Cross-organizational collaborative development of general technologies for artificial intelligence deep learning systems

Abstract

As the core engine driving the intelligent transformation of global industries, artificial intelligence (AI) deep learning systems have shown a cross-organizational development trend that breaks through the boundaries of a single organization and profoundly reshaping the global industrial ecosystem. The research findings are as follows: (1) The joint cost-sharing model and the aggregated integration decision-making model demonstrate outstanding effectiveness at the technical incentive mechanism level of AI deep learning. (2) When the initial innovation level of the product is relatively high, the AI deep learning system is positively correlated with the effort level and innovation ability of the participants, in the joint cost-sharing and aggregated integration decision-making model, various entities and the entire AI deep learning system have been optimized. (3) Under the aggregated and integrated decision-making mode, the effort level of participants and subjects towards the AI deep learning system is the highest, and thus the level of the AI deep learning system also reaches the optimal level. Previous studies have overlooked cooperative selection models of AI. This article can effectively enhance the depth of collaboration among entities, improve the level of output, and provide a theoretical path for the timely upgrading and development of AI systems.

Keywords

artificial intelligence deep learning systems common technology optimal decision cross-organization

Introduction

Common technology is a basic technology that can be widely used in many different industries or fields. The industrial common technology like building a solid “bridge” for the continuous innovation of subsequent complementary technologies.¹ Due to its platform effect, common technology plays a central role in promoting industrial development. Common technology has even become a powerful engine for economic progress and industrial upgrading in the country.^2,3 As the current disruptive technology leads to the development of AI, common technology, cross-organizational aggregation and integration decision-making research and development (R&D), optimal decision, and differential game fields, AI triggers a new wave revolution and becomes the core technology to generate new productivity.⁴ Under the perspective of industrial common technology, is a kind of cooperation mode with the development of technological innovation as the core, which has received increasing attention in recent years. Such models are typically composed of billions or even hundreds of billions of parameters, showing more robust performance and excellent generalization capabilities when dealing with complex tasks. The link between the AI deep learning systems and end-consumer enterprises has become even closer as the industry continues to develop. For example, Deepseek and Tsinghua University’s AI laboratory have developed the world’s first Longbench, a system for evaluating long text processing technology. In the consumer terminal field, Hisense TV and other enterprises have accessed Deepseek’s AI to provide more intelligent technical support for consumer terminals, thus bringing higher value-added to end-consumption enterprises. Therefore, we can analyze how AI deep learning-related enterprises can improve the level of aggregation and integration decision-making innovation and R&D efficiency from the perspective of common industrial technologies. A new model of cross-institutional cooperation in the R&D of common industrial technologies has been gradually established, which can not only effectively identify and solve the hidden “reefs” in the process of aggregation and integration decision-making R&D of supply chains of AI deep learning enterprises, but also cooperate with terminal consumer enterprises, and develops differentiated model choices for consumer enterprises based on different terminal enterprise application scenarios, continuously collects feedback from the terminal enterprise side to realize the rapid terminal application of the AI deep learning, and then promote the AI deep learning enterprises to achieve rapid iteration.

The environment for the development of AI deep learning enterprises is gradually “rising” as a series of activities, such as R&D, innovation, testing, and commissioning in the AI enterprises are in full swing, and their importance is increasingly magnified.⁵ Industrial security issues such as arithmetic bottlenecks, data shortages, and huge re-source consumption are becoming more prominent. These hidden dangers are not only in an isolated category but are very likely to trigger a “domino” shock effect, laying a very threatening potential pitfall.⁶ For AI deep learning enterprises, efficient use of R&D resources and industrial base, breakthroughs in key technological problems, promotion of industrial chain reinforcement and soundness, through cross-border cooperation to create AI deep learning enterprises common technology cross-institution cooperation R&D system, to enhance their core competitiveness, which is the core path to achieve win-win cooperation. Nowadays, AI deep learning systems are facing challenges in industry chain cooperation, application scene expansion and extension, effective cost control, business model innovation, and change. How to break the existing “bottlenecks” by restricting their development, and filling the current significant “short boards”, has become the core academic and practical problems that all the participants in the AI deep learning systems need to solve.

The research ideas of this article are as follows: based on the presented evidence, this paper starts from the perspective of cross-institutional common technology development. This paper applies to the differential game model, to explore the cross-institutional aggregation and integration decision-making R&D mechanism in a system composed of end-consumption enterprises, scientific research institutes, and AI deep learning enterprises. The goal is to provide theoretical guidance of practical value for the AI deep learning enterprises to enhance the benefits of synergism and realize the path of high-efficiency cooperation.

This paper presents the following innovations in the research process: (1) this paper considers the effect of time factors on the output of cross-institutional cooperation on common technologies in AI deep learning enterprises. Introducing a differential game approach to analyze coordination and technology-derived outcomes of cross-institutional cooperation in AI deep learning enterprises. (2) This paper develops a differential game-theoretic framework for cross-institutional distributed autonomous decision-making and aggregated and integrated decision-making among institutional members, and this paper considers three R&D cooperation models, analyzes, and calculates the benefit effects of technological innovation cooperation under different conditions. (3) This paper is based on the consumer-centric principle, the impact of feedback information from end-consumer enterprises on cross-institutional R&D collaboration coordination and optimal returns is considered in the model construction.

Literature review

Related research on technology innovation in the artificial intelligence deep learning

At present, AI deep learning enterprises problems such as arithmetic bottlenecks, data shortages, huge resource consumption, and other industrial security problems are becoming increasingly prominent.⁶ At the same time, cross-institutional cooperation in the R&D of common technologies for the AI deep learning enterprises has been established. The potential that AI deep learning enterprises technologies can provide for technology users to achieve the desired goals, can empower traditional manufacturing enterprises to achieve digital and intelligent transformation and upgrade with the uniqueness that distinguishes them from conventional technologies.⁷ For example, the potential to complete daily tasks for the enterprise through autonomous decision-making, autonomous learning, autonomous perception, and autonomous coordination, or the potential to achieve more natural, effective, and reliable in-formation sharing.⁸ Ao systematically reviewed the latest applications of deep learning across organizational departments, such as the demand for advanced preprocessing techniques in specific sensor environments and the deployment methods of deep learning in time series modeling.⁹ The AI deep learning system offers the potential to improve the digital transformation performance of manufacturing enterprises. However, the ability to produce the desired results also depends on the goal-oriented behavior these enterprises adopt.¹⁰

In addition, the AI deep learning enterprises have a natural ability to collaborate across organizations on the premise that the AI deep learning enterprises has the basic potential for cross-technological cooperation and possesses the characteristics of self-learning, reinforcement, and adaptability.¹¹ The common technical cooperation of AI deep learning systems is shown in the three aspects of storage, analysis, and recommendation, which not only reduces the cost between the cooperating subjects but also strengthens the stickiness of the cooperation between the subjects.¹² Sjödin et al., on the other hand, argued that the common technical cooperation of AI deep learning systems is reflected in the levels of autonomy and reinforcement. The former referred to the potential of AI deep learning systems to optimize strategic decisions and production processes, and the latter referred to the potential of AI deep learning system to reduce costs and increase efficiency by autonomously executing daily tasks.¹³

Research on cross-institutional research and development systems for common technologies

Common technologies are neither public goods in a purely economic concept nor exclusive at the commercial level. Because their R&D efforts face both technological and market uncertainties,^14,15 this situation led to a lack of sufficient R&D motivation for individual subjects, who often chose to take a wait-and-see attitude.¹⁶ This requires institutions with different strengths to break through institutional boundaries and engage in aggregation and integration decision-making R&D. Meanwhile, the following studies are mainly conducted for research of multi-subject common technologies.

