Choice Functions and Rejection Sets

Abstract

We establish an equivalent representation of coherent choice functions in terms of a family of rejection sets, and investigate how each of the coherence axioms translates into this framework. In addition, we show that this family allows to simplify the verification of coherence in a number of particular cases.

1 Introduction

Coherent choice functions constitute an uncertainty model that is more general than sets of desirable gambles, while still preserving some of their nice properties, such as being able to deal effectively with sets of probability zero when conditioning. One of their drawbacks is the technical difficulty of verifying the coherence axioms. In this paper, we try to remedy this situation somewhat by providing an equivalent representation of choice functions in terms of those option sets that allow a subject to reject the zero gamble, which may be interpreted as those option sets that he should consider preferable to the status quo. As we shall see, this representation, in addition to capturing more intuitively the ideas underlying coherence, also helps to simplify the verification of coherence in a number of particular cases.

This paper is organized as follows: in Sect. 2, we recall the basic aspects of coherent choice functions that we shall need in the rest of the paper. Our representation in terms of rejection sets is established in Sect. 3, where we also discuss two additional properties that seem of interest for choice functions. In Sect. 4, we look in more detail at a number of particular cases: choice functions on binary spaces (that is, when the experiment on which the outcomes of the options depend on can only take two values) and those defined by means of coherent sets of desirable gambles. The paper concludes with some additional remarks in Sect. 5.

2 Coherent Choice Functions

Let $\varOmega$ be a possibility space. A gamble on $\varOmega$ is a bounded map $f:\varOmega \rightarrow {\mathbb {R}}$. We denote by $\mathscr {L}$ the set of all gambles on $\varOmega$. Gambles will also be called options. For any two gambles f and g, we denote $f\le g$ if $f(\omega )\le g(\omega )$ for every $\omega$ in $\varOmega$, and we collect all the gambles f for which $f\le 0$ in $\mathscr {L}_{\le 0}$. We let $f<g$ if $f\le g$ and $f\ne g$, and collect all the gambles f for which $f<0$ in $\mathscr {L}_{<0}$, and the gambles f for which $f>0$ in $\mathscr {L}_{>0}$.

Choice functions are defined on finite collections of gambles. We collect all those collections in the set $\mathscr {Q}$.

Definition 2.1

A choice function $C_{}$ on a possibility space $\varOmega$ is a map

$$\begin{aligned} C_{}:\mathscr {Q}\rightarrow \mathscr {Q}\cup \left\{ \emptyset \right\} :A_{}\mapsto C_{}(A_{}) \,\text {such that}\, C_{}(A_{})\subseteq A_{}. \end{aligned}$$

We collect all the choice functions on $\varOmega$ in $\mathscr {C}\left( \varOmega \right)$, often denoted as $\mathscr {C}$ when the possibility space is clear from the context.

The idea underlying this simple definition is that a choice function $C_{}$ selects the set $C_{}(A_{})$ of ‘best’ options in the option set $A_{}$. Our definition resembles the one commonly used in the literature [1, 6, 8], except perhaps for an also not entirely unusual restriction to finite option sets [2, 5, 7].

Equivalently to a choice function $C_{}$, we may consider its associated rejection function $R_{}$, defined by $R_{}(A_{}):=A_{}\setminus C_{}(A_{})$ for all $A_{}$ in $\mathscr {Q}$. It returns the options $R_{}(A)$ that are rejected -not selected- by $C_{}$.

We focus here on a special class of choice functions, which we call coherent.

Definition 2.2

We call a choice function $C_{}$ on $\varOmega$ coherent if for all $A_{}$, $A_{1}$ and $A_{2}$ in $\mathscr {Q}$, all f and g in $\mathscr {L}$, and all $\lambda$ in ${{\mathbb {R}}_{>0}}$ ¹:

C$_1$.:

$C_{}\left( A_{}\right) \ne \emptyset$;

C$_2$.:

if $f < g$ then $\left\{ g\right\} =C_{}\left( \left\{ f,g\right\} \right)$;

C$_3$.:

a.:

if $C_{}\left( A_{2}\right) \subseteq A_{2}\setminus A_{1}$ and $A_{1}\subseteq A_{2}\subseteq A_{}$ then $C_{}\left( A_{}\right) \subseteq A_{}\setminus A_{1}$;

b.:

if $C_{}\left( A_{2}\right) \subseteq A_{1}$ and $A_{}\subseteq A_{2}\setminus A_{1}$ then $C_{}\left( A_{2}\setminus A_{}\right) \subseteq A_{1}$;

C$_4$.:

a.:

if $A_{1}\subseteq C_{}\left( A_{2}\right)$ then $\lambda A_{1}\subseteq C_{}\left( \lambda A_{2}\right)$;

b.:

if $A_{1}\subseteq C_{}\left( A_{2}\right)$ then $A_{1}+\left\{ f\right\} \subseteq C_{}\left( A_{2}+\left\{ f\right\} \right)$.

