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Social Choice Foundations for Simulation-Augmented Generation

Sonja Kraiczy, Smitha Milli, Ratip Emin Berker, Avinandan Bose, Brandon Amos, Jamelle Watson-Daniels, Maximilian Nickel, Edith Elkind, Ariel D. Procaccia

arXiv:2609.38287v1cs.GTcs.AIcs.LG

TL;DR

SAGE needs efficient inference-time routing that preserves representation of a larger population. The paper formalizes this with mPJR+ and adapts proportional clustering to select prompt-specific simulations. Across two domains, its method achieves higher mPJR+ satisfaction than k-means and random selection baselines.

  • Problem

    SAGE requires efficient inference-time routing without sacrificing representativeness of the full human population.

    Source: §1 · pp. 1–2

  • Method

    The paper formalizes representative routing with mPJR+ and adapts proportional clustering to select k prompt-dependent simulations from a sample of n individuals.

    Source: §1.1 · p. 2, §1.2 · p. 3

  • Results

    Across both datasets, Route-then-Simulate SEAR outperforms k-means and random methods in mPJR+ satisfaction for every k from 3 to 10.

    Source: §4.2 · p. 12

  • Takeaways & Limitations

    Approximate proportional representation of nH humans requires only n ≪ nH simulations and prompt-dependent querying of k ≪ n simulations.

    Source: §5 · p. 13

    The empirical evaluation uses two datasets, leaving generalization to other domains and populations for future work.

    Source: §5 · p. 13

Abstract

from arXiv · show

Simulation-augmented generation (SAGE) is a recent technical proposal in which models simulate individuals' viewpoints at inference time in order to provide more representative answers to contentious user queries. A core challenge for SAGE is making inference-time simulation efficient without sacrificing representation quality. We introduce the first formalization of this problem, based upon an axiom from proportional clustering known as metric proportional justified representation+ (mPJR+) which is the strongest proportionality axiom known to always be satisfiable by centroid-based clustering. We prove that to proportionally represent the viewpoints of a population of $n_H$ humans on a given prompt, we need only create simulations of $n \ll n_H$ individuals, and at inference time, need only dynamically route to $k \ll n$ of those simulations based upon the prompt. This twofold reduction still yields approximate proportional representation guarantees for the entire population. Empirically, across two domains-political questions and personal advice-our proposed routing algorithm achieves higher mPJR+ satisfaction rates than $k$-means-based or random selection baselines.

1 Introduction

The paper formalizes prompt-specific routing for SAGE: select a small number of simulations while preserving proportional representation of the larger human population. It adapts social-choice guarantees and proportional clustering to make this routing scalable.

1 Introduction

  • SAGE uses simulated individual viewpoints to produce responses representing a range of perspectives on socially contentious queries.Inference-time routing supports latency, context-window, and computational-cost constraints.Source: §1 · pp. 1–2
  • Routing must balance inference-time scalability with representation: construct n ≪ nH simulations, then select k ≪ n for each prompt while preserving guarantees for the full population.The motivating constraints are latency, compactness, and computational cost.Source: §1 · pp. 1–2

1.2 Our Contributions

  • The paper formalizes representative routing using metric proportional justified representation plus (mPJR+) from computational social choice [Kellerhals and Peters, 2024, He et al., 2026].The framework treats preferences over open-ended responses through individual reward functions.Source: §1.1 · p. 2, §1.2 · p. 3
  • The authors develop a faster SEAR implementation, with O(n^2 log n) amortized runtime versus the previous O(k^2n^4) bound.SEAR is a proportional clustering algorithm that satisfies the representation axiom studied.Source: §1.2 · p. 3
  • A random sample of n simulations suffices for approximate representation of nH humans, with n growing logarithmically in nH; under small reward-model dimension d, n can be independent of nH and k.At inference time, the method queries only k simulations, and representation depends on simulation fidelity.Source: §1.2 · p. 3
  • Across political-viewpoint and Reddit personal-advice domains, the proposed algorithm achieves higher mPJR+ satisfaction than k-means and uniform-random selection baselines.Experiments use black-box simulations and evaluate representativeness over the simulation pool.Source: §1.2 · p. 3

1.3 Related Work

  • Prior representative-viewpoint methods either require numerous LLM calls or use a fixed set of community models, whereas this work selects adaptively from a larger simulation population.The comparison is to Fish et al. [2026] and Boehmer et al. [2025] for the former, and Feng et al. [2024] for the latter.Source: §1.3 · pp. 3–4

2 Inference-Time Social Choice with Simulations

The section develops inference-time routing guarantees in successive settings: reward access, factorized geometric rewards, transfer from sampled humans to the population, and routing before simulation. Its methods reduce the number of simulations needed while retaining exact or approximate proportional representation under stated conditions.

