Assignment of Arrival Slots

Transcription

1 Assignment of Arrival Slots James Schummer Rakesh V. Vohra March 22, 2009 Abstract When inclement weather reduces airport landing capacity, the FAA first creates a new schedule of feasible landing slots, and allocates them to airlines as a function of the slots they had in the original schedule. Second, after airlines report which of their (newly allocated) slots they can feasibly use and which they cannot, the FAA runs an algorithm to efficiently re-allocate the slots each airline cannot use. In this paper, we focus on the second step of this procedure, which is a matching problem of allocating flights to landing slots when there is excess capacity. The main results of this paper concern incentives and core properties of the currently used Compression Algorithm (CA) and an algorithm we propose as an alternative. The results in this paper show that both algorithms possess good incentives properties, but unlike CA, our algorithm also chooses a core assignment. This latter property, commonly studied in the matching literature, reflects the attitude in the airline industry that airlines should be entitled to the property rights associated with any landing slots allocated to them at any time. 1 Introduction When an airport in the U.S. is subject to inclement weather that reduces visibility and/or the number of runways, the Federal Aviation Administration (FAA) institutes a ground delay program (GDP). Flights destined for the affected airport are issued Controlled Departure Times (CDT) at their departure point. Flights issued CDTs are not permitted to depart until their Kellogg School of Management, MEDS Department, Northwestern University. schummer@kellogg.northwestern.edu. Kellogg School of Management, MEDS Department, Northwestern University. r-vohra@kellogg.northwestern.edu. 1

2 Controlled Departure Time. CDTs are determined to ensure that the demand for arrival slots at the affected airport is equal to it s current available capacity. 1 A GDP reduces the number of landing slots at the affected airport and these scarce slots are rationed amongst those airlines with flights scheduled to arrive at the affected airport. For example, if under a GDP, the airport can accept 30 flights per hour, then 30 two-minute slots are created. Flights are then allocated to slots on a first-scheduled, first-served basis. This is called ration-by-schedule (RBS); see () for details. As revised arrival slots are allocated, changes in the status of flights, known only to the airlines, mean that a slot assigned to a scheduled flight will become vacant. This happens because a flight is unable to meet its arrival slot because of an earlier delay or cancelation. To fill the resulting gaps in the schedule, the FAA executes a compression algorithm that moves flights earlier in time (and never later) to fill vacant arrival slots. RBS and the compression algorithm currently in use to fill gaps were developed to avoid problems experienced under an earlier algorithm called Grover-Jack (). Grover-Jack rationed and filled vacant slots in a way that gave airlines an incentive to conceal which of the slots assigned to them had fallen vacant. This resulted in scarce arrival slots lying empty. See Vossen and Ball (2006b) for details. Arrival slots are indivisible goods indexed by time. The RBS determines an assignment of flights to slots. Slots assigned to the flights of a given airline, say, United, become the endowment of the airline. A slot that falls vacant can be used by the airline that owns it to move one of its later flights earlier. It can also be traded for mutual benefit with a slot owned by another airline. Phrased this way, the problem of reallocating slots after an initial endowment is determined is similar to the Shapley Scarf (1974) house trading problem. The environment considered here is more general for two reasons. First, unlike the original house trading model, agents have an initial endowment of homes. Second, agents are interested in consuming more than one home. Therefore this model is not a special case of, for instance, the recent extensive literatures on kidney exchange or on school assignments (see...). The main results of this paper are the following. 1. Under an appropriate model of preferences over arrival slots for the Airlines, we show that the compression algorithm is strategyproof. 1 Prior to 1981, the FAA allowed aircraft to take off whenever ready. If there was congestion at the destination, aircraft were placed in holding patterns until they were able to land or directed to an alternative airport if their fuel was spent. 2

