Piecewise additivity for non-expected utility

Abstract

A model of choice under purely subjective uncertainty, Piecewise Additive Choquet Expected utility, is introduced. PACE utility allows for optimism and pessimism simultaneously, but represents a minimal departure from expected utility. It can be seen as a continuous version of NEO-expected utility (Chateauneuf et al. in J Econ Theory 137:538–567, 2007) and, as such, is especially suited for applications with rich state spaces. The main theorem provides a preference foundation for PACE utility in the Savage framework of purely subjective uncertainty with an arbitrary outcome set.

1 Introduction

Economic life is rich with uncertainties, most of which look nothing like probabilistic risk. In the subjective expected utility theory of Savage (1954), the decision maker’s beliefs about uncertainty can be quantified probabilistically. All uncertainty reduces to risk. This theory also excludes probabilistic risk attitudes towards this subjective risk. Attitudes such as optimism and pessimism, for which there is considerable empirical support. This paper studies a simple modification of expected utility to allow for optimism and pessimism.

Cohen (1992) argued that the most important departures from expected utility can be explained by the security and potential factors. These are the worst and best possible outcomes of a decision (Lopes 1987). Axiomatic models have been developed by Gilboa (1988), Jaffray (1988), Cohen (1992), Essid (1997), Chateauneuf et al. (2007), and Schmidt and Zimper (2007). The NEO-expected utility (NEO-EU) model of Chateauneuf et al. (2007) evaluates choices using the formula:

$$\begin{aligned} \gamma \text{(worst } \text{ possible } \text{ utility) } + (1-\gamma -\delta )\text{(expected } \text{ utility) } + \delta \text{(best } \text{ possible } \text{ utility) }. \end{aligned}$$

NEO-EU is a special case of Choquet expected utility (Gilboa 1987; Schmeidler 1989; Wakker 1989) with a Non-Extremal-Outcome additive (NEO-additive) capacity. A NEO-additive capacity is a transformation of a probability measure that is linear for non-extreme probabilities, and departs from expected utility if and only if it is discontinuous at zero or one. As such, it retains subjective expected utility’s probabilistic sophistication property (Machina and Schmeidler 1992) whilst allowing for non-expected utility attitudes. Wakker (2001, 2005) gave model-free, behavioural definitions of pessimism and optimism using an uncertainty analogue of the common consequence effect (Allais 1953).¹ According to these definitions, a NEO-EU maximiser, with $\gamma$ and $\delta$ strictly positive, responds to the probabilistic beliefs by exhibiting both pessimism and optimism simultaneously, a condition called ambivalence or inverse-S behaviour. Ambivalence is the most prevalent uncertainty attitude observed in experiments (Wakker 2010: 203–243, 290–292).

NEO-EU has been applied extensively (Abdellaoui et al. 2010; Dominiak et al. 2012; Dominiak and Lefort 2013; Ford et al. 2013; Eichberger et al. 2012; Eichberger and Kelsey 2011, 2014; Ludwig and Zimper 2014; Romm 2014; Teitelbaum 2007; Zimper 2012). Its tractability stems from both: the probabilistic sophistication property, and the simple way such probabilities are used in the evaluation. Naively applied, it is not compatible with the ambiguity aversion examples of Ellsberg (1961). It has been argued, however, that source dependence of uncertainty attitudes is appropriate for explaining Ellsberg’s examples (Heath and Tversky 1991; Fox and Tversky 1995; Chow and Sarin 2001; Wakker 2001; Abdellaoui et al. 2011). For example, a British investor may respond more optimistically to his beliefs about the UK market than his beliefs about foreign markets, even if beliefs are probabilistic in both cases. By allowing for source dependence of uncertainty attitudes, probabilistic sophistication can be derived within each source of uncertainty, without requiring it to hold across each source (Chew and Sagi 2008). Allowing parameters of decision models to be source dependent adds further layers of difficulty for empirical applications. NEO-expected utility’s simple form, however, makes such an approach tractable, as demonstrated by Abdellaoui et al. (2011).

Chateauneuf et al. (2007) gave a preference foundation for NEO-EU in a purely subjective, Savage-style framework, in particular, using simple acts that map an arbitrary state space to a connected and separable outcome set. By deriving a continuous, cardinal utility on this rich outcome space, the subjective mixture techniques of Ghirardato et al. (2003) could be employed to give elegant preference axioms. In certain economic applications, most notably when outcomes are monetary, assuming rich topological structure on the outcome set is appropriate. In other applications, it is not natural to assume such a structure, for example, when the outcomes are: health outcomes, environmental outcomes, durable goods.

In this paper, the original Savage framework with an arbitrary outcome set is considered. The state space is at least countably infinite. There are several problems with NEO-EU in this framework. Because the probability transformation can be discontinuous, the axiomatic foundations are complicated. For risk, which is a special case of the Savage framework, Webb and Zank (2011) provided a preference foundation for NEO-expected utility. Complicated trade-off axioms, consistent optimism and consistent pessimism were used to measure and ensure consistency of the discontinuities, and additional structural assumptions were necessary to derive cardinal utility and unique parameters. A similar approach in the purely subjective setting would be no less complex. For arbitrary outcome sets, the NEO-EU parameters need not be uniquely determined (Webb and Zank 2011, 710, Example 8). For applications based on such a framework, it may be difficult to use NEO-EU to generate behavioural predictions related to comparative optimism or pessimism. Futhermore, applications of NEO-EU to frameworks with infinite states are problematic. Discontinuities in the evaluation formula mean that even simple applications are vulnerable to problems such as empty best responses, leading to non-existence of equilibria.²

To resolve both of the problems above, a continuous version of NEO-EU is developed in this paper. We start with NEO-EU on a finite state space, where the above-mentioned problems do not arise and consider the simplest extension to infinite states that has a continuous probability transformation function. The resulting theory is called Piecewise Additive Choquet Expected (PACE) utility. It will be shown that PACE utility admits a simple axiomatisation in the Savage framework and can be applied in cases where NEO-EU fails. If NEO-EU is the smallest departure from expected utility to allow inverse-S behaviour, then PACE utility is the smallest continuous departure from expected utility to allow inverse-S behaviour.

The remainder of the paper is structured as follows: Sect. 2 presents the theoretical background of this paper; Sect. 3 considers the problem of extending NEO-additive capacities to infinite state spaces and introduces PACE utility, and Sect. 4 studies the preference foundations in the Savage framework, with the paper’s main theorem delivering an axiomatic characterisation of those preferences that admit PACE utility representations.

2 Preliminaries

This section outlines the framework for choice under uncertainty and models for ambiguity. Let $\fancyscript{S}$ a set of states and $\fancyscript{E}$ be a $\sigma$-algebra of events. We allow for the case where $\fancyscript{E}$ is the power set of $\fancyscript{S}$. Some richness will later be imposed on $\fancyscript{S}$ via a solvability condition, which will imply that $\fancyscript{S}$ contains infinitely many states.³ Let $\fancyscript{X}$ be a set of outcomes. The set of outcomes can be finite or infinite. States and outcomes are the only primitives; from these all other definitions are derived.

An act is a function $f:\fancyscript{S}\rightarrow \fancyscript{X}$ that is measurable with respect to $\fancyscript{E}$. It is assumed that acts are simple, that is, they take only finitely many values. Acts will also be written $f=[A_1,f_2;\ldots ; A_n,f_n]$, denoting the act with outcome $f_i$ if the state belongs to event $A_i$. Acts are the objects of choice. By choosing act $f$, the decision maker receives outcome $f(s)$ if state $s$ obtains. The act results in outcome $x$ if the state belongs to $f^{-1}(x)\in \fancyscript{E}$. The set of acts is $\fancyscript{A}$. An act $f$ is constant if $f(s)=x$ for all $s \in \fancyscript{S}$ (we will write $f=x$). An act may be defined by its subacts. For $f,g \in \fancyscript{A}$ and $A \in \fancyscript{E}$, $f_A g := \{ h \in \fancyscript{A}: s\in A \Rightarrow h(s)=f(s), s\notin A \Rightarrow h(s)=g(s) \}$. An event $A\in \fancyscript{E}$ is null if $f_A h \sim g_A h$ for all $f,g,h\in \fancyscript{A}$, otherwise it is non-null.

A set function $\nu :\fancyscript{E}\rightarrow \mathbb {R}$ is normalised if $\nu (\emptyset ) = 0$ and $\nu (\fancyscript{S})=1$. It is monotonic if, for all $A,B\in \fancyscript{E}$, $A\subseteq B$ implies $\nu (A)\leqslant \nu (B)$. It is additive if, for all disjoint $A,B\in \fancyscript{E}$, $\nu (A\cup B) = \nu (A) + \nu (B)$. A capacity is a real-valued set function $\nu :\fancyscript{E}\rightarrow \mathbb {R}$ that is normalised and monotonic. A capacity $\nu :\fancyscript{E}\rightarrow [0,1]$ is convex-valued if for all $\alpha \in [0,1]$ there exists $A\in \fancyscript{E}$ such that $\nu (A)=\alpha$. A probability measure is an additive capacity. A capacity $\nu :\fancyscript{E}\rightarrow [0,1]$ is a probability transformation if there is a strictly increasing function $\phi :[0,1]\rightarrow [0,1]$ and a probability measure $p:\fancyscript{E}\rightarrow [0,1]$ such that $\nu =\phi \circ p$.