These axioms are a subset of the ones introduced in [6], duly translated from horse lotteries to gambles. We have omitted two of the coherence axioms from [6]: one is the Archimedean axiom, because it is not fully compatible with the idea of deriving choice functions from coherent sets of desirable gambles [9], which is one of the goals in our approach. The other one, which we shall consider later on, is the so-called convexity axiom. Although this axiom leads to a number of useful properties, and in particular to a connection with lexicographic probability systems [10], we have refrained from including it in the list of coherence axioms because it is not satisfied by some interesting choice functions.

Equivalent formulations of these axioms, better suited for our subsequent proofs, are the following:

$$\begin{aligned} (\mathrm {C}_3\mathrm {a})&\Leftrightarrow \left( \forall A_{},A_{}'\in \mathscr {Q},\forall f\in A_{}\right) {\left( f\in R_{}\left( A_{}\right) ,A_{}\subseteq A_{}'\right) \Rightarrow f\in R_{}\left( A_{}'\right) },\\ (\mathrm {C}_3\mathrm {b})&\Leftrightarrow \left( \forall A_{}\in \mathscr {Q},\forall f\in A_{}\right) {\left\{ 0,f\right\} \subseteq R_{}(A_{}) \Rightarrow 0 \in R_{}(A_{}\setminus \left\{ f\right\} )},\\ (\mathrm {C}_3\mathrm {a})&\Leftrightarrow \left( \forall A_{}\in \mathscr {Q},\forall \lambda >0\right) R_{}(\lambda A_{})=\lambda R_{}(A_{}),\\ (\mathrm {C}_3\mathrm {b})&\Leftrightarrow \left( \forall A_{}\in \mathscr {Q},\forall f\in \mathscr {L}\right) R_{}(A_{}+f)=R_{}(A_{})+f. \end{aligned}$$

3 A Representation in Terms of Rejection Sets

Next we give an equivalent representation of choice functions in terms of rejection sets. For any $f\in \mathscr {L}$ and any natural number i, we define

$$\begin{aligned} \mathbb {K}_{f}^{i}:=\{A_{}: f\in R_{}(A_{}), |A_{}|=i\} \ \text { and } \ \mathbb {K}_{f}:=\cup _{i\in {\mathbb {N}}}\mathbb {K}_{f}^{i}. \end{aligned}$$

(1)

We are going to characterize coherent choice functions in terms of these rejection sets. Our first result shows that we can restrict our attention to the case $f=0$:

Proposition 3.1

Let $C_{}$ be a choice function and consider the family of option sets $\{\mathbb {K}_{f}: f \in \varOmega \}$ it induces by means of Eq. (1). Then

$$\begin{aligned} C_{}\,\text {satisfies Axiom}~\mathrm {C}_4\mathrm {b} \Leftrightarrow \left( \forall f\in \mathscr {L}\right) \ \mathbb {K}_{0}+f=\mathbb {K}_{f}. \end{aligned}$$

Proof

For necessity, consider an option set $A_{}$ that includes 0. Then the option set $A_{}+f$ includes f, and since by C$_4$b it holds that $R_{}(A_{}+f)=R_{}(A_{})+f$, we conclude that $A_{}\in \mathbb {K}_{0}$ if and only if $A_{}+f\in \mathbb {K}_{f}$.

Conversely, for sufficiency, consider an option set $A_{}$ and a gamble f. Take any $g\in R_{}(A_{})$, then $A_{}\in \mathbb {K}_{g}$, whence by assumption $A_{}-g\in \mathbb {K}_{0}$ and as a consequence $A_{}-g+(f+g)=A_{}+f\in \mathbb {K}_{f+g}$. Then indeed $g+f\in R_{}(A_{}+f)$, whence Axiom C$_4$b holds. $\square$

Taking this result into account, in what follows we shall restrict our attention to rejection sets $\mathbb {K}_{}$ for which $\mathbb {K}_{0}+f=\mathbb {K}_{f}$ for every f in $\mathscr {L}$. We can then simplify the notation above to

$$\begin{aligned} K^{i}:=\mathbb {K}_{0}^{i}=\{A_{}: 0\in R_{}(A_{}), |A_{}|=i\} \ \text { and }K:=\mathbb {K}_{0}=\{A_{}: 0\in R_{}(A_{})\}, \end{aligned}$$

(2)

respectively, and denote $\mathscr {Q}_{0}$ the family of option sets that include the zero gamble. Our next result provides a characterisation of the different coherent axioms in terms of these sets:

Proposition 3.2

Let $C_{}$ be a choice function satisfying Axiom C$_4$b, and consider the sets $K^{i},K$ defined in Eq. (2).

• (a) $C_{}$ satisfies Axiom C$_1$ if and only if $\left( \forall A_{}\in \mathscr {Q}_0\right) \left( \exists f\in A_{}\right) A_{}-f\notin K$.

• (b) $C_{}$ satisfies Axiom C$_2$ if and only if $\left( \forall f \in \mathscr {L}_{>0}\right) \{f,0\}\subseteq K^2$.

