CASO BOTÍN

In this discussion, which follows that of J. Ingersoll, Jr. (1987), we do not derive results from the most primative assumptions, nor do we seek complete-

Elements of Utility Theory 99

ness or generality. Rather, we indicate how certain results of utility theory, for example, the idea of maximizing expected utility, may be derived from more primitive assumptions. Under the axiomatic approach, we define goods, pref- erences for different bundles of goods, and state fundamental axioms. As we shall see, in this context, the idea of an expected utility maximizing investor

follows as a logical consequenceof more fundamental axioms. The reader con- tent to accept the idea of an expected utility maximizing investor as apostulate

can skip this section.

We assume that there are goods:

Definition 4.1 (Bundle of goods) Let x denote a bundle, with xi units of

good i,x_∈X, whereX is a closed, convex subset of Rn _{that denotes the set}

of all bundles of goods.

The goods may represent tangible goods, such as cotton or steel. We define two types of preference relationships:

Definition 4.2 (Weakly preferred) The statementxz is read “x is weakly preferred to z.”

Definition 4.3 (Strictly preferred) x y (x is strictly preferred to y) if

xy, but not yx.

We define equivalence as follows:

Definition 4.4 (Equivalence) x _∼ y (x is equivalent to y) if x y, and

yx.

Now we state three natural axioms:

Axiom 1 (Completeness on bundles) For every pair of bundles, x∈X and

y∈X, eitherxy, or yx.

Axiom 2 (Reflexivity on bundles) For every bundle,x_∈X,xx.

Axiom 3 (Transitivity on bundles) For every pair of bundles, x ∈ X and

y_∈X, ifxy, andy z, thenxz.

We now introduce the notion of an ordinal utility function — a function that encodes the preference of the ordering relation:

Definition 4.5 (Ordinal utility function) Υ: X _→ R is an ordinal utility function if and only if, for bundles xand z∈X,

Υ(x)>Υ(z)if and only if xz, and

100 Utility-Based Learning from Data

We note that ordinal utility functions encode only the ordering of preferences, and not the magnitude of the preferences.3

The axioms stated above are not sufficient to insure the existence of an ordinal utility function. In order to insure the existence of an ordinal utility function, it is necessary to introduce an additional axiom. It can be shown that the following axiom, the continuity axiom, guarantees the existence of an ordinal utility function, as well as the continuity of the utility function.

Axiom 4 (Continuity on bundles) For every bundlex_∈X, the two subsets: (i) all bundles that are strictly preferred tox, and

(ii) all bundles that are strictly worse thanx

are open subsets of X.

That is, it can be shown4_that:

Theorem 4.1 (Existence of ordinal utility function) Under Axioms 1-4 above, there exists a continuous ordinal utility functionΥ mappingX to the real line that satisfies

Υ(x)>Υ(z)if and only if xz, and

Υ(x) = Υ(z)if and only if x∼z.

Observe that if Υ(x) is an ordinal utility, then so is θ[Υ(x)], whereθ(_·) is a strictly increasing function. That is, ordinal utility functions are equivalent up to strictly increasing monotone transformations. Also note that we have not yet introduced the notion of risk into our axiomatic discussion of utility theory. We do so now.

Definition 4.6 (Lottery) A lottery is a pair consisting of (i) a collection of bundles (x1_{, . . . , x}m_{), withx}i_{∈X, and}

(ii) the probability measure(π1_{, . . . , π}m_{)on these payoffs.}

We assume that there is a preference ordering on the lotteries, with strict preference and indifference defined as above, that satisfies the following ax- ioms:

Axiom 5 (Completeness on lotteries) For every pair of lotteries (L1, L2),

either L1L2, or L2L1.

Axiom 6 (Reflexivity on lotteries) For every lottery,L,LL.

3_{Below, we describe cardinal utility functions which encode more information about the} magnitude of the preferences than ordinal utility functions.

Elements of Utility Theory 101

Axiom 7 (Transitivity on lotteries) IfL1L2, andL2L3, thenL1L3.

It can be shown that these axioms are sufficient to guarantee that prefer- ences are consistent with an ordinal utility function defined on lotteries. We now state additional axioms from which it follows that a decision maker who subscribes to the full set of axioms will maximize an expected utility function.

Axiom 8 (Independence) LetL1 be the lottery with

(i) the collection of bundles (x1_{, . . . , x}v_{, . . . , x}m_{), and}

(ii) probability measure(π1_{, . . . , π}v_{, . . . , π}m_).

Let z denote a bundle of goods or another lottery. If z is a lottery, let it be the lottery with

(i) the collection of bundles (y1_{, . . . , y}n_{), and}

(ii) probability measure(p1_{, . . . , p}n_).

Let L2 be the lottery with

(i) the collection of bundles (x1_{, . . . , z, . . . , x}m_{), and}

(ii) probability measure(π1_{, . . . , π}v_{, . . . , π}m_).

If xv

∼ z, then L1 ∼ L2, whether z is a bundle or another lottery. If z is

another lottery, then L1∼L2∼L3, where L3 denotes the lottery with

(i) the collection of bundles (x1_{, . . . , x}v−1_{, y}1_{, . . . , y}n_{, x}v−1_{, . . . , x}m_{), and}

(ii) probability measure(π1_{, . . . , π}v−1_{, π}v_p1_{, . . . , π}v_pn_{, π}v−1_{, . . . , π}m_).

Under this axiom, only the preferences on final payoffs and final probabilities matter — not the path taken (through a single lottery or compound lottery). In that sense, preferences are independentof the path taken, hence the name of the axiom.

Axiom 9 (Continuity on lotteries) If x1 x2 x3, then there exists a probabilityπ,0_≤π_≤1, such thatx2

∼L, whereLdenotes the lottery with (i) the collection of bundles (x1_{, x}3_{), and}

(ii) probability measure(π,1₋π). This probability is unique unlessx1

∼x3_.

Axiom 10 (Dominance) Let L1 denote the lottery with

(i) the collection of bundles (x1_{, x}2_{), and}

102 Utility-Based Learning from Data

and letL2 denote the lottery with

(i) the collection of bundles (x1_{, x}2_{), and}

(ii) probability measure(p,1−p). If x1

x2_{, then L}

1L2 if and only ifπ > p.

We now state what is perhaps the main result from utility theory. It can be shown that5

Theorem 4.2 (Expected utility maximization) A decision maker, who sub- scribes to Axioms 1-10, facing two or more lotteries, will choose the lottery with maximum expected cardinal utility,Ψ(x).

The cardinal utility function of this theorem is called a von Neumann- Morgenstern utility function. For a cardinal utility function, the numerical value of the utility has a precise meaning (up to a linear transformation). Thus, unlike an ordinal utility function (which, when composed with any monotone function, produces another ordinal utility function), a cardinal utility function encodes information on preferences beyond rank. Two different cardinal utility functions can be consistent with the same ordinal utility function. Thus, two consumers who make the same choices under certainty may choose different lotteries.

4.2.1

Utility of Wealth

So far, we have expressed outcomes in terms of bundles of goods. In much of the finance literature and in the remainder of this book, utility functions are typically described in terms of wealth. In the case where the bundles consist of a single good — wealth — in varying quantity, the previous discussion leads directly to utility of wealth as a special case. In general, given market prices for the goods, we can define a utility function as a function of wealth as follows:

U(W;p) =max_{Ψ(x)_|pTx=W_}, (4.7) wherepis the price vector, i.e.,piis the price of goodi. Below, whenever we

use a utility function, we mean the utility of wealth, as defined here.