Decision Aggregation under Quantal Response
Authors: Zhihuan Huang
Yichong Xia
Yuqing Kong
Venue: ICLR 2026 Poster (2026)
Keywords: aggregation
quantal response
bounded rationality
large language models
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: The effectiveness of collective decision-making is often challenged by the bounded rationality and inherent stochasticity of individual agents. We investigate this by analyzing how to aggregate decisions from $n$ experts, each receiving a private signal about an unknown state. Assuming signals are conditionally independent and identically distributed, we depart from the fully rational paradigm and model expert behavior using quantal response—a stochastic choice model capturing bounded rationality. Within a minimax regret framework, we show that majority voting is the optimal robust aggregator when individual rationality falls below a certain threshold. Interestingly, such groups can outperform perfectly rational agents, as their decision randomness encodes weak but informative signals lost in deterministic behavior. We validate these findings using large language models (LLMs), which naturally exhibit quantal response via their temperature parameter. Aggregating moderately stochastic LLM outputs significantly improves accuracy on complex reasoning tasks, highlighting bounded rationality not as a limitation, but as a potential strength in collective intelligence.

Published as a conference paper at ICLR 2026
DECISION AGGREGATION UNDER QUANTAL RESPONSE
Zhihuan Huang, Yichong Xia & Yuqing Kong Center on Fronticrs of Computing Studics (CFCS)
School of Computer Science, Peking University {zhihuan huang, xiayc, yuqing. kong) @pku. edu. cn
ABSTRACT
The effectiveness of collective decision-making is often challenged by the bounded rationality and inherent stochasticity of individual agents. We investigate this by analyzing how to aggregate decisions from experts, each receiving a private signal about an unknown state. Assuming signals conditionally independent. and identically distributed, we depart from the fully rational paradigm and model expert behavior using quantal re- sponse-a stochastic choice model capturing bounded rationality. Within a minimax regret framework, we show that majority voting is the optimal robust aggregator when individual rationality falls below certain threshold. Interestingly, such groups can outperform perfectly rational agents, as their decision randomness encodes weak but informative signals lost in determinis- tic behavior. We validate these findings using large language models (LLMs), which naturally exhibit. quantal response via their temperature parameter. Aggregating moderately stochastic LLM outputs significantly improves ac- curacy on complex reasoning tasks, highlighting bounded rationality not as a limitation, but potential strength in collective intelligence.
INTRODUCTION
The market is very up and down. Your finger hovers over the BUY/SELI button. Seeking guidance, you consult two experts. Mia, renowned for her perfectly rational analysis, advises: "Sell." John, known for his sharp instincts but less predictable approach, urges: "Buy it. Trust me." " Obviously, you should follow rational Mia. But what you could consult multiple Mias Johns?
Let's frame this formally. Suppose we are deciding whether to BUY (X = 1) or SELL (X =) 0) function defined as w(Xww) = 1 if
stock, where the true state is either rise (w 1) or fall (w = 0). The utility w, and u(X -1 otherwise. We have two types of experts: Mia, who embodies perfect rationality, and John, who exhibits bounded rationality. Both receive private signals S E {0.1] that provide noisy information about the future state. Their decisions depend on the expected utility of buying conditioning private signals, denoted by E(u(1,w) S), and follow a quantal response function McKelvey and Palfrey 1995):
e (-B(u((10)s)]
Buy Probability Pr[X =1
|
Σxe(0.1) ed:R/u(aw))
buy
expected utility of buying
exv
Av exv
1 1 2Av
Let (px(e) 2x0. (x(v) is the probability of buying given the advisor's expected payoff 1) is v, with rationality parameter X. In this scenario, "good" and "bad" signals correspond to expected payoffs of e = 0.2 (still bad, but less severe) and v = -1 (strongly avoid), respectively. Mia (Perfectly Rational, À oo) always follows strict. logic if
from buying (X 1)

---

Published as a conference paper at ICLR 2026
0, she never buys; if O, she always buys. John's (Bounded Rational, À 2.5) decisions are probabilistic even when v suggests "don't 't buy," he might still take a chance.
25%
100%
0.2
BUY
Prior
50%
Posterior
B[u(L, 1]
(()
Response Function
75%
-1 1
SELL
a[](L,))SS =
World State
Signal
Expected Utility of BUY
Decision
aa)
Decision Process of Mia and John.
BUY
Signals
Decision
SELL
SELL
73%
50% BUY 50% SELL
8
100%
PMi !+) oloha!
SELL
BUY
27%
1
0.5
0.5
1
Expected Utility of BUY
Mia
John
Response Function (
Decision Outcomes of Mia and John
(b) Response Functions (xx(v) different
Figure 1: Comparison of rational (Mia) vs. bounded-rational (John) decision-making through unified quantal response functions.
Mia, the perfectly rational expert, makes the "correct" choice every time and earns average utility of 0.5. John, who sometimes gambles against the odds, earns lower 0.43.
Now, what if you ask bunch of Mias and bunch of Johns? Let's say each expert gets private signal conditionally independent on the underlying state. Interestingly, after perfectly combining team's 's judgments, team of Mias stays at 0.5 (always right, but never better), while a team of Johns actually improves to 0.51 oute-oupprrormiing the perfect experts. Why? Because John's occasional "mistakes" add information for the group's collective wisdom to uncover deeper truths.
Our example shows how imperfect experts outperform perfect ones but there's catch. To perfectly combine a team's judgments, you'd need to know their signal structures-joint distribution over signals and state. In reality, we almost never have this complete picture. This leads to our core question: How can we combine expert decisions effectively without knowledge of signal structures? Can imperfect experts still outperform perfect ones in this scenario?
This drives our research into robust aggregation methods that work without perfect informa- tion. We use a "worst-case scenario" framework. We assess different methods for combining opinions by evaluating their maximum "regret"- much worse they could be compared to the best possible method if we did know everything (Arieli et al. 2018 2023 Levy and Razin 2022 De Oliveira et al. 2021 Surprisingly, within this framework, we find that simply using majority vote works best when experts aren't perfectly rational and exhibit sufficiently bounded rationality. Interestingly, majority vote isn't 't the best when all experts perfectly rational. Furthermore, imperfect experts, when their opinions simply tallied by majority vote, can actually do better than perfect experts whose opinions are combined using the most sophisticated techniques.
2

---

Published as a conference paper at ICLR 2026
This advantage of bounded rationality aligns with observations in artificial intelligence: large language models (LLMs) exhibit similar quantal response behavior through their "temperature" parameter. Lower temperatures produce deterministic, Mia-like outputs, while higher temperatures induce John-like stochasticity. Just as in human groups lower- temperature outputs may yield individually accurate responses, but highen-temperature outputs leverage informational diversity to achieve superior collective decisions.
1.1 SUMMARY OF RESULTS
We study the aggregation of decisions from groups of boundedly rational experts modeled through quantal response functions. We consider experts receiving private signals that are conditionally independent and identically distributed (i..d.) given the true state. Our main contributions in Section 3 are as follows:
Optimal Robust Aggregator: We prove majority voting is optimally robust for groups whose rationality falls below a threshold dependent on group size (Sec- tion
Bounded Rationality Advantage: For a single expert (n = 1), rationality
maximizes aggregation outcomes. However, with multiple experts (n 2 2), bounded rationality can outperform perfect ralionality (Section A.3). Generalizing beyond specific scenario, numerical experiments (Figure 5) show that bounded rational experts consistently achieve lower worst-case regret than fully rational experts.
These theoretical insights rely on a dimension reduction result: any quantal response report structure under conditional independence equivalently be represented by simplified three-signal structure, significantly reducing analytical complexity.
In Section 4 we empirically validate our theoretical predictions using large language models (LLMs) as experts. The 'temperature" parameter, which adjusts randomness, naturally corresponds quantal response behavior.
Empirical analysis confirms LLMs exhibit quantal response behavior, with logistic regression aligning closely with theoretical models (Figure 2).
Quantal Response in LLMs:
Empirical Aggregation Advantage: Using LLMs with varied temperatures (t E {0, 0.5, 13) and aggregating decisions via majority voting with groups of size (1, we conduct two experiments: traditional box-ball tasks and multi- choice mathematical questions. Results confirm that bounded-rational experts with moderate temperature (e.g., f 0.5) collectively outperform purely rational experts (L ()=0) for n 2 3, even in non-binary settings (Figure 3).
Overall, our findings reveal that following the majority decisions of moderately irrational experts is effective, highlighting benefits of bounded rationality. This insight. impacts fields such as crowdsoureing, AT ensembles, economic decision-making, and product design, where incorporating diverse and less rational inputs can yield superior outcomes.
1.2 RELATED WORK
Information Aggregation and Robustness. Information aggregation the process of combining multiple information sources into collective decision is fundamental to collective intelligence (Foryyhhe and Lundholm 1990 Breiman 1996 Dietterich 2000). While
traditional models often assume perfect rationality and complete information (Winkler) 1981
Arieli et al.) 2018 2023), real-world experts frequently deviate due to cognitive limitations and inherent stochasticity (Simon 1990 Tversky et al. 1982) This reality necessitates robust aggregation methods capable of handling uncertain information structures. Recent literature has significantly expanded the frontiers robust aggregation to tackle unknown correlations (Levy and Razin 2022 De Oliveira et al. 2021), algorithric feasibility (Guo et al., 2025), second-order information in two-expert scenarios (Pan et al., 2024), and adversarial expert inputs (Guo and Kong 2025). Unlike prior studies that primarily focus perfectly
3

---

Published as a conference paper at ICLR 2026
rational or adversarial agents, we incorporate Quantal Response (QR) models McCKelvey and Palfrey 1995 1998) to capture bounded rationality through stochastic choice rules. By leveraging geometric dimension reduction, we prove that three signals suffice for optimal robust. aggregation, explicitly demonstrating how bounded rationality can paradoxically benefit collective outcomes.
Bounded Rationality and Strategic Voting. Bounded rationality models have found extensive applications across mechanism design [Braverman et al. 2018), information ion design (Yu et al. 2024), and robust prcision-makinn where cognitive biases, such base rate neglect, have been shown to yield surprising aggregation benefits (Kong et al. 2024 Gan al. 2023 Feng et al. 2024). Specifically, QR models are widely used azelyze strategic behavior in probabilistic voting and collective choice (Goeree ct al. 2002 2016 Hoppe and Schmitz 2013p. For instance, Quantal Response Equilibrium (QRE) has been applied to examine voter strategies, laboralory voling gaines, and the "wisdom of strategic voling" (McKelvey and Pally, 2006 Casella et al. 2006; Goerre el. al. 2005; Han el. al. 2023). However, this rich literature typically centers utility-based strategic manipulation where agents possess heterogeneous, outcome-dependent payoffs. In contrast, our research operates within shared-objective framework where experts lack strategic incentives to misreport. We focus entirely on the mechanism of collective intelligence, aggregating sancrr--uutbounddee QR-xxperts rather than navigating strategic utility landscapes.
Quantal Responses and Large Language Models. The emergence of Large Language
Models (LLMs) has spurred the application of behavioral economic models to Al agents (Jia
al. 2025 Kirshner et al., 2025 Concurrently, research on LLM temperature which inherently controls output randomness has investigated its impact on precision tasks (Renze and Guven 2024 code generation (Zhu al. 2024), and creativity (Peeperkorn al.) 20244]. Existing works mostly study how a single LLM behaves or how LLMs interact competitively, Differently, we formally link the temperature parameter to QR theory within an aggregation context. We demonstrate that, much like human bounded rationality, optimally tuned L.M stochasticity can significantly improve the accuracy aggregated group decisions.
2 MODEL SETUP
We consider a binary state Ω = 1) with a known prior probability Prfw = 1] E [o. 1].
There is one decision maker (DM) and experts indexed by I...., n. Each expert observes a private signal S, E S, drawn from a set of possible signals S, about the true state w. The joint distribution of the state and signals called the signal structure A(l x S*).
The set of all possible signal structures is θ.
To capture scenarios such as aggregating multiple responses from a large language model where each response is independently and identically generated from the same input we focus on signal structures where signals are conditionally independent and identi- cally distribuled (c.i.i.d)) given the state w. Formally, this means:
Pr[S $1...., Sn Sn w ww w] II Pr(S, st w w).
with the conditional distribution Pr[S, s set of all such c.i.id. signal structures is denoted by @ridd, This setting is standard and describes situations where each expert independently observes information about the state. Additionally, since experts anonymous from the DM's perspective, the DM treats each expert's signal distribution as identical.
w = w| identical across all experts. The
Each expert after observing the private signal St, reports binary decision X, = {0, to the DM. The utility for both DM and experts depends on correctly identifying the state:
X
u(X,w)
X X
4