Cross-organizational R&D of common technologies has a significant uplift effect on different industries. Capponi explored the core subjects in the common technology ecosystem in global mHealth and analyses their relationships with partners and projects.¹⁷ Lawniczuk et al. highlighted the key effectiveness of a common technology platform for photonic integrated circuits in mitigating supply failures and discussed in-depth the complexities of multi-party participation in the development of the platform, the construction of the institutional platform, and related complexity issues.¹⁸ Huang analyzed the slip effect of earthquakes using a new cross-organizational research method.¹⁹ Appio et al. revealed that diverse inputs of scientific and technological knowledge do not positively contribute to common technology R&D in all cases.²⁰

System administrators subsidize cross-organizational R&D on common technologies can effectively promote the effectiveness of R&D in the organization. Kokshagina’s empirical studied found that subsidies and science and technology policies effectively mitigate R&D failures.¹⁵ Kleer noted that government R&D subsidies can motivate enterprises and send signals to attract capital investment.²¹ Zhao et al. found that government subsidies do not always produce positive incentives for enterprise innovation.²² However, R&D subsidies, special programs, and other means of support can effectively compensate the enterprise for the common technology “prisoner’s dilemma” situation.^23,24

From the perspective of the realization path of cross-institutional cooperation, suggesting that the value interaction between enterprises and consumers is the fundamental way to reach value co-creation.^25,26 According to this interactive logic, enterprises no longer simply sell experiences to consumers but build service scenarios for consumers that enable personalization.²⁷ They also tap into customers’ latent needs by creating an AI deep learning system, which in turn leads to value co-creation based on ecological partnerships.²⁸ Based on the service-led logic of multi-subject cooperation, scholars believed that all exchange values are based on service.²⁹ In recent years, management studies have widely used this theory to explore the formation of multi-subject value creation.³⁰

Shortcomings and insights from existing research

Although existing research has pointed out that common technologies can improve the efficiency and output level of the whole systems through cross-organizational collaboration, such conclusions are mostly based on indirect evidence and lack direct verification and in-depth analysis of specific domains. However, in the emerging and complex field of AI deep learning systems, a crucial question is coming to the forefront: can the R&D activities of AI deep learning systems companies be effectively carried out in an aggregation and integration decision-making manner. Unfortunately, there is no research in the current literature that directly addresses this key question, which leaves an obvious research gap in the feasibility and operation mechanism of aggregation and integration decision-making R&D in this field.

Given this, this paper takes cross-institutional cooperation of common technologies of AI deep learning systems as an entry point, focuses on the heterogeneous impact on AI deep learning enterprises, further gives a reasonable picture of the future trend of dynamic diffusion of AI deep learning systems, and improves industrial co-operation capacity while alleviating the predicament of AI deep learning systems so that the benefits of different subjects and cross-institutional cooperation can achieve the effect of realizing progressive leapfrogging.

Problem description and model assumptions

Problem description

In the field of AI, deep learning systems can automatically extract high-level features of data and complete complex pattern recognition and predictive analysis by simulating the neural network mechanism of the human brain, thereby providing core support for intelligent decision-making and innovation to enhance the safety of intelligent driving.

Considering an AI deep learning systems system consisting of scientific research institutes, end-consumer enterprises, and AI deep learning enterprises over a continuous time $t \in [0, + \infty)$ . AI deep learning enterprises are oriented towards end-consumption enterprises and focus on developing new products and services for AI deep learning, realizing basic service unit functions in accordance with the actual business needs of end-consumption enterprises, and providing the developed new products and services to end-consumption enterprises. Scientific research institutions are closely integrated with AI deep learning enterprises to conduct R&D of AI deep learning enterprises-related technologies. They conduct scientific research based on the “difficulties” and “pain points” encountered by AI deep learning enterprises in their R&D practices. End-consumption enterprises face end-customers, mainly providing AI deep learning services to customers, enhancing their intuitive use experience and efficiency, and are responsible for the maintenance of AI deep learning units and the collection of customer feedback. Furthermore, this paper constructs a cross-organizational cooperative R&D mechanism diagram for the AI deep learning systems, as shown in Figure 1. As the leader in AI deep learning systems, the AI deep learning enterprises are the leading party in obtaining lasting core competitiveness through continuous knowledge sharing and innovation. In contrast, the scientific research institution and the end-consumer enterprise play the role of collaborators in the co-innovation system.

Figure 1.

A framework diagram for cross-institutional collaboration on artificial intelligence, deep learning, and enterprise general technologies.

Model assumptions

Based on the previous review and analysis of related issues, to provide theoretical presuppositions for the subsequent model construction and logical analysis framework, this paper intends to further propose the following research hypotheses:

Assumption 1: The R&D costs of scientific research institutes, terminal consumer enterprises and AI deep learning enterprises are intrinsically correlated with their effort levels. Moreover, the technology R&D costs show a convex function characteristic with respect to the effort costs, that is, as the degree of effort increases, the marginal increment of the technology R&D costs shows an upward trend. $γ_{i}$ represents the technology investment cost coefficient, $θ_{i}$ represents the technology cooperation cost coefficient, $χ_{i}$ represents the technology feedback cost coefficient, and $λ_{i}$ represents the technology application cost coefficient.³¹ R&D costs consider the following factors: technology investment, cooperation, feedback, and application costs. The cost of technology investment is $\frac{γ_{i}}{2} D_{i}^{2}$ , the cost of technology cooperation is $\frac{1}{θ_{i}} D_{i}^{2}$ , the cost of technology feedback is $\frac{1}{χ_{i}} D_{i}^{2}$ , and the cost of technology application is $\frac{λ_{i}}{2} D_{i}^{2}$ .

Assumption 2: $D_{P} (t)$ , $D_{K} (t)$ and $D_{Z} (t)$ represent the level of technological R&D efforts of momentary AI deep learning enterprises, scientific research institutions, and end-consumer enterprises, respectively, $E_{P} (t)$ , $E_{K} (t)$ , and $E_{Z} (t)$ represent the cost of technological R&D of momentary AI deep learning enterprises, scientific research institutions, and end-consumer enterprises, respectively. Referring to literatures,^32,33 and combining them with reality, we can conclude that the technology R&D costs of AI deep learning enterprises, scientific research institutes, and end-consumer enterprises now of $t$ are, respectively, as shown in equation (1).

E_{P} (t) = (\frac{γ_{P}}{2} + \frac{1}{θ_{P}} + \frac{1}{χ_{P}}) {D_{P}}^{2}, E_{K} (t) = (\frac{1}{θ_{K}} + \frac{1}{χ_{K}} + \frac{λ_{K}}{2}) {D_{K}}^{2}, E_{Z} (t) = (\frac{1}{χ_{Z}} + \frac{λ_{Z}}{2}) {D_{Z}}^{2}

(1)

Assumption 3: The product development process of AI deep learning enterprises has dynamic evolution characteristics. The effective advancement of this process depends on the continuous aggregation and integration decision-making interaction and innovative practices of all participants. Ultimately, the optimal state of product configuration is achieved through knowledge complementarity and resource integration.³⁴ In this paper, the level of R&D of AI deep learning products is used as a dynamic variable that each subject participant influences. If the level of technological R&D now $t$ is $M (t)$ , the R&D process of the AI deep learning enterprises satisfies equation (2).

M^{'} (t) = α_{P} D_{P} (t) + α_{K} D_{K} (t) + α_{Z} D_{Z} (t) - δ M (t)

(2)

In equation (2), the product’s initial innovation level is

M (0) = M_{0} \geq 0

, and

α_{i}

is the service technology innovation coefficient, which represents the degree of influence on the service technology innovation level.

δ

is the natural decay coefficient, representing the natural decay rate of technological innovations in AI deep learning product development,³⁵ and

δ > 0

Assumption 4: The total gain $π (t)$ for this AI deep learning enterprises at time $t$ , as shown in equation (3).

π (t) = β_{P} D_{P} (t) + β_{K} D_{K} (t) + β_{Z} D_{Z} (t) + η M (t)

(3)

In equation (3),

β_{i}

represents the marginal revenue coefficient

and η

is used to describe the contribution of labor input to the growth of the total system income.

η > 0

characterizes the intensity of the effect of technological innovation activities on the total revenue. The three together constitute the core parameters that describe the contribution degree of different elements in the model.

