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FairFRESH: A Neuro-Symbolic Framework for Fair and Explainable Household Food Recommendation

Kodur Sai Vinay Sathvik, Kakoli Bora, Tejaswini R. M

arXiv:2610.31506v1cs.AIcs.GT

TL;DR

Household food recommendation must allocate a finite shared pantry among people with different dietary rules, rather than rank foods independently. FairFRESH learns member utilities, allocates under welfare rules, filters dietary violations, and explains allocations; Nash allocation improves equity and accuracy over conflict resolution in the reported household evaluation.

  • Problem

    Household members share finite food stock and follow different dietary rules, but typical recommenders score each person independently and do not resolve resource conflicts or guarantee safety.

    Source: §1 · pp. 1–2

  • Method

    FairFRESH learns per-member utilities, allocates foods under explicit welfare rules, applies a deterministic dietary filter, and explains verified allocations with post-hoc attributions.

    Source: §1 · p. 2, §3.5 · p. 8, §8 · p. 15

  • Results

    42% lower Gini disparity under Nash allocation than conflict resolution, with higher Recall@20 and at most 1.5% of assignments changed after utility rescaling.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Within the evaluated setting, the allocation rule shapes the balance between accuracy and equity as much as the ranking model does.

    Source: §8 · p. 15

    For shared dishes, Stage 2 has no claimed approximation guarantee because shared goods break disjointness and the stock constraint is knapsack rather than matroid.

    Source: §6 · p. 13

Abstract

from arXiv · show

Recommender systems typically serve one user at a time. However, in households, members share a finite pantry and follow different dietary rules, making independent recommendations prone to resource conflicts and dietary violations. We formulate household food recommendation as constrained fair division and present FairFRESH, a neuro-symbolic framework that learns per-member utilities, allocates foods under explicit welfare objectives, enforces dietary rules through a deterministic filter, and explains recommendations using Shapley and Banzhaf attributions. Theoretically, we show that no finite penalty guarantees compliance near a dietary-rule threshold, establish approximation guarantees for greedy allocation under utilitarian and smoothed Nash objectives in the stock-free model before filtering, and characterize the scale invariance of Nash welfare. We also use the minimax theorem to analyze rotating shared meals. On NHANES-derived data, the learned utility model achieves Recall@20 of 0.293, outperforming five learned recommenders (at most 0.244) but not a previous-day repetition heuristic (0.321). On held-out four-person households, Nash allocation reduces Gini disparity by 42% relative to conflict resolution while improving Recall@20 from 0.180 to 0.197. It comes within 2% of an upper bound on the smoothed Nash optimum, and rescaling one member's utilities changes at most 1.5% of assignments. All recommendations satisfy the encoded dietary rules, while attribution scores correlate with utility loss upon component removal (r = 0.76). In simulation, shared meals reduce pantry use by about one-third, and weekly rotation increases the worst-off member's utility by 38% over the best fixed meal.

1 Introduction

Household food recommendation must allocate a shared, finite pantry among members with different preferences and dietary rules, rather than rank foods independently for each person. FairFRESH combines learned utilities, explicit welfare-based allocation, deterministic dietary filtering, and post-hoc explanations.

1 Introduction

  • Independent rankings can recommend the same scarce food to multiple household members, while soft dietary penalties can be outweighed by taste or popularity signals.The shared pantry makes recommendation an allocation problem, and dietary rules require a safety guarantee.Source: §1 · pp. 1–2
  • The three-stage pipeline scores member–food pairs, allocates foods under a utilitarian, max-min, or Nash rule, then filters dietary violations and explains verified recommendations.Shapley and Banzhaf attributions are post hoc and do not affect decisions; the deterministic filter alone decides safety.Source: §1 · p. 2
  • The evaluation tests availability signals, allocation equity and accuracy, dietary-constraint coverage, and the faithfulness of cooperative-game explanations.It uses NHANES-derived households of two to six members and compares five recommenders and five allocation strategies.Source: §1 · p. 2
  • FairFRESH frames household food recommendation as fair division of a shared, finite pantry under per-member dietary rules.The formulation allocates servings under member-specific safe sets and a cap of K items.Source: §1 · p. 2
  • 42% lower Gini disparity under Nash allocation than conflict resolution, with higher Recall@20 and at most 1.5% of assignments changed after utility rescaling.The utility model ranked above five learned recommenders, while a previous-day repetition heuristic performed better.Source: §1 · p. 2

2 Related Work

Prior food and group recommenders address personalization, nutrition, pantry features, or shared preferences, but do not divide scarce household stock under individual dietary constraints. FairFRESH connects learned utilities to fair-division allocation and household-level explanations.

