Parking Slot Assignment Problem

Department of Economics Boston College October 11, 2016

Motivation Research Question Literature Review What is the concern? Cruising for parking is drivers behavior that circle around an area for a parking space. It is mainly caused by under priced (compared to the off-street parking garages) public parking. Previous evidences show that cruising and walking time is almost same as driving time. While cruising, drivers waste fuel and time, as well as contribute to the traffic congestion and air pollution.

Motivation Research Question Literature Review Evidences from the Literature 1 Year Location % of traffic cruising Ave. cruising time (min.) 1927 Detroit 19 1927 Detroit 34 1960 New Haven 17 1965 London 6.1 1965 London 3.5 1965 London 3.6 1977 Freiburg 74 1984 Jerusalem 9.0 1985 Cambridge 30 11.5 1993 New York 8 7.9 1993 New York 10.2 1993 New York 13.9 1997 San Francisco 6.5 1 Source: Shoup (2005), Arnott (2005)

Motivation Research Question Literature Review Putting aside externalities and consequences of the cruising, finding a parking slot is difficult Centralized system to assign slots to drivers

Motivation Research Question Literature Review Finding a parking slot is difficult Centralized system to assign slots to drivers Residents spaces are wasted when residents drive out Include residents slots into system

Motivation Research Question Literature Review Finding a parking slot is difficult Centralized system to assign slots to drivers Residents spaces are wasted Include residents slots into system There exists a huge price gap between off-street parking and on-street parking Endogenous price in the mechanism

Motivation Research Question Literature Review Figure: I m talking about this,

Motivation Research Question Literature Review Figure: Not this.

Motivation Research Question Literature Review Research Question How to design a centralized mechanism to allocate public parking slots to drivers in downtown. Basic Static Model Matching with Contract Extension to a dynamic model (in progress) Top Trading Cycle with counter-offer (in a separate work)

Motivation Research Question Literature Review Parking Related Ayala et al. (2011), Parking slot assignment games. Xu et al. (2016), Private parking slot sharing. Shoup (2005), The high cost of free parking. Arnott (2005), Alleviating urban traffic congestion.

Motivation Research Question Literature Review 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

The Cruising Game Stability of Nash Equilibrium A parking slot assignment problem is a list (I, S, I, D) with I = {i 1,, i n } : a set of drivers at a given point of time. S = {s 1,, s m } : a set of parking slots at a given point of time. I = ( i1,, in ) : a list of drivers strict preferences over slots. D =(, d is, ) : a list of distances from each driver to each slot, which will serve as priorities of drivers at each slot.

The Cruising Game Stability of Nash Equilibrium By interpreting the distances as preferences of slots, we can treat this problem as a two-sided matching market. Reasonable since, if possible, we want to minimize the total distance traveled. P S = (P s1,, P sm ) : slots strict preferences over drivers, which satisfy ip s i iff d is < d i s

The Cruising Game Stability of Nash Equilibrium Matching The outcome of the parking slot assignment problem is a matching. A matching µ : I S is a function from the set of drivers to the set of slots such that no slot is assigned to more than one driver. Let µ(i) be the slot that driver i is assigned under matching µ, and µ 1 (s) be the driver that the slot s is matched to.

The Cruising Game Stability of Nash Equilibrium Stable matching A driver-slot pair (i, s) blocks matching µ if either (i) s i µ(i) and ip s µ 1 (s) (ii) s i µ(i) and µ 1 (s) = A matching µ is stable if there is no driver-slot pair (i, s) that blocks µ.

The Cruising Game Stability of Nash Equilibrium The cruising game In a decentralized parking market, drivers are facing a game situation, namely a cruising game, where the players are the drivers, I, strategies are the set of the slots, S, and the outcome is a matching. A driver chooses a slot to go, and park there if it remains empty when he arrives. If it is occupied, he chooses another slot to search.

The Cruising Game Stability of Nash Equilibrium A Nash Equilibrium of the cruising game is a strategy profile of the drivers such that No driver can get a preferred slot than he is assigned by changing his strategy alone while the others are playing the Nash equilibrium strategies. Next theorem shows the relationship between the Nash equilibrium outcome and the stable matching of the parking slot assignment problem.

The Cruising Game Stability of Nash Equilibrium Theorem The set of Nash equilibrium outcomes of the cruising game is equal to the set of stable matchings of the parking slot assignment problem.

The Cruising Game Stability of Nash Equilibrium In Boston Mechanism, priorities are respected only when the agents rank them high, especially to the top choice. In parking problem, priorities are given by the distances to the slots, because you can go faster if you are closer. If you don t go to the closest space, you lose your priority over that space. In that sense, a static parking problem is similar to the Boston Mechanism. Therefore, the theorem is a Corollary of the Theorem 1 in Ergin and Sönmez (2006), and the proof is almost identical.

