The impact of dual fairness concerns on bargaining game and its dynamic system stability

Abstract

This paper introduces dual fairness concerns into the classic two-level supply chain consisting of the fairness neutral supplier and fairness concerned retailers. The bargaining process is modeled under both simultaneous and sequential game to analyze the different situation of fairness concerns. The impact of dual fairness concerns is considered comprehensively on both short-term and long-term games. In short-term game, we conduct a sensitivity analysis on the optimal decision in a single cycle and find that the bargaining power and distributional fairness concern has opposite effects on the optimal solutions. Similarly, the impact of dual fairness concerns on that is also opposite. In long-term game, the dynamic system is constructed to investigate the influence of dual fairness concerns on system stability. At last, comparing the performance of the supplier and retailers, this paper explores the supplier’s timing choice based on the equilibrium point. The comparison shows that sequential game is more beneficial to the supplier because peer-induced fairness concern exacerbates the internal friction of retailers.

Appendix A: Proofs for main results

Proof of lemma 1

Given that $\overline{\pi }_{S,i} + \overline{\pi }_{{R_{i} }} = \pi_{S,i} + \pi_{{R_{i} }} = \pi_{{_{i} }}$, the Nash bargaining model of retailer $i$ can be written as: $\max [\pi_{{R_{i} }} - \lambda_{i} (\frac{{k_{i} }}{{1 - k_{i} }}(\pi_{i} - \overline{\pi }_{{R_{i} }} ) - \pi_{{R_{i} }} )]^{{k_{i} }} [(w_{i} - c)(a - bp_{i} )]^{{1 - k_{i} }}$. The reference profit can be obtained by deriving it.

Proof of proposition 1

In the simultaneous game, the bargaining process is carried out separately. The marginal utility of retailer $i$ is:

$$ \frac{{\partial U_{{R_{i} }} }}{{\partial p_{i} }} = (1 + \lambda_{i} )(a - 2bp_{i} + bw_{i} - \frac{{k_{i} \lambda_{i} }}{{k_{i} \lambda_{i} + \lambda_{i} + 1}}(a - 2bp_{i} + bc)) $$

(A.1)

By backward induction, the optimal solution can be obtained.

Proof of proposition 2

In the second stage of sequential bargaining game, the supplier negotiates with the retailer 2. The objective of the supplier is:

$$ \pi_{S,2} = (w_{2} - c)(a - bp_{2} ) $$

(A.2)

As the second mover, the retailer 2 has both distributional fairness concern and peer-induced fairness concern. The objective of the retailer 2 is:

$$ U_{{R_{2} }} = \pi_{{R_{2} }} - \lambda_{2} (\frac{{k_{2} }}{{1 - k_{2} }}\overline{\pi }_{S,2} - \pi_{{R_{2} }} ) - \theta (\frac{{k_{2} }}{{k_{1} }}\overline{\pi }_{{R_{1} }} - \pi_{{R_{2} }} ) $$

(A.3)

Considering that $\overline{\pi }_{S,2} + \overline{\pi }_{{R_{2} }} = \pi_{S,2} + \pi_{{R_{2} }} = \pi_{2}$, the bargaining process can be formulated as:

$$ \max [\pi_{{R_{2} }} - \lambda_{2} (\frac{{k_{2} }}{{1 - k_{2} }}(\pi_{2} - \overline{\pi }_{{R_{2} }} ) - \pi_{{R_{2} }} ) - \theta (\frac{{k_{2} }}{{k_{1} }}\overline{\pi }_{{R_{1} }} - \pi_{{R_{2} }} )]^{{k_{2} }} [(w_{2} - c)(a - bp_{2} )]^{{1 - k_{2} }} $$

(A.4)

The reference profit can be obtained by deriving it. based on the reference points, the utility of the retailer 2 is:

$$ U_{{R_{2} }} = (1 + \lambda_{2} + \theta )\pi_{{R_{2} }} - \frac{{k_{2} \lambda_{2} (k_{1} (1 + \theta + \lambda_{2} )\pi_{2} - \theta k_{2} \overline{\pi }_{{R_{1} }} )}}{{k_{1} (k_{2} \lambda_{2} + 1 + \theta + \lambda_{2} )}} - \frac{{\theta k_{2} \overline{\pi }_{{R_{1} }} }}{{k_{1} }} $$

