2.1 Representations of total trading systems

In this section, we show how to represent a total trading system by a (possibly nonstandard-valued) numerical function.

The following result follows easily from Definition 2.1.

Proposition 2.2. A numerical function U represents a total preorder $\precsim $ if and only if

(5) $$ \begin{align} X\prec Y\text{\ if and only if\ } U(X)<U(Y). \end{align} $$

Next, we introduce a class of numerical functions that represent total trading systems.

Proof. Assume that U is a standard-linear function on a standard-linear space $\mathcal {X}$ . Define the total preorder $\precsim $ on $\mathcal {X}$ by (2). For the first claim, we need to verify Assumptions 1–4. Assumption 1 follows because a preorder is reflexive. For Assumption 2, suppose that $Y-X=Y'-X'$ . Since U is standard-linear,

$$\begin{align*}U(Y)-U(X)=U(Y-X)=U(Y'-X')=U(Y')-U(X').\end{align*}$$

For Assumption 3, note that U being standard-linear implies that

$$\begin{align*}U(\alpha_1X_1+\alpha_2X_2)=\alpha_1U(X_1)+\alpha_2U(X_2),\end{align*}$$

for all standard $\alpha _1,\alpha _2$ and $X_1,X_2\in \mathcal {X}$ . For Assumption 4, note that $\ll $ is $\prec $ .

For the second claim, we need to prove that Assumption 5 holds if and only if U is monotone and respects dominance. For the “if” direction, assume that U is monotone and respects dominance. Since U is monotone, $X\leq Y$ implies $U(X)\leq U(Y)$ and $X\precsim Y$ . Since U respects dominance, $X\prec _{_{{\scriptsize \text{Dom}}}} Y$ implies $U(X)<U(Y)$ and $X\prec Y$ , so Assumption 5 holds. For the “only if” direction, assume that Assumption 5 holds. To see that U is monotone, assume that $X\leq Y$ . The first requirement of Assumption 5 is that $X\precsim Y$ , which implies that $U(X)\leq U(Y)$ , and U is monotone. To see that U respects dominance, assume that $X\prec _{_{{\scriptsize \text{Dom}}}} Y$ . The second requirement of Assumption 5 is that $X\prec Y$ , which implies that $U(X)<U(Y)$ , and U respects dominance.

2.2 Agreement, representation and extension

If $\precsim $ is a not a total preorder, then there can be no numerical function U such that (2) holds. The problem is the “if” direction of (2) rather than the “only if” direction. In other words, representation as defined in Definition 1.3 is not achievable for preorders that are not total. On the other hand, agreement with a preorder and with a strict partial order are possible.

When it comes to extension of a trading system, there are two modes of extension that are important to our analysis. One mode corresponds to adding more comparisons (amongst elements of a single set $\mathcal {X}$ ) to the preorder, bringing it closer to being total. The other mode corresponds to expanding the domain of definition of the preorder (from one set $\mathcal {X}$ to a larger set $\mathcal {X}'$ ). Along with the second mode of extension comes a corresponding concept of restricting the domain of definition.

To be clear, each binary relation and each trading system is both an extension $_1$ and an extension $_2$ of itself. The following result about extension $_2$ is key in our theorems on representation. Its proof appears in Appendix C.1.

Note that Lemma 2.6 above (as well as Lemma 2.7 and Theorem 2.9 below) have language about $\leq $ and $\prec _{_{{\scriptsize \text{Dom}}}}$ on a larger space being extensions $_2$ of $\leq $ and $\prec _{_{{\scriptsize \text{Dom}}}}$ on a smaller space. When $\mathcal {O}={\mathbb {R}}$ , this condition is met trivially. The language is included to allow us to use these same results in other cases after Section 4.1 where $\leq $ and $\prec _{_{{\scriptsize \text{Dom}}}}$ are defined in terms of each specific trading system.

2.3 Finding agreeing functions

Our main representation Theorem 2.8 states that a trading system $\mathcal {T}$ has a standard-linear function U that agrees with it and an extension $_1$ to a total trading system that is represented by U. Results from [Reference Giarlotta, Doumpos, Figueira, Greco and Zopounidis18, Reference Giarlotta and Greco19, Reference Nishimura and Ok28] give the extension $_1$ for a general preorder, but without the representing function and without attention to the properties of a trading system. The following result has both Theorems 2.8 and 2.9 as special cases, and its proof appears in Appendix C.3.

Example 5 (Continuation of Example 4).

Recall the smallest coherent trading system $\mathcal {T}=(\mathcal {X},\precsim ,\ll )$ from Example 4. This time, assume that $\prec _{_{{\scriptsize \text{Dom}}}}$ means weak dominance and that $\Omega $ is uncountable. Then $X\prec Y$ and $X\ll Y$ both mean that Y weakly dominates X. Here we show that $\precsim $ is non-Archimedean. Theorem 2.8 says that $\mathcal {T}$ can be extended $_1$ to a coherent total trading system $\mathcal {T}'=(\mathcal {X},\precsim ',\prec ')$ with a representing function U that agrees with $\mathcal {T}$ . Without loss of generality, assume that $U(0)=0$ and $U(1)=1$ . So $X\prec Y$ (i.e., $X\prec _{_{{\scriptsize \text{Dom}}}} Y$ ) implies $U(X)<U(Y)$ . For every nonempty finite subset $E=\{\omega _1,\dots ,\omega _n\}$ of $\Omega $ , dominance and standard-linearity imply that

(6) $$ \begin{align} 0<U(I_E)=\sum_{j=1}^nU(I_{\{\omega_j\}})<U(1)=1. \end{align} $$

