  Inspired by Strotz's consistent planning strategy, we formulate the infinite
horizon mean-variance stopping problem as a subgame perfect Nash equilibrium in
order to determine time consistent strategies with no regret. Equilibria among
stopping times or randomized stopping times may not exist. This motivates us to
consider the notion of liquidation strategies, which lets the stopping right to
be divisible. We then argue that the mean-standard deviation variant of this
problem makes more sense for this type of strategies in terms of time
consistency. It turns out that an equilibrium liquidation strategy always
exists. We then analyze whether optimal equilibrium liquidation strategies
exist and whether they are unique and observe that neither may hold.
