Endogenously choosing the timing of setting strategic contracts’ levels and content in a managerial mixed duopoly with welfare-based and sales delegation contracts

Abstract

This paper explores the full observable delay game—which involves the timing of setting one public firm’s and one private firm’s strategic contracts’ levels and content—in a managerial mixed duopoly with a welfare-based delegation à la Nakamura (Int Rev Econ Finance 35:262–277, 2015, J Ind Compet Trade 19:235–261, 2019b, J Ind Compet Trade 19:679–737, 2019c). In this paper, we first show that both the following types of market frameworks can be equilibrium market structures: (1) a market formation in which the manager of the public firm, which has a quantity contract, is the leader and manager of the private firm, which has a price contract and is the follower, and (2) a market configuration in which the manager of the public firm, which has a quantity contract, is the follower, and the private firm, which also has a quantity contract, is the leader. Second, we demonstrate that the form of highest social welfare is achieved under market structure (2). Hence, in such a managerial mixed duopoly, from the perspective of social welfare, it is not as necessary for the relevant authority (including the government) to regulate owners’ free determination of the timing of setting public and private firms’ strategic contracts’ levels and content.

Appendix

1.1 q–q game

Here, we give both firms’ equilibrium incentive parameters as well as their owners’ payoffs in the thin q–q game.

1.1.1 SSws case

In $(q^{1}_{0}, q^{1}_{1})$ and $(q^{2}_{0}, q^{2}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:³⁴

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{SSws}_{0qq} = - \dfrac{\delta (a_{1} - a_{0} \delta )}{a_{0} (8 + \delta ^2) - 3 a_{1} \delta }< 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{SSws}_{1qq} = 1 - \dfrac{2 \delta ^2 (a_{1} - a_{0} \delta )}{c_{1} (4 - 5 \delta ^2 + \delta ^4)} < 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(q^{1}_{0}, q^{1}_{1})$ and $(q^{2}_{0}, q^{2}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager more aggressive than the sole welfare maximizer, while firm 1’s owner makes his manager more aggressive than the sole profit maximizer. Then, in $(q^{1}_{0}, q^{1}_{1})$ and $(q^{2}_{0}, q^{2}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{SSws}_{0qq} = W^{SSws}_{qq} \\ \quad = \dfrac{4 a_{0}^2 (4 - 3 \delta ^2 + \delta ^4) - 2 a_{0} a_{1} \delta (12 - 5 \delta ^2 + \delta ^4) + a_{1}^2 (12 - 5 \delta ^2 + \delta ^4)}{2 \beta (4 \delta ^2)^2 (1 - \delta ^2)}, \\ U^{SSws}_{1qq} = \pi ^{SSws}_{1qq} = \dfrac{(4 (a_{1} - a_{0} \delta )^2}{\beta (4 - \delta ^2)^2 (1 - \delta ^2)}. \end{array}\right. } \end{aligned}$$

1.1.2 LFws case

In $(q^{1}_{0}, q^{2}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{LFws}_{0qq} = 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{LFws}_{1qq} = 1 - \dfrac{\delta ^2 (a_{1} - a_{0} \delta )}{2 c_{1} (2 - 3 \delta ^2 + \delta ^4)} < 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(q^{1}_{0}, q^{2}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager the sole welfare maximizer, while firm 1’s owner makes his manager more aggressive than the sole profit maximizer. Then, in $(q^{1}_{0}, q^{2}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{LFws}_{0qq} = W^{LFws}_{0qq} \\ \quad = \dfrac{a_{0}^2 (16 - 20 \delta ^2 + 7 \delta ^4) - 2 a_{0} a_{1} \delta (12 - 13 \delta ^2 + 4 \delta ^4) + a_{1}^2 (12 - 13 \delta ^2 + 4 \delta ^4)}{8 \beta (2 - \delta ^2)^2 (1 - \delta ^2)}, \\ U^{LFws}_{1qq} = \pi ^{LFws}_{1qq} = \dfrac{(a_{1} - a_{0} \delta )^2}{2 \beta (2 - \delta ^2) (1 - \delta ^2)}. \end{array}\right. } \end{aligned}$$

