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Ticket pricing for entertainment events under a dual-channel environment: a game-theoretical approach using uncertainty theory

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Abstract

The pricing of tickets is a crucial decision when managing entertainment events like concerts, operas, ballets, and theater performances. In this study, we examine the pricing decisions made by two major decision-makers before an entertainment event: the venue and the online retailer. In a game-theoretic approach with uncertain information, the interactions between decision-makers are studied in a single and dual-channel structure. A confidence-level-based uncertainty theory is utilized in order to formulate the amount of uncertainty, and the optimal solutions are analytically computed to maximize the total profit for both players. There are three distinct cases studied, including two single-channel market cases (one in which only the venue sells the tickets, and another in which only the online platform sells the tickets) as well as a dual-channel market case (in which both players sell tickets). Stackelberg and Nash games are used to illustrate the interactions between players in each game. The results indicate the optimal pricing, service level improvement, and total profit for both the venue and the online retailer. Additionally, it is shown that the amount of risk (confidence level) the decision-makers are willing to accept strongly affects the optimal solution. In some cases, the roles of the market participants do not have a significant impact on the optimal solution. The findings of this paper can help practitioners in the entertainment industry make better ticket pricing decisions in an uncertain and competitive environment.

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Acknowledgement

This research was supported by National Natural Science Foundation of China Project (72072021, 71772032).

Author information

Authors and Affiliations

  1. Department of Management and Marketing, East Tennessee State University, Johnson City, TN, USA

    Reza Maihami

  2. Department of Technology and Innovation, University of Southern Denmark, Odense, Denmark

    Devika Kannan

  3. School of Business, Woxsen University, Sadasivpet, Telangana, India

    Devika Kannan

  4. Newcastle Business School, Northumbria University, Newcastle Upon Tyne, UK

    Mohammad Fattahi

  5. School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, Sichuan, China

    Chunguang Bai

  6. School of Business, Howard University, Washington, DC, 20059, USA

    Iman Ghalehkhondabi

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  1. Reza Maihami
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  2. Devika Kannan
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Correspondence to Chunguang Bai.

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Appendix

Appendix

Proof of Lemma 1

The optimal selling price and service level are as follows:

In case 1, the venue, as the only ticket seller, wants to maximize its total profit, Eq. (5). The venue’s total profit depends upon the margin sale price \({m}_{V}\) and the level of service time \(L\). Thus, solving the following simultaneous equations systems will determine the optimal solution:

$$\left\{\begin{array}{c}\frac{\partial {\Pi }_{V}}{\partial {m}_{V}}=0\\ \frac{\partial {\Pi }_{V}}{\partial L}=0\end{array}\right.$$

Proof of Proposition 1

We will prove the second-order conditions with respect to \({m}_{V}\) and \(L\). Thus, we need to compute the Hessian matrix as follows:

$$\mathrm{H}=\left(\begin{array}{cc}\frac{{\partial }^{2}{\Pi }_{V}}{\partial {{m}_{V}}^{2}}& \frac{{\partial }^{2}{\Pi }_{V}}{\partial {m}_{V}\partial L}\\ \frac{{\partial }^{2}{\Pi }_{V}}{\partial \mathrm{L}\partial {m}_{V}}& \frac{{\partial }^{2}{\Pi }_{V}}{\partial {\mathrm{L}}^{2}}\end{array}\right)=\left(\begin{array}{cc}-2{\Phi }_{2V}^{-1}(\alpha )& {\Phi }_{3}^{-1}(1-\alpha )\\ {\Phi }_{3}^{-1}(1-\alpha )& -2k\end{array}\right)$$

A function will be concave if the first principal minor of H is negative; this means that \({H}^{1}< 0\). If the second principal minor of H is positive, \({H}^{2}>0\). Hence, we have:

$${H}^{1}=-2{\Phi }_{2V}^{-1}\left(\alpha \right)<0$$
$${H}^{2}=\mathrm{det}\left(H\right)=4k{\Phi }_{2V}^{-1}\left(\alpha \right)-{{(\Phi }_{3}^{-1}(1-\alpha ))}^{2}$$

To prove that \({H}^{2}>0\), \({\Phi }_{2V}^{-1}\left(\alpha \right)\) should be greater than \(\frac{{{(\Phi }_{3}^{-1}(1-\alpha ))}^{2}}{4k}\). This completes the proof.

Proof of Lemma 2

Equation (15) only depends on \({P}_{\mathrm{OR}}\). To obtain the optimal value, we take the first derivative of \({\Pi }_{J}\) with respect to \({P}_{\mathrm{OR}}\) and solve the resultant equation:

$$ \frac{{d\Pi _{J} }}{{dP_{{OR}} }} = 0 \Rightarrow P_{{{\text{OR}}}} = \frac{1}{2}(f + o + \frac{{\Phi _{1}^{{ - 1}} (1 - \alpha )}}{{\Phi _{{2OR}}^{{ - 1}} (\alpha )}}) $$

Proof of Proposition 2

We examine the second-order condition for \({\Pi }_{J}\) with respect to \({P}_{\mathrm{OR}}\): \(\frac{{d}^{2}{\Pi }_{J}}{d{{P}_{\mathrm{OR}}}^{2}}=-2{\Phi }_{2OR}^{-1}(\alpha )<0\)

This completes the proof.

