where γ is a positive parameter that affects the curvature and b is a positive parameter that sets the weight at p = 1 / 2 via ω (1 / 2) = b/ (1 + b ); b < 1 implies ω (1 / 2) < 1 / 2 reflecting an underweighting of an even chance. Empirical work in prospect theory typically finds b < 1 and γ < 1 implying pessimism and the overweighting of small probabilities and the underweighting of high ones. Figure S1b plots a typical decision weight form ω ( p ) as found in experiments, using b = 0 . 7 and γ = 0 . 6 . Also plotted is the weighting used in expected utility theory, ω = p , corresponding to b = 1 , γ = 1. Our binary gamble, where one bets on the outcome believed to be more likely (taken to be A here), implies that only decision weights ω ( p ) ≥ 1 / 2 are of concern. In this regime, ω is always lower than p until one reaches certainty so that ω has to increase at an accelerating rate in order to catch back up at p = 1. In other words, it has positive curvature ω ′′ > 0. To find the optimal wager, we maximize V ( w ) by setting V $^{′}$( w ) equal to zero obtaining which can be solved for the wager The risk aversion θ can now be expressed in terms of the probability p$_{max}$ at which an individual bets the maximum wager w$_{0}$ : The wager as a function of probability, w ( p ), is plotted in Figure S2. It is seen to be convex for both prospect and expected utility theories. Lowering the risk aversion u $^{′}$( w + w$_{0}$ ) u $^{′}$( w − w$_{0}$ ) = 1 − ω ( p ) ω ( p ) , w ( p ) = 1 2 θ log$^{(}$ ω ( p ) 1 − ω ( p )$^{)}$ . (S4) θ = 1 2 w$_{0}$ log$^{(}$ ω ( p$_{max}$ ) 1 − ω ( p$_{max}$ )$^{)}$ . (S5) S3