View source for Luce's choice axiom

In [[probability theory]], '''Luce's choice axiom''', formulated by [[R. Duncan Luce]] (1959), states that the [[probability]] of selecting one item over another from a pool of many items is not affected by the presence or absence of other items in the pool. Selection of this kind is said to have "[[independence of irrelevant alternatives|independence from irrelevant alternatives]]" (IIA).

==Overview==
{{see|Softmax function}}
Mathematically, the axiom states that the probability of selecting item ''i'' from a pool of ''j'' items is given by:

: <math>P(i) = \frac{w_i}{\sum_j{w_j}}</math>

where ''w'' indicates the weight (a measure of some typically salient property) of a particular item.

This function used elsewhere in mathematics and science, where it is known as the ''normalized exponential function'' or [[Softmax function|softmax]], and dates to the [[Boltzmann distribution]] in [[statistical mechanics]]; see {{slink|Softmax function|History}}.

==Applications==
The axiom is often encountered in [[economics]], where it can be used to model a [[consumer]]'s tendency to choose one brand of product over another. It is also found in [[psychology]], particularly in [[cognitive science]] where it is used to model approximately rational decision processes.

==External links==
*[http://www.scholarpedia.org/article/Luce%27s_choice_axiom Scholarpedia article Luce's Choice Axiom]

==References==
*{{cite book
  | author = Luce, R. D.
  | authorlink = R. Duncan Luce
  | title = Individual Choice Behavior: A Theoretical Analysis
  | year = 1959
  | isbn = 978-0-486-44136-8
  | location = New York
  | publisher = Wiley}}
*{{cite journal
  | author = Luce, R. D.
  | authorlink = R. Duncan Luce
  | title = The choice axiom after twenty years
  | url = http://www.socsci.uci.edu/~rdluce/bio/pre1990/1977/Luce_JMP_1977a.pdf
  | journal = Journal of Mathematical Psychology
  | volume = 15
  | issue = 3
  | pages = 215–233
  | year = 1977  | doi = 10.1016/0022-2496(77)90032-3
| citeseerx = 10.1.1.336.1366
  }}

[[Category:Decision theory]]