P[ :C]: Utility

Theory of Games

Some consider the conventional – precise – notion of probability to be justified by von Neumann and Morgenstern’s remarks on utility in their Theory of Games. Here we point out that, on the contrary, that theory assumes that preferences are completely ordered (3.6.1), which implies that probabilities are completely / totally ordered, and hence precise. Moreover, the notion of (imprecise) possible probabilities can be extended in a straightforward way to possible  / imprecise utilities.

Possibilistic utility

If P< :C> is a set of possible probabilities  for a context C, then for each p( ) in P< :C> there exists a system U(p, ), as in 3.6.1. . We may define U< :C> = {U(p, ) | p ∈ P< :C>}. This satisfies a weakened form of the axioms in 3.6.1: there is only a partial order. Conversely, anything that satisfies the weakened (partial order) form of 3.6.1 can be strengthened in many ways to form a complete order satisfying 3.6.1. Such a complete order is then determines a probability function. P< :C> is then the set of all such possibilities.

Preference elicitation

Von Neumann and Morgenstern’s simple example (3.3.2 fn) is this:

Assume that an individual prefers the consumption of a glass of tea to that of a cup of coffee, and the cup of coffee to a glass of milk. … Does he prefer a cup of coffee to a glass the content of which will be determined by a 50%-50% chance device as tea or milk[?]

This only sets limits on the relative utilities, not precise relative values.

[W]e have only postulated an individual intuition which permits decision as to which of two “events” is preferrable. … [This] information ought to be obtainable in a reproducible way by mere “questioning”. [Next fn.]

 This seems a very limited notion of preference. I am not sure that my own precise preferences would be reproducible in any meaningful sense. Faced with such a bizarre situation I would probably say ‘whatever’. Imprecise utilities seem to me to better reflect my actual preferences. But the situation gets yet more complex.

Instability of preferences

I attend a weekly club that has a break, where we have a choice of tea or coffee. After we have made our choice a volunteer is selected to make the drinks. Thus all have to choose between tea or coffee as it will be made by an unknown person, and the drinks maker tries to make the teas and coffees in a way that will not lead to too many complaints. As 4.6.2/3 note, one can expect some norms to emerge over time so that one has some idea what sort of teas or coffees one is likely to get, thus informing the ‘utilities’. But as 2.3.3 and 2.4 point out, social organisations are rarely absolutely stable: people come and go. It seems to me that one adapts one’s declared preferences to the circumstances as they arise, and it is not clear that I ever have precise preferences, much less that anyone would be able to deduce them – precisely – from choices.

Possibilistic preferences

There are also grounds for considering value as possibilistic, even when no uncertainty in outcomes is involved. The possibilistic utility is then the product of possibilistic values and probabilities:

U< :CV and CP> ≡ V< :CV> • P< :CP> ≡{v( ).p( ) | v( ) ∈ V< :CV>, p( ) ∈ P< :CP>}. 

Dave Marsay

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