Rethinking the Evaluation and Optimization of LLM-Based Social Simulation

# Rethinking the Evaluation and Optimization of LLM-Based Social Simulation
Source: [https://arxiv.org/html/2608.19689](https://arxiv.org/html/2608.19689)
## Rethinking the Evaluation and Optimization of LLM\-Based Social SimulationThanks:Project page:[https://yupei\-wang\.github\.io/rethinking\-social\-simulation/](https://yupei-wang.github.io/rethinking-social-simulation/)

Xu ChenJi\-Rong WenAffiliation:Gaoling School of Artificial Intelligence, Renmin University of ChinaEmail:[wang\_pei@ruc\.edu\.cn](mailto:)xu\.chen@ruc\.edu\.cnjrwen@ruc\.edu\.cn

###### Abstract

LLM\-based social simulation is a promising complement to traditional social science methods such as surveys and behavioral experiments\. A core question in this area is how to evaluate the fidelity of LLM\-simulated human behavior and, in turn, how to optimize LLMs toward it\. Prevailing practice evaluates by accuracy, checking whether the model selects the single response observed from a human, and accordingly trains the LLM to reproduce this one hard label\. However, human behavior is inherently subjective: the same person in the same situation may reasonably act in different ways, so an observed response is only one draw from an underlying response distribution, which renders accuracy\-based evaluation unreliable and hard\-label training misleading\. To address these problems, in this paper, we first introduce the*subjectivity coefficient*, an entropy\-based quantity that distinguishes objective tasks such as coding from subjective tasks such as social simulation, and then use it to systematically analyze how accuracy\-based evaluation and hard\-label training fail as subjectivity grows\. Based on the subjectivity coefficient, we further propose*Subjectivity\-Adaptive soft\-Label Training*\(SALT\): it pools observed outputs from semantically nearby inputs into soft distributional labels, with an aggregation radius adapted to the estimated subjectivity of each input; in the near\-objective limit the neighborhood shrinks, so SALT naturally falls back to standard single\-label training\. Moreover, since existing datasets record only single observed responses and thus cannot support distributional evaluation, we constructSubjSim, a benchmark of 19,300 contexts covering 193 annotators and 100 subjective questions\. Since real\-world data typically provide only a single observation per input, our experiments train models from single observed outputs while evaluating them against the full response distributions, thereby verifying the feasibility of our method in realistic settings\. Extensive results onSubjSimdemonstrate the advantages of our method\.

## 1Introduction

LLM\-based social simulation holds great potential for social science research, offering a low\-cost, controllable, and scalable way to study problems such as opinion dynamics, political polarization, market behavior, and policy interventions\([14](https://arxiv.org/html/2608.19689#bib.bib25);[30](https://arxiv.org/html/2608.19689#bib.bib28);[9](https://arxiv.org/html/2608.19689#bib.bib2);[11](https://arxiv.org/html/2608.19689#bib.bib3);[18](https://arxiv.org/html/2608.19689#bib.bib1);[23](https://arxiv.org/html/2608.19689#bib.bib14);[2](https://arxiv.org/html/2608.19689#bib.bib15)\)\. The core question in this area is how to evaluate the fidelity of LLM\-simulated behavior\([6](https://arxiv.org/html/2608.19689#bib.bib24);[7](https://arxiv.org/html/2608.19689#bib.bib27);[10](https://arxiv.org/html/2608.19689#bib.bib21);[29](https://arxiv.org/html/2608.19689#bib.bib22);[22](https://arxiv.org/html/2608.19689#bib.bib23)\), and then, under the established criterion, how to train the model toward it\. Most existing work approaches this question by directly following the practice of objective domains such as coding and math: collect one response for each person on each question, fine\-tune the model to reproduce that response as the correct label, and check with accuracy\-style metrics whether the model picks the same answer\.

For objective tasks this recipe is perfectly sound: there is a well\-defined correct answer, so a single label fully specifies the target, reproducing the label is exactly the desired behavior, and accuracy faithfully measures success\. However, social simulation breaks this premise\. Human behavior is stochastic rather than deterministic\([8](https://arxiv.org/html/2608.19689#bib.bib11);[3](https://arxiv.org/html/2608.19689#bib.bib9);[26](https://arxiv.org/html/2608.19689#bib.bib8);[33](https://arxiv.org/html/2608.19689#bib.bib10);[21](https://arxiv.org/html/2608.19689#bib.bib7);[16](https://arxiv.org/html/2608.19689#bib.bib5);[15](https://arxiv.org/html/2608.19689#bib.bib6)\): the same person may act differently across comparable occasions, and people with similar profiles may choose different actions\. An observed response is therefore only one draw from an underlying response distribution\. A single label no longer specifies the target\. During training, reproducing it collapses the model onto one action; at test time, whether the model hits that answer depends partly on chance rather than on how faithful the simulation is\.

To study the above problems systematically, we first define the*subjectivity coefficient*, an entropy\-based measure of the inherent randomness of human responses in a given decision context; under this definition, objective tasks \(e\.g\., coding\) have a near\-zero subjectivity coefficient, while subjective tasks \(e\.g\., social simulation\) have a much higher one\. Based on the subjectivity coefficient, we then formally show that the common recipe of accuracy evaluation plus single\-label training breaks down as subjectivity grows\. For evaluation, the recorded answer is itself only one random draw from the response distribution, so whether the model matches it reveals little about fidelity; for training, fitting the single label pulls the model away from the true response distribution\. Finally, we propose*Subjectivity\-Adaptive soft\-Label Training*\(SALT\)\. For each context, SALT collects the answers observed at semantically similar contexts that share the same options and turns them into a soft label, and the neighborhood size automatically adapts to how subjective the context is estimated to be\. When a context is nearly objective, the neighborhood shrinks and SALT falls back to standard single\-label training, so hard\-label supervision is recovered as the low\-subjectivity special case\.

Evaluating distributional fidelity requires reference human response distributions, but existing social\-simulation datasets typically record only a single response for each context\. We therefore constructSubjSim, a benchmark in which 193 annotators answer 100 subjective survey questions, yielding 19,300 annotator\-question pairs\. Each pair carries an elicited response\-propensity distribution obtained through probability\-ball allocation\. During training, models never see these distributions and receive only one derived hard action per pair; at test time, the full distributions are used to measure more accurately how faithfully the model reproduces human behavior\. With Qwen3\-8B as the backbone, SALT reduces aggregate KL divergence over standard supervised fine\-tuning \(SFT\) by 77\.6%, JSD by 45\.9%, TVD by 31\.2%, and MMD by 52\.7%, with the largest gains in mid\- and high\-subjectivity settings, where one observed response reveals the least about the full distribution\.

In summary, this paper makes contributions at three levels\. \(i\) At the problem level, we identify why the prevailing accuracy\-based evaluation and training practice is unreasonable for social simulation, and rigorously analyze this failure by introducing the subjectivity coefficient\. \(ii\) At the method level, we propose SALT, which aggregates observations from semantically similar contexts into soft distributional labels with a subjectivity\-adaptive radius\. \(iii\) At the data level, we constructSubjSim, a benchmark of 19,300 annotator\-question pairs with human\-annotated response distributions, and extensive experiments on it validate the effectiveness of SALT\.

## 2Preliminaries

### 2\.1Formulation of Social Simulation

We describe each individual in the target population by a persona vectoruu, which encodes demographic features, personality traits, prior beliefs, or other personal attributes\. A decision context is the tuplex:=\(u,s\)x:=\(u,s\), wheressis a situational description; we write𝒳\\mathcal\{X\}for the space of all decision contexts\. At contextxx, the individual chooses from a candidate action set𝒜⁡\(x\)=\{a\(1\),…,a\(K\)\}\\mathcal\{A\}\(x\)=\\\{a^\{\(1\)\},\\ldots,a^\{\(K\)\}\\\}, where each action is a natural\-language description andK=\|𝒜⁡\(x\)\|K=\|\\mathcal\{A\}\(x\)\|may vary across contexts\. For example,uumay describe a 26\-year\-old graduate student living in a big city,ssmay ask which factor matters most when choosing a job, and the corresponding action set is𝒜⁡\(x\)=\{“salary”,“stability”,“personal interest”,“work–life balance”\}\\mathcal\{A\}\(x\)=\\\{\\text\{\`\`salary''\},\\text\{\`\`stability''\},\\text\{\`\`personal interest''\},\\text\{\`\`work\-\-life balance''\}\\\}\. In another example,uumay describe a retired teacher living in a small town,ssmay ask how the individual mainly gets news, and𝒜⁡\(x\)=\{“social media”,“television”,“news apps”\}\\mathcal\{A\}\(x\)=\\\{\\text\{\`\`social media''\},\\text\{\`\`television''\},\\text\{\`\`news apps''\}\\\}\. Social simulation aims to use an LLM to mimic how these individuals respond and behave\.

### 2\.2The Ideal Optimization Target

Defining the ideal optimization target requires first understanding the nature of human behavior\. Many social science theories have shown that human behavior is not deterministic but contains an inherent random component\. Random utility theory\([15](https://arxiv.org/html/2608.19689#bib.bib6);[16](https://arxiv.org/html/2608.19689#bib.bib5)\), stochastic evidence\-accumulation models\([21](https://arxiv.org/html/2608.19689#bib.bib7)\), and preference construction research\([26](https://arxiv.org/html/2608.19689#bib.bib8)\)all model a person’s choice as a draw from a probability distribution rather than as a fixed answer\. This randomness is easy to see in daily life: the same person may pick a different dish from the same menu on different days, answer the same survey question differently when asked twice\([33](https://arxiv.org/html/2608.19689#bib.bib10)\), cooperate in one round of an economic game and defect in the next\([3](https://arxiv.org/html/2608.19689#bib.bib9)\), or be talkative at one gathering and quiet at another\([8](https://arxiv.org/html/2608.19689#bib.bib11)\)\. This stochasticity is part of the target itself rather than label noise\. Ideally, the model should therefore capture the uncertainty of human behavior: for each context, it should reproduce not only which action is most likely, but the full probability mass assigned to all plausible actions\. Formally, the target is the latent response\-propensity distributionϕ∗\(⋅∣x\)∈Δ\(𝒜\(x\)\)\\phi^\{\*\}\(\\cdot\\mid x\)\\in\\Delta\(\\mathcal\{A\}\(x\)\), namely the action distribution that would be observed if the same context could be measured repeatedly under comparable conditions, whereΔ\(𝒜\(x\)\)=\{\(q1,…,qK\):qk≥0,∑kqk=1\}\\Delta\(\\mathcal\{A\}\(x\)\)=\\\{\(q\_\{1\},\\ldots,q\_\{K\}\):q\_\{k\}\\geq 0,\\sum\_\{k\}q\_\{k\}=1\\\}is the probability simplex over the available actions and we writepk:=ϕ∗​\(k∣x\)p\_\{k\}:=\\phi^\{\*\}\(k\\mid x\)for brevity\. On the model side, an LLM with parametersθ\\thetainduces an action\-level distributionϕθ\(⋅∣x\)∈Δ\(𝒜\(x\)\)\\phi\_\{\\theta\}\(\\cdot\\mid x\)\\in\\Delta\(\\mathcal\{A\}\(x\)\), obtained by normalizing the generation probabilities of theKKcandidate actions, withϕθ​\(k∣x\)\\phi\_\{\\theta\}\(k\\mid x\)written analogously\. The ideal goal of social simulation is to learn a model whose induced action distribution is close toϕ∗\\phi^\{\*\}across contexts:

θ^=arg​minθ𝔼x\[D\(ϕ∗\(⋅∣x\),ϕθ\(⋅∣x\)\)\],\\hat\{\\theta\}=\\operatorname\{arg\\,min\}\_\{\\theta\}\\mathbb\{E\}\_\{x\}\\\!\\left\[D\\\!\\left\(\\phi^\{\*\}\(\\cdot\\mid x\),\\;\\phi\_\{\\theta\}\(\\cdot\\mid x\)\\right\)\\right\],\(1\)whereDDmeasures mismatch on the action simplex and can be instantiated as various distance functions, e\.g\., total variationDTV​\(P,Q\):=12​∑k=1K\|Pk−Qk\|D\_\{\\mathrm\{TV\}\}\(P,Q\):=\\frac\{1\}\{2\}\\sum\_\{k=1\}^\{K\}\|P\_\{k\}\-Q\_\{k\}\|, cross entropyDCE\(P,Q\):=−∑kPklogQkD\_\{\\mathrm\{CE\}\}\(P,Q\):=\-\\sum\_\{k\}P\_\{k\}\\log Q\_\{k\}, or KL divergenceDKL\(P∥Q\)D\_\{\\mathrm\{KL\}\}\(P\\\|Q\)\.

