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Register a new userDynamic Pricing and Mean Field Analysis for Controlling Age of Information
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Today many mobile users in various zones are invited to sense and send back real-time useful information (e.g., traffic observation and sensor data) to keep the freshness of the content updates in such zones. However, due to the sampling cost in sensing and transmission, a user may not have the incentive to contribute the real-time information to help reduce the age of information (AoI). We propose dynamic pricing for each zone to offer age-dependent monetary returns and encourage users to sample information at different rates over time. This dynamic pricing design problem needs to well balance the monetary payments as rewards to users and the AoI evolution over time, and is challenging to solve especially under the incomplete information about users arrivals and their private sampling costs. After formulating the problem as a nonlinear constrained dynamic program, to avoid the curse of dimensionality, we first propose to approximate the dynamic AoI reduction as a time-average term and successfully solve the approximate dynamic pricing in closed-form. Further, by providing the steady-state analysis for an infinite time horizon, we show that the pricing scheme (though in closed-form) can be further simplified to an $varepsilon$-optimal version without recursive computing over time. Finally, we extend the AoI control from a single zone to many zones with heterogeneous user arrival rates and initial ages, where each zone cares not only its own AoI dynamics but also the average AoI of all the zones in a mean field game system to provide a holistic service. Accordingly, we propose decentralized mean field pricing for each zone to self-operate by using a mean field term to estimate the average age dynamics of all the zones, which does not even require many zones to exchange their local data with each other.
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We consider a dynamic game with asymmetric information where each player observes privately a noisy version of a (hidden) state of the world V, resulting in dependent private observations. We study structured perfect Bayesian equilibria that use private beliefs in their strategies as sufficient statistics for summarizing their observation history. The main difficulty in finding the appropriate sufficient statistic (state) for the structured strategies arises from the fact that players need to construct (private) beliefs on other players private beliefs on V, which in turn would imply that an infinite hierarchy of beliefs on beliefs needs to be constructed, rendering the problem unsolvable. We show that this is not the case: each players belief on other players beliefs on V can be characterized by her own belief on V and some appropriately defined public belief. We then specialize this setting to the case of a Linear Quadratic Gaussian (LQG) non-zero-sum game and we characterize linear structured PBE that can be found through a backward/forward algorithm akin to dynamic programming for the standard LQG control problem. Unlike the standard LQG problem, however, some of the required quantities for the Kalman filter are observation-dependent and thus cannot be evaluated off-line through a forward recursion.
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Reachable set computation is an important technique for the verification of safety properties of dynamical systems. In this paper, we investigate reachable set computation for discrete nonlinear systems based on parallelotope bundles. The algorithm relies on computing an upper bound on the supremum of a nonlinear function over a rectangular domain, which has been traditionally done using Bernstein polynomials. We strive to remove the manual step of parallelotope template selection to make the method fully automatic. Furthermore, we show that changing templates dynamically during computations cans improve accuracy. To this end, we investigate two techniques for generating the template directions. The first technique approximates the dynamics as a linear transformation and generates templates using this linear transformation. The second technique uses Principal Component Analysis (PCA) of sample trajectories for generating templates. We have implemented our approach in a Python-based tool called Kaa and improve its performance by two main enhancements. The tool is modular and use two types of global optimization solvers, the first using Bernstein polynomials and the second using NASAs Kodiak nonlinear optimization library. Second, we leverage the natural parallelism of the reachability algorithm and parallelize the Kaa implementation. We demonstrate the improved accuracy of our approach on several standard nonlinear benchmark systems.
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