proposition 4.4 rely on voluntary utility disposal; the proof assumes f0∈F and Pareto optimality of leximin.

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Cite this review

Pith. "Pith review of Maximin Relative Improvement: Fair Learning as a Bargaining Problem." pith.science (2026). https://pith.science/paper/NXVXCD6H

@misc{pith2026260204155,
  author       = {Pith},
  title        = {Pith review of: Maximin Relative Improvement: Fair Learning as a Bargaining Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NXVXCD6H}},
  note         = {Machine review of arXiv:2602.04155}
}

read the original abstract

When deploying a single predictor across multiple subpopulations, we propose a fundamentally different approach: interpreting group fairness as a bargaining problem among subpopulations. This game-theoretic perspective reveals that existing robust optimization methods such as minimizing worst-group loss or regret correspond to classical bargaining solutions and embody different fairness principles. We propose relative improvement, the ratio of actual risk reduction to potential reduction from a baseline predictor, which recovers the Kalai-Smorodinsky solution. Unlike absolute-scale methods that may not be comparable when groups have different potential predictability, relative improvement provides axiomatic justification including scale invariance and individual monotonicity. We establish finite-sample convergence guarantees under mild conditions.

Figures

Figures reproduced from arXiv: 2602.04155 by the authors.

**Figure 1.** Minimax regret versus maximin relative improvement in a two-group simple linear regression example with heterogeneous predictability (β1 = 2, σ2 1 = 1 versus β2 = 7, σ2 2 = 9), yielding achievable risk reductions of 4 and 49 units, respectively. Left: risk landscape. Right: relative improvement space, where minimax regret yields (−56%, 87%) while maximin relative improvement achieves (69%, 69%). We establish that th… view at source ↗

**Figure 2.** Transformation from risk space to utility space. Left: risk space R(F) where groups aim to minimize risks. Right: utility space U = −R(F) where players aim to maximize utilities. To determine which risk profiles represent acceptable compromises, we need a notion of efficiency for comparing vectors. In the bargaining interpretation, an efficient outcome should not allow one group to improve without imposing additio… view at source ↗

**Figure 3.** Schematic diagram comparing group fairness methods in risk space, after assuming the Pareto Frontier. Each method selects a different solution on the Pareto frontier. Detailed characterization are provided in Appendix E.1. 3.3. Comparison with Alternative Fairness Approaches Our bargaining framework is not limited to relative improvement—it provides a unified lens for understanding existing robust fairness methods.… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Risk set R(F), its convex hull (left), and comprehensive closure (right) for the linear regression setting (in Section 2) with Θ = {θ : ∥θ∥ ≤ 1}, β1 = (0.4, 0), β2 = (0.4, 0.6), Σ1 = ( 1 0.5 0.5 1 ), Σ2 = ( 1 0 0 1 ), and σg = 1. Note that conv(R(F)) \ R(F) and comp(R(…

**Figure 5.** Figure 5: Transformation between risk space and relative improvement space. Left: Risk space with group-optimal risks r ∗ 1 = 0.1, r ∗ 2 = 0.4 (utopia point) and baseline disagreement point (1, 1). Right: Relative improvement space where the disagreement point maps to (0, 0) and…

**Figure 6.** Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 1 linked inside Pith

[1]

The constraintρ 1(f) =ρ 2(f)intersects the Pareto frontier at exactly one point
[2]

Lemma B.2(Berge’s Maximum Theorem (Berge, 1963)).Let X be a compact topological space and Θ a topological space

This unique point coincides with the maximin solutionmax f∈F ming∈{1,2} ρg(f). Lemma B.2(Berge’s Maximum Theorem (Berge, 1963)).Let X be a compact topological space and Θ a topological space. Let C: Θ⇒X be a compact-valued correspondence with C(θ)̸=∅ for all θ∈Θ . Let ξ:X×Θ→R be continuous. Define V(θ) = sup{ξ(x, θ) :x∈C(θ)}. IfCis continuous (upper and l...

