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Table 3

The equilibrium results for four strategies under the optimal subsidy.

Results	Stratege YD	Stratege YJ	Stratege YR	Stratege YC
s^*	$\frac{3 η}{2 ψ_{2}}$ $\frac{3\eta }{2{\psi }_2}$	$\frac{3 η}{2 ψ_{2}}$ $\frac{3\eta }{2{\psi }_2}$	$\frac{1}{2}$ $\frac{1}{2}$	$\frac{1}{2}$ $\frac{1}{2}$
$x_{ei}^{}$ ${x}_{{ei}}^{\mathrm{}}$	$\frac{4 ψ_{1} ψ_{2} ψ_{4}}{ψ_{9}}$ $\frac{4{\psi }_1{\psi }_2{\psi }_4}{{\psi }_9}$	$\frac{4 ψ_{1} ψ_{2} ψ_{4}}{ψ_{9}}$ $\frac{4{\psi }_1{\psi }_2{\psi }_4}{{\psi }_9}$	$\frac{4 ψ_{1} ψ_{2} ψ_{4}}{ψ_{9}}$ $\frac{4{\psi }_1{\psi }_2{\psi }_4}{{\psi }_9}$	$\frac{4 ψ_{1} ψ_{2} ψ_{4}}{ψ_{9}}$ $\frac{4{\psi }_1{\psi }_2{\psi }_4}{{\psi }_9}$
$x_{fi}^{}$ ${x}_{{fi}}^{\mathrm{}}$	$\frac{2 ϕ_{1} ϕ_{3} ϕ_{4}}{ϕ_{5}}$ $\frac{2{\phi }_1{\phi }_3{\phi }_4}{{\phi }_5}$	$\frac{2 ϕ_{1} ϕ_{2} ϕ_{4}}{ϕ_{6}}$ $\frac{2{\phi }_1{\phi }_2{\phi }_4}{{\phi }_6}$	$\frac{2 ϕ_{1} ϕ_{3} ϕ_{4}}{ϕ_{5}}$ $\frac{2{\phi }_1{\phi }_3{\phi }_4}{{\phi }_5}$	$\frac{2 ϕ_{1} ϕ_{2} ϕ_{4}}{ϕ_{6}}$ $\frac{2{\phi }_1{\phi }_2{\phi }_4}{{\phi }_6}$
$q_{ei}^{}$ ${q}_{{ei}}^{\mathrm{}}$	$\frac{3 γ ψ_{1}}{ψ_{9}}$ $\frac{3\gamma {\psi }_1}{{\psi }_9}$	$\frac{3 γ ψ_{1}}{ψ_{9}}$ $\frac{3\gamma {\psi }_1}{{\psi }_9}$	$\frac{3 γ ψ_{1}}{ψ_{9}}$ $\frac{3\gamma {\psi }_1}{{\psi }_9}$	$\frac{3 γ ψ_{1}}{ψ_{9}}$ $\frac{3\gamma {\psi }_1}{{\psi }_9}$
$q_{fi}^{}$ ${q}_{{fi}}^{\mathrm{}}$	$\frac{3 γ ϕ_{1}}{ϕ_{5}}$ $\frac{3\gamma {\phi }_1}{{\phi }_5}$	$\frac{3 γ ϕ_{1}}{ϕ_{6}}$ $\frac{3\gamma {\phi }_1}{{\phi }_6}$	$\frac{3 γ ϕ_{1}}{ϕ_{5}}$ $\frac{3\gamma {\phi }_1}{{\phi }_5}$	$\frac{3 γ ϕ_{1}}{ϕ_{6}}$ $\frac{3\gamma {\phi }_1}{{\phi }_6}$
$π_{Mi}^{}$ ${\pi }_{{Mi}}^{\mathrm{}}$	$\frac{γ ψ_{1}^{2} ψ_{10}}{ψ_{9}^{2}} + \frac{γ ϕ_{1}^{2} ϕ_{7}}{ϕ_{5}^{2}}$ $\frac{\gamma {\psi }_1^2{\psi }_{10}}{{\psi }_9^2}+\frac{\gamma {\phi }_1^2{\phi }_7}{{\phi }_5^2}$	$\frac{γ ψ_{1}^{2} ψ_{10}}{ψ_{9}^{2}} + \frac{γ ϕ_{1}^{2}}{ϕ_{6}}$ $\frac{\gamma {\psi }_1^2{\psi }_{10}}{{\psi }_9^2}+\frac{\gamma {\phi }_1^2}{{\phi }_6}$	$\frac{γ ψ_{1}^{2}}{ψ_{9}} + \frac{γ ϕ_{1}^{2} ϕ_{7}}{ϕ_{5}^{2}}$ $\frac{\gamma {\psi }_1^2}{{\psi }_9}+\frac{\gamma {\phi }_1^2{\phi }_7}{{\phi }_5^2}$	$\frac{γ ψ_{1}^{2}}{ψ_{9}} + \frac{γ ϕ_{1}^{2}}{ϕ_{6}}$ $\frac{\gamma {\psi }_1^2}{{\psi }_9}+\frac{\gamma {\phi }_1^2}{{\phi }_6}$
WS^*	$\frac{4 γ ψ_{1}^{2}}{ψ_{9}} + \frac{4 γ ϕ_{1}^{2} ϕ_{8}}{ϕ_{5}^{2}}$ $\frac{4\gamma {\psi }_1^2}{{\psi }_9}+\frac{4\gamma {\phi }_1^2{\phi }_8}{{\phi }_5^2}$	$\frac{4 γ ψ_{1}^{2}}{ψ_{9}} + \frac{4 γ ϕ_{1}^{2} ϕ_{8}}{ϕ_{6}^{2}}$ $\frac{4\gamma {\psi }_1^2}{{\psi }_9}+\frac{4\gamma {\phi }_1^2{\phi }_8}{{\phi }_6^2}$	$\frac{4 γ ψ_{1}^{2}}{ψ_{9}} + \frac{4 γ ϕ_{1}^{2} ϕ_{8}}{ϕ_{5}^{2}}$ $\frac{4\gamma {\psi }_1^2}{{\psi }_9}+\frac{4\gamma {\phi }_1^2{\phi }_8}{{\phi }_5^2}$	$\frac{4 γ ψ_{1}^{2}}{ψ_{9}} + \frac{4 γ ϕ_{1}^{2} ϕ_{8}}{ϕ_{6}^{2}}$ $\frac{4\gamma {\psi }_1^2}{{\psi }_9}+\frac{4\gamma {\phi }_1^2{\phi }_8}{{\phi }_6^2}$

Note: $ψ_{9} = 9 γ - 4 ψ_{2}^{2} ψ_{4}^{2}$ ${\psi }_9=9\gamma -4{\psi }_2^2{\psi }_4^2$ , $ψ_{10} = 9 γ - 4 ψ_{2} ψ_{3} ψ_{4}^{2}$ ${\psi }_{10}=9\gamma -4{\psi }_2{\psi }_3{\psi }_4^2$ , *∈{YD, YJ, YR, YC}.

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