# Online Prompt Pricing based on Combinatorial Multi-Armed Bandit and
Hierarchical Stackelberg Game

Meiling Li111These authors contributed equally to this work., Hongrun
Ren111These authors contributed equally to this work., Haixu Xiong, Zhenxing
Qian222Corresponding authors., Xinpeng Zhang222Corresponding authors.
Fudan University
Shanghai, 200433, China
{mlli20@,renhr20@,22110240113@m.,zxqian<EMAIL_ADDRESS>

###### Abstract

Generation models have shown promising performance in various tasks, making
trading around machine learning models possible. In this paper, we aim at a
novel prompt trading scenario, prompt bundle trading (PBT) system, and propose
an online pricing mechanism. Based on the combinatorial multi-armed bandit
(CMAB) and three-stage hierarchical Stackelburg (HS) game, our pricing
mechanism considers the profits of the consumer, platform, and seller,
simultaneously achieving the profit satisfaction of these three participants.
We break down the pricing issue into two steps, namely unknown category
selection and incentive strategy optimization. The former step is to select a
set of categories with the highest qualities, and the latter is to derive the
optimal strategy for each participant based on the chosen categories. Unlike
the existing fixed pricing mode, the PBT pricing mechanism we propose is more
flexible and diverse, which is more in accord with the transaction needs of
real-world scenarios. We test our method on a simulated text-to-image dataset.
The experimental results demonstrate the effectiveness of our algorithm, which
provides a feasible price-setting standard for the prompt marketplaces.

##  1 Introduction

The rise of text-to-image generation models has given birth to a new
marketplace. With proper prompts, users can generate any image they like.
However, creating high-quality images often involves a tedious process of
searching for prompt modifiers and evaluating the resulting images. This
process can be time-consuming and expensive due to the unclear impact of each
modifier. For instance, the logo of OctoSQL, a CLI project for database
querying, was designed using DALL·E 2, costing around $30 through a paid API
interaction111https://jacobmartins.com/posts/how-i-used-dalle2-to-generate-
the-logo-for-octosql/. Consequently, prompt engineers, specialists in crafting
effective prompts, have begun selling their prompts on platforms like
PromptBase [2], PromptSea[4], and NeutronField [1]. In these marketplaces,
customers can browse sample images generated by various prompts and purchase
desired prompts. PromptBase offers prompts with prices ranging from $2.99 to
$6.99, while for NeutronField, the prices difference between prompts vary
larger. Upon acquiring a prompt, customers can generate similar images or
tweak the prompt to create images in a similar style. The trend of prompt
trading marketplaces is on the rise, with PromptBase boasting tens of
thousands of registered users, and the user base continues to grow [35].

We first look at the prompt trading scenario of Promptbase website [3].
Concretely, prompt engineers upload their crafted prompts onto the website and
give the selling price. The viewers scan products on the platform. If
interested in the shown images, they can pay for the corresponding prompt.
Once the platform receives the payment from the consumer, it will take $20\%$
of the payment to make a profit, and give the remaining payment to the
corresponding seller. In this trading, three parties are involved, i.e.,
sellers, platform, and consumers. The procedure is as below:

  * 1.

Sellers post products onto the platform with a pre-defined selling price. If
several prompts are bundled for sale, the seller also needs to prescribe a
discount. Here we assume the final price of the products is $p_{s}$.

  * 2.

Consumers select desired products and pay the price $p_{s}$ to the platform
for purchase.

  * 3.

The platform transfers $80\%$ of the payment ($0.8*p_{s}$) to the seller and
retains the remaining $20\%$ ($0.2*p_{s}$) as its reward.

While the above trading method is simple and feasible, it has several
problems. First, the prices of the prompts are unilaterally determined by
sellers. It does not consider the value evaluation of the product by the
platform and the customer. This kind of trading mode puts the sellers in the
dominant position, which is not in line with the business model of the
“consumer first” in the trading market. Second, the revenue division between
the platform and the seller is fixed (i.e., 2:8), rather than flexibly
adjusted according to different products. Besides, this mode fails to
comprehensively consider the interests of sellers, platforms, and consumers.
In the long run, it is not conducive to the development of the prompt
marketplace.

Figure 1: Illustration of the proposed prompt bundle trading scenario.

In this paper, we consider a Prompt Bundle Trading (PBT) scenario, as shown in
Figure 1. In a prompt selling platform with a large number of prompt products,
customers tend to prefer top sellers with good reputations and high trading
volume when purchasing prompts. For popular sellers with a large number of
products, the platform can choose to recommend some bundled prompts to boost
profits. Meanwhile, consumers can select bundle deals to get multiple prompts
at a more cost-effective price. In this case, how to maximize the profit of
the three parties has become an important issue.

However, there exist two major challenges in the above PBT scenario. The first
is how to select categories with high quality for recommendation. This issue
can be seen as an online decision process. The platform can not only
repeatedly learn the categories’ qualities (exploration), but it can also
resort to the learned knowledge to make selection (exploitation). The platform
needs to strike a balance between the above two procedures, to obtain the
highest quality as much as possible. The second is how to derive an ideal
incentive strategy in a three-parties game. Note that all three parties can
influence others’ profits by manipulating their products or payments.
Therefore, they need to reach an optimal equilibrium to maximize each party’s
profit.

To address the above challenges and inspired by [6], we design a PBT mechanism
based on Combinatorial Multi-Armed Bandit (CMAB) and three-stage Hierarchical
Stackelberg (HS) game. First, we regard the selection of categories with
unknown qualities as a CMAB problem. Each category is taken as an arm, and its
quality is the corresponding reward. We design a greedy search-based policy to
solve the problem. Second, we introduce a three-stage HS game to find the
optimal incentive strategy. The consumer, the platform, and the seller are
taken as the first-tier leader, the second-tier leader, and the follower,
respectively. Afterward, we can derive an optimal incentive strategy via
backward deduction. The major contributions are as follows:

  * •

We propose a novel online Prompt Bundle Trading (PBT) task. As far as we know,
this is the first work focusing on prompt product pricing in prompt
marketplaces.

  * •

We propose a PBT mechanism by combining CMAB and HS, achieving profit
maximization of consumer, platform, and seller.

  * •

Extensive experiments based on simulated prompt products from text-to-image
datasets demonstrate the satisfactory performance of the proposed prompt
trading approach.

