Lotteries

We are all familiar with how a lottery works. Mathematically, it can be described as:

Let x be an "outcome" and let X be a set of outcomes. Let p be a simple probability measure on X, thus p = (p(x₁), p(x₂), ..., p(x_n)) where p(x_i) are probabilities of outcome x_i Î X occurring, i.e. p(x_i) ³ 0 for all i = 1, ..., n and å _i=1ⁿp(x_i) = 1. Note that for simple probability measures, there are finite elements x Î X for which p(x) > 0.  Define D (X) as the set of simple probability measures on X. A particular lottery p is a point in D (x).

Of course we can build a compound lottery, whereby a win in the first lottery results in tickets for another lottery. We can reduce compound lotteries into simple lotteries by combining the probabilities of the lotteries so that all we obtain is a single distribution over outcomes.

In the von Neumann-Morgenstern hypothesis, probabilities are assumed to be "objective" or exogenous and thus cannot be influenced by the agent. The problem of an agent under uncertainty is to choose among lotteries, and thus find the "best" lottery in D (X). One of von Neumann and Morgenstern’s major contributions to economics was to show that if an agent has preferences defined over lotteries, then there is a utility function U: D (X) ® R that assigns a utility to every lottery p Î D (X) that represents these preferences.

If lotteries are merely distributions, it might not seem to make sense that a person would "prefer" a particular distribution to another on its own. If we follow Bernoulli’s construction, we get a sense that what people really get utility from is the outcome or consequence, x Î X. We do not eat "probabilities", after all, we eat apples! Yet what von Neumann and Morgenstern suggest is that people's preferences are formed over lotteries and from these preferences over lotteries, combined with objective probabilities, we can deduce what the underlying preferences on outcomes might be. Thus, in von Neumann-Morgenstern's theory, unlike Bernoulli's, preferences over lotteries logically precede preferences over outcomes.

In other words the ony reason we prefer a lottery over another is due to the implied underlying outcomes, but the preferences are not defined over these outcomes but rather defined over lotteries. In other words, von Neumann and Morgenstern's great insight was to avoid defining preferences over outcomes and capturing everything in terms of preferences over lotteries. The essence of von Neumann and Morgenstern's expected utility hypothesis, then, was to confine themselves to preferences over distributions and then from that, deduce the implied preferences over the underlying outcomes.

With Von Neumann being a mathematician, he naturally structured his formulation by using an axiomatic approach. The four axioms used are comparability, transitivity, independence and certainty equivalence.

Axioms

 1. Comparability

An investor can state a preference among all alternative certain outcomes. Thus, if the investor has a choice of outcome A or B, a preference for A to B or B to A can be stated or indifference between them can be expressed. This is also referred to as completeness.

2. Transitivity

If an investor prefers A to B and B to C, then A is preferred to C.

3. Independence

If an investor is indifferent between two prospects X and Y, then they will be indifferent between the following two gambles (where Z is a third prospect):

X with probability P and Z with probability 1-P, and

Y with probability P and Z with probability 1-P.

4. Certainty Equivalent

• define and state axioms for preference relation ³ _h over simple lotteries, D (X);
• use this preference relation to construct a utility function on simple lotteries, U: D (X) ® R;
• prove that this utility function U has an "expected utility" structure, i.e. there is an underlying utility on outcomes u: X ® R that yields U(p) = å p(x)u(x).

Axioms of Preference

Let ³ _h be a binary relation over D (X), i.e. ³ _h Ì D (X) ´ D (X). Hence, we can write (p, q) Î ³ _h, or p ³ _h q to indicate that lottery p is "preferred to or equivalent to" lottery q. Naturally, Ø (p ³ _hq) = p <_h q, i.e. if p is not preferred to or equivalent to q, then we say q is strictly preferred to p. Of course, p ³ _h q and q ³ _h p implies p ~_h q, i.e. p is equivalent to q. We now state the four axioms for these preferences:

(A.1) ³ _h is complete, i.e. either p ³ _h q or q ³ _h q for all p, q Î D (X).

(A.2) ³ _h is transitive, i.e. if p ³ _h q and q ³ _h r then p ³ _h r for all p, q, r Î D (X)

(A.3) Archimedean Axiom: if p, q, r Î D (X) such that p >_h q >_h r, then there is an a , b Î (0, 1) such that a p + (1-a )r >_h q and q >_h b p + (1-b )r.

