# The Strategy Selection in Financial Fraud and Audit Supervision: A Study Based on a Three-Party Evolutionary Game Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Financial Fraud

#### 2.2. Auditing Supervision

## 3. Evolutionary Game Model

#### 3.1. Hypothesis of the Evolutionary Game Model

_{1}is the hypothesis of participants in the model. The participants are bounded rational players, including listed internet companies, accounting firms, and audit regulators.

_{2}is the hypothesis of strategy selections for participants. First, listed internet companies have two strategies: financial fraud and non-fraud. Second, accounting firms choose two strategies: loose audit and strict audit. Third, the two strategic selections of audit regulators are active supervision and negative supervision.

_{3}is the hypothesis of model parameters (revenues and costs of three parties).

_{1}, the extra revenue obtained through financial fraud of listed internet companies is represented by k, and the rent-seeking cost of listed internet companies bribing accounting firms is written as c

_{1}.

_{1}and c

_{2}, respectively. In the loose audit, the extra revenue and audit cost of accounting firms are represented as c

_{1}and c

_{3}in turn.

_{1}. The fine imposed on accounting firms is p

_{2}. If accounting firms receive bribes from listed internet companies, audit regulators will confiscate these bribes and impose a fine on listed internet companies. So, the active supervision revenue of audit regulators equals c

_{1}+ p

_{1}+ p

_{2}. At this point, the active supervision cost is c

_{4}. Otherwise, it is 0. Let us assume that the following relation holds: p

_{2}> c

_{4}; c

_{1}+ p

_{1}> c

_{4}; c

_{1}+ p

_{1}+ p

_{2}> c

_{4}.

_{1}) paid by listed internet companies. An important reason for accounting firms to choose a loose audit strategy is to obtain additional benefit (c

_{1}), which is assumed to be much larger than r

_{1}. The parameters of the evolutionary game model are shown in Table 1.

#### 3.2. Payoff Matrix

#### 3.3. Replicator Dynamics Equation

#### 3.3.1. Replicator Dynamics Equation of Listed Internet Companies

#### 3.3.2. Replicator Dynamics Equation of Accounting Firms

#### 3.3.3. Replicator Dynamics Equation of Audit Regulators

#### 3.4. Stability Analysis of Evolutionary Game Model

## 4. Numerical Simulation Analysis

_{1}+ p

_{1}− k < 0, −c

_{1}− c

_{2}+ c

_{3}+ p

_{2}< 0, and c

_{4}− c

_{1}− p

_{1}− p

_{2}< 0.

#### 4.1. The Evolutionary Game Analysis of Three-Party Strategy Selections

- For the red line (x = 0.5), when t equals 0.9153, the probability of financial fraud of listed internet companies equals 1.
- For the blue line (y = 0.5), when t equals 0.3911, the probability of strict audits of accounting firms equals 0.
- For the green line (z = 0.5), when t equals 0.4732, the probability of active supervision of audit regulators equals 1.

#### 4.2. The Evolutionary Game Analysis of Listed Internet Companies

- For the red line (x = 0.5; y = 0.1; z = 0.1), when t is 0.8545, the probability of financial fraud of listed internet companies equals 1.
- For the green line (x = 0.5; y = 0.5; z = 0.1), when t is 0.7912, the probability of financial fraud of listed internet companies equals 1.
- For the blue line (x = 0.5; y = 0.1; z = 0.5), when t is 0.9916, the probability of financial fraud of listed internet companies equals 1.
- For the cyan line (x = 0.5; y = 0.8; z = 0.8), when t is 0.8541, the probability of financial fraud of listed internet companies equals 1.

_{1}). When k is 2, 5, 25, and 30, respectively, the strategy selection of listed internet companies is shown in Figure 4.

- For the red line (k = 2), when t is 0.6034, the probability of financial fraud of listed internet companies equals 0.
- For the green line (k = 5), when t is 0.8429, the probability of financial fraud of listed internet companies equals 0.
- For the blue line (k = 25), when t is 0.3886, the probability of financial fraud of listed internet companies equals 1.
- For the cyan line (k = 30), when t is 0.2731, the probability of financial fraud of listed internet companies equals 1.

_{1}) is 1, 5, 15, and 20, respectively, the strategy selection of listed internet companies is shown in Figure 5.