##  2 Related Work

Establishing and promoting data pricing models can help govern and standardize
pricing behaviors in data markets. Depending on whether machine learning (ML)
is relevant to the trading scenario or not, current data trading scenarios can
be divided into traditional kinds and ML kinds.

###  2.1 Traditional Data Trading Scenarios

Several researchers have studied pricing approaches for specific data trading
scenarios. In the personal trading scenarios [10, 36, 30, 29], Shen et al.
[36] suggested a personal big data pricing model based on tuple granularity.
Niu et al. [29] introduced a novel context-aware dynamic pricing approach with
reserve price constraints to optimize cumulative revenue for data brokers. In
the mobile crowd-sensing scenarios [6, 21, 43], An et al. [6] devised a
streaming data trading system where transactions are facilitated between data
consumers and sellers via a platform. In cloud resource trading scenarios [8,
45, 40, 24], Tsai et al. [40] proposed a time-related intelligent data pricing
model to alleviate network congestion by offering different prices to network
users at various periods.

###  2.2 Machine Learning Trading Scenarios

In the machine learning transaction scenario, according to the trading
objects, the transactions can be briefly divided into data transactions and
model transactions [15], which are introduced below.

Data Pricing Mechanism In the machine learning market, data transactions
typically involve buyers and sellers [5], and transactional data usually fall
into two categories: training datasets and streaming data [27]. On the pricing
of training dataset transactions [25, 37, 14], Chen et al. [11] designed a
mechanism to price the data using expensive signals and considered the trade-
off between the cost of selling data and the accuracy of data quality
estimation. To achieve compensation allocation, Wang et al. [42] designed
federated Shapley value [34] for pricing datasets in collaborative machine
learning model training. This method guarantees that datasets receive rewards
in proportion to their contributions and considers the order of dataset
participation, reflecting its impact on the overall value. Building upon this,
Fan et al. [17] introduced complete federated Shapley value to address
potential fairness issues, ensuring fair reward allocation to data owners in
federated learning. For streaming data transactions, Xu et al. [44] designed
an online evaluation method to assess data value during model training. Then
accordingly, they determined the compensation for data providers.

Model Pricing Mechanism In machine learning model transactions, current
research can be divided into two categories [33]. One category focuses on the
model transactions between model sellers and buyers [9, 38]. Chen et al. [12]
proposed pricing machine learning models based on their performance.
Afterward, Chen et al. [11] introduced a formal pricing framework for machine
learning models. The other category simultaneously considers both the pricing
of training datasets and machine learning models [23, 39]. Liu et al. [23]
proposed a machine learning model market with differential privacy, where data
owners and model buyers transact through brokers. The broker first determines
the prices for a set of machine learning models to maximize revenue without an
arbitrage guarantee. Then, using Shapley values, the contributions of a set of
data owners’ data to the models are simulated, and data owners are compensated
accordingly. Sun et al. [39] designed a machine learning model market based on
differentially private federated learning and focused on an intermediary-
centered design.

In general, the existing data pricing approaches are mainly designed for
specific transaction scenarios. However, many of the aforementioned approaches
failed to consider the entire data product production process, which primarily
concentrated on unilateral pricing of transactions on data raw materials or
data products. Besides, these methods only focus on the economic interests of
brokers, neglecting the interests of other entities in the trading market.
Therefore, they are not applicable to the proposed pricing scenario of
prompts, which involves the participation of third-party entities during the
transaction. In conclusion, neither the market structure nor the transaction
mechanisms in the above focus on the prompt trading issue. Hence, it is
essential to encompass the entire data production process and devise efficient
pricing models for the proposed scenario.

##  3 The Prompt Bundle Trading Mechanism

The proposed prompt bundle trading mechanism mainly consists of two steps:
CMAB-based Unknown Category Selection and HS-based Optimal Incentive Strategy.
We first briefly introduce the Prompt Bundle Trading (PBT) system, and then
elaborate on the prompt bundle trading mechanism.

###  3.1 System Overview

We consider an online Prompt Bundle Trading (PBT) system, which contains a
platform, consumers, and sellers that provide prompts with various categories.
In the system, consumers want to buy a bundle of prompts with specific
categories from the sellers via the platform. The platform acts as a data
trading broker, aggregating prompt bundles for consumers in need. After the
consumer gives the payment, the platform will charge a part of the rewards to
compensate for its service, and pay the selected sellers the remaining rewards
as their income.

Suppose the platform includes prompts products with total $N$ categories
(e.g., scenery), and these products are related to various generative models.
Note that different generative models excel in various categories, and the
quality of each category’s prompts varies. For instance, GLIDE [28] is
unsuitable for generating human-related images. To increase profit, the prompt
trading website tends to recommend more profitable prompt bundles on the
homepage for display. They need to select $K$ high-quality categories out of a
total of $N$ categories and make prompt bundles of size $M$ from the selected
categories. Afterward, the three parties need to give their strategy
respectively to maximize the profit they obtain. For convenience, we give
major notations in Table 1.

Table 1: The summary of notations. Notation | Definition
---|---
$i,j,t$ | the indexes of category, prompt, and iteration, respectively
$N,K,M$ | the number of total categories, selected categories, and selected prompts in each category
$P^{i,t}_{j}$ |  the selected prompt $j$ in category $i$ in iteration $t$
$\chi_{i}^{t}$ |  indicates whether the category $i$ is selected in iteration $t$
$\bar{q}_{i}^{t}$ |  the learned quality of category $i$ in iteration $t$
$\bar{q}^{t}$ |  the overall mean quality of all selected categories in iteration $t$
$\varsigma_{i}^{t}$ |  the entity richness of category $i$ in iteration $t$
$p^{t}$ |  the unit price of the category in iteration $t$
$p^{G,t}$ |  the unit price of the category group in iteration $t$
${\varsigma_{i}^{t}}^{\ast}$ |  the strategy of the category $i$ in iteration $t$
${p^{t}}^{\ast},{p^{G,t}}^{\ast}$ |  the strategies of platform and consumer in iteration $t$
$\varphi_{j}^{i,t}$ |  the quality of the image generated by $P^{i,t}_{j}$
$\sigma_{j}^{i,t}$ |  the correlation of $P^{i,t}_{j}$ with generated image
$C_{i}(\cdot),a_{i},b_{i},c_{i}$ |  the cost function and parameters of category $i$
$C(\cdot),\gamma,\delta$ | the cost function and parameters of platform
$\nu(\cdot),\eta,\omega$ | the valuation function and parameter of consumer
$\Phi_{i}^{t}(\cdot)$ |  the profit function of category $i$ in iteration $t$
$\Psi^{t}(\cdot)$ |  the profit function of the platform in iteration $t$
$\Omega^{t}(\cdot)$ |  the profit function of the consumer in iteration $t$

