Background: over the course of iCarly, Carly Shay has gone through multiple boyfriends and love interests, as evidenced by this video. She's broken up with many people too, as evidenced here. Reading the comments, however, leads to horrifying insights.

"Carly is so shallow"

"Carly clearly has a type, and they all look like Freddie"

Now, this is the insight that any introductory economics student would make. This is a simple regression, perhaps logistic, of the probability of Carly dating a guy on a vector of his characteristics: hair color, height, muscle size, hobbies, talents, the number of competitors you have on the dating market, et cetera. However, this analysis is insufficient for gleaning the proper mechanisms behind why Carly dates whom she dates. Even if we add a lot of controls (and have enough data to avoid collinearity), instrument to fix reverse causality (the potential of dating Carly causes guys to change their characteristics), all we get are treatment effects: we don't have structural insights into why Carly chooses and breaks up with the guys she does, because those internal motivations are unobservable to the econometrician.

Dating is a dynamic decision-making process, and in our econometric analysis, we need to take that into account. Guiding our econometrics using economic theory will let us take account of Carly's aforementioned unobservable motivations. So let us proceed.

In each period t, Carly has two actions: she can choose to date a guy or not. Her action a gives her, in each period, utility U, which is composed of two additively separated components: u and epsilon.

u is a "direct" utility function that is affected by the characteristics of the guy she dates (assuming that she chooses to date him), the number of times she has dated him, and the utility parameters she places on those characteristics. The characteristics are represented by e, the parameters are represented by theta, and n is the number of times she has dated the guy in the past.

Epsilon are Type 1 Extreme Value, i.i.d. utility shocks in each period. There are epsilon shocks for when Carly chooses to date guys of certain characteristics, and epsilon shocks for when she chooses not to date. We normalize u = 0 when she chooses not to date.

All-in-all, during the dating process, this is what her utility looks like.

Now, suppose that Carly exits the dating market in order to enter a relationship with somebody else. We will safely assume that she sees the potential for marrying him, i.e. by sticking with him, she is maximizing her lifetime expected discounted utility. Let 0 < delta < 1 be her per-period discount factor. Then if her serious boyfriend has characteristics e^*, then this is her infinite-horizon discounted utility (just the geometric sum formula). We can safely assume that she will only make this decision for the last guy she's dated thus far.

Now, let X = {e, n} for conciseness. Let a capture the decisions to date or not to date in each period, separate from her decision to enter a relationship with the last guy she dated. Then this is our primary Bellman equation.

This is where the theoretical modeling ends. We have data on Carly's decisions to date in each period. We will assume conditional independence, and so if we define Carly's expected value function as EV, we can show that this fixed-point equation holds true. This enables us to derive Carly's conditional choice probability for each vector of characteristicse, and because the epsilons are Type 1 Extreme Value, if we let A include both a and her choice to start a relationship, then her choice probabilities look like this.

And using these probabilities to eventually derive theta using the polyalgorithm described in Rust (1988), we get the payoff: Carly Shay is not necessarily shallow, nor does she necessarily have a type! Note that her decisions depend on her discount factor, and the density function for the types of guys with characteristics e' that she will have in future periods. When we use nested fixed point algorithms to estimate parameters using this structural model, as opposed to simple estimation via regression, we account for Carly's unobserved decision-making processes.

On alleged shallowness: Carly breaks up with a lot of guys because her density function p(de'|a, X, theta) first-order stochastically dominates most normal people's. She has a famous web show, and hence many guys would want to date her. She has a distribution function for the dating market that many people could only dream about. Hence, in maximizing her long-run utility, she can afford to be picky and break up with a lot of guys because of the high probability that an even better guy will be waiting in the wings. The reason she dates anyone at all who doesn't reach the upper echelon is the discount factor and the potential for too many negative utility shocks when single.

On allegedly having a type: Carly doesn't simply have a type, she follows a multi-armed bandit learning process. The commenters allege that her boyfriends look like Freddie; however, this is not true. Note that the first guy Carly dated was Jake, who had blonde hair. After he turned her off by cheating on her, her subjective beliefs in the learning process caused her to switch to brunettes, meaning her conditional choice probabilities went up for brunettes and down for blondes. This is a perfectly rational dynamic choice process, and isn't reflected in a simple regression. For further empirical evidence, note the counterfactual episode in which Carly dates Nevel in the alternate dimension, an unfathomable proposition in the main show. This is clear evidence that she does not have a mere "type", but is following a learning process and updating her conditional choice probabilities with time. Note that we could make the learning process more explicit by defining a Markov process of states as in Arcidiacono & Miller (2011).

All in all, iCarly teaches us that we cannot run in naively with our empirical analyses of romance. We must be careful and consider people's unobservable internal choice processes.