Assumption 5: The total proceeds are aggregation and integration decision-making all agreed to be distributed between the scientific research institution, the end-consumer enterprise, and the AI deep learning enterprises, with the three parties having revenue-sharing coefficients of $μ$ , $ν$ , and $1 - μ - ν (0 < μ, 0 < ν, 1 - μ - ν < 1)$ , respectively. AI deep learning enterprises share R&D costs for scientific research institutions and end-consumer enterprises in the ratio of $φ$ and $ω (0 \leq φ, 0 \leq ω)$ . This ratio characterizes the incentive coefficient of AI deep learning enterprises for scientific research institutions and terminal consumer enterprises. The research assumes that all three types of entities adopt the same positive discount rate $ρ$ and all aim to maximize their own returns within the infinite time domain $t \in [0, + \infty)$ .At this point, the objective functions of each of the three types of subjects can be constructed.

The objective function of the end-consumer enterprise, as shown in equation (4).

\max_{D_{P} (t)} J_{P} = {\int_{0}^{\infty} e}^{- ρ t} [μ π (t) - (φ (t) - 1) (\frac{γ_{P}}{2} + \frac{1}{θ_{P}} + \frac{1}{χ_{P}}) {D_{P}}^{2} (t)] d t

(4)

The objective function of the scientific research institution, as shown in equation (5).

\max_{D_{K} (t)} J_{K} = \int_{0}^{\infty} e^{- ρ t} [ν π (t) - (ω (t) - 1) (\frac{1}{θ_{K}} + \frac{1}{χ_{K}} + \frac{λ_{K}}{2}) {D_{K}}^{2} (t)] d t

(5)

The objective function of the AI deep learning enterprises, as shown in equation (6).

\max_{D_{Z} (t)} J_{Z} = {\int_{0}^{\infty} e}^{- ρ t} [(1 - μ - ν) π (t) - (\frac{1}{χ_{Z}} + \frac{λ_{Z}}{2}) {D_{Z}}^{2} (t) - φ (t) (\frac{γ_{P}}{2} + \frac{1}{θ_{P}} + \frac{1}{χ_{P}}) {D_{P}}^{2} (t) - ω (t) (\frac{1}{θ_{K}} + \frac{1}{χ_{K}} + \frac{λ_{K}}{2}) {D_{K}}^{2} (t)] d t

(6)

Assumption 6: The AI deep learning systems contains the control variables $D_{P} (t)$ , $D_{K} (t)$ , $D_{Z} (t)$ , $φ$ ,and $ω$ as well as the dynamic state variable $M (t)$ . Given the technical constraints in model solving in dynamic situations, this paper sets all parameters as normal numbers that do not change over time. Based on the need for analytical simplification, this model performs steady-state processing on the dynamic process. At any point in the entire infinite time $t \in [0, + \infty)$ domain framework, it can represent any observation time point of this AI deep learning system during its long-term evolution process, the end-consumer enterprise, the scientific research institution, and the AI deep learning enterprises face the same game, restricting the strategy to a static one, from which a static feedback equilibrium solution can be further obtained.³⁶

Model construction and analysis

Based on previous research, this chapter will further analyze whether cost sharing can make optimal decisions for end-consumer enterprises, scientific research institutes, and AI deep learning enterprises and thus achieve the optimal level of cooperation. Suppose the optimal state of aggregation and integration decision-making cooperation cannot be achieved. In that case, the joint cost-sharing can analyze to see whether it can promote the optimal decision-making of end-consumption enterprises, scientific research institutes, and AI deep learning enterprises to reach Pareto improvement and its degree of improvement, to provide a decision-making basis for the aggregation and integration decision-making R&D of common technologies for AI deep learning systems through the cooperation of the three parties.

Distributed autonomous decision-making model

Under the framework of the distributed autonomous decision-making model, the three types of decision-making subjects have the characteristics of independence and equality. They all take maximizing their own benefits as the decision-making orientation and will simultaneously implement the selection of self-interest optimal strategies. Under this mode, the optimal strategy combinations of each entity have reached a Nash equilibrium state, which characterizes the stability of strategies in the multi-party decision-making interaction process. In this case, the payoff functions for end-consumer enterprises, scientific research institutions, and AI deep learning enterprises are $V_{P} (M)$ , $V_{K} (M)$ , and $V_{Z} (M)$ , respectively, continuously bounded and microscopic. At this time, the AI deep learning enterprises does not provide any subsidy incentives to scientific research institutions and end-consumption enterprises, that is, $φ = 0$ and $ω = 0$ . When the objective function equations of the end-consumption enterprises, the scientific research institutions, and the AI deep learning enterprises are, respectively, as shown in equations (7)–(9).

\begin{array}{l} ρ V_{P} (M) = \max_{D_{P} \geq 0} {μ π (t) - (\frac{γ_{P}}{2} + \frac{1}{θ_{P}} + \frac{1}{χ_{P}}) D_{P}^{2} + \\ V_{P}^{'} (M) (α_{P} D_{P} (t) + α_{K} D_{K} (t) + α_{Z} D_{Z} (t) - δ M (t))} \end{array}

(7)

\begin{array}{l} ρ V_{K} (M) = \max_{D_{K} \geq 0} {ν π (t) - (\frac{1}{θ_{K}} + \frac{1}{χ_{K}} + \frac{λ_{K}}{2}) D_{K}^{2} + \\ V_{K}^{'} (M) (α_{P} D_{P} (t) + α_{K} D_{K} (t) + α_{Z} D_{Z} (t) - δ M (t))} \end{array}

(8)

\begin{array}{l} ρ V_{Z} (M) = \max_{D_{Z} \geq 0} {(1 - μ - ν) π (t) - (\frac{1}{χ_{Z}} + \frac{λ_{Z}}{2}) D_{Z}^{2} + \\ V_{Z}^{'} (M) (α_{P} D_{P} (t) + α_{K} D_{K} (t) + α_{Z} D_{Z} (t) - δ M (t))} \end{array}

(9)

Apply the first-order partial derivative operation with respect to the variables $D_{P}$ , $D_{K}$ , and $D_{Z}$ to the right end of the system of equations (7)–(9) and construct the equilibrium condition by setting the derivative function to zero. The results obtained are shown in equations (10)–(12).

D_{P} = \frac{(V_{P}^{'} α_{P} + μ β_{P}) θ_{P} χ_{P}}{2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})}

(10)

D_{K} = \frac{(V_{K}^{'} α_{K} + ν β_{K}) θ_{K} χ_{F}}{2 χ_{F} + θ_{K} (2 + λ_{K} χ_{F})}

(11)

D_{Z} = \frac{[V_{Z}^{'} α_{Z} + (1 - μ - ν) β_{Z}] χ_{Z}}{2 + λ_{Z} χ_{Z}}

(12)

This can be obtained by bringing equations (10)–(12) into the objective function equations of the end-consumer enterprises, scientific research institutes, and AI deep learning enterprises and ordering $V_{P} (M) = j_{1} M + j_{2}$ , $V_{K} (M) = k_{1} M + k_{2}$ , and $V_{Z} (M) = l_{1} M + l_{2}$ , which we can further solve, as shown in equations (13)–(15).

\begin{array}{l} ρ (j_{1} M + j_{2}) = [μ η - δ j_{1}] M + \frac{{(j_{1} α_{P} + μ β_{P})}^{2} θ_{P} χ_{P}}{2 [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})]} + \\ \frac{[μ β_{K} + j_{1} α_{K}] (k_{1} α_{K} + ν β_{K}) θ_{K} χ_{F}}{2 χ_{K} + θ_{K} (2 + λ_{K} χ_{F})} + \frac{[l_{1} α_{Z} + (1 - μ - ν) β_{Z}] χ_{Z} [μ β_{Z} + j_{1} α_{Z}]}{2 + λ_{Z} χ_{Z}} \end{array}

