# Prompting for Numerical Sequences: A Case Study on
Market Comment Generation

###### Abstract

Large language models (LLMs) have been applied to a wide range of data-to-text
generation tasks, including tables, graphs, and time-series numerical data-to-
text settings. While research on generating prompts for structured data such
as tables and graphs is gaining momentum, in-depth investigations into
prompting for time-series numerical data are lacking. Therefore, this study
explores various input representations, including sequences of tokens and
structured formats such as HTML, LaTeX, and Python-style codes. In our
experiments, we focus on the task of Market Comment Generation, which involves
taking a numerical sequence of stock prices as input and generating a
corresponding market comment. Contrary to our expectations, the results show
that prompts resembling programming languages yield better outcomes, whereas
those similar to natural languages and longer formats, such as HTML and LaTeX,
are less effective. Our findings offer insights into creating effective
prompts for tasks that generate text from numerical sequences.

Keywords: Generation, Data-to-text, Large language model

Prompting for Numerical Sequences: A Case Study on
Market Comment Generation

Masayuki Kawarada, Tatsuya Ishigaki and Hiroya Takamura
---
Artificial Intelligence Research Center, AIST
<EMAIL_ADDRESS>
{ishigaki.tatsuya<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

Large language models (LLMs) have demonstrated remarkable performance in
various text-to-text natural language generation tasks, such as text
summarization Wang et al. (2023b); Zhang et al. (2023); Pham et al. (2023),
machine translation Wang et al. (2023a); Karpinska and Iyyer (2023); Vilar et
al. (2023); Agrawal et al. (2023), and dialogue systems Li et al. (2023); Jin
et al. (2023). Although LLMs have also been applied to data-to-text generation
tasks, their application has been limited to tasks where the input data are
structured and their components are represented as words, such as tables Saha
et al. (2023); Zhao et al. (2023) and structured data Jiang et al. (2023).
However, the question of how to effectively use LLMs for text generation from
numerical data remains unanswered. This paper addresses this question through
a case study on few-shot market comment generation from stock prices Murakami
et al. (2017); Aoki et al. (2018); Hamazono et al. (2021). In this task, the
stock price is provided as a numerical sequence, and the goal is to generate a
market commentary at a specific point in time. Figure 1 illustrates an example
of the market comment generation.

Figure 1: Example of market comment generation. We represent the time-series
numerical data using a line graph. The objective of this task is to generate
comments on the stock price movement at a specific time.

The key challenge in this study is to explore effective ways to represent
these numerical sequences as prompts for LLMs, because their characteristics
differ significantly from the text used to pretrain the models. We hypothesize
that better performance could be achieved by creating prompts with
characteristics similar to those used in the pretraining of LLMs. To test this
hypothesis, we compare various prompt designs with features that are both
similar to and different from the texts used for pretraining.

Specifically, we explore four categories of prompt designs for numerical
sequences: 1) directly providing a sequence of numbers; 2) treating the input
numerical sequence as a structured table and linearizing it; 3) representing
the input sequence as programming code (e.g., Python List, Python Dictionary,
HTML, and LaTeX formats), which is a common strategy in code generation
research; and 4) using templates to transform the input sequence into natural
language.

Contrary to our expectations, our experiments revealed that prompts using
expressions closer to programming languages performed better, whereas those
closer to natural languages performed worse. In addition, longer prompts, such
as HTML and LaTeX formats, were found to be less effective. These findings
suggest that converting numerical data into prompts that closely resemble the
format used during pretraining can lead to improved performance.

The main contributions of this study are threefold: 1) we present the
performance of zero-shot and few-shot approaches for generating market
commentary; 2) we compare various prompting strategies; and 3) we provide
insights into the effectiveness of different prompt designs for numerical
data.

Direct |  …9988.05 9982.06 9978.11 9972.66 9967.11, 9961.37 …
---|---
Column |  Time:15:00 14:55 14:50 14:45 14:40 14:35 …
|  Nikkei225:9988.05 9982.06 9978.11 9972.66 9967.11, 9961.37 …
Row |  Time Nikkei225\n15:00 9988.05\n14:55 9982.06\n14:50 9978.11\n14:45 9972.66\n14:40 9967.11\n14:35 9961.37 …
Python List |  Time = […, "15:00", "14:55", "14:50", "14:45", "14:40", "14:35", …]
|  Nikkei225 = […, 9988.05, 9982.06, 9978.11, 9972.66, 9967.11, 9961.37, …]
Python List (nested) |  Nikkei225 = […, [15:00, 9988.05], [14:55, 9982.06], [14:50, 9978.11], 14:45, 9972.66], [14:40,9967.11], [14:35,9961.37], …]
Python Dictionary |  Nikkei225 = {…, "15:00":9988.05, "14:55":9982.06, "14:50":9978.11, "14:45":9972.66, "14:40":9967.11, "14:35":9961.37, …}
HTML Table |

Time	Nikkei225
…(omitted)
15:00	9988.05

14:55	9982.06

14:50	9978.11

14:45	9972.66

14:40	9967.11

14:35	9961.37
 …

LaTeX Table |  \begin{table}[t] \begin{tabular} & \hline Timestamp & Nikkei225 \\\\\hline \hline 15:00 & 9988.05 \\\\\hline 14:55 & 9982.06 \\\\\hline 14:50 & 9978.11 \\\\\hline 14:45 & 9972.66 \\\\\hline 14:40 & 9967.11 \\\\\hline 14:35 & 9961.37 \\\\\hline … \end{tabular} \end{table}
Text (English) |  Nikkei225 as of 15:00 is 9982.06 yen.\nNikkei225 stock price as of 15:00 is 9988.05 yen.\nNikkei225 stock price as of 14:55 is 9982.06 yen.\nNikkei225 stock price as of 14:50 is 9978.11 yen.\nNikkei225 stock price as of 14:45 is 9972.66 yen.\nNikkei225 stock price as of 14:40 is 9967.11 yen.\nNikkei225 stock price as of 14:35 is 9961.37 yen….(omitted)
Text (Japanese) |  Nikkei225 as of 15:00 is 9982.06円.\n15:00時点のNikkei225は9988.05円.\n14:55時点のNikkei225は9982.06円.\n14:50時点のNikkei225は9978.11円.\n14:45時点のNikkei225は9972.66円.\n14:40時点のNikkei225は9967.11円.\n14:35時点のNikkei225は9961.37円….(omitted)
Table 1: Examples of [INPUT FORMAT (short-term)]. The actual prompts are written in Japanese except for Text (Japanese). Direct |  9988.05 9982.06 9978.11 9972.66 9967.11, 9961.37 9960.20
---|---
Column |  Date: 7DaysAgo 6DaysAgo 5DaysAgo 4DaysAgo 3DaysAgo 2DaysAgo 1DayAgo
|  Nikkei225: 9988.05 9982.06 9978.11 9972.66 9967.11, 9961.37 9960.20
Row |  Date Nikkei225\n7DaysAgo 9988.05\n6DaysAgo 9982.06\n5DaysAgo 9978.11\n4DaysAgo 9972.66\n3DaysAgo 9967.11\n2DaysAgo 9961.37 \n1DayAgo
Python List |  Time = ["7DaysAgo", "6DaysAgo", "5DaysAgo", "4DaysAgo", "3DaysAgo", "2DaysAgo", "1DayAgo"]
|  Nikkei225 = [9988.05, 9982.06, 9978.11, 9972.66, 9967.11, 9961.37, 9960.20]
Python List (nested) |  Nikkei225 = [["7DaysAgo", 9988.05], ["6DaysAgo" 9982.06], ["5DaysAgo", 9978.11], ["4DaysAgo", 9972.66], ["3DaysAgo",9967.11], ["2DaysAgo",9961.37], …]
Python Dictionary |  Nikkei225 = {"7DaysAgo":9988.05, "6DaysAgo":9982.06, "5DaysAgo":9978.11, "4DaysAgo":9972.66, "3DaysAgo":9967.11, "2DaysAgo":9961.37, … }
HTML |