(A.4) Independence Axiom: for all p, q, r Î D (X) and any a Î [0, 1], then p ³ _h q if and only if a p + (1-a )r ³ _h a q + (1-a )r.

The first two axioms (A.1) and (A.2) should be familiar from conventional theory. Together, (A.1) and (A.2) are sometimes referred to as the "weak order" axioms. The Archimedean Axiom (A.3) works like a continuity axiom on preferences. It effectively states that given any three lotteries strictly preferred to each other, p >_h q >_h r, we can combine the most and least preferred lottery (p and r) via an a Î (0, 1) such that the compound of p and r is strictly preferred to the middling lottery q and we can combine p and r via a b Î (0, 1) so that the middling lottery q is strictly preferred to the compound of p and r. Notice that one really needs D (X) to be a linear, convex structure to have (A.3).

The Independence Axiom (A.4) effectively claims that the preference between p and q is unaffected if they are both combined in the same way with a third lottery r. One can envisage this as a choice between a pair of two-stage lotteries. In this case, a p + (1-a )r is a two stage lottery which yields either lottery p with probability a and lottery r with probability (1-a ) in the first stage. Using the same interpretation for a q + (1-a )r, then since both mixtures lead to r with the same probability (1-a ) in the first stage and since one is equally well-off if this case occurs, then preferences between the two-stage lotteries ought to depend entirely on one's preferences between the alternative lotteries in the second-stage, p and q.

Note that these axioms, as stated, are derived from N.E. Jensen (1967) and are not exactly the original V-M axioms (in particular, they did not have an explicit independence axiom).

 Utility Function

We now want to proceed to the next step and derive the V-M utility function, U: D (X) ® R to represent preferences over lotteries, where by representation we mean that for any p, q Î D (X), p ³ _h q if and only if U(p) ³ U(q). Thus if lottery p is preferred or equivalent to q, then the utility from lottery p is greater than utility from lottery q and vice-versa. Let us then turn to the main existence theorem:

Theorem: Let D (X) be a convex subset of a linear space. Let ³ _h be a binary relation on D (X). Then ³ _h satisfies (A.1), (A.2), (A.3) and (A.4) if and only if there is a real-valued function U:D (X) ® R such that:

(a) U represents ³ _h (i.e. " p, q Î D (X), p ³ _h q Û U(p) ³ U(q))

(b) U is affine (i.e. " p, q Î D (X), U(a p + (1-a )q) = a U(p) + (1-a )U(q) for any a Î (0, 1))

Moreover, if V:D (X) ® R also represents preferences, then there is an b, c Î R (where b > 0) such that V = bU + c, i.e. U is unique up to a positive linear transformation.

Proof: 

We divide the proof into the following parts:

• axioms Þ utility representation and affinity;
• utility representation and affinity Þ axioms; and
• uniqueness of the utility function up to a positive linear transformation.

Part I: (Axioms Þ Representation and Affinity)

We proceed in three steps: firstly, we prove two lemmas on preferences; secondly, we prove that the theorem holds on a closed preference interval; finally, we extend this result to the entire D (X). So let us begin with the two lemmas:

Lemma: (L.1 - Mixture Monotonicity): For any p, q Î D (X), and a , b Î (0, 1) where p >_h q and a £ b , then b p + (1-b )q >_h a p + (1-a )q.

Proof: (i) Suppose a = 0. Note that p = b p + (1-b )p and q = b q + (1-b )q obviously. Now, by (A.4), p >_h q Þ b p + (1-b )p >_h b p + (1-b )q as we have b p on both sides. But, by (A.4) again, b p + (1-b )q >_h b q + (1-b )q as we now have (1-b )q in common on both sides. But note that this implies b p + (1-b )q >_h q = a p + (1-a )q when a = 0, and we are done. (ii) Suppose a > 0. Now, recall from (i) that b p + (1-b )q >_h q. Thus, defining r = b p + (1-b )q, then r >_h q. Now, define g = a /b . Then g r + (1-g )r >_h q. But, as r >_h q, then by (A.4), g r + (1-g )r >_h g r + (1-g )q where g r is in common on both sides. Or, by definition of r, g r + (1-g )r >_h g (b p + (1-b )q)) + (1-g )q. Then, rearranging, g r + (1-g )r >_h g b p + (1- b g )q. But, by definition of g , g b = a , thus g r + (1-g )r >_h a p + (1- a )q. But as r = g r + (1-g )r = b p + (1-b )q by definition, then b p + (1-b )q >_h a p + (1- a )q.

