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∈[N].¹ His (consumption) utility from consuming alternative i is: Ui=vi+ϵi.

• The individual observes both vi and ϵi, and selects the alternative that leads to the greatest consumption utility. That is, i∗=argmax

• 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.