Mean field game of optimal relative investment with jump risk

In this paper,we study the n-player game and the mean field game under the constant relative risk aversion relative performance on terminal wealth,in which the interaction occurs by peer competition.In the model with n agents,the price dynamics of underlying risky assets depend on a common noise and contagious jump risk modeled by a multi-dimensional nonlinear Hawkes process.With a continuum of agents,we formulate the mean field game problem and characterize a deterministic mean field equilibrium in an analytical form under some conditions,allowing us to investigate some impacts of model parameters in the limiting model and discuss some financial implications.Moreover,based on the mean field equilibrium,we construct an approximate Nash equilibrium for the n-player game when n is sufficiently large.The explicit order of the approximation error is also derived.

A Monte Carlo Neural Fictitious Self-Play approach to approximate Nash Equilibrium in imperfect-information dynamic games

Solving the optimization problem to approach a Nash Equilibrium point plays an important role in imperfect information games,e.g.,StarCraft and poker.Neural Fictitious Self-Play(NFSP)is an effective algorithm that learns approximate Nash Equilibrium of imperfect-information games from purely self-play without prior domain knowledge.However,it needs to train a neural network in an off-policy manner to approximate the action values.For games with large search spaces,the training may suffer from unnecessary exploration and sometimes fails to converge.In this paper,we propose a new Neural Fictitious Self-Play algorithm that combines Monte Carlo tree search with NFSP,called MC-NFSP,to improve the performance in real-time zero-sum imperfect-information games.With experiments and empirical analysis,we demonstrate that the proposed MC-NFSP algorithm can approximate Nash Equilibrium in games with large-scale search depth while the NFSP can not.Furthermore,we develop an Asynchronous Neural Fictitious Self-Play framework(ANFSP).It uses asynchronous and parallel architecture to collect game experience and improve both the training efficiency and policy quality.The experiments with th e games with hidden state information(Texas Hold^m),and the FPS(firstperson shooter)games demonstrate effectiveness of our algorithms.

Approximating Nash Equilibrium for Optimal Consumption in Stochastic Growth Model with Jumps

In this paper,we study a class of dynamic games consisting of finite agents under a stochastic growth model with jumps.The jump process in the dynamics of the capital stock of each agent models announcements regarding each agent in the game occur at Poisson distributed random times.The aim of each agent is to maximize her objective functional with mean-field interactions by choosing an optimal consumption strategy.We prove the existence of a fixed point related to the so-called consistence condition as the number of agents goes large.Building upon the fixed point,we establish an optimal feedback consumption strategy for all agents which is in fact an approximating Nash equilibrium which describes strategies for each agent such that no agent has any incentive to change her strategy.

Strong stability of Nash equilibria in load balancing games

We study strong stability of Nash equilibria in load balancing games of m(m 2)identical servers,in which every job chooses one of the m servers and each job wishes to minimize its cost,given by the workload of the server it chooses.A Nash equilibrium(NE)is a strategy profile that is resilient to unilateral deviations.Finding an NE in such a game is simple.However,an NE assignment is not stable against coordinated deviations of several jobs,while a strong Nash equilibrium(SNE)is.We study how well an NE approximates an SNE.Given any job assignment in a load balancing game,the improvement ratio(IR)of a deviation of a job is defined as the ratio between the pre-and post-deviation costs.An NE is said to be aρ-approximate SNE(ρ1)if there is no coalition of jobs such that each job of the coalition will have an IR more thanρfrom coordinated deviations of the coalition.While it is already known that NEs are the same as SNEs in the 2-server load balancing game,we prove that,in the m-server load balancing game for any given m 3,any NE is a(5/4)-approximate SNE,which together with the lower bound already established in the literature yields a tight approximation bound.This closes the final gap in the literature on the study of approximation of general NEs to SNEs in load balancing games.To establish our upper bound,we make a novel use of a graph-theoretic tool.