# Elementary Aspects of Choice in Universal Utility Models

Chapter

## Abstract

In this section we develop the universal utility model underlying the rest of this work.^{1} In section 6.2.1, the model employed in Hammond (1976a) was sketched out. His analysis is restricted to strong orderings. It is not appealing to define strong orderings over *infinite* budget sets. Either one uses weak orderings over infinite sets or, if this is not possible, strong orderings over finite sets. Accordingly, in Hammond the set of terminal nodes *X* is assumed to be finite.

### Keywords

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### Notes

- 1.Parts of this chapter draw upon v. Auer (1998); with kind permission from Kluwer Academic Publishers.Google Scholar
- 2.Compact sets are closed and bounded. They include all finite sets as well as the usual (infinite) budget sets with all prices greater than zero.Google Scholar
- 3.
- 4.In order to avoid sequences
*x*_{0},*x*_{1}, ⋯ converging to zero, one may impose a finite time horizon*T*.Google Scholar - 5.Recall that “infinite” indicates that life time wealth is infinitely divisible between the various periods. For consumption problems with infinitely many periods, a compact opportunity set (e.g. due to bounded life time wealth) must lead to a consumption profile which in some periods exhibits consumption levels arbitrarily close or equal to zero.Google Scholar
- 6.Hammond (1976a) uses an equivalent example featuring “drugs” instead of “TV” and “abstaining from drugs” instead of “reading a book”. Correspondingly, at
*n*_{1}, the choice maker prefers drugs to abstention. The snag with this example is that it suggests an*a priori*relationship between preferences and choices, even though no choice mechanism has yet been specified. The impression is that at*n*_{1}the drug addict must choose according to her addiction expressed in her preferences at*n*_{1}. However, there is no reason why we should restrict our attention only to those choice mechanisms in which choices at final nodes are executed in compliance with this node’s preferences.Google Scholar - 7.In decision trees we simply write
*aRb*instead of*aR*(*n*_{0})*b*or*aR*(*n*_{1})*b*. No confusion should arise, since in the trees preferences are stated right at the respective decision node.Google Scholar - 8.
- 9.We use
*e*,*f*, g, and*h*instead of*a*,*b*,*c*, and*d*, in order to avoid confusion with the options’ labels used in Definition 7.3.Google Scholar - 10.The notion of relevant options makes sense only for sophisticated choice mechanisms.Google Scholar
- 11.Recall that the decision at
*n*_{1}is part of a sequence of decisions which finally lead to the choice set*C*(*A*)(*n*_{0}).*C*(*A*)(*n*_{1}), in contrast, indicates the choice set of an agent who*starts*his decision process at node*n*_{1}and*who has got no recall of past preferences*(see section 7.2.3). We get*C*(*A*)(*n*_{1}) =*a*(*n*_{0}),*b*(n_{0}).Google Scholar - 12.This mechanism I owe to a suggestion by U. Schmidt.Google Scholar
- 13.An interesting alternative was pointed out to me by Prof. Seidl. He proposes a
*sanguine choice mechanism*. In Figure 7.10(b), for instance, the individual moves towards*n*_{1}, since the best of those options available at*n*_{1}and relevant at*n*_{0}— option*a*(*n*_{0}) — is better than option*c*(*n*_{0}):*aR*(*n*_{0})*c*. This mechanism shows some similarity to the notion of “optimistic behaviour” as used by Greenberg (1990, p.18). Greenberg’s notion of “conservative behaviour” is linked to the idea of cautious choice. However, he is not concerned with dynamic choice of a single agent but with*social*situations. He studies formal models of social behaviour in the spirit of game theory.Google Scholar - 14.If
*b*(*n*_{t}_{+1}) does not exist, then*b*(*n*_{t}) is the last decision node along*b*(*n*_{0}).Google Scholar - 15.As in Figure 7.6, we use
*e*,*f*,*g*, and*h*instead of*a*,*b*,*c*, and*d*, in order to avoid confusion with the options’ labels used in Definition 7.6.Google Scholar - 16.For an interesting alternative based upon the psychological notion of self-control see Thaler and Shefrin (1981). In their model the choice mechanism is endogenously determined. They formalize this idea by distinguishing between a doer’s and a planner’s utility function. Broadly speaking, the more resolute (the less naive) the mechanism, that is the more the planner interferes with the doer’s planned choices, the higher the costs. Such an approach requires some
*ad-hoc*utility function of the planner which weighs the gains from more resolution (they speak of self-control or control over the doer) against the costs of implementing it.Google Scholar - 17.The idea of subgame-perfection is formalized in Selten’s classic paper (1965).Google Scholar

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