(A.5)

The marginal utility of retailer 2 is:

$$ \frac{{\partial U_{{R_{2} }} }}{{\partial p_{2} }} = (1 + \lambda_{2} + \theta )(a - 2bp_{2} + bw_{2} - \frac{{k_{2} \lambda_{2} }}{{k_{2} \lambda_{2} + \lambda_{2} + \theta + 1}}(a - 2bp_{2} + bc)) $$

(A.6)

By backward induction, the optimal solution can be obtained.

Proof of proposition 3

The proof is similar to that of Proposition 2. In the first stage of sequential bargaining game, the supplier negotiates with the retailer 1. The objective of the supplier is:

$$ \pi_{S,1} = (w_{1} - c)(a - bp_{1} ) $$

(A.7)

As the first mover, the retailer 1 has only distributional fairness concern. The objective of the retailer 1 is:

$$ U_{{R_{1} }} = \pi_{{R_{1} }} - \lambda_{1} (\frac{{k_{1} }}{{1 - k_{1} }}\overline{\pi }_{S,1} - \pi_{{R_{1} }} ) $$

(A.8)

Considering that $\overline{\pi }_{S,1} + \overline{\pi }_{{R_{1} }} = \pi_{S,1} + \pi_{{R_{1} }} = \pi_{1}$, the bargaining process can be formulated as:

$$ \max [\pi_{{R_{1} }} - \lambda_{1} (\frac{{k_{1} }}{{1 - k_{1} }}\overline{\pi }_{S,1} - \pi_{{R_{1} }} )]^{{k_{1} }} [(w_{1} - c)(a - bp_{1} ) - \pi_{s}^{r} ]^{{1 - k_{1} }} $$

(A.9)

The reference profit can be obtained by deriving it. based on the reference points, the utility of the retailer 2 is:

$$ U_{{R_{1} }} = (1 + \lambda_{1} )\pi_{{R_{1} }} - \frac{{k_{1} \lambda_{1} ((1 + \lambda_{1} )(1 - k_{1} )\pi_{1} + k_{1} (1 + \lambda_{1} )\pi_{s}^{r} )}}{{(1 - k_{1} )(k_{1} \lambda_{1} + \lambda_{1} + 1)}} $$

(A.10)

The marginal utility of retailer 1 is:

$$ \frac{{\partial U_{{R_{1} }} }}{{\partial p_{1} }} = (1 + \lambda_{1} )(a - 2bp_{1} + bw_{1} - \frac{{k_{1} \lambda_{1} }}{{k_{1} \lambda_{1} + \lambda_{1} + \theta + 1}}(a - 2bp_{2} + bc)) $$

(A.11)

By backward induction, the optimal solution can be obtained.

Proof of proposition 4

In order to ensure that the retailer’s selling price is higher than the wholesale price, $a > bc$ should be satisfied. The first derivatives of the optimal solutions with respect to bargaining power are:

$$ \begin{gathered} \frac{{\partial w_{i}^{*} }}{{\partial k_{i} }} = - \frac{{(a - bc)\lambda_{i} (1 + \lambda_{i} )}}{{2b(1 + \lambda_{i} + k_{i} \lambda_{i} )^{2} }} < 0,\frac{{\partial \pi_{{R_{i} }}^{*} }}{{\partial k_{i} }} = \frac{{(a - bc)^{2} \lambda_{i} (1 + \lambda_{i} )}}{{8b(1 + \lambda_{i} + k_{i} \lambda_{i} )^{2} }} > 0 \hfill \\ \frac{{\partial \pi_{S,i}^{*} }}{{\partial k_{i} }} = - \frac{{(a - bc)^{2} \lambda_{i} (1 + \lambda_{i} )}}{{8b(1 + \lambda_{i} + k_{i} \lambda_{i} )^{2} }} < 0,\frac{{\partial U_{{R_{i} }}^{*} }}{{\partial k_{i} }} = - \frac{{(a - bc)^{2} \lambda_{i} (1 + \lambda_{i} )^{2} }}{{16b(1 + \lambda_{i} + k_{i} \lambda_{i} )^{2} }} < 0 \hfill \\ \end{gathered} $$

