Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dual-Channel Closed Loop Supply Chain Competition: A Stackelberg--Nash Approach

Gurkirat Wadhwa

Abstract

In many consumer electronics and appliance markets, manufacturers sell products through competing retailers while simultaneously relying on take-back programs to recover used items for remanufacturing. Designing such programs is challenging when firms compete on prices and consumers differ in their willingness to return products. Motivated by these settings, this paper develops a game theoretic framework to analyze pricing and take-back decisions in a dual-channel closed loop supply chain (CLSC) with two competing manufacturers and two competing retailers. Manufacturers act as Stackelberg leaders, simultaneously determining wholesale prices and consumer take-back bonuses, while retailers engage in Nash competition over retail prices. The model integrates three key elements: (i) segmented linear demand with cross-price effects, (ii) deterministic product returns, and (iii) an inertia responsiveness allocation mechanism governing the distribution of returned products between manufacturers. Closed form Nash equilibria are derived for the retailer subgame, along with symmetric Stackelberg equilibria for manufacturers. We derive a feasibility threshold for take-back incentives, identifying conditions under which firms optimally offer positive bonuses to consumers. The results further demonstrate that higher remanufacturing value or return rates lead the manufacturers to lower wholesale prices in order to expand sales and capture additional return volumes, while high consumer inertia weakens incentives for active collection. Numerical experiments illustrate and reinforce the analytical results, highlighting how consumer behavior, market structure and product substitutability influence prices, bonuses, and return volumes. Overall, the study provides managerial insights for designing effective take-back programs and coordinating pricing decisions in competitive circular supply chains.

Dual-Channel Closed Loop Supply Chain Competition: A Stackelberg--Nash Approach

Abstract

In many consumer electronics and appliance markets, manufacturers sell products through competing retailers while simultaneously relying on take-back programs to recover used items for remanufacturing. Designing such programs is challenging when firms compete on prices and consumers differ in their willingness to return products. Motivated by these settings, this paper develops a game theoretic framework to analyze pricing and take-back decisions in a dual-channel closed loop supply chain (CLSC) with two competing manufacturers and two competing retailers. Manufacturers act as Stackelberg leaders, simultaneously determining wholesale prices and consumer take-back bonuses, while retailers engage in Nash competition over retail prices. The model integrates three key elements: (i) segmented linear demand with cross-price effects, (ii) deterministic product returns, and (iii) an inertia responsiveness allocation mechanism governing the distribution of returned products between manufacturers. Closed form Nash equilibria are derived for the retailer subgame, along with symmetric Stackelberg equilibria for manufacturers. We derive a feasibility threshold for take-back incentives, identifying conditions under which firms optimally offer positive bonuses to consumers. The results further demonstrate that higher remanufacturing value or return rates lead the manufacturers to lower wholesale prices in order to expand sales and capture additional return volumes, while high consumer inertia weakens incentives for active collection. Numerical experiments illustrate and reinforce the analytical results, highlighting how consumer behavior, market structure and product substitutability influence prices, bonuses, and return volumes. Overall, the study provides managerial insights for designing effective take-back programs and coordinating pricing decisions in competitive circular supply chains.
Paper Structure (29 sections, 6 theorems, 84 equations, 5 figures, 1 table)

This paper contains 29 sections, 6 theorems, 84 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model Description
  3. Players, timing, and decisions
  4. Demand structure
  5. Return generation and allocation mechanism
  6. Costs, profits, and operating decisions
  7. Assumptions and market structure
  8. Equilibrium Analysis
  9. Retailer Nash Equilibrium
  10. Manufacturers’ Leader-Stage Conditions
  11. Symmetric Equilibrium
  12. Welfare Comparison
  13. Numerical Illustration and Discussion
  14. Baseline equilibrium outcomes
  15. Behavioral drivers of take-back incentives
  16. ...and 14 more sections

Key Result

lemma thmcounterlemma

If the net remanufacturing value satisfies $v_i-k_i\le 0$, then in any equilibrium of the game manufacturers optimally set zero take-back bonuses, i.e., and do not actively collect returned products. Consequently, the reverse channel is inactive and the model reduces to a forward-only Stackelberg pricing game.

Figures (5)

  • Figure 1: Dual-channel CLSC Model
  • Figure 2: Equilibrium take-back incentives. Left: Bonus $b^*$ as a function of consumer responsiveness $\beta$. Right: Bonus surface $b^*(\beta,\gamma_r)$ under inertia--responsiveness allocation.
  • Figure 3: Forward--reverse channel interactions. Left: Wholesale price $w^*$ as a function of remanufacturing value $v$. Right: Total return volume $Q_r(\beta,\gamma_r)$ under inertia--responsiveness allocation.
  • Figure 4: Effect of product substitutability $\varepsilon$ on equilibrium prices. Left: Wholesale price $w^*$. Right: Retail price $p^*$.
  • Figure 5: Effect of competition on return outcomes. Left: Total return volume $Q_r$ as a function of $\varepsilon$. Right: Joint effect of product substitutability $\varepsilon$ and consumer responsiveness $\beta$ on total returns.

Theorems & Definitions (6)

  • lemma thmcounterlemma: Reverse-channel viability
  • theorem thmcountertheorem: Retailer-stage uniqueness
  • theorem thmcountertheorem: Symmetric equilibrium and bonus threshold
  • theorem thmcountertheorem: Forward--reverse coupling under proportional allocation
  • corollary thmcountercorollary: Forward--reverse coupling under inertia--responsiveness
  • theorem thmcountertheorem: Welfare inefficiency