# Construction of Equilibria in Strategic Stackelberg Games in Multi-Period Supply Chain Contracts

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Problem Description

#### 3.1. Multi-Period Profit and Inventory Level Optimization Catering to Uncertain Demand

#### 3.2. A General Solution Scheme Based on Backward Induction

## 4. Demand Structure

#### 4.1. Demand’s Dependence on Time and Price

#### 4.2. Stochastic Demand

- ${\u03f5}_{k}$’s are independent and identically distributed random variables of arbitrary distribution,
- normalized such that $\mathrm{E}\left[{\u03f5}_{k}\right]=0$ and $\mathrm{Var}\left[{\u03f5}_{k}\right]=1,\forall k$ and supported on intervals with continuous density function, ${f}_{{\u03f5}_{k}}>0,$ a.e. on its support.
- ${r}_{k}=$ price per unit at period k.

#### 4.3. Memory-Based Uncertain Demand

## 5. Embedding the Memory-Based Demand in the Expected Profit Expression

**Proposition**

**1.**

**Proof.**

## 6. Equilibria in Decentralized Channels with Two Suppliers

#### 6.1. The Basic Model: The Game in Single-Period

- D = demand (random)
- w = wholesale price per unit (chosen by the manufacturer)
- r = retail price per unit (chosen by the retailer)
- q = order quantity (chosen by the retailer)
- ${c}_{m}$ = manufacturing cost per unit (fixed)
- s = salvage price per unit (fixed)
- ${\mathsf{\Pi}}^{r}$ = profit for the retailer
- ${\mathsf{\Pi}}^{m}$ = profit for the manufacturer

**Proposition**

**2.**

**Proof.**

#### 6.2. Multi-Period Vertical Contracting

#### 6.3. Multi-Period Games with Memory Functions

**Theorem**

**1.**

**Corollary**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

#### 6.4. The Infinite Horizon Case

## 7. Numerical Implementation of the Model

#### 7.1. Multi-Period Buyback Contracts

- Setting the model parameters

**Example**

**1.**

**Remark**

**4.**

**Example**

**2.**

#### 7.2. Cooperative Agents: Centralized Channel, No Double Marginalization

**Example**

**3.**

#### 7.3. Markets with Strategic Customers

**Example**

**4.**

#### 7.4. Positive Network Effects on Demand

**Example**

**5.**

## 8. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Profit Structures Compatible with the Use of Multiplicative Memory Functions

## Appendix B. The Expected Profit Structure in Supply Chain Coordination Problems

#### Appendix B.1. The Wholesale Price Contract

- Obtain the expression for the expected value of retailer’s profit as a function of r and q.
- Apply the First Order Condition (F.O.C.) with respect to q on ${\overline{\mathsf{\Pi}}}^{r}$, i.e., $\frac{\partial {\overline{\mathsf{\Pi}}}^{r}}{\partial q}=0$ to obtain ${q}^{*}$ as a function of r.
- Check the concavity of ${\overline{\mathsf{\Pi}}}^{r}$ with respect to ${q}^{*}\left(r\right)$.
- Substitute the obtained ${q}^{*}$ as a function of r in the expression for ${\overline{\mathsf{\Pi}}}^{r}$ and see if it is of the general structure in (A2).
- Substitute the obtained ${q}^{*}$ as a function of r in the expression for ${\overline{\mathsf{\Pi}}}^{m}$ and see if it is of the general structure in (A2).

#### Appendix B.2. The Buyback Contract

#### Appendix B.3. The Revenue-Sharing Contract

## Appendix C. Proof of Proposition 2

**Proof.**

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**MDPI and ACS Style**

Azad Gholami, R.; Sandal, L.K.; Ubøe, J.
Construction of Equilibria in Strategic Stackelberg Games in Multi-Period Supply Chain Contracts. *Games* **2022**, *13*, 70.
https://doi.org/10.3390/g13060070

**AMA Style**

Azad Gholami R, Sandal LK, Ubøe J.
Construction of Equilibria in Strategic Stackelberg Games in Multi-Period Supply Chain Contracts. *Games*. 2022; 13(6):70.
https://doi.org/10.3390/g13060070

**Chicago/Turabian Style**

Azad Gholami, Reza, Leif Kristoffer Sandal, and Jan Ubøe.
2022. "Construction of Equilibria in Strategic Stackelberg Games in Multi-Period Supply Chain Contracts" *Games* 13, no. 6: 70.
https://doi.org/10.3390/g13060070