# Luce choice axiom, question about conditional probability [closed]

I'm reading Luce (1959). Then I found this statement:

When a person chooses among alternatives, very often their responses appear to be governed by probabilities that are conditioned on the choice set. But ordinary probability theory with its standard definition of conditional probability does not seem to be quite what is needed. An example illustrates the difficulty. When deciding how to travel from home to another city, your choice may be by airplane (a), bus (b), or car (c). Let A, B, C denote the uncertain states of nature associated with  form of travel. Note that if one elects c all of the uncertainties of A and B remain because planes fly and buses run whether or not you are on them. However, if you elect either a or b , then your car remains in the garage and the set C is radically altered from when the car is driven. So there really is no universal event underlying the sources of uncertainty.

The choice axiom of chapter 1 was introduced as a first attempt to construct a probability-like theory of choice that by-passed the fixed, universal sample space assumption.

For me the probability measure is defined with the triplet $\Omega$, the sample space, a sigma-algebra $\mathcal{F}$ and finally a measure $P$.

With respect to the foregoing example what seems to be the problem if I define:

$\Omega = \{ \text{bus}, \text{car}, \text{airplane} \}$

One crucial assumption in common statistics is the ceteris paribus condition. Is this the reason we need to adjust basic probability theory in the context of choice behavior because the c.p. assumption is violated?

• Luce, R. D. 1959/2005. Individual Choice Behavior: A Theoretical Analysis. New York: Wiley. Reprinted by Dover Publications. May 27, 2013 at 7:58
• Yes that is the one. Thx for the reference. May 27, 2013 at 9:20
• It is a long time since I have read Luce, but I think you will find that he is not suggesting that probability theory needed to be adjusted in the context of choice behaviour, but was instead introducing an alternative model of choice behavior to those that existed prior to his own work.
– Tim
Sep 17, 2015 at 3:46