Exchangeable choice functions

TL;DR

This paper explores modeling exchangeability using choice functions, establishing a connection to de Finetti's theorem and providing a framework for finite and countable cases.

Contribution

It introduces a novel approach to represent exchangeability through choice functions, extending classical results to new contexts.

Findings

Exchangeability modeled via choice functions

De Finetti's Representation Theorem extended to finite and countable cases

Framework for structural assessment of uncertain sequences

Abstract

We investigate how to model exchangeability with choice functions. Exchangeability is a structural assessment on a sequence of uncertain variables. We show how such assessments are a special indifference assessment, and how that leads to a counterpart of de Finetti's Representation Theorem, both in a finite and a countable context.

Equations

$$I_{{\mathcal{P}}_n}:=\mathrm{span}\{f-{\pi }^{t}f:f\in \mathcal{L}({\mathcal{X}}^n),\pi \in {\mathcal{P}}_n\}.$$

$$\mathcal{N}^{n}\coloneqq\bigg{\{}m\in\mathbb{Z}_{\geq 0}^{\mathcal{X}}:\sum_{x\in\mathcal{X}}m_{x}=n\bigg{\}}.$$

$$T(x)=T(\pi x)\text{ for al x in Xn and π in Pn}.$$

$${\mathrm{inv}}_{{\mathcal{P}}_n}:\mathcal{L}({\mathcal{X}}^n)\to \mathcal{L}({\mathcal{X}}^n):f\mapsto {\mathrm{inv}}_{{\mathcal{P}}_n}(f):=\frac{1}{N!}\sum _{\pi \in {\mathcal{P}}_n}{\pi }^{t}f,$$

$${\mathrm{inv}}_{{\mathcal{P}}_n}(f)={\mathrm{inv}}_{{\mathcal{P}}_n}(g)\text{ if f=πtg for some π in Pn, for all f and g in L(Xn).}$$

$${\mathrm{inv}}_{{\mathcal{P}}_n}=\sum _{m\in {\mathcal{N}}^n}{\mathrm{H}}_n(\cdot |m){\mathbb{I}}_{[m]},$$

$${\mathrm{H}}_n(f|m):=\frac{1}{\left(\genfrac{}{}{0ex}{}{n}{m}\right)}\sum _{y\in [m]}f(y)\text{ for all f in L(Xn) and m in Nn.}$$

$${\mathrm{H}}_n:\mathcal{L}({\mathcal{X}}^n)\to \mathcal{L}({\mathcal{N}}^n):f\mapsto {\mathrm{H}}_n(f):={\mathrm{H}}_n(f|\cdot )$$

$${\tilde{\mathrm{H}}}_n:\mathcal{L}({\mathcal{X}}^n)/I_{{\mathcal{P}}_n}\to \mathcal{L}({\mathcal{N}}^n):\tilde{f}\mapsto {\mathrm{H}}_n(f)\text{ for any f in f~.}$$

$${{\tilde{\mathrm{H}}}_n}^{-1}:\mathcal{L}({\mathcal{N}}^n)\to \mathcal{L}({\mathcal{X}}^n)/I_{{\mathcal{P}}_n}:f\mapsto [\sum _{m\in {\mathcal{N}}^n}f(m){\mathbb{I}}_{[m]}].$$

$$\tilde{f}⪯\tilde{g}\iff (\exists f\in \tilde{f},\exists g\in \tilde{g})f\le g$$

$$C(O)=\{f\in O:{\mathrm{H}}_n(f)\in \tilde{C}({\mathrm{H}}_n(O))\}\text{ for all O in Q(L(Xn)).}$$

$${\mathrm{CoM}}_n:\mathcal{L}({\mathcal{N}}^n)\to {\mathcal{V}}^n({\Sigma }_{\mathcal{X}}):r\mapsto \sum _{m\in {\mathcal{N}}^n}r(m)B_m.$$

$$h_1{⪯}_B^n{h}_2\iff h_2-h_1\in \{0\}\cup \mathrm{posi}(\{B_m:m\in {\mathcal{N}}^n\})\text{ for all h1 and h2 in Vn(ΣX).}$$

$${\mathrm{M}}_n:\mathcal{L}({\mathcal{X}}^n)\to {\mathcal{V}}^n({\Sigma }_{\mathcal{X}}):f\mapsto {\mathrm{M}}_n(f|\theta ),$$

$${\tilde{\mathrm{M}}}_n:={\mathrm{CoM}}_n\circ {\tilde{\mathrm{H}}}_n,$$

$$C(O)=\{f\in O:{\mathrm{M}}_n(f)\in \tilde{C}({\mathrm{M}}_n(O))\}\text{ for all O in Q(L(Xn)).}$$

$$f^{\ast }(x_1,\dots ,x_n,\dots ):=f(x_1,\dots ,x_n)\text{ for all (x1,…,xn,…) in XN.}$$

$$\bar{\mathcal{L}}({\mathcal{X}}^{\mathbb{N}}):=\{f\in \mathcal{L}({\mathcal{X}}^{\mathbb{N}}):(\exists n\in \mathbb{N})f\in \mathcal{L}({\mathcal{X}}^n)\}=\bigcup _{n\in \mathbb{N}}\mathcal{L}({\mathcal{X}}^n).$$

$$I_{\mathcal{P}}:=\{f\in \bar{\mathcal{L}}({\mathcal{X}}^{\mathbb{N}}):(\exists n\in \mathbb{N})f\in I_{{\mathcal{P}}_n}\}=\bigcup _{n\in \mathbb{N}}{I}_{{\mathcal{P}}_n}.$$

$$h_1{⪯}_B{h}_2\iff h_2-h_1\in \{0\}\cup \mathrm{posi}(\{B_m:m\in {\mathcal{N}}^n,n\in \mathbb{N}\})$$

$$C_n(O)=\{f\in O:{\mathrm{M}}_n(f)\in \tilde{C}({\mathrm{M}}_n(O))\}\text{ for all O in Q(L(Xn)).}$$

$$f=\sum _{m\in {\mathcal{N}}^n}\mathrm{H}(f)(m){\mathbb{I}}_{[m]}=\sum _{m\in {\mathcal{N}}^n}\tilde{\mathrm{H}}(\tilde{f})(m){\mathbb{I}}_{[m]}.$$

$${\mathrm{H}}_n(g)(m^{\prime })=\frac{1}{\left(\genfrac{}{}{0ex}{}{n}{m^{\prime }}\right)}\sum _{y\in [m^{\prime }]}g(y)$$

$$=\sum _{m\in {\mathcal{N}}^n}f(m)\frac{1}{\left(\genfrac{}{}{0ex}{}{n}{m^{\prime }}\right)}\sum _{y\in [m^{\prime }]}{\mathbb{I}}_{[m]}(y)=\sum _{m\in {\mathcal{N}}^n}f(m){\mathbb{I}}_{\{m\}}(m^{\prime })=f(m^{\prime }),$$

$$\frac{1}{\left(\genfrac{}{}{0ex}{}{n}{m^{\prime }}\right)}\sum _{y\in [m^{\prime }]}{\mathbb{I}}_{[m]}(y)=\{\begin{array}{ll}1& \text{if m′=m,}\\ 0& \text{otherwise.}\end{array}$$

$$C(O)=\{f\in O:{\mathrm{H}}_n(f)\in \tilde{C}({\mathrm{H}}_n(O))\}\text{ for all O in Q(L(Xn)).}$$

$${\mathrm{CoM}}_n(f)\in \tilde{C}({\mathrm{CoM}}_n(O))\iff f\in C^{\prime \prime }(O)$$

$$(\forall\tilde{h}\in I_{\mathcal{P}})(\forall\tilde{O}\in\mathcal{Q}(\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})))\big{(}\{0,\tilde{h}\}\subseteq\tilde{O}\Rightarrow(0\in C(\tilde{O})\Leftrightarrow\tilde{h}\in C(\tilde{O}))\big{)}.$$

$$(\forall h\in I_{\mathcal{P}_{n}})(\forall O\in\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n})))\big{(}\{0,h\}\subseteq O\Rightarrow(0\in C_{n}(O)\Leftrightarrow h\in C_{n}(O))\big{)}.$$

$$C_n(O)=\{f\in O:{\mathrm{M}}_n(f)\in {\tilde{C}}_n({\mathrm{M}}_n(O))\}\text{ for all O in Q(L(Xn)),}$$

Full text

Keywords: Exchangeability; choice functions; indifference; sets of desirable gambles; representation.

1 Introduction

In this paper, we study how to model exchangeability, a structural assessment for uncertainty models that is important for inference purposes, in the framework of choice functions, an interesting approach to modelling uncertainty. This work builds on the work about exchangeability for lower previsions (see de Cooman et al. (2009)) and exchangeability for sets of desirable gambles (see de Cooman and Quaeghebeur (2012)).

Choice functions are related to the fundamental problem in decision theory: how to make a choice from within a set of available options. In their book, von Neumann and Morgenstern (1972) provide an axiomatisation of choice based on pairwise comparison between the options. Later on, Rubin (1987) generalised that idea and proposed a theory of choice functions based on choice between more than two elements. One of the aspects in Rubin (1987)’s theory is that, between any pair of options, the agent either prefers one of them, or he is indifferent between them, so two options can never be incomparable. However, for instance when the information available does not allow for a complete comparison of the options, the agent may be undecided between two options without being indifferent between them; this will for instance typically be the case when there is no relevant information available at all. This is one of the motivations for a theory of imprecise probabilities (see Walley (1991)), where incomparability and indifference are distinguished. Kadane et al. (2004) and Seidenfeld et al. (2010) generalise the axioms in Rubin (1987) to allow for incomparability.

Exchangeability is a structural assessment on a sequence of uncertain variables. Loosely speaking, making a judgement of exchangeability means that the order in which the variables are observed is considered irrelevant. This irrelevancy will be modelled through an indifference assessment. The first detailed study of exchangeability was de Finetti (1937). For a brief historical overview, we refer to Ref. (de Cooman and Quaeghebeur, 2012, Sec. 1).

In Sec. 2, we will recall the necessary tools for modelling indifference with choice functions. Next, in Sec. 3, we will derive de Finetti-like Representation Theorems for a finite sequence that is exchangeable. We will take this one step further in Sec. 4, where we consider a countable sequence and derive a representation theorem for such sequences. Because it will be useful to compare with de Cooman and Quaeghebeur (2012), we will also provide representation theorems for sets of desirable gambles.

