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The antagonism and cohesion of the upstream supply chain under information asymmetry

  • S.I.: Information- Transparent Supply Chains
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Abstract

Information asymmetry leads to short-term competitive antagonism between downstream manufacturers and upstream suppliers. This paper applies a master-slave game to reproduce the interest conflict between a manufacturer and a supplier, and proposes a cooperative decision-making model based on information sharing, considering the interest consistency of upstream supply chain members. Additionally, the overall profit coordination mechanism is constructed to ensure the continuation of the cooperative decision-making relationship. In addition, to ensure that the supplier participating in cooperative decision-making has the supply capacity desired by the manufacturer, a rational evaluation system combined with game theory is employed to determine the most suitable participating supplier. A numerical example is given to demonstrate the systematization and effectiveness of the proposed method. Finally, through sensitivity analysis and comparative analysis, managerial insights and recommendations are obtained.

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Acknowledgements

The study was supported by “Fundamental Research Funds for the Central Universities” (JB190606), “Major Theoretical and Practical Research Projects of Social Science in Shaanxi province” (2019C068).

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Authors and Affiliations

  1. Department of Management Engineering, School of Economics & Management, Xidian University, Xi’an, China

    Aijun Liu, Senhao Luo & Hongwei Qiu

  2. Pusan National University, Busan, South Korea

    Jian Mou

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  1. Aijun Liu
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  2. Senhao Luo
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Correspondence to Jian Mou.

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Appendices

Appendix A: Linguistic scale and entropy weight

Assuming that a group of linguistic variables of TFNs are represented by TFNs = {L, ML, M, MH, H}, the meaning is shown in Fig. 8.

Fig. 8
figure 8

Linguistic scale

Full size image

With m evaluation criteria and n evaluation objects, the original data matrix A is

$$ A = \left[ {\begin{array}{*{20}l} {a_{11} } \hfill & {a_{12} } \hfill & \ldots \hfill & {a_{1n} } \hfill \\ {a_{21} } \hfill & {a_{22} } \hfill & \ldots \hfill & {a_{2n} } \hfill \\ \vdots \hfill & \vdots \hfill & \vdots \hfill & \vdots \hfill \\ {a_{m1} } \hfill & {a_{m2} } \hfill & \ldots \hfill & {a_{mn} } \hfill \\ \end{array} } \right] $$
(40)

Matrix \( B = (b_{ij})_{m \times n }\) is determined after normalization of raw data matrix A. Where bij is the normalized value of the jth evaluation object in the ith evaluation criterion, \( b_{ij} \in [0,1] \), j = 1, 2, …, n. Moreover, for profitability indicators (positive indicators), there are

$$ b_{ij} = \frac{{a_{ij} - \mathop {\hbox{min} }\limits_{j} \left\{ {a_{ij} } \right\}}}{{\mathop {\hbox{max} }\limits_{j} \left\{ {a_{ij} } \right\} - \mathop {\hbox{min} }\limits_{j} \left\{ {a_{ij} } \right\}}} $$
(41)

For cost indicators (negative indicators), there are

$$ b_{ij} = \frac{{\mathop {\hbox{max} }\limits_{j} \left\{ {a_{ij} } \right\} - a_{ij} }}{{\mathop {\hbox{max} }\limits_{j} \left\{ {a_{ij} } \right\} - \mathop {\hbox{min} }\limits_{j} \left\{ {a_{ij} } \right\}}} $$
(42)

Among the evaluation problems with m evaluation criteria and n evaluation objects, the entropy definition of the ith evaluation criterion is

$$ E_{i} = - k\sum\limits_{j = 1}^{n} {c_{ij} \ln c_{ij} ,\quad i = 1,2, \ldots ,m.} $$
(43)

where

$$ c_{ij} = \frac{{b_{ij} }}{{\sum\limits_{j = 1}^{n} {b_{ij} } }},k = \frac{1}{\ln n}. $$
(44)

After defining the entropy of the ith evaluation criterion, the definition of the entropy weight of the ith evaluation criterion can be obtained.

$$ W_{Ei} = \frac{{1 - E_{i} }}{{m - \sum\limits_{i = 1}^{m} {E_{i} } }} $$
(45)

where

$$ 0 \le W_{Ei} \le 1,\sum\limits_{i = 1}^{m} {W_{Ei} } = 1. $$
(46)