### 2\.3The Current Accuracy\-Based Practice

In practice,ϕ∗\\phi^\{\*\}is not observable: each context is recorded only once, yielding a dataset𝒟=\{\(xi,a⁡\(xi\)\)\}\\mathcal\{D\}=\\\{\(x^\{i\},a\(x^\{i\}\)\)\\\}in whicha⁡\(x\)∈𝒜⁡\(x\)a\(x\)\\in\\mathcal\{A\}\(x\)is the single observed action, either one stochastic draw fromϕ∗\(⋅∣x\)\\phi^\{\*\}\(\\cdot\\mid x\)or a single\-action proxy derived from it\. Letj⁡\(x\)j\(x\)denote the index ofa⁡\(x\)a\(x\)and𝜹a⁡\(x\)\\bm\{\\delta\}\_\{a\(x\)\}the one\-hot distribution that puts all mass ona⁡\(x\)a\(x\)\. Prevailing practice trains and evaluates against this single observation\. On the training side, the model is trained to maximize the probability it assigns to the observed action:

θ~=argmaxθ𝔼x\[logϕθ\(j\(x\)∣x\)\]=arg​minθ𝔼x\[DCE\(𝜹a⁡\(x\),ϕθ\(⋅∣x\)\)\],\\tilde\{\\theta\}=\\arg\\max\_\{\\theta\}\\mathbb\{E\}\_\{x\}\\\!\\left\[\\log\\phi\_\{\\theta\}\(j\(x\)\\mid x\)\\right\]=\\operatorname\{arg\\,min\}\_\{\\theta\}\\mathbb\{E\}\_\{x\}\\\!\\left\[D\_\{\\mathrm\{CE\}\}\\\!\\left\(\\bm\{\\delta\}\_\{a\(x\)\},\\;\\phi\_\{\\theta\}\(\\cdot\\mid x\)\\right\)\\right\],\(2\)which is exactly objective \([1](https://arxiv.org/html/2608.19689#S2.E1)\) with the unobservable targetϕ∗\(⋅∣x\)\\phi^\{\*\}\(\\cdot\\mid x\)replaced by the one\-hot vector𝜹a⁡\(x\)\\bm\{\\delta\}\_\{a\(x\)\}andDDtaken as the cross entropy\. On the evaluation side, accuracy checks whether the model’s top\-probability action coincides with the same single observation:

Acc\(θ\)=𝔼x\[\[argmaxkϕθ\(k∣x\)=j\(x\)\]\],\\mathrm\{Acc\}\(\\theta\)=\\mathbb\{E\}\_\{x\}\\\!\\left\[\\mathbf\{1\}\\\!\\left\[\\textstyle\\arg\\max\_\{k\}\\phi\_\{\\theta\}\(k\\mid x\)=j\(x\)\\right\]\\right\],\(3\)replacing the divergence in \([1](https://arxiv.org/html/2608.19689#S2.E1)\) by a 0–1 comparison between the model’s mode and the single draw\. A natural question arises: how much do \([2](https://arxiv.org/html/2608.19689#S2.E2)\) and \([3](https://arxiv.org/html/2608.19689#S2.E3)\) actually capture of the ideal objective \([1](https://arxiv.org/html/2608.19689#S2.E1)\)? In the next section, we answer this question by rigorously analyzing the gap between them\.

## 3Misalignment Analysis of the Accuracy\-Based Practice

### 3\.1Evaluation Metric Analysis

Ideally, an evaluation metric should be consistent with true model quality: a model that scores better under the metric should also be closer to the true response distribution under the ideal objective \([1](https://arxiv.org/html/2608.19689#S2.E1)\)\. We now check whether accuracy \([3](https://arxiv.org/html/2608.19689#S2.E3)\) has this property\.

Consider ranking two modelsθ1\\theta\_\{1\}andθ2\\theta\_\{2\}by accuracy at a contextxxwith observed indexj=j⁡\(x\)j=j\(x\), and letk1k\_\{1\}andk2k\_\{2\}denote the two models’ top actions\. There are three cases\.

\(i\)k1=k2k\_\{1\}=k\_\{2\}: the two models receive the same score\. However, the true errorDTV​\(ϕθi,ϕ∗\)D\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\_\{i\}\},\\phi^\{\*\}\)depends on allKKprobability values rather than the top action alone, and the hidden gap can be nearly maximal\. Supposeθ1\\theta\_\{1\}matches the target exactly,ϕθ1=ϕ∗\\phi\_\{\\theta\_\{1\}\}=\\phi^\{\*\}, whileθ2\\theta\_\{2\}puts all its mass on the shared top actionk1k\_\{1\}, which is then the mode ofϕ∗\\phi^\{\*\}withpk1=pmaxp\_\{k\_\{1\}\}=p\_\{\\max\}\. Their true errors are

DTV​\(ϕθ1,ϕ∗\)=0,DTV​\(ϕθ2,ϕ∗\)=12​\[\(1−pmax\)\+∑k≠k1pk\]=1−pmax,D\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\_\{1\}\},\\phi^\{\*\}\)=0,\\qquad D\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\_\{2\}\},\\phi^\{\*\}\)=\\frac\{1\}\{2\}\\Bigl\[\(1\-p\_\{\\max\}\)\+\\sum\\nolimits\_\{k\\neq k\_\{1\}\}p\_\{k\}\\Bigr\]=1\-p\_\{\\max\},\(4\)which differ by1−pmax1\-p\_\{\\max\}and reach1−1/K1\-1/Kunder uniform behavior, yet accuracy scores the two models identically\.

\(ii\)k1≠k2k\_\{1\}\\neq k\_\{2\}and neither equalsjj: both models score zero, and accuracy again cannot tell them apart\. As in case \(i\), consider a pair of models where one is perfect and the other is a point\-mass model\. The observed actionjjis one random draw fromϕ∗\\phi^\{\*\}and may well miss the most probable action, so even the perfect modelϕθ1=ϕ∗\\phi\_\{\\theta\_\{1\}\}=\\phi^\{\*\}, whose top actionk1k\_\{1\}is the most probable one, can fall into this case, whileθ2\\theta\_\{2\}puts all its mass on another unobserved action,ϕθ2=𝜹k2\\phi\_\{\\theta\_\{2\}\}=\\bm\{\\delta\}\_\{k\_\{2\}\}\. Their true errors are

DTV​\(ϕθ1,ϕ∗\)=0,DTV​\(ϕθ2,ϕ∗\)=12​\[\(1−pk2\)\+∑k≠k2pk\]=1−pk2,D\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\_\{1\}\},\\phi^\{\*\}\)=0,\\qquad D\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\_\{2\}\},\\phi^\{\*\}\)=\\frac\{1\}\{2\}\\Bigl\[\(1\-p\_\{k\_\{2\}\}\)\+\\sum\\nolimits\_\{k\\neq k\_\{2\}\}p\_\{k\}\\Bigr\]=1\-p\_\{k\_\{2\}\},\(5\)so a perfect model and a model with error1−pk21\-p\_\{k\_\{2\}\}receive the same zero score\.

\(iii\)k1≠k2k\_\{1\}\\neq k\_\{2\}and exactly one equalsjj, sayk1=jk\_\{1\}=j: accuracy ranksθ1\\theta\_\{1\}higher, and this is the only case where it expresses a preference\. However, the preference can be exactly backward\. Supposeθ2\\theta\_\{2\}is the perfect model,ϕθ2=ϕ∗\\phi\_\{\\theta\_\{2\}\}=\\phi^\{\*\}, with its top actionk2k\_\{2\}the most probable one, whileθ1\\theta\_\{1\}puts all its mass on the drawj≠k2j\\neq k\_\{2\}\. Their true errors are

DTV​\(ϕθ1,ϕ∗\)=12​\[\(1−pj\)\+∑k≠jpk\]=1−pj\>0=DTV​\(ϕθ2,ϕ∗\),D\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\_\{1\}\},\\phi^\{\*\}\)=\\frac\{1\}\{2\}\\Bigl\[\(1\-p\_\{j\}\)\+\\sum\\nolimits\_\{k\\neq j\}p\_\{k\}\\Bigr\]=1\-p\_\{j\}\\;\>\\;0=D\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\_\{2\}\},\\phi^\{\*\}\),\(6\)so whenever the single draw misses the most probable action, accuracy prefers a strictly worse model over the perfect one, and the reversed gap1−pj1\-p\_\{j\}grows as behavior becomes more diffuse\.

In summary, the above counterexamples show that accuracy is not a reliable measure of behavioral fidelity: models with the same score can differ substantially in true quality, and in the worst case accuracy even prefers a strictly worse model over the perfect one\. A better accuracy score therefore does not imply a model closer to the true response distribution\.

### 3\.2Training Objective Analysis

We next examine the training objective \([2](https://arxiv.org/html/2608.19689#S2.E2)\)\. Its empirical signal at each training context is the one\-hot target𝜹a⁡\(x\)\\bm\{\\delta\}\_\{a\(x\)\}: the objective is monotonically increasing in the probability assigned to the observed action, so its optimum drivesϕθ​\(j∣x\)→1\\phi\_\{\\theta\}\(j\\mid x\)\\to 1on the training context, with no gradient signal that rewards distributing mass across multiple actions\.

###### Proposition 1\(Single\-observation training concentrates on one\-hot labels\)\.

Consider a finite training set in which each contextxix^\{i\}is observed once with hard labeljij^\{i\}\. If the closure of the model\-induced action distributions contains the probability simplex independently at each training context \(an idealized full\-capacity condition\), then any empirical\-risk minimizerθ~\\tilde\{\\theta\}of objective \([2](https://arxiv.org/html/2608.19689#S2.E2)\) in this closure assignsϕθ~\(k∣xi\)=𝟏\[k=ji\]\\phi\_\{\\tilde\{\\theta\}\}\(k\\mid x^\{i\}\)=\\mathbf\{1\}\[k=j^\{i\}\]for every training contextxix^\{i\}\. In other words, the closer a model is trained toward the minimum loss, the closer its predicted distribution comes to putting probability one on the single observed answer at each training context\.

The proof is given in[SectionC\.1](https://arxiv.org/html/2608.19689#A3.SS1)\. Under this point\-mass solution, the total\-variation error at a training context is:

DTV\(ϕθ~,ϕ∗\)=12∑k\|𝟏\[k=j\]−pk\|=1−pj,D\_\{\\mathrm\{TV\}\}\\\!\\left\(\\phi\_\{\\tilde\{\\theta\}\},\\phi^\{\*\}\\right\)=\\frac\{1\}\{2\}\\sum\_\{k\}\\bigl\|\\mathbf\{1\}\[k=j\]\-p\_\{k\}\\bigr\|=1\-p\_\{j\},\(7\)which equals the total probability mass thatϕ∗\\phi^\{\*\}assigns to actions other than the one observed\. Note that the observed answerjjis just one random draw: it tends to be a likely option but is not always the most likely one, so the error1−pj1\-p\_\{j\}is at least1−pmax1\-p\_\{\\max\}\. Even in the most favorable case where the draw hits the most likely option, diffuse behavior forces every probability, including the largest, to be small, so the error grows with subjectivity and reaches\(K−1\)/K\(K\-1\)/Kat the uniform distribution\.

The analysis above indicates, qualitatively, that the error of the single\-label optimum is ultimately governed by how diffuse the response distribution is\. To describe this relation quantitatively, we introduce a scalar that summarizes the dispersion of the entire distribution\.

###### Definition 1\(Subjectivity coefficient\)\.

The*subjectivity coefficient*of a decision contextxxis the negentropy of the target distribution:

κ⁡\(x\)≔∑k=1Kpk​log⁡pk∈\[−log⁡K,0\],\\kappa\(x\)\\;\\coloneqq\\;\\sum\_\{k=1\}^\{K\}p\_\{k\}\\log p\_\{k\}\\;\\in\\;\[\-\\log K,\\;0\],\(8\)wherepk=ϕ∗​\(k∣x\)p\_\{k\}=\\phi^\{\*\}\(k\\mid x\)\.

The coefficient equals zero whenϕ∗\\phi^\{\*\}is a point mass \(fully deterministic behavior\) and−log⁡K\-\\log Kwhen it is uniform \(maximum ambiguity\)\. Throughout, we mainly work with the normalized subjectivity coefficient

s⁡\(x\)≔−κ⁡\(x\)log⁡K=H\(ϕ∗\(⋅∣x\)\)log⁡K∈\[0,1\],s\(x\)\\coloneqq\-\\frac\{\\kappa\(x\)\}\{\\log K\}=\\frac\{H\(\\phi^\{\*\}\(\\cdot\\mid x\)\)\}\{\\log K\}\\in\[0,1\],\(9\)whereHHis the Shannon entropy, so largers⁡\(x\)s\(x\)means more subjective behavior\. Crucially,ssis a property of the decision contextxx, not of any model: it characterizes how inherently subjective a given scenario is\. Under this view, objective tasks such as coding and math occupy the near\-zero\-ssregime, subjective simulation tasks span the full range, and different task types differ only in their degree of subjectivity along this common axis\. Entropy is the standard measure of how uncertain a distribution is, and it takes allKKprobabilities into account rather than a single one\. We now show that the training error grows withssthrough an explicit lower bound\.