• (c) $C_{}$ satisfies Axiom C$_3$a if and only if $\left( \forall A_{}\in K,\forall A_{}'\in \mathscr {Q}_0\right) \left( A_{}\subseteq A_{}' \Rightarrow A_{}'\in K\right)$.

• (d) $C_{}$ satisfies Axiom C$_3$b if and only if $\left( \forall A_{}\!\in \!K,\forall f\in A_{}\right) \left( A_{}-f\in K\!\Rightarrow \!A_{}\!\setminus \!\left\{ f\right\} \!\in \!K\right)$.

• (e) $C_{}$ satisfies Axiom C$_4$a if and only if $\left( \forall A_{}\in \mathscr {Q}_0,\forall \lambda >0\right) \left( A_{}\in K \Leftrightarrow \lambda A_{}\in K\right)$.

Proof

• (a) Taking Axiom C$_4$b into account, Axiom C$_1$ holds if and only if $C_{}(A_{})\ne \emptyset$ for every $A_{}\in \mathscr {Q}_0$. This in turn is equivalent to $\left( \exists f\in A_{}\right) f\in C_{}(A_{})$, which by C$_4$b is equivalent to $0\in C_{}(A_{}-f)$ or, in other words, to $A_{}-f\notin K$.

• (b) Under Axiom C$_4$b, Axiom C$_2$ is equivalent to $\left( \forall f\in \mathscr {L}_{>0}\right) \left\{ f\right\} =C_{}(\{0,f\})$, or, in other words, to $\left( \forall f \in \mathscr {L}_{>0}\right) \{f,0\}\subseteq K^2$.

• (c) For necessity, consider any $A_{}$ in K and any $A_{}'$ in $\mathscr {Q}_0$ such that $A_{}'\supseteq A_{}$. Because $A_{}\in K$, $0\in R_{}\left( A_{}\right)$, whence, by Axiom C$_3$a, $0\in R_{}\left( A_{}'\right)$. Then indeed $A_{}'\in K$. Conversely, for sufficiency, consider any $A_{}$ and $A_{}'$ in $\mathscr {Q}_0$ such that $A_{}\subseteq A_{}'$, and any f in $R_{}(A_{})$. Then by Axiom C$_4$b, $0\in R_{}\left( A_{}-f\right)$, so $A_{}-f\in K$, whence also $A_{}'-f\in K$, because $A_{}'-f\supseteq A_{}-f$. Then $0\in R_{}\left( A_{}'-f\right)$, and applying again C$_4$b, indeed $f\in R_{}(A_{}')$.

• (d) For necessity, consider any $A_{}$ in K and f in $A_{}$ such that $A_{}-f\in K$. Then $0\in R_{}(A_{}-f)$, whence $f\in R_{}(A_{})$, by Axiom C$_4$b. Applying Axiom C$_3$b, we deduce that $0\in R_{}(A_{}\setminus \{f\})$, whence indeed $A_{}\setminus \{f\}\in K$. Conversely, for sufficiency, consider any $A_{}$ in $\mathscr {Q}$ and f in $A_{}$ such that $\left\{ 0,f\right\} \subseteq R_{}(A_{})$. Then $A_{}\in K$ and by Axiom C$_4$b, $f\in R_{}(A_{})$ implies that $0\in R_{}\left( A_{}-f\right)$, so $A_{}-f\in K$. Then $A_{}\setminus \{f\}\in K$, or, in other words, indeed $0\in R_{}(A_{}\setminus \{f\})$.

• (e) It suffices to note that under Axiom C$_4$b, Axiom C$_4$a is equivalent to $0\in C_{}(A_{}) \Leftrightarrow 0\in C_{}(\lambda A_{})$ for every $\lambda >0$ and every $A_{}\in \mathscr {Q}_{0}$. $\square$

An immediate consequence is:

Corollary 3.1

A choice function $C_{}$ is coherent if and only if it satisfies Axiom C$_4$b and the rejection set K it induces by Eq. (2) is increasing, scale invariant, includes $\{f,0\}$ for every $f \in \mathscr {L}_{>0}$ and it satisfies the following two properties:

• $\left( \forall A_{}\in \mathscr {Q}_0\right) \left( \exists f\in A_{}\right) A_{}-f\notin K$.

• $\left( \forall A_{}\in K,\forall f\in A_{}\right) \left( A_{}-f\in K\Rightarrow A_{}\setminus \left\{ f\right\} \in K\right)$.

Next we consider a couple of additional consistency axioms that were deemed interesting by [10]. The first one is the convexity axiom, which is given by:

C$_5$ :

if $A_{}\subseteq A_{1}\subseteq {CH}\left( A_{}\right)$ then $C_{}\left( A_{}\right) \subseteq C_{}\left( A_{1}\right)$, for all $A_{}$ and $A_{1}$ in $\mathscr {Q}$,

where $C_{}(A_{}):=\left\{ \sum _{i=1}^n\alpha _if_i: n\in \mathbb {N}, f_i\in A_{},\alpha _i\ge 0,\sum _{i=1}^n\alpha _i=1\right\}$ is the convex hull of $A_{}$.