###  3.2 CMAB-based Unknown Category Selection

Modeling and Formulation Selecting prompts with the largest qualities while
their qualities are unknown a prior is a crucial challenge in PBT system.
Given that this involves an online learning and decision-making procedure, we
frame it as a Combinatorial Multi-Armed Bandit (CMAB) problem. CMAB is a
commonly used reinforcement learning model for making online decisions in
environments with uncertainty [7]. We model the category selection as a
$K$-armed CMAB game, and each category’s quality is seen as the revenue of the
corresponding arm. Since the expected quality of each category is unknown, the
platform needs to go through iterations to select the optimal set of
categories. In each iteration $t$, after choosing a set of categories, the
platform randomly selects $M$ products from each category and separately packs
them for sale. The goal in this step is to maximize the total estimated
qualities of the selected categories for $T$ iterations:

$\begin{split}&Maximize:R(\bm{\chi})=\sum_{t=1}^{T}\sum_{i=1}^{N}\sum_{j=1}^{M}q_{j}^{i,t}\chi_{i}^{t},\\\
&s.t.,\sum_{i=1}^{N}\chi_{i}^{t}=K,\chi_{i}^{t}\in\\{0,1\\}~{}~{}(\forall
i\in\left[1,\dots,N\right],\forall t\in\left[1,T\right]).\end{split}$ (1)

CMAB-based Bandit Policy We adopt a greedy search-based mechanism for category
selection. That is, we always select the top $K$ categories with the highest
qualities. In each iteration $t$, we update every category’s estimated quality
based on the knowledge learned from previous iterations. Let $n_{i}^{t}$ be
the number of times that category $i$ has been learnt, then we have:

$\begin{split}\bar{q}^{t}_{i}=\left\\{\begin{aligned}
\frac{\bar{q}^{t-1}_{i}*{{n}^{t-1}_{i}}+\sum_{j=1}^{M}{q^{i,t}_{j}}}{{n}^{t-1}_{i}+M}&,&~{}\chi_{i}^{t}=1\\\
\bar{q}^{t-1}_{i}&,&~{}\chi_{i}^{t}=0\end{aligned}\right.,~{}~{}q_{j}^{i,t}=(\varphi_{j}^{i,t}+\sigma_{j}^{i,t})/2.\end{split}$
(2)

${n}^{t}_{i}=\left\\{\begin{aligned} {n}^{t-1}_{i}+M&,&~{}\chi_{i}^{t}=1\\\
{n}^{t-1}_{i}&,&~{}\chi_{i}^{t}=0\end{aligned}\right.,$ (3)

where $M$ represents the number of prompts in a bundle, also indicating that
once a category $i$ is selected, its quality will be learned $M$ times.

###  3.3 HS-based Optimal Incentive Strategy

Modeling and Formulation In the PBT system, the consumer will pay for the
bundled prompt products as compensation for the platform and the seller. Since
the seller’s products and the platform’s prompt bundles are both related to
the prompt itself, we propose a concept: entity richness $\varsigma_{i}^{t}$,
to evaluate the detail level of the prompt texts. Based upon this, in each
iteration $t$, the consumer determines a unit prompt bundle price $p^{G,t}$
for the platform, the platform determines a unit prompt price $p^{t}$ for the
categories, and each category $i$ is determined an entity richness
$\varsigma_{i}^{t}$. We call triple $<p^{G,t},p^{t},{\bm{\varsigma}^{t}}>$ as
an incentive strategy, and once it is fixed, all the payments will be settled.
Next, we define the profit functions of each party in iteration $t$.

1) The profit of each category $i$ is the difference between the payment from
the platform and prompt production cost:

$\begin{split}\Phi_{i}^{t}(p^{t},\varsigma_{i}^{t})&=p^{t}\varsigma_{i}^{t}\chi_{i}^{t}-C_{i}(\varsigma_{i}^{t},\bar{q}_{i}^{t},\varphi^{i,t})\chi_{i}^{t},\\\
C_{i}(\varsigma_{i}^{t},\bar{q}_{i}^{t},\varphi^{i,t})=&(a_{i}{\varsigma_{i}^{t}}^{2}+b_{i}\varsigma_{i}^{t})\bar{q}_{i}^{t}+c_{i}\varphi^{i,t},~{}~{}\varphi^{i,t}=\frac{1}{M}\sum_{j=1}^{M}\varphi_{j}^{i,t},\end{split}$
(4)

where the cost function
$C_{i}(\varsigma_{i}^{t},\bar{q}_{i}^{t},\varphi^{i,t})$ reflects the seller’s
effort level. We adopt a widely used quadratic cost function [13, 46].
Besides, we use an item $c_{i}\varphi^{i,t}$ to exclusively evaluate the
quality of images generated by $P_{j}^{i,t},j\in\left[1,\dots,M\right]$.
$a_{i}>0$ and $b_{i}\geq 0$ are pre-defined parameters.

2) The profit of the platform is the total payment from the consumer minus
payment to the seller and its prompt bundling cost:

$\begin{split}\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right)&=p^{G,t}\sum_{i=1}^{N}\varsigma_{i}^{t}\chi_{i}^{t}-p^{t}\sum_{i=1}^{N}\varsigma_{i}^{t}\chi_{i}^{t}-C(\bm{\varsigma^{t}}),\\\
C(\bm{\varsigma^{t}})&=\gamma\left({\sum_{i=1}^{N}\varsigma_{i}^{t}\chi_{i}^{t}}\right)^{2}+\delta\sum_{i=1}^{N}\varsigma_{i}^{t}\chi_{i}^{t},\end{split}$
(5)

where $\gamma>0$ and $\delta\geq 0$ are pre-defined parameters.

