Estimating the Loss of Efficiency due to Competition in Mobility on Demand Markets

Transcription

1 Estimating the Loss of Efficiency due to Competition in Mobility on Demand Markets Thibault Séjourné 1 Samitha Samaranayake 2 Siddhartha Banerjee 2 1 Ecole Polytechnique 2 Cornell University November 14, 2017 PGMO 2017 November 14, / 38

2 Overview 1 Introduction 2 Worst case study 3 Asymptotic study Description Warm-up: The case of the 2 Nodes Network Results for general networks and consequences PGMO 2017 November 14, / 38

3 Context PGMO 2017 November 14, / 38

4 Context Uber and Lyft drastically reduce the cost of taxis. PGMO 2017 November 14, / 38

5 Context Uber and Lyft drastically reduce the cost of taxis. Other MoD systems are appearing (Velib, Bixi, Car2Go, Autolib). PGMO 2017 November 14, / 38

6 Context Uber and Lyft drastically reduce the cost of taxis. Other MoD systems are appearing (Velib, Bixi, Car2Go, Autolib). 2/3 of the world population living in urban area by PGMO 2017 November 14, / 38

7 Context Uber and Lyft drastically reduce the cost of taxis. Other MoD systems are appearing (Velib, Bixi, Car2Go, Autolib). 2/3 of the world population living in urban area by In the USA, 6.9 bn hours lost in traffic jams, worth $160 bn. PGMO 2017 November 14, / 38

8 Context Uber and Lyft drastically reduce the cost of taxis. Other MoD systems are appearing (Velib, Bixi, Car2Go, Autolib). 2/3 of the world population living in urban area by In the USA, 6.9 bn hours lost in traffic jams, worth $160 bn. MoD systems has a great potential of being an efficient mean of transportation at a cheap price. PGMO 2017 November 14, / 38

9 Issue of competition Let s imagine the case of pooled demands : drop client 2 drop client 1 pick client 1 pick client 2 PGMO 2017 November 14, / 38

10 Issue of competition Let s imagine the case of pooled demands : drop client 2 drop client 1 pick client 1 pick client 2 Competition might considerably undermine the efficiency of MoD systems. PGMO 2017 November 14, / 38

11 Model The city is represented as a complete network of N stations. Hypothesis of an hourly steady state. Distance matrix D = (d ij ) (i,j) and travel time (τ ij ) (i,j) For each edge (i,j), customer demand Λ = (Λ ij ) Node total demand throughput Λ i = j (Λ ji Λ ij ) = (A.Λ) i We study the cost of rebalancing as a function RC(Λ) : min τ ij x ij (i,j) s.t. i, j (x ji x ij ) = Λ i x 0 PGMO 2017 November 14, / 38

12 First properties Properties RC is equal to its dual : RC is convex. max α i.λ i i s.t. (i, j), α i α j τ ij β corner point of the dual, c R, β + c is also a corner point and it yields the same score as β. Consequence: Without loss of generality, we can set any coordinate of β to zero. PGMO 2017 November 14, / 38

13 Worst case study Objective: Find the value of the worst split of demand in two parts, in order to check if the worst case is significant. The Price of Fragmentation (PoF) is: max RC(λ) + RC(Λ λ) s.t. 0 λ Λ PGMO 2017 November 14, / 38

14 Worst case study Objective: Find the value of the worst split of demand in two parts, in order to check if the worst case is significant. The Price of Fragmentation (PoF) is: max RC(λ) + RC(Λ λ) s.t. 0 λ Λ Property The PoF is convex. Thus, the optimal value is reached on a corner point. Proof: Consequence of the convexity of RC. PGMO 2017 November 14, / 38

15 Computing the worst PoF The PoF is a non-linear convex function, almost everywhere differentiable, over a bounded polyhedron isomorph to [0,1]. PGMO 2017 November 14, / 38

16 Computing the worst PoF The PoF is a non-linear convex function, almost everywhere differentiable, over a bounded polyhedron isomorph to [0,1]. Finding the best corner point is NP-hard. PGMO 2017 November 14, / 38