---

0.5 p 0.5 Published as a conference paper at ICLR 2026
This common payoff aligns their incentives towards correct decisions. We acknowledge that in strategic environments with heterogeneous utilities or pivotal incentives, sincere voting is not necessarily Nash equilibrium (Austen-Smith and Banks 1996 Han et al. 2023). However, our infornation-aggreaatioo setting features a shared-obeeclive structure where experts receive outcome-dependent payoffs. Under this setting, experts have strategic incentive to misreport, making sincere reporting natural and widely used modeling assumption (Arieli et al., 2018 Palley and Satopää 2023). This assumption is particularly appropriate when the experts are Large Language Models (LLMs), which do not possess stable, agent-specific utilities and do not benefit strategically from manipulating outcomes.
Experts choose their decisions according to a quantal response function, reflecting bounded rationality. We treat this as an "as-if" model rather than assuming experts perform precise Bayesian calculations followed by utllit--axiiizzatinn with noise. The randomness may arise from cognitive noise, imperfect internal signal processing, or intrinsic neural stochastic- S; s.] be expert's posterior belief after observing signal 5;- The expected utility of choosing X; E {0, 1] is:
ity (Knill and Pouget 2004). Let p = Pr|w
B(u(X,, S (2X; ()(2p 1) [-1,1].
Recall the quantal response function ς> |-1, + [0, 1]. with rationality level X 2 0, is given by the logistic (soflmax) form:
(a(v) = + 2A
V V [-1,1].
Note that ςa is equivalent to the softmax layer in large language models, showing the similarity between quantal response in decision theory and softmax outputs in machine
learning. Replacing v = E[u(X; = 1,w) | St] = 2p- - 1, the expert's decision probability becomes:
v사 = = 1) 1 21(1 2p)
P E 10.
Note that Ωx and a are equivalent representations; for convenience, we use 1i throughout the remainder of this paper.
The function 4i increases with p. When = 0, decisions are random; as X - oo, decisions become deterministic:
p > 0.5
x (p) =
p < 0.5
Given a signal structure 0 and response function thx; the report structure Ô e A(Ω x describes the joint distribution of the state and experts' reports. The mapping rep(f, v/x) defines the set of c.i.i.d. report structures:
@giid {0: 39 E ciid, rep(f,, 12)) =)-
2.1 DECISION AGGREGATION
The DM aggregates experts' decisions anonymously based only on the number of experts
X Σ X, reporting An aggregator function f [0. [0, 1] represents the DM's mixed strategy, where f(x) is the probability the DM guesses state w = 1 given reports of
The utility for aggregator f under report structure 0 is:
U(S,0) Σ x|2x=/ 1)(2f(x) 1).
5

---

Published as a conference paper at ICLR 2026
Omniscient Aggregator. An ideal benchmark, the omniscient aggregator maximizes expected utility knowing the true report structure 0:
1, Pralw = 1 X x| 0.5 Pralw = X = x| = =0.5.
(p)
0.5,
0,
Pry[== 1 X = x| < 0.5
Optimal Robust Aggregator. When facing uncertainty about 0, the DM aims to minimize worst-case regret, definod as the difference in utility between an aggregator f and the omniscient aggregator:
R(f. (opts. 0) Úf.
The optimal robust aggregator solves the min-max problem:
opt E arg min max R(f. 6),
f ôcô
thus ensuring robustness under uncertainty.
3 THEORETICAL RESULTS
This section introduces theoretical results for decision aggregation under bounded rationality. We focus setting where experts receive signals that are independent and identically distributed conditioning on the world state (c.i.id.).
Theorem 3.1 (Main Theorem). Consider a group of experts with c.i.i.d. signal structures A and quantal responses with rationality parameter X.
1. Optimal Robust Aggregator. When X S g(n), majority voting fmajj is the optimal robust aggregator:
R(Pmaj .)
min R(J, 0)
ò dgôuud
deóqud
using the decision rule:
0,
T. n/2
fmas (x)
1/2.
x = n/2
1,
x > n/2
The threshold function g(n) is defined the supremum of the set of X values that satisfy a specific inequality for all qo and q within a given ronge. Formally,
g(n) = sup
A
Agu. qı such that (0) S 90 S 91 S 0.5, the following inequality holds:
(qi(1 91 )))) - (1 2211 (go(1 - go)) 1 1(1 2qa)
2A + In(11 - qo) In(go)
x 2A In(1 - 9o) + In(go)
(1)
(1)
For n 2, majority voting remains optimal for any finite X, that is, the threshold is infinity g(n)
2. Bounded Rationality Advantage. While perfect rationality (A 0o) marimizes single-expert performance (n = 1), the group exhibits a different behavior:
For all > 2, there exist 0" E @cdd and finite X* such that under 0", the optimal ulility with bounded rationality level X strictly exceeds that with perfect ralionalily:
max U (f. rep (0" xxx-)) > max U (f, (0", (bo)))
max

---

Published as a conference paper at ICLR 2026
For all n > 2, this advantage of bounded rationality holds for majority voting: (fmos, rep (0", xxx)) > max (f, rep (0", voo))
In the above theorem, @ denotes the set of joint distributions over reported decisions induced by the c.i.d. signal structures Geidd through the quantal response function 1/x, i.e., Égiid rep(@cid vxx). The terms 1b (0) and (1) represent the probabilities of reporting X, when the posterior Pr(w 1 S.] is 0 or respectively. The inequality that defines g(n) illustrates that, in expectation, reporting 1 is optimal when exactly | experts report 0, based on the combination of the two symmetric report structures parameterized by 4o and 91-
Proof Sketch of Thcorem 3.1 shown in Section A
The proof structured in three main steps, which is
1. Dimension Reduction. To solve the min-max optimization over the intractable infinite-dimensional space of signal structures, we reduce its dimensionality. Since expert's report depends only on their posterior belicf [0, 1], encode each posterior as a point on a curve in R Valid report structures form the convex hull of this curve. We prove a key geometric lemmaa that four points on this curve are coplanar. Combined with Carathéodory's theorem, this shows any report structure can be represented by signal structure with at most three posteriors {0, p, 1}, simplifying the problem to managcable three-paramcter space.
2. Optimality of Majority Voting. Bounded rationality regularizes the space of report structures, smoothing out extreme structures that favor complex, non- monotonic aggregators. We prove that when S g(n), simple majority voting becomes the optimal robust aggregator. To establish this, prove pairwise optimality: for any symmetric pair of structures, majority voting minimizos their combined regret. We then show this implics global minimax optimality. The threshold g(n) is the largest for which pairwise optimality holds.
3. Bounded Rationality Advantage. Finally, we explain the counterintuitive result that bounded rationality can outperform perfect rationality in a group setting (n 2 2). While perfect rationality best for single expert, the randomness from bounded rationality introduces valuable diversity in a group. It allows weak but useful signals, which might be lost when fully rational experts all report the same decision, to influence the collective outcome. The proof is by construction. We design a signal structure 8 where perfectly rational experts unanimously report 0, yielding a utility of 0.5. We then show that with a finite rationality level X", their reports become stochastic. This variation, while noisy, is informative, allowing an aggregator to achieve utility strictly greater than 0.5. For 2, show that simple majority voting is sufficient to gain this advantage.
Numerical Study. To complement our theoretical analysis, we conducted numerical simulations to illustrate the key results. As shown in Section A.2 (Figure 5). we computed the worst-case regret for both majority voting and the optimal robust aggregator. The results confirm that for rationality levels À S g(n), the regrets are identical, verifying that majority voting (f)) is indeed the optimal robust aggregator. The plots also reveal that this threshold is a sufficient but not strictly necessary condition, as majority voting can remain optimal for some À g(n). However, for larger groups (e.g., rationality, more complex aggregator outperforms the majority rule. The U-shaped curve of the worst-case regret illustrates that choosing an appropriate, bounded level of rationality can be more effective than pursuing perfect rationality. This highlights that a moderate degree of "noise" or uncertainty can be beneficial for group decision-making.
5) and high
4 EMPIRICAL RESULTS
In this section, we describe two sets of studies designed to address the following two key research questions:
7

---

Published as a conference paper at ICLR 2026
1. Do the responses of large language model (LLM) conform to quantal response theory?
2. How do the number of experts (LLM instances) and their ralionality (Iemperature) affect the aggregation results, and how should these parameters be set to optimize performance?
Connection between LLMss and Quantal Response. Before detailing the experimental sctup, we formalize the connection between the quantal response (QR) model and the core outpul mechanism of LLMs. The QR model is structurally identical to the LLM's softmax function with temperature. ure. Let V, = R|u(X j, S) denole the expected utility of choosing action j. The QR model dictates that the probability of choosing action j is given by:
Aus
P)
Ex eduu
where À is the rationality parameter. Similarly, the probability of an LLM outputting choice j is determined by its internal logits 2j and the sampling temperature
ebilt
P(choice 5)
Σx exs/
These formulations are structurally identical. Assuming the LLM's internal logits are proportional to the expected utility under a given task, the rationality paramcter À is mathematically analogous the inverse temperature 1/t. A lower (increased stochasticity) directly corresponds to higher temperature L. (more randomness). This theoretical link makes LLMs ideal subjects for evaluating bounded-rational agents, an approach increasingly supported by recent literature modeling LLM decision-maiee making through behavioral game theory and QR frameworks (Jia et al., 2025
Kirshner et al., 2025).
These questions and theoretical connections guide the study setup, We simulate individual expert decisions by querying an LLM and using the single response as the expert's choico. To simulate the conditionally independent and identically distributed (c.i..d.) nature of expert decisions, we generate multiple responses to the same query independently. To balance performance and cost, selected gpl-4o-mini the model and used its API for querying.
4.1 BAYESIAN DECISION-MAKIN STUDY
The first study examines whether LLM responses follow quantal response theory in Bayesian decision framework.
Study Design We use the standard belief-updating task Phillips and Edwards (966);
Grether (1980) as the Bayesian Decision-Making Study. Specifically, large language models (LLMs) are tasked with inferring which of two boxes was selected based on the observed color of a ball drawn from the chosen box. The left box (w = 1) and the right box (w 0) contain red and blue balls in different proportions. The prior probability of selecting the left box Pr[w = 1], and the task is to infer the selected box based on the observed signal, S, The model is queried under three temperature settings (0, 0.5, and 1) to assess the impact decision-making, By discretization, sconario is repeated 20 times to capture stochastic variability, and the decision proportion is calculated the empirical probability of choosing the left box across all repetitions. The details are provided Scction
of model rationality
get 400 scenarios. Each
Examining the Quantal Response Logistic regression analysis was conducted to test whether the decision-making behavior of the LLM adheres to the quantal response model. The dependent variable is the model's decision proportion, while the independent variable is the posterior belief derived from the Bayesian framework. To ensure symmetry in the logistic regression, a symmetric processing approach was applicd: for each observation (posterior, proportion), corresponding symmetric observation (1-posterior, 1-proportion) was added. This transformation effectively doubled the dataset, expanding from 400 observations to 800, with each temperature condition. The regression coefficient. X corresponds to the rationality
8

---

Published as a conference paper at ICLR 2026
level in the quantal response, with higher values suggesting more deterministic responses and lower values reffecting increased randomness.
The results of the logistic regression, visualized in Figure reveal clear trend. As the temperature increases, the coefficient decreases, signifying a transition from more doterministic to more probabilistic docision-making. Specifically, À ranges from a value close to infinity (A + oo) at a temperature of 0.0, to 13.25 at 0.5, and further to 8.93 at 1.0. These results are statistically significant, as evidenced by z-values far exceeding the critical threshold of 1.96 (for 95% confidence level) and p-values effectively equal to 0 for all settings. These findings indicate that LLMs exhibit probabilistic decision-makinn consistent with the quantal response theory. A detailed summary of the regression results is provided in the appendix (see Table D.
t=0.0
t=0.5
t=1.0
Decision Proportin
1.0
Fitted
Decision Proportion
Fiited A 3325
Decsion Progortion
Fifted 8
0.8
0.8
0.5
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
1.0
0.0
0.6 Posterior Probubility
Posterior Probablitty
Posterior Frobability
Figure 2: Logistic Regression Results Across Temperatures. The plots illustrate the relationship between postcrior probability (x-axis) and decision proportion (y-axis) at three temperature settings (t = 0.0, 0.5, 1.0). Each gray dot represents a specific sconario, showing the decision proportion of GPT-40-mini for that scenario. The fitted quantal response curve (orange line) represents the model's predicted behavior, with the fitted coefficient Specifically, X + 30 at t t= 0.0, 13.25 at t = 0.5, and A = 8.93 at t = 1.0. The results demonstrate increased randomness as temperature rises, aligning with the quantal response model's predictions.
quantifying the rationality of decision-making
4.2 MULTIPLE-CHOICE QUESTION ANSWERING STUDY
The second study investigates the behavior of large language models (LLMs) when tasked with multiple-choice question answering, using the MathQA dataset (Amini et al. 2019). This study focuses on understanding the efficacy of plurality vote aggregation for problems involving multiple options and evaluates performance under varying levels of model rationality and the number of experts.
Study Design We evaluate aggregation performance by querying the model with 500 randomly selected questions from the MathQA dataset, under three temperature settings (1=0.0, 0.0, 0.5, 1.0). For each question, we generate 20 responses. Because the questions offer more than two options, This involves selecting the option with the highest frequency among the sampled responses. In the event of a tie, the aggregator randomly chooses among the tied options. Bootstrapping is used to assess aggregation robustness, repeated 1000 times for each temperature and expert group size (n 1,3,5), 5). Accuracy is the proportion of correct aggregated answers. The details are provided in Section B
use plurality voting (frlv), generalization of majority voting,
Evaluate Performance of Aggregators The performance of aggregation methods is evaluated under different temperature settings (t = 0.0, 0.5, 1.0) and varying numbers of experts (n 1, 5.). Specifically, we examine majority voting (f (fma) in the Bayesian Decision- Making Study and plurality voting (fr(r) in the Saltiplo-Choice Question Answering Study using the MathQA dataset. We use only odd values for n to avoid tics.
Across both studies (Figure 3/, when to lack of decision diversity. Deterministic expert behavior (t = 0.0) yields slightly higher utility compared to stochastic settings (t > 0.5), indicating that with only one expert,
1, the expected utility is generally lower due
9