3) The profit of the consumer is the difference between the valuation of the
purchased bundled prompt products and the payment to the platform:

$\begin{split}\Omega^{t}\left(p^{G,t},\bm{\varsigma^{t}}\right)&=\nu(\bm{\varsigma^{t}},\bm{\sigma^{t}},\bar{q}^{t})-p^{G,t}\sum_{i=1}^{N}\varsigma_{i}^{t}\chi_{i}^{t},\\\
\nu\left(\bm{\varsigma^{t}},\bm{\sigma^{t}},\bar{q}^{t}\right)=\eta\cdot\ln(1+&\bar{q}^{t}\sum_{i=1}^{N}\varsigma_{i}^{t}\chi_{i}^{t})+\omega\sum_{i=1}^{N}\sigma^{i,t}\chi_{i}^{t},~{}~{}\sigma^{i,t}=\frac{1}{M}\sum_{j=1}^{M}\sigma_{j}^{i,t},\end{split}$
(6)

where the value function $\nu(\bm{\varsigma^{t}},\bm{\sigma^{t}},\bar{q}^{t})$
is designed inspired by [46, 19], and
$\bar{q}^{t}=\frac{\sum_{i=1}^{N}\bar{q}_{i}^{t}\chi_{i}^{t}}{\sum_{i=1}^{N}\chi_{i}^{t}}$
is the mean of estimated qualities of selected categories in the $t$-th
iteration. Besides, we add an item
$\omega\sum_{i=1}^{N}\sigma^{i,t}\chi_{i}^{t}$ to evaluate the correlation
between prompt products and generated images. We believe that if the
correlation is large, the product’s value is high. $\eta>0$ and $\omega>0$ are
pre-defined parameters.

Three-stage HS Game To ensure that each participant can maximize its profit,
we need to find an optimal incentive strategy
$<{p^{G,t}}^{\ast},{p^{t}}^{\ast},{\bm{\varsigma}^{t}}^{\ast}>$, which
satisfies:

$\text{Stage 1
Consumer:~{}}{p^{G,t}}^{\ast}=\arg\max_{p^{G,t}}\Omega^{t}\left(p^{G,t},\bm{\varsigma^{t}}\right).$
(7)

$\text{Stage 2
Platform:~{}}{p^{t}}^{\ast}=\arg\max_{p^{t}}\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right).$
(8)

$\text{Stage 3 Category
$i$:~{}}{\varsigma_{i}^{t}}^{\ast}=\arg\max_{\varsigma_{i}^{t}}\Phi_{i}^{t}(p^{t},\varsigma_{i}^{t}).$
(9)

Then, we use the backward induction approach to derive the solutions for
Eq.(7-9). We first find each category’s optimal strategy
${\varsigma_{i}^{t}}^{\ast}$ given the platform’s strategy $p^{t}$ (Stage 3):

${\varsigma_{i}^{t}}^{\ast}=\frac{p^{t}-b_{i}\bar{q}_{i}^{t}}{2a_{i}\bar{q}_{i}^{t}}.$
(10)

Theorem 1. In Stage 3, given the unit prompt price $p^{t}$, the optimal entity
abundance ${\varsigma_{i}^{t}}^{\ast}$ of category $i$ can be determined by
Eq.(10).

_Proof._ Compute the first and second derivatives of the profit function
$\Phi_{i}^{t}(p^{t},\varsigma_{i}^{t})$ for each category with respect to
$\varsigma_{i}^{t}$ in the Eq.(4):

$\frac{\partial\Phi_{i}^{t}(p^{t},\varsigma_{i}^{t})}{\partial\varsigma_{i}^{t}}=p^{t}-2a_{i}\bar{q}_{i}^{t}\varsigma_{i}^{t}-b_{i}\bar{q}_{i}^{t},$
(11)

$\frac{\partial^{2}\Phi_{i}^{t}(p^{t},\varsigma_{i}^{t}))}{\partial({\varsigma_{i}^{t}})^{2}}=-2a_{i}\bar{q}_{i}^{t}<0.$
(12)

Therefore, it is evident that the second partial derivative of
$\Phi_{i}^{t}(p^{t},\varsigma_{i}^{t})$ to $\varsigma_{i}^{t}$ is negative,
indicating a strictly convex objective function and a unique global optimum.
Let the first partial derivative
$\frac{\partial\Phi_{i}^{t}(p^{t},\varsigma_{i}^{t})}{\partial\varsigma_{i}^{t}}=0$,
and the solution yields the optimal entity abundance
${\varsigma_{i}^{t}}^{\ast}$ of the category $i$. The result is shown in
Eq.(10).

Next, based on the categories’ strategies in Stage 3, we can derive the
platform’s optimal strategy ${p^{t}}^{\ast}$ given the consumer’s strategy
$p^{G,t}$ (Stage 2):

${p^{t}}^{\ast}=\frac{{p^{G,t}}A-(\delta A-2\gamma AB-B)}{2A(1+\gamma A)},$
(13)

where $A=\sum_{i=1}^{K}\frac{1}{2a_{i}\bar{q}_{i}^{t}}$, and
$B=\sum_{i=1}^{K}\frac{b_{i}}{2a_{i}}$. For Theorem 2 and corresponding Proof
details, please refer to Section A.1 in the Appendix.

Finally, based on the optimal strategies obtained in Stage 2 and Stage 3, we
determine the consumer’s optimal strategy ${p^{G,t}}^{\ast}$ (Stage 1):

${p^{G,t}}^{\ast}=\frac{3\bar{q}^{t}\Theta-2+\sqrt{\Delta}}{4\bar{q}^{t}\Gamma},$
(14)

where $\Delta=(2-\bar{q}^{t}\Theta)^{2}+8\eta\Gamma(\bar{q}^{t})^{2}$,
$\Gamma=\frac{A}{2(1+\gamma A)}$, $\Theta=\frac{\delta A-2\gamma
AB-B}{2(1+\gamma A)}+B$, $A=\sum_{i=1}^{K}\frac{1}{2a_{i}\bar{q}_{i}^{t}}$,
and $B=\sum_{i=1}^{K}\frac{b_{i}}{2a_{i}}$. Please refer to Section A.2 in
Appendix for Theorem 3 and corresponding Proof.

##  4 Implementation and Analysis

(1) SD

(2) LD

(3) GLIDE

(4) DALLE mini

Figure 2: Quality distributions of prompt products generated by different
generative models.