2 Choice functions, desirability and indifference

The material in this section is based on (Van Camp et al., 2017, Sec. 5). Consider a real vector space $\mathcal{V}$, provided with the vector addition and scalar multiplication. Elements $u$ of $\mathcal{V}$ are intended as abstract representations of options amongst which a subject can express his preferences, by specifying, as we will see below, choice functions. Mostly, options will be real-valued maps on the possibility space, interpreted as uncertain rewards, and therefore also called gambles. The set of all gambles on the possibility space $\mathcal{X}$ will be denoted as $\mathcal{L}(\mathcal{X})$. Given any subset $O$ of $\mathcal{V}$, we will define the linear hull $\operatorname*{span}(O)\coloneqq\{\sum_{k=1}^{n}\lambda_{k}u_{k}:n\in\mathbb{N},\lambda_{k}\in\mathbb{R},u_{k}\in O\}\subseteq\mathcal{V}$ and the positive hull $\operatorname*{posi}(O)\coloneqq\{\sum_{k=1}^{n}\lambda_{k}u_{k}:n\in\mathbb{N},\lambda_{k}\in\mathbb{R}_{>0},u_{k}\in O\}\subseteq\operatorname*{span}(O)$, where $\mathbb{R}_{>0}$ is the set of all (strictly) positive real numbers. A subset $O$ of $\mathcal{V}$ is called a convex cone if it is closed under positive finite linear combinations, i.e. if $\operatorname*{posi}(O)=O$. A convex cone $\mathcal{K}$ is called proper if $\mathcal{K}\cap-\mathcal{K}=\{0\}$. With any proper convex cone $\mathcal{K}\subseteq\mathcal{V}$, we associate an ordering $\preceq_{\mathcal{K}}$ on $\mathcal{V}$ as follows: $u\preceq_{\mathcal{K}}v\Leftrightarrow v-u\in\mathcal{K}$ for any $u$ and $v$ in $\mathcal{V}$. For any $u$ and $v$ in $\mathcal{V}$, we write $u\prec_{\mathcal{K}}v$ if $u\preceq_{\mathcal{K}}v$ and $u\neq v$. We collect all the options $u$ for which $0\prec_{\mathcal{K}}u$ in $\mathcal{V}_{{\succ}0}$. When we work with gambles, then $\mathcal{V}=\mathcal{L}(\mathcal{X})$ and the ordering will be the standard one $\leq$, given by $f\leq g\Leftrightarrow(\forall x\in\mathcal{X})f(x)\leq g(x)$. We collect the positive gambles—gambles $f$ for which $0<f$—in $\mathcal{L}(\mathcal{X})_{>0}$. Then $\leq$ corresponds to $\preceq_{\mathcal{K}}$ where we let $\mathcal{K}\coloneqq\mathcal{L}(\mathcal{X})_{>0}\cup\{0\}$.

We denote by $\mathcal{Q}(\mathcal{V})$ the set of all non-empty finite subsets of $\mathcal{V}$. Elements of $\mathcal{Q}(\mathcal{V})$ are the option sets amongst which a subject can choose his preferred options.

A choice function $C$ on $\mathcal{V}$ is a map $C\colon\mathcal{Q}\to\mathcal{Q}\cup\{\emptyset\}\colon O\mapsto C(O)$ such that $C(O)\subseteq O$. Not every such map represents a rational belief; only the coherent choice functions do. We call a choice function $C$ on $\mathcal{V}$ coherent111Our rationality axioms are based on those in Seidenfeld et al. (2010), slightly modified for use with sets of desirable gambles. if, for all $O$, $O_{1}$ and $O_{2}$ in $\mathcal{Q}(\mathcal{V})$, all $u$ and $v$ in $\mathcal{V}$, and all $\lambda$ in $\mathbb{R}_{>0}$:

1. C1.

$C(O)\neq\emptyset$; 2. C2.

if $u\prec v$ then $\{v\}=C(\{u,v\})$; 3. C3.

1. a.

if $C(O_{2})\subseteq O_{2}\setminus O_{1}$ and $O_{1}\subseteq O_{2}\subseteq O$ then $C(O)\subseteq O\setminus O_{1}$; 2. b.

if $C(O_{2})\subseteq O_{1}$ and $O\subseteq O_{2}\setminus O_{1}$ then $C(O_{2}\setminus O)\subseteq O_{1}$; 4. C4.

1. a.

if $O_{1}\subseteq C(O_{2})$ then $\lambda O_{1}\subseteq C(\lambda O_{2})$; 2. b.

if $O_{1}\subseteq C(O_{2})$ then $O_{1}+\{u\}\subseteq C(O_{2}+\{u\})$.

Consider two isomorphic vector spaces $\mathcal{V}$ and $\mathcal{W}$, a linear order isomorphism $\phi$ between $\mathcal{V}$ and $\mathcal{W}$, and a choice function $C$ on $\mathcal{V}$. Define the choice function $C^{\prime}$ on $\mathcal{W}$ as $u\in C(O)\Leftrightarrow\phi(u)\in C^{\prime}(\phi(O))$ for all $O$ in $\mathcal{Q}(\mathcal{V})$ and $u$ in $O$. Then, because $\phi$ is a bijection, $C$ satisfies Axioms C1 and C3 if and only if $C^{\prime}$ does; furthermore, because $\phi$ is order preserving, $C$ satisfies Axiom C2 if and only if $C^{\prime}$ does; and finally, because $\phi$ is linear, $C$ satisfies Axiom C4 if and only if $C^{\prime}$ does: such isomorphisms preserve coherence.

A set of desirable options (or gambles) $D\subseteq\mathcal{V}$ is essentially the restriction to pairwise comparison of a choice function: $D=\{u\in\mathcal{V}\setminus\{0\}:\{u\}=C(\{0,u\})\}$. We call $D$ coherent if $0\notin D$, $\mathcal{V}_{{\succ}0}\subseteq D$, $u\in D\Rightarrow\lambda u\in D$, and $u,v\in D\Rightarrow u+v\in D$ for all $u$ and $v$ in $\mathcal{V}$ and $\lambda$ in $\mathbb{R}_{>0}$. $D$ is coherent if the choice function $C$ it is based on, is coherent.

Since, as we will see, an exchangeability assessment amounts to a specific indifference assessment, we recall how to model an indifference assessment. For more information, we refer to (Van Camp et al., 2017, Sec. 5). Next to $C(O)$—the options that the agent strictly prefers from $O$—or $D$—the options that he strictly prefers to [math]—we consider the options that the agent considers to be equivalent to the zero option $I\subseteq\mathcal{V}$. We call $I$ coherent if, for all $u$ and $v$ in $\mathcal{V}$ and $\lambda$ in $\mathbb{R}$:

1. I1.

$0\in I$; 2. I2.

if $u\in\mathcal{V}_{{\succ}0}\cup\mathcal{V}_{{\prec}0}$ then $u\notin I$; 3. I3.

if $u\in I$ then $\lambda u\in I$; 4. I4.

if $u,v\in I$ then $u+v\in I$.

We collect all options that are indifferent to an option $u$ in $\mathcal{V}$ into the equivalence class $[u]\coloneq\{v\in\mathcal{V}:v-u\in I\}=\{u\}+I$. The set of all these equivalence classes is the quotient space $\mathcal{V}/{I}\coloneq\{[u]:u\in\mathcal{V}\}$, being a linear space itself. We provide it with the natural ordering inherited from $\mathcal{V}$: $\tilde{u}\preceq\tilde{v}\Leftrightarrow(\exists u\in\tilde{u},v\in\tilde{v})u\preceq v$ for all $\tilde{u}$ and $\tilde{v}$ in $\mathcal{V}/{I}$.

Consider any coherent set of indifferent options $I$. A choice function $C$ is then called compatible with $I$ if there is some representing choice function $C^{\prime}$ on $\mathcal{V}/{I}$ such that $C(O)=\{u\in O:[u]\in C^{\prime}(O/I)\}$ for all $O$ in $\mathcal{Q}(\mathcal{V})$. In that case, $C^{\prime}$ is uniquely determined by $C^{\prime}(O/I)=C(O)/I$ for all $O$ in $\mathcal{Q}(\mathcal{V})$, and, moreover, $C$ is coherent if and only if $C^{\prime}$ is. Equivalently, we find the following useful characterisation: $C$ is compatible with $I$ if and only if $0\in C(O)\Leftrightarrow u\in C(O)$ for all $u$ in $I$ and $O\supseteq\{0,u\}$ in $\mathcal{Q}(\mathcal{V})$, which corresponds to the definition of indifference in Ref. Seidenfeld (1988).

For desirability, compatibility with a coherent set of indifferent options $I$ is defined as follows. We call a set of desirable gambles $D$ compatible with $I$ if $D+I\subseteq D$, and this is equivalent to $D=\bigcup D^{\prime}$ where $D^{\prime}\subseteq\mathcal{V}/{I}$ is the representing set of desirable options. In that case, $D^{\prime}$ is uniquely given by $D^{\prime}=D/I$, and, moreover, $D$ is coherent if and only if $D^{\prime}$ is.

3 Finite exchangeability

Consider $n\in\mathbb{N}$ uncertain variables $X_{1}$, …, $X_{n}$ taking values in a non-empty set $\mathcal{X}$. The possibility space of the uncertain sequence $(X_{1},\dots,X_{n})$ is $\mathcal{X}^{n}$.

We denote by $x=(x_{1},\dots,x_{n})$ an arbitrary element of $\mathcal{X}^{n}$. For any $n$ in $\mathbb{N}$ we call $\mathcal{P}_{n}$ the group of all permutations $\pi$ of the index set $\{1,\dots,n\}$. There are $|\mathcal{P}_{n}|=n!$ such permutations. With any such permutation $\pi$, we associate a permutation of $\mathcal{X}^{n}$, also denoted by $\pi$, and defined by $(\pi x)_{k}\coloneqq x_{\pi(k)}$ for every $k$ in $\{1,\dots,n\}$, or in other words, $\pi(x_{1},\dots,x_{n})=(x_{\pi(1)},\dots,x_{\pi(n)})$. Similarly, we lift $\pi$ to a permutation ${\pi}^{t}$ on $\mathcal{L}(\mathcal{X}^{n})$ by letting ${\pi}^{t}f=f\circ\pi$, so $({\pi}^{t}f)(x)\coloneqq f(\pi x)$ for all $x$ in $\mathcal{X}^{n}$. Observe that ${\pi}^{t}$ is a linear permutation of the vector space $\mathcal{L}(\mathcal{X}^{n})$ of all gambles on $\mathcal{X}^{n}$.