Proof of Proposition 3

The second-order condition completes the proof as follows:

$$\frac{{d}^{2}{\Pi }_{OR}}{d{{m}_{OR}}^{2}}=-2{\Phi }_{2OR}^{-1}(\alpha )<0$$

Proof of Proposition 4

Like proposition 3, the second-order condition shows the concavity as follows:

$$\frac{{d}^{2}{\Pi }_{V}}{d{w}^{2}}=-2{\Phi }_{2OR}^{-1}(\alpha )$$

Proof of Lemma 3

\({\Pi }_{J}\) depends on \({P}_{V},L,\) and \({P}_{\mathrm{OR}}\). Thus, taking the first derivative with respect to these three variables and solving the resultant simultaneous equations systems will complete the proof:

$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial \Pi _{J} }}{{\partial P_{V} }} = 0} \\ {\frac{{\partial \Pi _{J} }}{{\partial {\text{L}}}} = 0} \\ {\frac{{\partial \Pi _{J} }}{{\partial P_{{{\text{OR}}}} }} = 0} \\ \end{array} } \right. \Rightarrow $$
$${P}_{V}={P}_{V31}=-(1/2{\Phi }_{3}^{-1}(\alpha )({\Phi }_{2OR}^{-1}(\alpha )(-4k{\Phi }_{2V}^{-1}\left(\alpha \right)+{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})+{\Phi }_{2V}^{-1}\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}+({\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha ))(-{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )+k({\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha )))))(-4k{\Phi }_{2OR}^{-1}(\alpha )+{({\Phi }_{3}^{-1}(\alpha ))}^{2})((-1+\rho ){\Phi }_{1}^{-1}\left(1-\alpha \right){\Phi }_{3}^{-1}(\alpha )-o{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )+o{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3OR}^{-1}\left(1-\alpha \right)+(o{\Phi }_{3}^{-1}(1-\alpha )-(f+o){\Phi }_{3}^{-1}(\alpha )){\Phi }_{3V}^{-1}(1-\alpha ))-({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )-2k({\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha )))({\Phi }_{2OR}^{-1}(\alpha )(2o{\Phi }_{3}^{-1}(1-\alpha )-(f+o){\Phi }_{3}^{-1}(\alpha ))+{\Phi }_{3}^{-1}(\alpha )(\rho {\Phi }_{1}^{-1}\left(1-\alpha \right)-o{\Phi }_{3V}^{-1}(1-\alpha )))$$
$$\begin{aligned} & L={L}_{31}=((o{\Phi }_{3OR}^{-1}\left(1-\alpha \right)(-{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )+{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3OR}^{-1}\\&\quad\left(1-\alpha \right))+(o{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )\\&\quad+(o{\Phi }_{3}^{-1}(1-\alpha )-(f+o){\Phi }_{3}^{-1}(\alpha )){\Phi }_{3OR}^{-1}\left(1-\alpha \right)){\Phi }_{3V}^{-1}(1-\alpha )\\&\quad-(f+o){\Phi }_{3}^{-1}(\alpha ){\left({\Phi }_{3V}^{-1}\left(1-\alpha \right)\right)}^{2}\\&\quad+{\Phi }_{2OR}^{-1}(\alpha )(2{\Phi }_{2V}^{-1}\left(\alpha \right)(-o{\Phi }_{3}^{-1}(1-\alpha )\\&\quad+(f+o){\Phi }_{3}^{-1}(\alpha ))+(f+o){\Phi }_{3}^{-1}(1-\alpha )(-{\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha )))\\&\quad+{\Phi }_{1}^{-1}\left(1-\alpha \right)(-2(-1+\rho ){\Phi }_{2OR}^{-1}(\alpha ){\Phi }_{3}^{-1}(1-\alpha )-2\rho {\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )\\&\quad+(\rho {\Phi }_{3}^{-1}(1-\alpha )+(-1+\rho ){\Phi }_{3}^{-1}(\alpha ))({\Phi }_{3OR}^{-1}\left(1-\alpha \right)\\&\quad+{\Phi }_{3V}^{-1}(1-\alpha )))))/(2({{\Phi }_{2OR}^{-1}(\alpha )}_{\alpha }(4k{\Phi }_{2V}^{-1}\left(\alpha \right)-{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})-{\Phi }_{2V}^{-1}\\&\quad\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}-({\Phi }_{3OR}^{-1}\left(1-\alpha \right)\\&\quad+{\Phi }_{3V}^{-1}(1-\alpha ))(-{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad+k({\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha )))))\end{aligned}$$
$$\begin{aligned} & {P}_{\mathrm{OR}}={P}_{OR31}=\left(\left(f+o\right){\Phi }_{2OR}^{-1}\left(\alpha \right)\left(4k{\Phi }_{2V}^{-1}\left(\alpha \right)-{\left({\Phi }_{3}^{-1}\left(1-\alpha \right)\right)}^{2}\right)+o{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}\right.\\ &\quad\left(1-\alpha \right){\Phi }_{3}^{-1}\left(\alpha \right)-2f{\Phi }_{2V}^{-1}\left(\alpha \right){\left({\Phi }_{3}^{-1}\left(\alpha \right)\right)}^{2}\\ &\quad-2o{\Phi }_{2V}^{-1}\left(\alpha \right){\left({\Phi }_{3}^{-1}\left(\alpha \right)\right)}^{2}\\ &\quad+2ko{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3OR}^{-1}\left(1-\alpha \right)-o{\left({\Phi }_{3}^{-1}\left(1-\alpha \right)\right)}^{2}{\Phi }_{3OR}^{-1}\\ &\quad\left(1-\alpha \right)+2f{\Phi }_{3}^{-1}\left(1-\alpha \right){\Phi }_{3}^{-1}\left(\alpha \right){\Phi }_{3OR}^{-1}\left(1-\alpha \right)\\ &\quad+2o{\Phi }_{3}^{-1}\left(1-\alpha \right){\Phi }_{3}^{-1}\left(\alpha \right){\Phi }_{3OR}^{-1}\left(1-\alpha \right)\\ &\quad-2fk{\left({\Phi }_{3OR}^{-1}\left(1-\alpha \right)\right)}^{2}-2ko{\left({\Phi }_{3OR}^{-1}\left(1-\alpha \right)\right)}^{2}\\ &\quad+\left(-2ko{\Phi }_{2V}^{-1}\left(\alpha \right)+\left(f+o\right)\left({\Phi }_{3}^{-1}\left(1-\alpha \right){\Phi }_{3}^{-1}\right.\right.\\ &\left.\left.\quad\left(\alpha \right)-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)\right)\right){\Phi }_{3V}^{-1}\\ &\quad\left(1-\alpha \right)+{\Phi }_{1}^{-1}\left(1-\alpha \right)\left(4k\rho {\Phi }_{2V}^{-1}\left(\alpha \right)\right.\\ &\quad-\rho {\left({\Phi }_{3}^{-1}\left(1-\alpha \right)\right)}^{2}\\ &\quad+\left(-1+\rho \right){\Phi }_{3}^{-1}\left(1-\alpha \right){\Phi }_{3}^{-1}\left(\alpha \right)\\ &\quad\left.\left.-2k\left(-1+\rho \right)\left({\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}\left(1-\alpha \right)\right)\right)\right)(1/(2({\Phi }_{2OR}^{-1}(\alpha )(4k{\Phi }_{2V}^{-1}\\ &\quad\left(\alpha \right)-{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})-{\Phi }_{2V}^{-1}\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}-({\Phi }_{3OR}^{-1}\\ &\quad\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha ))(-{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\ &\quad+k({\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha )))))) \end{aligned}$$