(A.12)

The profit difference between the supplier and the retailer with only the distributional fairness concern is: $\pi_{{R_{i} }}^{*} - \pi_{S,i}^{*} = \frac{{(a - bc)^{2} (3k_{1} \lambda_{1} - 1 - \lambda_{1} )}}{{16b(1 + \lambda_{1} + k_{1} \lambda_{1} )}}$. Denoting $\widehat{{k_{i} }} = \frac{{\lambda_{i} + 1}}{{3\lambda_{i} }}$, $\pi_{{R_{i} }}^{*}$ is higher than $\pi_{S,i}^{*}$ when $k_{i} > \widehat{{k_{i} }}$. Similarly, the profit difference between the supplier and the retailer with dual fairness concerns is: $\pi_{{R_{2} }}^{*} - \pi_{S,2}^{*} = \frac{{(a - bc)^{2} (3k_{2} \lambda_{2} - 1 - \lambda_{2} - \theta )}}{{16b(1 + \lambda_{2} + \theta + k_{2} \lambda_{2} )}}$. Denoting $\widehat{{k_{2} }} = \frac{{1 + \lambda_{2} + \theta }}{{3\lambda_{2} }}$, $\pi_{{R_{2} }}^{*}$ is higher than $\pi_{S,2}^{*}$ when $k_{2} > \widehat{{k_{2} }}$.

Proof of proposition 5

The first derivatives of the optimal solutions with respect to the distributional fairness concern are:

$$ \begin{gathered} \frac{{\partial w_{i}^{*} }}{{\partial \lambda_{i} }} = - \frac{{(a - bc)k_{i} }}{{2b(1 + \lambda_{i} + k_{i} \lambda_{i} )^{2} }} < 0,\frac{{\partial \pi_{{R_{i} }}^{*} }}{{\partial \lambda_{i} }} = \frac{{(a - bc)^{2} k_{i} }}{{8b(1 + \lambda_{i} + k_{i} \lambda_{i} )^{2} }} > 0 \hfill \\ \frac{{\partial \pi_{S,i}^{*} }}{{\partial \lambda_{i} }} = - \frac{{(a - bc)^{2} k_{i} }}{{8b(1 + \lambda_{i} + k_{i} \lambda_{i} )^{2} }} < 0,\frac{{\partial U_{{R_{i} }}^{*} }}{{\partial \lambda_{i} }} = \frac{{(a - bc)^{2} (1 + \lambda_{i} )(1 + k_{i} (\lambda_{i} - 1) + \lambda_{i} )}}{{16b(1 + \lambda_{i} + k_{i} \lambda_{i} )^{2} }} > 0 \hfill \\ \end{gathered} $$

(A.13)

From the perspective of the retailer with dual fairness concerns, the results are:

$$ \begin{gathered} \frac{{\partial w_{2}^{*} }}{{\partial \lambda_{2} }} = - \frac{{(a - bc)k_{2} (1 + \theta )}}{{2b(1 + \lambda_{2} + k_{2} \lambda_{2} + \theta )^{2} }} < 0,\frac{{\partial \pi_{{R_{2} }}^{*} }}{{\partial \lambda_{2} }} = \frac{{(a - bc)^{2} k_{2} (1 + \theta )}}{{8b(1 + \lambda_{2} + k_{2} \lambda_{2} + \theta )^{2} }} > 0, \\ \frac{{\partial \pi_{S,2}^{*} }}{{\partial \lambda_{2} }} = - \frac{{(a - bc)^{2} k_{2} (1 + \theta )}}{{8b(1 + \lambda_{2} + k_{2} \lambda_{2} + \theta )^{2} }} < 0, \\ \frac{{\partial U_{{R_{2} }}^{*} }}{{\partial \lambda_{2} }} = \frac{{\left[ \begin{gathered} (a - bc)^{2} [k_{2}^{2} (1 + \lambda_{1} )\theta (1 + \theta ) + k_{1}^{2} \lambda_{1} (1 + \lambda_{2} + \theta )((1 - k_{2} )(1 + \theta ) + \lambda_{2} (1 + k_{2} )) \hfill \\ + k_{1} [3k_{2}^{2} \lambda_{1} \theta (1 + \theta ) + (1 + \lambda_{1} )(1 + \lambda_{2} + \theta )(\lambda_{2} (1 + k_{2} ) + (1 - k_{2} )(1 + \theta ))]] \hfill \\ \end{gathered} \right]}}{{16bk_{1} (1 + \lambda_{1} + k_{1} \lambda_{1} )(1 + \lambda_{2} + k_{2} \lambda_{2} + \theta )^{2} }} > 0. \\ \end{gathered} $$

(A.14)

Considering the above results comprehensively, we can get proposition 5.

Proof of proposition 6

The first derivatives of the optimal solutions with respect to the peer-induced fairness concern are:

$$ \begin{gathered} \frac{{\partial w_{2}^{*} }}{\partial \theta } = \frac{{(a - bc)k_{2} \lambda_{2} }}{{2b(1 + \lambda_{2} + k_{2} \lambda_{2} + \theta )^{2} }} > 0,\frac{{\partial \pi_{{R_{2} }}^{*} }}{\partial \theta } = - \frac{{(a - bc)^{2} k_{2} \lambda_{2} }}{{8b(1 + \lambda_{2} + k_{2} \lambda_{2} + \theta )^{2} }} < 0, \\ \frac{{\partial \pi_{S,2}^{*} }}{\partial \theta } = \frac{{(a - bc)^{2} k_{2} \lambda_{2} }}{{8b(1 + \lambda_{2} + k_{2} \lambda_{2} + \theta )^{2} }} > 0, \\ \frac{{\partial U_{{R_{2} }}^{*} }}{\partial \theta } = \frac{{\left[ \begin{gathered} [k_{1}^{2} \lambda_{1} - k_{2} (1 + \lambda_{1} ) + k_{1} (1 + \lambda_{1} - 3k_{2} \lambda_{1} )](1 + \lambda_{2} + \theta )(1 + \lambda_{2} + k_{2} \lambda_{2} + \theta ) \hfill \\ - k_{2} \lambda_{2} [k_{2} (1 + \lambda_{1} )\theta - k_{1}^{2} \lambda_{1} (1 + \lambda_{2} + \theta ) - k_{1} (1 + \lambda_{2} + \theta + \lambda_{1} (1 + \lambda_{2} + \theta - 3k_{2} \theta ))] \hfill \\ \end{gathered} \right]}}{{16bk_{1} (1 + \lambda_{1} + k_{1} \lambda_{1} )(1 + \lambda_{2} + k_{2} \lambda_{2} + \theta )^{2} }} < 0. \\ \end{gathered} $$

(A.15)

Considering the above results comprehensively, we can get proposition 6.

Proof of proposition 7

Given the Nash equilibrium point $E^{*}$, the equilibrium solutions of the selling prices are:

$$ p_{1} = \frac{{bk_{1} \lambda_{1} (w_{1} - c) + (\lambda_{1} + 1)(a + bw_{1} )}}{{2b(\lambda_{1} + 1)}},p_{2} = \frac{{bk_{2} \lambda_{2} (w_{2} - c) + (1 + \theta + \lambda_{2} )(a + bw_{2} )}}{{2b(1 + \theta + \lambda_{2} )}} $$

(A.16)

By backward induction, the optimal wholesale prices are:

$$ w_{2}^{*} = \frac{{(a + bc)(1 + \theta + \lambda_{2} ) + 2bck_{2} \lambda_{2} }}{{2b(1 + \theta + \lambda_{2} + k_{2} \lambda_{2} )}},w_{1}^{*} = \frac{{a(1 + \lambda_{1} ) + bc(1 + \lambda_{1} + 2k_{1} \lambda_{1} )}}{{2b(k_{1} \lambda_{1} + \lambda_{1} + 1)}} $$