If a subject assesses that the sequence of variables $X$ in $\mathcal{X}^{n}$ is exchangeable, this means precisely that he is indifferent between any gamble $f$ on $\mathcal{X}^{n}$ and its permuted variant ${\pi}^{t}f$, for any $\pi$ in $\mathcal{P}_{n}$. This leads us to the following proposal for the corresponding set of indifferent gambles:

[TABLE]

Definition 1

A choice function $C$ on $\mathcal{L}(\mathcal{X}^{n})$ is called (finitely) exchangeable if it is compatible with $I_{\mathcal{P}_{n}}$. Similarly, a set of desirable gambles $D\subseteq\mathcal{L}(\mathcal{X}^{n})$ is called (finitely) exchangeable if it is compatible with $I_{\mathcal{P}_{n}}$.

Of course, so far, we do not yet know whether this notion of exchangeability is well-defined: indeed, we do not know yet whether $I_{\mathcal{P}_{n}}$ is a coherent set of indifferent gambles. In the next section, we will show that this is indeed the case.

3.1 Count vectors

In this section, we will provide the tools necessary to prove that $I_{\mathcal{P}_{n}}$ is a coherent set of indifferent gambles. In de Cooman et al. (2009) and de Cooman and Quaeghebeur (2012), all the maps we use here are defined.

The permutation invariant atoms $[x]\coloneqq\{\pi x:x\in\mathcal{X}^{n}\}$, $x$ in $\mathcal{X}^{n}$ are the smallest permutation invariant subsets of $\mathcal{X}^{n}$. We introduce the counting map $T\colon\mathcal{X}^{n}\to\mathcal{N}^{n}\colon x\mapsto T(x)$ where $T(x)$ is called the count vector of $x$. It is the $\mathcal{X}$-tuple with components $T_{z}(x)\coloneqq|\{k\in\{1,\dots,n\}:x_{k}=z\}|$ for all $z$ in $\mathcal{X}$, so $T_{z}$ is the number of times that $z$ occurs in the sequence $x_{1}$, …, $x_{n}$. The range of $T$—the set $\mathcal{N}^{n}$—is called the set of possible count vectors and is given by

[TABLE]

Remark that applying any permutation to $x$ leaves its result under the counting map unchanged:

[TABLE]

For any $x$ in $\mathcal{X}^{n}$, if $m=T(x)$ then $[x]=\{y\in\mathcal{X}^{n}:T(y)=m\}$, so the permutation invariant atom $[x]$ is completely determined by the count vector $m$ of all its elements, and is therefore also denoted by $[T(x)]=[m]$. Remark that $\{[m]:m\in\mathcal{N}^{n}\}$ partitions $\mathcal{X}^{n}$ into disjoint parts with constant count vectors, and that $|[m]|=\binom{n}{m}\coloneqq\frac{N!}{\prod_{z\in\mathcal{X}}m_{z}!}$.

In order to extend the idea of the count vectors for use with gambles, let us define the set of all permutation invariant gambles as $\mathcal{L}_{{\mathcal{P}_{n}}}(\mathcal{X}^{n})\coloneqq\{f\in\mathcal{L}(\mathcal{X}^{n}):(\forall\pi\in\mathcal{P}_{n}){\pi}^{t}f=f\}\subseteq\mathcal{L}(\mathcal{X}^{n})$, and a special transformation $\mathrm{inv}_{\mathcal{P}_{n}}$ of the linear space $\mathcal{L}(\mathcal{X}^{n})$

[TABLE]

which, as we will see, is closely linked with $\mathcal{L}_{{\mathcal{P}_{n}}}(\mathcal{X}^{n})$ (see de Cooman and Quaeghebeur (2012); Van Camp et al. (2017)).

Proposition 2

$\mathrm{inv}_{\mathcal{P}_{n}}$* is a linear transformation of $\mathcal{L}(\mathcal{X}^{n})$, and*

1. (i)

$\mathrm{inv}_{\mathcal{P}_{n}}\circ{\pi}^{t}=\mathrm{inv}_{\mathcal{P}_{n}}={\pi}^{t}\circ\mathrm{inv}_{\mathcal{P}_{n}}$* for all $\pi$ in $\mathcal{P}$;* 2. (ii)

$\mathrm{inv}_{\mathcal{P}_{n}}\circ\mathrm{inv}_{\mathcal{P}_{n}}=\mathrm{inv}_{\mathcal{P}_{n}}$; 3. (iii)

$\operatorname*{kern}(\mathrm{inv}_{\mathcal{P}_{n}})=I_{\mathcal{P}_{n}}$; 4. (iv)

$\operatorname*{rng}(\mathrm{inv}_{\mathcal{P}_{n}})=\mathcal{L}_{{\mathcal{P}_{n}}}(\mathcal{X}^{n})$.

So we see that $\mathrm{inv}_{\mathcal{P}_{n}}$ is a linear projection operator that maps any gamble to a permutation invariant counterpart.

As shown by de Cooman and Quaeghebeur (2012), the linear projection operator $\mathrm{inv}_{\mathcal{P}_{n}}$ renders a gamble insensitive to permutation by replacing it with the uniform average of all its permutations. As a result, it assumes the same value for all gambles that can be related to each other through some permutation:

[TABLE]

Furthermore, for any $f$ in $\mathcal{L}(\mathcal{X}^{n})$, its transformation $\mathrm{inv}_{\mathcal{P}_{n}}(f)$ is permutation invariant and therefore constant on the permutation invariant atoms $[m]$: $(\mathrm{inv}_{\mathcal{P}_{n}}(f))(x)=(\mathrm{inv}_{\mathcal{P}_{n}}(f))(y)$ if $[x]=[y]$, for all $x$ and $y$ in $\mathcal{X}^{n}$. We can use the properties of $\mathrm{inv}_{\mathcal{P}_{n}}$ to prove that $I_{\mathcal{P}_{n}}$ is suitable for the definition of exchangeability.

Proposition 3

Consider any $n$ in $\mathbb{N}$. Then $I_{\mathcal{P}_{n}}$ is a coherent set of indifferent gambles.

Since $I_{\mathcal{P}_{n}}$ is coherent, exchangeability is well-defined, and by the discussion in Sec. 2, the representing choice function $C^{\prime}$ is defined on $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$, and, similarly, the representing set of desirable gambles $D^{\prime}\subseteq\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$. So we shall focus on the quotient space and its elements, exchangeable equivalent classes of gambles.

But before we do that, it will pay to further explore the notions we have introduced thus far.

Consider any $f$ in $\mathcal{L}(\mathcal{X}^{n})$. What is the constant value that $\mathrm{inv}_{\mathcal{P}_{n}}(f)$ assumes on a permutation invariant atom $[m]$? To answer this question, consider any $x$ in $[m]$, then $(\mathrm{inv}_{\mathcal{P}_{n}}(f))(x)=\frac{1}{n!}\sum_{\pi\in\mathcal{P}_{n}}f(\pi x)=\frac{1}{n!}\frac{|\mathcal{P}_{n}|}{|[m]|}\sum_{y\in\{\pi x:\pi\in\mathcal{P}_{n}\}}f(y)=\frac{1}{\binom{n}{m}}\sum_{y\in[x]}f(y)=\frac{1}{\binom{n}{m}}\sum_{y\in[m]}f(y)$, whence

[TABLE]

where $\mathrm{H}_{n}({\cdot}|{m})$ is the linear expectation operator associated with the uniform distribution on the invariant atom $[m]$:

[TABLE]

It characterises a (multivariate) hyper-geometric distribution (see Ref. Johnson et al. (1997)), associated with random sampling without replacement from an urn with $n$ balls of types $\mathcal{X}$, whose composition is characterised by the count vector $m$.

The result of applying a gamble $f$ on $\mathcal{X}^{n}$ to the map

[TABLE]

is the gamble $\mathrm{H}_{n}(f)$ on $\mathcal{N}^{n}$ that assumes the value $\frac{1}{\binom{n}{m}}\sum_{y\in[m]}f(y)$ for every $m$ in $\mathcal{N}^{n}$.

3.2 Exchangeable equivalent classes of gambles

We already know that exchangeable choice functions are represented by choice functions on the quotient space $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$, and similar for sets of desirable gambles. In the quest for an elegant representation theorem, we thus need to focus on the quotient space $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ and its elements, which are exchangeable equivalent classes of gambles.

In this section we investigate how the representation of permutation invariant gambles helps us find a representation for exchangeable choice functions. To that end, the representation will use equivalence classes $[f]\coloneqq\{f\}+I_{\mathcal{P}_{n}}$ of gambles, for any $f$ in $\mathcal{L}(\mathcal{X}^{n})$. Recall that the quotient space $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}\coloneqq\{[f]:f\in\mathcal{L}(\mathcal{X}^{n})\}$ is a linear space itself, with additive identity $[0]=I_{\mathcal{P}_{n}}$, and therefore any element $\tilde{f}$ of $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ is invariant under addition of $I_{\mathcal{P}_{n}}$: $\tilde{f}+I_{\mathcal{P}_{n}}=\tilde{f}$. Elements of $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ will be generically denoted as $\tilde{f}$ or $\tilde{g}$.

Proposition 4

Consider any $f$ and $g$ in $\mathcal{L}(\mathcal{X}^{n})$. Then $[f]=[g]$ if and only if $\mathrm{H}_{n}(f)=\mathrm{H}_{n}(g)$

Therefore, it makes sense to introduce the map $\tilde{\mathrm{H}}_{n}$:

[TABLE]

Then Proposition 4 guarantees that elements of $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ are characterised using $\tilde{\mathrm{H}}_{n}$, in the sense that $\tilde{f}=\{f\in\mathcal{L}(\mathcal{X}^{n}):\mathrm{H}_{n}(f)=\tilde{\mathrm{H}}_{n}(\tilde{f})\}$ for all $\tilde{f}$ in $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$.

The map $\tilde{\mathrm{H}}_{n}$ takes as input an equivalence class of gambles, and maps it to some representing gamble on the count vectors. It will be useful later on to consider some converse map ${{\tilde{\mathrm{H}}_{n}}}^{-1}$:

[TABLE]

Proposition 5

The maps $\tilde{\mathrm{H}}_{n}$ as defined in Eq. (5) and ${{\tilde{\mathrm{H}}_{n}}}^{-1}$ as defined in Eq. (6) are each other’s inverses.