2.1 Representation under Reward Access

  • Simulate–then–Route queries every simulation, then uses REAR to select a k-response slate satisfying reward PJR+; REAR runs in O(mn log n) time.The exhaustive procedure requires n costly simulation queries to return k responses.Source: §2.1 · pp. 5–6

2.2 Geometric Representation under Factorized Rewards

  • With probability at least 1 − δ, any sampled-instance reward-PJR+ slate transfers to (ε, γβ)-approximate reward PJR+ for the full population, where γβ = min{π, 4β}.For nH = 8 billion, k = 10, δ = 0.05, and ε = 0.03, the stated sample size is roughly 9 × 10^3.Source: §2.2 · pp. 6–7
  • Under shared factorized rewards, proportional clustering routes to k candidates in the common response-embedding space, where mPJR+ provides the representation guarantee.Unlike balanced k-means, SEAR is stated to return k centroids satisfying mPJR+.Source: §2.2 · pp. 6–7
  • Under factorized rewards, reward PJR+ corresponds exactly to mPJR+, and corresponding REAR and SEAR rules select the same indexed candidates.SEAR can be implemented in O(n^2 log n) time.Source: §2.2 · pp. 7–8

2.3 Simulating and Routing Commute

  • When simulations return reward-maximizing responses, Route–then–Simulate selects the same indices as Simulate–then–Route and satisfies reward PJR+ with only k simulation queries.With nonzero fidelity gap bounded by α, the route-first slate instead satisfies γβ-approximate reward PJR+, where γβ = α + β + min{α, β}.Source: §2.3 · pp. 8–9

3 Representation under Viewpoint Embedding Access

This section considers routing when individual reward functions are unavailable and simulations can be accessed only as black boxes. It uses viewpoint embeddings to predict response geometry, allowing routing to make only k expensive generative calls while retaining approximate representation guarantees.

3 Representation under Viewpoint Embedding Access

  • The viewpoint embedding function ψ maps responses so that similar human beliefs or preferences are nearby, and exhaustive response embeddings provide an observable target for evaluating prediction error η.The fidelity gap α depends on unavailable prompt-conditioned preference embeddings, unlike η.Source: §3 · pp. 9–10, Figure 2
  • Routing on exhaustive viewpoint embeddings yields γβ-approximate reward PJR+, where γβ = β + 2α; when α = 0, the guarantee recovers exact reward PJR+.This exhaustive procedure requires generating every simulation response before routing.Source: §3 · p. 9
  • Predictive Route–then–Simulate predicts each simulation’s response embedding and queries only k selected simulations, without direct access to individual preference embeddings.The predictor estimates response embeddings, not response text or the unavailable preference embeddings.Source: §3 · pp. 9–10
  • With simulation fidelity gap at most α and VPE prediction error at most η, PRS satisfies eγβ-approximate reward PJR+, where eγβ = β + 2α + 4η.The VPE prediction error is defined against the embeddings of simulations’ generated responses.Source: §3 · p. 10

4 Experiments

The experiments assess whether Predictive Route-then-Simulate (PRS) can select a small representative set of simulations across political questions and personal advice. Across both domains, SEAR outperforms k-means and random routing, while performance declines as k increases.

4.1 Experimental set-up

  • The evaluation measures mPJR+ satisfaction relative to the simulation pool, treating generated response embeddings as both voters’ preferences and candidate responses.The experiments set the fidelity gap α to zero for this evaluation; the resulting measure does not directly establish representation of human preferences.Source: §4.1 · p. 11
  • The comparison includes an all-simulations oracle, PRS with SEAR, PRS with k-means, and uniform random selection.SEAR selects using proxy embeddings, whereas the k-means baseline chooses simulations closest to proxy-embedding cluster centers.Source: §4.1 · p. 11
  • Remesh tests political viewpoints using one simulation per participant; Reddit tests personal advice using HumanLM simulations conditioned on users’ Reddit histories.The Remesh study uses 310 participants and 1,000 prompts related to four original questions; the Reddit dataset comprises 992 posts and 4,567 users.Source: §4.1 · pp. 11–12