The Cruising Game Stability of Nash Equilibrium Proof of the Theorem 1. If µ is a Nash Equilibrium outcome, then it is stable. Let S = (S i1,, S in ) be a Nash Equilibrium strategy profile and µ be the resulting outcome. Assume that µ is not stable. Then there is a driver-slot pair (i, s) such that driver i prefers slot s to his assignment µ(i), and either slot s remains unmatched or i is closer to s than the driver i = µ 1 (s). If i changes his strategy to S i = s, then under the strategy profile S = (S i, S i ), driver i will be assigned s. Therefore, µ is not a Nash Equilibrium outcome, contradicting the assumption.

The Cruising Game Stability of Nash Equilibrium Proof of the Theorem 2. If µ is stable, then it is a Nash Equilibrium outcome. If each driver goes to the slot that they are assigned, i.e., if the strategy profile is S = (µ(1),, µ(n)), then the Cruising game ends at the first step and the resulting matching is µ. S 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 slot s to his matching µ(i), the one who is matched to s has higher priority than i, by stability.

The Cruising Game Stability of Nash Equilibrium 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

The Cruising Game Stability of Nash Equilibrium Problems of current system: Wasted spaces due to the lack of information Matching could be unstable due to the coordination failure Negative externalities of cruising-for-parking Introducing centralized mechanism: Complete parking information Assign better matching (stable) Drivers not cruising

The Cruising Game Stability of Nash Equilibrium DPDA 2 finds a stable matching Step 1 : Each driver i proposes to her 1st choice (among all acceptable choices). Each space s tentatively holds the closest proposal, if any, and reject the others.. Step k : Any driver who was rejected at step k-1 proposes to the best acceptable space which she hasn t yet made an offer. Each space holds the closest proposal 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

The Cruising Game Stability of Nash Equilibrium DPDA results in drivers-optimal stable matching, that is, all drivers prefer at least as well as any other stable matching. DPDA is strategy-proof for drivers. Best for the drivers under stability requirement. It is worst for parking spaces, implying 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 would be better than decentralized system, since the drivers will not be cruising for parking slots. Also, there are strategic issues in minimizing the total distance traveled.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Matching with contracts

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A parking slot assignment problem with contracts is a list (I, B, q, P, T, I, π) with I = {i 1,, i n } : a set of drivers. B = {b 1,, b m } : a set of parking blocks. q = (q b1,, q bm ) : a capacity vector. P = {p min,, p max } : a set of discrete prices. T = {r, l, s}: a set of terms. I = ( i1,, in ) : a list of drivers strict preferences over contracts. π =(π b1, π bm ) : a list of base priority rankings.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Including resident parking spaces We could build a model with visitors parking slots exclusively, treating residents and visitors parking as two distinct markets. However, by including residents parking slots into the system, we can allocate the parking spaces more efficiently, since the residents spaces are not always occupied. Any resident who wants to park at another slot will offer his own resident s slot to the system, until when he comes back.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Including resident parking spaces The set of driver I now includes residents, I = R V, where i represents a general driver, r R a resident, and v V a visitor. A resident has his own resident parking slot, and wants to park elsewhere as a visitor. To respect resident s priority, we need to consider how long each driver demands to park.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Terms T : {r, l, s} r : resident s parking, only the resident parking permit holder can be assigned a slot in that block with the term r. l : visitor s long-term parking, either long or short parking. s : visitor s short-term parking, only short-term parking. some of the slots only accept short-term parking to increase turnover in highly congested area. When a resident is assigned a slot other than his own, his resident s slot will be offered as a visitor parking slot.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Why blocks, rather than slots? If we treat the slots as resources to be allocated, there is no flexibility duration terms of the slots. By treating the blocks as resources, we can manage the durations within the block. For example, if there should always be a short-term parking slot in a given block, the block can assign different slot as a short-term depending on how the other slots are assigned.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism preference over contracts, not over slots A contract x = (i, b, p, t) specifies a driver i, a block b, the price p, and the term t. Let x I, x B, x P, and x T denote the driver i, the block b, the price p, and the contract term t of the contract x, respectively.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism How the system works Each driver submits their preference to the system (via smart-phone app). System assigns available spaces to the drivers every τ minutes. The spaces assigned to drivers will be reserved for them, and the parking fee will be charged immediately. If the system only gives recommendation or reservation without charging the fee, then drivers may not claim the space and resubmit their preferences, which makes the mechanism unreliable.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function Fix the priorities policy π. qb r and qs b slots are reserved for residents and short-term parking, respectively. Residents have priority at those qb r slots, and only short-term parking is acceptable at qb s slots The remaining slots, qb l = q b qb r qs b, are for long-term parking.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function r first, then l, and s last. any remaining slot from r or l will be assigned as s.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function Given a set of contracts X, a block b s choice C π b (X ) is obtained as follows: Phase 0 : Remove all the contracts for another block b and add them to the rejected set R π b (X ) and proceed with phase 1. Contracts survived phase 0 involves only block b. Phase 1 : For the first qb r potential elements of Cπ b (X ), choose any resident contract one at a time. If all contracts are considered in this phase, terminate the procedure; if qb r elements are chosen, set qb s = qs b and proceed with phase 2; if less than qr b elements are chosen, let qb s = qs b + (qr b (,,, r) Cπ b (X ) ) and proceed with phase 2.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function Phase 2 : For the next qb l potential elements, choose highest ranked long-term contracts one at a time according to π ranking. Remove all surviving long-term contracts adding them to Rb π(x ). If there is no surviving contract, then terminate the procedure. Otherwise, let qb s = qb s + (ql b (,,, l) Cπ b (X ) ) and proceed with phase 3. Phase 3 : For the last qb s potential elements of Cb π(x ), choose only the short-term contracts with highest π ranking one at a time, adding them to Cb π(x ). Remove all remaining contracts and terminate the procedure. Note that it is possible that some of the qb s slots remain unassigned..

