Assignment of Arrival Slots · 1:06 AA111 1:08 AF023 1:10 BA229 1:12 UA123..... 1:58 EI022 U.S. FAA now uses the Compression Algorithm (described later). Schummer & Vohra (Northwestern

Assignment of Arrival Slots

James Schummer Rakesh V. Vohra

Kellogg School of Management (MEDS)Northwestern University

March 2012

Schummer & Vohra (Northwestern Univ.) Assignment of Arrival Slots March 2012 1 / 38

Overview

Bad weather reduces available landing slots and causes flightcancelations.

Initially, scarce resources (slot rationing).Eventually, excess resources (flight cancelations).A (re-)matching problem.

Studied from operations perspective; little from mechanism designperspective.

FAA/airlines agree on desirable attributes (mechanism designaxioms).

Our goal: formalize attributes; critique existing mechanism; offersuperior alternative.


Overview

Bad weather reduces available landing slots and causes flightcancelations.

Initially, scarce resources (slot rationing).Eventually, excess resources (flight cancelations).A (re-)matching problem.

Studied from operations perspective; little from mechanism designperspective.

FAA/airlines agree on desirable attributes (mechanism designaxioms).

Our goal: formalize attributes; critique existing mechanism; offersuperior alternative.


Ground Delay Program (part 1 of 2)

Rationing a scarce supply of slots to scheduled flights.

Slot Flight

1:00 BA0271:01 UA3011:02 UA0811:03 AA1111:04 AF0231:05 BA2291:06 UA123

......

=⇒

Slot Flight

1:001:021:041:06

......

U.S. F.A.A. now uses Ration-by-Schedule: first-come first-served,based on position in previous schedule.


Ground Delay Program (part 1 of 2)

Rationing a scarce supply of slots to scheduled flights.

Slot Flight

1:00 BA0271:01 UA3011:02 UA0811:03 AA1111:04 AF0231:05 BA2291:06 UA123

......

=⇒

Slot Flight

1:00 BA0271:02 UA3011:04 UA0811:06 AA111

......

U.S. F.A.A. now uses Ration-by-Schedule: first-come first-served,based on position in previous schedule.


GDP (part 2 of 2): Reassignment

Rationing/bad weather/mechanical problems may cause an airline tocancel a flight, leaving excess resources (slots). Part 2 of GDP is tomove flights up, making efficiency use of landing capacity.

Slot Flight

1:00 BA0271:02 UA3011:04 UA0811:06 AA1111:08 AF0231:10 BA2291:12 UA123

......

1:58 EI022

U.S. FAA now uses the Compression Algorithm (described later).Schummer & Vohra (Northwestern Univ.) Assignment of Arrival Slots March 2012 4 / 38



Slot Flight

1:001:02 UA3011:041:06 AA1111:08 AF0231:10 BA2291:12 UA123

......

1:58 EI022




Slot Flight

1:001:02 UA3011:041:06 AA1111:08 AF0231:10 BA2291:12 UA123

......

1:58 EI022


A matching problem

Our objective: study only “GDP part 2” (the reassignmentproblem).We take part 1’s rationing (RBS) as given.

Any improvement we offer over the current algorithms becomesmore striking, since adapting to RBS is a design constraint.In actuality, RBS is executed just once, while ‘part 2’ is repeatedlyiterated over time: too complex to make reassignment a function ofprevious rounds’ trades.


Industry Objectives – Incentives

The FAA and airlines agree on various objectives.

“If an airline reported a mechanical delay on a flight, thesystem would re-project its arrival time. If [rationing withGrover–Jack were done] at that time, that flight would likelyreceive an additional delay on top of its mechanical delay.These effects were known as the ‘Double Penalty’issue. . . The airlines would simply not send in information thatwould [harm themselves]. RBS removes this disincentive."

– FAA website (http://cdm.fly.faa.gov/ad/rbs.html)

That is, the FAA likes strategy-proofness: incentive to report feasiblearrival times truthfully.



The FAA and airlines agree on various objectives.

“If an airline reported a mechanical delay on a flight, thesystem would re-project its arrival time. If [rationing withGrover–Jack were done] at that time, that flight would likelyreceive an additional delay on top of its mechanical delay.These effects were known as the ‘Double Penalty’issue. . . The airlines would simply not send in information thatwould [harm themselves]. RBS removes this disincentive."

