Exploring the Continuous Motivation of Algorithm Engineers under Multiple Objectives: A Mixed-Methods Study

Abstract

There is a multi-objective coordination relationship between online platform enterprises and algorithm engineers. Based on principal–agent theory, this study builds a multi-objective coordination incentive model for the two using a mixed-methods approach. Qualitative analysis reveals three main attributes of algorithm items: completion time, difficulty, and quality. The quantitative analysis had two results: first, the level of effort of algorithm engineers on the three indicators—time, difficulty coefficient, and quality—is correlated positively with their own technical competence and negatively with the change rate of their marginal effort costs. Second, the company’s incentive coefficient for algorithm engineers depends on two factors: (1) comprehensive technical level, risk aversion coefficient, and marginal effort cost change rate of each algorithm engineer; and (2) the importance of the project for the company. The research findings suggest that enterprises adopt different incentive methods for different projects and enact hierarchical incentives for algorithm engineers with different levels of technical competence.

1. Introduction

With the rapid development of big data and artificial intelligence, algorithm engineers are playing an increasingly important role in enterprises. They are involved in the innovative development of business, especially in online platform enterprises. However, the algorithm project itself involves great uncertainty, and the incentives of algorithm engineers and performance evaluation of algorithm projects have become practical problems faced by online platform enterprises. The question of how to effectively motivate algorithm engineers has become the management focus of online platform enterprises.

There is a principal–agent relationship between algorithm engineers and online platform enterprises. Enterprises entrust algorithm projects to algorithm engineers to motivate them to achieve higher project performance. Existing studies have explored similar relationships, as moral risks and adverse selection are common issues. Many scholars have conducted extensive research on the moral risks of the single-objective principal–agent model. However, in reality, the principal–agent relationship is often multi-objective. In this situation, the single-objective incentive will become invalid. Therefore, studying the multi-objective principal–agent model has practical value and research significance.

However, the existing literature, while discussing multi-objective incentive problems, adopt the linear weighting approach. Although this simplistic approach helps us to multitask due to the importance of sorting, it ignores that the task cannot completely replace the reality. The linear model allows a utility of zero, but the output of the enterprise’s total utility is not zero. This assumption is not consistent with the principle of “one vote for code quality” in evaluating algorithm performance in practice. Therefore, the rationality of the linear model is worth discussing. In the field of sustainable performance management, how to effectively integrate sustainability in project management systems is still an emerging question, and its role in supporting sustainability has not been well studied within organizations. In addition, sustainability performance management and control are not discussed in a structured way; there is only a detailed description of the concept.

Based on the analysis of the previous literature, the research question proposed in this study is “In Internet enterprises, when algorithm engineering is faced with multi-objective tasks, how can enterprises motivate among various objectives to achieve sustainable performance management for algorithm engineers?” Therefore, this study introduces the multi-task principal–agent model [1], and constructs the multi-task principal–agent model between enterprise and employees based on the multi-task of task evaluation index of algorithm engineers. Because the existing literature on the task measurement of algorithm engineers is relatively simple, there is a deviation from the practice. We adopted a combination of qualitative and quantitative tests based on the existing literature and enterprise interview analysis. We also summed up three indicators to measure the task quality of algorithm engineers: Project Completion Time, Project Difficulty, and Project Quality. Quantitative analysis was performed in order to incorporate these three indicators into the analysis model of this study.

The research contribution of this study is based on the multi-task principal–agent, an incentive model for Internet enterprises and algorithm engineers. It is constructed and combined with the characteristics of algorithm projects, and an extended Cobb-Douglas production function is used to reflect the output under the conditions of task completion time, difficulty, and quality of multi-task effort input. The results show that the effort level that algorithm engineers put into task completion time, difficulty, and quality is positively correlated with the comprehensive technical level, and negatively correlated with the change rate of marginal effort cost of each task. Meanwhile, the relative effort level of the three tasks is determined by the relative importance of the task and the change rate of the relative marginal cost. The more important the task, the lower the change rate of the relative marginal effort cost, and the greater the relative effort invested. The results are helpful for algorithm engineers in reasonably allocating their own resources and making the corresponding level of effort in order to achieve certain task requirements. This study also contributes to the research on performance management in sustainable development, and analyzes the factors that influence the optimal incentive coefficient to promote the sustainable development of the project. Moreover, the multi-task principal–agent model developed by this study is a theoretic model, and the applicability of the results will be improved by further empirical analysis.

2. Theoretical Background

2.1. Multi-Task Principal–Agent Theory

The principal–agent theory has been widely applied in the research of information asymmetry and incentive issues, and it can be used to study the incentive problems of task allocation in enterprise. Based on the principal–agent theory, this study will study how enterprises design incentive mechanisms to optimize the overall performance of algorithm projects. Regarding the incentive problem of algorithm engineers, scholars mainly attribute them to the impact of incentive on project performance, the establishment of the incentive model, and assignment between the three aspects of research tasks.

The first issue to be discussed is the role of incentives. Holmstrom and Milgrom [1] studied the substantial impact of inter-task externalities, or spillover effect, on the incentive contract design via a multi-task principal–agent model. This model assumes that the principal cannot observe the agent’s effort, but the result of the agent’s effort can be observed as a performance signal. Afterward, many scholars began to follow this thinking. Baker et al. [2] studied the impact of performance evaluation on multi-task principal–agent incentive systems. Schnedler [3] studied the incentive effect on different task difficulties. Edmans et al. [4] studied the optimal compensation contract, using which managers can improve current business performance through normal efforts and profit manipulation behaviors.

The second is the model setting of the incentive mechanism. In terms of model construction, the existing studies are generally analyzed based on principal–agent theory [1,2,3,4]. In the multi-objective incentive model, the task is generally assumed to be unobservable, but the results of the agent can be measured according to certain observable performance indicators or signals. At present, many studies focus on the issue of how agents’ efforts are allocated among different indicators. Some scholars have found that using multiple performance indicators to measure a job, as well as linking the agent’s remuneration contract with the total value of multiple performance indicators, can not only solve the incentive issues, but also alleviate the distortion of the agent’s effort allocation on multiple tasks [5,6]. The study performed by Gibbons [7] shows that using multiple performance indicators to measure an agent’s output can improve the efficiency of the agent’s effort allocation between different tasks. Baker’s study [8] on the multi-task model also shows that the degree of consistency between performance indicators and the agent tasks can be measured by the cosine value between output sensitivity and performance indicator sensitivity, but the model assumes that the effort cost of different tasks is the same.

The third is the study of assignment among tasks. According to Holmstrom, Milgrom (1991), and Baker (1992) [1,8], there is much research on multi-tasking in a principal–agent environment. Unlike in a single-task environment, tasks are clear, and there is mutual influence between tasks in multi-task environment. The literature studied the interaction between different tasks, as well as how to optimize the assignment of tasks among different professional agents [9,10]. The dynamic model proposed to study how to handle multi-task situations when employees have a certain specialty was also investigated [11,12]. However, the above studies did not consider the case that the incentive object can be difficult to replace among different tasks, and the incentive object itself is relatively homogenous.

However, the existing literature, while discussing multi-objective incentive problems, adopted the linear weighting approach. Although this simplistic approach helps to multitask due to the importance of sorting, it ignores that the task cannot completely replace the reality. A basic assumption of the above studies is that the performance indicators of different task objectives are equally important to the agent, and that different objective performance indicators can replace each other. The research process for this study revealed that different task objectives may vary for the company, and that they cannot completely replace each other. This study used the Cobb-Douglas production function to reflect the impact of the importance of different measurement indicators on the final output. Based on the interview data, three indicators were extracted to measure the utility of algorithm engineers: time, difficulty, and quality of project completion. The three indicators were analyzed in order to obtain the mutual relationship among the indicators. According to the principal–agent model, an incentive model was constructed, which ultimately optimized the existing incentive plan of Online Platforms K.