17 Computing the worst PoF The PoF is a non-linear convex function, almost everywhere differentiable, over a bounded polyhedron isomorph to [0,1]. Finding the best corner point is NP-hard. Approximated evaluation via projected subgradient descent. Result: Increase of 567% of the rebalancing cost due to the split. PGMO 2017 November 14, / 38

18 PoF under Stochastic Demand Splitting In practice the worst case will never happen, and there are random fluctuations. PGMO 2017 November 14, / 38

19 PoF under Stochastic Demand Splitting In practice the worst case will never happen, and there are random fluctuations. Use a stochastic model to compute an average loss with respect to the monopoly, ie: γ = E[RC(λ) + RC(Λ λ)] RC(Λ) With λ being a splitting r.v. respecting an exogenous market shares ratio ρ, first assumed to be homogeneous. PGMO 2017 November 14, / 38

20 PoF under Stochastic Demand Splitting In practice the worst case will never happen, and there are random fluctuations. Use a stochastic model to compute an average loss with respect to the monopoly, ie: γ = E[RC(λ) + RC(Λ λ)] RC(Λ) With λ being a splitting r.v. respecting an exogenous market shares ratio ρ, first assumed to be homogeneous. Initial idea: Use a binomial split s.t. λ B (Λ, ρ), assuming Λ has integer coordinates. PGMO 2017 November 14, / 38

21 Properties of the Binomial process Property: Fluid limit Under binomial splitting, Let s define θ N such that λ θ B (θλ, ρ). We have when θ : E[RC(λ θ )]/θ = E[RC(λ θ /θ)] RC(E[λ]) Thus, when the demand is scaled we get that γ 0. Proof: Consequence of the central limit theorem, the continuity of g and the decomposition of binomial into a sum of bernoulli random variables. PGMO 2017 November 14, / 38

22 Consequence of this property The previous property implies that there is no loss of efficiency when the number of demands tends to infinity. Questions If the efficiency loss disappears when the demand is scaled to infinity, then how much does the loss depends on the demand volume? How fast do we converge to the fluid limit? PGMO 2017 November 14, / 38

23 Consequence of this property The previous property implies that there is no loss of efficiency when the number of demands tends to infinity. Questions If the efficiency loss disappears when the demand is scaled to infinity, then how much does the loss depends on the demand volume? How fast do we converge to the fluid limit? We will then focus on the rescaled PoF which depends on θ N: γ θ = PoF (θ) θ = E[RC(λθ ) + RC(θΛ λ θ )] RC(θΛ) θ PGMO 2017 November 14, / 38

24 The example of the 2 nodes network Dual variables for each station α 1 = 0 and α 2, Total demand λ 12 = λ and λ 21 = µ, Random split X N (θρλ, θλs) and Y N (θρµ, θµs), s = ρ(1 ρ) Distances by d 12 = 1 and d 21 = a 1. d λ 12 = λ 12 = λ 21 = µ d 21 = a PGMO 2017 November 14, / 38

25 The example of the 2 nodes network Dual variables for each station α 1 = 0 and α 2, Total demand λ 12 = λ and λ 21 = µ, Random split X N (θρλ, θλs) and Y N (θρµ, θµs), s = ρ(1 ρ) Distances by d 12 = 1 and d 21 = a 1. d λ 12 = λ 12 = λ 21 = µ d 21 = a Formula: RC for one company RC(λ, µ) = max{a.(µ λ), (λ µ)} = [ (a 1).(µ λ) + (a + 1). µ λ ]/2 PGMO 2017 November 14, / 38

26 Two numerical examples Let s set λ = 50, µ = 70, a = 10, ρ = 0.5 Then E[X ] = 25, E[Y ] = 35 γ = g(x, Y ) + g(λ X, µ Y ) g(λ, µ) d λ = = 1 µ = d 21 = 10 PGMO 2017 November 14, / 38

27 Two numerical examples Let s set λ = 50, µ = 70, a = 10, ρ = 0.5 Then E[X ] = 25, E[Y ] = 35 γ = g(x, Y ) + g(λ X, µ Y ) g(λ, µ) d λ = = 1 µ = d 21 = 10 Example 1: X = 30 and Y = 40 Company 1: g(30, 40) = = 100 Company 2: g(20, 30) = = 100 Monopoly: g(50, 70) = 200 PGMO 2017 November 14, / 38

