# Optimal Timing Strategies in the Evolutionary Dynamics of Competitive Supply Chains

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

Paper | Modeling Technique | No. of Players | Period | Applied Industry | |
---|---|---|---|---|---|

Information update and evolution | [11] | Newsvendor model | One manufacturer and one supplier | Two | General |

[12] | MMFE | One manufacturer and one supplier | Multiple | General | |

[7] | Newsvendor model with MMFE | One manufacturer and one supplier | Multiple | Toy and fashion | |

[15] | Nash Game | Two retailers and one supplier | Multiple | General | |

Asymmetric and imperfect information | [16] | Nash Game | One manufacturer and one supplier | Two | Toy |

[17] | Cournot Game | One supplier and multiple retailers | 4 periods | General | |

[18] | Dynamic programming | One manufacturer and one supplier | Multiple | General | |

[19] | Signaling game | One manufacturer and one supplier | Single | Electronics | |

Endogenous timing of orders | [23] | Stackelberg game | One manufacturer and one buyer | Two | General |

[28] | Stackelberg game | One manufacturer and one buyer | Single | General | |

[24] | Stackelberg and Cournot game | One manufacturer and one retailer | Two | Movie | |

[25] | Nash Game | Two retailers | Two | Retailing | |

[29] | Stackelberg and Cournot game | Two retailers and one supplier | Single | Fashion and apparel | |

[26] | Bertrand game | Two retailers | Single | Fashion |

## 3. The Model

## 4. Baseline Model Analysis: The IIM

**Lemma**

**1.**

**Proposition**

**1.**

- (i)
- At the Early point, the better-informed firm initiates an order. The quantity of the order is as follows:

- (ii)
- At the Later point, the less-informed firm observes the order of the better-informed firm and then orders at the Later point. Its order quantity is as follows:

- (iii)
- The profits of the two firms are as follows:

## 5. Extension: Analysis of the IEM

**Lemma**

**2.**

#### 5.1. The Early Order by the Better-Informed Firm, Followed by the Later Order by the Less-Informed Firm: (E, L)

_{H}= 30, A

_{L}= 10, and p = 0.5. In this case, the threshold ${\alpha}^{*}$ is calculated to be 0.87. Similarly, when A

_{H}= 50, A

_{L}= 10, and p = 0.5, ${\alpha}^{*}$ is determined to be 0.95. Hence, when $\alpha $ is greater than or equal to ${\alpha}^{\mathrm{*}}$, and the better-informed firm opts to shift from the Early point to the Later point, it prompts the less-informed firm to reassess its belief structure, deducing that it indicates a high-demand state. Consequently, the game transitions into a scenario of complete information. Conversely, if $\alpha $ is less than ${\alpha}^{*}$, the less-informed firm remains unable to adjust its belief structure due to both high- and low-type better-informed firms favoring late orders. Ultimately, the deviation of the better-informed firm is advantageous below ${\alpha}^{*}$. Thus, it can be concluded that with a threshold value of ${\alpha}^{*}$, the better-informed firm tends to place orders early; conversely, below ${\alpha}^{*}$, it deviates towards the Later point. Employing the same rationale, deviation of the less-informed firm from the Later point to the Early point is deemed non-profitable, given its inability to gather valuable information. Consequently, we verify that the equilibrium of the Early point order of the better-informed firm and the Later point order of the less-informed firm (E, L) holds if $\alpha $ exceeds the threshold value ${\alpha}^{*}$; otherwise, it fails to constitute an equilibrium.