2.2. Performance Management Systems and Sustainable Development

It is important to manage sustainability issues for the success and survival of organizations [13,14]. Existing research indicates that integrating sustainability issues and stakeholder concerns into an organization’s strategic objectives, management systems, and operations is key to achieving long-term business success [13,15]. Research also shows that the focus of sustainability has shifted, and the focus of sustainability management research has shifted from why companies should pay attention to sustainability issues to how to integrate sustainability issues into organizational systems and processes [14,16]. However, these new studies are largely based on external reports of sustainability and the factors that lead to the implementation of sustainability practices [17,18]. There is a lack of research on how to make the project development sustainable and how to promote the sustainable contribution of employees through the incentive of project tasks in the process of project execution.

Performance management system is an important tool for integrating sustainability issues into enterprise management, and economic, social, and environmental goals into organizational missions, strategies, and actions [15]. However, how to effectively integrate sustainability in project management systems is still an emerging field, and its role in supporting sustainability has not been well-studied within organizations [19,20]. In addition, sustainability performance management and control are not discussed in a structured way; there is only a detailed description of the concept [21]. Therefore, the literature has defined sustainability performance management and measurement as the interface between measuring performance and managing business, society, and the environment [20]. Schaltegger (2011) [21] further constructed the management and control of sustainable development from the perspective of the system, and expanded the concept of balanced scorecard for sustainable development to describe the financial oriented, market-oriented, process-oriented, knowledge, learning oriented, and non-market-oriented sustainability MCSs. In general, the question of how to start from the specific project management in the project management system to explore how to effectively manage a single project, in order to achieve the sustainable development of the enterprise, still lacks discussion.

3. Mixed-Methods Design

In this study, we used a mixed-methods combined qualitative and quantitative study with two sequential stages: a qualitative study to find the variables affecting the work of algorithm engineers, and a quantitative study to construct and analyze models. The mixed-methods approach benefits from the advantages of both qualitative and quantitative research designs, and thus obtains richer and more reliable research results [22]. The qualitative design facilitates exploratory research that can be used to find variables of context-specific constructs in models [23]. The quantitative components facilitate an expanded phase in which the relationship between variables can be analyzed by the research model [23,24,25,26]. The mixed-methods approach is particularly suitable for studying new issues in emerging fields that require context-specific constructs, which can then be analyzed. Given the paucity of studies on algorithm research and development context, and the lack of clarity regarding the factors that influence algorithm engineers, the mixed-methods approach is suited to our study. As this illustrates, the two methods complement one another; cases provide a rich source of novel phenomena and theoretical insight, while analytical models can weave these insights into theory in a precise and rigorous way.

An outlook which would help to explain multi-task incentives to algorithm engineers, and help them to understand the field of multi-task principal-agent, is lacking. At this stage, either qualitative or quantitative methods would be insufficient to answer the research questions of this study. Thus, we used a mixed-methods design for development purposes, whereby we used findings from the qualitative phase to inform the research design of the quantitative phase. Figure 1 presents the mixed-methods design of this study. First, we conducted a qualitative study (Phase 1) to explore evaluation dimensions of task performance of algorithm engineers. The results were incorporated into the theoretical model. Next, we conducted a quantitative study (Phase 2) to empirically investigate the effects of incentive effect between different dimensions of the algorithm engineer task.

Qualitative Analysis and Findings

The enterprise in this study is an online music platform (K for short). We interviewed 21 executives, project managers, and algorithm engineers of K and obtained a detailed understanding of its incentive challenges. The profiles of the survey respondents are shown in

Table 1. Profiles of respondents.

Objects	Executives	Project Managers	Algorithm Engineers
Job Titles	Chief Executive Officer (CEO), Executive Assistant, Vice President	Project managers (A1–A8)	Algorithm engineers (Virtual online project development) (B1–B4) Algorithm engineers (Game Project Development) (B5–B14)
No. of interviewees	3 s	8	14
Recording Time	135 min	270 min	487 min

. Currently, K’s incentives for algorithm engineers are based on project completion. The score is set according to the difficulty of the project, with higher scores indicating higher project difficulty. The company records the scores based on the completion of each algorithm engineer’s project, and divides the algorithm engineers into four levels (SABC) according to the scores obtained each month. These ratings are referenced for both salary increases and promotion. This leads to a situation in system implementation in which large numbers of algorithm engineers choose to complete projects with lower difficulty and lower scores, in order to achieve victory by quantity, while those who select projects with more difficulty may not complete them within the necessary time, so they are unable to earn points.

4. Results

4.1. Qualitative Data Analysis

This study adopted an inductive method to analyze the interview data [27]. Before the data analysis, all interview documents were coded as ‘Ix’ (for instance, ‘I1′ indicates the first interviewee). The main steps in the data analysis process were as follows:

Round 1: Transcribing the recording into words, identifying and encoding the key sentences and phrases. We read all the interviewee documents thoroughly to identify all sentences and phrases related to our research questions.

Round 2: Classifying coding results. We integrated sentences and phrases with similar meanings together and grouped them into one classification. We further created a list of keywords to describe the categorized interview documents.

Round 3: Identifying key constructs according to the interview data, summarizing the core variables in the algorithm work. In this step, we reassembled all the interview documents. Meanwhile, we tried to understand potential relationships between constructs by reviewing interview data and relevant literature.

As shown in

Table 2. Selected quotes from interviewees.

Category	Construct	Selected Quote
Project Completion Time	Project Completion Requirements	‘Each project has a defined completion cycle, which is determined during project planning’. (I7)
	Perceived Urgency	‘There is uncertainty in the project development process, and every time we think about the completion time, it makes us nervous’. (I12)
Project Difficulty	Difficulty of Project Planning	‘The first step of the project is to make a complete plan, which needs to be negotiated by the team’. (I9)
	Difficulty of Project Implementation	‘The implementation process often deviates from the planning, which requires constant adjustment of the program and solution of project problems’. (I2)
Project Quality	Quality of Algorithmic Code	‘The same project may have different algorithmic solutions where the length and efficiency of the program are important’. (I9)
	Quality of Algorithmic Application	‘Another important criterion of the project is the efficiency of the algorithm and the frequency of errors. We want the algorithm to run well and fast’. (I1)

, the qualitative analysis yielded key variables: project completion time, difficulty, and quality. We consider these three variables as core metrics for algorithm engineers. The qualitative analysis results can provide evidence on the conceptualization of constructs and hypothesis development [23].

4.2. Problem Description and Model Assumptions

Following the principal–agent model, this study regards the online platform K as the principal and the algorithm engineers in its technical department as the agents. It analyzes the K’s algorithm engineers’ incentive model and provides solutions for the above issues.

With reference to the interview data, this study assumes that the project completion effect of K’s algorithm engineers will be evaluated mainly by three indicators: the time it takes the algorithm engineer to complete the project, the degree of difficulty of the project, and the completion quality. High-quality completion involves the algorithm engineer using less code to complete the work; that is, high completion quality means lower completion cost. The level of effort (LOE) of the algorithm engineer in time completion is set as ${\mathrm{e}}_1$, that in the completion difficulty is set as ${\mathrm{e}}_2$, and that in the completion quality is set as ${\mathrm{e}}_3$. The three values are all greater than 0.