28 Two numerical examples Let s set λ = 50, µ = 70, a = 10, ρ = 0.5 Then E[X ] = 25, E[Y ] = 35 γ = g(x, Y ) + g(λ X, µ Y ) g(λ, µ) d λ = = 1 µ = d 21 = 10 Example 2: X = 30 and Y = 25 Company 1: g(30, 25) = 1 5 = 5 Company 2: g(20, 45) = = 250 Monopoly: g(50, 70) = 200 PGMO 2017 November 14, / 38

29 2 Nodes Network: Theoretical result Theorem When θ, A R such that : λ = µ γ θ = Aθ 1/2 ρ(λ µ) 2 λ µ γ θ = Aθ 3/2 e 2(1 ρ)(λ+µ).θ + o(θ 3/2 e Proof: Calculus via the folded normal distribution. ρ(λ µ)2 2(1 ρ)(λ+µ).θ ) PGMO 2017 November 14, / 38

30 2 Nodes Network: Theoretical result Theorem When θ, A R such that : λ = µ γ θ = Aθ 1/2 ρ(λ µ) 2 λ µ γ θ = Aθ 3/2 e 2(1 ρ)(λ+µ).θ + o(θ 3/2 e Proof: Calculus via the folded normal distribution. Some comments: ρ(λ µ)2 2(1 ρ)(λ+µ).θ ) PGMO 2017 November 14, / 38

31 2 Nodes Network: Theoretical result Theorem When θ, A R such that : λ = µ γ θ = Aθ 1/2 ρ(λ µ) 2 λ µ γ θ = Aθ 3/2 e 2(1 ρ)(λ+µ).θ + o(θ 3/2 e Proof: Calculus via the folded normal distribution. Some comments: ρ(λ µ)2 2(1 ρ)(λ+µ).θ ) Corner points matter: If we keep the same one for all companies (see Ex. 1) then there is no loss. PGMO 2017 November 14, / 38

32 2 Nodes Network: Theoretical result Theorem When θ, A R such that : λ = µ γ θ = Aθ 1/2 ρ(λ µ) 2 λ µ γ θ = Aθ 3/2 e 2(1 ρ)(λ+µ).θ + o(θ 3/2 e Proof: Calculus via the folded normal distribution. Some comments: ρ(λ µ)2 2(1 ρ)(λ+µ).θ ) Corner points matter: If we keep the same one for all companies (see Ex. 1) then there is no loss. We distinguish two regimes: PGMO 2017 November 14, / 38

33 2 Nodes Network: Theoretical result Theorem When θ, A R such that : λ = µ γ θ = Aθ 1/2 ρ(λ µ) 2 λ µ γ θ = Aθ 3/2 e 2(1 ρ)(λ+µ).θ + o(θ 3/2 e Proof: Calculus via the folded normal distribution. Some comments: ρ(λ µ)2 2(1 ρ)(λ+µ).θ ) Corner points matter: If we keep the same one for all companies (see Ex. 1) then there is no loss. We distinguish two regimes: Balanced demands: square root decay. PGMO 2017 November 14, / 38

34 2 Nodes Network: Theoretical result Theorem When θ, A R such that : λ = µ γ θ = Aθ 1/2 ρ(λ µ) 2 λ µ γ θ = Aθ 3/2 e 2(1 ρ)(λ+µ).θ + o(θ 3/2 e Proof: Calculus via the folded normal distribution. Some comments: ρ(λ µ)2 2(1 ρ)(λ+µ).θ ) Corner points matter: If we keep the same one for all companies (see Ex. 1) then there is no loss. We distinguish two regimes: Balanced demands: square root decay. Imbalanced demands: Exponential decay. PGMO 2017 November 14, / 38

35 PoF in General Networks Since the change of corner point induces the loss of efficiency we thus define: C α = { λ s.t. RC(λ) = α λ } PGMO 2017 November 14, / 38

