Language Models Trained to do Arithmetic Predict Human Risky and Intertemporal Choice
Under review as a conference paper at ICLR 2025

LANGUAGE MODELS TRAINED TO DO ARITHMETIC
PREDICT HUMAN RISKY AND INTERTEMPORAL
CHOICE

Anonymous authors
Paper under double-blind review

ABSTRACT

The observed similarities in the behavior of humans and Large Language Mod-
els (LLMs) have prompted researchers to consider the potential of using LLMs
as models of human cognition. However, several significant challenges must be
addressed before LLMs can be legitimately regarded as cognitive models. For in-
stance, LLMs are trained on far more data than humans typically encounter, and
may have been directly trained on human data in specific cognitive tasks or aligned
with human preferences. Consequently, the origins of these behavioral similarities
are not well understood. In this paper, we propose a novel way to enhance the util-
ity of language models as cognitive models. This approach involves (i) leveraging
computationally equivalent tasks that both a language model and a rational agent
need to master for solving a cognitive problem and (ii) examining the specific
task distributions required for a language model to exhibit human-like behaviors.
We apply this approach to decision-making – specifically risky and intertemporal
choice – where the key computationally equivalent task is the arithmetic of ex-
pected value calculations. We show that a small language model pretrained on an
ecologically valid arithmetic dataset, which we call Arithmetic-GPT, predicts hu-
man behavior better than many traditional cognitive models. Pretraining language
models on ecologically valid arithmetic datasets is sufficient to produce a strong
correspondence between these models and human decision-making. Our results
also suggest that language models used as cognitive models should be carefully
investigated via ablation studies of the pretraining data.

1

INTRODUCTION

Scientists studying the behavior of Large Language Models (LLMs) in cognitive tasks typically per-
formed by humans have found substantial evidence that LLMs produce performance similar to that
of human participants (Binz & Schulz, 2023b; Horton, 2023; Zhu & Griffiths, 2024; Dasgupta et al.,
2022; Frank, 2023b; Marjieh et al., 2023; Webb et al., 2023; Coda-Forno et al., 2024).1 Like hu-
mans, LLMs often make judgments and decisions that deviate from rational norms (Binz & Schulz,
2023b; Horton, 2023; Zhu & Griffiths, 2024; Coda-Forno et al., 2024). For instance, GPT-3 demon-
strates human-like biases in risky choice, such as risk aversion and loss aversion (Binz & Schulz,
2023b), and the statistical properties of probability judgments generated by LLMs align qualitatively
with those of humans (Zhu & Griffiths, 2024). LLMs also make errors in other related settings; for
example, when trained on the task of predicting the next token in sequences involving arithmetic
operations, they fail to learn precise arithmetic operands and instead approximate the correct results
up to a certain input range (Nogueira et al., 2021; Lee et al., 2023).

In this paper, we focus on risky and intertemporal choice as exemplar domains for comparing the be-
havior of language models and humans (see Figure 1). Central to both domains is the computational
challenge of calculating expectations. To assess the benefits of engaging in a gamble, an intelligent

1LLMs can also display significant deviations from human behavior (e.g., Chen et al., 2023; Hagendorff

et al., 2023).

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Under review as a conference paper at ICLR 2025

(A) Pre-training and evaluation pipelines. We began by generating a synthetic dataset
Figure 1:
comprised of mathematical equations including addition, subtraction, multiplication, and exponen-
tiation. Arithmetic-GPT was pretrained on this synthetic dataset. After training, we froze model
weights and extracted embeddings from the pretrained model, which then processes stylized choice
tasks as input. These embeddings were subsequently compared with human choice data to evaluate
their correspondence. (B) Ecological distributions of probabilities and values. In the top panel,
English probability-describing phrases (black bars) can be modeled using a Beta(0.27, 0.27) distri-
bution. In the bottom panel, the value distribution of debits from UK bank accounts (scatterpoints)
follows a power-law distribution. Figures adapted from Zhu et al. (2020) and Stewart et al. (2006).

system must be able to calculate the expected value (EV) of the gamble, typically represented as:

EV (A) =

(cid:88)

pi ×

xi

(1)

i
∈
where each outcome i of gamble A is associated with a payoff xi and a probability pi, with the
constraint that (cid:80)
i pi = 1. Similarly, in considering an intertemporal choice the computation of the
present value (PV) of future outcomes in A is crucial:
(cid:88)

A

(2)

P V (A) =

dt

xt

×

t
∈
where the value xt is realized at time t and is discounted by a factor of d, reflecting the time prefer-
ence of the decision-maker. Note that a risk-neutral and time-consistent agent should always select
the option that maximizes EV and PV. However, extensive research in economics and psychology
demonstrates that people systematically deviate from this maximizer model (Kahneman & Tversky,
2013; Gigerenzer & Gaissmaier, 2011; Laibson, 1997; Zhu et al., 2020).

A

Pretrained, off-the-shelf LLMs, such as the GPT and LLaMA series, have demonstrated behavioral
similarities to humans in tasks involving risky and intertemporal choices (Horton, 2023; Binz &
Schulz, 2023b; Manning et al., 2024). However, the embeddings generated by these LLMs are
not by default able to account for human data. For example, embeddings from the LLaMA-1-65B
model, when not finetuned on human risky choices, poorly predict those choices (Binz & Schulz,
2023a). Therefore, understanding what enables LLMs to exhibit human-like decision-making be-
havior remains an unresolved challenge.

We propose a hypothesis about how human-like decision patterns might be produced in language
models for risky and intertemporal choice: such human-like behaviors might arise from a lan-
guage model trained on ecologically valid calculations of expectations. This suggests that human-
like biases could result from the training task and numerical reasoning, independent of natural
language supervision. As a corollary, this hypothesis also implies that deviations from rational
choice in humans could be primarily explained by computational errors during the EV or PV
calculations. To test this hypothesis, we generate a series of synthetic datasets that contain ex-
pected value computations; examples include 0.5*100=+50, 0.8*1+0.8ˆ2*10=+7.2, and

