286 research outputs found
Stochastic Graphon Mean Field Games with Jumps and Approximate Nash Equilibria
We study continuous stochastic games with inhomogeneous mean field
interactions on large networks and explore their graphon limits. We consider a
model with a continuum of players, where each player's dynamics involve not
only mean field interactions but also individual jumps induced by a Poisson
random measure. We examine the case of controlled dynamics, with control terms
present in the drift, diffusion, and jump components.
We introduce the graphon game model based on a graphon controlled stochastic
differential equation (SDE) system with jumps, which can be regarded as the
limiting case of a finite game's dynamic system as the number of players goes
to infinity. Under some general assumptions, we establish the existence and
uniqueness of Markovian graphon equilibria. We then provide convergence results
on the state trajectories and their laws, transitioning from finite game
systems to graphon systems. We also study approximate equilibria for finite
games on large networks, using the graphon equilibrium as a benchmark. The
rates of convergence are analyzed under various underlying graphon models and
regularity assumptions.Comment: 37 page
Linear Quadratic Graphon Field Games
Linear quadratic graphon field games (LQ-GFGs) are defined to be LQ games
which involve a large number of agents that are weakly coupled via a weighted
undirected graph on which each node represents an agent. The links of the graph
correspond to couplings between the agents' dynamics, as well as between the
individual cost functions, which each agent attempts to minimize. We formulate
limit LQ-GFG problems based on the assumption that these graphs lie in a
sequence which converges to a limit graphon. First, under a finite-rank
assumption on the limit graphon, the existence and uniqueness of solutions to
the formulated limit LQ-GFG problem is established. Second, based upon the
solutions to the limit LQ-GFG problem, epsilon-Nash equilibria are constructed
for the corresponding game problems with a very large but finite number of
players. This result is then generalized to the case with random initial
conditions. It is to be noted that LQ-GFG problems are distinct from the class
of graphon mean field game (GMFG) problems where a population is hypothesized
to be associated with each node of the graph [Caines and Huang CDC 2018, 2019]
Master equation of discrete time graphon mean field games and teams
In this paper, we present a sequential decomposition algorithm equivalent of
Master equation to compute GMFE of GMFG and graphon optimal Markovian policies
(GOMPs) of graphon mean field teams (GMFTs). We consider a large population of
players sequentially making strategic decisions where the actions of each
player affect their neighbors which is captured in a graph, generated by a
known graphon. Each player observes a private state and also a common
information as a graphon mean-field population state which represents the
empirical networked distribution of other players' types. We consider
non-stationary population state dynamics and present a novel backward recursive
algorithm to compute both GMFE and GOMP that depend on both, a player's private
type, and the current (dynamic) population state determined through the
graphon. Each step in computing GMFE consists of solving a fixed-point
equation, while computing GOMP involves solving for an optimization problem. We
provide conditions on model parameters for which there exists such a GMFE.
Using this algorithm, we obtain the GMFE and GOMP for a specific security setup
in cyber physical systems for different graphons that capture the interactions
between the nodes in the system.Comment: 26 pages, 6 figures. arXiv admin note: text overlap with
arXiv:1905.0415
- …