When No Price Is Right

In this paper, we show how to represent a non-Archimedean preference over a set of random quantities by a nonstandard utility function. Non-Archimedean preferences arise when some random quantities have no fair price. Two common situations give rise to non-Archimedean preferences: random quantities whose values are required to be greater than every real number, and strict preferences between random quantities that are deemed closer in value than every positive real number. We also show how to extend a non-Archimedean preference to a larger set of random quantities. The random quantities that we consider include real-valued random variables, horse lotteries in the sense of Anscombe and Aumann (1963), and acts in the theory of Savage (1954). In addition, we weaken the state-independent utility assumptions made by the existing theories and give conditions under which the utility that represents preference is the expected value of a state-dependent utility with respect to a probability over states.


Introduction 1.Motivation
The primary goal of this paper is to extend three well-known theories of decision making (described in Section 3 below) to allow for non-Archimedean (unbounded and/or discontinuous) preferences as defined in Definition 1.
Definition 1.Let ≾ be a binary relation on a set .
1. We call ≾ a preorder if it is both reflexive and transitive.2. A preorder ≾ on a set  is called total if, for all , ∈ , ( ≾ ) ∨ ( ≾ ).
Archimedean preferences have a continuity at = 0 for mixtures of the form in the first clause of (1).Also, if is worth infinitely less than which in turn is just a little bit less valuable than , then (1) can fail.
All three Archimedean theories that we extend also assume that preferences are state-independent, are expressed via total preorders, and satisfy a linearity assumption.We relax the state-independence and total-preorder assumptions and drop altogether the assumption that preferences are Archimedean.Allowing utility to be state-dependent is particularly important in financial applications where different states of the world can entail different exchange rates between currencies as in Schervish, Seidenfeld and Kadane (1990) and/or different relative prices for commodities.We maintain a linearity assumption in order to achieve an expected-utility representation.
The use of lexicographies to represent discontinuous preference is widespread, e.g., Blume, Brandenburger and Dekel (1991b) use lexicographies in game theory to allow conditioning on events that would otherwise have zero probability and Rekola (2003); Gelso and Peterson (2005) use lexicographies to help model ecological preferences where certain options are essentially infinitely more valuable than others.See also Hausner (1954); Halpern (2010); Petri and Voorneveld (2016) for some theoretical considerations.The following is adapted from Definition 3.1 of Blume, Brandenburger and Dekel (1991a) and Definition 2.2 of Halpern (2010).
Definition 2. Let Ω be a set with a field Σ of subsets.A well-ordered set  = { } ∈ℵ of finitely-additive probabilities on (Ω, Σ) is called a lexicographic probability system.Let  be a set of standard-valued functions defined on Ω such that ( ) is a finitely-additive expectation of with respect to for each ∈ ℵ.The lexicographic preference (derived from ) is defined as follows.For each , ∈ ,
Examples 3.3 and 4.8 of Halpern (2010) show that there are cases of non-Archimedean preferences that cannot be modeled via lexicographic probability systems while showing that all lexicographic preferences can be represented by a nonstandard utility.See also Rizza (2015).In this way, a nonstandard representation is a strict generalization of lexicographic preferences.In this paper we use nonstandard-valued functions to represent non-Archimedean preferences.

Standard versus Nonstandard Numbers
For the remainder of the paper, we refer to the familiar real numbers in the set IR as standard numbers to distinguish them from the nonstandard numbers that we describe in Appendix A and use liberally throughout the paper.We call a function numerical if it takes either standard or nonstandard values.In all cases, the calculations that are part of an agent's expressions of preference involve only standard numbers.We use nonstandard numbers to represent an agent's preferences after the fact and to infer a probability and utility to express that representation.Since we use multiple number systems, we need to be careful about what we mean by "linear" in various settings.Definition 3. A space  of functions is a standard-linear space if + ∈  for all standard , and all , ∈ .A nonstandard-valued function on a standard-linear space  is called a standard-linear function if ( + ) = ( ) + ( ), for all , ∈  and all standard , .The standard-linear span of a set is the smallest standard-linear space containing the set.
Notice that (0) = 0 for every standard-linear function .Definition 3 restricts the coefficients in linear combinations to be standard even though the values of might be nonstandard.Readers desiring a more thorough understanding of nonstandards than we present in Appendix A could read one of the many treatments such as Nelson (1987); Robinson (1996).
Other treatments of probability and/or decision theory that make use of nonstandard numbers include Pedersen (2012); Duanmu and Roy (2017); Benci, Horsten and Wenmackers (2013); Wenmackers (2019).Section 3.2 of Pedersen (2012) has extensive references along with some details of some of the attempts to make use of nonstandardvalued probabilities.The same author, in Pedersen (2014), investigates representations of non-Archimedean coherent preference over unconditional real-valued gambles.Our representation incorporates coherent conditional preferences in Section 4. The approach of Benci et al. (2013); Wenmackers (2019) is primarily to define probabilities that take infinitesimal values.For a probability on a set Ω to be a "non-Archimedean probability," in their terminology, they impose a condition that requires all singletons { } ∈ Ω to have probabilities that are standard multiples of a common infinitesimal .That is, there is an infinitesimal such that for every ∈ Ω, there is a standard > 0 such that ({ }) = . This assumption places severe restrictions on the forms of non-Archimedean preferences that can be expressed.
For those familiar with nonstandard models of the reals, all of our analysis is external rather than internal. 1The main reason for an external analysis is that the nonstandards are non-Archimedean from an external perspective, but are Archimedean from an internal perspective.An internal perspective is used by Duanmu and Roy (2017), who start with the familiar decision theory setup (loss functions, bayes rules, minimax rules, admissible rules, etc.) and obtain new results by allowing probabilities to take nonstandard values.Narens (1974) develops a non-Archimedean theory of measurement.The theory leads to measurements whose values lie in nonstandard models of the reals.Narens' measurement systems have a number of features in common with probability and preference, so it is not surprising that nonstandard numbers are useful for representing non-Archimedean preference structures.See Halpern (2010) for a more thorough comparison of the uses of lexicographic preferences and nonstandard numbers in representing preferences.

Some Notation
Throughout this paper, Ω denotes a state space,  denotes a set of random quantities, which are functions from Ω to a space  of outcomes.Subsets of Ω are called events.When  is a set of random variables, the space  will be the standard numbers IR.For other cases, both  and  will be more complicated sets that are constructed later.We will make much use of the following concepts: Definition 4. Let ≾ be a binary relation on a set .
The following results follow easily from Definition 4.
Proposition 1.Let be a numerical function defined on a set .
• represents a unique preorder ≾ on , defined via (2) and ≾ is total.
• represents a preorder ≾ if and only if + represents ≾ for all positive and all .