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Synthetic datasetArithmetic-GPTPlease select option A or BA: $16 with certainty B: $1 with probability 0.6      $40 with probability 0.4pre-trainfreeze weightsRisky choice problems10*0.2-4*0.8=-1.2 0.1*7+0.2*5+0.7*0=+1.7 0.8^2=+0.64 0.39*10-0.63*6=+0.12 92*0.23-0.21*77=+4.99 -214*0.85+164=-17.9stylize the calculation for EV diﬀerences16-1*0.6+40*0.4=Please select option A or BA: $5 today B: $10 in 4 daysIntertemporal choice problems5-10*0.99^4=embeddingslogistic regressionstylize the calculation for PV diﬀerencesEcological distributions<latexit sha1_base64="88sNs7RbTf2b7n04b3if8vgndxM=">AAACFnicbVBLS0JBFJ5rL7OX1bLNkAQKJfeKaJtAatPSIB+gInPHow7OfTBzbiQXf0Wb/kqbFkW0jXb9m8bHoqwPzuHj+85h5nxuKIVG2/6yEiura+sbyc3U1vbO7l56/6Cug0hxqPFABqrpMg1S+FBDgRKaoQLmuRIa7uhq6jfuQGkR+Lc4DqHjsYEv+oIzNFI3fVbNthHuMQ5V4DJXSIHjSe5irl0CsknWzhfKp3Tac910xs7bM9C/xFmQDFmg2k1/tnsBjzzwkUumdcuxQ+zETKHgEiapdqQhZHzEBtAy1Gce6E48O2tCT4zSo/1AmfKRztSfGzHztB57rpn0GA71sjcV//NaEfbPO7HwwwjB5/OH+pGkGNBpRrQnFHCUY0MYV8L8lfIhU4yjSTJlQnCWT/5L6oW8U8qXboqZSnERR5IckWOSJQ4pkwq5JlVSI5w8kCfyQl6tR+vZerPe56MJa7FzSH7B+vgGxledwg==</latexit>P(probability)=Beta(0.27,0.27)<latexit sha1_base64="bmpBHLfELzTOrsmx+494hrfZRP0=">AAACFXicbVDJSgNBFOyJW4xb1KOXwSBE0DAjMeot4MVjBLNAJoaezpukSc9C95tgGPITXvwVLx4U8Sp482/sLIcYLWgoqt7jdZUbCa7Qsr6N1NLyyupaej2zsbm1vZPd3aupMJYMqiwUoWy4VIHgAVSRo4BGJIH6roC6278e+/UBSMXD4A6HEbR82g24xxlFLbWzJ5W8g/CAyYCKGEbHTiTDCENzXrxPTq3CVfF81M7mrII1gfmX2DOSIzNU2tkvpxOy2IcAmaBKNW0rwlZCJXImYJRxYgURZX3ahaamAfVBtZJJqpF5pJWO6YVSvwDNiTq/kVBfqaHv6kmfYk8temPxP68Zo3fZSngQxQgBmx7yYmHq2OOKzA6XwFAMNaFMcv1Xk/WopAx1kRldgr0Y+S+pnRXsUqF0W8yVi7M60uSAHJI8sckFKZMbUiFVwsgjeSav5M14Ml6Md+NjOpoyZjv75BeMzx+k+Z8Q</latexit>P(value)/value0.945ABevaluate on human choice dataUnder review as a conference paper at ICLR 2025

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30*0.79-261*0.83=-192.93. Subsequently, we train a small, randomly-initialized language
model (approximately 10M parameters) on these datasets. After training, we extract embeddings
from the now pretrained language model and analyze how well they can account for human choices.

We conduct carefully ablated experiments on different aspects of the synthetic data to isolate the
factors that result in embeddings that better predict human choice patterns. Our findings reveal that
when the synthetic dataset reflects the ecological distributions of probabilities and values–mirroring
real-world frequencies–the resulting embeddings best predict human choices. With this pretraining,
models based on the derived embeddings outperform many existing behavioral models of risky and
intertemporal choice.

2 BACKGROUND

The question of whether language models can serve as models of human cognition has sparked in-
tense debate within the fields of machine learning and cognitive science (Frank, 2023b; Messeri &
Crockett, 2024; Griffiths et al., 2023; Horton, 2023). Although there are behavioral similarities be-
tween off-the-shelf LLMs and humans, these do not inherently qualify LLMs as effective cognitive
models (c.f. the Clever Hans effect) (Shiffrin & Mitchell, 2023). There are compelling reasons why
current LLMs may not be suitable as cognitive models. First, LLMs are trained on datasets vastly
larger than those available to human learners (Frank, 2023a). Second, LLMs may have already been
trained on the test questions, particularly if the training data is undisclosed and poorly controlled
(Jiang et al., 2024). Third, the inclusion of value alignment steps, such as Reinforcement Learning
from Human Feedback (Ziegler et al., 2019) and Direct Preference Optimization (Rafailov et al.,
2024), may artificially enhance human-like behaviors in leading LLMs. Finally, the immense size
of deep neural networks and the proprietary nature of leading LLMs hinder detailed investigation
into their internal representations.

Researchers have taken steps to address some of the concerns associated with using LLMs as cogni-
tive models. One branch of research looks at the potential benefits of fine-tuning off-the-shelf LLMs
to better understand human cognition. For instance, fine-tuning the LLaMA-1-65B model (Touvron
et al., 2023) on datasets of human choices results in a model that can predict human data more accu-
rately than traditional cognitive models (Binz & Schulz, 2023a). Although the specific mechanisms
underlying the finetuned LLM’s ability to replicate human choices remain unclear, this work serves
as a proof-of-concept and suggests that the embeddings learned from extensive pretraining on Inter-
net text and/or from the value alignment process may offer valuable insights into human cognitive
processes that complement those provided by traditional cognitive models.

Another line of research emphasizes the importance of the composition of synthetic datasets, which
are critical for enhancing certain capabilities of LLMs (Lee et al., 2023). These studies typically
assess LLMs based on their problem-solving abilities rather than their human-likeness. For exam-
ple, it has been found that larger language models tend to perform arithmetic tasks, such as addition
and multiplication, better than their smaller counterparts (Yuan et al., 2023). Moreover, the order of
input and the inclusion of intermediate information about task decomposition have been shown to fa-
cilitate chain-of-thought reasoning, which significantly helps smaller language models in mastering
arithmetic tasks (Lee et al., 2023).

Finally, there is a precedent for the idea that pretraining machine learning models on synthetic
datasets can improve performance in predicting human decisions. Bourgin et al. (2019) showed
that a model pretrained on choice data generated from a psychological theory could perform ex-
tremely well in predicting human decisions when fine-tuned with a small amount of human data.
Our approach builds on this idea, but reduces it to the most primitive components – rather than pre-
training on data generated from a psychological theory, we pretrain on a task that captures the basic
computations required to make rational decisions.

3 ARITHMETIC-GPT: A SMALL LANGUAGE MODEL TRAINED TO PERFORM

ARITHMETIC

In this paper, we confront the challenges of making language models as cognitive models head-on.
First, we define a data generation algorithm to produce synthetic datasets, thereby gaining complete

3

Under review as a conference paper at ICLR 2025

control over the training data for language models and addressing issues related to data gaps and
contamination. Second, we have direct access to the neural activation patterns that are crucial for
decision-making processes. This approach allows us to more thoroughly evaluate and understand
the capabilities and limitations of language models.

