Affine utility on probability measures

An affine utility on probability measures functional satisfies $U_0(p\lambda +(1-p)\mu )=p{U}_0(\lambda )+(1-p)U_0(\mu )$ on mixtures of probability measures. Strict preferences are represented by comparing its real values, and indifference means equal values. This is cardinal utility on lotteries rather than merely an ordinal ordering of deterministic outcomes.

Set $U(x)=U_0({\delta }_x)$. On a finite measurable space, points in the same atom of a sigma-algebra give the same Dirac measure, so $U$ is a measurable function and constant on atoms. Every probability measure is a finite mixture of representative Dirac measures with its atom masses as weights. Affineness then proves the expected-utility representation, even when the sigma-algebra is smaller than the power set.

Two real affine functionals on a convex set of lotteries that represent the same strict preferences differ by $V=aU+b$ with $a>0$. Equal utility values must first be shown to give equal values of the other functional. The induced function on the real interval of utility values preserves mixtures; anchor interpolation, including exterior points represented through an anchor, makes it linear throughout. Constant representations are handled separately and still admit a positive slope.

For three strictly ordered affine-utility values $u>v>w$, the middle lottery is indifferent to the mixture of the extreme lotteries with weight $p=(v-w)/(u-w)$ on the best one. This weight lies strictly between zero and one. It is the exact mixture-indifference property underlying cardinal utility comparisons.

Mixing two lotteries with the same third lottery and the same positive weight preserves their strict preference order. For affine utility on probability measures, the difference of the two mixed utilities is the original difference multiplied by that weight. At weight zero the strict preference disappears, so positivity is essential.