36 PoF in General Networks Since the change of corner point induces the loss of efficiency we thus define: C α = { λ s.t. RC(λ) = α λ } Property α E, C α is such that : It is a closed cone. The intersection of two cones is a plane. Proof: The cones structure comes from the homogeneity of RC. The closedness is due to the fact that C α = (RC(λ) α λ) 1 ({0}). PGMO 2017 November 14, / 38

37 Geometrical intuition λ 2 λ 2 α α β β γ γ λ 1 λ 1 Exponential decay Square root decay PGMO 2017 November 14, / 38

38 Main Theorem: Concepts ρ is not necessarily homogeneous. ρ Λ C α, (1 ρ) Λ C β and Λ C η, i.e. α, β and η are the optimal corner points for respectively company 1, company 2 and the monopoly. The function f used in the main theorem will denote a speed decay which is faster than a square root (i.e. the exponential decay). PGMO 2017 November 14, / 38

39 Main Theorem Theorem Let s suppose that: There is a r.v. ξ such that λ θ = θρ Λ + θσξ with E[ ξ 1 ] < P( ξ > t) = O(f (t)) with f (t) = O(t) Let s define: Then we have: L = α ρ Λ + β (1 ρ) Λ η Λ α C α and β C β γ θ = L + O(f ( θ)) α C α \ C α or β C β \ C β γ θ = L + Θ(θ 1/2 ) PGMO 2017 November 14, / 38

40 Intuition of the proof λ 2 λ 2 α α β β γ γ λ 1 λ 1 Exponential decay Square root decay PGMO 2017 November 14, / 38

41 Interesting cases Applies for both the binomial process and the Poisson process: Binomial process: a R, P( ξ > t) = O(e at2 ) Poisson process: a R, P( ξ > t) = O(e at ) If ρ is homogenous, then α = β = η L = 0 PGMO 2017 November 14, / 38

42 Sufficient condition on the square root regime We know that the square root regime is critical because it decays much slower. But When does it happen? PGMO 2017 November 14, / 38

43 Sufficient condition on the square root regime We know that the square root regime is critical because it decays much slower. But When does it happen? Theorem: Sufficient condition for the square root decay Let s suppose that: The constraint matrix of the dual is not singular E H = {(i, j) x ij > 0} contains at least two connected components. Then the dual optimal point is defined by two different corner points. PGMO 2017 November 14, / 38

44 Sufficient condition on the square root regime We know that the square root regime is critical because it decays much slower. But When does it happen? Theorem: Sufficient condition for the square root decay Let s suppose that: The constraint matrix of the dual is not singular E H = {(i, j) x ij > 0} contains at least two connected components. Then the dual optimal point is defined by two different corner points. Proof: Let s define EĤ = {(i, j) α i α j = τ ij }. Starting from EĤ = E H, we define a dual corner point β which can be slightly modified to saturate new dual constraints and defining another dual corner point. PGMO 2017 November 14, / 38

45 Example: an imbalanced graph with square root decay Λ 1 = [2, 3, 4, 1] Λ 2 = [2, 3, 1, 4] PGMO 2017 November 14, / 38

46 Example: an imbalanced graph with square root decay Λ 1 = [2, 3, 4, 1] Λ 2 = [2, 3, 1, 4] PGMO 2017 November 14, / 38

47 Example: an imbalanced graph with square root decay Λ 1 = [2, 3, 4, 1] Λ 2 = [2, 3, 1, 4] Λ 3 = [2, 3, 3, 2] is optimal for two corner points. PGMO 2017 November 14, / 38

48 Numerical simulations with the previous network Simulations with Λ 1 : Imbalanced demand. Simulations with Λ 3 : Balanced demand. PGMO 2017 November 14, / 38

49 Numerical simulations with real data log(γ θ ) depending on log(θ). TLC Data clustered into 40 stations. PGMO 2017 November 14, / 38

50 Frequency of a square root decay Probability of having two connected components on two months period from TLC dataset. PGMO 2017 November 14, / 38

51 Numerical simulations comparing random processes log(γ θ ) depending on log(θ). Comparison of binomial and Poisson processes. PGMO 2017 November 14, / 38

