Outsourcing sustainability: a game-theoretic modeling approach

Abstract

As a response to stakeholders’ interest in sustainable products and services, an organization may change its approach to sustainability issues, from isolated social and environmental projects to corporate sustainability strategies and practices that are part of their core business. However, many of the efforts associated with these sustainability strategies cannot be directly exerted by focal organizations. We consider the situation in which a focal organization (sustainability buyer) outsources sustainability efforts to another organization (sustainability seller). While buyers cannot directly exert sustainability efforts, they can provide economic or technical support to their sellers in order to incentivize these efforts. We investigate how the effort and support decisions change according to characteristics of stakeholders, buyers, and sellers. Additionally, the presence of sustainability-minded stakeholders may lead to buyers’ competition on the sustainability effort exerted by their sellers. Therefore, we extend our analysis of sustainability efforts and incentives to the case of two competing buyers, and we determine conditions under which sharing a seller is preferred by the buyers to having a separate seller for each buyer.

Appendix

Proof of Proposition 1

Anticipating the seller’s response function given by (2), the buyer chooses the support level u ≥ 0 that maximizes her benefit function

$$ \pi^{B} \left( u \right) = \left\{ {\begin{array}{*{20}c} {A^{0} \left( {a - bA^{0} } \right) + \left( {a - 2bA^{0} - c^{b} /\gamma } \right)\gamma u - b\gamma^{2} u^{2} } \hfill & { {\text{if}}\,u \ge - A^{0} /\gamma } \hfill \\ { - c^{b} u} \hfill & {{\text{if}}\,u < - A^{0} /\gamma } \hfill \\ \end{array} } \right. $$

If $A^{0} > 0$, any positive u will increase seller’s effort from A ⁰ to $A^{0} + \gamma u$. Therefore, the buyer maximizes $A^{0} \left( {a - bA^{0} } \right) + \left( {a - 2bA^{0} - c^{b} /\gamma } \right)\gamma u - b\gamma^{2} u^{2}$ subject to u ≥ 0. The KKT conditions are given by

$$ \begin{array}{ll}\gamma \left( {a - 2bA^{0} - c^{b} /\gamma } \right) - 2b\gamma^{2}u \le 0 \hfill \\ \left[ {\gamma \left( {a - 2bA^{0} - c^{b} /\gamma} \right) - 2b\gamma^{2} u} \right]u = 0\hfill \\\end{array} $$

The second-order condition is $- 2b\gamma^{2} < 0$. Solving for u and replacing in the seller’s best response function, we obtain the optimal solution

$$ u^{*} = \left\{ {\begin{array}{*{20}c} {\frac{{a - c^{b} /\gamma - 2bA^{0} }}{2b\gamma }} \hfill & {{\text{if}}\,a - c^{b} /\gamma > 2bA^{0} } \hfill \\ 0 \hfill & { {\text{if}}\,a - c^{b} /\gamma \le 2bA^{0} } \hfill \\ \end{array} } \right. $$

$$ x^{*} = \left\{ {\begin{array}{*{20}c} {\frac{{a - c^{b} /\gamma }}{2b}} \hfill & { {\text{if}}\,a - c^{b} /\gamma > 2bA^{0} } \hfill \\ {A^{0} } \hfill & {{\text{if}}\,a - c^{b} /\gamma \le 2bA^{0} } \hfill \\ \end{array} } \right. $$

If $A^{0} \le 0$, the seller will exert a positive effort only if $u \ge - A^{0} /\gamma$. Thus, the buyer finds $u$ that maximizes the net benefit $A^{0} \left( {a - bA^{0} } \right) + \left( {a - 2bA^{0} - c^{b} /\gamma } \right)\gamma u - b\gamma^{2} u^{2}$ subject to $u \ge - A^{0} /\gamma$. Additionally, this net benefit must be positive. Otherwise, the buyer will provide no support. The KKT conditions are given by

$$ \gamma \left( {a - 2bA^{0} - c^{b} /\gamma } \right) - 2b\gamma^{2} u + \lambda = 0 $$

$$ u + A^{0} /\gamma \ge 0 $$

$$ \left( {u + A^{0} /\gamma } \right)\lambda = 0 $$

$$ \lambda \ge 0 $$

On the one hand, if $u = - A^{0} /\gamma$, then $\lambda = - \gamma a + c^{b}$. This is feasible if $a \le c^{b} /\gamma$. However, $\pi^{B} \left( { - A^{0} /\gamma } \right) = A^{0} a < 0$, so the buyer will prefer to provide no support. On the other hand if $\lambda = 0$, then $u = \left( {a - c^{b} /\gamma - 2bA^{0} } \right)/2b\gamma$, which is feasible if $a > c^{b} /\gamma$. However, $\pi^{B} \left( {\left( {a - c^{b} /\gamma - 2bA^{0} } \right)/2b\gamma } \right) \ge 0$ only if $a - c^{b} /\gamma > \left( { - 4bc^{b} A^{0} /\gamma } \right)^{1/2}$. Thus,