###  4.1 Experimental Settings

Dataset Construction We first simulate a prompt product dataset using four
popular pre-trained text-to-image generative models, i.e., Stable Diffusion
v1.5 (SD) [32], Latent Diffusion (LD) [32], GLIDE [28], and DALL·E mini [16].
We take $20,000$ captions from MSCOCO [22] dataset as prompts for sale,
finally getting $80,000$ prompt-image pairs as products. These products
belongs to $80$ categories pre-defined by MSCOCO, so we split the whole
dataset into $80$ sub-datasets for experiments (Please refer to A.3 in the
Appendix for more details about data construction). For each category, we set
a fixed generative model for better comparison. The statistics of the dataset
are shown in Table 2. For image quality assessment, we use BRUSQUE [26] and
ClipIQA [41]. The similarity between prompts and corresponding generated
images is evaluated by CLIPScore [20]. We then visualize the quality
distribution of simulated prompt products from each model. As displayed in
Figure 2, the quality of prompt products from different models yields distinct
distribution characteristics, with SD and LD producing higher-quality images
than GLIDE and DALLE-mini in general.

Table 2: The statistics of the text-to-image dataset1.

Categories | Generative Model |  ClipIQA $\uparrow$ |  BRUSQUE $\downarrow$ |  CLIPScore $\uparrow$ |  Quality $\uparrow$ 2
---|---|---|---|---|---
1 $\sim$ 20 | SD | 0.894 | 16.459 | 26.729 | 0.778
21 $\sim$ 40 | LD | 0.907 | 19.155 | 25.539 | 0.775
41 $\sim$ 60 | GLIDE | 0.486 | 45.330 | 21.976 | 0.548
61 $\sim$ 80 | DALLE mini | 0.667 | 23.599 | 24.254 | 0.668

  * 1

All the calculated values in the table are the mean values of the specific
scope of categories.

  * 2

Quality=Scaled(2*ClipIQA + ClipScore*1.0/100-Brisque*1.0/100).

Implementation Details The cost function parameters of categories $a_{i}$,
$b_{i}$ are randomly set in $\left[0.01,0.5\right]$ and $\left[0.01,1\right]$,
respectively. Similarly, we set the cost function parameters of the platform
as $\gamma\in\left[0.1,1\right]$, $\delta\in\left[0.01,0.05\right]$, and the
valuation function parameters of the consumer as $\eta\in\left[1,10\right]$
and $\omega\in\left[1,5\right]$. We set
$\gamma=0.1,\delta=0.01,\eta=1,\omega=1$ by default. The Python version we use
is $3.8.19$. The default value of $N,K,M$ is $80,4,4$, respectively. When the
settings are as default, it takes about $1$ hour to complete a $10,000$
iteration round. The experiments were conducted on an Ubuntu 18.04.6 LTS
system with the random seed fixed as $42$.

Baselines During the unknown category selection process, we compare our
mechanism with several commonly used algorithms [18, 31]. Specifically, the
Random algorithm randomly selects arms in all iterations. The $\epsilon$-first
algorithm randomly selects arms in the first $\epsilon*T$ iterations and
selects the current top-$K$ categories with the highest estimated qualities
for the remaining $(1-\epsilon)*T$ iterations. The optimal algorithm, which
knows the quality of the categories in advance, always selects the same
top-$K$ categories in all the iterations. Besides, we also compare CUCB
algorithm proposed in [6].

Evaluation Metrics We take total revenue, strategy, and profit as evaluation
metrics like [6], and use PoC, SoC, PoP, SoP, PoS, and SoS to denote the
Profit and Strategy of the Consumer, Platform, and Seller’s categories(s),
respectively. Besides, we define $\Delta$-PoC, $\Delta$-PoP, and
$\Delta$-PoS(s) to measure the difference in profit between the optimal and
other algorithms.

Figure 3: Effect of $T$ on revenue.

Figure 4: Effect of $N$ on revenue.

Figure 5: Effect of $K$ on revenue.

Figure 6: Effect of $T$ on PoS.

Figure 7: Effect of $N$ on PoP.

Figure 8: Effect of $K$ on PoC.

###  4.2 Evaluation of greedy search-based CMAB

First, we fix $N=80,K=4$, and evaluate the effect of iteration number $T$ on
the total revenue and profits. As seen from Figure 8, as the iteration $T$
increases, the total revenue rises for all algorithms. The greedy search-based
approach outperforms others, with its curve closest to the curve of optimal.
We can observe that more iterations with exploitation tend to achieve near-
optimal results: greedy search performs better than $\epsilon$-first and
random algorithms. In Figure 8, with $T$ growing, $\Delta$-PoS decreases
gradually, demonstrating the estimated quality becomes more accurate, and the
election results closer to the optimal algorithm.

Then, we set $T=5000,K=4$, and change the number of candidate categories $N$.
According to Figure 8, the total revenue decreases as $N$ increases, this is
because the average qualities of the latter categories are proven to be lower,
as displayed in Figure 2. Even though, the greedy search-based selection
method can still tightly approach optimal method. For $\Delta$-PoP from Figure
8, although the appearance of new candidate categories makes some impact, the
trend is overall stable except for some fluctuations caused by the uncertainty
of estimated qualities.

Finally, we set $T=5000,N=80$, and evaluate the number of selected categories
$K$. We notice that the total revenues climb along with $K$ for all the
algorithms (Figure 8). The reason is the number of times the revenue
accumulated is proportional to the number of selected categories $K$ when $N$
is a fixed value. Meanwhile, there exists more dispersion between curves of
different algorithms, because as $K$ increases, the total quality estimation
error also rises. Overall, K’s growth will not have a significant impact on
$\Delta$-PoC (Figure 8).

The complete experimental results on CMAB evaluation are provided in Section
A.4 in the Appendix.

Figure 9: SoC vs. PoC/PoP.

Figure 10: SoC vs. PoS(s).

Figure 11: SoP vs. PoC/PoP.

Figure 12: SoP vs. PoS(s).

Figure 13: SoS vs. PoC/PoP.

Figure 14: SoS vs. PoS(s).

Figure 15: $\gamma$ vs. SoC/SoP.

Figure 16: $\gamma$ vs. SoS(s).

Figure 17: $a_{37}$ vs. SoC/SoP.

Figure 18: $a_{37}$ vs. SoS(s).

Figure 19: $a_{37}$ vs. PoC/PoP.

Figure 20: $a_{37}$ vs. PoS(s).

###  4.3 Evaluation of three-stage HS game

As the optimal incentive strategy in each iteration is similar, we select
iteration $t=100$ to evaluate the profit and strategy of each participant.

The effect of SoC on PoP/PoC/PoS(s). We first see how SoC ($p^{G}$) impacts
the each part’s profit. In terms of Eq.(10) and Eq.(13), the value of SoC
affects the optimal value of SoP and SoS(s), demonstrating the consumer as the
first-tier leader. We can see from Figure 20 and Figure 20 that as $p^{G,100}$
increases, the consumer obtains less profit, while the platform and all the
categories harvest more profits. That’s because the growth of $SoC$ leads to
the increase of $SoP$ and $SoS(s)$, hence bringing more profits to the other
two participants.