Expressed Preference
In our approach, preference amongst random quantities is expressed by willingness to trade.
Definition 5. Let and be elements of a standard-linear space .If an agent is willing to trade to receive , we write ≾ .If both ≾ and ≾ , we say that the agent is indifferent between and , which we express by ∼ .If ( ≾ ) ∧ [¬( ≾ )] we write ≺ .
We deliberately give no name to the relation ≺ for reasons that will become apparent in Example 1 below.The first assumption that we make merely avoids the two extremes in which the agent either is willing to make no trades or is willing to make all trades.
Our next assumption expresses the idea that willingness to trade depends only on the agent's net change in fortune, which we state formally as follows.
Assumption 2. Suppose that , ′ , , ′ ∈  and − = ′ − ′ .The agent is willing to give to get if and only if the agent is willing to give ′ to get ′ .
Our next assumption is the trading analog to de Finetti's assumption that an agent is willing to accept all finite sums of fair gambles.
construct new objects which play the roles of nonstandard numbers.These constructed objects are numbers when looked at internally, i.e., as objects that satisfy the Peano postulates and to which the Zermelo-Fraenkel axioms of set theory can be applied.When the constructed objects are looked at externally, i.e., as functions of standard objects, what counts as a set is defined in terms of the sets of standard objects from which they are built.Some of the most useful of these external sets include the standard numbers, the standard natural numbers and the infinitesimal numbers.These are sets when viewed externally, i.e. from the point of view of the standard objects from which they are defined.However, they do not satisfy the definition of "set" according to the Zermelo-Frankel axioms applied to the nonstandard numbers when viewed internally.
Assumption 3. Suppose that ≾ for = 1, 2 and 1 , 2 are positive standard numbers.Then . Proposition 2 states some straightforward properties of the first three assumptions.
Proposition 2. Suppose that ≾ satisfies Assumptions 1-3.Then ≾ is a preorder, ≺ is a strict partial order, and ∼ is an equivalence relation.
A general preorder might not be total, and hence may leave some elements of  uncompared, i.e., neither ≾ nor ≾ .
Example 1 (Consensus).Let ℵ be a set, and let {≾ } ∈ℵ be a collection, indexed by ℵ, of total preorders on a standardlinear space .Our agent might think of ℵ as indexing a set of experts whose opinions the agent wants to adopt to the extent that they agree.Define the binary relation ≾ on  by ≾ if, for all ∈ ℵ, ≾ .If each ≾ satisfies Assumptions 1-3, then so does ≾, which will also be a preorder, but not necessarily total.In general, each instance of ≾ partitions ℵ into two sets ℵ ∼ and ℵ ≺ as follows: , and ∈ ℵ ≺ if ≺ .If either of the two sets is empty, there is unanimity about how the experts would trade and .For example, if ℵ ∼ = ∅, the agent is willing to trade to get and will refuse to trade to get .If both are nonempty, the agent is willing to trade to get but has expressed neither willingness nor refusal to trade to get .For example the agent might want to look more closesly at which experts lie in each of the sets ℵ ∼ and ℵ ≺ before deciding whether to trade to get .⋄ As other authors have done, e.g.Giarlotta and Creco (2013); Giarlotta (2019); Nishimura and Ok (2020) we find it useful to allow an agent to distinguish preferences like the two cases that appear at the end of Example 1. Definition 6.Let ≾ satisfy Assumptions 1-3.For each case of ≺ , the agent can express whether this is an unambiguous one-way preference, which we denote ≪ or an ambiguous one-way preference, which we denote ⊲ .
In order for an "unambiguous" one-way preference to mean what it sounds like, we impose the following assumption.
Assumption 4. The relations ≪ and ⊲ satisfy the following: • ( ≪ ) ∧ ( ≾ ) implies ≪ , and If the second bullet in Assumption 4 were violated, the agent would be willing to trade to get and would be willing to contemplate trading to get , which would violate the understanding of ≪ as unambiguous willingness to trade one-way.A similar violation arises if the third bullet is violated.The first claim in Proposition 3 is a direct consequence of Theorem 3.4 of Giarlotta and Creco (2013), and the second claim is straightforward.Proposition 3. If ≾ satisfies Assumptions 1-4, then ≾ is a consensus as in Example 1.In Example 1, choosing ≪ to be ≺ satisfies Assumption 4, as does choosing ≪ to be empty.
We are now ready to formalize our model for trading.Definition 7. Let Ω be a set, and for each ∈ Ω let  be a standard-linear space.Let  Ω = ∏ ∈Ω  and let  ⊆  Ω be a standard-linear space of functions with domain Ω.Let ≾ and ≪ be binary relations on .If ≾ and ≪ satisfy Assumptions 1-4, we call  = (, ≾, ≪) a trading system.If ≾ is a total preorder and ≪ is ≺, then  is a total trading system.The sum of finitely many terms of the form ( − ), where ∼ and is standard is called a fair trade.The sum of finitely many terms of the form ( − ), where ≾ and > 0 is standard is called an acceptable trade.Denote the set of acceptable trades as   .
Proposition 4 states some straightforward properties of trading systems.
Proposition 4. Suppose that  = (, ≾, ≪) is a trading system.The set   of all acceptable trades is a convex cone, and it is the set of all trades such that 0 ≾ .The set of all fair trades is a standard-linear space, and it is the equivalence class (under ∼) that contains the trade 0. Finally, ∈   if and only if for all ∈ , ≾ + .

Dominance and Coherence (Part One)
Suppose that a (possibly nonstandard-valued) function on  represents a total trading system (, ≾, ≺).There is a necessary condition for ( ) to be expressed as the expected value of the utility of ( ) with respect to a probability over Ω. Loosely speaking, the condition is the following: Let , ∈ .If for all , ( ) is at least as valuable as ( ) when state occurs, then ( ) ≤ ( ).
In the theory of de Finetti (1974), where  is a linear space of standard-valued random variables, we can be more precise about the above condition.For each standard number and each ∈ Ω and each random variable such that ( ) = , is assumed to be the utility value to the agent, when the state occurs, of receiving the random variable .The condition then becomes " ( ) ≤ ( ) for all implies ( ) ≤ ( )." In more general theories, where each ( ) may be some non-numerical object ∈  (the codomain of ) and the utility of each object in  might vary with , the utility to the agent of receiving ( ) = could depend on both and .Later (Definition 16 in Section 4.1) we define what we mean by ( ) ≤ ( ) and ( ) < ( ) when  ≠ IR.
Regardless of what are the objects in , there are several ways in which ≤ but ≠ .Definition 8. Let , ∈ .
• We say that weakly dominates or is weakly dominated by if ( ) ≤ ( ) for all ∈ Ω and there is ∈ Ω such that ( ) < ( ).
• We say that strictly dominates or is strictly dominated by if ( ) < ( ) for all ∈ Ω.
• We say that uniformly dominates or is uniformly dominated by if there exists a standard > 0 such that ( ) ≤ ( ) − for all ∈ Ω.
It is trivial to see that weak dominance is an extension of strict dominance which, in turn, is an extension of uniform dominance.Many of our results do not depend on which version of dominance an agent chooses.For those results that depend on the form of dominance (primarily in Section 4), we are explicit about which form is needed.We use ≺ Dom to denote " dominates " in whichever sense the agent chooses.In de Finetti (1974), dominance means uniform dominance.Our next assumption formalizes the idea that more is better.
When  = IR, note that dominance is defined on all of IR Ω , while Assumption 5 pertains only to elements of .Until we can state Definition 16, Assumption 5 makes sense only when  = IR.In the meantime, we state some results with clauses such as "If  is coherent . . ." Those results that are not preceded by such clauses apply more generally.

Representations of Total Trading Systems
In this section, we show how to represent a total trading system by a (possibly nonstandard-valued) numerical function.
Definition 10.If  = (, ≾, ≺) is a total trading system and represents ≾ (recall Definition 4) then we say that represents  .
The following result follows easily from Definition 10.

Proposition 5. A numerical function represents ≾ if and only if
≺ if and only if ( ) < ( ). (3) Next, we introduce a class of numerical functions that represent total trading systems.
A monotone standard-linear function is said to respect dominance if ≺ Dom implies ( ) < ( ).
Lemma 1.Let be a standard-linear function defined on a standard-linear space  of functions defined on a state space Ω.Then represents a total trading system  = (, ≾, ≺).Also  is coherent if and only if is monotone and respects dominance.
Proof.Assume that is a standard-linear function on a standard-linear space .Define the total preorder ≾ on  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 − = ′ − ′ .Since is standard-linear, For Assumption 3, note that being standard-linear implies that for all standard 1 , 2 and 1 , 2 ∈ .For Assumption 4, note that ≪ is ≺.
For the second claim, we need to prove that Assumption 5 holds if and only if is monotone and respects dominance.For the "if" direction, assume that is monotone and respects dominance.Since is monotone, ≤ implies ( ) ≤ ( ) and ≾ .Since respects dominance, ≺ Dom implies ( ) < ( ) and ≺ , so Assumption 5 holds.For the "only if" direction, assume that Assumption 5 holds.To see that is monotone, assume that ≤ .The first requirement of Assumption 5 is that ≾ , which implies that ( ) ≤ ( ), and is monotone.To see that respects dominance, assume that ≺ Dom .The second requirement of Assumption 5 is that ≺ , which implies that ( ) < ( ), and respects dominance.

Agreement, Representation and Extension
If ≾ is a not a total preorder, then there can be no numerical function such that (2) holds.The problem is the "if" direction of (2) rather than the "only if" direction.In other words, representation in the sense of (2) is not achievable for preorders that are not total.On the other hand, a weaker version of (2) is available.
Definition 12. Let  = (, ≾, ≪) be a trading system.A numerical function on  agrees with  if ≾ implies ( ) ≤ ( ), and (4) 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 ) 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  to a larger set  ′ .)Along with the second mode of extension comes a corresponding concept of restricting the domain of definition.
Definition 13.Let  and  ′ be standard-linear spaces with  ⊆  ′ .Let be a binary relation on , and let ′ be a binary relation on  ′ .
• If  =  ′ and implies ′ , we say that ′ is an extension 1 of .
• If ′ is an extension 2 of , we say that is the restriction of ′ to .
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 Appedix C.1.
Lemma 2. Let  and  ′ be standard-linear spaces of functions with domain Ω and such that  ⊆  ′ .Let  = (, ≾, ≪) be a trading system.There exists a trading system  ′ = ( ′ , ≾ ′ , ≪ ′ ) that is an extension 2 of  .If it is not desired that  ′ be coherent,  ′ can be chosen such that   ′ =   .If  is coherent and it is desired that  ′ be coherent, assume that ≤ and ≺ Dom are defined on  ′ and are extensions 2 of ≤ and ≺ Dom on .Then a coherent  ′ can be chosen so that for every ′ ∈   ′ there is ∈   such that ≤ ′ .
In addition to the proof of Lemma 2, Appendix C contains the lengthier proofs of results to be stated later.Note that Lemma 2 above (as well as Lemma 3 and Theorem 2 below) have language about ≤ and ≺ Dom on a larger space being extensions 2 of ≤ and ≺ Dom on a smaller space.When  = IR, 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 ≤ and ≺ Dom are defined in terms of each specific trading system.