Date	Nikkei225

7DaysAgo	9988.05

6DaysAgo	9982.06

5DaysAgo	9978.11

4DaysAgo	9972.66

3DaysAgo	9967.11

2DaysAgo	9961.37	 …

LaTeX |  \begin{table}[t] \begin{tabular} & \hline Timestamp & Nikkei225 \\\\\hline \hline 7DaysAgo & 9988.05 \\\\\hline 6DaysAgo & 9982.06 \\\\\hline 5DaysAgo & 9978.11 \\\\\hline 4DaysAgo & 9972.66 \\\\\hline 3DaysAgo & 9967.11 \\\\\hline 2DaysAgo & 9961.37 \\\\\hline … \end{tabular} \end{table}
Text (English) |  Nikkei225 as of 7 days ago was 9982.06 yen.\nNikkei225 closing stock price as of 6 days ago was 9988.05 yen.\nNikkei225 closing stock price as of 5 days ago was 9982.06 yen.\nNikkei225 closing stock price as of 4 days ago was 9978.11 yen.\nNikkei225 closing stock price as of 3 days ago was 9972.66 yen.\nNikkei225 closing stock price as of 2 days ago was 9967.11 yen.\nNikkei225 closing stock price as of yesterday was 9961.37 yen. …
Text (Japanese) |  7日前のNikkei225終値は9982.06円.\n6日前のNikkei225終値は9988.05円.\n5日前のNikkei225終値は9982.06円.\n4日前のNikkei225終値は9978.11.\n3日前のNikkei225終値は9972.66円.\n2日前のNikkei225終値は9967.11円.\n2日前のNikkei225終値は9961.37円… …
Table 2: Examples of [INPUT FORMAT (long-term)]. The actual prompts are
written in Japanese except for Text (English). Output the market comment at
the current time in the form of a <comment>market comment</comment>. ###
Input: [INPUT FORMAT (short-term)] [INPUT FORMAT (long-term)] Output:
Nikkei225 closes at large, rebounding yen strength pushes mainstay stocks
higher ### Input: [INPUT FORMAT (short-term)] [INPUT FORMAT (long-term)]
Output: Table 3: The template we use for the few-shot setting.

## 2\. Related Work

Existing data-to-text generation settings can be categorized based on the
input and output types. The input types include tables Puduppully et al.
(2019); Lebret et al. (2016), graphs Bai et al. (2022); Konstas et al. (2017),
RDF triples Gardent et al. (2017), and time-series numerical sequence Murakami
et al. (2017); Chang et al. (2022); Ishigaki et al. (2021); Kantharaj et al.
(2022). From the output types, existing studies generate e.g., bibliographies
Lebret et al. (2016), summaries of sporting events Puduppully et al. (2019),
and commentaries Ishigaki et al. (2021); Zhang et al. (2022); Chang et al.
(2022). Despite the large amount of time-series numerical data in the real
world, many existing settings deal with tabular and graphical data owing to
public benchmark datasets such as WebNLG, E2E, and RotoWire. Our study aims to
accelerate the research on time-series numerical data.

Various tasks have been proposed as time-series numerical-data-to-text
settings, such as commentary generation from user gameplay Ishigaki et al.
(2021), explaining a drone’s movement from sensor data Chang et al. (2022) and
explaining price changes in a market Murakami et al. (2017); Hamazono et al.
(2021); Aoki et al. (2018).

Two primary model architectures prevail in data-to-text settings: the encoder-
decoder and the decoder-only. A conventional approach is fine-tuning
pretrained encoder-decoder models, for example, BART Lewis et al. (2020) and
T5 Raffel et al. (2020). Recently, interest has shifted to zero-shot or few-
shot generation using decoder-only models such as GPTs Brown et al. (2020).
Several prompting methods have been proposed for the table-to-text and graph-
to-text settings Axelsson and Skantze (2023); Lorandi and Belz (2023);
however, none have been proposed for the time-series numerical-data-to-text
setting.

Time-series numerical data can be converted into tables by aligning numerical
values and timestamps. In this case, the linearization method proposed in the
existing table-to-text research can be applicable. In addition, expressions in
programming languages such as Python have been used in research on automatic
code generation. In this study, we propose using time-series numerical
sequences as such representations.

## 3\. Task

This section describes the task settings. Figure 1 presents an example of an
input and its corresponding output market comments. We use two time-series
numerical sequences derived from the Nikkei Stock Average (Nikkei225): 1) a
long-term series that records daily closing prices over the last seven days,
and 2) a short-term series that captures daily price fluctuations at five min
intervals from market opening to closing. The system generates a market
comment on the target timestamp based on these two inputs.

## 4\. Methods

This section compares the prompting methods. We assume that each number in
sequences can be aligned to a timestamp. In the example shown in Table 4, a
value 9988.05 yen in an input time-series numerical sequence can be aligned
to, for example, the timestamp of 15:00, at which the value was tracked.

Table 3 lists the prompt templates we use for zero- and few-shot generations.
In practice, we replace [INPUT FORMAT(short-term)] and [INPUT FORMAT(long-
term)] with the respective input formats in Tables 1 and 2. This prompt
contains two placeholders, i.e., [INPUT FORMAT] (short-term) and [INPUT
FORMAT] (long-term), which are replaced by one of the nine prompts listed in
Tables 1 and 2. These prompts are divided into four categories: 1) direct
prompts; 2) converting the sequence into a table and then linearizing it; 3)
converting the sequence into expressions used for computer programs; and 4)
filling a template that produces human-like language.

### 4.1. Direct Prompt (Baseline)

This method simply combines adjacent numerical values with a space to
represent a time-series numerical sequence as a sequence of space-separated
numerical values.

### 4.2. Linearized Table

Timestamp | Nikkei225
---|---
15:00 | 9988.05
14:55 | 9982.06
14:50 | 9978.11
14:45 | 9972.66
14:40 | 9967.11
14:35 | 9961.37
Table 4: Example of a table converted from a numerical sequence of market
prices.

A direct prompt does not use the information about the alignment between a
value and its timestamp. Thus, we promptly included richer information by
adding more information about the alignments. We convert a time-series
numerical sequence into a table as shown in Table 4. Each value in the input
sequence is aligned with the timestamp at which the market price is recorded.
The converted table has two rows: the first row contains a sequence of
timestamps, and the second row represents a sequence of market prices. Each
row has the headings “Time” and the name of the market index, i.e.,
“Nikkei225”, respectively.

Next, we apply linearization inspired by existing table-to-text studies. In
this category, we compare two linearization methods: Column and Row as
explained below:

Column

This method extracts the heading and values for each column and concatenates
them by adding spaces. This process is performed for two columns, and the two
token sequences are combined by placing a space to obtain the final column.

