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On the Stackelberg knapsack game

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  • Pferschy, Ulrich
  • Nicosia, Gaia
  • Pacifici, Andrea
  • Schauer, Joachim

Abstract

In this work we consider a bilevel knapsack problem, in which one player, the follower, decides on the optimal utilization of a bounded resource. The second player, the leader, can offer incentives, or shared profits, so that the follower chooses options attractive also for the leader. Formally, each of the two players is associated with a subset of the knapsack items. The leader may offer profits for its items as incentives to the follower, before the follower selects a subset of all items in order to maximize its overall profit. The leader receives as pay-off only the profits from those of its items that are included by the follower in the overall knapsack solution. This pay-off is then reduced by the profits offered to the follower. The resulting setting is a Stackelberg strategic game. The leader has to resolve the trade-off between offering high profits as incentives to the follower and offering low profits to gain high pay-offs.We analyze the problem for the case in which the follower solves the resulting knapsack problem to optimality and obtain a number of strong complexity results. Then we invoke a common assumption of the literature, namely that the follower’s computing power is bounded. Under this condition, we study several natural Greedy-type heuristics applied by the follower. The solution structure and complexity of the resulting problems are investigated and solution strategies are derived, in particular an Integer Linear Programming (ILP) model, but also pseudopolynomial and polynomial algorithms, when possible.

Suggested Citation

  • Pferschy, Ulrich & Nicosia, Gaia & Pacifici, Andrea & Schauer, Joachim, 2021. "On the Stackelberg knapsack game," European Journal of Operational Research, Elsevier, vol. 291(1), pages 18-31.
    • Handle: RePEc:eee:ejores:v:291:y:2021:i:1:p:18-31
    DOI: 10.1016/j.ejor.2020.09.007
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    References listed on IDEAS

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    1. Nicosia, Gaia & Pacifici, Andrea & Pferschy, Ulrich, 2017. "Price of Fairness for allocating a bounded resource," European Journal of Operational Research, Elsevier, vol. 257(3), pages 933-943.
    2. van Hoesel, Stan, 2008. "An overview of Stackelberg pricing in networks," European Journal of Operational Research, Elsevier, vol. 189(3), pages 1393-1402, September.
    3. M. Hosein Zare & Oleg A. Prokopyev & Denis Sauré, 2020. "On Bilevel Optimization with Inexact Follower," Decision Analysis, INFORMS, vol. 17(1), pages 74-95, March.
    4. Fischetti, Matteo & Monaci, Michele & Sinnl, Markus, 2018. "A dynamic reformulation heuristic for Generalized Interdiction Problems," European Journal of Operational Research, Elsevier, vol. 267(1), pages 40-51.
    5. Darmann, Andreas & Nicosia, Gaia & Pferschy, Ulrich & Schauer, Joachim, 2014. "The Subset Sum game," European Journal of Operational Research, Elsevier, vol. 233(3), pages 539-549.
    6. Alberto Caprara & Margarida Carvalho & Andrea Lodi & Gerhard J. Woeginger, 2016. "Bilevel Knapsack with Interdiction Constraints," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 319-333, May.
    7. Chen, Lv & Shen, Yang, 2018. "On A New Paradigm Of Optimal Reinsurance: A Stochastic Stackelberg Differential Game Between An Insurer And A Reinsurer," ASTIN Bulletin, Cambridge University Press, vol. 48(2), pages 905-960, May.
    8. Matteo Fischetti & Ivana Ljubić & Michele Monaci & Markus Sinnl, 2019. "Interdiction Games and Monotonicity, with Application to Knapsack Problems," INFORMS Journal on Computing, INFORMS, vol. 31(2), pages 390-410, April.
    9. Ensthaler, Ludwig & Giebe, Thomas, 2014. "Bayesian optimal knapsack procurement," European Journal of Operational Research, Elsevier, vol. 234(3), pages 774-779.
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