(13)

\begin{array}{l} ρ (k_{1} M + k_{2}) = [ν η - δ k_{1}] M + \frac{[ν β_{P} + k_{1} α_{P}] (j_{1} α_{P} + μ β_{P}) θ_{P} χ_{P}}{2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})} + \\ \frac{{(k_{1} α_{K} + ν β_{K})}^{2} θ_{K} χ_{F}}{2 [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{F})]} + \frac{[l_{1} α_{Z} + (1 - μ - ν) β_{Z}] χ_{Z} [ν β_{Z} + k_{1} α_{Z}]}{2 + λ_{Z} χ_{Z}} \end{array}

(14)

\begin{array}{l} ρ (l_{1} M + l_{2}) = [(1 - μ - ν) η - δ l_{1}] M + \frac{[(1 - μ - ν) β_{P} + l_{1} α_{P}] (j_{1} α_{P} + μ β_{P}) θ_{P} χ_{P}}{2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})} + \\ \frac{[(1 - μ - ν) β_{K} + l_{1} α_{K}] (k_{1} α_{K} + ν β_{K}) θ_{K} χ_{F}}{2 χ_{K} + θ_{K} (2 + λ_{K} χ_{F})} + \frac{{[l_{1} α_{Z} + (1 - μ - ν) β_{Z}]}^{2} χ_{Z}}{2 [2 + λ_{Z} χ_{Z}]} \end{array}

(15)

Theorem 1: The dynamic feedback Nash equilibrium strategies of the three parties in the case of a decentralized cooperation model between end-consumer enterprises, scientific research institutes, and AI deep learning enterprises are, respectively, as shown in equations (16)–(18).

{D_{P}}^{*} = \frac{μ [η α_{P} + β_{P} (ρ + δ)] θ_{P} χ_{P}}{[2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] (ρ + δ)}

(16)

{D_{K}}^{*} = \frac{ν [η α_{K} + β_{K} (ρ + δ)] θ_{K} χ_{K}}{[2 χ_{K} + θ_{K} (2 + λ_{K} χ_{F})] (ρ + δ)}

(17)

{D_{Z}}^{*} = \frac{(1 - μ - ν) [η α_{Z} + β_{Z} (ρ + δ)] χ_{Z}}{(2 + λ_{Z} χ_{Z}) (ρ + δ)}

(18)

According to the optimal effort strategy, the optimal revenue functions for end-consumer enterprises, scientific research institutions, and AI deep learning enterprises we can obtain as shown in equations (19)–(21), respectively.

\begin{array}{l} V_{P 1} {(M)}^{*} = \frac{μ η}{ρ + δ} M + \frac{μ^{2} θ_{P} χ_{P} {[η α_{P} + β_{P} (ρ + δ)]}^{2}}{2 ρ {(ρ + δ)}^{2} [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})]} + \\ \frac{μ ν {[β_{K} (ρ + δ) + η α_{K}]}^{2} θ_{K} χ_{K}}{ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{μ (1 - μ - ν) {[η α_{Z} + β_{Z} (ρ + δ)]}^{2} χ_{Z}}{ρ (2 + λ_{Z} χ_{Z}) {(ρ + δ)}^{2}} \end{array}

(19)

\begin{array}{l} V_{K 1} {(M)}^{*} = \frac{ν η}{ρ + δ} M + \frac{μ ν {[β_{P} (ρ + δ) + η α_{P}]}^{2} θ_{P} χ_{P}}{ρ {(ρ + δ)}^{2} [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})]} + \\ \frac{ν^{2} {[η α_{K} + β_{K} (ρ + δ)]}^{2} θ_{K} χ_{K}}{2 ρ {(ρ + δ)}^{2} [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})]} + \frac{ν (1 - μ - ν) {[η α_{Z} + β_{Z} (ρ + δ)]}^{2} χ_{Z}}{ρ [2 + λ_{Z} χ_{Z}] {(ρ + δ)}^{2}} \end{array}

(20)

\begin{array}{l} V_{Z 1} {(M)}^{*} = \frac{(1 - μ - ν) η}{ρ + δ} M + \frac{μ (1 - μ - ν) {[β_{P} (ρ + δ) + η α_{P}]}^{2} θ_{P} χ_{P}}{ρ [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] {(ρ + δ)}^{2}} + \\ \frac{ν (1 - μ - ν) {[β_{K} (ρ + δ) + η α_{K}]}^{2} θ_{K} χ_{K}}{ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{{(1 - μ - ν)}^{2} {[η α_{Z} + β_{Z} (ρ + δ)]}^{2} χ_{Z}}{2 ρ [2 + λ_{Z} χ_{Z}] {(ρ + δ)}^{2}} \end{array}

(21)

Furthermore, we can draw the conclusion that the total benefit of the entire AI deep learning system under the distributed autonomous decision-making model is equation (22).

\begin{array}{l} V_{1} {(M)}^{*} = \frac{η}{ρ + δ} M + \frac{μ (2 - μ) θ_{P} χ_{P} {[η α_{P} + β_{P} (ρ + δ)]}^{2}}{2 ρ {(ρ + δ)}^{2} [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})]} + \\ \frac{ν (2 - ν) {[η α_{K} + β_{K} (ρ + δ)]}^{2} θ_{K} χ_{K}}{2 ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{[1 - {(μ + ν)}^{2}] {[η α_{Z} + β_{Z} (ρ + δ)]}^{2} χ_{Z}}{2 ρ (2 + λ_{Z} χ_{Z}) {(ρ + δ)}^{2}} \end{array}

(22)

Inference 1: In the distributed autonomous decision-making model, there is $χ_{i}$ significant positive correlation between the global benefit optimal value of the AI deep learning system and the technical feedback cost, the coefficient of impact on the system $β_{i}$ , and the coefficient of service innovation $α_{i}$ and negatively correlated with the coefficient of the cost of technology application $λ_{i}$ . This describes the dynamic optimal payoff strategies of end-consumer enterprises, scientific research institutions, and AI deep learning enterprises in a decentralized cooperative game. This reflects that in the cross-organizational aggregation and integration decision-making R&D process of AI deep learning systems, the strategic choices of all participants are based on maximizing their own interests.

Joint cost-sharing model

In the joint cost-sharing model, AI deep learning enterprises are the cross-organizational leaders of the common technologies of this system, so to encourage the development of the AI deep learning technology enterprises, the AI deep learning enterprises will cost incentives for scientific research institutions and end-consumption enterprises, respectively, as $φ$ , $ω (0 \leq φ, ω \leq 0)$ . At this time, the end-consumer enterprises play a driving role in technology cooperation and R&D. The end-consumer enterprises provide suggestions for the AI deep learning enterprises R&D, and the AI deep learning enterprises give the end-consumer enterprises a certain percentage of preferences. From a long-term and dynamic perspective, terminal consumer enterprises, scientific research institutions and AI deep learning enterprises form a differentiated decision-making model involving AI deep learning enterprises and terminal consumer enterprises. In this scenario, the objective functions for end-consumer enterprises, scientific research institutions, and AI deep learning enterprises are as follows, respectively, as shown in equations (23)–(25).

\begin{array}{l} ρ V_{P} (M) = \max_{D_{P} \geq 0} {μ π (t) - (1 - φ) (\frac{γ_{P}}{2} + \frac{1}{θ_{P}} + \frac{1}{χ_{P}}) D_{P}^{2} + \\ V_{P}^{'} (M) (α_{P} D_{P} (t) + α_{K} D_{K} (t) + α_{Z} D_{Z} (t) - δ M (t))} \end{array}

(23)

\begin{array}{l} ρ V_{K} (M) = \max_{D_{K} \geq 0} {ν π (t) - (1 - ω) (\frac{1}{θ_{K}} + \frac{1}{χ_{K}} + \frac{λ_{K}}{2}) D_{K}^{2} + \\ V_{K}^{'} (M) (α_{P} D_{P} (t) + α_{K} D_{K} (t) + α_{Z} D_{Z} (t) - δ M (t))} \end{array}

(24)

\begin{array}{l} ρ V_{Z} (M) = \max_{D_{Z} \geq 0} {(1 - μ - ν) π (t) - (\frac{1}{χ_{Z}} + \frac{λ_{Z}}{2}) D_{Z}^{2} - φ (\frac{γ_{P}}{2} + \frac{1}{θ_{P}} + \frac{1}{χ_{P}}) D_{P}^{2} - \\ ω (\frac{1}{θ_{K}} + \frac{1}{χ_{K}} + \frac{λ_{K}}{2}) D_{K}^{2} + V_{Z}^{'} (M) (α_{P} D_{P} (t) + α_{K} D_{K} (t) + α_{Z} D_{Z} (t) - δ M (t))} \end{array}

(25)

Apply the first-order partial derivative operation with respect to the variables $D_{P}$ , $D_{K}$ , and $D_{Z}$ to the right end of the system of equations (23)–(25) and construct the equilibrium condition by setting the derivative function to zero. The results obtained are shown in equations (26)–(28).