The projects of Company K are divided into basic and new projects. The basic projects are routine tasks undertaken by every algorithm engineer, while the new projects are performed by algorithm engineers according to their personal situations. Therefore, algorithm engineers’ income is divided into basic salary and additional earnings generated by completion of new projects. The output function of algorithm engineers’ efforts is expressed by the expanding-type Cobb-Douglas production function:$\mathrm{B}\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right)={\mathrm{Ae}}_1{}^{\mathrm{p}}{\mathrm{e}}_2{}^{\mathrm{q}}{\mathrm{e}}_3{}^{1-\mathrm{p}-\mathrm{q}}+\mathsf{\epsilon }$—namely, the revenue function generated by algorithm engineers’ certain LOE. In this function, $\mathrm{E}>0$, which can reflect the algorithm engineer’s own competence level, expressing the competence of the algorithm engineer to transfer their efforts into the company revenue. $\mathrm{p}$ represents the importance attached to the project completion time by Company K. $\mathrm{q}$ represents the importance attached to the project degree of difficulty by Company K, and $\left(1-\mathrm{p}-\mathrm{q}\right)$ stands for the importance attached to the project completion quality by Company K. When the value of $\mathrm{p}$, q or $\left(1-\mathrm{p}-\mathrm{q}\right)$ is larger, it indicates that Company K attaches more importance to this indicator, and, therefore, the algorithm engineer will increase the LOE on this indicator. $\mathsf{\epsilon }$ is a normally distributed disturbance term, and $\mathsf{\epsilon }~\mathrm{N}(0,{ \mathsf{\delta }}^2)$ represents the impact of other factors unrelated to algorithm engineers’ LOE on business revenue, which reflects the uncertainty of the external environment. The greater the value of ${\mathsf{\delta }}^2$, the greater the uncertainty. $\mathrm{B}\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right)$ satisfies the economics assumption that marginal returns decrease, and is a strictly increasing concave function: $\frac{\partial \mathrm{B}}{\partial {\mathrm{e}}_{\mathrm{i}}}>0$,$\frac{{\partial }^2\mathrm{B}}{\partial {\mathrm{e}}_{\mathrm{i}}{}^2}<0$.

The effort algorithm engineers invest in the three indicators requires a certain effort cost. It is assumed that, in Company K, more effort put into one certain indicator will reduce the LOE in other indicators, but the effort cost on each indicator will not change. Based on the above analysis, in order to simplify the model and satisfy the basic reality of Company K, it is assumed that multi-task effort costs are mutually independent and can be expressed as a function: $\mathrm{C}\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right)=\frac{1}{2}\left({\mathrm{c}}_1{\mathrm{e}}_1{}^2+{\mathrm{c}}_2{\mathrm{e}}_2{}^2+{\mathrm{c}}_3{\mathrm{e}}_3{}^2\right)$, in which ${\mathrm{c}}_1$, ${ \mathrm{c}}_2$, ${ \mathrm{and} \mathrm{c}}_3$ are greater than 0, representing the project completion time of the algorithm engineer, the project difficulty coefficient, and the change rate of marginal effort cost of the project quality, respectively. $\mathrm{C}\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right)$ is a strictly increasing convex function, and $\frac{\partial {\mathrm{C}}_{\mathrm{i}}}{\partial {\mathrm{c}}_{\mathrm{i}}}>0$.

K motivates its algorithm engineer according to the output $\mathrm{B}\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right)$, and the payroll function is expressed in linear form: $\mathrm{S}=\mathsf{\alpha }+\mathsf{\beta }·\mathrm{B}\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right)$, in which $\mathsf{\alpha } \mathrm{is} \mathrm{the} \mathrm{fixed}$ salary obtained by completing the basic project arranged by the company. $\mathsf{\beta }$ is the income incentive coefficient of additional incentives obtained by completing the new projects. On one hand, the linear income function helps to improve the operability of the incentive mechanism, which makes Company K able to reward or punish the algorithm engineer based on the completion of the new projects; this is in conformity with the algorithm engineer’s performance appraisal in actual operation. On the other hand, principal–agent models generally use a linear payroll function; thus, the use of a linear payroll function in this study will aid model analysis and solution.

Following the general agency analytical framework, it is assumed that K is risk-neutral, and algorithm engineers are risk-averse with a utility function $\mathrm{M}=-\exp \left(-\mathsf{\rho }\mathrm{w}\right)$, in which the value of the absolute risk aversion is constant. In this function, $\mathsf{\rho }$ is the absolute risk-aversion coefficient—namely, Arrow–Pratt absolute risk aversion—and $\mathrm{w}$ represents the actual income.

It is assumed that K cannot observe the algorithm engineer’s LOE, but can evaluate their effort through the results of the algorithm engineer’s project completion. At the same time, the company can clearly identify projects with different difficulty coefficients and the effort the algorithm engineer puts into them when the project is not completed.

5. Model Construction

There is a principal–agent relationship between Internet enterprises and algorithm engineers based on algorithm projects. As the client, Internet enterprises encourage algorithm engineers to contribute their knowledge in the hope that algorithm engineering can complete tasks with high quality and efficiency. As the executor of the project, the algorithm engineer is the main body of the project implementation. A principal–agent relationship is established when an enterprise requires an algorithm engineer to complete a project task, as shown in Figure 2. In addition, there is information asymmetry between Internet companies and algorithm engineers. Algorithmic engineers have information advantages that Internet companies do not. Therefore, this paper draws on the model setting of the classical principal–agent theory in order to propose the conditions for the compatibility of participation constraints and incentives.

Company K is risk-averse. According to the above hypotheses, algorithm engineer income E is equal to the mean value of actual income minus the cost of effort and the risk cost of income. Therefore, the income function of the company algorithm engineer is $\mathrm{S}=\mathsf{\alpha }+\mathsf{\beta }·\mathrm{B}\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right)+\mathsf{\epsilon }, \mathsf{\epsilon }~\mathrm{N}(0,{\mathsf{\delta }}^2)$, and the algorithm engineer’s risk cost is $\frac{1}{2}\mathsf{\rho }\mathrm{Var}\left(\mathrm{s}\right)=\frac{1}{2}\mathsf{\rho }\mathrm{Var}\left(\mathsf{\alpha }+\mathsf{\beta }·{\mathrm{Aa}}_1{}^{\mathrm{q}}{\mathrm{a}}_2{}^{1-\mathrm{q}}+\mathsf{\epsilon }\right)=\frac{1}{2}{\mathsf{\rho }\mathsf{\beta }}^2{\mathsf{\delta }}^2$.

Therefore, the risk cost of the algorithm engineer’s income is affected by the algorithm engineer’s own absolute risk aversion $\mathsf{\rho }$ and the uncertainty of the external environment ${\mathsf{\delta }}^2$. The low risk aversion here mainly means that the algorithm engineer may not be able to complete projects with high difficulty coefficients. Therefore, projects with smaller difficulty coefficients are selected, and, due to the lower external uncertainty, the value of ${\mathsf{\delta }}^2$ is smaller. The income of the algorithm engineer can be expressed as follows:

$$\mathrm{E}=\mathsf{\alpha }+\mathsf{\beta }·\mathrm{B}\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right)-\frac{1}{2}\left({\mathrm{c}}_1{\mathrm{e}}_1{}^2+{\mathrm{c}}_2{\mathrm{e}}_2{}^2+{\mathrm{c}}_3{\mathrm{e}}_3{}^2\right)-\frac{1}{2}{\mathsf{\rho }\mathsf{\beta }}^2{\mathsf{\delta }}^2$$

(1)

when the company revenue minus the algorithm engineer income is

$$\mathrm{U}=\left(1-\mathsf{\beta }\right){\mathrm{Ae}}_1{}^{\mathrm{p}}{\mathrm{e}}_2{}^{\mathrm{q}}{\mathrm{e}}_3{}^{1-\mathrm{p}-\mathrm{q}}-\mathsf{\alpha }.$$

(2)