###### Proposition 2\(Subjectivity lower\-bounds the error of single\-label training\)\.

Letpmax=maxk⁡pkp\_\{\\max\}=\\max\_\{k\}p\_\{k\},s=s⁡\(x\)s=s\(x\), andK≥3K\\geq 3\. Any point\-mass model incurs training error

DTV​\(ϕθ,ϕ∗\)≥1−pmax≥s​log⁡K−log⁡2log⁡\(K−1\),D\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\},\\phi^\{\*\}\)\\;\\geq\\;1\-p\_\{\\max\}\\;\\geq\\;\\frac\{s\\log K\-\\log 2\}\{\\log\(K\-1\)\},\(10\)where the first inequality follows from[Equation7](https://arxiv.org/html/2608.19689#S3.E7)sincepj≤pmaxp\_\{j\}\\leq p\_\{\\max\}, and the second follows from Fano’s inequality \(Appendix[C\.2](https://arxiv.org/html/2608.19689#A3.SS2)\)\. Hence the more subjective the context, i\.e\., the largerss, the larger the unavoidable error of any point\-mass model; the lower bound is approximatelyssfor largeKK, so the unavoidable error is at least roughly the normalized subjectivity of the context\. At the uniform distribution, wheres=1s=1, the exact error1−pmax1\-p\_\{\\max\}equals1−1/K1\-1/K\.

The essential cause behind the failures on both the evaluation and the training side is the same: one observed answer per context carries too little information about a diffuse response distribution\. Ideally, if each context were annotated many times, the empirical answer frequencies would recover the true distribution and both problems would disappear\. This is unrealistic, however: reliable model ranking alone would requireΩ⁡\(K2​s\)\\Omega\(K^\{2s\}\)repeated observations per context \(Appendix[C\.3](https://arxiv.org/html/2608.19689#A3.SS3)\), while real data provide exactly one\. A classical alternative is to directly merge the observed answers of similar contexts, as in local smoothing\([17](https://arxiv.org/html/2608.19689#bib.bib19);[31](https://arxiv.org/html/2608.19689#bib.bib20)\)\. Merging, however, involves a trade\-off: pooling more neighbors supplies more distributional information, but neighbors are only similar rather than identical, so pooling also mixes in answers from different distributions and biases the target\. In the next section, we analyze this trade\-off theoretically and derive an algorithm with a context\-adaptive merging radius\.

## 4Subjectivity\-Adaptive Soft\-Label Training \(SALT\)

SALT replaces each one\-hot target with a soft label aggregated from semantically similar contexts, and chooses the neighborhood size to balance the trade\-off above\. Concretely, given𝒟=\{\(xi,a⁡\(xi\)\)\}i=1n\\mathcal\{D\}=\\\{\(x^\{i\},a\(x^\{i\}\)\)\\\}\_\{i=1\}^\{n\}, SALT outputs a soft labelϕ^\(⋅∣x\)\\hat\{\\phi\}\(\\cdot\\mid x\)for each context\. Because actions are comparable only within a shared candidate set, we first partition contexts into action\-space groups:

𝒢g=\{x∈𝒳:𝒜\(x\)=𝒜g\},g=1,…,G,\\mathcal\{G\}\_\{g\}\\;=\\;\\bigl\\\{x\\in\\mathcal\{X\}:\\mathcal\{A\}\(x\)=\\mathcal\{A\}\_\{g\}\\bigr\\\},\\qquad g=1,\\ldots,G,\(11\)where𝒜⁡\(x\)\\mathcal\{A\}\(x\)is the candidate action set of contextxxdefined in[Section2\.1](https://arxiv.org/html/2608.19689#S2.SS1)and𝒜g\\mathcal\{A\}\_\{g\}is the shared candidate set of groupgg; in survey data, for instance, all contexts answering the same question with the same options form one group\. Within each group, contexts are embedded with a pretrained encoder and compared byℓ2\\ell\_\{2\}distancedd\. The neighborhood ofx∈𝒢gx\\in\\mathcal\{G\}\_\{g\}is

𝒩⁡\(x\)=\{x′∈𝒢g:d⁡\(x′,x\)≤r⁡\(x\)\},\\mathcal\{N\}\(x\)\\;=\\;\\bigl\\\{x^\{\\prime\}\\in\\mathcal\{G\}\_\{g\}:d\(x^\{\\prime\},x\)\\leq r\(x\)\\bigr\\\},\(12\)where the radiusr⁡\(x\)r\(x\)is adaptive; how to choose it is the key design question, addressed below\. Since all neighbors share𝒜g\\mathcal\{A\}\_\{g\}, their hard actions define a local empirical distribution:

ϕ^\(k∣x\)≔1\|𝒩⁡\(x\)\|∑x′∈𝒩⁡\(x\)𝟏\[a\(x′\)=a\(k\)\],k=1,…,K,\\hat\{\\phi\}\(k\\mid x\)\\;\\coloneqq\\;\\frac\{1\}\{\|\\mathcal\{N\}\(x\)\|\}\\sum\_\{x^\{\\prime\}\\in\\mathcal\{N\}\(x\)\}\\mathbf\{1\}\[a\(x^\{\\prime\}\)=a^\{\(k\)\}\],\\qquad k=1,\\ldots,K,\(13\)and the model is trained to match this distributional target:

ℒθagg=∑x∈𝒟DKL\(ϕ^\(⋅∣x\)∥ϕθ\(⋅∣x\)\),\\mathcal\{L\}^\{\\mathrm\{agg\}\}\_\{\\theta\}\\;=\\;\\sum\_\{x\\in\\mathcal\{D\}\}D\_\{\\mathrm\{KL\}\}\\\!\\left\(\\hat\{\\phi\}\(\\cdot\\mid x\)\\;\\big\\\|\\;\\phi\_\{\\theta\}\(\\cdot\\mid x\)\\right\),\(14\)whereϕθ\(⋅∣x\)∈Δ\(𝒜\(x\)\)\\phi\_\{\\theta\}\(\\cdot\\mid x\)\\in\\Delta\(\\mathcal\{A\}\(x\)\)is obtained by normalizing generation probabilities across candidates\. Note that this differs from prior grouping\-based methods\([12](https://arxiv.org/html/2608.19689#bib.bib16);[4](https://arxiv.org/html/2608.19689#bib.bib26)\), which partition samples into disjoint groups \(e\.g\., by demographic attributes\) and let all samples in a group share one target distribution, whereas SALT centers a neighborhood at each context, so every context receives its own soft label\.

The key remaining question is how to determine the radiusr⁡\(x\)r\(x\)\. To answer it, we bound the error of aggregation within one action\-space group𝒢g\\mathcal\{G\}\_\{g\}, writingd𝒳d\_\{\\mathcal\{X\}\}for the intrinsic dimension of its context space,ng=\|𝒢g\|n\_\{g\}=\|\\mathcal\{G\}\_\{g\}\|for the number of contexts in the group, andKKfor the number of actions\. The dimension enters through the volume of a neighborhood: a ball of radiusrrin ad𝒳d\_\{\\mathcal\{X\}\}\-dimensional space holds a fraction≍rd𝒳\\asymp r^\{d\_\{\\mathcal\{X\}\}\}of the contexts, so\|𝒩⁡\(x\)\|≍ng⋅rd𝒳\|\\mathcal\{N\}\(x\)\|\\asymp n\_\{g\}\\cdot r^\{d\_\{\\mathcal\{X\}\}\}\. The bound rests on a smoothness assumption that formalizes the intuition behind SALT, namely that semantically nearby contexts induce similar response distributions; it is stated as anLL\-Lipschitz condition in the embedding distancedd\([1](https://arxiv.org/html/2608.19689#Thmassumption1)in Appendix[C\.4](https://arxiv.org/html/2608.19689#A3.SS4)\)\. The statistical part of the bound depends on how many actions carry substantial probability, captured by the effective action count\.

###### Definition 2\(Effective number of actions\)\.

For a contextxx, the*effective number of actions*is

Keff∗​\(x\)≔\(∑k=1Kϕ∗​\(k∣x\)\)2∈\[1,K\],K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)\\;\\coloneqq\\;\\Bigl\(\\sum\_\{k=1\}^\{K\}\\sqrt\{\\phi^\{\*\}\(k\\mid x\)\}\\Bigr\)^\{2\}\\;\\in\\;\[1,K\],\(15\)which attains its lower endpoint whenϕ∗\(⋅∣x\)\\phi^\{\*\}\(\\cdot\\mid x\)is a point mass and its upper endpoint whenϕ∗\(⋅∣x\)\\phi^\{\*\}\(\\cdot\\mid x\)is uniform\. It further satisfiesKeff∗​\(x\)≥Ks⁡\(x\)K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)\\geq K^\{s\(x\)\}, so it increases with the subjectivity of the context; this bound and the remaining properties used below are established in[Remark1](https://arxiv.org/html/2608.19689#Thmremark1)\.

Combining the decomposition with statistical and bias bounds yields the main tradeoff\.

###### Theorem 1\(Aggregation–Estimation Tradeoff\)\.

Supposeϕ∗\\phi^\{\*\}isLL\-Lipschitz in the embedding distancedd\(Appendix[C\.4](https://arxiv.org/html/2608.19689#A3.SS4)\)\. With\|𝒩⁡\(x\)\|≍ng⋅rd𝒳\|\\mathcal\{N\}\(x\)\|\\asymp n\_\{g\}\\cdot r^\{d\_\{\\mathcal\{X\}\}\}, where≍\\asymp\(≲\\lesssim\) denotes equality \(inequality\) up to constant factors, letεopt≔𝔼\[DTV\(ϕθ\(⋅∣x\),ϕ^\(⋅∣x\)\)\]\\varepsilon\_\{\\mathrm\{opt\}\}\\coloneqq\\mathbb\{E\}\[D\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\}\(\\cdot\\mid x\),\\hat\{\\phi\}\(\\cdot\\mid x\)\)\]\. The expected error satisfies

𝔼\[DTV\(ϕθ\(⋅∣x\),ϕ∗\(⋅∣x\)\)\]≲εopt⏟opt\. error\+L⋅r⏟bias\+Keff∗​\(x\)ng⋅rd𝒳⏟stat\. error\.\\mathbb\{E\}\\\!\\left\[D\_\{\\mathrm\{TV\}\}\\\!\\left\(\\phi\_\{\\theta\}\(\\cdot\\mid x\),\\;\\phi^\{\*\}\(\\cdot\\mid x\)\\right\)\\right\]\\;\\lesssim\\;\\underbrace\{\\varepsilon\_\{\\mathrm\{opt\}\}\}\_\{\\text\{opt\.\\ error\}\}\\;\+\\;\\underbrace\{L\\cdot r\}\_\{\\text\{bias\}\}\\;\+\\;\\underbrace\{\\sqrt\{\\frac\{K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)\}\{n\_\{g\}\\cdot r^\{d\_\{\\mathcal\{X\}\}\}\}\}\}\_\{\\text\{stat\.\\ error\}\}\.\(16\)The bias increases inrrwhile the statistical error decreases inrr\(since\|𝒩⁡\(x\)\|\|\\mathcal\{N\}\(x\)\|grows withrd𝒳r^\{d\_\{\\mathcal\{X\}\}\}\)\. When the aggregation loss is optimized so thatεopt\\varepsilon\_\{\\mathrm\{opt\}\}is negligible, balancing the remaining two terms gives the bias–variance optimized radius and the corresponding optimized upper bound:

r∗≍\(Keff∗​\(x\)ng​L2\)1/\(d𝒳\+2\),𝔼⁡\[DTV\]≲εopt\+\(Ld𝒳⋅Keff∗​\(x\)ng\)1/\(d𝒳\+2\)\.r^\{\*\}\\;\\asymp\\;\\left\(\\frac\{K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)\}\{n\_\{g\}L^\{2\}\}\\right\)^\{\\\!1/\(d\_\{\\mathcal\{X\}\}\+2\)\},\\qquad\\mathbb\{E\}\[D\_\{\\mathrm\{TV\}\}\]\\;\\lesssim\\;\\varepsilon\_\{\\mathrm\{opt\}\}\\;\+\\;\\left\(\\frac\{L^\{d\_\{\\mathcal\{X\}\}\}\\cdot K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)\}\{n\_\{g\}\}\\right\)^\{\\\!1/\(d\_\{\\mathcal\{X\}\}\+2\)\}\.\(17\)In the worst case \(Keff∗​\(x\)=KK\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)=K\), the bias–statistical term becomes\(Ld𝒳​K/ng\)1/\(d𝒳\+2\)\(L^\{d\_\{\\mathcal\{X\}\}\}K/n\_\{g\}\)^\{1/\(d\_\{\\mathcal\{X\}\}\+2\)\}\.

