1 Problem description

(1) Participants playing the game: A, B, C are representative of distribution channel: traditional network channel, mobile Internet channel and non-network channel. *i* ∈ {1, 2, 3}, respectively on behalf of player: A, B, C, three channels corresponding to different values.

(2) Assumption: game players are rational, no one has the advantage of distribution, each channel is independent, distribution in different channel happens simultaneously;

(3) Parameters: *T* is total supply of the three channels (*T* = *t*_{1} + *t*_{2}, *t*_{3}), *C* is the distribution cost, which is related with the distribution scale in that channel, *C*_{i}(*t*_{i}) means the cost of channel *i*, *M* is market size; *I* is profit.

2 Static game model

*I* is profit, *I*_{1}, *I*_{2}, *I*_{3} is profit respectively for three distribution channels:
$$\begin{array}{ll}{I}_{1}\hfill & =(M-T)*{t}_{1}-{C}_{1}({t}_{1})\hfill \\ \hfill & =(M-{t}_{1}-{t}_{2}-{t}_{3})*{t}_{1}-{C}_{1}({t}_{1})\hfill \end{array}$$(1)
$$\begin{array}{ll}{I}_{2}\hfill & =(M-T)*{t}_{2}-{C}_{2}({t}_{2})\hfill \\ \hfill & =(M-{t}_{1}-{t}_{2}-{t}_{3})*{t}_{2}-{C}_{2}({t}_{2})\hfill \end{array}$$(2)
$$\begin{array}{ll}{I}_{3}\hfill & =(M-T)*{t}_{3}-{C}_{3}({t}_{3})\hfill \\ \hfill & =(M-{t}_{1}-{t}_{2}-{t}_{3})*{t}_{3}-{C}_{3}({t}_{3})\hfill \end{array}$$(3)

For maximizing the profit, let $\frac{\partial {C}_{1}({t}_{1})}{\partial {t}_{1}}{{C}^{\prime}}_{1}$, $\frac{\partial {C}_{2}({t}_{2})}{\partial {t}_{2}}{{C}^{\prime}}_{2}$, $\frac{\partial {C}_{3}({t}_{3})}{\partial {t}_{3}}{{C}^{\prime}}_{3}$
$$\{\begin{array}{l}\frac{\partial {I}_{1}({t}_{1})}{\partial {t}_{1}}=M-2{t}_{1}-{t}_{2}-{t}_{3}-{{C}^{\prime}}_{1}\\ \frac{\partial {I}_{2}({t}_{2})}{\partial {t}_{2}}=M-{t}_{1}-2{t}_{2}-{t}_{3}-{{C}^{\prime}}_{2}\\ \frac{\partial {I}_{3}({t}_{3})}{\partial {t}_{3}}=M-{t}_{1}-{t}_{2}-2{t}_{3}={{C}^{\prime}}_{3}\end{array}$$(4)

Set $\frac{\partial {I}_{1}({t}_{1})}{\partial {t}_{1}}=0$, $\frac{\partial {I}_{2}({t}_{2})}{\partial {t}_{2}}=0$, $\frac{\partial {I}_{3}({t}_{3})}{\partial {t}_{3}}=0$, we get equations.
$$\{\begin{array}{l}M-2{t}_{1}-{t}_{2}-{t}_{3}={{C}^{\prime}}_{1}\\ M-{t}_{1}-2{t}_{2}-{t}_{3}={{C}^{\prime}}_{2}\\ M-{t}_{1}-{t}_{2}-2{t}_{3}={{C}^{\prime}}_{3}\end{array}$$(5)
$$\{\begin{array}{l}{t}_{1}=\frac{M-{t}_{2}-{t}_{3}-{{C}^{\prime}}_{1}}{2}\\ {t}_{2}=\frac{M-{t}_{1}-{t}_{3}-{{C}^{\prime}}_{2}}{2}\\ {t}_{3}=\frac{M-{t}_{1}-{t}_{2}-{{C}^{\prime}}_{3}}{2}\end{array}$$(6)

The solution is:
$$\{\begin{array}{l}{t}_{1}=\frac{M-3{{C}^{\prime}}_{1}+{{C}^{\prime}}_{2}+{{C}^{\prime}}_{3}}{4}\\ {t}_{2}=\frac{M+{{C}^{\prime}}_{1}-3{{C}^{\prime}}_{2}+{{C}^{\prime}}_{3}}{4}\\ {t}_{3}=\frac{M+{{C}^{\prime}}_{1}+{{C}^{\prime}}_{2}-3{{C}^{\prime}}_{3}}{4}\end{array}$$(7)

The equation result shows that distribution scale *t*_{i} is related with ${C}_{1}^{\text{'}},{C}_{2}^{\text{'}},{C}_{3}^{\text{'}}.{C}_{i}^{\text{'}}$ “is lower” means the cost per distribution for quality of channel *i* is lower, the circulation of channel *i* is more efficient than the other channel. The difference in the distribution scale between distribution channel, ${t}_{2}-{t}_{1}={C}_{1}^{\text{'}}-{C}_{2}^{\text{'}}$, ${t}_{3}-{t}_{1}={C}_{1}^{\text{'}}-{C}_{{3}^{\prime}}$, ${t}_{3}-{t}_{2}={C}_{2}^{\text{'}}-{C}_{{3}^{\prime}}$. It can be known that distribution scale difference between different distribution channels depends on cost change per distribution quality in that distribution channel. When ${C}_{1}^{\text{'}}={C}_{2}^{\text{'}}={C}_{3}^{\text{'}}={C}^{\prime}$, $T=\frac{3}{4}(M-{C}^{\prime})$. In the early period of transition of traditional publishing industry, the cost of digital content production and distribution is higher than traditional distribution channels. More digital products are distributed in non-network channel. With the gradual advancement of digital transition, the cost of network distribution is basically stable, and one can increase distribution in network channel, both of traditional network channel and mobile network channels. The research shows if traditional publishing did not finish the industry transition quickly, there would be obstacles to obtain high industry profits, and would leave digital publishing industry lags behind. So, one must make efforts to finish industry transformation.

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