The importance of Prop. 5 lies in the fact that now, $\tilde{\mathrm{H}}_{n}$ is a bijection between $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ and $\mathcal{L}(\mathcal{N}^{n})$, and therefore, exchangeable equivalence classes of gambles are in a one-to-one correspondence with gambles on count vectors.

\mathcal{L}(\mathcal{X}^{n})$$\mathcal{L}(\mathcal{N}^{n})$$\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$$\mathrm{H}_{n}$$[\cdot]$$\tilde{\mathrm{H}}_{n}

The commuting diagram shows the surjections $[\cdot]\colon\mathcal{L}(\mathcal{X}^{n})\to\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}\colon f\mapsto[f]$ and $\mathrm{H}_{n}$ (indicated with a single arrow), and the bijection $\tilde{\mathrm{H}}_{n}$ (indicated with a double arrow). Since the representing choice function $C^{\prime}$ is defined from $C$ through $[\cdot]$—working point-wise on sets—this already suggests that $C^{\prime}$ can be transformed into a choice function on $\mathcal{L}(\mathcal{N}^{n})$. To prove that they preserve coherence, there is only one missing link: the ordering between $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ and $\mathcal{L}(\mathcal{N}^{n})$ should be preserved.

Therefore, to define the ordering $\preceq$ on $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$, as usual, we let $\preceq$ be inherited by the ordering $\leq$ on $\mathcal{L}(\mathcal{X}^{n})$:

[TABLE]

for all $\tilde{f}$ and $\tilde{g}$ in $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$, turning $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ into an ordered linear space. It turns out that this vector ordering on $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ can be represented elegantly using $\tilde{\mathrm{H}}_{n}$:

Proposition 6

Consider any $\tilde{f}$ and $\tilde{g}$ in $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$, then $\tilde{f}\preceq\tilde{g}$ if and only if $\tilde{\mathrm{H}}_{n}(\tilde{f})\leq\tilde{\mathrm{H}}_{n}(\tilde{g})$.

Props. 5 and 6 imply that $\mathrm{H}_{n}$ is a linear order isomorphism.

3.3 A representation theorem

Now that we have found a linear order isomorphism $\tilde{\mathrm{H}}_{n}$ between $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ and $\mathcal{L}(\mathcal{N}^{n})$, we are ready to represent coherent and exchangeable choice functions.

Theorem 7** (Finite Representation)**

Consider any choice function $C$ on $\mathcal{L}(\mathcal{X}^{n})$. Then $C$ is exchangeable if and only if there is a unique representing choice function $\tilde{C}$ on $\mathcal{L}(\mathcal{N}^{n})$ such that

[TABLE]

Furthermore, in that case, $\tilde{C}$ is given by $\tilde{C}(\mathrm{H}_{n}(O))=\mathrm{H}_{n}(C(O))$ for all $O$ in $\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n}))$. Finally, $C$ is coherent if and only if $\tilde{C}$ is.

Similarly, consider any set of desirable gambles $D\subseteq\mathcal{L}(\mathcal{X}^{n})$. Then $D$ is exchangeable if and only if there is a unique representing set of desirable gambles $\tilde{D}\subseteq\mathcal{L}(\mathcal{N}^{n})$ such that $D=\bigcup{{\tilde{\mathrm{H}}_{n}}}^{-1}(\tilde{D})$. Furthermore, in that case, $\tilde{D}$ is given by $\tilde{D}=\mathrm{H}_{n}(D)$. Finally, $D$ is coherent if and only if $\tilde{D}$ is.

The number of occurrences of any outcome in a sequence $(x_{1},\dots,x_{n})$ is fixed by its count vector $m$ in $\mathcal{N}^{n}$. If we impose an exchangeability assessment on it, then we see, using Theorem 7, that the joint model on $\mathcal{X}^{n}$ is characterised by a model on $\mathcal{L}(\mathcal{N}^{n})$. So an exchangeable choice function $C$ essentially represents preferences between urns with $n$ balls of types $\mathcal{X}$ with different compositions $m$: the choice $C(O)$ between the gambles in $O$ is based upon the composition $m$.

3.4 Finite representation in terms of polynomials

In Sec. 4, we will prove a similar representation theorem for infinite sequences. Since it no longer makes sense to count in such sequences, we first need to find a equivalent representation theorem in terms of something that does not depend on counts. More specifically, we need, for every $n$ in $\mathbb{N}$ another order-isomorphic linear space to $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$, that allows for embedding: the linear space for $n_{1}<n_{2}$ must be a subspace of the one for $n_{2}$.

All the maps we use here have been introduced by de Cooman et al. (2009). Moreover, we use their idea and work with polynomials on the $\mathcal{X}$-simplex $\Sigma_{\mathcal{X}}\coloneqq\{\theta\in\mathbb{R}^{\mathcal{X}}:\theta\geq 0,\sum_{x\in\mathcal{X}}\theta_{x}=1\}$. We consider the special subset $\mathcal{V}(\Sigma_{\mathcal{X}})$ of $\mathcal{L}(\Sigma_{\mathcal{X}})$: $\mathcal{V}(\Sigma_{\mathcal{X}})$ are the polynomial gambles $h$ on $\Sigma_{\mathcal{X}}$, which are those gambles that are the restriction to $\Sigma_{\mathcal{X}}$ of a multivariate polynomial $p$ on $\mathbb{R}^{\mathcal{X}}$, in the sense that $h(\theta)=p(\theta)$ for all $\theta$ in $\Sigma_{\mathcal{X}}$. We call $p$ then a representation of $h$. It will be useful to introduce a notation for polynomial gambles with fixed degree $n$ in $\mathbb{N}$: $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ is the collection of all polynomial gambles that have at least one representation whose degree is not bigger than $n$. Both $\mathcal{V}(\Sigma_{\mathcal{X}})$ and $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ are linear subspaces of $\mathcal{L}(\Sigma_{\mathcal{X}})$, and, as wanted, for $n_{1}<n_{2}$, $\mathcal{V}^{n_{1}}(\Sigma_{\mathcal{X}})$ is a subspace of $\mathcal{V}^{n_{2}}(\Sigma_{\mathcal{X}})$.

Some special polynomial gambles are the Bernstein gambles:

Definition 8** (Bernstein gambles)**

Consider any $n$ in $\mathbb{N}$ and any $m$ in $\mathcal{N}^{n}$. Define the Bernstein basis polynomial $B_{m}$ on $\mathbb{R}^{\mathcal{X}}$ as $B_{m}(\theta)\coloneqq\binom{n}{m}\prod_{x\in\mathcal{X}}\theta_{x}^{m_{x}}$ for all $\theta$ in $\mathbb{R}^{\mathcal{X}}$. The restriction to $\Sigma_{\mathcal{X}}$ is called a Bernstein gamble, which we also denote as $B_{m}$.

As shown in de Cooman and Quaeghebeur (2012); De Bock et al. (2016), the set of all Bernstein gambles forms a basis of the linear space $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$:

Proposition 9

Consider any $n$ in $\mathbb{N}$. The set of Bernstein gambles $\{B_{m}:m\in\mathcal{N}^{n}\}$ constitutes a basis of the linear space $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$.

As we have seen, we need linear order isomorphisms to preserve coherence. So we wonder whether there is one between $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ and $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$. In Sec. 3.2 we have seen that there is one between $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ and $\mathcal{L}(\mathcal{N}^{n})$, namely $\tilde{\mathrm{H}}_{n}$. Therefore, it suffices to find one between $\mathcal{L}(\mathcal{N}^{n})$ and $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$. Consider the map

[TABLE]

Before we can establish that $\mathrm{CoM}_{n}$ is a linear order isomorphism, we need to provide the linear space $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ with an order $\preceq_{B}^{n}$. We use the proper cone $\{0\}\cup\operatorname*{posi}(\{B_{m}:m\in\mathcal{N}^{n}\})$ to define the order $\preceq_{B}^{n}$:

[TABLE]

The following proposition is shown in de Cooman and Quaeghebeur (2012).

Proposition 10

Consider any $n$ in $\mathbb{N}$. Then the map $\mathrm{CoM}_{n}$ is a linear order isomorphism between the ordered linear spaces $\mathcal{L}(\mathcal{N}^{n})$ and $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$.

The linear order isomorphism $\mathrm{CoM}_{n}$ helps us to define a linear order isomorphism between the linear spaces $\mathcal{L}(\mathcal{X}^{n})$ and $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$, a final tool needed for a representation theorem in terms of polynomial gambles. Indeed, consider for the map $\mathrm{M}_{n}\coloneqq\mathrm{CoM}_{n}\circ\mathrm{H}_{n}$:

[TABLE]

where $\mathrm{M}_{n}(f|\theta)\coloneqq\sum_{m\in\mathcal{N}^{n}}\sum_{y\in[m]}f(y)\prod_{x\in\mathcal{X}}\theta_{x}^{m_{x}}$ is the expectation of $f$ associated with the multinomial distribution whose parameters are $n$ and $\theta$. We introduce its version

[TABLE]

mapping $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ to $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$. There is an immediate connection between $\mathrm{M}_{n}$ and $\tilde{\mathrm{M}}_{n}$: they are both compositions of two linear order isomorphisms, and are therefore linear order isomorphisms themselves. Due to Prop. 4, considering any $\tilde{f}$ in $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$, $\mathrm{M}_{n}$ is constant on $\tilde{f}$, and the value it takes on any element of $\tilde{f}$ is exactly $\tilde{\mathrm{M}}_{n}(\tilde{f})$.

\mathcal{L}(\mathcal{X}^{n})$$\mathcal{L}(\mathcal{N}^{n})$$\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$$\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$$\mathrm{H}_{n}$$\mathrm{M}_{n}$$[\cdot]$$\mathrm{CoM}_{n}$$\tilde{\mathrm{M}}_{n}$$\tilde{\mathrm{H}}_{n}

The commuting diagram shows the surjections $[\cdot]$, $\mathrm{H}_{n}$ and $\mathrm{M}_{n}$, and the bijections $\tilde{\mathrm{H}}_{n}$, $\tilde{\mathrm{M}}_{n}$ and $\mathrm{CoM}_{n}$. It shows that both $\mathcal{L}(\mathcal{N}^{n})$ and $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ are order-isomorphic to $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$, so they are both suitable to define a representing choice function on. In Theorem 7, we used the space $\mathcal{L}(\mathcal{N}^{n})$. Here, we will use the other equivalent space $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$.