Finding Agreeing Functions
Our main representation Theorem 1 states that a trading system  has a standard-linear function that agrees with it and an extension 1 to a total trading system that is represented by .Results from Giarlotta and Creco (2013); Giarlotta (2019); Nishimura and Ok (2020) 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 1 and 2 as special cases, and its proof appears in Appendix C.3.Lemma 3. Assume the following structure: •  and  are standard-linear spaces of functions defined on Ω with  a proper subset of .
where *IR is a nonstandard model of the reals.
Then can be extended to a standard-linear function ′ ∶  → *IR ′ , where *IR ′ contains *IR and such that ′ represents a total trading system  ′ = (, ≾ ′ , ≺ ′ ) that is an extension 2 of   .Also, if   is coherent and ≤ and ≺ Dom are defined on  so as to be extensions 2 of ≤ and ≺ Dom on , then  ′ can be chosen to be coherent.
Theorem 1.Let  be a trading system.There exists a standard-linear function that agrees with  and total trading system  ′ that is an extension 1 of  such that represents  ′ .If  is coherent,  ′ can be chosen to be coherent.
Theorem 2 of Skala (1974) shows that every total preorder can be represented by a nonstandard-valued function, but the standard-linear nature that we need is not proven in that paper.

Extending 2 a Trading System
Let  = (, ≾, ≪) be a (coherent) trading system.Theorem 1 says that there exists a standard-linear function that agrees with  and such that represents a (coherent) total trading system  ′ that is an extension 1 of  .Extension 2 is also possible if  is a subspace of a larger standard-linear space, as stated in Theorem 2.
Theorem 2. Let  = (, ≾, ≺) be a total trading system on a standard-linear space  with standard-linear representing function .Let  ′ be a standard-linear space of functions that includes  as a proper subset.Then there is a total trading system  ′ = ( ′ , ≾ ′ , ≺ ′ ) that is an extension 2 of  and a standard-linear function ′ on  ′ that represents  ′ and extends to  ′ .If  is coherent and ≤ and ≺ Dom are defined on  ′ so that they are extensions 2 of ≤ and ≺ Dom on , then  ′ can be chosen to be coherent.
Proof.Apply Lemma 3 with   =  ,  =  ′ , and being the in the statement of Theorem 2. The  ′ and ′ that result from Lemma 3 satisfy the conclusion of Theorem 2.
There are many examples of standard-linear spaces with proper supersets.One can easily imagine an agent determining a set of preferences over a small set  of objects and then being offered additional options in a set  ′ .
Here is an example of a situation that might seem of a different nature, but which still fits the setup of Theorem 2.
Example 2 (Refining a State Space).Suppose that an agent has a trading system  = (, ≾, ≪) where each element of  is a function from Ω to a space .At some point, the agent realizes that the elements of Ω are not atomic.That is, each element appears to be, itself, a subset of a different set Ω ′ .To be specific, for each ∈ Ω, there is a subset ⊆ Ω ′ such that the distinct elements of  = { ∶ ∈ Ω} form a partition of Ω ′ .Let ∼ Ω be the equivalence relation on Ω ′ that corresponds to , i.e., for all , ∈ Ω ′ , ∼ Ω means that there is ∈ Ω such that , ∈ . In this way, Ω is the quotient space Ω ′ ∕ ∼ Ω .Each element ∈  corresponds to a function ( ) ∶ Ω ′ →  defined by ( )( ) = ( ) where ∈ .Each such ( ) is constant on each element of .If at least one has more than one element, then there are additional functions defined on Ω ′ that do not have the form ( ) for ∈ .Let  ′ be a standard-linear space of functions defined on Ω ′ that contains  = { ( ) ∶ ∈ } as a subset.A trading system  = (, ≾, ≪) maps to a corresponding trading system Even if  is total, the resulting  ′ is not total.Theorem 2 can be used to extend 2  ′ to a total trading system  * on  ′ .If  is coherent and  = IR, then ≤ and ≺ Dom are defined on  ′ so as to be extensions 2 of ≤ and ≺ Dom on .In fact, ( ) ≺ Dom ( ) in  ′ if and only if ≺ Dom in . ⋄

Three Decision Theories
In this section, we show how the structure of Section 1 and the results of Section 2 extend three well-known theories of decision making.

Previsions for Random Variables
The first Archimedean theory to which our results apply is the theory of previsions of de Finetti (1974), which begins with an arbitrary set of standard-valued random variables.For each in that set, an agent chooses a standard value ( ) (called the prevision of ) such that the agent is willing to trade away either or ( ) in order to receive the other one.Specifically, the change in fortune [ − ( )] is considered a fair gamble for all standard .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 8 and 9.) In de Finetti (1974), the agent is willing to accept every finite sum of fair gambles.In particular, the agent is willing to accept for all standard .If ( ) = ( ), the right-hand side of ( 5) is ( − ), and the agent is implicitly willing to trade to get or to trade to get .If ( ) ≠ ( ), there is an implicit strict preference in one direction, e.g., if ( ) > ( ) and < 0, the fair trade ( 5) is strictly smaller than | |( − ), so the agent is willing to trade to get , but not the other way.In addition, willingness to accept all finite sums of fair trades implies that a coherent prevision on an arbitrary set  of random variables extends uniquely to a coherent prevision on the linear span  of .Define the total preorder ≾ on  defined " ≾ if and only if ( ) ≤ ( )."It follows that (, ≾, ≺) is a total trading system that is represented by the linear function .Our theory extends that of de Finetti (1974) by dropping the requirement that every element of  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 Fishburn implies that there is no standard-valued prevision that ranks these bets in the order of the stated preferences.See also Debreu (1954).But a nonstandard-valued function can represent such preferences.Random variables with infinite previsions are also cases in which fair prices are not available.See Seidenfeld, Schervish and Kadane (2009) for some surprising examples.

Horse Lotteries
The second theory to which our results apply is that of Anscombe and Aumann (1963);VonNeumann and Morgenstern (1947) for decisions about horse lotteries, which are functions from Ω to the set of simple lotteries over a set of prizes.

Horse Lotteries in General
Definition 14.For each ∈ Ω, let  be the set of prizes available in state .A simple lottery is a probability on a finite subset ( ) ⊆  .Let  be the convex set of simple lotteries available in state . 2 For ease of notation, let  = ⋃ ∈Ω  and  = ⋃ ∈Ω  be respectively the sets of all prizes available in at least one state and all lotteries available in at least one state.Let  Ω = ∏ ∈Ω  , which is a subset of  Ω .A horse lottery is a function ℎ ∈  Ω , i.e, ℎ( ) ∈  for every ∈ Ω.Let  stand for the set of horse lotteries under consideration, which we assume to be a convex subset of  Ω . 3  In each application, the set  of horse lotteries can be different, but each such  must be a convex subset of  Ω .For horse lotteries is not a linear space.Next, we show how to create a linear space that is equivalent to  in an appropriate sense.

A Linear Space for Horse Lotteries
The set  of horse lotteries is a convex subset of  Ω , but is not a linear space.Hausner (1954) (Sections 2-4) assumes that ≾ ′ is a total preorder that satisfies the following axiom, which is part of the theory of Anscombe and Aumann (1963);VonNeumann and Morgenstern (1947): Hausner (1954) shows how to create a standard-linear space  0 with a preorder ≾ that satisfies our Assumptions 2 and 3 in Section 1.4 above.This is done as follows.For each ∈ Ω, let  be the set of all simple signed measures 4 on  that give measure 0 to the whole set  .Let  = ⋃ ∈Ω  , and let Hausner (1954) (Section 4) shows that ≾ is well defined and satisfies Assumptions 2 and 3.The theory of Anscombe and Aumann (1963);VonNeumann and Morgenstern (1947) satisfies Assumption 4 vacuously since ≾ is a total preorder.Dominance and coherence are not issues that arise in the theory of Anscombe and Aumann (1963);VonNeumann and Morgenstern (1947) as horse lotteries are not numerically comparable without further assumptions.The state-independence assumption of Anscombe and Aumann (1963); VonNeumann and Morgenstern (1947) implies that all  sets are the same.Our theory is general enough to include cases in which the  sets might all be the same or might be different.We also drop the Archimedean axiom and allow ≾ ′ to not be total as do other authors such as Aumann (1962); Dubra, Maccheroni and Ok (2004); Baucells and Shapley (2008).Our weaker state-independence Assumption 7 is stated in Section 4.1.

Representing Horse Lotteries
For the remainder of this paper, when we refer to the horse-lottery case, we will assume that  is the standard-linear space  0 defined in Section 3.2.2.(The case in which  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  as its domain rather than  0 .This is easily arranged.Let  = ( 0 , ≾, ≺) be a total trading system in a horse-lottery case with standard-linear representing function .Let  be the set of horse lotteries that corresponds to  0 as in Section 3.2.2.For each ∈  0 , we can write = (ℎ 1 − ℎ 2 ) with > 0 standard.Then 0 ≾ is equivalent to ℎ 2 ≾ ′ ℎ 1 for a total preorder ≾ ′ on .Let ℎ 0 ∈  be arbitrary, and define It follows that (ℎ 0 ) = 0 and Also, represents ≾ ′ and satisfies for all ℎ 1 , ℎ 2 ∈  and all standard ∈ [0, 1].

Savage-style Acts
The third theory to which our results apply is that of Savage (1954).This theory makes some assumptions (including state-independence) about preferences amongst acts (functions) from a state space Ω to a set  of consequences (prizes) and then proves an expected-utility representation for those preferences.Lemma 4 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 (1954) 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 (1954).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 (1954) 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 4 is in Appendix C.4.Lemma 4. Let  be a set of functions from Ω to , and let ≾ ′ be a total preorder on  that satisfies the seven postulates (P1-P7) of Savage Savage (1954).Let  be the set of finite mixtures of elements of  .Then ≾ ′ extends to a total preorder on , and the  0 and ≾ constructed from  and ≾ ′ in Section 3.2.2form a total trading system ( 0 , ≾, ≺) that satisfies Assumptions 1-4.
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  of Savage-style acts with a preorder (not necessarily total) ≾ ′ .Then embed  into the convex set  of Lemma 4 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.