2 Related Work

  • Food recommenders model individual users, adding health objectives [38], food knowledge graphs, multimodal input, or pantry availability as a ranking feature.These approaches do not divide a pantry that one member’s choice depletes for another.Source: §2 · p. 2
  • Group recommenders aggregate preferences or seek proportionality and envy-freeness for a shared package, but treat shared items as public goods rather than private servings drawn from common stock.Every member receives the same items in these systems.Source: §2 · pp. 2–3
  • Fair-division results cover envy-free up to one good, Nash-optimal, and max-min allocations, but hardness and cardinality limits restrict available guarantees.Round robin finds EF1 allocations in polynomial time.Source: §2 · p. 3
  • Related constrained allocation work maximizes the quality of the most disadvantaged paper under reviewer-load constraints, while public-decision work assumes each agent consumes the chosen item [12] [14].This literature assumes known valuations and is not connected to learned utilities or dietary constraints.Source: §2 · p. 3
  • Neuro-symbolic methods can let strong taste or popularity scores outweigh relaxed symbolic constraints, and post-hoc Shapley or Banzhaf methods explain single predictions [26, 3].FairFRESH instead uses a deterministic filter for compliance and defines attributions over household allocation.Source: §2 · p. 3

3 Methodology

FairFRESH treats household food recommendation as allocation of a shared pantry under member-specific dietary rules. It combines learned member–food utilities with explicit welfare objectives, a deterministic safety filter, and post-hoc explanations.

3.1 Preliminaries and Problem Definition

  • The model assigns each member a disjoint bundle of at most K foods from a shared pantry, using additive utilities over member–food pairs.Most catalog foods remain unallocated because nK is far smaller than the catalog size.Source: §3.1 · p. 3

3.2 Utility Model

  • Taste, retrieval-augmented dietary evidence, and previous-day food history provide the three signals fused to score each member–food pair.The pantry encoder uses previous-day consumption as an availability proxy because NHANES records no household inventory.Source: §3.2 · pp. 4–5
  • For any finite penalty weight, a food just over a dietary threshold can outrank a safe food, so the learned penalty cannot certify near-threshold compliance.The required logit margin vanishes as the violation approaches zero.Source: §3.2 · p. 5

3.3 Allocation Rules and Greedy Allocation

  • Greedy Nash and utilitarian allocation achieve at least 1/3 of the smoothed Nash optimum and 1/2 of the utilitarian optimum, respectively, before filtering and without stock limits.No guarantee is claimed for greedy egalitarian allocation; the bounds do not generally transfer to filtered or stock-constrained outputs.Source: §3.3 · p. 6
  • Nash-optimal allocations are invariant to positive rescaling of each member’s utilities, unlike the utilitarian and egalitarian rules.Greedy Nash is scale-free once each member holds an item; the remaining first-pick dependence vanishes as ϵ approaches zero.Source: §3.3 · p. 7

3.4 Symbolic Safety Filter

  • The deterministic filter removes every food that violates a member’s encoded rules, while post-filter bundles may contain fewer than K items because removed foods are not replaced.In the held-out allocation runs, the soft penalty already kept all 219,539 recommendations within the rules.Source: §3.4 · p. 7

3.5 Post-Hoc Explainability Module

  • Post-hoc Shapley and Banzhaf attributions treat Taste, Pantry, RAG, Popularity, and Fairness as players while keeping dietary safety active in every coalition.The language model renders and checks explanations only after decisions are final, so explanations cannot change ranking, allocation, or safety.Source: §3.5 · p. 8

4 Experimental Setup

The evaluation uses NHANES-derived data to compare individual utility prediction and household allocation. Shared utility scores allow allocation strategies to be compared, with household size, pantry availability, and multiple accuracy, equity, and dietary-coverage metrics specified.