$$ u^{*} = \left\{ {\begin{array}{*{20}c} {\frac{{a - c^{b} /\gamma - 2bA^{0} }}{2b\gamma }} \hfill & { {\text{if}}\,a - c^{b} /\gamma > \left( { - 4bc^{b} A^{0} /\gamma } \right)^{1/2} } \hfill \\ 0 \hfill & { {\text{if}}\,a - c^{b} /\gamma \le \left( { - 4bc^{b} A^{0} /\gamma } \right)^{1/2} } \hfill \\ \end{array} } \right. $$

$$ x^{*} = \left\{ {\begin{array}{*{20}c} {\frac{{a - c^{b} /\gamma }}{2b}} \hfill & {{\text{if}}\,a - c^{b} /\gamma > \left( { - 4bc^{b} A^{0} /\gamma } \right)^{1/2} } \hfill \\ 0 \hfill & {{\text{if}}\,a - c^{b} /\gamma \le \left( { - 4bc^{b} A^{0} /\gamma } \right)^{1/2} } \hfill \\ \end{array} } \right. $$

The optimal benefits are obtained by replacing the optimal effort and support in the net benefit functions (1) and (3).

Proof of Proposition 2

The symmetric equilibrium is given by $u_{1} = u_{2} = K/2$. We can obtain $u_{1}^{*}$ and $u_{2}^{*}$ using the expression for K obtained in Eq. (10). Thus, we obtain $x^{*}$ by substituting $u_{1}^{*}$ and $u_{2}^{*}$ in Eq. (8). The optimal net benefits can be obtained by substituting $x^{*}$, $u_{1}^{*}$ and $u_{2}^{*}$ in Eqs. (7) and (9).

Proof of Proposition 3

From (17), we have a system of two equations. Solving for u ₁ and u ₂, we obtain $u_{1}^{*}$ and $u_{2}^{*}$. Then, we substitute these in Eq. (15) to obtain $x_{1}^{*}$ and $x_{2}^{*}$. Finally, we substitute the optimal efforts and supports in Eqs. (13) and (16) to obtain the respective optimal net benefits for sellers and buyers.

Proof of Proposition 4

• 1) By directly comparing $x^{*}$ from Eq. (11) with $x_{1}^{*} + x_{2}^{*}$ from Eq. (18), we can determine that the effort in the shared-seller network is greater than the total effort in the separate-sellers network if $a \le - 2c^{b} /\gamma$. However, this is not possible given our assumption that $\tilde{K} > 0$.

• 2) From Eqs. (11) and (18), we can determine that the effort in the shared-seller network is greater than the effort of each seller in the separate-sellers network if $a \ge \frac{{4c^{b} }}{\gamma }$. Additionally, the seller’s net benefit function in both networks can be written as $\tilde{r}\left[ {1 - \left( {1 + \alpha x} \right)e^{ - \alpha x} } \right]$ (because $\delta = 0$ and $\hat{r} = \tilde{r}$). Because this function is increasing in the effort x, the seller’s net benefit is greater in the shared-seller network relative to the separate-sellers network if $a \ge \frac{{4c^{b} }}{\gamma }$.

• 3) By direct comparison of the optimal buyer’s net benefit from Eqs. (12) and (19), we can determine that this net benefit is higher in the shared-seller network if $a \ge \frac{{4c^{b} }}{\gamma } + 36b{ \ln }\left( {\frac{{\alpha \tilde{r} }}{{c^{0} }}} \right)$.

1.1 One buyer–one seller with alternative price function

In Sect. 4, we found that the seller’s best response function is given by the following:

$$ x\left( u \right) = \left\{ {\begin{array}{*{20}c} {A^{0} + \gamma u, } \hfill & {{\text{if}}\,u > - \frac{{A^{0} }}{\gamma },} \hfill \\ {0,} \hfill & {{\text{otherwise}}.} \hfill \\ \end{array} } \right. $$