---

Published as a conference paper at ICLR 2026
0.850
0.8650
A-0
1-0.0
t-0,5,
1325
Exolcted Acaraay
0.845
t-0.5
Bioedeed Utilty
0,840
0.8600
0.835
0.8575
0.830
0.8550
0.8525
0.820
0.815
Number Experts
Number of Experts
(a) Bayesian Decision-Making Study
(b) Multiple-Choice QA Study
Figure Performance of MaajorityPPluralle Voting A Aggregation. Panel (a) presents the expected utility of the majority-vote aggregator (fmaa) in the Bayesian decision-making task, where different temperature settings (t 0.0, 0.5, 1.0) and numbers of experts (n 1,3, 5) are evaluated. Panel (b) shows the accuracy of the plurality-vote aggregator (frlu) when applied to the multiple-choice question answering task using the MathQA datasot. In both pancls, error bars represent the standard error of the mean (SEM), which quantifies the uncertainty in the estimated mean. Both studies show the same pattern: increasing the number of experts (n) improves aggregation performance. When = 1, higher temperature
(t) decreases performance due to increased randomness. When n 2 3, higher temperature improves performance as diversity enhances aggregation effectiveness. These results highlight that while randomness degrades individual decisions, it benefits collective decision-making when properly aggregated, which aligns with our theoretical findings.
randomness negatively impacts docision reliability. However, when n 3,5, the utility of majority (plurality) voting interestingly increases with higher temperatures. This outcome aligns with our theoretical predictions, suggesting that randomness can enhance aggregation performance by providing additional informational diversity. Additionally, we evaluate simple random follow rule, which performs equivalently to majority voting with = 1, and find its performance consistently inferior or equal majority voting with larger groups.
These findings align closcly with thcoretical expectations and ensemble learning literature, emphasizing the value of aggregating multiple stochastic responses from LLMs improve decision accuracy.
5 DISCUSSION AND FUTURE WORK
This paper shows that bounded rationality plays a crucial role in shaping decision aggro- gation. Majority voting reaches minimax optimality when group rationality À is below critical Uhreshold (g(n)). In particular, experts with moderate levels of stochastic behavior (X 0o) sometimes perform perfectly rational experts. This phenomenon plained theoretically in robust aggregation framework and is observed empirically in large language models (LLMs), where semperature-controllee randomness improves the accuracy of aggregated decision. These results suggest that introducing imperfections may enhance collective intelligence.
be
While these findings offer practical implications for domains such as investment (where "noisy" retail investors are common) and AI ensembles (where temperature diversity can be optimized), several assumptions require relaxing for real-world applications. Our current framework assumes conditionally independent and identically distributed (i.i.d.) experts, neglecting potential correlations and adversarial conditions. Furthermore, the homogeneous rationality model does not account for groups with diverse rationality. Future research should address signal dependencics through graph-based aggregation methods and explore the impact of heterogencons rationality on decision-making including real-world testing in critical areas like medical diagnostics.
This work is supported by National Natural Science Foundation of China award number 62372007.
10

---

Published as a conference paper at ICLR 2026
REFERENCES
Aida Amini, Saadia Gabriel, Peter Lin, Rik Koncol-Kadziorski, Yejin Choi, and Hannanch Hajishirzi. Mathqa: Towards interpretable math word problem solving with operation- based formalisms arXiv preprint orXie: 13319, 2019.
Ttai Arieli, Yakov Babichenko, and Rann Smorodinsky. Robust forecast aggregation. Pro- ceedings of the National Academy of Sciences, 112133) E12143, 2018. doi: 10.1073/ 1813934115
pnas.1813934112 URL https: /ttw//wwww// pnas http://pww...ggg/o////////......
Itai Arieli, Yakov Babichenko, Inbal Cohen,, and Konstantin Zabarnyi. Universally robust information aggregation for binary decisions. In Proceedings of the 24th ACM Conference on Economics and Computation, pages 118-118, 2023.
David Austen-Smith and Jeffrey Banks. Information aggregation, rationality, and the condorcet jury theorem. American Political Science Revicu, 90(1):34-45, 1996.
Mark Braverman, Jieming Mao, Jon Schneider, and Matt Weinberg. Selling to a no-regret buyer. In Proceedings of the 2018 ACM Conference on Economics and Compulation, pages 523-538 2018.
Leo Breiman. Bagging predictors. Machine learning, 24:123-1483 1996.
Constantin Carathéodory. Über den variabilitätsbereich der fourier's schhen konstanten von positiven harmonischen funktionen. Rendiconti Del Circolo Matematico di Palermo (1884- 1940), (2(11)193-211 217, 1911.
Alessandra Casella, Andrew Gelman, and Thomas R Palfrey. An experimental al study of storable votes. Games and Economic Behavior, 57(1):123-11 11111-11444 2006.
Henrique De Oliveira, Yuhta Ishüi, and Xiao Lin. Robust merging of information. In Proceedings of the 22nd ACM Conference on Economics and Compulation, pages 341-342, 2021.
Thomas G Dietterich. Ensemble methods in machine learning. In International workshop on multiple classifier systems, pages 1-15. Springer, 2000.
Yiding Feng, Chien-Ju Ho, and Wei Tang, Rationality-robust information design: Bayesian persuasion under quantal response. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 501-546. SIAM, 2024.
Robert Forsythe and Russell Lundholm. Information aggregation in an experimental market. Econometrica: Journal of the Econometric Society, pages 309-347, 1990.
Jiarui Gan, Minbiao Han, Jibang Wu, and Haifeng Xu. Robust stackelberg equilibria. arXiv preprint 149999 2023.
Jacob K Goeree, Charles A Holt, and Thomas R Palfrey. Quantal response equilibrium and overbidding in private-value auctions. Journal of Economic Theory, 104(1):247 272, 2002.
Jacob K Goeree, Charles A Holt, and Thomas R Palfrey. Regular quantal response equilibrium. Experimentol Economics, 8(4):347-36 2005.
Jacob K. Goeree, Charles A. Holt, and Thomas R. Palfrey. Applications to political science. In Quantal Response Equilibrium: A Stochastic Theory of Games. Princeton University Press, 06 2016. ISBN 7770691124230 URL httht::://daaa org/10 209..33933220
doi: 10.23943/ 10.23943/princetonn /901/99700001299000000 pttp///etto//7/009111220300
0008
David M Grether. Bayes rule as a descriptive model: The representativeness heuristic. The Quarterly journal of economics, 9533)5537-557, 1980.
Yongkang Guo and Yuqing Kong. Robust aggregation with adversarial experts. In Proceedings of the ACM Web Conference 2025 (WWW), 2025.
11

---

Published as a conference paper at ICLR 2026
Yongkang Guo, Jason D. Hartline, Zhihuan Huang, Yuging Kong, Anant Shah, and Fang- Yi Yu. Algorithmic Robust Forecast Aggregation. Working Poper, 2023.
Yongkang Guo, Jason D. Hartline, Zhihuan Huang, Yuqing Kong, Anant Shah, and Fang Yi Yu. Algorithmic robust forccast aggrogation. In Proceedings of the 26th ACM Conference on Economics and Compulation (BC), pages 1110 1129, 2025.
Qishen Han, Grant Schoenebeck, Biaoshuai Tao, and Fang-Y Yi Wang. The wisdom of strategic voting. In Proceedings of the 24th ACM Conference on Economics and Computation, pages 885 905, 2023.
Eva Hoppe and Patrick W Schmitz. Contracting under incomplete information and social preferences: An experimental study. Review of Economic Studies, 80(4):1516 1544,1544444 2013.
Jingru Jia, Zehua Yuan, Junhao Pan, Paul E McNamara, and Deming Chen. Lim strate- gie reasoning: Agentic study through behavioral game theory. In Advances in Neural Information Processing Systems, volume 38, 2025.
Samuel N Kirshner, Yiwen Pan, and Jason Xianghua Wu. Prosocial when simple and cold-hearted when complex: How task difficully shapes Il behavior. Decision Analysis, 2025.
David C Knill and Alexandre Pouget. The bayesian brain: the role of uncertainty in neural coding and computation. Trends in Neurosciences, 27(12):712 719, 2004.
Yuqing Kong, Shu Wang, and Ying Wang. The surprising benefits of base rate neglect in robust aggregation. 2024.
Gilat Levy and Ronny Razin. Combining forecasts in the presence of ambiguity over correlation structures. Journal of Economic Theory, 199:105075, 2022. ISSN 0022-0531. doi: https:://do.orr//100/ /ttp://o..rr....../.......... 2020. 2020.100075 URI. https: / /ww. sciencedirect. com/
/ttiam///arrrccle//////222223331 and Model Uncertainty.
Symposium Issue on Ambiguity, Robustness,
Richard D McKelvey and Thomas R Palfrey. Quantal response equilibria for normal form games. Games and economic behavior, 10(1):6 38, 1995.
Richard D McKelvey and Thomas R Palfrey. Quantal response equilibria for extensive form games. Experimental economics, 1:9-41, 1998.
Richard D McKelvey and John W Patty. A theory of voting in large elections. Games and Economic Behavior, 511:11155-180, 2006.
Asa B Palley and Ville A Satopää. Boosting the wisdom of crowds based on peer predictions. Management Science, (9(11):6636-666 2023.
Yugi Pan, Zhaohua Chen, and Yuging Kong. Robust decision aggregation with second-order information. In Proceedings of the ACM on Web Conference 2024, pages 77 77-88, 2024.
Max Peeperkorn, Tom Kouwenhoven, Dan Brown, and Anna Jordanous. Is temperature the creativity parameter of large language models? arXiv preprint arXiv:2405.000 00492, 2024.
Lawrence D Phillips and Ward Edwards. Conservatism in simple probability inference task. Journal of experimental psychology, 72(3):346, 1966.
Matthew Renze and Erhan Guven. The effect sampling temperature on problem solving in large language models. arXiv preprint arxiv:2402. 05201, 2024.
Walter Rudin et al. Principles of mathematical analysis, volume 3. McGraw-hill New York, 1964.
Herbert A Simon. Bounded rationality. Utility and probability, pages 15-18, 1990.
Amos Tversky, Daniel Kahneman, and Paul Slovic. Judgment under uncertainty: Heuristics and biases. Cambridge, 1982.
12

---

Published as a conference paper at ICLR 2026
Robert L Winkler. Combining probability distributions from dependent information sources. Management Science, 27(4):4794488) 1981.
Guanghui Yu, Wei Tang, Saumik Narayanan, and Chien-Ju Ho. Encoding human behavior in information design through deep learning. Advonces in Neural Information Processing Systems, 36, 2024.
Yuqi Zhu, Jia Li, Ge Li, YunFei Zhao, Zhi Jin, and Hong Mei. Hot or cold? adaptive temperature sampling for code generation with large language models. In Proceedings of the AAAI Conference on Artificial Intelligence, volune 38, pages 437 445, 2024.
13

---

Published as a conference paper at ICLR 2026
A PROOFS FOR SECTION 3
A.1 DIMENSION REDUCTION
To prove Section 3
the primary challenge is to reduce the complexity of the signal space. Since, by definition the respouse function, the report depends only on the posterior given A, we slightly reduce complexity by grouping signals that yield identical posteriors. We use Sg to denote the set of signals that can be reccived by cach expert under 6. The above obscrvation is formalized as follows:
Observation A.1 (Signal Grouping). Let Sg = (o(1), g()) (.).....s(m) be the signal set of the signal structure 0 Ocid, For any two signals 8(0), ((k) E E with identical posteriors, i.e.,
Pr[w= =
| = =0) | Pr(w = 11 S: = =(()),
we define new signal g(ivk) representing the event S: = زنا) or S = s( That is,
S = g(vM) = S, = 2 S₁ (k)
By grouping such signals, we construct a new signal structure O' with smaller signal set that induces the same report structure A:
AY> 0, rep(0, v2) rep(O', xbx).
After grouping signals with identical posteriors, we establish a one-to-one correspondence between signals and posteriors. This allows represent each signal by its corresponding
posterior value. For instance, a signal s where Pr(w = S; = = 0.8 can be represented simply as 0.8.
Now, let's define 3-signal space signal structure, denoted @ This structure comprises signal structures where the signal contains at most three distinct posteriors:
3 = {0 Geidd Bp (0, 1) such that Sa S 10)
As defined, any @3eid has a signal Sy containing most three clements, cach representing a posterior. At most one of these posteriors, p, can lie strictly between 0 and 1; the others must be and We can parameterize each signal structure E @3eild with three values: p, Yo, and 71. These parameters define the joint distribution:
S. =
S, = p
=
0
70
7111 p)
0
0
PPP
70
7
This 3-signal space structure represents a significant simplification compared to the general signal space. The following lemma will demonstrate the sufficiency of considering only this simplified structure.
Lemma A.2 (Dimension Reduction). For all quantal response rationality levels X O, the report structure induced by the 3-signal space structure is equivalent to that induced by the general signal space. Formally:
rep( ,b))) = G
We now formalize the dimension reduction technique that allows to compress the signal space.
Proof Sketch of Lemma A.2
The proof consists of three main sleps:
1. 3- 3-Dimensional Encoding of Signal and Report Structures.
We begin by encoding signal structures and report structures into a 3-dimensional space. Under the response function thx, each signal s E [0, corresponds to point
v(s) ((s, (s)) (s))
=(Prww St s), Pr|X, = S = =], Pr 1, S; s))
14