52 On the necessity of a market maker Those properties allow to draw conclusions on how we can best treat the total demand. We have those three different regimes: PGMO 2017 November 14, / 38

53 On the necessity of a market maker Those properties allow to draw conclusions on how we can best treat the total demand. We have those three different regimes: Highly inhomogenous demand: PoF (θ) L.θ Balanced demand: PoF (θ) θ Imbalanced demand: PoF (θ) e aθ PGMO 2017 November 14, / 38

54 On the necessity of a market maker Those properties allow to draw conclusions on how we can best treat the total demand. We have those three different regimes: Highly inhomogenous demand: PoF (θ) L.θ Balanced demand: PoF (θ) θ Imbalanced demand: PoF (θ) e aθ Phase transition: either lim x PoF (θ) = 0 or + PGMO 2017 November 14, / 38

55 On the necessity of a market maker Those properties allow to draw conclusions on how we can best treat the total demand. We have those three different regimes: Highly inhomogenous demand: PoF (θ) L.θ Balanced demand: PoF (θ) θ Imbalanced demand: PoF (θ) e aθ Phase transition: either lim x PoF (θ) = 0 or + Thus we deduce that to enhance the efficiency we need to: Homogenize demands between companies. Imbalance demands at each node for both companies. PGMO 2017 November 14, / 38

56 Numerical simulations of the two decaying regimes θγ θ depending on θ for the demands Λ 1 and Λ 3 PGMO 2017 November 14, / 38

57 On the necessity of a market maker Though, pricing policies might already aim at balancing demands, thus generating balanced nodes and inducing a square root decay. PGMO 2017 November 14, / 38

58 On the necessity of a market maker Though, pricing policies might already aim at balancing demands, thus generating balanced nodes and inducing a square root decay. But is it possible to transform a balanced demand into two imbalanced one? PGMO 2017 November 14, / 38

59 On the necessity of a market maker Though, pricing policies might already aim at balancing demands, thus generating balanced nodes and inducing a square root decay. But is it possible to transform a balanced demand into two imbalanced one? We have that Λ 1 + Λ 2 = Λ, and they need to keep the same corner point to have at most a square root decay. Assuming the cones are convex, if they have the same corner points then so has the monopoly. Conversely, if the monopoly is optimized on several corner points, thus each company should be optimized on different corner points. PGMO 2017 November 14, / 38

60 On the necessity of a market maker Though, pricing policies might already aim at balancing demands, thus generating balanced nodes and inducing a square root decay. But is it possible to transform a balanced demand into two imbalanced one? We have that Λ 1 + Λ 2 = Λ, and they need to keep the same corner point to have at most a square root decay. Assuming the cones are convex, if they have the same corner points then so has the monopoly. Conversely, if the monopoly is optimized on several corner points, thus each company should be optimized on different corner points. Conclusion: It seems that the demand itself needs to be modified so as to make it imbalanced. PGMO 2017 November 14, / 38

61 Ideas to go further The previous conclusion draws attention on the need of a more complex model, or other approaches: PGMO 2017 November 14, / 38

62 Ideas to go further The previous conclusion draws attention on the need of a more complex model, or other approaches: Queuing theory to study the availability of nodes. On-line Linear Programs to have an unsteady demand. Lumped model instead of clustering the city into stations. Game theoretical models to consider pricing or social welfare. PGMO 2017 November 14, / 38

63 Conclusion We proved that there were dramatically different behaviours of the loss due to competition. PGMO 2017 November 14, / 38

64 Conclusion We proved that there were dramatically different behaviours of the loss due to competition. Those results imply that settling a market maker would be beneficial for the system. PGMO 2017 November 14, / 38

65 Conclusion We proved that there were dramatically different behaviours of the loss due to competition. Those results imply that settling a market maker would be beneficial for the system. Though the models shows some limits and the subject needs further developments. Other questions need to be answered: PGMO 2017 November 14, / 38

66 Conclusion We proved that there were dramatically different behaviours of the loss due to competition. Those results imply that settling a market maker would be beneficial for the system. Though the models shows some limits and the subject needs further developments. Other questions need to be answered: Is there a splitting policy that could dramatically enhance the system s efficiency? PGMO 2017 November 14, / 38

