The optimal coalition structure can be obtained by means of convex optimization. Other applications of game theory include packet forwarding in ad hoc networks, distributed cooperative source coding, routing problems, and localization algorithms, which will be more elaborated in the next chapter. The expansion and enhancement of wireless and mobile devices has aroused the demand of context-aware applications, in which location is often viewed as one of the most important contexts.
Those applications include pervasive medical care, wireless sensor network surveillance, mobile peer-to-peer computing etc.
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The essential purpose of wireless sensor networks WSN is to provide information about observed events. Furthermore, location information can be used to improve the communication system itself. Geo-location information can serve as complementary data to estimate and predict critical parameters for improving wireless communication networks, such as setting up location dependent load balancing schemes Yanmaz E. Several studies have shown how the efficiency of available radio resources can be improved by the availability of position information to provide accurate scheduling and link adaptation Tang S.
Additionally, localizing the nodes can help reduce power consumption in multi hop wireless networks. However their accuracy strongly depends on the scenario. Especially in dense urban or indoor environments, navigation based on GNSS becomes inaccurate or impossible, since the necessary amount of 4 directly visible satellites is not reached. In order to provide accurate MT position estimation, the MT position shall be estimated with alternative techniques focusing on radio signals which are provided by the terrestrial RANs itself.
The rapid deployment of WLAN and WPAN technologies, especially in dense indoor environments, made it another compelling choice for localization, relying only on the existing network infrastructure. Generally, the localization process assumes a number of location aware nodes, called anchors.
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In a typical two-stage positioning system, the first phase is the ranging phase, where nodes estimate the distances to their neighbors by observing time of arrival, received signal strength or some other distance dependent signal metric. In the second phase, nodes use the ranging information and the known anchor position for calculation of their coordinates.
Each estimated distance represents the radius of a circle centered at the corresponding reference node. For 2-D positioning, measurements from at least three reference nodes are required, and the location is obtained as the intersection of circles. This method is also used for GPS. Having in mind the errors in estimated distances to the anchors, the geometrical trilateration technique can only provide a region of uncertainty, instead of a single point.
Therefore the solution is based on iterative algorithms to obtain the node position by formulating and solving a set of nonlinear equations. The availability of positioning information depends on the existing infrastructure such as GPS satellites or base stations.
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Cooperative positioning techniques are used in scenarios where non-cooperative solutions are not feasible, or do not perform well in terms of accuracy, cost and complexity. The challenge is to allow nodes which are not in range of a sufficient number of anchors to be located, and hereby increase localization performance in terms of both accuracy and coverage. This can be achieved by means of iterative multilateration, among other solutions.
Iterative multilateration is a way to expand localization coverage throughout the network in a step-by-step fashion, allowing also nodes which are not in range of a sufficient number of references to be localized. In this sense, coverage is the fraction of nodes that have an accurate position estimate.
It follows an iterative scheme: once an unknown node estimates its position, it becomes an anchor and broadcasts its position estimate to all neighboring nodes. The process is repeated until all nodes that can have three or more reference nodes obtain a position estimate. As a newly localized node is becoming new anchor for its neighbors, the estimation error of the first node can propagate to other nodes and eventually get amplified.
Over iterations the error could spread throughout the network, leading to abundant error in large topologies. The number of actively participating nodes should be kept to a minimum, and therefore an appropriate cooperation subset has to be chosen, while the other nodes can be ignored. Such a restrictive and selective use of references is crucial in networks with limited resources. A frequently used method is to select the nearest k anchor nodes. However, this method does not take into account node geometry.
Therefore other metrics such as geometric dilution of precision, Cramer Rao lower bound or stochastic observability are more appropriate. The geometric conditioning on localization accuracy is derived in the GDOP geometric dilution of precision metric Spirito M.
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In brief, when reference nodes are well separated around the target, the GDOP is lower. Localization can be defined as an estimation problem where measurements like wireless signal strength, angle or time of arrival are provided to an estimator i. In case of localization, the CRLB captures information about node geometry and ranging quality, i.