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function A block has all 3 types of slots, R for residents, L for long-term visitor, and S for short-term visitor. There are six contracts in the block s choice set, one resident contract, three long-term visitor contracts, and two short-term visitor contracts.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function The block first choose the resident contract, assign it to a R slot.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function After that, long-term visitor contracts are chosen one by one up to its capacity, according to the priority ordering.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function Lastly, short-term visitor contracts are chosen.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function Any vacant resident slots are assigned to short-term visitor contracts according to the priority ordering.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function The final result of the block s choice function. Note that, a short-term visitor contract is occupying one vacant resident slot.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism A block s choice function In this choice function, any vacant residents parking spaces are assigned only to short-term visitors. This will ensure the residents taking the slot again when they come back. There will be efficiency loss if a resident gets a long-term visitor parking and her space is not assigned when there is no short-term demand for the slot. However, this is essential for the mechanism to respect resident s priority while producing a stable allocation. This issue will be discussed further later.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Property of the choice function Contracts are substitutes for block b if X, X X, X X R b (X ) R b (X ).

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Property of the choice function Contracts are not substitutes under block priority Suppose the capacity of the block b is qb l = 1, qs b = 1. Let X = {x, x } and X = {x, x, y}, where x = (i, b, p, l), x = (i, b, p, s), and y = (i, b, p, l), i.e., X has two contracts of driver i, long-term and short-term, and X has one more contracts, which involves another driver i. Also, suppose that π y > π x. Then, C b (X ) = {x}, R b (X ) = {x }, C b (X ) = {x, y}, violating the substitutes condition. R b (X ) = {x},

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Property of the choice function Contracts are unilateral substitutes for block b if a contract x = (i, b, p, t) is rejected from a set X when x is the only contract for driver i, then the contract x is also rejected from a larger set X X.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Property of the choice function Proposition: Contracts are unilateral substitutes for every block, and the block priorities satisfy the IRC condition and the LAD condition.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Cumulative Offer Algorithm Step 1 : One (randomly chosen) driver offers her most preferred contract x 1 = (i(1), b(1), p, t). The block that is offered the contract, b(1), holds the contract if x 1 C b(1) ({x 1 }) and rejects it otherwise. Let A b(1) (1) = {x 1 } and A b (1) = for all b b(1). In general, Step k : One of the drivers without contract held by any block offers her most preferred contract among the ones that are not previously rejected, x k = (i(k), b(k), p, t). Block b(k) holds the contract if x k C b(k) (A b(k) (k 1) {x k }) and rejects it otherwise. Let A b(k) (k) = A b(k) (k 1) {x k } and A b (k) = A b (k 1) for all b b(k).

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Cumulative Offer Algorithm The Algorithm terminates when every driver is matched to a block or every unmatched driver has no remaining acceptable contracts. Since there is a finite number of contracts, the algorithm terminates in some finite number K of steps. All contracts held at step K are finalized resulting in allocation b B C b (A k ).

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Definition An allocation X is stable if (1) i I C i (X ) = X. (2) b B C b (X ) = X, (3) there exist no driver i, block b, and contract x = (i, b, p, t) X \ X s.t. {x} = C i (X {x}) and x C b (X {x}).

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Theorem The cumulative offer algorithm produces a stable allocation under the block priorities π. Moreover, this allocation is weakly preferred by any driver to all stable allocation.

