A paradox in time-consistency in the mean-variance problem? (Q1711723)

In the context of financial investment, agents seldom keep their preferences the same over time. In contrast, they usually make an investment decision that may only be optimal with respect to their current utility, regardless of what their future preferences might be. This myopic decision may lead to substantial burdens in the long run. This phenomenon is described as time-inconsistency in decision making. For a time-inconsistent problem, the optimal choice for an agent's contemporary preference would no longer be optimal for his future preference. Exploring this phenomenon, the authors establish new conditions under which a constrained (no short-selling) time-consistent equilibrium strategy, starting at a certain time, will beat the unconstrained counterpart, as measured by the magnitude of their corresponding equilibrium mean-variance value functions. It is proved that the pure strategy of solely investing in a risk-free bond can sometimes simultaneously dominate both constrained and unconstrained equilibrium strategies. Under a precommitment approach, the value function of an investor increases with the size of the admissible sets of strategies. However, this may fail to be true under the game-theoretic paradigm, as the constraint of time-consistency itself affects the value function differently when short-selling is and is not prohibited.