The proof is given in[SectionC\.6](https://arxiv.org/html/2608.19689#A3.SS6)\. The result has two implications\. First, with negligible optimization error, oracle aggregation improves asngn\_\{g\}grows, whereas a local point\-mass fit still incurs error at least1−pmax1\-p\_\{\\max\}on the context \([Equation10](https://arxiv.org/html/2608.19689#S3.E10)\)\. Second,r∗r^\{\*\}increases withKeff∗​\(x\)K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\): the more subjective a context, the larger its neighborhood should be\.

#### Practical implementation of the oracle radius\.

[Theorem1](https://arxiv.org/html/2608.19689#Thmtheorem1)gives an oracle radiusr∗≍\(Keff∗​\(x\)/\(ng​L2\)\)1/\(d𝒳\+2\)r^\{\*\}\\asymp\(K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)/\(n\_\{g\}L^\{2\}\)\)^\{1/\(d\_\{\\mathcal\{X\}\}\+2\)\}\. SinceKeff∗​\(x\)K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)is defined throughϕ∗\(⋅∣x\)\\phi^\{\*\}\(\\cdot\\mid x\), it is unobserved in training\. In practice we evaluate the*same*expression at the model’s current distribution, which gives the model\-based estimateK^eff​\(x\)≔\(∑kϕθ​\(k∣x\)\)2\\hat\{K\}\_\{\\mathrm\{eff\}\}\(x\)\\coloneqq\\bigl\(\\sum\_\{k\}\\sqrt\{\\phi\_\{\\theta\}\(k\\mid x\)\}\\bigr\)^\{2\}\. This is reasonable for two reasons\. The aggregation loss \([Equation14](https://arxiv.org/html/2608.19689#S4.E14)\) is itself a distribution\-matching objective, so it drivesϕθ\(⋅∣x\)\\phi\_\{\\theta\}\(\\cdot\\mid x\)towardϕ∗\(⋅∣x\)\\phi^\{\*\}\(\\cdot\\mid x\), and the expression is continuous in the distribution, so the estimate returns to the true count as the optimization error vanishes\. Exact recovery is moreover not required, becauser⁡\(x\)r\(x\)depends on the count only through the power1/\(d𝒳\+2\)1/\(d\_\{\\mathcal\{X\}\}\+2\), so a multiplicative error in the count is damped into a much smaller relative change in the radius\.[Remark2](https://arxiv.org/html/2608.19689#Thmremark2)makes both precise\. The Lipschitz constantLLand other constants are absorbed into a tunable hyperparameterCC, giving

r⁡\(x\)=C⋅\(K^eff​\(x\)ng\)1/\(d𝒳\+2\)\.r\(x\)\\;=\\;C\\cdot\\left\(\\frac\{\\hat\{K\}\_\{\\text\{eff\}\}\(x\)\}\{n\_\{g\}\}\\right\)^\{1/\(d\_\{\\mathcal\{X\}\}\+2\)\}\.\(18\)The estimate is refreshed periodically during training\. This substitution preserves the oracle bound’s monotone dependence on action dispersion, and we evaluate it through fixed\-KeffK\_\{\\mathrm\{eff\}\}, fixed\-neighborhood, and radius\-scale ablations\. The theory thus does not merely justify aggregation; it specifies when to aggregate more: contexts with diffuse predicted behavior require larger neighborhoods to reduce statistical error, while concentrated contexts should remain close to their observed hard label to avoid unnecessary smoothing\. In the zero\-subjectivity limit,K^eff​\(x\)→1\\hat\{K\}\_\{\\mathrm\{eff\}\}\(x\)\\to 1and the radius contracts toward its minimum, so the soft label concentrates on the observed action and SALT approaches standard hard\-label training\. Hard\-label supervision is therefore recovered as the low\-subjectivity special case of SALT\. More details and the complete algorithm can be found in Appendix[D](https://arxiv.org/html/2608.19689#A4)\.

## 5Experiments

### 5\.1SubjSim Benchmark

Evaluating distributional fidelity requires the true response distributions as ground truth, which existing single\-response datasets cannot provide\. We therefore constructSubjSim, where 193 annotators first answer 30 demographic questions that define their persona vectors and then annotate 100 subjective survey questions, yielding 19,300 persona\-question contexts\. For each context, the annotator allocates probability balls across the candidate options\([28](https://arxiv.org/html/2608.19689#bib.bib12);[5](https://arxiv.org/html/2608.19689#bib.bib13)\), producing a response\-propensity distribution that is used only for evaluation\. All training methods, including SALT’s aggregation, see just one answer per context: the option receiving the largest share of that annotator’s balls, with ties resolved in favor of the lowest\-indexed option\. The questions are organized into eight topic domains: economy, politics, technology, social issues, culture, health, environment, and education\. Their subjectivity coefficients cover the full range from near\-deterministic to near\-uniform, enabling evaluation across low, mid, and high subjectivity regimes \(see[Figure12](https://arxiv.org/html/2608.19689#A7.F12); construction details in Appendix[G](https://arxiv.org/html/2608.19689#A7)\)\.

### 5\.2Experimental Setup

We train on the samples of about 85% of the respondents and test on those of the remaining ones, which gives 16,500 training pairs and 2,800 test pairs\. All experiments use Qwen3\-8B\([32](https://arxiv.org/html/2608.19689#bib.bib29)\)as the backbone\. We compare SALT with the untrained zero\-shot backbone, SFT, DPO, PPO, and DSA\([12](https://arxiv.org/html/2608.19689#bib.bib16)\)\. DPO constructs preference pairs and PPO derives reward signals from the hard labels described in[Section5\.1](https://arxiv.org/html/2608.19689#S5.SS1), so that all methods observe identical training information\. DSA is the most recent distribution\-level baseline, which fine\-tunes the LLM to match the response distributions of demographic groups and to align distribution shifts across groups\. At test time, models output distributions over candidate options via generation\-probability normalization\. We report KL, JSD, TVD, and linear\-kernel MMD computed on the full test set; see Appendix[D](https://arxiv.org/html/2608.19689#A4)for further experimental settings\.

### 5\.3Main Results

The main results are shown in[Table1](https://arxiv.org/html/2608.19689#S5.T1)\. The pretrained backbone performs poorly, and neither DPO nor PPO improves over it; DPO in fact degrades KL substantially\. We attribute this to the construction of the preference signal from a single hard action: the observed action serves as the positive and the remaining options as negatives, yet under subjective behavior these options may themselves carry substantial probability, so responses the respondent might well choose are suppressed as negatives, making the resulting signal even noisier than fitting the label directly\. SFT is the strongest baseline, suggesting that directly fitting the observed labels already captures a substantial part of the underlying behavioral structure\. DSA ranks between the preference\-based methods and SFT, clearly outperforming the former, which confirms the benefit of distribution\-alignment training\. However, DSA groups respondents by only a few discrete background attributes, and all individuals within a group share a single target distribution; the supervision is therefore noisy at the individual level, leaving DSA behind SFT\. SALT controls this noise with its adaptive radius, pooling only sufficiently similar contexts while still collecting enough answers for a reliable distribution estimate\. It performs best on all four metrics in every domain, and over the full test set it reduces KL over SFT by 77\.6%, JSD by 45\.9%, TVD by 31\.2%, and MMD by 52\.7%\. A paired bootstrap over test contexts confirms that all of these improvements are statistically significant \(Appendix[F](https://arxiv.org/html/2608.19689#A6)\)\.

Table 1:Main results on SubjSim\. All metrics are divergences \(lower is better\)\. Best values are inboldon abluebackground; second\-best values are underlined on asandbackground\.Figure 1:Subjectivity\-stratified comparison between SFT and SALT onSubjSim, over low\-, mid\-, and high\-subjectivity questions\. All metrics are divergences \(lower is better\), and percentages give SALT’s relative change over SFT\.
### 5\.4Performance Comparison across Subjectivity Levels

A core design of SALT is that the aggregation is calibrated by the estimated subjectivity of each context, and we now examine experimentally whether this design indeed leads to better performance\. We compute the normalized subjectivity coefficients⁡\(x\)s\(x\)in \([9](https://arxiv.org/html/2608.19689#S3.E9)\) directly from the annotated response\-propensity distributions\. We then partition the questions into low, mid, and high strata at the1/31/3and2/32/3quantiles ofs⁡\(x\)s\(x\)and compare SFT and SALT within each stratum \([Figure1](https://arxiv.org/html/2608.19689#S5.F1)\)\. The results match the analysis in[Section3](https://arxiv.org/html/2608.19689#S3)\. SFT is competitive in near\-deterministic contexts but degrades steadily as targets become diffuse, whereas SALT’s advantage grows with subjectivity; in the mid and high strata SALT outperforms SFT on all metrics, and in the high stratum it reduces KL by 96\.9%\. In the lowest\-subjectivity regime the comparison is mixed, with SALT improving KL and MMD but falling behind SFT on JSD and TVD\. This is what we would expect, because such questions have a clear majority answer, so there is little for aggregation to add and pooling neighbors can only blur a target that is already sharp\.

### 5\.5Ablation Studies

The core design of SALT is the context\-adaptive merging radius, so we ask whether such adaptivity is really necessary\. We build three baselines that differ from SALT only in how neighbors are selected: Top\-NNgives every context the same number of nearest neighbors; fixedKeffK\_\{\\text\{eff\}\}replacesK^eff​\(x\)\\hat\{K\}\_\{\\text\{eff\}\}\(x\)in the radius formula by one shared constant, so every context gets the same radius; Global\-Freq drops similarity altogether and treats everyone who answered the same question as a neighbor\.

From the results shown in[Figure2](https://arxiv.org/html/2608.19689#S5.F2), we draw three conclusions\. First, every variant improves substantially over SFT, which shows that aggregation is already useful on its own because it replaces a single observed answer with a distributional target\. Second, how neighbors are selected matters: Top\-NNdegrades asNNgrows and dissimilar contexts enter the neighborhood, fixedKeffK\_\{\\text\{eff\}\}stops improving and stays worse than SALT on every metric, and the adaptiveK^eff​\(x\)\\hat\{K\}\_\{\\text\{eff\}\}\(x\)is best on all four\. Third, Global\-Freq is clearly worse than SALT, so similarity is necessary: pooling everyone who answered the same question mixes dissimilar respondents and biases the soft label\.

Figure 2:Ablation over neighbor selection\. Top\-NNvariesNN\(teal, bottom axis\); fixedKeffK\_\{\\text\{eff\}\}substitutes a constant forK^eff​\(x\)\\hat\{K\}\_\{\\text\{eff\}\}\(x\)\(navy, top axis\)\. Both share theyyaxis and reference lines; SFT lies far above the plotted range and is reported as text\.Figure 3:Sensitivity of SALT to the radius scaleCC\(divergences, lower is better\)\.
### 5\.6Parameter Analysis

The main hyperparameter of SALT is the radius scaleCCin \([18](https://arxiv.org/html/2608.19689#S4.E18)\)\. The theory in[Section4](https://arxiv.org/html/2608.19689#S4)fixes only how the radius should scale withK^eff​\(x\)\\hat\{K\}\_\{\\mathrm\{eff\}\}\(x\)andngn\_\{g\}; the remaining constants, including the Lipschitz constantLL, are absorbed intoCC, which is therefore chosen empirically\.[Figure3](https://arxiv.org/html/2608.19689#S5.F3)reports performance forC∈\{0\.10,0\.15,0\.20,0\.25,0\.30\}C\\in\\\{0\.10,0\.15,0\.20,0\.25,0\.30\\\}\. All metrics improve monotonically asCCincreases from 0\.10 to 0\.20: a larger radius admits more genuinely similar neighbors, so the aggregated soft labels become more reliable\. Beyond 0\.20 the improvement stops and fluctuates slightly, as a wider radius includes less similar contexts and mild oversmoothing offsets the gains; the trade\-off is best aroundC=0\.20C=0\.20, which we use as the default\. Importantly, the variation across the whole range is small—about 0\.003 in KL and at most 0\.002 in the other metrics—so SALT does not rely on a finely tuned radius constant\.

## 6Conclusion

In this paper, we studied how to evaluate and train LLMs for social simulation when human behavior is subjective\. We introduced the subjectivity coefficient, which places objective and subjective tasks on a common axis and explains why accuracy\-based evaluation and hard\-label training become unreliable as subjectivity grows\. We then proposed SALT, which turns each observed response into a soft distributional label aggregated over a subjectivity\-adaptive neighborhood, and builtSubjSim, whose annotators provide full response distributions for 19,300 persona\-question contexts, making distributional evaluation possible\. SALT improves distributional alignment over all baselines, with the largest gains in high\-subjectivity regimes\. Future work includes extending this framework to sequential, multi\-agent, and open\-ended settings\.