In terms of rejection sets, it is characterized by the following proposition:

Proposition 3.3

Let $C_{}$ be a choice function satisfying Axiom C$_4$b. Then $C_{}$ satisfies Axiom C$_5$ if and only if $\left( \forall A_{1}\in K,\forall A_{}\in \mathscr {Q}_0\right) \left( A_{}\subseteq A_{1}\subseteq {CH}(A_{})\right. \left. \Rightarrow A_{}\in K\right)$.

Proof

For necessity, application of Axiom C$_5$ tells us that, whenever $A_{}\subseteq A_{1}\subseteq {CH}\left( A_{}\right)$ hold, $0\in R_{}(A_{1})$ implies that $0\in R_{}(A_{})$, or, in other words, $A_{1}\in K$ implies that $A_{}\in K$.

Conversely, for sufficiency, consider two option sets $A_{}$ and $A_{1}$ such that $A_{}\subseteq A_{1} \subseteq {CH}\left( A_{}\right)$, and let us show that $C_{}(A_{})\subseteq C_{}(A_{1})$. Assume ex absurdo that there is some $f\in A_{}$ such that $f\in R_{}(A_{1})$ and $f\in C_{}(A_{})$. Then since $A_{}-f\subseteq A_{1}-f\subseteq {CH}(A_{}-f)$, we can apply axiom C$_4$b and assume that, without loss of generality, $f=0$. But then we obtain that $A_{1}\in K$ while $A_{}\notin K$, a contradiction. $\square$

A weaker property that is also useful is the so-called separate homogeneity, which means that for all n in ${\mathbb {N}}$, all $f_1$, $f_2$, ..., $f_n$ in $\mathscr {L}$ and all $\mu _1$, $\mu _2$, ...$\mu _n$ in ${{\mathbb {R}}_{>0}}$:

$$\begin{aligned} 0\in C_{}\left( \left\{ 0,f_1,f_2,\dots ,f_n\right\} \right) \Leftrightarrow 0\in C_{}\left( \left\{ 0,\mu _1 f_1,\mu _2 f_2,\dots ,\mu _n f_n\right\} \right) . \end{aligned}$$

(3)

This property follows from axioms C$_3$a, C$_4$a, C$_5$ [10, Proposition 1]. Moreover, and unlike C$_5$ that is linked to lexicographic choice functions, separate homogeneity is compatible with maximality as a decision rule, and therefore better suited for connecting choice functions with desirability. Furthermore, separate homogeneity is strictly weaker: there are classes of interesting coherent choice functions that satisfy Eq. (3) but not Axiom C$_5$. In terms of the rejection sets, it is trivial to prove that it can be expressed in the following manner:

Proposition 3.4

Let $C_{}$ be a choice function satisfying Axiom C$_4$b. It satisfies separate homogeneity if and only if for all n in ${\mathbb {N}}$, all $f_1$, $f_2$, ..., $f_n$ in $\mathscr {L}$ and all $\mu _1$, $\mu _2$, ...$\mu _n$ in ${{\mathbb {R}}_{>0}}$, $\left\{ 0,f_1,f_2,\dots ,f_n\right\} \in K\Leftrightarrow \left\{ 0,\mu _1 f_1,\mu _2 f_2,\dots , \mu _n f_n\right\} \in K$.

4 Particular Cases

In this section, we consider a number of particular cases of choice functions for which the representation in terms of rejection sets simplifies somewhat.

4.1 Coherent Choice Functions Defined via Maximality

We begin by considering choice functions defined via Walley’s notion of maximality [9]. A set of gambles ${\mathscr {D}}$ is called coherent when it is a convex cone that includes all non-negative gambles and does not include the zero gamble. We refer to [3, 4, 11] for a study of the notion of desirability and its variants. In particular, any coherent set of desirable gambles can be used to define a coherent choice function, by means of the formula

$$\begin{aligned} C_{\mathscr {D}}(A_{}):=\left\{ f\in A_{}:\left( \forall g\in A_{}\right) g-f\notin \mathscr {D}\right\} . \end{aligned}$$

(4)

Unlike general choice functions, the ones defined in the manner above are uniquely determined by binary comparisons. Thus, it is not surprising that for them the representation in terms of rejection sets takes a simpler form:

Proposition 4.1

Let $\mathscr {D}$ be a coherent set of gambles and let $C_{\mathscr {D}}$ be the coherent choice function it induces by Eq. (4). Then $K=\left\{ A_{}\in \mathscr {Q}_0:\left( \exists A_{1}\in K^2\right) \right. \left. A_{1}\subseteq A_{}\right\}$ and $K^2=\left\{ \left\{ 0, f\right\} :f\in \mathscr {D}\right\}$.

