Equilibrium Metrics for Dynamic Supply-Demand Networks Fan Zhou Department of Biostatistics, University of North Carolina at Chapel Hill AI Labs, DiDi Chuxing Based on joint works with Jieping Ye, Hongtu Zhu

Supply-Demand Imbalance Equilibrium Metric Supply-Demand Imbalance Equilibrium Metric AI digital product： A metric for the level of supply and demand balancing Predicting core metrics on the platform Providing objective function for dispatching strategy, pricing, and incentive optimizations Motivation Lack of a metric to represent the supply-demand situation Need a metric to evaluate the efficiency of the entire platform Number of People in Line Expected Waiting Time Equilibrium Metric Heatmap Policy Evaluation Pricing incentive City-wise Dispatching Deploying Growth

Equilibrium Metric Notations River Notations Undirected (or directed) graph 𝐺=(𝑉, 𝐸); weight matrix 𝑊= 𝑤 𝑖𝑗 ; 𝑉={ 𝑣 1 , …, 𝑣 𝑁 } is a set of |𝑉| = 𝑁 vertices, and 𝐸 is a set of edges. If 𝑣 𝑖 , 𝑣 𝑗 ∉𝐸, 𝑤 𝑖𝑗 =∞. (2) The transport cost on graph 𝐺 from 𝑣 𝑖 to 𝑣 𝑗 is defined as 𝑐 𝑖𝑗 = min 𝐾≥0, 𝑖 𝑘 𝑘=0 𝐾 : 𝑣 𝑖 → 𝑣 𝑗 { 𝑘 𝑤 𝑖 𝑘 , 𝑖 𝑘+1 :∀𝑘∈ 0, 𝐾−1 , 𝑣 𝑖 𝑘 , 𝑣 𝑖 𝑘+1 ∈𝐸} We can introduce a transport cost matrix on (𝐺, 𝑊), denoted as 𝐶= 𝑐 𝑖𝑗 ∈ 𝑅 𝑁 ×𝑁 , which is asymmetric when the graph is directed, and then we define the weighted graph structure as 𝐺, 𝑊, 𝐶 . (3) Two discrete Lebesgue measures 𝜇, 𝜈∈ 𝑀 + (𝑉) with locally finite mass. We define 𝜇 𝑗 =𝜇( 𝑣 𝑗 ) and 𝜈 𝑗 =𝜈( 𝑣 𝑗 ) as the point masses at vertex 𝑣 𝑗 for the two. 𝝁 = 𝑖=1 𝑁 𝜇 𝑖 and 𝝂= 𝑖=1 𝑁 𝜈 𝑖 may be unequal to each other.

Equilibrium Metric Traditional Metrics Hellinger Distance Wasserstein Distance 𝐷 𝐻 𝜇, 𝜈 = 1 2 𝑋 𝑑𝜇 𝑑𝜆 − 𝑑𝜈 𝑑𝜆 2 𝑑𝜆 Weakness A B within the range of order dispatching A B Case 1: Fail to consider transport from nearby grids Fail to consider the absolute difference between the two sides For the right side, the Wasserstein value is 0 for both since no transport is required at this moment. However, the difference between demand and supply is obviously different. Case 2: Problem by normalization

Equilibrium Metric Unbalanced Metrics Limitations on Graphs A directed edge from vertex A to vertex B, but not from vertex B to vertex A One `demand' unit at vertex C cannot be matched (the upper line) since the transport from vertex A to C is not allowed in this case, whereas they can be transported for Wasserstein distance

Equilibrium Metric Unbalanced Optimal Transport Problem Definitions Let 𝑐: 𝑉 × 𝑉→ 𝑅∪ {∞} be a function and 𝛾∈ 𝑀 + (𝑉 × 𝑉) be an nonnegative transport. Our EM on the weighted graph structure (𝐺, 𝑊, 𝐶) is defined as 𝜌 𝜆 𝜇,𝜈 𝐺, 𝐶 = min 𝜇 ∈ 𝑀 + 𝑉 , 𝛾∈ 𝑀 + (𝑉 × 𝑉) { 𝜈− 𝜇 +𝜆 𝑉 × 𝑉 𝑐 𝑑𝛾 } subject to an equality constraint and two transport constraints given by 𝜇 = 𝜇 , 𝑃 #1 𝑉 𝛾 𝑣 𝑖 = 𝑣 𝑗 ∈ 𝒩 𝑖 𝛾 𝑣 𝑖 , 𝑣 𝑗 = 𝜇 𝑖 𝑎𝑛𝑑 𝑃 #2 𝑉 𝛾 𝑣 𝑖 = 𝑣 𝑖 ∈ 𝒩 𝑗 𝛾 𝑣 𝑗 , 𝑣 𝑖 = 𝜇 𝑖 where 𝒩 𝑖 denotes the neighboring set of 𝑣 𝑖 in 𝑉. Our proposed EM is equivalent to solve a discrete optimization problem as follows: Unbalanced Optimal Transport Problem 𝜌 𝜆 𝜇,𝜈 𝐺, 𝐶 = min 𝛾∈Γ { 𝜈− 𝜈 1 +𝜆 𝑣 𝑖 ∈𝑉 𝑣 𝑗 ∈𝑉 𝑐 𝑖𝑗 𝛾 𝑖𝑗 } 𝑠.𝑡. 𝑣 𝑗 ∈ 𝒩 𝑖 𝛾 𝑖𝑗 = 𝜇 𝑖 , 𝑣 𝑗 ∉ 𝒩 𝑖 𝛾 𝑖𝑗 =0 , 𝑎𝑛𝑑 𝑣 𝑖 ∈ 𝒩 𝑗 𝛾 𝑗𝑖 = 𝜇 𝑖 𝑓𝑜𝑟 ∀ 𝑣 𝑖 ∈𝑉

