Anscombe and Auman Expected Utility
AA approach appears to follow two stages: first, &quot;nature&quot; provides which event obtains, which results in the given lottery; second, the lottery is resolved, therefore revealing the intrinsic probabilities. AA suggests to resolve by backward induction. Under usual conditions for vNM utility, agent may define a function for her utility.
I found a nice paper comparing Savage vs. AA approaches, from which I got this construction, e.g. see Fig. 1 and 2. I hope it helps.
https://link.springer.com/article/10.1007/s11166-018-9273-7