D_{P} = \frac{(V_{P}^{'} α_{P} + μ β_{P}) θ_{P} χ_{P}}{(1 - φ) [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})]}

(26)

D_{K} = \frac{(V_{K}^{'} α_{K} + ν β_{K}) θ_{K} χ_{K}}{(1 - ω) [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})]}

(27)

D_{Z} = \frac{[V_{Z}^{'} α_{Z} + (1 - μ - ν) β_{Z}] χ_{Z}}{2 + λ_{Z} χ_{Z}}

(28)

The solution can obtain substitute equations (26)–(28) into equation (25), solving for the right end portion to maximize the conditional equation (25), and solving for the first-order partial derivatives of $φ$ and $ω$ , respectively, and making them equal to zero. Thus, equations (29) and (30) can be obtained.

φ = \frac{[(2 V_{Z}^{'} - V_{P}^{'}) α_{P}] + (2 - 3 μ - 2 ν) β_{P}}{(V_{P}^{'} + 2 V_{Z}^{'}) α_{P} + (2 - μ - 2 ν) β_{P}}

(29)

ω = \frac{[(2 V_{Z}^{'} - V_{K}^{'}) α_{K}] + (2 - 2 μ - 3 ν) β_{K}}{(V_{K}^{'} + 2 V_{Z}^{'}) α_{K} + (2 - 2 μ - ν) β_{K}}

(30)

This can obtain bring equations (26)–(30) into the objective function equations of the end-consumer enterprises, scientific research institutes, and AI deep learning enterprises and ordering $V_{P} (M) = j_{1} M + j_{2}$ , $V_{K} (M) = k_{1} M + k_{2}$ , and $V_{Z} (M) = l_{1} M + l_{2}$ , which in turn can further solve, as shown in equations (31)–(33).

\begin{array}{l} ρ (j_{1} M + j_{2}) = [μ η - δ j_{1}] M + \frac{θ_{P} χ_{P} [μ β_{P} + j_{1} α_{P}] θ_{P} χ_{P} [(j_{1} + 2 l_{1}) α_{P} + (2 - μ - 2 ν) β_{P}]}{4 [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})} + \\ \frac{θ_{K} χ_{K} [μ β_{K} + j_{1} α_{K}] [(k_{1} + 2 l_{1}) α_{K} + (2 - 2 μ - ν) β_{K}]}{2 [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})]} + \frac{[μ β_{Z} + j_{1} α_{Z}] [l_{1} α_{Z} + (1 - μ - ν) β_{Z}] χ_{Z}}{2 + λ_{Z} χ_{Z}} \end{array}

(31)

\begin{array}{l} ρ (k_{1} M + k_{2}) = [ν η - δ k_{1}] M + \frac{[ν β_{P} + k_{1} α_{P}] θ_{P} χ_{P} [(j_{1} + 2 l_{1}) α_{P} + (2 - μ - 2 ν) β_{P}]}{2 [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})]} + \\ \frac{[(k_{1} α_{K} + ν β_{K})] θ_{K} χ_{K} [(k_{1} + 2 l_{1}) α_{K} + (2 - 2 μ - ν) β_{K}]}{8 [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})]} + \frac{[ν β_{Z} + k_{1} α_{Z}] [l_{1} α_{Z} + (1 - μ - ν) β_{Z}] χ_{Z}}{2 + λ_{Z} χ_{Z}} \end{array}

(32)

\begin{array}{l} ρ (l_{1} M + l_{2}) = [(1 - μ - ν) η - δ l_{1}] M + \frac{{[θ_{P} χ_{P} (j_{1} + 2 l_{1}) α_{P} + (2 - μ - 2 ν) β_{P}]}^{2}}{8 [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})]} + \\ \frac{θ_{K} χ_{K} {[(k_{1} + 2 l_{1}) α_{K} + (2 - 2 μ - ν) β_{K}]}^{2}}{8 [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})]} + \frac{{[l_{1} α_{Z} + (1 - μ - ν) β_{Z}]}^{2} χ_{Z}}{2 [2 + λ_{Z} χ_{Z}]} \end{array}

(33)

Theorem 2: The dynamic feedback Nash equilibrium strategies and optimal subsidy incentive coefficients of the three parties in the case of the joint cost-sharing model among end-consumption enterprises, scientific research institutions, and AI deep learning enterprises, as shown in equations (34)–(38).

{D_{P}}^{*} = \frac{(2 - μ - 2 ν) θ_{P} χ_{P} [η α_{P} + β_{P} (ρ + δ)]}{2 [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] (ρ + δ)}

(34)

{D_{K}}^{*} = \frac{(2 - 2 μ - ν) θ_{K} χ_{K} [η α_{K} + β_{K} (ρ + δ)]}{2 [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] (ρ + δ)}

(35)

{D_{Z}}^{*} = \frac{χ_{Z} [(1 - μ - ν)] [η α_{Z} + β_{Z} (ρ + δ)]}{(2 + λ_{Z} χ_{Z}) (ρ + δ)}

(36)

\begin{array}{l} φ^{*} = \frac{2 - 3 μ - 2 ν}{2 - μ - 2 ν}, 3 μ + 2 ν < 2 \\ 0, 3 μ + 2 ν \geq 2 \end{array}

(37)

\begin{array}{l} ω^{*} = \frac{2 - 2 μ - 3 ν}{2 - 2 μ - ν}, 2 μ + 3 ν < 2 \\ 0, 2 μ + 3 ν \geq 2 \end{array}

(38)

According to the optimal effort strategy, the following equations (39)–(41) can be obtained.

\begin{array}{l} V_{P 2} {(M)}^{*} = \frac{μ η}{ρ + δ} M + \frac{θ_{P} χ_{P} μ (2 - μ - 2 ν) {[β_{P} (ρ + δ) + η α_{P}]}^{2}}{4 ρ [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] {(ρ + δ)}^{2}} + \\ \frac{μ θ_{K} χ_{K} (2 - 2 μ - ν) {[β_{K} (ρ + δ) + η α_{K}]}^{2}}{2 ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{μ χ_{Z} (1 - μ - ν) {[β_{Z} (ρ + δ) + η α_{Z}]}^{2}}{ρ [2 + λ_{Z} χ_{Z}] {(ρ + δ)}^{2}} \end{array}

(39)

\begin{array}{l} V_{K 2} {(M)}^{*} = \frac{ν η}{ρ + δ} M + \frac{ν θ_{P} χ_{P} (2 - μ - 2 ν) {[β_{P} (ρ + δ) + η α_{P}]}^{2}}{2 ρ [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] {(ρ + δ)}^{2}} + \\ \frac{ν θ_{K} χ_{K} (2 - 2 μ - ν) {[η α_{K} + β_{K} (ρ + δ)]}^{2}}{4 ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{ν χ_{Z} (1 - μ - ν) {[β_{Z} (ρ + δ) + η α_{Z}]}^{2}}{ρ [2 + λ_{Z} χ_{Z}] {(ρ + δ)}^{2}} \end{array}