Theorem 11** (Finite Representation)**

Consider any choice function $C$ on $\mathcal{L}(\mathcal{X}^{n})$. Then $C$ is exchangeable if and only if there is a unique representing choice function $\tilde{C}$ on $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ such that

[TABLE]

Furthermore, in that case, $\tilde{C}$ is given by $\tilde{C}(\mathrm{M}_{n}(O))=\mathrm{M}_{n}(C(O))$ for all $O$ in $\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n}))$. Finally, $C$ is coherent if and only if $\tilde{C}$ is.

Similarly, consider any set of desirable gambles $D\subseteq\mathcal{L}(\mathcal{X}^{n})$. Then $D$ is exchangeable if and only if there is a unique representing set of desirable gambles $\tilde{D}\subseteq\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ such that $D=\bigcup\tilde{\mathrm{M}}_{n}^{-1}(\tilde{D})$. Furthermore, in that case, $\tilde{D}$ is given by $\tilde{D}=\mathrm{M}_{n}(D)$. Finally, $D$ is coherent if and only if $\tilde{D}$ is.

4 Countable exchangeability

In the previous section, we assumed a finite sequence $X_{1}$,…, $X_{n}$ to be exchangeable, and inferred representation theorems. In this section, we will consider the whole sequence $X_{1}$, …, $X_{n}$, …to be exchangeable, and derive representation theorems for such assessments. We will call $\mathcal{X}^{\mathbb{N}}\coloneqq\bigtimes_{j\in\mathbb{N}}\mathcal{X}$, the set of all possible countable sequences where each variable takes values in $\mathcal{X}$.

First, we will need a way to relate gambles on different domains. Let $f$ be some gamble on $\mathcal{X}^{n}$, and let ${f}^{*}$ be its cylindrical extension, defined as

[TABLE]

Formally, ${f}^{*}$ belongs to $\mathcal{L}(\mathcal{X}^{\mathbb{N}})$ while $f$ belongs to $\mathcal{L}(\mathcal{X}^{n})$. However, they contain the same information, and therefore, are indistinguishable from a behavioural point of view. In this paper, we will identify $f$ with its cylindrical extension ${f}^{*}$. Using this convention, we can for instance identify $\mathcal{L}(\mathcal{X}^{n})$ with a subset of $\mathcal{L}(\mathcal{X}^{\mathbb{N}})$, and, as an other example, for any $\mathcal{A}\subseteq\mathcal{L}(\mathcal{X}^{\mathbb{N}})$, regard $\mathcal{A}\cap\mathcal{L}(\mathcal{X}^{n})$ as those gambles in $\mathcal{A}$ that depend upon the first $n$ variables only.

4.1 Marginalisation

Using the notational convention we just discussed, we can very easily define what marginalisation means for choice functions. Given any choice function $C$ on $\mathcal{L}(\mathcal{X}^{\mathbb{N}})$ and any $n$ in $\mathbb{N}$, its $\mathcal{X}^{n}$-marginal $C_{n}$ is determined by $C_{n}(O)\coloneqq C(O)$ for all $O$ in $\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n}))$.

Similarly, given any set of desirable gambles $D\subseteq\mathcal{L}(\mathcal{X}^{\mathbb{N}})$ and any $n$ in $\mathbb{N}$, its $\mathcal{X}^{n}$-marginal $D_{n}$ is defined by $D_{n}\coloneqq D\cap\mathcal{L}(\mathcal{X}^{n})$.

Coherence is preserved under marginalisation [it is an immediate consequence of the definition; see, amongst others, (de Cooman and Miranda, 2012, Proposition 6) for sets of desirable gambles]

Proposition 12

Consider any coherent choice function $C$ on $\mathcal{L}(\mathcal{X}^{\mathbb{N}})$ and any coherent set of desirable gambles $D\subseteq\mathcal{L}(\mathcal{X}^{\mathbb{N}})$. Then for every $n$ in $\mathbb{N}$, their $\mathcal{X}^{n}$-marginals $C_{n}$ and $D_{n}$ are coherent.

4.2 Gambles of finite structure

Before we can explain what it means to assess a countable sequence to be exchangeable, we need to realise that now there are (countably) infinite many variables. However, we do not regard it useful from a behavioural point of view to choose between gambles that depend upon an infinite number of variables. Indeed, since we will never be able to know the actual outcome, gambles will never be actually paid-off, and hence every assessment is essentially without any risk. Instead, we believe that it makes sense to only consider choices between gambles of finite structure: gambles that depend upon a finite number of variables only. See De Bock et al. (2016) for more information.

Definition 13** (Gambles of finite structure)**

We will call any gamble that depends only upon a finite number of variables a gamble of finite structure. We collect all such gambles in $\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$:

[TABLE]

$\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$ is a linear space, with the usual ordering $\leq$: for any $f$ and $g$ in $\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$, $f\leq g\Leftrightarrow f(x)\leq g(x)$ for all $x$ in $\mathcal{X}^{\mathbb{N}}$.

Due to our finitary context, we can even establish a converse result to Prop. 12, whose proof for the part about sets of desirable gambles can be found in (De Bock et al., 2016, Proposition 4), and for the part about choice functions is omitted since it is a straight-forward check of all the axioms.

Proposition 14

Consider any choice function $C$ on $\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$, and any set of desirable gambles $D\subseteq\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$. If for every $n$ in $\mathbb{N}$, its $\mathcal{X}^{n}$-marginal $C_{n}$ on $\mathcal{L}(\mathcal{X}^{n})$ is coherent, then $C$ is coherent. Similarly, if for every $n$ in $\mathbb{N}$, its $\mathcal{X}^{n}$-marginal $D_{n}\subseteq\mathcal{L}(\mathcal{X}^{n})$ is coherent, then $D$ is coherent.

4.3 Set of indifferent gambles

If a subject assesses the sequence of variables $X_{1}$, …, $X_{n}$, …to be exchangeable, this means that he is indifferent between any gamble $f$ in $\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$ and its permuted variant ${\pi}^{t}f$, for any $\pi$ in $\mathcal{P}_{n}$, where $n$ now is the (finite) number of variables that $f$ depends upon: his set of indifferent gambles is

[TABLE]

If we want to use $I_{\mathcal{P}}$ to define countable exchangeability, it must be a coherent set of indifferent gambles.

Proposition 15

The set $I_{\mathcal{P}}$ is a coherent set of indifferent gambles.

Countable exchangeability is now easily defined, similar to the definition for the finite case.

Definition 16

A choice function $C$ on $\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$ is called (countably) exchangeable if $C$ is compatible with $I_{\mathcal{P}}$. Similarly, a set of desirable gambles $D\subseteq\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$ is called (countably) exchangeable if it is compatible with $I_{\mathcal{P}}$.

This definition is closely related to its finite counterpart.

Proposition 17

Consider any coherent choice function $C$ on $\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$. Then $C$ is exchangeable if and only if for every choice of $n$ in $\mathbb{N}$, the $\mathcal{X}^{n}$-marginal $C_{n}$ of $C$ is exchangeable. Similarly, consider any coherent set of desirable gambles $D\subseteq\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$. Then $D$ is exchangeable if and only if for every choice of $n$ in $\mathbb{N}$, the $\mathcal{X}^{n}$-marginal $D_{n}$ of $D$ is exchangeable.

4.4 A representation theorem for countable sequences

We will look for a similar representation result. However, since we no longer deal with finite sequences of length $n$, now the representing choice function won’t be defined on $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$, but instead on $\mathcal{V}(\Sigma_{\mathcal{X}})$.

\mathcal{L}(\mathcal{X}^{n})$$\mathcal{L}(\mathcal{N}^{n})$$\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$$\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$$\mathcal{V}(\Sigma_{\mathcal{X}})$$\mathrm{H}_{n}$$\mathrm{M}_{n}$$[\cdot]$$\mathrm{CoM}_{n}$$\tilde{\mathrm{M}}_{n}$$\tilde{\mathrm{H}}_{n}

In the commuting diagram, a dashed line represents an embedding: indeed, for every $n$ in $\mathbb{N}$, $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ is a subspace of $\mathcal{V}(\Sigma_{\mathcal{X}})$. That shows the importance of the polynomial representation.

As we have seen, in order to define coherent choice functions on some linear space, we need to provide it with a vector ordering. Similar to what we did before, we use the proper cone $\{0\}\cup\operatorname*{posi}(\{B_{m}:m\in\mathcal{N}^{n},n\in\mathbb{N}\})$ to define the order $\preceq_{B}$ on $\mathcal{V}(\Sigma_{\mathcal{X}})$:

[TABLE]

for all $h_{1}$ and $h_{2}$ in $\mathcal{V}(\Sigma_{\mathcal{X}})$.

Keeping Props. 12 and 14 in mind, the following result is not surprising.

Proposition 18

Consider any choice function $C^{\prime}$ on $\mathcal{V}(\Sigma_{\mathcal{X}})$. Then $C^{\prime}$ is coherent if and only if for every $n$ in $\mathbb{N}$ the choice function $C^{\prime}_{n}$, given by $C^{\prime}_{n}(O)\coloneqq C^{\prime}(O)$ for all $O$ in $\mathcal{Q}(\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$ is coherent.

Theorem 19** (Countable Representation)**

Consider any choice function $C$ on $\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$. Then $C$ is exchangeable if and only if there is a unique representing choice function $\tilde{C}$ on $\mathcal{V}(\Sigma_{\mathcal{X}})$ such that, for every $n$ in $\mathbb{N}$, the $\mathcal{X}^{n}$-marginal $C_{n}$ of $C$ is determined by

[TABLE]

Furthermore, in that case, $\tilde{C}$ is given by $\tilde{C}(O)\coloneqq\bigcup_{n\in\mathbb{N}}\tilde{C}_{n}(O\cap\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$ for all $O$ in $\mathcal{Q}(\mathcal{V}(\Sigma_{\mathcal{X}}))$, with $\tilde{C}_{n}(\mathrm{M}_{n}(O))\coloneqq\mathrm{M}_{n}(C_{n}(O))$ for every $O$ in $\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n}))$, and where we let $\tilde{C}_{n}(\emptyset)\coloneqq\emptyset$ for notational convenience. Finally, $C$ is coherent if and only if $\tilde{C}$ is.