Intuitively, if lottery p is preferred to lottery q, then if we construct two compound lotteries with different weights, then we prefer the compound lottery in which lottery p is given the relatively greater weight.

Lemma: (L.2 - Unique Solvability): If p, q, r Î D (X) and p ³ _h q ³ _h r and p >_h r, then there is a unique a * Î [0, 1] such that q ~_h a *p + (1-a *)r.

Proof: (i) If p ~_h q, then a * = 1 and we are done. (ii) if r ~_h q, then a * = 0, and we are done. (iii) if p >_h q >_h r, then define the set Q³ = {a Î (0, 1) ½ q ³ _h a p + (1-a )r}. This set is non-empty because a = 0 is an element of it and it is bounded above by a £ 1. Thus, there is a supremum (least upper bound) of Q³ . Let a * = sup Q³ . Then we can consider two violating cases. Case 1: p >_h q >_{h a} *p + (1-a *)r. Then, by (A.3), there is a b Î [0, 1] such that q >_h b (a *p + (1-a *)r) + (1-b )p. Or, rearranging, q >_h [1 - b (1-a *)]p + b (1-a *)r. But, as b (1-a *) < (1-a *), then (1-b (1-a *)) > a *. But then a * is not a supremum of Q³ . A contradiction. Case 2: a *p + (1-a *)r >_h q. We can proceed the same way, i.e. by (A.3) we can find some g Î [0, 1] such that [1 - g (1-a *)]p + g (1-a *)r >_h q, which implies that a * is not a supremum - thus a contradiction. Consequently, it must be that neither Case 1 or Case 2 can apply, thus a *p + (1-a *)r ~_h q. Finally, by mixture monotonicity (L.1), a * is unique.

Given a lottery q, we can construct a compound lottery which yields the same utility as q by appropriately combining any lottery p which is preferred to q with any lottery r to which q is preferred.

Now, let us return to the main proof. Consider first the following case: suppose that, for any p, q Î D (X), we have p ~_h q (all lotteries are equivalent) In this case, U is constant, i.e. U(p) = c for all p Î D (X), which is of course real-valued and affine. Thus, this trivial case is easily disposed with. But consider now the following. Suppose s, r Î D (X) where s >_h r. Define RS = {p Î D (X) ½ s ³ _h p ³ _h r}, which is a closed and convex subset of D (X) (by (A.4)). For each p Î RS, define ¦ (p) as a number such that p ~_h ¦ (p)s + (1-¦ (p))r. By unique solvability (L.2), such a ¦ (p) exists and is unique. We now make two claims:

Proposition (Representation): ¦ (.) represents preferences on RS, i.e. for all p, q Î RS, ¦ (p) ³ ¦ (q) if and only if ¦ (p)s + (1-¦ (p))r ³ _h ¦ (q)s + (1-¦ (q))r.

To prove this, consider that by mixture monotonicity (L.1), s >_h r and ¦ (p) ³ ¦ (q) implies that ¦ (p)s + (1-¦ (p))r >_h ¦ (q)s + (1-¦ (q))r. But, by the definition of ¦ (p) and ¦ (q), (i.e. p ~_h ¦ (p)s + (1-¦ (p))r and q ~_h ¦ (q)s + (1-¦ (q))r), we can note immediately by transitivity (A.2) that this implies that p >_h q. The same argument works in reverse. Thus, ¦ (p) ³ ¦ (q) Û p >_h q, i.e. ¦ (.) represents preferences ³ _h on RS, and we are done. Q.E.D.

Proposition (Affinity): ¦ (.) is affine for all p, q Î RS, i.e. ¦ (a p + (1-a )q) = a ¦ (p) + (1-a )¦ (q).