(40)

\begin{array}{l} V_{Z 2} {(M)}^{*} = \frac{(1 - μ - ν) η}{ρ + δ} M + \frac{θ_{P} χ_{P} {(2 - μ - 2 ν)}^{2} {[η α_{P} + β_{P} (ρ + δ)]}^{2}}{8 ρ [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] {(ρ + δ)}^{2}} + \\ \frac{θ_{K} χ_{K} {(2 - 2 μ - ν)}^{2} {[η α_{K} + β_{K} (ρ + δ)]}^{2}}{8 ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{χ_{Z} {(1 - μ - ν)}^{2} {[η α_{Z} + β_{Z} (ρ + δ)]}^{2}}{2 ρ [2 + λ_{Z} χ_{Z}] {(ρ + δ)}^{2}} \end{array}

(41)

Based on the above content, we can further solve the total revenue of the entire AI deep learning system under joint cost-sharing conditions, as shown in equation (42).

\begin{array}{l} V_{2} {(M)}^{*} = \frac{η}{ρ + δ} M + \frac{θ_{P} χ_{P} [4 - {(μ + 2 ν)}^{2}] {[β_{P} (ρ + δ) + η α_{P}]}^{2}}{8 ρ [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] {(ρ + δ)}^{2}} + \\ \frac{θ_{K} χ_{K} [4 - {(2 μ + ν)}^{2}] {[β_{K} (ρ + δ) + η α_{K}]}^{2}}{4 ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{χ_{Z} [1 - {(μ + ν)}^{2}] {[β_{Z} (ρ + δ) + η α_{Z}]}^{2}}{2 ρ [2 + λ_{Z} χ_{Z}] {(ρ + δ)}^{2}} \end{array}

(42)

Inference 2: Under the model of joint cost-sharing model and aggregation and integration decision-making, there is $θ_{i}$ significant positive correlation between the optimal total benefit level of the AI deep learning system and the technical cooperation cost coefficient, as well as the influence coefficient b of the effort level on the system $β_{i}$ and negatively correlated with the coefficient of the cost of application of technology $λ_{i}$ . This explores a three-way dynamic feedback equilibrium strategy where AI deep learning enterprises subsidize scientific research institutes for solving AI deep learning attacking technologies, and concessions are given to end-consumer enterprises. This can obviously be seen as subsidy support from AI deep learning enterprises and end-consumer enterprise strategies interact.

Aggregation and integration decision-making model

Under the aggregated and integrated decision-making model, AI deep learning enterprises, research institutions, and terminal consumer enterprises exhibit aggregation and integration decision-making and interactive characteristics. These three types of entities take maximizing the overall benefits of the AI deep learning system as the core goal and jointly establish the optimal benefit function through strategy coordination and goal coupling. The cost incentives of the AI deep learning enterprises for both are internal funds transfers, and the cost incentive coefficients $φ$ and $ω$ are arbitrary values of 0–1. At this time, AI deep learning enterprises are improving the competitiveness of their products, scientific research institutions are promoting the transformation of project results, and end-consumer enterprises are enhancing the consumption experience of end users. This section will explore the coordination and cooperation among the three parties in the context of the AI deep learning system composed of AI deep learning enterprises, scientific research institutions and terminal consumer enterprises jointly discussing how to conduct cross-organizational aggregation and integration decision-making R&D. In this context, the objective function of the AI deep learning system is shown in the following equation (43).

\begin{array}{l} \max_{D_{P} (t), D_{K} (t), D_{Z} (t)} J = {\int_{0}^{\infty} e}^{- ρ t} [π (t) - (\frac{γ_{P}}{2} + \frac{1}{θ_{P}} + \frac{1}{χ_{P}}) D_{P}^{2} (t) - \\ (\frac{1}{θ_{K}} + \frac{1}{χ_{K}} + \frac{λ_{K}}{2}) D_{K}^{2} (t) - (\frac{1}{χ_{Z}} + \frac{λ_{Z}}{2}) D_{Z}^{2} (t)] d t \end{array}

(43)

There exists an optimal function $V (M)$ in an AI deep learning system, and the optimal function is continuously bound and differentiable. For $M \geq 0$ the following HJB equation is satisfied, as shown in equation (44).

\begin{array}{l} ρ V (M) = \max_{D_{P} (t) π \geq 0, D_{K} (t) \geq 0, D_{Z} (t) \geq 0} π (t) - (\frac{γ_{P}}{2} + \frac{1}{θ_{P}} + \frac{1}{χ_{P}}) D_{P}^{2} (t) - \\ (\frac{1}{θ_{K}} + \frac{1}{χ_{K}} + \frac{λ_{K}}{2}) D_{K}^{2} (t) - (\frac{1}{χ_{Z}} + \frac{λ_{Z}}{2}) D_{Z}^{2} (t) + \\ V_{P}^{'} (M) [α_{P} D_{P} (t) + α_{K} D_{K} (t) + α_{Z} D_{Z} (t) - δ M (t)] \end{array}

(44)

The solution is solve the right end part of the HJB equation by taking the first-order partial derivatives of pairs $D_{P}$ , $D_{K}$ , and $D_{Z}$ , respectively, and letting them be zero, as shown in equations (45)–(47).

D_{P} = \frac{(V^{'} α_{P} + β_{P}) θ_{P} χ_{P}}{2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})}

(45)

D_{K} = \frac{(V^{'} α_{K} + β_{K}) θ_{K} χ_{K}}{2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})}

(46)

D_{Z} = \frac{(V^{'} α_{Z} + β_{Z}) χ_{Z}}{2 + λ_{Z} χ_{Z}}

(47)

Let $V_{P} (M) = j_{1} M + j_{2}$ , $V_{K} (M) = k_{1} M + k_{2}$ ,and $V_{Z} (M) = l_{1} M + l_{2}$ , and substitute equations (45)–(47) into the objective function equation, and solve to obtain equation (48).

\begin{array}{l} ρ (j_{1} M + j_{2}) = [η - δ j_{1}] M (t) + \frac{θ_{P} χ_{P} {[β_{P} + j_{1} α_{P}]}^{2}}{2 [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})]} + \\ \frac{θ_{K} χ_{K} {[β_{K} + j_{1} α_{K}]}^{2}}{2 [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})]} + \frac{χ_{Z} {[β_{Z} + j_{1} α_{Z}]}^{2}}{2 (2 + λ_{Z} χ_{Z})} \end{array}

(48)

Theorem 3: In the case of aggregation and integration decision-making cooperative games, the dynamic feedback equilibrium strategies of the three are, respectively, as shown in equations (49)–(51).

{D_{P 3}}^{*} = \frac{θ_{P} χ_{P} [η α_{P} + β_{P} (ρ + δ)]}{[2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] (ρ + δ)}

(49)

{D_{K 3}}^{*} = \frac{θ_{K} χ_{K} [η α_{K} + β_{K} (ρ + δ)]}{[2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] (ρ + δ)}

(50)

{D_{Z 3}}^{*} = \frac{χ_{Z} [η α_{Z} + β_{Z} (ρ + δ)]}{(2 + λ_{Z} χ_{Z}) (ρ + δ)}

(51)

Based on the analysis of the optimal effort strategy, the global optimal utility function of the AI deep learning system can be mathematically represented, and its formal definition is shown in equation (52).

\begin{array}{l} V_{3} {(M)}^{*} = \frac{(1 - μ - ν) η}{ρ + δ} M + \frac{(1 - μ - ν) θ_{P} χ_{P} {[β_{P} (ρ + δ) + η α_{P}]}^{2}}{2 ρ [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] {(ρ + δ)}^{2}} + \\ \frac{(1 - μ - ν) θ_{K} χ_{K} {[β_{K} (ρ + δ) + η α_{K}]}^{2}}{2 ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{(1 - μ - ν) χ_{Z} {[β_{Z} (ρ + δ) + η α_{Z}]}^{2}}{2 ρ (2 + λ_{Z} χ_{Z}) {(ρ + δ)}^{2}} \end{array}

(52)

The optimal revenue functions for end-consumer enterprises, scientific research institutes, and AI deep learning enterprises are, respectively, as shown in equations (53)–(55).

