The function-space approach to expected utility

A initial demonstration of why the standard approaches will not work and why new inter-context conditions are necessary is provided by the following discussion. We consider a theorem for finite alternative and state spaces by Adams (1965) and Fishburn (1979). It will show that in our setting, where a complete contingent plan for each p is the objective, the theorem of Fish- burn gives a family of additive representations, one for each p. However, as the condition has nothing to say about how preferences vary with p, it is not sufficient to ensure linearity inp.

Let As denote the set of outcomes in state s of each of actions in A. The

next theorem defines preferences on ˆn s_“1As.

Example 1.4. In example 1.3 recall that

A:“ twalk, bike, busu ” ta, b, cu, and

S:“ train, sun, snow{iceu ” tr, s, tu.

It is not only arrays of outcomes such as par, as, atq (which may be identified

with the alternative “walk” lie in Ar ˆ As ˆAt) that lie in the domain of

preferences; the idealistic but unfeasible array pcr, as, btq is also there. As a

result even in a simple problem such as this the cardinality of the space Arˆ

AsˆAt is 27, whereas AˆS “9.

Definition 1.1 (Fishburn (1979): Equivalence relation on ˆn s“1As).

pα1

, . . . , αJ_qE

Jpβ1. . . βJq if and only if J ą 1, αj, βj P ˆns“1As for j “

1. . . , J, and it is true for eachs thatα1

s, . . . , αJs is a permutation ofβ

1

s, . . . βsJ.

For a fixed, suppressed context we have the following result.

Theorem 1.1 (Scott and Suppes (1958), Fishburn (1979)). The relation ą.

onˆn

s“1As satisfies the following condition: for all J-sequences,tαjuandtβju

in ˆn s“1As, and all J “2,3, . . . (F) if pα1 , . . . , αJ_qE Jpβ1, . . . , βJq and pβj ą.αjq for j “1, . . . , J´1, then pαJ _ą .βJq,

if and only if there exist real valued functions u1, . . . , un onA1, . . . , An respec-

tively such that, for all α, β P ˆn s“1As, p4q αą_.β ô ÿ sPS ` uspαsq ´uspβsq ˘ ą0.

One problem with the condition (F) in the above theorem is that it is somewhat difficult to evaluate without case by case verification. Moreover verification is by no means an easy task: in example (1.3) where there are three alternatives and three states, in order to verify the condition, we have to make `272

˘

“351 comparisons.

Another problem with the above representation is that it gives us no informa- tion about how preferences vary with the context that is the current belief. Presumably, for context preferencespą_p, pP∆pSqq, the theorem holds for any given p, and the representation is as follows:

p5q αą_p β ô ÿ s_PS ` uspαs, pq ´uspβs, pq ˘ ą0.

If we seek a representation that has more structure, and thereby resembles a plan of action we need more conditions on how preferences vary with context. Chapters 2 and 3 provide two approaches to dealing with this problem.

By contrast, the classical approach to adding more structure has been to sup- press reference topand focus on state-preferences. In the Anscombe-Aumann (1963) approach, the first step is to restrict the space of outcomes to be the same in each state and define lotteries over these outcomes.11

The second step is to impose that the ordering of the set of outcomes in each state is the same across states. This state-independence allows us to identify the actual proba- bility measurep that represents the information context (beliefs) of the agent at decision time. Thus the Anscombe-Aumann approach says nothing about how preferences vary with the information context, and hencep. Perhaps it is best to view this model as capturing behaviour at stage two in the time-line we describe in remark (1.1) above.

The same may be said of the models of Savage (1954), Scott and Suppes (1958), and Debreu (1959). They seem to be more appropriate for addressing the problem of an agent making a decision in the heat of the moment and at a particular information set. By contrast, the present approach is closer in this sense to [vNM] where a complete, contingent plan of action is outlined in the first stage.

11

Or at least ensure that there exist two outcomes that are common to all states (Fishburn (1979)).