For each $\omega \in \Omega $ either $U(I_{\{\omega \}})$ is infinitesimal or it is greater than $1/n$ for some standard integer n. There can be no more than $n-1$ values of $U(I_{\{\omega \}})>1/n$ or (6) would be violated for a finite set E containing n such $\omega $ points. Hence, all but at most countably many $\omega $ have $U(I_{\{\omega \}})$ infinitesimal. Let $\omega _1,\omega _2\in \Omega $ be such that $U(I_{\{\omega _j\}})$ is infinitesimal for $j=1,2$ . Let $X=-1$ , $Y=I_{\{\omega _1,\omega _2\}}$ , and $Z=I_{\{\omega _1\}}$ . Then, $Z\prec _{_{{\scriptsize \text{Dom}}}} Y$ , $U(Z)>0$ , and $U(\alpha X+[1-\alpha ]Y)<0$ if $0<\alpha \leq 1$ . It follows that

$$\begin{align*}\alpha X+(1-\alpha)Y\prec_{_{{\scriptsize \text{Dom}}}} Z,\text{ for all}\ 0<\alpha\leq1,\end{align*}$$

but $Z\prec _{_{{\scriptsize \text{Dom}}}} Y$ , which violates (1). Hence the preference is non-Archimedean. It also follows that for each coherent extension $_1 \mathcal {T}'$ , $\precsim '$ is also non-Archimedean. For example, we might want all $I_{\{\omega \}}$ to be indifferent to each other. In this case, we can construct a representing function U as follows. For each constant function $X\equiv c$ , $U(X)=c$ , and for each nonconstant function $X=h_0+\sum _{j=1}^nh_jI_{\{\omega _j\}}$ , $U(X)=h_0+\epsilon \sum _{j=1}^nh_j$ . Note that this U is standard-linear and monotone, and it respects weak dominance, so Lemma 2.4 says that it represents a coherent total trading system which extends $_1 \mathcal {T}$ .

Theorem 2 of [Reference Skala40] shows that every total preorder can be represented by a nonstandard-valued function, but the standard-linear property that we need is not proven in that paper.

2.4 Extending $_2$ a trading system

Let $\mathcal {T}=(\mathcal {X},\precsim ,\ll )$ be a (coherent) trading system. Theorem 2.8 says that there exists a standard-linear function U that agrees with $\mathcal {T}$ and such that U represents a (coherent) total trading system $\mathcal {T}'$ that is an extension $_1$ of $\mathcal {T}$ . Extension $_2$ of $\mathcal {T}'$ is also possible if $\mathcal {X}$ is a subspace of a larger standard-linear space, as stated in Theorem 2.9.

Example 6 (Continuation of Example 5).

The linear space $\mathcal {X}$ used in Example 5 is the set of all standard-valued functions on an uncountable set $\Omega $ that are constant except at possibly finitely many points. Because weak dominance is defined on the set of all functions from $\Omega $ to ${\mathbb {R}}$ , Theorem 2.9 says that we could extend $_2$ each of the non-Archimedean coherent total trading systems from Example 5 to non-Archimedean coherent total trading systems on larger linear spaces $\mathcal {X}'$ of standard-valued functions.

For example, suppose that $\Omega =[0,1]$ , the closed unit interval and that we have chosen $\mathcal {T}$ to make $U(I_{\{\omega \}})=\epsilon $ for all $\omega $ . It is possible to extend $_2\ \mathcal {T}$ to the set $\mathcal {X}'$ of all simple functions that are measurable with respect to the smallest field $\Sigma $ of subsets of $\Omega $ that contains all intervals (closed, open, half-open, or degenerate). For example, the degenerate interval $[\omega ,\omega ]$ is the singleton $\{\omega \}$ . (We do not allow $[\omega ,\omega )$ to be called an interval.) The field $\Sigma $ consists of all unions of finitely many disjoint intervals. The simple functions that are measurable with respect to $\Sigma $ have the form

(7) $$ \begin{align} \sum_{j=1}^nh_jI_{J_j}, \end{align} $$

for finite n, standard numbers $h_1,\ldots ,h_n$ , and disjoint intervals $J_1,\ldots ,J_n$ . For a nondegenerate interval J define $U'(I_J)$ to be the length of J minus $\epsilon $ for each missing endpoint, and $U'(I_{\{\omega \}})=\epsilon $ . Although the given representation of functions in $\mathcal {X}$ is not unique, it is straightforward to show that, for X being the function in (7),

$$\begin{align*}U'(X)=\sum_{j=1}^nh_jU'(I_{J_j}),\end{align*}$$

is well defined and extends the function U defined in Example 5. The function $U'$ is standard-linear and monotone, and it respects weak dominance. Hence, by Lemma 2.4, $U'$ represents a coherent total trading system which extends $_2\ \mathcal {T}$ because $U'$ extends U.

The following example is modified from [Reference Seidenfeld, Schervish and Kadane38]. It involves random variables with a familiar distribution in which externally infinite nonstandard numbers are needed to represent non-Archimedean preferences amongst them.

Example 7. Let $\mathbb {Z}^+$ stand for the set of positive integers, and let $\Omega =\mathbb {Z}^+\times \{1,2\}$ . Let P be the following probability on $\Omega $ : $P(\{(n,j)\})=2^{-n-1}$ for $n\in \mathbb {Z}^+$ and $j\in \{1,2\}$ , which is countably-additive. Let $\mathcal {X}$ be the set of all bounded functions defined on $\Omega $ , each of which has a finite expected value $E(X)$ with respect to P. The function $E(X)$ represents a (weakly, strictly, and uniformly) coherent total trading system $\mathcal {T}=(\mathcal {X},\precsim ,\prec )$ .