The decision maker exists only to maximise a preference relation $\succcurlyeq$ defined over $\fancyscript{A}$. A utility function $U$ over acts $\fancyscript{A}$ is a real-valued function that represents preferences such that $f\succcurlyeq g$ if and only if $U(f)\geqslant U(g)$. Preferences $\succcurlyeq$ over acts $\fancyscript{A}$ conform to subjective expected utility if they are represented by:

$$\begin{aligned} E(p,u)(f) = \sum _{x\in \fancyscript{X}} p(f^{-1}(x))u(x) \end{aligned}$$

where $p:\fancyscript{E}\rightarrow [0,1]$ is a probability measure and $u:\fancyscript{X}\rightarrow \mathbb {R}$ is a utility function for outcomes. Additivity of the probability measure ensures that, for all $x\in \fancyscript{X}$, the following holds:

$$\begin{aligned} p(f^{-1}(x))=p \left( \bigcup _{y\succcurlyeq x} f^{-1}(y)\right) - p \left( \bigcup _{y\succ x} f^{-1}(y)\right) . \end{aligned}$$

The subjective expected utility formula can therefore be written as:

$$\begin{aligned} E(p,u)(f) = \sum _{x\in \fancyscript{X}} \left[ p \left( \bigcup _{y\succcurlyeq x} f^{-1}(y)\right) - p \left( \bigcup _{y\succ x} f^{-1}(y)\right) \right] u(x). \end{aligned}$$

This exercise helps one clearly distinguish between subjective expected utility and the following model, Choquet expected utility (Schmeidler 1989; Wakker 1989), in which the probability measure $p$ in the above expression is replaced with $\nu$, a (possibly) non-additive capacity. Preferences $\succcurlyeq$ over acts $\fancyscript{A}$ conform to Choquet expected utility if they are represented by:

$$\begin{aligned} E(\nu ,u)(f) = \sum _{x\in \fancyscript{X}}\left[ \nu \left( \bigcup _{y\succcurlyeq x} f^{-1}(y)\right) -\nu \left( \bigcup _{y\succ x} f^{-1}(y)\right) \right] u(x) \end{aligned}$$

where $\nu :\fancyscript{E}\rightarrow [0,1]$ is a capacity and $u:\fancyscript{X}\rightarrow \mathbb {R}$ is a utility function for outcomes. Notice that we use the shorthand $E(\nu ,u)(f)$ for the Choquet expected utility of act $f\in \fancyscript{A}$ using capacity $\nu$ and utility for outcomes $u$.

Choquet expected utility allows for non-neutral attitudes to ambiguity. A capacity $\nu :\fancyscript{E}\rightarrow [0,1]$ is convex if $\nu (A\cup B)-\nu (B)$ is non-decreasing as $B$ $\supseteq$-increases. Similarly, a capacity is concave if $\nu (A\cup B)-\nu (B)$ is non-increasing as $B$ $\supseteq$-increases. Wakker (2001, 2005) gave behavioural definitions of pessimism and optimism and characterised such behaviour in context of Choquet expected utility. Under Wakker’s definitions, a Choquet expected utility maximiser is pessimistic if and only if the capacity is convex, and optimistic if and only if the capacity is concave. The most prevalent attitude found in experiments is ambivalence, a composition of pessimistic and optimistic responses to uncertainty. Because such capacities are initially concave and then convex, ambivalence is often called inverse-S behaviour.

A special case of Choquet expected utility, presented and axiomatised by Chateauneuf et al. (2007), is the NEO-additive capacities. These are discussed further in the next section. The Choquet expected utility of an act $f$ with respect to a NEO-additive capacity can be shown to be a convex combination of subjective expected utility, the utility of the act’s best outcome, $u(f^*)$, and utility of the act’s worst outcome, $u(f_*)$:

$$\begin{aligned} \hbox {NEO}(f) = \gamma u(f_*) + (1-\gamma -\delta )E(p,u)(f) + \delta u(f^*) \end{aligned}$$

with $\gamma ,\delta \geqslant 0$ and $\gamma + \delta < 1$. Here, $\gamma +\delta$ dictates the extent of the departure from expected utility; the degree of ambiguity. It is apparent from the representation that two acts with identical (or indifferent) best and worst outcomes will be ranked according to their expected utilities. Choquet expected utility using NEO-additive capacities is sometimes called NEO-expected utility (NEO-EU).

3 PACE utility

In this section, the problem of extending NEO-additive capacities to Savage’s infinite state space is addressed. Consider a NEO-additive capacity $\omega$ defined over a finite set of states $S\subset \fancyscript{S}$, $S=\{s_1,\ldots , s_n\}$. Let $\fancyscript{E}_S= \{\emptyset , A_1, \ldots , A_i, \ldots , \fancyscript{S} \}$ denote the set of $2^n$ events, formed by all subsets of $S$. Suppose, for the sake of presentational simplicity, that all non-empty events in $\fancyscript{E}_S$ occur with positive probability, it is known that such NEO-additive capacities have the following form:

$$\begin{aligned} \omega :&\fancyscript{E}_S\rightarrow [0,1], \\&\emptyset \mapsto 0, \\&A_1 \mapsto (1-\gamma -\delta )p(A_1) + \delta , \\&\vdots \\&A_i \mapsto (1-\gamma -\delta )p(A_i) + \delta , \\&\vdots \\&S \mapsto 1, \end{aligned}$$

where $\gamma ,\delta \geqslant 0$ and $\gamma +\delta < 1$.

The NEO-expected utility model is tractable for economic applications assuming finitely many states. Consider the following example, which will be used throughout this section:

Example 3.1

(A simple insurance model with finite states) An agent with monetary wealth $w$ faces a loss of $l$ with probability $(1-p)$. Full insurance is available at premium $z$, which is actuarially fair, $w-z = pw + (1-p)(w-l)$. The agent chooses the probability $q$ that she receives full insurance.⁴ Suppose there are finitely many options: $0 = q_1 < \cdots < q_{n-1} < q_n =1$, the problem is choosing $q$ to make the lottery $(q, w-z; (1-q)p, w; (1-q)(1-p), w-l)$ as preferable as possible. The agent is a CEU maximiser with a strictly increasing and strictly concave utility for money $u$ and probability transformation $\omega =\phi \circ p$. If $\phi$ is the identity (expected utility), then $u(w-z)> pu(w)+(1-p)u(w-l):= e$, and full insurance is strictly preferred to no insurance. Maximising $qu(w-z) + (1-q)e$ yields $q = q_n = 1$ as the unique solution. If $\phi$ is NEO-additive then, for $q\in (0,1)$, the utility is given by $\gamma u(w-l) + (1-\gamma -\delta )[qu(w-z) + (1-q)e] + \delta u(w)$, which is strictly increasing in $q$. It is possible, however, that $q=1$ is not optimal. Utility for $q\in (0,1)$ is bounded below by $\gamma u(w-l) + (1-\gamma -\delta )e + \delta u(w)$ which is greater than $u(w-z)$ for $\delta$ sufficiently close to one. In that case, when the agent has a high degree of optimism, optimality occurs where $q=q_{n-1}$.

An extension of $\omega$ to the infinite state space $\fancyscript{S}$, is a capacity $\nu :\fancyscript{E}\rightarrow [0,1]$ that coincides with $\omega$ wherever $\omega$ is defined, $\nu |_{\fancyscript{E}_S} = \omega$. One extension of a NEO-additive capacity to consider is the probability transformation such that, for all $A\in \fancyscript{E}$:

$$\begin{aligned} \nu (A) = \left\{ \begin{array}{ll} 0 &{} \text{ if } p(A) = 0, \\ (1-\gamma -\delta ) p(A) + \delta &{} \text{ if } 0< p(A) < 1, \\ 1 &{} \text{ if } p(A)= 1, \end{array} \right. \end{aligned}$$

with $\gamma ,\delta \geqslant 0$ and $\gamma + \delta < 1$. That is, $\nu =\phi \circ p$ with the transformation $\phi$ that is strictly increasing everywhere, linear for all probabilities between zero and one, but possibly discontinuous at zero and/or at one. The probability transformation above is certainly the most obvious extension of a NEO-additive capacity to infinite states. Indeed, we call such $\phi$ a NEO-additive transformation function. There are, however, some problems with the NEO-expected utility model that results. In particular, the discontinuity present in this transformation function presents difficulties for even simple applications. In the following example, there is a continuum (a compact and connected set) of states, which is typical of many economic applications:

Example 3.2

(A simple insurance model with a continuum of states) Consider the model of example above, except the problem now involves choosing $q\in [0,1]$. If $\phi$ is the identity, maximising $qu(w-z) + (1-q)e$ yields $q = 1$ as the unique solution. If $\phi$ is continuous, the problem has at least one solution. If $\phi$ is NEO-additive then, utility is strictly increasing for $q\in (0,1)$. If $\delta$ sufficiently close to one, $q=1$ is not optimal. In that case there is no well-defined solution to the agent’s problem.