1.1.3 FLws case

In $(q^{2}_{0}, q^{1}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{FLws}_{0qq} = 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{FLws}_{1qq} = - \dfrac{\delta (a_{1} - a_{0} \delta )}{a_{0} (8 - 3 \delta ^2) - 3 a_{1} \delta } < 0, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(q^{2}_{0}, q^{1}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager the sole welfare maximizer, while firm 1’s owner makes his manager more aggressive than the sole profit maximizer. Then, in $(q^{2}_{0}, q^{1}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{FLws}_{0qq} = W^{FLws}_{0qq} = \dfrac{a_{0}^2 (2 - \delta ^2)^2 - a_{0} a_{1} (6 \delta - 4 \delta ^3) + a_{1}^2 (3 - 2 \delta ^2)}{2 \beta (1 - \delta ^2) (4 - 3 \delta ^2)}, \\ U^{FLws}_{1qq} = \pi ^{FLws}_{1qq} = \dfrac{(a_{1} - a_{0} \delta )^2 (2 - \delta ^2)^2}{\beta (4 - 3 \delta ^2)^2 (1 - \delta ^2)}. \end{array}\right. } \end{aligned}$$

1.2 p–p game

Here, we give both firms’ equilibrium incentive parameters as well as their owners’ payoffs in the thin p–p game.

1.2.1 SSws case

In $(p^{1}_{0}, p^{1}_{1})$ and $(p^{2}_{0}, p^{2}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:³⁵

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{SSws}_{0pp} = \dfrac{\delta (a_{1} - a_{0} \delta )}{a_{0} (8 - 5 \delta ^2) - a_{1} \delta (5 - 2 \delta ^2)}> 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{SSws}_{1pp} = 1 + \dfrac{\delta ^2 (a_{1} - a_{0} \delta ) (2 - \delta ^2)}{c_{1} (4 - 5 \delta ^2 + \delta ^4)} > 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(p^{1}_{0}, p^{1}_{1})$ and $(p^{2}_{0}, p^{2}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager the sole welfare maximizer, while firm 1’s owner makes his manager less aggressive than the sole profit maximizer. Then, in $(p^{1}_{0}, p^{1}_{1})$ and $(p^{2}_{0}, p^{2}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{SSws}_{0pp} = W^{SSws}_{pp} = \dfrac{ a_{0}^2 (16 - 28 \delta ^2 + 12 \delta ^4 - \delta ^6) - 2 a_{0} a_{1} \delta (12 - 21 \delta ^2 + 9 \delta ^4 - \delta ^6) + a_{1}^2 (12 - 21 \delta ^2 + 9 \delta ^4 - \delta ^6)}{2 \beta (4 - 5 \delta ^2 + \delta ^4)^2}, \\ U^{SSws}_{1pp} = \pi ^{SSws}_{1pp} = \dfrac{(a_{1} - a_{0} \delta )^2 (2 - \delta ^2)^2}{\beta (4 - 5 \delta ^2 + \delta ^4)^2}. \end{array}\right. } \end{aligned}$$

1.2.2 LFws case

In $(p^{1}_{0}, p^{2}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{LFws}_{0pp} = 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{LFws}_{1pp} = 1 + \dfrac{\delta ^2 (a_{1} - a_{0} \delta )}{4 c_{1} (1 - \delta ^2)} > 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(p^{1}_{0}, p^{2}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager the sole welfare maximizer, while firm 1’s owner makes his manager less aggressive than the sole profit maximizer. Then, in $(p^{1}_{0}, p^{2}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{LFws}_{0pp} = W^{LFws}_{pp} = \dfrac{a_{1}^2 (12 - 13 \delta ^2) - a_{0} a_{1} (24 \delta - 26 \delta ^3) + a_{0}^2 (16 - 20 \delta ^2 + 3 \delta ^4)}{32 \beta (1 - \delta ^2)^2}, \\ U^{LFws}_{1pp} = \pi ^{LFws}_{1pp} = \dfrac{(a_{1} - a_{0} \delta )^2 (2 - \delta ^2)}{8 \beta (1 - \delta ^2)^2}. \end{array}\right. } \end{aligned}$$