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Since the variance of position estimates is associated to the mean error, the lower bound on variance can be seen as the upper bound on accuracy. Recently game theory has been applied in localization algorithms, mainly for modeling the cost-performance trade-off and for selection of reference nodes. The work in Ghassemi F. The localization process has been modeled as a game belonging to the class of weighted-graph games.
Introducing Game Theory and its Applications
For such a representation, the vertices correspond to the players, and the coalition value can be obtained by summing the weights of the edges that connect a pair of vertices in the coalition with self-loop edges only considered with half of their weights. Basic idea is to allocate more measurements to nodes that contribute more to the localization process. The allocation algorithm has been integrated into a Bayesian estimator. In Ghassemi F. Additionally, a price for transmission is included to account for the current energy level in the nodes, and the energy needed for data transmission.
The algorithm proposed in Moragrega A. Anchors transmitting with lower energy can provide coverage to a smaller number of nodes; aim is to minimize power consumption at the anchor nodes, while assuring desired localization accuracy. The metric for positioning quality is the GDOP. The problem has been formulated as a noncooperative game, using Nash equilibrium as solution concept.
In Bejar B. Using only a subset of available reference nodes does not necessarily degrade the accuracy, since some of them provide redundant information. In some situations it might be even useful to discard ranging information from some reference nodes, after they have been identified as unreliable due to biases in the measurements.
This paper the localization problem has been defined as a coalitional NTU game, where coalitions are formed based on the merge and split procedure. The utility function is defined to account for both a quality and cost indicator. The target tracking task based on coalition formation has been implemented using a Kalman filter. For the coalition formation approach a higher mean estimation error has been observed than for grand coalition, i.
Nevertheless, in terms of communication costs the proposed scheme provides significant savings. Ghareshiran O. Assuming that nodes in sleep mode do not record any measurements and thereby save energy in both sensing and transmitting data, the optimization problem is formulated to maximize the average sleep time of all nodes in the network, assuring that targets are localized with desired accuracy.
An important contribution is exploitation of spatial correlation of sensor readings. The characteristic function is formulated in a way that larger coalitions of sensors do not necessarily lead to longer sleep times. This is mainly due to the fact that the B-FIM, depending on both relative angles and distances of sensors to the target, does not automatically increase as the number of sensor nodes in a coalition goes up.
The trade-off between performance and average sleep time allocated in the network is demonstrated via Monte Carlo simulations. As stated in the previous section, a typical localization process consists of the ranging phase, where nodes estimate the distances to their neighbors, and a second phase where nodes use the ranging information and the known anchor position to calculate their coordinates. In a dense network one can assume a large number of available anchor nodes. However, transmitting and processing all the obtainable information would consume immense power, without necessarily leading to better localization performance.
This is due to the fact that not all the anchors provide reliable measurements, what leads to erroneous distance estimates. Furthermore, the geometry of selected reference nodes shows to have significant impact on localization accuracy, what will be extensively elaborated in our work.
Assuming that at each time instant a target has several neighboring anchor nodes in near vicinity, and different coalitions can be formed, the considered scenario is illustrated in Fig. We propose an algorithm for reference node selection based on coalitional games.
We model the localization process as a cooperative game, and formulate the corresponding utility function. Position estimates are obtained using the linearized least squares algorithm trilateration. We assume that the distance estimates between nodes are obtained using RSS measurements. RSS based distance estimates are obtained using the lognormal model:. The following parameters are relevant for reference node selection: number of references, quality of range estimates and geometry. Therefore we propose a node selection mechanism based on the Cramer Rao Lower Bound.
Besides the quality indicator, utility function also has to reflect the cost. Structure Partners Scientific Board Secretariat. Search form Search. You are here Home. Event format:. Event website. It will seem that, surely, choosing the safe bridge straight away would be a mistake, since that is just where she will expect you, and your chances of death rise to certainty.
So perhaps you should risk the rocks, since these odds are much better. But wait … if you can reach this conclusion, your pursuer, who is just as rational and well-informed as you are, can anticipate that you will reach it, and will be waiting for you if you evade the rocks. So perhaps you must take your chances with the cobras; that is what she must least expect. But, then, no … if she expects that you will expect that she will least expect this, then she will most expect it.
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