# Dynamic Game Analysis on Cooperative Advertising Strategy in a Manufacturer-Led Supply Chain with Risk Aversion

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## Abstract

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## 1. Introduction

## 2. Literature Review

#### 2.1. Cooperative Advertising in Supply Chain

#### 2.2. Risk Aversion in Supply Chain

#### 2.3. Complexity in Supply Chain Game

## 3. Supply Chain Model

#### 3.1. Model Description

#### 3.2. Model Assumption

- (1)
- The unit production cost of the manufacturer and unit handling cost of the retailer incurred in addition to purchasing cost are assumed to be equal to zero.
- (2)
- Cooperative advertising is divided into the manufacturer’s national advertising expenditure ${A}_{m}$ and the retailer’s local advertising Due to different formula editors, some formula formats are not uniform. We will adjust the format of this part in later stage of production. ${A}_{r}$, which are decision variables for the supply chain participants.
- (3)
- In this model, the manufacturer and retailer are risk-averse, which revenues are represented by exponential utility functions and follow a normal distribution.

#### 3.3. Model Construction

#### 3.4. The Stability of the System

## 4. Dynamics Analysis and Numerical Simulation

#### 4.1. The Dynamics with Respect to Adjustment Speeds ${\alpha}_{1}$ and ${\alpha}_{2}$

#### 4.2. The Influence of the Manufacturer Participation Rate $\mu $

#### 4.3. The Influence of Risk Tolerance Levels ${\lambda}_{1}$ and ${\lambda}_{2}$

#### 4.4. The Influence of Advertising Expenditure Effect Coefficients ${k}_{m}$ and ${k}_{r}$

## 5. Chaos Control

## 6. Contribution and Implication

- (1)
- If the enterprises keep adjustment speeds within the stability region, no matter what initial advertising expenditures are chosen by the enterprises, the market will eventually reach the Nash equilibrium after a series of games. As a result, the enterprises can obtain optimal strategies. Otherwise, if the adjustment speeds of advertising expenditure are too fast, the system loses stability and enters chaotic states, which lead to a rapid decrease in expected utilities. The manufacturer and retailer will control their adjustment speeds to maintain the stability of the supply chain market for the maximum revenue.
- (2)
- The stability range of the system expands with an increase in the participation rate of local advertising expenditure by the manufacturer. As the manufacturer increases the participation rate of local advertising, the retailer spends more on local advertising expenditures. However, the national advertising expenditure of the manufacturer only has a slight downward adjustment with the increase in the participation rate. The expected utility of the manufacturer decreases, while the retailer increases slightly on the whole by an increase in the participation rate. For market stability and maximizing revenue, the manufacturer should adjust the participation rate appropriately, avoiding too high or too low values.
- (3)
- The stable region has no variation with risk tolerance levels of the manufacturer and retailer. The risk tolerance level of the manufacturer has no effect on the national advertising expenditure by the manufacturer and local advertising expenditure by the retailer. However, the manufacturer’s national advertising expenditure rises and the retailer’s local advertising expenditure reduces with an increase in the risk tolerance level of the retailer when the system is in a stable state. The expected utility of the manufacturer decreases and the expected utility of retailer remains unchanged with an increase in the risk tolerance level of the manufacturer. The expected utility of the manufacturer rises and the expected utility of the retailer decreases with an increase in the risk tolerance level of the retailer. Hence, the manufacturer will try to reduce their own risk tolerance level for economic revenue. In order to make neither too much advertising expenditure nor too little expected utility, the retailer will appropriately adjust the risk tolerance level to adapt to their own development according to their own enterprise strategy in the supply chain market competition.
- (4)
- With increasing effect coefficients of advertising expenditure, the stable region remains unchanged. The advertising expenditures and expected utilities rise with an increase in effect coefficients of advertising expenditure. The increase in its effect coefficient has a greater impact on its own advertising investment. In order to obtain more expected utilities and less advertising expenditures, both the manufacturer and retailer will reduce their own effect coefficients of advertising expenditure; meanwhile, they will attempt to increase their opponent’s effect coefficients to gain the most revenue.
- (5)
- It is profitable to keep the relevant factors in a certain range for manufacturer and retailer. Once the system falls into chaos, the chaos phenomenon can be delayed or eliminated effectively by the parameters control method.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**The expected utilities and the average expected utilities with respect to ${\alpha}_{1}$ and ${\alpha}_{2}$.