## Ethics Statement

SubjSiminvolves human annotation\. All 193 annotators participated voluntarily; before starting the annotation task, they were informed of the purpose of the study, the type of data collected, and their right to withdraw at any time\. Annotation compensation was calculated so that the resulting hourly rate is guaranteed to exceed the highest applicable local hourly wage standard\. To protect privacy, no directly identifying information \(such as names or contact details\) was collected, all responses were recorded under anonymized identifiers, and the released data contain only demographic attribute values and annotated response distributions with no link back to individuals\. The survey questions themselves were screened for cultural suitability for the annotator population during dataset construction, and questions flagged as unsuitable were removed \(Appendix[G](https://arxiv.org/html/2608.19689#A7)\)\. The representativeness limits of our annotator pool are discussed in Appendix[G\.3](https://arxiv.org/html/2608.19689#A7.SS3)\.

## AI Use Statement

We used large language models only as writing and figure assistants in the preparation of this paper: ChatGPT was used to check grammar and correct typos in the manuscript, and Codex was used to assist in producing the figures\. No LLM was used to generate research ideas, analyses, results, or claims\. Separately, and as part of the research methodology itself rather than paper preparation, DeepSeek\-chat was used to translate and pre\-screen survey questions during the construction ofSubjSim, as documented in Appendix[G](https://arxiv.org/html/2608.19689#A7)\.

## Reproducibility Statement

We have taken several measures to make our results reproducible\. All theoretical results are stated with their assumptions in[Sections3](https://arxiv.org/html/2608.19689#S3)and[4](https://arxiv.org/html/2608.19689#S4), and complete proofs are given in[SectionsC\.1](https://arxiv.org/html/2608.19689#A3.SS1),[C\.2](https://arxiv.org/html/2608.19689#A3.SS2),[C\.3](https://arxiv.org/html/2608.19689#A3.SS3),[C\.4](https://arxiv.org/html/2608.19689#A3.SS4),[C\.5](https://arxiv.org/html/2608.19689#A3.SS5)and[C\.6](https://arxiv.org/html/2608.19689#A3.SS6)\. The full SALT procedure is specified in[Algorithm1](https://arxiv.org/html/2608.19689#alg1), with implementation details, the theory\-to\-implementation mapping \([Table2](https://arxiv.org/html/2608.19689#A4.T2)\), and all training hyperparameters \([Table3](https://arxiv.org/html/2608.19689#A4.T3)\) in[AppendixD](https://arxiv.org/html/2608.19689#A4)\. The construction ofSubjSim, including source surveys, screening and translation pipeline, the exact screening prompt, annotation protocol, and annotator demographics, is documented in[AppendixG](https://arxiv.org/html/2608.19689#A7)\. Evaluation is deterministic, and the backbone model \(Qwen3\-8B\), context encoder \(Qwen3\-embedding\-8b\), and training stack \(LLaMA\-Factory, DeepSpeed ZeRO\-2 on 8×\\timesH20\) are publicly available\. We will release theSubjSimdataset, the annotation platform specification, and the training and evaluation code upon publication\.

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Appendix

## Appendix ALimitations

SubjSimuses elicited response propensities rather than direct repeated\-choice frequencies\. The annotator\-level split tests unseen personas within known questions rather than transfer to entirely new decision contexts\. SALT also depends on behavioral smoothness in the representation space, so poor embeddings or discontinuous response patterns can make aggregation harmful\.

## Appendix BRelated Work and Positioning

#### LLM agents for social simulation\.

Recent work uses LLM agents to simulate individuals, groups, and social systems, ranging from interactive generative agents to opinion dynamics, polarization, market behavior, and historical or policy simulations\([18](https://arxiv.org/html/2608.19689#bib.bib1);[9](https://arxiv.org/html/2608.19689#bib.bib2);[11](https://arxiv.org/html/2608.19689#bib.bib3);[23](https://arxiv.org/html/2608.19689#bib.bib14);[2](https://arxiv.org/html/2608.19689#bib.bib15)\)\. This line of work establishes that LLMs can produce plausible social behavior and can be embedded in multi\-agent environments\. Our focus is different: we ask what behavioral target such agents should be trained and evaluated against\. Rather than treating simulation quality as matching a single observed response, we formulate the target as recovering the full distribution over plausible actions for a persona\-context pair\.

#### Single\-action imitation and preference optimization\.

The dominant way to adapt LLM agents is supervised imitation of observed actions, often evaluated by point\-prediction accuracy\. Preference optimization methods such as DPO and PPO provide a stronger alignment toolkit by learning from chosen/rejected comparisons\([20](https://arxiv.org/html/2608.19689#bib.bib17);[24](https://arxiv.org/html/2608.19689#bib.bib18)\)\. However, when the data provide only one realized action per context, both supervised and preference objectives still receive hard, mode\-like supervision\. They therefore learn which action was observed, or which action should be preferred to alternatives, rather than the entire behavioral frequency structure\. This is the same hard\-supervision gap that motivates our misalignment analysis, though our formal result focuses on the single\-observation SFT objective\.

#### Distributional evaluation and distribution alignment\.

A separate line of work changes the evaluation target: instead of reporting only accuracy, it compares model outputs to response distributions using KL, JSD, TVD, Wasserstein distance, or related survey\-response divergences\. DSA is closest to our work in target space: it also aims to match survey response distributions, but it does so through distribution\-shift alignment rather than context\-level hard\-label aggregation\([12](https://arxiv.org/html/2608.19689#bib.bib16)\)\. Our setting differs in the supervision assumption\. DSA and related distributional alignment methods rely on distributional supervision that is available or can be constructed at the population/question level\. SALT targets the stricter single\-observation regime: it does not train on the persona\-specific probability\-ball distribution, but constructs distributional supervision by pooling hard observations from semantically similar contexts, following the repeated\-measurement intuition in the main analysis\.

#### Human label variation\.

NLP research on annotation has similarly argued that disagreement among annotators carries signal rather than noise, advocating soft labels, annotator\-aware models, and evaluation against label distributions instead of a single gold label\([1](https://arxiv.org/html/2608.19689#bib.bib32);[19](https://arxiv.org/html/2608.19689#bib.bib33);[27](https://arxiv.org/html/2608.19689#bib.bib34)\)\. These works concern classification\-style NLP annotations and model variation at the label or annotator level\. Our setting differs in both object and ground truth: the estimation target is a persona\-conditional behavioral distribution for social simulation, andSubjSimmeasures it directly through probability\-ball allocation rather than approximating it from a handful of discrete annotations\.

#### Context aggregation and nonparametric estimation\.

SALT is related to nonparametric smoothing and kernel regression, where local neighborhoods are used to estimate a conditional response function\([17](https://arxiv.org/html/2608.19689#bib.bib19);[31](https://arxiv.org/html/2608.19689#bib.bib20);[25](https://arxiv.org/html/2608.19689#bib.bib4)\)\. The key difference is the object being smoothed and the constraint imposed by social\-simulation data\. We aggregate over contexts that share an action space and are close in persona\-context embedding space, producing a soft label over discrete actions from otherwise hard observations\. The adaptive radius is not a generic hyperparameter search: it is derived from the subjectivity\-controlled bias–variance tradeoff, so more diffuse behavioral distributions receive broader aggregation while nearly deterministic contexts remain local\.

## Appendix CTheoretical Proofs

### C\.1Proof of[Proposition1](https://arxiv.org/html/2608.19689#Thmproposition1)\(Single\-Observation SFT Concentrates on Empirical One\-Hot Labels\)

###### Proof\.

For each training contextxix^\{i\}, the empirical SFT contribution is−log⁡ϕθ​\(ji∣xi\)\-\\log\\phi\_\{\\theta\}\(j^\{i\}\\mid x^\{i\}\)at the action level\. Over the action\-distribution simplex, this term is minimized by maximizingϕθ​\(ji∣xi\)\\phi\_\{\\theta\}\(j^\{i\}\\mid x^\{i\}\), whose largest possible value is11\. Because the statement is made in the closure of the model\-induced action distributions, this boundary point is included and can be selected independently for every training context, yieldingϕθ\(k∣xi\)=𝟏\[k=ji\]\\phi\_\{\\theta\}\(k\\mid x^\{i\}\)=\\mathbf\{1\}\[k=j^\{i\}\]for allii\. Since−log⁡p\-\\log pis uniquely minimized atp=1p=1, every empirical\-risk minimizer must saturate the per\-context term at every training context, so any minimizerθ~\\tilde\{\\theta\}in this closure satisfiesϕθ~\(k∣xi\)=𝟏\[k=ji\]\\phi\_\{\\tilde\{\\theta\}\}\(k\\mid x^\{i\}\)=\\mathbf\{1\}\[k=j^\{i\}\]for allii\. For finite softmax parameterizations, the same boundary point may be approached only in the limit: any sequence with empirical risk approaching the infimum must haveϕθ​\(ji∣xi\)→1\\phi\_\{\\theta\}\(j^\{i\}\\mid x^\{i\}\)\\to 1and henceϕθ​\(k∣xi\)→0\\phi\_\{\\theta\}\(k\\mid x^\{i\}\)\\to 0for everyk≠jik\\neq j^\{i\}\. This is an empirical\-risk statement for finite single\-observation data; with repeated observations from the same context, the population cross\-entropy optimum would instead match the conditional label distribution\. ∎

### C\.2Proof of the Fano Step in[Proposition2](https://arxiv.org/html/2608.19689#Thmproposition2)

###### Proof\.

Writee:=1−pmaxe:=1\-p\_\{\\max\}andH:=H\(ϕ∗\(⋅∣x\)\)=−κH:=H\(\\phi^\{\*\}\(\\cdot\\mid x\)\)=\-\\kappa\. Fano’s inequality for the error of the optimal single guess \(the mode\) statesH≤h⁡\(e\)\+e​log⁡\(K−1\)H\\leq h\(e\)\+e\\log\(K\-1\), whereh⁡\(⋅\)h\(\\cdot\)is the binary entropy function\. Sinceh⁡\(e\)≤log⁡2h\(e\)\\leq\\log 2, we obtainH≤log⁡2\+e​log⁡\(K−1\)H\\leq\\log 2\+e\\log\(K\-1\), and rearranging gives

1−pmax=e≥H−log⁡2log⁡\(K−1\)=−κ−log⁡2log⁡\(K−1\)=s​log⁡K−log⁡2log⁡\(K−1\)1\-p\_\{\\max\}=e\\;\\geq\\;\\frac\{H\-\\log 2\}\{\\log\(K\-1\)\}=\\frac\{\-\\kappa\-\\log 2\}\{\\log\(K\-1\)\}=\\frac\{s\\log K\-\\log 2\}\{\\log\(K\-1\)\}\(19\)forK≥3K\\geq 3, where the last step usess=−κ/logKs=\-\\kappa/\\log Kfrom[Equation9](https://arxiv.org/html/2608.19689#S3.E9)\. Combining withDTV​\(ϕθ,ϕ∗\)≥1−pmaxD\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\},\\phi^\{\*\}\)\\geq 1\-p\_\{\\max\}from[Equation7](https://arxiv.org/html/2608.19689#S3.E7)yields[Equation10](https://arxiv.org/html/2608.19689#S3.E10)\. The bound is vacuous whens​log⁡K<log⁡2s\\log K<\\log 2, i\.e\., for weakly subjective contexts, which is consistent with the analysis: the failures of the accuracy\-based pipeline concentrate in the high\-subjectivity regime\. ForK=2K=2, Fano’s inequality reduces toH≤h⁡\(e\)H\\leq h\(e\), givinge≥h−1​\(H\)e\\geq h^\{\-1\}\(H\)instead; the conclusion that the error grows with subjectivity is unchanged\. ∎

### C\.3Proof of[Proposition3](https://arxiv.org/html/2608.19689#Thmproposition3)\(Local Lower Bound under Near\-Uniform Behavior\)

###### Proposition 3\(Local lower bound under near\-uniform behavior\)\.

Fix aKK\-action context and consider true distributions in a near\-uniform neighborhood of radiusO⁡\(1/K\)O\(1/K\), so thate−2​κ=Θ⁡\(K2\)e^\{\-2\\kappa\}=\\Theta\(K^\{2\}\)\. There exist two alternativesP,QP,Qin this neighborhood and two fixed point\-mass models whoseDTVD\_\{\\mathrm\{TV\}\}\-error ranking is reversed underPPandQQ, whileDKL\(P∥Q\)=O\(1/K2\)D\_\{\\mathrm\{KL\}\}\(P\\\|Q\)=O\(1/K^\{2\}\)\. Consequently, for any fixedη\>0\\eta\>0, any procedure that identifies the correct ranking fromnni\.i\.d\. observations with error probability at most12−η\\frac\{1\}\{2\}\-\\etain this local problem requiresn=Ω⁡\(K2\)=Ω⁡\(e−2​κ\)n=\\Omega\(K^\{2\}\)=\\Omega\(e^\{\-2\\kappa\}\)samples\.