Assume that instead of a linear price function, the price that stakeholders are willing to pay for sustainability efforts is given by $p\left( x \right) = {a \mathord{\left/ {\vphantom {a {(1 + bx)}}} \right. \kern-0pt} {(1 + bx)}}$. Thus, anticipating the seller’s response function, the buyer chooses the support level u ≥ 0 that maximizes her benefit function

$$ \pi^{B} \left( u \right) = \left\{ {\begin{array}{*{20}c} {\frac{{a(A^{0} + \gamma u)}}{{1 + b(A^{0} + \gamma u)}} - c^{b} u} \hfill & {{\text{if}}\,u \ge - A^{0} /\gamma } \hfill \\ { - c^{b} u} \hfill & {{\text{if}}\,u < - A^{0} /\gamma } \hfill \\ \end{array} } \right. $$

If A ⁰ > 0, any positive u will increase seller’s effort from A ⁰ to $A^{0} + \gamma u$. Therefore, the buyer maximizes $\frac{{a(A^{0} + \gamma u)}}{{1 + b(A^{0} + \gamma u)}} - c^{b} u$ subject to u ≥ 0. It can be checked that the second-order condition is negative. Solving for u and replacing in the seller’s best response function, we obtain the optimal solution

$$ u^{*} = \left\{ {\begin{array}{*{20}c} {\frac{{\left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} - bA^{0} - 1}}{b\gamma }} \hfill & {{\text{if}}\,\left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} > bA^{0} + 1} \hfill \\ 0 \hfill & {{\text{if}}\,\left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} \le bA^{0} + 1} \hfill \\ \end{array} } \right. $$

$$ x^{*} = \left\{ {\begin{array}{*{20}c} {\frac{{\left( {\frac{{a\gamma }}{{c^{b} }}} \right)^{{1/2}} - 1}}{b}} \hfill & {{\text{if}}\left( {\frac{{a\gamma }}{{c^{b} }}} \right)^{{1/2}} > bA^{0} + 1} \hfill \\ {A^{0} } \hfill & {{\text{if}}\left( {\frac{{a\gamma }}{{c^{b} }}} \right)^{{1/2}} \le bA^{0} + 1} \hfill \\ \end{array} } \right. $$

If $A^{0} \le 0$, the seller will exert a positive effort only if $u \ge - A^{0} /\gamma$. Thus, the buyer finds u that maximizes the net benefit $\frac{{a(A^{0} + \gamma u)}}{{1 + b(A^{0} + \gamma u)}} - c^{b} u$ subject to $u \ge - A^{0} /\gamma$. Additionally, this net benefit must be positive. Finding the KKT conditions and solving for u, we obtain

$$ u^{*} = \left\{ {\begin{array}{*{20}c} {\frac{{\left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} - bA^{0} - 1}}{b\gamma }} \hfill & {{\text{if}}\, \left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} > \left( { - bA^{0} } \right)^{1/2} + 1} \hfill \\ 0 \hfill & {{\text{if}}\, \left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} \le \left( { - bA^{0} } \right)^{1/2} + 1} \hfill \\ \end{array} } \right. $$

$$ x^{*} = \left\{ {\begin{array}{*{20}c} {\frac{{\left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} - 1}}{b}} \hfill & {{\text{if}}\, \left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} > \left( { - bA^{0} } \right)^{1/2} + 1} \hfill \\ {A^{0} } \hfill & {{\text{if}}\, \left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} \le \left( { - bA^{0} } \right)^{1/2} + 1} \hfill \\ \end{array} } \right. $$

We can see that the optimal effort and support decisions, $x^{*}$ and $u^{*}$, vary for four different regions characterized by $A^{0}$, a, c ^b, and $\gamma$:

$$ \Upomega_{1} = \left\{ {\left( {A^{0} , a, c^{b} , \gamma } \right):A^{0} > 0, \, \left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} > bA^{0} + 1} \right\} $$

$$ \Upomega_{2} = \left\{ {\left( {A^{0} , a, c^{b} , \gamma } \right):A^{0} \le 0, \, \left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} > \left( { - bA^{0} } \right)^{1/2} + 1} \right\} $$

$$ \Upomega_{3} = \left\{ {\left( {A^{0} , a, c^{b} , \gamma } \right):A^{0} > 0, \, \left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} \le bA^{0} + 1} \right\} $$

$$ \Upomega_{4} = \left\{ {\left( {A^{0} , a, c^{b} , \gamma } \right):A^{0} \le 0, \, \left( {\frac{a\gamma }{{c^{b} }}} \right)^{1/2} \le \left( { - bA^{0} } \right)^{1/2} + 1} \right\} $$

In regions ${{\Upomega}}_{1}$ and ${{\Upomega}}_{2}$, both the support and the effort are positive. In region ${{\Upomega}}_{3}$, the effort is positive but the support is zero. Finally, in region ${{\Upomega}}_{4}$, both the effort and the support are equal to zero. These results have a similar structure to those presented in Proposition 1, for the linear price case.