3.1 MODEL DETAILS

Our small language model employs a standard architecture for a Generative Pretrained Transformer
(GPT) model (Vaswani et al., 2017; Radford et al., 2019). Detailed specifications of the model
architecture are provided in Table 1.

Table 1: Arithmetic-GPT 10M: a small language model pretrained to do arithmetic

Pre-training hyperparameters Value
Hidden size
Layers
Heads
Context length
Vocabulary size
Attention variant
Dropout
Biases

320
8
8
26
320
Causal self attention
0.2
None

Tokenizer. To handle a domain-specific vocabulary dedicated for arithmetic equations, we built
a custom tokenizer on the sub-word level. The vocabulary size is 320, containing special tokens
(e.g., <AMB>,<PAD>), arithmetic operators (e.g., +,-,*,.,ˆ,=), and all the integers from 0 to
300 which are designed to split numbers into individual digits. The vocabulary can cover most EV
calculations in risky choice and PV calculations in intertemporal choice tasks.

Positional embedding. Absolute positional embedding was learned during training through an
embedding layer. Each position had a corresponding 320 dimensional embedding vector.

Attention Mask. To ensure that attention is only applied to the left in the input sequence, we
incorporated a causal mask to prevent attending to future tokens when predicting the next one.

3.2 SYNTHETIC DATASETS

At the heart of the EV and PV computations lies the multiplication of two real numbers. Typically,
each outcome appears in the computation as either a probability multiplied by a value, or a discount
factor multiplied by a value. Probabilities are real numbers ranging from 0 to 1, represented with a
precision of up to two decimal places. Similarly, values are real numbers that range from 0 to 300,
also with a maximum of two decimal places. We selected these ranges to align with the numerical
scope of human experimental studies. A single training example involves either the addition or
subtraction of two simulated outcomes, which together constitute the left-hand side of an equation.
We then compute the corresponding result and place it on the right-hand side of the equation. In
total, we randomly simulated 1M such equations.

In our experiment, we evaluate four variants of the data generation algorithm by manipulating the
frequency of probabilities and values (see Table 3). The Uniform synthetic data generates proba-
bilities and values with maximum uncertainty; probabilities are uniformly distributed between 0 and
1 (i.e., U [0, 1]), and values range from 0 to 300 (i.e., U [0, 300]). Conversely, the Ecological syn-
thetic data generates probabilities following a Beta distribution Beta(0.27, 0.27) (Zhu et al., 2020)
and values according to a power-law distribution with an exponent of
0.945 for the same range
(Stewart et al., 2006). These distributions are chosen because they have been shown to closely match
the natural frequencies of probabilities and values in real-world scenarios (Stewart et al., 2006; Zhu
et al., 2020) (see Figure 1B for details). For both uniform and ecological synthetic datasets, we
also created a matching dataset where the answers on the right-hand side are generated with a 50%
chance of displaying the incorrect sign (i.e., the ablated variants in Table 3). In each of the four

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synthetic datasets, we randomly masked 10% of the probability values using a special <AMB> token
to denote unknown probabilities.

3.3 PRETRAINING DETAILS

We trained our Arithmetic-GPT models from scratch with a context length of 26. Batch size was
set to 2048 lines of mathematical equations, randomly sampled from the synthetic datasets, and the
3. To optionally terminate the training process, we designated 90% of the
learning rate was 10−
synthetic dataset as the training set, reserving the remaining 10% for validation. We used cross-
entropy loss to measure the discrepancy between the predicted and target sequences. Training was
stopped when the validation loss plateaued, with validation loss evaluated every 100 epochs (see
Figure A1 in Appendix C for an example learning curve). The AdamW optimizer was used.

3.4 HUMAN TARGETS

To investigate whether Arithmetic-GPT contains information pertinent
to explaining human
decision-making, we reanalyzed recent experiments in which people were asked to make risky and
intertemporal choices (Peterson et al., 2021; Erev et al., 2017; Gershman & Bhui, 2020; Agrawal
et al., 2023). The primary reason for examining these particular types of human choices is that
calculations of expected value are integral to making rational decisions (see Equations 1 and 2). In
other words, they are computationally equivalent tasks under the assumption of rationality.

As summarized in Table 2, we sampled four existing datasets from the literature, which included two
large-scale experiments (Peterson et al., 2021; Agrawal et al., 2023). In experiments involving risky
choices (Peterson et al., 2021; Erev et al., 2017), participants were often presented with two options,
each fully describing the details of a gamble (see Figure 1A risky choices).
In cases involving
ambiguous gambles where probabilities are unknown, we used the special token <AMB> to denote
gambles with unknown probabilities. We excluded decision-with-feedback trials, as these would
require additional assumptions about how individuals respond to feedback (but see Table A2 of
Appendix B for a comparison based on the entire choices13k dataset). In intertemporal choice
tasks (Gershman & Bhui, 2020; Ch´avez et al., 2017; Agrawal et al., 2023), participants were also
presented with two options, typically offering a choice between a smaller, sooner payoff and a larger,
later payoff (see Figure 1A intertemporal choices). Without loss of generality, we fixed the annual
discount factor at dyear = 0.85 throughout the paper. This also corresponds to a monthly discount
factor of dmonth = 0.98 and a daily discount factor of dday = 0.99. We rescaled the values in both
options by the same factor to fit within the specified range.

Table 2: Overview of human experiments and data sources

No. Participants No. Problems
Paper
15,153
Peterson et al. (2021)
Erev et al. (2017)
446
Gershman & Bhui (2020) gershman20 Intertemporal choices 221
Agrawal et al. (2023)

Dataset
choices13k Risky choices
cpc18
Risky choices

agrawal23 Intertemporal choices 12,906

13,006
270
4,794
9,853

Domain

Note. In our analysis of the choices13k and cpc18 datasets, we excluded trials involving risky
choices made with feedback. Modeling how individuals respond to feedback requires additional
cognitive mechanisms beyond the scope of this work.

