2Preferences

In a complete financial market model, the price of a contingent claim is determined by arbitrage arguments, without involving the preferences of economic agents. In an incomplete model, such claims may carry an intrinsic risk which cannot be hedged away. In order to determine desirable strategies in view of such risks, the preferences of an investor should be made explicit, and this is often done in terms of an expected utility criterion.

The paradigm of expected utility is the theme of this chapter. We begin with a general discussion of preference relations on a set X of alternative choices and their numerical representation by some functional U on X . In the financial context, such choices can usually be described as payoff profiles. These are defined as functions X on an underlying set of scenarios with values in some set of payoffs. Thus we are facing risk or even uncertainty. In the case of risk, a probability measure is given on the set of scenarios. In this case, we can focus on the resulting payoff distributions. We are then dealing with preferences on “lotteries”, i.e., on probability measures on the set of payoffs.

In Sections 2.2 and 2.3 we discuss the conditions – or axioms – under which such a preference relation on lotteries μ can be represented by a functional of the form

where u is a utility function on the set of payoffs. This formulation of preferences on lotteries in terms of expected utility goes back to D. Bernoulli [26]; the axiomatic theory was initiated by J. von Neumann and O. Morgenstern [216]. Section 2.4 characterizes uniform preference relations which are shared by a given class of functions u. This involves the general theory of probability measures on product spaces with given marginals which will be discussed in Section 2.6.

In Section 2.5 we return to the more fundamental level where preferences are defined on payoff profiles, and where we are facing uncertainty in the sense that no probability measure is given a priori. Savage [243] clarified the conditions under which such preferences on a space of functions X admit a representation of the form

where Q is a “subjective” probability measure on the set of scenarios. We are going to concentrate on a robust extension of the Savage representation which was introduced by Gilboa and Schmeidler [144] and later extended by Maccheroni, Marinacci, and Rustichini [205]. Here the utility functional is of the form

It thus involves a whole class Q of probability measures Q, which are taken more or less seriously according to their penalization α(Q). The axiomatic approach to the robust Savage representation is closely related to the construction of coherent and convex risk measures, which will be the topic of Chapter 4.

2.1Preference relations and their numerical representation

Let X be some nonempty set. An element x ∈ X will be interpreted as a possible choice of an economic agent. If presented with two choices x, y ∈ X , the agent might prefer one over the other. This will be formalized as follows.

Definition 2.1. A preference order (or preference relation) on X is a binary relation with the following two properties.

–Asymmetry: If x y, then y ⊁ x.

–Negative transitivity: If x y and z ∈ X , then either x z or z y or both must hold.

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Negative transitivity states that if a clear preference exists between two choices x and y, and if a third choice z is added, then there is still a choice which is either least preferable (y if z y) or most preferable (x if x z).

Definition 2.2. A preference order on X induces a corresponding weak preference order ≻ defined by

and an indifference relation ∼ given by

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Thus, x y means that either x is preferred to y or there is no clear preference between the two.

Remark 2.3. It is easy to check that the asymmetry and the negative transitivity of are equivalent to the following two respective properties of :

(a) Completeness: For all x, y ∈ X , either y x or x y or both are true.

(b) Transitivity: If x y and y z, then also x z.

Conversely, any complete and transitive relation ≻ induces a preference order via the negation of , i.e.,

The indifference relation ∼ is an equivalence relation, i.e., it is reflexive, symmetric and transitive.

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Exercise 2.1.1. Prove the assertions in the preceding remark.

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Definition 2.4. A numerical representation of a preference order is a function U : X → ℝ such that

Clearly, (2.1) is equivalent to

Note that such a numerical representation U is not unique: If f is any strictly increasing function, then (x) := f (U(x)) is again a numerical representation.

Definition 2.5. Let be a preference relation on X . A subset Z of X is called order dense if for any pair x, y ∈ X such that x y there exists some z ∈ Z with x z y.

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The following theorem characterizes those preference relations for which there exists a numerical representation.

Theorem 2.6. For the existence of a numerical representation of a preference relation it is necessary and sufficient that X contains a countable, order dense subset Z . In particular, any preference order admits a numerical representation if X is countable.

