A study of general and security Stackelberg game formulations

International audienceIn this paper, we analyze different mathematical formulations for general Stackelberg games (GSGs) and Stackelberg security games (SSGs). We consider GSGs in which a single leader commits to a utility maximizing strategy knowing that one of p possible followers optimizes its own utility taking this leader strategy into account. SSGs are a type of GSG that arise in security applications where the strategies of the leader consist in protecting subsets of targets and the strategies of the p followers consist in attacking a single target. We compare existing mixed integer linear programming (MILP) formulations for GSGs, sorting them according to the tightness of their linear programming (LP) relaxations. We show that SSG formulations are projections of GSG formulations and exploit this link to derive a new SSG MILP formulation that i) has the tightest LP relaxation known among SSG MILP formulations and ii) its LP relaxation coincides with the convex hull of feasible solutions in the case of a single follower. We present computational experiments empirically comparing the difficulty of solving the formulations in the general and security settings. The new SSG MILP formulation is computationally efficient, in particular as the problem size increases

Leadership in Singleton Congestion Games: What is Hard and What is Easy

We study the problem of computing Stackelberg equilibria Stackelberg games whose underlying structure is in congestion games, focusing on the case where each player can choose a single resource (a.k.a. singleton congestion games) and one of them acts as leader. In particular, we address the cases where the players either have the same action spaces (i.e., the set of resources they can choose is the same for all of them) or different ones, and where their costs are either monotonic functions of the resource congestion or not. We show that, in the case where the players have different action spaces, the cost the leader incurs in a Stackelberg equilibrium cannot be approximated in polynomial time up to within any polynomial factor in the size of the game unless P = NP, independently of the cost functions being monotonic or not. We show that a similar result also holds when the players have nonmonotonic cost functions, even if their action spaces are the same. Differently, we prove that the case with identical action spaces and monotonic cost functions is easy, and propose polynomial-time algorithm for it. We also improve an algorithm for the computation of a socially optimal equilibrium in singleton congestion games with the same action spaces without leadership, and extend it to the computation of a Stackelberg equilibrium for the case where the leader is restricted to pure strategies. For the cases in which the problem of finding an equilibrium is hard, we show how, in the optimistic setting where the followers break ties in favor of the leader, the problem can be formulated via mixed-integer linear programming techniques, which computational experiments show to scale quite well

Network-constrained models of liberalized electricity markets: the devil is in the details

Numerical models for electricity markets are frequently used to inform and support decisions. How robust are the results? Three research groups used the same, realistic data set for generators, demand and transmission network as input for their numerical models. The results coincide when predicting competitive market results. In the strategic case in which large generators can exercise market power, the predicted prices differed significantly. The results are highly sensitive to assumptions about market design, timing of the market and assumptions about constraints on the rationality of generators. Given the same assumptions the results coincide. We provide a checklist for users to understand the implications of different modelling assumptions

Stationary Strong Stackelberg Equilibrium in Discounted Stochastic Games

International audienceIn this work we study Stackelberg equilibria for discounted stochastic games. We consider two solution concepts for these games: Stationary Strong Stackelberg Equlibrium (SSSE) and Fixed Point Equilibrium (FPE) solutions. The SSSE solution is obtained by explicitly solving the Stackelberg equilibrium conditions, while the FPE can be computed efficiently using value or policy iteration algorithms. However, previous work has overlooked the relationship between these two different solution concepts. Here we investigate the conditions for existence and equivalence of these solution concepts. Our theoretical results prove that the FPE and SSSE exist and coincide for important classes of games, including Myopic Follower Strategy and Team games. This however does not hold in general and we provide numerical examples where one of SSSE or FPE does not exist, or when they both exist, they differ. Our computational results compare the solutions obtained by value iteration, policy iteration and a mathematical programming formulations for this problem. Finally, we present a discounted stochastic Stackelberg game for a security application to illustrate the solution concepts and the efficiency of the algorithms studied

Network-constrained models of liberalized electricity markets: the devil is in the details

Numerical models for electricity markets are frequently used to inform and support decisions. How robust are the results? Three research groups used the same, realistic data set for generators, demand and transmission network as input for their numerical models. The results coincide when predicting competitive market results. In the strategic case in which large generators can exercise market power, the predicted prices differed significantly. The results are highly sensitive to assumptions about market design, timing of the market and assumptions about constraints on the rationality of generators. Given the same assumptions the results coincide. We provide a checklist for users to understand the implications of different modelling assumptions.Market power, Electricity, Networks, Numeric models, Model comparison

Transforming Energy Networks via Peer to Peer Energy Trading: Potential of Game Theoretic Approaches

Peer-to-peer (P2P) energy trading has emerged as a next-generation energy management mechanism for the smart grid that enables each prosumer of the network to participate in energy trading with one another and the grid. This poses a significant challenge in terms of modeling the decision-making process of each participant with conflicting interest and motivating prosumers to participate in energy trading and to cooperate, if necessary, for achieving different energy management goals. Therefore, such decision-making process needs to be built on solid mathematical and signal processing tools that can ensure an efficient operation of the smart grid. This paper provides an overview of the use of game theoretic approaches for P2P energy trading as a feasible and effective means of energy management. As such, we discuss various games and auction theoretic approaches by following a systematic classification to provide information on the importance of game theory for smart energy research. Then, the paper focuses on the P2P energy trading describing its key features and giving an introduction to an existing P2P testbed. Further, the paper zooms into the detail of some specific game and auction theoretic models that have recently been used in P2P energy trading and discusses some important finding of these schemes.Comment: 38 pages, single column, double spac

Jamming Games in the MIMO Wiretap Channel With an Active Eavesdropper

This paper investigates reliable and covert transmission strategies in a multiple-input multiple-output (MIMO) wiretap channel with a transmitter, receiver and an adversarial wiretapper, each equipped with multiple antennas. In a departure from existing work, the wiretapper possesses a novel capability to act either as a passive eavesdropper or as an active jammer, under a half-duplex constraint. The transmitter therefore faces a choice between allocating all of its power for data, or broadcasting artificial interference along with the information signal in an attempt to jam the eavesdropper (assuming its instantaneous channel state is unknown). To examine the resulting trade-offs for the legitimate transmitter and the adversary, we model their interactions as a two-person zero-sum game with the ergodic MIMO secrecy rate as the payoff function. We first examine conditions for the existence of pure-strategy Nash equilibria (NE) and the structure of mixed-strategy NE for the strategic form of the game.We then derive equilibrium strategies for the extensive form of the game where players move sequentially under scenarios of perfect and imperfect information. Finally, numerical simulations are presented to examine the equilibrium outcomes of the various scenarios considered.Comment: 27 pages, 8 figures. To appear, IEEE Transactions on Signal Processin