4 OTHER MODELS

We also conduct a model comparison on human data, evaluating the following approaches: (i) clas-
sical behavioral models such as prospect theory (Kahneman & Tversky, 2013) and the hyperbolic
discounting model (Laibson, 1997); (ii) a neural network directly trained on the human datasets; (iii)
an untrained Arithmetic-GPT model; and (iv) an off-the-shelf, open-weight LLM, LLaMA-3-70B-
Instruct.2

2https://llama.meta.com/llama3/

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Classical behavioral models. To explain human risky choices, prospect theory proposes an S-
shaped utility function that is concave for gains, convex for losses, and steeper for losses than for
gains (reflecting loss aversion). The utility function is represented mathematically as:

U (x) =

(cid:26)xα,

λxβ,

−

if x
0
≥
if x < 0

(3)

where x denotes the value, while the shape parameters α and β define the curvature of the utility
1 reflects loss aversion. The theory further suggests that individuals
function. The parameter λ
possess a distorted perception of probabilities, modeled as follows:

≥

w(p) =

pγ

(pγ + (1

p)γ)1/γ

−

(4)

where p is the objective probability, and γ is a parameter controlling the curvature of the weighting
function. Consequently, the utility of a gamble is formally expressed as:

U (A) =

(cid:88)

A

i

∈

w(pi)

×

U (xi)

(5)

Contrary to the consistent risk preferences implied by Equation 1 or other monotonic transformations
of value, prospect theory suggests that risk preferences are inconsistent across different values and
probabilities. This inconsistency results in incoherent choices when individuals are faced with risky
decisions (Tversky & Kahneman, 1992).

To capture human time preferences, particularly the impact of present bias, the hyperbolic discount-
ing model suggests that future values should be discounted as follows:

P V (xt) =

xt
1 + kt

(6)

where xt is the value to be received at future time t, and k is the discount factor that quantifies the
degree of time preference. In contrast to the consistent time preferences implied by Equation 2, the
hyperbolic discounting model suggests that time preferences are inconsistent across different time
horizons, leading to stronger preference to immediate over future rewards (Laibson, 1997).

MLPs directly trained on human choices. We also implemented Multilayer Perceptrons (MLPs)
with a single hidden layer containing 320 neurons and using the sigmoid activation function. These
MLPs were directly trained on each of the four human datasets, using the stimuli features as input
to predict choice rates. These MLPs potentially capture an upper bound of the explainable vari-
ance within human data (Agrawal et al., 2020). However, they may have overlooked significant
constraints from the original psychological experiments, as these models use task features as input
rather than the actual stimuli presented in texts or the stylized representations required for rational
agents, as in our Arithmetic-GPT.

Choice probabilities and embeddings from open-weight LLMs. To further investigate the impact
of different input formats and compare with off-the-shelf LLMs, we evaluated the performance of
LLaMA-3-70B-Instruct on human choice data. Unlike Arithmetic-GPT, the LLaMA3 model not
only excels in arithmetic tasks but also comprehends and generates human-like text. This capability
results from its training on extensive text data and a significantly larger number of model parameters.
In short, LLaMA3 is a more versatile and powerful model, also based on the transformer architecture
and trained autoregressively.

Given these features of LLaMA3, we presented each choice problem in two different formats: text-
based and arithmetic equation-based. The text format converts each choice problem into a descrip-
tive narrative, simulating the stimuli presented to human participants (see Appendix A for a detailed
description of the text prompts). We instructed the model to report its selection between the two
options, using the log probability of the chosen option to determine the model’s predicted choice
rates for the corresponding option. In contrast, the arithmetic-equation format presents the choice
problems as a series of arithmetic computations required by a rational agent. Note that this format
is identical to the input used for Arithmetic-GPT. We obtained the embeddings from LLaMA3 for
each choice problem represented in the arithmetic-equation format.

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Table 3: Proportion of the variance of human choices explained (R2) by embeddings from the
pretrained Arithmetic-GPT model compared to other computational models. Bold numbers indicate
the best models within each group.

Training data
Model
Synthetic (unif.)a
Arith.-GPT
Synthetic (unif. abl.)b
Arith.-GPT
Synthetic (eco.)c
Arith.-GPT
Synthetic (eco. abl.)d
Arith.-GPT
Nonee
Arith.-GPT
LLaMA3 (txt.)f
Undisclosedg
LLaMA3 (txt emb.)h Undisclosedg
Undisclosedg
LLaMA3 (arith.)i
Human choices
Prospect theory
Hyperbolicj
Human choices
Expected valuek
None
Human choices
MLP

Risky choices

Intertemporal choices

choices13k cpc18 gershman20 agrawal23

69.3%
57.5%
70.8%
61.4%
21.0%
14.2%
66.2%
63.6%
51.5%
N/A
31.6%
82.7%

63.2%
37.9%
65.5%
33.8%
28.4%
8.3%
55.7%
34.8%
54.2%
N/A
43.9%
97.8%

64.0%
59.8%
67.8%
60.7%
10.8%
4.0%
66.5%
69.3%
N/A
53.4%
41.7%
60.6%

96.1%
81.1%
95.5%
80.1%
21.1%
3.0%
98.0%
96.0%
N/A
36.1%
26.1%
94.0%

Note. aUniform synthetic datasets. bThe same uniform synthetic datasets but with the answers on
the right-hand side of the equations removed and the signs randomized between positive and
negative. cEcological synthetic datasets. dThe same ecological synthetic datasets but with the
answers on the right-hand side of the equations removed and the signs randomized between
positive and negative. eEmbeddings from an untrained Arithmetic-GPT model with randomly
initialized weights. f Log probabilities of LLaMA3 elicited from text descriptions of choice
problems. gThe training data is not publicly available, but Meta has disclosed some summary
statistics of the training corpus. hEmbeddings of LLaMA3 elicited from text descriptions of choice
problems. iEmbeddings of LLaMA3 elicited from arithmetic equations of choice problems. jThe
hyperbolic discounting model for intertemporal choices. kLogistic regression results when applied
to the expected value difference between the two choice options.

5 EXPERIMENTAL RESULTS

5.1 MODEL COMPARISONS

−

In this section, we present the experimental results from our model comparisons. We first obtained
embeddings from Arithmetic-GPT and LLaMA3 models, evaluating versions of the model that were
pretrained on each of our four distinct synthetic datasets as well as a version without any training.
Specifically, we extracted embeddings for the expected values of the two options, denoted as eA
and eB. Additionally, we obtained embeddings for the difference in expected values, denoted as
B. All embeddings were derived from the representation in the final layer before the autoregres-
eA
sive prediction. We then performed a logistic regression using eA, eB, and eA
B as independent
variables, with human choice probabilities as the dependent variable. Adjusted R2 values were used
for all logistic regressions, including those for Arithmetic-GPT, LLaMA3, and EV results. All other
R2 values were reported as the squared Pearson’s r between model predictions and human data.
These results indicate that the embeddings from the Arithmetic-GPT model pretrained on ecologi-
cally valid synthetic datasets most accurately capture human choices. This model also outperforms
the embeddings obtained from the LLaMA3 model (the 7th row of Table 3), suggesting that pre-
training on synthetic datasets is sufficient to create a strong correspondence between LLMs and
human decision-making. However, the LLaMA3 model performs comparably to Arithmetic-GPT
in predicting intertemporal choices. The log probabilities from the same LLaMA3 model perform
poorly in comparison to human data (the 6th row of Table 3), replicating previous findings using
choice rates reported from LLaMA1 (Binz & Schulz, 2023a). The R2 values between Arithmetic-
GPT pretrained on uniform and ecological synthetic datasets are small yet consistent. This may be
due to the limited range tested in the human datasets.

