Supply chain dynamics with assortative matching

Abstract

This article studies the evolutionarily stable equilibria of one-manufacturer and one-retailer supply chains. Each agent chooses to be either shareholder-oriented or stakeholder-oriented based on its own preference, then gives its pricing decision. Supply chains are formed by two types of matching processes: uniform random matching and assortative matching. Results indicate that, under uniform random matching, only one evolutionarily stable equilibrium exists, namely, the strict Nash equilibrium where both manufacturer and retailer choose shareholder strategy. Under assortative matching, the strict Nash equilibrium may not be evolutionarily stable under sign-compatible dynamics. The equilibrium where both manufacturer and retailer choose stakeholder strategy may be evolutionarily stable for certain values of the indices of assortativity. Furthermore, an interior equilibrium is observed with assortative matching, and the boundary equilibrium may be an evolutionarily stable equilibrium in some special cases.

Appendices

Appendix A

1.1 The proof of Proposition 1

According to Definition 3, an equilibrium $s^{*}=(s_{m}^{*}, s_{r}^{*})$ of differential equation (12) is an evolutionarily stable equilibrium if and only if there exists an open neighborhood O(s^∗) satisfying

$$ ((s_{m}^{*},s_{r}^{*})-(x, y))(G_{m}(x, y), G_{r}(x, y))^{\text{T}} > 0\quad \text{for all} (x, y)\in O(s^{*})\backslash\{s^{*}\}. $$

(13)

Due to the evolutionary dynamic (12) being sign-compatible, Eq. 13 holds if and only if

$$ ((s_{m}^{*},s_{r}^{*})-(x, y))({\varDelta} w_{m}(x, y), {\varDelta} w_{r}(x, y))^{\text{T}} > 0 \quad \text{for all}\quad (x, y)\in O(s^{*})\backslash\{s^{*}\}. $$

When s^∗ = (0, 0),

$$ \begin{array}{l} ((0, 0)-(x, y))({\varDelta} w_{m}, {\varDelta} w_{r})^{\text{T}}= -x{\varDelta} w_{m}-y{\varDelta} w_{r} \end{array} $$

(14)

Because (x, y) ∈ O((0, 0))∖{(0, 0)}, Eq. 14 is positive if $\pi _{m}^{Th}-\pi _{m}^{Hh}+\alpha (\pi _{m}^{Tt}-\pi _{m}^{Th}) < 0$ and $\pi _{r}^{Ht}-\pi _{r}^{Hh}+\beta (\pi _{r}^{Tt}-\pi _{r}^{Ht}) < 0$.

By the direct computation, as $\beta < \frac {2k_{r} A_{3}}{A_{4}(1-k_{r})^{2}}$ and $\alpha < \frac {{k_{m}^{2}} (1-k_{r})A_{3}}{4k_{r}(1-k_{m})A_{1}}$, the above equation (13) at s^∗ = (0, 0) holds. Hence, the equilibrium (0, 0) is an evolutionarily stable equilibrium.

Similarly, we can prove the other results in Proposition 1.

Appendix B

The other dynamics of games with multiple evolutionarily stable strategy profiles.

The asymmetric equilibrium derived in Corollary 1(3) is presented in Fig. 12, where manufacturers and retailers choose opposite strategies in the evolutionarily stable equilibrium. For the initial parameter values in Corollary 1(4), equilibrium is characterized by stakeholder-oriented manufacturers and both types of retailers (Fig. 13), whereas Corollary 1(5) characterizes the equilibrium with stakeholder-oriented retailers and both types of manufacturers (Fig. 14). Figures 15 and 16 present equilibria with three evolutionarily stable strategy profiles.

Fig. 12

Phase portrait emerging from Corollary 1(3) when k_m = .77, k_r = .31, α = .96, β = .55

Fig. 13

Phase portrait emerging from Corollary 1(4) when k_m = .72, k_r = .28, α = .92, β = .6

Fig. 14

Phase portrait emerging from Corollary 1(5) when k_m = .72, k_r = .28, α = .83, β = .62

Fig. 15

Phase portrait emerging from Corollary 1(6) when k_m = .782, k_r = .333, α = .883, β = .592

Fig. 16

Phase portrait emerging from Corollary 1(7) when k_m = .77, k_r = .31, α = .94, β = .57