Row

As with the Column method, a sequence of space-delimited tokens is obtained
for each column, and the two series are joined by a breakline symbol (“\n”).
This includes the information that the two token sequences are generated from
different columns.

| 0-shot | 5-shot | 10-shot
---|---|---|---
| BLEU | METEOR | BERTScore | BLEU | METEOR | BERTScore | BLEU | METEOR | BERTScore
Direct | 0.01 | 0.48 | 60.30 | 8.26 | 25.22 | 73.50 | 9.39 | 26.55 | 73.96
Column | 0.38 | 14.06 | 65.33 | 8.30 | 24.99 | 73.35 | 9.49 | 26.00 | 73.65
Row | 0.42 | 8.86 | 64.83 | 9.16 | 26.33 | 73.76 | 10.49 | 27.88 | 74.31
Python List | 0.36 | 16.16 | 65.01 | 8.32 | 25.32 | 73.54 | 9.59 | 26.51 | 73.87
Python List (nested) | 0.40 | 8.94 | 65.77 | 9.15 | 26.77 | 74.01 | 9.86 | 27.42 | 74.15
Python Dictionary | 0.44 | 9.60 | 65.40 | 9.17 | 26.42 | 73.96 | 10.41 | 28.25 | 74.56
HTML Table | 0.35 | 12.26 | 63.93 | 8.30 | 26.10 | 73.92 | 8.45 | 26.44 | 74.08
LaTeX Table | 0.44 | 15.11 | 67.59 | 8.36 | 26.10 | 73.76 | 9.53 | 27.56 | 74.02
Text (English) | 0.30 | 15.66 | 60.13 | 8.49 | 25.92 | 73.97 | 9.10 | 26.95 | 74.52
Text (Japanese | 0.03 | 0.95 | 55.35 | 8.51 | 26.55 | 74.00 | 9.26 | 27.60 | 74.21
Table 5: Comparison of methods in terms of BLEU, METEOR, and BERTScore. | BLEU | METEOR | BERTScore
---|---|---|---
EncDec | 11.41 | 30.90 | 75.94
Direct | 9.39 | 26.55 | 73.96
Column | 9.49 | 26.00 | 73.65
Row | 10.49 | 27.88 | 74.31
Python List | 9.59 | 26.51 | 73.87
Python List (nested) | 9.86 | 27.42 | 74.15
Python Dictionary | 10.41 | 28.25 | 74.56
HTML Table | 8.45 | 26.44 | 74.08
LaTeX Table | 9.53 | 27.56 | 74.02
Text (English) | 9.10 | 26.95 | 74.52
Text (Japanese | 9.26 | 27.60 | 74.21
Table 6: Comparison of prompts using 10-shot and EncDec in terms of BLEU,
METEOR, and BERTScore.

### 4.3. Programming Language-like Prompts

We compare the idea of adopting the representation methods used in programming
languages, such as Python List and Dictionary. LLMs are also pre-trained on
source code extracted from repositories such as GitHub. The inclusion of
source-code in a prompt is common, especially in source code generation tasks,
in which high performance in both understanding and generation has been
reported. Therefore, it is beneficial to convert an input time-series
numerical sequence into a form similar to that of a programming language.

Python List

This method first creates two python-like lists named “Time” and “Nikkei225”.
The former list contains the timestamps as string values, e.g., ["15:00",
"14:55", "14:40", … "14:35"]. The latter list contains the numerical stock
prices as floating values [9988.05, 9982.06, …, 9961.37]. Finally, these two
lists are concatenated by a space.

Python List (nested)

This prompting method first creates a python-like list with two elements for
each timestamp and price pair, e.g., [15:00, 9988.05] means that the price is
9988.05 at 15:00. This method iteratively adds the created list into another
python list Nikkei225, as listed in Table 1.

Python Dictionary

This method converts a numerical sequence into a Python Dictionary format with
key-value pairs, where each key represents the timestamp and each value
corresponds to a stock price e.g., {"15:00": 9988.05, "14:55": 9982.06, …}.

HTML

As with the  Column method,  HTML method assumes that the time-series
numerical sequence is represented by a two-row table. This method represents a
table as an HTML code, as shown in Table 1.

LaTeX

As another format for representing a table, we also compare the Latex format.

### 4.4. Language Template-based Prompt

LLMs are trained mainly on natural language text on the Web. We hypothesize
that prompts resembling natural language work well. Thus, we propose a
template-based method for converting a time-series numerical sequence into a
natural language-like-prompt. Specifically, we use two templates:  Text
(English) and  Text (Japanese), as explained below:

Text (Japanese)

This setup uses the template “AAA時点での日経平均はBBB。”, which means “Nikkei225 as of
AAA is BBB yen”. In this template, AAA refers to the timestamp and BBB is the
corresponding market price. For each time-price pair, we obtain a sentence by
filling the slots. All obtained sentences are finally combined by a breakline
symbol (“\n”).

Text (English)

For many released LLMs, a large portion of the training data for pretraining
is in English. Thus, we also compare the English version of the prompt, i.e.,
“Nikkei225 as of AAA is BBB yen”.

## 5\. Experiments

Here, we describe our experiments; dataset, comparison methods, and
evaluations.

### 5.1. Dataset

We use the dataset in Murakami et al. (2017); Aoki et al. (2018); Hamazono et
al. (2021). The dataset contains 18,489 pairs of time-series numerical
sequences and market comments. The time-series numerical sequences are
obtained from IBISquare111http://www.ibi-square.jp/index.html. Stock price
data are collected from December 2010 to September 2016. The comments are
provided by Nikkei Quick News222”We have released the source code for
preprocessing at https://github.com/aistairc/market-reporter.. We split this
dataset into training, validation, and testing sets, comprising 15,035, 1,759,
and 1,695 data points, respectively. For this study, we define “short-term
sequence” as the sequence of stockvalues recorded every five minutes and
“long-term sequence” as the closing prices for the stock over the last seven
days. The Nikkei market operates in two sessions: 1) a morning session from
9:00 to 11:30, and 2) an afternoon session from 12:30 to 15:00. Consequently,
the short-term sequence comprised a maximum of 62 prices per day, whereas the
long-term sequence comprises seven days.

### 5.2. Compared Methods

We compare our zero- and few-shot models with an existing fine-tuned encoder-
decoder model Murakami et al. (2017).

#### Existing Encoder-decoder Model (EncDec)

We use an extended implementation of an existing model by Murakami et al.
(2017). The original implementation directly encodes long- and short-term
time-series numerical sequences using multilayer perceptrons (MLPs), and then,
the encoded vectors are fed into an LSTM-based encoder-decoder. Our extended
implementation uses a pretrained BART
333https://huggingface.co/stockmark/bart-base-japanese-news for the encoder-
decoder, which has been used more frequently in recent data-to-text settings.
Our implementation uses multilayer perceptrons (MLPs) to convert both short-
term and long-term input vectors into fixed-size vectors of size 768. This
size is selected because the embedding layer of the BART is 768. Finally, BART
then receives the two vectors and outputs the comments. This model is fine-
tuned on the abovementioned dataset to minimize the cross-entropy loss.

Figure 2: BLEU scores of different prompts on different numbers of shots.
Figure 3: BLEU scores of different prompts on different numbers of shots.

#### Proposed Zero- and Few-shot Models

We use an instruct-tuned GPT Brown et al. (2020)444In particular, we use
gpt-3.5-turbo in OpenAI’s API.. In the zero-shot setting, the prompt shown in
Table 3 is used as is. In the few-shot setting, we append a reference comment
to the prompt after Output: in the prompt. We compare up to 10 shots in our
experiments. We set the maximum number to 10 owing to the length limits of the
GPT. In preliminary experiments, we found that the performance varied
significantly depending on the instances used for the shots. Therefore, we
randomly select the instances for shots, repeat the experiments 10 times, and
report the averaged scores.