- For the red line (p
_{1}= 1), when t is 0.5360, the probability of financial fraud of listed internet companies equals 1. - For the green line (p
_{1}= 5), when t is 0.9153, the probability of financial fraud of listed internet companies equals 1. - For the blue line (p
_{1}= 15), when t is 1.3445, the probability of financial fraud of listed internet companies equals 0. - For the cyan line (p
_{1}= 20), when t is 0.6917, the probability of financial fraud of listed internet companies equals 0.

_{1}. Otherwise, it converges to 1. Specifically, when p

_{1}decreases from 5 to 1, the convergence speed of the probability gradually accelerates to 1. However, the probability of financial fraud of listed internet companies approaches 0 at a fast convergence rate, when p

_{1}increases from 15 to 20. Therefore, the smaller fine for financial fraud found by audit regulators promotes listed internet companies to choose a financial fraud strategy. Comparing Figure 4 and Figure 5, we can see that listed internet companies choose a financial fraud strategy under the higher extra revenue (k) and the smaller fine (p

_{1}).

#### 4.3. The Evolutionary Game Analysis of Accounting Firms

- For the red line (x = 0.1; y = 0.5; z = 0.1), when t is 0.4011, the probability of strict audits of accounting firms equals 0.
- For the green line (x = 0.5; y = 0.5; z = 0.1), when t is 0.3553, the probability of strict audits of accounting firms equals 0.
- For the blue line (x = 0.1; y = 0.5; z = 0.5), when t is 0.4605, the probability of strict audits of accounting firms equals 0.
- For the cyan line (x = 0.8; y = 0.5; z = 0.8), when t is 0.3756, the probability of strict audits of accounting firms equals 0.

_{2}) and the fine after loose audits found by audit regulators (p

_{2}). When c

_{2}is 2, 4, 8, and 12, respectively, the strategy selection of accounting firms is shown in Figure 7.

- For the red line (c
_{2}= 2), when t is 1.1391, the probability of strict audits of accounting firms equals 0. - For the green line (c
_{2}= 4), when t is 0.8338, the probability of strict audits of accounting firms equals 0. - For the blue line (c
_{2}= 8), when t is 0.5336, the probability of strict audits of accounting firms equals 0. - For the cyan line (c
_{2}= 12), when t is 0.3911, the probability of strict audits of accounting firms equals 0.

_{2}. However, it should be noted that the change in c

_{2}changes the evolution time of the model. When the strict audit cost (c

_{2}) is larger, the convergence speed is faster and the simulation time is shorter. For example, when c

_{2}increases from 2 to 12, accounting firms spend less time choosing a loose audit strategy.

_{2}) is 4, 9, 25, and 35, respectively, the strategy selection of accounting firms is shown in Figure 8.

- For the red line (p
_{2}= 4), when t is 0.4144, the probability of strict audits of accounting firms equals 0. - For the green line (p
_{2}= 9), when t is 0.6196, the probability of strict audits of accounting firms equals 0. - For the blue line (p
_{2}= 25), when t is 0.8527, the probability of strict audits of accounting firms equals 1. - For the cyan line (p
_{2}= 35), when t is 0.3627, the probability of strict audits of accounting firms equals 1.

_{2}= 4 and p

_{2}= 9), and the probability of accounting firms choosing a strict audit strategy approaches 0. However, accounting firms are inclined to choose a strict audit strategy with the increase in p

_{2}. When p

_{2}increases from 25 to 35, the evolution time (t) is shortened from 0.8527 to 0.3627. Thus, a larger fine after loose audits found by audit regulators (p

_{2}) prompts accounting firms to choose a strict audit strategy. Otherwise, they make a loose audit strategy with a small p

_{2}. Comparing Figure 7 and Figure 8, it can be seen that the change in the strict audit cost (c

_{2}) does not change the strategy selection of accounting firms, but a larger fine after loose audits found by audit regulators (p

_{2}) prompts accounting firms to create a strict audit strategy.

#### 4.4. The Evolutionary Game Analysis of Audit Regulators

- For the red line (x = 0.1, y = 0.1, z = 0.5), when t is 0.6902, the probability of active supervision of audit regulators equals 1.
- For the green line (x = 0.5, y = 0.1, z = 0.5), when t is 0.4835, the probability of active supervision of audit regulators equals 1.
- For the blue line (x = 0.1, y = 0.5, z = 0.5), when t is 0.6508, the probability of active supervision of audit regulators equals 1.
- For the cyan line (x = 0.8, y = 0.8, z = 0.5), when t is 0.4272, the probability of active supervision of audit regulators equals 1.