4.1 Data

  • The utility model trains on 1,000 respondents’ first-day foods to predict second-day foods, using an 8,704-item catalog derived from NHANES recalls.Allocation experiments use 1,000 other respondents; the individual comparison uses 400 further test users.Source: §4.1 · p. 8
  • Because NHANES has no household inventory, previous-day foods proxy member availability, while allocation stock uses 2–7 units of each of the 100 most frequent food-description words.Strategies B–E consume the required ingredient-word units when allocating a food.Source: §4.1 · p. 9

4.2 Baselines

  • The study compares five learned recommenders and two heuristics for individual accuracy, then compares five household strategies using the same utility model.Independent top-K lists may duplicate assignments; conflict resolution and the three welfare rules allocate each food to at most one member.Source: §4.2 · p. 9

4.3 Evaluation Protocol and Metrics

  • Evaluation reports Recall@20 and NDCG@20, pantry utilization, utility-based equity, and dietary coverage; each member receives K = 20 items.Equity uses predicted utilities, whose cross-member comparability is tested separately; confidence intervals use 2,000 bootstrap resamples.Source: §4.3 · p. 9

5 Results

Across household experiments, Nash allocation improves equity while retaining strong accuracy, and results also show how availability, dietary constraints, and component choices shape recommendations. Attribution tests find that Shapley values track component-removal utility losses more closely than Banzhaf values.

5.1 RQ-A: Does Availability Provide Useful Signal?

  • Zeroing availability drops Recall@20 from 0.293 to 0.083 and NDCG@20 from 0.311 to 0.091; 96.6% of recommendations are previous-day foods with availability enabled.With availability zeroed, only 8.5% of the top 20 are previous-day foods.Source: §5.1 · p. 9

5.2 Comparison with Recommender Baselines

  • Repeat-D1 achieves Recall@20 0.321, above the utility model’s 0.293; the utility model exceeds every learned baseline, led by LightGCN at 0.244.With Repeat-D1 utilities, 2.7% of allocated foods violate dietary rules and post-filter members with diabetes or hypertension receive 17.8 rather than 20 items; learned utilities keep allocations within rules before filtering.Source: §5.2 · p. 9

5.3 RQ-B: Does Fairness Improve Household Recommendation?

  • Nash allocation lowers Gini by 42% from 0.192 to 0.112 and raises Recall@20 from 0.180 to 0.197 versus conflict resolution.It also raises NSW from 8.54 to 10.98; the separate Table 2 comparison covers 100 held-out four-person households.Source: Table 2 · p. 10, §5.3 · p. 10

5.4 Scaling with Household Size

  • As household size rises from n = 2 to n = 6, conflict resolution’s minimum utility falls 61%, while Nash’s falls 33%; Nash has the highest NSW at every size.The figure plots Gini and minimum utility by household size; Nash Recall@20 stays within 0.002 of the most accurate coordinated strategy, while CSR remains 1.00.Source: §5.4 · pp. 10–11, Figure 3 · p. 11

5.5 RQ-C: Retrieval-Augmented Constraint Coverage

  • Before filtering, retrieval-augmented encoding attains constraint satisfaction of 100% for diabetes and hypertension, 96% for CKD, and 98% for CVD.Evidence-relevance scores are 0.95, 0.94, 0.88, and 0.91, respectively; these assess retrieval and evidence quality, not clinical validity.Source: §5.5 · p. 11

5.6 Component Ablation

  • Removing availability cuts Recall@20 from 0.197 to 0.064 and NSW from 10.98 to 5.58, the largest component-ablation loss.Removing medical constraints leaves Recall@20 nearly unchanged but makes 14.4% of recommendations violate a dietary limit; replacing Nash with utilitarian allocation raises Gini from 0.112 to 0.146.Source: Table 3 · p. 11, §5.6 · p. 11