### 5.3. Automatic and Human Evaluations

In this subsection, we describe the evaluation.

#### 5.3.1. Metrics for Automatic Evaluation

For the scores, we adopt the commonly used BLEU and F1-score from the
BERTScore. BLEU has been employed in many existing studies; however, further
evaluation is required because it considers only surface words. Therefore,
BERTScore, which uses neural network embedding to capture semantic
similarities, is employed.

Such automatic evaluation metrics only capture the similarity between
automatically generated and reference comments but do not capture the
correctness of the generated text.

#### 5.3.2. Evaluation by Human Judges

Therefore, we also evaluate the generated comments by human judges. For each
comparison method, we present 30 comments of time-series numerical data to a
human evaluator, who is a native Japanese speaker, and ask the evaluator to
evaluate whether the output text is consistent or inconsistent with the
reference. The evaluator can also refer to the short-term and long-term
sequences, if needed. We report the number of consistent and inconsistent
results for each method.

## 6\. Results and Discussions

The results are presented in the following section.

### 6.1. Automatic Evaluation

Figure 2 shows the changes in BLEU scores with the number of shots. The
detailed scores of BLEU, METEOR, and BERTScore are shown in Table 6. Overall,
the scores generally increase with the number of shots, except for the HTML
prompt, as shown in the bottom-left graph in Figure 2. All methods on the
zero-shot setting achieve very low scores in terms of BLEU, METEOR, and
BERTScore, implying that the multitasking capability of the LLM is not
sufficient for the market comment generation. Direct prompt does not work well
for this task, and all other prompts outperform it.

A further look at the scores for prompts other than the Direct prompt yielded
three findings: 1) Python Dictionary, Python List (nested), and Row work
better than the other prompts, 2) Text unexpectedly works worse; and 3) among
programming language-based prompts, HTML performs the worst. Each finding is
explained in detail below.

### Finding 1: Python Dictionary, Python List (nested), and Row perform better

Table 6 lists the BLEU, METEOR, and BERTScores for each prompt in all
categories. Owing to the space limitation, we show the values obtained in the
setting of 0, 5, and 10 shots. Overall, we observe that there are three best-
performing models among all models: Python Dictionary, Python List (nested),
and Row. These best-performing prompts share common characteristics: 1) they
have similar characteristics to the data used for pretraining, and 2) the
timestamps and market prices are written closely to each other in a prompt.
Python codes are also used for the pretraining of the LLMs, thus, better
results may be reasonable.  Row also achieves better results because this
approach is used for dataset creation for pretraining to linearize HTML tables
on the web.

Furthermore, the timestamps and market prices are written closely to each
other, for example, [15:00, 9988.05] in the  Python List (nested) and
15:00":9988.05 in the  Python Dictionary. Other programming language-like
prompts, such as  Python List,  HTML, and  LaTeX, with lower performance write
the timestamps and the market prices far away from each other. One possible
reason for this is that GPT was able to accurately capture the correspondence
between the prices and a time.

### Finding 2:  Text unexpectedly performs worse

We hypothesized that  Text (English) and  Text (Japanese) would perform
better; however, Table 3 reveals that these methods perform worse than the
best-performing programming language-like prompts, that is,  Python Dictionary
and  Python List (nested). Specifically, the performances of Text (English),
represented by a red line, are lower than those of other prompts, that is, Row
and Python Dictionary, which are defined by orange and green lines,
respectively, except for 7-shot setting for the Row prompt.

One possible reason for this lower performance is that the template generates
repetition of the same expression. For example, the English template generates
“Nikkei225 as of 15:00 is 9982.06 yen” and repeatedly generates similar
expressions. Such text is unnatural and may not be included in the web texts
used for pretraining.

Interestingly, the bottom left graph in Table 2 shows that Text (English)
performs better than Text (Japanese) although the task is to generate a
comment in Japanese. A possible reason could be that English texts are used
more frequently during pretraining.

ipxma  Reference 日経平均、続伸 前引けは421円高の１万6349円 (The Nikkei 225 extended its gains,
closing up 421 yen at 16,349 at the midday break) EncDec 東証前引け、反落 米株安や円安一服で
利益確定売りも (Tokyo stocks fall at midday close, retreating on Wall Street declines
and a pause in yen weakening, with profit-taking also in play.) Direct 日経平均、反落
前引けは80円安の１万6349円 (The Nikkei 225 reversed course, falling 80 yen to close at
16,349 at the midday break.) Column 日経平均、反落 前引けは421円安の１万6349円 (The Nikkei 225
reversed course, falling 421 yen to close at 16,349 at the midday break.) Row
日経平均、反発 前引けは421円高の１万6349円 (The Nikkei 225 rebounded, closing up 421 yen at
16,349 at the midday break.) Python List 日経平均、反落 前引けは421円安の１万6349円 (The Nikkei
225 reversed course, falling 421 yen to close at 16,349 at the midday break.)
Python List (nested) 日経平均、大幅続落 前引けは421円安の１万6349円 (The Nikkei 225 continued to
drop sharply, falling 421 yen to close at 16,349 at the midday break.) Python
Dictionary 日経平均、反落 前引けは421円安の１万6349円 (The Nikkei 225 reversed course, falling
421 yen to close at 16,349 at the midday break.) HTML 日経平均、続伸
前引けは421円高の１万6349円 (The Nikkei 225 extended its gains, closing up 421 yen at
16,349 at the midday break.) LaTeX 日経平均、続伸 前引けは200円超の１万6349円 (The Nikkei 225
extended its gains, closing up 200 yen at 16,349 at the midday break.) Text
(English) 日経平均、反落 前引けは121円安の１万6349円 (The Nikkei 225 reversed course, falling
121 yen to close at 16,349 at the midday break.) Text (Japanese) 日経平均、反落
前引けは421円安の１万6349円 (The Nikkei 225 reversed course, falling 421 yen to close at
16,349 at the midday break.)

Table 7: Comparison of text output by each method, with ’movement terms’
highlighted in red in the table.

### Finding 3: Among programming language-based prompts,  HTML and  LaTeX
perform worse

The bottom left graph in Figure 2 shows each model’s BLEU scores in the
program language-like prompts category. Among the prompts in this category,
HTML and  LaTeX perform worse than other prompts. In particular, the
performance of  HTML has not improved after 5-shot and has already converged,
whereas the performances of the other prompts are still improving.

According to the statistics provided by
GitHub555https://madnight.github.io/githut, one of the pretraining sources,
the proportion of HTML formatted files is less than 0.1%. One possible reason
may be the small percentage of data used for the pretraining. Furthermore, the
HTML prompt is inherently longer than the others, which makes reasoning using
by the GPT difficult.

| # Consistent | # Inconsistent |
---|---|---|---
EncDec | 7 | 23 |
Direct | 8 | 22 |
Row | 14 | 16 |
Python Dictionary | 12 | 18 |
Text (English) | 12 | 18 |
Table 8: Results of human evaluation on EncDec, Direct, Row, Python
Dictionary, and Text English.

### 6.2. Evaluation by Human Judges

We conducted human evaluations of the encoder-decoder method and the method
with the highest BLEU score for each category: Direct, Row, Python Dictionary,
and Text (English).

Table 8 presents the results of the human evaluation. Across all methods, we
discovered that the evaluator judges the generated comments more frequently
than comments consistent with the reference comments. This includes many cases
in which the reference mentions that the stock price is rising, but the
generated comment states that it is falling, or vice versa. Table 7 shows the
output of each method. In this table, the ‘movement term’, referring to the
stock price movement such as ‘rising’ or ‘falling’, is highlighted in red.