Probability and Expected Utility
In this section, we explore the relationship between finitely-additive expectation and the representing function of a total trading system.Throughout the section,  = (, ≾, ≺) denotes a total trading system on a standard-linear space  of functions from Ω to a standard-linear space .Let be a standard-linear function that represents  .Let Σ be a field of subsets of Ω.In order to construct a probability from a trading system, the indicators of elements of Σ must play a role in the elements of .We make the following assumption about a total trading system  = (, ≾, ≺) throughout this section.Assumption 6.For all ∈ Σ and all ∈ , ∈ , where If Assumption 6 is not met, one can apply Theorem 2 to find an extension 2 of  to This extension needs to be done before attempting to infer a probability on (Ω, Σ) or attempting to interpret a representing function as an expected utility.Once Assumption 6 is met, we can define conditional preference.
Definition 15.Let  be a standard-linear space of functions defined on a set . Let Γ be a field of subsets of .Suppose that ∈  for each ∈ Γ and ∈ .Let ≾ be a total preorder on .For , ∈ , if 0 ≺ ( − ) , we write ≺ | and we say that is conditionally preferred to given .If 0 ≾ ( − ) , we write

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 statedependent) utility function defined on the codomain  of the elements of .
For the remainder of this section, we assume that  is the space  0 defined in Section 3.2.2 and  is a set of simple signed measures on subsets of the prize set  and that assign signed measure 0 to .As such, each element of  0 is a function from Ω to . 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 that consists entirely of null states, i.e., every ∈ is a null state.Then a state-by state comparison of two elements and of  given the elements of tells us nothing about how and compare (or should compare) given .The reason is that ∼ |{ } for every and and every null state .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 (1963) (Assumption 1,) Savage (1954) (P3-P4,) and Blume et al. (1991a) (Axioms 5 and 5 ′ .)Assumption 7 below allows varying degrees of state dependence for the utilities of prizes.
Assumption 7.There exists a partition  ⊆ Σ of Ω into non-null events such that, for each ∈ , • for all ∈ ,  is the same set ( ), • for each ∈ ( ), there is ∈  such that ( ) = for all ∈ , and • for each ∈ ( ), 0 ≺ | if and only if 0 ≺ | for every non-null ⊆ .For each , we will use to denote the element of  that contains .
The second bullet assumes that certain functions defined on  Ω are in .If these functions are not in  and the first bullet is satisfied, we can apply Theorem 2 to find an extension 2 of  to 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  ′ 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 (1963); Savage (1954); Blume et al. (1991a) correspond to the extreme case of Assumption 7 in which  = {Ω}, i.e., utility is independent of state for the whole state space.The opposite extreme case of Assumption 7 has  = {{ } ∶ ∈ Ω}, i.e., every state is non-null, in which case all three bullets of Assumption 7 are tautological because each is a singleton.There are cases between these two extremes, such as Example 5 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  is the set of states with a fixed specification of exchange rates.
Definition 16 below defines ≤ 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 9).As in the random-variable case, whichever (if any) sense of dominance is reflected in the agent's preferences, we express " dominates " by ≺ Dom .
As in Definition 8 for the random-variable case, Definition 16 defines dominance on a larger set of objects than just .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 Σ. (See Lemma 6 in Section 4.2.)Lemma 5. Let  = (, ≾, ≺) be a total trading system that satisfies Assumptions 6 and 7. Let ∈  Ω , and let 0 ∈ Ω.There exists , 0 ∈  such that , 0 ( ) = ( 0 ) for all ∈ 0 .Proof.Because = ( 0 ) ∈ ( 0 ), Assumption 7 says that there is ∈  such that ( ) = = ( 0 ) for all ∈ 0 .Rename to be , 0 .Lemma 5 gives us what we need to define dominance on all of  Ω .
• If ≤ and there exists 0 such that ( 0 ) < ( 0 ), we say that weakly dominates .Now, the definition of "coherent" (Definition 9) applies in the horse-lottery case.We did not define "uniform dominance" because the < symbol in Definition 16 is not a relation between numerical values, but rather a nonnumerical preference relation.Once we define a state-dependent utility function (Definition 18 in Section 4.3) we can define uniform dominance in the horse-lottery case.

Numeraires and Probability
Definition 17. Assume the conditions stated in Definition 15.Let ∈ .If 0 ≺ and 0 ≾ | for all , we call a numeraire for Γ.If, in addition, 0 ≺ | for every non-null , we call a strong numeraire for Γ.
In the random-variable case with a coherent trading system, every non-negative function in  that is strictly preferred to 0 is a numeraire.Nevertheless, even positive constants might not be strong numeraires.
Example 3. Let Ω = ZZ + , the positive integers, and let Σ be the finite/cofinite field, i.e., the collection of all finite subsets of Σ and their complements.Let ∶ Ω → IR be ( ) = .Let  be the standard-linear span of all standardvalued bounded functions and the functions { ∶ ∈ Σ}.Each ∈  can be written uniquely as where is bounded, is standard, and = {11, 12, …}.5 Define Also, ∶  → IR is standard-linear (and standard-valued.)If ≺ Dom (uniform or strict, but not weak dominance) then ( ) < ( ), so represents a coherent (with uniform or strict dominance) total trading system  = (, ≾, ≺) with ≾ meaning ( ) ≤ ( ).Since (1) = 1, 0 ≺ 1.Since 0 ≤ 1, 0 ≾ 1| for all ∈ Σ, and 1 is numeraire for Σ.Since 0 ≺ | , is a non-null event.On the other hand, 0 ∼ 1| , so 1 is not a strong numeraire for Σ.The non-null events are all cofinite sets and all elements of Σ that are supersets of the singletons {1}, … , {10}.For each = 1, … , 10, 0 ≺ |{ }, and for each cofinite set , 0 ≺ | , so is a strong numeraire for Σ. ⋄ In Section 4.5, we show how different numeraires relate to each other within the same trading system.We take probability to be finitely additive in Lemma 7 below and elsewhere due to the difficulty in defining countable sums of nonstandard numbers.(See the results and examples in Appendix A.4.)In general, we need a numeraire in order to derive a probability on (Ω, Σ) 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.Lemma 6.Let  = (, ≾, ≺) be a coherent total trading system in the horse-lottery case that satisfies Assumptions 6 and 7.There is a coherent extension 2  ′ = ( ′ , ≾ ′ , ≺ ′ ) of  such that • there is a numeraire for Σ, • is constant on each element of  (from Assumption 7,) and Proof.First, we show that, for each ∈ , there is ∈ ( ) such that 0 ≺ | , where is defined in the second bullet of Assumption 7. Let ∈ , which, by definition, is non-null.So there is ∈  such that 0 ≺ | .For each ∈ , let = ( ).If ≾ 0| for every ∈ , then ≤ 0 and ≾ 0| by coherence.This contradicts 0 ≺ | , so there must be 0 ∈ such that 0 ≺ | for = 0 .Define ∶ Ω → *IR as follows: For all ∈  and all ∈ , let ( ) = .By construction 0 ≺ | for all ∈  and 0 ≺ Dom by either definition of dominance.
For the remainder of the paper, assume that each trading system  in the horse-lottery case is an extension 2 from Lemma 6. Lemma 7. Let  = (, ≾, ≺) be a coherent total trading system that satisfies Assumption 6 (and Assumption 7 in the horse-lottery case.)Let be a numeraire for Σ, and let be a standard-linear function that represents  .Define ( ) = ( )∕ ( ) for each ∈ Σ.Then is a finitely-additive (possibly nonstandard-valued) probability on Σ.

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  are in units of dollars.A European foreign-exchange trader might be more comfortable comparing units of euros rather than dollars.If ( ) is the exchange rate (in dollars per euro) in state , ( )∕ ( ) is the utility of in state to the European trader measured in the preferred currency units of euros.We need something analogous to ∕ for the horse-lottery case.
Definition 18.Let Σ be a field of subsets of Ω, and let  be a coherent total trading system that satisfies Assumption 6 (and Assumption 7 in the horse-lottery case.)Let ∶  → *IR be a standard-linear function that represents  .In the horse-lottery case, let be a numeraire of the sort obtained through Lemma 6.In the random-variable case, let be a strictly positive numeraire.Define, for each ∈ , each ∈ Ω, and each ∈  : in the random-variable case, in the horse-lottery case.
We call * the state-dependent utility function (relative to the numeraire .)Let , ∈ .If there exists a standard > 0 such that * ( , ( )) ≤ * ( , ( )) − for all , we say that is uniformly dominated by relative to or uniformly dominates relative to .
As Definition 18 makes explicit, which elements of  uniformly dominate each other depends on which numeraire is used.Example 6 illustrates this fact.

