by lowering p$_{max}$ decreases the curvature while maintaining convexity. The convexity of w ( p ) implies the concavity of the subjective probability as a function of the wager, p ( w ), as required by RIA-based polarization. We can show that w ( p ) is convex in general for prospect theory in the regime of interest by considering the curvature directly as given by the second derivative w $^{′′}$( p ). The first derivative is and the second derivative is The terms in the numerator are positive because as noted above: in the first term, the curvature of the decision weight function ω $^{′′}$( p ) is positive; and for the second term ω ( p ) > 1 / 2. Therefore, w $^{′′}$( p ) > 0, which implies that the inverse function p ( w ) is concave, consistent with RIA-based polarization. S4 w $^{′}$( p ) = 1 2 θ ω ′ ω (1 − ω ) , (S6) w $^{′′}$( p ) = 1 2 θ ω $^{′′}$(1 − ω ) + ω ′ $^{2}$(2 ω − 1) ω $^{2}$(1 − ω ) 2 . (S7)