The fixed salary $\mathsf{\alpha }$ only affects the allocation of total income between K and the algorithm engineer, which is determined by the algorithm engineer’s retained utility, but does not affect $\mathsf{\beta }$ and $\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right).$ If $\left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right)$ is fixed, $\mathsf{\alpha }$ is determined by the retained utility of the algorithm engineer. Therefore, the incentive issue of Online Platforms K becomes how to determine the incentive coefficient $\mathsf{\beta }$ to maximize the total company profits TE—namely, the sum of certainty equivalent profit of the Online Platforms K—and the certainty equivalent profits of the algorithm engineer:

$$\mathrm{TE}={\mathrm{Ae}}_1{}^{\mathrm{p}}{\mathrm{e}}_2{}^{\mathrm{q}}{\mathrm{e}}_3{}^{1-\mathrm{p}-\mathrm{q}}-\frac{1}{2}\left({\mathrm{c}}_1{\mathrm{e}}_1{}^2+{\mathrm{c}}_2{\mathrm{e}}_2{}^2+{\mathrm{c}}_3{\mathrm{e}}_3{}^2\right)-\frac{1}{2}{\mathsf{\rho }\mathsf{\beta }}^2{\mathsf{\delta }}^2.$$

(3)

Therefore, under the condition of information asymmetry, the incentive problem of Internet companies for algorithm engineers is equivalent to choosing the $\mathsf{\beta }$ to maximize the total certainty equivalent profit under the condition of incentive compatibility constraints (IC)—that is, solving the optimization problem (4):

$$\begin{array}{c}\mathrm{P}\text{:} \mathrm{maxTE}\text{:} {\mathrm{Ae}}_1{}^{\mathrm{p}}{\mathrm{e}}_2{}^{\mathrm{q}}{\mathrm{e}}_3{}^{1-\mathrm{p}-\mathrm{q}}-\frac{1}{2}\left({\mathrm{c}}_1{\mathrm{e}}_1{}^2+{\mathrm{c}}_2{\mathrm{e}}_2{}^2+{\mathrm{c}}_3{\mathrm{e}}_3{}^2\right)-\frac{1}{2}{\mathsf{\rho }\mathsf{\beta }}^2{\mathsf{\delta }}^2\\ \mathrm{s}.\mathrm{t}\text{:} \mathsf{\alpha }+{\mathsf{\beta }\mathrm{Ae}}_1{}^{\mathrm{p}}{\mathrm{e}}_2{}^{\mathrm{q}}{\mathrm{e}}_3{}^{1-\mathrm{p}-\mathrm{q}}-\frac{1}{2}\left({\mathrm{c}}_1{\mathrm{e}}_1{}^2+{\mathrm{c}}_2{\mathrm{e}}_2{}^2+{\mathrm{c}}_3{\mathrm{e}}_3{}^2\right)-\frac{1}{2}{\mathsf{\rho }\mathsf{\beta }}^2{\mathsf{\delta }}^2\ge \mathsf{\alpha }\end{array}$$

(4)

(Participation constraints.)

$$\mathrm{IC}\text{:} \left({\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3\right)\in \mathrm{argmax}\left[\mathsf{\alpha }+\mathsf{\beta }·{\mathrm{Ae}}_1{}^{\mathrm{p}}{\mathrm{e}}_2{}^{\mathrm{q}}{\mathrm{e}}_3{}^{1-\mathrm{p}-\mathrm{q}}-\frac{1}{2}\left({\mathrm{c}}_1{\mathrm{e}}_1{}^2+{\mathrm{c}}_2{\mathrm{e}}_2{}^2+{\mathrm{c}}_3{\mathrm{e}}_3{}^2\right)-\frac{1}{2}{\mathsf{\rho }\mathsf{\beta }}^2{\mathsf{\delta }}^2\right]$$

(Incentive compatibility constraints.)

6. Model Analysis and Insights

If all of ${\mathrm{e}}_{\mathrm{i}}$ are greater than 0, then a certain value is given to $\mathsf{\beta }$. The solution of Equation (4) is to solve the extreme value under the condition of IC. In the IC equation, the first-order derivative is obtained as follows:

$$\{\begin{array}{c}\mathsf{\beta }\mathrm{A}\mathrm{p}{\mathrm{e}}_1^{\mathrm{p}-1}·{\mathrm{e}}_2^{\mathrm{q}}·{\mathrm{e}}_3^{1-\mathrm{p}-\mathrm{q}}={\mathrm{c}}_1{\mathrm{e}}_1\\ \mathsf{\beta }\mathrm{A}\mathrm{q}{\mathrm{e}}_1^{\mathrm{p}}·{\mathrm{e}}_2^{\mathrm{q}-1}·{\mathrm{e}}_3^{1-\mathrm{p}-\mathrm{q}}={\mathrm{c}}_2{\mathrm{e}}_2\\ \mathsf{\beta }\mathrm{A}\left(1-\mathrm{p}-\mathrm{q}\right){\mathrm{e}}_1^{\mathrm{p}}·{\mathrm{e}}_2^{\mathrm{q}}·{\mathrm{e}}_3^{1-\mathrm{p}-\mathrm{q}}={\mathrm{c}}_3{\mathrm{e}}_3\end{array}$$

By solving the above equations, the stagnation point of the function IC is M(${\mathrm{e}}_1^{*},{\mathrm{e}}_2^{*},{\mathrm{e}}_3^{*}$), and ${\mathrm{e}}_1^{*},{\mathrm{e}}_2^{*},{\mathrm{e}}_3^{*}$ are respectively equivalent to

$$\{\begin{array}{c}{\mathrm{e}}_1^{*}=\mathsf{\beta }\mathrm{A}{\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{-\frac{1+\mathrm{p}}{2}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{-\frac{\mathrm{q}}{2}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{-\frac{1-\mathrm{p}-\mathrm{q}}{2}}\\ {\mathrm{e}}_2^{*}=\mathsf{\beta }\mathrm{A}{\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{-\frac{\mathrm{p}}{2}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{-\frac{1+\mathrm{q}}{2}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{-\frac{1-\mathrm{p}-\mathrm{q}}{2}}\\ {\mathrm{e}}_3^{*}=\mathsf{\beta }\mathrm{A}{\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{-\frac{\mathrm{p}}{2}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{-\frac{\mathrm{q}}{2}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{-\frac{2-\mathrm{p}-\mathrm{q}}{2}}\end{array}$$

(5)

Via constant derivation of the objective function in IC at the stagnation point M (${\mathrm{e}}_1^{*},{\mathrm{e}}_2^{*},{\mathrm{e}}_3^{*}$), the second-order Hesse (H) matrix is shown below:

$$\left[\begin{array}{ccc}\mathsf{\beta }\mathrm{A}\mathrm{p}(\mathrm{p}-1){\mathrm{e}}_1^{\mathrm{p}-2}·{\mathrm{e}}_2^{\mathrm{q}}·{\mathrm{e}}_3^{1-\mathrm{p}-\mathrm{q}}-{\mathrm{c}}_1& \mathsf{\beta }\mathrm{A}\mathrm{p}\mathrm{q}{\mathrm{e}}_1^{\mathrm{p}-1}·{\mathrm{e}}_2^{\mathrm{q}-1}·{\mathrm{e}}_3^{1-\mathrm{p}-\mathrm{q}}& \mathsf{\beta }\mathrm{A}\mathrm{p}(1-\mathrm{p}-\mathrm{q}){\mathrm{e}}_1^{\mathrm{p}-1}·{\mathrm{e}}_2^{\mathrm{q}}·{\mathrm{e}}_3^{-\mathrm{p}-\mathrm{q}}\\ \mathsf{\beta }\mathrm{A}\mathrm{p}\mathrm{q}{\mathrm{e}}_1^{\mathrm{p}-1}·{\mathrm{e}}_2^{\mathrm{q}-1}·{\mathrm{e}}_3^{1-\mathrm{p}-\mathrm{q}}& \mathsf{\beta }\mathrm{A}\mathrm{q}(\mathrm{q}-1){\mathrm{e}}_1^{\mathrm{p}}·{\mathrm{e}}_2^{\mathrm{q}-2}·{\mathrm{e}}_3^{1-\mathrm{p}-\mathrm{q}}-{\mathrm{c}}_2& \mathsf{\beta }\mathrm{A}\mathrm{q}(1-\mathrm{p}-\mathrm{q}){\mathrm{e}}_1^{\mathrm{p}}·{\mathrm{e}}_2^{\mathrm{q}-1}·{\mathrm{e}}_3^{-\mathrm{p}-\mathrm{q}}\\ \mathsf{\beta }\mathrm{A}\mathrm{p}(1-\mathrm{p}-\mathrm{q}){\mathrm{e}}_1^{\mathrm{p}-1}·{\mathrm{e}}_2^{\mathrm{q}}·{\mathrm{e}}_3^{-\mathrm{p}-\mathrm{q}}& \mathsf{\beta }\mathrm{A}\mathrm{q}(1-\mathrm{p}-\mathrm{q}){\mathrm{e}}_1^{\mathrm{p}}·{\mathrm{e}}_2^{\mathrm{q}-1}·{\mathrm{e}}_3^{-\mathrm{p}-\mathrm{q}}& \mathsf{\beta }\mathrm{A}(-\mathrm{p}-\mathrm{q})(1-\mathrm{p}-\mathrm{q}){\mathrm{e}}_1^{\mathrm{p}}·{\mathrm{e}}_2^{\mathrm{q}}·{\mathrm{e}}_3^{-\mathrm{p}-\mathrm{q}-1}-{\mathrm{c}}_3\end{array}\right]$$

Through the analysis of the H matrix, since $\mathrm{p},\mathrm{q}\in \left(0,1\right),$ the first-order Sequential Principal Minor of H (M)$\left(\mathrm{p}-2\right){\mathrm{c}}_1<0$, the second-order Sequential Principal Minor $2\left(2-\mathrm{p}-\mathrm{q}\right){\mathrm{c}}_1{\mathrm{c}}_2>0$, and the third-order Sequential Principal Minor $-4{\mathrm{c}}_1{\mathrm{c}}_2{\mathrm{c}}_3<0$. Thus, H(M) is negative definite, and M(${\mathrm{e}}_1^{*},{\mathrm{e}}_2^{*},{\mathrm{e}}_3^{*}$) is the maximum point of IC. Substituting M into the required optimal solution in function P, we can obtain the following:

$$\mathrm{TE}=-\frac{1}{2}{\mathsf{\beta }}^2\left[{\mathrm{A}}^2{\left(\frac{\mathrm{P}}{{\mathrm{c}}_1}\right)}^{\mathrm{p}}{\left(\frac{\mathrm{q}}{{\mathrm{c}}_2}\right)}^{\mathrm{q}}{\left(\frac{1-\mathrm{p}-\mathrm{q}}{{\mathrm{c}}_3}\right)}^{1-\mathrm{p}-\mathrm{q}}+{\mathrm{p}\mathsf{\delta }}^2\right]+{\mathsf{\beta }\mathrm{A}}^2{\left(\frac{\mathrm{P}}{{\mathrm{c}}_1}\right)}^{\mathrm{p}}{\left(\frac{\mathrm{q}}{{\mathrm{c}}_2}\right)}^{\mathrm{q}}{\left(\frac{1-\mathrm{p}-\mathrm{q}}{{\mathrm{c}}_3}\right)}^{1-\mathrm{p}-\mathrm{q}}$$

Solving the first-order derivative of the above equation, and letting the first-order derivative be 0; then, the optimal incentive coefficient ${\mathsf{\beta }}^{*}$ can be obtained:

$${\mathsf{\beta }}^{*}=\frac{1}{1+{\mathsf{\rho }\mathsf{\delta }}^2{\mathrm{A}}^{-2}{\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{\mathrm{p}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{\mathrm{q}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{1-\mathrm{p}-\mathrm{q}}}$$

(6)

Through calculating the second derivative of the TE formula, the second-order derivative is less than 0, so ${\mathsf{\beta }}^{*}$ is the optimal incentive coefficient. Substituting the optimal incentive coefficient ${\mathsf{\beta }}^{*}$ into the original equation, the LOE under the optimal incentive coefficient can be obtained:

$$\left\{\begin{array}{c}{\mathrm{e}}_1^{*}=\frac{\mathrm{A}{\left(\frac{\mathrm{p}}{{\mathrm{c}}_1}\right)}^{\frac{\mathrm{p}+1}{2}}{\left(\frac{\mathrm{q}}{{\mathrm{c}}_2}\right)}^{\frac{\mathrm{q}}{2}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{\frac{\mathrm{p}+\mathrm{q}+1}{2}}}{{\mathrm{c}}_3+{\mathrm{c}}_3\frac{\mathrm{p}+\mathrm{q}}{1-\mathrm{p}-\mathrm{q}}+\mathsf{\rho }{\mathsf{\delta }}^2{\mathrm{A}}^{-2}{\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{\mathrm{p}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{\mathrm{q}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{2-\mathrm{p}-\mathrm{q}}}\hfill \\ {\mathrm{e}}_2^{*}=\frac{\mathrm{A}{\left(\frac{\mathrm{p}}{{\mathrm{c}}_1}\right)}^{\frac{\mathrm{p}}{2}}{\left(\frac{\mathrm{q}}{{\mathrm{c}}_2}\right)}^{\frac{\mathrm{q}+1}{2}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{\frac{\mathrm{p}+\mathrm{q}+1}{2}}}{{\mathrm{c}}_3+{\mathrm{c}}_3\frac{\mathrm{p}+\mathrm{q}}{1-\mathrm{p}-\mathrm{q}}+\mathsf{\rho }{\mathsf{\delta }}^2{\mathrm{A}}^{-2}{\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{\mathrm{p}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{\mathrm{q}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{2-\mathrm{p}-\mathrm{q}}}\hfill \\ {\mathrm{e}}_3^{*}=\frac{\mathrm{A}{\left(\frac{\mathrm{P}}{{\mathrm{c}}_1}\right)}^{\frac{\mathrm{p}}{2}}{\left(\frac{\mathrm{q}}{{\mathrm{c}}_2}\right)}^{\frac{\mathrm{q}}{2}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{\frac{\mathrm{p}+\mathrm{q}}{2}}}{{\mathrm{c}}_3+{\mathrm{c}}_3\frac{\mathrm{p}+\mathrm{q}}{1-\mathrm{p}-\mathrm{q}}+\mathsf{\rho }{\mathsf{\delta }}^2{\mathrm{A}}^{-2}{\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{\mathrm{p}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{\mathrm{q}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{2-\mathrm{p}-\mathrm{q}}}\hfill \end{array}\right.$$

(7)

From the above equations, we can obtain

$$\{\begin{array}{c}{\mathrm{e}}_1^{*}=\sqrt{\frac{{\mathrm{c}}_3\mathrm{p}}{{\mathrm{c}}_1\left(1-\mathrm{p}-\mathrm{q}\right)}}{\mathrm{e}}_3^{*}\\ {\mathrm{e}}_2^{*}=\sqrt{\frac{{\mathrm{c}}_3\mathrm{q}}{{\mathrm{c}}_1\left(1-\mathrm{p}-\mathrm{q}\right)}}{\mathrm{e}}_3^{*}\end{array}$$

(8)

6.1. LOE Analysis of Algorithm Engineers with Different Ways of Working

Conclusion 1: The optimal LOE on the three indicators (time, difficulty coefficient, and quality) of algorithm engineers is positively related to the engineer’s own technical competence and negatively to the change rate of marginal effort costs for each indicator. In order to calculate the solution of the above model Equation (5), we can obtain $\frac{\partial {\mathrm{e}}_{\mathrm{i}}^{*}}{\partial \mathrm{A}}>0$,$\frac{\partial {\mathrm{e}}_{\mathrm{i}}^{*}}{\partial {\mathrm{c}}_{\mathrm{i}}}<0$. This means that, as the algorithm engineer’s own technical competence enhances, the engineer’s efforts will be better transferred into revenue. When the change rate of algorithm engineers’ marginal effort cost in the three indicators (time, difficulty coefficient, and quality) becomes higher—that is, the algorithm engineer’s effort cost for the same job becomes higher—it will affect the algorithm engineer’s work enthusiasm and motivation, and the LOE at work will also decrease accordingly.