These issues do not arise for probability transformations that are continuous. Therefore, we now seek the simplest extension of a NEO-additive capacity that has a continuous transformation function. Recall that the NEO-additive transformation function above linearly transforms all non-extreme probabilities:

$$\begin{aligned} A\mapsto (1-\gamma -\delta ) p(A) + \delta \quad \text{ if } 0< p(A) < 1. \end{aligned}$$

Because of this, a NEO-additive transformation function $\phi$ is continuous if and only if it is the identity. Consider the following, minor weakening of the above requirement. Let $\kappa \in [\frac{1}{2},1]$ and consider a capacity $\nu =\phi \circ p$ that satisfies:

$$\begin{aligned} A\mapsto (1-\gamma -\delta ) p(A) + \delta \quad \text{ if } 1-\kappa < p(A) < \kappa . \end{aligned}$$

By taking $\kappa$ close to one, this capacity is empirically indistinguishable from a NEO-additive capacity. Under this weaker requirement, however, continuity of $\phi$ can be retained. To do so, we must specify $\phi$ on $[0,1-\kappa ]$ and $[\kappa ,1]$. The simplest assumption, and therefore most in keeping with the NEO-EU spirit, is that $\phi$ is also strictly increasing and linear on these intervals. The only capacity to achieve all of this is the capacity $\nu$ such that, for all $A\in \fancyscript{E}$:

$$\begin{aligned} \nu (A) = \left\{ \begin{array}{ll} (1-\gamma - \frac{\delta \kappa }{1-\kappa }) p(A) &{} \text{ if } p(A) \leqslant 1-\kappa \\ (1-\gamma -\delta ) p(A) + \delta &{} \text{ if } 1-\kappa \leqslant p(A) \leqslant \kappa \\ \frac{1-[(1-\gamma -\delta )\kappa +\delta ]}{1-\kappa }p(A) + \frac{(1-\gamma -\delta )\kappa + \delta -\kappa }{1-\kappa } &{} \text{ if } \kappa \leqslant p(A) \end{array} \right. \end{aligned}$$

with $p:\fancyscript{E}\rightarrow [0,1]$ a probability measure, $\kappa \in [\frac{1}{2},1]$, $\kappa \leqslant \frac{1-\gamma }{1-\gamma -\delta }$ and $\kappa \leqslant \frac{1-\delta }{1-\gamma -\delta }$ and $\gamma +\delta < 1$. That is, $\nu =\phi \circ p$ with the transformation $\phi$ being continuous and strictly increasing everywhere, and linear on $[0,1-\kappa ]$, $[1-\kappa ,\kappa ]$ and $[\kappa ,1]$. We call such capacities piecewise additive. Notice that $\gamma ,\delta \geqslant 0$ is not required, hence departures from additivity can, but need not, be of the inverse-S variety. Hence, piecewise additive capacities may be considered continuous versions of the generalised NEO-additive capacities of Eichberger et al. (2012).

Definition 3.3

(Piecewise Additive Choquet Expected (PACE) utility) PACE utility holds if preferences are represented by Choquet expected utility with a piecewise additive capacity.

The PACE utility model can be thought of as “expected utility with kinks”, hence the $\kappa$. In applications, it is easier to deal with “kinks” than to deal with “jumps”. In the insurance example developed above, the decision maker’s best response will always be non-empty if $\phi$ is continuous. The problem encountered with NEO-EU in Example 2 cannot arise under PACE utility. Also, PACE utility remains tractable enough to obtain a closed-form solution:

Example 3.4

(A simple insurance model with a continuum of states) Suppose, in the example above, that the agent chooses $q\in [0,1]$ and her capacity is piecewise additive. Utility in this case varies continuously with $q\in [0,1]$, hence the problem has a well-defined solution. It is possible for the solution to differ from the expected utility case. To see this, let $\kappa > p$, and let $\delta$ be close to one. Then, utility increases with $q$ on $[0,\frac{\kappa -p}{1-p}]$ and decreases with $q$ on $[\frac{\kappa -p}{1-p},1]$. Optimality occurs where $q = \frac{\kappa -p}{1-p}$. At this $q$, the probability of getting at least $w-z$ is $\kappa$, and the agent is more sensitive to the unlikely, best outcome $w$. This agent prefers full insurance to no insurance, but most prefers to gamble on being insured.

A NEO-EU maximiser stratifies events into “impossible”, “uncertain” and “certain”, being probability zero, probability in $(0,1)$, and probability one respectively, and behaves as an expected utility maximiser within each class, but not across the classes. PACE utility is based on a similar trichotomy, buts allows the classes to be subjective. Events are now stratified into “unlikely”, “moderate” and “likely”, corresponding to probability “low enough”, probability “not too low or too high”, and probability “high enough”, respectively. To operationalise this idea, we used a personal parameter, $\kappa \in [\frac{1}{2},1]$, such that “unlikely”, “moderate” and “likely” correspond to probability not greater than $1-\kappa$, in $[1-\kappa ,\kappa ]$, and not less than $\kappa$, respectively. For example, if $\kappa = 2/3$, then events occurring with probability less than $1/3$ are dubbed “unlikely” and events occurring with probability 2/3 or greater are dubbed “likely”.⁵ PACE utility reduces to expected utility within the unlikely, moderate, and likely classes of events. It permits departures from expected utility when comparing events from different classes.

The main theorem of this paper, Theorem 4.6, presents a preference foundation for PACE utility under purely subjective uncertainty. The axiomatic foundations of Choquet expected utility with NEO-additive capacities are complicated in the rich state space, arbitrary outcome set framework (Webb and Zank 2011). PACE utility will be derived here, however, from a simple weakening of expected utility’s sure-thing principle.

4 A preference foundation

Here we recall the axioms for subjective expected utility. We assume there are at least three outcomes $x,y,z\in \fancyscript{X}$ such that $x\succ y \succ z$.

Axiom 1

(Ordering) Preferences $\succcurlyeq$ over acts $\fancyscript{A}$ are a weak order.

Axiom 2

(Monotonicity) For acts $f,g\in \fancyscript{A}$, $f(s)\succcurlyeq g(s)$ for all $s\in \fancyscript{S}$ implies $f\succcurlyeq g$.

An Archimedean axiom is required. Here, a rank-dependent axiom is used. A simpler axiom could be used at this point. But rank dependence is required later. The benefit of using a slightly more complicated Archimedean axiom is that only one such axiom will be used throughout the paper. For an act $f$ and event $A$, the event $B$ dominates $A$ under $f$ if, for all $\tilde{s}\in B$ and $s\in A$, $f(\tilde{s})\succcurlyeq f(s)$. The rank of an event $A$ under $f$, denoted $\mathcal {R}(A,f)$, is the largest event that dominates $A$ under $f$. Hence, $f(\tilde{s})\succcurlyeq f(s)$ holds for all $\tilde{s}\in \mathcal {R}(A,f)$ and $s\in A$ holds, and also $f(\tilde{s})\preccurlyeq f(s)$ holds for all $\tilde{s}\notin \mathcal {R}(A,f)$ and $s\in A$ because $\mathcal {R}(A,f)$ is the $\supseteq$-maximal dominating event. For a simple act, $f=[A_1,f_1; \ldots ; A_n, f_n]$, we may label the outcomes of $f$ so that $f_1 \prec \cdots \prec f_n$. Then, the rank of an event $A_i$ under $f$ is given by $\mathcal {R}(A_i, f) = \cup _{j=i}^n A_j$.

Axiom 3

(Archimedeanity) If $S = A^1, A^2, \ldots ,$ is a sequence of non-null events such that:

$$\begin{aligned} x_{A^1} f \succ x_{A^1} g \;\; \text{ and }\;\; x_{A^i} f \sim x_{A^{i+1}} g \end{aligned}$$

with $\mathcal {R}(A^i, x_{A^i} f) = \mathcal {R}(A^i, x_{A^i} g)$, for all $i=1,2,\ldots ,$ then $S$ is finite.

The following condition, solvability, relates to the structure of $\fancyscript{E}$, so is separated from the preference conditions above. Solvability holds if, for all acts $f,g,h\in \fancyscript{A}$ with $f\succ g \succ h$, there is an event $A$ such that $g \sim f_A h$.

Let $\fancyscript{B}$ denote the set of binary acts, acts taking at most two values. Given $x\in \fancyscript{X}$, $A\in \fancyscript{E}$ and $f\in \fancyscript{B}$, the act $x_A f$ takes at most three values. The following axiom concerns consistency of a likelihood order revealed using such acts:

Axiom 4

(Comparative likelihood consistency) For all $f,g,\tilde{f},\tilde{g}\in \fancyscript{B}$, all $A,B\in \fancyscript{E}$, $x,y,\tilde{x},\tilde{y}\in \fancyscript{X}$ with $x\prec y$ and $\tilde{x}\prec \tilde{y}$, the implication:

$$ \begin{aligned} x_A f \sim x_B g, \;\; \tilde{x}_A \tilde{f} \sim \tilde{x}_B \tilde{g}, \;\; \& \;\; y_A f \succcurlyeq y_B g \;\; \Rightarrow \;\; \tilde{y}_A \tilde{f} \succcurlyeq \tilde{y}_B \tilde{g}, \end{aligned}$$

holds if $\mathcal {R}(A,j)$ and $\mathcal {R}(B,k)$ are constant, where $j = x_Af, \tilde{x}_A\tilde{f}, y_A f, \tilde{y}_A\tilde{f}$ and $k= x_B g, \tilde{x}_B \tilde{g}, y_B g, \tilde{y}_B \tilde{g}$.

Axiom 4 implies Savage’s axiom P4.⁶ Axiom 4 is the well-known P2* axiom of Gilboa (1987), restricted to ranked, three-outcome acts.⁷ Abdellaoui and Wakker (2005) provides an extensive treatment of such axioms. Define an order $\succ$ over events, written $A\succ ^L B$ and read, “$A$ is, subjectively, more likely than $B$,” whenever $x_A y \succ x_B y$ for some (for all, by axiom 4) $x,y\in \fancyscript{X}$ with $x\succ y$. Definition 2.1 of Abdellaoui and Wakker (2005) referred to the same condition as revealed more likely in a basic sense. Define $\prec ^L$, $\succcurlyeq ^L$, $\preccurlyeq ^L$, and $\sim ^L$ in the usual way. It can be shown that $\succcurlyeq ^L$ is a well-defined weak order over $\fancyscript{E}$.

Subjective expected utility preferences and PACE utility preferences both satisfy axioms 1–4. The following axiom, the sure-thing principle, when combined with axioms 1–4, characterises subjective expected utility:

Axiom 5

(The sure-thing principle) For all events $A\in \fancyscript{E}$ and acts $f,\tilde{f},g,h \in \fancyscript{A}$, the following implication holds: $f_A g \succcurlyeq f_A h \;\Rightarrow \; \tilde{f}_A g \succcurlyeq \tilde{f}_A h$.

