Anscombe and Auman Expected Utility I would like to hopefully get some insights on the Anscombe and Aumann Expected utility.

I've read some proofs and understood the Expected Utility Theorem (VNM) which allows us to approach consumers preferences over not pre-determined lotteries or risky outcomes.

However, I don't manage to rationalize this way the Anscombe and Aumann expected utility, although I believe I understand the proof.

Particularly, I would like to know how I can rely heuristically the 'state space' and 'probability spaces' (other than being antecedent and image of the Aumann act).

Also, sometimes 'objective probabilities' are evoked as well as an explanation with horses, that I don't get.

Sorry for this confusing message and thanks a lot!
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