– FAA website (http://cdm.fly.faa.gov/ad/rbs.html)

That is, the FAA likes strategy-proofness: incentive to report feasiblearrival times truthfully.



“Once arrival slots have been allocated, the Compressionalgorithm performs schedule updates by an inter-airline slotexchange, which aims to provide airlines with an incentive toreport flight cancelations and delays.”

–Vossen and Ball (2006)

Note, two incentive conditions:Incentive to report delays (arrival times). (strategy-proofness)Incentive to report cancelations (infinite arrival time?).

No formalization in this trans/ops literature.



“If an airline sits on a slot that it is not planning to use, is thereany way for [the system] to detect this and to take this slotaway from the airline? Should this be done?"

–US DOT internal memo, 1996.

Concern about failure to truthfully report cancelations.


Industry Objectives – Property Rights

Under the FAA’s Facility Operation and Administration handbook,airlines can rearrange their own portion of the schedule.

“[An airline] can reassign its flight-slot assignment. . . It shouldbe emphasized that this notion of slot ownership is one of themain tenets of the CDM paradigm: there is a generalconsensus among airlines that this is indeed a fair method ofrationing arrival capacity."

–Vossen and Ball (2006)

The CDM paradigm results from collaboration between FAA andairlines.We view this as an endorsement of the normative idea that airlineshave a degree of property rights associated with some degree of“ownership" of the slots assigned to them by the RBS procedure.


Industry Objectives – Money?

No.


Overview of Talk

-2 Explanation of GDP-1 Discussion of objectives0 Overview1 Model2 Compression Algorithm

characteristics3 TradeCycle algorithm

Connection to previous literaturecharacteristics


Model

Set of slots S = {1,2, . . . , |S|}. (s, t ∈ S)Set of airlines A. (A,B ∈ A)Set of flights F . (f ,g ∈ FA ⊂ F )earliest arrival time for f : ef ≤ |S|.Landing Schedule Π: F → S (feasible).Slot ownership function Φ: S → A (consistent with Π).We call (Π,Φ) an assignment.

An Instance (or Economy) is I = (S,A, (FA)A,e,Π,Φ). (|F | < |S|)A matching function maps I → Π′.


Model

Set of slots S = {1,2, . . . , |S|}. (s, t ∈ S)Set of airlines A. (A,B ∈ A)Set of flights F . (f ,g ∈ FA ⊂ F )earliest arrival time for f : ef ≤ |S|.Landing Schedule Π: F → S (feasible).Slot ownership function Φ: S → A (consistent with Π).We call (Π,Φ) an assignment.

An Instance (or Economy) is I = (S,A, (FA)A,e,Π,Φ). (|F | < |S|)A matching function maps I → Π′.


Example of an Instance

Slot Flight Airline feasible arrival time ef

1 vacant A2 f2 B 13 f3 C 14 f4 A 2

Slot 1: vacant and owned by airline A.Slot 2: owned by B, occupied by f2 ∈ FB which could arrive in slot 1.


Example of an Instance


1 vacant A2 f2 B 13 f3 C 14 f4 A 2

5 vacant B6 f6 C 57 f7 A 58 f8 B 6

9 vacant C10 f10 A 911 f11 B 912 f12 C 10

Slot 1: vacant and owned by airline A.Slot 2: owned by B, occupied by f2 ∈ FB which could arrive in slot 1.


Formalizing Objectives: preferences

We are concerned with matching functions satisfying. . .Strategy-proofness;Non-manipulability by withholding slots;property rights (via IR, core, etc.).

When is an airline better off?

A “weak" model: An airline gains only if each flight gains.

Why use?Airlines already have some ability to make tradeoffs, by swappingflights within their own subschedule.Negative results are stronger.














Formalizing Objectives

We are concerned with matching functions that are. . .Strategy-proof: no airline can gain by misreporting the arrivaltimes of its flights.Non-manipulable by slot destruction: no airline can gain bywithholding (destroying) a vacant slot from the mechanism.(Weak) core selecting: no coalition of airlines could all (strictly)gain by simply trading amongst themselves.