\begin{array}{l} V_{P 3} {(M)}^{*} = \frac{η}{ρ + δ} M + \frac{θ_{P} χ_{P} {[β_{P} (ρ + δ) + η α_{P}]}^{2}}{2 ρ [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] {(ρ + δ)}^{2}} + \\ \frac{θ_{K} χ_{K} {[β_{K} (ρ + δ) + η α_{K}]}^{2}}{2 ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{χ_{Z} {[β_{Z} (ρ + δ) + η α_{Z}]}^{2}}{2 ρ (2 + λ_{Z} χ_{Z}) {(ρ + δ)}^{2}} \end{array}

(53)

\begin{array}{l} V_{K 3} {(M)}^{*} = \frac{μ η}{ρ + δ} M + \frac{μ θ_{P} χ_{P} {[β_{P} (ρ + δ) + η α_{P}]}^{2}}{2 ρ [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] {(ρ + δ)}^{2}} + \\ \frac{μ θ_{K} χ_{K} {[β_{K} (ρ + δ) + η α_{K}]}^{2}}{2 ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{μ χ_{Z} {[β_{Z} (ρ + δ) + η α_{Z}]}^{2}}{2 ρ (2 + λ_{Z} χ_{Z}) {(ρ + δ)}^{2}} \end{array}

(54)

\begin{array}{l} V_{Z 3} {(M)}^{*} = \frac{(1 - μ - ν) η}{ρ + δ} M + \frac{(1 - μ - ν) θ_{P} χ_{P} {[β_{P} (ρ + δ) + η α_{P}]}^{2}}{2 ρ [2 χ_{P} + θ_{P} (2 + γ_{P} χ_{P})] {(ρ + δ)}^{2}} + \\ \frac{(1 - μ - ν) θ_{K} χ_{K} {[β_{K} (ρ + δ) + η α_{K}]}^{2}}{2 ρ [2 χ_{K} + θ_{K} (2 + λ_{K} χ_{K})] {(ρ + δ)}^{2}} + \frac{(1 - μ - ν) χ_{Z} {[β_{Z} (ρ + δ) + η α_{Z}]}^{2}}{2 ρ (2 + λ_{Z} χ_{Z}) {(ρ + δ)}^{2}} \end{array}

(55)

Inference 3: In the aggregation and integration decision-making model, the global Pareto optimal solution of the deep learning system has a monotonically increasing relationship with the policy input intensity factor of subsystem $β_{i}$ ,the coefficient of the effect of technological innovations on the total benefits $η$ , the coefficient of the distribution of benefits among the three parties $1 - μ - ν$ , and negatively correlated with the coefficient of the cost of application of technology $λ_{i}$ . This explores the dynamic feedback equilibrium strategies of end-consumer enterprises, scientific research institutions and AI deep learning enterprises in this scenario. This inference suggests that in the framework of cooperative gaming, the three parties work together to promote cooperative cross-institutional R&D in common technologies through cooperation.

Comparative analysis of models

Comparing the optimal payoff function, optimal effort level, and optimal payoff function of the AI deep learning system for end-consumer enterprises, scientific research institutes, and the game behavior of AI deep learning enterprises in the three scenarios, this study deduces the following core propositions:

Proposition 1: The optimal returns for all three types of subjects are greater in the joint cost-sharing model than in the no-incentive model. That is $V_{P 2} {(M)}^{*} > V_{P 1} {(M)}^{*}$ , $V_{K 2} {(M)}^{*} > V_{K 1} {(M)}^{*}$ , and $V_{Z 2} {(M)}^{*} > V_{Z 1} {(M)}^{*}$ .

Proposition 2: The optimal R&D benefit level of AI deep learning systems reaches its maximum value in the aggregation and integration decision-making mode, followed by the joint cost-sharing mode, and is at its minimum value in the distributed autonomous decision-making mode. That is $V_{1} {(M)}^{*} < V_{2} {(M)}^{*} < V_{3} {(M)}^{*}$ .

Proposition 3: Compared with the distributed autonomous decision-making model, under the framework of the cost-sharing joint decision-making model, the efforts of AI deep learning enterprises remain stable. The extent to which the efforts of scientific research institutions and terminal consumer enterprises have improved is exactly equivalent to the subsidy ratio of AI deep learning enterprises for their costs, i.e. the optimal incentive coefficient. The aggregation and integration decision-making model have the highest level of effort for all three types of subjects. That is ${D_{P 1}}^{*} < {D_{P 2}}^{*} < {D_{P 3}}^{*}$ , ${D_{K 1}}^{*} = {D_{K 2}}^{*} < {D_{K 3}}^{*}$ , and ${D_{Z 1}}^{*} = {D_{Z 2}}^{*} < {D_{Z 3}}^{*}$ .

Proposition 4: When the initial level of innovation $M_{0}$ is high, and the level of R&D effort $D_{i}$ and the service innovation coefficient $α_{i}$ of the participating agents are low, the cross-institutional aggregation and integration decision-making R&D of common technologies for AI deep learning systems will not lead to an increase in system revenue, but rather to a decrease in benefits over time $t$ .

Numerical simulation analysis

In recent times, the government has successively introduced several policies to encourage the overall development of AI deep learning systems. Meanwhile, the increasing maturity of the cross-institutional collaboration model for common technologies has also deeply driven the optimization, transformation and upgrading of the R&D model for AI deep learning systems. By means of numerical model simulation, the dynamic evolution trend of the interests and concerns of each participating entity over time can be intuitively analyzed, as well as the mechanism by which changes in core variables affect the interests and concerns. This not only verifies the validity of the model but also provides theoretical and empirical support for the strategy optimization and adjustment of AI deep learning systems from the perspective of cross-institutional cooperation of common technologies.

Therefore, this section takes the relevant data of the AI deep learning system as the research sample and uses numerical simulation methods to model and analyze the changes in the interests of terminal consumer enterprises, scientific research institutes and AI deep learning enterprises under three modes. At the same time, this explores the dynamic influence mechanism of effort cost and innovation cost on the system interests in the aggregated and integrated decision-making mode. The research team conducted on-site investigations and analyses of enterprises engaged in cross-institutional R&D cooperation on general technologies of AI deep learning systems. They concluded that the range of the impact of the efforts of innovation entities on innovation benefits is 0.6 to 0.8, the range of the cost coefficient of technology input and cooperation is 0.2 to 0.3, and the range of the cost coefficient of technology feedback and application is 0.2 to 0.4. The range of the tripartite benefit-sharing ratio is 0.2 to 0.3. In addition, they are coupled with the real scenarios of cross-institutional aggregation and general technology integration decision-making R&D. This paper completes the initial parameter assignment by referring to the parameter simulation logic in Refs 37 and 38. For specific parameter configuration, please refer to Table 1.

Table 1.

Parameter assignment.

Parameters	Assignment	Parameters	Assignment
$M (0)$	0.1	$M_{0}$	0.1
$α_{P}$	0.4	$α_{K}$	0.5
$α_{Z}$	0.3	$β_{P}$	0.7
$β_{K}$	0.8	$β_{Z}$	0.6
$γ_{P}$	0.2	$θ_{P}$	0.3
$θ_{K}$	0.2	$χ_{P}$	0.2
$χ_{K}$	0.3	$χ_{Z}$	0.2
$λ_{K}$	0.4	$λ_{Z}$	0.3
$δ$	0.1	$η$	0.4
$μ$	0.3	$ν$	0.2
$ρ$	0.1

Comparison of the benefits of the three models

In the joint cost-sharing and aggregation and integration decision-making mode, the benefits to end-consumer enterprises, scientific research institutions and AI deep learning enterprises all increase over time $t$ . The increase in R&D is large in the first period and gradually increases in the later period, as shown in Figure 2. In an AI deep learning system, the cooperative model is superior to the joint cost-sharing model, and both are superior to the distributed autonomous decision-making model. This simulation result is consistent with the conclusions of Propositions 1–2.

Figure 2.

Optimal functions of different subjects. (a) The optimal return function of the end-consuming enterprise; (b) The optimal return function of scientific research institutes; (c) The optimal return function of scientific research institutes; and (d) The optimal return function of the total system.