Define the following three unbounded functions on $\Omega $ :

$$ \begin{align*} X_1((n,j))&=\begin{cases} 1,&\text{if } j=1,\\ 2^n,&\text{if }j=2,\end{cases}\\ X_2((n,j))&=\begin{cases} 1,&\text{if }j=2,\\ 2^n,&\text{if }j=1,\end{cases}\\ W((n,j))&=2^{n-1}. \end{align*} $$

It follows that all three of these functions have the same distribution, namely, each of them takes the value $2^k$ with probability $2^{-k}$ for $k=1,2,...$ . Each of them also has infinite expected value. Let $\mathcal {X}'$ be the linear span of $\mathcal {X}\bigcup \{X_1,X_2,W\}$ . All three forms of dominance are defined on $\mathcal {X}'$ and are extensions $_2$ of the same form of dominance on $\mathcal {X}$ . We can use Theorem 2.9 to extend $_2\ \mathcal {T}$ to a coherent total trading system $\mathcal {T}'=(\mathcal {X},\precsim ',\prec ')$ with representing function $U'$ . Since each of $X_1$ , $X_2$ , and W has infinite expected value under P, it is necessary that all three of $U'(X_1)$ , $U'(X_2)$ , and $U'(W)$ must be externally infinite nonstandard numbers. However, we show next that, despite having identical distributions, $X_1$ , $X_2$ , and W cannot be indifferent in a (weakly, strictly, or uniformly) coherent trading system.

First, note that

(8) $$ \begin{align} X_1(\omega)+X_2(\omega)=2W(\omega)+1, \end{align} $$

for all $\omega $ . Hence there is a uniform dominance relation that cannot be respected by a preference in which all three are indifferent. We could, for example, have $W\prec ' X_1\sim ' X_2$ by choosing an externally infinite x and setting $U'(X_1)=U'(X_2)=x$ and $U'(W)=x-1/2$ . There are other possible choices for $U'(X_1)$ and $U'(X_2)$ , but they must satisfy $U'(X_1)+U'(X_2)=2U'(W)+1$ because of (8). Finally, $U'$ is well defined on all of $\mathcal {X}'$ which includes all linear combinations of $X_1$ , $X_2$ , and W with both positive and negative coefficients.Footnote ⁵

There are many other examples of standard-linear spaces with proper supersets. One can easily imagine an agent determining a set of preferences over a small set $\mathcal {X}$ of objects and then being offered additional options in a set $\mathcal {X}'$ . Here is an example of a situation that might seem of a different nature, but which still fits the setup of Theorem 2.9. It is related to the concepts of awareness growth and reverse Bayesianism.Footnote ⁶

Example 8 (Awareness growth).

Suppose that an agent has a total trading system $\mathcal {T}=(\mathcal {X},\precsim ,\prec )$ where each element of $\mathcal {X}$ is a function from $\Omega $ to a space $\mathcal {O}$ . At some point, the agent realizes that the “actual” state space is a different set $\Omega '$ . There are two common models for this realization.

1. Refinement: The agent learns that the elements of $\Omega $ are not atomic. That is, each element $\omega $ appears to be a subset of $\Omega '$ .

2. Expansion: The agent learns that $\Omega $ is merely a proper subset of the “actual” state space $\Omega '$ .

In either case, one recommendation (called reverse Bayesianism) is for the agent to preserve those preferences included in $\mathcal {T}$ when extending the preference relation to include any new objects that need to be considered. Theorem 2.9 can be applied in both the refinement and expansion cases to achieve that goal. Specifically, there is a one-to-one mapping T between $\mathcal {X}$ and a set $\mathcal {X}^*=\{T(X):X\in \mathcal {X}\}$ of functions from $\Omega '$ to a standard-linear space $\mathcal {O}'$ such that $\mathcal {T}^*=(\mathcal {X}^*,\precsim ^*,\prec ^*)$ is a total trading system with $X\precsim Y$ if and only if $T(X)\precsim ^* T(Y)$ . If $\mathcal {T}$ is coherent and $\mathcal {O}={\mathbb {R}}$ , then $\leq $ and $\prec _{_{{\scriptsize \text{Dom}}}}$ are defined on $\mathcal {X}'$ so as to be extensions $_2$ of $\leq $ and $\prec _{_{{\scriptsize \text{Dom}}}}$ on $\mathcal {X}$ .

In the refinement case, for each $\omega \in \Omega $ , there is a subset $C_\omega \subseteq \Omega '$ such that the distinct elements of $\mathcal {C}=\{C_\omega :\omega \in \Omega \}$ form a partition of $\Omega '$ . For each $X\in \mathcal {X}$ and $\omega '\in \Omega '$ , define $T(X)(\omega ')=X(\omega )$ where $\omega '\in C_\omega $ . Each such $T(X)$ is constant on each element of $\mathcal {C}$ . If $\mathcal {T}$ is coherent, then $X\prec _{_{{\scriptsize \text{Dom}}}} Y$ in $\mathcal {T}$ if and only if $T(X)\prec _{_{{\scriptsize \text{Dom}}}} T(Y)$ in $\mathcal {T}^*$ for all three forms of dominance.

In the expansion case, the additional functions in $\mathcal {X}'$ are defined on the whole set $\Omega '$ and may take values that are not in the codomain $\mathcal {O}$ of the functions in $\mathcal {X}$ . Let $\mathcal {O}'$ be the codomain of the functions defined on $\Omega '$ that the agent wants be able to trade. For each $X\in \mathcal {X}$ and $\omega \in \Omega $ define

$$\begin{align*}T(X)(\omega)=\begin{cases} X(\omega),&\text{if }\omega\in\Omega,\\ 0,&\text{if }\omega\in\Omega'\setminus\Omega.\end{cases}\end{align*}$$

All such $T(X)$ are identical on the part of $\Omega '$ of which the agent recently became aware. If $\mathcal {T}$ is weakly coherent, $X\prec _{_{{\scriptsize \text{Dom}}}} Y$ if and only if $T(X)\prec _{_{{\scriptsize \text{Dom}}}} T(Y)$ . There are no cases of $X\prec _{_{{\scriptsize \text{Dom}}}} Y$ in $\mathcal {T}^*$ for the other two forms of dominance, but if $\mathcal {T}$ is uniformly or strictly coherent, $\mathcal {T}^*$ is both uniformly and strictly coherent.

