Supplier development investment strategies: a game theoretic evaluation

Abstract

Supplier development is a critical competitive endeavor for organizations and their supply chains. Investigation into this area has increased over the past few years, yet further understanding of suppler development is needed. Specifically, recent supplier development investment practices have shown a shift from standalone actions by organization or supplier to joint actions between organization and suppliers. Faced with this phenomenon, the goal of this paper is to develop a theoretic model based on the Cobb–Douglas production function. The study focuses on determining optimal supplier development investment strategies with respect to joint actions that increase supplier production capability for the benefit of both the focal organization and its suppliers. The supplier development investment strategies mainly refer to joint actions between an organization and multiple suppliers through capital resources (tangible) investments, knowledge (intangible) investments, and sharing cost of capital resources (tangible) investments. Using various game theoretic models, we reveal how supplier development investment strategies and profits of all the members are affected by various buyer-supplier relationships and investment returns to scale reasons. Whether the focal organization (buyer) has any incentives to share cost of capital resources (tangible) investments is also investigated. Our first finding is that supplier development investment activities motivation is derived from increases in supply volume for the “increasing returns to scale” situation, and derives from increasing the organization and suppliers’ marginal profit in the “decreasing returns to scale” situation. Secondly, the cooperative relationship is more economically beneficial to the supply chain, but it also requires more capital resources and knowledge expenditures (investments) than a non-cooperative relationship. Thirdly, through numerical analysis it is found that the cooperative relationship can not obtain Pareto efficiency for all the members of the supply chain when using the Nash bargaining model. Additional gaming insights and implications are also provided from parametric analysis. Opportunities for further research are also presented.

Appendix: Proof of results

1.1 Proof of feasibility conditions

• (1) Nash game:

  $$\begin{aligned}&\rho _O (\eta x^{*\alpha }y^{*\beta })-x^{*}=\left( {\eta \alpha ^{\alpha }\beta ^{\beta }\rho _O ^{1-\beta }\left( {\sum _{i=1}^n {\rho _{S\,i} } } \right) ^{\beta }} \right) ^{\frac{1}{1-\alpha -\beta }}(1-\alpha )>0\\&\left( {\sum _{i=1}^n {\rho _{S\,i} } } \right) (\eta x^{*\alpha }y^{*\beta })-y^{*}=\left( {\eta \alpha ^{\alpha }\beta ^{\beta }\rho _O ^{\alpha }\left( {\sum _{i=1}^n {\rho _{S\,i} } } \right) ^{1-\alpha }} \right) ^{\frac{1}{1-\alpha -\beta }}(1-\beta )>0 \end{aligned}$$

• (2) Stackelberg game:

  $$\begin{aligned}&\rho _O (\eta x^{**\alpha }y^{**\beta })-x^{**}-t^{**}y^{**}\\&\quad =\left( \eta \alpha ^{\alpha }\beta ^{\beta }\left( \rho _O +\beta \left( {\sum _{i=1}^n {\rho _{S\,i} } } \right) \right) \right) ^{\frac{1}{1-\alpha -\beta }}(1-\alpha -\beta )>0 \end{aligned}$$

• (3) Cooperative game

  $$\begin{aligned}&\left( {\rho _O +\sum _{i=1}^n {\rho _{S\,i} } } \right) (\eta \overline{{x}}^{\alpha }\overline{{y}}^{\beta })-\overline{{x}}-\overline{{y}}\\&\quad =\left( {\eta \alpha ^{\alpha }\beta ^{\beta }\left( {\rho _O +\left( {\sum _{i=1}^n {\rho _{S\,i} } } \right) } \right) } \right) ^{\frac{1}{1-\alpha -\beta }}(1-\alpha -\beta )>0 \end{aligned}$$

1.2 Proof of Proposition 5

To arrive at Proposition 5, a relationship proposition on investment quantities for each of the three types of games, we begin by first using expressions (19) to arrive at the following general relationships:

$$\begin{aligned} x^{**}=\left\{ {{\begin{array}{ll} x_1^{**} =\left( {\eta \beta ^{\beta }\alpha ^{1-\beta }\left( {\rho _O +\beta \left( {\sum _{i=1}^n {\rho _{S\,i} } } \right) } \right) } \right) ^{\frac{1}{1-\alpha -\beta }},&{}\quad { if}\,\rho _O >(1-\beta )\left( {\sum _{i=1}^n {\rho _{S\,i} } } \right) \\ x_2^{**} =\left( {\left( {\frac{\alpha \rho _O }{1-\beta }} \right) ^{1-\beta }\eta \left( {\beta \left( {\sum _{i=1}^n {\rho _{S\,i} } } \right) } \right) ^{\beta }} \right) ^{\frac{1}{1-\alpha -\beta }},&{}\quad { if}\,\rho _O \le (1-\beta )\left( {\sum _{i=1}^n {\rho _{S\,i} } } \right) \\ \end{array} }} \right. \end{aligned}$$

Then we know that the following relationships hold:

$$\begin{aligned} \overline{{x}}-x_1^{**}&= \left( {\eta \beta ^{\beta }\alpha ^{1-\beta }\left( {\rho _O +\left( {\sum \limits _{i=1}^n {\rho _{S\,i} } } \right) } \right) } \right) ^{\frac{1}{1-\alpha -\beta }}-\left( {\eta \beta ^{\beta }\alpha ^{1-\beta }\left( {\rho _O +\beta \left( {\sum \limits _{i=1}^n {\rho _{S\,i} } } \right) } \right) } \right) ^{\frac{1}{1-\alpha -\beta }} \\&= (\eta \beta ^{\beta }\alpha ^{1-\beta })^{\frac{1}{1-\alpha -\beta }} \left( {\left( {\rho _O +\left( {\sum \limits _{i=1}^n {\rho _{S\,i} } } \right) } \right) ^{\frac{1}{1-\alpha -\beta }}-\left( {\rho _O +\beta \left( {\sum \limits _{i=1}^n {\rho _{S\,i} } } \right) } \right) ^{\frac{1}{1-\alpha -\beta }}} \right) >0 \\ x_2^{**} -x^{*}&= \left( {\left( {\frac{\alpha \rho _O }{1-\beta }} \right) ^{1-\beta }\eta \left( {\beta \left( {\sum \limits _{i=1}^n {\rho _{S\,i} } } \right) } \right) ^{\beta }} \right) ^{\frac{1}{1-\alpha -\beta }}-\left( {\eta \alpha ^{1-\beta }\beta ^{\beta }\rho _O^{1-\beta } \left( {\sum \limits _{i=1}^n {\rho _{S\,i} } } \right) ^{\beta }} \right) ^{\frac{1}{1-\alpha -\beta }} \\&= \left( {\eta \alpha ^{1-\beta }\beta ^{\beta }\rho _O^{1-\beta } \left( {\sum \limits _{i=1}^n {\rho _{S\,i} } } \right) ^{\beta }} \right) ^{\frac{1}{1-\alpha -\beta }}\left( {\left( {\frac{1}{1-\beta }} \right) ^{\frac{1-\beta }{1-\alpha -\beta }}-1} \right) >0 \end{aligned}$$

From expression (15), we know that the focal organization’s knowledge investment quantity $x$ positively correlates with the organization’s marginal profit $\rho _O$. Given this positive correlation, we know that $x_1^{**} \ge x_2^{**}$. Then, the general relationships for knowledge investment equilibrium quantities is: $\overline{{x}}>x^{**}>x^{*}$. We also can get $\overline{{y}}>y^{**}>y^{*}$ with same process.