# Examples of Analysis of Competition in a Latent Product Characteristic Space

Competition between firms offering heterogenous products is often an exercise of constructing the right kind of spatial model.

There is the famous Hotelling model and many other papers that model competition in a product characteristic space. I can think of examples where cars are differentiated by size, power, gas mileage, etc. These are directly measurable and interpretable features.

However, are there any examples of analysis where latent features/product characteristics (interpretable or not) are used? I am thinking of the kinds of latent features that might be the product of using alternating least squares as an example, though it might be less common in economics.

Below I expand on matrix factorization, which is sometimes used for recommendations. Example article.

We might have a matrix of movie reviews by users, $$R = \left( r_{ij} \right)_{i\in I, j\in J}.$$

Then features are extracted based on finding $$R\approx W F^T$$.

Let $$R$$ be $$n\times m$$ for $$n$$ users and $$m$$ movies. A number of latent features $$k$$ can be chosen exogenously to extract. Then $$W$$ is $$n \times k$$, $$w_{il}$$ giving the weight user $$i$$ attaches to feature $$l$$. A movie's $$l$$-ness is then given by $$f_{jl}$$. The user scores movie $$j$$ according to $$r_{ij} = \sum_{l=1}^k w_{il}f_{jl}.$$

Here, $$l$$-ness is uninterpretable, simply a result of the matrix factorization and then weights $$w$$ simply give linear utility weights for these features. Similar uninterpretability might also arise if many non-latent features are condensed via dimensionality reduction.

• Do you have any examples of latent features in products? Designed artefacts like products seldom have latent variables, unless they are poorly designed e.g. nobody thought of reliability or security when they designed the thingy. Even then there's usually a way to measure it directly post factum. Apr 11, 2019 at 21:12
• @Fizz I edited the question to expand a bit. Products will have latent characteristics insofar as they are discoverable in the data or the result of reducing the dimensionality of a feature space. The matrix factorization bit seems common for building a model for movie recommendations. Would you imagine a different way of deriving features from movies? Apr 12, 2019 at 21:54
• It seems the keyword you're looking for is "latent separability" ideas.repec.org/p/ags/aaea03/21892.html Apr 12, 2019 at 22:59
• or "latent comparative advantage" openknowledge.worldbank.org/handle/10986/15838 Apr 12, 2019 at 23:17