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Parking Space Assignment Problem:A Matching Mechanism Design Approach

Jinyong Jeong

Boston CollegeITEA 2017, Barcelona

June 23, 2017

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

Motivation

Cruising for parking is drivers’ behavior that circle around an areafor a parking space.

While cruising, drivers waste fuel and time, as well as contribute tothe traffic congestion and air pollution.

Evidences from the Literature1

Year Location % of traffic cruising Ave. cruising time (min.)1927 Detroit 191927 Detroit 341960 New Haven 171965 London 6.11965 London 3.51965 London 3.61977 Freiburg 741984 Jerusalem 9.01985 Cambridge 30 11.51993 New York 8 7.91993 New York 10.21993 New York 13.91997 San Francisco 6.5

1Source: Shoup (2005), Arnott (2005)Jinyong Jeong (Boston College ITEA 2017, Barcelona)Parking Space Assignment Problem: A Matching Mechanism Design ApproachJune 23, 2017 3 / 43

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

Overview

Cruising game

Parking problem as matching

Mechanism design

Policy suggestions

Figure: I’m talking about this,

Figure: Not this.

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.

Literature: matching

Ergin and Sonmez (2006), Games of school choice under theBoston mechanism

Hatfield and Milgrom (2005), Matching with contractsHatfield and Kojima (2010), Substitutes and stability for matchingwith contracts

Sonmez (2013), Bidding for Army Career Specialties

The Model

A setup of the parking space assignment problem is:

I = {i1, · · · , in} : a set of drivers with unit demand,S = {s1, · · · , sm} : a set of available parking spaces,�I = (�i1 , · · · ,�in ) : a list of individuals’ strict preferences.D =(d11, · · · ,dnm) : a list of distances from each driver to each

space.

Cruising game

In a decentralized parking market, drivers are facing a gamesituation, namely a cruising game, where

the players are the drivers, I,each driver’s strategy is σi ∈ 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 emptywhen he arrives.

Cruising game

A driver i will be assigned a space s if

σi = s and,dis < djs for all j with σj = s.

In words,

i chooses to go to the space s,and i is closer to any driver who goes to s.

Cruising game

A driver i will be assigned a space s if

σi = s and,dis < djs for all j with σj = s.

In words,

i chooses to go to the space s,and i is closer to any driver who goes to s.

Cruising game

Let Ai (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 thecruising game if for all i and σi ,

Ai (σ∗) �i Ai (σi , σ∗−i )

where σ∗−i denotes the strategy that all drivers except i follows theequilibrium strategy.

Example

s2 �1 s1

s1 �2 s2

Example: Nash equilibrium

s2 �1 s1

s1 �2 s2

Example: Nash equilibrium

s2 �1 s1

s1 �2 s2

Matching

In a (one-sided) matching problem, there are

I = {i1, · · · , in} : a set of agents with unit demand,S = {s1, · · · , sm} : 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.

Matching

Priority �s is a binary relation that determines who has a higherclaim 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 dis < djs

Matching

A matching µ : I → S is a function from the set of drivers to the setof 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.

Stable matching

DefinitionA 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.

Stable matching

DefinitionA 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.

Example: stable matching

s2 �1 s1

s1 �2 s2

Example: stable matching

s2 �1 s1

s1 �2 s2

TheoremThe set of Nash equilibrium outcomes of the cruising game is equal tothe set of stable matchings of the parking space assignment problem.

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 theresulting outcome. Assume that µ is not stable.Then there is a driver-space pair (i , s) such that driver i prefersspace s to his assignment µ(i), and either space s remainsunmatched 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 profileσ′ = (σ′i , σ

∗−i ), driver i will be assigned s.

Therefore, µ is not a Nash equilibrium outcome, contradicting theassumption.

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 thestrategy profile is σ = (µ(1), · · · , µ(n)), then the Cruising gameends 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 onewho is matched to s has higher priority than i , by stability.

Mechanism Design

Problems of current decentralized system:Wasted spaces due to the lack of informationMatching 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 informationAssign better matching (stable)Drivers not cruising

Mechanism Design

Problems of current decentralized system:Wasted spaces due to the lack of informationMatching 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 informationAssign better matching (stable)Drivers not cruising

Mechanism Design

Examples include;

First-come-first-served serial dictatorshipRandom serial dictatorshipRandom assignmentAuction

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 toeach problem,i.e., φ : (�I ,�S)→ M, where M is the set of all matchings.

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 toeach problem,i.e., φ : (�I ,�S)→ M, where M is the set of all matchings.

Mechanism Design

A mechanism φ : (�I ,�S)→ M induces

Preference revelation game for drivers.How to collect this information is practically important in this parkingproblem.

Priority decision problem for the policy makersPriority 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 partof the talk.

Mechanism Design

Mechanism: DPDA2

Step 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.

...Step 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 offersincluding it was holding, and rejects the others.

If no rejections occurs, finalize the mechanism and match the”holding” offers.

2Drivers Proposing Deferred AcceptanceJinyong Jeong (Boston College ITEA 2017, Barcelona)Parking Space Assignment Problem: A Matching Mechanism Design ApproachJune 23, 2017 31 / 43

Mechanism: DPDA2

Mechanism: DPDA

Example 1

s1 �1 s2 �1 ∅s1 �2 s2 �2 ∅

Mechanism: DPDA

Example 1

s1 �1 s2 �1 ∅s1 �2 s2 �2 ∅

Mechanism: DPDA

Example 1

s1 �1 s2 �1 ∅s1 �2 s2 �2 ∅

Mechanism: DPDA

Example 2

s2 �1 ∅s2 �2 s1 �2 ∅

Mechanism: DPDA

Example 2

s2 �1 ∅s2 �2 s1 �2 ∅

Mechanism: DPDA

Example 2

s2 �1 ∅s2 �2 s1 �2 ∅

Mechanism: DPDA

DPDA produces drivers-optimal stable matching, that is, all driversprefer the assignment at least as well as any other stablematching.

⇒ 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 mostamong all stable matchings.We wanted to minimize the negative externality of driving, so itwould be better if we could minimize distance traveled.

Note that, however, it is far better than decentralized system, sincethe drivers will not be cruising for parking spaces.Also, there are strategic issues in minimizing the total distancetraveled.

Mechanism: DPDA

Mechanism: SPDA

Spaces proposing deferred acceptance is the same as DPDA, withspaces and drivers change their roles in the mechanism. (hencespaces proposing)

SPDA results in spaces-optimal stable matching, so the totaldistance traveled is minimized.

However, drivers now have incentive to manipulate theirpreferences to get a preferred outcome.

Mechanism: SPDA

Other issues

Endogenous price can be done byMatching with contract modelCumulative offer algorithm (extension of DA)

Including resident spacesConcerns regarding property rightsMatching with claim (in progress)

This is a static modelDynamic concerns. (driving closer to destination before submittingthe preferences.)Some part of the dynamic issues can be addressed by prioritydesign.

Other issues

How to submit preferences

It is not feasible to have drivers submit their full list of preferences.

safety concerns while drivinglack of information

Ask minimal information to construct the preference lists,

level down the strategic filed,complete information.

It is not feasible to have drivers submit their full list of preferences.

safety concerns while drivinglack of information

Ask minimal information to construct the preference lists,

level down the strategic filed,complete information.

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 preferencefor all drivers;

Restricting preference to single peaked,Assuming constant rate of substitution between walking and paying.

One suggestion is GDP.G oal: final destination;D istance: that the driver is willing to walk more for the unit price

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?

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