Similarly, consider any set of desirable gambles $D\subseteq\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$. Then $D$ is exchangeable if and only if there is a unique representing $\tilde{D}\subseteq\mathcal{V}(\Sigma_{\mathcal{X}})$ such that, for every $n$ in $\mathbb{N}$, the $\mathcal{X}^{n}$-marginal $D_{n}$ is given by $D_{n}=\bigcup\tilde{\mathrm{M}}_{n}^{-1}(\tilde{D}\cap\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$. Furthermore, in that case, $\tilde{D}$ is given by $\tilde{D}=\bigcup_{n\in\mathbb{N}}\mathrm{M}_{n}(D_{n})$. Finally, $D$ is coherent if and only if $\tilde{D}$ is.

5 Conclusion

We studied exchangeability and we have found counterparts to de Finetti’s finite and countable representation results, in the general setting of choice functions. We have shown that an exchangeability assessment is a particular indifference assessment, where we identified the set of indifferent options. The main idea that made (finite) representation possible is the linear order isomorphism ${{\tilde{\mathrm{H}}_{n}}}^{-1}$ between the quotient space and the set of gambles on count vectors, indicating that (finitely) exchangeable choice functions can be represented by a choice function that essentially represents preferences between urns with $n$ balls of types $\mathcal{X}$ with different compositions $m$. Alternatively, for the countable case, we have shown that there is a polynomial representation.

Choice functions form a belief structure (see Van Camp et al. (2017)). Therefore, any infimum of coherent choice functions is a coherent choice function itself. Since any infimum of choice functions compatible with some fixed set of indifferent options $I$, is compatible with $I$ as well (see Van Camp et al. (2017)), our results indicate that, using choice functions, it is conceptually easy to reason about exchangeable sequences: infima of exchangeable and coherent choice functions will be exchangeable and coherent as well.

A possible future goal is to investigate how exchangeability behaves under updating. It is shown, in de Cooman and Quaeghebeur (2012), that, for exchangeable sets of desirable gambles, updating can be done directly for the representing set of desirable gambles in the count space. We expect this to be the case for choice functions as well.

Acknowledgments

Gert de Cooman’s research was partly funded through project number 3G012512 of the Research Foundation Flanders (FWO). The authors would like to thank Enrique Miranda for his comments on a draft of the paper.

A Proofs of some results

**Proof **[Proof of Prop. 3] For Axiom I1, since $I_{\mathcal{P}_{n}}$ is a linear span, [math] is included in $I_{\mathcal{P}_{n}}$. For Axiom I2, consider any $f$ in $I_{\mathcal{P}_{n}}$ and assume ex absurdo that $f\in\mathcal{L}(\mathcal{X}^{n})_{>0}\cup\mathcal{L}(\mathcal{X}^{n})_{<0}$. If $f\in\mathcal{L}(\mathcal{X}^{n})_{>0}$ then ${\pi}^{t}f\in\mathcal{L}(\mathcal{X}^{n})_{>0}$ for all $\pi$ in $\mathcal{P}_{n}$, and therefore $\mathrm{inv}_{\mathcal{P}_{n}}(f)>0$, a contradiction with Prop. 2(iii). If $f\in\mathcal{L}(\mathcal{X}^{n})_{<0}$ then, similarly $\mathrm{inv}_{\mathcal{P}_{n}}(f)<0$, again a contradiction with Prop. 2(iii). Axioms I3 and I4 are satisfied because $I_{\mathcal{P}_{n}}$ is a linear span.

**Proof **[Proof of Prop. 4] Infer the following equivalences. Start with $[f]=[g]$, what, due to the definition of equivalence classes, is equivalent to $f+h_{1}=g+h_{2}$, and equivalently, $g-f=h_{1}-h_{2}$, for some $h_{1}$ and $h_{2}$ in $I_{\mathcal{P}_{n}}$. In turn, since $I_{\mathcal{P}_{n}}$ is a linear space, that is equivalent to $g-f\in I_{\mathcal{P}_{n}}$. Because $\ker(\mathrm{inv}_{\mathcal{P}_{n}})=I_{\mathcal{P}_{n}}$ [by Proposition 2(iii)], we find equivalently that $\mathrm{inv}_{\mathcal{P}_{n}}(g-f)=0$, and, due to the linearity of $\mathrm{inv}_{\mathcal{P}_{n}}$, equivalently $\mathrm{inv}_{\mathcal{P}_{n}}(f)=\mathrm{inv}_{\mathcal{P}_{n}}(g)$. Use Lemma 20 to find that, indeed, equivalently $\mathrm{H}_{n}(f)=\mathrm{H}_{n}(g)$.

Lemma 20

Consider any $f$ and $g$ in $\mathcal{L}(\mathcal{X}^{n})$. Then $\mathrm{inv}_{\mathcal{P}_{n}}(f)=\mathrm{inv}_{\mathcal{P}_{n}}(g)$ if and only if $\mathrm{H}_{n}(f)=\mathrm{H}_{n}(g)$.

**Proof **[Proof of Lemma 20] Infer the following equivalences. Start with $\mathrm{inv}_{\mathcal{P}_{n}}(f)=\mathrm{inv}_{\mathcal{P}_{n}}(g)$, what, by Eq. (2), is equivalent to $\sum_{m\in\mathcal{N}^{n}}\mathrm{H}_{n}({f}|{m})\mathbb{I}_{[m]}=\sum_{m\in\mathcal{N}^{n}}\mathrm{H}_{n}({g}|{m})\mathbb{I}_{[m]}$. Equivalently, we find that $\mathrm{H}_{n}({f}|{m})=\mathrm{H}_{n}({g}|{m})$ for all $m$ in $\mathcal{N}^{n}$. In turn, by Eq. (4), that is equivalent to $\mathrm{H}_{n}(f)=\mathrm{H}_{n}(g)$.

**Proof **[Proof of Prop. 5] This proof is structured as follows: we show that (i) ${{\tilde{\mathrm{H}}_{n}}}^{-1}\circ\tilde{\mathrm{H}}_{n}=\mathrm{id}_{\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}}$, and (ii) $\tilde{\mathrm{H}}_{n}\circ{{\tilde{\mathrm{H}}_{n}}}^{-1}=\mathrm{id}_{\mathcal{L}(\mathcal{N}^{n})}$, together implying that $\tilde{\mathrm{H}}_{n}$ and ${{\tilde{\mathrm{H}}_{n}}}^{-1}$ are each other’s inverses.

For (i), consider any $\tilde{f}$ in $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$. We need to show that then ${{\tilde{\mathrm{H}}_{n}}}^{-1}(\tilde{\mathrm{H}}_{n}(\tilde{f}))=\tilde{f}$. Let $h$ be an arbitrary element of $\tilde{f}$, and $f\coloneqq\mathrm{inv}_{\mathcal{P}_{n}}(h)$. Then $\mathrm{inv}_{\mathcal{P}_{n}}(f)=\mathrm{inv}_{\mathcal{P}_{n}}(h)$ by Prop. 2(ii), and therefore, using Lemma 20, $\mathrm{H}_{n}(f)=\mathrm{H}_{n}(h)$, so Prop. 4 implies that $f\in\tilde{f}$ as well. Then $\tilde{\mathrm{H}}_{n}(\tilde{f})$ assumes the value $\mathrm{H}_{n}(f)(m)=\frac{1}{\binom{n}{m}}\sum_{y\in[m]}f(y)$ on every $m$ in $\mathcal{N}^{n}$. But $f$ is constant on every permutation atom $[m]$, so $\frac{1}{\binom{n}{m}}\sum_{y\in[m]}f(y)=\frac{1}{\binom{n}{m}}|[m]|f(x)=f(x)$ for every $x$ in $[m]$, and therefore

[TABLE]

Then indeed ${{\tilde{\mathrm{H}}_{n}}}^{-1}(\tilde{\mathrm{H}}_{n}(\tilde{f}))=[\sum_{m\in\mathcal{N}^{n}}\tilde{\mathrm{H}}(\tilde{f})(m)\mathbb{I}_{[m]}]=[f]=\tilde{f}$, where the first equality follows from Eq. (6), the second one from Eq. (10), and the last one from the fact that $f\in\tilde{f}$ and $[f]=\{f\}+I_{\mathcal{P}_{n}}$.

For (ii), consider any $f$ in $\mathcal{L}(\mathcal{N}^{n})$. We need to show that then $\tilde{\mathrm{H}}_{n}({{\tilde{\mathrm{H}}_{n}}}^{-1}(f))=f$. Let $g\coloneqq\sum_{m\in\mathcal{N}^{n}}f(m)\mathbb{I}_{[m]}$, being a gamble on $\mathcal{X}^{n}$. Then ${{\tilde{\mathrm{H}}_{n}}}^{-1}(f)=[g]$ by Eq. (6), so $\tilde{\mathrm{H}}_{n}({{\tilde{\mathrm{H}}_{n}}}^{-1}(f))=\tilde{\mathrm{H}}_{n}([g])$, and since $g\in[g]$, we find using Eq. (5) that $\tilde{\mathrm{H}}_{n}([g])=\mathrm{H}_{n}(g)$ and therefore $\tilde{\mathrm{H}}_{n}({{\tilde{\mathrm{H}}_{n}}}^{-1}(f))=\mathrm{H}_{n}(g)$. The proof is finished if we show that $\mathrm{H}_{n}(g)=f$. Consider any $m^{\prime}$ in $\mathcal{N}^{n}$, and infer that

[TABLE]

where the first equality follows from Eq. (3) and the penultimate from the fact that

[TABLE]

Therefore indeed $\tilde{\mathrm{H}}_{n}({{\tilde{\mathrm{H}}_{n}}}^{-1}(f))=\mathrm{H}_{n}(g)=f$.

**Proof **[Proof of Prop. 6] For necessity, assume that $\tilde{f}\preceq\tilde{g}$. Then, by Eq. (7), $f\leq g$ for some $f$ in $\tilde{f}$ and $g$ in $\tilde{g}$. Consider any $m$ in $\mathcal{N}^{n}$, and infer that $\mathrm{H}_{n}(f)(m)=\frac{1}{\binom{n}{m}}\sum_{y\in[m]}f(y)\leq\frac{1}{\binom{n}{m}}\sum_{y\in[m]}g(y)=\mathrm{H}_{n}(g)(m)$. Then $\mathrm{H}_{n}(f)\leq\mathrm{H}_{n}(g)$, and therefore, by Eq. (5), indeed $\tilde{\mathrm{H}}_{n}(\tilde{f})\leq\tilde{\mathrm{H}}_{n}(\tilde{g})$.