Interestingly, although the fine-tuned encoder-decoder method achieved high
scores in the automatic evaluation, it received lower ratings in the human
evaluation. By contrast, the prompt-based method yielded comments that were
more consistent with the reference comments. We hypothesize that this
discrepancy arises because the fine-tuned comments are stylistically similar
to the references but do not accurately capture numerical movements. In
contrast, the prompt-based method excels in understanding numerical trends but
is limited in covering all text styles, given its reliance on at most ten
examples as shots. To verify this hypothesis, a separate evaluation focusing
on the accuracy of references to text style and numerical movements is
necessary, which indicates a potential direction for future research.

## 7\. Conclusion

In this study, we address the use of LLMs for text generation from numerical
sequences through the generation of market comments. We approached the problem
by converting numerical sequences into ten different formats across four
categories, and conducted experiments to compare these methods with the
existing encoder-decoder approach based on BART. Our results indicated that
prompts based on programming languages yielded strong performance, whereas
textual prompts and those based on HTML and LaTeX tables, were less effective.
Furthermore, a human evaluation assessing the consistency with reference
revealed that the prompt-based methods outperformed the encoder-decoder
method. This highlights the potential of prompt-based approaches to generate
market comment and similar data-to-text tasks.

Although our research focused on market comments as a representative task with
numerical sequence inputs, many other tasks share comparable characteristics.
In future work, it will be valuable to conduct experiments on datasets with
other types of numerical sequence inputs to further validate and extend our
findings.

## Acknowledgements

This study is based on the results obtained from a project JPNP20006,
commissioned by the New Energy and Industrial Technology Development
Organization (NEDO). Ideas related to time-aware prompts are obtained from
discussions in the BRIDGE (Program for Bridging the Gap Between R&D and Ideal
Society (Society5.0) and Generating Economic and Social Value) supported by
Cabinet Office. For experiments, the computational resource of the AI Bridging
Cloud Infrastructure (ABCI) provided by the National Institute of Advanced
Industrial Science and Technology (AIST) were used.

##

  * (1)

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††thanks: Corresponding author

# Quantum strategy in moving frames

Jian-Chuan Tan<EMAIL_ADDRESS>Quantum Theory Group, Department of
Modern Physics
University of Science and Technology of China, Hefei, 230026, P.R.China  An
Min Wang<EMAIL_ADDRESS>Quantum Theory Group, Department of Modern
Physics
University of Science and Technology of China, Hefei, 230026, P.R.China

###### Abstract

We investigate quantum strategy in moving frames by considering Prisoner’s
Dilemma and propose four thresholds of $\gamma$ for two players to determine
their Nash Equilibria. Specially, an interesting phenomenon appears in
relativistic situation that the quantum feature of the game would be enhanced
and diminished for different players whose particle’s initial spin direction
are respectively parallel and antiparallel to his/her movement direction, that
is, for the former the quantum feature of the game is enhanced while for the
latter the quantum feature would be diminished. Thus a classical latter could
still maintain his/her strictly dominant strategy (classical strategy) even if
the game itself is highly entangled.

###### pacs:

03.67.-a, 03.65.Bz, 02.50.Le, 03.75.-b, 03.65.Pm

Strategy theory (or Game theory) is a branch of applied mathematics devised to
analyze certain situations in which there is an interplay between parties that
may have similar, opposed, or mixed interests. It draws broad attention
because of its practical application in Economics, Politics, and other fields
which involve cooperation or conflict Game theory . As an applied mathematical
theory, strategy theory inevitably possesses its own physical properties. It
is not surprising, since a game should be played through some strategies, and
these strategies must be put in practice to some physical carriers. Thus the
traits of the carriers under some certain physical conditions would affect the
result of a game. Based on this consideration, to explore how to gain as much
as reward in a game in some particular physical situations has been a popular
research aspect in recent years.

In 1999, Eisert et al. proposed a novel model of quantum game in terms of the
famous nonzero sum game— Prisoners’ Dilemma, in which the physical carriers
are two spin-$\frac{1}{2}$ particles, and players could adopt some unitary
quantum operations as strategies. Although this model was criticized for not
possessing the dominance over a classical game Enk' , we find it is actually
go beyond a classical game and worth studying based on the considerations that
it is important for us to distinguish the difference between the equivalence
of payoffs and the equivalence of strategies, and that to understand the
essences of a cooperative game and a noncooperative game is of high
significance in studying a game with a physical background. More
interestingly, a physical carrier possesses not only quantum traits but also
relativistic ones. So we are concerning on this effect by using Eisert et
al.’s model. In this model, two particles (start in a produce state
$\left|CC\right\rangle$) are initially entangled by a gate $\hat{J}$ to form a
pairs of physical carriers of this game, and then be distributed to two
players, Alice and Bob, who independently chooses a quantum strategy

$\hat{U}(\theta,\phi)=\left(\begin{array}[]{ccc}e^{{\rm
i}\phi}\cos{\theta/2}&\sin{\theta/2}\\\ -\sin{\theta/2}&e^{-{\rm
i}\phi}\cos{\theta/2}\\\ \end{array}\right),$ (1)

with $0\leq\theta\leq\pi$ and $0\leq\phi\leq\pi/2$. Finally, a disentangling
gate $\hat{J}^{\dagger}$ is carried out and the carrier pair is measured in
the computational basis. In terms of game theory, it exists a new Nash
Equilibrium (NE), that is, both of the players choose strategy
$\hat{Q}=\hat{U}(0,\pi/2)$, because strategy $\hat{Q}$ has the property of
being Pareto optimal, and help players escape the dilemma in classical game
Game theory ; Eisert .

Let us restrict the physical carriers to be two spin-$\frac{1}{2}$ particles
and denote the states of the particles as:
$\left|\frac{1}{2}\right\rangle=\left|\textit{C}\right\rangle=\left(\begin{array}[]{c}1\\\
0\end{array}\right)$, and
$\left|-\frac{1}{2}\right\rangle=\left|\textit{D}\right\rangle=\left(\begin{array}[]{c}0\\\
1\end{array}\right)$. Meanwhile, an arbiter is needed to determine each
player’s payoff by measuring the state of the two particles with a physical
measurement device, and the principle of the determination is well known to
both players. The players could only gain expected payoff since quantum
mechanics itself is a probabilistic theory. Alice’s and Bob’s expected payoffs
are given by

$\begin{array}[]{c}\$_{A}=\textit{r}\textit{P}_{CC}+\textit{p}\textit{P}_{DD}+\textit{t}\textit{P}_{DC}+\textit{s}\textit{P}_{CD},\\\
\$_{B}=\textit{r}\textit{P}_{CC}+\textit{p}\textit{P}_{DD}+\textit{s}\textit{P}_{DC}+\textit{t}\textit{P}_{CD},\\\
\end{array}$ (2)

where $P_{ab}=\left|\left\langle ab|\psi_{f}\right\rangle\right|^{2}$
($a,b=C,D$) is the joint probability that the arbiter’s measure device would
display $a,b$. We take $\textit{t}=5,\textit{r}=3,\textit{p}=1$ and
$\textit{s}=0$ in this model Eisert . In this game, we assume that the arbiter
moves in the _x_ direction, Alice’s particle moves in the _z_ direction, and
Bob’s the -_z_ direction. Thus their movements cause boosts in the direction
of _x_ , _z_ , and -_z_ , respectively. Thus, Alice’s and Bob’s movement
directions are respectively parallel and antiparallel to their particles’
initial spin directions. We denote the boosts with each’s rapidity as $\alpha$
for the arbiter, $\delta_{A}$ for Alice, and $\delta_{B}$ for Bob.