**Corollary**

**1.**

#### 5.2. The Later Order by the Better-Informed Firm, and the Early Order by the Less-Informed Firm: (L, E)

#### 5.3. The Early Order by Both Firms: (E, E)

#### 5.4. The Later Order by Both Firms: (L, L)

**Proposition**

**2.**

**Case**

**A.**

- (i)
- At the Early point, the better-informed firm initiates an order. The quantity of the order is as follows:

- (ii)
- At the Later point, the less-informed firm observes the order of the better-informed firm and then orders. Its order quantity is as follows:

- (iii)
- The profits of both firms are as follows:

**Case**

**B.**

- (i)
- At the Early point, the less-informed firm initiates an order. The quantity of the order is as follows:

- (ii)
- At the Later point, the better-informed firm then orders after acquiring an accurate picture of demand

- (iii)
- The profits of profits of both firms are as follows:

_{H}/A

_{L}) increases, the region of (L, E) also increases. Hence, when Ѳ is low (A

_{H}= 30, A

_{L}= 10) in Figure 4a indicating a correspondingly low market potential, the likelihood of the better-informed firm preempting the first mover’s advantage increases. However, if there is even a slight possibility that the market potential is high (A

_{H}= 50, A

_{L}= 10) in Figure 4b, the right strategy is to delay the order by observing the evolution of information.

#### 5.5. Supply-Chain Performance with Two Equilibria

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof of Lemma**

**1.**

**Proof of Lemma**

**2.**

#### Appendix A.1. Equilibrium Analysis of Possible Scenarios

#### Appendix A.2. The Early Order by Both Firms: (E, E)

## Notes

1 | When $\alpha =1$, the game becomes a perfect information game. |

2 | “The early point” also refers to the point at which a first mover’s advantage can be taken, emphasizing the importance of placing orders before competitors. Achieving a first mover’s advantage over competitor entails setting order timing to the earliest possible moment (the first period in the case of a multiple period model), constituting an equilibrium. Conversely, for obtaining more accurate market information, setting order timing to the latest possible moment (the later point) is optimal, with period N being the optimal timing in an N-period model. |

3 | We index a well-informed retailer as the “incumbent (i)”, the less-informed retailer as the “entrant (e)”, and the supplier (s), respectively. High-type and low-type actors are denoted by H and L, respectively. |

4 | $E\left[{\pi}_{iH}\right]={\left[\frac{\alpha p{A}_{H}+\left(1-\alpha \right)\left(1-p\right){A}_{L}}{3\left\{\alpha p+\left(1-\alpha \right)\left(1-p\right)\right\}}\right]}^{2}<{\left[\frac{\alpha p{A}_{H}+\left(1-\alpha \right)\left(1-p\right){A}_{L}}{2\left\{\alpha p+\left(1-\alpha \right)\left(1-p\right)\right\}}-\frac{\mu}{6}\right]}^{2}$ |

5 | If the less-informed firm deviates from the Early point to the Later point, the game transitions into a scenario of simultaneous moves with incomplete information. The anticipated profit for the less-informed firm then equals ${\frac{\mu}{9}}^{2}$, indicating a distinct reduction compared to the initial profit. |