Proof of Proposition 5

. In this case, the joint total profit depends upon three variables: \({P}_{V},L\), and \({P}_{\mathrm{OR}}\). Thus, the corresponding Hessian matrix is as follows:

$$\mathrm{H}=\left(\begin{array}{ccc}\frac{{\partial }^{2}{\Pi }_{J}}{\partial {{P}_{V}}^{2}}& \frac{{\partial }^{2}{\Pi }_{J}}{\partial {P}_{V}\partial L}& \frac{{\partial }^{2}{\Pi }_{J}}{\partial {P}_{V}\partial {P}_{\mathrm{OR}}}\\ \frac{{\partial }^{2}{\Pi }_{J}}{\partial L\partial {P}_{V}}& \frac{{\partial }^{2}{\Pi }_{J}}{\partial {\mathrm{L}}^{2}}& \frac{{\partial }^{2}{\Pi }_{J}}{\partial \mathrm{L}\partial {P}_{\mathrm{OR}}}\\ \frac{{\partial }^{2}{\Pi }_{J}}{\partial {P}_{\mathrm{OR}}\partial {P}_{V}}& \frac{{\partial }^{2}{\Pi }_{J}}{\partial {P}_{\mathrm{OR}}\partial \mathrm{L}}& \frac{{\partial }^{2}{\Pi }_{J}}{\partial {{P}_{OR}}^{2}}\end{array}\right)$$
$$H=\left(\begin{array}{ccc}-2{\Phi }_{2V}^{-1}\left(\alpha \right)& {\Phi }_{3}^{-1}(1-\alpha )& {\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha )\\ {\Phi }_{3}^{-1}(1-\alpha )& -2k& -{\Phi }_{3}^{-1}(\alpha )\\ {\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha )& -{\Phi }_{3}^{-1}(\alpha )& -2{\Phi }_{2OR}^{-1}(\alpha )\end{array}\right)$$

The first principal minor of \(H\) is \({H}^{1}=-2{\Phi }_{2V}^{-1}\left(\alpha \right)<0\). The second principal minor of \(H\) is \({H}^{2}=det\left(\begin{array}{cc}-2{\Phi }_{2V}^{-1}\left(\alpha \right)& {\Phi }_{3}^{-1}(1-\alpha )\\ {\Phi }_{3}^{-1}(1-\alpha )& -2k\end{array}\right)\), and the third principal minor of \(H\) is \({H}^{3}=\mathrm{det}(H)=2({\Phi }_{2OR}^{-1}(\alpha )(-4k{\Phi }_{2V}^{-1}\left(\alpha \right)+{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})+{\Phi }_{2V}^{-1}\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}+({\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha ))(-{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )+k({\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha ))))\). If \({\Phi }_{2V}^{-1}\left(\alpha \right)\)>\(\frac{{{(\Phi }_{3}^{-1}(1-\alpha ))}^{2}}{4k}\) holds true, then \({H}^{2}>0\) and \({H}^{3}<0\). Thus, alternating algebraic signs has been proven for the principal minors of \(H\). This shows that \(H\) is negative definite and the joint total profit is concave.