To prove this, consider any p, q Î RS and define p¢ = a p + (1-a )q. As RS is convex, then p¢ Î RS for any a Î (0, 1). Thus, by unique solvability (L.2) there is a real number ¦ (p¢ ) such that p¢ ~_h ¦ (p¢ )s + (1-¦ (p¢ ))r. But as p¢ = a p + (1-a )q and p ~_h ¦ (p)s + (1-¦ (p))r by (L.2), then p¢ ~_h a [¦ (p)s + (1-¦ (p))r] + (1-a )q by the independence axiom (A.4). Doing the same for q ~_h ¦ (q)s + ((1-¦ (q))r, then we obtain p¢ ~_h a [¦ (p)s + (1-¦ (p))r] + (1-a )[¦ (q)s + ((1-¦ (q))r]. Rearranging a bit, we obtain that p¢ ~_h [a ¦ (p) + (1-a )¦ (q)]s + [a (1-¦ (p)) + (1-a )(1-¦ (q))]r, thus p¢ is equivalent to another convex combination of s and r. But, by unique solvability (L.2), there is only one a * such that p¢ ~_h a *s + (1-a *)r. Thus, it must be that a * = ¦ (p¢ ) = [a ¦ (p) + (1-a )¦ (q)], or, by the definition of p¢ , ¦ (a p + (1-a )q) = a ¦ (p) + (1-a )¦ (q). This is the definition of affinity.

Let us now enter on our third stage and extend the representation and affinity results from RS to the entire set. To do so, we first need to prove the following claim:

Proposition: (Order-Preservation): If ¦ represents ³ _h and is affine, then g = a + b¦ where b > 0 also (i) represents ³ _h and (ii) is affine.

The proof is simple. (i) For any p, q Î D (X), then p ³ _h q Þ ¦ (p) ³ ¦ (q) by representation of ¦ . Thus, if b > 0, then this implies a + b¦ (p) ³ a + b¦ (q), thus g(p) ³ g(q) by definition. (ii) As ¦ is affine, then ¦ (a p + (1-a )q) = a ¦ (p) + (1-a )¦ (q). Now by definition, g(a p + (1-a )q) = a + b¦ (a p + (1-a )q)) = a + b[a ¦ (p) + (1-a )¦ (q)] = a a + (1-a )a + ba ¦ (p) + b(1-a )¦ (q) = a [a + b¦ (p)] + (1-a )[a + b¦ (q)] = a g(p) + (1-a )g(q). Q.E.D..

Let us return to the extension of RS. By the definition of ¦ , s ~_h ¦ (s)s + (1-¦ (s))r and r ~_h ¦ (r)s + (1-¦ (r))r, thus ¦ (s) = 1 and ¦ (r) = 0. Now, Define RS₁ = {p Î D (X) ½ s₁ ³ _h p ³ _h r₁} where s₁ >_h s and r >_h r₁, so obviously RS Ì RS₁. Now, let us define ¦ ₁ over RS₁ a manner analogous to before, so that for any p Î RS₁, then p ~_h ¦ ₁(p)s₁ + (1-¦ ₁(p))r₁ and ¦ ₁ is affine. Let us now find a₁ and a b₁ > 0 and thus a function g₁ = a₁ + b_{1¦ 1} such that g₁(s) = a₁ + b_{1¦ 1} (s) = 1 and g₁(r) = a₁ + b_{1¦ 1}(r) = 0. If we think of D (X) as the real line and preferences increasing along it, then ¦ ₁ and the adjustment to g₁ can be represented as in Figure 2.

Now, define RS₂ = {p Î D (X) | s₂ ³ _h p ³ _h r₂} where s₂ >_h s and r >_h r₂, so we again obtain RS Ì RS₂. Defining ¦ ₂ the same way as before on RS₂, we can thus find now find a₂ and b₂ > 0 such that g₂(s) = a₂ + b_{2¦ 2}(s) = 1 and g₂(r) = a₂ + b_{2¦ 2}(r) = 0. Thus, g₁(r) = g₂(r) = 0 and g₁(s) = g₂(s) = 1

We now show that for any p Î RS₁ Ç RS₂ Þ g₁(p) = g₂(p). As p is in the intersection, then either p is inside, above or below RS. In other words, one of the following three cases will be true:

(i) s ³ _h p ³ _h r: Þ by unique solvability (L.2), $ a such that p ~_h a s + (1-a )r

(ii) p >_h s >_h r: Þ by unique solvability (L.2), $ a such that s ~_h a p + (1-a )r

(iii) s >_h r >_h p: Þ by unique solvability (L.2), $ a such that r ~_h a s + (1-a )p