_{4}). When c

_{4}is 1, 3, 20, and 25, respectively, the simulation results of the strategy selection of audit regulators are shown in Figure 10.

- For the red line (c
_{4}= 1), when t is 0.4341, the probability of active supervision of audit regulators equals 1. - For the green line (c
_{4}= 3), when t is 0.5119, the probability of active supervision of audit regulators equals 1. - For the blue line (c
_{4}= 20), when t is 1.0748, the probability of active supervision of audit regulators equals 0. - For the cyan line (c
_{4}= 25), when t is 0.4856, the probability of active supervision of audit regulators equals 0.

_{4}, the probability of active supervision of audit regulators converges to 0 or 1. Specifically, when the active supervision cost is larger (such as c

_{4}= 20 and c

_{4}= 25), audit regulators will choose a negative supervision strategy, and the probability of active supervision converges to 0. When the active supervision cost (c

_{4}) increases from 20 to 25, the evolution time (t) is shortened from 1.0748 to 0.4856. When the active supervision cost (c

_{4}) gradually decreases, the probability of active supervision of audit regulators gradually increases. For example, when the active supervision cost takes a smaller value (such as c

_{4}= 1 and c

_{4}= 3), audit regulators choose an active supervision strategy, and the probability of active supervision converges to 1. If c

_{4}is reduced from 3 to 1, the evolution time (t) will be shortened from 0.5119 to 0.4341.

_{4}) directly affects the strategy selection of audit regulators. When the cost of active supervision is higher, audit regulators conduct negative supervision. Conversely, the lower supervision cost promotes them to conduct active supervision.

## 5. Discussion

_{1}), the audit cost in strict audits of accounting firms (c

_{2}), the fine after loose audits found by audit regulators (p

_{2}), and the active supervision cost of audit regulators (c

_{4}). The previous studies demonstrated that the primary motivation for listed internet companies to commit financial fraud was to obtain higher revenues [8], and the strict audit was conducive to preventing financial fraud [9,16]. These conclusions are consistent with our findings. The larger extra revenue from financial fraud (k) prompts listed internet companies to choose a financial fraud strategy in less time. Moreover, the larger fine after loose audits found by audit regulators (p

_{2}) enables accounting firms to apply a strict audit strategy to prevent the financial fraud of listed internet companies. Through numerical simulation analyses, we discuss the dynamic evolution mechanism of three-party strategy selections. This paper expands the existing research framework of financial fraud and audit supervision, and builds an evolutionary game model based on listed internet companies, accounting firms, and audit regulators. The research methods and conclusions complement and enrich the existing literature.

_{3}, which involves the revenues and costs of the three parties. Although the expected revenue function and the replicator dynamics equation are the critical factors in finding the stable equilibrium point, we only consider the model parameters that affect three-party strategy selections. Some factors that affect the revenues or costs of one party and do not affect the strategic selections of other parties are not considered in the evolutionary game model. Therefore, our research framework depends on model assumptions and parameter settings. Under different market environments and regulatory systems, the stable equilibrium strategies of the three parties are probably different. The future research direction can be summarized as follows. First, the model parameters should be optimized based on the analysis of financial fraud cases. Many cases of financial fraud are affected by the revenues and costs of all parties. With the changes in revenues and costs, the strategic selections of listed internet companies, accounting firms, and audit regulators change accordingly. Through a practical case, the model parameters can be further revised and the three-party evolutionary game model can be further optimized. Second, an early warning mechanism for financial fraud of listed internet companies should be established. We find that the initial probability of the strategy selection does not affect the final strategy of the three parties. However, other model parameters related to the revenues and costs can directly affect the final strategies of the three parties. Thus, we should further analyze the influence of different model parameters on the probabilities of three-party strategy selections through many numerical simulations and create warning plans for the financial fraud of listed internet companies in advance.

