Wireless mobile backhaul networks have been proposed as a substitute in cases in which wired alternatives are not available due to economical or geographical reasons. In this work, we study the location problem of base stations in a given region where mobile terminals are distributed according to a certain probability density function and the base stations communicate through microwave backhaul links. Using results of optimal transport theory, we provide the optimal asymptotic distribution of base stations in the considered setting by minimizing the total power over the whole network.
Location games have been introduced by Hotelling @cite_0 , who modeled the spatial competition along a street between two firms for persuading the largest number of customers which are uniformly distributed. Problems similar to location games, as for example the maximum capture problem, have been analyzed by @cite_4 @cite_5 and references therein.
@cite_0 @cite_5 @cite_4
2072113937 2136651023 2597894505 2810102121
In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria. We consider a competitive facility location problem on a network, in which consumers are located on the vertices and wish to connect to the nearest facility. Knowing this, competitive players locate their facilities on vertices that capture the largest possible market share. In 1991, Eiselt and Laporte established the first relation between Nash equilibria of a facility location game in a duopoly and the solutions to the 1-median problem. They showed that an equilibrium always exists in a tree because a location profile is at equilibrium if and only if both players select a 1-median of that tree [4]. In this work, we further explore the relations between the solutions to the 1-median problem and the equilibrium profiles. We show that if an equilibrium in a cycle exists, both players must choose a solution to the 1-median problem. We also obtain the same property for some other classes of graphs such as quasi-median graphs, median graphs, Helly graphs, and strongly-chordal graphs. Finally, we prove the converse for the latter class, establishing that, as for trees, any median of a strongly-chordal graph is a winning strategy that leads to an equilibrium. In a pursuit-evasion game, a team of pursuers attempt to capture an evader. The players alternate turns, move with equal speed, and have full information about the state of the game. We consider the most restrictive capture condition: a pursuer must become colocated with the evader to win the game. We prove two general results about this adversarial motion planning problem in geometric spaces. First, we show that one pursuer has a winning strategy in any compact CAT(0) space. This complements a result of Alexander, Bishop and Ghrist, who provide a winning strategy for a game with positive capture radius. Second, we consider the game played in a compact domain in Euclidean two-space with piecewise analytic boundary and arbitrary Euler characteristic. We show that three pursuers always have a winning strategy by extending recent work of Bhadauria, Klein, Isler and Suri from polygonal environments to our more general setting. We consider multi-player repeated games involving a large number of players with large strategy spaces and enmeshed utility structures. In these ldquolarge-scalerdquo games, players are inherently faced with limitations in both their observational and computational capabilities. Accordingly, players in large-scale games need to make their decisions using algorithms that accommodate limitations in information gathering and processing. This disqualifies some of the well known decision making models such as ldquoFictitious Playrdquo (FP), in which each player must monitor the individual actions of every other player and must optimize over a high dimensional probability space. We will show that Joint Strategy Fictitious Play (JSFP), a close variant of FP, alleviates both the informational and computational burden of FP. Furthermore, we introduce JSFP with inertia, i.e., a probabilistic reluctance to change strategies, and establish the convergence to a pure Nash equilibrium in all generalized ordinal potential games in both cases of averaged or exponentially discounted historical data. We illustrate JSFP with inertia on the specific class of congestion games, a subset of generalized ordinal potential games. In particular, we illustrate the main results on a distributed traffic routing problem and derive tolling procedures that can lead to optimized total traffic congestion.