The sure-thing principle is will be suitably modified to account for the type of uncertainty attitudes permitted under PACE utility. It is useful to first consider the how PACE utility compares with NEO-EU. Under NEO-EU, a decision maker will conform to expected utility when comparing acts with common best and worst outcomes. Indeed, even if acts do not have common best and worst outcomes, a sure-thing principle holds whenever common outcomes are changed in a way that leaves best and worst outcomes unaffected. For an act $f\in \fancyscript{A}$, let $b(f)$ and $w(f)$ denote the ranks of the best and worst outcomes of $f$, respectively. Of course, $w(f)\sim ^L \fancyscript{S}$. Then, NEO-EU necessarily satisfies the following condition:

Definition 4.1

(The NEO-sure-thing principle) For all events $A\in \fancyscript{E}$ and acts $f,\tilde{f},g,h \in \fancyscript{A}$, the implication:

$$\begin{aligned} f_A g \succcurlyeq f_A h \;\;\Rightarrow \;\; \tilde{f}_A g\succcurlyeq \tilde{f}_A h \end{aligned}$$

holds if $w(j) \succ ^L \mathcal {R}(A,j) \succ ^L b(j)$, for all $j=f_A g , f_A h , \tilde{f}_A g, \tilde{f}_A h$.

The NEO-sure-thing principle is necessary but, when combined with axioms 1–4, it is not sufficient for NEO-EU. See example 10 of Webb and Zank (2011:711). To pin down NEO-EU, Webb and Zank (2011) employed further axioms, consistent optimism and consistent pessimism. The derivation of PACE utility, however, will require only a weakening of the sure-thing principle.

Consider a PACE utility representation, and let $K$ be an event with $p(K)=\kappa$. The key properties of PACE utility will be sure-thing principles, the above implication, that hold whenever:

• 1. Events are ranked as likely: $\mathcal {R}(A,j) \succcurlyeq ^L K$ for all $j$.

• 2. Events are ranked as unlikely: $K^{\complement } \succcurlyeq ^L \mathcal {R}(A,j)$ for all $j$.

• 3. Events are ranked as moderate: $K \succcurlyeq ^L \mathcal {R}(A,j) \succcurlyeq ^L K^{\complement }$ for all $j$.

In each case above, $j=f_A g , f_A h , \tilde{f}_A g, \tilde{f}_A h$. Sure-thing principles based on the above conditions are necessary for PACE utility. They are not, however, falsifiable axioms because they assume a priori knowledge of an event $K$ with the required properties. A preference axiom, based on falsifiable conditions, is now developed that will imply both the existence of such a $K$ and the corresponding behaviour within in each class of events. This axiom will be called the piecewise sure-thing principle. Before stating the piecewise sure-thing principle, it is necessary to formulate various local versions of the sure-thing principle. For a given event, $A\in \fancyscript{E}$, we define upper, lower, outer and inner sure-thing principles that hold “at $A$”.

Definition 4.2

(Upper sure-thing principle at $A$ ) For $A\in \fancyscript{E}$, the implication:

$$\begin{aligned} f_B g \succcurlyeq f_B h \;\;\Rightarrow \;\; \tilde{f}_B g\succcurlyeq \tilde{f}_B h \end{aligned}$$

holds if $\mathcal {R}(B,j) \succcurlyeq ^L A$, for all $j=f_B g , f_B h , \tilde{f}_B g, \tilde{f}_B h$.

The upper sure-thing principle at $A$ implies the sure-thing principle holds for outcomes ranked likelier than $A$. This is a simple and testable condition. Conforming to the standard sure-thing principle is often seen as normatively desirable. One might appeal to a weaker criterion such as this, when violations of the sure-thing principle are permitted. The upper sure-thing principle at $K$ necessarily holds under PACE utility.

Definition 4.3

(Lower sure-thing principle at $A$ ) For $A\in \fancyscript{E}$, the implication:

$$\begin{aligned} f_B g \succcurlyeq f_B h \;\;\Rightarrow \;\; \tilde{f}_B g\succcurlyeq \tilde{f}_B h \end{aligned}$$

holds if $A \succcurlyeq ^L \mathcal {R}(B,j)$, for all $j=f_B g , f_B h , \tilde{f}_B g, \tilde{f}_B h$.

The lower sure-thing principle at $A$ implies the sure-thing principle holds for outcomes ranked less likely than $A$. This seems to carry the same normative content as the upper sure-thing principle. Under PACE utility, events are considered unlikely only if they are no more likely than $K^{\complement }$. Then, the lower sure-thing principle at $K^{\complement }$ necessarily holds under PACE utility. Consider an event $A$, with $A$ likelier than its complement $A^{\complement }$. Given their apparently equivalent normative status, if one conforms to the upper sure-thing principle at $A$, then conforming to the lower sure-thing principle at $A^{\complement }$ is reasonable. We call this the outer sure-thing principle:

Definition 4.4

(Outer sure-thing principle at $A$) The upper sure-thing principle at $A$ and the lower sure-thing principle at $A^{\complement }$ both hold, or the upper sure-thing principle at $A^{\complement }$ and the lower sure-thing principle at $A$ both hold.

Under PACE utility, preferences must satisfy the outer sure-thing principle at $K$. That is, the sure-thing principle holds for likely outcomes and unlikely outcomes. The third class, moderate likelihood, is covered by the following condition:

Definition 4.5

(Inner sure-thing principle at $A$ ) For $A\in \fancyscript{E}$, with $A\succcurlyeq ^L A^{\complement }$, the implication:

$$\begin{aligned} f_B g \succcurlyeq f_B h \;\;\Rightarrow \;\; \tilde{f}_B g\succcurlyeq \tilde{f}_B h \end{aligned}$$

holds if $A \succcurlyeq ^L \mathcal {R}(B,j) \succcurlyeq ^L A^{\complement }$, for all $j=f_B g , f_B h , \tilde{f}_B g, \tilde{f}_B h$.

The inner sure-thing principle applies to events with ranks of moderate likelihood. Under PACE utility, the inner sure-thing principle at $K$ necessarily holds, where $K$ and $K^{\complement }$ are labelled so that $K\succcurlyeq ^L K^{\complement }$. In probability terms, this amounts to expected utility when the outcomes are ranked with probability in the “inner” interval $[1-\kappa , \kappa ]$. The outer sure-thing principle refers to the “outer” intervals $[0,1-\kappa ]$ and $[\kappa ,1]$.

The key axiom for PACE utility can now be stated:

Axiom 5 $\kappa$ (The Piecewise Sure-Thing Principle): For all $A\in \fancyscript{E}$, at least one of the inner sure-thing principle at $A$ or the outer sure-thing principle at $A$ holds.

The piecewise sure-thing principle takes the content of the above conditions, then adds a simplifying assumption. It forces every event to fall into at least one of three categories: a set of events where the upper sure-thing principle holds, a set of events where the lower sure-thing principle holds, or a set of events where the inner sure-thing principle holds. The following theorem characterises PACE utility:

Theorem 4.6

Let solvability hold. Then, the following statements are equivalent:

• 1. The preference relation $\succcurlyeq$ satisfies axioms 1, 2, 3, 4 and 5$\kappa$ (weak order, monotonicity, Archimedeanity, comparative likelihood consistency, and the piecewise sure-thing principle).

• 2. There exists a convex-valued, piecewise additive capacity $\nu _{\kappa }$ over $\fancyscript{E}$ and a real-valued, strictly $\succcurlyeq$-increasing utility function $u$ over outcomes $\fancyscript{X}$ such that $\succcurlyeq$ is represented by $E(\nu _{\kappa },u)$. That is, PACE utility holds.

In statement 2, the capacity is unique and utility is cardinal.

5 Closing comments

This paper has presented a simple way of integrating optimism and pessimism into subjective expected utility. To get PACE utility, ‘kinks’ were incorporated into expected utility. In rough terms, approximating ‘inverse-S’ with a ‘Z’. An intuitive weakening of the sure-thing principle called the piecewise sure-thing principle was introduced. Theorem 4.6 provided a behavioural foundation for PACE utility.

Chateauneuf et al. (2007) have shown how NEO-expected utility can be applied to resolve well-known phenomena that are difficult to reconcile with expected utility, such as the coexistence of gambling and insurance and the equity premium puzzle. Since, for finite state spaces, NEO-expected utility is obtained as a special case, PACE utility can generate the same results. In applications with rich state spaces, PACE utility is a more tractable alternative.

Appendix

1.1 General results on additive separability

The proof of Theorem 4.6 uses the general results of Wakker (1991) so the main result of that paper is repeated here. We restrict attention to the case of three or more coordinates here, although Wakker (1991) allows for two coordinates.

Let $\fancyscript{Y}$ be a non-empty set of outcomes, and $\succcurlyeq '$ a weak order on $\fancyscript{Y}$. Any $n$-tuple $\mathbf {y} = (y_1,\ldots ,y_n) \in \fancyscript{Y}^n$ is called a rank-ordered alternative if $y_1 \succcurlyeq ' \cdots \succcurlyeq ' y_n$. The set of rank-ordered alternatives is $\fancyscript{Y}_{\succcurlyeq '}^{n}$. An outcome $y$ is minimal if $y\succ ' x$ for no $x\in \fancyscript{Y}$ and is maximal if $x\succ ' y$ for no $x\in \fancyscript{Y}$. An extreme alternative $\mathbf {y}$ has its first (so every) coordinate a minimal outcome, or its last (so every) coordinate a maximal outcome.