Proof

Consider an option set $A_{}$ in K. By Eq. (4), $0\in R_{\mathscr {D}}(A_{})$ if and only if $A_{}\cap \mathscr {D}\ne \emptyset$. If $|A_{}|=2$, then $A_{}=\left\{ 0, f\right\}$ for some f in $\mathscr {D}$, and as a consequence $K^2\supseteq \left\{ \left\{ 0, f\right\} :f\in \mathscr {D}\right\}$. Conversely, consider any $A_{}'\in K^2$. Then $A_{}'=\left\{ 0, g\right\}$ for some g in $\mathscr {L}$. But since $0\in R_{\mathscr {D}}\left( A_{}'\right)$, we have $g\in \mathscr {D}$, so $K^2\subseteq \left\{ \left\{ 0, f\right\} :f\in \mathscr {D}\right\}$, proving that indeed $K^2=\left\{ \left\{ 0, f\right\} :f\in \mathscr {D}\right\}$. If, on the other hand, $|A_{}|\ge 3$, then $A_{}\supseteq \left\{ 0, f\right\}$ for some f in $\mathscr {D}$. But then $0\in R_{\mathscr {D}}\left( \left\{ 0, f\right\} \right)$, so $A_{}\supseteq A_{}'$ for some $A_{}'\in K^2$, and therefore indeed $K=\left\{ A_{}\in \mathscr {Q}_0:\left( \exists A_{1}\in K^2\right) A_{1}\subseteq A_{}\right\}$. $\square$

4.2 Coherent Choice Functions on Binary Spaces

Next, we consider coherent choice functions defined on binary spaces. It turns out that, under separate homogeneity, they are determined by rejection sets of cardinality two or three:

Proposition 4.2

Let $C_{}$ be a coherent choice function on $\varOmega =\{a, b\}$. If $C_{}$ satisfies Eq. (3), then

$$\begin{aligned} K=\left\{ A_{}\in \mathscr {Q}_0:\left( \exists A_{1}\in K^2\cup K^3\right) A_{1}\subseteq A_{}\right\} . \end{aligned}$$

Proof

Let us prove that for every $A_{}$ in K there exists a $A_{1}$ in $K^2\cup K^3$ for which $A_{1}\subseteq A_{}$.

Consider thus $A_{}$ in K. By Axiom C$_2$, we find that $A_{}\cap \mathscr {L}_{<0}\subseteq R_{}\left( A_{}\cap \mathscr {L}_{\le 0}\right)$, so Axiom C$_3$a implies that then $A_{}\cap \mathscr {L}_{<0}\subseteq R_{}\left( A_{}\right)$. Since $A_{}\in K$ and therefore also $0\in R_{}\left( A_{}\right)$, by Axiom C$_3$b we find that then $0\in \left( A_{}\cap \mathscr {L}_{<0}^c\right)$, so we can assume without loss of generality that $A_{}\cap \mathscr {L}_{<0}=\emptyset$. There are two possibilities.

If $A_{}\cap \mathscr {L}_{>0}\ne \emptyset$, then for any f in $A_{}\cap \mathscr {L}_{>0}$ it follows from Axiom C$_2$ that $0\in R_{}(\{0, f\})$, whence the set $\left\{ 0, f\right\} \subseteq A_{}$ belongs to $K^2$. So we find indeed that $A_{1}:=\left\{ 0, f\right\}$ in $K^2$ for which $A_{1}\subseteq A_{}$.

If $A_{}\cap \mathscr {L}_{>0}=\emptyset$, then we can denote $A_{}=\{f_1,\dots ,f_n,g_1,\dots ,g_m\}$ for some $n\ge 0$ and $m\ge 0$ but $\max \left\{ m,n\right\} \ge 1$, where $f_i$ belongs to the second quadrant (i.e., $f_i(a)<0<f_i(b)$) for every i in $\left\{ 1,\dots ,n\right\}$ and $g_j$ belongs to the fourth quadrant (i.e., $g_j(a)>0>g_j(b)$) for every j in $\left\{ 1,\dots ,m\right\}$. Let $\lambda _i:=\frac{-1}{f_i(a)}$ and $\mu _j:=\frac{1}{g_j(a)}$ for every i in $\left\{ 1,\dots ,n\right\}$ and j in $\left\{ 1,\dots ,m\right\}$. Then, applying Eq. (3),

$$\begin{aligned} 0\in R_{}(\left\{ 0,\lambda _1 f_1,\dots ,\lambda _n f_n, \mu _1 g_1,\dots ,\mu _m g_m\right\} ). \end{aligned}$$

Infer that $\lambda _if_i(a)=-1$ for every $i\!\in \!\left\{ 1,\dots ,n\right\}$. Letting $i^*\!:=\!\arg \max \left\{ \lambda _if_i(b): \right. \left. i\in \left\{ 1,\dots ,n\right\} \right\}$, we infer that

$$\begin{aligned} \lambda _k f_k(b)<\lambda _{i^*} f_{i^*}(b) \Rightarrow \lambda _k f_k \in R_{}(\left\{ \lambda _k f_k, \lambda _{i^*} f_{i^*}\right\} ) \Rightarrow \lambda _k f_k \in R_{}(A_{}), \end{aligned}$$