Proof of Lemma 4

$$\left\{\begin{array}{c}\frac{\partial {\Pi }_{V}}{\partial {m}_{V}}=0\\ \frac{\partial {\Pi }_{V}}{\partial \mathrm{L}}=0\\ \frac{\partial {\Pi }_{V}}{\partial \mathrm{w}}=0\end{array}\right. \Rightarrow $$
$$\begin{aligned} & {m}_{V}={m}_{V32}=({\Phi }_{2OR}^{-1}(\alpha )(o({\Phi }_{2V}^{-1}\left(\alpha \right){\left({\Phi }_{3}^{-1}\left(\alpha \right)\right)}^{2}+{\Phi }_{3OR}^{-1}\left(1-\alpha \right)(-{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad+2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)))-((f+o){\left({\Phi }_{3}^{-1}\left(1-\alpha \right)\right)}^{2}-6ko{\Phi }_{3OR}^{-1}\left(1-\alpha \right)){\Phi }_{3V}^{-1}(1-\alpha )\\&\quad+{\Phi }_{1}^{-1}\left(1-\alpha \right)(-8k(-1+\rho ){\Phi }_{2OR}^{-1}(\alpha )-\rho {\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )-{\left({\Phi }_{3}^{-1}\left(\alpha \right)\right)}^{2}\\&\quad+\rho {\left({\Phi }_{3}^{-1}\left(\alpha \right)\right)}^{2}+2k\rho {\Phi }_{3OR}^{-1}\left(1-\alpha \right)+6k\rho {\Phi }_{3V}^{-1}(1-\alpha ))+{\Phi }_{2OR}^{-1}(\alpha )(-8ko{\Phi }_{2V}^{-1}\left(\alpha \right)\\&\quad+(f+o)({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)\\&\quad+2k{\Phi }_{3V}^{-1}(1-\alpha )))))(1/(2{\Phi }_{2OR}^{-1}\left(\alpha \right)\left({\Phi }_{2OR}^{-1}\left(\alpha \right)\left(8k{\Phi }_{2V}^{-1}\left(\alpha \right)-2{\left({\Phi }_{3}^{-1}\left(1-\alpha \right)\right)}^{2}\right)\right.\\&\quad\left.-{\Phi }_{2V}^{-1}\left(\alpha \right){\left({\Phi }_{3}^{-1}\left(\alpha \right)\right)}^{2}+{\Phi }_{3OR}^{-1}\left(1-\alpha \right)\left({\Phi }_{3}^{-1}\left(1-\alpha \right){\Phi }_{3}^{-1}\left(\alpha \right)-k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)\right)\right)\\&\quad+6{\Phi }_{2OR}^{-1}\left(\alpha \right)\left({\Phi }_{3}^{-1}\left(1-\alpha \right){\Phi }_{3}^{-1}\left(\alpha \right)-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)\right){\Phi }_{3V}^{-1}\left(1-\alpha \right)\\&\quad-2\left(k{\Phi }_{2OR}^{-1}\left(\alpha \right)+{\left({\Phi }_{3}^{-1}\left(\alpha \right)\right)}^{2}\right){\left({\Phi }_{3V}^{-1}\left(1-\alpha \right)\right)}^{2}))\end{aligned}$$
$$\begin{aligned} & L={L}_{32}=({\Phi }_{2OR}^{-1}(\alpha )(-4{\Phi }_{2OR}^{-1}(\alpha )((-1+\rho ){\Phi }_{1}^{-1}\left(1-\alpha \right)\\&\quad+o{\Phi }_{2V}^{-1}\left(\alpha \right)){\Phi }_{3}^{-1}(1-\alpha )+2(-\rho {\Phi }_{1}^{-1}\left(1-\alpha \right)\\&\quad+(f+o){\Phi }_{2OR}^{-1}(\alpha )){\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )\\&\quad-((f+o){\Phi }_{2OR}^{-1}(\alpha ){\Phi }_{3}^{-1}(1-\alpha )\\&\quad+o{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )+{\Phi }_{1}^{-1}\left(1-\alpha \right)({\Phi }_{3}^{-1}(\alpha )\\&\quad-\rho ({\Phi }_{3}^{-1}(1-\alpha )\\&\quad+{\Phi }_{3}^{-1}(\alpha )))){\Phi }_{3OR}^{-1}\left(1-\alpha \right)+o{\Phi }_{3}^{-1}(1-\alpha ){({\Phi }_{3OR}^{-1}\left(1-\alpha \right))}^{2})\\&\quad+{\Phi }_{2OR}^{-1}(\alpha )((f+o){\Phi }_{2OR}^{-1}(\alpha ){\Phi }_{3}^{-1}(1-\alpha )\\&\quad+3o{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )+3{\Phi }_{1}^{-1}\\&\quad\left(1-\alpha \right)(\rho {\Phi }_{3}^{-1}(1-\alpha )\\&\quad+(-1+\rho ){\Phi }_{3}^{-1}(\alpha ))-(-3o{\Phi }_{3}^{-1}(1-\alpha )\\&\quad+(f+o){\Phi }_{3}^{-1}(\alpha )){\Phi }_{3OR}^{-1}\left(1-\alpha \right)){\Phi }_{3V}^{-1}(1-\alpha )\\&\quad-{\Phi }_{3}^{-1}(\alpha )(2\rho {\Phi }_{1}^{-1}\left(1-\alpha \right)\\&\quad+(f+o){\Phi }_{2OR}^{-1}(\alpha )+2o{\Phi }_{3OR}^{-1}\\&\quad\left(1-\alpha \right)){({\Phi }_{3V}^{-1}(1-\alpha ))}^{2})(1/(2{\Phi }_{2OR}^{-1}(\alpha )({\Phi }_{2OR}^{-1}(\alpha )(8k{\Phi }_{2V}^{-1}\\&\quad\left(\alpha \right)-2{\Phi 3}_{1-\alpha }^{2})-{\Phi }_{2V}^{-1}\\&\quad\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}\\&\quad+{\Phi }_{3OR}^{-1}\left(1-\alpha \right)({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad-k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)))+6{\Phi }_{2OR}^{-1}(\alpha )({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)){\Phi }_{3V}^{-1}(1-\alpha )\\&\quad-2(k{\Phi }_{2OR}^{-1}(\alpha )+{({\Phi }_{3}^{-1}(\alpha ))}^{2}){({\Phi }_{3V}^{-1}(1-\alpha ))}^{2}))\end{aligned}$$
$$\begin{aligned} & w={w}_{32}=(2{\Phi }_{2OR}^{-1}(\alpha )(-(f-o){\Phi }_{2OR}^{-1}(\alpha )(4k{\Phi }_{2V}^{-1}\left(\alpha \right)-{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})\\&\quad+{\Phi }_{1}^{-1}\left(1-\alpha \right)(4k\rho {\Phi }_{2V}^{-1}\left(\alpha \right)+{\Phi }_{3}^{-1}(1-\alpha )(-\rho {\Phi }_{3}^{-1}(1-\alpha )\\&\quad+(-1+\rho ){\Phi }_{3}^{-1}(\alpha ))-2k(-1+\rho ){\Phi }_{3OR}^{-1}\\&\quad\left(1-\alpha \right))+o({\Phi }_{2V}^{-1}\left(\alpha \right)({\Phi }_{3}^{-1}(1-\alpha )-{\Phi }_{3}^{-1}(\alpha )){\Phi }_{3}^{-1}(\alpha )\\&\quad+(2k{\Phi }_{2V}^{-1}\left(\alpha \right)+{\Phi }_{3}^{-1}(1-\alpha )(-{\Phi }_{3}^{-1}(1-\alpha )\\&\quad+{\Phi }_{3}^{-1}(\alpha ))){\Phi }_{3OR}^{-1}\left(1-\alpha \right)\\&\quad-k{({\Phi }_{3OR}^{-1}\left(1-\alpha \right))}^{2}))\\&\quad+({\Phi }_{1}^{-1}\left(1-\alpha \right)(-4k(-1+\rho ){\Phi }_{2OR}^{-1}(\alpha )\\&\quad+{\Phi }_{3}^{-1}(\alpha )(\rho {\Phi }_{3}^{-1}(1-\alpha )+{\Phi }_{3}^{-1}(\alpha )\\&\quad-\rho {\Phi }_{3}^{-1}(\alpha ))-2k\rho {\Phi }_{3OR}^{-1}\left(1-\alpha \right))+{\Phi }_{2OR}^{-1}(\alpha )(-4ko{\Phi }_{2V}^{-1}\\&\quad\left(\alpha \right)-3(f-o)({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )-2k{\Phi }_{3OR}^{-1}\\&\quad\left(1-\alpha \right)))+o(-{\Phi }_{2V}^{-1}\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}\\&\quad+{\Phi }_{3OR}^{-1}\left(1-\alpha \right)({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)))){\Phi }_{3V}^{-1}(1-\alpha )\\&\quad+(2k\rho {\Phi }_{1}^{-1}\left(1-\alpha \right)+2fk{\Phi }_{2OR}^{-1}(\alpha )\\&\quad+(f-o){({\Phi }_{3}^{-1}(\alpha ))}^{2}+2ko{\Phi }_{3OR}^{-1}\left(1-\alpha \right)){({\Phi }_{3V}^{-1}(1-\alpha ))}^{2})(1/(2{\Phi }_{2OR}^{-1}(\alpha )({\Phi }_{2OR}^{-1}(\alpha )(8k{\Phi }_{2V}^{-1}\\&\quad\left(\alpha \right)-2{\Phi 3}_{1-\alpha }^{2})-{\Phi 2\mathrm{V}}_{\alpha }{\Phi 3}_{\alpha }^{2}\\&\quad+{\Phi 3\mathrm{OR}}_{1-\alpha }({\Phi 3}_{1-\alpha }{\Phi 3}_{\alpha }-k{\Phi 3\mathrm{OR}}_{1-\alpha }))\\&\quad+6{\Phi }_{2OR}^{-1}(\alpha )({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)){\Phi }_{3V}^{-1}(1-\alpha )-2(k{\Phi }_{2OR}^{-1}(\alpha )\\&\quad+{({\Phi }_{3}^{-1}(\alpha ))}^{2}){({\Phi }_{3V}^{-1}(1-\alpha ))}^{2}))\end{aligned}$$

