a game-theoretic modeling approach to air traffic...

A Game-Theoretic Modeling Approach to Air Traffic Forecasting

Vikrant Vaze, Reed Harder

Thayer School of Engineering Dartmouth College

ATM2017

Motivation: Modeling Airline Capacity Decisions

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Airlines allocate capacity (frequency and seats-per-flight) across a network of airports, in the face of competition.

Capacity decisions affect operating costs as well as fare revenues, but are also interdependent with decisions of other carriers, and thus have significant implications for overall system performance.

WN Network UA Network


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Misallocation of capacity has been implicated in… Costs worth billions of dollars annually (Ball et al., 2010), Wastage of system resources (Morisset and Odoni, 2011), Passenger inconvenience (Barnhart, Fearing and Vaze, 2014;

Wittman, 2014), Environmental damages (Schumer and Maloney, 2008).

Frequency competition in particular implicated in increased airport congestion (Vaze and Barnhart, 2012a).


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Aviation planners leverage airport traffic forecasts in evaluation of technological improvements, airport capacity expansion, workforce planning.

Want to model these decisions and competitive dynamics, evaluate different scenarios in a changing environment.

US Airways Network, Q1 2007


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Aviation planners leverage airport traffic forecasts in evaluation of technological improvements, airport capacity expansion, workforce planning.

Want to model these decisions and competitive dynamics, evaluate different scenarios in a changing environment.

US Airways Network, Q4 2007

Why a game-theoretic approach? Capacity and fare decisions of different airlines are

interdependent: interaction determines market share, and ultimately profits.

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Why a two-stage game?

Capacity decisions made under approximate knowledge of fares. Capacity decided in Stage I and fares in Stage II.

Literature Review Single-stage frequency-only game (Hansen, 1990; Vaze and Barnhart,

2012a; 2012b; 2015). Single-stage frequency-seat game (Hong and Harker, 1992; Adler and

Berechman, 2001; Wei and Hansen, 2007). Single-stage capacity-fare game (Adler, 2001; 2005; Adler, Pels and

Nash, 2010; Brueckner, 2010). Treatment of two-stage capacity-fare games more limited,

largely focuses on single-market duopoly (Dobson and Lederer, 1993; Schipper, Rietveld and Nijkamp, 2003; Brueckner and Flores-Fillol, 2007; Hansen and Liu, 2015). Little work that bridges analytical, computational, and empirical

results.

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Overview

Introduce two-stage game model.

Analysis of simplified model: demonstrate that it has properties that ensure a tractable and credible solution.

Use numerical approximations of 2nd stage payoff functions to show that these properties extend to more realistic scenarios.

Empirical calibration and validation using data from U.S. air transportation network.

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Two-Stage Game Model: Frequency versus Seats-per-Flight

Assume seats-per-flight held constant because… Limited effect on passenger demand, and Shows lower variability across time and segments.

For 10-year data across the U.S. (2005-2014)

Airline Coefficient of Variation: Across Years

Coefficient of Variation: Across Segments

Seats-per-Flight Frequency Seats-per-Flight Frequency American Airlines (AA) 4.7% 12.7% 13.3% 60.2% Delta Airlines (DL) 6.3% 21.7% 15.9% 67.9% United Airlines (UA) 5.3% 27.3% 21.7% 67.1% US Airways (US) 6.3% 16.9% 15.4% 58.6% Southwest Airlines (WN) 3.5% 16.6% 1.8% 70.0%

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Two-Stage Frequency-Fare Model Multinomial logit model of market share: Consistent with

several prior studies. Two most common formulations:

