ARTICLE IN PRESS JID: EOR [m5G;September 26, 2020;17:35] European Journal of Operational Research xxx (xxxx) xxx Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Discrete Optimization On the Stackelberg knapsack game Ulrich Pferschy a,∗ , Gaia Nicosia b , Andrea Pacifici c , Joachim Schauer d a Department of Statistics and Operations Research, University of Graz, Austria b Dipartimento di Ingegneria, Università degli Studi Roma Tre, Italy c Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università degli Studi di Roma “Tor Vergata”, Italy d Department of Statistics and Operations Research, University of Graz, Austria a r t i c l e i n f o Article history: Received 23 October 2019 Accepted 4 September 2020 Available online xxx Keywords: Complexity theory Knapsack problem Stackelberg game Bilevel optimization a b s t r a c t 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 play- ers is associated with a subset of the knapsack items. The leader may offer profits for its items as in- centives 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 fol- lower 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 of- fering 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. © 2020 Elsevier B.V. All rights reserved. 1. Introduction A Stackelberg game (named after the market model due to von Stackelberg, 1934) is a strategic game in which there are two in- teracting players at two distinct levels. First, one player L, called the leader, makes its choice by choosing some elements or setting certain parameters. Then, in view of the leader’s decision, the other player F , called the follower, chooses its response. Both players aim at optimizing their own objectives, which are usually conflicting or at least not positively correlated. To this purpose, the leader needs to anticipate the optimal response of the follower. In this setting, it is usually assumed that the players have complete and mutual knowledge about each other’s models. Stackelberg games well rep- resent the behavior of distinct individuals acting in a market and therefore play a relevant role in the field of economics in general. More specifically, they were used in revenue management for set- ∗ Corresponding author. E-mail addresses: pferschy@uni-graz.at (U. Pferschy), gaia.nicosia@uniroma3.it (G. Nicosia), andrea.pacifici@uniroma2.it (A. Pacifici), joachim.schauer@uni-graz.at (J. Schauer). ting prices and for deciding tolls in road networks (see e.g. van Hoesel, 2008). Stackelberg games may be viewed as special Bilevel Program- ming (BP) problems, that is, optimization problems in which some of the decision variables in an upper level problem must be opti- mal to some other, lower level problem. A BP problem deals with the optimization of scenarios whose outcome depends on the in- terplay of two decision makers. In the Stackelberg context, the up- per level optimization problem is commonly referred to as the leader’s problem, while the lower level one as the follower’s prob- lem. BP problems are, in general, computationally difficult to solve: Jeroslow (1985) showed that they are N P-hard even when the ob- jective functions and constraints are linear. The authors of Labbé and Violin (2013) present a number of complexity results and state that most solution techniques for BP have been developed focusing on special cases in which convenient properties, such as linearity or convexity, can be exploited to develop efficient solution meth- ods. In this paper we analyze a new Stackelberg-type leader-follower scenario for the classical binary knapsack problem (KP). In KP we are given a discrete finite set N of items, each having a nonnegative weight and a profit, and a knapsack with bounded weight capacity. https://doi.org/10.1016/j.ejor.2020.09.007 0377-2217/© 2020 Elsevier B.V. All rights reserved. Please cite this article as: U. Pferschy, G. Nicosia, A. Pacifici et al., On the Stackelberg knapsack game, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2020.09.007