−

To benchmark the performance of Arithmetic-GPT models in explaining human data, we directly
fit behavioral models and MLPs on each of the human choice datasets. Prospect theory and the

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Figure 2: Embeddings from ecologically pretrained Arithmetic-GPT for inputs including (A) prob-
abilities, (B) values, and (C) discount factors. Inputs are shown along the horizontal axes and em-
beddings are shown on the vertical axes. The embeddings, shown as black dots, were reduced to 1D
using multidimensional scaling. Embeddings for probabilities and discount factors are normalized
between 0 and 1 (see Appendix D for details). The red curves represent the best-fitting behavioral
economic models: (A) the probability weighting function from PT with best-fitting γ = 0.58, (B)
the utility function from PT with best-fitting α = 0.42, β = 0.45, λ = 1.4, and (C) the hyperbolic
discount function with best-fitting k = 0.08.

hyperbolic discounting model are leading behavioral models in risky and intertemporal choices, re-
spectively (Kahneman & Tversky, 2013; Laibson, 1997). The behavioral models have interpretable
mechanisms and contain few free parameters. However, they do not explain human data as well as
the embeddings from both LLaMA3 and Arithmetic-GPT, although they still outperform a simple
choice model based on EV difference. Moreover, prospect theory and the hyperbolic discounting
model do not generalize to both risky and intertemporal choices, resulting in N/A values in their
respective rows of Table 3.

In contrast, fitting an MLP directly on human data potentially reveals the ceiling performance of any
model in explaining the data (Agrawal et al., 2020; Peterson et al., 2021). Except for the intertem-
poral choice tasks, MLPs outperform all other models. It is important to note that MLP training
was based on task features rather than text descriptions that simulated participants’ experiences or
arithmetic equations that mimic a rational agent’s computations. Consequently, these differences
in input formats could also lead to diverging performance. The arithmetic format of intertemporal
choice may provide a better fit for human data. Indeed, there is increasing evidence from experimen-
tal economics suggesting that the complexity of discounting values, even in the absence of actual
payoff delays, influences intertemporal choices (Enke et al., 2023). We include a robustness check
using 10-fold cross-validation in Appendix B.

5.2

IMPLICIT FUNCTIONS OF PROBABILITIES, VALUES, AND DISCOUNT FACTORS

To understand why Arithmetic-GPT, pretrained to calculate expected values with ecological distri-
butions of probabilities and values, can capture human choices, we examined the implicit functions
of probabilities, values, and times derived from the model embeddings. We extracted embeddings
for probabilities ranging from 0.0 to 1.0, values ranging from
300 to +300, and discount factors
ranging from 0.990 to 0.9930. These high-dimensional embeddings were then reduced to a single
dimension using multidimensional scaling with Euclidean distance. Additionally, for probabilities
and discount factors, we normalized the embeddings to a range between 0 and 1.

−

We find that the embeddings from ecologically pretrained Arithmetic-GPT replicate classical find-
ings from behavioral economics, including value and probability weighting functions from the
prospect theory (Tversky & Kahneman, 1992) and the hyperbolic discounting function (Laibson,
1997). Specifically, probabilities close to 0.5 are more similar to each other than to probabilities
close to either 0 or 1 (see Figure 2A and Equation 4). Embeddings for values illustrate concavity
for positive values, convexity for negative values, and a steeper slope for negative values than for
positive values (see Figure 2B and Equation 3). These features reflect risk aversion for gains, risk
seeking for losses, and loss aversion (Kahneman & Tversky, 2013; Tversky & Kahneman, 1992).
Moreover, embeddings for the discount factor demonstrate that distant times are more similar (see

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Figure 2C and Equation 6), enabling a present-bias (Meier & Sprenger, 2010). However, embed-
dings from Arithmetic-GPT pretrained on non-ecological datasets (i.e., ablated ecological, uniform,
and ablated uniform) and those from LLaMA3 exhibit fewer human-like distortions in their implicit
functions (see Appendix E for details).

For the probability weighting function, the curvature parameter (γ) of human participants was found
to be 0.61 for gains and 0.69 for losses (Tversky & Kahneman, 1992). Follow-up replications by
(Wu & Gonzalez, 1996) found γ values around 0.7. Moreover, the utility curvature parameters (α
and β) typically range between 0.5 and 0.9, while the loss aversion parameter (λ) is approximately
2.25 (Tversky & Kahneman, 1992). Regarding human intertemporal choices, typical k values range
between 0.01 and 0.1 for small magnitude studies (Odum, 2011). Comparing to these best-fitting
parameters from human experiments, we observe that the implicit functions derived from the embed-
dings of Arithmetic-GPT also quantitatively match those observed in humans (see Figure 2). The
implicit functions, however, exhibit discontinuities, suggesting that the smooth functions derived
from human theories may not be directly applicable to explain the embeddings of Arithmetic-GPT.

6 DISCUSSION

We introduced an approach to transforming language models into cognitive models by pretraining
them on tasks that mirror the computational process performed by a rational agent. Specifically, we
pretrained Arithmetic-GPT to calculate expected values using ecologically distributed probabilities
and values. This pretraining allows the model to better capture human risky and intertemporal
choices compared to classical behavioral models and, in some cases, even surpass the performance
of the general-purpose LLaMA3 model. These results suggest that the observed cognitive biases
in LLMs may stem from the training task, architecture, and numerical reasoning abilities, without
requiring natural language supervision. These results have implications for a number of questions in
cognitive science and machine learning, although we also note the limitations of our current work.