---

Published as a conference paper at ICLR 2026
in thrce-dimensional space. Here, xs(s) Pr[X, 1 U.SS
represents the
probability of taking action 1 given posterior s, while (1)(s) = Pr[X; =,, = S s] represents the joint probability of taking action 1 and the action matching the true state of the world. As 8 varies over the interval [0, 1], this mapping traces out curve, which we denote by C, in the three-dimensional space.
Each report structure also corresponds to point in three-dimensional space:
v(0) = (Pr[w = 1], Pr|x, Pr(x; 1, = 1J)
Critically, the point corresponding to a report structure 0 rep(O. vxx) is a convex combination of the points corresponding to the signals in the signal set So of O. Consequently, all possible report structures correspond to points within the convex hull of the curve C.
Thus, to establish the equivalence of report structures induced by the 3-signal space and the general signal space, it suffices to show that their respective convex hulls are equivalent. Specifically, we must demonstrate that any point v(0) within the convex hull of C can be expressed as convex combination of at most three points v(0), (p), (1) on C, corresponding to signals of 0, p, and 1, for some p E [].
2. No Four Points Coplanar.
To prove this, we need the following key result: four points on the curve C are coplanar. We establish this by considering a 4x4 matrix where each row takes the form [1, s, (s),,, stix (s)] for four values of
b < c d. Assuming (p) is the quantal response function, we demonstrate that the determinant of this matrix maintains consistent. sign (either always positive always negalive) regardless of the values of a, b,c, d, which implies the result.
3. Representing Any Report Structure v(0) with v(0), v(p), w(1).
Using the "no four points coplanar" result, and by repeatedly applying it along with the intermediate value theorem (Rudin et al. 1964). first establish preliminary result: any point on the line segment connecting two points v(j), u(k) on C can be expressed as convex combination of at most three points v(0), v(p), v(1) C, corresponding to signals 0, p, and for some p E [0, 1]. We then extend this result to encompass any point in the convex hull of C through an iterative reduction argument.
Proof of Lemma A.2
3-Dimensional Encoding of Signal and Report Structures.
We need to first analyze the relationship between the report structure and the signal structure.
To analyze the report structure â of a specific expert, we observe that three quantitics can uniquely determine a report structure of expert i: Pralw 1), Pra(X; = 1], and Pra(X; = 1,w = 1]. These three quantities can be considered as a point in R We define v(0) as the point in R² corresponding to ô. Formally,
v()) (Pr(w 1). Pr(X, 1), Pr|x 1,w = 1))
Using this perspective, we can establish the connection between the signal structure and the report structure. Consider a signal e [0, 1] with posterior Pr[w = | S = s] = We
represent this signal as point in R using the following coordinates:
v(s) = (s, (ss), ss)(s)
= (Pr[w = 1 S. = (Pr(w = 5; = =]), = 1 S; = sPr = 1| S = =|))
This point represents the report structure when only this specific signal s is considered. Here, Pr[w = S, = s] is the postcrior probability that the true state w = 1 given the signal s,
(Pr[w S. s]) is the probability that the expert reports 1 given this posterior, and
Pr[w = 1 S: = s]rl( = 1| S; = s)) is the joint probability that w = 1 and the expert reports 1.
15

---

Published as a conference paper at ICLR 2026
Given the signal S; the state and the decision X, conditionally independent. This conditional independence allows us to express the overall report structure as a linear combination of the points corresponding to different signals.
Lemma A.3. For corresponding signal structure 0 and report structure 0, the point of report structure 0 for eapert can be represented conver combination of the points corresponding to the individual signals S in the signal structure 0. That is,
v
Σ SES
Pr[S: = s]o(s)
Proof of Lemma A.3 Let be the corresponding signal structure of 0, and Sq be the set of all possible signals that expert can receive. The probability of receiving signal is denoted by Pro[S; = s]. The three quantities defining the overall report structure can be expressed expectations over these points:
Prafw = 1] is the expected posterior probability:
Pr(w 1]
Σ
Pr(S:
s)
b Pr S, = s].
SESp
Pra[X; 1] the expected reporting probability:
Pr[X; 1]
Σ Pr(X; = 1, =
0
sC5p
Σ
Pr[S: = SJU(P|W = 1 Sy =s).
=
sESp
Prg(X: = 1, = is the expected joint probability:
Pr[X; 1,
Σ
Pr[X; = s,
SESy
Σ
Pr(S, Pr|X; 1,w 1| 8 s]
SESg
Σ sES,
Pr[S; Pr[w 1 1, s) Pr[X; = 1 SS =
(Conditional Independenco)
Σ sES
Pr(S, s] Pr|w 1 S spr 1 S. = =s).
Each of these expectations weighted sum of the corresponding coordinates of the points v(s) in RS, with weights given by the probabilities Pro[S. s]. Thus, the point of report structure v00) convex combination of the points v(s).
0 = Σ Pr(S: = s)v(s)
SESp
Define curve C be the set of all points signals in 3R3 that is, C {(s, (())s))())) I E [o, 4]}- Let conv(C) be the convex hull of curve C. Since the point of report structure can be represented as a convex combination of the points in C, conv(C) the set all valid report structures.
According to Carathéodory 1911
is theorem, in R3, any point inside the convex hull of a set be represented as combination of at most points from the set. Thus, we only
16

---

Published as a conference paper at ICLR 2026
need 4 signals to represent all report structures. However, in the next subsection, will further prove that when (ss) is a quantal response function, the signal structures in @ can express all report structures.
No Four Points Coplanar.
Then we prove that, given X O, there no four points coplanar in the curve C {(s, vx(s), s4x(()) E [0. Formally,
Lemma A.4. Given > O, any four different points on
v(s) (s.
Jo, 1])
are not coplanar.
{ b (b)
(a
anix
a4ixx0 (b)
Proof of Lemma A.4 Let M(a
=
1 c x'
anix (c)
1 d (d(d)
ad' (d)
We need to prove Va, b.
(a, 0.
Let B 4), d C, U =
w
B.
x ==0 0
In
h(z)
g0z)
x 0>
0'
In(e - 1) In
x
ze 2
Then we have
b,, >0
2220226420-22 2d 2)
II (2+ dct())
- (d- () 48 e0 (E - e) (d a)(c (e e ) ca8)
- uu( 1e 1) > v(u + v w) 1) (e 00
h(v)h(u + + ww h(u)h(w)
When x > O, we define t = fx > 0.
Then
f'(x) 0
3 38x
8 8z 33x
3x
DC
3
+
re 2
-
2
+ 2
> 0
2
2
3 + Bx
+2>0 0
tol +1 2e +2>0
400 th
h!
Then we know h(u + v + w) > h(u + w) and h(v) > h(0) = B. Since lim h(x) - g(0), h(x) is continuous when 0. Because vz 2 0, hzxx)))))))))))())) > 0.
(o))(u+vut u > h())h(w)
(w)(+))h() 2 h()))((r))
In(h(uu + w)) + In(h(0)) > In(h(u)) In(Www) g(u + w) + (()) 2 g(4) +
Since lim g(x) = g(),, g(x) is continuous when x 2 0. When I > 0, we define Br 0.
17

---

Published as a conference paper at ICLR 2026
Then
("(x) >0
B25
>
x²
(e0z 1)2
(c 1)22 > m cm
em + t > m
t00
2m2 (2k)
> 0
k=2
Then we know g(x) is convex on +00). So g(u + w) + g(0) 2 g(u) + g(w), which means that the theorem is proved.
Representing Any Report Structure v(0) with v(0), v()), v().
In the following Lemmas A.5 and A.6 we will first show that for any convex combination of two points on the curve, the resulting point can be expressed as convex combination of the points v(0), up)), v(1).
Lemma A.5. When dh (p) is a continuous function such that no four distinct points on the curve C = {(s,, (s),, sb(s)) s E [0, 1]} are coplanar. Then, v0 < < P2 < 1,0 5 S 1, s.t.
qo(p1) (1 ()))) v(p), v(0), v(1) coplanar.
(1)
There only one p [0, satisfies eq.
[1] and p E [21.Pe]
1 (a((a
arlx (a)
1 b (b)
(b)
Proof of Lemma A.5. Let b(,
1 c (c)
(c)
1 d (x(d)
an)
Since no four distinct points in the curve ((s, (8)), 81{()) E [0. 1}} are coplanar, we have for all distinct a, b, c, d E [0, 1], with 0 S b < d 1, that the determinant of the matrix M (a, b, c, df is constant sign:
det(M(a, b, c, d)) > 0 or dct(M b, d)) < 0.
This means that the determinant is either always positive always negative.
(0.0P..2. P1. x, 1)) + (1- - q) (M p2, x, 1)). The signs of h(x) on the intervals (0. p1 and [p2, are opposite. According to the Intermediate Value Theo- rem Rudin et al. 1964)) because h(x) is contimuous on [0. 1] and h(p)) x h(p2) S 0, there exists p [Pi, Pa] satishes h(p) O. Then qu(pi) (1- (p))) v(p), v(0), v(1) coplanar. If there exists another p' + satisfies eq. 1 then v(p), v(), v(1), o(p') are coplanar, which impossible.
Then we define h(z) = q (ed((M()))))) (0,
Lemma A.6. Let 0 < P1 P2 < 1 and < 4 S 1. Then there cxist functions p. y. R3 +R R such that the following conditions hold:
qu(p1) + (1 - g)v(p2) = = () P2, q)) + y(1..P,, ())( + =(1 (1) with 1111. P2, + y(1, P2. 9) 2(1. 1, that is,
(p(p)) + (1 () (p((1 P2, 9)), v v0), x(1) are coplanor.
And the following inequalilies hold:
x(p1, P2. 2 0, u(P, P2, 2 0, =(P1,P2, 2 0.
Proof of Lemma
A.6 We first prove that s(p1,P2 P2, 9) 0.
To show 2(p1, P2, q) 2 O, define po(g) p(p1, P2. and zo(g) x(pr, P2, 9). Consider the function:
= (e PL, 1)) (1 4) (0 1))
18

---

Published as a conference paper at ICLR 2026
Since h(pu(g), 9) 0 and h(s,g) is continuous [0, 1]2, we have: for any q"
lim h(po(g). 9) = h( lim po(g). q") = 0. 4-
This implies li Po(g) = po(a"). which means Po is continnous. Next, we express zo (g)
0(0)
det qo(pi) + ()v(p2)
v(1)
z0(9)
v(0)
det
(pp( v(1)
Since the determinant. in the denominator is non-zero for distinct pr and p2, xn(4) is continuous [0, 1]. Given that (0) = and 2o(1) = if zo(g) < 0 for some q E (0, 1), by the Intermediate Value Theorem (Rudin et al., 1964,, there exists such that zo(() = 0. This leads to (0), v(p2), v(1) being coplanar, contradicting the condition.
Next, we will prove y(p,,P2,9) 1) 2 0.
When q 0, we have p(p1: P2, P2, PP,,,P2,91 P2, q) 1, and y(p,, P2, 9) P2. 0. When 1, we have p(p1. P2: q) = Pt, x(P1, P2. q) = 1, and y(P ,2,,,22,4) 9) = z(P,, =.
For 0 < q < 1, we consider the case that pl
eq.
then p = p2-
So we can define p(0, (0,,2,9) 4) = p2, x(0,22, v) = =1 -1.4.00. = 9. =00, = 0,
and have 4A Pt P2 1,0 < q 1, R RR 8.l. gp(p))
(1 (p2( = 2(P1,P2, (p(((pL.P2.0) + plP..p2.g)e((0) 2pI. xt) P2. p(P..P2. + =P1,.22.)) =
Then we fix and p2 and define:
Po(pi) = Mp..P2.00 y0(p1) = y(p1.22.9) 92.9).
We then analyze the function:
p(s,Pt) (0 8, 1)) (1 2) det (M 8)).
Since h(po(p)). 0 and h(s,pi) is continuous, we have:
lim h(po(P1), = h( lim (o(P)), = 0.
P1
pi -i
This implies limp, - Po(P (o(p)) means po is continuous. We can express vo(p1)
(go(p)) + (1 -g)((p2) ((po(p))) v(1)
det
(o(p))
v(0)
det (ppo(p))) v(1)
Thus, (p1) is contimuous on [0, 1). Given that (0) = 2> > 0, if (u(p1) < 0 for some Pi E (0, 1), then by the Intermediate Value Theorem RRddin et al. 1964). there exists t such that (t) = O. This leads to w(p), v(pi), 0(22), v11) being coplanar, contradicting the condition.
Finally, we will prove =(1,P22, 4) 2 0.
When q we have p(P). P2. P2, e(11, q) and y(pi. 9) 2(p), P2. 9) = 0. When 1, we have p(p), = Pl, x(pi P2, g) = 1, and P2, 4) = (p)) P2, 4) = 0.
For 0 < g < 1, consider the case that P2 in eq- Then p == P1-
19

---

Published as a conference paper at ICLR 2026
So define p(21, 1, P2: z(p1.1.9) 1, 0. =(11,1,9)
and we have v0 < pl < P2 S 1,0 < < 1,3p, y. 2,2 R2 + R, s.t.qu((p))
(1 - ()p(p2)
z(P P2. q) (((pppp.. + (2.1.22.9)0(00) + 2()) y(21,P2,0) 9) + =(p1, P2, !) =
Then fix q and pr and define:
po(p2) p(p1.p2.0) z0(p2) = =p1, 9)-
We analyze the function:
h(s, = q de 1)) (1 q) de p2, 1)).
Since h(po(p2), p2) 0 and h(s, is continuous, we have:
lim (PP))22).22 = h( lim po(p2).16) = 0. p2-p2/
P2 p2-
This implics limpa- ppp Po (p(2) (o(p)),whicl means po continuous.
We can express 20 (P22 as:
v()) (PPo(P2) =0(p2) = (p1) + (1 (p2)
det
v(0) det (Po()))) v(1)
This shows that (p(2 is continuous [0, 1). Given that 20(0) =1 9> 0, if 20(22) < 0 for some P2 E (0, 1), then by the Intermediate Value Theorem Rudin et al. 1964), there exists t such that (t) This leads to v(p), v(p)),((p)) v(1) being coplanar, contradicting the condition.
Thus, we conclude that s(v1, p2, 9) 0, y(p,,2,,1 P2, ②) 2 0, (p,. 0.
Finally, prove that rep(O, v/x) 0, where 0 Aaeiid and Ô Ôgitt, defines bijection between the sets @deidd and @ We first verify the surjectivity the mapping:
vo dxin 30 33iid such that rep(O, v) -
yorid E Gridd 303rid O aniid such that r0piiddidd w) = rep(Oriid v)
Aocid E Gotid 303did E θ 3eiid such that Vi E [n],
Σ Pr [S s)v( Pr genid
w = 1 S; s))
Σ
Pr [S, s)u( Pr [w 1 S = s)). @
Hell
gaesid
SESp
#E (0.p..1)
In order to establish surjectivity, we first show that every signal set can be reduced to canonical form, as captured in the following lemma.
Lemma A.7 (Canonical Form Transformation). Let
= = (1)) (2)) ( s....,,ss!! (
be a finite set of signals. Then there erists an iterative transformation that maps Sa to the cononical signal set {0. p, 1} with
Pr(w 0 Pr(w=1 = 1 S, = = and Pr[w= 1
0
Moreover, this transoormation preserves the equivalence of the corresponding poinl, that is,
Σ Pr(S: = = |S, = =10))
Σ
Pr(St = x] =))
(0.p.1)
20