1.2.3 FLws case

In $(p^{2}_{0}, p^{1}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{FLws}_{0pp} = \dfrac{\delta (a_{1} - a_{0} \delta )}{a_{0} (8 - 9 \delta ^2) - a_{1} \delta (5 - 6 \delta ^2)} > 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{FLws}_{1pp} = 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(p^{2}_{0}, p^{1}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager more aggressive than the sole welfare maximizer, while firm 1’s owner makes his manager the sole profit maximizer. Then, in $(p^{2}_{0}, p^{1}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{FLws}_{0pp} = W^{FLws}_{0pp} = \dfrac{4 a_{0}^2 - 6 a_{0} a_{1} \delta + 3 a_{1}^2}{2 \beta (4 - 3 \delta ^2)}, \\ U^{FLws}_{1pp} = \pi ^{FLws}_{1pp} = \dfrac{4 (a_{1} - a_{0} \delta )^2}{\beta (4 - 3 \delta ^2)^2}. \end{array}\right. } \end{aligned}$$

1.3 p–q game

Here, we give both firms’ equilibrium incentive parameters as well as their owners’ payoffs in the thin p–q game.

1.3.1 SSws case

In $(p^{1}_{0}, q^{1}_{1})$ and $(p^{2}_{0}, q^{2}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:³⁶

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{SSws}_{0pq} = - \dfrac{\delta (a_{1} - a_{0} \delta )}{a_{0} (8 - 3 \delta ^2) - 3 a_{1} \delta } < 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{SSws}_{1pq} = 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(p^{1}_{0}, q^{1}_{1})$ and $(p^{2}_{0}, q^{2}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager more than the sole welfare maximizer, while firm 1’s owner makes his manager the sole profit maximizer. Then, in $(p^{1}_{0}, q^{1}_{1})$ and $(p^{2}_{0}, q^{2}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{SSws}_{0pq} = W^{SSws}_{pq} = \dfrac{a_{0}^2 (2 - \delta ^2)^2 - a_{0} a_{1} (6 \delta - 4 \delta ^3) + a_{1}^2 (3 - 2 \delta ^2)}{2 \beta (1 - \delta ^2) (4 - 3 \delta ^2)}, \\ U^{SSws}_{1pq} = \pi ^{SSws}_{1pq} = \dfrac{(a_{1} - a_{0} \delta )^2 (2 - \delta ^2)^2}{\beta (4 - 3 \delta ^2)^2 (1 - \delta ^2)}. \end{array}\right. } \end{aligned}$$

1.3.2 LFws case

In $(p^{1}_{0}, q^{2}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{LFws}_{0pq} = 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{LFws}_{1pq} = 1 + \dfrac{\delta ^2 (a_{1} - a_{0} \delta )}{4 c_{1} (1 - \delta ^2)} > 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(p^{1}_{0}, q^{2}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager the sole welfare maximizer, while firm 1’s owner makes his manager less aggressive than the sole profit maximizer. Then, in $(p^{1}_{0}, q^{2}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{LFws}_{0pq} = W^{LFws}_{pq} = \dfrac{a_{0}^2 (16 - 20 \delta ^2 + 3 \delta ^4) - a_{0} a_{1} (24 \delta - 26 \delta ^3) + a_{1}^2 (12 - 13 \delta ^2)}{32 \beta (1 - \delta ^2)^2}, \\ U^{LFws}_{1pq} = \pi ^{LFws}_{1pq} = \dfrac{(a_{1} - a_{0} \delta )^2 (2 - \delta ^2)}{8 \beta (1 - \delta ^2)^2}. \end{array}\right. } \end{aligned}$$

1.3.3 FLws case

In $(p^{2}_{0}, q^{1}_{1})$, both firms’ FJSV incentive parameters are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{FLws}_{0pq} = - \dfrac{\delta (a_{1} - a_{0} \delta )}{a_{0} (8 - 3 \delta ^2) - 3 a_{1} \delta } < 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{FLws}_{1pq} = 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(p^{1}_{0}, q^{2}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager less aggressive than the sole welfare maximizer, while firm 1’s owner makes his manager the sole profit maximizer. Then, in $(p^{2}_{0}, q^{1}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{FLws}_{0pq} = W^{FLws}_{pq} = \dfrac{a_{0}^2 (2 - \delta ^2)^2 - a_{0} a_{1} (6 \delta - 4 \delta ^3) + a_{1}^2 (3 - 2 \delta ^2)}{2 \beta (1 - \delta ^2) (4 - 3 \delta ^2)}, \\ U^{FLws}_{1pq} = \pi ^{FLws}_{1pq} = \dfrac{(a_{1} - a_{0} \delta )^2 (2 - \delta ^2)^{2}}{\beta (4 - 3 \delta ^2)^2 (1 - \delta ^2)}. \end{array}\right. } \end{aligned}$$

1.4 q–p game

Here, we give both firms’ equilibrium incentive parameters as well as their owners’ payoffs in the thin q–p game.

