# American Institue of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

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## Journal of Dynamics & Games

2016 , Volume 3 , Issue 4

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Konstantin Avrachenkov , Giovanni Neglia and  Vikas Vikram Singh
2016, 3(4): 303-318 doi: 10.3934/jdg.2016016 +[Abstract](46) +[PDF](393.4KB)
Abstract:
Network formation games have been proposed as a tool to explain the topological characteristics of existing networks. They assume that each node is an autonomous decision-maker, ignoring that in many cases different nodes are under the control of the same authority (e.g. an Autonomous System) and then they operate as a team. In this paper we introduce the concept of network formation games for teams of nodes and show how very different network structures can arise also for some simple games studied in the literature. Beside extending the usual definition of pairwise stable networks to this new setting, we define a more general concept of stability toward deviations from a specific set $\mathcal{C}$ of teams' coalitions ($\mathcal{C}$-stability). We study then a trembling-hand dynamics, where at each time a coalition of teams can create or sever links in order to reduce its cost, but it can also take wrong decisions with some small probability. We show that this stochastic dynamics selects $\mathcal{C}$-stable networks or networks from closed cycles in the long run as the error probability vanishes.
2016, 3(4): 319-334 doi: 10.3934/jdg.2016017 +[Abstract](53) +[PDF](353.5KB)
Abstract:
In the framework of mean field game theory, a new optimization problem is presented by adding an additional player, called the principal. After introducing a proper payoff for the principal, continuity and existence of minimum is proved. Some considerations about uniqueness and possible ways of continuing the analysis of this problem are given.
2016, 3(4): 335-354 doi: 10.3934/jdg.2016018 +[Abstract](24) +[PDF](754.9KB)
Abstract:
A qualitative criterion for a pursuer to intercept a target in a class of differential games is obtained in terms of future cones: Topological cones that contain all attainable trajectories of target or interceptor originating from an initial position. An interception solution exists after some initial time iff the future cone of the target lies within the future cone of the interceptor. The solution may be regarded as a kind of Nash equilibrium. This result is applied to two examples:
1. The game of Two Cars: The future cone condition is shown to be equivalent to conditions for interception obtained by Cockayne.
2. Satellite warfare: The future cone for a spacecraft or direct-ascent antisatellite weapon (ASAT) maneuvering in a central gravitational field is obtained and is shown to equal that for a spacecraft which maneuvers solely by means of a single velocity change at the cone vertex.
The latter result is illustrated with an analysis of the January 2007 interception of the FengYun-1C spacecraft.
Pablo Sánchez and  Jaume Sempere
2016, 3(4): 355-369 doi: 10.3934/jdg.2016019 +[Abstract](60) +[PDF](395.6KB)
Abstract:
This paper develops a model where agents can create private property rights on a resource by making appropriative activities. We show that the value of the resource has a non-monotonic effect on the emergence of private property. When the resource is sufficiently valuable, agents have an incentive to leave a sharing agreement and private property can appear. However if the value of the resource increases beyond a given threshold, deviations from the sharing agreement leads to a very costly confrontation. In this case, private property is not sustainable. Our analysis also finds that population size has an important effect on the size of the parameter set in which private property is sustainable.
Lisa C Flatley , Robert S MacKay and  Michael Waterson
2016, 3(4): 371-398 doi: 10.3934/jdg.2016020 +[Abstract](148) +[PDF](831.3KB)
Abstract:
We characterize cost-minimizing operating strategies for an energy store over a given interval of time $[0,T]$. The cost functional here can represent, for example, a traditional economic cost or a penalty for time-variation of the output from a storage-assisted wind farm or more general imbalance between supply and demand. Our analysis allows for leakage, operating inefficiencies and general cost functionals. In the case where the cost of a store depends only on its instantaneous power output (or input), we present an algorithm to determine the optimal strategies. A key feature is that this algorithm is localized in time, in the sense that the action of the store at a time $t\in[0,T]$ requires cost information over only some usually much shorter subinterval of time $[t,t_k]\subset[t,T].$