Proof of Proposition 6

The second-order condition shown below completes the proof:

$$\frac{{d}^{2}{\Pi }_{OR}}{d{{m}_{OR}}^{2}}=-2{\Phi }_{2OR}^{-1}(\alpha )<0$$

Proof of Proposition 7

We compute the Hessian matrix as follows:

$$\mathrm{H}=\left(\begin{array}{ccc}\frac{{\partial }^{2}{\Pi }_{V}}{\partial {{m}_{V}}^{2}}& \frac{{\partial }^{2}{\Pi }_{V}}{\partial {m}_{V}\partial L}& \frac{{\partial }^{2}{\Pi }_{V}}{\partial {m}_{V}\partial \mathrm{w}}\\ \frac{{\partial }^{2}{\Pi }_{V}}{\partial L\partial {m}_{V}}& \frac{{\partial }^{2}{\Pi }_{V}}{\partial {\mathrm{L}}^{2}}& \frac{{\partial }^{2}{\Pi }_{V}}{\partial \mathrm{L}\partial \mathrm{w}}\\ \frac{{\partial }^{2}{\Pi }_{V}}{\partial \mathrm{w}\partial {m}_{V}}& \frac{{\partial }^{2}{\Pi }_{V}}{\partial \mathrm{w}\partial \mathrm{L}}& \frac{{\partial }^{2}{\Pi }_{V}}{\partial {\mathrm{w}}^{2}}\end{array}\right)$$
$$H=\left(\begin{array}{ccc}-2{\Phi }_{2V}^{-1}\left(\alpha \right)& {\Phi }_{3}^{-1}(1-\alpha )& {\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha )\\ {\Phi }_{3}^{-1}(1-\alpha )& -2k& -{\Phi }_{3}^{-1}(\alpha )\\ 0& -{\Phi }_{3}^{-1}(\alpha )& -2{\Phi }_{2OR}^{-1}(\alpha )\end{array}\right)$$

Similar to proposition 5, we have \({H}^{1}=-2{\Phi }_{2V}^{-1}\left(\alpha \right)<0\), \({H}^{2}=det\left(\begin{array}{cc}-2{\Phi }_{2V}^{-1}\left(\alpha \right)& {\Phi }_{3}^{-1}(1-\alpha )\\ {\Phi }_{3}^{-1}(1-\alpha )& -2k\end{array}\right)\), and \({H}^{3}=\mathrm{det}\left(H\right)=2{\Phi }_{2OR}^{-1}(\alpha )(-4k{\Phi }_{2V}^{-1}\left(\alpha \right)+{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})+{\Phi }_{3}^{-1}(\alpha )(2{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )-{\Phi }_{3}^{-1}(1-\alpha )({\Phi }_{3OR}^{-1}\left(1-\alpha \right)+{\Phi }_{3V}^{-1}(1-\alpha )))\). If \({\Phi }_{2V}^{-1}\left(\alpha \right)\)>\(\frac{{{(\Phi }_{3}^{-1}(1-\alpha ))}^{2}}{4k}\) holds true, then \({H}^{2}>0\) and \({H}^{3}<0\). This completes the proof.