1.4.1 SSws case

In $(q^{1}_{0}, p^{1}_{1})$ and $(q^{2}_{0}, p^{2}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:³⁷

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{SSws}_{0qp} = \dfrac{\delta (a_{1} - a_{0} \delta )}{a_{0} (8 - 9 \delta ^2) - a_{1} \delta (5 - 6 \delta ^2)} > 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{SSws}_{1qp} = 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(q^{1}_{0}, p^{1}_{1})$ and $(q^{2}_{0}, p^{2}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager more aggressive than the sole welfare maximizer, while firm 1’s owner makes his manager the sole profit maximizer. Then, in $(q^{1}_{0}, p^{1}_{1})$ and $(q^{2}_{0}, p^{2}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{SSws}_{0qp} = W^{SSws}_{qp} = \dfrac{4 a_{0}^2 + 3 a_{1}^2 - 6 a_{0} a_{1} \delta }{2 \beta (4 - 3 \delta ^2)}, \\ U^{SSws}_{1qp} = \pi ^{SSws}_{1qp} = \dfrac{4 (a_{1} - a_{0} \delta )^2}{\beta (4 - 3 \delta ^2)^2}. \end{array}\right. } \end{aligned}$$

1.4.2 LFws case

In $(q^{1}_{0}, p^{2}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{LFws}_{0qp} = 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{LFws}_{1qp} = 1 - \dfrac{\delta ^2 (a_{1} - a_{0} \delta )}{2 c_{1} (2 - 3 \delta ^2 + \delta ^4)} < 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(q^{1}_{0}, p^{2}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager the sole welfare maximizer, while firm 1’s owner makes his manager more aggressive than the sole profit maximizer. Then, in $(q^{1}_{0}, p^{2}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{LFws}_{0qp} = W^{LFws}_{qp} = \dfrac{a_{0}^2 (16 - 20 \delta ^2 + 7 \delta ^4) - 2 a_{0} a_{1} \delta (12 - 13 \delta ^2 + 4 \delta ^4) + a_{1}^2 (12 - 13 \delta ^2 + 4 \delta ^4)}{8 \beta (2 - \delta ^2)^2 (1 - \delta ^2)}, \\ U^{LFws}_{1qp} = \pi ^{LFws}_{1qp} = \dfrac{(a_{1} - a_{0} \delta )^2}{2 \beta (2 - \delta ^2) (1 - \delta ^2)}. \end{array}\right. } \end{aligned}$$

1.4.3 FLws case

In $(q^{2}_{0}, p^{1}_{1})$, firm 0’s welfare-based incentive parameter and firm 1’s FJSV incentive parameter are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^{FLws}_{0qp} = \dfrac{\delta (a_{1} - a_{0} \delta )}{a_{0} (8 - 9 \delta ^2) - a_{1} \delta (5 - 6 \delta ^2)} > 0, &{}\forall \delta \in \left( 0, 1 \right) . \\ \theta ^{FLws}_{1qp} = 1, &{}\forall \delta \in \left( 0, 1 \right) . \end{array}\right. } \end{aligned}$$

Thus, in $(q^{2}_{0}, p^{1}_{1})$, for any $\delta \in \left( 0, 1 \right)$, firm 0’s owner makes his manager more aggressive than the sole welfare maximizer, while firm 1’s owner makes his manager the sole profit maximizer. Then, in $(q^{2}_{0}, p^{1}_{1})$, both firms’ owners’ payoffs are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} U^{FLws}_{0qp} = W^{FLws}_{qp} = \dfrac{4 a_{0}^2 - 6 a_{0} a_{1} \delta + 3 a_{1}^2}{2 \beta (4 - 3 \delta ^2)}, \\ U^{FLws}_{1qp} = \pi ^{FLws}_{1qp} = \dfrac{4 (a_{1} - a_{0} \delta )^2}{\beta (4 - 3 \delta ^2)^2}. \end{array}\right. } \end{aligned}$$

1.5 Ranking orders of both firms’ owners’ payoffs among the three cases in the q–q game

In the q–q game, we obtain the following ranking orders of both firms’ owners’ payoffs among the three cases:

• 1. $U^{FLws}_{0qq}> U^{SSws}_{0qq} > U^{LFws}_{0qq}$ if $\delta \in \left( 0, 1 \right)$.