where last implication follows from Axiom C$_3$a. Similarly, $\mu _jg_j(a)=1$ for every $j\in \{1,\dots ,m\}$, and letting $j^*:=\arg \max \left\{ \mu _jg_j(b):j\in \{1,\dots ,m\}\right\}$, we infer that

$$\begin{aligned} \mu _j g_j(b)<\mu _{j^*} g_{j^*}(b) \Rightarrow \mu _j g_j \in R_{}(\left\{ \mu _j g_j,\mu _{j^*} g_{j^*}\right\} ) \Rightarrow \mu _j g_j \in R_{}(A_{}), \end{aligned}$$

where again last implication follows from Axiom C$_3$a. If we now apply C$_3$b, we deduce that $0 \in R_{}\left( \left\{ 0,\lambda _{i^*} f_{i^*},\mu _{j^*} g_{j^*}\right\} \right)$, whence $0\in R_{}\left( \left\{ 0,f_{i^*},g_{j^*}\right\} \right)$, applying Eq. (3). Thus, there is a subset of $A_{}$ with cardinality three that also belongs to K. $\square$

A key property in the proof of Proposition 4.2 is that separate homogeneity, together with Axiom C$_2$, allows to assume without loss of generality that an option set $A_{}$ that includes the zero gamble has at most one gamble f in the second quadrant (for which $f(a)<0<f(b)$) and one g in the fourth quadrant (for which $g(a)>0>g(b)$). Let us show that this does not necessarily happen without separate homogeneity:

Example 4.1

Consider $\varOmega =\left\{ a,b\right\}$ and let $\mathscr {D}$ be the coherent set of gambles $\mathscr {D}:=\left\{ f\in \mathscr {L}:f(a)<0<f(b)\text { and }f(a)+f(b)>0\right\} \cup \mathscr {L}_{>0}$. Let $C_{}$ be the choice function determined by the rejection function

$$\begin{aligned} 0\in R_{}(A_{})\Leftrightarrow {A_{}\cap \mathscr {D}\ne \emptyset \text { or }\left( \exists \lambda _1>\lambda _2>0\right) \left\{ (-\lambda _1,\lambda _1),(-\lambda _2,\lambda _2)\right\} \subseteq A_{}} \end{aligned}$$

(5)

for all $A_{}$ in $\mathscr {Q}_0$. We extend the domain of $R_{}$ to $\mathscr {Q}$ by letting $f\in R_{}(A_{}) \Leftrightarrow 0 \in R_{}(A_{}-f)$ for all $A_{}$ in $\mathscr {Q}$ and f in $A_{}$. Remark already that $(-\lambda ,\lambda )$ lies on the border of $\mathscr {D}$ for every $\lambda >0$: indeed, for every g in $\mathscr {D}$ we have that $(-\lambda ,\lambda )+g\in \mathscr {D}$.

Let us show that $C_{}$ is a coherent choice function. Taking into account the last part of the definition, we see that C$_4$b holds, and we can restrict our attention to option sets in $\mathscr {Q}_0$. We show that $C_{}$ satisfies Axioms C$_2$, C$_3$a, C$_3$b, C$_4$a, and C$_1$, in this order.

For Axiom C$_2$, consider any f in $\mathscr {L}_{>0}$. Then $f\in \mathscr {D}$, so indeed $0\in R_{}\left( \left\{ 0, f\right\} \right)$.

For Axiom C$_3$a, consider any $A_{}$ and $A_{}'$ in $\mathscr {Q}_0$ such that $A_{}\subseteq A_{}'$, and any f in $R_{}\left( A_{}\right)$. Using Axiom C$_4$b, then $0\in R_{}\left( A_{}-f\right)$, whence $\left( A_{}-f\right) \cap \mathscr {D}\ne \emptyset$ or $\left\{ (-\lambda _1,\lambda _1),(-\lambda _2,\lambda _2)\right\} \subseteq A_{}-f$ for some $\lambda _1>\lambda _2>0$. But $A_{}'-f\supseteq A_{}-f$, so also $\left( A_{}'-f\right) \cap \mathscr {D}\ne \emptyset$ or $\left\{ (-\lambda _1,\lambda _1),(-\lambda _2,\lambda _2)\right\} \subseteq A_{}'-f$, and therefore $0\in R_{}\left( A_{}'-f\right)$, whence, again by Axiom C$_4$b, indeed $0\in R_{}\left( A_{}'\right)$.