###### Proof\.

We give an explicit two\-point construction\. Letuube the uniform distribution onKKactions and chooseε=cK−3/2\\varepsilon=cK^\{\-3/2\}for a sufficiently small constantc\>0c\>0\. Define

P1\\displaystyle P\_\{1\}=1K\+ε,\\displaystyle=\\frac\{1\}\{K\}\+\\varepsilon,P2\\displaystyle P\_\{2\}=1K−ε,\\displaystyle=\\frac\{1\}\{K\}\-\\varepsilon,Pk\\displaystyle P\_\{k\}=1K\(k≥3\),\\displaystyle=\\frac\{1\}\{K\}\\quad\(k\\geq 3\),\(20\)Q1\\displaystyle Q\_\{1\}=1K−ε,\\displaystyle=\\frac\{1\}\{K\}\-\\varepsilon,Q2\\displaystyle Q\_\{2\}=1K\+ε,\\displaystyle=\\frac\{1\}\{K\}\+\\varepsilon,Qk\\displaystyle Q\_\{k\}=1K\(k≥3\)\.\\displaystyle=\\frac\{1\}\{K\}\\quad\(k\\geq 3\)\.\(21\)For large enoughKK, both distributions are valid and lie in anO⁡\(1/K\)O\(1/K\)neighborhood ofuu\. Their entropy satisfiesH⁡\(P\)=H⁡\(Q\)=log⁡K−O⁡\(K​ε2\)=log⁡K−O⁡\(1/K2\)H\(P\)=H\(Q\)=\\log K\-O\(K\\varepsilon^\{2\}\)=\\log K\-O\(1/K^\{2\}\), soκ=−log⁡K\+O⁡\(1/K2\)\\kappa=\-\\log K\+O\(1/K^\{2\}\)and thereforee−2​κ=Θ⁡\(K2\)e^\{\-2\\kappa\}=\\Theta\(K^\{2\}\)\.

Consider two point\-mass models,θ1\\theta\_\{1\}assigning all mass to action11andθ2\\theta\_\{2\}assigning all mass to action22\. For any distributionRR, the total variation error of the point mass at actionjjis

DTV​\(δj,R\)=1−Rj\.D\_\{\\mathrm\{TV\}\}\(\\delta\_\{j\},R\)=1\-R\_\{j\}\.\(22\)Hence, underPP,DTV​\(δ1,P\)=1−P1<1−P2=DTV​\(δ2,P\)D\_\{\\mathrm\{TV\}\}\(\\delta\_\{1\},P\)=1\-P\_\{1\}<1\-P\_\{2\}=D\_\{\\mathrm\{TV\}\}\(\\delta\_\{2\},P\), soθ1\\theta\_\{1\}is better thanθ2\\theta\_\{2\}\. UnderQQthe inequality is reversed\. Any procedure that identifies the correct ranking in this local problem therefore distinguishes whether the samples came fromPPorQQ\.

It remains to bound the statistical distance between the two hypotheses\. Leta=1/Ka=1/Kandt=ε/a=c/Kt=\\varepsilon/a=c/\\sqrt\{K\}\. The per\-sample KL divergence is

DKL\(P∥Q\)\\displaystyle D\_\{\\mathrm\{KL\}\}\(P\\\|Q\)=\(a\+ε\)​log⁡a\+εa−ε\+\(a−ε\)​log⁡a−εa\+ε\\displaystyle=\(a\+\\varepsilon\)\\log\\frac\{a\+\\varepsilon\}\{a\-\\varepsilon\}\+\(a\-\\varepsilon\)\\log\\frac\{a\-\\varepsilon\}\{a\+\\varepsilon\}\(23\)=2​ε​log⁡1\+t1−t\.\\displaystyle=2\\varepsilon\\log\\frac\{1\+t\}\{1\-t\}\.\(24\)Sincet<1/2t<1/2for large enoughKK,log⁡\(\(1\+t\)/\(1−t\)\)≤C​t\\log\(\(1\+t\)/\(1\-t\)\)\\leq Ctfor a universal constantCC, and thus

DKL\(P∥Q\)≤2Cε2a=O\(Kε2\)=O\(1/K2\)\.D\_\{\\mathrm\{KL\}\}\(P\\\|Q\)\\leq 2C\\frac\{\\varepsilon^\{2\}\}\{a\}=O\(K\\varepsilon^\{2\}\)=O\(1/K^\{2\}\)\.\(25\)
LetPnP^\{n\}andQnQ^\{n\}denote thenn\-sample product distributions\. By tensorization and Pinsker’s inequality,

DTV​\(Pn,Qn\)≤DKL\(Pn∥Qn\)2=nDKL\(P∥Q\)2≤C′​nK2\.D\_\{\\mathrm\{TV\}\}\(P^\{n\},Q^\{n\}\)\\leq\\sqrt\{\\frac\{D\_\{\\mathrm\{KL\}\}\(P^\{n\}\\\|Q^\{n\}\)\}\{2\}\}=\\sqrt\{\\frac\{n\\,D\_\{\\mathrm\{KL\}\}\(P\\\|Q\)\}\{2\}\}\\leq C^\{\\prime\}\\sqrt\{\\frac\{n\}\{K^\{2\}\}\}\.\(26\)Le Cam’s lemma gives minimax error at least12​\(1−DTV​\(Pn,Qn\)\)\\frac\{1\}\{2\}\(1\-D\_\{\\mathrm\{TV\}\}\(P^\{n\},Q^\{n\}\)\)\. Therefore, if a ranking procedure achieves error probability at most12−η\\frac\{1\}\{2\}\-\\etafor fixedη\>0\\eta\>0, thenDTV​\(Pn,Qn\)≥2​ηD\_\{\\mathrm\{TV\}\}\(P^\{n\},Q^\{n\}\)\\geq 2\\eta, which requiresn≥cη​K2n\\geq c\_\{\\eta\}K^\{2\}\. Sincee−2​κ=Θ⁡\(K2\)e^\{\-2\\kappa\}=\\Theta\(K^\{2\}\)in this construction, the required sample size isΩ⁡\(e−2​κ\)\\Omega\(e^\{\-2\\kappa\}\)\. ∎

### C\.4Supporting Lemmas for Context Aggregation

We first record the properties of the effective action count that motivate[Definition2](https://arxiv.org/html/2608.19689#Thmdefinition2)\.

The analysis rests on the following smoothness assumption\.

###### Assumption 1\(LL\-Lipschitz behavioral distribution\)\.

There existsL\>0L\>0such that for allx,x′∈𝒳x,x^\{\\prime\}\\in\\mathcal\{X\}:DTV\(ϕ∗\(⋅∣x\),ϕ∗\(⋅∣x′\)\)≤L⋅d\(x,x′\)D\_\{\\mathrm\{TV\}\}\(\\phi^\{\*\}\(\\cdot\\mid x\),\\,\\phi^\{\*\}\(\\cdot\\mid x^\{\\prime\}\)\)\\leq L\\cdot d\(x,x^\{\\prime\}\)\.

This assumption should be read as behavioral smoothness in the chosen representation space\. It requires that, within a fixed action\-space group, nearby persona\-context embeddings induce similar response distributions; the ablations over fixed neighborhoods, global frequency labels, and radius scale test whether this approximation is useful inSubjSim\.

###### Lemma 1\(Three\-term decomposition\)\.

For any contextxx, letϕ¯\(⋅∣x\)≔1\|𝒩⁡\(x\)\|∑x′∈𝒩⁡\(x\)ϕ∗\(⋅∣x′\)\\bar\{\\phi\}\(\\cdot\\mid x\)\\coloneqq\\frac\{1\}\{\|\\mathcal\{N\}\(x\)\|\}\\sum\_\{x^\{\\prime\}\\in\\mathcal\{N\}\(x\)\}\\phi^\{\*\}\(\\cdot\\mid x^\{\\prime\}\)denote the population mean ofϕ∗\\phi^\{\*\}within the neighborhood ofxx\. Then:

DTV\(ϕθ\(⋅∣x\),ϕ∗\(⋅∣x\)\)\\displaystyle D\_\{\\mathrm\{TV\}\}\\\!\\left\(\\phi\_\{\\theta\}\(\\cdot\\mid x\),\\;\\phi^\{\*\}\(\\cdot\\mid x\)\\right\)≤DTV\(ϕθ\(⋅∣x\),ϕ^\(⋅∣x\)\)⏟εopt\+DTV\(ϕ^\(⋅∣x\),ϕ¯\(⋅∣x\)\)⏟εstat\+DTV\(ϕ¯\(⋅∣x\),ϕ∗\(⋅∣x\)\)⏟εbias\.\\displaystyle\\quad\\leq\\;\\underbrace\{D\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\}\(\\cdot\\mid x\),\\;\\hat\{\\phi\}\(\\cdot\\mid x\)\)\}\_\{\\varepsilon\_\{\\mathrm\{opt\}\}\}\+\\;\\underbrace\{D\_\{\\mathrm\{TV\}\}\(\\hat\{\\phi\}\(\\cdot\\mid x\),\\;\\bar\{\\phi\}\(\\cdot\\mid x\)\)\}\_\{\\varepsilon\_\{\\mathrm\{stat\}\}\}\+\\;\\underbrace\{D\_\{\\mathrm\{TV\}\}\(\\bar\{\\phi\}\(\\cdot\\mid x\),\\;\\phi^\{\*\}\(\\cdot\\mid x\)\)\}\_\{\\varepsilon\_\{\\mathrm\{bias\}\}\}\.\(30\)

###### Lemma 2\(Statistical and bias bounds\)\.

Under[1](https://arxiv.org/html/2608.19689#Thmassumption1), with neighborhood radiusrr:

𝔼⁡\[εstat\]≤12​Keff\|𝒩⁡\(x\)\|,εbias≤L⋅r,\\mathbb\{E\}\\\!\\left\[\\varepsilon\_\{\\mathrm\{stat\}\}\\right\]\\;\\leq\\;\\frac\{1\}\{2\}\\sqrt\{\\frac\{K\_\{\\mathrm\{eff\}\}\}\{\|\\mathcal\{N\}\(x\)\|\}\},\\qquad\\varepsilon\_\{\\mathrm\{bias\}\}\\;\\leq\\;L\\cdot r,\(31\)whereKeff=Keff\(ϕ¯\(⋅∣x\)\)K\_\{\\mathrm\{eff\}\}=K\_\{\\mathrm\{eff\}\}\(\\bar\{\\phi\}\(\\cdot\\mid x\)\)is the effective action count of the neighborhood\-averaged population distributionϕ¯\\bar\{\\phi\}of[Lemma1](https://arxiv.org/html/2608.19689#Thmlemma1)\. Under[1](https://arxiv.org/html/2608.19689#Thmassumption1),ϕ¯\(⋅∣x\)\\bar\{\\phi\}\(\\cdot\\mid x\)lies withinL​rLrin total variation ofϕ∗\(⋅∣x\)\\phi^\{\*\}\(\\cdot\\mid x\), so by \([29](https://arxiv.org/html/2608.19689#A3.E29)\) it differs fromKeff∗​\(x\)K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)of[Definition2](https://arxiv.org/html/2608.19689#Thmdefinition2)only through the same smoothness that already controls the bias term; the main statements are written withKeff∗​\(x\)K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)for readability\. As a worst case one may substituteKeff=KK\_\{\\mathrm\{eff\}\}=K\.

###### Corollary 1\(Aggregation eliminates the structural error of standard training\)\.

In the single\-observation regime, standard SFT fits one hard label per context and therefore incurs the point\-mass errorDTV​\(ϕθSFT,ϕ∗\)≥1−pmaxD\_\{\\mathrm\{TV\}\}\(\\phi\_\{\\theta\}^\{\\mathrm\{SFT\}\},\\phi^\{\*\}\)\\geq 1\-p\_\{\\max\}at that context\. Context\-aggregation achieves errorεopt\+\(Ld𝒳​Keff∗​\(x\)/ng\)1/\(d𝒳\+2\)\\varepsilon\_\{\\mathrm\{opt\}\}\+\(L^\{d\_\{\\mathcal\{X\}\}\}K\_\{\\mathrm\{eff\}\}^\{\*\}\(x\)/n\_\{g\}\)^\{1/\(d\_\{\\mathcal\{X\}\}\+2\)\}up to constants; when optimization error is negligible, this upper bound tends to zero asng→∞n\_\{g\}\\to\\infty\.