Appendix B: Profits under non-cooperative game

$$ \begin{aligned} \prod\nolimits_{S}^{*} = - F_{1} - P_{U1} \left( {a - \left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right.} \right. \hfill \\ \_\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \hfill \\ +\, bP_{U1} ( - \frac{{\sqrt[3]{{27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} + \sqrt {\left( {27b{\rm O}_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( { - bP_{U1} O_{C1} + aO_{C2} - bP_{U1} O_{C2} } \right)^{3} } }}}}{{3\sqrt[3]{2}S_{C2} }} \hfill \\ \left. { -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right)/ \hfill \\ \left( {2\left( { -\, \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \hfill \\ \left. {\left. {\_\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right)} \right) \hfill \\ + \left( {\left( {a\left( { -\, \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right.} \right. \hfill \\ \_\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \hfill \\ +\, bP_{U1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \hfill \\ \left. {\_\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right) \hfill \\ \left( {a - \left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right.} \right. \hfill \\ -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \hfill \\ + bP_{U1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \hfill \\ -\, \left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right)/ \hfill \\ \left( {2\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }} - } \right.} \right. \hfill \\ \end{aligned} $$
$$ \begin{aligned} \left. {\left. {\left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right)} \right)} \right)/ \hfill \\ \left( {2b\left( { -\, \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \hfill \\ -\, \left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right) \hfill \\ + \frac{1}{2}\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \hfill \\ - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)S_{C1} \hfill \\ - \left( {O_{C1} \left( {a - \left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right.} \right.} \right. \hfill \\ -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \hfill \\ + bP_{UC1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \hfill \\ \left. { -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right)/ \hfill \\ \left( {2\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \hfill \\ - \left. {\left. {\left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right)} \right)} \right)/ \hfill \\ -\, \left( {\frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \hfill \\ -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} \prod\nolimits_{M}^{*} & = - F_{2} - \frac{1}{2}S_{C2} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \\ & \quad - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad +\, \frac{{c\left( {P^{ - } \left( {\frac{{ - \gamma \theta^{ - } + \gamma \theta - P^{ - } + P_{U2} }}{{2P^{ - } }} + 1} \right) - P_{U2} } \right)\left( {\gamma \left( {\theta - \theta^{ - } } \right) + \frac{1}{2}\left( {\gamma \theta^{ - } - \gamma \theta + P^{ - } - P_{U2} } \right)} \right)}}{{\gamma \left( {\theta - \theta^{ - } } \right)}} \\ & \quad \_\left( {O_{C2} \left( {a - \left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right.} \right.} \right. \\ & \quad -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad + bP_{U1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \\ & \quad \left. {\left. { -\, \frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right)/ \\ & \quad \left( {2\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \\ & \quad -\, \left. {\left. {\left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right)} \right)} \right)/ \\ & \quad \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \\ & \quad -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad -\, \left( {\left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right.} \right. \\ & \quad -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad +\, bP_{U1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \\ & \quad \left. { - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right) \\ & \quad \left( {a - \left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }} - } \right.} \right.} \right. \\ \end{aligned} $$
$$ \begin{aligned} \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \hfill \\ +\, bP_{U1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \hfill \\ - \left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right)/ \hfill \\ \left( {2\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \hfill \\ \left. {\left. {\left. { -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right)} \right)} \right)/ \hfill \\ -\, \left( {2b\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \hfill \\ \left. { -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right) \hfill \\ \end{aligned} $$

