Resolving Time Inconsistency of Decision Problem with Non-expectation Operator: From Internal Conflict to Internal Harmony by Strategy of Self-Coordination

When a stochastic decision problem is time inconsistent, the decision maker would be puzzled by his conflicting decisions optimally derived from his time-varying preferences at different time instants (with different time horizons). While the long-run self (LR) of the decision maker pursues the long-term optimality, the short-run selves (SRs) of the decision maker at different time instants bow to short-term temptations. While the literature began to recognize the importance to strike a balance between LR's and SRs' interests, the existing results are not applicable to situations where the decision maker's preferences involve non-expectation operators. We propose an operable unified two-tier dual-self game model with commitment by punishment, which can cope with general time inconsistent stochastic decision problems with both expectation and non-expectation operators in the objective function. By attaching punishment terms to both the preferences of LR and SRs which quantitatively evaluate the internal conflict among different selves, our game model aligns the interests of the LR and SRs to a certain degree. The equilibrium strategy, termed strategy of self-coordination, achieves some degree of internal harmony among various selves. We successfully apply the model to the investment and consumption problem with quasi-hyperbolic discounting and the dynamic mean-variance portfolio selection problem.