A weak order $\succcurlyeq$ is defined on $Y\subseteq \fancyscript{Y}_{\succcurlyeq '}^{n}$, a subset of the rank-ordered alternatives. A constant alternative $(\alpha ,\ldots ,\alpha )$ is identified with the outcome $\alpha \in \fancyscript{Y}$. It is therefore ensured that $\succcurlyeq$ and $\succcurlyeq '$ are in agreement in the following sense: $(\alpha ,\ldots ,\alpha ) \succ (\beta ,\ldots ,\beta ) \Rightarrow \alpha \succ ' \beta$, and $(\alpha ,\ldots ,\alpha ) \sim (\beta ,\ldots ,\beta ) \Rightarrow \alpha \sim ' \beta$.

We write $\alpha _i \mathbf {y}$ for the alternative $\mathbf {y} \in Y$ with coordinate $y_i$ replaced by $\alpha$. Replacing more than one coordinate of $\mathbf {y}$, the notation $\alpha _i\beta _j \mathbf {y}$ is clear. Replacing a set of coordinates $I\subseteq \{1,\ldots ,n\}$ write $\alpha _I \mathbf {y}$. Solvability holds if, for all alternatives $\alpha _i \mathbf {z}, \mathbf {y}, \gamma _i \mathbf {z} \in \fancyscript{Y}$ with $\alpha _i\mathbf {z} \succ \mathbf {y} \succ \gamma _i\mathbf {z}$ there is an outcome $\beta$ such that $\beta _i\mathbf {z} \sim \mathbf {y}$. An order is monotonic if the following equivalence holds: $\alpha \succcurlyeq \beta \Leftrightarrow \alpha _i \mathbf {y} \succcurlyeq \beta _i \mathbf {y}$ for all $\alpha _i \mathbf {y}, \beta _i \mathbf {y} \in Y$. The order is coordinate independent if the following holds: $\alpha _i \mathbf {y} \succcurlyeq \alpha _i \mathbf {z} \Leftrightarrow \beta _i \mathbf {y} \succcurlyeq \beta _i \mathbf {z}$ when all the alternatives are in $Y$.

Call a sequence $\alpha ^1,\alpha ^2,\ldots$ a standard sequence if, for some $j\in \{1,\ldots ,n\}$, $\alpha ^1_j \mathbf {y} \succ \alpha ^1_j \mathbf {z}$ and $\alpha ^k_j \mathbf {y} \sim \alpha ^{k+1}_j \mathbf {z}$ for $k=1,2,\ldots$. Standard sequences could be finite or infinite. A standard sequence $\alpha ^1,\alpha ^2,\ldots$ is bounded if for all $k$ there exist outcomes $\beta ,\gamma$ such that $(\beta ,\ldots ,\beta ) \succcurlyeq (\alpha ^k,\ldots ,\alpha ^k)\succcurlyeq (\gamma ,\ldots ,\gamma )$ and there are alternatives $\mathbf {y},\mathbf {z}\in Y$ with $y_i=\beta$ and $z_i=\gamma$. An order is Archimedean if every bounded standard sequence is finite.

The extended reals is $\overline{\mathbb {R}} := \mathbb {R}\cup \{-\infty ,\infty \}$. A preference relation $\succcurlyeq$ on $Y$ has an extended additive representation if: there are functions $V_1,\ldots , V_n$ so that $\sum _{j=1}^{n} V_j: Y \rightarrow \overline{\mathbb {R}}$ represents $\succcurlyeq$ on $Y$. To characterise those orders for which an extended additive representation exists, Wakker (1991) offered the following theorem:

Theorem 6.1

(Wakker 1991) Let $\fancyscript{Y}$ be a non-empty set, and $\succcurlyeq$ a preference relation on a set $Y\subseteq \fancyscript{Y}^n_{r}$ ($n\geqslant 3$) of rank-ordered alternatives, where rank-ordering is with respect to the weak order $\succcurlyeq '$ on $\fancyscript{Y}$ agreeing with the binary relation $\succcurlyeq$ on $Y$ restricted to constant alternatives. Let $\succcurlyeq$ satisfy solvability on $Y$. The following two statements are equivalent:

• 1. The preference relation $\succcurlyeq$ on $Y$ is an Archimedean, monotonic, weak order satisfying coordinate independence.

• 2. The preference relation $\succcurlyeq$ on $Y$ is represented by an extended additive function that is real valued for all non-extreme alternatives.

  $$\begin{aligned} \mathbf {y} \succcurlyeq \mathbf {z} \quad \Leftrightarrow \quad \sum _{i=1}^{n}V_i(y_i) \geqslant \sum _{i=1}^{n}V_i(z_i) \end{aligned}$$

Further, the additive representation is an interval scale on $Y\setminus \{extremes\}$ unless $\succcurlyeq$ on $Y$ has exactly two equivalence classes and $n>3$.

1.2 Proof of Theorem 4.6

If preferences are represented by PACE utility, with a convex-valued capacity, one can verify weak order, the piecewise sure-thing principle, monotonicity, comparative likelihood consistency and Archimedeanity by substitution of the preference functional. Hence, we assume statement 1 of the theorem and derive statement 2. For finite $X\subseteq \fancyscript{X}$, the set of acts with outcomes only in $X$ is written $\fancyscript{A}_X$. Enumerate the outcomes in $X$ to be increasing in terms of preference: $X=\{x_0,\ldots ,x_n\}$ with $x_0 \prec \cdots \prec x_n$. For $f\in \fancyscript{A}_X$ the decumulative representation, written $F$ as follows:

$$\begin{aligned} F=(F_1,\ldots ,F_n) \quad \text{ where } F_i = \bigcup _{x_j \succcurlyeq x_i }f^{-1}(x_j) \end{aligned}$$

so each $F_i$ is the union of all events for which the act $f$ yields $x_i$ or any other preferred outcome. Clearly, $F_n=f^{-1}(x_n)$ for all $f\in \fancyscript{A}$ as no outcome in $X$ is preferred to $x_n$. Also, one would find $F_0 = \fancyscript{S}$ for all acts. Therefore, we drop the $F_0$ from the notation and write $F=(F_1,\ldots ,F_n)$. No relevant information is lost when transforming an act to its decumulative representation, that is, $F=G$ iff $f=g$. The set of all decumulative acts is $\fancyscript{A}^*$. The preference relation $\succcurlyeq$ over $\fancyscript{A}$ naturally induces a preference relation over $\fancyscript{A}^*$, that we also write $\succcurlyeq$. As we proceed we will refer to elements of $\fancyscript{A}^*_X$ as $F,F',G,G'$ and so on without referring to the underlying acts $f,f',g,g'$ unless necessary.

Comparative likelihood consistency, axiom 4, affects the decision maker’s preferences for acts with, at most, three possible outcomes. For $X$ with $|X|=3$, preferences over $\fancyscript{A}_X$ satisfy the axioms of Theorem 4.1.4 (and Corollary 4.1.4) of Gilboa (1987). Hence, preferences admit a Choquet expected utility representation on $\fancyscript{A}_X$, with utility $u_X$ and convex-valued capacity $\nu _X$. The utility is cardinal and the capacity is unique.

Let $\fancyscript{L}$ denote the set of events $A$ for which $\succcurlyeq ^L$ satisfies the lower sure-thing principle at $A$, let $\fancyscript{U}$ denote the set of events $A$ for which $\succcurlyeq ^L$ satisfies the upper sure-thing principle at $A$, and let $\fancyscript{M}$ denote the set of events $A$ for which $\succcurlyeq ^L$ satisfies the inner sure-thing principle at $A$. By the piecewise sure-thing principle, every event belongs to at least one of $\fancyscript{U}, \fancyscript{M}$ or $\fancyscript{L}$.

An order satisfies $\fancyscript{Z}$-cancellation if the following holds:

$$\begin{aligned}&[A, B, A\cup C, B\cup C \in \fancyscript{Z}, A\cap C = B\cap C = \emptyset , A_i F \succcurlyeq B_i G] \\&\quad \Rightarrow (A\cup C)_i F\succcurlyeq (B\cup C)_i G \end{aligned}$$

An order satisfies $\fancyscript{K}$-cancellation if it satisfies $\fancyscript{Z}$-cancellation for $\fancyscript{Z} \in \{ \fancyscript{U}, \fancyscript{M},\fancyscript{L} \}$.

Lemma 6.2

Preferences $\succcurlyeq$ over $\fancyscript{A}^*$ satisfy $\fancyscript{K}$-cancellation.

Proof

The details for $\fancyscript{U}$-cancellation are given, the other conditions follow similar reasoning. Let $i\geqslant 1$. Let $h$ and $\tilde{h}$ be the acts associated with decumulative acts $A_i F$ and $B_i G$. Define $E:= h^{-1}(x_{i-1})$ and $\tilde{E} := \tilde{h}^{-1}(x_{i-1})$. Suppose that, for $C\in \fancyscript{E}$ with $A\cap C = B \cap C = \emptyset$, the decumulative acts $(A \cup C)_i F$ and $(B \cup C)_i G$ are well defined. Then it must be⁸ that $E \supseteq C$ and $\tilde{E}\supseteq C$. So, both acts coincide on the event $C$ (with outcome $x_{i-1}$). By assumption, $\mathcal {R}( h^{-1}(x_i), h ) \succcurlyeq ^L K$ and $\mathcal {R}(\tilde{h}^{-1}(x_i), \tilde{h} ) \succcurlyeq ^L K$, so $\mathcal {R}( E, h ) = F_{i-1} \succcurlyeq ^L K$ and $\mathcal {R}( \tilde{E}, \tilde{h} ) = G_{i-1} \succcurlyeq ^L K$. By the upper sure-thing principle at $K$, we may replace $x_{i-1}$ with $x_i$ on an event $C \subset E$, without reversing the expressed preference:

$$\begin{aligned} {x_{i}}_{C} {x_{i-1}}_{E\setminus C} f \succcurlyeq {x_{i}}_{C} {x_{i-1}}_{E\setminus C} g. \end{aligned}$$

The map that transforms these acts to their decumulative representation is as follows: ${x_{i}}_{C} {x_{i-1}}_{E\setminus C} f \mapsto (F_1,\ldots , F_{i-1}, A \cup C, F_{i+1}, \ldots F_n)$ and ${x_{i}}_{C} {x_{i-1}}_{E\setminus C} g \mapsto (G_1,\ldots , G_{i-1}, B\cup C, G_{i+1}, \ldots , G_n)$, as required. $\square$

A capacity $\nu$ is $\fancyscript{Z}$-modular if the following holds:

$$\begin{aligned}&[A, B, A\cup C, B\cup C \in \fancyscript{Z}, A\cap C = B\cap C = \emptyset ] \\&\quad \Rightarrow \nu (A\cup C) - \nu (A) = \nu (B\cup C) - \nu (B). \end{aligned}$$

The capacity is $\fancyscript{K}$-modular if it is $\fancyscript{Z}$-modular for $\fancyscript{Z}\in \{ \fancyscript{U}, \fancyscript{M}, \fancyscript{L} \}$. The following lemma establishes the $\fancyscript{K}$-modularity of the $\nu _X$ obtained above.