**Figure 8.**The 2D bifurcation diagrams with respect to ${\alpha}_{1}$ and ${\alpha}_{2}$ with different $\mu $.

**Figure 9.**The Bifurcation diagram of ${A}_{m}$ and ${A}_{\mathrm{r}}$ with respect to $\mu $ with different ${\alpha}_{1}$ and ${\alpha}_{2}$.

**Figure 10.**The expected utilities with respect to $\mu $ with ${\alpha}_{1}=3.5$ and ${\alpha}_{2}=3.5$.

**Figure 11.**The 2D bifurcation diagram with respect to ${\alpha}_{1}$ and ${\alpha}_{2}$ with different ${\lambda}_{1}$ and ${\lambda}_{2}$.

**Figure 12.**The 3D bifurcation diagrams of ${A}_{m}$ and ${A}_{\mathrm{r}}$ with respect to ${\alpha}_{1}$ with (

**a**,

**b**) ${\lambda}_{1}=1,2,3,4$ and (

**c**,

**d**) ${\lambda}_{2}=2,3,4,5$.

**Figure 13.**Nash equilibrium points and the expected utilities with ${\alpha}_{1}=3.5{\alpha}_{2}=3.5$.

**Figure 14.**The 2D bifurcation diagram with respect to ${\alpha}_{1}$ and ${\alpha}_{2}$ with different ${k}_{m}$ and ${k}_{r}$.

**Figure 15.**The 3D bifurcation diagrams of ${A}_{m}$ and ${A}_{\mathrm{r}}$ with respect to ${\alpha}_{1}$ with ${\alpha}_{2}=3.5$.

**Figure 16.**Nash equilibrium points and the expected utilities with ${\alpha}_{1}=3.5{\alpha}_{2}=3.5$.

**Figure 17.**The bifurcation diagram of ${A}_{m}$ and ${A}_{\mathrm{r}}$ and the largest Lyapunov exponent with respect to $m$.

**Figure 18.**The 2D bifurcation diagrams with respect to ${\alpha}_{1}$ and ${\alpha}_{2}$ with different $m$.

Notation | Explanation |
---|---|

$p$ | Price of retailer, $p>w>0$ |

$v$ | Baseline demand, $v>0$ |

${A}_{m}$ | National advertising expenditure of manufacturer, ${A}_{m}\ge 0$ |

${A}_{r}$ | Local advertising expenditure of retailer, ${A}_{r}\ge 0$ |

${k}_{m}$ | Effect coefficient of national advertising expenditure, ${k}_{m}>0$ |

${k}_{r}$ | Effect coefficient of local advertising expenditure, ${k}_{r}>0$ |

$w$ | Wholesale price (manufacturer’s price to the retailers) |

$\mu $ | Manufacturer participation rate in percentage, $0\le \mu \le 1$ |

${\alpha}_{1}$ | Adjustment speed of manufacturer advertising expenditure, ${\alpha}_{1}>0$ |

${\alpha}_{2}$ | Adjustment speed of retailer advertising expenditure, ${\alpha}_{2}>0$ |

${\lambda}_{1}$ | Risk tolerance level of manufacturer, ${\lambda}_{1}>0$ |

${\lambda}_{2}$ | Risk tolerance level of retailer, ${\lambda}_{2}>0$ |

**Table 2.**Nash equilibrium points $({A}_{m},{A}_{r})$ and expected utilities $(E({U}_{m}),E({U}_{r}))$ with respect to ${\lambda}_{1}$ and ${\lambda}_{2}$ with ${\alpha}_{1}=3.5$ and ${\alpha}_{2}=3.5$.

