Parking Space Assignment Problem: A Matching Mechanism Design Approach

Transcription

1 Parking Space Assignment Problem: A Matching Mechanism Design Approach Jinyong Jeong Boston College ITEA 2017, Barcelona June 23, 2017 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

2 Motivation Cruising for parking is drivers behavior that circle around an area for a parking space. While cruising, drivers waste fuel and time, as well as contribute to the traffic congestion and air pollution. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

3 Evidences from the Literature 1 Year Location % of traffic cruising Ave. cruising time (min.) 1927 Detroit Detroit New Haven London London London Freiburg Jerusalem Cambridge New York New York New York San Francisco Source: Shoup (2005), Arnott (2005) Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

4 Overview Difficult to find a parking space Centralized system to assign spaces to drivers Wasted residents spaces Include residents spaces into system Price gap between off-street parking and on-street parking Endogenous price in the mechanism Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

5 Overview Cruising game Parking problem as matching Mechanism design Policy suggestions Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

6 Figure: I m talking about this, Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

7 Figure: Not this. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

8 Literature: parking related Ayala et al. (2011), Parking space assignment games. Xu et al. (2016), Private parking space sharing. Shoup (2005), The high cost of free parking. Arnott (2005), Alleviating urban traffic congestion. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

9 Literature: matching Ergin and Sönmez (2006), Games of school choice under the Boston mechanism Hatfield and Milgrom (2005), Matching with contracts Hatfield and Kojima (2010), Substitutes and stability for matching with contracts Sönmez (2013), Bidding for Army Career Specialties Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

10 The Model A setup of the parking space assignment problem is: I = {i 1,, i n } : a set of drivers with unit demand, S = {s 1,, s m } : a set of available parking spaces, I = ( i1,, in ) : a list of individuals strict preferences. D =(d 11,, d nm ) : a list of distances from each driver to each space. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

11 Cruising game In a decentralized parking market, drivers are facing a game situation, namely a cruising game, where Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

12 Cruising game In a decentralized parking market, drivers are facing a game situation, namely a cruising game, where the players are the drivers, I, each driver s strategy is σ i S, strategies of all drivers are denoted by σ, and the outcome is an assignment A(σ; I, S) : I S. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

13 Cruising game In a decentralized parking market, drivers are facing a game situation, namely a cruising game, where the players are the drivers, I, each driver s strategy is σ i S, strategies of all drivers are denoted by σ, and the outcome is an assignment A(σ; I, S) : I S. A driver chooses a space to go, and park there if it remains empty when he arrives. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

14 Cruising game A driver i will be assigned a space s if σ i = s and, d is < d js for all j with σ j = s. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

15 Cruising game A driver i will be assigned a space s if σ i = s and, d is < d js for all j with σ j = s. In words, i chooses to go to the space s, and i is closer to any driver who goes to s. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

16 Cruising game Let A i (σ) be a space assigned to driver i when drivers strategy is σ. Definition (Nash Equilibrium) A strategy profile σ = {σ 1,, σ n} is a Nash equilibrium of the cruising game if for all i and σ i, A i (σ ) i A i (σ i, σ i ) where σ i denotes the strategy that all drivers except i follows the equilibrium strategy. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

17 Example s 2 1 s 1 s 1 2 s 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

18 Example: Nash equilibrium s 2 1 s 1 s 1 2 s 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

19 Example: Nash equilibrium s 2 1 s 1 s 1 2 s 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

20 Matching In a (one-sided) matching problem, there are I = {i 1,, i n } : a set of agents with unit demand, S = {s 1,, s m } : a set of resources to be assigned, I = ( i1,, in ) : a list of agents strict preferences over S, S =( s1,, sm ) : a list of priorities at each s over agents. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

21 Matching Priority s is a binary relation that determines who has a higher claim at the space s. An agent i has higher claim than j at space s, if i s j. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

22 Matching Priority s is a binary relation that determines who has a higher claim at the space s. An agent i has higher claim than j at space s, if i s j. Priority structure reflects various values, e.g., senior priority, first-come-first-served, affirmative action, random lottery number, etc. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

23 Matching Priority s is a binary relation that determines who has a higher claim at the space s. An agent i has higher claim than j at space s, if i s j. Priority structure reflects various values, e.g., senior priority, first-come-first-served, affirmative action, random lottery number, etc. In the parking problem, we first consider distance priority. i s j iff d is < d js Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

24 Matching A matching µ : I S is a function from the set of drivers to the set of spaces such that no space is assigned to more than one driver. Let µ(i) be the space that driver i is assigned under matching µ, and µ 1 (s) be the driver that the space s is matched to. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