Example 6 (Continuation of Example 5).
The state-dependent utility that corresponds to the numeraire ℎ is, for each so that * ℎ ( , ℎ ( )) = 1 for all , and ℎ uniformly dominates 0 relative to 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 ℎ ( ) = 1∕2 for all so that ℎ is constant.Let ℎ ′ be another horse lottery with ℎ ′ ( ) = 1 − for all , so that ℎ ′ is ∧-shaped and * ℎ ′ ( , ℎ ′ ( )) = 1 for all .Then ℎ ′ uniformly dominate 0 relative to ℎ for ≥ 1∕2, so ℎ ′ does not uniformly dominate 0 relative to ℎ .⋄ To see that * is well defined in the horse-lottery case, note that for each ∈  of Assumption 7, ( ) > 0 because 0 ≺ | .In general * is a nonstandard-valued function defined on the set The interpretation of * ( , ) is the utility to the agent of in state measured in units of numeraire when ( ) = .In Section 4.4, Theorem 3 shows that ( ) can be interpreted as an expected value of the state-dependent utility of with respect to the probability on (Ω, Σ) that corresponds to the chosen numeraire via Lemma 7.

Expected Utility
The following definition is a generalization of the concept of Daniell integral to the finitely-additive nonstandardvalued case.See Schervish, Seidenfeld and Kadane (2014) for discussion of the finitely-additive standard-valued case.We give additional motivation for this definition in Appendix B.
Definition 19.Let *IR be a nonstandard model of the reals, and let  be a set.Let  be a standard-linear space that is a subset of (*IR)  , and that contains all standard constants.Let ∶  → *IR be a standard-linear function that satisfies

•
(1) = 1, and • for 1 , 2 ∈ , 1 ≤ 2 implies ( 1 ) ≤ ( 2 ) (i.e., is monotone.) Then acts as an expected value on .Suppose, in addition, that Γ is a field of subsets of  and for every ∈ Γ and ∈ , ∈ .Define ( ) = ( ) for ∈ Γ so that is a finitely-additive probability on (, Γ).Then acts as an expected value on  with respect to .For each ∈ , we also say that ( ) is an expected value of with respect to .We also use the notation ( ) to denote ( ).
It should be apparent that Definition 19 agrees with the familiar countably-additive definition of expected value for simple probabilities when *IR = IR.Furthermore, all finite standard-valued countably-additive expected values satisfy Definition 19, but they also have an additional continuity property that does not carry over to the nonstandard-valued case.
Theorem 3. Assume the conditions from and terms defined in Definition 18.Let ( ) = * ( , ( )) for each ∈  and ∈ Ω, and let be the probability on (Ω, Σ) from Lemma 7 using as the numeraire for Σ.Let  = { ∶ ∈ }.Then •  is a standard-linear space of functions from Ω to *IR, • ( ) = ( )∕ ( ) defines an expected value of with respect to , and • if dominance means weak dominance, then every non-empty element of Σ is non-null and has positive probability.
Proof.First, we show that ≤ implies ≤ .This is immediate in the random-variable case.In the horselottery case, if ≤ , then for each ∈  and each 0 ∈ , , 0 ≾ , 0 | in the notation of Lemma 5 and Definition 16.It follows that ≤ .
Corollary 1. Assume the conditions of Theorem 3 in the random-variable case with numeraire = 1.Then  = , = for all ∈ , and acts as an expected value on  with respect to .

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 rather than elements of  .Let  and * be as in Theorem 3. Let ℎ 0 ∈  be arbitrary, and define * ∶ ⋃ ∈Ω { } ×  → *IR by * ( , ) = * ( , − ℎ 0 ( )). (11) This has the effect of shifting the utility in state so that ℎ 0 ( ) has value 0. It follows that, for 1 , 2 ∈  , * ( , which gives us equivalent ways to express state-dependent utility in both  and  0 .Equivalent ways to express marginal utilities were given in ( 6) and ( 7) in Section 3.2.3.Finally, for each ∈ Ω and ∈  , let stand for the simple lottery that assigns probability 1 to the prize , and define * 0, ( , ) = * ( , ).
A useful consequence of the above notation is the following simple corollary of Theorem 3.
Our final goal, in this section, is to show that * ( , ) can be interpreted as an expected value of * 0, ( , ⋅) with respect to the simple probability .
Corollary 2 and Lemma 8 combine to say that (ℎ) is an iterated expected value of * 0, ( , ) where, for each ∈ Ω, the inner expected value is with respect to the probability ℎ( ) on (ℎ( )), and the outer expected value is with respect the the probability on Ω.To be precise, ( ) = ℎ( )[ * 0, ( , ⋅)] is the inner expected value, and (ℎ) = [ (⋅)] is the outer expected value.

The Original Savage-style Acts
In this section, we assume that a coherent total trading system  = ( 0 , ≾, ≺) that satisfies Assumptions 1-7 was generated from a set  of Savage-style acts by creating the set  of horse lotteries that are simple mixtures of elements of  .The probability that results from Lemma 7 is a function from Σ to *IR, and as such can be associated with the set  , regardless of how its existence was proved.Similarly, the function of (6), when restricted to the elements of  , represents the preorder ≾ ′ on  that is the restriction of ≾.Also, the state-dependent utility of act ∈  in state is * 0, ( , ( )) from ( 13).Hence the expected-utility interpretation of Corollary 2, when restricted to elements of  gives an interpretation of as an expected value of the state-dependent utility * 0, ( , ( )) with respect to .

Existence of Coherent Trading Systems in the Horse-Lottery Case
In the random-variable case, all forms of dominance are defined independently of an agent's willingness to trade.Furthermore, a respect for a chosen form of dominance can be enforced while the agent is stating preferences.The same is not true for the horse-lottery case, where dominance is defined based on an existing total trading system.If that trading system does not respect dominance, the agent either has to start over or can try to modify the stated preferences.
There is a way for the agent to enforce respect for a chosen form of dominance, but it places restrictions on the order in which preferences can be stated.To mimic the random-variable case, for each , ∈ , the agent needs to be able to determine, prior to saying which trades between and are acceptable, whether ≺ Dom or ≺ Dom or neither.To do this,one needs a partition  of Ω into non-null events that satisfy the first two bullets of Assumption 7. The agent also needs to be willing to require preferences to satisfy the third bullet.
If the agent wishes to respect weak dominance, then all states will be non-null, and  can be taken to be {{ } ∶ ∈ Ω}.The agent would first determine whether ≺ |{ } or ≺ |{ } or neither.Such comparisons reduce to checking, for each , ∈  , how { } and { } compare.Once all such determinations are made, weak dominance is defined on all of  Ω , and all instances of ≺ Dom can be labelled as ≪ before the rest of the trading system is determined.
If the agent wishes only to respect strict dominance, the agent needs to determine, for each ∈  and each ( , ) ∈ ( ) 2 , whether ≺ | .If so, the agent then declares that ≺ | for every non-null subset of .As above, strict dominance is now defined on all of  Ω , and all instances of ≺ Dom can be labelled as ≪ before the rest of the trading system is determined.

Conditional Trading
Conditional preference (see Definition 15) can be interpreted as a willingness to trade given that some event occurs.Throughout this section, assume the conditions of Definition 18 and Theorem 3. In particular: •  = (, ≾, ≺) is a coherent total trading system with standard-linear representing function .
Σ is a field of events for which Assumption 6 holds.
• A numeraire for Σ exists with corresponding probability as in Lemma 7.
• There exists a state-dependent utility The following result is straightforward from standard-linearity of .
In the spirit of conditioning on an event, the following result shows how to restrict a trading system on a state space Ω to a smaller state space consisting of an element of Σ with ( ) > 0. The restriction | of a function to a subset of its domain is defined to be the function | ( ) = ( ) for all ∈ , which maps into the codomain of .The corresponding restriction of a field Σ is Σ = { ∩ ∶ ∈ Σ}.
Lemma 9. Let be a non-null element of Σ, and define ), and which satisfies Assumption 6.If ( ) > 0, then (a) | is a numeraire for Σ with associated probability ( ) = ( ∩ )∕ ( ), and (b) there is an expected state-dependent utility representation for with respect to .Finally, if dominance in  means weak dominance, then  is coherent with weak dominance.
Proof.It is straightforward that ≾ is a well-defined total preorder on  as is the fact that Assumptions 1-4 hold in  .Since is standard-linear and the operation of restriction to commutes with linear combinations, it follows that is also standard-linear.Since ≾ | means ≾ , it is clear that represents  .For Assumption 6, the appropriate field is Σ , and the assumption holds in  .
Next, assume that ( ) > 0 so that 0 ≾ | and | is a numeraire for Σ .Since ( ) = ( ) and ( ∩ ) = ( ∩ ), the probability associated with | is as stated.Also, for each ∈ , As a corollary, we have a version of the law of total probability/expectation for conditional trading systems.Note that, if is a non-empty null event then ( ) = 0 and ( | ) = 0 for all ∈ .
Corollary 3. Let be a standard finite integer, and let 1 , … , be a partition of Ω into non-empty events.Then, for each ∈ , The following example illustrates why coherence of  in Lemma 9 requires ( ) > 0 for uniform and strict dominance.