5.7 RQ-D: Are Post-Hoc Explanations Faithful?

  • Shapley attribution correlates with utility loss after component removal at r = 0.763, versus r = 0.597 for Banzhaf; pantry receives the most credit under both.These tests reflect agreement with the pipeline’s own removal behavior, not stronger causal faithfulness; automated explanation checks report 98.7% text faithfulness and 100.0% evidence-citation accuracy.Source: §5.7 · p. 12
  • Fairness has Shapley value 0.064 but Banzhaf index 0.007, below the 0.020 threshold implied if its mean marginal contribution were non-negative at every coalition size.The authors infer that fairness lowers coalition utility at some intermediate sizes and earns Shapley credit in the smallest and largest coalitions.Source: §5.7 · p. 12

6 Extension: Shared Household Meals

The extension models cooked-once shared dishes alongside private servings, choosing shared meals for the worst-off member and rotating them when pantry stock makes a fixed weekly pair infeasible. Shared meals complete meals while reducing pantry use, and stock-constrained rotation improves worst-off utility over a fixed pair.

Setting

  • The shared-meal evaluation uses four-person households, K = 6, and Indian recipes with simulated utilities or NHANES foods with learned utilities.The study evaluates 100 households under five seeds in each setting.Source: §6 · p. 12

Two-stage allocation

  • The method first selects an in-stock grain–dal pair safe for every member by maximizing worst-off utility, then fills the remaining K − 2 slots by Nash allocation.If no safe pair is in stock, it skips the shared-dish stage and uses Nash allocation within safe sets; each dish is allocated to at most one member per meal.Source: §6 · p. 13
  • Stage 1 exactly enumerates safe pairs in O(n|G||L|) time, but Stage 2 has no approximation guarantee because shared dishes break disjointness and stock limits form a knapsack constraint.Both stages choose only from safe sets, so no post-hoc filter is needed.Source: §6 · p. 13

Rotating shared meals

  • Rotation treats safe grain–dal pairs as planner strategies and household members as adversarial payoff columns; the optimal rotation and member weights solve a linear program and its dual.The minimax result says rotation can only help the worst-off member, and the best rotation is computed exactly.Source: §6 · p. 13

Results

  • Shared meals complete every meal and use 32% less stock with Indian recipes and 31% less with NHANES foods than individual meals.Gini falls from 0.030 to 0.024 and from 0.045 to 0.030, while mean utility falls by 4% and 3%, respectively; all three differences are significant in both settings.Source: §6 · p. 13, Table 4 · p. 14

Rotation

  • In the Indian-recipe setting, over a seven-day plan with 68,523 to 464,610 safe pairs per household, the rounded optimal rotation raises worst-off average utility from 1.329 to 1.365 and improves on the fixed pair in 88.6% of households.It uses 2.6 distinct pairs on average.Source: §6 · p. 14
  • 38%: Under stock limits, the seven-day integer rotation gives the worst-off member utility 1.334 versus 0.978 for the best pair cookable all week.It improves on that pair in every household and lies within 0.005 of the linear-programming bound; every instance solves to optimality.Source: §6 · p. 14
  • With stock limits, the integer rotation improves worst-off utility over the best all-week pair by 43% for NHANES households that have such a pair; every household has a feasible rotation.Only 150 of 500 NHANES households have a pair cookable all week.Source: §6 · p. 14

7 Discussion

The discussion attributes reported equity differences to the allocation rule, contrasts utility-sensitive Nash allocation with ranking-based round robin, and notes strategic-reporting and evaluation limitations.

