An incentive compatible, efficient market for air traffic flow management
Introduction
Air Traffic Flow Management (ATFM) is a challenging operations research problem whose importance keeps escalating with the growth of the airline industry. In the presence of inclement weather, the problem becomes particularly serious and leads to substantial monetary losses and delays,3 Yet, despite massive efforts on the part of the U.S. Federal Aviation Administration (FAA), airline companies, and even the academia, the problem remains largely unsolved.
In a nutshell, the reason for this is that any viable solution needs to satisfy several conflicting requirements, e.g., in addition to ensuring efficiency the solution also needs to be viewed as “fair” by all parties involved. Indeed, [8] state that “ ... While this work points at the possibility of dramatically reducing delay costs to the airline industry vis-a-vis current practice, the vast majority of these proposals remain unimplemented. The ostensible reason for this is fairness ... .” It also needs to be computationally efficient – even moderate sized airports today handle hundreds of flights per day, with the 30 busiest ones handling anywhere from 1000 to 3000 flights per day. The full problem involves scheduling flight-landings simultaneously for multiple airports over a large period of time, taking into consideration inter-airport constraints. Yet, according to [7], current research has mostly remained at the level of a single airport because of computational tractability reasons.
Building on a sequence of recent ideas that were steeped in sound economic theory, and drawing on ideas from game theory and the theory of algorithms, we present a solution that has a number of desirable properties. Our solution for allocating flights to landing slots at a single airport is based on the principle of a free market, which is known to be fair and a remarkably efficient method for allocating scarce resources among alternative uses (sometimes stated in the colorful language of the “invisible hand of the market” [26]). We define a market in which goods are delays and buyers are airline companies; the latter pay money to the FAA to buy away the desired amount of delay on a per flight basis and we give a notion of equilibrium for this market. W.r.t. equilibrium prices, the total cost (price paid and cost of delay) of each agent, i.e., flight, is minimized.
This involves a multi-objective optimization, one for each agent, just like all market equilibrium problems. Yet, for some markets an equilibrium can be found by optimizing only one function. As an example, consider the linear case of Fisher's market [10] for which an optimal solution to the remarkable Eisenberg–Gale [16] convex formulation gives equilibrium prices and allocations. For our market, we give a special LP whose optimal solution gives an equilibrium.
Using results from matching theory, we show how to find equilibrium allocations and prices in strongly polynomial time. Moreover, using [21] it turns out that our solution is incentive compatible in dominant strategy, i.e., the players (airline companies) will not be able to game the final allocation to their advantage by misreporting their private information.
We note that the ATFM problem involves several issues that are not of a game-theoretic or algorithmic nature, e.g., the relationship between long term access rights (slot ownership or leasing) and short term access rights on a given day of operations, e.g., see [5]. Our intention in this paper is not to address the myriad of such issues. Instead, we have attempted to identify a mathematically clean, core problem that is amenable to the powerful tools developed in the theories stated above, and whose solution could form the core around which a practical scheme can be built.
Within academia, research on this problem started with the pioneering work of Odoni [23] and it flourished with the extensive work of Bertsimas et al.; we refer the reader to [7][9] for thorough literature overviews and references to important papers. These were centralized solutions in which the FAA decides a schedule that is efficient, e.g., it decides which flights most critically need to be served first in order to minimize cascading delays in the entire system.
A conceptual breakthrough came with the realization that the airlines themselves are the best judge of how to achieve efficiency,4 thus moving away from centralized solutions. This observation led to a solutions based on collaborative decision making (CDM) which is used in practice [28], [4], [29].
More recently, a market based approach was proposed by Castelli, Pesenti and Ranieri [11]. Although their formulation is somewhat complicated, the strength of their approach lies in that it not only leads to efficiency but at the same time, it finesses away the sticky issue of fairness – whoever pays gets smaller delays, much the same way as whoever pays gets to fly comfortably in Business Class! Paper [11] also gave a tatonnement-based implementation of their market. Each iteration starts with FAA announcing prices for landing slots. Then, airlines pick their most preferred slots followed by FAA adjusting prices, to bring parity between supply and demand, for the next iteration. However, they are unable to show convergence of this process and instead propose running it a pre-specified number of times, and in case of failure, resorting to FAA's usual solution. They also give an example for which incentive compatibility does not hold.
Our market formulation is quite different and achieves both efficient running time and incentive compatibility. We believe that the simplicity of our solution for this important problem, and the fact that it draws on fundamental ideas from combinatorial optimization and game theory, should be viewed as a strength rather than a weakness.
In Section 2 we give details of our basic market model for allocating a set of flights to landing slots for one airport. This set of flights is picked in such a way that their actual arrival times lie in a window of a couple of hours; the reason for the latter will be clarified in Section 4. The goods in our market are delays and buyers are airline companies; the latter pay money to the FAA to buy away the desired amount of delay on a per flight basis. Typically flights have a myriad of interdependencies with other flights – because of the use of the same aircraft for subsequent flights, passengers connecting with other flights, crew connecting with other flights, etc. The airline companies, and not FAA, are keenly aware of these and are therefore in a better position to decide which flights to avoid delay for. The information provided by airline companies for each flight is the dollar value of delay as perceived by them.