𝑀𝑀𝑀𝑀𝑎𝑎,𝑚𝑚 = 𝑒𝑒α 𝑙𝑙𝑙𝑙 𝑓𝑓𝑎𝑎 − 𝛽𝛽𝑝𝑝𝑎𝑎,𝑚𝑚

𝑁𝑁𝑚𝑚+∑ 𝑒𝑒α 𝑙𝑙𝑙𝑙 𝑓𝑓𝑖𝑖,𝑚𝑚 − 𝛽𝛽𝑝𝑝𝑖𝑖,𝑚𝑚𝑖𝑖∈𝐴𝐴𝑚𝑚

𝑀𝑀𝑀𝑀𝑎𝑎,𝑚𝑚 = 𝑒𝑒−φ𝑓𝑓𝑎𝑎,𝑚𝑚−𝑟𝑟− 𝛽𝛽𝑝𝑝𝑎𝑎,𝑚𝑚

𝑁𝑁𝑚𝑚+∑ 𝑒𝑒−φ𝑓𝑓𝑖𝑖,𝑚𝑚−𝑟𝑟− 𝛽𝛽𝑝𝑝𝑖𝑖,𝑚𝑚

𝑖𝑖∈𝐴𝐴𝑚𝑚

Payoff function (π𝑎𝑎):

π𝑎𝑎 = � min(𝑀𝑀𝑚𝑚 𝑀𝑀𝑀𝑀𝑎𝑎,𝑚𝑚 , 𝑓𝑓𝑎𝑎,𝑚𝑚𝑠𝑠𝑎𝑎,𝑚𝑚) 𝑝𝑝𝑎𝑎,𝑚𝑚 −𝑐𝑐𝑎𝑎,𝑚𝑚 𝑓𝑓𝑎𝑎,𝑚𝑚𝑚𝑚∈𝐾𝐾𝑎𝑎

“S-Curve Formulation”

“Schedule Delay Formulation”

Revenue Operating Cost

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Sub-game Perfect Pure Nash Equilibrium For a given frequency decision, fare decision modeled by

second-stage pure strategy Nash equilibrium. With knowledge of fare equilibria, players choose frequency. Predict frequency decisions as 1st-stage outcomes of

subgame-perfect pure strategy equilibrium. Many games are quite difficult to solve. Is this a reasonable solution concept in this case?

Starting assumptions (strong): Two airlines, single segment No connecting passengers Unlimited seating capacity No no-fly alternative

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Analytical Results Under these strong assumptions, for either formulation: Proposition 1: Second stage game has a unique equilibrium.

Supports the credibility and practical tractability of second

stage equilibrium.

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Analytical Results Under these strong assumptions, for either formulation: Proposition 2: Each airline’s payoff in the first stage game is a

submodular function in the overall strategy space, i.e. 𝜕𝜕2𝜋𝜋𝑖𝑖𝜕𝜕𝑓𝑓1𝜕𝜕𝑓𝑓2

< 0 for i ϵ {1, 2}.

Corollary: The game is supermodular: can transform strategy

space such that 𝜕𝜕2𝜋𝜋𝑖𝑖

𝜕𝜕𝑓𝑓1𝜕𝜕𝑓𝑓2> 0 for i ϵ {1, 2}

Ensures the convergence of several iterative learning dynamics, and the existence of first stage equilibrium.

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Analytical Results Under these strong assumptions, for either formulation: Proposition 3: Each airline’s payoff in the first stage game is

concave in its own strategy, i.e. 𝜕𝜕2𝜋𝜋𝑖𝑖𝜕𝜕𝑓𝑓𝑖𝑖

2 < 0 for i ϵ {1, 2} in

plausible parameter ranges: α < 2.44 in s-curve model. Empirical range: α = 1.3 - 1.7 (Belobaba, 2016) Ensures that individual payoff maximization problems are

easy to solve (even for large scale networks), existence of 1st stage equilibrium.

Not guaranteed for single stage frequency game (Hansen, 1990).

Simple model: nice analytical properties. But these results rest on strong assumptions. What if we relax them?

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Numerical Extensions

Approach: numerically approximate second stage payoffs with polynomials, evaluate approximation function properties.

Relax earlier assumptions and evaluate a wide variety of scenarios: for 1-player, 2-player and 3-player situations in the presence of a no-fly alternative with finite seating with connecting passengers with various values of 𝛼𝛼, 𝛽𝛽, 𝑁𝑁, 𝜑𝜑, r utility parameters with various seating capacities

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1) Heuristic to compute 2nd stage equilibria

Player 1 fare equilibria, 𝛼𝛼=1.29, 𝛽𝛽=0.0045, N=0.5


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π𝑎𝑎,𝑚𝑚 = min(𝑀𝑀𝑚𝑚 𝑀𝑀𝑀𝑀𝑎𝑎,𝑚𝑚 , 𝑓𝑓𝑎𝑎,𝑚𝑚𝑠𝑠𝑎𝑎,𝑚𝑚) 𝑝𝑝𝑎𝑎,𝑚𝑚−𝑐𝑐𝑎𝑎,𝑚𝑚 𝑓𝑓𝑎𝑎,𝑚𝑚