In both the refinement and expansion cases, the agent can determine what set $\mathcal {X}'$ (containing $\mathcal {X}^*$ ) of functions defined on $\Omega '$ are available for trading. Theorem 2.9 then allows extension $_2$ of $\mathcal {T}^*$ to a total trading system $\mathcal {T}'=(\mathcal {X}',\precsim ',\prec ')$ . If $\mathcal {T}$ is coherent, the extension $_2$ of $\mathcal {T}^*$ to $\mathcal {T}'$ can be done so that $\mathcal {T}'$ retains the same form of coherence.

3.1 Previsions for random variables

The first Archimedean theory to which our results apply is the theory of previsions of [Reference de Finetti11], which begins with an arbitrary set of standard-valued random variables. For each X in that set, an agent chooses a standard value $P(X)$ (called the prevision of X) such that the agent is willing to trade away either X or $P(X)$ in order to receive the other one. Specifically, the change in fortune $\alpha [X-P(X)]$ is considered a fair gamble for all standard $\alpha $ . Although de Finetti’s theory deals only in fair trades (indifference), there is an implicit assumption that “more is better,” which is built into his notion of coherence (corresponding to uniform dominance in Definitions 1.11 and 1.12). In [Reference de Finetti11], the agent is willing to accept every finite sum of fair gambles. In particular, the agent is willing to accept

(9) $$ \begin{align} \alpha[X-P(X)]-\alpha[Y-P(Y)]=\alpha(X-Y)+\alpha[P(Y)-P(X)], \end{align} $$

for all standard $\alpha $ . If $P(X)=P(Y)$ , the right-hand side of (9) is $\alpha (X-Y)$ , and the agent is implicitly willing to trade X to get Y or to trade Y to get X. If $P(X)\ne P(Y)$ , there is an implicit strict preference in one direction, e.g., if $P(Y)>P(X)$ and $\alpha <0$ , the fair trade (9) is strictly smaller than $\vert \alpha \vert (Y-X)$ , so the agent is willing to trade X to get Y, but not the other way. In addition, willingness to accept all finite sums of fair trades implies that a coherent prevision P on an arbitrary set $\mathcal {Y}$ of random variables extends uniquely to a coherent prevision on the linear span $\mathcal {X}$ of $\mathcal {Y}$ . Define the total preorder $\precsim $ on $\mathcal {X}$ defined “ $X\precsim Y$ if and only if $P(X)\leq P(Y)$ .” It follows that $(\mathcal {X},\precsim ,\prec )$ is a total trading system that is represented by the linear function P. Our theory extends that of [Reference de Finetti11] by dropping the requirement that every element of $\mathcal {X}$ be indifferent to some standard constant.

A simple example of a random variable that is not indifferent to a standard constant arises with an “almost-fair” coin. For an even-money bet (odds equal 1) the agent strictly prefers the bet that pays on heads over the bet that pays on tails. But, for every bet that is not at even money (i.e., odds are different from 1), the agent strictly prefers the side of the bet that pays the larger amount. Theorem 3.1 of [Reference Fishburn15] implies that there is no standard-valued prevision that ranks these bets in the order of the stated preferences. See also [Reference Debreu, Thrall, Coombs and Davis12]. But a nonstandard-valued function can represent such preferences. Random variables with infinite previsions (such as Example 7) are also cases in which fair prices are not available. The nonstandard-valued functions that represent these preferences are not fair prices, but they provide a numerical representation of preference in the manner of expected utility.

3.2 Horse lotteries

The second theory to which our results apply is that of [Reference Anscombe and Aumann1, Reference Von Neumann and Morgenstern41] for decisions about horse lotteries, which are functions from $\Omega $ to the set of simple lotteries over a set of prizes.

3.2.2 A linear space for horse lotteries

The set $\mathcal {H}$ of horse lotteries is a convex subset of $\mathcal {R}_\Omega $ , but is not a linear space. Hausner [Reference Hausner, Thrall, Coombs and Davis21] (Sections 2–4) assumes that $\precsim '$ is a total preorder that satisfies the following axiom, which is part of the theory of [Reference Anscombe and Aumann1, Reference Von Neumann and Morgenstern41]:

Independence Axiom: Let $\precsim '$ be a preorder on a convex set $\mathcal {H}$ of horse lotteries. For all $h_1,h_2,g\in \mathcal {H}$ and standard $0<\alpha <1$ , $h_1\precsim ' h_2$ if and only if $\alpha h_1+(1-\alpha )g\precsim '\alpha h_2+(1-\alpha )g$ .

Hausner [Reference Hausner, Thrall, Coombs and Davis21] shows how to create a standard-linear space $\mathcal {K}_0$ with a preorder $\precsim $ that satisfies our Assumptions 2 and 3 in Section 1.4 above. This is done as follows. For each $\omega \in \Omega $ , let $\mathcal {O}_\omega $ be the set of all simple signed measuresFootnote ⁹ on $\mathcal {P}_\omega $ that give measure 0 to the whole set $\mathcal {P}_\omega $ . Let $\mathcal {O}=\bigcup _{\omega \in \Omega }\mathcal {O}_\omega $ , and let $\mathcal {O}_\Omega =\prod _{\omega \in \Omega }\mathcal {O}_\omega $ . Then

$$\begin{align*}\mathcal{K}_0=\{\alpha(h_1-h_2): h_1,h_2\in\mathcal{H}\text{\ and\ } \alpha\in{\mathbb{R}}\}\subseteq\mathcal{O}_\Omega\end{align*}$$

is a standard-linear space. Define $\precsim $ on $\mathcal {K}_0$ as follows. For each $k_1,k_2\in \mathcal {K}_0$ , express $k_2-k_1=\alpha (h_2-h_1)$ with $\alpha>0$ and $h_1,h_2\in \mathcal {H}$ . Then say that $k_1\precsim k_2$ if $h_1\precsim 'h_2$ . Hausner [Reference Hausner, Thrall, Coombs and Davis21] (Section 4) shows that $\precsim $ is well defined and satisfies Assumptions 2 and 3. The theory of [Reference Anscombe and Aumann1, Reference Von Neumann and Morgenstern41] satisfies Assumption 4 vacuously since $\precsim $ is a total preorder. Dominance and coherence are not issues that arise in the theory of [Reference Anscombe and Aumann1, Reference Von Neumann and Morgenstern41] as horse lotteries are not numerically comparable without further assumptions.

