I'm working in a decision making topic where a product (e.g., a hotel) is described by some attributes, that is: $p=(p_1,\ldots,p_n)$. An attribute $p_i$ can either be numeric (e.g., the room average price), ordinal (e.g., the hotel category: $1\star,\ldots,5\star$) or nominal (e.g., the room color: red, green, blue).
We compute the utility of a product $p$ as a weighted sum of partial utilities $f_i(x)$, that is: $Util(p)=\sum_{i=1}^{n}w_i \times f_i(p_i)$. Here $f_i(p_i)$ is a value function or partial utility defined on the $ith$ product attribute.
I'm looking for a formal partial utility definition $f_i(x)$ for the case of a nominal attribute. There are several examples in the case of numeric and ordinal attributes. The problem with nominal attributes is that no preference order on the attribute values is known in advance. This is not the case of numeric or ordinal attribute values, where for example we know that $1\star < \ldots < 5\star$.
I'm not considering the case when the utility of every single nominal value is manually defined (e.g., $f_i(x_1)=v_1,\ldots,f_i(x_k)=v_k$, where $x_t$, $t=1\dots k$ are the different nominal attribute values). I'm looking for a general function definition. I have looked at some related information in books, papers, etc., but nothing useful has been found so far.
Any help will be really appreciated, thanks in advance. Harold
So if I understand you right, your problem is the estimation of a discrete choice model : you want to find the specification of the utility function that matches best the data you have on people's choices.
Essentially, this is a question of applied statistics, much more than a mathematical one. I think your question should be migrated to http://stats.stackexchange.com. You will probably get better answers there : they already have a set of question about the estimation of discrete choice models.
There are different possible ways to tackle your problem. To witness the breath of approaches, have a look at the results of a google search for "estimation of discrete choice models", e.g.
http://biogeme.epfl.ch/v18/tutorialv18.pdf http://www.youtube.com/watch?v=Ok534E1lf_k http://statisticalinnovations.com/products/choice_tutorial1.pdf
Your choice should depend on the data you have, the assumptions you are ready to make, the time you can put into conducting the analysis and the level of sophistication you are able to deal with.