Conclusion 2: The optimal LOE of the three indicators is determined by the relative importance of the task and the relative change rate of the marginal cost. The more important the task is, the lower the change rate of the relative marginal cost, and the greater LOE put into it by the algorithm engineer.

By calculating the above Equation (8), we obtain ${\mathrm{e}}_1^{*}=\sqrt{\frac{{\mathrm{c}}_3\mathrm{p}}{{\mathrm{c}}_1\left(1-\mathrm{p}-\mathrm{q}\right)}}{\mathrm{e}}_3^{*}$,${ \mathrm{e}}_2^{*}=\sqrt{\frac{{\mathrm{c}}_3\mathrm{q}}{{\mathrm{c}}_1\left(1-\mathrm{p}-\mathrm{q}\right)}}{\mathrm{e}}_3^{*}$. $\mathsf{\mu }=\sqrt{\frac{{\mathrm{c}}_3\mathrm{p}}{{\mathrm{c}}_1\left(1-\mathrm{p}-\mathrm{q}\right)}}$ and $\mathsf{\omega }=\sqrt{\frac{{\mathrm{c}}_3\mathrm{p}}{{\mathrm{c}}_2\left(1-\mathrm{p}-\mathrm{q}\right)}}$, $\mathsf{\mu }\left(\mathrm{p},\mathrm{q},{\mathrm{c}}_1,{\mathrm{c}}_2,{\mathrm{c}}_3\right)$ and $\mathsf{\omega }\left(\mathrm{p},\mathrm{q},{\mathrm{c}}_1,{\mathrm{c}}_2,{\mathrm{c}}_3\right)$ can be regarded as two-variable functions of $\mathrm{p}$ and $\mathrm{q}$ when given $\left({\mathrm{c}}_1,{\mathrm{c}}_2,{\mathrm{c}}_3\right)$ a certain value. Letting ${\mathrm{c}}_1={\mathrm{c}}_3$ and ${\mathrm{c}}_2={\mathrm{c}}_3$, the graphs are shown as below:

Figure 3 shows that, when p is gradually close to 0—that is, when the importance of timely completion becomes lower—the algorithm engineer’s LOE in completing tasks on time is also low. With the increase of p and invariability of q (or vice versa), $\mathsf{\mu }$ is also increasing; that is, completing the company project on time is more important than ensuring the quality of the project. Under this circumstance, the algorithm engineer’s effort to complete the project on time will gradually increase. When p increases to close to 1, it means that completing the project on time is very important. Meanwhile, the LOE on project quality and difficulty coefficient will be significantly reduced, which will eventually lead to a situation in which quality is ignored to complete the project on time. The actual survey revealed that, when there is a temporary emergency situation at the company and multiple projects must be completed within a certain period, algorithm engineers often tend to choose projects with lower difficulty. At the same time, when new projects must be completed in a very short time, the completion result is often unsatisfactory and includes many loopholes.

Figure 4 shows that, when q becomes gradually close to 0—that is, when the company does not care about the difficulty coefficient of the projects selected by the algorithm engineer—the algorithm engineer will put less effort into it, tending to choose the simpler projects. When q gradually increases to close to 1, it means that the company attaches great importance to the difficulty coefficient of the project selected by the algorithm engineer, so algorithm engineers will tend to select the project with a higher difficulty coefficient. At the same time, the LOE on project completion time and quality will be significantly reduced. In the actual survey, there will be cases in which algorithm engineers significantly increase the completion time and reduce the completion quality to complete projects with higher difficulty coefficients. What is more, there will be cases in which the algorithm engineer may not be able to complete the project because of its high difficulty coefficient.

When ${\mathrm{c}}_1\ne {\mathrm{c}}_3$ and ${\mathrm{c}}_2\ne {\mathrm{c}}_3$, the charts expand or contract on the vertical axis in a ratio of $\sqrt{\frac{{\mathrm{c}}_3}{{\mathrm{c}}_1}}$ and $\sqrt{\frac{{\mathrm{c}}_3}{{\mathrm{c}}_1}}$, respectively. The smaller the ratio, the lower the algorithm engineer’s LOE in project completion time and difficulty coefficient compared to quality. Research has revealed that, when Company K demands high completion quality of the projects, algorithm engineers’ LOE in time and difficulty coefficient are relatively low, which leads to a phenomenon in which, when the time spent on the project is increased, algorithm engineers are more willing to complete projects with lower difficulty coefficients. This conforms to the model analysis in this study.

In summary, when measuring the completion of an algorithm engineer’s project with the three indicators (time, difficulty coefficient, and quality) in the model assumed in this study, it is difficult to meet the three requirements at the same time. Therefore, a strategy that can be adopted is to hire more skilled algorithm engineers or provide technical training to the existing algorithm engineers in order to improve their technical competence when the company has a high demand for different project indicators. To be more specific, when time is short, the company demands fast completion with a guarantee of high quality, or, when the project has a high difficulty coefficient, the company demands completion both on time and with a guarantee of high quality.

6.2. Analysis of the Optimal Incentive Coefficient

Conclusion 3: The optimal incentive coefficient is correlated positively with the algorithm engineer’s own competence and negatively with the risk aversion coefficient and external uncertainty. The formula of the optimal incentive coefficient is ${ \mathsf{\beta }}^{*}=\frac{1}{1+{\mathsf{\rho }\mathsf{\delta }}^2{\mathrm{A}}^{-2}{\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{\mathrm{p}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{\mathrm{q}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{1-\mathrm{p}-\mathrm{q}}}$. For the convenience of calculation, we assume H = ${\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{\mathrm{p}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{\mathrm{q}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{1-\mathrm{p}-\mathrm{q}}$ and derivate $\mathrm{E}, \mathsf{\rho },{ \mathrm{and} \mathsf{\delta }}^2,$ respectively, in the following functions:

$$\{\begin{array}{c}\frac{\partial {\mathsf{\beta }}^{*}}{\partial \mathsf{\rho }}=-\frac{{\mathsf{\delta }}^2{\mathrm{A}}^{-2}\mathrm{H}}{\left(1+{\mathsf{\rho }\mathsf{\delta }}^2{\mathrm{A}}^{-2}\mathrm{H}\right)}\\ \frac{\partial {\mathsf{\beta }}^{*}}{\partial {\mathsf{\delta }}^2}=-\frac{{\mathsf{\rho }\mathrm{A}}^{-2}\mathrm{H}}{\left(1+{\mathsf{\rho }\mathsf{\delta }}^2{\mathrm{A}}^{-2}\mathrm{H}\right)}\\ \frac{\partial {\mathsf{\beta }}^{*}}{\partial \mathrm{A}}=\frac{2{\mathsf{\rho }\mathsf{\delta }}^2\mathrm{H}}{\left(1+{\mathsf{\rho }\mathsf{\delta }}^2{\mathrm{A}}^{-2}\mathrm{H}\right)}\end{array}$$

From the above formula, we obtain $\frac{\partial {\mathsf{\beta }}^{*}}{\partial \mathsf{\rho }}<0, \frac{\partial {\mathsf{\beta }}^{*}}{\partial {\mathsf{\delta }}^2}<0,\frac{\partial {\mathsf{\beta }}^{*}}{\partial \mathrm{A}}>0$.

