发表在 Production and operations management, 2021. DOI: https://doi.org/10.1111/poms.13424.
Key words: overconfidence; transshipment; behavioral operations management; managerial bias
这篇文章研究了过度自信这一行为对带转运的报童模型的影响。
Consider . . . the most important concepts of behavioral economics: overconfidence . . .
—— Richard Thaler (2016)
Of these numerous types of bias, overconfidence is “one of the most consistent, powerful, and widespread” cognitive biases.
Overconfident newsvendor
过度自信的 newsvendor，他们会更多的认为需求靠近均值，如果需求是 $D$，那么因为这种 cognitive bias，在过度自信的人眼里，需求是： $$ X = \alpha \mu + (1-\alpha) D $$ 其中 $\mu = {E}[D]$。如果 $\alpha > 0$，说明有过度自信存在，$\alpha=1$ 是最大程度的自信。
过度自信降低了需求估计的偏差，因为 ${E}[X] = {E}[D], \operatorname{Var}(X) = (1-\alpha)^2 \operatorname{Var}(D)$ .
Transshipment
考虑两个独立的newsvendor在两个不同的地点，它们面临同样的需求分布 $D_i\, (i=1,2)$，订货成本 $c$，单位利润 $r$，在这种 basic setting 下每个 newsvendor 的利润是： $$ \pi_{i}^{N}:=E\left\{r \min \left(D_{i}, q_{i}\right)-c q_{i}\right\} $$ 假如 newsvendor $i$ 处有多余的库存，同时 newsvendor $j$ 处有未满足的需求，这时候可以从 $i$ 到 $j$ 转运货物，假设这种转运的成本（售价）是 $\tau \leq r$， $$ \begin{aligned} \pi_{i}^{T}\left(q_{i}, q_{j}\right) &= E\left\{r \min \left(D_{i}, q_{i}\right)-c q_{i}\right\} \\ &+ E\left\{(r-\tau) \min \left[\left(D_{i}-q_{i}\right)^{+},\left(q_{j}-D_{j}\right)^{+}\right]\right.\\ & + \left.\tau \min \left[\left(D_{j}-q_{j}\right)^{+},\left(q_{i}-D_{i}\right)^{+}\right]\right\} \end{aligned} $$ 在均衡状态下，带转运的 $(q_1^T, q_2^T)$ 带来的利润 $\pi_T$ 不低于 $\pi_N$，这是文章的 lemma1。
Symmetric Overconfidence
在相同的过度自信水平下，订货量 $\hat{q}^{N}:=\alpha \mu+(1-\alpha) q^{N}$，从而最终的利润是： $$ \hat{\pi}_{N}(\alpha):=E\left\{r \min \left(D_{i}, \hat{q}^{N}\right)-c \hat{q}^{N}\right\} $$ 文章接着给出了一个 Observation，就是当 $0 < \alpha < 1$ 时，有可能过度自信会带来利润的降低。
Taken together, when the transshipment price is low and the profit margin is high, the second effect can dominate the first effect so that the biased newsvendors eventually suffer from transshipment.
Asymmetric Overconfidence
这一部分，文章研究不同过度自信水平造成的影响。不妨设 $\alpha_1 < \alpha_2$。
给出的结论是：如果对方的过度自信水平更高，那么自己的利润越低！
Empirical Evidence
Using the experiment data from Zhao et al. (2020)，这一部分文章做了三步统计分析
- Use MLE to estimate the overconfidence parameter
- Use MLE to fit the reduced model
- Perform a likelihood ratio test to assert the existence of overconfidence by rejecting null hypothesis
Note their MLE for $\alpha$ is approximately 0.65.
Robustness Check
In this section, more behavioral factors are incorporated.
Fairness Concern
With fairness taken into account, this paper models the utility of newsvendor $i$ by: $$ U_{i}=\pi_{i}-\lambda\left(\pi_{j}-\pi_{i}\right) $$ Where $\lambda \geq 0$ is a fairness parameter. Typically, $\lambda=0$ implies fairness-neutral newsvendor.
This paper argues that transshipment may still hurt overconfident newsvendors with fairness concerns when the overconfidence level is moderate $(0.25 < \alpha < 0.42)$
Demand- and Supply-side Thinking
Loss Aversion
Demand Anchoring
Conclusion
文章给出了一个不同寻常的结论：the biased newsvendor does not necessarily receive a higher expected profit under transshipment, particularly when the transshipment price is low and profit margin is high.