The effect of SoP on PoP/PoC/PoS(s). Then we fix SoC as the optimal value and
analyze the impact of the platform’s strategy $p$ on the three parts’ profits.
According to Eq.(10), the optimal strategy of each category is influenced by
the platform’s strategy $p$, so SoS(s) change along with SoP, indicating the
platform as the second-tier leader. From Figure 20, the increase of $p^{1}00$
directly results in the platform giving more rewards to the seller for
collecting products, hence reducing the platform’s profit. And the consumer’s
profit also decreases, by raising the payment to the platform. We can tell
from Figure 20 that as $p^{1}00$ rises, the profits of all the categories also
increase. The reason can be obtained from Eq.(4): when the platform gives the
seller a higher unit product price, the seller receives more profit
accordingly.

The effect of SoS on PoP/PoC/PoS(s). Later, we fix SoP and SoC as the optimal
values to explore the effect of SoS on the profits of the three participants.
Take the strategy of category 37 ($\tau_{37}$) as an example. Figure 20 shows
that both the profits of the consumer and platform first increase and later
decline, reaching a maximum point respectively. Besides, from Figure 20, when
$\tau_{37}^{100}$ takes the optimal value, category 37 can get the highest
profit correspondingly. The trends between the different categories are
independent of each other, which is consistent with Eq.(4). Note that our
purpose is to achieve all participants’ profit maximization, so we need to
consider the whole profits of all three parts, rather than any single
participant.

The effect of hyper-parameters on strategies and profits. Here we focus on two
parameters in platform’s cost function ($\gamma$) and category’s cost function
($a_{i}$). For $\gamma$, when it increases, the platform’s cost rises, so the
consumer enhances price $p^{G}$, while the platform reduces $p$ to guarantee
profit, as illustrated in Figure 20. To deal with this situation, the seller
reduces each categories’ entity richness to ensure profit (Figure 20).
Afterwards, we examine how $a_{i}$ make a difference to the strategy-making
process. Suppose at iteration $t$, the chosen categories are
$\left[20,30,31,37\right]$. We evaluate how the parameter $a_{37}$ in category
37’s cost function impacts the strategy and profit of the consumer, platform,
and categories. According to Figure 20 and Figure 20, with $a_{37}$
increasing, category 37’s cost increases, the strategy of category 37 is
inversely proportional to $a_{37}$ (c.f. Eq.(10). Meanwhile, SoC and SoP are
proportional to the change of $a_{37}$, as the consumer and the platform have
to raise $p^{G}$ and $p$. As for profit (Figure 20 and Figure 20), PoC
declines sharply due to higher $p^{G}$, while PoP increases and then
approaches to a flat. Moreover, the profit of category 37 decreases as
$a_{37}$ grows, which can be explained by Eq.(4), where larger $a_{37}$ leads
to more cost. The platform, as a broker, when obtains more payment from the
consumer and pays less to the seller, it will earn more.

The above results verify the effectiveness of the three-stage HS game.

##  5 Conlusion

In this paper, we consider a novel prompt trading scenario and correspondingly
propose a prompt bundle trading (PBT) mechanism. We break the PBT issue into
two steps, adopt greedy search-based Combined Multi-Arm Bandit (CMAB) for
unknown categories selection and apply a three-stage hierarchical Stackelburg
game to derive the best strategies for the three participants in the trading.
In this way, we can maximize the profits between the consumer, the platform,
as well as the seller. Compared to the pricing method used by the real prompt
trading website, our mechanism is more flexible and thoughtful. We hope this
work can prompt to create a good prompt trading environment.

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##  Appendix A Appendix / supplemental material

###  A.1 Determining the platform’s optimal strategy

Theorem 2. In Stage 2, based on the optimal strategy
${\varsigma_{i}^{t}}^{\ast}$ for the seller’s prompt $i$ given in Stage 3, the
optimal strategy ${p^{t}}^{\ast}$ for the platform can be calculated using the
following formula:

${p^{t}}^{\ast}=\frac{{p^{G,t}}A-(\delta A-2\gamma AB-B)}{2A(1+\gamma A)},$
(15)

where $A=\sum_{i=1}^{K}\frac{1}{2a_{i}\bar{q}_{i}^{t}}$, and
$B=\sum_{i=1}^{K}\frac{b_{i}}{2a_{i}}$.

_Proof._ Substitute Eq.(4) into Eq.(5) and solve for the first and second
derivatives of $\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right)$ with
respect to $p^{t}$.

Firstly, by substituting
${\varsigma_{i}^{t}}^{\ast}=\frac{p^{t}-b_{i}\bar{q}_{i}^{t}}{2a_{i}\bar{q}_{i}^{t}}$
into Eq.(5), we can obtain:

$\begin{split}\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right)=p^{G,t}\sum_{i=1}^{N}\frac{p^{t}-b_{i}\bar{q}_{i}^{t}}{2a_{i}\bar{q}_{i}^{t}}\chi_{i}^{t}-p^{t}\sum_{i=1}^{N}\frac{p^{t}-b_{i}\bar{q}_{i}^{t}}{2a_{i}\bar{q}_{i}^{t}}\chi_{i}^{t}-\gamma\left({\sum_{i=1}^{N}\frac{p^{t}-b_{i}\bar{q}_{i}^{t}}{2a_{i}\bar{q}_{i}^{t}}\chi_{i}^{t}}\right)^{2}-\delta\sum_{i=1}^{N}\frac{p^{t}-b_{i}\bar{q}_{i}^{t}}{2a_{i}\bar{q}_{i}^{t}}\chi_{i}^{t}\end{split}$
(16)

Then, solve for the first derivative
$\frac{\partial\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right)}{\partial
p^{t}}$ and the second derivative
$\frac{\partial^{2}\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right)}{\partial(p^{t})^{2}}$
of $\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right)$ with respect to
$p^{t}$:

$\begin{split}\frac{\partial\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right)}{\partial
p^{t}}={p^{G,t}}\sum_{i=1}^{K}\frac{1}{2a_{i}\bar{q}_{i}^{t}}-2\sum_{i=1}^{K}\frac{p^{t}}{2a_{i}\bar{q}_{i}^{t}}+\sum_{i=1}^{K}\frac{b_{i}}{2a_{i}}\\\
-\left[2\gamma\sum_{i=1}^{K}\frac{p^{t}-b_{i}\bar{q}_{i}^{t}}{2a_{i}\bar{q}_{i}^{t}}\sum_{i=1}^{K}\frac{1}{2a_{i}\bar{q}_{i}^{t}}+\delta\sum_{i=1}^{K}\frac{1}{2a_{i}\bar{q}_{i}^{t}}\right],\end{split}$
(17)

$\frac{\partial^{2}\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right)}{\partial(p^{t})^{2}}=-2A-2\gamma
A^{2}<0.$ (18)

Therefore, the second partial derivative of
$\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right)$ concerning $p^{t}$ is
strictly negative, indicating that the objective function is strictly convex
and that there is a unique global optimum. Let the first partial derivative
$\frac{\partial\Psi^{t}\left(p^{G,t},p^{t},\bm{\varsigma^{t}}\right)}{\partial
p^{t}}=0$, and the solution yields the optimal unit price of the seller’s
prompt ${p^{t}}^{\ast}$ for the prompt platform. The result is shown in
Eq.(15).

###  A.2 Determining the consumer’s optimal strategy

Theorem 3. In Stage 1, based on the optimal strategy ${p^{t}}^{\ast}$ for the
platform given in Stage 2, the optimal strategy ${p^{G,t}}^{\ast}$ for the
consumer can be calculated using the following formula:

${p^{G,t}}^{\ast}=\frac{3\bar{q}^{t}\Theta-2+\sqrt{\Delta}}{4\bar{q}^{t}\Gamma}$
(19)

where $\Delta=(2-\bar{q}^{t}\Theta)^{2}+8\eta\Gamma(\bar{q}^{t})^{2}$,
$\Gamma=\frac{A}{2(1+\gamma A)}$, $\Theta=\frac{\delta A-2\gamma
AB-B}{2(1+\gamma A)}+B$, $A=\sum_{i=1}^{K}\frac{1}{2a_{i}\bar{q}_{i}^{t}}$,
and $B=\sum_{i=1}^{K}\frac{b_{i}}{2a_{i}}$.

_Proof._ Substitute Eq.(10) and Eq.(13) into Eq.(6) and solve for the first
and second derivatives of $\Omega^{t}\left(p^{G,t},\bm{\varsigma^{t}}\right)$
with respect to $p^{G,t}$.

Firstly, by substituting
${\varsigma_{i}^{t}}^{\ast}=\frac{p^{t}-b_{i}\bar{q}_{i}^{t}}{2a_{i}\bar{q}_{i}^{t}}$
into Eq.(6), we can obtain:

$\begin{split}\Omega^{t}\left(p^{G,t},\bm{\varsigma^{t}}\right)=\eta\cdot\ln\left(1+\bar{q}^{t}\sum_{i=1}^{N}\frac{p^{t}-b_{i}\bar{q}_{i}^{t}}{2a_{i}\bar{q}_{i}^{t}}\chi_{i}^{t}\right)+\omega\sum_{i=1}^{N}\sigma^{i,t}\chi_{i}^{t}-p^{G,t}\sum_{i=1}^{N}\frac{p^{t}-b_{i}\bar{q}_{i}^{t}}{2a_{i}\bar{q}_{i}^{t}}\chi_{i}^{t}\\\
=\eta\cdot\ln\left(1+\bar{q}^{t}p^{t}A-\bar{q}^{t}B\right)+\omega\sum_{i=1}^{N}\sigma^{i,t}\chi_{i}^{t}-p^{G,t}(p^{t}A-B)\end{split}$
(20)

Then, by substituting ${p^{t}}^{\ast}=\frac{{p^{G,t}}A-(\delta A-2\gamma
AB-B)}{2A(1+\gamma A)}$ into Eq.(20), we can obtain:

$\begin{split}\Omega^{t}\left(p^{G,t},\bm{\varsigma^{t}}\right)=\eta\cdot\ln\left(1+\bar{q}^{t}\frac{{p^{G,t}}A-(\delta
A-2\gamma AB-B)}{2A(1+\gamma
A)}A-\bar{q}^{t}B\right)+\omega\sum_{i=1}^{N}\sigma^{i,t}\chi_{i}^{t}\\\
-p^{G,t}(\frac{{p^{G,t}}A-(\delta A-2\gamma AB-B)}{2A(1+\gamma
A)}A-B)\end{split}$ (21)

Let $\Gamma=\frac{A}{2(1+\gamma A)}$, $\Theta=\frac{\delta A-2\gamma
AB-B}{2(1+\gamma A)}+B$. Eq.(21) can be rewritten:

$\begin{split}\Omega^{t}\left(p^{G,t},\bm{\varsigma^{t}}\right)=\eta\ln\left(1+\bar{q}^{t}p^{G,t}\Gamma-\bar{q}^{t}\Theta\right)+\omega\sum_{i=1}^{N}\sigma^{i,t}\chi_{i}^{t}-(p^{G,t})^{2}\Gamma+p^{G,t}\Theta\end{split}$
(22)

Let $\Upsilon=\Theta-p^{G,t}\Gamma<0$, and convert optimal $P^{{G,t}^{\ast}}$
determination problem into optimal $\Upsilon^{\ast}$ determination problem:

$\Omega^{t}(\Upsilon)=\eta\ln(1-\bar{q}^{t}\Upsilon)+\omega\sum_{i=1}^{N}\sigma^{i,t}\chi_{i}^{t}+\frac{\Upsilon(\Theta-\Upsilon)}{\Gamma}$
(23)

compute the first-order derivative of $\Omega^{t}(\Upsilon)$:

$\frac{\partial\Omega^{t}(\Upsilon)}{\partial\Upsilon}=\frac{-\eta\bar{q}^{t}}{1-\bar{q}^{t}\Upsilon}+\frac{\Theta-2\Upsilon}{\Gamma}=\frac{2\bar{q}^{t}\Upsilon^{2}-(2+\bar{q}^{t}\Theta)\Upsilon+\Theta-\eta\bar{q}^{t}\Gamma}{\Gamma(1-\bar{q}^{t}\Upsilon)}$
(24)