3 Thus, our paper is the first to make precise the notion that the compression algorithm eliminates the incentives to conceal information about vacant slots. 2. Under the same model of preferences, we show that the compression algorithm need not return an outcome in the core. Experience, both experimental and empirical, from centralized exchanges that match workers with jobs, suggest that producing core outcomes are crucial for the stability of such exchanges. Otherwise, participants have the incentive to circumvent the exchange and strike side deals. For details see Roth and Peranson (1999) as well Kagel and Roth (2000). 3. Under the same model of preferences we describe a mechanism (called TRADECYCLE) that is both strategy-proof and core selecting. Despite the fact that our model has differences with the house allocation problem, our mechanism has a connection with classes of mechanisms described by Abdulkadiroğlu and Sönmez (1999) and Pápai (2000). We use this connection to prove our incentives results in a straightforward way. The next section introduces the notation and definitions used in this paper. The following sections discuss the compression algorithm and it s properties as well the mechanism TRADECYCLE. We conclude with some open problems and unresolved issues. 2 Notation and Definitions There is a set of arrival Slots S = {1, 2, 3,..., s}. We interpret the labels of these slots as (ordinal) representations of time: for s, t S, s < t means that slot s is earlier than slot t. There is a set of airlines A and, for each airline A A, a set of flights F A. We interpret F A as the set of flights that airline A has not canceled at the time of a ground delay. Let F = A F A be the set of all flights. We consider situations in which F < s. The earliest feasible arrival time of flight f F is denoted e f {1,..., s}. Hence, flight f can be feasibly assigned to slot j only if e f j. A Landing Schedule is formalized as a feasible assignment of flights to slots. Specifically, it is a function Π: F S that satisfies the following two properties for any f, f F : (injective) f f implies Π(f) Π(f ), and (feasible) Π(f) e f. That is, distinct flights get distinct slots, and flights are assigned to slots no earlier than their feasible arrival time. Landing schedules 3

4 do not specify the ownership of vacant slots. Therefore we introduce the concept of a slot ownership function, which is a function Φ: S A that satisfies consistency with some Π: for all f F, f F A implies Φ(Π(f)) = A. An assignment is a pair of functions (Π, Φ) that jointly satisfy the above conditions. 2.1 Preferences Airlines wish to get their flights on the ground as early as possible, subject to feasibility. For a single flight, this means it is better to be assigned slot e f, second-best to be assigned slot e f +1, etc. Being assigned any slot s < e f is unacceptable since the flight cannot arrive in time to use it. To express airlines preferences over landing schedules involves the schedule of multiple flights. As a first approach in this paper, we suppose that an airline is made better off only if all of its flights weakly move up in the schedule. To formalize this, consider two Landing Schedules Π and Π. We say that airline A strictly prefers Π to Π if Π(f) Π (f) for every flight f F A, and Π(f) < Π (f) for at least one f F A. That is, airlines prefer assignments that never move a flight down, i.e., later and move their flights up, i.e, earlier. On the other hand, if assignment Π adjusts Π by simultaneously delaying some of A s flights and moving some earlier, then Π is preferenceincomparable with Π. 2 This specification of preferences rules out the possibility that an airline might wish to switch flights. That is, move a flight with a lot of passengers earlier by delaying an earlier flight with fewer passengers. This is reasonable (in some circumstances) because of the follow on effect such a switch may have on other parts of the schedule. For example, the earlier flight may be needed at the destination to ferry a large number of passengers elsewhere. 2.2 Core schedules In our model, airlines arrive with initial property rights in the form of slot ownership. We consider how to adjust an initial assignment after airlines report arrival times (e f s) and create vacancies in some slots. Therefore, in order to examine incentives, and discuss fairness of various reallocation 2 Analysis that considers these tradeoffs is worthy of future work. Our approach represents an approximation of a reality in which an airline does not want the reputation of intentionally delaying some of its flights. 4