(A.17)

Correspondingly, the supplier’s profit and the retailer 2’s utility in the second stage of sequential bargaining game are:

$$ \pi_{S,2}^{*} = \frac{{(a - bc)^{2} (1 + \lambda_{2} + \theta )}}{{8b(1 + \theta + \lambda_{2} + k_{2} \lambda_{2} )}},\pi_{{R_{2} }}^{*} = \frac{{(a - bc)^{2} (1 + \lambda_{2} + \theta { + }3k_{2} \lambda_{2} )}}{{16b(1 + \theta + \lambda_{2} + k_{2} \lambda_{2} )}} $$

(A.18)

The supplier’s profit and the retailer 1’s utility in the first stage of sequential bargaining game are:

$$ \pi_{S,1}^{*} = \frac{{(a - bc)^{2} (1 + \lambda_{1} )}}{{8b(k_{1} \lambda_{1} + \lambda_{1} + 1)}},\pi_{{R_{1} }}^{*} = \frac{{(a - bc)^{2} (1 + \lambda_{1} + 3k_{1} \lambda_{1} )}}{{16b(k_{1} \lambda_{1} + \lambda_{1} + 1)}} $$

(A.19)

Given that, the difference of the supplier’s profits in the negotiation with different retailers is:

$$ \Delta \pi_{S} { = }\pi_{S,2}^{*} - \pi_{S,1}^{*} = \frac{{(a - bc)^{2} (k_{1} \lambda_{1} - k_{2} \lambda_{2} + k_{1} \lambda_{1} \lambda_{2} - k_{2} \lambda_{1} \lambda_{2} + k_{1} \lambda_{1} \theta )}}{{8b(k_{1} \lambda_{1} + \lambda_{1} + 1)(1 + \theta + \lambda_{2} + k_{2} \lambda_{2} )}} $$

(A.20)

Denoting that $\widehat{\theta } = \frac{{ - k_{1} \lambda_{1} + k_{2} \lambda_{2} - k_{1} \lambda_{1} \lambda_{2} + k_{2} \lambda_{1} \lambda_{2} }}{{k_{1} \lambda_{1} }}$, when $\theta > \widehat{\theta }$, $\Delta \pi_{S} > 0$.

Proof of proposition 8

The difference of the supplier’s profits in the negotiation with different retailers is:

$$ \begin{aligned}\Delta U_{R} &= U_{{R_{2} }}^{*} - U\pi_{{R_{1} }}^{*} \\&= \frac{{(a - bc)^{2} \left[ \begin{gathered} (1 - k_{1} )[k_{1}^{2} \lambda_{1} - k_{2} (\lambda_{1} + 1) + k_{1} (1 + \lambda_{1} - 3k_{2} \lambda_{1} )]\theta^{2} + [2k_{1}^{3} \lambda_{1} (\lambda_{1} - \lambda_{2} ) \hfill \\ - k_{2} (1 + \lambda_{1} )(1 + \lambda_{2} ) + k_{1} (1 + \lambda_{1} )(1 - \lambda_{1} + 2\lambda_{2} ) + k_{1} k_{2} (1 - 2\lambda_{1} )(1 + \lambda_{2} ) \hfill \\ + k_{1}^{2} (\lambda_{1}^{2} - 1 - 2\lambda_{2} + \lambda_{1} (2 + 3k_{2} (1 + \lambda_{2} )))]\theta + k_{1} [(1 - k_{1} )(1 - k_{2} + \lambda_{2} )\lambda_{2} \hfill \\ - \lambda_{1}^{2} (1 + (1 + \lambda_{2} )\lambda_{2} - 2k_{1}^{2} (1 + \lambda_{2} ) - k_{1} (1 + \lambda_{2} + k_{2} \lambda_{2} )) \hfill \\ - \lambda_{1} (1 + 2k_{2} \lambda_{2} - \lambda_{2}^{2} - 2k_{1} (1 + \lambda_{2} + k_{2} \lambda_{2} ) - k_{1}^{2} (1 - \lambda_{2}^{2} ))] \hfill \\ \end{gathered} \right]}}{{16b(k_{1} \lambda_{1} + \lambda_{1} + 1)(1 + \theta + \lambda_{2} + k_{2} \lambda_{2} )k_{1} (1 - k_{1} )}}\end{aligned} $$

