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# It's a Love Story

14 Feb 2021

As a rule, I try not to do Freakonomics-esque posts - but it is Valentine’s Day and Love Story (Taylor’s Edition) did just come out two days ago, so let’s examine what economic theory can tell us about dating, relationships and love! Firstly, we will consider what we want to see out of a potential partner. Secondly, we will examine the process of matching with a potential partner. And thirdly, we will look at whether we can get stable matches of different people.

I’ll be cleaning up bottles with you on New Year’s Day

For those unacquainted, there’s a concept by Gary Chapman known as “love languages” - in effect, it describes five ways in which people express and experience love: acts of service, gift giving, physical touch, quality time, and words of affirmation. In discussing this with a friend, I realised that although we had mostly similar pairwise rankings, I prioritised words of affirmation a lot less than her, instead putting quality time as the most important one. How can I explain why I find words of affirmation so irrelevant?

Suppose there are two types of people perceived as being potential partners - the parameter $\theta$ measures how much they care about you. One is genuinely romantically interested $(\theta=2)$ while the other is simply being friendly $(\theta=1)$. Unfortunately, it is difficult to differentiate between the two in practice, because we cannot simply find out people’s true parameter values.

Insofar as we want to reciprocate and make our investments into relationships equal to how much the other person cares about us, this makes it a struggle. If all we know is that a proportion $\alpha$ of the population is just being friendly, we would reciprocate equal to the average amount people care i.e. $1 \alpha + 2(1-\alpha)=2-\alpha$. This is an inefficient outcome, because someone who is genuinely romantically interested will be underappreciated, while those who are simply casual friends are overappreciated. So we would like a way to separate the two.

One metric to do so is by considering the amount of time they spend with you, which unlike an abstract parameter value, we can measure perfectly. Suppose that the cost of spending time $T$ with you depends on how much someone cares about you and is given by $\frac{T}{\theta}$ - the actual potential partner faces cost $\frac{T}{2}$, while the casual friend faces cost $\frac{T}{1}$. The time someone spends on you can therefore serve as a proxy for how much they care about you.

Now suppose we decide that the amount we reciprocate $W$ is based on a rule $W(T)=1+F(T \geq T^\ast)$. That is, how much we care about them is dependent on a constant $1$ and a function $F$ based on whether they spend more time on us than some target $T^\ast$, with $F=1$ if they do and $F=0$ if they don’t.

Recognise that for a perceived potential partner i.e. both those who are genuinely interested and those who are just friendly, they face the optimisation problem $\max u(T) = W(T) - c(T)$, where $c(T)$ simply reflects the cost function of spending time we described earlier. They want to maximise how much you care about them minus how much effort they’d have to put in.

In this case, two possible choices dominate. Between $0$ and $T^\ast$, it is optimal to pick $T=0$, since they know we will reciprocate at $W=1$ regardless. Between $T^\ast$ and $\infty$, it is optimal to pick $T=T^\ast$, since they know we will reciprocate at $W=2$ regardless. So there are two dominant choices. For potential partners, they face the following choice. If $2-\frac{T^\ast}{2}>1$, they will choose to spend $T^\ast$ in time with us, while if $2-\frac{T^\ast}{2} \leq 1$, they will choose to spend no time with us. As for casual friends, they will choose to spend $T^\ast$ in time with us if $2-T^\ast>1$ and no time if $2-T^\ast \leq 1$.

To ensure we get a separating equilibrium where it makes sense for those who are potential partners to differentiate themselves from those who are casual friends, we need to set a $T^\ast$ such that $2-\frac{T^\ast}{2}>1$ and $2-T^\ast \leq 1$. That is, we must have $T^\ast\in (1,2)$. Notice that this means we will also end up spending the exact amount of effort as the other person, reciprocating perfectly.

So this is why I care more about quality time than words of affirmation, because talk is cheap but time is precious. And in a world of imperfect information, a costly signal is a useful mechnaism. If you liked this, you may want to check out “Job Market Signalling” (Spence 1973), where he explains how education is a form of signalling in the job market. Indeed, this line of work is what won him the 2001 Nobel Prize in Economics!

