# Anscombe-Aumann Acts and Lotteries

Notation: Throughout I will let $\Delta X$ denote the set of probability distributions over the set $X$.

I have been studying expected utility theory, and especially Savage Acts and Anscombe-Aumann Acts. However I am new to it and am not sure if I have the terminology correct.

Let $S$ be the finite set of possible states, let $Z$ be the set of possible outcomes. An Anscombe-Aumann act is defined as a function $f:S \to \Delta Z$. Let the space of acts be denoted $X$. Preferences are defined over acts.

If preferences have a state independent expected utility representation (SIEU) we have that there is a function $u: Z \to \mathbb{R}$ and a distribution over states $p \in \Delta S$ such that for any two acts $f,g \in X$, $$f \precsim g \iff \sum_{s \in S} p(s) \sum_{z \in Z} u(z) f(s,z) \leq \sum_{s \in S} p(s) \sum_{z \in Z} u(z) g(s,z)$$

Suppose I want to define a lottery $L \in \Delta Z$ in this situation. For simplicity, suppose $L$ is the lottery that assigns probability $1$ to outcome $z^{*}$ and probability $0$ to all other outcomes.

My Question is: Is this lottery (or any lottery for that matter) a compound lottery first over states ($p$) and then over anscombe -aumann acts ($f$)? Or how is a lottery defined in this situation?

Let me know if any clarification is required.

Let's assume that $Z$ is finite, $Z=\{z_1,\cdots,z_n\}$ and that $L$ is the objective lottery that puts weight $l_i$ on the prize $z_i$, with $l_i \geq 0$ and $\sum_{i=1}^{n}{l_i}=1$.
You can identify the lottery $L$ with the act $f:s \rightarrow \Delta Z$ such that $f(s)=L$ for any $s \in S$. That is, for any $s \in S$, $f(s)$ is the objective lottery that puts weight $l_i$ on the prize $z_i$, exactly as $L$ does.