Basic Literature

Our model is similar to housing problems (Shapley-Scarf). Differencesare:

Airlines can own multiple flights.Vacant slots (houses) belong to airlines.Preferences (of flights) are restricted:ef � ef + 1 � · · · � s � 1 ∼ · · · ∼ ef − 1

Nevertheless we will see a connection to previous housing-marketgeneralizations,e.g. Abdulkadiroglu and Sonmez (1999); Papai (2000).


Basic Literature

Our model is similar to housing problems (Shapley-Scarf). Differencesare:

Airlines can own multiple flights.Vacant slots (houses) belong to airlines.Preferences (of flights) are restricted:ef � ef + 1 � · · · � s � 1 ∼ · · · ∼ ef − 1

Nevertheless we will see a connection to previous housing-marketgeneralizations,e.g. Abdulkadiroglu and Sonmez (1999); Papai (2000).


Compression Algorithm

Compression: If an airline can’t use a slot, it trades with the nextscheduled flight that can use it.

Step 0 Begin with an initial assignment (Π,Φ).Set V = the set of vacant slots.

Step 1 If V = ∅, end the algorithm at (Π,Φ).Otherwise make ‘active’ the earliest vacant slot s ∈ V .

Step 2 Let A = Φ(s) denote the airline that owns s. If A has a laterflight f ∈ FA that could feasibly use slot s, move f to s, make f ’soriginal slot ‘active,’ and repeat Step 2. Otherwise go to Step 3.

Step 3 If any other airline B has a later flight that could feasibly useslot s, let f be the earliest such flight.Move f to s, assign f ’s original slot to A, make it active, andreturn to Step 2.Otherwise delete useless slot s from V and return to Step 1.


Compression Example


1 vacant A2 f2 B 13 f3 C 14 f4 A 2

5 vacant B6 f6 C 57 f7 A 58 f8 B 6

9 vacant C10 f10 A 911 f11 B 912 f12 C 10


Compression Example


1 f2 B 12 vacant A3 f3 C 14 f4 A 2

5 vacant B6 f6 C 57 f7 A 58 f8 B 6

9 vacant C10 f10 A 911 f11 B 912 f12 C 10


Compression Example


1 f2 B 12 f4 A 23 f3 C 14 vacant A

5 vacant B6 f6 C 57 f7 A 58 f8 B 6

9 vacant C10 f10 A 911 f11 B 912 f12 C 10


Compression Example


1 f2 B 12 f4 A 23 f3 C 14 vacant A

5 f6 C 56 vacant B7 f7 A 58 f8 B 6

9 vacant C10 f10 A 911 f11 B 912 f12 C 10


Compression Example


1 f2 B 12 f4 A 23 f3 C 14 vacant A

5 f6 C 56 f8 B 67 f7 A 58 vacant B

9 vacant C10 f10 A 911 f11 B 912 f12 C 10


Compression Example


1 f2 B 12 f4 A 23 f3 C 14 vacant A

5 f6 C 56 f8 B 67 f7 A 58 vacant B

9 f10 A 910 vacant C11 f11 B 912 f12 C 10


Compression Example


1 f2 B 12 f4 A 23 f3 C 14 vacant A

5 f6 C 56 f8 B 67 f7 A 58 vacant B

9 f10 A 910 f12 C 1011 f11 B 912 vacant C


Compression: incentives

TheoremThe compression algorithm is strategy-proof: it cannot be manipulatedby misreporting an arrival time.

Proof.Suppose f gets s by reporting ef honestly, but reports e′

f .• If ef < e′

f ≤ s, then the outcome of the algorithm cannot be affected.Whenever f is the flight chosen in Steps 2 or 3 when ef is reported, fwould still be chosen when e′

f is reported, because f never moved intoa slot earlier than s.• If s < e′

f , then f would have to end up in a slot strictly worse than s,since the Compression Algorithm never places a flight in a slot earlierthan its reported earliest arrival time.• If e′

f < ef , then the only way this misreport can change the outcomeis to assign f to a slot earlier than ef , which is infeasible for f .


Compression: slot destruction

PropositionThe Compression Algorithm is manipulable by slot destruction.

Proof.


1 vacant A2 vacant B3 vacant A4 f4 C 25 f5 B 46 f6 A 47 f7 B 1




Proof.