Of course, the arbiter’s boost $\alpha$ respect to a player could also be
equivalent to the player emitting the particle to the arbiter with a rapidity
$-\alpha$ (that is, with $\alpha$ in the -_x_ direction). In this case, we
could further think that the arbiter is at rest, and the two players are far
away from the arbiter, so they have to take part in this game by emitting
their own particles to the arbiter, and the rapidity of each particle will
sort of determine how much payoff the players would attain. Thus, at what
speed the particle is emitted could be controlled by the player, and we name
this speed-control as a relativistic operation. From our point of view, this
model should be worth studying since it is a well guidance to long-distance
games, and even in the near future when interstellar travel comes true, this
model would also be useful.

Now we set out our game model and its process is illustrated in Fig.1, in
which the Lorentz boost is introduced in Refs.Alsing ; Ahn , and $\gamma$ is a
monotonic function with the measure of entanglement, indicating how much the
two particles entangle. The degree of entanglement between the two particles
would decrease if their momentum have distributions, say, with width. So
tracing out the momentum from the Lorentz-transformation density matrix
destroys some of the entanglement Gingrich . We assume the momentum of both
particles to be exact, namely no distributions, thus their degree of
entanglement would remain invariant under Lorentz transformation, and so does
$\gamma$. When $\gamma=0$, the game’s players are separable and the game does
not display any features which go beyond the classical game.

Figure 1: Process of the game model. $\hat{J}=\exp({\rm
i}\gamma\hat{D}\otimes\hat{D}/2)$, $\gamma\in\left[0,\pi/2\right]$,
$\hat{D}=\hat{U}(\pi,0)$, is defined in Eisert  to make the two particles
entangle. $R(\Lambda,\bm{p})$ is the Wigner rotation applied to a particle.
$\hat{U}_{A}$ and $\hat{U}_{B}$ are operations Alice and Bob applies to her
and his own particle respectively.

The Lorentz transformation $\Lambda$ results in a unitary transformation on
states in the Hilbert space that
$\mbox{$\left|{\Psi}\right\rangle$}\rightarrow
U(\Lambda)\mbox{$\left|{\Psi}\right\rangle$}$. Thus, the state of entangled
particles under the Lorentz transformation is given by

$U(\Lambda)(\hat{U}_{A}\otimes\hat{U}_{B})\hat{J}\left|\bm{p}_{A},\\!C;\bm{p}_{B},\\!C\right\rangle=\sum_{a,b=C,D}k_{ab}\mbox{$\left|{\psi_{ab}}\right\rangle$},$
(3) $\left|{\psi_{CC}}\right\rangle$ $\displaystyle=$
$\displaystyle\\!\\!\sqrt{\frac{(\Lambda\bm{p}_{A})^{0}}{\bm{p}_{A}^{0}}}\sqrt{\frac{(\Lambda\bm{p}_{B})^{0}}{\bm{p}_{B}^{0}}}\sum_{\sigma,\sigma^{\prime}}{D}_{\sigma,\frac{1}{2}}^{(\frac{1}{2})}\left(\emph{R}(\Lambda_{A})\right)$
(4)
$\displaystyle{D}_{\sigma^{\prime},\frac{1}{2}}^{(\frac{1}{2})}\left(\emph{R}(\Lambda_{B})\right)a^{\dagger}(\bm{p}_{A_{\Lambda}},\sigma)a^{\dagger}(\bm{p}_{B_{\Lambda}},\sigma^{\prime})\mbox{$\left|{\psi_{0}}\right\rangle$},$
$\left|{\psi_{CD}}\right\rangle$ $\displaystyle=$
$\displaystyle\\!\\!\sqrt{\frac{(\Lambda\bm{p}_{A})^{0}}{\bm{p}_{A}^{0}}}\sqrt{\frac{(\Lambda\bm{p}_{B})^{0}}{\bm{p}_{B}^{0}}}\sum_{\sigma,\sigma^{\prime}}{D}_{\sigma,\frac{1}{2}}^{(\frac{1}{2})}\left(\emph{R}(\Lambda_{A})\right)$
(5)
$\displaystyle{D}_{\sigma^{\prime},-\frac{1}{2}}^{(\frac{1}{2})}\left(\emph{R}(\Lambda_{B})\right)a^{\dagger}(\bm{p}_{A_{\Lambda}},\sigma)a^{\dagger}(\bm{p}_{B_{\Lambda}},\sigma^{\prime})\mbox{$\left|{\psi_{0}}\right\rangle$},$
$\left|{\psi_{DC}}\right\rangle$ $\displaystyle=$
$\displaystyle\\!\\!\sqrt{\frac{(\Lambda\bm{p}_{A})^{0}}{\bm{p}_{A}^{0}}}\sqrt{\frac{(\Lambda\bm{p}_{B})^{0}}{\bm{p}_{B}^{0}}}\sum_{\sigma,\sigma^{\prime}}{D}_{\sigma,-\frac{1}{2}}^{(\frac{1}{2})}\left(\emph{R}(\Lambda_{A})\right)$
(6)
$\displaystyle{D}_{\sigma^{\prime},\frac{1}{2}}^{(\frac{1}{2})}\left(\emph{R}(\Lambda_{B})\right)a^{\dagger}(\bm{p}_{A_{\Lambda}},\sigma)a^{\dagger}(\bm{p}_{B_{\Lambda}},\sigma^{\prime})\mbox{$\left|{\psi_{0}}\right\rangle$},$
$\left|{\psi_{DD}}\right\rangle$ $\displaystyle=$
$\displaystyle\\!\\!\sqrt{\frac{(\Lambda\bm{p}_{A})^{0}}{\bm{p}_{A}^{0}}}\sqrt{\frac{(\Lambda\bm{p}_{B})^{0}}{\bm{p}_{B}^{0}}}\sum_{\sigma,\sigma^{\prime}}{D}_{\sigma,-\frac{1}{2}}^{(\frac{1}{2})}\left(\emph{R}(\Lambda_{A})\right)$
(7)
$\displaystyle{D}_{\sigma^{\prime},-\frac{1}{2}}^{(\frac{1}{2})}\left(\emph{R}(\Lambda_{B})\right)a^{\dagger}(\bm{p}_{A_{\Lambda}},\sigma)a^{\dagger}(\bm{p}_{B_{\Lambda}},\sigma^{\prime})\mbox{$\left|{\psi_{0}}\right\rangle$}.$

where $\left|{\psi_{0}}\right\rangle$ is the Lorentz invariant vacuum state,
and

$\displaystyle k_{CC}$ $\displaystyle=$ $\displaystyle e^{{\rm
i}(\phi_{A}+\phi_{B})}c_{\theta_{A}}c_{\theta_{B}}c_{\gamma}+{\rm
i}s_{\theta_{A}}s_{\theta_{B}}s_{\gamma},$ (8) $\displaystyle k_{CD}$
$\displaystyle=$ $\displaystyle-e^{{\rm
i}\phi_{A}}c_{\theta_{A}}s_{\theta_{B}}c_{\gamma}+{\rm i}e^{-{\rm
i}\phi_{B}}s_{\theta_{A}}c_{\theta_{B}}s_{\gamma},$ (9) $\displaystyle k_{DC}$
$\displaystyle=$ $\displaystyle-e^{{\rm
i}\phi_{B}}s_{\theta_{A}}c_{\theta_{B}}c_{\gamma}+{\rm i}e^{-{\rm
i}\phi_{A}}c_{\theta_{A}}s_{\theta_{B}}s_{\gamma},$ (10) $\displaystyle
k_{DD}$ $\displaystyle=$ $\displaystyle
s_{\theta_{A}}s_{\theta_{B}}c_{\gamma}+{\rm i}e^{-{\rm
i}(\phi_{A}+\phi_{B})}c_{\theta_{A}}c_{\theta_{B}}s_{\gamma}.$ (11)