67 Conclusion We proved that there were dramatically different behaviours of the loss due to competition. Those results imply that settling a market maker would be beneficial for the system. Though the models shows some limits and the subject needs further developments. Other questions need to be answered: Is there a splitting policy that could dramatically enhance the system s efficiency? If there is, then could it be feasibly implemented in practice? PGMO 2017 November 14, / 38

68 Conclusion We proved that there were dramatically different behaviours of the loss due to competition. Those results imply that settling a market maker would be beneficial for the system. Though the models shows some limits and the subject needs further developments. Other questions need to be answered: Is there a splitting policy that could dramatically enhance the system s efficiency? If there is, then could it be feasibly implemented in practice? Which economic consequences would it have? Would it preserve the benefits of competition? PGMO 2017 November 14, / 38

69 Conclusion We proved that there were dramatically different behaviours of the loss due to competition. Those results imply that settling a market maker would be beneficial for the system. Though the models shows some limits and the subject needs further developments. Other questions need to be answered: Is there a splitting policy that could dramatically enhance the system s efficiency? If there is, then could it be feasibly implemented in practice? Which economic consequences would it have? Would it preserve the benefits of competition? Thank you! PGMO 2017 November 14, / 38

70 Proof for the exponential decay (1/3) We suppose for sake of simplicity that ρ is homogenous. We define x = ρλ, y = (1 ρ)λ, and δ such that: Where P (α,β) = C α C β. δ = min{min α E d(x, P (α,β)), min α E d(y, P (α,β))} We also denote: η = max η E η Furthermore we have: λ θ C α1 and (θλ λ θ ) C α2 PGMO 2017 November 14, / 38

71 Proof for the exponential decay (2/3) γ θ = θ 1.E[α 1 λθ + α 2 (θλ λθ ) η θλ] = θ 1.E[(ρα 1 + (1 ρ)α 2 η) θλ + (α 1 α 2 ) θσξ] 2 η.( i Λ i ).P( i, ( θσξ) i > θδ) + θ 1 α 1 α 2.E[ θσξ 1.1 { i, ( θσξ)i >θδ} ] 2 η.( i Λ i ).P( i, ( θσξ) i > θδ) + 2θ 1 η θσ 1.E[ ξ 1 i, ( θσξ) i > θδ].p( i, ( θσξ) i > θδ) PGMO 2017 November 14, / 38

72 Proof for the exponential decay (3/3) Thanks to the law of total expectation: E[ ξ 1 i, ( θσξ) i > θδ] < Furthermore: P( i, ( θσξ) i > θδ) P( i, θ σ 1. ξ i > θδ) N 2.P( σ 1. ξ i > δ θ) = O(f ( θ)) PGMO 2017 November 14, / 38

73 Proof for the square root decay (1/2) Demonstration of the lower bound: γ θ = θ 1.E[g(λ θ ) + g(θλ λ θ ) g(θλ)] θ 1.E[(α λ θ + β (θλ λ θ ) β θλ).1 {α σξ β σξ}] + θ 1.E[(β λ θ + α (θλ λ θ ) α θλ).1 {β σξ α σξ}] = θ 1/2.E[(α β) σξ.1 {α σξ β σξ}] + θ 1/2.E[(α β) σξ.1 {β σξ α σξ}] = θ 1/2.E[ (α β) σξ ] = Ω(θ 1/2 ) PGMO 2017 November 14, / 38

74 Proof for the square root decay (2/2) Demonstration of the higher bound: γ θ = θ 1.E[(g(λ θ ) + g(θλ λ θ ) g(θλ))1 { i, ( θσξ)i θδ} ] + θ 1.E[(g(λ θ ) + g(θλ λ θ ) g(θλ))1 { i, ( θσξ)i >θδ} ] θ 1/2.E[ (α β) σξ 1 { i, ( θσξ)i θδ} ] + O(f ( θ)) θ 1/2.E[ (α β) σξ ] + O(f ( θ)) = O(θ 1/2 ) Thus we finally have: γ θ = Θ( θ) PGMO 2017 November 14, / 38