Statistically Comparing (Heterogeneous) Preferences Are there any statistical methods that can compare (heterogeneous) preferences?
For instance, suppose there are two people (Person 1 and Person 2). Both of these people make a list of their favorite sports (from most favorite to least favorite):
 person_1
[1] "tennis"   "hockey"   "soccer"   "baseball" "swimming" "golf"     "boxing"  

person_2
 [1] "squash"            "basketball"        "soccer"            "tennis"            "polo"              "american football" "track and field"   "swimming"          "boxing"            "snowboarding" 

By visually inspecting both of these lists, we can roughly conclude that both of these people like "soccer and tennis" , and both of these people generally dislike "boxing". But apart from visual inspection, are there "statistical methods" (e.g. hypothesis testing) which can be used to compare these preferences between both people? Is this considered a "ranking" problem?
Thanks
 A: First I will caution that sometimes human preferences are intransitive in that if they prefer A over B, and they prefer B over C, they do not necessarily prefer A over C...
But assuming you essentially have a list of items for each person representing the total order of their preferences, then ranking them 1,...,n and computing their (Spearman) correlation should be fine. This assumes that the lists are of equal length.
If the lists of items are not equal, you can turn to set similarity measures. Since the order of the items in the lists represent an order relation, and order relations are subsets of cartesian products of sets, we can consider going back to that Cartesian product representation.
Example
In my own work I call this 'setification of an order'.

Galen: $(A,B,C) \mapsto G =\{(A,B), (A,C), (B,C)\}$
Sarah: $(B,A,C) \mapsto S = \{(B,A), (B,C), (A,C)\}$

With your data converted into sets, there are various set-based similarity measures including:

*

*Jaccard Index

*Tversky Index

*Dice Coefficient

*Simple Matching Score

*...

Some of these are generalizations of others. Taking the Jaccard similarity as an example.
Example

$$J(G,S) = \frac{|\{ (A,C), (B,C)  \}|}{|\{ (A,B), (A,C), (B,C), (B,A) \}|} = \frac{2}{4} = \frac{1}{2}$$

This approach works on any finite relation (that your computer can handle anyway) for comparing their similarity.
Now you mentioned hypothesis testing. A choice of null hypothesis may be difficult to identify. In this case the order of the elements in the list is what is important. Perhaps a null hypothesis that any order is equally likely? In order to obtain a p-value you could bootstrap the statistic above by randomly permuting the lists and seeing how many times you get a similarity at least as extreme as the one you observed.

Kozolavska is right: you can also look at the concordant pairs. Indeed, your could compute Kendall's $\tau$ when defined as:
$$\tau = 2\frac{C-D}{n^2-n}$$
where C is the number of concordant pairs, D is the number of discordant pairs, and n is the sample size.