25 Stable matching Definition A matching µ is stable if, i) for all i, µ(i) i, ii) there does not exist a driver-space pair (i, s), where s i µ(i) and i s µ 1 (s). Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

26 Stable matching Definition A matching µ is stable if, i) for all i, µ(i) i, ii) there does not exist a driver-space pair (i, s), where s i µ(i) and i s µ 1 (s). i) is individual rationality, ii) is called no justified envy. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

27 Example: stable matching s 2 1 s 1 s 1 2 s 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

28 Example: stable matching s 2 1 s 1 s 1 2 s 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

29 Theorem The set of Nash equilibrium outcomes of the cruising game is equal to the set of stable matchings of the parking space assignment problem. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

30 Proof of the Theorem 1. If µ is a Nash Equilibrium outcome, then it is stable. Let σ be a Nash equilibrium strategy profile and µ be the resulting outcome. Assume that µ is not stable. Then there is a driver-space pair (i, s) such that driver i prefers space s to his assignment µ(i), and either space s remains unmatched or i is closer to s than the driver j = µ 1 (s). If i changes his strategy to σ i = s, then under the strategy profile σ = (σ i, σ i ), driver i will be assigned s. Therefore, µ is not a Nash equilibrium outcome, contradicting the assumption. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

31 Proof of the Theorem 2. If µ is stable, then it is a Nash equilibrium outcome. If each driver goes to the space that they are assigned, i.e., if the strategy profile is σ = (µ(1),, µ(n)), then the Cruising game ends at the first step and the resulting matching is µ. σ is a Nash equilibrium, hence µ is a Nash equilibrium outcome, since no driver can profitably change his strategy from S. If a driver i prefers another space s to his matching µ(i), the one who is matched to s has higher priority than i, by stability. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

32 Mechanism Design Problems of current decentralized system: Wasted spaces due to the lack of information Matching could be unstable due to the coordination failure Hard to result in a Nash equilibrium Negative externalities of cruising-for-parking behavior Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

33 Mechanism Design Problems of current decentralized system: Wasted spaces due to the lack of information Matching could be unstable due to the coordination failure Hard to result in a Nash equilibrium Negative externalities of cruising-for-parking behavior Introducing centralized mechanism: Complete parking information Assign better matching (stable) Drivers not cruising Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

34 Mechanism Design Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

35 Mechanism Design Examples include; First-come-first-served serial dictatorship Random serial dictatorship Random assignment Auction Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

36 Mechanism Design Fix I and S. Then the parking space assignment problem, or simply a problem, is given by ( I, S ). Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

37 Mechanism Design Fix I and S. Then the parking space assignment problem, or simply a problem, is given by ( I, S ). A mechanism φ is a systematic procedure to find a matching to each problem, i.e., φ : ( I, S ) M, where M is the set of all matchings. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

38 Mechanism Design A mechanism φ : ( I, S ) M induces Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

39 Mechanism Design A mechanism φ : ( I, S ) M induces Preference revelation game for drivers. How to collect this information is practically important in this parking problem. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

40 Mechanism Design A mechanism φ : ( I, S ) M induces Preference revelation game for drivers. How to collect this information is practically important in this parking problem. Priority decision problem for the policy makers Priority will be set depending on the policy goals. Now this can be far more general than the distance priority. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

41 Mechanism Design A mechanism φ : ( I, S ) M induces Preference revelation game for drivers. How to collect this information is practically important in this parking problem. Priority decision problem for the policy makers Priority will be set depending on the policy goals. Now this can be far more general than the distance priority. Due to the time limit, these will be briefly addressed in the last part of the talk. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

42 Mechanism: DPDA 2 tep 1 : Each driver i proposes to her 1st choice. Each space s tentatively holds the one with highest priority, if any, and reject the others. 2 Drivers Proposing Deferred Acceptance inyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

43 Mechanism: DPDA 2 tep 1 : Each driver i proposes to her 1st choice. Each space s tentatively holds the one with highest priority, if any, and reject the others.. tep k : Any driver who was rejected at step k-1 proposes to the best space among which she hasn t yet made an offer. Each space holds the highest priority one among all the offers including it was holding, and rejects the others. 2 Drivers Proposing Deferred Acceptance Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

44 Mechanism: DPDA 2 tep 1 : Each driver i proposes to her 1st choice. Each space s tentatively holds the one with highest priority, if any, and reject the others.. tep k : Any driver who was rejected at step k-1 proposes to the best space among which she hasn t yet made an offer. Each space holds the highest priority one among all the offers including it was holding, and rejects the others. If no rejections occurs, finalize the mechanism and match the holding offers. 2 Drivers Proposing Deferred Acceptance Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