## References

- Fildes, R.; Ma, S.; Kolassa, S. Retail forecasting: Research and practice. Int. J. Forecast.
**2019**, 38, 1283–1318. [Google Scholar] [CrossRef] - Wen, X.; Choi, T.-M.; Chung, S.-H. Fashion retail supply chain management: A review of operational models. Int. J. Prod. Econ.
**2019**, 207, 34–55. [Google Scholar] [CrossRef] - GAP Inc. 2020 Annual Report; GAP Inc.: San Francisco, CA, USA, 2020. [Google Scholar]
- Ren, S.; Chan, H.L.; Siqin, T. Demand forecasting in retail operations for fashionable products: Methods, practices, and real case study. Ann. Oper. Res.
**2020**, 291, 761–777. [Google Scholar] [CrossRef] - Raguseo, E.; Vitari, C.; Pigni, F. Profiting from big data analytics: The moderating roles of industry concentration and firm size. Int. J. Prod. Econ.
**2020**, 229, 107758. [Google Scholar] [CrossRef] - Ghani, N.A.; Hamid, S.; Hashem, I.A.T.; Ahmed, E. Social media big data analytics: A survey. Comput. Hum. Behav.
**2019**, 101, 417–428. [Google Scholar] [CrossRef] - Wang, T.; Atasu, A.; Kurtuluş, M. A Multiordering Newsvendor Model with Dynamic Forecast Evolution. Manuf. Serv. Oper. Manag.
**2012**, 14, 472–484. [Google Scholar] [CrossRef] - Biçer, I.; Seifert, R.W. Optimal Dynamic Order Scheduling under Capacity Constraints Given Demand-Forecast Evolution. Prod. Oper. Manag.
**2017**, 26, 2266–2286. [Google Scholar] [CrossRef] - Suarez, F.; Lanzolla, G. The half-truth of first-mover advantage. Harv. Bus. Rev.
**2005**, 83, 121–127. [Google Scholar] - Gal-Or, E. First Mover Disadvantages with Private Information. Rev. Econ. Stud.
**1987**, 54, 279. [Google Scholar] [CrossRef] - Ferguson, M.E.; DeCroix, G.A.; Zipkin, P.H. Commitment decisions with partial information updating. Nav. Res. Logist. (NRL)
**2005**, 52, 780–795. [Google Scholar] [CrossRef] - Lu, X.; Song, J.-S.; Regan, A. Inventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds. Oper. Res.
**2006**, 54, 1079–1097. [Google Scholar] [CrossRef] - Ziarnetzky, T.; Mönch, L.; Uzsoy, R. Rolling horizon, multi-product production planning with chance constraints and forecast evolution for wafer fabs. Int. J. Prod. Res.
**2018**, 56, 6112–6134. [Google Scholar] [CrossRef] - Shen, B.; Zhang, T.; Xu, X.; Chan, H.; Choi, T. Preordering in luxury fashion: Will additional demand information bring negative effects to the retailer? Decis. Sci.
**2022**, 53, 681–711. [Google Scholar] [CrossRef] - Demirag, O.C.; Xue, W.; Wang, J. Retailers’ Order Timing Strategies under Competition and Demand Uncertainty. Omega
**2021**, 101, 102256. [Google Scholar] [CrossRef] - Taylor, T.A. Sale Timing in a Supply Chain: When to Sell to the Retailer. Manuf. Serv. Oper. Manag.
**2006**, 8, 23–42. [Google Scholar] [CrossRef] - Shin, H.; Tunca, T.I. Do Firms Invest in Forecasting Efficiently? The Effect of Competition on Demand Forecast Investments and Supply Chain Coordination. Oper. Res.
**2010**, 58, 1592–1610. [Google Scholar] [CrossRef] - Oh, S.; Özer, Ö. Mechanism Design for Capacity Planning under Dynamic Evolutions of Asymmetric Demand Forecasts. Manag. Sci.
**2013**, 59, 987–1007. [Google Scholar] [CrossRef] - Zhang, J.; Li, S.; Zhang, S.; Dai, R. Manufacturer encroachment with quality decision under asymmetric demand information. Eur. J. Oper. Res.
**2019**, 273, 217–236. [Google Scholar] [CrossRef] - Chen, J.; Pun, H.; Zhang, Q. Eliminate demand information disadvantage in a supplier encroachment supply chain with information acquisition. Eur. J. Oper. Res.
**2023**, 305, 659–673. [Google Scholar] [CrossRef] - Shen, B.; Choi, T.-M.; Minner, S. A review on supply chain contracting with information considerations: Information updating and information asymmetry. Int. J. Prod. Res.
