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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.

source: http://www.scholarpedia.org/article/Luce's_choice_axiom

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?

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    $\begingroup$ Luce, R. D. 1959/2005. Individual Choice Behavior: A Theoretical Analysis. New York: Wiley. Reprinted by Dover Publications. $\endgroup$
    – Nick Cox
    May 27, 2013 at 7:58
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    $\begingroup$ Yes that is the one. Thx for the reference. $\endgroup$
    – Druss2k
    May 27, 2013 at 9:20
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    $\begingroup$ 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. $\endgroup$
    – Tim
    Sep 17, 2015 at 3:46

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I see no reason that probability theory would have any difficulty framing this situation, or any variation on it. If the choice probabilities are conditioned on the choice set, then presumably the choice-set can be made an object in the analysis, and you can then specify conditional probabilities based on possible values of the choice-set. Also, the choice of car use is not fundamentally different from the others -- regardless of the choice made, there will be some causal consequences on the types of transport used now or in the future (e.g., if you don't take a bus then the bus company gets less money and it decides to reduce its services). The mere fact that actions have causal consequences, and there are counterfactual possibilities, does not seem to me to give rise to any problems in probability theory.

I always find descriptions of cases like this to be ill-posed. It is very easy to pose a complex situation, and then pose a simplistic probability framework that fails to capture the situation properly. That is not a deficiency of probability theory - it is just a case of not using it correctly.

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