Consider now the consequences of the different cases: Case (i) implies that g₁(p) = a g₁(s) + (1-a )g₁(r) = a by construction of g₁. But it is also true that g₂(p) = a g₂(s) + (1-a )g₂(r) = a again by construction, thus g₁(p) = g₂(p) = a . Case (ii) implies that 1 = g₁(s) = a g₁(p) + (1-a )g₁(r) = a g₁(p), so g₁(p) = 1/a . But similarly, 1 = g₂(s) = a g₂(p) + (1-a )g₂(r) = a g₂(p), so g₂(p) = 1/a . Thus, once again g₁(p) = g₂(p) = 1/a . Finally, Case (iii) implies that 0 = g₁(r) = a g₁(s) + (1-a )g₁(p) = a + (1-a )g₁(p), so g₁(p) = a /(a -1). Similarly, 0 = g₂(r) = a g₂(s) + (1-a )g₂(p) = a + (1-a )g₂(p), so g₂(p) = a /(a -1). Thus, again g₁(p) = g₂(p) = a /(a -1).

Thus, for every p Î RS₁ Ç RS₂, g₁(p) = g₂(p). Consider now an increasing sequence RS Ì RS₁ Ì RS₂ Ì RS₃ Ì ...Ì D (X). At each step, we can define g_i that represents preferences over RS_i, but g_i(p) = g_i-1(p) = g_i-2(p) = ... for all p Î RS_i-1. Thus, let us define this common value g_i(p) = g_i-1(p) = U(p). We can thereby construct a U that represents preferences over the entire set D (X). Thus the first important part of the proof, the derivation of a utility function U:D (X) ® R from axioms (A.1)-(A.4) is finished.

Part II: (Representation and Affinity Þ Axioms).

If U: D (X) ® R is affine and represents preferences, then we want so show that the axioms (A.1)-(A.4) hold. Completeness is clear enough: as U is defined over D (X), then for any pair p, q Î D (X) then either U(p) ³ U(q) or U(p) £ U(q) or both. By representation, this implies (A.1). Similarly, for any triple, p, q, r Î D (X), by representation, U(p) ³ U(q) and U(q) ³ U(r) implies p ³ _h q and q ³ _h r. It then follows from the properties of the real number line that U(p) ³ U(r), thus p ³ _h r, so transitivity (A.2) is done. The Archimedean axiom (A.3) is just as simple. By representation, U(p) ³ U(q) ³ U(r) implies p ³ _h q ³ _h r. We know by the properties of the real line (called the Archimedian axiom in fact), there is an a Î (0, 1) such that a U(p) + (1-a )U(r) ³ U(q). As, by affinity, a U(p) + (1-a )U(q) = U(a p + (1-a )q), then U(a p + (1-a )r) ³ U(q) so, by representation, a p + (1-a )r ³ _h q. The same reasoning applies when choosing a b so b U(p) + (1-b )U(r) £ U(q), etc., thus we are done. Finally, for the independence axiom (A.4), note that if U(p) ³ U(q), then p ³ _h q. Notice also that this implies that for a Î (0, 1), that a U(p) ³ a U(q). Thus, adding (1-a )U(r) from both sides a U(p) + (1-a )U(r) ³ a U(q) + (1-a )U(r). By affinity, as U(a p + (1-a )r) = a U(p) + (1-a )U(r) and U(a q + (1-a )r) = a U(q) + (1-a )U(r), thus U(a p + (1-a )r) ³ U(a q + (1-a )r) so, by representation a p + (1-a )r ³ _h a q + (1-a )r. The reverse also applies by the same reasoning. Thus, the Archimedean axiom is finished.

Part III: (Uniqueness)

We now wish to turn to the "moreover" remark and prove that if both U: D (X) ® R and V: D (X) ® R represent preferences, then there is a c and b > 0 such that V = bU + c. Let us get rid of the trivial case first: if for all p, q Î D (X), p ~_h q, then U(p) = k and V(p) = k¢ , thus V(p) = U(p) - (k-k¢ ), so c = (k - k¢ ) and b = 1. Now, suppose there is s, p, r Î D (X) such that s >_h p >_h r. Then define the following: H^U(p) = [U(p) - U(r)]/[U(s) - U(r)] and H^V(p) = [V(p) - V(r)]/[V(s) - V(r)]. By unique solvability (L.2), there is an a Î (0, 1) such that p ~_h a s + (1-a )r. Thus H^U(p) = [U(a s + (1-a )r) - U(r)]/[U(s) - U(r)] or, by affinity:

H^U(p) = [a U(s) + (1-a )U(r) - U(r)]/[U(s) - U(r)] = a

Similarly, as H^V(p) = [V(a s + (1-a )r) - V(r)]/[V(s) - V(r)], then H^V(p) = a . This implies, then, that H^U(p) = H^V(p). Thus,

[U(p) - U(r)]/[U(s) - U(r)] = [V(p) - V(r)]/[V(s) - V(r)]

cross-multiplying:

[U(p) - U(r)][V(s) - V(r)] = [U(s) - U(r)][V(p) - V(r)]

or:

U(p)[V(s) - V(r)] - U(r)[V(s) - V(r)] = [U(s) - U(r)]V(p) - [U(s) - U(r)]V(r)

or simply:

V(p) = U(p)[V(s) - V(r)]/[U(s) - U(r)] - U(r)[V(s) - V(r)]/ [U(s) - U(r)] + V(r)

thus letting b = [V(s) - V(r)]/[U(s) - U(r)] and c = - U(r)[V(s) - V(r)]/ [U(s) - U(r)] + V(r), then:

V(p) = bU(p) + c

which is the form we wanted.

The Expected Utility Representation

We have now obtained the utility function U:D (X) ® R on the basis of the four axioms set forth earlier. However, we have not finished in proving the expected utility hypothesis, namely, that a utility function U:D (X) ® R has a representation:

U(p) = å _{xÎ Supp(p)} p(x)u(x)

where u: X ® R is a elementary utility function on the underlying outcomes X. Note that as D (X) is the set of simple probability distributions on X, then if p Î D (X), then p has a finite support denoted Supp(p) Ì X. D (X), of course, is a convex set. Finally, we should note that by convexity, for any p, q Î D (X), a p + (1-a )q Î D (X) for any a Î (0, 1) and that, if p and q are simple probability distributions, then (a p + (1-a )q)(x) = a p(x) + (1-a )q(x) for any x Î X.

We now state the expected utility representation as a corollary to the earlier V-M theorem:

Corollary: (Expected Utility Representation) Let D (X) be the set of all simple probability distributions on X. Let ³ _h be a binary relation on D (X). Then ³ _h satisfies (A.1)-(A.4) if and only if there is a function u: X ® R such that for every p, q Î D (X):

p ³ _h q if and only if å _{xÎ Supp(p)} p(x)u(x) ³ å _{xÎ Supp(q)} q(x)u(x).

Moreover, v: X ® R represents ³ _h in the above sense if and only if there exist c and b > 0 such that v = bu + c.

Proof: From the V-M theorem, there is a U: D (X) ® R which represents preferences ³ _h on D (X) and is affine. Now, define the function d _x: X ® {0, 1} as d _x(y) = 1 if y = x and d _x(y) = 0 otherwise. This implies that for every x Î X, d _x is a degenerate distribution, thus d _x Î D (X). Let U(d _x) = u(x). Now consider a distribution p = [p(x), p(y)]. Obviously, we can write this out as a convex combination of degenerate distributions d _x and d _y, i.e. p = p(x)d _x + p(y)d _y. Thus, U(p) = U(p(x)d _x + p(y)d _y) = p(x)U(d _x) + p(y)U(d _y) = p(x)u(x) + p(y)u(y) by affinity and our definition of u(x) and u(y). Thus, more generally, any distribution p with finite support can be written out as a convex combination of degenerate distributions, p = å _{xÎ Supp(p)} p(x)d _x, and thus we obtain U(p) = U(å _{xÎ Supp(p)}p(x)d (x)) = å _{xÎ Supp(p)} p(x)u(x) which is the expected utility representation of U(p). Thus as p ³ _h q iff U(p) ³ U(q) by the von Neumann-Morgenstern theorem, then equivalently, p ³ _h q iff å _{xÎ Supp(p)} p(x)u(x) ³ å _{xÎ Supp(q)} q(x)u(x).

(i) Cardinality

Conventional, non-stochastic utility functions u: X ® R are generally assumed to be ordinal, i.e. they are order-preserving indexes of preferences. By this we mean that the numerical magnitudes we give to u are irrelevant, as long as they preserve preference orderings. However, when facing the V-M expected utility decomposition U(p) = å _x p(x)u(x), it is common to fall into the misleading conclusion that utility is cardinal, i.e. that utility here is a measure of preferences.