Proof of Lemma 5

$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial \Pi _{V} }}{{\partial m_{V} }} = 0} \\ {\frac{{\partial \Pi _{V} }}{{\partial {\text{L}}}} = 0} \\ {\frac{{\partial \Pi _{V} }}{{\partial {\text{w}}}} = 0} \\ {\frac{{\partial \Pi _{{OR}} }}{{\partial {\text{m}}_{{OR}} }} = 0} \\ \end{array} \Rightarrow } \right. $$
$$\begin{aligned} &{m}_{V}={m}_{V33}=({\Phi }_{2OR}^{-1}(\alpha )(o({\Phi }_{2V}^{-1}\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}\\&\quad+{\Phi }_{3OR}^{-1}\left(1-\alpha \right)(-{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad+2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)))-((f+o){({\Phi }_{3}^{-1}(\alpha ))}^{2}\\&\quad-4ko{\Phi }_{3OR}^{-1}\left(1-\alpha \right)){\Phi }_{3V}^{-1}(1-\alpha )\\&\quad+{\Phi }_{1}^{-1}\left(1-\alpha \right)(-6k(-1+\rho ){\Phi }_{2OR}^{-1}(\alpha )\\&\quad-\rho {\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )-{({\Phi }_{3}^{-1}(\alpha ))}^{2}\\&\quad+\rho {({\Phi }_{3}^{-1}(\alpha ))}^{2}+2k\rho {\Phi }_{3OR}^{-1}\\&\quad\left(1-\alpha \right)+4k\rho {\Phi }_{3V}^{-1}(1-\alpha ))+{\Phi }_{2OR}^{-1}(\alpha ){\Phi }_{2OR}^{-1}(\alpha )(-6ko{\Phi }_{2V}^{-1}\\&\quad\left(\alpha \right)+(f+o)({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)+2k{\Phi }_{3V}^{-1}(1-\alpha )))))(1/({\Phi }_{2OR}^{-1}(\alpha )(3{\Phi }_{2OR}^{-1}(\alpha )(4k{\Phi }_{2V}^{-1}\\&\quad\left(\alpha \right)-{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})-2({\Phi }_{2V}^{-1}\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}\\&\quad+{\Phi }_{3OR}^{-1}\left(1-\alpha \right)(-{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad+k{\Phi }_{3OR}^{-1}\left(1-\alpha \right))))+4{\Phi }_{2OR}^{-1}(\alpha )({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)){\Phi }_{3V}^{-1}(1-\alpha )-(2k{\Phi }_{2OR}^{-1}(\alpha )+{({\Phi }_{3}^{-1}(\alpha ))}^{2}){({\Phi }_{3V}^{-1}(1-\alpha ))}^{2})\end{aligned}$$
$$\begin{aligned} & L={L}_{33}=({\Phi }_{2OR}^{-1}(\alpha )({\Phi }_{2OR}^{-1}(\alpha ){\Phi }_{2V}^{-1}\left(\alpha \right)(-3o{\Phi }_{3}^{-1}(1-\alpha )\\&\quad+2(f+o){\Phi }_{3}^{-1}(\alpha ))-((f+o){\Phi }_{2OR}^{-1}(\alpha ){\Phi }_{3}^{-1}(1-\alpha )\\&\quad+o{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )){\Phi }_{3OR}^{-1}\left(1-\alpha \right)\\&\quad+o{\Phi }_{3}^{-1}(1-\alpha ){({\Phi }_{3OR}^{-1}\left(1-\alpha \right))}^{2}\\&\quad+{\Phi }_{1}^{-1}\left(1-\alpha \right)(-3(-1+\rho ){\Phi }_{2OR}^{-1}(\alpha ){\Phi }_{3}^{-1}(1-\alpha )\\&\quad-2\rho {\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )\\&\quad+(\rho {\Phi }_{3}^{-1}(1-\alpha )\\&\quad+(-1+\rho ){\Phi }_{3}^{-1}(\alpha )){\Phi }_{3OR}^{-1}\left(1-\alpha \right)))+{\Phi }_{2OR}^{-1}(\alpha )((f+o){\Phi }_{2OR}^{-1}(\alpha ){\Phi }_{3}^{-1}(1-\alpha )\\&\quad+2o{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(\alpha )\\&\quad+2{\Phi }_{1}^{-1}\left(1-\alpha \right)(\rho {\Phi }_{3}^{-1}(1-\alpha )\\&\quad+(-1+\rho ){\Phi }_{3}^{-1}(\alpha ))-(-2o{\Phi }_{3}^{-1}(1-\alpha )+(f+o){\Phi }_{3}^{-1}(\alpha )){\Phi }_{3OR}^{-1}\left(1-\alpha \right)){\Phi }_{3V}^{-1}(1-\alpha )\\&\quad-{\Phi }_{3}^{-1}(\alpha )(\rho {\Phi }_{1}^{-1}\\&\quad\left(1-\alpha \right)+(f+o){\Phi }_{2OR}^{-1}(\alpha )\\&\quad+o{\Phi }_{3OR}^{-1}\left(1-\alpha \right)){({\Phi }_{3V}^{-1}(1-\alpha ))}^{2})(1/(3{({\Phi }_{2OR}^{-1}(\alpha ))}^{2}(4k{\Phi }_{2V}^{-1}\left(\alpha \right)\\&\quad-{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})-{({\Phi }_{3}^{-1}(\alpha ))}^{2}{({\Phi }_{3V}^{-1}(1-\alpha ))}^{2}\\&\quad-2{\Phi }_{2OR}^{-1}(\alpha )({\Phi }_{2V}^{-1}\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}-{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )({\Phi }_{3OR}^{-1}\\&\quad\left(1-\alpha \right)+2{\Phi }_{3V}^{-1}(1-\alpha ))+k({({\Phi }_{3OR}^{-1}\left(1-\alpha \right))}^{2}\\&\quad+4{\Phi }_{3OR}^{-1}\left(1-\alpha \right){\Phi }_{3V}^{-1}(1-\alpha )+{({\Phi }_{3V}^{-1}(1-\alpha ))}^{2})))))\end{aligned}$$