The state-independence assumption of [Reference Anscombe and Aumann1, Reference Von Neumann and Morgenstern41] implies that all $\mathcal {R}_\omega $ sets are the same. Our theory is general enough to include cases in which the $\mathcal {R}_\omega $ sets might all be the same or might be different. We also drop the Archimedean axiom and allow $\precsim '$ to not be total as do other authors such as [Reference Aumann2, Reference Baucells and Shapley3, Reference Dubra, Maccheroni and Ok14]. Our weaker state-independence Assumption 7 is stated in Section 4.1.

3.2.3 Representing horse lotteries

For the remainder of this paper, when we refer to the horse-lottery case, we will assume that $\mathcal {X}$ is the standard-linear space $\mathcal {K}_0$ defined in Section 3.2.2. (The case in which $\mathcal {X}$ is a linear space of standard-valued random variables will be called the random-variable case.) In the horse-lottery case, it would be easier on the intuition if each representing function of a trading system had $\mathcal {H}$ as its domain rather than $\mathcal {K}_0$ . This is easily arranged. Let $\mathcal {T}=(\mathcal {K}_0,\precsim ,\prec )$ be a total trading system in a horse-lottery case with standard-linear representing function U. Let $\mathcal {H}$ be the set of horse lotteries that corresponds to $\mathcal {K}_0$ as in Section 3.2.2. For each $k\in \mathcal {K}_0$ , we can write $k=\alpha (h_1-h_2)$ with $\alpha>0$ standard. Then $0\precsim k$ is equivalent to $h_2\precsim 'h_1$ for a total preorder $\precsim '$ on $\mathcal {H}$ . Let $h_0\in \mathcal {H}$ be arbitrary, and define

(10) $$ \begin{align} V(h)=U(h-h_0). \end{align} $$

It follows that $V(h_0)=0$ and

(11) $$ \begin{align} U(\alpha[h_1-h_2])=\alpha[V(h_1)-V(h_2)]. \end{align} $$

Also, V represents $\precsim '$ and satisfies

$$\begin{align*}V(\beta h_1+[1-\beta]h_2)=\beta V(h_1)+(1-\beta)V(h_2),\end{align*}$$

for all $h_1,h_2\in \mathcal {H}$ and all standard $\beta \in [0,1]$ .

3.3 Savage-style acts

The third theory to which our results apply is that of Savage [Reference Savage35]. This theory makes some assumptions (including state-independence) about preferences amongst acts (functions) from a state space $\Omega $ to a set F of consequences (prizes) and then proves an expected-utility representation for those preferences. Lemma 3.2 below starts with those same assumptions and shows that there is a set of lotteries over the acts with an implied willingness to trade that satisfies the assumptions that appear in Section 1.4 of this paper. We then weaken the original assumptions of Savage [Reference Savage35] and show how to use our results for the horse-lottery case to represent non-Archimedean and state-dependent preferences over the acts of Savage [Reference Savage35]. Whenever we refer to “the horse-lottery case” in this paper, we implicitly include the theory of Savage in that case.

At this point, we can show how a non-Archimedean version of the theory of Savage [Reference Savage35] becomes a special case of trading systems in the horse-lottery case without any additional assumptions or choices by the agent. The proof of Lemma 3.2 is in Appendix C.4.

Despite the fact that Savage worked hard to avoid making the assumption that his set of acts contained the mixtures that we assume, his postulates are sufficient to show that his preorder extends to a total trading system that satisfies our assumptions without any further choices needed from the agent. For the purposes of this paper, instead of assuming a subset of P1–P7 or some weakened versions of them, assume only that there is a set $\mathcal {F}$ of Savage-style acts with a preorder (not necessarily total) $\precsim '$ . Then embed $\mathcal {F}$ into the convex set $\mathcal {H}$ of Lemma 3.2 which is a special case of a set of horse lotteries. We then proceed with the same analysis and assumptions as in Sections 3.2.2 and 1.4. In particular, we make Assumptions 1–4. By so doing, we implicitly weaken some of Savage’s postulates so as to allow non-Archimedean preferences, In addition, all of the extension and representation results in Section 2 above apply to the resulting trading system, as well as the results in Section 4 below. In the end, if the agent does not want to think about mixtures of Savage-style acts, we show (in Section 4.7) how to restrict the results of Section 4 to the original Savage-style acts.

4.1 Dominance and coherence (part two)

In this section, we extend the concepts of dominance and coherence to certain horse-lottery cases. This extension is useful in Section 4.4 where we show how, in both the random-variable and horse-lottery cases, a standard-linear function that represents a coherent trading system can be interpreted as an expected value of a (possibly state-dependent) utility function defined on the codomain $\mathcal {O}$ of the elements of $\mathcal {X}$ .