Proof. Suppose first that we are given a countable order dense subset Z of X . For x ∈ X , let

The relation x y implies that Z(x) ⊆ Z(y) and Z≺(x) ⊇ Z≺(y). If the strict relation x y holds, then at least one of these inclusions is also strict. To see this, pick z ∈ Z with x z y, so that either x z y or x z y. In the first case, z ∈ Z≺(x)≺(y), while z ∈ Z(y)(x) in the second case.

Next, take any strictly positive probability distribution μ on Z , and let

By the above, U(x) > U(y) if and only if x y so that U is the desired numerical representation.

For the proof of the converse assertion take a numerical representation U and let J denote the countable set

For every interval I ∈ J we can choose some zI ∈ X with U(zI ) ∈ I and thus define the countable set

At first glance it may seem that A is a good candidate for an order dense set. However, it may happen that there are x, y ∈ X such that U(x) < U(y) and for which there is no z ∈ X with U(x) < U(z) < U(y). In this case, an order dense set must contain at least one z with U(z) = U(x) or U(z) = U(y), a condition which cannot be guaranteed by A.

Let us define the set C of all pairs (x, y) which do not admit any z ∈ A with y z x:

Then (x, y) ∈ C implies the apparently stronger fact that we cannot find any z ∈ X such that y z x: Otherwise we could find a, b ∈ ℚ such that

so I := [a, b] would belong to J, and the corresponding zI would be an element of A with y zI x, contradicting the assumption that (x, y) ∈ C.

It follows that all intervals U(x), U(y)) with (x, y) ∈ C are disjoint and nonempty. Hence, there can be only countably many of them. For each such interval J we pick now exactly one pair (xJ , yJ) ∈ C such that U(xJ) and U(yJ) are the endpoints of J, and we denote by B the countable set containing all xJ and all yJ.

Finally, we claim that Z := A ∪ B is an order dense subset of X . Indeed, if x, y ∈ X  with y x, then either there is some z ∈ A such that y z x, or (x, y) ∈ C. In the latter case, there will be some z ∈ B with U(y) = U(z) > U(x) and, consequently, y z x.

The following example shows that even in a seemingly straightforward situation, a given preference order may not admit a numerical representation.

Example 2.7. Let be the usual lexicographical order on X := [0, 1] × [0, 1], i.e., (x1, x2) (y1, y2) if and only if either x1 > y1, or if x1 = y1 and simultaneously x2 > y2. One easily checks that is asymmetric and negative transitive, and hence a preference order. We show now that does not admit a numerical representation. To this end, let Z be any order-dense subset of X . Then, for x ∈ [0, 1] there must be some (z1, z2) ∈ Z such that (x, 1) (z1, z2) (x, 0). It follows that z1 = x and that Z is uncountable. Theorem 2.6 thus implies that there cannot be a numerical representation of the lexicographical order .

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Definition 2.8. Let X be a topological space. A preference relation is called continuous if for all x ∈ X

are open subsets of X.

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Remark 2.9. Every preference order that admits a continuous numerical representation is itself continuous. Under some mild conditions on the underlying space X , the converse statement is also true; see Theorem 2.15 below.

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Example 2.10. The lexicographical order of Example 2.7 is not continuous: If (x1, x2) ∈ [0, 1] × [0, 1] is given, then

which is typically not an open subset of [0, 1] × [0, 1].

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Recall that a topological space X is called a topological Hausdorff space if any two distinct points in X have disjoint open neighborhoods. In this case, all singletons {x} are closed. Clearly, every metric space is a topological Hausdorff space.

Proposition 2.11. Let be a preference order on a topological Hausdorff space X . Then the following properties are equivalent.

(a) is continuous.

(b) The set {(x, y) | y x} is open in X ×X .

(c) The set {(x, y) | y x} is closed in X ×X .

Proof. (a)⇒(b): We have to show that for any pair

there exist open sets U, V ⊂ X such that x0 ∈ U, y0 ∈ V, and U × V ⊂ M. Consider first the case in which there exists some z ∈ B(x0)∩B≺(y0) for the notation B(x0) and B≺(y0) introduced in (2.2). Then y0 z x0, so that U := B≺(z) and V := B(z) are open neighborhoods of x0 and y0, respectively. Moreover, if x ∈ U and y ∈ V, then y z x, and thus U × V ⊂ M.