5 procedures, we must start with a designated initial assignment, which we denote (Π I, Φ I ). We focus on one of the primary concepts that has appeared in various literatures (including matching): the core. With respect to an initial assignment (Π I, Φ I ), a landing schedule Π is a core-schedule if no subgroup of airlines could reallocate their initial slots (from Φ I ) to each other in order to make themselves better off than in Π. Formally, Π is in the core if there exists no other landing schedule Π and set of airlines B A such that (i) for all f A B F A, Φ I (Π (f)) B, and (ii) each airline A B strictly prefers Π to Π. 3 We give an example to show that the core can be multi-valued. Suppose S = {1, 2, 3,..., 12}, and that the initial assignment is as shown below. Each row lists a slot, the flight that is initially assigned it, the airline owning that flight, and the flight s earliest arrival time. Example 1 The initial assignment (Π I, Φ I ) is described by the following table. Slot Flight Airline feasible arrival time e f 1 vacant A 2 f 2 B 1 3 f 3 C 1 4 f 4 A 2 5 vacant B 6 f 6 C 5 7 f 7 A 5 8 f 8 B 6 9 vacant C 10 f 10 A 9 11 f 11 B 9 12 f 12 C 10 For instance, slot 1 is vacant and is owned by airline A (i.e. Φ I (1) = A). Slot 2 is owned by airline B and is occupied by B s flight f 2 F B (Π I (f 2 ) = 2). Flight f 2 could feasibly arrive in slot 1 (e f2 = 1). The following landing schedule Π is a core schedule with respect to the 3 This is what is known as the weak core, since every deviating airline is required to be strictly better off in Π. Here, as is well known in models with indifference, a stronger core notion may fail existence. 5

6 initial assignment in Example 1. Π(f 2 ) = 1 Π(f 6 ) = 5 Π(f 10 ) = 9 Π(f 3 ) = 3 Π(f 7 ) = 7 Π(f 11 ) = 11 Π(f 4 ) = 2 Π(f 8 ) = 6 Π(f 12 ) = 10 To see that Π in the core, consider the coalition of airlines B = {A, B}. They do not initially own slot 9 (i.e. Φ I (9) = C). Hence any schedule they attempt to construct on their own must make flight f 10 arrive later than it does under Π, hence airline A cannot be made better off. A symmetric argument applies to any other pair of airlines, and the remaining arguments are trivial. The following landing schedule Π is also in the core. Π (f 2 ) = 1 Π (f 6 ) = 5 Π (f 10 ) = 9 Π (f 3 ) = 2 Π (f 7 ) = 6 Π (f 11 ) = 10 Π (f 4 ) = 3 Π (f 8 ) = 7 Π (f 12 ) = 11 We leave it to the reader to verify this, using similar arguments. 3 The Compression Algorithm The Compression Algorithm currently used in practice is a vast improvement over previous methods for allocating slots. We refer the reader to Vossen and Ball (2006a) for details, but the primary advance this algorithm has over previous methods is that it rewards airlines for giving up slots they cannot use. When an airline gives up a slot it considers useless, it trades the slot for a later slot owned by the airline that eventually takes possession of the earlier slot. The Compression Algorithm is formally defined in Figure 3. In the rest of the section we discuss its properties. The compression algorithm relies on two items of information: the reported earliest arrival times (e f ) of each flight and the report of which slots are vacant. An airline could conceivably misreport either of these. Given the confines of the model we can rule out the possibility of an airline claiming that a flight f can arrive earlier than e f. If such a misreport resulted in a flight actually being assigned to a slot earlier than e f, the misreport could be discovered and the offending airline could be penalized. 4 4 In reality, an airline could claim, for example, mechanical problems subsequent to the 6

7 Step 0 Initialize the current assignment as Φ = Φ I, Π = Π I. Let V = S \ Π I (F ) denote the set of vacant slots. Step 1 If V =, end the algorithm at the assignment (Π, Φ). Otherwise pick the earliest vacant slot s V and declare it active. Step 2 Let A = Φ(s) denote the airline that owns s. Check whether airline A has a flight f F A that both (i) occupies a later slot Π(f) > s and (ii) could feasibly use slot s (e f s). If so, let f be the earliest such flight, denote its slot t = Π(f), and go to Step 4. Otherwise go to Step 3. Step 3 Check whether any airline has a flight f that both (i) occupies a later slot Π(f) > s and (ii) could feasibly use slot s (e f s). If so, let f be the earliest such flight, denote its slot t = Π(f), and go to Step 4. Otherwise remove slot s from V and return to Step 1. Step 4 Move flight f from slot t to slot s: set Π(f) = s and set Φ(s) equal to the airline of flight f. Set Φ(t) = A. Remove s from V and add t to V. Return to Step 2 using t as the new active slot s. Figure 1: The Compression Algorithm 7

