In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function described by a Hamilton-Jacobi-Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the meanfield best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean-Vlasov dynamics.2 Not all behavior, however, can be captured by risk-neutral cost functions. One way of capturing risk-seeking or risk-averse behavior is by exponentiating loss functions before expectation (see [2], [18] and the references therein).The particular risk-sensitive mean-field stochastic differential game that we consider in this paper involves an exponential term in the stochastic long-term cost function. This approach was first taken by Jacobson in [18], when considering the risk-sensitive Linear-Quadratic-Gaussian (LQG) problem with state feedback. Jacobson demonstrated a link between the exponential cost criterion and deterministic linear-quadratic differential games. He showed that the risk-sensitive approach provides a method for varying the robustness of the controller and noted that in the case of no risk, or risk-neutral case, the well known LQR solution would result (see, for followup work on risk-sensitive stochastic control problems with noisy state measurements, [35], [6],[27]).In this paper, we examine the risk-sensitive stochastic differential game in a regime of large population of players. We first present a mean-field stochastic differential game model where the players are coupled not only via their risk-sensitive cost functionals but also via their states. The main coupling term is the mean-field process, also called the occupancy process or population profile process. Each player reacts to the mean field or a subset of the mean field generated by the states of the other players in an area, and at the same time the mean field evolves according to a controlled Kolmogorov forward equation.Our contribution can be summarized as follows. Using a particular structure of state dynamics, we derive the mean-field limit of the individual state dynamics leading to a non-linear controlled macroscopic McKean-Vlasov equation; see [21]. Combining this with a limiting risk-sensitive cost functional, we arrive at the mean-field response framework, and establish its compatibility with the density distribution using the controlled Fokker-Planck-Kolmogorov forward equation.The mean-field equilibria are characterized by coupled backward-forwar...

Underwater communication remains a challenging technology via communication cables and the cost of underwater sensor network (UWSN) deployment is still very high. As an alternative, underwater wireless communication has been proposed and have received more attention in the last decade. Preliminary research indicated that the Radio Frequency (RF) and Magneto-Inductive (MI) communication achieve higher data rate in the near field communication. The optical communication achieves good performance when limited to the line-of-sight positioning. The acoustic communication allows long transmission range. However, it suffers from transmission losses and time-varying signal distortion due to its dependency on environmental properties. These latter are salinity, temperature, pressure, depth of transceivers, and the environment geometry. This paper is focused on both the acoustic and magneto-inductive communications, which are the most used technologies for underwater networking. Such as acoustic communication is employed for applications requiring long communication range while the MI is used for real-time communication. Moreover, this paper highlights the trade-off between underwater properties, wireless communication technologies, and communication quality. This can help the researcher community by providing clear insight for further research.

In this article, we investigate the competitive interaction between electrical vehicles or hybrid oilelectricity vehicles in a Cournot market consisting of electricity transactions to or from an underlying electricity distribution network. We provide a mean field game formulation for this competition, and introduce the set of fundamental differential equations ruling the behavior of the vehicles at the feedback Nash equilibrium, referred here to as the mean field equilibrium. This framework allows for a consistent analysis of the evolution of the price of electricity as well as of the instantaneous electricity demand in the power grid. Simulations precisely quantify those parameters and suggest that significant reduction of the daily electricity peak demand can be achieved by appropriate electricity pricing. I. INTRODUCTION Electrical vehicles (EV) and plug-in hybrid electrical vehicles (PHEV) have been recognized as natural components of future electricity distribution networks, known as smart grids [1], [2], [3]. As opposed to classical vehicles, EV and PHEV are equipped with batteries which can be charged or discharged by using a simple plug-in connector compatible with the local electricity distribution grid. Thus, EV and PHEV can be conceived as both energy consuming devices and mobile energy sources [4], [5], [6], [7].In the former case, EV and PHEV can be seen as devices straining the energy demand of energy suppliers and, thus, adding a new constraint to reliably distribute the electricity. In the latter case, EV and PHEV R. Couillet is with SUPELEC. 3 rue Joliot-Curie,

We consider a noncooperative interaction among a large population of mobiles that interfere with each other through many local interactions. The first objective of this paper is to extend the evolutionary game framework to allow an arbitrary number of mobiles that are involved in a local interaction. We allow for interactions between mobiles that are not necessarily reciprocal. We study 1) multiple-access control in a slotted Aloha-based wireless network and 2) power control in wideband code-division multiple-access wireless networks. We define and characterize the equilibrium (called evolutionarily stable strategy) for these games and study the influence of wireless channels and pricing on the evolution of dynamics and the equilibrium.