Abstract: In several time-inconsistent problems, the time-inconsistency is due to the fact the underlying problem is multi-variate in some sense. For example, the mean-risk problem has two underlying objective functions; maximizing performance measures often involves a ratio of two functions; Nash equilibria in a dynamic game might not be unique and attain different values; a dynamic risk measure in a market with frictions is set-valued.
What unifies these examples is that one can formulate these problems with a set-valued value function that accounts for these different types of multi-variateness.
And it can be shown that this value function is recursive under mild assumptions. Thus, these problems are actually time-consistent in a set-valued sense. Practical implications and economic interpretations are discussed. Examples are given, where this set-valued Bellman's principle is implemented to numerically solve the problem.