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Computing pure Nash equilibria in network revenue management games

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Abstract

We present a mixed-integer model to optimize a competitor’s behavior in a network revenue management game. Our model is based on a well-known deterministic linear program for the single-airline network revenue management problem. Assuming that the competitors make decisions based on our model, we present an algorithm to compute a pure Nash equilibrium (NE) in a two-player game through an iterative search for best responses. If the algorithm gets stuck in a loop without finding an NE, additional constraints are added to the models to control the search. The complete algorithm finds an NE with certainty if one exists in the game. The players’ price vectors are modified so that their models have unique solutions, and the search follows a unique path. This makes sure that both players end up in the same NE even if uniqueness is not guaranteed or cannot be proved. A computational study shows the algorithm’s performance which can compute NE in networks of realistic size in acceptable time.

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Acknowledgements

This work was done with financial support from the German Research Foundation (DFG) under Grant No. KI 1272/2-2. We thank the area editor and two anonymous referees for useful comments about an earlier version of the paper.

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Authors and Affiliations

  1. Chair of Logistics and Operations Research, Mercator School of Management, University of Duisburg-Essen, Lotharstr. 65, 47048, Duisburg, Germany

    W. Grauberger & A. Kimms

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  1. W. Grauberger
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  2. A. Kimms
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Correspondence to A. Kimms.

Appendix: Computational results for perturbation order “alphabetic by O&D name, then by fare class”

Appendix: Computational results for perturbation order “alphabetic by O&D name, then by fare class”

In the following we show the computational results for the instances with perturbation order “alphabetic by O&D name, then increasing by fare class.” Figures 10, 11, and 12 show the average computation times, while Figs. 13, 14, and 15 show the average iterations needed to reach an NE. As in the other perturbation order, there is a positive correlation between the computation time and number of iterations needed to reach an NE. The larger the networks and the higher the competition intensity, the more time and iterations were needed to find an NE.

Fig. 10
figure 10

Average computation times in minutes with \(\mu = 2\)

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Fig. 11
figure 11

Average computation times in minutes with \(\mu = 4\)

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Fig. 12
figure 12

Average computation times in minutes with \(\mu = 6\)

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Fig. 13
figure 13

Average numbers if iterations with \(\mu = 2\)

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Fig. 14
figure 14

Average numbers if iterations with \(\mu = 4\)

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Fig. 15
figure 15

Average numbers if iterations with \(\mu = 6\)

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Table 9 Average payoff ratios with \(\mu = 2\)
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Table 10 Average payoff ratios with \(\mu = 4\)
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Table 11 Average payoff ratios with \(\mu = 6\)
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Tables 9, 10, and 11 show the average payoff ratios. The higher the CI, the more it pays off to take competition into account. This circumstance is indicated by the shrinking ratio of NE and non-competitive payoffs (NC/NE). Tables 12, 13, and 14 represent the average payoff ratios with linear vs. algorithmic binary variables. Again, no definite statement can be made about which concept dominates which.

Table 12 Average solution time ratios with linear versus logarithmic binary variables with \(\mu = 2\)
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Table 13 Average solution time ratios with linear versus logarithmic binary variables with \(\mu = 4\)
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Table 14 Average solution time ratios with linear versus logarithmic binary variables with \(\mu = 6\)
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Grauberger, W., Kimms, A. Computing pure Nash equilibria in network revenue management games. OR Spectrum 40, 481–516 (2018). https://doi.org/10.1007/s00291-018-0507-5

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