For sufficiency, assume that $\tilde{\mathrm{H}}_{n}(\tilde{f})\leq\tilde{\mathrm{H}}_{n}(\tilde{g})$. Then, by Eq. (5) and Prop. 4, $\mathrm{H}_{n}(f)\leq\mathrm{H}_{n}(g)$ for all $f$ in $\tilde{f}$ and $g$ in $\tilde{g}$. Consider any $f$ in $\tilde{f}$ and $g$ in $\tilde{g}$ and let $f^{\prime}\coloneqq\mathrm{inv}_{\mathcal{P}_{n}}(f)$ and $g^{\prime}\coloneqq\mathrm{inv}_{\mathcal{P}_{n}}(g)$. Then $\mathrm{inv}_{\mathcal{P}_{n}}(f^{\prime})=\mathrm{inv}_{\mathcal{P}_{n}}(f)$ and $\mathrm{inv}_{\mathcal{P}_{n}}(g^{\prime})=\mathrm{inv}_{\mathcal{P}_{n}}(g)$ by Prop. 2(ii), so Lemma 20 and Prop. 4 together imply that $f^{\prime}\in\tilde{f}$ and $g^{\prime}\in\tilde{f}$, so $\mathrm{H}_{n}(f^{\prime})\leq\mathrm{H}_{n}(g^{\prime})$. Then, by Eqs. (4) and (3), $\frac{1}{\binom{n}{m}}\sum_{y\in[m]}f^{\prime}(y)\leq\frac{1}{\binom{n}{m}}\sum_{y\in[m]}g^{\prime}(y)$ for every $m$ in $\mathcal{N}^{n}$. But $f^{\prime}$ and $g^{\prime}$ are constant on every $[m]$, so $f^{\prime}(y)\leq g^{\prime}(y)$ for every $y$ in $[m]$ and every $m$ in $\mathcal{N}^{n}$. Then $f^{\prime}\leq g^{\prime}$, and therefore indeed $\tilde{f}\preceq\tilde{g}$.

**Proof **[Proof of Theorem 7] We begin with the representation of choice functions. For the first statement, note that $C$ is exchangeability is equivalent to compatibility with $I_{\mathcal{P}_{n}}$ [by Def. 1], and equivalently, there is some representing choice function $C^{\prime}$ on $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ such that $C(O)=\{f\in O:[f]\in C^{\prime}(O/I_{\mathcal{P}_{n}})\}$ for all $O$ in $\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n}))$. We use the linear order isomorphism $\tilde{\mathrm{H}}_{n}$ to define a choice function $\tilde{C}$ on $\mathcal{L}(\mathcal{N}^{n})$: we let $[f]\in C^{\prime}(O/I_{\mathcal{P}_{n}})\Leftrightarrow\tilde{\mathrm{H}}_{n}([f])\in\tilde{C}(\tilde{\mathrm{H}}_{n}(O/I_{\mathcal{P}_{n}}))$ for all $f$ in $\mathcal{L}(\mathcal{X}^{n})$ and $O$ in $\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n}))$. Since $f\in[f]$, use Prop. 4 and Eq. (5) to infer that $\tilde{\mathrm{H}}_{n}([f])=\mathrm{H}_{n}(f)$. Similarly, infer that $\tilde{\mathrm{H}}_{n}(O/I_{\mathcal{P}_{n}})=\{\tilde{\mathrm{H}}_{n}([g]):g\in O\}=\{\mathrm{H}_{n}(g):g\in O\}=\mathrm{H}_{n}(O)$, so $[f]\in C^{\prime}(O/I_{\mathcal{P}_{n}})\Leftrightarrow\mathrm{H}_{n}(f)\in\tilde{C}(\mathrm{H}_{n}(O))$. Then indeed

[TABLE]

To show that $\tilde{C}$ is unique, use that $C^{\prime}$ is unique and $\tilde{\mathrm{H}}_{n}$ is a bijection to infer that $\tilde{C}$ is unique too.

For the second statement, consider any $O$ in $\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n}))$ and infer, using the definition of $\tilde{C}$, that $\tilde{C}(\tilde{\mathrm{H}}_{n}(O/I_{\mathcal{P}_{n}}))=\tilde{\mathrm{H}}_{n}(C^{\prime}(O/I_{\mathcal{P}_{n}}))$ , and therefore $\tilde{C}(\mathrm{H}_{n}(O))=\tilde{\mathrm{H}}_{n}(C^{\prime}(O/I_{\mathcal{P}_{n}}))$. Since $C^{\prime}$ is given by $C^{\prime}(O/I_{\mathcal{P}_{n}})=C(O)/I_{\mathcal{P}_{n}}$, we find $\tilde{\mathrm{H}}_{n}(C^{\prime}(O/I_{\mathcal{P}_{n}}))=\tilde{\mathrm{H}}_{n}(C(O)/I_{\mathcal{P}_{n}})=\mathrm{H}_{n}(C(O))$, and therefore indeed $\tilde{C}(\mathrm{H}_{n}(O))=\mathrm{H}_{n}(C(O))$.

For the third statement, by the compatibility with $I_{\mathcal{P}_{n}}$ guarantees that $C$ is coherent if and only if $C^{\prime}$ on $\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ is coherent. But since $\tilde{C}$ is defined from $C^{\prime}$ using the linear order isomorphism $\mathrm{H}_{n}$, we have immediately that $\tilde{C}$ is coherent if and only if $C^{\prime}$ is coherent. Therefore indeed $C$ is coherent if and only if $\tilde{C}$ is coherent.

We now turn to the representation for sets of desirable gambles. Since $D$ is compatible with $I_{\mathcal{P}_{n}}$, there is some representing set of desirable gambles $D^{\prime}\subseteq\mathcal{L}(\mathcal{X}^{n})/{I_{\mathcal{P}_{n}}}$ such that $D=\bigcup D^{\prime}$. Using the linear order isomorphism $\tilde{\mathrm{H}}_{n}$ we can transform $D^{\prime}$ to $\tilde{D}\coloneqq\tilde{\mathrm{H}}_{n}(D^{\prime})$ on the isomorphic space $\mathcal{L}(\mathcal{N}^{n})$. Then $D^{\prime}={{\tilde{\mathrm{H}}_{n}}}^{-1}(\tilde{D})$, so indeed $D=\bigcup{{\tilde{\mathrm{H}}_{n}}}^{-1}(\tilde{D})$. Because compatibility with $I_{\mathcal{P}_{n}}$ guarantees that $D^{\prime}$—and therefore also $\tilde{D}$—is unique, we have shown the first statement to be true.

For the second statement, infer that indeed $\tilde{D}=\tilde{\mathrm{H}}_{n}(D^{\prime})=\mathrm{H}_{n}(D)$ by Eq. (5).

For the third statement, compatibility with $I_{\mathcal{P}_{n}}$ guarantees that $D$ is coherent if and only if $D^{\prime}$ is coherent. But since $\tilde{D}$ is defined from $D^{\prime}$ using the linear order isomorphism $\mathrm{H}_{n}$, we have immediately that $\tilde{D}$ is coherent if and only if $D^{\prime}$ is coherent. Therefore indeed $D$ is coherent if and only if $\tilde{D}$ is coherent.

**Proof **[Proof of Theorem 11] Let $C^{\prime\prime}$ on $\mathcal{L}(\mathcal{N}^{n})$ and $D^{\prime\prime}\subseteq\mathcal{L}(\mathcal{N}^{n})$ be the representing choice function and set of desirable gambles from Theorem 7, and let $\tilde{C}$ be defined by

[TABLE]

for all $O$ in $\mathcal{Q}(\mathcal{L}(\mathcal{N}^{n}))$, and $\tilde{D}\coloneqq\mathrm{CoM}_{n}(D^{\prime\prime})$. Since $\mathrm{CoM}_{n}$ is a linear order isomorphism, then $\mathrm{M}_{n}(f)\in\tilde{C}(\mathrm{M}_{n}(O))\Leftrightarrow\mathrm{H}_{n}(f)\in C^{\prime\prime}(\mathrm{H}_{n}(O))$ for all $O$ in $\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n}))$, and $\tilde{D}=\mathrm{CoM}_{n}(\mathrm{H}_{n}(D))=\mathrm{M}_{n}(D)$, and all the coherence properties are preserved, from which the statements follow.

**Proof **[Proof of Prop. 15] For Axiom I1, since, by Prop. 3, $0\in I_{\mathcal{P}_{n}}$ for every $n$ in $\mathbb{N}$, also $0\in I_{\mathcal{P}}$. For Axiom I2, consider any $f$ in $I_{\mathcal{P}}$, then there is some $n$ in $\mathbb{N}$ for which $f\in I_{\mathcal{P}_{n}}$ By Prop. 3, we infer that indeed $f\not<0$ and $f\not>0$. For Axioms I3 and I4, consider any $f_{1}$, $f_{2}$ and $f_{3}$ in $I_{\mathcal{P}}$ and any $\lambda$ in $\mathbb{R}$. Then there are $n_{i}$ in $\mathbb{N}$ such that $f_{i}\in I_{\mathcal{P}_{n_{i}}}$, for every $i$ in $\{1,2,3\}$. Let $n\coloneqq\max\{n_{1},n_{2},n_{3}\}$. Then $f_{1}$, $f_{2}$ and $f_{3}$ are elements of $I_{\mathcal{P}_{n}}$, so $\lambda f_{1}\in I_{\mathcal{P}_{n}}$ and $f_{2}+f_{3}\in I_{\mathcal{P}_{n}}$ by Prop. 3. Then indeed $\lambda f_{1}\in I_{\mathcal{P}}$ and $f_{2}+f_{3}\in I_{\mathcal{P}}$.

**Proof **[Proof of Prop. 17] The proof for sets of desirable gambles, in a more general context, can be found in (De Bock et al., 2016, Proposition 18).

We give the proof for choice functions. For necessity, assume that $C$ is exchangeable, or equivalently, that $C$ is compatible with $I_{\mathcal{P}}$. Use Ref. (Van Camp et al., 2017, Proposition 31) to infer that then, equivalently,

[TABLE]

Consider any $n$ in $\mathbb{N}$. We need to prove that then $C_{n}$ is compatible with $I_{\mathcal{P}_{n}}$, or equivalently, that

[TABLE]

So consider any $O$ in $\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n}))$ such that $0\in O$, and any $h$ in $O$. Then $O$ is an element of $\mathcal{Q}(\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}}))$ and $h$ an element of $\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}})$, so $0\in C(O)\Leftrightarrow h\in C(O)$. Therefore, by marginalisation, $0\in C_{n}(O)\Leftrightarrow h\in C_{n}(O)$, whence $C_{n}$ is compatible with $I_{\mathcal{P}_{n}}$, and therefore indeed exchangeable.

