# Choice Models 101

Choice models – the theory that studies how individual decision makers (e.g., consumers) make decisions when faced with a collection of alternatives – is of interest to several research disciplines. The canonical choice model is as follows.

• A single decision-maker chooses one among a collection $$N$$ alternatives, indexed by $$i \in [N]$$.1 His (consumption) utility from consuming alternative $$i$$ is: $U_i = v_i + \epsilon_i.$

• The individual observes both $$v_i$$ and $$\epsilon_i$$, and selects the alternative that leads to the greatest consumption utility. That is, $i^* = \arg\max_{i \in [N]} U_i.$

• The researcher/analyst observes $$v_i$$,2 the eventual choice $$i^*$$, but not $$\epsilon_i$$. The most common assumption (for analytical tractability) 3 is to assume that the user samples his $$\epsilon_i$$ as: $\epsilon_i\sim_{i.i.d} Gumbel.$ This gives the classic MNL model – a probabilistic choice model – where the individual chooses alternative $$i$$ with probability: $p_i := \Pr[\text{alternative } i \text{ is chosen}] = \frac{\exp(v_i)}{\sum_{j \in [N]}\exp(v_j)}.$

• If the alternatives are symmetric, i.e., $$v_i = v_j$$ for each $$(i,j)$$ pair, then, $$p_i = \frac{1}{N}$$.

This is what is taught in a typical undergraduate ECON course.

# Reality is (Slightly) Different

While a number of applications fit the above assumption (reasonably well), it is virtually impossible for a single decision-maker to know everything about every alternative he evaluates. A central assumption – that the decision-maker knows $$U_i$$ for each $$i$$ – feels like a stretch.

This got me thinking about how I evaluate alternatives, especially in situations where I know very little about these alternatives. Two example fit this situation very well in my own life.

• As a vegetarian living in the US, it is virtually hard for me to find a place that provides a good vegetarian and tasty meal. So, every time I am faced with a task of finding a good lunch/dinner spot, I decide to evaluate each alternative (that my friends suggest) based on the pictures, reviews, menu options, etc., on Yelp/Google Reviews. This is obviously an intensive process. What complicates this more is a need to not visit a previously-visited place.
• How I pick movies to watch on Amazon Prime Video: I know very little about a movie I plan to watch, and don’t watch a movie I already watched. So, I read about the movie, IMDB/Rotten Tomato ratings, reviews, etc. Will I perfectly realize the utility I stand to gain from making a particular choice of movie? Highly unlikely.

1. Here, $$[N]$$ denotes the set $$\{1, 2, \ldots, N\}$$.↩︎

2. Or proposes a model of $$v_i$$.↩︎

3. This simplification is due to the property that i.i.d Gumbels are closed under addition.↩︎