Allocation rules

  • Because the utility model is fixed while allocation rules vary, reported equity differences come from allocation alone, and conflict resolution is the least equitable tested rule.Round robin is EF1 by construction and nearly matches Nash in experiments, while greedy Nash has higher NSW at the same accuracy: 10.63 against 10.47.Source: §7 · p. 14
  • Nash allocation uses utility intensities to trade off member utilities, whereas round robin uses rankings alone and is safer when only rankings can be trusted.Nash’s exact optimum maximizes NSW and is Pareto-optimal; round robin cannot distinguish strong from weak preferences.Source: §7 · p. 14

Strategic reporting

  • Rescaling reported utilities can benefit a member under utilitarian or egalitarian allocation but not exact Nash; scale invariance does not prevent other misreports.The impossibility result is conditional: additive grain–dal utilities restrict Stage 1’s preference domain.Source: §7 · p. 14

Limitations

  • Households, availability, and stock are proxies; utility training lacked a validation set, and baselines were not tuned.Equity is measured on predicted rather than realized utilities, and the shared-meal extension uses approximate pantry quantities and simulated Indian-recipe utilities.Source: §7 · pp. 14–15

8 Conclusion and Future Work

FairFRESH frames household food recommendation as constrained fair division, with learned utilities, explicit allocation, and deterministic dietary filtering. The conclusion reports results on survey-derived households and shared-meal simulations, while identifying evaluation scope and future work.

8 Conclusion and Future Work

  • Conflict resolution was least equitable, while Nash allocation improved accuracy, removed most inequity, came within a few percent of exact optima, and was EF1 in every household.Round robin performed almost as well; evaluations used held-out households of two to six members with exclusive allocation.Source: §8 · p. 15
  • The utility model ranked above five learned recommenders but below a heuristic that repeats previous-day foods.The conclusion says previous-day consumption carried most of the predictive signal.Source: §8 · p. 15
  • Nash welfare is scale-invariant under natural weights, and shared dishes saved about a third of the pantry; rotating them usually made a weekly plan feasible when a fixed meal could not.The authors characterize the shared-meal findings as simulation results.Source: §8 · p. 15
  • The evaluation uses households assembled from individual survey respondents, predicted utilities, and dietary rules not clinically reviewed.Future work includes validating cohorts against observed households and clinically reviewing dietary rules.Source: §8 · p. 15

A Derivations and Proofs

  • The appendix states that several propositions specialize known results, with contributions in the formulation and the conditions under which results apply.It identifies Proposition 2, Proposition 3(a), and Proposition 5(a) as specializations.Source: A Derivations and Proofs · p. 17

A.1 The Soft Constraint Penalty

  • Under independent logistic rule uncertainty, the penalty has a probabilistic reading; its probability interpretation is near equality when relevance probabilities are small.The text also gives the logistic-noise assumption and describes the relevance-and-safety interpretation under independence.Source: §A.1 · pp. 17–18

A.2 Proof of Proposition 1

The proof shows that penalized ranking can still rank an unsafe food above a safe one, and that no finite penalty weight certifies safety arbitrarily close to a rule threshold.

A.2 Proof of Proposition 1

  • As a rule violation approaches zero, the sufficient penalty condition fails, so no finite penalty weight guarantees safety near the threshold.The proof also states that a sufficiently large relevance-logit advantage can make the unsafe food rank higher than the safe one.Source: §A.2 · p. 18

A.3 Proof of Proposition 2

The proof establishes greedy approximation guarantees for smoothed Nash and utilitarian allocation over two matroids, and identifies a polynomial-time exact comparator for the utilitarian objective.

A.3 Proof of Proposition 2

  • The smoothed Nash objective is normalized, monotone, and submodular, so greedy allocation achieves at least 1/3 of its optimum over the two matroids.The guarantee follows from the stated 1/(p + 1) bound with p = 2.Source: §A.3 · p. 18
  • Utilitarian greedy allocation achieves at least 1/2 of the optimum over the two matroids, while a maximum-weight bipartite b-matching computes the exact optimum in polynomial time.The exact allocation gives each food to at most one member and each member at most K foods.Source: §A.3 · p. 18
  • With K ≥ m, the unconstrained max-min allocation problem is NP-hard to approximate within any factor below two.Source: §A.3 · p. 19

A.4 Proof of Proposition 3

The proof characterizes how utility rescaling affects Nash welfare and allocation rules, and compares exact allocation before and after safety filtering.