1 f7 B 12 vacant B3 vacant A4 f4 C 25 f5 B 46 f6 A 47 vacant A




Proof.


1 f7 B 12 f4 C 23 vacant A4 vacant B5 f5 B 46 f6 A 47 vacant A




Proof.


1 f7 B 12 f4 C 23 vacant A4 f5 B 45 vacant B6 f6 A 47 vacant A




Proof.


1 f7 B 12 f4 C 23 vacant A4 f5 B 45 f6 A 46 vacant B7 vacant A




Proof.






Proof.






Proof.


1 vacant A2 f7 B 13 vacant A4 f4 C 25 f5 B 46 f6 A 47 vacant B




Proof.


1 vacant A2 f7 B 13 f4 C 24 vacant A5 f5 B 46 f6 A 47 vacant B




Proof.


1 vacant A2 f7 B 13 f4 C 24 f6 A 45 f5 B 46 vacant A7 vacant B


The Core

DefinitionAn assignment is in the core if no coalition of airlines can strictly gainby simply trading amongst themselves from their original endowments.

Loosely speaking, if an outcome is not in the core, then some airlinesare not receiving the ‘best’ slots to which they are entitled, based ontheir slot endowments. In this sense, the core provides a form ofproperty rights.

The Strong Core can be empty. (E.g., airline with a vacant slot and noflights.)


The Core

DefinitionAn assignment is in the core if no coalition of airlines can strictly gainby simply trading amongst themselves from their original endowments.

Loosely speaking, if an outcome is not in the core, then some airlinesare not receiving the ‘best’ slots to which they are entitled, based ontheir slot endowments. In this sense, the core provides a form ofproperty rights.

The Strong Core can be empty. (E.g., airline with a vacant slot and noflights.)


Core: Example revisited


1 vacant A2 f2 B 13 f3 C 1

The strong core can be empty.Schummer & Vohra (Northwestern Univ.) Assignment of Arrival Slots March 2012 22 / 38



1 vacant A2 f2 B 13 f3 C 14 f4 A 2

Unique (strong) core matching (a là Shapley–Scarf).Schummer & Vohra (Northwestern Univ.) Assignment of Arrival Slots March 2012 22 / 38



1 vacant A2 f2 B 13 f3 C 14 f4 A 2

5 vacant B6 f6 C 57 f7 A 58 f8 B 6

9 vacant C10 f10 A 911 f11 B 912 f12 C 10

The core can be multi-valued.Schummer & Vohra (Northwestern Univ.) Assignment of Arrival Slots March 2012 22 / 38

Compression and Core

Apply Compression to the following example.


1 vacant A2 vacant B3 f3 C 14 f4 B 15 f5 A 2

f3 assigned to slot 1.Slot 3 is active; f5 fills it.Slot 2 is active; f4 fills it.However, this is not in the core.





1 f3 C 12 vacant B3 vacant A4 f4 B 15 f5 A 2






1 f3 C 12 vacant B3 f5 A 24 f4 B 15 vacant A






1 f3 C 12 f4 B 13 f5 A 24 vacant B5 vacant A



Top trading cycles

Housing marketsEach airline has exactly one flight/slot.No vacant slots.However, preferences are arbitrary.

Shapley & Scarf describe the top trading cycle algorithm to find theunique core outcome.

Each flight points to its favorite slot.Cycles exist, and are cleared.Repeat.


Top trading cycles

Housing marketsEach airline has exactly one flight/slot.No vacant slots.However, preferences are arbitrary.

Shapley & Scarf describe the top trading cycle algorithm to find theunique core outcome.

Each flight points to its favorite slot.Cycles exist, and are cleared.Repeat.


TradeCycle Algorithm

Step 0 Take as input an initial assignment, and declare all slots andflights “active."

Step 1 If the set of active flights is empty, the algorithm ends. Otherwise,construct a graph as follows.

Step 1a Introduce a node for each active slot and each active flight.Step 1b From each flight f , draw a directed edge to the earliest active slot

that f can occupy.Step 1c From each occupied slot, draw a directed edge to the flight that

occupies it.Step 1c From each vacant slot owned by any airline A, draw a directed edge

to (i) the earliest scheduled active flight in FA, if one exists; (ii) theearliest scheduled active flight in F , otherwise.