4.2 Results

  • For both datasets, Route-then-Simulate SEAR outperforms k-means and random for every tested k from 3 to 10; satisfaction degrades as k increases across all methods.Oracle Simulate-then-Route always achieves a 100% satisfaction rate.Source: §4.2 · p. 12
  • For one Remesh prompt, proxy-embedding SEAR satisfies mPJR+ and selects responses similar to oracle SEAR, whereas k-means and random violate it.The example’s groups warranting representation at k = 4 must comprise at least 25% of the population; selecting disagreement outliers leaves the larger agreement cluster underrepresented.Source: §4.2 · p. 13, Figure 4 · p. 13

5 Discussion

The discussion frames the work as a formalization of scalable, representative routing for SAGE, with approximate representation achieved by sampling simulations and dynamically querying a smaller subset. It also identifies boundaries on the evidence and cautions that routing alone is not a complete trustworthy SAGE system.

5 Discussion

  • The paper formalizes representative routing for SAGE and argues that n simulations can represent nH humans while querying only k simulations at inference time, with approximate population-level guarantees.The discussion characterizes both n ≪ nH and k ≪ n.Source: §5 · p. 13
  • The empirical evaluation covers two datasets; generalization to other domains and populations remains future work.The choice of k is application-dependent because it determines the minimum group size for representation guarantees.Source: §5 · p. 13
  • Future work could explore other axioms and routing that accounts for simulations with different reliability across domains or population subgroups.The discussion names the scalability–representativeness Pareto frontier and heterogeneous fidelity as open directions.Source: §5 · p. 14
  • The paper addresses only representative simulation routing; trustworthy SAGE also requires high-fidelity simulation, confidence estimation, and appropriate user interfaces.The authors caution against deploying the routing methods in isolation.Source: §5 · p. 14

A.1 Proofs Omitted from Section 2.1

The appendix establishes representation and runtime results for the routing procedures, including a fast SEAR implementation. Its analysis uses budget thresholds to select candidates and charge voters, and states the broader setting in which the algorithm is applied.

A.1 Proofs Omitted from Section 2.1

  • Simulate-then-Route returns a slate satisfying reward PJR+.Its proof treats generated responses as candidates and applies ResponseRouter to them.Source: §A.1 · p. 18
  • The general formulation allows separate indexed voter and candidate multisets, including distinct objects at the same point, and assumes k ≤ min{m, n}.The proof’s budget accounting updates candidate-prefix sums when voters pay and advances each candidate’s distance-ordered threshold index at most n times.Source: §A.1 · pp. 18, 20
  • SEAR selects the unchosen candidate whose expanding ball first reaches remaining voter budget n/k, then charges voters in that ball until the threshold is paid.Each voter begins with budget 1, and ties are resolved by index order.Source: §A.1 · pp. 18, 20
  • SEAR can be implemented in O(mn log n) time for m candidates and n voters, without a separate multiplicative dependence on k.The runtime combines distance sorting with amortized budget and threshold bookkeeping.Source: §A.1 · pp. 18, 20

A.2 Details Omitted from Section 2.2

The omitted examples show that both ordinary and balanced k-means can violate mPJR+, while the population-transfer result and reward–metric equivalence establish formal guarantees for sampled response slates.

k-means example

  • Ordinary k-means can have zero objective yet violate mPJR+: its slate selects v1, v5, and v6, while the four candidates at the origin warrant two representatives.Only v1 in the slate meets the group's required zero-distance condition.Source: §A.2 · p. 20

Balanced k-means example

  • A balanced k-means slate selects candidates at A, B, and C but none at O, so the three-candidate group at O receives no required representation.The group is 1-large, and the slate has no selected candidate at distance 0 from its members.Source: §A.2 · p. 21

Population transfer

  • With probability at least 1 − δ, a size-k slate satisfying reward PJR+ on a uniformly sampled instance satisfies (ε, γβ)-approximate reward PJR+ for the full population, where γβ = min{π, 4β}.The sampling is from the n-element subsets of the human population.Source: §A.2 · p. 21

Reward–metric equivalence

  • Reward PJR+ holds for a response slate exactly when its corresponding indexed embedding slate satisfies mPJR+, and the corresponding REAR and SEAR procedures select the same indexed candidates under matching rules.The selection equivalence assumes corresponding tie-breaking and payment rules.Source: §A.2 · p. 22

A.3 Proofs Omitted from Section 2.3

The proofs establish exact agreement between routing before and after simulation under a common tie-breaking rule, and approximate representation when simulations have bounded fidelity gap.