**2019**, 57, 4898–4936. [Google Scholar] [CrossRef] - Vosooghidizaji, M.; Taghipour, A.; Canel-Depitre, B. Supply chain coordination under information asymmetry: A review. Int. J. Prod. Res.
**2020**, 58, 1805–1834. [Google Scholar] [CrossRef] - Ferguson, M.E. When to commit in a serial supply chain with forecast updating. Nav. Res. Logist. (NRL)
**2003**, 50, 917–936. [Google Scholar] [CrossRef] - Liu, B.; Ma, S.; Guan, X.; Xiao, L. Timing of sales commitment in a supply chain with manufacturer-quality and retailer-effort induced demand. Int. J. Prod. Econ.
**2018**, 195, 249–258. [Google Scholar] [CrossRef] - Perdikaki, O.; Kostamis, D.; Swaminathan, J.M. Timing of service investments for retailers under competition and demand uncertainty. Eur. J. Oper. Res.
**2016**, 254, 188–201. [Google Scholar] [CrossRef] - Zhang, Q.; Chen, J.; Lin, J. Interaction between innovation choice and market-entry timing in a competitive fashion supply chain. Int. J. Prod. Res.
**2023**, 61, 1606–1623. [Google Scholar] [CrossRef] - Luo, H.; Niu, B. Impact of competition type on a competitive manufacturer’s preference of decision timing. Int. J. Prod. Econ.
**2022**, 251, 108548. [Google Scholar] [CrossRef] - Arcelus, F.J.; Kumar, S.; Srinivasan, G. Pricing, rebate, advertising and ordering policies of a retailer facing price-dependent stochastic demand in newsvendor framework under different risk preferences. Int. Trans. Oper. Res.
**2006**, 13, 209–227. [Google Scholar] [CrossRef] - Kim, Y. Retailers’ endogenous sequencing game and information acquisition game in the presence of information leakage. Int. Trans. Oper. Res.
**2021**, 28, 809–838. [Google Scholar] [CrossRef] - Mailath, G.J. Endogenous Sequencing of Firm Decisions. J. Econ. Theory
**1993**, 59, 169–182. [Google Scholar] [CrossRef] - Tomaru, Y.; Kiyono, K. Endogenous timing in mixed duopoly with increasing marginal costs. J. Institutional Theor. Econ. (JITE)/Z. Gesamte Staatswiss.
**2010**, 166, 591–613. [Google Scholar] [CrossRef] - Normann, H.-T. Endogenous Timing with Incomplete Information and with Observable Delay. Games Econ. Behav.
**2002**, 39, 282–291. [Google Scholar] [CrossRef] - Song, J.; Yao, D.D. Supply Chain Structures: Coordination, Information and Optimization; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013; Volume 42. [Google Scholar]
- Efron, B. Bayes’ Theorem in the 21st Century. Science
**2013**, 340, 1177–1178. [Google Scholar] [CrossRef] [PubMed] - Gibbons, R.S. Game Theory for Applied Economists; Princeton University Press: Princeton, NJ, USA, 1992. [Google Scholar]
- Cachon, G.P.; Swinney, R. Purchasing, Pricing, and Quick Response in the Presence of Strategic Consumers. Manag. Sci.
**2009**, 55, 497–511. [Google Scholar] [CrossRef] - Choi, T. Quick response in fashion supply chains with retailers having boundedly rational managers. Int. Trans. Oper. Res.
**2017**, 24, 891–905. [Google Scholar] [CrossRef]

**Figure 5.**Illustration of the two equilibria in the IEM and expected payoff comparison (parameter values: A

_{H}= 50, A

_{L}= 10. In figure (iii), the solid line represents the (E, L) equilibrium (i), while the dotted line represents the (L, E) equilibrium (ii)).

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**MDPI and ACS Style**

Kim, Y.
Optimal Timing Strategies in the Evolutionary Dynamics of Competitive Supply Chains. *Systems* **2024**, *12*, 114.
https://doi.org/10.3390/systems12040114

**AMA Style**

Kim Y.
Optimal Timing Strategies in the Evolutionary Dynamics of Competitive Supply Chains. *Systems*. 2024; 12(4):114.
https://doi.org/10.3390/systems12040114

**Chicago/Turabian Style**

Kim, Yongjae.
2024. "Optimal Timing Strategies in the Evolutionary Dynamics of Competitive Supply Chains" *Systems* 12, no. 4: 114.
https://doi.org/10.3390/systems12040114