We will present the key insights and ideas leading to a proof of Theorem 1. All the definitions and proofs can be found in the formalization. Since we use the setup introduced in the previous section all assumptions and notations carry over. In particular \(\succeq \) will denote the previously introduced relation \(\mathcal {R}\).
Theorem 1 is proved by showing two implications. Both directions can be found in the formalization. However, we will discuss the more difficult direction. That is, a preference relation satisfying (1), (3), and (4) admits expected utility representation.
The set of degenerate lotteries is finite, trivially there exists at least one most preferred element (with respect to \(\mathcal {R}\)). Moreover, we can prove Lemma 1.
Lemma 1
Every bestFootnote 4 degenerate lottery \(B_{deg}\) is at least as good as any other lottery in \(\mathcal {P}\).
$$\begin{aligned} \forall y \in \mathcal {P}. \; B_{deg}\; \succeq \; y \end{aligned}$$
The same can be shown for the worst (least preferred) elements. Thus proving that there exists at least one best \(\mathcal {B}\) and one worst \(\mathcal {W}\) element in \(\mathcal {P}\) such that
$$\begin{aligned} \forall x \in \mathcal {P}. \; \mathcal {B}\succeq x \wedge x \succeq \mathcal {W}. \end{aligned}$$
(1)
If \(\mathcal {B}\approx \mathcal {W}\) any constant function would represent the preference relation \(\mathcal {R}\), thus proving Theorem 1 for this special case. Hence, we will assume \(\mathcal {B}\succ \mathcal {W}\).
From the assumption of continuity and Property 1, we know that \(\forall p \in \mathcal {P}\),
$$\begin{aligned} \exists \alpha . \; \alpha \;\mathcal {B} + (1-\alpha )\;\mathcal {W} \approx p. \end{aligned}$$
Moreover, we can show that such an \(\alpha \) is unique. If it was not, we could create two distinct lotteries \(p = \alpha \;\mathcal {B} + (1-\alpha )\;\mathcal {W}\) and \(q=\beta \;\mathcal {B} + (1-\beta )\;\mathcal {W}\) with \(\alpha > \beta \) and \(p \approx q\). However, since \(\mathcal {B}\succ \mathcal {W}\) and p has a higher chance of the best outcome than q, we deduce \(p \succ q\), a contradiction. This shows that for all lotteries \(p \in \mathcal {P}\), there exists a unique calibration probability \(\alpha \), such that, \(\alpha \mathcal {B}+ (1-\alpha ) \mathcal {W}\approx p\).
The key idea is to define a function that assigns the unique calibration probability to every lottery in \(\mathcal {P}\). This is realised with the utility function util. Given a pmf p its unique calibration \(\alpha \) is obtained (using the indefinite choice operator SOME) and returned.
The next lemma shows that util indeed is a utility function as per Definition 2.
Lemma 2
For all p and q in \(\mathcal {P}\),
$$\begin{aligned} p \succeq q \iff {{\mathtt {util}}}(p) \ge {{\mathtt {util}}}(q) \end{aligned}$$
Lemma 2 is already an important result. However, since we are not only interested in general utility functions, but utility functions that adhere to expected utility form (Definition 5), we also need to prove the following Lemma.
Lemma 3
\({{\mathtt {util}}}\) is linear. That is, for all p, q in \(\mathcal {P}\),
$$ {{\mathtt {util}}}( \alpha \;p + (1-\alpha )\;q) = \alpha \;{{\mathtt {util}}}(p) + (1-\alpha )\;{{\mathtt {util}}}(q) $$
Proof
Outline. First, we generate two lotteries that have the same preference as p and q using util, \(\mathcal {B}\), and \(\mathcal {W}\). After substituting these generated lotteries in the left hand side of the equation, we can distribute \(\alpha \), rearrange the terms and apply the definition of util to derive the right hand side. For a detailed account of this lemma, we refer to the formalization. \(\square \)
One of the most prominent modern books on game theory [11] defines von-Neumann-Morgenstern utility functions simply as linear functions which util indeed is (Lemma 3). Since linearity is the defining property of expected utility functions Lemma 4 can be proven. Note, that util has the wrong type \('\alpha \ pmf \Rightarrow real\). Therefore, we simply define the Bernoulli utility function u with the following lambda abstraction \((\lambda x.\ {{\texttt {util}}}({{\texttt {return\_pmf}}}\ x))\) of type \('\alpha \Rightarrow real\).
Lemma 4
Given a \(p \in \mathcal {P}\)
$$\begin{aligned} \mathcal {U}(p) = \sum _{x \in \mathcal {O}} p(x) * u(x) \end{aligned}$$
This shows the existence of an expected utility function assuming (1), (3), and (4), thus proving one direction of Theorem 1.