2) Compute payoffs




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π𝑎𝑎,𝑚𝑚 = min(𝑀𝑀𝑚𝑚 𝑀𝑀𝑀𝑀𝑎𝑎,𝑚𝑚 , 𝑓𝑓𝑎𝑎,𝑚𝑚𝑠𝑠𝑎𝑎,𝑚𝑚) 𝑝𝑝𝑎𝑎,𝑚𝑚−𝑐𝑐𝑎𝑎,𝑚𝑚 𝑓𝑓𝑎𝑎,𝑚𝑚

2) Compute payoffs



3) Fit payoffs to polynomial function of frequency profile

Quadratic Approximations Fit polynomial approximations of profits as functions of frequencies of all players: 1-player non-stop only game: π1 ~ 𝛽𝛽0 + 𝛽𝛽1𝑓𝑓1 + 𝛽𝛽2𝑓𝑓1

2 2-player non-stop only game: π1 ~ 𝛽𝛽0 + 𝛽𝛽1𝑓𝑓1 + 𝛽𝛽2𝑓𝑓2 +

𝛽𝛽3𝑓𝑓12 + 𝛽𝛽4𝑓𝑓2

2 + 𝛽𝛽5𝑓𝑓1𝑓𝑓2 3-player non-stop only game: π1 ~ 𝛽𝛽0 + 𝛽𝛽1𝑓𝑓1 + 𝛽𝛽2𝑓𝑓2 +

𝛽𝛽3𝑓𝑓3 + 𝛽𝛽4𝑓𝑓12 + 𝛽𝛽5𝑓𝑓2

2 + 𝛽𝛽6𝑓𝑓32 + 𝛽𝛽7𝑓𝑓1𝑓𝑓2 + 𝛽𝛽8𝑓𝑓1𝑓𝑓3 + 𝛽𝛽9𝑓𝑓2𝑓𝑓3

2-player non-stop and one-stop game: π1 ~ 𝛽𝛽0 + 𝛽𝛽1𝑓𝑓11 +𝛽𝛽2𝑓𝑓12 + 𝛽𝛽3𝑓𝑓21 + 𝛽𝛽4𝑓𝑓22 + 𝛽𝛽5𝑓𝑓11

2 + 𝛽𝛽6𝑓𝑓122 + 𝛽𝛽7𝑓𝑓21

2 + 𝛽𝛽8𝑓𝑓222 +

𝛽𝛽9𝑓𝑓11𝑓𝑓21 + 𝛽𝛽10𝑓𝑓21𝑓𝑓22 Etc.

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Quadratic Approximations: Results Model parameters varied one-at-a-time over ranges found in

literature or practice: N (0 to 1), α (1 to 2), r (.1 to 1), 𝜑𝜑 (1 to 10),𝛽𝛽 (-0.001 to -0.01) and seats (25-250), players (1-3)

In nearly all cases, excellent fit: R2 of model > 0.9, ranging from 0.91-0.9998. Additionally, signs of all estimated parameter values ensure submodularity and concavity.

𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝜷𝜷𝟎𝟎 𝜷𝜷𝟏𝟏 𝜷𝜷𝟐𝟐 𝜷𝜷𝟑𝟑 𝜷𝜷𝟒𝟒 𝜷𝜷𝟓𝟓 R2 250 122200.0 18135.0 -17856.0 -494.4 686.1 -533.0 95.5% 225 122250.0 18130.0 -17861.0 -494.2 686.2 -532.9 95.5% 200 122400.0 18115.0 -17876.0 -493.9 686.6 -532.4 95.5% 175 122640.0 18095.0 -17901.0 -493.4 687.2 -531.6 95.5% 150 123470.0 18030.0 -17989.0 -492.1 689.6 -529.0 95.5% 100 129430.0 17885.0 -18838.0 -495.5 716.3 -514.1 93.9%

75 136710.0 18277.0 -20301.0 -512.8 773.0 -518.1 92.8% 25 104880.0 27814.0 -18929.0 -743.5 773.0 -846.6 96.9%

For 2-player s-curve, non-stop passengers case with 𝛼𝛼=1.29, 𝛽𝛽=0.0045, N=0.5, and different seats-per-flight values.