That’s how you get the girl

Having considered what we might be looking for while dating, how does one actually meet potential partners? Oftentimes, people use dating apps - so let’s consider why people might use Bumble, which is unique because it only lets female users make first contact with male users. (I apologise in advance for the fact that this will be incredible heteronormative and gendered, but this is for the sake of easily modelling what Bumble does.)

Suppose a one-period model where there are some male and female users. The male users are indexed along the unit interval $[0,1]$, while the female users are also indexed along the unit interval $[0,1]$. At the beginning of the period, everyone is unmatched. That means the number of unattached male users $u_t$ is 1. We will denote the number of unattached male users at the end of the period as $u’_t$. Because it is female users who are able to message first, we can think of them as controlling the amount of vacancies for potential relationships $v_t$. Couples get matched into relationships $R_t$ by a matching function $R_t=\psi M(u_t,v_t)$, where $\psi$ is a efficiency parameter of matching. We assume this matching function faces constant returns to scale.

For an unattached male user, they receive a baseline level of utility $b$ if they are not in a relationship and gain $w_t$ in utility from their partner if they are in a relationship. Their chance of finding a relationship is $f_t=\frac{R_t}{u_t}=\psi M\left(1, \frac{v_t}{u_t}\right)$. As such, their utility function is given by $U_m = f_t w_t + (1-f_t)b$. For female users, it takes effort to message first - as such, they face the cost $k$ if they initiate the possibility for matching by messaging. They have a chance $q_t=\frac{R_t}{v_t}$ of finding a relationship from messaging. And they face a utility function of $U_f = q_t(z_t-w_t) + (1-q_t)b - k$, where they gain $z_t$ from their partner but also have to expend $w_t$ for their partner.

We assume that there is a free entry condition, where the utility minus the cost of sending a message for female users who decide to message a male user must go to 0 - that is, they are indifferent between sending a message and not sending a message. Otherwise, more female users would message male users until it reached 0. That means we can write $q_t(z_t-w_t) + (1-q_t)b - k=0$. We can solve for $q_t=\frac{k-b}{z_t-w_t-b}$. It is clear that the value of $w_t$ affects how much of the value generated by being matched goes to the man and the woman - as such, we can think of it as the result of some process of bargaining between the man and the woman. In particular, we can consider the how the two parties split the payoff from being matched that is in excess of their outside option of simply receiving $b$. This is done via them engaging in Nash bargaining to decide the $w_t$ that maximises this total payoff. That is, each match faces the optimisation problem of $\max_{w_t} (z_t-w_t-b)^\chi (w_t-b)^{1-\chi}$. The parameter $\chi$ reflects the bargaining power of each side. The first-order condition for this is $-\chi(z_t-w_t-b)^{\chi-1}(w_t-b)^{1-\chi}+(1-\chi)(z_t-w_t-b)^{\chi} (w_t-b)^{-\chi}=0$.

We can rearrange this to produce $(1-\chi)(z_t-w_t-b)=\chi(w_t-b)$ and therefore that $w_t=(1-\chi)(z_t-b)+\chi b$. In other words, it is a linear combination of the gains to the woman beyond her outside option and the man’s outside option. We can substitute this into our expression for $q_t$ to find the vacancy filling rate $q_t=\frac{k-b}{\chi(z_t-2b)}$. Notice that as $\chi \to 0$, the woman has less bargaining power and we see the man capturing all the value from the match since $w\to z_t-b$. The corollary to this is that it also leads to $q_t \to \infty$, where because the woman doesn’t get anything out of this match but it costs her to send the message, she drops out of this market, resulting in a filling rate of infinity.