Matching with Contracts Centralized System Choice Functions Drivers Optimal Stable Mechanism Theorem Induced DOSM under block priorities π is strategy-proof. Definition A matching is equal-duration for residents if the term of the contract that a resident is assigned (other than his own resident one) is equal to the term of the contract that his resident slot is matched to. Theorem DOSM under block priorities π produces an equal-duration for residents matching.

Brief Overview We should assign the vacant resident s slot to a short-term contract. It will be better if we can match the durations of the contract that a resident is assigned and the contract that his slot is matched to. i.e., when a resident gets a long-term visitor slot, his resident s slot can accept a long-term contract as well.

Brief Overview The idea is a resident s dummy application. When a resident demands a short-term visitor parking along the algorithm, he applies to his own slot as a resident-short contract too, which is a dummy application. The dummy application does not occupy a slot, but will indicate that the slot is now only for short-term parking. We need to restrict the preferences to produce a stable matching that is equal-duration for residents.

Brief Overview Equal-duration algorithm r is a resident permit holder at block b. When he demands a long-term parking at another block b, the contract is considered as a visitor long-term. Since r gets a long-term contract, a visitor v can demand a long-term visitor parking at block b.

Brief Overview Equal-duration algorithm If r demands a short-term visitor contract, then he sends a dummy application r s to his own block. As a result, only a visitor who demands a short-term visitor parking can apply to the slot.

Brief Overview Equal-duration algorithm As a result, R slot at block b is assigned to a visitor v as a short-term contract, while the resident r is assigned a short-term contract at another block b.

Brief Overview Equal-duration algorithm When r demands a short-term contract at another block b, then the dummy application r s will reject any long-term contract at block b.

Brief Overview Equal-duration algorithm As a result, r is assigned a short-term visitor contract at block b, while the visitor v is remain unmatched.

Brief Overview Theorem There is no mechanism that is stable and produces equal-duration matching for the residents.

Brief Overview Definition A matching µ is wastefull if there is a unmatched space s and a driver i who prefers s to µ(i). Theorem If short-term visitor contracts has higher priority in residents parking slots to long-term visitor contracts, then the equal-duration algorithm produces a less wastefull allocation than the original cumulative offer algorithm.

This is a static model Dynamic extension where available parking slots and drivers change over time. is it a dominant strategy for drivers to apply as soon as they start seeking a parking space?

Dirvers and parking spaces arrive at each period. The static parking assignment algorithm repeats at every period with new drivers and spaces. How to design the priority structure?

Definition A driver postpones to participate if he submits his preference after period τ, when there are acceptable parkings for him at τ. Combined with truthful reporting of the preferences, we can define a dynamic version of strategy proofness. Definition A parking slot assignment mechanism is delay proof if it is a dominant strategy for each driver not to postpone to participate and truthfully report the preferences.

A dynamic parking slot assignment problem with contracts at each period τ is a list (I τ, B, q τ, P, T, I τ, π τ ) with 1. a set of drivers at each period I τ, 2. a set of parking blocks B = {b 1,, b m }, 3. a capacity vector q = (q τ b 1,, q τ b m ), 4. a set of discrete prices P = {p min,, p max }, 5. a set of terms T = {r, l, s}, 6. a list of drivers strict preferences over contracts I τ, 7. a list of priority rankings π τ = (π τ b 1, π τ b m ).

Naive algorithm repeating the static cumulative algorithm at every period. finalize the assignment at each period, so that if a driver who is assigned a contract wants to participate again in next period, he should give up the contract and pay the price for it. the priority rule does not change across period.

Naive algorithm 1. a set of drivers at each period I τ = {i τ 1, iτ 2, }, 6. a list of drivers strict preferences over contracts I τ = { τ 1, τ 2, }, 7. a list of priority rankings π τ = (π b1, π bm ).

Naive algorithm Theorem Naive dynamic mechanism is not delay proof. A driver who thinks there will be a better parking space next period, he may find it profitable to delay.

Sophisticated algorithm the system will run the static cumulative algorithm every period, with updated priority orderings based on the assignments in previous period contract x that is assigned at a block x B in round τ will get a highest priority at block x B in period τ + 1. If a driver x I decides to participate again in period τ + 1, therefore, he will be guaranteed a contract at least as good as the contract x.

Sophisticated algorithm let A τ I CI τ (X) be the set of drivers matched at period τ, but participate the system again in next period. let πb τ (τ 1) is block b s priority such that the contracts assigned to b at period τ 1 has highest priority at each corresponding space.

Sophisticated algorithm 1. a set of drivers at each period I τ = A τ 1 I {i τ 1, iτ 2, }, 6. a list of drivers strict preferences over contracts I τ = { A τ 1, τ 1, τ 2, }, 7. a list of priority rankings π τ = (π τ b 1 (τ 1), π τ b m (τ 1)).

Sophisticated algorithm Theorem The sophisticated dynamic mechanism for a parking slot assignment problem is delay proof.