  In this paper, a Nash-type fictitious game framework is introduced to handle
a time-inconsistent linear-quadratic optimal control. The Nash-type game in
this framework is called fictitious as it is between the decision maker (called
real player) and an auxiliary control variable (called fictitious player) with
the real player and fictitious player looking for time-consistent policy and
precomitted optimal policy, respectively. Namely, the fictitious-game framework
is actually an auxiliary-variable-based mechanism where the fictitious player
is our particular design. Noting that the real player's cost functional is
revised in accordance with that of fictitious player, the equilibrium policy of
real player is called an open-loop self-coordination control of original
linear-quadratic problem. As a generalization, a time-inconsistent nonzero-sum
stochastic linear-quadratic dynamic game is investigated, where one player is
to look for precommitted optimal policy and the other player is to search
time-consistent policy. Necessary and sufficient conditions are presented to
ensure the existence of open-loop equilibrium of the nonzero-sum game, which
resort to a set of Riccati-like equations and linear equations. By applying the
developed theory of nonzero-sum game, open-loop self-coordination control of
the linear-quadratic optimal control is fully characterized, and multi-period
mean-variance portfolio selection is also investigated. Finally, numerical
simulations are presented, which show the efficiency of the proposed
fictitious-game framework.