(A.21)

The threshold $\widehat{\theta }^{\prime }$ can be obtained by solving $\Delta U_{R} = 0$. Next, we prove the uniqueness of zero point. Since that $\frac{{(a - bc)^{2} }}{{16b(k_{1} \lambda_{1} + \lambda_{1} + 1)(1 + \theta + \lambda_{2} + k_{2} \lambda_{2} )k_{1} (1 - k_{1} )}} > 0$, it can also be transformed into the discussion of zero point of $f(\theta )$. Where,

$$ \begin{gathered} f(\theta ) = (1 - k_{1} )[k_{1}^{2} \lambda_{1} - k_{2} (\lambda_{1} + 1) + k_{1} (1 + \lambda_{1} - 3k_{2} \lambda_{1} )]\theta^{2} + [2k_{1}^{3} \lambda_{1} (\lambda_{1} - \lambda_{2} ) - k_{2} (1 + \lambda_{1} )(1 + \lambda_{2} ) \\ + k_{1} (1 + \lambda_{1} )(1 - \lambda_{1} + 2\lambda_{2} ) + k_{1} k_{2} (1 - 2\lambda_{1} )(1 + \lambda_{2} ) + k_{1}^{2} (\lambda_{1}^{2} - 1 - 2\lambda_{2} + \lambda_{1} (2 + 3k_{2} (1 + \lambda_{2} )))]\theta \\ + k_{1} [(1 - k_{1} )(1 - k_{2} + \lambda_{2} )\lambda_{2} - \lambda_{1}^{2} (1 + (1 + \lambda_{2} )\lambda_{2} - 2k_{1}^{2} (1 + \lambda_{2} ) - k_{1} (1 + \lambda_{2} + k_{2} \lambda_{2} )) \\ - \lambda_{1} (1 + 2k_{2} \lambda_{2} - \lambda_{2}^{2} - 2k_{1} (1 + \lambda_{2} + k_{2} \lambda_{2} ) - k_{1}^{2} (1 - \lambda_{2}^{2} ))] \\ \end{gathered} $$

(A.22)

The first derivative of it is:

$$ \begin{gathered} f^{\prime}(\theta ) = - k_{1} (1 + \lambda_{1} )[(1 - k_{1} )(1 + \lambda_{1} ) - 2k_{1}^{2} \lambda_{1} ] - (1 - k_{1} )(1 + \lambda_{2} + \theta )[k_{2} (1 + \lambda_{1} + 3k_{1} \lambda_{1} )] \\ - (1 - k_{1} )[k_{2} (1 + \lambda_{1} + 3k_{1} \lambda_{1} )\theta - k_{1} (1 + \lambda_{1} + k_{1} \lambda_{1} )(1 + \lambda_{2} + \theta )] \\ \end{gathered} $$

(A.23)

Since that $f^{\prime}(\theta ) < 0$,$f(0) > 0$,$f(1) < 0$, there is a unique zero point.

Proof of proposition 9

The difference of the supplier’s profits in both games is:

$$ \Delta \pi_{S}^{\prime } = \pi_{{\text{S}}}^{Seq} - \pi_{{\text{S}}}^{Sim} = \frac{{(a - bc)^{2} k_{2} \lambda_{2} \theta }}{{8b(1 + \lambda_{2} + k_{2} \lambda_{2} )(1 + \lambda_{2} + k_{2} \lambda_{2} + \theta )}} > 0 $$

(A.24)

Therefore, the supplier can profit more from the sequential game than from the simultaneous game.