Lemma 6.3

The capacity $\nu _X$ is $\fancyscript{K}$-modular.

Proof

Let $X=\{x_0,x_1,x_2\}$ be enumerated in increasing order of preference: $x_0 \prec x_1 \prec x_2$. We prove that $\nu _X$ is $\fancyscript{U}$-modular. The remaining conditions are entirely similar. In the decumulative framework, an act $f$ translates to $(A,E)$, where $E = f^{-1}(x_2)$ and $A = E \cup f^{-1}(x_1)$. Take any $A, B, C$ such that $A\cap C = B\cap C= \emptyset$ and $A, B, A\cup C, B\cup C\in \fancyscript{U}$. Let $\nu _X(A) \geqslant \nu _X(B)$. We require: $\nu _X(A\cup C) - \nu _X(A) = \nu _X(B\cup C) - \nu _X(B)$. Without loss of generality, we fix $u_X(x_0)=0$. Define $d := [\nu _X(A) - \nu _X(B)]u_X(x_1) >0$ and $e:= \nu _X(B)[u_X(x_2)-u_X(x_1)]$. We separate the proof of step 1 into two cases.

Case 1: $d \leqslant e$. We know $\nu _X(A)u_X(x_1) \geqslant \nu _X(B)u_X(x_1)$. This holds if and only if, $(A, \emptyset ) \succcurlyeq (B, \emptyset )$. Also, $d\leqslant e$ if and only if $\nu _X(A)u_X(x_1) \leqslant \nu _X(B)u_X(x_1) + \nu _X(B)[u_X(x_2)-u_X(x_1)]$, equivalent to $(A, \emptyset ) \preccurlyeq (B,B)$. Appealing to convex-valuedness of the $\nu _X$ functions, there exists an $E$ (which could be $\emptyset$) such that $(A,\emptyset ) \sim (B,E)$. Furthermore, by $\fancyscript{K}$-cancellation (Lemma 6.2) applied twice, the above indifference holds only if $(A\cup C,\emptyset ) \sim (B\cup C, E)$. Substituting the additive representation for these two indifferences yields:

$$\begin{aligned} \nu _X(A )u_X(x_1)= \nu _X(B)u_X(x_1) + \nu _X(E)[u_X(x_2)-u_X(x_1)], \end{aligned}$$

and

$$\begin{aligned} \nu _X(A \cup C)u_X(x_1) = \nu _X(B \cup C)u_X(x_1) + \nu _X(E)[u_X(x_2)-u_X(x_1)], \end{aligned}$$

which jointly imply: $\nu _X(A) - \nu _X(B) = \nu _X(A \cup C) - \nu _X(B \cup C)$. Hence, $\nu _X$ is $\fancyscript{U}$-modular.

Case 2: $d > e$. This is the more difficult case, in that the proof is indirect. We reduce the proof to case 1 holding for some other, appropriately constructed, $\tilde{A},\tilde{B}$.

We now have, $d> e$, or: $\nu _X(A)u_X(x_1) > \nu _X(B)u_X(x_1) + \nu _X(B)[u_X(x_2)-u_X(x_1)]$. Equivalently, $(A,\emptyset ) \succ (B, B) \succcurlyeq (B, \emptyset )$, the second preference being implied by monotonicity. By convex-valuedness, there exists $Z \in \fancyscript{E}$ with $A \succcurlyeq ^L Z \succcurlyeq ^L B$ such that $(Z,\emptyset ) \sim (B,B)$. Let $Y$ be defined so that $A\setminus Y = Z$. So, $(A\setminus Y, \emptyset ) \sim (B, B)$. One can verify that $B\cup Y$ and $B\cup Y \cup C$ are in $\fancyscript{U}$. By $\fancyscript{K}$-cancellation, $(A\setminus Y \cup C, \emptyset ) \sim (B \cup C, B)$. Again, by $\fancyscript{K}$-cancellation, $(A, \emptyset ) \sim (B \cup Y, B)$. And, again, by $\fancyscript{K}$-cancellation, $(A\cup C, \emptyset ) \sim (B \cup Y \cup C, B)$. Substituting the Choquet expected utility representation for each of these four indifferences, subtracting the obtained equations from each other and cancelling the common utility terms, the following equations can be established:

$$\begin{aligned} \nu _X(A) - \nu _X(B) = \nu _X(B\cup Y) - \nu _X(A\setminus Y), \end{aligned}$$

and,

$$\begin{aligned} \nu _X (A\cup C) - \nu _X (B\cup C) = \nu _X (B\cup Y \cup C) - \nu _X(A\setminus Y \cup C). \end{aligned}$$

Hence, given the last two equations, we can establish the $\fancyscript{U}$-modularity property for the chosen $A, B, C$ only if the $\fancyscript{U}$-modularity property can be established using $\tilde{A},\tilde{B},C$ with $\tilde{A} = A\setminus Y$ and $\tilde{B} = B\cup Y$. Define $\tilde{d}$ and $\tilde{e}$ in the same manner as $d$ and $e$, but using $\tilde{A}$ and $\tilde{B}$ where $A$ and $B$ were used. Then, it is immediate that $\tilde{d} = e$ and $\tilde{e} >e$, hence case 1 holds ($\tilde{d}\leqslant \tilde{e}$) and $\fancyscript{U}$-modularity is proved. $\square$

The following lemma contains an implication of the piecewise sure-thing principle. This lemma is crucial in deriving the ‘kinks’ endogenously. By the piecewise sure-thing principle, every event belongs to at least one of $\fancyscript{U}, \fancyscript{M}$ or $\fancyscript{L}$. It is apparent from the definitions that $A\in \fancyscript{U}$ and $B\succcurlyeq ^L A$ only if $B\in \fancyscript{U}$. Also, by the piecewise sure-thing principle, $A^{\complement } \in \fancyscript{L}$ and, then, $B^{\complement } \in \fancyscript{L}$ and also $C\preccurlyeq ^L A$ only if $C\in \fancyscript{L}$. Suppose there exist $A\in \fancyscript{L}$ and $B \in \fancyscript{U}$ with $A \succ ^L B$ ($\fancyscript{U}$ and $\fancyscript{L}$ “overlap”). Then, $\nu _X$ is modular within $\fancyscript{U}$ and $\fancyscript{L}$, hence also in their intersection, so $\nu _X$ must be modular globally. Then $\nu _X$ is additive, hence the following holds:

Lemma 6.4

There exist $A\in \fancyscript{L}$ and $B \in \fancyscript{U}$ with $A \succ ^L B$ ($\fancyscript{U}$ and $\fancyscript{L}$ “overlap”) if and only if the sure-thing principle holds.

To avoid repeatedly adding ‘if expected utility does not hold’ qualifiers to everything that follows, we assume for the remainder of the paper that preferences are non-expected utility unless otherwise stated. Equivalently, we assume, $\fancyscript{U},$ $\fancyscript{M}$ and $\fancyscript{L}$ do not overlap. Let $K$ be the $\succcurlyeq ^L$-infimum of $\fancyscript{U}$. It must be that $K \succcurlyeq ^L K^{\complement }$, or else $\fancyscript{U}$ and $\fancyscript{L}$ would overlap. $K$ is the $\succcurlyeq ^L$-supremum of $\fancyscript{M}$, $K^\complement$ is the $\succcurlyeq ^L$-supremum of $\fancyscript{L}$, and $K^\complement$ is the $\succcurlyeq ^L$-infimum of $\fancyscript{M}$. Clearly, the inner and outer sure-thing principles at $K$ hold. The same holds for $J\in \fancyscript{E}$ if and only if $J\sim ^L K$.

We now let $X$ be finite and have at least four outcomes. The following lemma establishes coordinate independence for decumulative acts:

Lemma 6.5

Preferences $\succcurlyeq$ over $\fancyscript{A}^*$ are coordinate independent.

