Alibaba Pricing

\\item We propose a \\emph{semi-parametric} structural model that taking advantage of both predictability and interpretability for predicting counterfactual demand, and present a \\emph{multi-period} dynamic pricing approach which takes the demand uncertainty into consideration, and finally design a very efficient two stage pricing algorithm to solve it.

\\item We successfully apply the proposed pricing algorithm into the real-world e-commerce fresh retail. The results show great advantage.

The function $g(d_i; L_i, \\bm{\\theta})$ is the parametric price-elasticity model of product $i$ and its parameter $\\bm{\\theta} \\in \\mathbb{R}^{m+1}$ represents the price elasticity vector, which is shared by all products. The function $h(d_i^o, \\bm{x}_i)$ is nonparametric preditive model whose role is to predict the log ratio of sales  at the base discount $d_i^o$. Note that let $d_i=d_i^o$,  we have
\\begin{equation}
\\mathbb{E}[\\ln (Y_i/Y_i^{nor}) |d_i^o] = h(d_i^o, \\bm{x}_i).
\\end{equation}
Model (\\ref{semi_param}) is semi-parametric model since it consists of the parametric model $g(\\cdot)$ and the non-parametric model $h(\\cdot)$. It provides both high interpretability through parametric econometric model and high predictability through nonparametric machine learning model. We respectively present these two models in the following.

\\subsection{Base Sales Forecasting} \\label{sales_forecast}
The fundamental building block $h(d_i^o, \\bm{x}_i)$ aims to forecast the sales at base discount $d_i^o$ using the observational data $\\{ \\bm{x}_{i,t}, Y_{i,t}^{obs}\\}$, where $t$ represents the index of data points. A naive approach may treat the discount as one of input variables, and train a predictive model using observational data, then predict the counterfactual outcome by varying the discount, i.e., $E[Y_i|\\text{do}(d_i)] = h(d_i, \\bm{x}_i)$.  However, this method does not work.
Because transactional features, such as historical sales, are affected by the historical discount.
If we do an intervention on the discount $d_i$, i.e. varying the discount, we also need to change the relevant historical features $\\bm{x}_i(d_i)$ which is impossible. To resolve this problem, we dissociate the treatment from the sales-related features, and divide model into two parts
as presented in model (\\ref{semi_param}). We emphasize that the sales forecasting module only predicts the sales at an average of historical discounts $d_i^o$, i.e.,
\\begin{equation} \\label{non_param_predictive}
\\begin{split}
& h(\\cdot) \\gets \\{\\bm{x}_{i,t}, Y_{i, t}^{obs}/Y_{i,t}^{nor}\\}_{t=1,\\dots, T, i=1,\\dots, N},   \\\\
& \\ln (Y_i^{o}/Y_{i, T+1}^{nor})  = h(d_i^o, \\bm{x}_{i, T+1}), i=1,\\dots, N.
\\end{split}
\\end{equation}
We call $(d_i^o, Y_i^o)$ as base discount and base sales pair of product $i$.
Many fundamental non-parametric predictive model can be applied to learn $h(\\cdot)$, such as deep neural networks \\cite{lecun2015deep} and boosting tree model \\cite{chen2016xgboost}.

\\section{Multi-Period Price Optimization}

\\begin{figure}[tbp!]
	\\centering
	\\includegraphics[width=0.9\\linewidth]{figs/shops.pdf}
	\\caption{An example of the activated shops and index of multiple periods. The dotted blue vertical line represents the current time. The dotted grey blocks represent the inactivate days, and solid blue block represent the activate days.}
	\\label{fig:shops}
\\end{figure}

\\begin{figure}[tbp!]
	\\centering
	\\includegraphics[width=0.9\\linewidth]{figs/mdp.pdf}
	\\caption{An example of the Markov decision process of shop $j$ with $4$ states. Circles represent the states and the arrows represent the possible transition directions with action $d_{t}$.
		For simplicity, we only draw the arrows of blue circles.}
	\\label{fig:mdp}
\\end{figure}