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π1 ~ 𝛽𝛽0 + 𝛽𝛽1𝑓𝑓1 + 𝛽𝛽2𝑓𝑓2 + 𝛽𝛽3𝑓𝑓12 + 𝛽𝛽4𝑓𝑓2

2 + 𝛽𝛽5𝑓𝑓1𝑓𝑓2

Payoff Approximations: Further Extensions

Results suggests model is highly tractable, and can be efficiently solved by an iterative best response heuristic.

Can we make this more rigorous?

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Quadratic fit Quartic fit 𝑅𝑅2 = 0.95 𝑅𝑅2 = 0.99

Payoff Approximations: Further Extensions Proposition 4: Our N-player game, with quadratic, concave,

and submodular payoff functions and non-negative strategy spaces, assuming interaction terms are same within an airline, belongs to the class of quasi-aggregative games (as defined by Jensen, 2010), and the myopic best response algorithm converges to the set of Nash Equilibrium

Estimated coefficients are consistent with guaranteed unique first-stage equilibrium according to Rosen’s diagonal strict concavity condition (Rosen, 1965), with a few exceptions (high α, >1.7).

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Airline Network Case Study: Description 11-airport network in the Western U.S. In Q1 2007: 4 carriers, 68 carrier-segments (each with one

frequency value to be predicted). Actual BTS (Bureau of Transportation Statistics) data.

AS UA US WN

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Airline Network Case Study: Description Solved using iterative optimization of quadratic payoffs. Payoffs initialized with fitted functions above, transformed in each

market with cost/demand data. Additionally, enforce aircraft availability constraints.

AS UA US WN

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Convergence Results Model converges within 6-7 iterations (four optimizations each,

one per airline) in <1 second.

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Model Calibration and Results

engineering.dartmouth.edu NetEcon 2016

Minimize the Mean Absolute Percentage Error (MAPE) 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = ∑ �̂�𝑓𝑐𝑐𝑚𝑚−𝑓𝑓𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚∈𝐶𝐶𝐶𝐶

∑ 𝑓𝑓𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚∈𝐶𝐶𝐶𝐶 between predicted and observed

frequencies (similar to Vaze and Barnhart, 2012a; Cadarso et al. 2015)

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Segment-carriers grouped as: ‘3-player’, ‘2-player hub-to-hub’, ‘other 2-player’, and ‘1-player’. Calibrate 11 total coefficients,

transformed by demand/cost Adjusted using a stochastic

gradient approximation algorithm.

MAPE = 18.4% 49% abs. errors < 1, 78% < 2

Out-of-Sample Testing: Example Train coefficients on past data, test on future data, with new

demand, cost, market and player attributes. E.g.: Train 11 coefficients on 2007 Q1 data. Test on 2007 Q4. Out-of-Sample Testing MAPE = 20.6% 47% abs. errors < 1, 73% < 2

Error Adjusted MAPE = 11.2% 72% abs. errors < 1, 92% < 2

Market PDX-SFO does not exist in 2007 Q1, only in Q4. Using Q1-trained coefficients, predict frequencies of this new

market:

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Note: Compare to single airport MAPEs in 14%-20% range from previous research as benchmark (Vaze and Barnhart, 2012).

PDX-SFO True Frequency

Estimated Frequency

AS 3.02 3.52

UA 6.11 7.28

Multi-Year Validation Results, 2007-2015

Average MAPE < 20% for up to a couple years of look-ahead. Increases almost monotonically as the look-ahead period

increases. Reasonable prediction tool for short-to-medium term horizon.