---

Published as a conference paper at ICLR 2026
Proof of Lemma A.7 We prove the lemma by describing iterative procedure that trans- forms Sg into the canonical form {0.p.1}.
Step (Group Identical Posteriors):
For any two signals 20)) and s(h) in Sg satisfying
Pr(w 1 S = 1
-
we can group these signals by summing their marginal prob- abilities. Thus, without loss of generality, we may assume that every signal in So corresponds to unique posterior probability.
Recall theorem A.1
Step (Select and Replace Pair of Intermediate Signals): Choose two distinct signals s6) and ((k) with posterior probabilities
0 < Pr(w = 1 S = 200 < Pr[w = S; = =s( < 1.
Define
P1 = Pr(w = 1 -1 and Pk: = Pr|w 1 S = =((k).
By applying Lemma A.6 there exists unique intermediate value p" [pj. pk] and nonnegative coefficients x, v, (satisfying + y Pr(S, = s0)) + Pr(S, (k)] such that
Pr[S: 5 v(ps) + Pr[S; = s14)] = x + (v0)))
In this step, the pair (80), s } is replaced by new composite signal corresponding to the intermediate value p", while the signals corresponding to 0 and remain intact.
Step (Iterate Until Canonical Form Is Achieved):
Replace the pair (80), s(k) in Sp with the new set of signals {0,p", (noting that the signals corresponding to 0 and 1 may already exist in Sg). Since each replacement reduces the number of signals with posterior probabilities strictly between 0 and and because Sg is finite, this iterative process terminates after finitely many steps. The final signal set will therefore be in the canonical form
{0,p, 1}:
where p is the unique intermediate posterior (if any) between 0 and
This completes the proof.
Lemma A.7 demonstrates that for every report structure Ô Ggiid there exists equivalent signal structure structure 0 E @leidd satislying
rep tx)) =
Thus, the mapping is surjective. Next, we verify the injectivity of the mapping:
40, 0' 3erld
rep(0, vb) rep(0', (1) 0= 0'.
21

---

Published as a conference paper at ICLR 2026
Since Profw = 1 S = 0] Præ ( 1| S; 0] and Pro(w 1 S; 1] Prø (w
S = = 1, we have
rep(0, 16) rep(0', 1p) Σ
Pr(S: si (Prlw 1 s:)) Σ
Pr(S; st)o(Pr(w 1 | s.))
8
8'
1.(.0.).1)
(.E(0.p.1)
Σ
Pr[S; =
= 1 S = si))
Σ
Pr[S: = s:Ju(Pr(w
1 S = s:.) = 0
0
0r
6"
p.(()p.1))
#EE(0.p.1}
(Pr(s: = 0] - = ()))()) + (P:(5: = 1| Pr(S, = 1))m(1)
Pr(s:
+ Pr[[, = pl - 1 S = p - Pr(S: - plu(Pr = St pp) =
0'
Pro|s: 0] Præ [S, 0] 0
Prol[S: = 1] - Præ [S; = 1] =0
[0(Pro -W 1 S; pl) v(Pro 1 -pl)
Pro = = ) PralsS pl 0
Pro|SS pl Præ (S. pl =0
-
Together, these results confirm that 0 rep(0, vx) is both injective and surjective, thereby defining bijection between 3 and G
Thus we proved
rep Scisd
3oitd ,xxx) = Ogidd
A.2 OPTIMAL ROBUST AGGREGATOR
We start to identify the optimal aggregator when experts exhibit bounded rationality. We prove that the simple majority voting is the optimal robust aggregator under certain conditions. In particular, for a given number experts n, when the rationality level À is below certain threshold ( the majority voting fmas minimizes the regret. The threshold function g(n) (for n to 20) visualized in fig.
2.64
g(n)
1.74
1.38
1.18
1
1.04
0.95
0.87
0.81
0.77
5
7
9
11
13
15
17
19
Figure 4: Threshold g(n) vs. Group Size n (n=33 to 20) The plot begins at n = 3 because when < 2, the majority voting is always the optimal robust aggregator, independent of X. For n E [3, 20), g(n) decreases with increasing n, with every even n satisfying g(n) = g(n 1). These properties indicate that as group size grows, the bounded rationality threshold becomes stricter, representing lower rationality level to preserve the optimality of the majority voting rulc.
Intuition for the Sike Threshold g(n): As observed in fig. the threshold function exhibits step-like behavior where g(2k) g(2k 1) for any integer k ' 2. This occurs
22

---

Published as a conference paper at ICLR 2026
because the expected utility of the majority vote aggregator f is mathematically identical for a group of 2k experts and a group of 2 - 1 experts. Specifically, for an odd group of 2k - 1 experts, no ties are possible. For even group of 2k experts, ties are broken by a 0.5 random coin flip. The decision process for 2k experts is equivalent in expectation to randomly removing one expert and taking the majority vote of the remaining 2k: 1 experts. If the 2k experts are not in a tie, removing one expert does not change the majority outcome. If they are in a perfect vs. tie, removing one expert results in 2k group whose majority vote will be '1' or '0' with 0.5 probability, matching the tic- tic-breaking rulc exactly. Since the expected utilities are identical, their optimal rationality thresholds g(n) are also identical. Conversely, this symmetry breaks when moving from 2k to 2k + (i.e., g(2k + 1) & ((kk), because a near-lie (k + 1, outcome is deterministic for 2k- + 1 experts, but removing an expert can force a stochastic tie-breaking outcome in the 2k case, causing their expected utilities to diverge.
Proof Sketch of Theorem 3.1 Optimal Robust Aggregator.
The proof consists of two main steps:
1. Pairwise Optimality of Majority Voting.
We aim to show fmaj is the optimal solution to the following optimization problem under certain conditions:
pmaj E arg min max R(S,0). dg@eid
A direct approach to showing the worst-case regret optimality of the majority voting, fmaj, would involve maximizing over the entire space of possible report structures. To avoid this global maximization, we will employ approaches that consider each signal strueture individually.
One such approach would be to prove the stronger condition that fmajj is universally optimal, meaning fmajj E arg miny R(S, 0) for all report structures 0 E @ While this would immediately imply worst-casc regret optimality, fmai is not, in fact, universally optimal.
Instead, we establish a weaker, yet sufficient, condition. We prove that, under certain conditions, for any pair of symmetric signal structures and 0' within the restricted set C Scidd fmaj minimizes the sum of expected regret:
pmaj E arg min (R(f, vx)) + r(f,rop(0" bx)))
We then demonstrate how this pairwise optimality implics the desired worst-case regret optimality of fmai, effectively avoiding the need to maximize over the entire space of signal structures.
2. Regret Minimization Condition and Threshold Derivation.
First, we expand the formula R(f, rep(O, tbx)) + R(f, rep(6', xbx)) so that we can analyze each term of f(x) separately. Then we proved fmass minimizes the sum of the expected regret and only
n 1
|
Pr
w w-1 X
< 0.5,
2
where X is the number of experts reporting 1. This further simplifies the question. Finally, derive the threshold function g(n) given the above condition. We prove that when S g(n), the majorily voting minimizes the sum of the expected regret under the pair of signal structures (0,0% 0'.
Proof of Theorem 3.7 Optimal Robust Aggregalor.
Pairwise Optimality of Majority Voting.
23

---

Published as a conference paper at ICLR 2026
Consider the following symmetric signal structures 0, 0' A with n experts
S 0
S:=p
S =1
w 0
(1 - (1-1)(1- - (po)
(I 4opo
1
μp1
111 P1)
0'
W 0
pl1 - -pI)
up1
0
W
(1 plpo
(1 (1) Po)
Let p
HP1
We define
pp. (1-(ppp p)ps
= Pr(x; = w = = (1-p)(((()) - Po))(p);
Q1 Pr[X, 1 w= = (1- (i)xx)) (p))
and we have the joint distribution of the two report structures
=
X;
w 0
(1 - (1 qu)
(1 - 14
[1(1 Q1)
paq
0
p=1 91)
w
(1 - 10)40
(1 11) go)
Since the signal structures 0, 0' are odd symmetric, we have
Uoppg 0) U(oplg, 0*)
Thus, to find the best response to {0, 0') minimizing the regret is equivalent to maximizing the utility, i.c.,
arg min max R(S,0) (0.000))
arg min max U(opts, 0) - (f, 6E(000))
arg max min U(
f dE(0,0")
Then, we need to prove fmaj is the best response to {0, 0'}, that is,
min U(fmaj, 0) max min U(f 0E(0,01
de(0,0))
Regret Minimization Condition and Threshold Derivation.
We consider mu (U, + (ff,''))
U(S,))+ U( = Σ
(2f(x)- ())()) (0 =x xww=
z=0
+
Σ
(2f(x) -1) pr = =-== 2f(x)) pr x, =0)=
(1)
=1 + 2
Σ
x f(x) (Juff(1 n-= (1 -.)//==(( =1 90)
-1-1)g6(1 yo)n== Lnd 1 - (1-1))²)
Next, we will prove that whon fmai maximize the utility, majority voting is an optimal robust aggregator. Specially, since and Ø' are odd symmetric, we have Uffmaj, 0) = U Ufmajj amaj,67) 8¹). Thus prove
24

---

Published as a conference paper at ICLR 2026
Lemma A.8. If for any signal structure 0 E A the majority voting frnn satisfy smaj E arg m) U(U,)))
Then the majority voting fras is a best response to @ i.e.,
pmas E arg max min U(f
degoid
Proof of Lemma A.8
min U(S0)) U(f,0) + U(F,0) (E(0,0")
2
(U 0) + (ff.
max min
U(f < max f 0E(0,8'
f
max min U(f U(mmaj 0) + (. 0'
I (E(0,0))
2
max min Uff S min (E(0,0')
U(fmas, 0)
f (E(0)))
max min U(f f (0,00))
min Uffmaj .0) 0E(0,07
(optg 0) max min f 0Ef0,0"
(ff 0) ('pptto 0)
min U(fmas, 0)
0E(0,0)
max min R(f 0) (E(8)))))
min H(fmajj .0) (E(000')
fma) arg min R(f,0)
0E10,07
Finally, we prove that when X < g(n), the best response to Acidd is pmaj,
R(S 2 max 40, max oEBaid
R(J, O)
0E(0,0")
vO, min max f dEgrid
R(f 0) 2 min max R(f. 0) f (E
v0, min max deanid
R(f,)) 2 R(smaj 0)
min max
R(f,0) <
max R( pmos 0) DE
f dgeeii
min max
max R( pmas 0) Bggend R(S00
f eseend
pmaj arg max min U(f
f
Let f E arg
maxx (U
+ U T();)))))((((>))==)(()))) = ( 1))0 '(1 90)= (1
1) (1 -go)"== pap = q1 ); we have
0,
T(x) < 0, T(z)
(x)
any value in [0, 1].
1,
T(x) > 0.
25

---

Published as a conference paper at ICLR 2026
Note that μaq (1-11) are odd symmetric with respect to the midpoint i == n 2
91) -=)aa==- (1 -
90)" and (1 -1)0( (1 -00"~~= "q""""("" 91)²
Thus, we have
pmat E mrg + U(S,0)
if and only if for all integer < n. the incquality
ugi -2) q=)== + (1 - 10 qo)" S (1 :2)0 (1 20)== + uqq 14qii(111 - 91 )a
holds.
To formalize this condition, we begin with the following lemma.
Lemma A.9. go S q1, then fmas E arg (UX,((UU))))UU((())) iff the following inequality
holds:
((91(1- - ( 2qn) s (1 - (000 -1))((1-))))+ (1-2go).
Proof of Lemma A.9. To establish the equivalence between the condition in the lemma and the statement frany E arg max,, we need to prove the equivalence of
For all integer i i< n 2 pof (1- + +(1-1)06 1)96 "(1----1- =0 s (1 (1-w) 11- = n- 100-0111100 (1-
and
((an(1 - 11)) - 2n) S (1 - (g) - (0)) (1 - 2go)
Let I(n) 1, is odd
then we have
2, n is even'
u(g1 (1 (1)) ( - (1 241) S (1)
-200
p(gi (1 - (1)) ( ((1 - )1) ) (m)
[(m))
S (1 - () q0)) 1 ((1-001)) -q0) %
[(m)
41
pal | (1- - q1) | + (1 4440
1+1
(1 (00 1
s (1 4)90
(1 9o) البلة
[+
(1 91) ال21
# (1 91) [n)*( 11-91
1-n < (1 p)g0 2 (1 go)
4 1-
n is even
90
1- 00
uni (1 q1)
30 1-9
1-91
< (1 p)go
3
(1
90)
1 4n
n is odd
21
m
go
When i < 2: 0 < 90 5 11 5 0.5, we define to =
1-90 90
4
1-91
7 - i. We know
9:
1 1 < to. And
pa (1 q1 (1)r*== nース + (1 "0/0o"(11 90)= S (1- - m)g (1 n~= + + =11 91)²
- ووم # (1 41) (t - ( < (1 - -1990
4 (1 90) 3 (to t)²
(5)
When is even, we know 0 < وو (1 g1)
(4 4 s (1 n2 v - (1o 1o
26