1963
[3]

Translation Invariance (TI).For any constants {cg}m g=1, the affine transformation eRg(f) =R g(f) +c g preserves the solution structure: eRg( ef) =R g(f) +c g for allg, where efdenotes the solution under the transformed problem
[4]

Independence of Irrelevant Alternatives with Ideal point (IIIA)If comp(S1)⊆comp(S 2) with the same ideal pointI(comp(S 1)) =I(comp(S 2)), and iff(S 2,d) =g(comp(S 2),d)∈comp(S 1), then f(S 1,d) =g(comp(S 1),d) =g(comp(S 2),d) =f(S 2,d)
[5]

sup f∈F bRg(f)−R g(f) # ≤ 2 ng EX,Y,ε

Modified Individual Monotonicity (IM’)If comp(S1)⊆comp(S 2) and the relevant projections for player i coincide, icomp(S1) = icomp(S2)(equivalently, comp( iS1) =comp( iS2)), then fi(S1,d) =g i(comp(S1),d)≤g i(comp(S2),d) =f i(S2,d). Our relative improvement maximizer fRI operates on function classes F rather than abstract feasible sets S under the mapping ...

2000
[6]

Therefore,(ρ ∗ 1, ρ∗ 1)lies on both the Pareto frontier and the fairness diagonal, establishing part (i)

=ρ ∗ 1 Geometrically, since one group’s optimal point lies above and the other below the fairness diagonal, continuity of the frontier guarantees an intersection (see Figure 5). Therefore,(ρ ∗ 1, ρ∗ 1)lies on both the Pareto frontier and the fairness diagonal, establishing part (i). For part (ii), we show that this point is the maximin solution. Because o...

1975
[9]

This is weaker than (PO), which requires that no solution strictly dominates in all components while weakly improving at least one

Weak Pareto Optimality (WPO).There exists no f ′ ∈ Fsuch that ρg(f ′)> ρg(f) for all g∈ G. This is weaker than (PO), which requires that no solution strictly dominates in all components while weakly improving at least one
[10]

Independence of Irrelevant Alternatives (IIA).If F1 ⊆ F2 with the same baseline f0 and disagreement point, and the solution underF 2 satisfiesf 2 ∈ F1, thenf 1 =f 2

Show all 17 references

[12]

Strong Monotonicity (SM).If F1 ⊆ F2 with the same baseline f0, then ρg(f2)≥ρ g(f1) for all g∈ G, where f1 and f2 denote the solutions underF 1 andF 2, respectively
[13]

Nash Bargaining SolutionThe Nash bargaining solution (Nash et al., 1950) maximizes the product of utility gains: maxu∈S Qm i=1(ui −d i)

Strong Monotonicity other than Ideal Point (SMON).If F1 ⊆ F2 with the same group-optimal risks Rg(f ∗ g ) for all g, thenR g(f1)≥R g(f2)for allg∈ G, wheref 1 andf 2 denote the solutions underF 1 andF 2, respectively. Nash Bargaining SolutionThe Nash bargaining solution (Nash e...

1950
[14]

First, maximize the minimum relative improvement: F1 = arg maxf∈F min g∈G ρg(f) = arg maxf∈F ρ(1)(f)
[15]

Among predictors inF 1, maximize the second-smallest relative improvement: F2 = arg maxf∈F 1 ρ(2)(f)
[16]

, m, Fk = arg maxf∈F k−1 ρ(k)(f)

Continue sequentially: fork= 3, . . . , m, Fk = arg maxf∈F k−1 ρ(k)(f)
[17]

The set of leximin solutions isF m. 23 Maximin Relative Improvement: Fair Learning as a Bargaining Problem 0 1 2 3 4 5 6 u1 0 1 2 3 4 5 u2 d umax 1 umax 2 (umax 1 , umax 2 ) uKS Feasible set S Pareto frontier Disagreement d Utopia point KS solution 0 1 2 3 4 5 6 u1 0 1 2 3 4 5...