Language Models as Cognitive Models. There is a growing research effort focused on exploring
LLMs as scientific tools for understanding the human mind (Binz & Schulz, 2023a; Frank, 2023b;
Horton, 2023). Despite the challenges outlined in Section 2, LLMs offer unique opportunities for
investigating cognitive processes in ways that are not feasible with human participants. Recent
studies have even demonstrated that fine-tuning a LLaMA1 model on human data allows the model
to outperform classical models in explaining this data (Binz & Schulz, 2023a). While the LLaMA
series’ model architecture and weights are publicly available, researchers lack access to the training
data, which makes crucial scientific inference practically impossible.
In contrast, we explicitly
manipulate the training data for our Arithmetic-GPT, thereby uncovering a key factor that contributes
to the model’s ability to explain human choices. This targeted approach, however, comes at the cost
of the broader versatility inherent in off-the-shelf LLMs, which are designed to handle a wider
range of tasks. Notably, language models, including Arithmetic-GPT, do not directly model human
cognitive processes. Instead, our work suggests that these models are better suited for probing the
computational problems that human cognition aims to solve.

Bayesian Models of Cognition, Meta-learning, and Pre-training. Bayesian models of cognition
have been instrumental in understanding human performance across a variety of cognitive tasks by
providing optimal solutions to the inductive inference problem. Recently, it has been argued that
Bayesian models and neural network models can be viewed as complementary to one another (Grif-
fiths et al., 2023). Neural networks that are meta-trained have also been shown to exhibit properties
consistent with Bayesian models (Lake & Baroni, 2023; McCoy & Griffiths, 2023). Moreover, pre-
training a neural network model for improved performance on downstream tasks can be seen as
a form of meta-learning or the acquisition of useful inductive biases, similar to Bayesian models
(Hsu et al., 2018). Our work makes the implicit priors learned by Arithmetic-GPT more explicit by
specifying the synthetic dataset on which it was pre-trained.

Computationally Equivalent Tasks for Cognitive Modeling. Modeling human cognition is chal-
lenging because the hypothesis space of possible cognitive mechanisms that can explain human data
equally well is vast. This makes principles of rationality desirable, as assuming people are rational in
some sense greatly constrains the hypothesis space, thereby making scientific inference more effec-
tive. However, human rationality has been a subject of debate in economics and psychology for over
a century (von Neumann & Morgenstern, 1944; Kahneman & Tversky, 2013; Simon, 1997). While

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significant progress has been made in understanding human rationality (e.g., Lieder & Griffiths,
2020; Gershman et al., 2015), the advent of LLMs seems to challenge the need for rational theories.
Simply training LLMs to predict the next word appears sufficient to produce human-like behaviors,
suggesting that we can model human cognition without the constraints imposed by rational theories.
However, our experimental results suggest an alternative route to enhancing the correspondence be-
tween behaviors produced by LLMs and humans: pretraining on computationally equivalent tasks
that a rational agent would need to master. Future research should investigate the impact of different
assumptions about the nature of rationality on task content and distributions, and explore whether
there are more effective assumptions for pre-training models to explain human behavior.

Implications for Theories of Human Risk and Time Preferences. The success of Arithmetic-
GPT in explaining human risky and intertemporal choices has significant implications for theoretical
work on human risk and time preferences. While the Homo economicus portrayal of human beings
as perfectly rational and self-interested agents has been inadequate in describing human choices
(Gigerenzer & Gaissmaier, 2011; Kahneman & Tversky, 2013), existing behavioral models that
deviate from rationality do not generalize well across different task domains. For instance, it is
challenging to use prospect theory to model time preferences or to use the hyperbolic discounting
model to explain risk preferences. In contrast, Arithmetic-GPT has demonstrated substantial trans-
ferability across task domains. The same model, in principle, can be adapted to other judgment
and decision-making tasks such as social choice, probability judgment, and belief updating. The
key factor enabling language models to generalize across a wide range of tasks is the presence of a
language interface, which underpins a significant range of cognitive tasks.

The Importance of Training Data Disclosure. Our results demonstrate that the training data used
for LLMs is crucial for understanding their emergent capabilities and the inductive biases they ac-
quire during pretraining. Adjusting the distribution of, or ablating, the training data can significantly
affect the degree to which LLMs correspond with human behaviors. These findings suggest that
existing off-the-shelf LLMs, whether proprietary or open-weight, including the GPT series (Brown
et al., 2020) and the LLaMA series (Touvron et al., 2023), are less effective as models of human
cognition. This is primarily because their training data is rarely disclosed, making it difficult for
scientists to control for data contamination and thereby precisely identify the sources of human-like
behaviors in these models.

Limitations and Future Research. While we have made progress in addressing many challenges
associated with using language models as cognitive models, some issues remain unresolved. To ad-
dress the data gap between LLMs and human learners, we limited the scope of our synthetic dataset
to the arithmetic of expected value calculations. Despite this, LLMs still require a substantial amount
of training data to perform arithmetic accurately within a limited range of input values. Moreover,
it is unrealistic for human learners to process 1M randomly generated mathematical equations, as
Arithmetic-GPT did, to acquire the skill of computing expected values. Further research is needed
to continue bridging this data gap (but see Appendix F for an initial attempt).

Additional ablation studies could be performed on model architectures and training objectives. Our
work is fundamentally limited to autoregressive training and a decoder-only transformer architec-
ture. We believe that alternative training mechanisms and model architectures could potentially
yield better embeddings from language models. One area for future research is estimating the lower
bounds on model size necessary to achieve a certain level of correspondence with human behavior.
Another area of future work involves leveraging interpretability techniques to distill novel cognitive
mechanisms from Arithmetic-GPT.

Conclusion. Large language models have opened new horizons for research on human cognition,
but also introduce a new set of challenges based on the volume of training data, the content of those
data, the influence of value alignment, and limited access to the training regimes and weights of
these models. We have proposed an approach to addressing these challenges, based on training
small language models with datasets that are generated based on tasks that are hypothesized to be
computationally related to the problem that human minds face. Our results show that this approach
is extremely effective in predicting human decisions, where training on arithmetic results in repre-
sentations that can be used to predict human choices better than both existing psychological models
and large language models trained on broader datasets. This approach is easily generalizable to other
cognitive tasks that primarily rely on language interface, and can even be used with other kinds of
foundation models to study human perception.

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A PROMPTS

EXAMPLE OF A RISKY CHOICE DESCRIBED IN TEXT

You are participating in a gambling game where you will be

System:
shown two options, Gamble A and Gamble B. Your task is to choose
between the two. You must select one option.

Gamble A offers a 10% chance to win $10 and a 90% chance to

User:
lose $12. Gamble B offers a 40% chance to lose $13 and a 60%
chance to win $22. Please limit your answer to either ‘A’ or ‘B’.

EXAMPLE OF AN INTERTEMPORAL CHOICE DESCRIBED IN TEXT

System: You are participating in an intertemporal choice experiment.
You will be presented with two options, A and B, and your task is
to choose between them.