Step 2 Within any (directed) cycle in the graph: Permanently assign eachflight to the slot it points to in the cycle; declare the flight and itsassigned slot inactive. (Newly vacated slots within a cycle remainactive.) Return to Step 1.




















Example

Apply TC to this example.

Slot Flight Airline feasible arrival time ef1 vacant A2 vacant B3 f3 C 14 f4 B 15 f5 A 2


Example




Example




Example




Example




Example




Link to past work

PropositionTradeCycle is algorithmically equivalent to a fixed endowmenthierarchical exchange rule (Pápai 2000) where

1 flights are treated as individual agents, and2 each slot has an inheritance structure that prioritizes flights in the

following order:1 the flight that occupies it (if any),2 other flights of the same airline, by ETA (if any),3 flights of other airlines, by ETA.


Incentives

TheoremTradeCycle is strategy-proof: it cannot be manipulated by misreportingan arrival time.

Proof.From Pápai 2000, FEHE rules for assignment problems are groupstrategy-proof even when arbitrary preferences over slots arepermitted. Here, an airline is a coalition of flights, and (flight)preferences are restricted (ef � ef + 1, etc.). Since groupstrategy-proofness is preserved under domain reduction, the resultfollows.


Main Point

TheoremTradeCycle returns an assignment in the weak core.

Proof.Suppose B blocks the assignment via Π′

B.If B flights point to Π′

B, assignment stays same (gsp).Consider first round in which B loses a slot to Bc .A cycle is formed where a flight in Bc is pointed to by a vacant slotfrom B.Its owner must already be fully assigned, receiving its favorite slotfor each airline, not strictly better off at Π′

B, contradiction.


Tradecycle: slot destruction

PropositionTradeCycle is manipulable by slot destruction.

Proof.


1 vacant C2 f2 A 13 vacant A4 vacant D5 f5 B 36 f6 C 57 f7 A 58 f8 D 7




Proof.


1 f2 A 12 vacant A3 f5 B 34 vacant D5 f6 C 56 vacant C7 f7 A 58 f8 D 7




Proof.


1 f2 A 12 vacant A3 f5 B 34 vacant D5 f6 C 56 f7 A 57 f8 D 78 vacant C




Proof.






Proof.






Proof.


1 vacant C2 f2 A 13 vacant A4 f5 B 35 f7 A 56 f6 C 57 f8 D 78 vacant D




Proof.


1 vacant C2 f2 A 13 vacant A4 f5 B 35 f7 A 56 f6 C 57 f8 D 78 vacant D




Proof.


1 f2 A 12 vacant C3 vacant A4 f5 B 35 f7 A 56 f6 C 57 f8 D 78 vacant D


“Slot Destruction" in other matching models

Theorem (Atlamaz and Klaus, 2007)Using our terminology, suppose airlines have arbitrary preferencesover all subsets of slots. Then

[Efficient + Individually Rational]⇓

[Manipulable by Endowment Destruction]

No implication to our model, due to our preference restriction.


Wrap-up

Application of matching theory to another real world problem:reassignment of landing slots.FAA’s current algorithm may fall outside of weak core.Our algorithm has the same incentives properties, and selectsfrom the core.Our motivation for core rests more on property rights than onstability (e.g. unraveling in doctor/hospital markets).

Strategy-proofness: obtainable (weak definition of prefs.)Core: obtainable.Airline incentive to keep useless slot. (proof of concept; is itwidespread?)


Wrap-up

Application of matching theory to another real world problem:reassignment of landing slots.FAA’s current algorithm may fall outside of weak core.Our algorithm has the same incentives properties, and selectsfrom the core.Our motivation for core rests more on property rights than onstability (e.g. unraveling in doctor/hospital markets).

Strategy-proofness: obtainable (weak definition of prefs.)Core: obtainable.Airline incentive to keep useless slot. (proof of concept; is itwidespread?)


Assignment of Arrival Slots · 1:06 AA111 1:08 AF023 1:10 BA229 1:12 UA123..... 1:58 EI022 U.S. FAA now uses the Compression Algorithm (described later). Schummer & Vohra (Northwestern

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