发表在 Management Science, 2022. DOI: https://doi.org/10.1287/mnsc.2022.4317
Keywords: fairness; price discrimination; personalization; social welfare
文章主要围绕两个问题展开：
- How do we model fairness in the context of pricing?
- What is the impact of adding fairness constraints to social welfare?
考虑商家将一个单位成本为 $c$ 的产品销售给两类消费者，$X \in \{0, 1\}$，以价格 $p_i, (i = 0, 1)$ 。
记 $d_i$ 是第 $i$ 类消费者的数量，$V_i \sim F_i( \cdot)$ 是该类消费者对于该产品的 valuation distribution，于是 $d_i \bar{F}_i(p_i)$ 是该类消费者的总需求。
商家的总利润为 $R_0(p_0) + R_1(p_1)$，其中 $R_i(p_i) = (p_i - c) d_i \bar{F}_i(p_i)$；在没有公平性约束的情况下，记 $p_i^\ast = \arg \max_{p_i} R_i(p_i)$ .
平均消费者福利为 $S_i(p_i) = \mathbb{E}[(V_i - p_i)^{+}]$，no-purchase valuation 为 $N_i(p_i) = \mathbb{E}[V_i \mid V_i < p_i]$
Fairness Definitions
文章定义了四种 fairness
- Price fairness, which is measured by $\left|p_0-p_1\right|$.
- Demand fairness, which is measured by $\mid \bar{F}_0\left(p_0\right)-$ $\bar{F}_1\left(p_1\right) \mid$.
- Surplus fairness, which is measured by $\mid S_0\left(p_0\right)-$ $S_1\left(p_1\right) \mid$.
- No-purchase valuation fairness, which is measured by $\left|N_0\left(p_0\right)-N_1\left(p_1\right)\right|$.
设 $M$ 是任意一种 fairness measure，则 $\left|M_0\left(p_0^\ast\right)-M_1\left(p_1^\ast\right)\right|$ 是在最优定价策略下的 fairness gap。
引入参数 $\alpha \in [0, 1]$ 来刻画公平程度，得到优化问题： $$ \begin{aligned} \mathcal{R}(\alpha):= & \max _{p_0, p_1 \geq 0}\, R_0\left(p_0\right)+R_1\left(p_1\right) \\ & \text { s.t. }\left|M_0\left(p_0\right)-M_1\left(p_1\right)\right| \leq(1-\alpha)\left|M_0\left(p_0^\ast\right)-M_1\left(p_1^\ast\right)\right|\\ & \end{aligned} $$