In the distributed autonomous decision-making model, the benefits of the cross-institutional co-operative R&D system for the three types of subjects and the AI deep learning systems diminish with time $t$ and finally level off gradually, which is related to Proposition 4. When the initial level of innovation ( $M_{0}$ ) is high for an AI deep learning product, using a distributed autonomous decision-making model results in diminishing returns over time $t$ for the tripartite subjects. If the initial innovation level ( $M_{0}$ ) is low, all three modes of cooperation increase returns to the three parties over time $t$ . The AI deep learning R&D is knowledge-intensive and has a high level of initial innovation in its products. Therefore, in this industry environment, the joint cost-sharing model is more suitable than the aggregation and integration decision-making model, which can bring more abundant value to the parties involved in value co-creation.

Impact of important parameters in three models

From equation (52), $\frac{\partial V_{3} (M) *}{\partial α_{i}} > 0$ , the influence of innovation capability coefficient ( $α_{i}$ ) on the benefit $V_{3} {(M)}^{*}$ of AI deep learn systems is positive, the front-end effect of the innovation cycle is weak, but it shows an increasing trend of marginal influence as the R&D process deepens, as shown in Figure 3. There is a time lag in the transmission of innovation capacity input to the development of AI deep learning systems, thus forming the “time lag effect” of innovation impact capacity, in the early stage of innovation resource input, the role of innovation capacity on the AI deep learning systems of cross-organizational co-innovation of common technologies shows editing incremental characteristics. And the innovation capacity of scientific research institutions has the greatest impact on the returns. The innovation capacity of AI deep learning enterprises has the least impact on the returns.

Figure 3.

The influence of $α_{i}$ and $t$ on $V_{3} {(M)}^{*}$ . (a) Trends·in·optimal returns for the end-consumer enterprise and scientific research institution; (b) Trends·in·optimal returns for the artificial intelligence deep learning enterprises.

AI deep learning systems should continuously invest innovative resources in AI deep learning to reach the threshold of technological development and application, thereby leading to the technological iteration of deep learning in AI and the coupling of general technologies.

From equation (52), $\frac{\partial V_{3} (M) *}{\partial α_{i}} > 0$ , as technology cooperation cost coefficient ( $θ_{i}$ ) increases, $V_{3} {(M)}^{*}$ carries on decreasing to a small extent, and technology cooperation cost coefficient ( $θ_{i}$ ) does not show a significant response to the economic rate of return. This result suggests that the combination of common technology R&D of AI deep learning systems and new cooperation models will result in higher technical cooperation costs, this will only reduce the final benefits by a small amount, as shown in Figure 4. Although technical cooperation will produce significant aggregation and integration decision-making costs, through the communication transmission mechanism of the end-consumer enterprises, we can promote the information flow among the upstream and downstream entities of cross-organizational R&D of common technologies among the themes of AI deep learning systems, and ultimately to meet the market’s diversified needs and the value of the product value-added.

Figure 4.

The influence of $θ_{i}$ and $t$ on $V_{3} {(M)}^{*}$ . (a) Trends·in·optimal returns for the artificial intelligence deep learning enterprises; (b) Trends·in·optimal returns for the scientific research institution.

When developing AI deep learning products, the artificial intelligence deep learning system should abandon the technology-driven thinking that focuses on the feasibility of concepts, and build a strategic framework oriented towards value creation. The market potential of the product should be evaluated based on users’ perception of practicality.

From equation (52), $\frac{\partial V_{3} (M) *}{\partial θ_{i}} < 0$ , the effect of technology application cost coefficient ( $λ_{i}$ ) on the return $V_{3} {(M)}^{*}$ of the model AI deep learning system is negative, the impact of technology application cost on the return of the scientific research institutes and the AI deep learning enterprises is small, and the effect of the end-consuming enterprises on the return is the largest, which is deepened with the impact of time $t$ , as shown in Figure 5. The technology application cost coefficient ( $λ_{i}$ ) has a smaller effect on earnings in the early stages of the cooperation and a larger impact on earnings in the later stages of the cooperation. Scientific research institutions and AI deep learning enterprises mostly focus on the construction of the underlying architecture of AI deep learning technology and are less directly involved in the application of terminal scenarios, while end-consumption enterprises are in sharp contrast, which are directly oriented to customer demand and assume a key role in the landing of the application of AI deep learning technology. With the in-depth development of cross-organizational cooperation on common technologies of AI deep learning systems, scene integration, technology adaptation and other aspects of the complexity of the enhancement, which has led to a rising trend in the cost of technology application, and the end-consumer enterprises have the greatest impact.

Figure 5.

The influence of $λ_{i}$ and $t$ on $V_{3} {(M)}^{*}$ . (a) Trends·in·optimal returns for the scientific research institution; and (b) Trends·in·optimal returns for the End-consumer enterprise.

AI deep learning systems should enhance the technical application level of AI deep learning systems, enabling them to better adapt to the constantly emerging new changes, new requirements, and new scenarios in the market, achieve AI deep learning innovation capabilities that match the market, and improve the innovation and profitability of the entire system.

Results and implications

Main results

Facing a new blue ocean, this paper starts from the perspective of cross-institutional cooperation in common technologies. With the help of the differential game method, analyzing the stability and optimization problems of common technology R&D in the AI deep learning systems and the key factors affecting the stability of cross-institutional cooperation, and verifying them through evolutionary simulation. Based on the above analysis, the main conclusions of this paper are as follows:

(1) The joint cost-sharing and aggregation and integration decision-making models are highly effective in terms of incentives. To varying degrees, they have increased the participants’ effort. The level of effort demonstrated by the three types of subjects is at its highest in the aggregation and integration decision-making model. In contrast, the incentive intensity of the joint cost-sharing model is equivalent to the proportion of subsidy given by the platform to the costs of manufacturers and developers.

(2) If the product’s initial level of innovation is high and the participants’ effort and innovation capacity are low. In this case, product development will not increase AI deep learning systems benefits but rather decrease them over time.

(3) The joint cost-sharing and aggregation and integration decision-making model has a strong positive correlation between the level of participation and feedback from the end-consumer enterprises on the final optimal revenue outcome for the institution, compared to the distributed autonomous decision-making model.

(4) The aggregation and integration decision-making model have higher subjective benefits and the benefits of a cross-institutional cooperative system than the other two models, reaching a Pareto equilibrium. The best solution and development direction for constructing cross-institutional cooperative R&D of common technologies in AI deep learning systems. However, when produce undesirable results if the benefits are not distributed.

Research implications

The findings of this paper provide important insights into optimizing the cooperative relationship between AI deep learning enterprises and other subjects, and boosting the development of the AI deep learning systems:

(1) Reasonably reduce the cost of aggregation and integration decision-making, AI deep learning enterprises to improve AI deep learning products oriented to the actual application needs, scientific research institutions focus on basic research, end-consumption enterprises focus on the application in practice and product feedback, to promote cross-institution cooperation of common technologies of the AI deep learning systems oriented to the forward-looking strategic needs.

(2) Strengthening the construction of interaction mechanisms between AI deep learning enterprises and end-consumption enterprises, enable end-consumption enterprises to transform from passive recipients to active participants. This model adjustment can significantly increase the depth of participation of end-consumption enterprises, and thus strongly promote the efficient process of R&D in the AI deep learning systems.

(3) Cross-institutional cooperation enterprises of common technologies in AI deep learning systems should abandon traditional game thinking. Even if they cannot adopt the aggregation and integration decision-making mode, they should maintain the effectiveness of the cooperation commitment. This is because under the aggregation and integration decision-making mode, when the negotiation parameters are within a certain range, the respective profits of the members of the commonality technology cross-institutional channel are not smaller than the respective profits under the decentralized decision-making mode.

Footnotes

Acknowledgments

The authors acknowledge the National Social Science Fund of China (Grant: 22BJY015).

ORCID iDs

Qianli Ma

Yixin Wang

Ailian Qiu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Social Science Fund of China (Grant: 22BJY015).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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