使用市场边际价值来解决干扰偏差

As illustrated in Figure 2, when supply is abundant, the marginal value of having rider R1 matches its face value, which is $6. However, in a low supply scenario where resources are limited, the resource is allocated to rider R2. Consequently, both the marginal and face values for R1 become zero. For rider R2, the face value is $10, but its marginal value is only the additional $4 gained by having rider R2. This demonstrates how the marginal value inherently accounts for resource contention. By aggregating the marginal values across both the treatment and control groups and calculating the difference, one can derive an unbiased estimator of the average treatment effect.

如图2所示，当供应充足时，骑手R1的边际价值与其面值相匹配，为$6。然而，在资源有限的低供应场景中，资源被分配给骑手R2。因此，R1的边际价值和面值都变为零。对于骑手R2，面值为$10，但其边际价值仅为拥有骑手R2所获得的额外$4。这表明边际价值本质上考虑了资源争夺。通过汇总治疗组和对照组的边际价值并计算差异，可以得出平均治疗效果的无偏估计。

How to compute MMVs?

如何计算MMV？

As previously mentioned, shadow prices in the dispatch optimization problem can be used to obtain the MMVs. The primal dispatch problem can be described as follows:

如前所述，调度优化问题中的影子价格可以用来获得MMV。原始调度问题可以描述如下：

Where xᵢⱼ is a variable that takes the value of 1 if the driver j got matched to that rider i, and 0 otherwise. πᵢⱼ represents the score (e.g., profit) of matching driver j to rider i. The first constraint ensures that a driver is matched with at most one ride per a matching cycle (more on this later), and the second constraint indicates that a rider can have at most one driver. Solving this optimization gives the optimal matching of drivers to riders. We can relax the last constraint into xᵢⱼ ≥ 0, and obtain a linear relaxation of the above problem for which we can compute the dual as:

其中xᵢⱼ是一个变量，如果司机j与乘客i匹配，则取值为1，否则为0。πᵢⱼ表示将司机j与乘客i匹配的得分（例如，利润）。第一个约束确保每个匹配周期内每个司机最多匹配一个行程（稍后会详细说明），第二个约束表示每个乘客最多只能有一个司机。解决此优化问题可以得到司机与乘客的最佳匹配。我们可以将最后一个约束放宽为xᵢⱼ ≥ 0，并获得上述问题的线性松弛，从而可以计算对偶为：

The dual variable μⱼ is associated to the driver constraint (first primal constraint), and λᵢ is associated with the rider constraint. This means that for each driver j, there is a...