Conference Agenda

( paper https://arxiv.org/abs/2408.01175v1 ) We study mean-field games where common noise dynamics are described by integer-valued random measures, for instance Poisson random measures, in addition to Brownian motions. In such a framework, we describe Nash equilibria for mean-field portfolio games of both optimal investment and hedging under relative performance concerns with respect to exponential (CARA) utility preferences. Agents have independent individual risk aversions, competition weights and initial capital endowments, whereas their liabilities are described by contingent claims which can depend on both common and idiosyncratic risk factors. Liabilities may incorporate, e.g., compound Poisson-like jump risks and can only be hedged partially by trading in a common but incomplete financial market, in which prices of risky assets evolve as Itô-processes. Mean-field equilibria are fully characterized by solutions to suitable McKean-Vlasov forward-backward SDEs with jumps, for whose we prove existence and uniqueness of solutions, without restricting competition weights to be small.

For many Markovian decision problems, it is reasonable to consider several statistically equal decision makers operating simultaneously on the same state space and interacting with each other (e.g. maintenance of identical machines in a production site, population of potentially infected persons). Depending on the model, the state transition and the profit of the individual may depend on the empirical distribution of the decision makers across the states. In the limiting case, as the number $N$ of decision makers tends to infinity, we show that the resulting mean-field model describes a classical deterministic control problem, for which the limit state process is characterized by a controlled ordinary differential equation. We show that an optimal control of the mean-field model yields an asymptotically optimal control for the model with $N$ decision makers. In the end we discuss some applications. The corresponding paper is joint work with N. Bäuerle and appeared in Applied Mathematics & Optimization 90, 12 (2024), https://doi.org/10.1007/s00245-024-10154-1.

We consider a symmetric search game with the following features: each player chooses a searching area, any player's search can be successful at only one location and successively the reward (yield) at any location is assigned at random to one of the players searching at this location. We derive the mean-field version of the game by letting the number of players converge to infinity. Within the mean-field version of the game we obtain a concise characterization of equilibrium strategies. Based on this we show sufficient conditions on the reward function for the existence and uniqueness of equilibria. We illustrate with an example that the equilibrium may not be Pareto optimal, suggesting that the intervention of a central planner is useful for every player.

In this study, we consider a large-scale bipartite queueing model for mobile edge computing (MEC). MEC is a network architecture designed to bring computing power and data storage closer to the data's origin, typically at the network edge or on mobile devices, to enhance processing speed and reduce latency.

Despite the potential of MEC to enable real-time communication, minimize latency, and improve data processing, task offloading is one of the challenging issues in MEC. The end users decide whether to offload to edge servers or to process the job locally based on the status such as interaction among the large number of users, network conditions, and task requirements. Thus, it is crucial to find the optimal offloading decision policy for the users. There are many related works to deal with the concerns. To overcome the difficulties, existing studies considered game theoretic approach, reinforcement learning, Lyapunov optimization, and stochastic optimization. Besides, to achieve effective task offloading, load balancing aims to utilize computational resources across the network efficiently. Under certain assumptions, distributing tasks to shorter queues has been shown to result in optimal load balancing. Although both the join-the-shortest-queue (JSQ) policy and the power of-$d$ choices policy (also referred to as JSQ($d$)) are considered to be optimal strategies, many studies have overlooked the impact of delays when monitoring queue lengths. Specifically, implementing JSQ policy requires the dispatcher to have real-time information on the length of each server's queue, which can cause significant communication overheads and hinder scalability in environments with a large number of servers.

To the best of our knowledge, there is no work to simultaneously deal with the large number of user interactions, queueing dynamics and the resource allocations, including the delay in checking the number of queue lengths in the MEC context. We minimize the response time (sojourn time in queueing theory) and identify the optimal offloading decisions by applying the mean-field theory. Mean-field theory is a powerful methodology that simplifies the analysis of complex queueing networks by approximating their behavior when the number of entities (servers or queues) becomes very large. We adapt the threshold-based offloading and explore the efficiency of this offloading policy by applying the mean-field technique to a large-scale bipartite queueing model, which is the abstraction of the resource allocation under specific conditions related to graph connectivity between users and edge servers. Numerical experiments show the sojourn time of the tasks and the optimal threshold of the users for some task offloading algorithms, including random allocation, JSQ, and JSQ($d$). The experiments also demonstrate the consistency between the numerical analysis by mean-field theory and the simulation data.