---

Published as a conference paper at ICLR 2026
Since
t f
22
112
1541-2k
2k
1 2
1
0
k=1
2h ---11
k=1
to to
40 to
eq. holds.
When is odd, we know 0 < pap "
(1 q1 (t7
(1-))) (1 20) 8 b1.)
Since
1 17 8
1 4
1 bbott)
2 k=1
+ 1o
2
10 10
t to
eq. 5 holds.
When 0.5 & qu I 91 S 1, we use (4, q1) to replace (1 (1-AA n, -2, 9o). The equation is completely the same.
When qn 5 0.5 S Q1, we have
n
pgf (1 - (1)" n-² S u0/ (1 - )1
Vi
2 (1 u)g0 (1 40)a < (1 u)gØ (1 g0)===
Vi
n
2 ud (1 -1)===- nー= + (1 1)/00 2 (1 -0000 (1 11)6(1 -0)=~ + 11p==((1111000 91)" )5
And let = اااحي J, we have
/tt (1 911) n- + - 1
(1 - q0) f= < (1 - 1) (1 - qo) "ー + 141 "1 91)) )"
udi البية (1 91) | - (1 11)0
a+1
(1 90) 1
< (1 (1- ا (1 90) 1 uh
ا
(1 q1) 1''''
Thus, the condition always holds.
27

---

Published as a conference paper at ICLR 2026
The above lemma characterizes the condition under which the majority voting is the optimal robust aggregator in terms of the quantities qu and 91. In order to connect these conditions with the parameter A, we now provide a reduction result.
Lemma A.10. Given n, X, we have
(1-2q)) Vp, Po. P1 E [0, 1], (1q1(1 - 91))) (1)) 201)= (1- - ()0
(1- -
-0))((1))))))==== - (1-90))
Vgo: Qu such that (0) < 90 S 91 S 0.5, the following inequality holds:
(11 91)) (1) Q1
(1-211) 2q1) < (go(1 -
(10)) - (1 - 24o)
2 + In(1 (1 - ς0) - Im((0)
4x(11 20
2> In( 1 40 In(g))
Proof of Lemma A.10 Since qu S 4 (p) S q1, it follows that qu > 91 is impossible.
Consider the case when qo < 0.5 < q1 We have the following inequalities:
[1(2 - 91)) )))) (1 2q1) 0 (1 (1- (1) 20) 2g0)
(1)(1-91))1 = ((1-2 0 < (9=(1_ (1-20)))))))) x 2- 1-
2A+lm(1-90)
(1(1)
-1(1)-90 -00
Next, we consider the case when qo S q1 S 0.5 or 0.5 90 S q1. Define the following:
n'pp
n' = 1- - ui p6 pb, pi = po, 1' = n'y' + (1 - =1')r''
d = (1 - -6))x + (p') of =1 (1--1) (1) (1)++1
We observe that %6 = 1 - a1 and 11 1 - 20- Thus, we have
Vo. P1 E 10. if 0.5 S qu S q1 s (1)),
(( - -
(0011122))))))) S (1 - ()((0(1 (1-1)(9001 - q0)) - 290)
You, ②1, 14, Po. E 10, 0.5 9o I 91 I (1(1),
(1-
(1)1(1))) (1-20)))
241) S (1 - (111)))6(11111 - %6)) ( [1 226)
10. if (0) S %0 S )1 S 0.5,
u(q)(1- ()) 2q.)= s (1 - () - = | (1 220))
Thus, we conclude that
Po. E [0, u(gi (1 .1)) - 2q1) S (1 - () - -q0)) (1 -20o)
Yp. E [0. if (0) S q S 9: < 0.3,
we have u(q (1 (1) 241 < (1 p) - ( (1=) (1 -200).
When qu- = (0), we have م = 0, p1 = 0, or pu = 0.
If p 0, then
u(qi (1 91 )) 1 - (1-2 0 (1 (1 -200).
If p1 = 0, then qı = vbx(1) (1). Thus,
p(q (1 )) 1 (1 10 2411)
- ( -qu)) (1 - -2q0).
If Po 0, then p and 91 = (1)) Thus,
p(q (1 91)) " (1 - 241) s0 p) - (1-
(1 - -10))(1
-200).
Therefore, when qo = (0), we have
(1 )) " " 1(1 2nn S (1 - (1) - -qo))("²²" (1 2go)
28

---

Published as a conference paper at ICLR 2026
Combining all cases, we conchude that
Po. E [0, 1]. plqn - (11)-))))))1111111110111111)))1- 2q.) - (1 (1- p)(90(1 (1-))(o(111 - -00))( (1 - -2go)
Po. E [0. if (x) < 20 2 00.,,
we have ulqı (1 - (1) - 2q1) - (1 p)) --0 (1 290).
Since 4y is strictly monotonically increasing on []. for x0 9u S 9 S 0.5, we know
p E T (qo). 35 (q1),, where 55 is the inverse function of thy For all (0) ^ 90 14 91 S 0.5 (go),
and p E (go)) 2x (q:)), let
(1) 91 (1 - (p))
90 - (0) 4x(p) (0)
PoP
P1
Po
/
PoP + Pi p)
Then n4 4.PoPP E [0, 1] and
p =pi-(1-a)po²
1P:
90 = (1 - (POW>())))
Q1 = (1 - (1) + (ID)(p).
Therefore, we can conclude that:
Yp, Po, Pı E [0, ((n
11 (1-2g) S (1-x))6 (1-))<1))(00))) (1)))) - 220)
You, such that 40) < 90 S 11 0.., the following incquality holds
Vp E (5T + (q)),, (lqn - ()) - ( 20r S (1 - (1) - ( (1 220)
Finally, we define hx)
2h - -hk
Since 40 = - -p) + (o)a(D) and q1 =
(1 (1-P1) (11))exx(1)- + (1)) we have
x
Po = 90 - 1/ (0)
(1(1) 91
4x (0)'
+x(1) 4' (p))
Then,
p
P
JP1
(1 P)PP
(1)x(1) an (p)x(()) - (O
(1 10(200 VA (O)) vVx())
So,
90 (0)
2M(1-)) - -2((1p))
p
(1( 91
1-P
22pp - - 2Ap
90 1/1 (0)
h(1 - p)
(1) 91
h(p)
29

---

Published as a conference paper at ICLR 2026
2(2
h strictly monotonically increasing on [0, +00). Because R is strictly monotonically increasing on [0, 1] and qu S (x(p),
Since h(x)
E
(2k+1J]
90 x0)
h(1-P)
(1) - q1 h(p)
90 - x0x(0)
x h(1 (90))
(1(1) 91
h(05 ((o))
(1x((1) - (2 - 1n(1 - 40) + In g0
(1x(1) (2A In(1 40) In 40
Especially, when
(go); (go);
(1bx - (2 In(1 - 40) + In 70
1
1 1 (vx(1) - ()) (2X In(1 40) In 40)
So,
Mp, Po. Pı E [0,1], p(qi - -- 22g1) s (1- ()) - 9n)) = 290)
Yqu Qn such that ()) @u $ 91 s 0.5,, the following inequality holds
Up E (1o), a5 (2:),, p(g - -2)) | 2n) s (1 (2) (1 qo)) الل (1 24o
Yga qi such that (0) go S 9) S 0.5, the following inequality holds
(01 (1 11) -2n) (1o(1 10)) - 240) 2A + In(1 40) - In(40)
x 2A In qu) In((g))
(1) - q1
(1 qu
Lemma A.10 is crucial because it reformulates the condition from Lemma A.9 in terms of the parameter À and the transformed quantities (0) and (1). This reformulation paves the way for defining threshold function g(n) that governs the optimality of the majority voting.
Before generalizing the case 2, we now examine the special scenario when n S
Lemma A.11. Given n S 2 and X, if qu, satisfy vb) qu S 91 I 2 the following inequality holds:
(q q1)) J(1 291) (go(1 - (7" ( 2go)
2A + In(1 go) In go)
x
(1x
(1x 40
2 In[1 9o) + In(go)
Proof of Lemma A.11 Recall h(z)) Σt +oo 2(2 26
(2k+1T) in the proof of Lemma A.10 h is
h
strictly monotonically increasing on [0, +00). Because thx is strictly monotonically increasing
30

---

Published as a conference paper at ICLR 2026
on [0, 1] and qu S xx()).
1
90 x5 (0)
h(1-p) p)
(1 91
h(p)
90 (0)
X h(1 - S/ (90))
(1(1) 01
11 (1o) (10))
(1x(1) (2A - 1n(1 qu) + In
(1) 91 (2A + 1n(1 qo)
In 40
(1)
(1) 91
0.5 90
0.5
1 - 2/o
1 291
Thus we proved y(1 2Ω1) S (1 - 10)(1 - 2go)
Lemma A. confirms that when there are at most two experts, the transformed inequality always holds. This simple case is important for understanding the behavior of the system in small dimensions. With these insights, we now turn to the more general case.
Lemma A.12. Given n > 2 and X, let
g(n) = ssup { x ygu, Q such that 1bx (0) S qo S 92 S 0.5, the following inequality holds:
(g1(1- - --)) 7 ((1 221) (go(1 - 90)) 7 (1 2go)
2A + In - go) In(g))
xxx(1)
(1) 40
2X - In(1 - (00( + In (o))
Then, when A S g()), if go, satisfy (0) < 90 S Φ1 - 1, the following inequality holds:
(1=(1-9))) - (1)g))=)++)1 291) (go(1 - 90)) 1 (1 2go) x1x 90
2A + In - go) - In(go)
x
2/a (1) 91
2 In go) In(go)
Proof of Lemma 12 Note that the term (1)-9 - strictly monotoni-
(x(1) t0 2X- 100on(l10)=+o(n))
cally decreasing in A. Therefore, for any Ao < X1 (n), if the inequality holds for À = A
with parameters (n, p. go. q1), it will also hold for Ao with the same parameters.
Lemma, A.12
introduces the threshold function g(n) for the general case when n. >> 2. This function serves as critical cutoff: long X S g(n), the inequality (and hence the optimality conditions) established in the previous lemmas will hold.
Collecting the results from Lemmas A.9 A.10 A.1l A.12 and A.8 we conclude that when < g(n) the majority voting maj is indeed optimal.
This completes the proof.
Numerical Results. To demonstrate the relationship between regret and rationality level X, conducted numerical experiments and calculated the regret of majority voting and optimal robust aggregator when the number of experts n = 1, 5 and is in [0, 5], as shown in Figure 5 Regret is the maximum regret over all report structures induced by all c.i.id. signal structures. The optimal robust aggregator is computed using an online learning algorithm Guo et. al. 2023
31

---

Published as a conference paper at ICLR 2026
n-1
n
n
Optimal Robust
Majority Vate
Regret
g(n)
Regret
Regret
0.3
0.3
0.2
0.2
6.1
0.1
0.1
Figure 5: Regret Comparison: Majority Voting vs. Optimal Robust Aggregator (Varying Rationality and Group Size). These plots illustrate the relationship between the rationality level (on the x-axis) and the regret (on the y-axis) for three different group sizes: n 1,3, and 5. Regret is the maximum regret over all report structures induced by all c.i.id. signal structures. The solid curves represent the regret incurred by the majority voting, fmaj, and the optimal robust aggregator, opt The dotted line indicates the threshold function g(n). When the performance of majority voting matches that the optimal robust aggregator (i.e., their regrets are equal), majority voting is optimal. With single expert, any aggregation rulc that follows that export's 's decision, including majority voting, is trivially optimal. As À increases, reflecting increased rationality, the regret decreascs. When and À S g(n), majority voling remains optimal. However, the threshold g(n) is not tight; majority voling may still be optimal even when À exceeds g(n). Nevertheless, for sufficiently large X, majority voting is no longer optimal. Furthermore, as À increases, the regret of both majority voting and the optimal robust aggregator initially decreases and then increases. This suggests that moderate degree of bounded rationality can, in fact, improve aggregation performance.
3
and
A.3 BOUNDED RATIONALITY ADVANTAGE
This subsoction explores the impact of varying levels of rationality among experts on the aggregation outcomes. Specifically, we examnne whether bounded rationality can sometimes lead to superior results compared to full rationality.
Proof Sketch of Theorem 3.1 Bounded Rationality Advantage.
The proof consists of three main steps:
1. Single Expert Case
In the case of single expert, the optimal aggregation rule is to follow the expert's 's decision regardless of rationality level Hence, the overall utility of the aggregator is the same the expected utility of the expert. Since a single expert attains the highest expected payoff when behaving in a fully rational manner, follows that for any signal structure 0 @rid, the maximal achievable utility is obtained under full rationality + Co.
2. Exo-Expert Case (n=2).
When n = 2, we use the example mentioned in the introduction as the specific signal structure 0- Under full rationality (A oo), experts always report the same value (here, X; = 0) and the optimal aggregator yields a utility of 0.5. In contrast, when experts exhibit bounded rationality (with, say, À = 2.5), their reporting behaviors differ, and we obtain
U (Ophoop(or (0 rep(0", V/ v2=25) R 0.507674 > 0.5 = U Uoporeepp0 sh sax- (0p( (a-n)))
This establishes that for n 2 bounded rationality can lead to improved outcomes.
3. Extension to > 2 Experts.
Finally, we extend the argument to larger number of experts under 8* Still, under full rationality (A - xo), experts always report the same value and the optimal aggregator yields a utility 0.5. For odd n, we prove that the majority voting yields utility striclly greater than 0.5 when experts exhibit bounded rationality. For
32

