A note on the complexity of integer programming games
aa r X i v : . [ c s . CC ] J u l A NOTE ON THE C OMPLEXITY OF INTEGER PROGR AMMINGGAMES
A P
REPRINT
Margarida Carvalho
CIRRELT and Département d’informatique et de recherche opérationnelleUniversité de MontréalMontreal, QC H3C 3J7 [email protected]
July 30, 2019 A BSTRACT
In this brief note, we prove that the existence of Nash equilibria on integer programming games is Σ p -complete. K eywords Computational Complexity · Game Theory · Integer Programming · Nash Equilibria
Integer programming games (IPGs) model games in which there is a finite set of players M = { , . . . , m } and eachplayer p ∈ M has a set of feasible strategies X p given by lattice points inside a polytopes described by finite systemsof linear inequalities. Therefore, each players aims to solve max x p ∈ X P Π p ( x p , x − p ) , (1)where x p is player p ’s strategy and x − p is the vector of strategies of all players, except player p .A vector x ∈ Q p ∈ M X p is called a pure profile of strategies . If a pure profile of strategies x solves the optimizationproblem (1) for all players, it is called pure Nash equilibrium . A game may fail to have pure equilibria and, therefore,a broader solution concept for a game must be introduced, the Nash equilibria . Under this concept, each playercan assign probabilities to her pure strategies. Let ∆ p denote the space of Borel probability measures over X p and ∆ = Q p ∈ M ∆ p . Each player p ’s expected payoff for a profile of strategies σ ∈ ∆ is Π p ( σ ) = Z X p Π p ( x p , x − p ) dσ. (2)A Nash equilibrium (NE) is a profile of strategies σ ∈ ∆ such that Π p ( σ ) ≥ Π p ( x p , σ − p ) , ∀ p ∈ M ∀ x p ∈ X p . (3)In [1], the authors discuss the existence of Nash equilibria for integer programming games. It is proven that the exis-tence of pure Nash equilibria for IPGs is Σ p -complete and that even the existence of Nash equilibria is Σ p -complete.However, the later proof seems incomplete in the “proof of only if”, since it does not support why we can concludethat the leader cannot guarantee a payoff of 1. In the following section, we provide a correct proof using a completelynew reduction. In what follows, we show that even in the simplest case, linear integer programming games with two players, theexistence of Nash equilibria is a Σ p -complete problem. PREPRINT - J
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Theorem 2.1
The problem of deciding if an IPG has a Nash equilibrium is Σ p -complete problem.Proof. The proof that this decision problem belongs to Σ p can be found in [1].It remains to show that it is Σ p -hard. We will reduce the following to Σ p -complete problem to it (see[2]): Problem: S UBSET -S UM -I NTERVAL
INSTANCE
A sequence q , q , . . . , q k of positive integers; two positive integers R and r with r ≤ k . QUESTION
Does there exist an integer S with R ≤ S < R + 2 r such that none ofthe subsets I ⊆ { , . . . , k } satisfies P i ∈ I q i = S ?Our reduction starts from an instance of S UBSET -S UM -I NTERVAL . We construct the following instance of IPG • The game has two players, M = { Z, W } . • Player Z solves max z z + k X i =1 q i z i + Qz (2 w − z ) (4a) s.t. z + k X i =1 q i z i ≤ z (4b) z , z , . . . , z k ∈ { , } (4c) R ≤ z ≤ R + 2 r − , z ∈ N (4d)Add binary variables y ∈ { , } r and make z = R + P r − i =0 i y i . Note that z = Rz + P r − i =0 i y i z . Thus,replace y i z by h i and add the respective McCormick constraints [3]. In this way, we can equivalently linearizethe previous problem: max z,y,h z + k X i =1 q i z i + 2 Qzw − QRz − r − X i =0 i h i (5a) s.t. z + k X i =1 q i z i ≤ z (5b) z , z , . . . , z k ∈ { , } (5c) R ≤ z ≤ R + 2 r − , z ∈ N (5d) z = R + r − X i =0 i y i (5e) y , y , . . . , y r − ∈ { , } (5f) h i ≥ i = 0 , . . . , r − (5g) h i ≥ z + ( R + 2 r − y i − i = 0 , . . . , r − (5h) h i ≤ z + R ( y i − i = 0 , . . . , r − (5i) h i ≤ ( R + 2 r − y i i = 0 , . . . , r − (5j)For sake of simplicity, we consider the quadratic formulation (4). The linearization above serves the purposeof showing that the proof is valid even under linear utility functions for the players. • Player W solves max w (1 − z ) w (6a) s.t. R ≤ w ≤ R + 2 r − , z ∈ N (6b) w ∈ R (6c)2 PREPRINT - J
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30, 2019(Proof of if). Assume that the S
UBSET -S UM -I NTERVAL instance has answer YES. Then, there is R ≤ S < R + 2 r such that for all subsets I ⊆ { , . . . , k } , we have P i ∈ I q i = S . Let player W strategy be w ∗ = S and w ∗ = 0 . Notethat the term Qz (2 w − z ) in player Z ’s utility is dominant and attains a maximum when z is equal to w . Thus, make z ∗ = w ∗ = S and since P ki =1 q i z i is at most S − , make z ∗ = 1 . Choose z ∗ i such that the remaining utility of player Z is maximized. By construction, player Z is selecting her best response to ( w ∗ , w ∗ ) . Sinze z ∗ = 1 , then player W isalso selecting an optimal strategy. Therefore, we can conclude that there is an equilibrium.(Proof of only if). Assume that the S UBSET -S UM -I NTERVAL instances has answer NO. Then, for all R ≤ S < R + 2 r ,there is a subset I ⊆ { , . . . , k } such that P i ∈ I q i = S . In this case, player Z will always make z = 0 which givesincentive for player W to choose w as large as possible. Since w has no upper bound, there is no equilibrium for thegame. (cid:3) Acknowledgements
The authors wish to thank Sriram Sankaranarayanan for bringing to our attention the incompleteness of our previousproof in [1].We wish acknowledge the support of the Institut de valorisation des données and the Canadian Natural Sciences andEngineering Research Council under the discovery grants program.
References [1] Margarida Carvalho, Andrea Lodi and João Pedro Pedroso. Existence of Nash Equilibria on Integer Program-ming Games. In
Congress of APDIO 2017, the Portuguese Operational Research Society , pages 11–23. SpringerInternational Publishing, 2018.[2] Christian Eggermont and Gerhard J. Woeginger. Motion planning with pulley, rope, and baskets. In
Proceed-ings of the 29th International Symposium on Theoretical Aspects of Computer Science (STACS’2012) , LeibnizInternational Proceedings in Informatics 14, 374–383. 2012.[3] Garth P. McCormick. Computability of global solutions to factorable nonconvex programs: Part I - convex under-estimating problems.