If B(x0) ∩ B≺(y0) = ∅, we let U := B≺(y0) and V := B(x0). If (x, y) ∈ U × V, then y0 x and y x0 by definition. We want to show that y x in order to conclude that U × V ⊂ M. To this end, suppose that x y. Then y0 y by negative transitivity, hence y0 y x0. But then y ∈ B(x0) ∩B≺(y0) ≠ ∅, and we have a contradiction.

(b)⇒(c): First note that the mapping ϕ(x, y) := (y, x) is a homeomorphism of X ×X . Then observe that the set {(x, y) | y x} is just the complement of the open set ϕ({(x, y) | y x}).

(c)⇒(a): Since X is a topological Hausdorff space, {x} ×X is closed in X ×X , and so is the set

Hence {y | y x} is closed in X , and its complement {y | x y} is open. The same argument applies to {y | y x}.

Example 2.12. For x0, y0 ∈ ℝ with x0 < y0 consider the set X := (−∞, x0] ∪ [y0,∞) endowed with the usual order > on ℝ. Then, with the notation introduced in (2.2), B≺(y0) = (−∞, x0] andB(x0) = [y0,∞). Hence,

despite y0 x0, a situation we had to consider in the preceding proof.

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Recall that the topological space X is called connected if X cannot be written as the union of two disjoint and nonempty open sets. Assuming that X is connected will rule out the situation occurring in Example 2.12.

Proposition 2.13. Let X be a connected topological space with a continuous preference order . Then every dense subset Z of X is also order dense in X . In particular, there exists a numerical representation of if X is separable.

Proof. Take x, y ∈ X with y x, and consider B(x) and B≺(y) as defined in (2.2). Since y ∈ B(x) and x ∈ B≺(y), neither B(x) nor B≺(y) are empty sets. Moreover, negative transitivity implies that X = B(x) ∪ B≺(y). Hence, the open sets B(x) and B≺(y) cannot be disjoint, as X is connected. Thus, the open set B(x) ∩B≺(y) must contain some element z of the dense subset Z , which then satisfies y z x. Therefore Z is an order dense subset of X .

Separability of X means that there exists a countable dense subset Z of X , which then is order dense. Hence, the existence of a numerical representation follows from Theorem 2.6.

Remark 2.14. Consider the situation of Example 2.12, where X := (−∞, x0]∪[y0, ∞), and suppose that x0 and y0 are both irrational. Then Z := ℚ ∩ X is dense in X , but there exists no z ∈ Z such that y0 z x0. Hence, Z is not order dense in X . This example shows that the assumption of topological connectedness is essential for Proposition 2.13.

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Theorem 2.15. Let X be a topological space which satisfies at least one of the following two properties:

–X has a countable base of open sets.

–X is separable and connected.

Then every continuous preference order on X admits a continuous numerical representation.

For a proof we refer to [80, Propositions 3 and 4]. For our purposes, namely for the proof of the von Neumann–Morgenstern representation in the next section and for the proof of the robust Savage representation in Section 2.5, the following lemma will be sufficient.

Lemma 2.16. Let X be a connected metric space with a continuous preference order . If U : X → ℝ is a continuous function, and if its restriction to some dense subset Z is a numerical representation for the restriction of to Z , then U is also a numerical representation for on X .

Proof. We have to show that y x if and only if U(y) > U(x). In order to verify the “only if” part, take x, y ∈ X with y x. As in the proof of Proposition 2.13, we obtain the existence of some z0 ∈ Z with y z0 x. Repeating this argument yields such that Now we take two sequences (zn) and in Z with zn → y and By continuity of , eventually

and thus

The continuity of U implies that U(zn) → U(y) and U(zʹn) → U(x), whence

For the proof of the converse implication, suppose that x, y ∈ X are such that U(y) > U(x). Since U is continuous,

are both nonempty open subsets of X. Moreover,U≺(y)∪U(x) = X . Connectedness of X implies that U≺(y) ∩ U(x) ≠ ∅. As above, a repeated application of the preceding argument yields z0, such that

Since Z is a dense subset of X ,we can find sequences (zn) and in Z with zn → y and as well as with U(zn) > U(z0) and Since U is a numerical representation of on Z , we have

Hence, by the continuity of , neither z0 y nor can be true, and negative transitivity yields y x.