Appendix C: Profits under cooperative decision-making

$$ \begin{aligned} \tau_{1} (f) & = \frac{1}{2}\left( { - 2F_{1} + \frac{1}{2}S_{C1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \\ & \quad -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad +\, \frac{1}{2}S_{C2} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \\ & \quad -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad - \,P_{U1} \left( {a - \left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right.} \right. \\ & \quad -\, \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + \\ \end{aligned} $$
$$ \begin{aligned} bP_{U1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \hfill \\ - \left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right)/ \hfill \\ \left( {2\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \hfill \\ \left. {\left. { - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right)} \right) \hfill \\ + \frac{{c\left( {P^{ - } \left( {\frac{{ - \gamma \theta^{ - } + \gamma \theta - P^{ - } + P_{UC2} }}{{2P^{ - } }} + 1} \right) - P_{U2} } \right)\left( {\gamma \left( {\theta - \theta^{ - } } \right) + \frac{1}{2}\left( {\gamma \theta^{ - } - \gamma \theta + P^{ - } - P_{U2} } \right)} \right)}}{{\gamma \left( {\theta - \theta^{ - } } \right)}} \hfill \\ + \frac{{c\left( {P^{ - } \left( {\frac{{ - \gamma \theta^{ - } + \gamma \theta - P^{ - } + P_{U1} + P_{U2} }}{{2P^{ - } }} + 1} \right) - P_{U1} - P_{U2} } \right)\left( {\gamma \left( {\theta - \theta^{ - } } \right) + \frac{1}{2}\left( {\gamma \theta^{ - } - \gamma \theta + P^{ - } - P_{U1} - P_{U2} } \right)} \right)}}{{\gamma \left( {\theta - \theta^{ - } } \right)}} \hfill \\ - \left( {O_{C1} \left( {a - \left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right.} \right.} \right. \hfill \\ - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \hfill \\ + bP_{U1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \hfill \\ - \left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right)/ \hfill \\ \left( {2\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \hfill \\ \left. {\left. {\left. { - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right)} \right)} \right)/ \hfill \\ \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \hfill \\ - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \hfill \\ \end{aligned} $$
$$ \begin{aligned} \tau_{2} (f) = \frac{1}{2}\left( { - 2F_{2} - \frac{1}{2}S_{C1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. - \hfill \\ - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \hfill \\ - \frac{1}{2}S_{C2} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \hfill \\ - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \hfill \\ + P_{U1} \left( {a - \left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }} - } \right.} \right.} \right. \hfill \\ \end{aligned} $$
$$ \begin{aligned} & \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad + bP_{U1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \\ & \quad - \left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right)/ \\ & \quad \left( {2\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \\ & \quad + \left. {\left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right)} \right) \\ & \quad + \left( {O_{C1} \left( {a - \left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right.} \right.} \right. \\ & \quad - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad + bP_{U1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \\ & \quad - \left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right)/ \\ & \quad \left( {2\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \\ & \quad - \left. {\left. {\left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right)} \right)} \right)/ \\ & \quad \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \\ & \quad - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad + \left( {O_{C2} \left( {a - \left( {a\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }} - } \right.} \right.} \right.} \right. \\ \end{aligned} $$
$$ \begin{aligned} & \quad \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad + bP_{U1} \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \\ & \quad - \left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) + bO_{C1} } \right)/ \\ & \quad \left( {2\left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right.} \right. \\ & \quad - \left. {\left. {\left. {\left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right)} \right)} \right)} \right)/ \\ & \quad \left( { - \frac{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}{{3\sqrt[3]{2}S_{C2} }}} \right. \\ & \quad - \left. {\frac{{\sqrt[3]{2}\left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)}}{{\sqrt[3]{{\sqrt {\left( {27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} } \right)^{2} - 108S_{C2}^{3} \left( {aO_{C2} - bO_{C1} P_{U1} - bO_{C2} P_{U1} } \right)^{3} } + 27bO_{C1}^{2} S_{C2}^{2} + 54bO_{C1} O_{C2} S_{C2}^{2} }}}}} \right) \\ & \quad + \frac{{c\left( {P^{ - } \left( {\frac{{ - \gamma \theta^{ - } + \gamma \theta - P^{ - } + P_{U1} + P_{U2} }}{{2P^{ - } }} + 1} \right) - P_{U1} - P_{U2} } \right)\left( {\gamma \left( {\theta - \theta^{ - } } \right) + \frac{1}{2}\left( {\gamma \theta^{ - } - \gamma \theta + P^{ - } - P_{U1} - P_{U2} } \right)} \right)}}{{\gamma \left( {\theta - \theta^{ - } } \right)}} \\ & \quad - \left. {\frac{{c\left( {\gamma \left( {\theta - \theta^{ - } } \right) + \frac{1}{2}\left( {\gamma \theta^{ - } - \gamma \theta + P^{ - } - P_{U2} } \right)} \right)\left( {P^{ - } \left( {\frac{{ - \gamma \theta^{ - } + \gamma \theta - P^{ - } + P_{U2} }}{{2P^{ - } }} + 1} \right) - P_{U2} } \right)}}{{\gamma \left( {\theta - \theta^{ - } } \right)}}} \right) \\ \end{aligned} $$

Appendix D: Pseudocodes of algorithm core process

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Liu, A., Luo, S., Mou, J. et al. The antagonism and cohesion of the upstream supply chain under information asymmetry. Ann Oper Res 329, 527–572 (2023). https://doi.org/10.1007/s10479-020-03881-5

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  • DOI: https://doi.org/10.1007/s10479-020-03881-5

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