Resolving Time Inconsistency in Financial Decision Problems With Non-Expectation Operator: From Internal Conflict to Internal Harmony by Strategy of Self-Coordination

Abstract

When a stochastic decision problem is time inconsistent, the decision maker would always be troubled by his conflicting decisions “optimally” derived from his time-varying preferences at different time instants. The long-run self (LR) of the decision maker pursues the long-term optimality and thus would like to take the pre-committed strategy. On the other hand, the short-run selves (SRs) of the decision maker at different time instants lack foresight, bow to short-term temptations and would like to take the so-called time consistent strategy. As these two extreme strategies have been criticized for their incomplete viewpoints in dealing with time inconsistency, the emerging literature of self-control game models recognizes an importance to strike a balance between the LR’s and SRs’ interests. While most risk measures in financial decisions are of a non-expectation nature, the existing results of self-control game models are confined to situations where the decision maker’s preferences involve only expectation operators. We take a step forward to develop in this study an operable unified two-tier dual-self game model with commitment by punishment in continuous time setting, which can cope with general time inconsistent stochastic financial decision problems with both expectation and non-expectation operators in the objective function. By attaching to both the preferences of LR and SRs certain punishment terms which quantitatively evaluate the internal conflict among different selves, our game model can align the interests of the LR and SRs to any desired degree. The equilibrium strategy, termed strategy of self-coordination in this paper, achieves some degree of internal harmony among various selves. We demonstrate in the paper successful applications of our dual-self game model to two long standing challenges in the literature, the continuous-time investment and consumption problem with quasi-hyperbolic discounting and the continuous-time mean-variance portfolio selection problem.