For Axiom C$_3$b, consider any $A_{}$ in $\mathscr {Q}_0$ and any f in $A_{}$ such that $\left\{ 0, f\right\} \subseteq R_{}\left( A_{}\right)$. We need to prove that then $0\in R_{}\left( A_{}\setminus \left\{ f\right\} \right)$. Since $f\in R_{}\left( A_{}\right)$, then

$$\begin{aligned} \text {(i) }\mathscr {D}\cap \left( A_{}-f\right) \ne \emptyset \text {, or (ii)}\left\{ (-\lambda _1,\lambda _1),(-\lambda _2,\lambda _2)\right\} \!\subseteq \!A_{}-f \, \text {for some} \, \lambda _1\!>\!\lambda _2\!>\!0. \end{aligned}$$

Furthermore, since $0\!\in \!R_{}\left( A_{}\right)$, then $\mathscr {D}\cap A_{}\ne \emptyset$, or $\left\{ (-\lambda _1,\lambda _1),(-\lambda _2,\lambda _2)\right\} \!\subseteq \!A_{}$ for some $\lambda _1>\lambda _2>0$. If $f\notin \mathscr {D}\cap A_{}$ and $f\notin \left\{ (-\lambda _1,\lambda _1),(-\lambda _2,\lambda _2)\right\}$, then also $\mathscr {D}\cap A_{}\setminus \left\{ f\right\} \ne \emptyset$ or $\left\{ (-\lambda _1,\lambda _1),(-\lambda _2,\lambda _2)\right\} \subseteq A_{}\setminus \left\{ f\right\}$, whence $0\in R_{}\left( A_{}\setminus \left\{ f\right\} \right)$. So assume that (a) $f\in \mathscr {D}$ or (b) $f=(-\lambda ,\lambda )$ for some $\lambda >0$.

If (a) $f\in \mathscr {D}$, then (i) or (ii) must be the case. If (i) occurs, then there is some gamble g in $\left( \mathscr {D}+f\right) \cap A_{}$, whence $g-f\in \mathscr {D}$ for some g in $A_{}$. But since $f\in \mathscr {D}$, also $g=f+g-f\in \mathscr {D}$, and therefore $0\in R_{}\left( \left\{ 0, g\right\} \right)$, whence by Axiom C$_3$a indeed $0\in R_{}\left( A_{}\setminus \left\{ f\right\} \right)$. If (ii) occurs, then there are $\lambda _1>\lambda _2>0$ such that $f+(-\lambda _1,\lambda _1),f+(-\lambda _2,\lambda _2)\in A_{}$, whence, since $f\in \mathscr {D}$, by construction also $f+(-\lambda _1,\lambda _1)\in \mathscr {D}$. Therefore $0\in R_{}\left( \left\{ 0, f+(-\lambda _1,\lambda _1)\right\} \right)$, whence by Axiom C$_3$a, also $0\in R_{}\left( A_{}\setminus \left\{ f\right\} \right)$.

If (b) $f=(-\lambda ,\lambda )$ for some $\lambda >0$, then, similarly, (i) or (ii) must be the case. If (i) occurs, then there is some g in $A_{}$ such that $g-f\in \mathscr {D}$. Therefore by construction also $g=f+g-f\in \mathscr {D}$, whence $0\in R_{}\left( \left\{ 0, g\right\} \right)$, and then by Axiom C$_3$a, also $0\in R_{}\left( A_{}\setminus \left\{ f\right\} \right)$. If (ii) occurs, then there is some $\lambda _1>0$ and $\lambda _2>0$ for which $\left\{ f+(-\lambda _1,\lambda _1),f+(-\lambda _2,\lambda _2)\right\} =\left\{ (-\lambda -\lambda _1,\lambda +\lambda _1),f+(-\lambda -\lambda _2,\lambda +\lambda _2)\right\} \subseteq A_{}$. Letting $\lambda _1':=\lambda +\lambda _1$ and $\lambda _2':=\lambda +\lambda _2$, we find that $\left\{ (-\lambda _1',\lambda _1'),f+(-\lambda _2',\lambda _2')\right\} \subseteq A_{}\setminus \left\{ f\right\}$, whence $0\in R_{}\left( A_{}\setminus \left\{ f\right\} \right)$.

Axiom C$_4$a follows from Eq. (5), taking into account that $\mathscr {D}$ is a cone.

Finally, for Axiom C$_1$, assume ex absurdo that $C_{}\left( A_{}\right) =\emptyset$ for some $A_{}$ in $\mathscr {Q}_0$. Then $A_{}=R_{}\left( A_{}\right)$ whence, by Axiom C$_3$b, $0\in R_{}\left( \left\{ 0\right\} \right)$. But $0\notin \mathscr {D}$ and $(-\lambda ,\lambda )\notin \left\{ 0\right\}$ for every $\lambda >0$, so $0\notin R_{}\left( \left\{ 0\right\} \right)$, a contradiction.

On the other hand, it follows by Eq. (5) that, given the option set $A_{}=\{0,(-1,1),(-2,2)\}$, we obtain $C_{}(A_{})=\{(-1,1),(-2,2)\}$. However, the same equation implies that $0\in C_{}(\{0,(-1,1)\})$. This shows that $C_{}$ does not satisfy separate homogeneity, and also that we cannot reduce the intersection with the second quadrant to only one gamble.

On the other hand, Proposition 4.2 also depends crucially on the assumption that $|\varOmega |=2$, as our next example shows:

Example 4.2

Consider a ternary space $\varOmega$, some n in ${\mathbb {N}}$, and let $f_k$ be the gamble given by $f_k:=(-1,\frac{k}{n},-\frac{k^2}{n^2})$, for all k in $\left\{ 1,\dots ,n\right\}$. Let us show that for each k we can find a probability measure whose expectation operator $P_k$ (called linear prevision in Walley’s terminology) satisfies $P_k(f_k)>0>P_k(f_j)$ for every $j\in \left\{ 1,\dots ,n\right\} \setminus \left\{ k\right\}$.