### C\.5Proof of[Lemma2](https://arxiv.org/html/2608.19689#Thmlemma2)\(Statistical and Bias Bounds\)

###### Proof\.

Statistical bound\.Writenx=\|𝒩⁡\(x\)\|n\_\{x\}=\|\\mathcal\{N\}\(x\)\|and, forx′∈𝒩⁡\(x\)x^\{\\prime\}\\in\\mathcal\{N\}\(x\),px′,k=ϕ∗​\(k∣x′\)p\_\{x^\{\\prime\},k\}=\\phi^\{\*\}\(k\\mid x^\{\\prime\}\)\. Each coordinateϕ^k\\hat\{\\phi\}\_\{k\}is an average of independent Bernoulli variables with possibly different meanspx′,kp\_\{x^\{\\prime\},k\}\. Letϕ¯k=nx−1​∑x′∈𝒩⁡\(x\)px′,k\\bar\{\\phi\}\_\{k\}=n\_\{x\}^\{\-1\}\\sum\_\{x^\{\\prime\}\\in\\mathcal\{N\}\(x\)\}p\_\{x^\{\\prime\},k\}\. Then

Var​\(ϕ^k\)=1nx2​∑x′∈𝒩⁡\(x\)px′,k​\(1−px′,k\)≤ϕ¯knx\.\\text\{Var\}\(\\hat\{\\phi\}\_\{k\}\)=\\frac\{1\}\{n\_\{x\}^\{2\}\}\\sum\_\{x^\{\\prime\}\\in\\mathcal\{N\}\(x\)\}p\_\{x^\{\\prime\},k\}\(1\-p\_\{x^\{\\prime\},k\}\)\\leq\\frac\{\\bar\{\\phi\}\_\{k\}\}\{n\_\{x\}\}\.\(32\)By linearity of expectation and Jensen’s inequality \(𝔼⁡\[\|X\|\]≤𝔼⁡\[X2\]\\mathbb\{E\}\[\|X\|\]\\leq\\sqrt\{\\mathbb\{E\}\[X^\{2\}\]\}\):

𝔼⁡\[εstat\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\varepsilon\_\{\\mathrm\{stat\}\}\\right\]=12​∑k=1K𝔼⁡\[\|ϕ^k−ϕ¯k\|\]≤12​∑k=1KVar​\(ϕ^k\)\\displaystyle=\\frac\{1\}\{2\}\\sum\_\{k=1\}^\{K\}\\mathbb\{E\}\\\!\\left\[\|\\hat\{\\phi\}\_\{k\}\-\\bar\{\\phi\}\_\{k\}\|\\right\]\\;\\leq\\;\\frac\{1\}\{2\}\\sum\_\{k=1\}^\{K\}\\sqrt\{\\text\{Var\}\(\\hat\{\\phi\}\_\{k\}\)\}\(33\)≤12​∑k=1Kϕ¯knx=12​nx​∑k=1Kϕ¯k\.\\displaystyle\\leq\\frac\{1\}\{2\}\\sum\_\{k=1\}^\{K\}\\sqrt\{\\frac\{\\bar\{\\phi\}\_\{k\}\}\{n\_\{x\}\}\}\\;=\\;\\frac\{1\}\{2\\sqrt\{n\_\{x\}\}\}\\sum\_\{k=1\}^\{K\}\\sqrt\{\\bar\{\\phi\}\_\{k\}\}\.\(34\)By \([27](https://arxiv.org/html/2608.19689#A3.E27)\),∑kϕ¯k=Keff\(ϕ¯\(⋅∣x\)\)\\sum\_\{k\}\\sqrt\{\\bar\{\\phi\}\_\{k\}\}=\\sqrt\{K\_\{\\mathrm\{eff\}\}\(\\bar\{\\phi\}\(\\cdot\\mid x\)\)\}, so𝔼⁡\[εstat\]≤12​Keff/nx\\mathbb\{E\}\[\\varepsilon\_\{\\mathrm\{stat\}\}\]\\leq\\frac\{1\}\{2\}\\sqrt\{K\_\{\\mathrm\{eff\}\}/n\_\{x\}\}\. Note that by Cauchy–Schwarz,∑kϕ¯k≤K​∑kϕ¯k=K\\sum\_\{k\}\\sqrt\{\\bar\{\\phi\}\_\{k\}\}\\leq\\sqrt\{K\\sum\_\{k\}\\bar\{\\phi\}\_\{k\}\}=\\sqrt\{K\}\(since∑kϕ¯k=1\\sum\_\{k\}\\bar\{\\phi\}\_\{k\}=1\), soKeff≤KK\_\{\\mathrm\{eff\}\}\\leq Kalways holds\. Whenϕ¯\(⋅∣x\)\\bar\{\\phi\}\(\\cdot\\mid x\)is a point mass,∑kϕ¯k=1\\sum\_\{k\}\\sqrt\{\\bar\{\\phi\}\_\{k\}\}=1soKeff=1K\_\{\\mathrm\{eff\}\}=1; whenϕ¯\(⋅∣x\)\\bar\{\\phi\}\(\\cdot\\mid x\)is uniform,∑k1/K=K\\sum\_\{k\}\\sqrt\{1/K\}=\\sqrt\{K\}soKeff=KK\_\{\\mathrm\{eff\}\}=K\.

Bias bound\.For the overlapping neighborhood used by SALT, everyx′∈𝒩⁡\(x\)x^\{\\prime\}\\in\\mathcal\{N\}\(x\)satisfiesd⁡\(x′,x\)≤rd\(x^\{\\prime\},x\)\\leq r\. By convexity ofDTVD\_\{\\mathrm\{TV\}\}and[1](https://arxiv.org/html/2608.19689#Thmassumption1):

εbias\\displaystyle\\varepsilon\_\{\\mathrm\{bias\}\}=DTV\(ϕ¯\(⋅∣x\),ϕ∗\(⋅∣x\)\)\\displaystyle=D\_\{\\mathrm\{TV\}\}\(\\bar\{\\phi\}\(\\cdot\\mid x\),\\phi^\{\*\}\(\\cdot\\mid x\)\)\(35\)≤1\|𝒩⁡\(x\)\|∑x′∈𝒩⁡\(x\)DTV\(ϕ∗\(⋅∣x′\),ϕ∗\(⋅∣x\)\)\\displaystyle\\leq\\frac\{1\}\{\|\\mathcal\{N\}\(x\)\|\}\\sum\_\{x^\{\\prime\}\\in\\mathcal\{N\}\(x\)\}D\_\{\\mathrm\{TV\}\}\(\\phi^\{\*\}\(\\cdot\\mid x^\{\\prime\}\),\\phi^\{\*\}\(\\cdot\\mid x\)\)\(36\)≤1\|𝒩⁡\(x\)\|​∑x′∈𝒩⁡\(x\)L⋅d⁡\(x′,x\)≤L​r\.\\displaystyle\\leq\\frac\{1\}\{\|\\mathcal\{N\}\(x\)\|\}\\sum\_\{x^\{\\prime\}\\in\\mathcal\{N\}\(x\)\}L\\cdot d\(x^\{\\prime\},x\)\\;\\leq\\;Lr\.\(37\)∎

### C\.6Proof of[Theorem1](https://arxiv.org/html/2608.19689#Thmtheorem1)\(Aggregation–Estimation Tradeoff\)

###### Proof\.

Substituting\|𝒩⁡\(x\)\|≍ng​rd𝒳\|\\mathcal\{N\}\(x\)\|\\asymp n\_\{g\}r^\{d\_\{\\mathcal\{X\}\}\}into[Lemma2](https://arxiv.org/html/2608.19689#Thmlemma2)and combining via[Lemma1](https://arxiv.org/html/2608.19689#Thmlemma1)gives

𝔼⁡\[DTV\]≲εopt\+L​r\+Keffng​rd𝒳\.\\mathbb\{E\}\[D\_\{\\mathrm\{TV\}\}\]\\;\\lesssim\\;\\varepsilon\_\{\\mathrm\{opt\}\}\+Lr\+\\sqrt\{\\frac\{K\_\{\\mathrm\{eff\}\}\}\{n\_\{g\}r^\{d\_\{\\mathcal\{X\}\}\}\}\}\.\(38\)The optimization term is independent of the neighborhood radius in this tradeoff\. The bias term increases inrrand the statistical term decreases inrr\. To find the optimalrr, we differentiate with respect torrand set the result to zero:

L=d𝒳2⋅1r⋅Keffng​rd𝒳,L\\;=\\;\\frac\{d\_\{\\mathcal\{X\}\}\}\{2\}\\cdot\\frac\{1\}\{r\}\\cdot\\sqrt\{\\frac\{K\_\{\\mathrm\{eff\}\}\}\{n\_\{g\}r^\{d\_\{\\mathcal\{X\}\}\}\}\},\(39\)which givesL2​rd𝒳\+2≍Keff/ngL^\{2\}r^\{d\_\{\\mathcal\{X\}\}\+2\}\\asymp K\_\{\\mathrm\{eff\}\}/n\_\{g\}, yielding

r∗≍\(Keffng​L2\)1/\(d𝒳\+2\)\.r^\{\*\}\\;\\asymp\\;\\left\(\\frac\{K\_\{\\mathrm\{eff\}\}\}\{n\_\{g\}L^\{2\}\}\\right\)^\{1/\(d\_\{\\mathcal\{X\}\}\+2\)\}\.\(40\)Substitutingr∗r^\{\*\}back: the bias term isL​r∗=L⋅\(Keff/\(ng​L2\)\)1/\(d𝒳\+2\)=\(Ld𝒳​Keff/ng\)1/\(d𝒳\+2\)Lr^\{\*\}=L\\cdot\(K\_\{\\mathrm\{eff\}\}/\(n\_\{g\}L^\{2\}\)\)^\{1/\(d\_\{\\mathcal\{X\}\}\+2\)\}=\(L^\{d\_\{\\mathcal\{X\}\}\}K\_\{\\mathrm\{eff\}\}/n\_\{g\}\)^\{1/\(d\_\{\\mathcal\{X\}\}\+2\)\}, and the statistical term is of the same order, giving the optimized upper bound in \([17](https://arxiv.org/html/2608.19689#S4.E17)\)\. ∎

## Appendix DSALT Implementation Details

OnSubjSim, SALT groups contexts by survey question, embeds each context with Qwen3\-embedding\-8b, retrieves adaptive\-radius neighbors within the group, and trains on the resulting soft labels\. The remainder of this section specifies each of these steps\.

Algorithm 1Context\-Aggregation Training with Adaptive Radius0:Dataset

𝒟\\mathcal\{D\}; pretrained context encoder; radius function

ρ\\rho; refresh interval

TT; number of epochs

EE
0:Trained model

ϕθ\\phi\_\{\\theta\}
1:Action\-space partitioning:group all contexts by action space:

𝒢g=\{x:𝒜⁡\(x\)=𝒜g\}\\mathcal\{G\}\_\{g\}=\\\{x:\\mathcal\{A\}\(x\)=\\mathcal\{A\}\_\{g\}\\\}
2:Embed all contexts using the pretrained encoder

3:Initialize

ϕθ\\phi\_\{\\theta\}from a pretrained LLM

4:forepoch

=1,…,E=1,\\ldots,Edo

5:foreach optimizer stepdo

6:ifthe step index is a multiple of

TTthen

7:foreach context

xxdo

8:Compute adaptive radius

r⁡\(x\)←ρ⁡\(ϕθ,x\)r\(x\)\\leftarrow\\rho\(\\phi\_\{\\theta\},x\)
9:Retrieve neighborhood

𝒩⁡\(x\)←\{x′∈𝒢g:d⁡\(x′,x\)≤r⁡\(x\)\}\\mathcal\{N\}\(x\)\\leftarrow\\\{x^\{\\prime\}\\in\\mathcal\{G\}\_\{g\}:d\(x^\{\\prime\},x\)\\leq r\(x\)\\\}
10:Construct soft label

ϕ^\(k∣x\)←\|𝒩\(x\)\|−1∑x′∈𝒩⁡\(x\)𝟏\[a\(x′\)=a\(k\)\]\\hat\{\\phi\}\(k\\mid x\)\\leftarrow\|\\mathcal\{N\}\(x\)\|^\{\-1\}\\sum\_\{x^\{\\prime\}\\in\\mathcal\{N\}\(x\)\}\\mathbf\{1\}\[a\(x^\{\\prime\}\)=a^\{\(k\)\}\]
11:endfor

12:endif

13:Update

θ\\thetaon the current batch by minimizing

DKL\(ϕ^\(⋅∣x\)∥ϕθ\(⋅∣x\)\)D\_\{\\mathrm\{KL\}\}\(\\hat\{\\phi\}\(\\cdot\\mid x\)\\\|\\phi\_\{\\theta\}\(\\cdot\\mid x\)\)
14:endfor

15:endfor

Table 2:Mapping between theoretical concepts, notation, and their SubjSim implementation\.#### Context embeddings\.

We use Qwen3\-embedding\-8b\([34](https://arxiv.org/html/2608.19689#bib.bib30)\)as the context encoder, which produces 4096\-dimensional vectors for Chinese text\. Embeddings are precomputed once before training and cached on disk; they are not updated during training\. All contexts within the same action\-space group share the same situational descriptionss\. As a result, the embeddings primarily capture persona similarity within each group rather than situational variation\.

#### Neighborhood Construction

Pairwise distances between context embeddings are computed using the Euclidean \(ℓ2\\ell\_\{2\}\) distance\. The full distance matrix is precomputed once at training startup using SciPy on CPU and cached for reuse\. We estimate the intrinsic dimensiond𝒳d\_\{\\mathcal\{X\}\}of the persona embedding space via PCA with a variance threshold of0\.900\.90, yieldingd𝒳=28d\_\{\\mathcal\{X\}\}=28\. The adaptive radiusr⁡\(x\)r\(x\)is then computed per context according to Equation[18](https://arxiv.org/html/2608.19689#S4.E18), with scale factorC=0\.2C=0\.2\.

