  The relative arbitrage portfolio outperforms a benchmark portfolio over a
given time-horizon with probability one. With market price of risk processes
depending on the market portfolio and investors, this paper analyzes the
multi-agent optimization of relative arbitrage opportunities in the coupled
system of market and wealth dynamics. We construct a well-posed market
dynamical system of McKean-Vlasov type under an empirical measure of investors,
where each investor seeks for relative arbitrage with respect to a benchmark
dependent on market and all the agents. We show the conditions to guarantee
relative arbitrage opportunities among competitive investors through the
Fichera drift. Under mild conditions, we derive the optimal strategies for
investors and the unique Nash equilibrium that depends on the smallest
nonnegative solution of a Cauchy problem.