Proof

We show that, for all $F_i G, F_i H, \tilde{F}_i G, \tilde{F}_i H\in \fancyscript{A}^*_X$, $F_i G \succcurlyeq F_i H$ only if $\tilde{F}_i G\succcurlyeq \tilde{F}_i H$. Let $\tilde{F}_i = F_i \cup A$ for some $A$ with $A\cap F_i = \emptyset$, so that $\tilde{F}_i \succcurlyeq ^L F_i$. Then, there are six cases to consider:

i. {$\tilde{F}_i \succcurlyeq ^L F_i \succcurlyeq ^L K$}. $F_i G \succcurlyeq F_i H$ only if $\tilde{F}_i G\succcurlyeq \tilde{F}_i H$ by $\fancyscript{U}$-cancellation. ii. {$\tilde{F}_i \succcurlyeq ^L K \succcurlyeq ^L F_i \succcurlyeq ^L K^{\complement }$}. $F_i G \succcurlyeq F_i H$ only if $K_i G\succcurlyeq K_i H$ by $\fancyscript{M}$-cancellation. Then, $K_i G \succcurlyeq K_i H$ only if $\tilde{F}_i G\succcurlyeq \tilde{F}_i H$ by $\fancyscript{U}$-cancellation. iii. {$\tilde{F}_i \succcurlyeq ^L K \succcurlyeq ^L K^{\complement }\succcurlyeq ^L F_i$}. $F_i G \succcurlyeq F_i H$ only if $K^{\complement }_i G\succcurlyeq K^{\complement }_i H$ by $\fancyscript{L}$-cancellation. Then, $K^{\complement }_i G \succcurlyeq K^{\complement }_i H$ only if $K_i G\succcurlyeq K_i H$ by $\fancyscript{M}$-cancellation. Finally, $K_i G\succcurlyeq K_i H$ only if $\tilde{F}_i G\succcurlyeq \tilde{F}_i H$ by $\fancyscript{U}$-cancellation. iv. {$K \succcurlyeq ^L \tilde{F}_i \succcurlyeq ^L F_i \succcurlyeq ^L K^{\complement }$}. $F_i G \succcurlyeq F_i H$ only if $\tilde{F}_i G\succcurlyeq \tilde{F}_i H$ by $\fancyscript{M}$-cancellation. v. {$K \succcurlyeq ^L \tilde{F}_i \succcurlyeq ^L K^{\complement }\succcurlyeq ^L F_i$}. $F_i G \succcurlyeq F_i H$ only if $K^{\complement }_i G\succcurlyeq K^{\complement }_i H$ by $\fancyscript{L}$-cancellation. Then, $K^{\complement }_i G\succcurlyeq K^{\complement }_i H$ only if $\tilde{F}_i G\succcurlyeq \tilde{F}_i H$ by $\fancyscript{M}$-cancellation. vi. {$K^{\complement } \succcurlyeq ^L \tilde{F}_i \succcurlyeq ^L F_i$}. $F_i G \succcurlyeq F_i H$ only if $\tilde{F}_i G\succcurlyeq \tilde{F}_i H$ by $\fancyscript{L}$-cancellation. $\square$

By construction, $\fancyscript{A}^*$ is the rank-ordered set $\fancyscript{S}^n_{\supseteq }$ with $n\geqslant 3$. The preference relation $\succcurlyeq$ restricted to constant acts agrees with $\supseteq$. Weak ordering of $\succcurlyeq$ on $\fancyscript{A}^*$ is inherited from the same property on $\fancyscript{A}$. Lemmas B.1 and B.3 of Abdellaoui and Wakker (2005: 44–45) ensure that monotonicity and solvability in the decumulative framework hold. Lemma 6.5 established the coordinate independence of $\succcurlyeq$ on $\fancyscript{A}^*$. Our Archimedean axiom (A5) translates to the following: bounded standard sequences on coordinate $i$, that is $A^1_i F \succ A^1_i G$ and $A^k_i F \sim A^{k+1}_i G$ for $k=1,2,\ldots$, are finite for $i=1,\ldots ,n$. That is, preferences over $\fancyscript{A}^*$ are Archimedean in the sense of Wakker (1991). We now invoke Theorem 6.1, so there exist functions $V_1,\ldots ,V_n$ such that $\sum _{j=1}^{n}V_j$ represents $\succcurlyeq$ on $\fancyscript{A}^*$. Each $V_j$ is real-valued, for $j=2,\ldots ,n-1$. It is possible that $V_1(\emptyset )=-\infty$ and/or $V_n(\fancyscript{S})=+\infty$. But, $\succcurlyeq$ has a real-valued, additive representation on $\fancyscript{A}^* \setminus \{extremes\}$.

Choquet expected utility holds over $\fancyscript{A}_{X}$ with $|X|=3$. We now extend this to arbitrary, finite $X$ by a routine argument appealing to uniqueness properties on overlapping outcome subsets. To see this, consider the four-outcome case, $X= \{ x_0, x_1, x_2, x_3 \}$. We know preferences over decumulative acts $\fancyscript{A}^*_X$ are represented by $V_1 + V_2 + V_3$. Let $Y = \{ x_0, x_1, x_3 \}$ and $Z = \{ x_0, x_2, x_3 \}$. We then obtain Choquet expected utility representations with utilities $u_{Y}, u_{Z}$ and capacities $\nu _Y, \nu _Z$. Rescale these functions so that $u_Y(x_3) = u_Z(x_3) =1$ and $u_Y(x_0) = u_Z(x_0) = 0$. Rescale the additive representation so that $V_j(\emptyset )= 0$ for $j=1,2,3$ and $\sum _{j=1}^3 V_j(\fancyscript{S})=1$. Since $\sum _j V_j$ represents preferences when restricted to $\fancyscript{A}^*_Y$ and $\fancyscript{A}^*_Z$, we have $V_j (\cdot ) = \nu _Y(\cdot )[u_Y(x_j) - u_Y(x_{j-1})] = \nu _Z(\cdot )[u_Z(x_j) - u_Z(x_{j-1})]$ for $j=1,2,3$. So, $\nu _Y = \nu _Z$ and then $u_Y=u_Z$. Let $\nu =\nu _Y$ and define $u$ so that $u|_Y = u_Y$ and $u|_Z = u_Z$. Then, Choquet expected utility with utility $u$ and capacity $\nu$ represents preferences over $\fancyscript{A}_X$. In the same manner, Choquet expected utility on $\fancyscript{A}_X$ can be obtained for all finite $X$. We now extend this to preferences $\succcurlyeq$ over $\fancyscript{A}$, the set of simple acts over $\fancyscript{X}$. If $\fancyscript{X}$ has finitely many equivalence classes under $\sim$ then, passing to the quotient, the proof above applies. If $\fancyscript{X}$ has infinitely many equivalence classes under $\sim$, the representation can be obtained as in Abdellaoui and Wakker (2005, pp. 60–61).

The capacity $\nu$ is ordinally additive if the following holds:

$$\begin{aligned}&[A, B, A\cup C, B\cup C \in \fancyscript{E}, A\cap C = B\cap C = \emptyset , \nu (A) < \nu (B )]\\&\quad \Rightarrow \nu (A\cup C) < \nu (B\cup C). \end{aligned}$$

Notice that ordinal additivity concerns events within and across the $\fancyscript{K}$ sets, over all of $\fancyscript{E}$. The following lemma establishes the ordinal additivity of $\nu$.

Lemma 6.6

The capacity $\nu$ is ordinally additive.

Proof

Let $A,B,C$ satisfy the prerequisite conditions of the ordinally additive implication. If $A, B, A\cup C, B\cup C\in \fancyscript{Z}$ for some $\fancyscript{Z}\in \{\fancyscript{U}, \fancyscript{M}, \fancyscript{L} \}$ then the implication follows from $\fancyscript{K}$-cancellation established in Lemma 6.2. There are various cases to consider, although the proofs are sufficiently similar to fully explain the proof when $A \in \fancyscript{M}$ and $A\cup C, B, B\cup C\in \fancyscript{U}$. Then, $\nu (A) \leqslant \nu (K) \leqslant \nu (B)$, with at least one inequality strict, so we take $\nu (A)<\nu (K)$. There exists, by convex-valuedness, $E$ such that $\nu (A\cup (C\setminus E)) = \nu (K)$. Then, by $\fancyscript{U}$ modularity and the capacity’s monotonicity, $\nu (A\cup (C\setminus E)\cup E) = \nu (A\cup C) \leqslant \nu (B\cup E) \leqslant \nu (B\cup E \cup (C\setminus E)) = \nu (B\cup C)$, as required. $\square$

Lemma 6.2 can be amended now, $\succcurlyeq ^L$ satisfies cancellation, so it is a qualitative probability order (Savage 1954: 32). We now show that $\succcurlyeq ^L$ is a quantitative probability order. That is, for all $A, B\in \fancyscript{E}$, with $A\succ ^L B$, there exists a finite partition $\{C_1,\ldots , C_n\}$ of $\fancyscript{S}$ such that $A\succ ^L B\cup C_i$ for all $i=1,\ldots ,n$. Savage (1954) showed that any quantitative probability order may be represented by a convex-valued, finitely additive probability measure, with such a probability uniquely determined (see also Krantz et al. 1971: 202–208). In showing that $\succcurlyeq ^L$ is a quantitative probability, we prove:⁹

Lemma 6.7

There exists a convex-valued and finitely additive probability measure $p$ on $\fancyscript{E}$ and a real-valued, strictly increasing function $\phi :[0,1]\rightarrow [0,1]$ such that $\nu = \phi \circ p$. Such $\phi$ and $p$ are unique.

Proof

Let $A\succ ^L B$. We show there exists a finite partition $\{C_1,\ldots , C_n\}$ of $\fancyscript{S}$ such that $A\succ ^L B\cup C_i$ for all $i=1,\ldots ,n$. The order $\succcurlyeq ^L$ is tight if $A\cup C \succcurlyeq ^L B$ for all disjoint $C\succ ^L \emptyset$ and $B\cup D \succcurlyeq ^L A$ for all disjoint $D\succ ^L \emptyset$ jointly imply $A\sim ^L B$. The order is fine if, for all $A\succ ^L \emptyset$, there exists a finite partition $\{C_1,\ldots , C_n\}$ of $\fancyscript{S}$ such that $A\succ ^L C_i$ for all $i=1,\ldots ,n$. An equivalent formulation, due to Theorem 4 of Savage (1954: 38), of the finite partition condition is that $\succcurlyeq ^L$ is fine and tight. If $A\succ ^L B$, then the existence of disjoint $C$ such that $A\succ B\cup C$ can be established using convex-valuedness of $\nu$. Then, $A\cup C \succcurlyeq ^L B$ for all disjoint $C\succ ^L \emptyset$ with $A\succ ^L B$ is excluded and $\succcurlyeq ^L$ is tight.