Example 8 (Continuation of Examples 3 and 4).
Recall that the trading system  is coherent using either uniform or strict dominance, and that the set = {11, 12, …} is non-null.The two probabilities and computed in Example 4 differ most notably by the fact that 1 ( ) = 0 while ( ) > 0. Suppose that we try to restrict the trading system to the set as is done in Lemma 9.With = 1 as numeraire, 1 ( ) = 0, so Lemma 9 doesn't apply.In particular, 0 ∼ 1| which violates all forms of dominance.With = as numeraire, ( ) > 0, but * ( , ) = ∕ .In order for to uniformly dominate 0 there must be a standard > 0 such that ( )∕ > for all .A necessary condition for this is > 0, so only some unbounded functions uniformly dominate 0, and all ∈  with > 0 satisfy 0 ≺ | .So  ′ is coherent using uniform dominance with numeraire .Finally, recall that  ′ is not coherent using strict dominance because 0 ∼ 1| , despite the fact that  is coherent using strict dominance.⋄

Discussion
The major contributions of this paper are • a systematic representation of coherent preferences amongst random variables or horse lotteries regardless of how strong is the form of dominance that one wishes to respect, • the use of nonstandard models of the reals to represent non-Archimedean preferences, • an extension theorem from one standard-linear space of random variables to a larger space, • an expected utility interpretation for the nonstandard representation in special cases (including those of existing theories,) and • a derivation of conditional preferences.
Some, but not all, of the examples of non-Archimedean preferences arise from the use of weak dominance in the definition of coherence.For example, it is impossible to have an Archimedean preference structure that respects weak dominance on the set of random variables defined on an uncountable state space.
Weak dominance is the weakest of the three dominance concepts in Definition 8. Weak dominance is the same as the form of dominance used to define inadmissibility in statistical decision theory.The strongest of the three dominance concepts is the one used in de Finetti's theory, namely uniform dominance.Strict dominance is intermediate to the other two.Since dominance is used to prevent calling a trading system coherent, the stronger the dominance condition, the weaker the sense of coherence, i.e., the easier it is to call a trading system coherent.Since some of our results use the weakest form of dominance, those results use the strongest form of coherence.
There is room for future work.Assumption 7, in the presence of the other assumptions, is sufficient to prove the existence of an expected-utility representation of preference, but it is not necessary.If we define the finitely-additive signed measure ( ) = ( ) for ∈ Σ, then * (⋅, (⋅)) behaves like a Radon-Nikodym derivative of with respect to in Theorem 3. The missing necessary condition would be equivalent to a Radon-Nikodym theorem for nonstandard-valued finitely-additive signed measures.A standard-valued finitely-additive Radon-Nikodym theorem for bounded measures was proved by Maynard (1979), but it is heavily dependent on standard real numbers.Here is an example of a total trading system with an expected-utility representation that fails Assumption 7.
Theorem 7.1 of Comfort and Negrepontis (1974) then says that if  has the finite intersection property, then there is an ultrafilter  ⊇  .

A.2. Ultraproducts
In this section we introduce a popular method of constructing and extending nonstandard models of the reals.The construction and extension of nonstandard models are viturally identical.In what follows, the initial construction uses  = IR, the standard reals.Later extensions (if needed) use  equal to an existing nonstandard model.
Let  be an infinite set, let  be an ultrafilter on , and let  be a nonstandard model of the reals (possibly IR itself.)The elements of   are functions from  to  .Define a binary relation of   by It is easy to see that ∼  is an equivalence relation because the intersection of finitely many elements of  is in  .Call *  =   ∕ ∼  (the set of equivalence classes corresponding to ∼  ) the ultraproduct corresponding to  ,  and  .The embedding of  into *  (which we call the natural embedding) is ↦ [ ]  , where (for ∈  ) ∈   is the constant function ( ) = for all ∈ .We take the liberty of using the symbol to stand for [ ]  when ∈  .If  is a principal ultrafilter with { 0 } ∈  , then *IR  is essentially  because ∈ [ ( 0 ) ]  for every ∈   .The classic ultraproducts are those that start with  = IR,  = Z Z (the positive integers,) and  a non-principal ultrafilter on .
The following examples of ultraproducts are non-Archimedean extensions of the standard real numbers, and we will use one of them the first time that we need to construct nonstandard numbers.
Example 11.Let  = ZZ, let  be a non-principal ultrafilter of subsets of  that contains { , + 1, …} for every integer , and let  = IR.Let *IR  denote the ultraproduct nonstandard model *  in the above construction.Let ( ) = 1∕ for all .It is easy to check that [ ]  < for every strictly positive standard real .Just note that each set of the form { ∶ ( ) < } equals { , + 1, + 2, …} where is the first integer such that > 1∕ .At the other extreme, let ( ) = for all .Then [− ]  < < [ ]  for every standard .Finally, let be an arbitrary finite standard.For every standard < and every standard > , we have < [ − ]  < < [ + ] < .In words, *IR  includes numbers that are squeezed in between and every standard less than (or greater than) .In this way *IR  is non-Archimedean.⋄ Definition 23.A nonstandard such that | | < for every positive standard is called infinitesimal.A nonstandard such that | | > for every positive standard is called externally infinite.A nonstandard that is not externally infinite is called externally finite.
The infinitesimals and standard reals are externally finite, as are hybrid nonstandards such as 1 + , where is infinitesimal.
Lemma 10.The ultraproducts constructed at the start of this section are nonstandard models of the reals.
Proof.The extension of All of these extensions are well-defined because  is closed under finite intersections of its elements.The additive identity (zero element) in *  is [0]  where 0 is the additive identity in  .Similarly, the multiplicative identity The additive and multiplicative inverses of an element [ ]  are respectively [− ]  and [1∕ ]  , the latter applying only when is not the zero element of  .
It is straigtforward that each ultraproduct extension of a non-Archimedean nonstandard model of the reals is also non-Archimedean.
Proof.If is itself standard, no other standard is closer to , so ℜ( ) = .For the remainder of the proof, assume that is not itself standard.Let = { ∈ IR ∶ < } and = { ∈ ∶ < }.Then both and are nonempty, ∩ = ∅, and ∪ = IR.Also, each element of is an upper bound for and each element of is a lower bound for .Since both and are sets of standards, sup = inf , and we call the common value ′ , which is standard.We now show that ℜ( ) = ′ is the nearest standard to .Let < ′ so that < ( + ′ )∕2 < ′ .Since ( + ′ )∕2 is closer to than , no standard less than ′ is closer to .Similarly, let > ′ so that ( + ′ )∕2 is closer to than , so no < ′ is closer to .
We call ℜ( ) the standard part of .Note that and have the same standard part if and only if − is infinitesimal.For convenience, we say that the standard part of an externally infinite is infinite and express the fact as ℜ( ) = ∞ or ℜ( ) = −∞ as appropriate.Note that Lemma 11 doesn't depend on how the nonstandard model is constructed.
The externally finite nonstandards have upper and lower bounds that are externally infinite, but there is neither a least upper bound nor a greatest lower bound.An example of a bounded external set of nonstandards that has no supremum or infimum is the following.
Example 12. Let 0 be a standard real, and let = { ∶ ℜ( ) = 0 }.It is clear that, for every ∈ , + ∈ for every positive infinitesimal , so no element of is an upper bound for .Similarly, no element of is a lower bound for .Hence, every upper bound for has standard part ℜ( ) > 0 .For every such , ( 0 + ℜ( ))∕2 is an upper bound for that is smaller than .Similarly, for every lower bound for , ( 0 + ℜ( ))∕2 is a larger lower bound for .Hence, has neither a greatest lower bound nor a least upper bound.⋄ In the theorems of Section 2, we assign numerical values to objects in a trading system sequentially.There are two different situations when the number system  we are using does not have a value that is appropriate for the next object.One situation arises when the next object to be assigned a value is strictly preferred (dispreferred) to every element of a set and each number in  is already less (greater) than or equal to the value assigned to an element of .In this case, we need to expand  to include values that are larger (smaller) than everything already in  .The second situation arises when the next object needs to be assigned a value strictly between two non-empty sets 1 and 2 that already partition  .The following lemma shows how to extend a number system  in each of those situations.
Lemma 12. Let  be a nonstandard model of the reals.
Proof.If  = IR, the nonstandard model of Example 11 satisfies both claims 1 and 2. If  is already nonstandard, start with claim 1.Let  =  , and let  be the collection of all subsets of  of the form { ∈  ∶ > 0 } for some 0 ∈ .
The collection  has the finite intersection property because every finite subset of  has a maximum element.Let  be an ultrafilter of subsets of  that contains , so that  is non-principal.Let * be the corresponding ultraproduct nonstandard model.The only case remaining is claim 2 when 1 has no least upper bound.In this case, let  = 2 with  being the collection of sets of the form { ∈  ∶ < 0 } for some 0 ∈ .Then  has the finite intersection property.Let  be a (non-principal) ultrafilter that contains , and let ( ) = for all ∈ .The equivalence class [ ]  > 1 for all 1 ∈ 1 because every ( ) > 1 .Also, [ ]  < 2 for all 2 ∈ 2 because { ∶ ( ) < 2 } ∈  .Some of our results rely on the possibility of applying Lemma 12 infinitely many times, in a well-ordered manner.To be specific, let Γ be an ordinal.Let  0 = IR.For each successor ≤ Γ, let  be the result of applying Lemma 12 to  −1 .For each limit ≤ Γ (if any), let  − = ⋃ <  .It is straightforward to show that  − is a nonstandard model of the reals when is a limit ordinal.