You must select one option.

User:
immediately. Please limit your answer to either ‘A’ or ‘B’.

Option A offers $80 after 30 days.

Option B offers $13

B ROBUSTNESS CHECK

We supplement our main results with a robustness check using 10-fold cross-validation. Each dataset
of human choices was randomly partitioned into a 90% training set and a 10% validation set. For
the gershman20 dataset, a stratified train/validation split was employed due to the certainty equiv-
alence design of the experiment (Gershman & Bhui, 2020), ensuring that choices made within the
same problem were grouped together.

Embeddings from Arithmetic-GPT models and LLaMA-3-70B-Instruct were fitted using logistic
regression with a LASSO penalty on the training set. MLPs were fitted to task features in the
training set. Performance for all models was assessed on the validation set, with R2 calculated as the
squared Pearson’s r. This process was repeated 10 times, each with a random train/validation split.
The mean and standard error (SE) of R2 are summarized in Table A1, demonstrating a replication
of our main results presented in Table 3. Finally, we conducted a model comparison using the entire
choices13k dataset, including the decision-from-feedback trials (see Table A2). We observe that
LLaMA3 can outperform MLP on the agrawal23 dataset. While the two neural network models
use different task formats as inputs, which may contribute to the differences in performance, it is
also possible that the dataset lacks sufficient statistical power to reliably distinguish between the two
models.

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Under review as a conference paper at ICLR 2025

Table A1: Cross-validation results. Propotion of the variance in human choices explained (R2) on
the validation set for each model. Numbers in parentheses represent standard errors from 10-fold
cross-validation.

Model

Training data

Arith.-GPT

Synthetic (unif.)

Arith.-GPT

Synthetic (unif. abl.)

Arith.-GPT

Synthetic (eco.)

Arith.-GPT

Synthetic (eco. abl.)

Arith.-GPT

None

LLaMA3 (txt emb.) Undisclosed

LLaMA3 (arith.)

Undisclosed

MLP

Human choices

Risky choices

Intertemporal choices

choices13k cpc18 gershman20 agrawal23

56.4%
(0.4%)
38.5%
(1.5%)
57.7%
(0.4%)
38.5%
(1.5%)
33.1%
(1.5%)
54.0%
(0.8%)
38.6%
(1.5%)
62.3%
(1.2%)

46.3%
(3.1%)
21.9%
(4.6%)
51.7%
(4.9%)
33.6%
(5.0%)
23.5%
(3.8%)
35.0%
(5.0%)
20.8%
(3.0%)
50.3%
(3.9%)

56.7%
(1.1%)
51.1%
(0.7%)
57.1%
(1.2%)
55.3%
(1.2%)
15.3%
(1.7%)
52.7%
(1.1%)
54.9%
(1.2%)
58.7%
(1.3%)

81.5%
(0.4%)
54.0%
(0.3%)
83.7%
(0.3%)
68.3%
(0.2%)
50.4%
(5.1%)
89.9%
(0.1%)
87.6%
(0.1%)
84.9%
(2.4%)

Note. The bold numbers represent the top-performing variants of the Arithmetic-GPT models, as
measured by R2. In cases where multiple models are highlighted, the differences in R2 values are
not statistically significant at the p = 0.05 level.

Table A2: Proportion of the variance in human choices explained (R2) on the entire choices13k
dataset for each model. Bold numbers indicate the best models within each group.

Training data
Model
Synthetic (unif.)
Arith.-GPT
Synthetic (unif. abl.)
Arith.-GPT
Synthetic (eco.)
Arith.-GPT
Synthetic (eco. abl.)
Arith.-GPT
LLaMA3 (txt.)
Undisclosed
LLaMA3 (txt emb.) Undisclosed
Undisclosed
LLaMA3 (arith.)
Human choices
Prospect theory
None
Expected value
Human choices
MLP

Entire choices13k
64.0%
49.2%
64.3%
52.1%
14.6%
65.6%
60.6%
45.6%
40.2%
73.8%

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Under review as a conference paper at ICLR 2025

C LEARNING CURVE

(A) Training loss decreases over the course of training epochs for Arithmetic-GPT
Figure A1:
trained on ecological synthetic dataset. (B) Histogram displaying the differences between the top-1
responses of the pretrained Arithmetic-GPT model and the actual expected values in the ecological
synthetic dataset.

D DIMENSIONALITY REDUCTION DETAILS

We used multidimensional scaling to reduce the dimensionality of embeddings to 1D, with Euclidean
distance serving as the dissimilarity metric for forming the 1D manifolds. Embeddings of proba-
bilities and discount factors were normalized using max-min normalization to ensure the resultant
values fell within the range of [0, 1].

E ADDITIONAL IMPLICIT FUNCTIONS

Figure A2: Visualizations of embeddings from LLaMA-3-70B-Instruct.

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Figure A3: Visualizations of embeddings from Arithmetic-GPTs pretrained on different synthetic
datasets. (A) Uniform synthetic dataset. (B) Ablated uniform synthetic dataset. (C) Ablated ecolog-
ical synthetic dataset.

F VARIATIONS IN MODEL SIZES, DATA QUANTITY, AND DATA

DISTRIBUTIONS

In this section, we explore how adjustments to key hyperparameters of Arithmetic-GPT impact the
quality of its pretrained embeddings. Specifically, we investigate the effects of reducing the model’s
hidden size from 320 (as reported in the main text) to 104 and 16. Moreover, we examine the in-
fluence of decreasing the size of the ecological synthetic dataset, reducing it from 1M mathematical
equations (as reported in the main text) to subsets of 100K and 10K equations. Apart from these
modifications, we adhered to the same pretraining procedure and evaluated the resulting embeddings
on the two largest datasets, choices13k for risky choices (Peterson et al., 2021) and agrawal23
(Agrawal et al., 2023) for intertemporal choices, using 10-fold cross-validation.

The results, summarized in Table A3, reveal an intriguing pattern: reducing the size of the ecological
synthetic dataset to as few as 10K equations does not significantly impair the ability of the 10M
model’s embeddings to predict human choices. It is important to note that Arithmetic-GPT models
were never exposed to human data during pretraining. However, significant reductions in the model

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A0.01.00.50.250.75ProbabilitiesEmbeddings-300+300+100+200Embeddings-100-2000ValuesDiscount factorsEmbeddings0.99^00.99^300.99^100.99^20Arithmetic-GPT pretrained on uniform synthetic datasetB0.01.00.50.250.75ProbabilitiesEmbeddings-300+300+100+200Embeddings-100-2000ValuesDiscount factorsEmbeddings0.99^00.99^100.99^20Arithmetic-GPT pretrained on ablated uniform synthetic datasetC0.01.00.50.250.75ProbabilitiesEmbeddings-300+300+100+200Embeddings-100-2000ValuesDiscount factorsEmbeddings0.99^00.99^100.99^20Arithmetic-GPT pretrained on ablated ecological synthetic dataset0.99^300.99^30Under review as a conference paper at ICLR 2025

size – particularly 30K parameters – lead to a decline in performance, highlighting the critical role
of model capacity in embedding quality.

