Subgroup fairness in two-sided markets

Abstract

It is well known that two-sided markets are unfair in a number of ways. For example, female drivers on ride-hailing platforms earn less than their male colleagues per mile driven. Similar observations have been made for other minority subgroups in other two-sided markets. Here, we suggest a novel market-clearing mechanism for two-sided markets, which promotes equalization of the pay per hour worked across multiple subgroups, as well as within each subgroup. In the process, we introduce a novel notion of subgroup fairness (which we call Inter-fairness), which can be combined with other notions of fairness within each subgroup (called Intra-fairness), and the utility for the customers (Customer-Care) in the objective of the market-clearing problem. Although the novel non-linear terms in the objective complicate market clearing by making the problem non-convex, we show that a certain non-convex augmented Lagrangian relaxation can be approximated to any precision in time polynomial in the number of market participants using semidefinite programming, thanks to its "hidden convexity". This makes it possible to implement the market-clearing mechanism efficiently. On the example of driver-ride assignment in an Uber-like system, we demonstrate the efficacy and scalability of the approach and trade-offs between Inter- and Intra-fairness.

Fig 1. Experimental results of four formulations implemented for 5 trials (4 × 5 runs in total), in DOcplex, with different batch of jobs from NYC taxi dataset and different dt(i,j) for each trial.

The mean values of six indices for the 20 implementations are denoted as bars, while the black vertical line on top of each bar denotes the mean ± one standard deviation.

Fig 2. An illustration of the trade-off between Intra- and Inter-fairness in a single trial.

The position of each dot represents the value of Intra- and Inter-fairness from one experiment using the implementation of the augmented Lagrangian formulation (10) in tssos. Red dots suggest the use of γ⁽¹⁾ = 1, γ⁽²⁾ = 0, as in [11]. Green dots suggest the use of Intra 5 + Inter 3 (L). The Pareto front is shown by a black curve. See the Section of Supporting information for further details.

Fig 3. All pairs of trade-offs between Intra- and Inter-fairness from 250 trials of formulation Intra 5 + Inter 3 (L).

Each trial uses distinct batch of jobs and its parameters (γ₁, γ₂) uniformly vary from (0.5, 0.5) to (0.9, 0.1) with the interval of 0.1. Each dot in a subplot represents the values of the corresponding Intra- and Inter-fairness measured post hoc for the result of one trial. The histograms on the top and the right sides show the distribution of the Intra- and Inter-fairness among the 250 trials, respectively.

Fig 4. The runtime of formulations “Sühr et al. 2019” and our method Intra 5 + Inter 3 using CP Optimizer (CP), CPLEX (MILP), and an SDP solver (SDPA), with or without augmented Lagrangian relaxation, respectively.

The four types of red curves denote the runtime of formulation “Sühr et al. 2019” (with △ markers) and its Lagrangian variant (with ▽ markers) against the number of variables. The four types of green curves denote that of formulations Intra 5 + Inter 3 and its variant. Subplots on the right present the mean runtime and mean ± one standard deviations across 5 runs by curves with shaded error bands. The subplots on the left give a zoom-in effect of the right ones, without shaded error bands.