  This paper considers finitely many investors who perform mean-variance
portfolio selection under a relative performance criterion. That is, each
investor is concerned about not only her terminal wealth, but how it compares
to the average terminal wealth of all investors (i.e., the mean field). At the
inter-personal level, each investor selects a trading strategy in response to
others' strategies (which affect the mean field). The selected strategy
additionally needs to yield an equilibrium intra-personally, so as to resolve
time inconsistency among the investor's current and future selves (triggered by
the mean-variance objective). A Nash equilibrium we look for is thus a tuple of
trading strategies under which every investor achieves her intra-personal
equilibrium simultaneously. We derive such a Nash equilibrium explicitly in the
idealized case of full information (i.e., the dynamics of the underlying stock
is perfectly known), and semi-explicitly in the realistic case of partial
information (i.e., the stock evolution is observed, but the expected return of
the stock is not precisely known). The formula under partial information
involves an additional state process that serves to filter the true state of
the expected return. Its effect on trading is captured by two degenerate Cauchy
problems, one of which depends on the other, whose solutions are constructed by
elliptic regularization and a stability analysis of the state process. Our
results indicate that partial information alone can reduce investors' wealth
significantly, thereby causing or aggravating systemic risk. Intriguingly, in
two different scenarios of the expected return (i.e., it is constant or
alternating between two values), our Nash equilibrium formula spells out two
distinct manners systemic risk materializes.