For the remainder of this section, we assume that $\mathcal {X}$ is the space $\mathcal {K}_0$ defined in Section 3.2.2 and $\mathcal {O}$ is a set of simple signed measures on subsets of the prize set $\mathcal {P}$ and that assign signed measure 0 to $\mathcal {P}$ . As such, each element of $\mathcal {K}_0$ is a function from $\Omega $ to  $\mathcal {O}$ . Dominance for horse lotteries is defined in terms of conditional preference on a state-by-state basis. Suppose that there is a non-null event B that consists entirely of null states, i.e., every $\omega \in B$ is a null state. Then a state-by-state comparison of two elements X and Y of $\mathcal {X}$ given the elements of B tells us nothing about how X and Y compare (or should compare) given B. The reason is that $X\sim Y\vert \{\omega \}$ for every X and Y and every null state $\omega $ . To circumvent this problem, we make an assumption that is a generalization (weaker assumption) of the state-independent utility assumptions made by Anscombe and Aumann [Reference Anscombe and Aumann1] (Assumption 1), Savage [Reference Savage35] (P3–P4), and Blume, Brandenburger, and Dekel [Reference Blume, Brandenburger and Dekel6] (Axioms 5 and 5 $'$ ). Assumption 7 below allows varying degrees of state dependence for the utilities of prizes.

The second bullet assumes that certain functions defined on $\mathcal {O}_\Omega $ are in $\mathcal {X}$ . If these functions are not in $\mathcal {X}$ and the first bullet is satisfied, we can apply Theorem 2.9 to find an extension $_2$ of $\mathcal {T}$ to $\mathcal {T}'=(\mathcal {X}',\precsim ',\prec ')$ where $\mathcal {X}'$ is the standard-linear span of $\mathcal {X}\bigcup \{X_xI_BI_A:B\in \mathcal {B},x\in \mathcal {O}(B),A\in \Sigma \}$ . Such an extension $_2$ will continue to satisfy Assumption 6 and the first bullet of Assumption 7. We will assume that such an extension $_2$ has been done for the remainder of the paper. Note that there is no guarantee that such a $\mathcal {T}'$ satisfies the third bullet of Assumption 7. But we cannot even check whether the third bullet is satisfied until the second bullet is satisfied.

The state-independence assumptions made by Anscombe and Aumann; Savage; and Blume, Brandenburger, and Dekel [Reference Anscombe and Aumann1, Reference Blume, Brandenburger and Dekel6, Reference Savage35] correspond to the extreme case of Assumption 7 in which $\mathcal {B}=\{\Omega \}$ , i.e., utility is independent of state for the whole state space. The opposite extreme case of Assumption 7 has $\mathcal {B}=\{\{\omega \}:\omega \in \Omega \}$ , i.e., every state is non-null, in which case all three bullets of Assumption 7 are tautological because each B is a singleton. There are cases between these two extremes, such as Example 13 in Section 4.2. A more concrete example would be the following.

• • Each state consists of a specification of exchange rates between a set of currencies and a specification of a set of meteorological conditions.

• • The values of the prizes depend only on the exchange rates and not on the meteorological conditions.

• • Each set in the partition $\mathcal {B}$ is the set of states with a fixed specification of exchange rates.

Definition 4.3 below defines $\leq $ and dominance in those horse-lottery cases that satisfy Assumption 7. It allows us to talk about coherent trading systems in the horse-lottery case with no modifications to Assumption 5 or the definition of coherence (Definition 1.12). As in the random-variable case, whichever (if any) sense of dominance is reflected in the agent’s preferences, we express “Y dominates X” by $X\prec _{_{{\scriptsize \text{Dom}}}} Y$ .

As in Definition 1.11 for the random-variable case, Definition 4.3 defines dominance on a larger set of objects than just $\mathcal {X}$ . The reason is that we may need elements of that larger set of objects in order to infer the existence of a probability on the field $\Sigma $ . (See Lemma 4.5 in Section 4.2.)

Lemma 4.2 gives us what we need to define dominance on all of $\mathcal {O}_\Omega $ .

For trading systems in the horse-lottery case that satisfy Assumptions 6 and 7, the definition of “coherent” (Definition 1.12) applies. We did not define “uniform dominance” because the $<$ symbol in Definition 4.3 is not a relation between numerical values, but rather a non-numerical preference relation. Once we define a state-dependent utility function (Definition 4.7 in Section 4.3) we can define uniform dominance in the horse-lottery case.

4.2 Numeraires and probability

In the random-variable case with a coherent trading system, every non-negative function in $\mathcal {X}$ that is strictly preferred to 0 is a numeraire.

Example 10 (Continuation of Example 9).

Consider the set $\mathcal {X}'$ of simple functions that are measurable with respect to the field $\Sigma $ generated by the interval subsets of $\Omega =[0,1]$ in Examples 6 and 9. Every strictly positive function $X\in \mathcal {X}'$ is a strong numeraire. The probability associated with the constant function 1 is $P_1(E)=U'(I_E)$ . For an arbitrary positive function X of the form (7),

$$\begin{align*}P_X(E)=\frac{\sum_{j=1}^n h_jU'(I_{E\cap J_j})}{\sum_{j=1}^n h_jU'(I_{J_j})}\end{align*}$$

is another probability that corresponds to X being the numeraire.

Sometimes, even positive constants might not be strong numeraires.

Example 11. Let $\Omega =\mathbb {Z}^+$ , the positive integers, and let $\Sigma $ be the finite/cofinite field, i.e., the collection of all finite subsets of $\Sigma $ and their complements. Let $G:\Omega \rightarrow {\mathbb {R}}$ be $G(\omega )=\omega $ . Let $\mathcal {X}$ be the standard-linear span of all standard-valued bounded functions and the functions $\{GI_B:B\in \Sigma \}$ . Each $X\in \mathcal {X}$ can be written uniquely as

(12) $$ \begin{align} X=X_b+\alpha_XGI_E, \end{align} $$

where $X_b$ is bounded, $\alpha _X$ is standard, and $E=\{11,12,\ldots \}$ .Footnote ¹⁰ Define

(13) $$ \begin{align} U(X)=\frac{1}{10}\sum_{\omega=1}^{10}X(\omega)+2\alpha_X. \end{align} $$