Exact commutation

  • Under a common tie-breaking rule, Simulate–then–Route and Route–then–Simulate select the same simulation indices, so Route–then–Simulate satisfies reward PJR+ while querying only k simulations.The proof uses reward-maximizing simulations whose embeddings coincide with the corresponding human preference embeddings.Source: §A.3 · p. 22

Approximate commutation

  • When Simgap(p) ≤ α, Route–then–Simulate satisfies γβ-approximate reward PJR+, with γβ = α + β + min{α, β}.The proof bounds representation using both an unselected candidate and the diameter of the cohesive group.Source: §A.3 · p. 23

Approximate commutation proof

  • The proof concludes that at least ℓ responses meet the group’s approximate reward threshold by combining the two bounds and choosing the better one.It establishes the guarantee for arbitrary ℓ, unselected response, and qualifying group.Source: §A.3 · p. 23

A.4 Proof Omitted from Section 3

The predictive routing proof extends approximate representation to predicted embeddings, with its guarantee depending on both simulation fidelity and prediction error.

Predictive representation

  • If Simgap(p) ≤ α and VPE prediction error is at most η, predictive Route–then–Simulate satisfies eγβ-approximate reward PJR+, where eγβ = β + 2α + 4η.The proof routes on predicted embeddings and transfers the representation guarantee to generated responses.Source: §A.4 · pp. 23–24

Proof

  • The proof applies mPJR+ to predicted embeddings using an unselected predicted candidate, bounding its distance to the cohesive group by β + α + 2η.This bound combines prediction error, simulation fidelity, and the group’s reward threshold.Source: §A.4 · p. 24
  • The final response bound adds simulation-fidelity and prediction-error terms to the routed-embedding distance, yielding the theorem’s approximate reward guarantee.The proof uses the angular triangle inequality to relate human preferences, simulated responses, and predicted embeddings.Source: §A.4 · p. 24

B Additional Sampling Results

Additional sampling results establish population transfer for geometrically defined and arbitrary cohesive groups, and extend transfer guarantees to stratified samples.

Dimension-dependent population transfer

  • For Euclidean populations, the uniform-sampling result transfers mPJR+ representation to ball-induced groups, with sample complexity depending on dimension rather than population size or slate size.The proof uses the VC dimension bound for Euclidean balls, at most d + 1.Source: B Additional Sampling Results · pp. 24, 26
  • For a range family with VC dimension h, a sample size n = Ω(ε^-2(h log(1/ε) + log(1/δ))) suffices for failure probability at most δ.The finite-population bound applies to sampling without replacement.Source: B Additional Sampling Results · pp. 24–25
  • For arbitrary cohesive groups, the alternative population-transfer guarantee incurs a factor-3 approximation in distance and has sample complexity independent of k and nH.Its sample complexity depends on d, ε, and δ.Source: B Additional Sampling Results · p. 26

Stratified sampling

  • Conditioning on a uniformly random sample being stratified extends the uniform-sampling guarantee, at a polynomial cost in the failure probability.The resulting guarantee is 4-approximate mPJR+ population transfer for every qualifying group with probability at least 1 − δ.Source: B Additional Sampling Results · pp. 27, 30
  • Under stratified sampling, every mPJR+ slate for the sample satisfies the stated 4-approximate population-transfer guarantee with probability at least 1 − δ.The result applies to groups of size at least (ℓ/k + ε)nH.Source: B Additional Sampling Results · pp. 27, 30

C.1 Remesh Dataset Details

The Remesh appendix describes generating opinion prompts in two stages: first deriving diverse topics from participants’ campus-protest answers, then creating questions for each topic.

C.1.1 Topic Generation Prompt

  • Campus-protest answers were used to prompt Claude Opus 4.6 to generate ten diverse, predictable topics.The prompt requested JSONL topic descriptions intended to guide later question generation.Source: §C.1.1 · p. 30

C.1.2 Prompt Generation Prompt

  • Claude Opus 4.6 was then prompted to generate 100 stand-alone, open-ended opinion questions per topic, mixing contentious and non-contentious questions.Questions were requested in JSONL format.Source: §C.1.2 · p. 30

C.1.3 Generated Topics

The appendix lists generated evaluation topics and describes how demographic profiles and prior responses shape roleplayed agents, whose answers are later condensed into participant viewpoint summaries.