$$\begin{aligned} & w={w}_{33}=({\Phi }_{2OR}^{-1}(\alpha )(-(f-2o){\Phi }_{2OR}^{-1}(\alpha )(4k{\Phi }_{2V}^{-1}\left(\alpha \right)-{({\Phi }_{3V}^{-1}(1-\alpha ))}^{2}))\\&\quad+{\Phi }_{1}^{-1}\left(1-\alpha \right)(4k\rho {\Phi }_{2V}^{-1}\left(\alpha \right)+{\Phi }_{3}^{-1}(1-\alpha )(-\rho {\Phi }_{3}^{-1}(1-\alpha )+(-1+\rho ){\Phi }_{3}^{-1}(\alpha ))\\&\quad-2k(-1+\rho ){\Phi }_{3OR}^{-1}\left(1-\alpha \right))+o({\Phi }_{2V}^{-1}\left(\alpha \right)({\Phi }_{3}^{-1}(1-\alpha )\\&\quad-2{\Phi }_{3}^{-1}(\alpha )){\Phi }_{3}^{-1}(\alpha )+(2k{\Phi }_{2V}^{-1}\left(\alpha \right)-{({\Phi }_{3}^{-1}(1-\alpha ))}^{2}\\&\quad+2{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )){\Phi }_{3OR}^{-1}\left(1-\alpha \right)\\&\quad-2k{({\Phi }_{3OR}^{-1}\left(1-\alpha \right))}^{2}))-{\Phi }_{2OR}^{-1}(\alpha )(4k(-1+\rho ){\Phi }_{1}^{-1}\left(1-\alpha \right)+4ko{\Phi }_{2V}^{-1}\left(\alpha \right)\\&\quad +(f-3o)({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right))){\Phi }_{3V}^{-1}(1-\alpha )+(-o{({\Phi }_{3}^{-1}(\alpha ))}^{2}\\&\quad+2k(\rho {\Phi }_{1}^{-1}\left(1-\alpha \right)+f{\Phi }_{2OR}^{-1}(\alpha )\\&\quad+o{\Phi }_{3OR}^{-1}\left(1-\alpha \right))){({\Phi }_{3V}^{-1}(1-\alpha ))}^{2}))(1/({\Phi }_{2OR}^{-1}(\alpha )(3{\Phi }_{2OR}^{-1}(\alpha )(4k{\Phi }_{2V}^{-1}\\&\quad\left(\alpha \right)-{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})-2({\Phi }_{2V}^{-1}\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}+{\Phi }_{3OR}^{-1}\left(1-\alpha \right)(-{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad+k{\Phi }_{3OR}^{-1}\left(1-\alpha \right))))+4{\Phi }_{2OR}^{-1}(\alpha )({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)){\Phi }_{3V}^{-1}(1-\alpha )\\&\quad-(2k{\Phi }_{2OR}^{-1}(\alpha )+{({\Phi }_{3}^{-1}(\alpha ))}^{2}){({\Phi }_{3V}^{-1}(1-\alpha ))}^{2}))) \end{aligned}$$
$$\begin{aligned} & {m}_{\mathrm{OR}}={m}_{\mathrm{OR}33}\\&\quad=({\Phi }_{2OR}^{-1}(\alpha )(-(f+o){\Phi }_{2OR}^{-1}(\alpha )(4k{\Phi }_{2V}^{-1}\left(\alpha \right)-{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})\\&\quad+o{\Phi }_{2V}^{-1}\left(\alpha \right){\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )+o(2k{\Phi }_{2V}^{-1}\left(\alpha \right)-{({\Phi }_{3}^{-1}(1-\alpha ))}^{2}){\Phi }_{3OR}^{-1}\left(1-\alpha \right)\\&\quad+{\Phi }_{1}^{-1}\left(1-\alpha \right)(4k\rho {\Phi }_{2V}^{-1}\left(\alpha \right)+{\Phi }_{3}^{-1}(1-\alpha )(-\rho {\Phi }_{3}^{-1}(1-\alpha )+(-1+\rho ){\Phi }_{3}^{-1}(\alpha ))\\&\quad-2k(-1+\rho ){\Phi }_{3OR}^{-1}\left(1-\alpha \right)))+({\Phi }_{1}^{-1}\left(1-\alpha \right)\\&\quad(2k(-1+\rho ){\Phi }_{2OR}^{-1}(\alpha )+{\Phi }_{3}^{-1}(\alpha )(\rho {\Phi }_{3}^{-1}(1-\alpha )\\&\quad+{\Phi }_{3}^{-1}(\alpha )-\rho {\Phi }_{3}^{-1}(\alpha ))-2k\rho {\Phi }_{3OR}^{-1}\left(1-\alpha \right))\\&\quad+2{\Phi }_{2OR}^{-1}(\alpha )(ko{\Phi }_{2V}^{-1}\left(\alpha \right)\\&\quad-(f+o)({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)))\\&\quad+o(-{\Phi }_{2V}^{-1}\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}+{\Phi }_{3OR}^{-1}\left(1-\alpha \right)({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)))){\Phi }_{3V}^{-1}(1-\alpha )\\&\quad+((f+o){({\Phi }_{3}^{-1}(\alpha ))}^{2}-2k(\rho {\Phi }_{1}^{-1}\left(1-\alpha \right)+o{\Phi }_{3OR}^{-1}\\&\quad\left(1-\alpha \right))){({\Phi }_{3V}^{-1}(1-\alpha ))}^{2}))(1/({\Phi }_{2OR}^{-1}(\alpha )(3{\Phi }_{2OR}^{-1}(\alpha )(4k{\Phi }_{2V}^{-1}\left(\alpha \right)-{({\Phi }_{3}^{-1}(1-\alpha ))}^{2})\\&\quad-2({\Phi }_{2V}^{-1}\left(\alpha \right){({\Phi }_{3}^{-1}(\alpha ))}^{2}+{\Phi }_{3OR}^{-1}\\&\quad\left(1-\alpha \right)(-{\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )+k{\Phi }_{3OR}^{-1}\left(1-\alpha \right))))\\&\quad+4{\Phi }_{2OR}^{-1}(\alpha )({\Phi }_{3}^{-1}(1-\alpha ){\Phi }_{3}^{-1}(\alpha )\\&\quad-2k{\Phi }_{3OR}^{-1}\left(1-\alpha \right)){\Phi }_{3V}^{-1}(1-\alpha )\\&\quad-(2k{\Phi }_{2OR}^{-1}(\alpha )+{({\Phi }_{3}^{-1}(\alpha ))}^{2}){({\Phi }_{3V}^{-1}(1-\alpha ))}^{2})))\end{aligned}$$

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Maihami, R., Kannan, D., Fattahi, M. et al. Ticket pricing for entertainment events under a dual-channel environment: a game-theoretical approach using uncertainty theory. Ann Oper Res 331, 503–542 (2023). https://doi.org/10.1007/s10479-023-05192-x

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