A new axiom for preference orderings over lotteries, called the projective independence axiom, is formulated. Given suitable continuity and monotonicity assumptions, the axiom implies that utility is either in the weighted utility class or is quadratic in probabilities. The betweeness axiom is...

First-order risk aversion happens when the risk premium a decision maker is willing to pay to avoid the lottery $t\cdot {\tilde \epsilon }, E[{\tilde \epsilon }]=0,$ is proportional, for small t, to t. Equivalently, $\partial \pi /\partial t\mid_{t=0^{+}} 0.$ We show that first-order risk...