45 Mechanism: DPDA Example 1 s 1 1 s 2 1 s 1 2 s 2 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

46 Mechanism: DPDA Example 1 s 1 1 s 2 1 s 1 2 s 2 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

47 Mechanism: DPDA Example 1 s 1 1 s 2 1 s 1 2 s 2 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

48 Mechanism: DPDA Example 2 s 2 1 s 2 2 s 1 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

49 Mechanism: DPDA Example 2 s 2 1 s 2 2 s 1 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

50 Mechanism: DPDA Example 2 s 2 1 s 2 2 s 1 2 Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

51 Mechanism: DPDA DPDA produces drivers-optimal stable matching, that is, all drivers prefer the assignment at least as well as any other stable matching. Best for the drivers under stability requirement. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

52 Mechanism: DPDA DPDA produces drivers-optimal stable matching, that is, all drivers prefer the assignment at least as well as any other stable matching. Best for the drivers under stability requirement. DPDA is strategy-proof for drivers. Truthful reporting is the dominant strategy for every driver. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

53 Mechanism: DPDA DPDA produces drivers-optimal stable matching, that is, all drivers prefer the assignment at least as well as any other stable matching. Best for the drivers under stability requirement. DPDA is strategy-proof for drivers. Truthful reporting is the dominant strategy for every driver. It is spaces-pessimal, the total distance traveled is the most among all stable matchings. We wanted to minimize the negative externality of driving, so it would be better if we could minimize distance traveled. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

54 Mechanism: DPDA DPDA produces drivers-optimal stable matching, that is, all drivers prefer the assignment at least as well as any other stable matching. Best for the drivers under stability requirement. DPDA is strategy-proof for drivers. Truthful reporting is the dominant strategy for every driver. It is spaces-pessimal, the total distance traveled is the most among all stable matchings. We wanted to minimize the negative externality of driving, so it would be better if we could minimize distance traveled. Note that, however, it is far better than decentralized system, since the drivers will not be cruising for parking spaces. Also, there are strategic issues in minimizing the total distance traveled. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

55 Mechanism: SPDA Spaces proposing deferred acceptance is the same as DPDA, with spaces and drivers change their roles in the mechanism. (hence spaces proposing) Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

56 Mechanism: SPDA Spaces proposing deferred acceptance is the same as DPDA, with spaces and drivers change their roles in the mechanism. (hence spaces proposing) SPDA results in spaces-optimal stable matching, so the total distance traveled is minimized. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

57 Mechanism: SPDA Spaces proposing deferred acceptance is the same as DPDA, with spaces and drivers change their roles in the mechanism. (hence spaces proposing) SPDA results in spaces-optimal stable matching, so the total distance traveled is minimized. However, drivers now have incentive to manipulate their preferences to get a preferred outcome. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

58 Other issues Endogenous price can be done by Matching with contract model Cumulative offer algorithm (extension of DA) Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

59 Other issues Endogenous price can be done by Matching with contract model Cumulative offer algorithm (extension of DA) Including resident spaces Concerns regarding property rights Matching with claim (in progress) Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

60 Other issues Endogenous price can be done by Matching with contract model Cumulative offer algorithm (extension of DA) Including resident spaces Concerns regarding property rights Matching with claim (in progress) This is a static model Dynamic concerns. (driving closer to destination before submitting the preferences.) Some part of the dynamic issues can be addressed by priority design. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

61 How to submit preferences It is not feasible to have drivers submit their full list of preferences. safety concerns while driving lack of information Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

62 How to submit preferences It is not feasible to have drivers submit their full list of preferences. safety concerns while driving lack of information Ask minimal information to construct the preference lists, level down the strategic filed, complete information. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

63 How to submit preferences One suggestion is GDP. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

64 How to submit preferences One suggestion is GDP. G oal: final destination; D istance: that the driver is willing to walk more for the unit price reduction; P rice: the maximum willingness to pay if park at the destination. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

65 How to submit preferences One suggestion is GDP. G oal: final destination; D istance: that the driver is willing to walk more for the unit price reduction; P rice: the maximum willingness to pay if park at the destination. With GDP information, one can construct the full list of preference for all drivers; Restricting preference to single peaked, Assuming constant rate of substitution between walking and paying. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

66 How to design priorities To maximize revenue of the parking authority, price only priority. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

67 How to design priorities To maximize revenue of the parking authority, price only priority. To minimize the total distance traveled, distance only priority. Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43

68 How to design priorities To maximize revenue of the parking authority, price only priority. To minimize the total distance traveled, distance only priority. Or, mixture of the two? Jinyong Jeong (Boston College ITEA 2017, Barcelona) Parking Space Assignment Problem: A Matching Mechanism Design June 23, Approach / 43