---

Published as a conference paper at ICLR 2026
show that the expected utility of majority voting remains the same in the odd case (n-1). Combining these two parts, we conclude that when n 2 2, bounded rationality leads to superior aggregation performance compared to full rationality.
Proof of Theorem 3.1 Bounded Rationality Advantage.
Single Expert Case (n 1).
We begin by handling the = 1 case.
Lemma A.13. For 1, for ony signal structure @cidd and any rationality level X, holds that
max U (f, rep (0, xx)) U (pmas, rep (0, x=x)))
Proof of Lemma A.13 For the aggregator f depends solely on the report X; of the single expert. For any signal structure 8, we have
U (J, (0, vx)) - vss) vA) u([(X:),
Σ
PPr) = =1 115;==s|-11
si] -1) (2Pr|=-1|s: =
S; - ss ss] Pr]s s.]
S
2Pr 1 S s.] 1 si).
Recall that the response function of rational expert is
1, > 2
Sb (p) =
0.5, = 2'
0,
p 2
and the majority voting is given by
mma) x.
Thus, the utility of fmaj under rational
U (mmas rep (0, )))+
Prrww = 11 1 = =s:1) -1) 2PP = 1|S; S = s;] 1-l-1 Pr(S: = s.]
= Σ
200
(
8, E.
Σ
2Pr(w = 11S, sie - Pr[S, =s.
s,SS,
This immediately yields the desired incquality.
Two-Expert Case (n = 2).
Next, consider the case 2. Define the signal structure 0* with the following joint distribution:
=
S, =1
0r w = 0
3 W
3
0
so that
Pr[w 1 S 0 = 0, Pr|w 1 1| S 2
Under absolute rationality (A + oo), the experts always report X; 0 and the optimal aggregator (which, in this example, is the majority voting) yields
(opto rep(O", xxx=)) 0.5.
33

---

Published as a conference paper at ICLR 2026
When experts are bounded rational with X 5, the report distributions differ. In the case n 2, letting
1
1
go Pr(X; = = 0|
Ω Pr[X, = - 1
2 2e
2 2e'
and choosing the aggregator
0, x S 1,
(x)
1, I = 2,
one obtains
reptx. v(>=2)) 0.507674 U rep("
rep(O" x2)+)))).
This shows that for 2,
max U rep (f"", xxx=25)) max U (f, rep (f", x/-xx))
f
Extension to n 2 Experts.
Finally, we consider the case when n > 2. When is odd, the following lemma holds
Lemma A.14. When n odd, we have
U (fmms, rep((", (xx=))) 0.5
Proof.
U (P"ai,rep(0"", rrpp(0', ,xxx)))))))))))))))))))))))))))) (2 - 1) (Pr[w 1, = Pr[w = 0, =])
==0
(Pr(w = 0| (Pr(X= = w= 0] - Pr[X = n - x | = 0])
Pr(w = 1] (Pr(X = x w= Pr[X = n - x w -- 1)))
(0.75 (9) -0)=~> % n-=(11 - 90))))
0.25 (qf (1 41b "- of f1 9 )r)
0.75
4 (1 90)
90)m~ n-
40 (1 g0)x
T
0.25
Σ
n
o (1 (1) n-
91 R- (1 91)
2=0
= 0.5 + 0.25
11
(1 91) )a +
9 " n- (1 91)²
z=0
- 0.75
Σ
11
n 2 (1
90
40 2 (1 70)
0=0
When is odd, we have
n=1
771
[
and the expression becomes:
34

---

Published as a conference paper at ICLR 2026
U "raa,,ep(("" ; = 0.5 +
n
(0.5g) 1 (1 91)" 1.5%% "(1 - (1-90)²)
==0
n-1
0.5 +
11
(0.50 (1 -q1) (11-22-1111 - 21)=-1)
1.5g (1
qu)
(90 2x (1 -9u)==1)
2 0.5
n
(0.5gf 41) 1.5gh [0)) 9 1---1{ (1 90 -=)== --
> 0.5.
(since 0.5gf(1 Q1) 1.5gg 2o) 0)
And when n is even, the expected utility of the majority voting is equal to the case when there are n experts:
Lemma A.15. Let n be even, and let 0n-1 E Goid denote the report structure with n 1 experts, and 0n E Gridd denote the report structure with experts. If the joint distribution of θn-1 and 0g for certain expert is identicol, then the utility of the majority voling remains the same. Formally, we have:
Yz,, w E {0,1}, Pr [X, x2 Pr[X, 2,,
On-1
U(fma 0n-1)= U(pmaj 0m)
Since is odd, and Lemma A.14 have already proven that the utility is greater than 0.5 when the number of experts is odd, it follows that the utility is also greater than 0.5 when n even.
Proof of Lemma
A.15 Let
Pro, [w 1], go Pr, [X = 1 3 0).
111 Prg. [X; = || 1].
35

---

Published as a conference paper at ICLR 2026
And we have
U(f pn)
n
(2fmai (x) 11)
Pr[w = 1, = 2) -
Pr[[w
= 0, X = =x]
On
Pr[w = 0] On
Pr|x x = 0]
Pr[x
n2
0]
6n.
Pr[w 0n
1]
Pr[x
x
w 11
Pr[x
n
Bn
6n
8
2p + 1
2
12
11
z (1
n1) )z
+
n
h
z=0
x
?
4-1
(1 11)
2
90
(1
90)= ) +
n
90
(1
90)
1
n
n 1
=1 2p 10
20 +2
+
1 (1 41) 0r +2
9i (1 91
x
n 2
--1
n (n-1)
n
n 1
- (1 11)
2% +2
10 (1 00) )a +2
40 go)
x
x
n
r=l
=1 2p 1
2
n2
41 n1 (1 )1)² + 2
Σ
1 n nー (1 ạ1) )"
1
z=0
2=1
- (1 (1-M) 11)
2
n1
90 n
(1 90) 1z +2 Σ
n 1
46 (1 20)
z=0
x=1
4-1
n 1
=1 2p
2 Σ
(on) n-
(1 - -1) - (1 1)g0 1----11- (1 20 )0)
x
x=0
Pr w
0]
Pr [X
0]
Pr [X
--
An
8, 1
0, 1
Pr w 1]
Pr [X w=
Pr X - 2
0--1
=U( mmaj 0n-
Thus, we have shown that when n > 2,
samms,rre((e xx=x))) > max U (frre" v22+))))
f
Numerical Results. To further substantiate theoretical findings, we performed mu- merical simulations to assess the performance of two aggregation methods on report structure rep(0*, v)x): the majority voting (fmas) and the omniscient aggregator (opl sox))- These simulations explore how their utility varies with the ronaliyyyy level X and the number of
36

---

Published as a conference paper at ICLR 2026
n=1
n=2
nm3
Omniscient Aggregater Majarity
Ubiley
0.500
2.500
UNilit
0.500
0.475
2,415
0.415
0.450
Figure Utility Comparison: Majority Voting vs. Omniscient Aggregator
(Varying Rationality and Group Size) The utility comparison is performed under the specific signal structure 8" illustrated the introduction. Under full rationality (A oo), the maximum achievable utility is 0.5. For n 1, the omniscient aggregator mainlains constant utility of 0.5, while the majority voling converges to 0.5 as - 0o. For n = 2, the omniscient aggregator achieves utility greater than 0.5 for certain values of X. Majority voting yields the same utility for n and n = 2. However, this omniscient aggregator requires knowledge of the signal structure, whereas the majority voting does not. For n = 3, the benefits of bounded rationality become more apparent. The majority voting, which requires knowledge of the signal structure, exceed utility of 0.5, reaching its peak performance at intermediate values of X.
experts n. The results are summarized in fig. These numerical results support our theoret- ical conclusions. In particular, under the specific signal structure 0* and when 2 2, they demonstrate that bounded rationality can enhance the decision decision-making performance, enabling the aggregators to surpass the maximum utility of 0.5 attainable under full rationality.
B DETAILED STUDY SETUP FOR SECTION
B.1 LICENSES AND TERMS OF USE
MathQA Dataset The MathQA datasot (Amini 2019 is publicly available under the Apache License 2.0. It can be accessed at https:: http//uuggnnnggann co/datasets/ aalaaastsssallllaaalaaaaaa allenai/
math_qa
GPT-4o-mini Model The GPT-44-mini-202444 by OpenAL. Its usage is governed by OpenAl's Terms of Use and Service Terms, available at ltttps:/openaal www.:/owwiisseetreessoooooaeeeee -usse and hhtpp::////// /ttpss//opeeni.. com/policies/ service-terms! respoctively.
model used in this study is provided
B.2 BAYESIAN RECISION-MAKING STUDY
Study Design The Bayesian Decision-Making Study is framed follows: There two boxes, the left box (w = 1) and the right box (w = 0), each containing red and blue balls in specific proportions. In the right box (w = 0), the proportion of red balls is Pr[S, = w 0), and the proportion of blue balls is Pr[S, = =b w 0). Similarly, in the left box (w 1), the corresponding proportions are Pr(S, =
= I] and Pr[S, = b w 1]. A box is selected randomly with a prior probability μ = = Pr(w = 1] of choosing the left box. A ball is then drawn from the chosen box, and its observed color, S. (r. b), represents the signal, where S; = denotes red ball and S; a blue ball. The task is to infer which box was selected based on the observed signal S,, the provided prior Pr(w = 1] and the conditional probabilities Pr[S, w).
Because the number of scenarios we can query is finite, the probabilities are discretized as follows: the prior probability Pr|w = 1]. the probability of observing red ball given the left box Pr[S, = r = 1), and the probability of observing a red ball given the right box Pr[S, r Specifically, these probabilities take values from the set {0/N, 1ON,,//N,2 2/N,.,, discretization creates set of decision-makiin tasks that span range of prior and conditional
0] are each expressed as fractions with N 5 the denominator.
This
37

---

Published as a conference paper at ICLR 2026
probabilities. After removing the illegal and impossible cases, we have a total of 400 possible discretized scenarios.
The study utilizes the oot-qo-mini model as the expert decision-maker. The model is queried under three distinct temperature settings {0, 0.5, 1}, each representing different levels of rationality. A Chain of Thought (CoT) prompting strategy is employed, requiring the model to explain its reasoning before delivering a binary decision ("L" for the left box, X; =,=1, "R" for the right box, X; = =0). To capture variability in the model's stochastic behavior, each decision-making scenario is repeated 20 times for each temperature set setting, The decision proportion for given scenario and temperature is defined as the empirical probability of choosing the left box ("L") across all repetitions.
Temperature
Coefficients (A) Standard Errors (SE)
z-values (z)
p-values (p)
0.0
0.5
26.49
10.23
2.59
<0.001
1.0
17.86
5.44
3.28
<0.001
Table 1: Logistic regression results for different temperature scttings. The regression coefficient X, corresponding to the rationality level the quantal response: higher À values indicate more deterministic behavior, while lower values suggest increased randomness. For the case with temperature 0.0, the observed perfect determinism in choices leads to complete separation in the logistic regression, resulting in theoretically infinite coefficients. The standard error (SF) reflects the precision of X, with smaller values indicating greater accuracy. z-values and p-values confirm the statistical significance of the findings.
B.3 SELTIPLE-CHOIGE QUESTION ANSWERING STUDY
Dataset Description The MathQA dataset (Amini et al. 2019 is a collection of complex mathematical reasoning problems presented as multiple-choice questions. Fach problem consists of a problem statement and a set of five options (A, B, C, D, F.,, with one correct answer. The dataset designed to test logical reasoning, arithmetic, algebra, and other mathematical concepts, making it challenging benchmark for evaluating LLMs. For this study, we use a subset of 5000 MathQA problems to ensure computational fcasibility, focusing on tasks that require probabilistic rcasoning and decision-making.
Study Design To evaluate LLM performance, deploy the following experimental sctup. The LLM is queried under three distinet temperature scttings (t = 0.0, 0.5, 1.0) to simulate varying levels of rationality. For each question, the model generates 20 responses per uemperature setting. Because the questions offer more than two options, we use plurality voting (frla), generalization of majority voting. This involves selecting the option with the highest frequency among the sampled responses. In the event of a tie, the aggregator randomly chooses among the tied options.
Bootstrapping To assess the robustness of aggregation, we perform bootstrapping by randomly sampling n responses (without replacement) from the 20 generated responses per question. This bootstrapping procedure is repeated 1000 times for each combination of temperature setting i and expert group size n = 1,3,5 5. Accuracy is defined as the proportion of aggregated answers that match the ground truth, with error bars representing the standard error of the mean across bootstrap iterations.
An Example We present example in our study of how increasing temperature (i.e., adding randomness) counterintuitively increases expected accuracy. In our study, we ask gpt-4o-mini the following MathQA question.
"An alloy of copper and zinc contains copper and zinc in the ratio 3:5. Another alloy contains copper and zine in the ratio 6:2. In what ratio should the two alloys be mixed to achieve equal proportions of copper and zinc?" The options are "1 2", "2 2", "2 5", "2 :2:6" 6", "2 7".
38

---

Published as a conference paper at ICLR 2026
At 0.0 (deterministic mode), after querying LLM, 19/20 responses selecting the incorrect option B (*2 2*). Consequently, plurality voting always selects the incorrect option B. However, at == 1.0, 4/20 responses correctly identify A ("1: 2") Despite most responses being incorrect (16/20 chose B), the increased variance allows the aggregation to sometimes identify the correct answer.
One possible explanation is ambiguity in the question regarding the order of the mixing ratio (first alloy to second, or second to first). In deterministic mode = 0.0), the LLM frequently calculates the correct 2:1 ratio (first to second), but then fails to select the ground truth answer of 1:2. Some responses even explicitly justify this error, stating, "However, since the options provided do not include 2:1, we will look for the closest match. The closest option to 2:1 **B) 2:2** which represents equal proportions.".
However, the increased randomness in stochastic mode (t = 1.0) allows the LLM to exhibit greater "intelligence." Some responses cleverly recognize the intended meaning, acknowlodg- ing the discrepancy and stating, "None of the options list a ratio of 2:1 directly. There is no exact match, but the closest option that can relate context of inverse." This suggests that the added randomness enables the LLM to explore a wider range of interpretations and, in some cases, deduce the correct answer despite the question's 's ambiguity. See the Section C.2 for more details.
option 1:2 we consider in
C PROMPTS FOR LLMs
This section provides detailed descriptions of the prompls employed in our studies with Large Language Models (LLMss). These prompts are used to assess various aspects of hallucination detection, including prior answer generation, semantic clustering, and posterior answer generation. Our intention is to provide transparency and reproducibility for our method and those compare against, particularly in terms of how LLMs interact with different prompt structures.
These prompts were developed specifically for use in studies described in the main body of the paper. While they have been designed to provide effective results, we note that they may still be open to refinement for further improvements in performance and reliability.
39

