A Model of Moderation: Finding Skiba Points on a Slippery Slope

A simple model is considered that rewards ”moderation” - finding the right balance between sliding down either of two ”slippery slopes”. Optimal solutions are computed as a function of two key parameters: (1) the cost of resisting the underlying uncontrolled dynamics and (2) the discount rate. Analytical expressions are derived for bifurcation lines separating regions where it is optimal to fight to stay balanced, to give in to the attraction of the ”left” or the ”right”, or to decide based on one’s initial state. The latter case includes situations both with and without so-called Dechert- Nishimura-Skiba (DNS) points defining optimal solution strategies. The model is unusual for having two DNS points in a one-state model, having a single DNS point that bifurcates into two DNS points, and for the ability to explicitly graph regions within which DNS points occur in the 2-D parameter space. The latter helps give intuition and insight concerning conditions under which these interesting points occur.