Table A3: Variants of Arithmetic-GPT models pretrained on ecological synthetic datasets of vary-
ing sizes. In each cell, R2 results (in percentage) are reported as choices13k (left of ‘/’) and
agrawal23 (right of ‘/’). Numbers in parentheses indicate standard errors obtained from 10-fold
cross-validation.

10M
(hidden size=320)

Model Sizes
1M
(hidden size=104)

30K
(hidden size=16)

1M 57.7 (0.4) / 83.7 (0.3) 54.7 (0.6) / 83.4 (0.3) 30.6 (0.4) / 56.8 (0.4)
Data quantity 100K 57.2 (0.6) / 83.6 (0.4) 55.5 (0.5) / 81.6 (0.4) 31.4 (0.8) / 63.4 (0.2)
10K 57.7 (0.5) / 83.6 (0.3) 50.0 (0.5) / 84.1 (0.3) 29.1 (0.6) / 63.0 (0.4)

Note. The bold numbers represent the top-performing variants of the Arithmetic-GPT models, as
measured by R2. In cases where multiple models are highlighted, the differences in R2 values are
not statistically significant at the p = 0.05 level.

We conducted an exploratory analysis using the original Arithmetic-GPT model architecture and
synthetic datasets of 1M equations to investigate the effects of varying data distributions. Specif-
ically, we independently manipulated the distributions of probabilities and values. For probability
distributions, we considered Beta(0.27, 0.27) (ecological), Beta(1, 1) (uniform), and Beta(2, 2). For
value distributions, we examined power-law distributions with exponents of 0 (uniform), -0.945
(ecological), and -2. All Arithmetic-GPT models were randomly initialized and trained following
the procedure outlined in Section 3.1.

As shown in Table A4, aligning probability distributions with ecological patterns, such as Beta(0.27,
0.27), generally enhances the quality of embeddings for predicting risky choices. Conversely, adopt-
ing a power-law distribution for values with an exponent of -2 tends to improve the model’s perfor-
mance in predicting intertemporal choices.

Table A4: Variants of Arithmetic-GPT models pretrained on different data distributions. In each
cell, R2 results (in percentage) are reported as choices13k (left of ‘/’) and agrawal23 (right
of ‘/’). Numbers in parentheses indicate standard errors obtained from 10-fold cross-validation.

Distribution of probabilities
Beta(1,1)
56.5 (0.5) / 82.5 (0.4) 56.4 (0.4) / 81.5 (0.4) 56.0 (0.3) / 77.5 (0.4)

Beta(0.27,0.27)

Beta(2,2)

0

Distribution of values
(exponent of power-law)

-0.945 57.7 (0.4) / 83.7 (0.3) 56.7 (0.3) / 78.4 (0.5) 56.5 (0.3) / 77.6 (0.4)

-2

55.7 (0.3) / 85.9 (0.3) 56.3 (0.5) / 82.6 (0.4) 55.5 (0.4) / 82.2 (0.4)

Note. The bold numbers represent the top-performing variants of the Arithmetic-GPT models, as
measured by R2.

G COMPUTATIONAL ABILITIES OF LANGUAGE MODELS

We evaluate the abilities of LLaMA-3-70B-Instruct and Arithmetic-GPT in computing expected
values using two newly generated test datasets. Both test sets consist of 20K synthetically generated
equations involving expected value calculations. In one test set, the probabilities and values in the
equations are sampled from a uniform distribution, while the other test set features probabilities and
values derived from ecological distributions, as illustrated in Figure 1. The train-test relationships
for these datasets are summarized in Table A5.

The computational ability of Arithmetic-GPT models was assessed by calculating the probability
of generating correct expected values (rounded to two decimal places). Formally, the correct prob-
ability is computed as Pmodel(true expected values
the left-hand side of the equation), using token
probabilities. Arithmetic-GPT, when pretrained on ecological distributions, demonstrated higher

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Under review as a conference paper at ICLR 2025

probabilities of generating correct values across both uniform and ecological test sets. When com-
paring performance on the same model across the two test sets, expected values in the ecological
test set were found to be easier to predict. One possible explanation for the improved performance
of Arithmetic-GPT models pretrained on ecological distributions is that these distributions are more
“bursty,” characterized by multiple occurrences of similar numerical values (cf. Chan et al., 2022).
Pretraining on bursty sequences has been shown to facilitate emergent behaviors in LLMs, such as
in-context learning (Chan et al., 2022).

To evaluate LLaMA3’s ability to compute expected values, we directly prompted the model to solve
arithmetic equations while setting the temperature to 0. The integer part of the expected values
was extracted from LLaMA3’s responses by filtering out irrelevant tokens. The correct probability
was calculated as the relative frequency of correctly predicting the integer part of the true expected
values across 20K questions. As shown in Table A5, LLaMA3 performed poorly compared to
Arithmetic-GPT, achieving only 20.97% and 38.79% accuracy on the uniform and ecological test
sets, respectively.

The model’s behavioral performance on the ecological test set broadly predicts the ranking of its
embeddings in modeling human data. However, directly interpreting a language model’s perfor-
mance on arithmetic tasks in relation to model-fitting results that leverage its embeddings can be
challenging. Therefore, further investigation is necessary to establish a robust connection between
a language model’s arithmetic capabilities and the predictive power of its embeddings in predicting
human choices.

Table A5: Evaluation of language models’ probabilities of generating correct expected values across
different train-test distributions. Numbers in parentheses represent standard errors.

Model
Training data
Arith.-GPT Synthetic (unif.) Uniform

Arith.-GPT Synthetic (eco.) Uniform

Test data Correct probability

0.8261 (0.0023)
Ecological 0.9336 (0.0014)
0.9388 (0.0013)
Ecological 0.9780 (0.0007)
Uniform
Ecological

0.2097
0.3879

LLaMA3 Undisclosed

H IMPLEMENTATION DETAILS

Here we detail the hyperparameter setup for experiments. All computations for synthetic datasets
are run on single Nvidia RTX 3060 GPU, and those for LLaMA3 embeddings are run on single
A100 GPU.

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