# Luce’s Choice Axiom

R. DUNCAN LUCE The Choice Axiom after Twenty Years JOURNAL OF MATHEMATICAL PSYCHOLOGY 15, 215-233 (1977)

## MATHEMATICAL AND STATISTICAL RESULTS

Anyone reading this article is probably already familiar with the choice axiom (axiom I of Luce, 1959), so I confine myself to a brief reminder. One part says that if a choice set

Scontains two elements,aandb, such thatais never chosen overbwhen the choice is restricted to justaandb; thenacan be deleted fromSwithout affecting any of the choice probabilities. …

## Introduction

(Here out of order).

Although it is clear from many experiments that the conditions under which the choice axiom holds are surely delicate, the need for simple, rational underpinnings in complex theories, as in economics and sociology, leads one to accept assumptions that are at best approximate. And the third section concerns alternative, more general theories which, in spirit, are much like the choice axiom.

After some attempts to apply choice models to psychophysical phenomena (discussed below in its proper place), I was led to conclude that it is not a very promising approach to, these data, and so I have not been actively studying any aspect of the choice axiom.

## GENERALIZATIONS

As we have seen, there is much evidence that choices are not generally governed by axioms as simple as the choice axiom. Indeed, the difficulty pervades the whole broad class of models exhibiting simple scalability.

Consider nearly equivalent trips to Paris and to Rome, and let the added symbol + denote some minor added benefit, such as a slight reduction in price. It is evident that for a = either Paris or Rome, [a+ is definitely preferred to a], but that it is quite possible [that neither Paris+ is definitely preferred to Rome or that Rome+ is definitely preferred to Paris].

## CONCLUSION

After somewhat less than 20 years, where does the choice axiom stand ? As a descriptive tool, it is surely imperfect; sometimes it works well, other times not very well. As Debreu and Restle made clear and as has been repeatedly demonstrated experimentally, it fails to describe choice behavior when the stimulus set is structured in such a way that several alternatives are treated as substantially the same. It probably also fails whenever the experimental subjects have the belief-

which I fear may often be the case-that they should employ the response alternatives about equally often. This tends to introduce some form of responsebiaswhich can well differ from one experimental run to another. Keep in mind that once we enter the path of strict rejection of models on the basis of statistically significant differences, little remains.To the best of my knowledge, the only property of general choice probabilities that has never been empirically disconfirmed is regularity-a decrease in the choice set does not decrease the probability of choosing any of the remaining alternatives-but even that looks suspect in a Gedanken experiment.Despite these empirical difficulties, there remains a tendency to invoke the choice axiom in many behavioral models-often implicitly. This is partly because it is so simple and the resulting computations are so easy. Perhaps the greatest strength of the choice axiom, and one reason it continues to be used, is as a canon of probabilistic rationality.It is a natural probabilistic formulation of K. J. Arrow’s famed principle of the independence of irrelevant alternatives, and as such it is a possible underpinning for rational, probabilistic theories of social behavior.Thus, in the development of economic theory based on the assumption of probabilistic individual choice behavior, it can play a role analogous to the algebraic rationality postulates of the traditional theory. However long the choice axiom may prove useful, at least during the 1960s and 1970s it contributed to the interplay of ideas about choices that arose in economics, psychology, and statistics.

## My Comments

First, some asides:

- The ‘choice axiom’ of mathematical psychology is not to be confused with ‘the axiom of choice’ of mathematical logic.
- If the choice axiom “is a natural probabilistic formulation of K. J. Arrow’s famed principle of the independence of irrelevant alternatives” then it is worth noting Arrow’s impossibility theorem, which suggests to me that this principle is widely violated. (Conversely, if the choice axiom fails then so does Arrow’s version.)

Now, my substantive comments.

Decision theories are of various types:

- Those which are purely nominal. We are at liberty to choose axioms at will, to see what the implications are.
- Those which are intended as normative, i.e. as a guide to ‘right action’.
- Those which are intended as descriptive, at least of ‘normal’ behaviour.

Even 40 year after this paper I am often left wondering in which sense mathematical psychologists intend their work.

The italics in the comments are mine. It seems to be widely accepted that nominal mathematical theories are not very good as descriptions, which motivates the search for alternatives. But the use of terms like ‘fear’ and ‘bias’ concern me. It as if the nominal theories were regarded as somehow normative. Or else why would one care that people departed from some wrong theory?

The Paris/Rome example seems to me one that is very common in selecting city breaks or holidays, The implication of the paper seems to be that one should have a definite preference between the two. But often I do not, and I see no reason to regard myself as irrational for not doing so. Instead I regard the situation as uncertain, and may look for further information and options (e.g. ‘What’s on’) before forming a preference, and even then I don’t see why I should be required to justify any such preference in terms of Luce’s theories: it would simply be a ‘best guess’ and I am often happy to let others break any tie.

As Luce makes clear, the attraction of the choice axiom, particularly the principle of regularity, is that to doubt it is to doubt various social and economic theories, a belief in which have been widely regarded as essential to modern life viewed as a social construct.

If we regard the principle of regularity as purely nominal then we can ask under what circumstances it might hold. We can apply some straightforward logic: If the arguments such as those which Luce cites indicate circumstances in which the ‘axioms’ hold, then we should look elsewhere. The Paris/Rome example seems a good start.

An implicit assumption of Luce’s work is that a decision is a once and for all thing. But suppose that I would prefer going on particular flight to Rome and staying in a particular hotel to a similar opportunity in Paris? Does this mean that I will prefer a booking to Rome to a booking to Paris? If a booking necessarily leads to the actuality then presumably there should be no difference. But, living in the UK, it seems to me that if my local airport was closed (e.g., due to bad weather) I could easily get to Paris by other means, whereas Rome would be more problematic. Hence I might prefer to book a trip to Paris even though I might (marginally) prefer to actually go to Rome.

Game theory can be applied to strategies as well as one-off actions. In this case the minimax action might be to buy tickets to Paris when there was a fall-back option of getting there by train but otherwise to buy tickets to Rome. Thus game theory seems a better guide to such uncertain situations even when there is no adversary with the ability to close airports.

It seems to me that where game theory is appropriate and gives a different result to the kind of theory that many other psychologists consider normative, Luce would ‘fear’ that people would follow the appropriate theory and consider their actions to be ‘biased’ and even ‘irrational’. I would not.