As $\Omega^{t}(\Upsilon)$ is not standardly concave, we need to analyze its
monotonicity. We find the numerator term of
$\frac{\partial\Omega^{t}(\Upsilon)}{\partial\Upsilon}$ is a quadratic formula
of $\Upsilon$, so let
$2\bar{q}^{t}\Upsilon^{2}-(2+\bar{q}^{t}\Theta)\Upsilon+\Theta-\eta\bar{q}^{t}\Gamma=0$
and seek for the roots of it as follows:

$\Delta=(2+\bar{q}^{t}\Theta)^{2}-8\bar{q}^{t}(\Theta-\eta\bar{q}^{t}\Gamma)=(2-\bar{q}^{t}\Theta)^{2}+8\eta\Gamma(\bar{q}^{t})^{2}>(2-\bar{q}^{t}\Theta)^{2}\geq
0$ (25)

Accordingly, the numerator term in Eq.24 has two roots. When
$2-\bar{q}^{t}\Theta>0,\textit{i.e.},\Theta<\frac{2}{\bar{q}^{t}}$, the two
roots can be computed:

$\Upsilon_{1}=\frac{2+\bar{q}^{t}\Theta-\sqrt{\Delta}}{4\bar{q}^{t}}<\frac{2+\bar{q}^{t}\Theta-(2-\bar{q}^{t}\Theta)}{4\bar{q}^{t}}<\frac{1}{\bar{q}^{t}}$
(26)

$\Upsilon_{2}=\frac{2+\bar{q}^{t}\Theta+\sqrt{\Delta}}{4\bar{q}^{t}}>\frac{2+\bar{q}^{t}\Theta+(2-\bar{q}^{t}\Theta)}{4\bar{q}^{t}}=\frac{1}{\bar{q}^{t}}$
(27)

Else when $2-\bar{q}^{t}\Theta\leq
0,\textit{i.e.},\Theta\geq\frac{2}{\bar{q}^{t}}$, the two roots can be
computed:

$\Upsilon_{1}=\frac{2+\bar{q}^{t}\Theta-\sqrt{\Delta}}{4\bar{q}^{t}}<\frac{2+\bar{q}^{t}\Theta-(2-\bar{q}^{t}\Theta)}{4\bar{q}^{t}}=\frac{1}{\bar{q}^{t}}$
(28)

$\Upsilon_{2}=\frac{2+\bar{q}^{t}\Theta+\sqrt{\Delta}}{4\bar{q}^{t}}>\frac{2+\bar{q}^{t}\Theta+(2-\bar{q}^{t}\Theta)}{4\bar{q}^{t}}\geq\frac{1}{\bar{q}^{t}}$
(29)

According to Eq.(26)-Eq.(29),
$\Upsilon_{1}\leq\frac{1}{\bar{q}^{t}}\leq\Upsilon_{2}$ always holds. Hence,
$\Omega^{t}(\Upsilon)$ monotonically increases in $(-\infty,\Upsilon_{1})$,
decreases in $(\Upsilon_{1},\frac{1}{\bar{q}^{t}})$, increases in
$(\frac{1}{\bar{q}^{t}},\Upsilon_{2})$, and decreases in
$(\Upsilon_{2},+\infty)$. Since $\Upsilon<0$, we choose $\Upsilon_{1}$ as the
optimal point of $\Omega^{t}(\Upsilon)$. Then ${p^{G,t}}^{\ast}$ can be
calculated as:

${p^{G,t}}^{\ast}=\frac{\Theta-\Upsilon_{1}}{\Gamma}=\frac{3\bar{q}^{t}\Theta-2+\sqrt{\Delta}}{4\bar{q}^{t}\Gamma}$
(30)

###  A.3 Data Construction

As a prompt may relate to various categories, while splitting sub-datasets, we
classify the prompt into all the categories related to it. Therefore, the sub-
dataset scale of different categories varies. Besides, the whole categories we
considered in the experiments are listed below.

[person, bicycle, car, motorcycle, airplane, bus, train, truck, boat, traffic
light, fire hydrant, stop sign, parking meter, bench, bird, cat, dog, horse,
sheep, cow, elephant, bear, zebra, giraffe, backpack, umbrella, handbag, tie,
suitcase, frisbee, skis, snowboard, sports ball, kite, baseball bat, baseball
glove, skateboard, surfboard, tennis racket, bottle, wine glass, cup, fork,
knife, spoon, bowl, banana, apple, sandwich, orange, broccoli, carrot, hot
dog, pizza, donut, cake, chair, couch, potted plant, bed, dining table,
toilet, tv, laptop, mouse, remote, keyboard, cell phone, microwave, oven,
toaster, sink, refrigerator, book, clock, vase, scissors, teddy bear, hair
drier, toothbrush]

###  A.4 Experimental Results on CMAB

Figure32 to Figure 32 illustrate the experimental results of CMAB-based unknow
category selection, which demostrate the effectiveness and feasibility of our
method.

Figure 21: $T$ vs. revenue.

Figure 22: $T$ vs. $\Delta$-PoC.

Figure 23: $T$ vs. $\Delta$-PoP.

Figure 24: $T$ vs. $\Delta$-PoS(s).

Figure 25: $N$ vs. revenue.

Figure 26: $N$ vs. $\Delta$-PoC.

Figure 27: $N$ vs. $\Delta$-PoP.

Figure 28: $N$ vs. $\Delta$-PoS(s).

Figure 29: $K$ vs. revenue.

Figure 30: $K$ vs. $\Delta$-PoC.

Figure 31: $K$ vs. $\Delta$-PoP.

Figure 32: $K$ vs. $\Delta$-PoS(s).

###  A.5 Limitations

Due to the limited experimental resources and other factors, this work has the
following limitations. First, The Prompt Bundle Trading (PBT) scenario
proposed in this paper is only part of the whole prompt trading scenario,
which constrains the application of the proposed methodology. Besides,
considering the copyright of the real prompt products, we did not scrape the
real data on the selling websites, nor did we purchase a large number of real
prompt products due to restricted funding. Instead, we considered several
large text-to-image generative models, constructed a dataset using caption
texts from the MSCOCO dataset as prompts, and generated corresponding images
offline. Moreover, We didn’t verify our methodology on prompt products related
to large language models. Therefore, due to the distribution difference
between the simulated data and the real prompt products in the marketplace,
the experimental results may deviate to a certain extent.

###  A.6 Societal Impact

This paper focuses on the pricing scenario of prompt bundle deals in prompt
trading websites. This is a practical problem in real-world application
scenarios. The solution to this problem can achieve a win-win-win situation
for consumers, platforms, and sellers, which is conducive to creating a good
prompt trading environment.

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