Equilibrium Metric Distribution of cost 𝒄 𝒊𝒋 𝑐 𝑖𝑗 = 𝑖 𝑘 ∈ 𝑝 𝑖𝑗 𝑤 𝑖 𝑘 , 𝑖 𝑘+1 Density Function The unbalanced optimal transport problem is modified as 𝑐 𝑖𝑗 𝛿 0 = 𝑥< 𝛿 0 𝑥 𝑓 𝑖𝑗 𝑥 𝑑𝑥 𝜌 𝜆, 𝛿 0 𝜇,𝜈 𝐺, 𝐶 = min 𝛾∈Γ { 𝜈− 𝜈 1 +𝜆 𝑣 𝑖 ∈𝑉 𝑣 𝑗 ∈𝑉 𝑐 𝑖𝑗 𝛿 0 𝛾 𝑖𝑗 } 𝑠.𝑡. 𝑣 𝑗 ∈ 𝒩 𝑖 𝛾 𝑖𝑗 = 𝜇 𝑖 , 𝑣 𝑗 ∉ 𝒩 𝑖 𝛾 𝑖𝑗 =0 , 𝑎𝑛𝑑 𝑣 𝑖 ∈ 𝒩 𝑗 𝛾 𝑗𝑖 ⋅ 𝜏 𝑖𝑗 𝛿 0 = 𝜇 𝑖 𝑓𝑜𝑟 ∀ 𝑣 𝑖 ∈𝑉 𝑄 𝑐 𝑖𝑗 𝜏 𝑖𝑗 𝛿 0 = 𝛿 0

Equilibrium Metric Application: Matching Estimation Data Description: We used the supply-demand data set of Nanjing from April 21st to May 21st, 2018 on DiDi platform. We divided the whole urban area into 1000 non-overlapping hexagonal grids with radius of 1 km. The time scale is 10 min. Transport is only allowed between nearby vertexes. Experiment: We compare the dynamic coherence between supply and demand networks measured by different metrics with the true matching rate of the two systems. We define the estimated un-matched demand value at 𝑣 𝑖 at time 𝑡 as ( 𝑜 𝑖𝑡 − 𝑑 𝑖𝑡 )∗ 1( 𝑑 𝑖𝑡 ≤ 𝑜 𝑖𝑡 ) + 0∗ 1( 𝑑 𝑖𝑡 > 𝑜 𝑖𝑡 ) Four measurements are used to quantify the similarity between ground truth 𝑂 𝑢𝑛 and estimation 𝑂 𝑢𝑛 : Frobenius Distance: 𝐹 𝑂 𝑢𝑛 , 𝑂 𝑢𝑛 = 𝑇𝑟( 𝑂 𝑢𝑛 − 𝑂 𝑢𝑛 𝑂 𝑢𝑛 − 𝑂 𝑢𝑛 𝑇 ) PCA Similarity: 𝑇𝑟(𝐿 𝑀 𝑇 𝑀 𝐿 𝑇 ) where 𝐿 and M are the matrices contain the first 𝑘 PC scores of 𝑂 𝑢𝑛 and 𝑂 𝑢𝑛 Hellinger Distance and 𝑳 𝟏 Distance

Equilibrium Metric Application: Matching Estimation

Equilibrium Metric Application: Policy Evaluation We evaluate the dispatching policy of a real ride-sharing business platform by computing the distance between the measure calculated based on real dispatching history and the measure based on our EM. Specifically, we introduce an efficiency ratio of a real matching policy as 𝑅 𝑡 ={‖ 𝑜 𝑡 − 𝑑 𝑡 ‖ 1 +𝜆 𝑣 𝑖 ∈𝑉 𝑣 𝑗 ∈𝑉 𝑐 𝑖𝑗 𝛾 𝑖𝑗𝑡 } / 𝜌 𝜆 ( 𝑑 𝑡 , 𝑜 𝑡 |𝐺,𝐶) Figure below compares the global order answer rate with 𝑅 𝑡 on two selected days (April 22th-23th) at each 1-min time interval. The red and blue lines correspond to the real global order answer rate and the computed ratio 𝑅 𝑡

THANKS