C.1.3 Generated Topics

  • Ten generated topics include free speech, Israel–Palestine, university leadership, activism, police use of force, media coverage, and institutional divestment.The list also covers foreign policy, antisemitism and Islamophobia, and generational differences in political engagement.Source: §C.1.3 · p. 31

C.1.4 Simulated Agent Prompt

  • Simulated agents roleplay specific participants using demographic descriptions, with optional prior responses and statements they agreed or disagreed with as context.The prompt asks for a genuine, specific-perspective response in two to three sentences.Source: §C.1.4 · pp. 31–32

C.1.5 Viewpoint Summary Prompt

  • For each participant, Claude Opus 4.6 summarizes training-prompt responses by topic in at most three sentences, synthesizing key views, attitudes, and reasoning.The per-topic summaries are concatenated into one participant viewpoint summary, formatted by topic.Source: §C.1.5 · p. 32
  • An example viewpoint summary preserves a participant’s topic-specific positions and reasoning, including tensions between support for historical civil disobedience and objections to lawbreaking.The example also acknowledges that history may vindicate protesters while expressing skepticism about disruptive tactics.Source: §C.1.5 · p. 33

C.2 Proxy Embedding Training Details

The proxy embedding model predicts simulated-response embeddings from participant context and prompts, while its training objective combines embedding accuracy with preservation of pairwise distances.

C.2 Proxy Embedding Training Details

  • The preference embedding model ψ reflects preference distances, while proxy model θ predicts a participant’s ground-truth response embedding from user context and prompt.The proxy is a fine-tuned copy of the preference embedding model.Source: §C.2 · p. 34
  • Training combines elementwise mean-squared error on embeddings with Pearson correlation loss on pairwise-distance structure, using λ = 1.The batch consists of users sampled for a prompt.Source: §C.2 · p. 35
  • The model is fine-tuned for one epoch with LoRA adapters on query and value attention modules, using AdamW at learning rate 10^-5.Training uses batches of 16 responses to the same prompt, eight H100 GPUs, and a maximum sequence length of 2,048 tokens.Source: §C.2 · p. 35

C.3 Qualitative Examples

The qualitative appendix illustrates selected responses and embeddings across Remesh political questions and Reddit advice prompts, while noting that Remesh projections are generally easier to interpret.

C.3 Qualitative Examples

  • Up to 76% of variance is explained by the first two PCs for Remesh prompts, versus up to 38% for Reddit; Remesh examples are generally easier to interpret.Examples were selected from prompts in each dataset’s top ten for variance explained by the first two PCs.Source: §C.3 · p. 35
  • For Reddit human YTA judgments, the HumanLM simulator predicts NTA 87% of the time among response pairs with NTA/YTA judgments.The confusion matrix axes compare ground-truth human judgments with simulated judgments, against a dataset skew toward NTA.Source: Figure 5 · p. 36

C.3.1 Remesh Qualitative Examples

  • The Remesh pension-investment example includes selected responses supporting ethical considerations and others prioritizing financial returns.The selected-response table is paired with an embedding visualization for the prompt.Source: Table 1 · p. 36, §C.3.1 · pp. 37–38
  • The Remesh encampment and student-activism examples show selected responses taking different positions on protest and its conditions.The examples concern university encampments and high-school participation in political activism and civil disobedience.Source: Table 2 · p. 39, §C.3.1 · pp. 40, 42, Table 3 · p. 41

C.3.2 Reddit Qualitative Examples

  • The Reddit wedding-dress example presents selected judgments strongly validating the poster’s reaction and autonomy.The appendix pairs the selected responses with an embedding visualization.Source: Table 4 · p. 44, §C.3.2 · p. 45
  • For the Reddit dad-joke prompt, selected responses range from treating the exchange as playful to emphasizing the daughter’s hurt and the need to acknowledge it.These responses illustrate divergent readings of the same interaction.Source: Table 5 · p. 46, §C.3.2 · p. 47
  • For the pregnancy-intentions prompt, the selected Reddit responses largely judge disclosure to the boyfriend as justified and condemn the friend’s alleged deception.The prompt’s four selected responses appear in Table 6, with embeddings shown in Figure 10.Source: Table 6 · p. 48, §C.3.2 · p. 49

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