A.3. Some Notes About Infinity
The use of the symbol ∞ to stand for "larger than every standard number" has a long history, and rarely causes trouble when discussing standard reals.Certain conventions allow some arithmetic with ∞.For example, • for all finite , ∞ + = ∞, and • for all finite, non-zero , ∞ equals ±∞, with the sign matching that of .
However, there is no place for standard infinity in a nonstandard model of the reals.Externally infinite, but internally finite, nonstandards replace standard infinity, and they require no special conventions to allow internally finitary arithmetic.Whenever we need to represent something in a nonstandard model that is larger than every number in the model, we appeal to Lemma 12 which essentially iterates the ultraproduct construction to produce a larger nonstandard model that contains internally finite numbers to represent what we need.

A.4. Countable Additivity
Nonstandard models of the reals are designed to have all of the finitary properties of the reals along with non-Archimedean structure.One must be careful not to expect everything that one knows about infinite sets of standard reals to apply to nonstandards.We already saw examples of bounded sets with no suprema or infima.(See Example 12.) A related property is that externally countable sums do not behave the same in nonstandard arithmetic as they do in standard arithmetic.See Section 4.3 of Halpern (2010) for more discussion of this point.For example, when the sequence of finite partial sums of a standard positive countable sequence is bounded, one can define the sum of the entire sequence to be the supremum of the partial sums.The same is not always possible for nonstandards.
Example 13.Let > 0 be infinitesimal, and let = ∕2 for each standard positive integer .One might think that that ∑ ∞ =1 = .However, we show next that, despite the fact that the sequence of finite partial sums is bounded, there is no least upper bound.Clearly, each finite partial sum = ∑ =1 < for every standard positive integer , so that is an upper bound on the finite partial sums.Let be an arbitrary finite upper bound on the finite partial sums.Next, we show that there is a smaller upper bound than .
Let = 2 so that < for every standard positive integer .It follows that, for each standard positive integer , Hence − is a smaller upper bound on the sequence of partial sums.⋄ The remainder of this section is devoted to exploring the extent to which one can make sense of the sum of externally countably many elements of a nonstandard model of the reals.First, we note that the sequence of finite partial sums = ∑ =1 ∞ =1 of an externally countable sequence { } ∞ =1 in a nonstandard model of the reals converges to an element of the same model if and only if {| − |} ∞ =1 converges to 0. So, we state our results in terms of externally countable sequences of strictly positive numbers converging to 0. To say that a strictly positive externally countable sequence { } ∞ =1 converges to 0 means that, for every > 0 in the model, there is a standard integer such that for all > , < .To show that such a sequence fails to converge, we need to find an > 0 in the model such that > for infinitely many .The first step is the following.Proof.Let { } ∞ =1 be an externally countable sequence of strictly positive elements of *IR.Represent each = [( ,1 , ,2 , …)]  , where  is the appropriate ultrafilter.each > 0, we can assume without loss of generality that , > 0 for all , .Construct another strictly positive nonstandard = [( 1 , 2 , …)]  as follows.Let 0 < 1 < 1,1 , and for each > 1, let 0 < < min{ 1, , ⋯ , , }.For each , Hence, 0 < < for all .The next step is to extend the conclusion of Lemma 13 to further applications of Lemma 12.
Lemma 14.Let  be a nonstandard model of the reals such that no externally countable sequence of strictly positive elements of  converges to 0. Apply Lemma 12 to create * .Then no externally countable sequence of strictly positive elements of * converges to 0.
Proof.Let  and  be the set and ultrafilter used in the extension of  to * .Let {[ ]  } ∞ =1 be an externally countable sequence of strictly positive elements of * .For each ∈ , { ( )} ∞ =1 is an externally countable sequence of strictly positive elements of  , hence there exists ( ) > 0 in  such that ( ) > ( ) for all .Then [ ]  > [ ]  > 0 for all , and {[ ]  } ∞ =1 does not converge to 0 in * .Recall that convergence of an externally countable sum to an element of a nonstandard model of the reals requires convergence to 0 of the difference between and the finite partial sums.Since convergence to 0 of a countable sequence in nonstandard models of the reals is so difficult to ensure, we assume only that probabilities are finitely additive.Furthermore, the non-Archimedean nature of the preferences that we model leads to utilities being nonstandardvalued.Integrating a nonstandard-valued function with respect to countably-additive probability runs into the same problems we just exhibited for nonstandard-valued probabilities.The limits required to define the integral of a nonsimple nonstandard-valued function do not exist in general.The next section discusses a finitely-additive definition of expected value that is suitable for use with nonstandard-valued functions and/or probabilities.

B. Expected Values for Nonstandard-Valued Functions
In the theory of countably-additive probability and expected value for standard-valued functions, each probability on a measurable space (, Γ) has a unique extension from indicators of elements of Γ to an expected value for all bounded measurable functions and for all non-negative measurable functions (where ∞ may be the resulting expected value.)Finally, there is a unique extension to all functions whose positive and negative parts don't both have ∞ as their expected valued.The first step is the trivial extension to the simple functions, those that assume only finitely many values.This trivial extension applies equally well in the nonstandard-valued and/or the finitely-additive cases, namely The extension of expected value to a bounded measurable function is done by a sequence of uniform approximations of by a sequence of pairs of simple functions {( ,< , ,> } ∞ =1 where ,< ( ) ≤ ,> ( ) and ,> ( )−1∕ ≤ ( ) ≤ ,< ( )+1∕ for all ∈  and all ∈ Z Z + .The sequences of expected values { ( ,> )} ∞ =1 and { ( ,>) } ∞ =1 both converge to the same number, and that number is ( ).This feature applies in the finitely-additive standard-valued case, but not so much to nonstandard-valued cases.In particular, we can get uniform approximations of bounded functions by simple functions to within each positive standard value, which allows us to pin down the standard part of ( ), but we cannot uniquely determine the value of ( ) from these simple functions alone.To get a uniform approximation to within an infinitesimal amount requires a "simple" function with an externally infinite nonstandard integer number of terms.The sum of countably many nonstandard values is generally not possible to define, and countably-additive probabilities are not generally additive over an externally infinite nonstandard number of values.
The standard-valued countably-additive extension to a non-negative measurable function is done by which is still possible in the standard-valued finitely-additive case.In the nonstandard-valued cases, there are many sets of finite numbers, even bounded sets of finite numbers, for which no supremum (least upper bound) exits, as noted in Appendix A. So, one cannot use ( 14) to define the expected value of a non-negative nonstandard-valued function.Instead, ( ) must be at least as large as ( ) for every simple ≤ .This makes expected values of unbounded standard-valued functions and general nonstandard-valued functions non-unique extensions of the underlying probabilities, be they countably-additive or merely finitely-additive.Example 8 in Section 4.9 is a case of a countably-additive probability with a finitely-additive extension to unbounded functions.
The implications of the non-uniqueness of extensions are handled as follows.In the finitely-additive standardvalued case, it is coherent, in the sense of de Finetti (1974) and for a single unbounded non-negative function , to assign a value to ( ) that equals the right-hand side of ( 14) plus for > 0. However, doing so has implications for the expected values of other unbounded non-negative functions.Our extension Theorems 1 and 2 are set up to take into account all of those implications if and when they arise.The reader should also note that, (14) often forces ( ) = ∞ because there can be simple functions ≤ with arbitrarily large ( ).In our nonstandard approach, we would assign a nonstandard externally infinite expected value as ( ).In fact, the very idea of what counts as "bounded" or "unbounded" changes when nonstandard externally infinite numbers are being used.
For the above reasons, we use Definition 19 to define expected values of nonstandard-valued functions with respect to a finitely-additive (or even a countably-additive) probability .Essentially, an expected-value functional with respect to is a standard-linear mapping ∶  → *IR, where  is a standard-linear space of (possibly nonstandardvalued) functions that includes the indicators of the sets on which is defined as well as other functions for which one desires expected values.The functional needs to have two additional properties: (i) ≤ implies ( ) ≤ ( ) (monotonicity) and (ii) (1) = 1 (normalized.)Each finitely-additive probability has multiple extensions to each  that includes non-simple (even unbounded) functions.Each extension involves the space of functions whose expected values need to be computed as well as the specific expected values assigned to the functions.For each finitely-additive probability and each standard-linear space  of functions whose expected values we want, there is a convex set  , of possible extensions of to .
Our assumptions refer to an agent's willingness to engage in various trades amongst elements of a set .If the agent's willingness to trade satisfies our assumptions, then there is a (possibly nonstandard-valued) function on  that represents the trades that the agent is willing to make.We then show that ( ) can be interpreted as an expected value of a state-dependent utility of the value of in state with respect to a probability over (Ω, Σ).Unless we impose more restrictions on which trades an agent should be willing to make, i.e., make more restrictive assumptions, the inferred expected-value functionals could be arbitrary elements of  , .