8 It is more plausible that an airline could declare a later feasible arrival time for one or more of its flights. Fortunately, the compression algorithm provides no incentive for an airline to do this. Theorem 1 The compression algorithm cannot be manipulated by reporting a later feasible arrival times for a single flight f. Proof: Suppose that by reporting e f honestly, flight f is assigned to slot s. Clearly, e f s. A misreport e f could be of three types. (I) If e f < e f s, then the outcome of the algorithm cannot be affected. Whenever f is the flight chosen in Steps 2 or 3 when e f is reported, f would still be chosen when e f is reported, because f never moved into a slot earlier than s. (II) 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 earlier than its reported earliest arrival time. (III) If e f < e f, then the only way this misreport can change the outcome of the algorithm is to assign f to a slot earlier than e f. In (I) the algorithm is unaffected. In (II) and (III), flight f receives a later slot than when reporting e f. Hence its airline cannot gain under our model of preferences. The second form of manipulation we consider is the destruction of vacant slots. Suppose an airline has a flight in some slot s that is to be canceled, and such that e f = s. By delaying the cancelation announcement sufficiently long, the airline in effect destroys the existence of this slot. If the algorithm perceives f to be a non-canceled flight, f stays in slot s, while its presence does not affect the slots of any other flights. Suppose that e f < s. Then s is available to be exchanged during the execution of the compression algorithm. The only benefit this can confer is that f s airline may gain possession of an earlier slot during an execution of Step 3. Such a slot is valuable only if this airline has another flight that can be feasibly assigned to s. However, this slot would be allocated to the airline anyway during the execution of Step 3. Therefore, consider destruction of a vacant slot s to be the insertion of a dummy flight f into slot s such that e f = s. assignment to create a reason why the flight could not make it to the slot. Repeated occurrences of this, however, would presumably raise suspicions and be detected, so as a practical matter we ignore such misreports. 8

9 Theorem 2 The compression algorithm cannot be manipulated by destruction of a vacant slot. Proof: When a slot becomes active, the compression algorithm gives priority to the airline that owns the slot to fill it. When that airline cannot fill it, priority goes to the earliest flight in the current schedule of any other airline that can fill it. Hence, the benefit from destroying a vacant slot, must come from preventing a another airline s flight moving ahead of one s own flight. Denote by Π k the assignment of flights to slot at the end iteration k of the compression algorithm. Π 0 is the initial assignment. We shall say a reversal takes place at the end of iteration k if there are f, g F not in the same airline such that Π k 1 [f] < Π k 1 [g] and Π k [f] > Π k [g]. A reversal takes place because some slot j < Π k 1 [f] becomes active, but f cannot fill it and g can. A reversal at iteration k involves airline A if f F A. How can a reversal at iteration k involving airline A harm it? Suppose at some subsequent iteration, r > k, say, a slot i becomes vacant such that: 1. The airline that owns slot i cannot fill it. 2. i < Π r 1 [g] < Π r 1 [f]. 3. e f, e g i. 4. No other flight earlier than f can fill slot i. By Step 3 of compression, flight g would be selected to fill slot i instead of flight f. However, if the reversal has not taken place, flight f would have received priority. We now argue that preventing this reversal cannot benefit airline A. Since the reversal occurred at the end of iteration k, it means that at the start of iteration k a slot j < Π k 1 [f] < Π k 1 [g] was declared active. At the end of iteration k, flight g was assigned to slot j, i.e., Π k [g] = j. Furthermore, since this reversal could have been prevented by airline A it means that slot i belongs to airline A. If slot i belonged to airline A and e f j, then by Step 2, slot j would have been assigned to f. Hence e f > j. Since j = Π k [g] > i it follows that e f > i as well. Hence, flight f cannot be assigned to slot i. 3.1 Core While the Compression Algorithm satisfies some incentives properties described above, we now give an example showing that it can return a landing schedule that is not in the core. 9

