Dynamic Nonlinear Pricing Model Based on Adaptive and Sophisticated Learning

Table 1

List of symbols (𝑖=1,2).

𝐴^𝑎𝑖,𝐴^𝑠𝑖	𝐴^𝑎𝑖,𝐴^𝑠𝑖 denote, respectively, the propensity of adaptive and sophisticated consumers to good 𝐴^𝑎𝑖,𝐴^𝑠𝑖∈[0,∞)
𝜃^𝑎𝑖,𝜃^𝑠𝑖	𝜃^𝑎𝑖,𝜃^𝑠𝑖 are the adaptive and sophisticated consumers’ estimate for the quality or goodwill of good 𝑖, respectively. 𝜃^𝑎𝑖,𝜃^𝑠𝑖∈[0,∞) and 𝜃^𝑎𝑖,𝜃^𝑠𝑖≥𝑝𝑖
𝑝𝑖	The price of good 𝑖. 𝑝𝑖∈[0,∞)
𝜂^𝑎,𝜂^𝑠	𝜂^𝑎,𝜂^𝑠 are the adaptive and sophisticated consumers’ relative quality or goodwill of two goods, respectively. 𝜂^𝑎=𝜃^𝑎1−𝜃^𝑎2;𝜂^𝑠=𝜃^𝑠1−𝜃^𝑠2
𝜌	𝜌=𝑝1−𝑝2 is the relative price of two goods.
𝑞^𝑎𝑖,𝑞^𝑠𝑖	𝑞^𝑎𝑖,𝑞^𝑠𝑖 are the probability that the adaptive consumers, sophisticated consumers choose good 𝑖, respectively. 0≤𝑞^𝑎𝑖,𝑞^𝑠𝑖≤1
𝛽	𝛽 measures the sensitivity of consumers for goods. 𝛽∈[0,∞].
𝜋𝑖(𝑡)	𝜋𝑖(𝑡) is the utility which is got by consuming good 𝑖 at time 𝑡,
𝛿1	𝛿1 is the “recency” parameter of preference for purchased goods. 0<𝛿1≤1
𝛿2	𝛿2 is the “recency” parameter of preference for goods which are not purchased.
𝜔	𝜔=𝛿2/𝛿1 is the ratio of “recency” parameters described by 𝛿1 and 𝛿2. 0≤𝜔≤1
𝜆	𝜆 respects the influence coefficient from other consumers to sophisticated consumers. 𝜆≥0
𝜎	𝜎=𝜆/𝛿1 is the ratio of influence coefficient 𝜆 and “recency” parameter 𝛿1. 𝜎≥0