For simplicity, we denote

$c_{x}=\cos\frac{x}{2},\quad s_{x}=\sin\frac{x}{2},$ (12)

where $x$ can be taken as $\theta_{A}$, $\theta_{B}$ and $\gamma$ as well as
so-called Wigner angle $\Omega_{A}$ and $\Omega_{B}$ respectively with Alice’s
and Bob’s particles. Note that a particle’s Wigner angle is determined by the
rapidities of itself ($\delta$) and the arbiter ($\alpha$) Alsing Ahn ,

$\Omega_{\tau}=\arctan\frac{\sinh\alpha\sinh\delta_{\tau}}{\cosh\alpha+\cosh\delta_{\tau}},\tau=A,B.$
(13)

The final state measured by the arbiter is
$\left|\psi_{f}\right\rangle=\hat{J}^{\dagger}U(\Lambda)(\hat{U}_{A}\otimes\hat{U}_{B})\hat{J}\left|\bm{p}_{A},C;\bm{p}_{B},C\right\rangle$.
We have

$\left(\begin{array}[]{c}\mathfrak{p}_{1}\\\ \mathfrak{p}_{2}\\\
\mathfrak{p}_{3}\\\
\mathfrak{p}_{4}\end{array}\right)=\left(\begin{array}[]{cccc}\omega_{1}&\omega^{*}_{2}&-\omega^{*}_{3}&-\omega_{4}\\\
-\omega^{*}_{2}&\omega_{1}&\omega_{4}&-\omega_{3}\\\
\omega^{*}_{3}&\omega_{4}&\omega_{1}&-\omega_{2}\\\
-\omega_{4}&\omega^{*}_{3}&-\omega^{*}_{2}&\omega_{1}\end{array}\right)\left(\begin{array}[]{c}k_{CC}\\\
k_{CD}\\\ k_{DC}\\\ k_{DD}\end{array}\right),$ (14)

where $\omega_{1}=c_{\gamma}c_{\Omega_{A}}c_{\Omega_{B}}+{\rm
i}s_{\gamma}s_{\Omega_{A}}s_{\Omega_{B}}$,
$\omega_{2}=c_{\gamma}c_{\Omega_{A}}s_{\Omega_{B}}+{\rm
i}s_{\gamma}s_{\Omega_{A}}c_{\Omega_{B}}$,
$\omega_{3}=c_{\gamma}s_{\Omega_{A}}c_{\Omega_{B}}+{\rm
i}s_{\gamma}c_{\Omega_{A}}s_{\Omega_{B}}$, and
$\omega_{4}=c_{\gamma}s_{\Omega_{A}}s_{\Omega_{B}}+{\rm
i}s_{\gamma}c_{\Omega_{A}}c_{\Omega_{B}}$, and $*$ denotes complex
conjugation. Thus we get ${P}_{CC}=\left|\mathfrak{p}_{1}\right|^{2}$,
${P}_{CD}=\left|\mathfrak{p}_{2}\right|^{2}$,
${P}_{DC}=\left|\mathfrak{p}_{3}\right|^{2}$, and
${P}_{DD}=\left|\mathfrak{p}_{4}\right|^{2}$.

Actually, how much the two particles are initially entangled would be
essential to this game model, since $\gamma$ induces some features which go
beyond the classical game. Du et al. found two thresholds of $\gamma$ in the
Quantum Prisoners’ Dilemma— $\gamma_{th1}=\arcsin{\sqrt{1/5}}$ and
$\gamma_{th2}=\arcsin{\sqrt{2/5}}$, which separate the game into three
regions: classical region ($\gamma\in\left[0,\gamma_{th1}\right)$),
intermediate region ($\gamma\in\left[\gamma_{th1},\gamma_{th2}\right)$), and
fully quantum region ($\gamma\in\left[\gamma_{th2},\pi/2\right]$), see Ref.Du
Du' . According to Du, the classical region means in this domain, the game
behaves classically, i.e., the NE of the game is $\hat{D}\otimes\hat{D}$; in
the quantum region, the game is similar to the maximally entangled one in
Eisert’s Letter Eisert  that $\hat{Q}\otimes\hat{Q}$ becomes the new NE and
has the property to be _Pareto Optimal_ ; while the intermediate region
possesses compatibility to $\hat{D}$ and $\hat{Q}$, where
$\hat{D}\otimes\hat{D}$ is no longer the NE because each player could improve
his/her payoff by unilaterally deviating from the strategy $\hat{D}$, thus two
Nash Equilibria (NE’s) $\hat{D}\otimes\hat{Q}$ and $\hat{Q}\otimes\hat{D}$
emerge Du .

In order to explore the relativistic-quantum features of this game, we take
four situations as examples, in which $4$-kinds of payoffs are considered for
each player— (a) Alice moves at low speed (AL) & Bob moves at low speed (BL),
(b) Alice moves at low speed (AL) & Bob moves at high speed (BH), (c) Alice
moves at high speed (AH) & Bob moves at low speed (BL), and (d) Alice moves at
high speed (AH) & Bob moves at high speed (BH); and
$G_{1}:=\$(\hat{D}\otimes\hat{D})$, $G_{2}:=\$(\hat{Q}\otimes\hat{D})$,
$G_{3}:=\$(\hat{D}\otimes\hat{Q})$, and $G_{4}:=\$(\hat{Q}\otimes\hat{Q})$.
And we concentrate our discussion to a simple but typical strategy set
$S=\\{\hat{D},\hat{Q}\\}$, since $\hat{D}=\hat{U}(\pi,0)$ is a classical spin-
rotating operation which could be implemented by sort of classical equipments,
while $\hat{Q}=\hat{U}(0,\pi/2)$ is a purely phase-controlling operation which
could only be implemented by a quantum gate. It is an essential difference
between these two strategies. Thus, there are at most six thresholds of
$\gamma$ ($\gamma_{\mu\nu},\mu,\nu=1,2,3,4$, with $\mu<\nu$, where
$\gamma_{\mu\nu}$ is the point where $G_{\mu}=G_{\nu}$) for each player’s
payoff in each situation. Among these $\gamma_{\mu\nu}$, there are two
thresholds are essential for each player— for Alice, they are $\gamma_{12}$
and $\gamma_{34}$, we denote them as $\gamma^{A}_{12}$ and $\gamma^{A}_{34}$;
similarly, for Bob, they are $\gamma^{B}_{13}$ and $\gamma^{B}_{24}$. These
four thresholds are essential because they demonstrate Alice’s and Bob’s
strictly dominant strategies (SDS) for different
$\gamma\in\left[0,\frac{\pi}{2}\right]$ Game theory . Fig.2 illustrates
Alice’s and Bob’s payoffs in the four situations. Here, as for low and high
speed we can respectively take $\Omega_{\tau}=\frac{\pi}{16}$ and
$\Omega_{\tau}=\frac{7\pi}{16}$. As is mentioned in Ref.Ahn , $\Omega_{\tau}$
is a monotonic function with player $\tau$’s and the arbiter’s speeds. Thus in
this example, $\Omega_{\tau}=\frac{\pi}{16}$ corresponds to arbiter’s speed
0.01c and $\tau$’s speed 0.001c, while $\Omega_{\tau}=\frac{7\pi}{16}$
corresponds to arbiter’s speed 0.97c and $\tau$’s speed 0.908c, where c is the
light-speed. The arbiter’s speed is equivalent to the same speed that the
player emits his/her particle in the -_x_ direction, as mentioned above.