Based on the analysis of the above formula, the optimal incentive coefficient is an increasing function corelated to the algorithm engineer’s own competence, as well as a decreasing function corelated to the algorithm engineer’s absolute risk aversion coefficient and the uncertainty of the external environment. As for a certain incentive intensity, the greater $\mathsf{\beta }, \mathsf{\rho },{ \mathrm{and} \mathsf{\delta }}^2$ are, the higher the risk cost of the company’s algorithm engineer. In order to reduce the risk cost, the company demands the smaller optimal incentive coefficient $\mathsf{\beta }$. The higher the marginal productivity of company algorithm engineers, the greater the optimal incentive coefficient $\mathsf{\beta }$ required to better motivate them.

The optimal incentive coefficient is a decreasing function of the absolute risk aversion coefficient, which means that, as for algorithm engineers with different risk preferences, the company should adopt different incentive mechanisms. If an algorithm engineer is risk-averse, the engineer tends to accept a stable income, so the incentive coefficient is small. If the algorithm engineer’s risk tolerance is high, the engineer should be motivated. Our survey revealed that new employees or employees with low technical competence are generally risk-averse. They often choose less difficult projects to earn points, while algorithm engineers who have certain work experience and a clearer understanding of the risks tend to take on tasks with greater difficulty.

The negative correlation between the optimal incentive coefficient and the external environment uncertainties means that as an external environment is more uncertain, it will become more difficult to determine whether the company revenue comes from the algorithm engineer’s efforts or from the impact of the random change of the external environment. Thus, the information asymmetry between companies and the algorithm engineer is increased. In short, high company revenue is not necessarily due to the efforts of algorithm engineers, and low company revenue is not necessarily due to the lack of effort by algorithm engineers, since it is closely related to changes in the overall environment. In this case, it is difficult to motivate algorithm engineers to work harder by increasing the incentive coefficient. However, in Company K, the degree of information asymmetry is low, and the external environment has less influence on the optimal incentive coefficient.

The optimal incentive coefficient is an increasing function of the employee’s own competence, which means that the stronger the algorithm engineer’s own competence and, at the same level, the greater the revenue brought in, the harder the algorithm engineer will work, and the corresponding incentive intensity will also increase accordingly.

Algorithm engineers’ risk preference is often positively related to their own technical level. The more experienced and the higher the technical level, the stronger the risk tolerance. Therefore, the higher the overall quality of the algorithm engineer, the greater the incentive intensity should be.

At present, the points system adopted by Company K is based on the projects completed by algorithm engineers. Completing projects with low difficulty coefficients will earn fewer points, and completing projects with high difficulty coefficients will earn more points. Ultimately, salary increases and promotion will be determined by the points earned through project completion. As a consequence, this incentive method will cause each algorithm engineer to tend to complete simpler projects in order to earn points rather than participate in projects with higher difficulty coefficients. Based on the above analysis, Company K should adopt different incentive methods for algorithm engineers with different technical levels. For employees with high technical levels and more years working in the company, the company should further increase the incentive intensity for completing projects with larger difficulty coefficients. For new employees or employees with low technical level, the company should maintain the original incentive intensity (namely, evaluate the algorithm engineer based on the result of the project completion).

Based on the findings in this study, Internet companies should motivate algorithm engineers of different levels hierarchically. For algorithm engineers with relatively weak competence, Internet companies should maintain the original incentive system, while, for algorithm engineers with relatively high competence, Internet companies should increase their income for completing projects with high difficulty coefficients. This would encourage them to create greater value of work to achieve a win-win situation between the company and themselves. At the same time, for algorithm engineers who choose to complete projects with high difficulty coefficients, but cannot complete the projects, Internet companies should evaluate their final completion results using two methods. One should be to give more time to complete the project, but reduce the incentive intensity; the other is to provide incentives based on completion. In this case, the incentive intensity may be higher than the incentives for choosing simple projects, or it may be lower than the incentives for completing simple projects. This helps the company to create an atmosphere where everyone is willing to solve more difficult projects.

Conclusion 4: Under the same conditions, at the extreme point, the sum of higher change rate of marginal effort cost in time, difficulty coefficient, and quality leads to a lower optimal incentive coefficient and vice versa.

Since ${\mathsf{\beta }}^{*}=\frac{1}{1+{\mathsf{\rho }\mathsf{\delta }}^2{\mathrm{A}}^{-2}{\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{\mathrm{p}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{\mathrm{q}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{1-\mathrm{p}-\mathrm{q}}}$ and H=${\left(\frac{{\mathrm{c}}_1}{\mathrm{q}}\right)}^{\mathrm{q}}{\left(\frac{{\mathrm{c}}_2}{1-\mathrm{q}}\right)}^{1-\mathrm{q}}$, the larger H is, the smaller the optimal incentive coefficient is. $\mathrm{H}(\mathrm{p},\mathrm{q},{\mathrm{c}}_1,{\mathrm{c}}_2,{\mathrm{c}}_3)$ is a power-exponential function of $\mathrm{q}$, while the variable $({\mathrm{c}}_1,{\mathrm{c}}_2,{\mathrm{c}}_3)$ is the parameter. Deriving the function H, we can obtain the following:

$$\{\begin{array}{c}{\mathrm{H}}_1=\frac{\partial \mathrm{H}}{\partial \mathrm{p}}={\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{\mathrm{p}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{\mathrm{q}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{1-\mathrm{p}-\mathrm{q}}\ln \frac{{\mathrm{c}}_1\left(1-\mathrm{p}-\mathrm{q}\right)}{{\mathrm{c}}_3\mathrm{p}}\\ {\mathrm{H}}_2=\frac{\partial \mathrm{H}}{\partial \mathrm{q}}={\left(\frac{{\mathrm{c}}_1}{\mathrm{p}}\right)}^{\mathrm{p}}{\left(\frac{{\mathrm{c}}_2}{\mathrm{q}}\right)}^{\mathrm{q}}{\left(\frac{{\mathrm{c}}_3}{1-\mathrm{p}-\mathrm{q}}\right)}^{1-\mathrm{p}-\mathrm{q}}\ln \frac{{\mathrm{c}}_2\left(1-\mathrm{p}-\mathrm{q}\right)}{{\mathrm{c}}_3\mathrm{q}}\end{array}$$

Letting the first-order derivative be 0, the extreme point of H can be obtained. $({\mathrm{p}}^{*},{\mathrm{q}}^{*})=\left(\frac{{\mathrm{c}}_1}{{\mathrm{c}}_1+{\mathrm{c}}_2{\mathrm{c}}_3},\frac{{\mathrm{c}}_2}{{\mathrm{c}}_1+{\mathrm{c}}_2{\mathrm{c}}_3}\right).$ Then, we solve the second-order derivative of H. Based on the solution of the extreme value of the two-variable functions, $({\mathrm{p}}^{*},{\mathrm{q}}^{*})$ is the maximum point of H, and the maximum value of H is ${\mathrm{H}}^{*}={\mathrm{c}}_1+{\mathrm{c}}_2+{\mathrm{c}}_3$. The optimal incentive coefficient ${\mathsf{\beta }}^{*}$ is a decreasing function of H, so, when H obtains the maximum value, ${\mathsf{\beta }}^{*}$ takes on a minimum value. Based on the above analysis, at the extreme point, with a higher sum of the marginal effort cost of the three indicators (time, difficulty coefficient, and quality), the algorithm engineer’s effort cost will rise sharply. In this case, Internet companies should adopt a weaker incentive intensity.