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Adjusted MAPE, individual frequencies MAPE in new markets

Out of Sample Aggregated Results

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Airport Observed Flights Q1

Observed Flights Q4

Predicted Flights Q4

LAX 137 128 132

SJC 60 64 61

LAS 154 167 168

SAN 101 108 110

SMF 70 71 70

SEA 98 105 104

PDX 33 41 43

SFO 67 92 99

ONT 61 61 62

PHX 159 159 153

OAK 88 88 85

Aggregated Measure

MAPE calculation Adj. MAPE (Avg. Error)

Carrier ∑ ∑ 𝑓𝑓𝑎𝑎,𝑚𝑚𝑚𝑚∈𝐾𝐾𝑎𝑎 − ∑ 𝑓𝑓𝑎𝑎,𝑚𝑚𝑚𝑚∈𝐾𝐾𝑎𝑎𝑎𝑎∈𝐴𝐴

∑ ∑ 𝑓𝑓𝑎𝑎,𝑚𝑚𝑚𝑚∈𝐾𝐾𝑎𝑎𝑎𝑎∈𝐴𝐴

1.5% (2.11)

Coefficient category

∑ ∑ 𝑓𝑓𝑐𝑐𝑐𝑐𝑒𝑒𝑓𝑓,𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚∈𝐶𝐶𝐶𝐶 − ∑ 𝑓𝑓𝑐𝑐𝑐𝑐𝑒𝑒𝑓𝑓,𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚∈𝐶𝐶𝐶𝐶𝑐𝑐𝑐𝑐𝑒𝑒𝑓𝑓∈𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶

∑ ∑ 𝑓𝑓𝑐𝑐𝑐𝑐𝑒𝑒𝑓𝑓,𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚∈𝐶𝐶𝐶𝐶𝑐𝑐𝑐𝑐𝑒𝑒𝑓𝑓∈𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶

2.5% (3.42)

Market ∑ ∑ 𝑓𝑓𝑎𝑎,𝑚𝑚𝑎𝑎∈𝐴𝐴𝑚𝑚 − ∑ 𝑓𝑓𝑎𝑎,𝑚𝑚𝑎𝑎∈𝐴𝐴𝑚𝑚𝑚𝑚∈𝐾𝐾

∑ ∑ 𝑓𝑓𝑎𝑎,𝑚𝑚𝑎𝑎∈𝐴𝐴𝑚𝑚𝑚𝑚∈𝐾𝐾

6.3% (0.86)

Airport ∑ ∑ 𝑓𝑓𝑎𝑎𝑎𝑎,𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚∈𝐶𝐶𝐶𝐶 − ∑ 𝑓𝑓𝑎𝑎𝑎𝑎,𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚∈𝐶𝐶𝐶𝐶𝑎𝑎𝑎𝑎∈𝐴𝐴𝐴𝐴

∑ ∑ 𝑓𝑓𝑎𝑎𝑎𝑎,𝑐𝑐𝑚𝑚𝑐𝑐𝑚𝑚∈𝐶𝐶𝐶𝐶𝑎𝑎𝑎𝑎∈𝑎𝑎𝐴𝐴

2.6% (2.59)

Out of sample predictions of daily flights deployed at all airports in network, Q4 2007. Calibrated on Q1 2007 data.


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Observed Flights Q4


LAX 137 128 132

SJC 60 64 61

LAS 154 167 168

SAN 101 108 110

SMF 70 71 70

SEA 98 105 104

PDX 33 41 43

SFO 67 92 99

ONT 61 61 62

PHX 159 159 153

OAK 88 88 85



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Observed Flights Q4


LAX 137 128 132

SJC 60 64 61

LAS 154 167 168

SAN 101 108 110

SMF 70 71 70

SEA 98 105 104

PDX 33 41 43

SFO 67 92 99

ONT 61 61 62

PHX 159 159 153

OAK 88 88 85


Airport level aggregated MAPE values (adjusted for calibration error) for varying look-ahead durations

Takeaways

Two-stage frequency-fare games are behaviorally consistent with the actual airline decision process. For simple cases, the analytical properties indicate very

well-behaved games: existence, uniqueness, convergence, stability. These nice properties extend to more complex cases too. Model predictions approximate actual airline decisions

reasonably well within a short-to-medium term horizon.

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Takeaways

Theory-motivated framework for informing 2-stage game-theoretic predictive model. Refinements of model could serve as a scenario analysis

tool to aid in planning and policy-making decision-support. Future work: extension to larger networks, more flexible

payoff parameterizations to capture market and airline characteristics, integration with statistical forecasting models, deeper understanding of schedule competition

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