A.4 Proof of Proposition 3

  • Multiplying each member’s utility by a positive constant changes the Nash product by a positive constant, leaving its maximizers unchanged.Source: §A.4 · p. 19
  • In the examples, doubling one member’s utilities changes the utilitarian allocation, while halving them changes the egalitarian greedy allocation and its utility.The examples show distinct sensitivity to scaling for the utilitarian and egalitarian rules.Source: §A.4 · p. 19
  • For the greedy Nash rule, rescaling changes the order in which members with empty bundles pick, but with fixed order and no ties, sufficiently small epsilon yields the same allocation under every scaling.Source: §A.4 · p. 19

A.5 Proof of Proposition 4

  • With exact maximizers, allocating within safe sets yields welfare at least as high as allocating first and filtering afterward; for greedy allocation, this inequality is expected but not guaranteed instance by instance.Source: §A.5 · p. 19

A.6 Proof of Proposition 5

The proof uses minimax equality to relate the planner’s mixed-strategy problem to its dual, then bounds the support of an optimal strategy. Complementary slackness identifies a binding member constraint when that member receives positive dual weight.

A.6 Proof of Proposition 5

  • Minimax equality makes the planner’s maximum over meal mixtures equal the minimum over member weights of the maximum weighted payoff.The proof then formulates the planner’s problem and its dual as linear programs.Source: §A.6 · pp. 19–20
  • An optimal meal mixture uses at most n nonzero components, plus the number of binding stock constraints when stock constraints are present.The support bound follows from an optimal basic feasible solution with at most n + 1 nonzero variables, including the positive optimum t.Source: §A.6 · p. 20
  • If member i has positive optimal dual weight, that member’s constraint binds.This is the complementary-slackness condition stated for the primal–dual pair.Source: §A.6 · p. 20

B Additional Results

Additional results examine greedy allocations against exact optima, report fairness properties, and extend the evaluation to shared meals using NHANES-derived and simulated settings.

Implementation details

  • The utility model combines 384-dimensional food and evidence embeddings with a 128-dimensional latent space and attention-based evidence and feature fusion.It is trained for three epochs without a validation set; the final epoch is retained.Source: B Additional Results · p. 20

Exact optima and EF1

  • Utilitarian, egalitarian, and Nash allocations and round robin are EF1 in every household, while conflict resolution is EF1 in 99%, 98%, and 92% of households for n = 2, 3, and 4.For four-person households, round robin has Recall@20 0.216, Gini 0.097, and NSW 10.47, versus Nash allocation’s 0.218, 0.106, and 10.63.Source: B Additional Results · p. 20
  • 95–99% of the optimum is attained by greedy allocation across household sizes in the exact-optimum comparison.The Nash comparison uses an upper bound from a piecewise-linear relaxation of the logarithm.Source: Table 5 · p. 20, B Additional Results · p. 20

Shared-meal settings

  • Shared-meal experiments use 100 four-person households under each of five seeds, with K = 6 and ϵ = 10^-5.The pantry tracks ingredient quantities, and a shared dish consumes one batch’s requirements.Source: B Additional Results · p. 20

Shared-meal allocation

  • The shared-meal procedure first selects a safe grain-and-legume pair for everyone, then fills remaining individual slots with Nash-greedy picks.The common pair is selected from in-stock pairs; individual picks are restricted to each member’s safe set and remaining stock.Source: B Additional Results · p. 21

Simulated recipe setting

  • The simulated recipe corpus contains 3,597 Indian recipes, with member utilities generated from correlated random ingredient-affinity vectors.Because the corpus lacks interaction histories, the paper does not train its learned utility model on these recipes.Source: B Additional Results · p. 21

NHANES setting

  • NHANES food descriptions lack ingredient lists, so ingredients are approximated with the 100 most frequent description words, averaging 2.9 per food.The evaluation stocks each food with 2 to 7 units, and dietary safety uses the same filter criterion as the main evaluation.Source: B Additional Results · p. 21

Shared-meal example

  • In one held-out household, shared bread and beans are safe for all members and reduce pantry use from 14.35% to 10.71%.The four individual foods per member match their first four single-serving picks; total utilities’ spread narrows from 4.74–5.85 to 4.09–4.33.Source: B Additional Results · p. 21, Table 7 · p. 22

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