Different Policy Objectives of the Road Pricing Problem – a Game Theory Approach

Using game theory we investigate a new approach to formulate and solve optimal tolls with a focus on different policy objectives of the road authority. The aim is to gain more insight into determining optimal tolls as well as into the behavior of users after tolls have been imposed on the network. The problem of determining optimal tolls is stated and defined using utility maximization theory, including elastic demand on the travelers’ side and different objectives for the road authority. Game theory notions are adopted regarding different games and players, rules and outcomes of the games played between travelers on the one hand and the road authority on the other. Different game concepts (Cournot, Stackelberg and monopoly game) are mathematically formulated and the relationship between players, their payoff functions and rules of the games are defined for very simplistic cases. The games are solved for different scenarios and different objectives for the road authority, using the Nash equilibrium concept. Using the Stackelberg game concept as being most realistic for road pricing, a few experiments are presented illustrating the optimal toll design problem subject to different pricing policies considering different objectives of the road authority. Results show different outcomes both in terms of optimal tolls as well as in payoffs for travelers. There exist multiple optimal solutions and objective function may have a non- continuous shape. The main contribution is the two-level separation between of the users from the road authority in terms of their objectives and influences.

Stackelberg Network Pricing Games

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of $m$ priceable edges in a graph. The other edges have a fixed cost. Based on the leader's decision one or more followers optimize a polynomial-time solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the leader's prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions, and the problem is to find revenue maximizing prices. Our model extends several known pricing problems, including single-minded and unit-demand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a $(1+\epsilon) \log m$-approximation for any $\epsilon >0$. This can be extended to provide a $(1+\epsilon)(\log k + \log m)$-approximation for the general problem and $k$ followers. The latter result is essentially best possible, as the problem is shown to be hard to approximate within \mathcal{O(\log^\epsilon k + \log^\epsilon m). If followers have demands, the single-price algorithm provides a $(1+\epsilon)m^2$-approximation, and the problem is hard to approximate within \mathcal{O(m^\epsilon) for some $\epsilon >0$. Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex cover, which is based on non-trivial max-flow and LP-duality techniques. Our results can be extended to provide constant-factor approximations for any constant number of followers

Playing Stackelberg Opinion Optimization with Randomized Algorithms for Combinatorial Strategies

From a perspective of designing or engineering for opinion formation games in social networks, the "opinion maximization (or minimization)" problem has been studied mainly for designing subset selecting algorithms. We furthermore define a two-player zero-sum Stackelberg game of competitive opinion optimization by letting the player under study as the first-mover minimize the sum of expressed opinions by doing so-called "internal opinion design", knowing that the other adversarial player as the follower is to maximize the same objective by also conducting her own internal opinion design. We propose for the min player to play the "follow-the-perturbed-leader" algorithm in such Stackelberg game, obtaining losses depending on the other adversarial player's play. Since our strategy of subset selection is combinatorial in nature, the probabilities in a distribution over all the strategies would be too many to be enumerated one by one. Thus, we design a randomized algorithm to produce a (randomized) pure strategy. We show that the strategy output by the randomized algorithm for the min player is essentially an approximate equilibrium strategy against the other adversarial player

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

On the Two-user Multi-carrier Joint Channel Selection and Power Control Game

In this paper, we propose a hierarchical game approach to model the energy efficiency maximization problem where transmitters individually choose their channel assignment and power control. We conduct a thorough analysis of the existence, uniqueness and characterization of the Stackelberg equilibrium. Interestingly, we formally show that a spectrum orthogonalization naturally occurs when users decide sequentially about their transmitting carriers and powers, delivering a binary channel assignment. Both analytical and simulation results are provided for assessing and improving the performances in terms of energy efficiency and spectrum utilization between the simultaneous-move game (with synchronous decision makers), the social welfare (in a centralized manner) and the proposed Stackelberg (hierarchical) game. For the first time, we provide tight closed-form bounds on the spectral efficiency of such a model, including correlation across carriers and users. We show that the spectrum orthogonalization capability induced by the proposed hierarchical game model enables the wireless network to achieve the spectral efficiency improvement while still enjoying a high energy efficiency.Comment: 31 pages, 13 figures, accepted in IEEE Transactions on Communication