10 Consider the following initial assignment of slots to flights. Slot Flight Airline feasible arrival time e f 1 vacant A 2 vacant B 3 f 3 C 1 4 f 4 B 1 5 f 5 A 2 In Step 1 of the Compression Algorithm, slot 1 is declared active. Slot 1 belongs to A, but the only flight in F A (f 5 ) cannot be assigned to it. Therefore, in Step 3 of the algorithm, flight f 3 is assigned slot 1. This updates the original assignment to the following one. Slot Flight Airline feasible arrival time e f 1 f 3 C 1 2 vacant B 3 vacant A 4 f 4 B 1 5 f 5 A 2 By Step 4 of the compression algorithm slot 3 is declared active. 5 Since flight f 5 can feasibly fill it, subsequent steps of the algorithm result in the following assignment. Slot Flight Airline feasible arrival time e f 1 f 3 C 1 2 vacant B 3 f 5 A 2 4 f 4 B 1 5 vacant A When slot 5 becomes active, no flight can be assigned to it, so it is discarded. Returning to Step 1, slot 2 is declared active, and since f 4 can fill it, we have the following assignment. Slot Flight Airline feasible arrival time e f 1 f 3 C 1 2 f 4 B 1 3 f 5 A 2 4 vacant B 5 vacant A 5 If slot 2 is declared active next, instead, the same result would obtain. 10

11 Slot 4 has no further use, so the algorithm completes with the above schedule. However, airlines A and B can achieve the following schedule using only their own resources amongst themselves. Slot Flight Airline feasible arrival time e f 1 f 4 B 1 2 f 5 A 2 3 f 3 C 1 4 vacant B 5 vacant A Notice that airlines A and B are each strictly better off in this assignment than in the one given by the Compression Algorithm. One can check that this schedule is the unique one in the core of this example. The intuition behind this example is as follows. The Compression algorithm benefits airlines when, for example, if airline A cannot use a slot A owns, then the algorithm trades it to another airline in exchange for some other slot. However, A cannot decide exactly which other airline it trades with. In the example above, A would like to trade with B. However, the Compression algorithm does not allow this. 6 In the next section we introduce an algorithm that does take into account A s implicit preference with whom to trade. Our algorithm has the same incentive properties as the Compression algorithm but has the additional property that it returns an outcome in the core. 4 TRADECYCLE We describe in Figure 4 an algorithm called TRADECYCLE that returns an allocation in the core. It iteratively adjusts the original assignment of flights to slots until no flight can use any vacant slot. In each iteration, a subset of flights and slots are permanently reassigned, i.e. they are inactive for all subsequent rounds of the algorithm. Assume that the initial schedule is feasible, i.e. that all flights can feasibly arrive at their initially assigned slot. Then in step 2, there are two kinds of cycles that can be formed: trivial loops, consisting of a flight and its 6 This is tangentially related to the issue of the order of slot compression under the Compression algorithm. See section of Vossen and Ball (2006a), who show that order matters. 11

12 Step 0 Take as input an initial assignment, and declare all slots and flights 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-eta active flight in F A, if one exists; (ii) the earliest-eta flight in F, otherwise. Step 2 Within any (directed) cycle in the graph: Assign each flight to the slot it points to in the cycle; declare the flight and its assigned slot inactive. (Newly vacated slots within a cycle remain active.) Return to Step 1. Figure 2: The tradecycle algorithm. 12