• 2. $U^{FLws}_{1qq}> U^{LFws}_{1qq} > U^{SSws}_{1qq}$ if $\delta \in \left( 0, 1 \right)$.

In the q–q game, social welfare (which equals firm 0’s owner’s payoff) is the highest in the FLws case, since producer surplus is sufficiently high in the FLws case for any $\delta \in \left( 0, 1 \right)$.³⁸ In the q–q game, consumer surplus is sufficiently high in the SSws case for any $\left( 0, 1 \right)$, while producer surplus is sufficiently low for any $\delta \in \left( 0, 1 \right)$. Then, since the latter effect is stronger than the former effect on the ranking order of social welfare (which equals firm 0’s owner’s payoff), social welfare is higher in the SSws case than in the FLws case for any $\delta \in \left( 0, 1 \right)$.³⁹ Thus, in the q–q game, the ranking order of social welfare is mainly explained not by that of consumer surplus, but by that of producer surplus for any $\delta \in \left( 0, 1 \right)$.

By contrast, in the q–q game, firm 1’s profit (which equals firm 1’s payoff) is the lowest for any $\delta \in \left( 0, 1 \right)$, since firm 1’s price in the SSws case is the lowest in the SSws case among all three cases for any $\delta \in \left( 0, 1 \right)$. Meanwhile, firm 1’s profit (which equals its owner’s payoff) is not the highest in the LFws case among all three cases, since firm 1’s quantity is the lowest in the LFws case among all three cases. As a result, firm 1’s profit (which equals its owner’s payoff) is the highest in the FLws case for any $\delta \in \left( 0, 1 \right)$ among all three cases, since firm 1’s quantity is the highest, and firm 1’s price is the second highest, in the FLws case for any $\delta \in \left( 0, 1 \right)$ among all three cases.

1.6 Ranking orders of both firms’ owners’ payoffs among the three cases in the p–p game

In the p–p game, we obtain the following ranking orders of both firms’ owners’ payoffs among the three games:

• 1.

  • (a) $U^{LFws}_{0pp} \ge U^{FLws}_{0pp} > U^{SSws}_{0pp}$ if $\delta \in \left( 0, 0.942809 \right]$.

  • (b) $U^{FLws}_{0pp}> U^{LFws}_{0pp} > U^{SSws}_{0pp}$ otherwise.

• 2. $U^{LFws}_{1pp}> U^{SSws}_{1pp} > U^{FLws}_{1pp}$ if $\delta \in \left( 0, 1\right)$.

In the p–p game, social welfare (which equals firm 0’s owner’s payoff) is the highest in the LFws case among all three cases, since consumer surplus in the LFws case is sufficiently high when $\delta \in \left( 0, 0.510794 \right]$, while social welfare is the highest in the FLws case among all three cases, since consumer surplus in the FLws case is sufficiently high when $\delta \in \left[ 0.942809, 1 \right)$.⁴⁰ Thus, when $\delta \in \left( 0.510794, 0.942809 \right)$, although social welfare is the highest in the LFws case among all three cases, its fact is not supported by the ranking orders of consumer and producer surpluses. Considering that both consumer surplus and producer surplus are the second highest in the LFws case when $\delta \in \left( 0.510794, 0.942809 \right)$, the fact that social welfare is the highest in the LFws case is explained. By contrast, in the p–p game, the ranking order of firm 1’s profit (which equals its owner’s payoff) coincides with that of firm 1’s quantity.⁴¹

1.7 Ranking orders of both firms’ owners’ payoffs among the three cases in the p–q game

In the p–q game, we obtain the following ranking orders of both firms’ owners’ payoffs among the three games:

• 1. $U^{SSws}_{0pq} = U^{FLws}_{0pq} > U^{LFws}_{0pq}$ if $\delta \in \left( 0, 1 \right)$.