For sufficiency, assume that $C_{n}$ is exchangeable for every $n$ in $\mathbb{N}$—so it satisfies Eq. (12) for every $n$ in $\mathbb{N}$. We need to prove that then $C$ is exchangeable. Using Eq. (11), it suffices to consider any $\tilde{O}$ in $\mathcal{Q}(\bar{\mathcal{L}}(\mathcal{X}^{\mathbb{N}}))$ such that $0\in\tilde{O}$, and any $\tilde{h}$ in $\tilde{O}$, and prove that $0\in C(\tilde{O})\Leftrightarrow\tilde{h}\in C(\tilde{O})$. Since $\tilde{O}\cup\{\tilde{h}\}$ consist of gambles of finite structure, there is some (sufficiently large) $n$ in $\mathbb{N}$ for which $\tilde{O}\in\mathcal{Q}(\mathcal{L}(\mathcal{X}^{n}))$ and $\tilde{h}\in\mathcal{L}(\mathcal{X}^{n})$. Then by Eq. (12), $0\in C_{n}(\tilde{O})\Leftrightarrow\tilde{h}\in C_{n}(\tilde{O})$, so $0\in C(\tilde{O})\Leftrightarrow\tilde{h}\in C(\tilde{O})$, whence $C$ is compatible with $I_{\mathcal{P}}$, and therefore indeed exchangeable.

**Proof **[Proof of Prop. 18] We only prove sufficiency, since necessity is trivial. So consider any $C^{\prime}$ on $\mathcal{V}(\Sigma_{\mathcal{X}})$ such that for every $n$ in $\mathbb{N}$, $C^{\prime}_{n}$ is coherent. We prove that then $C^{\prime}$ is coherent.

For Axiom C1, consider any $O$ in $\mathcal{V}(\Sigma_{\mathcal{X}})$. Then every polynomial in $O$ has a certain degree; let $n$ be the maximum of those degrees. Then $O\in\mathcal{Q}(\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$, whence indeed $C^{\prime}(O)=C^{\prime}_{n}(O)\neq\emptyset$, since $C^{\prime}_{n}$ is coherent.

For Axiom C2, consider any $h_{1}$ and $h_{2}$ in $\mathcal{V}(\Sigma_{\mathcal{X}})$ such that $h_{1}\preceq_{B}h_{2}$. Then $h_{2}-h_{1}\in\operatorname*{posi}(\{B_{m}:m\in\mathcal{N}^{n},n\in\mathbb{N}\})$. Let $n_{1}$ be the degree of $h_{1}$ and $n_{2}$ the degree of $h_{2}$, and let $n\coloneqq\max\{n_{1},n_{2}\}$. Then the degree of the polynomial $h_{2}-h_{1}$ is not higher than $n$, so $h_{2}-h_{1}\in\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ and therefore $h_{2}-h_{1}\in\operatorname*{posi}(\{B_{m}:m\in\mathcal{N}^{n}\})$. That guarantees that $h_{1}\preceq_{B}^{n}h_{2}$, whence indeed $\{h_{2}\}=C^{\prime}_{n}(\{h_{1},h_{2}\})=C^{\prime}(\{h_{1},h_{2}\})$, since $C^{\prime}_{n}$ is coherent.

For Axiom C3, consider any $O$, $O_{1}$ and $O_{2}$ in $\mathcal{Q}(\mathcal{V}(\Sigma_{\mathcal{X}}))$, and let $n$ be the highest degree that appears in $O\cup O_{1}\cup O_{2}$. Then $O$, $O_{1}$ and $O_{2}$ all are elements of $\mathcal{Q}(\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$. For Axiom C3C3a, assume that $C^{\prime}(O_{2})\subseteq O_{2}\setminus O_{1}$ and $O_{1}\subseteq O_{2}\subseteq O$. Then $C^{\prime}_{n}(O_{2})\subseteq O_{2}\setminus O_{1}$ and therefore, since $C^{\prime}_{n}$ is coherent, indeed $C^{\prime}(O)=C^{\prime}_{n}(O)\subseteq O\setminus O_{1}$. For Axiom C3C3b, assume that $C^{\prime}(O_{2})\subseteq O_{1}$ and $O\subseteq O_{2}\setminus O_{1}$. Then $C^{\prime}_{n}(O_{2})\subseteq O_{2}\setminus O_{1}$ and therefore, since $C^{\prime}_{n}$ is coherent, indeed $C^{\prime}(O_{2}\setminus O)=C^{\prime}_{n}(O_{2}\setminus O)\subseteq O_{1}$.

For Axiom C4, consider any $h$ in $\mathcal{V}(\Sigma_{\mathcal{X}})$, any $\lambda$ in $\mathbb{R}_{>0}$ and any $O_{1}$ and $O_{2}$ in $\mathcal{Q}(\mathcal{V}(\Sigma_{\mathcal{X}}))$, and let $n$ be the highest degree that appears in $\{h\}\cup O_{1}\cup O_{2}$. Then $h\in\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ and $O_{1}$ and $O_{2}$ both are elements of $\mathcal{Q}(\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$. Assume that $O_{1}\subseteq C^{\prime}(O_{2})=C^{\prime}_{n}(O_{2})$, then, since $C^{\prime}_{n}$ is coherent, indeed $\lambda O_{1}\subseteq C^{\prime}_{n}(\lambda O_{2})=C^{\prime}(\lambda O_{2})$ and $O_{1}+\{h\}\subseteq C^{\prime}_{n}(O_{2}+\{h\})=C^{\prime}(O_{2}+\{h\})$.

**Proof **[Proof of Theorem 19] We first prove the representation theorem for choice functions. That $C$ is exchangeable is, by Prop. 17, equivalent to, $C_{n}$ is exchangeable, for every $n$ in $\mathbb{N}$. Therefore, for all $n$ in $\mathbb{N}$, by Theorem 11, that is equivalent to

[TABLE]

where $\tilde{C}_{n}$ is uniquely given by

[TABLE]

Consider any choice function $\tilde{C}$ on $\mathcal{Q}(\mathcal{V}(\Sigma_{\mathcal{X}}))$ such that $\tilde{C}(\mathrm{M}_{n}(O))=\tilde{C}_{n}(\mathrm{M}_{n}(O))$. Since $\mathrm{M}_{n}$ is a surjection, we find that then $\tilde{C}(O^{\prime})=\tilde{C}_{n}(O^{\prime})$ for all $O^{\prime}$ in $\mathcal{Q}(\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$. But every $O^{\prime}$ in $\mathcal{Q}(\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$ can be identified with $O\cap\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ for some $O$ in $\mathcal{Q}(\mathcal{V}(\Sigma_{\mathcal{X}}))$, so $\tilde{C}(O)=\tilde{C}_{n}(O\cap\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$ for all $O$ in $\mathcal{Q}(\mathcal{V}(\Sigma_{\mathcal{X}}))$, where, again, we use the assumption that $\tilde{C}_{n}(\emptyset)=\emptyset$, and thus making $\tilde{C}$ unique. Then $\tilde{C}(\mathrm{M}_{n}(O))=\bigcup_{n\in\mathbb{N}}\tilde{C}_{n}(\mathcal{V}^{n}(\Sigma_{\mathcal{X}})\cap\mathrm{M}_{n}(O))=\tilde{C}_{n}(\mathrm{M}_{n}(O))$, so indeed

[TABLE]

proving the first and second statement.

For the third statement, infer from Prop. 12 and Prop. 14 that $C$ is coherent if and only if $C_{n}$ is coherent for every $n$ in $\mathbb{N}$. Infer from Theorem 11 that, for every $n$ in $\mathbb{N}$, $C_{n}$ is coherent if and only if $\tilde{C}_{n}$ on $\mathcal{V}^{n}(\Sigma_{\mathcal{X}})$ is coherent. Infer that, by construction, $\tilde{C}(O)=\tilde{C}_{n}(O)$ for every $O$ in $\mathcal{Q}(\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$, then Prop. 18 tells us that indeed $\tilde{C}_{n}$ is coherent if and only if $\tilde{C}$ on $\mathcal{V}(\Sigma_{\mathcal{X}})$ is coherent.

We now turn to the representation theorem for sets of desirable gambles. That $D$ is exchangeable is, by Prop. 17, equivalent to, $D_{n}$ is exchangeable, for every $n$ in $\mathbb{N}$. Therefore, for any $n$ in $\mathbb{N}$, by Theorem 11, that is equivalent to $D_{n}=\bigcup\tilde{\mathrm{M}}_{n}^{-1}(\tilde{D}_{n})$ where $\tilde{D}_{n}$ is uniquely given by $\tilde{D}_{n}=\mathrm{M}_{n}(D_{n})$. Infer that $D_{n}=\bigcup D_{n}/I_{\mathcal{P}_{n}}$ by the compatibility of $D_{n}$ with $I_{\mathcal{P}_{n}}$. Using Eq. (5), infer that $D_{n}/I_{\mathcal{P}_{n}}=\tilde{\mathrm{M}}_{n}^{-1}(\mathrm{M}_{n}(D_{n}))$, whence $D_{n}=\bigcup\tilde{\mathrm{M}}_{n}^{-1}(\mathrm{M}_{n}(D_{n}))$, which is equal to $\tilde{\mathrm{M}}_{n}^{-1}(\bigcup_{k\in\mathbb{N}}\mathrm{M}_{k}(D_{k})\cap\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$. This shows that $D_{n}$ equals $\tilde{\mathrm{M}}_{n}^{-1}(\tilde{D}\cap\mathcal{V}^{n}(\Sigma_{\mathcal{X}}))$ for some $\tilde{D}$, and that $\tilde{D}$ is exactly given by $\tilde{D}=\bigcup_{n\in\mathbb{N}}\mathrm{M}_{n}(D_{n})=\bigcup_{n\in\mathbb{N}}\tilde{D_{n}}$, also proving its uniqueness. This proves the first and the second statement.

For the third statement, since $\tilde{D}=\bigcup_{n\in\mathbb{N}}\mathrm{M}_{n}(D_{n})$, we see that clearly $\tilde{D}$ is coherent if and only every $\mathrm{M}_{n}(D_{n})$ is coherent, which is, by Theorem 11 equivalent to $D_{n}$ is coherent for every $n$ in $\mathbb{N}$. Now use Props. 12 and 14 to infer that this is equivalent to $D$ is coherent, proving the third statement.