To find such expectation operators, let $P$ be the expectation operator associated with the mass function $(0,\frac{2k}{n+2k},\frac{n}{n+2k})$. Then $P(f_k-f_j) =\frac{k-j}{n(n+2k)}(2k-(k+j))$, whence $P(f_k-f_j)>0$ if $k \ne j$. Moreover, $P(f_k) =\frac{k^2}{n(n+2k)}>0$.

If we now consider any $\lambda \in (0,1)$ and define $P_k$ as the expectation operator associated with the mass function $(\lambda ,(1-\lambda )\frac{2k}{n+2k},(1-\lambda )\frac{n}{n+2k}),$ we obtain $P(f_k-f_j)=(1-\lambda )P(f_k-f_j)>0$ whenever $k\ne j$. Moreover,

$$\begin{aligned} P_k(f_k)=-\lambda +(1-\lambda ) P(f_k)>0 \Leftrightarrow \lambda <\frac{P(f_k)}{1+P(f_k)}, \end{aligned}$$

and similarly

$$\begin{aligned} P_k(f_j)=-\lambda +(1-\lambda ) P(f_j)<0 \Leftrightarrow \lambda >\frac{P(f_j)}{1+P(f_j)}. \end{aligned}$$

Since, for every $j\in \left\{ 1,\dots ,n\right\} \setminus \left\{ k\right\}$, $\frac{P(f_j)}{1+P(f_j)}<\frac{P(f_k)}{1+P(f_k)}$ because $P(f_j)<P(f_k)$, we let

$$\begin{aligned} \lambda \in \left( \max _{j\in \left\{ 1,\dots ,n\right\} \setminus \left\{ k\right\} } \frac{P(f_j)}{1+P(f_j)}, \frac{P(f_k)}{1+P(f_k)}\right) , \end{aligned}$$

and for this $\lambda$ we obtain $P_k(f_k)>0>P_k(f_j)$ for every $j\ne k$.

Now, let $\mathscr {D}_k$ be the coherent set of gambles given by $\mathscr {D}_k:=\left\{ f\in \mathscr {L}:P_k(f)>0\right\}$, and let $C_{\mathscr {D}_k}$ be the coherent choice function it induces by Eq. (4). Then the choice function $C_{}$ given by $C_{}(A_{}):=\bigcup _{k=1}^{n} C_{\mathscr {D}_k}(A_{})$ is also coherent [9, Proposition 3], and it can be checked to satisfy separate homogeneity because all $C_{\mathscr {D}_k}$ do. If we now consider the option set $A_{}=\left\{ 0,f_1,\dots ,f_n\right\}$, we get that $C_{\mathscr {D}_k}(A_{})=\left\{ f_k\right\}$ for every k, since $P_k(f_k)>0>P_k(f_j)$ implies that $f_k,f_k-f_j \in \mathscr {D}_k$ for every j. As a consequence, we obtain $C_{}(A_{})=\left\{ f_1,\dots ,f_n\right\}$, whence $A_{}\in K$. However, for every k it holds that $C_{\mathscr {D}_k}(A_{}\setminus \left\{ f_k\right\} )=\left\{ 0\right\}$, using again that $P_k(f_j)<0$ for every $j\ne k$, and therefore $C_{}(A_{}\setminus \{f_k\})=A_{}\setminus \{f_k\}$. Thus, $A_{}$ has no proper subset that also belongs to the rejection class K.

5 Conclusions

It is a consequence of coherence that a choice function is uniquely determined by those option sets that allow us to reject the zero gamble, i.e., those that are considered preferable to the status quo. In this paper, we have investigated the structure of these sets and shown that the coherence axioms can be expressed more intuitively in terms of these sets. In addition, we have shown that all the necessary information is given by option sets of cardinality two when the choice function is defined via maximality, and with cardinality two or three in most (but not all) cases of interest when the possibility space is binary. Moreover, we have shown that this last result does not extend to larger possibility spaces; thus, determining an analogous representation for arbitrary spaces would be the main open problem for the future.

On Pedro Gil

I was lucky to meet Pedro Gil in many different roles: first as a teacher, then as head of department, later as a supervisor, and finally, and all throughout, as a friend. He helped me in my first research work, on robust statistics for my final project during my BsC; and we taught together in a course on Mathematics for everyday life until a few weeks prior to his death.

He was brilliant as a professor, generous as a researcher, and charismatic as a leader. Few bring together unanimity in the manner he did; and still, or perhaps because of it, he was always unassuming: he would treat you in the same manner irrespective of your position, and would make you feel that, whatever your problems, someone cared and would try to help. Because above all his many qualities, he had one that few possess: he was a good man.

He has not left a void because he is still very much present.

Enrique Miranda, April 2017