Figure 2: (Color online) Alice’s and Bob’s payoffs in 4 situations— (a) AL &
BL, (b) AL & BH, (c) AH & BL, and (d) AH & BH.

In Fig.2, we name the region where $\gamma^{A}_{12}<\gamma<\gamma^{A}_{34}$
Alice’s transition region ($\mathcal{T}_{A}$), and where
$\gamma^{B}_{13}<\gamma<\gamma^{B}_{24}$ Bob’s transition region
($\mathcal{T}_{B}$). If $\gamma$ is on the left side of $\mathcal{T}_{\tau}$,
then $\tau$’s SDS is $\hat{D}$ (purely classical strategy); if $\gamma$ is on
the right side of $\mathcal{T}_{\tau}$, the SDS is $\hat{Q}$ (purely quantum
strategy); while if $\gamma$ is in $\mathcal{T}_{\tau}$, $\tau$ would have no
SDS, but the NE still exist. Game theory proves that the combination of each
player’s SDS must be the NE of the game, but a NE may not be the combination
of each’s SDS Game theory . From Fig.2, we could see that in some situations,
$\mathcal{T}_{A}$ and $\mathcal{T}_{B}$ overlap partially with each other, and
in the overlapping region, two new NE’s $\hat{D}\otimes\hat{Q}$ and
$\hat{Q}\otimes\hat{D}$ appear, although there is no SDS exists for each
player. On the other hand, if $\gamma$ is in $\mathcal{T}_{A}$ but not in
$\mathcal{T}_{B}$, Bob has SDS $\hat{D}$ or $\hat{Q}$, but Alice has not, in
this case, the NE is $\hat{Q}\otimes\hat{D}$ or $\hat{D}\otimes\hat{Q}$, that
is to say, Alice should choose the strategy opposite to Bob’s SDS. It is
similar to the case that $\gamma$ is in $\mathcal{T}_{B}$ but not in
$\mathcal{T}_{A}$. What is noteworthy is the highly relativistic situation in
Fig.2.(d): $\Omega_{A}=\Omega_{B}=\frac{7\pi}{16}$. In this case, there is no
transition region for Bob, and for all
$\gamma\in\left[0,\frac{\pi}{2}\right]$, Bob’s SDS is $\hat{D}$, that is to
say, when Alice’s and Bob’s particles both move at very high speed, the game
behaves classically for Bob, even if he is highly entangled with Alice. It is
an interesting phenomenon that the relativistic operations would diminish the
quantum feature of the game. Fig.3 shows the area where Bob’s SDS is $\hat{D}$
for all $\gamma\in\left[0,\frac{\pi}{2}\right]$, i.e., where the relativistic
operation entirely eliminate the quantum feature of the game for Bob.

Figure 3: The shadowed area indicates the situation in which Bob’s SDS is
always $\hat{D}$ in spite of how much the two particles are entangled.

In fact, the four thresholds vary with $\Omega_{A}$ and $\Omega_{B}$ as

$\displaystyle\gamma^{A}_{12}\\!=\\!\arcsin\\!{\sqrt{\frac{c^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}\\!-\\!2s^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!+\\!2c^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!-\\!s^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}}{5c^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}\\!-\\!5s^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!+\\!3c^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!+\\!2s^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}}}},$
(15)
$\displaystyle\gamma^{A}_{34}\\!=\\!\arcsin\\!{\sqrt{\frac{2c^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}\\!-\\!s^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!+\\!c^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!-\\!2s^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}}{5c^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}\\!-\\!5s^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!+\\!3c^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!+\\!2s^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}}}},$
(16)
$\displaystyle\gamma^{B}_{13}\\!=\\!\arcsin\\!{\sqrt{\frac{c^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}\\!-\\!2s^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!-\\!c^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!+\\!2s^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}}{5c^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}\\!-\\!5s^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!-\\!3c^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!-\\!2s^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}}}},$
(17)
$\displaystyle\gamma^{B}_{24}\\!=\\!\arcsin\\!{\sqrt{\frac{2c^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}\\!-\\!s^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!-\\!2c^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!+\\!s^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}}{5c^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}\\!-\\!5s^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!-\\!3c^{2}_{\Omega_{A}}s^{2}_{\Omega_{B}}\\!-\\!2s^{2}_{\Omega_{A}}c^{2}_{\Omega_{B}}}}}.$
(18)

always with $\gamma^{A}_{12}<\gamma^{A}_{34}$ and
$\gamma^{B}_{13}<\gamma^{B}_{24}$. We plot these four thresholds in Fig.4. In
particular, when Alice, Bob and the arbiter are all at rest, i.e.,
$\Omega_{A}=\Omega_{B}=0$, $\mathcal{T}_{A}$ and $\mathcal{T}_{B}$ overlap
entirely with each other. In this case,
$\gamma^{A}_{12}=\gamma^{B}_{13}=\gamma_{th1}$ in Du’s paper Du , and
$\gamma^{A}_{34}=\gamma^{B}_{24}=\gamma_{th2}$, thus two NE’s emerge in the
overlapping region.

Figure 4: The four thresholds $\gamma^{A}_{12}$, $\gamma^{A}_{34}$,
$\gamma^{B}_{13}$ and $\gamma^{B}_{24}$, which divide the game into three
regions respectively according to $\gamma$, and determine the Nash Equilibrim
of this game.

Finally, we could see in Fig.4.(b) that for Alice,
$\gamma^{A}_{34}<\frac{\pi}{2}$ in all situations, and
$\gamma^{A}_{34}\rightarrow 0$ when $\Omega_{A}\rightarrow\frac{\pi}{2}$,
i.e., when Alice’s particle moves at very high speed, her SDS would be
$\hat{Q}$ even if the two particles are entirely separable; while in
Fig.4.(c), $\gamma^{B}_{13}>\frac{\pi}{2}$ in some situations, where the
quantum feature of the game is entirely eliminated for Bob, so his SDS is
$\hat{D}$ even if the two particles are entirely entangled. That is to say, in
the same game, the relativistic operations enhance the quantum feature of the
game for Alice, but diminish it for Bob.

In summary, we have demonstrated that some new and interesting features appear
if classical games such as Prisoners’ Dilemma are extended to the quantum and
relativistic domain, in which the initial symmetry of this game is broken by
the respect movements of the two players. We also propose four thresholds for
Alice and Bob, which divide the game into three regions in which different
strictly dominant strategies emerge, and how Nash Equilibrium is determined in
different situations. Moreover, a interesting phenomenon appears in
relativistic situation that the relativistic operations could enhance the
quantum feature of the game for the player whose particle’s initial spin
direction is parallel to its movement direction (Alice), but diminish it for
the one whose particle’s initial spin direction is antiparallel to its
movement direction (Bob), i.e., the respect movements of Alice, Bob and the
arbiter determine “how quantum” the game is for each player. We believe these
properties would be useful to guide remote games in the future and that
extending game theory to quantum and relativistic domain would lead us to
understand the physical essence of game theory.

We are grateful to all the collaborators of our quantum theory group in the
institute for theoretical physics of our university. We thank Prof. Lewenstein
and Zeyang Liao for triggering and useful discussion. This work was supported
by the National Natural Science Foundation of China under Grant No. 60573008.

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