Simulations¶
Prest offers two ways in which one can obtain information about the probability distribution of various variables of interest (e.g. axiom violations; model distance scores) when a large number of artificial subjects are assumed to make uniform-random choices from menus that are derived from a finite set of general choice alternatives.
From these probability distributions one can then identify human subjects whose choice behaviour cannot be distinguished from random behaviour for a given level of statistical significance.
This procedure therefore allows one to perform a power test for their model-estimation and consistency-analysis computations on general datasets, as was first suggested in Bronars [1987] for budgetary datasets.
Generating random datasets and choices¶
Under “Menu distribution options” select one of the following:
“Exhaustive (each possible menu once)”. Choices are made from all \(2^n-1\) menus that are derived from the specified set with \(n\) elements.
“Random sample with replacement”. Choices are made from a random selection of the \(2^n-1\) menus that are derived from the specified set with \(n\) elements, possibly with repetitions.
“All binary menus”. Choices are made from the \({n}\choose{2}\) binary menus that are derived from the underlying set with \(n\) alternatives.
“Default alternative”. Select between “None” and “Uniformly random”; in the latter case every feasible alternative in every menu is equally likely to be the default.
If “Default alternative -> None” was selected above, then one can also select one of the following under “Choice mode”/”Observations without default alternatives”:
“Forced choice”: some alternative is chosen from every menu (deferral/outside option not feasible).
“Non-forced choice”: choices can be empty-valued (deferral/outside option feasible).
Note
In both these cases one can allow multiple alternatives to be chosen by checking the “Multi-valued choice” box.
The corresponding simulated choice probabilities are as follows:
Menu with `k` alternatives |
Single-valued choice |
Multi-valued choice |
|---|---|---|
\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\) Forced choice |
||
Probability for any alternative |
\(\frac{1}{k}\) |
\(\frac{1}{2}\frac{2^k}{2^k-1}\) |
Probability for any submenu |
Not defined |
\(\frac{1}{2^k-1}\) |
\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\) Non-forced choice |
||
Probability for any alternative (excluding deferral/outside option) |
\(\frac{1}{k+1}\) |
\(\frac{1}{2}\frac{2^k}{2^k-1}\frac{k}{k+1}\) |
Probability for deferral/outside option |
\(\frac{1}{k+1}\) |
\(\frac{1}{k+1}\) |
Probability for any submenu |
Not defined |
\(\frac{1}{2^k-1}\frac{k}{k+1}\) |
If “Default alternative -> Uniform” was selected above, then one can also select one of the following under “Choice mode”/”Observations with default alternatives”:
“Unbiased”: all alternatives (including the default) are equally likely to be chosen.
“Default-biased”: this adapts the structure of “Non-forced choice” simulations to an environment where a default/status quo option is present and replaces the deferral/outside option; however, because the default/status quo option is one of the \(k\) alternatives in the menu now, this adaptation generates a choice probability distribution that is biased towards that option.
Note
In both these cases one can again allow multiple alternatives to be chosen by checking the “Multi-valued choice” box.
In this case the corresponding simulated choice probabilities are as follows:
Menu with `k` alternatives |
Single-valued choice |
Multi-valued choice |
|---|---|---|
\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\) Unbiased |
||
Probability for any alternative except the default/status quo option |
\(\frac{1}{k}\) |
\(\frac{1}{2}\frac{2^k}{2^k-1}\) |
Probability for the default/status quo option |
\(\frac{1}{k}\) |
\(\frac{1}{2}\frac{2^k}{2^k-1}\) |
Probability for any submenu |
Not defined |
\(\frac{1}{2^k-1}\) |
\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\) Default-biased |
||
Probability for any alternative except the default/status quo option |
\(\frac{1}{k+1}\) |
\(\frac{1}{2}\frac{2^k}{2^k-1}\frac{k}{k+1}\) |
Probability for the default/status quo option |
\(\frac{2}{k+1}\) |
\(\frac{1}{k+1}+\frac{1}{2}\frac{2^k}{2^k-1}\frac{k}{k+1}=\frac{2^kk+2(2^k-1)}{2(2^k-1)(k+1)}\) |
Probability for any submenu |
Not defined |
\(\frac{k}{(k+1)(2^k-1)}\) |
Note
“Single-valued choice” here refers to the case where “Multi-valued choice” at the bottom of the dialog box is not selected, and results in up to one alternative being chosen from each menu.
Note
The probability of an alternative being chosen under the “Multi-valued choice” mode is interpreted as the probability that this belongs to the chosen submenu of the relevant menu. Assuming “Forced choice” and considering an arbitrary menu \(A\) with \(k\) alternatives, every nonempty weak submenu of \(A\) is chosen with probability \(\frac{1}{2^k-1}\). Since each of the \(k\) alternatives belongs to exactly \(\frac{2^k}{2}\) of these submenus, it follows that each of them is chosen with probability \(\frac{2^k}{2(2^k-1)}\). If “Non-forced choice” is selected instead, then since some nonempty submenu of \(A\) is chosen with probability \(\frac{k}{k+1}\) (because the deferral/outside option is chosen with probability \(\frac{1}{k+1}\)), the corresponding choice probability for each of the \(k\) alternatives is adjusted accordingly.
Generating random choices based on an existing dataset¶
