Abstract
Recent literature has examined the problem facing decision makers with bounded awareness, who may be unaware of some states of nature. A question that naturally arises here is whether a value of awareness (VOA), analogous to value of information (VOI), can be attributed to changes in awareness. In this paper, such a value is defined. It is shown that the sum VOA \(+\) VOI is constant and, except for scale effects, independent of the choice set. It follows that the larger is VOA, the smaller is VOI. This point is illustrated for a simple two-state case, then proved for general classes of compact convex choice sets, and for alternative interpretations of the concept of unawareness.
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Notes
Laffont (1989, Sect. 5.4) provides a summary.
For simplicity in statement of the results, we consider only the maximum value of awareness and information, obtained in the transition from minimal awareness and information to full awareness and information. The extension to the case of more limited changes in awareness, analogous to the work of Blackwell (1951) and Laffont (1989), is left for future work.
In the context of individual decision, considered in the present paper, the difference between complete unawareness and the attribution of zero probability is essentially a matter of interpretation.
By contrast, in the case of extensive-form games, the rationalizability of outcomes differs depending on which of these concepts is applicable. As Heifetz et al. (2013) observe,
With rationalizability, generalized games are necessary for properly modeling unawareness; trying to model unawareness by having the unaware player assigning probability zero to the contingency of which she is unaware might give rise to a completely different rationalizable behavior, which does not square with unawareness in the proper sense of the word.
Unawareness may also be represented using a propositional approach, beginning with a set of elementary propositions and developing modal operators to characterize knowledge and awareness. There exists a canonical equivalence between state-space and propositional approaches. Given a set of elementary propositions P, the truth table for P determines a state space \(\varOmega =2^{P}.\) Thus, the choice between propositional and state-space approaches is one of the analytical conveniences. For present purposes the state-space approach is more convenient.
Alternatively, we may interpret the results as applying to expected utility preferences where the choice set is assumed to be in utility space. The effect of shifting from outcome space to utility space is discussed below.
This point is less clear if restriction is interpreted as attaching zero probability to a state. On this interpretation, agents presumably consider the consequences of an act in all states \(\omega \in \varOmega ,\) even those they regard as having zero probability.
This assumption is made for simplicity. It is natural when all states are equally probable.
Note that, by the definition of \(\mathcal {X},\) if this condition holds for one s it must hold for all s.
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John Quiggin is indebted to Simon Grant for helpful comments and criticism.
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Quiggin, J. The value of information and the value of awareness. Theory Decis 80, 167–185 (2016). https://doi.org/10.1007/s11238-015-9496-x
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DOI: https://doi.org/10.1007/s11238-015-9496-x