1983
[69]

Agarwal, A., Dud ´ık, M., and Wu, Z

PMLR, 2018. Agarwal, A., Dud ´ık, M., and Wu, Z. S. Fair regression: Quantitative definitions and reduction-based algorithms. InInternational conference on machine learning, pp. 120–

2018
[129]

Berge, C.Topological spaces: including a triatment of mltivalued functions, vector spaces and convexity

PMLR, 2019. Berge, C.Topological spaces: including a triatment of mltivalued functions, vector spaces and convexity. Oliver and Boyd, 1963. Chen, M. A. Individual monotonicity and the leximin solu- tion.Economic Theory, 15(2):353–365, 2000. Chun, Y . The equal-loss principle f...

2019
[2015]

gov/data-research/research-reports/ data-point-credit-invisibles/

URL https://www.consumerfinance. gov/data-research/research-reports/ data-point-credit-invisibles/. Diana, E., Gill, W., Kearns, M., Kenthapadi, K., and Roth, A. Minimax group fairness: Algorithms and experiments. InProceedings of the 2021 AAAI/ACM Conference on AI, Ethics, an...

2021 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

The constraintρ 1(f) =ρ 2(f)intersects the Pareto frontier at exactly one point

[2] [2]

Lemma B.2(Berge’s Maximum Theorem (Berge, 1963)).Let X be a compact topological space and Θ a topological space

This unique point coincides with the maximin solutionmax f∈F ming∈{1,2} ρg(f). Lemma B.2(Berge’s Maximum Theorem (Berge, 1963)).Let X be a compact topological space and Θ a topological space. Let C: Θ⇒X be a compact-valued correspondence with C(θ)̸=∅ for all θ∈Θ . Let ξ:X×Θ→R be continuous. Define V(θ) = sup{ξ(x, θ) :x∈C(θ)}. IfCis continuous (upper and l...

1963

[3] [3]

Translation Invariance (TI).For any constants {cg}m g=1, the affine transformation eRg(f) =R g(f) +c g preserves the solution structure: eRg( ef) =R g(f) +c g for allg, where efdenotes the solution under the transformed problem

[4] [4]

Independence of Irrelevant Alternatives with Ideal point (IIIA)If comp(S1)⊆comp(S 2) with the same ideal pointI(comp(S 1)) =I(comp(S 2)), and iff(S 2,d) =g(comp(S 2),d)∈comp(S 1), then f(S 1,d) =g(comp(S 1),d) =g(comp(S 2),d) =f(S 2,d)

[5] [5]

sup f∈F bRg(f)−R g(f) # ≤ 2 ng EX,Y,ε

Modified Individual Monotonicity (IM’)If comp(S1)⊆comp(S 2) and the relevant projections for player i coincide, icomp(S1) = icomp(S2)(equivalently, comp( iS1) =comp( iS2)), then fi(S1,d) =g i(comp(S1),d)≤g i(comp(S2),d) =f i(S2,d). Our relative improvement maximizer fRI operates on function classes F rather than abstract feasible sets S under the mapping ...

2000

[6] [6]

Therefore,(ρ ∗ 1, ρ∗ 1)lies on both the Pareto frontier and the fairness diagonal, establishing part (i)

=ρ ∗ 1 Geometrically, since one group’s optimal point lies above and the other below the fairness diagonal, continuity of the frontier guarantees an intersection (see Figure 5). Therefore,(ρ ∗ 1, ρ∗ 1)lies on both the Pareto frontier and the fairness diagonal, establishing part (i). For part (ii), we show that this point is the maximin solution. Because o...

1975

[7] [9]

This is weaker than (PO), which requires that no solution strictly dominates in all components while weakly improving at least one

Weak Pareto Optimality (WPO).There exists no f ′ ∈ Fsuch that ρg(f ′)> ρg(f) for all g∈ G. This is weaker than (PO), which requires that no solution strictly dominates in all components while weakly improving at least one

[8] [10]

Independence of Irrelevant Alternatives (IIA).If F1 ⊆ F2 with the same baseline f0 and disagreement point, and the solution underF 2 satisfiesf 2 ∈ F1, thenf 1 =f 2