---

Published as a conference paper at ICLR 2026
C.1 BAYESIAN DECISION-MAKING STUDY
User Prompt
### Scenario:
You are given tvo boxes of balls. Each box contains 100 balls, which can be either red or blue. A box is selected at random, and ball is draun from the selected box. Hosever, the identity of the selected box not revealed.
You are provided with the following information:
Left Box contains (left_red) red balls and (left_blue) blue balls.
Right Box contains (right_red) red balls and (right_blue) blue
balls.
The probability of selecting the Left Box is probabiioy
3%.
The probability of selecting the Right Box { right.prooabilityy..
### Question:
If the ball drawn is (color), which box has been selected?
Please first briefly show the calculations in the <reason></
reason> section, and then provide your final ansver in the answer></ansver> section.
should **strictly** be single uppercase letter: "L" for the Left Box or "R" for the Right Box. No additional text, no explanations. Your response must contain definitive anseer of either "L" or "R" in it.
Your in Vanswer></ansver
section with
Example response:
<reason ># Briefly show your calculations here, using the provided information.o iifornation//rresonn <ansver> single uppercase letter: L R</answer
40

---

Published as a conference paper at ICLR 2026
Example Input
### Scenario:
You are given two boxes of balls. Each box contains 100 balls, which can be either red blue. A box is selected at random, and ball is draun from the selected box. However, the identity of the selected box not revealed.
You are provided with the following information:
Left Bor contains 100 red balls and 0 blue balls.
Right Box contains 50 red balls and 50 blue balls.
The probability of selecting the Left Box 20.0%. The probability of selecting the Right Box is 80.0%.
*** Question:
If the ball drawn red, which box has been selected?
Please first briefly show the calculations in the <reason></ reason> section, and then provide your final answer in the < answer></ansver> section.
Your answer in <<nserr><ansswe>>>> should **strictly** be single uppercase letter: "L" for the Left Box or "R" for the Right Box. No additional text, no explanations. Your response must contain <ansver></ansver definitive answer either "L" or "R" in it.
section with
Example response:
<reason ># Briefly show your calculations here, using the provided information.</reason
<answer> ># single uppercase letter: or R</answer>
41

---

Published as a conference paper at ICLR 2026
Example Output
<reason>To find the probability selecting each box given that a red ball is drawn, we can use Bayes' theorem.
Let:
(( P(L) 0.2 V) (probability
selecting the Left Box)
(( P(R) 0.8 A) (probability of selecting the Right Box)
\( P(Atext [ = V) (probability of drawing a red ball
from the Left Box)
V( (Red) R) 0.5 V (probability drawing red
ball from the Right Box)
Using the law of total probability, we calculate (( P(Atext(Ree NI
}) \)
(Red)) (Redext(Red] Aedot P(L) P(Atext(Re)) | R)
Aedot P(R) = Aedot 0.2 0.5 Aedot 0.8 0.2 0.4 0.6
AJ
Now, we can find N( PLL \text(Red)) using Bayes' theorem: A[
PL
| L) Vedot P(L)) Red))} = \frac(1 Aedot 0.2) 6) = (0rac(..H00.66 = Afrac (1)(3)
\frac(P(\texttRed) (Red))
A]
Similarly, find A( P(R I \text( V): AL
P(R (Red) Vrrcc((( R) Vedot (P Red))} (0.ac Aedot 0. 0.8)(0.6} (0 \frac (2}{}
A]
Since \( (R (tex)( P (L (tex)( W) is more likely that the Right Box was selected if red ball is drawn </reason>
<answer>R< >R</ answer>
C.2 MULTIPLE-CHOICE QUESTION ANSWERING STUDY
We first provide the template of prompt, then we provide the example input and two example
outputs. The input and outputs correspond to the example in Section B.3
42

---

Published as a conference paper at ICLR 2026
User Prompt of Math
### Scenario:
Please review the following math question carefully:
#** Question: {question_atr)
#*# Options:
{options_str}
### Instructions:
1. Provide your step-by-step calculations using your math knowledge, enclosed within <reason></reaso tags.
2. Select the best answer from the options provided, and place the uppercase letter of your final answer within <answer></ answer> tags.
Please ensure your reasoning clearly supports the selected and that your final
is the provided options. If none the options are precisely correct, choose the closest one.
Example response:
<reason>> ># Your reasoning and calculations go <answer ># single uppercase letter: the option you choose</ answer>
43

---

Published as a conference paper at ICLR 2026
Example Input
### Scenario:
Please review the following math question carefully:
#*# Question:
alloy of copper and zinc contains copper and zinc in the ratio 5 another alloy of copper and zinc contains copper and zinc in the ratio 6 2 in what ratio should the two alloys be mixed so that the resultant alloy contains equal proportions of copper and zinc
### Options:
A)
1
2
B) 2
2
C)
D) 2
E) 2
### Instructions:
1. Provide your step-by-step calculations using your math knowledge, enclosed within <reason><<reason tags.
2. Select the best ansver from the options provided, and place the uppercase letter of your final answer within <answer></ answer> tags.
Please ensure your reasoning clearly aupports the selected answer and that your final answer is one of the provided options. If none of the options precisely correct, choose closest one.
Example response:
<reason ># Your reasoning and calculations go here </reason> <answer># single uppercase letter: the option you choose</ answer>
44

---

Published as a conference paper at ICLR 2026
Example Output When Equals
<reason>
Let 's denote the first alloy (with a copper to zinc ratio of 3:5) as Alloy A and the second alloy (with copper to zinc ratio of 6:2) Alloy B.
1. **Determine the composition of Alloy A:** The total parts in Alloy A = (copper) 5 (zinc) 8 parts.
Copper in Alloy Zinc Alloy A 5/8 the total weight.
3/8 of the total weight.
2. **Deermiine the composition of Alloy B:*. The total parts in Alloy B (copper) 2 (zinc) parts.
Copper in Alloy
6/8 3/4 of the total weight.
Zinc in Alloy B = 2/8 = 1/4 of the total weight.
3. **Let the weights of Alloy and Alloy B be and y respectively.
The total copper from both alloys (3/8)x + (3/4)
The total zinc from both alloys = (5/8) (1/4)
4. **Set the equation for equal proportions copper and zinc :**
A[
(3/8)x+ (3/4)y = (5/8)x (1/4)
\]
5. Clear the fractions by multiplying through by 8:**
A
3x
6y 5x 2y
\]
6. ** #Rarranngi gives:** ***
\[
3x
5x 6y 2y \implies -2x 4y Aimplies 2x 4y
limplies x/y 4/2 2/1
7. **hhus, the ratio of Alloy A to Alloy B is 2:1.4x ** This means need mix Alloy A and Alloy B the ratio of 2:1.
Since the question asks for the ratio of the two alloys, can express this 2:2 (which simplifies to 1:1) 2:1. However since the options provided do not include 2:1, will look for the closest match.
The closest option to 2:1 is *+B) 2:2**, which represents equal proportions.
Therefore, the final ansver is: </reason> <answer >B</ anaver>
45

---

Published as a conference paper at ICLR 2026
Example Output When Equals
<reason>To >To solve the problem, we need to first determine the composition of each alloy in terms of copper and zinc.
1. **First Alloy (Ratio 3:5)+*; **:
The total parts in the first alloy
parts.
Copper in the first alloy (3/8) 100% 37.5%
Zinc in the first alloy (5/8) * 100% 62.5%
2. ** Alloy (Ratio ++
The total parts the second alloy 6 Copper in the second alloy (6/8) 100% 75%
parts.
Zinc in the second alloy (2/8) 100% 25%
Let denote the first alloy as A (copper 37.5%, zinc 62.5%) and the second alloy as (copper 75%, zinc 25%) want to mix these two alloys in the ratio x:y such that the resultant alloy has equal proportions of copper and zinc.
We will express the copper zinc contents the resultant mizture:
Copper from A \(x Atimes 0.375\)
Zine from \(x Atimes 0.625\)
Copper from B = A(y \tizes 0.75
Zinc from B = M(y \times 0. 0.25
The total copper content in the mixture B) is: A[
Atext Copper) = 0.375x 0.75y A]
The total zinc content in the mixture (A is:
A[
Atext(Total Zinc] = 0 625x 0.25y AJ
We want these tvo quantities to be equal: A[
375x + 7. = 625x + 0.25y
A]
Rearranging this equation, we get: \[
0.375x 625x 0.75y 0. 25y =
A]
N[
0.255x + 0.5y = NJ
N[
0.5y \]
25x
46

---

Published as a conference paper at ICLR 2026
Example Output When Equals
y 0.5x
So, the ratio of A(x y\) be written A(x 0.5x\) simplifying gives:
y : \]
This means we need to mix the alloys in a ratio of 2:1 (2 parts of A for every 1 part of B).
Now, should
which option matches this finding:
None of the options list ratio of 2:1 directly. There is no exact match, but the closest option that can relate to 2 is option A: 1:2 if we consider it in context of inverse.
Since the required ratio of the alloys results in copper and zinc being equal, let interpret the problem in inverse terms as it signifies that for choosing A and B: To maintain balance of ratios, the best option that reflects appropriate mixing **option A** which provides a guideline for consideration, perhaps in taking more from the second one (due to its higher concentration of copper).
So, based this reasoning:
</reason>
Sanswer>A// answer>
47


Reviews:
[{"review_id": "weOw2nMjk5", "reviewer": "Reviewer_vc8K", "summary": "This work shows that when experts make rational, stochastic decisions (modeled by quantal response), majority voting optimally aggregates their judgments under minimax regret. Surprisingly, such partially random groups can outperform perfectly rational agents, as their randomness preserves informative variation. Experiments with LLMs confirm that aggregating moderately stochastic outputs improves reasoning, revealing bounded rationality as a strength in collective intelligence.", "questions": "See above", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. Theoretical results: the authors proved interesting theoretical results showing that bounded rationality can outperform perfect rationality;\n2. Interesting experiments: the authors leverage LLMs as conditional independent agents, which provides a scenario where the condition of the theoretical results can be easily satisfied.", "weaknesses": "1. It seems not straightforward to calculate g(n) in the main theorem. Appendix A2 provides a plot of g(n), but mostly observations. May provide more insight on why g(n) = g(n-1) for even n's. \n2. (minor) the authors refer to Fig 5 in line 340, but Fig 5 is in the appendix. Better refer to both the appendix section and the figure.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6hhwkz0yHT", "reviewer": "Reviewer_shYv", "summary": "This paper presents theoretic results on how decisions are aggregated under uncertainty while quantal response is considered. The main theorem is in 3.1, which proves that (1) if the rational parameter is bounded under a threshold, then majority voting is the optimal aggregation rule and (2) the group collective decision can be benefited from bounded rationality of individuals.", "questions": "The theoretical results in this paper generally look great and insightful. However, I wonder the authors could comment more about why LLMs are considered quantal best responder, as section 4.1 only concerns on one domain. Also, is here every LLM considered to have the same \\lambda?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The technical contribution is sound. The results are quite interesting.", "weaknesses": "It is not very clear why LLMs are generally considered as quantal best responder", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Popf0XiYG6", "reviewer": "Reviewer_eia5", "summary": "This paper examines information aggregation under bounded rationality. A decision maker gathers reports from agents receiving noisy signals about a ground truth. Instead of sincere voting, agents follow a quantal response modeled by a logistic link between their posterior and report. The authors show that (1) majority voting with an optimal threshold minimizes regret, and (2) with more agents, quantal responders can outperform sincere ones. Experiments further confirm that (1) large language models exhibit quantal response behavior via temperature, and (2) the theoretical results hold empirically.", "questions": "1. How do you justify sincere voting?\n2. Do the same results extend to various settings?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The paper offers a valuable contribution to the information aggregation literature by applying a rational framework to analyze boundedly rational agents. The results are both elegant and insightful, providing a simple yet positive perspective on a complex problem. The experimental findings are also convincing.", "weaknesses": "I am not fully convinced by the second main result (bounded rational advantage). The idea behind it is not as surprising as it may seem. Quantal response preserves randomness even when rational (I'd rather call sincere) agents become deterministic. This randomness helps the majority vote to operate processes like the Condorcet Jury Theorem, constructing another form of informative voting (voting as their signal suggests) and reaching the ground truth with high probability.\n\nThe thing is, this observation is based on the assumption that agents vote sincerely. Personally, I don't quite buy this assumption. On the simplicity side, conducting a Bayesian update and maximizing expected utility is not an easy task for voters who don't really know Bayesian statistics. On the rational side, sincere voting is not necessarily a Nash Equilibrium [Austen-Smith and Banks, 1996; Han et. al, 2023]. Consequently, I am curious to see if the same result extends to various settings. Nevertheless, I am overall positive about this paper. \n\nAusten-Smith D, Banks J S. Information aggregation, rationality, and the Condorcet jury theorem[J]. American Political Science Review, 1996, 90(1): 34-45.\nHan Q, Schoenebeck G, Tao B, et al. The Wisdom of Strategic Voting[C]//Proceedings of the 24th ACM Conference on Economics and Computation. 2023: 885-905.\n\nP.S. You might want to update some of the references, whose status seems out-of-date.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)