Lemma C.3 of Abdellaoui and Wakker (2005) guarantees, for all $\epsilon >0$, the existence of a partition $\{C_1,\ldots , C_n \}$ of $\fancyscript{S}$ such that $\nu (\bigcup _{j=i}^{n} C_j) - \nu (\bigcup _{j=i-1}^{n} C_j) < \epsilon$ for all $i=1,\ldots ,n$.¹⁰ To show $\succcurlyeq ^L$ is fine, let $A\succ ^L \emptyset$, choose $\epsilon < \min \{ \nu (A), \frac{\nu (K^{\complement })}{2}\}$ and take such a partition. Clearly $\nu (C_n) < \epsilon$. Given $\nu (C_{n-1}\cup C_n)-\nu (C_n)<\epsilon$, exists $E \in \fancyscript{E}$ disjoint from $C_n$ such that $\nu (C_n \cup E) = \nu (C_n) +\epsilon < 2\epsilon < \nu (K^{\complement })$. By choosing $\epsilon$ as stated, $C_n\cup E \in \fancyscript{L}$, which includes $\emptyset$, and by $\fancyscript{L}$-modularity (equivalent to additivity), $\nu (E)=\epsilon$. By ordinal additivity, $\nu (C_{n-1}\cup C_n) < \nu (C_n \cup E)$ holds only if $\nu (C_{n-1})< \nu (E) = \epsilon$ as required. Continuing in this way, $\nu (C_i)<\epsilon$ holds for all $i=1,\ldots ,n$, and $\succcurlyeq ^L$ is fine.

The capacity induces an order which satisfies the Savage axioms for a probability representation, hence the capacity is a strictly increasing transformation of such a probability measure. Uniqueness of $p$ is given by Savage’s result. Uniqueness of $\phi$ follows from that of $\nu$. $\square$

Lemma 6.8

The capacity $\nu$ is piecewise additive.

Proof

According to lemma 6.7, the capacity $\nu$ and the probability measure $p$ obtained above both represent $\succcurlyeq ^l$, the likelihood order over $\fancyscript{E}$. By additivity, $p(A\cup B) - p(B) = p(A)$ for all $A, B$ with $A\cap B=\emptyset$, hence $p$ is $\fancyscript{K}$-modular.

Let $\fancyscript{Z} \in \{ \fancyscript{U}, \fancyscript{M}, \fancyscript{L} \}$ and take $A, B, C, D, A\cup D, B \cup C \in \fancyscript{Z}$ with $A\cap D=B \cap C=\emptyset$. We show $p(A)-p(B)=p(C)-p(D)$ if and only if $\nu (A)-\nu (B)=\nu (C)-\nu (D)$. Assume the first equation holds and, without loss of generality, that $p(B)< p(A) < p(C)$. Then, there exists $E$ such that $p(A\cup E)=p(C)$, so $p(D)=(B\cup E)$ by $\fancyscript{K}$-modularity, so $\nu (D)=\nu (B\cup E)$. There also exists $\tilde{E}$ such that $p(A)= p(B\cup \tilde{E})$ and $B\cap \tilde{E}=E\cap \tilde{E} = \emptyset$. By $\fancyscript{K}$-modularity of $\nu$: $\nu (B \cup \tilde{E}) - \nu (B) = \nu (B\cup \tilde{E} \cup E) - \nu (B\cup E)$, which is equivalent to $\nu (A)-\nu (B)=\nu (C)-\nu (D)$. As solvability holds, the $\fancyscript{Z}$ sets are sufficiently rich to ensure that $\nu$ and $p$ are affinely related. It is readily verified, for example, that the conditions of Theorem 4.2 of Krantz et al. (1971) hold.¹¹ Then, there are $a_0,a_1,a_3,b_1,b_2\in \mathbb {R}$ such that:

$$\begin{aligned} \phi (A) = \left\{ \begin{array}{ll} a_0 p(A) &{} \text{ if } A \in \fancyscript{L} \\ a_1 p(A) + b_1 &{} \text{ if } A \in \fancyscript{M} \\ a_2 p(A) + b_2 &{} \text{ if } A \in \fancyscript{U} \end{array} \right. \end{aligned}$$

with $a_1,a_2 ,a_3 > 0$, and $b_1,b_2 \in \mathbb {R}$ such that monotonicity and normalisation hold. We have already shown the following:

$$\begin{aligned} \fancyscript{U}:= & {} \{ A\in \fancyscript{E}: A\succcurlyeq ^L K \} \\ \fancyscript{M}:= & {} \{ A\in \fancyscript{E}: K\succcurlyeq ^L A \succcurlyeq ^L K^{\complement }\} \\ \fancyscript{L}:= & {} \{ A\in \fancyscript{E}: K^{\complement } \succcurlyeq ^L A \} \end{aligned}$$

Define $\kappa := p(K)$. Because $\fancyscript{L}$ and $\fancyscript{M}$ share a common element $K^{\complement }$, for $\nu$ to be well defined, we must have $a_0(1-\kappa ) = a_1(1-\kappa ) + b_1$. Similarly, because $\fancyscript{U}$ and $\fancyscript{M}$ share a common element $K$, for $\nu$ to be well defined, we must have $a_1\kappa + b_1 = a_2\kappa +b_2$. Now uniquely define $\gamma$ and $\delta$ such that $\gamma =1+a_1+b_1$ and $\delta = b_1$, and we are done.

It is now apparent that preferences $\succcurlyeq$ over $\fancyscript{A}$ are represented by Choquet expected utility with capacity $\nu$, with piecewise additive. That is, PACE utility holds on $\fancyscript{A}$. $\square$

1.3 Further comments on Theorem 4.6

Theorem 4.6 characterises a special case of Choquet expected utility, with a convex-valued (solvable) capacity. Applying Theorem 6 of Wakker (1996) gives the following observation:

Observation 6.9

If, in statement 1 of Theorem 4.6, the piecewise sure-thing principle (axiom 5$\kappa$) is replaced with the sure-thing principle (axiom 5), then the capacity obtained in statement 2 is additive (subjective expected utility holds).

Consider the following weakening of the sure-thing principle:

Axiom 6

(The rank-dependent sure-thing principle) For all outcomes $x,y\in \fancyscript{X}$ and acts $g$, $h \in \fancyscript{A}$, the implication:

$$\begin{aligned} x_A g \succcurlyeq x_A h \;\;\Rightarrow \;\; y_A g\succcurlyeq y_A h \end{aligned}$$

holds whenever $\mathcal {R}(A,x_A g) = \mathcal {R}(A,x_A h)=\mathcal {R}(A,y_A g) = \mathcal {R}(A,y_A h)$.

The rank-dependent sure-thing principle asserts that the sure-thing principle holds whenever the common outcomes have common ranks and are changed in a way that preserves the ranks. By a direct translation of the above definition, it can be shown that the rank-dependent sure-thing principle for preferences over acts is equivalent to the coordinate independence condition for preferences over acts in their decumulative representation. Therefore, Lemma 6.5 in Appendix 6.2 has shown that the rank-dependent sure-thing principle can be derived from axioms 1–4 and 5$\kappa$. Following that Lemma, as part of the proof of Theorem 4.6, a Choquet expected utility representation was proved for preferences satisfying axioms 1–4 over acts, and satisfying coordinate independence for acts in their decumulative representation. Hence the following observation is proved:

Observation 6.10

If, in statement 1 of Theorem 4.6, the piecewise sure-thing principle (axiom 5$\kappa$) is replaced with the rank-dependent sure-thing principle (axiom 6), then Choquet expected utility with a convex-valued capacity holds in statement 2.

In the representation obtained in Theorem 4.6, the capacity is a probability transformation: $\nu _{\kappa } = \phi \circ p$ with $\phi$ strictly increasing and continuous. As in Savage (1954), however, the probability measure $p$ is not, in general, countably additive.¹² Under solvability, the power set of $\fancyscript{S}$ is too large a set of events to derive a convex-valued, countably additive probability measure. One must restrict $\fancyscript{E}$ to be some other, sufficiently small, $\sigma$-algebra. Having done so, one may apply the monotone continuity axiom:

Axiom 7

(Monotone continuity) Given acts $f,g\in \fancyscript{A}$ with $f\succ g$, outcome $x\in \fancyscript{X}$, and countable collection of events $E_1,E_2,\ldots$, with $E_1 \supseteq E_2 \supseteq \cdots$, and $\cap _{i=1}^{\infty } E_i = \emptyset$, there exists $j\geqslant 1$ such that $x_{E_j}f \succ g$ and $f\succ x_{E_j}g$.

Following Villegas (1964), Arrow (1970) introduced monotone continuity in order derive subjective expected utility with a countably additive probability measure. The same axiom was used by Chateauneuf et al. (2005) to characterise countable additivity of all probability measures in a set of priors associated with a multiple priors representation. Machina and Schmeidler (1992:771) also noted that this axiom would lead to countable additivity of their probabilistic sophistication model. PACE utility is a special case of probabilistic sophistication, and collapses to expected utility in the two-outcome case, hence we note the following:

Observation 6.11

If, in statement 1 of Theorem 4.6, the set of events $\fancyscript{E}$ is restricted to be a sufficiently small $\sigma$-algebra, the set of acts $\fancyscript{A}$ is restricted to those acts measurable with respect to the restricted set of events, the axioms in statement 1 hold on this restricted set of acts, and the monotone continuity axiom is further assumed, then the capacity obtained in statement 2 is a strictly increasing and continuous transformation of a convex-valued, countably additive probability measure.