• 2. $U^{LFws}_{1pq} > U^{SSws}_{1pq} = U^{FLws}_{1pq}$ if $\delta \in \left( 0, 1 \right)$.

In the p–q game, the ranking order of social welfare (which equals firm 0’s owner’s payoff) coincides with that of consumer surplus for any $\delta \in \left( 0, 1 \right)$, while that of social welfare is the reverse of that of its price.⁴² By contrast, the ranking order of firm 1’s profit (which equals its owner’s payoff) is the same as that of firm 1’s price and producer surplus for any $\delta \in \left( 0, 1 \right)$.⁴³

1.8 Ranking orders of both firms’ owners’ payoffs among the three cases in the q–p game

In the q–p game, we obtain the following ranking orders of both firms’ owners’ payoffs:

• 1. $U^{LFws}_{0qp} > U^{SSws}_{0qp} = U^{FLws}_{0qp}$ if $\delta \in \left( 0, 1 \right)$.

• 2. $U^{LFws}_{1qp} > U^{SSws}_{1qp} = U^{FLws}_{1qp}$ if $\delta \in \left( 0, 1 \right)$.

In the q–p game, the ranking order of social welfare (which equals firm 0’s owner’s payoff) coincides with those of both firms’ quantities and consumer surplus for any $\delta \in \left( 0, 1 \right)$, while the ranking order of firm 1’s profit (which equals firm 1’s owner’s payoff) is the same as that of firm 1’s quantity for any $\delta \in \left( 0, 1 \right)$.⁴⁴

1.9 Proof of Proposition 1

The proof of this proposition is composed of the following eight claims on the optimal strategy of firm i provided that the strategy of its rival, firm j, is fixed, $\left( i, j = 0, 1; i \ne j \right)$.

Claim 1

For a given firm 1’s choice $q^{1}_{1}$: $U^{FLws}_{0qq} = U^{FLws}_{0pq} = U^{SSws}_{0pq} > U^{SSws}_{0qq}$for any $\delta \in \left( 0, 1 \right)$.

The first equality is derived from Lemma 6, the second equality is derived from Lemma 3, and the third inequality is derived from Result 1. Thus, when firm 1’s owner’s strategy is $q^{1}_{1}$, firm 0’s optimal strategy is $q^{2}_{0}$, $p^{1}_{0}$, or $p^{2}_{0}$. In this case, firm 0’s owner is indifferent in terms of choosing between $q^{2}_{0}$, $p^{1}_{0}$, and $p^{2}_{0}$.

Claim 2

For a given firm 1’s choice $q^{2}_{1}$: $U^{SSws}_{0pq}> U^{SSws}_{0qq}> U^{LFws}_{0qq} > U^{LFws}_{0pq}$for any $\delta \in \left( 0, 1 \right)$.

The first inequality is derived from Result 1, the second inequality is derived from Lemma 1, and the third inequality is derived from Lemma 5. Thus, when firm 1’s owner’s strategy is $q^{2}_{1}$, firm 0’s owner’s optimal strategy is $p^{2}_{0}$.

Claim 3

For a given firm 1’s choice $p^{1}_{1}$: $U^{SSws}_{0qp} = U^{FLws}_{0qp} = U^{FLws}_{0pp} > U^{SSws}_{0pp}$for any $\delta \in \left( 0, 1 \right)$.

When firm 1’s strategy is $p^{1}_{1}$, the first equality is derived from Lemma 4, the second equality is derived from Lemma 6, and the third inequality is derived from Lemma 2. Thus, when firm 1’s owner’s strategy is $p^{1}_{1}$, firm 0’s owner’s optimal strategy is $q^{1}_{0}$, $q^{2}_{0}$, or $p^{2}_{0}$. In this case, firm 0’s owner is indifferent in terms of choosing between $q^{1}_{0}$, $q^{2}_{0}$, and $p^{2}_{0}$.

Claim 4

For a given firm 1’s choice $p^{2}_{1}$:

• 1. $U^{LFws}_{0qp}> U^{SSws}_{0qp} \ge U^{LFws}_{0pp} > U^{SSws}_{0pp}$ if $\delta \in \left[ 0.942809, 1 \right)$.

• 2. $U^{LFws}_{0qp}> U^{LFws}_{0pp}> U^{SSws}_{0qp} > U^{SSws}_{0pp}$otherwise.