Of course, the elementary utility function u: X ® R within U(p) is cardinal. Why this is so is clear enough: if expected utility is obtained by adding up probabilities multiplied by elementary utilities, then the precise measure of the elementary utilities matters very much indeed. More explicitly, if u represents preferences over outcomes, then it is unique up to any linear transformation, i.e. if v represents preferences over outcomes, then there is a b> 0 and a such that v = bu + c. This can be regarded as the "definition" of cardinality.

What is important to realize is that even though the elementary utility function u is a cardinal utility measure on outcomes, the utility function over lotteries U is not a cardinal utility function. This is because the elementary utilities on outcomes are not primitives, rather the preferences on lotteries are the primitives. The utility function over lotteries U thus logically precedes the elementary utility function: the latter is derived from the former.

Suppose we have an agent whose ordinal utility function is known. Indeed, suppose that it’s our river-crossing fugitive. Let’s assign him the following ordinal utility function:

Escape 4

Death by shooting 3

Death by rockfall 2

Death by snakebite 1

Now, we know that his preference for escape over any form of death is likely to be stronger than his preference for, say, shooting over snakebite. This should be reflected in his choice behavior in the following way. In a situation such as the river-crossing game, he should be willing to run greater risks to increase the relative probability of escape over shooting than he is to increase the relative probability of shooting over snakebite. This bit of logic is the crucial insight behind the V-M solution to the cardinalization problem.

Begin by asking our agent to pick, from the available set of outcomes, a best one and a worst one. ‘Best’ and ‘worst’ are defined in terms of rational choice: a rational agent always chooses so as to maximize the probability of the best outcome -- call this W -- and to minimize the probability of the worst outcome -- call this L. Now consider prizes intermediate between W and L. We find, for a set of outcomes containing such prizes, a lottery over them such that our agent is indifferent between that lottery and a lottery including only W and L. In our example, this would be a lottery having shooting and rockfall as its possible outcomes. Call this lottery T . We define a utility function q = u(T) such that if q is the expected prize in T , the agent is indifferent between winning T and winning a lottery in which W occurs with probability u(T) and L occurs with probability 1  u( T).

We now construct a compound lottery T* over the outcome set {W, L} such that the agent is indifferent between T and T*. A compound lottery is one in which the prize in the lottery is another lottery. This makes sense because, after all, it is still W and L that are at stake for our agent in both cases; so we can then analyze T* into a simple lottery over W and L. Call this lottery r. It follows from transitivity that T is equivalent to r. The rational agent will now choose the action that maximizes the probability of winning W. The mapping from the set of outcomes to u(r) is a von Neumann-Morgenstern utility function (VNMuf).

What exactly have we done here? We’ve simply given our agent choices over lotteries, instead of over prizes directly, and observed how much extra risk he’s willing to run to increase the chances of winning escape over snakebite relative to getting shot or clobbered with a rock. A VNMuf yields a cardinal, rather than an ordinal, measure of utility. Our choice of endpoint-values, W and L, is arbitrary, as before; but once these are fixed the values of the intermediate points are determined. Therefore, the VNMuf does measure the relative preference intensities of a single agent. However, since our assignment of utility values to W and L is arbitrary, we can’t use VNMufs to compare the cardinal preferences of one agent with those of another. Furthermore, since we are using a risk-metric as our measuring instrument, the construction of the new utility function depends on assuming that our agent’s attitude to risk itself stays constant from one comparison of lotteries to another.

Consequently, the utility function U:D (X) ® R itself is an ordinal utility function since any increasing transformation of U will preserve the ordering on the lotteries. In other words, if U represents preferences on lotteries, then so does V = ¦ (U) where ¦ is an increasing monotonic transformation of U, e.g. if U(p) is a representation of the preference ordering on lotteries D (X), then V(p) = [U(p)]² = [å _xp(x)u(x)]² is also a representation of the preference ordering on lotteries. However, there is one sense in which U still carries an element of cardinality. Namely, given U(p) = å _x p(x)u(x), then v: X ® R will generate an ordinally equivalent V(p) = å _x p(x)v(x) if and only if v = bu + c for b > 0 - and thus V = bU + c. Thus, in V-M theory, we have a "cardinal utility which is ordinal".