Conclusion 5: When the external influences and algorithm engineers’ risk preferences are determined, and when the relative importance of time, quality, and difficulty coefficient is close to the marginal effort cost change ratio of the three indicators to the total effort cost change rate, the optimal incentive intensity is relatively weak. By calculation, the maximum point of H is $({\mathrm{p}}^{*},{\mathrm{q}}^{*})=\left(\frac{{\mathrm{c}}_1}{{\mathrm{c}}_1+{\mathrm{c}}_2+{\mathrm{c}}_3},\frac{{\mathrm{c}}_2}{{\mathrm{c}}_1+{\mathrm{c}}_2+{\mathrm{c}}_3}\right)$; i.e., the relative importance of time, difficulty coefficient, and quality equals the ratio of marginal effort cost change rate of the three indicators to the total effort cost change rate. H obtains the maximum value and has a very small optimal incentive coefficient.

As an example, Figure 5, Figure 6 and Figure 7 show the function images when ${\mathrm{c}}_1={\mathrm{c}}_2={\mathrm{c}}_3=1;$ ${\mathrm{c}}_1=1,{\mathrm{c}}_2=2,{\mathrm{c}}_3=3;$ and ${\mathrm{c}}_1=1,{\mathrm{c}}_2=3,{\mathrm{c}}_3=2$.

Based on the analysis of the above, the optimal incentive coefficient ${\mathsf{\beta }}^{*}$ is a decreasing function of H, so, when H obtains the maximum value, ${\mathsf{\beta }}^{*}$ takes on a minimum value. As is shown in the figure, around the extreme point of H, a ‘weak incentive region’ is formed. In this region, given a higher H value of the algorithm engineer’s absolute risk aversion and external environmental uncertainty, the optimal incentive coefficient ${\mathsf{\beta }}^{*}$ will be smaller.

According to the above conclusion, there are different combinations to measure the importance of algorithm engineers’ project indicators; thus, the optimal incentive coefficient’s weak incentive region will continuously increase. For example, when the relative importance of the project is high and the difficulty coefficient is low, the algorithm engineer is willing to put in more effort to complete the task and ultimately achieve the effect of ‘fast and good’. As for algorithm engineers, they should negotiate with the company to improve on a certain indicator, which is helpful for the algorithm engineer.

Based on comprehensive analyses of the three conclusions related to the optimal incentive coefficient, the optimal incentive coefficient is mainly related to two factors. One is the algorithm engineer’s own factors, including the engineer’s own comprehensive quality, risk aversion coefficient, and marginal effort cost change rate; the other is company-related factors, including the company’s emphasis on different project indicators and the uncertainty of the external environment. Therefore, in the actual incentive mechanism design of Internet companies, it is necessary to consider the situation, not only of one party, but of both the company and the algorithm engineer. By analyzing the correlation between the parameters of the two, the optimal incentive coefficient that satisfies both the company’s project requirements and the algorithm engineer’s income improvement can be determined. The incentive balance of three dimensions of algorithm project is helpful to realize the sustainable incentive for algorithm engineers.

7. Conclusions

With reference to the data of Company K, this study constructs an incentive model for Internet companies and algorithm engineers based on multi-task principal–agent theory, intended, on the basis of this, to deepen the understanding of sustainable management in multi-task projects. According to the survey results, project evaluations were divided into three indicators: completion time, difficulty coefficient, and quality. The study adopted an expanding-type Cobb-Douglas production function to express the multi-task project output of efforts in time, difficulty coefficient, and quality, and provided an overall combined incentive strategy from the perspective of multi-objective coordination. The conclusion of this study is to contribute to the research of multi-tasking principal–agent and sustainable performance management.

7.1. Theoretical Contribution

This study contributes to the research of multi-task principal-agent theory, expanding the hypothesis that multitasking can be interchangeable. The existing literature, while discussing multi-objective incentive problems, has adopted the linear weighting approach. Although this simplistic approach helps to multitask due to the importance of sorting, it ignores that the task cannot completely replace the reality. The linear model allows a utility of zero, but the output of the enterprise’s total utility is not zero. This assumption is not consistent with the principle of “one vote for code quality” in evaluating algorithm performance in practice. The research results show that algorithm engineers’ LOE on the three indicators (time, difficulty coefficient, and quality) is correlated positively with their own technical competence and negatively with their marginal effort cost change rate. The algorithm engineer’s relative LOE on the three indicators depends on the relative importance of the project and the relative marginal cost change rate; the more important the project, the lower the relative marginal effort cost change rate, and the greater the effort invested by the algorithm engineer. The research findings help algorithm engineers to evaluate projects according to their own situation, and to allocate time and resources reasonably.

This study contributes to the research of performance management in sustainable development and analyzes the factors that influence the optimal incentive coefficient to promote the sustainable development of the project. How to effectively integrate sustainability in project management systems is still an emerging field, and its role in supporting sustainability has not been well-studied within organizations (Durden, 2008; Herzig et al., 2012). In addition, sustainability performance management and control are not discussed in a structured way, but only in a detailed description of the concept (Schaltegger, 2011). There is a lack of research on how to make the project development sustainable and promote the sustainable contribution of employees through the incentive of project tasks in the process of project execution. Additionally, the company’s optimal incentive coefficient for algorithm engineers is affected by two factors. The first is the impact of the project itself and the external environment, mainly the importance of the project to the company and the uncertainty of the environment. The higher the project difficulty coefficient or the more complex the environment, the harder it is to measure the LOE of algorithm engineers. However, in Internet companies, the uncertainty is lower, so algorithm engineers should be given stronger incentives. The other factor is the algorithm engineers themselves, including their overall skill level, their risk aversion coefficient (related to their technical level), and the change rate of the marginal cost effort. For algorithm engineers with a higher comprehensive technical level, the company should adopt strong incentive intensity, and weaker incentive intensity should be adopted for algorithm engineers with a low comprehensive technical level. In order to improve the overall efficiency of the company, algorithm engineers should be regularly trained to improve their technical level. When the sum of marginal effort cost change rate raises, the optimal incentive intensity becomes weaker.

This study analyzes the optimal incentive coefficient for algorithm engineers. According to the factors affecting the optimal incentive coefficient, Internet companies can adopt different incentive methods for projects with different requirements and levels of importance, and adopt differentiated motivation according to the technical level differences among algorithm engineers. In this way, both the company revenue and the algorithm engineer’s income can increase, achieving sustainable performance management.

7.2. Practical Implications

For enterprise project management, selection of the appropriate incentive intensity and the enterprise and the algorithm engineer’s reasonable risk sharing is an important guarantee for the success of the project. In this paper, the influence factors of the optimal incentive coefficient are identified based on the difficulty of the enterprise, according to the project, time requirements and algorithm engineer characteristics. This allows them to choose a reasonable level of incentive intensity. Thus, algorithm engineers are encouraged to work hard to achieve a win-win situation.

Sustainable enterprise performance management, when achieving the Sustainable Development Goals (SDGS), can be achieved by setting incentives for different indicators. Enterprises can link the sustainability performance evaluation and the reward system. At the same time, a compensation committee and a sustainable development committee can be set up so that the key performance indicators of the project can be linked to the performance related to sustainable development. This allows for greater accountability for such issues, and for greater monitoring and control. In addition, KPIs must be regularly assessed and evaluated in order to ensure that they are aligned with key objectives. In addition, research to understand the impact of key performance indicators on behaviour and performance will contribute to more effective and efficient implementation of the SDGS.

7.3. Limitations

This study has several limitations. It only considers the one-time LOE choice of the algorithm engineers, so it is a static model. Additionally, in the course of a project, the effort of the algorithm engineer will change, so this study fails to dynamically optimize and control the algorithm engineer’ LOE. This study also assumes that the LOEs of the three selected measurement indicators are independent of each other, but, in fact, the effort levels among the three will affect each other. For example, the increase in time effort will increase the project quality effort cost. Therefore, the interrelationship and the influence between multi-task indicators of the same project require further study.