#### Soft\-label refresh schedule\.

Soft labels are refreshed every 30 training steps using the current model checkpoint\. Per\-epoch refresh adapts slowly in early training, whereas per\-step refresh is computationally prohibitive and unstable\. The 30\-step interval balances label responsiveness with training efficiency\.

#### Training hyperparameters\.

Table[3](https://arxiv.org/html/2608.19689#A4.T3)reports the main training and method\-specific hyperparameters for each method\. All methods are trained on 8 GPUs; SALT uses per\-device batch size 1 with 16 gradient\-accumulation steps\. Both the DPO and PPO policies are initialized from the SFT checkpoint\. SALT uses a maximum sequence length of 1024 for evaluation \(ca\_eval\_max\_length\)\.

Table 3:Main training and method\-specific hyperparameters for all methods\.
#### Reward Model Construction \(PPO\)\.

The reward model is initialized from the SFT checkpoint\. For each context, the action with the highest probability in the annotator’s empirical distribution, i\.e\., the hard label defined in[Section5\.1](https://arxiv.org/html/2608.19689#S5.SS1), is treated as the chosen response\. Each remaining candidate is paired individually as a rejected response, yieldingK−1K\-1preference pairs per context\.

#### Computational Cost\.

SALT and SFT are trained on eight NVIDIA H20 GPUs using DeepSpeed ZeRO\-2\. Context embeddings for all 16,500 training samples are precomputed in 40 seconds\. Each training epoch takes approximately 35 minutes for SALT and 31 minutes for SFT\.

## Appendix EPer\-Domain Results

[Figures1](https://arxiv.org/html/2608.19689#S5.F1),[2](https://arxiv.org/html/2608.19689#S5.F2)and[3](https://arxiv.org/html/2608.19689#S5.F3)in the main text aggregate over the whole test set\. This section repeats the same three analyses within each of the eight topic domains ofSubjSim, so that the subjectivity trend, the neighbor\-selection ablation, and the radius\-scale sensitivity can each be checked domain by domain\.

\(a\)Subjectivity\-stratified comparison between SFT and SALT\(b\)Ablation over neighbor selection\(c\)Sensitivity to the radius scaleCC
Figure 4:Per\-domain results on theEconomydomain, mirroring[Figure1](https://arxiv.org/html/2608.19689#S5.F1),[Figure2](https://arxiv.org/html/2608.19689#S5.F2), and[Figure3](https://arxiv.org/html/2608.19689#S5.F3)of the main text\. All metrics are divergences \(lower is better\)\.\(a\)Subjectivity\-stratified comparison between SFT and SALT\(b\)Ablation over neighbor selection\(c\)Sensitivity to the radius scaleCC
Figure 5:Per\-domain results on thePoliticaldomain, mirroring[Figure1](https://arxiv.org/html/2608.19689#S5.F1),[Figure2](https://arxiv.org/html/2608.19689#S5.F2), and[Figure3](https://arxiv.org/html/2608.19689#S5.F3)of the main text\. All metrics are divergences \(lower is better\)\.\(a\)Subjectivity\-stratified comparison between SFT and SALT\(b\)Ablation over neighbor selection\(c\)Sensitivity to the radius scaleCC
Figure 6:Per\-domain results on theTechnologydomain, mirroring[Figure1](https://arxiv.org/html/2608.19689#S5.F1),[Figure2](https://arxiv.org/html/2608.19689#S5.F2), and[Figure3](https://arxiv.org/html/2608.19689#S5.F3)of the main text\. All metrics are divergences \(lower is better\)\.\(a\)Subjectivity\-stratified comparison between SFT and SALT\(b\)Ablation over neighbor selection\(c\)Sensitivity to the radius scaleCC
Figure 7:Per\-domain results on theSocialdomain, mirroring[Figure1](https://arxiv.org/html/2608.19689#S5.F1),[Figure2](https://arxiv.org/html/2608.19689#S5.F2), and[Figure3](https://arxiv.org/html/2608.19689#S5.F3)of the main text\. All metrics are divergences \(lower is better\)\.\(a\)Subjectivity\-stratified comparison between SFT and SALT\(b\)Ablation over neighbor selection\(c\)Sensitivity to the radius scaleCC
Figure 8:Per\-domain results on theCulturedomain, mirroring[Figure1](https://arxiv.org/html/2608.19689#S5.F1),[Figure2](https://arxiv.org/html/2608.19689#S5.F2), and[Figure3](https://arxiv.org/html/2608.19689#S5.F3)of the main text\. All metrics are divergences \(lower is better\)\.\(a\)Subjectivity\-stratified comparison between SFT and SALT\(b\)Ablation over neighbor selection\(c\)Sensitivity to the radius scaleCC
Figure 9:Per\-domain results on theHealthdomain, mirroring[Figure1](https://arxiv.org/html/2608.19689#S5.F1),[Figure2](https://arxiv.org/html/2608.19689#S5.F2), and[Figure3](https://arxiv.org/html/2608.19689#S5.F3)of the main text\. All metrics are divergences \(lower is better\)\.\(a\)Subjectivity\-stratified comparison between SFT and SALT\(b\)Ablation over neighbor selection\(c\)Sensitivity to the radius scaleCC
Figure 10:Per\-domain results on theEnvironmentdomain, mirroring[Figure1](https://arxiv.org/html/2608.19689#S5.F1),[Figure2](https://arxiv.org/html/2608.19689#S5.F2), and[Figure3](https://arxiv.org/html/2608.19689#S5.F3)of the main text\. All metrics are divergences \(lower is better\)\.\(a\)Subjectivity\-stratified comparison between SFT and SALT\(b\)Ablation over neighbor selection\(c\)Sensitivity to the radius scaleCC
Figure 11:Per\-domain results on theEducationdomain, mirroring[Figure1](https://arxiv.org/html/2608.19689#S5.F1),[Figure2](https://arxiv.org/html/2608.19689#S5.F2), and[Figure3](https://arxiv.org/html/2608.19689#S5.F3)of the main text\. All metrics are divergences \(lower is better\)\.
## Appendix FStatistical Significance

Evaluation is deterministic \([Section5\.2](https://arxiv.org/html/2608.19689#S5.SS2)\), so the only source of uncertainty in the reported metrics is the finiteness of the test set\. We therefore quantify it with a paired bootstrap over test contexts: we resample the 2,800 test contexts with replacement 1,000 times, apply the same resampled index set to every method, and recompute each metric on every replicate\.[Table4](https://arxiv.org/html/2608.19689#A6.T4)reports the point estimates with 95% percentile intervals\. The intervals of SALT and every baseline are disjoint on every metric; testing the paired differences directly, the 95% interval of SALT minus each comparator excludes zero on all four metrics, including the strongest ablation variant \(fixedKeffK\_\{\\mathrm\{eff\}\}: KL difference−0\.0273\-0\.0273, CI\[−0\.0307,−0\.0237\]\[\-0\.0307,\-0\.0237\]\)\. SALT’s improvements are therefore statistically significant rather than an artifact of the test split\.

Table 4:Point estimates and 95% paired\-bootstrap confidence intervals on the full test set \(1,000 resamples over contexts\)\. All metrics are divergences \(lower is better\)\.
## Appendix GSubjSim Dataset Details

#### Why elicited propensities\.

Exact repeated measurement of the same person is not a viable route to the distributional ground truth, since repetition can change memory, reflection, fatigue, and demand effects\.SubjSimtherefore operationalizes the target via elicited subjective response propensities: respondents allocate plausibility across the same action space, so the hidden distributional target is available for evaluation while training methods receive only a single hard action\.

Figure 12:Overview of the SubjSim dataset\. \(a\) Topics\. \(b\) Option countKK\. \(c\) Subjectivity coefficient\.
### G\.1Question Pool and Filtering

#### Source Surveys\.

Questions are drawn from seven internationally standardized social survey programs: the American Trends Panel \(ATP\), General Social Survey \(GSS\), World Values Survey \(WVS\), American National Election Studies \(ANES\), Chinese General Social Survey \(CGSS\), European Social Survey \(ESS\), and International Social Survey Programme \(ISSP\)\. These programs were selected for their broad topical coverage, institutional authority, and diversity of question types\. The initial candidate pool contains 5,964 items in total\.

#### Translation\.

All non\-Chinese items are translated into Chinese using DeepSeek\-chat\([13](https://arxiv.org/html/2608.19689#bib.bib31)\)\. CGSS items are retained in their original Chinese form\.

#### Automated Screening\.

Each candidate item is evaluated by DeepSeek\-chat\([13](https://arxiv.org/html/2608.19689#bib.bib31)\)along three dimensions:

- •Cultural suitability:whether the item is appropriate for Chinese respondents, considering cultural sensitivity, social norms, privacy boundaries, linguistic conventions, and regional variation\.
- •Option ordinality:whether the response options follow a logical order \(ordinal\) or are unordered categories \(nominal\)\. Only nominal items are retained, as ordinal scales introduce additional measurement assumptions that complicate distributional evaluation with metrics such as TVD and JSD\.
- •Question objectivity:whether the item has a factually verifiable answer \(objective\) or depends on personal attitudes, feelings, or preferences \(subjective\)\.

The system prompt used for automated screening is as follows:

Screening PromptYou are a professional survey design expert\. Please perform a three\-dimensional annotation analysis on the given survey question\.Annotation Dimensions1\. Cultural Suitability•Determine whether the question is appropriate for Chinese respondents\.•Consider: cultural sensitivity, social norms, privacy boundaries, linguistic conventions, and regional variation\.•Output: suitable / caution / unsuitable, with a brief explanation\.2\. Option Ordinality•Determine whether the response options have a logical order\.•Ordinal: options exhibit a clear gradient, ranking, or sequence \(e\.g\., very dissatisfied→\\rightarrowvery satisfied; 18\-\-25→\\rightarrow26\-\-35→\\rightarrow36\+\)\.•Nominal: options are parallel with no inherent order \(e\.g\., red/blue/green; football/basketball/swimming\)\.•Output: ordinal / nominal, with justification\.3\. Question Objectivity•Determine whether the question is objective or subjective\.•Objective: the answer is factually verifiable and does not depend on personal feelings or opinions \(e\.g\., age, household size, education level, occupation, home ownership, weekly exercise frequency\)\.•Subjective: the answer depends on personal attitudes, feelings, evaluations, or preferences \(e\.g\., satisfaction, sense of identity, importance ratings, willingness, brand preference\)\.•Output: objective / subjective, with justification\.

After automated screening, 109 items are labeled as suitable, nominal, and subjective\. These items proceed to manual review\. The remaining 1,012 suitable objective items serve as the pool for demographic attribute dimensions\.

#### Manual Review and Localization\.

We manually review the 109 subjective items, remove duplicates, verify subjectivity, and adapt phrasing to the Chinese cultural context\. This yields the final 100 questions\.

#### Demographic Attribute Dimensions\.

From the 1,012 objective items, we manually select those most relevant to the 100 subjective questions, yielding 30 demographic attribute dimensions\.

Table 5:Source distribution of the 100 subjective questions in SubjSim\.

### G\.2Question and Demographic Attribute Examples

Table 6:Examples of Demographic Attributes and Subjective Questions
### G\.3Annotator Demographics

We recruited 193 annotators via a university online forum, including both students and non\-students\. Figure[13](https://arxiv.org/html/2608.19689#A7.F13)summarizes their distributions across age, gender, academic status, and field of study\. The pool skews toward young, university\-educated individuals, with a roughly balanced gender ratio and a mix of STEM and Humanities backgrounds\. This limits claims about population\-level representativeness, but it is less central to the benchmark’s main target: evaluating distributional alignment for individual behavioral tendencies\.

Figure 13:Demographic distributions of annotators across gender, age, academic status, and field of study\.
### G\.4Probability Ball Allocation Protocol

![Refer to caption](https://arxiv.org/html/2608.19689v1/figs/annotation_page.png)Figure 14:Screenshot of the annotation interface\. The platform was deployed in Chinese for native Chinese\-speaking annotators; the interface shown here is an English translation for presentation purposes\.In the probability\-ball protocol, annotators distribute a fixed number of balls across all available options and must allocate all balls before submission\. The fraction assigned to each option represents the annotator’s subjective probability for that option\. This avoids known issues with Likert\-scale ratings such as scale bias and cross\-item incomparability\.

The number of balls scales withKKto balance resolution and cognitive load: 10 balls forK=2K=2, 12 forK=3K=3andK=4K=4, 15 forK=5K=5, and 20 otherwise\.

We implemented a web\-based annotation platform where annotators adjust ball counts per option via sliders or\+/−\+/\-buttons\. The remaining ball count is displayed in real time, and submission is blocked until all balls are allocated\. Figure[14](https://arxiv.org/html/2608.19689#A7.F14)shows a screenshot of the interface\.