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.
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Notes
Parts of this chapter draw upon v. Auer (1998); with kind permission from Kluwer Academic Publishers.
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.
In Example 2.1, for instance, Xt was coffee with milk, black coffee.
In order to avoid sequences x0, x1, ⋯ converging to zero, one may impose a finite time horizon T.
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.
Hammond (1976a) uses an equivalent example featuring “drugs” instead of “TV” and “abstaining from drugs” instead of “reading a book”. Correspondingly, at n1, 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 n1 the drug addict must choose according to her addiction expressed in her preferences at n1. 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.
In decision trees we simply write aRb instead of aR(n0)b or aR(n1)b. No confusion should arise, since in the trees preferences are stated right at the respective decision node.
That is a(nt) is not the last decision node along a(n0).
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.
The notion of relevant options makes sense only for sophisticated choice mechanisms.
Recall that the decision at n1 is part of a sequence of decisions which finally lead to the choice set C(A)(n0). C(A)(n1), in contrast, indicates the choice set of an agent who starts his decision process at node n1 and who has got no recall of past preferences (see section 7.2.3). We get C(A)(n1) = a(n0), b(n0).
This mechanism I owe to a suggestion by U. Schmidt.
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 n1, since the best of those options available at n1 and relevant at n0 — option a(n0) — is better than option c(n0): aR(n0)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.
If b(nt+1) does not exist, then b(nt) is the last decision node along b(n0).
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.
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.
The idea of subgame-perfection is formalized in Selten’s classic paper (1965).
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© 1998 Springer-Verlag Berlin Heidelberg
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von Auer, L. (1998). Elementary Aspects of Choice in Universal Utility Models. In: Dynamic Preferences, Choice Mechanisms, and Welfare. Lecture Notes in Economics and Mathematical Systems, vol 462. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58879-2_7
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