Axioms of Utility in Stochastic Environments

We now elaborate our economic environment to include a stochastic relationship between an agent's choices and the resulting economic outcomes. Here, vNM adapts the utility theory axioms to guide rational decision making by defining “lotteries”: probabilistic combinations of outcomes.

Elements

1. 2.3.aCompleteness over Lotteries

   The ability to determine a preference between two lotteries $A$ and $B$. E.g., prefer $A$ over $B$, $B$ over $A$, or indifference.

2. 2.3.bTransitivity over Lotteries

   The ability to select among lotteries in a consistent manner. E.g., if $A$ is preferred over $B$, and $B$ over $C$, then $A$ should be preferred over $C$.

3. 2.3.cIndependence over Lotteries

   The ability to remain consistent in preferences between pairs of lotteries regardless of the presence of other alternatives. E.g., if $A$ is preferred to $B$, introducing a third lottery $C$ should not change this preference.