When firm 1’s strategy is $p^{2}_{1}$, although the ranking order of firm 0’s owner’s payoff depends on the level of $\delta$, firm 0’s owner’s optimal strategy is unique for any $\delta \in \left( 0, 1 \right)$ as follows:

• 1. $U^{LFws}_{0qp} > U^{SSws}_{0qp}$ is derived from Lemma 4.

• 2. $U^{LFws}_{0pp} > U^{SSws}_{0pp}$ is derived from Lemma 2.

• 3. $U^{LFws}_{0qp} > U^{LFws}_{0pp}$ is derived from Lemma 5.

• 4. $U^{SSws}_{0qp} > U^{SSws}_{0pp}$ is derived from Result 1.

In addition, from easy calculations, $U^{SSws}_{0qp} \ge U^{LFws}_{0pp}$ if $\delta \in \left( 0, 0.942809 \right]$, while $U^{LFws}_{0pp} > U^{SSws}_{0qp}$ otherwise. Thus, when firm 1’s owner’s strategy is $p^{2}_{1}$, firm 0’s owner’s optimal strategy is $q^{1}_{0}$.

Claim 5

For a given firm 0’s choice $q^{1}_{0}$: $U^{LFws}_{1qq} = U^{LFws}_{1qp}> U^{SSws}_{1qp} > U^{SSws}_{1qq}$for any $\delta \in \left( 0, 1 \right)$.

The first equality is derived from Lemma 5, the second inequality is derived from Lemma 4, and the third inequality is derived from Result 1. Thus, when firm 0’s owner’s strategy is $q^{1}_{0}$, firm 1’s owner’s optimal strategy is $q^{2}_{1}$ or $p^{2}_{1}$. In this case, firm 1’s owner is indifferent in terms of choosing between choosing $q^{2}_{1}$ and $p^{2}_{1}$.

Claim 6

For a given firm 0’s choice $q^{2}_{0}$: $U^{FLws}_{1qq}> U^{SSws}_{1qq} > U^{SSws}_{1qp} = U^{FLws}_{1qp}$for any $\delta \in \left( 0, 1 \right)$.

The first inequality is derived from Lemma 1, the second inequality is derived from Result 1, and the third inequality is derived from Lemma 4. Thus, when firm 0’s owner’s strategy is $q^{2}_{0}$, firm 0’s owner’s optimal strategy is $q^{1}_{1}$. Firm 1’s owner is indifferent in terms of choosing $q^{1}_{1}$.

Claim 7

For a given firm 0’s choice $p^{1}_{0}$: $U^{LFws}_{1pq} = U^{LFws}_{1pp}> U^{SSws}_{1pp} > U^{SSws}_{1pq}$for any $\delta \in \left( 0, 1 \right)$.

The first equality is derived from Lemma 5, the second inequality is derived from Lemma 2, and the third inequality is derived from Result 1. Thus, when firm 0’s owner’s strategy is $p^{1}_{0}$, firm 1’s owner’s optimal strategy is $q^{1}_{1}$ or $p^{1}_{1}$.

Claim 8

For a given firm 0’s choice $p^{2}_{0}$: $U^{SSws}_{1pp}> U^{SSws}_{1pq} = U^{FLws}_{1pq} > U^{FLws}_{1pp}$for any $\delta \in \left( 0, 1 \right)$.

The first inequality is derived from Result 1, the second equality is derived from Lemma 3, and the third inequality is derived from Lemma 6. Thus, when firm 0’s strategy is $p^{2}_{0}$, firm 1’s owner’s optimal strategy is $p^{2}_{1}$.

In sum, taking firm i’s owner’s optimal strategy into account provided that firm j’s owner’s strategy is given, the market structure under which both firms’ owners’ optimal strategies coincide are observed in the equilibrium as the two pairs of their strategies: (1) $\left( q^{2}_{0}, q^{1}_{1} \right)$ and (2) $\left( q^{1}_{0}, p^{2}_{1} \right)$. This implies that both (1) $(q^{2}_{0}, q^{1}_{1})$ and (2) $(q^{1}_{0}, p^{2}_{1})$ can be seen as equilibrium market structures, $\left( i, j = 0, 1; i \ne j \right)$. $\square$