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.

Choices with Simultaneous Search

I provide an illustration below (this is my own take on the problem).

Let the true type (utility) of the \(i^{th}\) alternative be sampled from: \[U_i \sim \mathcal{N}(\mu, \sigma^2).\] A decision maker has, at his disposal, \(R\) resources (e.g., the budget of time units spent investigating making a decision). He chooses to allot these \(R\) resources to investigate these \(N\) alternatives. A unit resource can investigate exactly one alternative. The decision-maker chooses \[\mathbf{r} = (r_1, r_2, \ldots, r_N) \\ \text{ s.t. } \mathbf{r} \ge \boldsymbol{0} \text{ and } \mathbf{r} \cdot \boldsymbol{1} \le R. \] Each resource provides a noisy unbiased signal of the alternative it investigates. That is, \[s_r \sim \mathcal{N}(U_{i_r}, \sigma_R^2).\] The decision-maker’s posterior about alternative \(i\) is: \[U_i | r_i \sim \mathcal{N}\left(\alpha (r_i) \overline{s}_i + (1-\alpha(r_i)) \mu, \left(\sigma^2 + \frac{\sigma_R^2}{r_i}\right) \alpha_i (1-\alpha_i)\right) \\ \text{ where } \alpha(r_i) = \frac{\sigma^2}{\sigma^2 + \frac{\sigma_R^2}{r_i}} \quad \text{ and } \overline{s}_i = \frac{\sum_{r: i_r =i} s_r}{r_i}\]

For convenience, refer to \(\alpha(r_i)\) as \(\alpha_i\). The principal chooses the alternative with the highest posterior mean, i.e., \[i^* = \arg\max_{i \in [N]} \mathbb{E}[U_i | r_i]. \] The principal’s objective simplifies to: \[\mu + \sigma \max_{\mathbf{r} \ge \boldsymbol{0}} \mathbb{E}_{\mathbf{s|r}} [\max_i \gamma_i z_i] \\ \text{ where } \quad \gamma_i = \frac{1}{\sqrt{1 + \frac{\sigma_R^2}{{\sigma^2} r_i}}} \quad \text{ and } z_i \sim_{i.i.d} \mathcal{N}(0, 1).\] Normalize \(\mu = 0\) and \(\sigma = 1\). It suffices to solve: \[ \max_{\mathbf{r} } \mathbb{E}_{\mathbf{s | r}} [\max_i \gamma_i z_i] \\ \text{ where } \gamma_i = \frac{1}{\sqrt{1 + \frac{\sigma_R^2}{r_i}}} \] Solving this problem analytically can be done for some special cases (e.g., \(N = 2\), or the comparison of some heuristics, e.g., the fully symmetric vs. fully asymmetric solution). I provide a numerical comparison below.