C. Lengthy Proofs
This appendix contains the lengthier proofs of the results in the main paper.

C.1. Proof of Lemma 2
If  ′ = , then  ′ =  satisfies the conclusions of the lemma.For the remainder of the proof, assume that  is a proper subset of  ′ .
We start with the case in which it is not required that  ′ be coherent.Define ≾ ′ on  ′ as follows: For For Assumption 3, suppose that ≾ ′ for and > 0 is standard for = 1, 2. Then For Assumption 4 on  ′ , only the final two bullets need to be proven.To that end, If  is coherent and it is desired that  ′ be coherent, define ≾ ′ on  ′ as follows: For , ∈  ′ , ≾ ′ if there is ∈   such that ≤ − .Then ≾ ′ extends 2 ≾ and   ′ satisfies the final claim of the lemma.Define ≪ ′ on  ′ as follows: For , ∈  ′ , ≪ ′ if there is ∈   such that either ≺ Dom − or 0 ≪ ≤ − .This makes  ′ satisfy Assumption 5. Since 0 ∈   and 0 ≤ − , Assumption 1 holds.For Assumption 2, suppose that , , ′ , ′ ∈  ′ with − = ′ − ′ .Then ≾ ′ if and only if there is ∈   such that ≤ − = ′ − ′ if and only if ′ ≾ ′ ′ .For Assumption 3, suppose that ≾ ′ and > 0 is standard for = 1, 2. For = 1, 2, let ∈   be such that ≤ − .Then = 1 1 + 2 2 ∈   , and For Assumption 4 on  ′ , the first bullet is immediate from the definition of ⊲ ′ .For the last two bullets, ≪ ′ , and there are several things that could lead to the left-hand clauses: Similarly, there are two ways to achieve the right-hand clause(s): (vii) there is 7 ∈   such that 7 ≺ Dom − , (viii) there is 8 ∈  such that 0 ≪ 8 ≤ − .

C.2. Lemma 15 and Its Proof
The proofs of Theorems 1 and 2 are transfinite inductions.Lemma 15 is a template for the successor ordinal steps in the transfinite inductions.Lemma 3 (whose proof is in Appendix Appendix C.3) is the remainder of the transfinite induction, including the limit ordinal steps.
The proof of Lemma 15 uses an argument that resembles the proof of de Finetti's fundamental theorem of prevision.The main step is constructing bounds for the possible values of the agreeing function (prevision in de Finetti's case, in Theorem 1) at a new object given previously chosen values of the agreeing function.In de Finetti's theorem, one uses existing previsions of random variables for which either ≤ or ≤ .In Theorem 1, we replace prevision by an agreeing function , and we replace ≤ by a combination of ≾ , ≪ , and/or ≺ Dom .Additional steps are needed to deal with strict preferences of a non-Archimedean nature and with sets of nonstandards that don't have suprema and/or infima.
Lemma 15.Assume the following structure: •  and  are linear spaces of functions from Ω to  with  a proper subset of .
•  = (, ≾  , ≺  ) is a total trading system that is represented by the standard-linear function ∶  → *IR, where *IR is a nonstandard model of the reals.
•   = (, ≾  , ≪  ) is the extension 2 of  obtained from Lemma 2. Let ∈ .Let  be the standard-linear span of  ⋃ { }.Then can be extended to a standard-linear function ′ ∶  → *IR ′ , where *IR ′ contains *IR and such that ′ represents a total trading system  ′ = (, ≾ ′ , ≺ ′ ) that is an extension 2 of  .Also, if   is coherent, then  ′ can be chosen to be coherent.
Proof.If coherence is an issue, note that dominance has the same type (uniform, strict, or weak) in both  and , so we will use the same notation ≺ Dom to mean that dominates regardless of whether , are both in , both in  or one in each.We have It is straightforward to show that the representation of elements of  ⧵  in ( 15) is unique.Define ′ ( ) = ( ) for ∈ .
Start with the case in which there is ∈  such that ∼  .In this case, set ′ ( ) = ( ), *IR ′ = *IR, and for all other elements of .Set ≾ ′ to be the total preorder on  that ′ represents.The only thing that remains to show, in this case, is that For the remainder of the proof, assume that for all ∈ , ¬( ∼ ).We start by choosing a value for ′ ( ).After that, we make ′ standard-linear by defining it through (16).Then, we show that setting ≪ ′ to ≺ ′ satisfies Assumption 4. Finally, we prove that the trading system  ′ that ′ represents (recall Lemma 1) is coherent if   is coherent.Since ′ extends ,  ′ extends  .
When we attempt to choose a value for ′ ( ), we need to attend to instances of ≪  , if any.
The definition of ≪  and the fact that represents  guarantee that, for = 1, 2, < for all ∈  and ∈  .Also,  2 ⊆  1 and  2 ⊆  1 .(If ≪  is empty, then  2 =  2 = ∅.) There are several cases (and subcases) to handle: (a) Both  1 and  1 are nonempty, and (a)(i) there is ∈ *IR such that ≤ ≤ for all ∈  1 and ∈  1 , and at least one such satisfies ∉  2 ⋃  2 , or (a)(ii) there is no as described in case (a)(i).
(b)  1 is empty,  1 is nonempty, and (b)(i) there is ∈ *IR such that ≤ for all ∈  1 , and at least one such satisfies ∉  2 , or (b)(ii) there is no as described in case (b)(i).
(c)  1 is empty,  1 is nonempty, and (c)(i) there is ∈ *IR such that ≤ for all ∈  1 , and at least one such satisfies ∉  2 , or (c)(ii) there is no as described in case (c)(i).
(d) Both  1 and  1 are empty.
In case (c)(ii), apply claim 1 of Lemma 12 to find an extension *IR ′ of *IR and ∈ *IR ′ such that > for all ∈  1 , and set ′ ( ) = .
In case (d), we have two choices.One choice is to let ′ ( ) ∈ *IR and set *IR ′ = *IR.The other choice is to apply either claim of Lemma 12 to find an extension *IR ′ of *IR and let ∈ *IR ′ .By construction, we have that ′ extends from  to .
It follows that ≾ ′ extends 2 ≾  from  to .If ≪  is empty, the proof is over.

C.3. Proof of Lemma 3
The proof proceeds by transfinite induction on  ⧵ .Let Λ be an ordinal, and let { } 0< <Λ be a well-ordering of the elements of  ⧵ .Let  0 = ,  0 =   , *IR 0 = *IR, and 0 = .Then the following induction hypothesis holds for = 0: Induction hypothesis: Let < Λ be an ordinal.There is a total trading system  = ( , ≾ ) such that • ≾ is a total preorder and is an extension 2 of ≾  to  , •  is represented by a standard-linear function ∶  → *IR , where *IR is a nonstandard model of the reals that contains *IR for each < , and Next, we deal with an arbitrary successor ordinal .Assume that the induction hypothesis holds for = − 1.We must prove that the induction hypothesis holds for = .Apply Lemma 15 with  =  −1 , = , = −1 , ≾  =≾ −1 , and *IR = *IR −1 .Then  is the  in Lemma 15.Let and *IR be respectively the ′ and *IR ′ that result from Lemma 15.Then, the induction hypothesis holds for = .
Finally, we prove that the induction hypothesis holds for each limit ordinal .We start by creating objects to play the roles of , , ≾  , and *IR in the statement of Lemma 15.Define Clearly, *IR < is a nonstandard model of the reals that contains *IR for all < .For ∈  < , let < ( ) = ( ), where is the first ordinal such that ∈  .Each such is strictly less than .This makes < ∶  < → *IR < .Define ≾ < on  < by ≾ < if ≾ for being the first ordinal such that both , ∈  .Then < and < represents ≾ < on  < .To see that < is standard-linear, let 1 , 2 ∈  < .Let be the first ordinal for which both 1 and 2 are in  .Then < and < ( ) = ( ) for = 1, 2. Since is standard-linear, so < is standard-linear.

C.4. Proof of Lemma 4
Note that each function ∈  is a special case of a horse-lottery ℎ for which each lottery ℎ( ) puts probability 1 on a single prize (consequence) ( ).In this way, we can think of  as a subset of a set of horse lotteries.Savage (1954) proves that there is a probability on Ω and a utility ∶  → IR such that for all , ∈  , ≾ ′ if and only if [ ( (⋅))] ≤ [ ( (⋅))].Let  be the set of finite mixtures of elements of  , and let  ′ be the set of finite mixtures of elements of .Define ′ on  ′ by ′ ∑ =1 = ∑ =1 ( ).Next, we show that ′ is well defined.Suppose that with all and all ′ strictly positive.A necessary condition for ( 19) is that the set of distinct be the same as the set of distinct ′ .Another necessary condition is that, if = ′ , the sums of the and/or ′ corresponding to repeated values of and/or ′ must be equal.This makes ′ well defined.Hence, ′ [ℎ( )] is well defined for every and where ′ , ( , ) = * ( , ) ( ) for =∈ and ∈  .Finally, assume that dominance in  means weak dominance.If | ≺ Dom | in  , then | ≤ | , and there is ∈ such that | ( ) < | ( ) so it follows that ≺ Dom and ≺ in  .Hence | ≺ | , and Assumption 5 holds in  .

Lemma 13 .
Let *IR be the nonstandard model in Example 11.No externally countable sequence of strictly positive elements of *IR converges to 0.
Then  is naturally embedded in * .Let ( ) = define a function in   .The equivalence class [ ]  is greater than every element of  by construction and [− ]  is smaller than every element of  .For claim 2, if 1 has a least upper bound , then is also a greatest lower bound for 2 .In this case = + [1∕ ]  satisfies claim 2 ∈ 1 , and = − [1∕ ]  satisfies claim 2 if ∈ 2 .