13 currently assigned slot (i.e. when a flight is already assigned to its most preferred slot), and loops containing at least one vacant slot. Readers familiar with the matching literature 7 may notice that our algorithm has some of the flavor of the Top Trading Cycle algorithm (Shapley and Scarf (1974)) which finds core assignments in one-to-one housing markets. In fact, TRADECYCLE corresponds, at least algorithmically, to an element of the class of fixed endowment hierarchical exchange rules described by Pápai (2000). Pápai s model differs from ours in two ways. First, using our terminology, each flight behaves as an individual strategic agent in Pápai s model. However, since she obtains results on group strategyproofness, strategyproofness can be obtained in our model. Second, Pápai s model allows general preferences over slots (arbitrary orderings). The structure of our airport landing slots environment suggests that not all orderings of slots should be considered; by announcing an ETA e f, it is implicitly understood that a flight improves by moving closer to, but not beyond, that slot. A standard result in implementation theory, however, tells us that any strategyproof (group or not) rule on a larger domain of preferences yields a strategyproof rule when projected to a smaller domain of preferences. We prove the next result in an Appendix, since the models technically differ as we have just described. Proposition 1 TRADECYCLE is algorithmically equivalent to a fixed endowment hierarchical exchange rule (Pápai (2000)) in which (i) flights are treated as individual agents, and (ii) each slot has an inheritance structure whereby it prioritizes flights as follows. First is the flight that occupies it in the initial Landing Schedule. Following that are the flights of the same airline, ordered by ETA. Finally are the flights of the remaining airlines, ordered by ETA. Proof: see Appendix (TO BE ADDED). Our algorithm shares the same fundamental incentives property that the Compression Algorithm was shown to satisfy in Theorem 1. The proof is straightforward in light of Proposition 1. Theorem 3 TRADECYCLE cannot be manipulated by reporting later earliest arrival times. Proof: This follows from the Theorem in Pápai (2000), stating that FEHE rules for assignment problems are group strategy-proof even when arbitrary 7 See Roth and Sotomayor (1990) for an introductory survey. 13

14 preferences over slots are permitted. Here, each flight announces an ETA e f, which implicitly represents preferences which rank the slots in the order e f, e f + 1,..., s, 1, 2,..., e f 1. The next result shows that our algorithm also provides the same incentives to reveal vacant slots as the Compression Algorithm was shown to in Theorem 2. Theorem 4 TRADECYCLE cannot be manipulated by destruction of a vacant slot. Proof: (incomplete sketch) Consider airline A owning vacant slot s. Let j be the first iteration in which TRADECYCLE identifies a cycle C containing slot s. For A to gain from destroying s, A must have at least one flight f which, at iteration j, is not assigned to slot e f. Therefore, there is such a flight f which is part of C; it points to slot t, and is pointed to by s. Any other active slot that f can feasibly fill will be later than t. Hence, airline A cannot gain by destroying slot s. Finally, we show that our algorithm has the additional property of choosing core assignments. Theorem 5 TRADECYCLE returns an assignment in the core. Proof: (Sketch) Suppose, in contradiction to the theorem, that a coalition of airlines B could gain by reallocating their initially scheduled slots amongst themselves. Consider a manipulation of the mechanism in which each airline in the coalition reports, for each of their flights, an ETA which precisely matches the slot they receive in the coalitional reallocation. It is straightforward to show that the algorithm would implement the reallocation that makes the coalition better off. However, this violates Theorem 3. 5 Conclusion To be written. References Abdulkadiroglu, A., and T. Sonmez (1999): House Allocation With Existing Tenants, Journal of Economic Theory 88,

15 Ball, M., R. Hoffman and T. Vossen (2002): An analysis of resource rationing methods for collaborative decision making, ATM-2002, Capri, Italy. Ball, M.O., Vossen, T. and Hoffman, R., (2001): Analysis of demand uncertainty in ground delay programs, in Proceedings of 4th USA/Europe Air Traffic Management R&D Seminar. Kagel, J and A. E. Roth (2000): The Dynamics of Reorganization in Matching Markets: a labaratory experiment motivated by a natural experiment, Quarterly J. of Economics 115:1, Pápai, S. (2000): Strategyproof Assignment by Hierarchical Exchange, Econometrica 68:6, Roth, A. E. and E. Peranson (1999): The Redesign of the Matching Market for American Physicians: some engineering aspects of economic design, Amer Economic Review 89, Roth, A. E. and M. Sotomayor (1990): Two-sided matching: A study in game-theoretic modeling and analysis, Econometric Society Monographs 18, Cambridge University Press. Shapley, L. and H. Scarf (1974): On Cores and Indivisibility, Journal of Mathematical Economics, 1:1, Vossen, T. and M.O. Ball (2006): Optimization and mediated bartering models for ground delay programs, Naval Research Logistics 53:1, Vossen, T. and M.O. Ball (2006):, Slot Trading Opportunities in Collaborative Ground Delay Programs, Transportation Science, 40, Wambsganss, M. (1996): Collaborative decision making through dynamic information transfer, Air Traffic Control Quarterly 4,

16 6 Appendix To be added. 16