$({A}_{m},{A}_{r})\phantom{\rule{0ex}{0ex}}(E({U}_{m}),E({U}_{r}))$ | ${\lambda}_{1}=2$ | ${\lambda}_{1}=3$ | ${\lambda}_{1}=4$ | ${\lambda}_{1}=5$ | ${\lambda}_{1}=6$ |

${\lambda}_{2}=2$ | (406.8, 193.7) (19,020, 9356) | (406.8, 193.7) (17,420, 9356) | (406.8, 193.7) (15,820, 9356) | (406.8, 193.7) (14,220, 9356) | (406.8, 193.7) (12,620, 9356) |

${\lambda}_{2}=3$ | (407, 177.9) (19,810, 8593) | (407, 177.9) (18,210, 8593) | (407, 177.9) (16,610, 8593) | (407, 177.9) (15,010, 8593) | (407, 177.9) (13,410, 8593) |

${\lambda}_{2}=4$ | (407.2, 162.8) (20,610, 7863) | (407.2, 162.8) (19,010, 7863) | (407.2, 162.8) (17,401, 7863) | (407.2, 162.8) (15,810, 7863) | (407.2, 162.8) (14,210, 7863) |

${\lambda}_{2}=5$ | (407.4, 148.3) (21,400, 7165) | (407.4, 148.3) (19,800, 7165) | (407.4, 148.3) (18,200, 7165) | (407.4, 148.3) (16,600, 7165) | (407.4, 148.3) (15,000, 7165) |

${\lambda}_{2}=6$ | (407.6, 134.6) (22,190, 6499) | (407.6, 134.6) (20,590, 6499) | (407.6, 134.6) (18,990, 6499) | (407.6, 134.6) (17,390, 6499) | (407.6, 134.6) (15,790, 6499) |

**Table 3.**Nash equilibrium points $({A}_{m},{A}_{r})$ and expected utilities $(E({U}_{m}),E({U}_{r}))$ with respect to ${k}_{m}$ and ${k}_{r}$ with ${\alpha}_{1}=3.5$ and ${\alpha}_{2}=3.5$.

$({A}_{m},{A}_{r})\phantom{\rule{0ex}{0ex}}(E({U}_{m}),E({U}_{r}))$ | ${k}_{m}=0.3$ | ${k}_{m}=0.5$ | ${k}_{m}=0.7$ | ${k}_{m}=0.9$ |

${k}_{r}=0.3$ | (234.3, 365) (19,090, 7694) | (650.1, 400.4) (19,500, 8439) | (1272, 456.3) (20,130, 9618) | (2098, 536.4) (20,960, 11,310) |

${k}_{r}=0.5$ | (251.9, 1148) (19,970, 8193) | (697.3, 1262) (20,420, 9013) | (1359, 1444) (21,090, 10,310) | (2230, 1704) (21,970, 12,160) |

${k}_{r}=0.7$ | (280.6, 2749) (21,470, 9072) | (772.7, 3038) (21,960, 10,030) | (1459, 3494) (22,700, 11,530) | (2429, 4141) (23,670, 13,660) |

${k}_{r}=0.9$ | (320.9, 6148) (23,830, 10,570) | (874.3, 6834) (24,390, 11,750) | (1665, 7903) (25,220, 13,590) | (2650, 9396) (26,290, 16,160) |

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**MDPI and ACS Style**

Liu, J.; Li, C.
Dynamic Game Analysis on Cooperative Advertising Strategy in a Manufacturer-Led Supply Chain with Risk Aversion. *Mathematics* **2023**, *11*, 512.
https://doi.org/10.3390/math11030512

**AMA Style**

Liu J, Li C.
Dynamic Game Analysis on Cooperative Advertising Strategy in a Manufacturer-Led Supply Chain with Risk Aversion. *Mathematics*. 2023; 11(3):512.
https://doi.org/10.3390/math11030512

**Chicago/Turabian Style**

Liu, Jia, and Cuixia Li.
2023. "Dynamic Game Analysis on Cooperative Advertising Strategy in a Manufacturer-Led Supply Chain with Risk Aversion" *Mathematics* 11, no. 3: 512.
https://doi.org/10.3390/math11030512