Also, $U:\mathcal {X}\rightarrow {\mathbb {R}}$ is standard-linear (and standard-valued). If $X\prec _{_{{\scriptsize \text{Dom}}}} Y$ (uniform or strict, but not weak dominance) then $U(X)<U(Y)$ , so U represents a coherent (with uniform or strict dominance) total trading system $\mathcal {T}=(\mathcal {X},\precsim ,\prec )$ with $X\precsim Y$ meaning $U(X)\leq U(Y)$ . Since $U(1)=1$ , $0\prec 1$ . Since $0\leq 1$ , $0\precsim 1\vert B$ for all $B\in \Sigma $ , and $1$ is numeraire for $\Sigma $ . Since $0\prec G\vert E$ , E is a non-null event. On the other hand, $0\sim 1\vert E$ , so $1$ is not a strong numeraire for $\Sigma $ . The non-null events are all cofinite sets and all elements of $\Sigma $ that are supersets of the singletons $\{1\},\ldots ,\{10\}$ . For each $\omega =1,\ldots ,10$ , $0\prec G\vert \{\omega \}$ , and for each cofinite set B, $0\prec G\vert B$ , so G is a strong numeraire for $\Sigma $ .

In Section 4.5, we show how different numeraires relate to each other within the same trading system. In general, we need a numeraire in order to derive a probability on $(\Omega ,\Sigma )$ from a trading system. Since 1 is a numeraire in the random-variable case, the following result is needed only in the horse-lottery case.

For the remainder of the paper, assume that each trading system $\mathcal {T}$ in the horse-lottery case is an extension $_2$ from Lemma 4.5.

We take probability to be finitely additive in Lemma 4.6 and elsewhere due to the lack of countably additive probabilities in some relatively simple situations. (See Example 18 in Section A.3.)

Example 13. Let $\Omega =(0,1)$ , the unit interval with $\Sigma $ the Borel $\sigma $ -field. Let $\mathcal {P}=\{a,b\}$ . Let $\mathcal {R}$ be the set of all lotteries over $\mathcal {P}$ . Each element $r\in \mathcal {R}$ is characterized by $r(\{b\})$ . Let $\mathcal {H}$ be that subset of $\mathcal {R}^\Omega $ for which $g_h(\omega )=h(\omega )(\{b\})$ is a Borel-measurable function of $\omega $ . In this example, for states $\omega <1/2$ , prize a is better than b, and the situation is the reverse for all $\omega>1/2$ . We include the null event $\{1/2\}$ with $\omega>1/2$ for the rest of the example. Define

(14) $$ \begin{align} V(h)=\int_0^{1/2}[1-g_h(\omega)]d\omega+\int_{1/2}^1g_h(\omega)d\omega. \end{align} $$

Each element of $\mathcal {K}_0$ has the form $\alpha (h_1-h_2)$ for standard $\alpha \geq 0$ and $h_1,h_2\in \mathcal {H}$ . Define

$$\begin{align*}U(\alpha[h_1-h_2])=\alpha[V(h_1)-V(h_2)].\end{align*}$$

The trading system represented by U satisfies Assumption 7 with $\mathcal {B}=\{(0,1/2),[1/2,1)\}$ . There are many numeraires for $\Sigma $ . Let $g_{h_0}(\omega )=I_{(0,1/2)}(\omega )$ , so that $V(h_0)=0$ . Each h with $g_h(\omega )<1$ for $\omega <1/2$ and $g_h(\omega )>0$ for $\omega \geq 1/2$ has $V(h)>0$ and can be combined with $h_0$ to define a numeraire for $\Sigma $ as follows. Define $Z_h=(h-h_0)/V(h)$ . Since $U(Z_hI_A)\geq 0$ for each Borel set A, $0\precsim Z_h\vert A$ . Also, $0\prec Z_h$ , so each such $Z_h$ is a numeraire for $\Sigma $ . As (14) would suggest, the probability derived from $Z_h$ has the following density with respect to Lebesgue measure:

$$\begin{align*}f_h(\omega)=\frac{1}{V(h)}\left([1-g_h(\omega)]I_{(0,1/2)}(\omega) +g_h(\omega)I_{[1/2,1)}(\omega)\right).\end{align*}$$

Since $f_h>0$ , $Z_h$ is a strong numeraire for $\Sigma $ .

4.3 State-dependent utility

In the random-variable case, one can think about a numeraire as the units that correspond to numerical values of the random variables. For example, suppose that the random variables in a trading system $\mathcal {T}$ are in units of dollars. A European foreign-exchange trader might be more comfortable comparing units of euros rather than dollars. If $G(\omega )$ is the exchange rate (in dollars per euro) in state $\omega $ , $X(\omega )/G(\omega )$ is the utility of X in state $\omega $ to the European trader measured in the preferred currency units of euros. We need something analogous to $X/G$ for the horse-lottery case.

Definition 4.7. Let $\Sigma $ be a field of subsets of $\Omega $ , and let $\mathcal {T}$ be a coherent total trading system that satisfies Assumption 6 (and Assumption 7 in the horse-lottery case). Let $U:\mathcal {X}\rightarrow ^{\ast}\hspace{-2.5pt}{\mathbb {R}}$ be a standard-linear function that represents $\mathcal {T}$ . In the horse-lottery case, let Z be a numeraire of the sort obtained through Lemma 4.5. In the random-variable case, let Z be a strictly positive numeraire. Define, for each $X\in \mathcal {X}$ , each $\omega \in \Omega $ , and each $x\in \mathcal {O}_\omega $ :

$$ \begin{align*} U^*_Z(\omega,x)=\begin{cases}\displaystyle\frac{x}{Z(\omega)},&\text{in the random-variable case,}\\[.11in] \displaystyle\frac{U(xI_{B_\omega})}{U(ZI_{B_\omega})},&\text{in the horse-lottery case.}\end{cases} \end{align*} $$

We call $U^*_Z$ the state-dependent utility function (relative to the numeraire Z). Let $X,Y\in \mathcal {X}$ . If there exists a standard $\epsilon>0$ such that $U^*_Z(\omega ,X(\omega ))\leq U^*_Z(\omega ,Y(\omega ))-\epsilon $ for all $\omega $ , we say that X is uniformly dominated by Y relative to Z or Y uniformly dominates X relative to Z.