## 6. Conclusions

_{1}). Afterward, we analyze the strategy selection of accounting firms under the impact of the audit cost in strict audits (c

_{2}) and the fine after loose audits found by audit regulators (p

_{2}). In the end, we study the strategy selection of audit regulators under the impact of the active supervision cost (c

_{4}). The conclusions are summarized as follows.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Players | Parameter | Description |
---|---|---|

Listed internet companies | e_{1} | The revenue of listed internet companies under normal operation (the default is 0) |

k | The extra revenue from financial fraud | |

c_{1} | The rent-seeking cost of listed internet companies bribing accounting firms | |

p_{1} | The fine after the financial fraud is discovered by audit regulators | |

Accounting firms | r_{1} | The normal revenue from strict audits |

c_{1} | The extra revenue from loose audits (that is the bribe from listed internet companies) | |

c_{2} | The audit cost in strict audits | |

c_{3} | The audit cost in loose audits | |

p_{2} | The fine after loose audits is found by audit regulators | |

Audit regulators | c_{4} | The active supervision cost (the negative supervision cost defaults to 0) |

c_{1}+ p_{1}+ p_{2} | The active supervision revenue |

Three Parties | Accounting Firms | Audit Regulators | ||
---|---|---|---|---|

Active Supervision (z) | Negative Supervision (1 − z) | |||

Listed internet companies | Non-fraud (1 − x) | Strict audit (y) | (0, r_{1} − c_{2}, −c_{4}) | (0, r_{1} − c_{2}, 0) |

Loose audit (1 − y) | (0, r_{1} − c_{3} − p_{2}, p_{2} − c_{4}) | (0, r_{1} − c_{3}, 0) | ||

Financial Fraud (x) | Strict audit (y) | (k − p_{1}, r_{1} − c_{2}, c_{1} + p_{1} − c_{4}) | (k, r_{1} − c_{2}, 0) | |

Loose audit (1 − y) | (k − c_{1} − p_{1}, r_{1} + c_{1} − c_{3} − p_{2}, c_{1} + p_{1} + p_{2} − c_{4}) | (k − c_{1}, r_{1} + c_{1} − c_{3}, 0) |

Equilibrium Point | Eigenvalues $\left({\mathit{\lambda}}_{1},{\mathit{\lambda}}_{2},{\mathit{\lambda}}_{3}\right)$ | Symbols of $\left({\mathit{\lambda}}_{1},{\mathit{\lambda}}_{2},{\mathit{\lambda}}_{3}\right)$ | Stability |
---|---|---|---|

C(0, 0, 1) | (k − p_{1} − c_{1}, c_{3} − c_{2} + p_{2}, c_{4} − p_{2}) | $-,+,-$ | Instability |

D(0, 1, 1) | (k − p_{1}, − c_{3} + c_{2} − p_{2}, c_{4}) | $-,+,+$ | Instability |

F(1, 0, 1) | (c_{1} + p_{1} − k, − c_{1} − c_{2} + c_{3} + p_{2}, c_{4} − c_{1} − p_{1} − p_{2}) | $-,-,-$ | ESS |

G(1, 1, 1) | (p_{1} − k, c_{1} + c_{2} − c_{3} − p_{2}, c_{4} − c_{1} − p_{1}) | $-,+,-$ | Instability |

Parameters | k | p_{1} | p_{2} | c_{1} | c_{2} | c_{3} | c_{4} |
---|---|---|---|---|---|---|---|

Initial values | 18 | 5 | 3 | 8 | 12 | 2 | 2 |

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**MDPI and ACS Style**

Wu, B.; Yang, J.; Fu, G.; Zhang, M.
The Strategy Selection in Financial Fraud and Audit Supervision: A Study Based on a Three-Party Evolutionary Game Model. *Systems* **2022**, *10*, 173.
https://doi.org/10.3390/systems10050173

**AMA Style**

Wu B, Yang J, Fu G, Zhang M.
The Strategy Selection in Financial Fraud and Audit Supervision: A Study Based on a Three-Party Evolutionary Game Model. *Systems*. 2022; 10(5):173.
https://doi.org/10.3390/systems10050173

**Chicago/Turabian Style**

Wu, Binghui, Jing Yang, Guanhao Fu, and Mengjiao Zhang.
2022. "The Strategy Selection in Financial Fraud and Audit Supervision: A Study Based on a Three-Party Evolutionary Game Model" *Systems* 10, no. 5: 173.
https://doi.org/10.3390/systems10050173