For finding equilibrium allocations and prices in our market, we give a special LP in which parameters can be set according to the prevailing conditions at the airport and the delay costs declared by airline companies. We arrived at this LP as follows. Consider a traffic network in which users selfishly choose paths from their source to destination. One way of avoiding congestion is to impose tolls on roads. [13] showed the existence of such tolls for minimizing the total delay for the very special case of one source and one destination, using Kakutani's fixed point theorem. Clearly, their result was highly non-constructive. In a followup work, [18] gave a remarkable LP whose optimal solution yields such tolls for the problem of arbitrary sources and destinations and moreover, this results in a polynomial time algorithm. Their LP, which was meant for a multi-commodity flow setting, was the starting point of our work. One essential difference between the two settings is that whereas they sought a Nash equilibrium, we seek a market equilibrium; in particular, the latter requires the condition of market clearing.
We observe that the underlying matrix of our LP is totally unimodular and hence it admits an integral optimal solution. Such a solution yields an equilibrium schedule for the set of flights under consideration and the dual of this LP yields equilibrium price for each landing slot. Equilibrium entails that each flight is scheduled in such a way that the sum of the delay price and landing price is minimum possible. We further show that an equilibrium can be found via an algorithm for the minimum weight perfect b-matching problem and hence can be computed combinatorially in strongly polynomial time. In hindsight, our LP resembles the b-matching LP, but there are some differences.
Since the b-matching problem reduces to the maximum matching problem, our market is essentially a matching market. Leonard [21] showed that the set of equilibrium prices of a matching market with minimum sum corresponds precisely to VCG payments [22], thereby showing that the market is incentive compatible in dominant strategy. Note that, unlike general assignment game, in our setting only the flights (operators) are strategic while the landing slots are just goods, and therefore incentive compatibility is with respect to the flights, i.e., flights will report their delay costs truthfully. Since equilibrium prices form a lattice [27], [15], [25], the one minimizing sum has to be simultaneously minimum for all goods. For our market, we give a simple, linear-time procedure that converts arbitrary equilibrium prices to ones that are simultaneously minimum for all slots. Incentive compatibility with these prices follows. An issue worth mentioning is that the total revenue, or the total cost, of VCG-based incentive compatible mechanisms has been studied extensively, mostly with negative results [2], [20], [17], [14], [19]. In contrast, since the prices in our natural market model happened to be VCG prices, we have no overhead for making our mechanism incentive compatible.
The next question is how to address the scheduling of landing slots over longer periods at multiple airports, taking into consideration inter-airport constraints. Airlines can and do anticipate future congestion and delay issues and take these into consideration to make complex decisions. However, sometimes unexpected events happening even at a few places are likely to have profound cascading effects at geographically distant airports, making it necessary to make changes dynamically. For such situations, in Section 4, we propose a dynamic solution by decomposing this entire problem into many small problems, each of which will be solved by the method proposed above. The key to this decomposition is the robustness of our solution for a single set of flights at one airport: we have not imposed any constraints on delay costs, not even monotonicity. Therefore, airline companies can utilize this flexibility to encode a wide variety of inter-airport constraints.
We note that this approach opens up the possibility of making diverse types of travelers happy through the following mechanism: the additional revenues generated by FAA via our market gives it the ability to subsidize landing fees for low budget airlines. As a result, both types of travelers can achieve an end that is most desirable to them, business travelers and casual/vacation travelers. The former, in inclement weather, will not be made to suffer delays that ruin their important meetings and latter will get to fly for a lower price (and perhaps sip coffee for an additional hour on the tarmac, in inclement weather, while thinking about their upcoming vacation).
To the best of our knowledge, ours is the first work to give a simple LP-based efficient solution for the ATFM problem. We note that an LP similar to ours is also given in [1]. This paper considers two-sided matching markets with payments and non-quasilinear utilities. They show that the lowest priced competitive equilibria are group strategy proof, which induces VCG payments for the case of quasilinear utilities. Another related paper is [12], which considers a Shapley–Shubik assignment model for unit-demand buyers and sellers with one indivisible item each. Buyers have budget constraint for every item. This sometimes prevents a competitive equilibrium from existing. They give a strongly polynomial-time algorithm to check if an equilibrium exists or not, and if it does exist, then it computes the one with lowest prices. However, they do not ensure incentive compatibility.
Section snippets
The market model
In this section we will consider the problem of scheduling landings at one airport only. Let A be the set of all flights, operated by various airlines, that land in this airport in a given period of time. We assume that the given period of time is partitioned into a set of landing time slots, in a manner that is most convenient for this airport; let S denote this set. Each slot s has a capacity specifying the number of flights that can land in this time slot. As mentioned in [5] the
Strongly polynomial implementation
As discussed in the previous section, LP (1) has an integral optimal solution as its underlying matrix is totally unimodular. In this section, we show that the problem of obtaining such a solution can be reduced to a minimum weight perfect b-matching problem,7 and hence can be found in strongly polynomial time; see [24] Volume A. The equilibrium prices, i.e.,
Dealing with multiple airports
In this section, we suggest how to use the above-stated solution to deal with unexpected events that result in global, cascading delays. Our proposal is to decompose the problem of scheduling landing slots over a period of a day at multiple airports into many small problems, each dealing with a set of flights whose arrival times lie in a window of a couple of hours – the window being chosen in such a way that all flights would already be in the air and their actual arrival times, assuming no
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2022, Journal of High Speed Networks
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Supported by NSF Grant CCF-1833617.
- 2
Supported by NSF Grant CCF-1216019.