As Definition 4.7 makes explicit, which elements of $\mathcal {X}$ uniformly dominate each other depends on which numeraire Z is used. Example 14 illustrates this fact.

Example 14 (Continuation of Example 13).

The state-dependent utility that corresponds to the numeraire $Z_h$ is, for each $X=\alpha (h_1-h_2)\in \mathcal {K}_0$ ,

$$\begin{align*}U^*_h(\omega,X(\omega))=\frac{\alpha}{f_h(\omega)}\begin{cases} g_{h_2}(\omega)- g_{h_1}(\omega),&\text{for }\omega<1/2,\\ g_{h_1}(\omega)-g_{h_2}(\omega),&\text{for }\omega\geq1/2,\end{cases}\end{align*}$$

so that $U^*_h(\omega ,Z_h(\omega ))=1$ for all $\omega $ , and $Z_h$ uniformly dominates 0 relative to $Z_h$ . The same could be said for each numeraire constructed in the same fashion. However, what “uniformly dominates” 0 depends on which numeraire is used to construct the probability. To be specific, suppose that $g_h(\omega )=1/2$ for all $\omega $ so that $f_h$ is constant. Let $h'$ be another horse lottery with $g_{h'}(\omega )=1-\omega $ for all $\omega $ , so that $f_{h'}$ is $\wedge $ -shaped and $U^*_{h'}(\omega ,Z_{h'}(\omega ))=1$ for all $\omega $ . Then $h'$ uniformly dominate 0 relative to $Z_{h'}$ , but

$$\begin{align*}U^*_h(\omega,Z_{h'}(\omega))=\begin{cases}\omega,&\text{for }\omega<1/2,\\ 1-\omega,&\text{for }\omega\geq1/2,\end{cases}\end{align*}$$

so $h'$ does not uniformly dominate 0 relative to $Z_h$ .

To see that $U^*_Z$ is well defined in the horse-lottery case, note that for each $B\in \mathcal {B}$ of Assumption 7, $U(ZI_B)>0$ because $0\prec Z\vert B$ . In general $U^*_Z$ is a nonstandard-valued function defined on the set $\bigcup _{\omega \in \Omega }(\{\omega \}\times \mathcal {O}_\omega )$ . The interpretation of $U^*_Z(\omega ,x)$ is the utility to the agent of X in state $\omega $ measured in units of numeraire Z when $X(\omega )=x$ . In Section 4.4, Theorem 4.9 shows that $U(X)$ can be interpreted as an expected value of the state-dependent utility of X with respect to the probability P on $(\Omega ,\Sigma )$ that corresponds to the chosen numeraire Z via Lemma 4.6.

4.4 Expected utility

The following definition is a generalization of the concept of Daniell integral to the finitely additive nonstandard-valued case. See [Reference Schervish, Seidenfeld and Kadane37] for discussion of the finitely additive standard-valued case. We give additional motivation for this definition in Appendix B.

It should be apparent that Definition 4.8 agrees with the familiar countably additive definition of expected value for simple probabilities when $^{\ast }{\mathbb {R}}={\mathbb {R}}$ . Furthermore, all finite standard-valued countably additive expected values satisfy Definition 4.8, but they also have an additional continuity property that does not carry over to the nonstandard-valued case.

4.5 Changes of numeraire

Assume the conditions stated in Definition 4.7. Let $Z_1,Z_2$ be two numeraires of the sort described there with corresponding probabilities $P_{Z_1},P_{Z_2}$ . Then

$$\begin{align*}\mathcal{W}_{Z_j}=\{U^*_{Z_j}(\cdot,X(\cdot)):X\in\mathcal{X}\},\end{align*}$$

for $j=1,2$ . And $W_{Z_j}:\mathcal {W}_{Z_j}\rightarrow ^{\ast }\hspace{-2.5pt}{\mathbb {R}}$ , defined by

$$\begin{align*}W_{Z_j}(U^*_{Z_j}(\cdot,X(\cdot)))=U(X)/U(Z_j),\end{align*}$$

acts as an expected value on $\mathcal {W}_{Z_j}$ with respect to $P_{Z_j}$ . Note that

$$ \begin{align*} U^*_{Z_2}(\cdot,Z_1(\cdot))&=\frac{1}{U^*_{Z_1}(\cdot,Z_2(\cdot))}, \text{\ and}\\ \mathcal{W}_{Z_2}&=\left\{wU^*_{Z_2}(\cdot,Z_1(\cdot)):w\in\mathcal{W}_{Z_1}\right\}. \end{align*} $$

For each $w\in \mathcal {W}_{Z_2}$ ,

$$\begin{align*}W_{Z_2}(w)=W_{Z_1}\left[wU^*_{Z_2}(\cdot,Z_1(\cdot))\right] \frac{U(Z_1)}{U(Z_2)}.\end{align*}$$

In other words, $U^*_{Z_2}(\cdot ,Z_1(\cdot ))U(Z_1)/U(Z_2)$ has the defining feature of a Radon–Nikodym derivative of $W_{Z_2}$ with respect to $W_{Z_1}$ (when restricted to indicators of events), without referring to absolute continuity.

4.6 Another layer of expected utility

As in Section 3.2.3, in the horse-lottery case, it would be easier on the intuition if state-dependent utility could be expressed as a function of lotteries in state $\omega $ rather than elements of $\mathcal {O}_\omega $ . Let $\mathcal {W}$ and $U^*_Z$ be as in Theorem 4.9. Let $h_0\in \mathcal {H}$ be arbitrary, and define $V^*_Z: \bigcup _{\omega \in \Omega }\left (\{\omega \}\times \mathcal {R}_\omega \right )\rightarrow ^{\ast}\hspace{-2.5pt}{\mathbb{R}}$ by