This also pins down the number of women who send messages, because we know that $q_t=\psi M\left(\frac{u_t}{v_t},1\right)$. We can solve this explicitly by ascribing a functional form on the matching function, taking it to be of Cobb-Douglas form $R_t=\psi u^{1-\rho}_t v^{\rho}_t$. Given we know $u_t=1$, this means can solve $\psi v^{\rho-1}=\frac{k-b}{\chi(z_t-2b)}$ to get $v_t = \left(\frac{\psi \chi(z_t-2b)}{k-b}\right)^{\frac{1}{1-\rho}}$. We can see that a female user will be more likely to message if the matching efficiency $\psi$ is improved, if she has more bargaining power $\chi$ or if the gains from being in a relationship $z_t$ are greater. And unsurprisingly, her propensity to message is decreasing in the cost of sending a message $k$.

From this, we can find the match finding rate for men of $f_t=\frac{R}{u}=\psi u^{-\rho}_t v^{\rho}_t=\psi \left(\frac{\psi \chi(z_t-2b)}{k-b}\right)^{\frac{\rho}{1-\rho}}$. Again, the comparative statics make sense. And since everyone starts unmatched, we know that the unmatched rate at the end of the period is just $u’_t=1-f_t=1-\psi \left(\frac{\psi \chi(z_t-2b)}{k-b}\right)^{\frac{\rho}{1-\rho}}$. That is, the number of people who remain unmatched is decreasing in the efficiency of matching and in the total gains from a match.

Is this an efficient outcome? Consider a benevolent planner who picks $v_t$ to maximise the gains from a match plus the value of the outside option minus the cost of messaging i.e. $\mathcal{U}=z_t \psi v^\rho_t+(1-\psi v^\rho_t)2b-kv_t$. The FOC is $\frac{d\mathcal{U}}{dv_t}=0$, which is solved as $z_t \psi \rho v^{\rho-1}_t-2\psi b \rho v^{\rho-1}_t=k$. That is, $\psi \rho v_t^{\rho-1} = \frac{k}{z_t-2b}$.

Since we know from before that free entry requires that $q_t=\frac{k-b}{\chi(z_t-2b)}$, the only way the equilibrium is efficient is if $q_t \chi = \psi \rho v_t^{\rho-1}$. And because we know that $q_t$ is defined as $\frac{\psi M}{v_t}=\psi v_t^{\rho-1}$, we can see that the equilibrium is only efficient if $\chi=\rho$. Tthat is, if the bargaining power of women equals the elasticity of the matching function to the number of messages by women. If they have too much bargaining power, they will message too often, creating a congestion externality.

So this has illustrated some useful comparative static results. For example, we can see that an improvement in match efficiency, such as by using an application like Bumble instead of occasionally stumbling into people you might be into in real life, allows for a lower number of people to remain unmatched. A similar effect occurs if you put more information on your profile, since it makes it easier and lowers the cost of sending a message to you. If you liked the set up of having one side posting vacancies and the other side matching, the models described in Diamond (1982), Mortensen and Pissarides (1994) and Pissarides (1985) are the canonical choices for using this approach to explain features of the labour market. Indeed, this DMP model is what won these three the 2010 Nobel Prize in Economics.

You belong with me

Finally, we would like to recognise the heterogeneity in the dating market - that is to say, does the fact that there are different people with different preferences mean that it is never possible to reach an optimal set of pairings? More formally, this is the stable matching problem. Consider a group of men $M$ and women $W$, who all start off initially free. Each person has a preference ranking over all of the opposite gender. A matching function $\mu$ ensures everyone is either matched to themselves i.e. single, or is matched to someone of the opposite gender. A matching is stable if there isn’t any match $(M,W)$ for which both prefer each other to their current partner. The question is, can a group of $n$ men and $n$ women be put into matches which are all stable?

The deferred acceptance algorithm involves each man proposing a match with their first choice. Each woman can hold their preferred proposed match offer while rejecting all other proposers. Then each man who has been rejected proposes to his next highest choice, regardless of whether they have an offer. Again, the women can choose to hold an offer and are able to drop previously held offers. This is repeated until no more proposals are made. What was shown in Gale and Shapley (1962) was that this would always produce a stable matching where everyone is matched, though it would result in the outcome that is best for the men (and trivially a female-proposing world would result in the best outcome for the women).

We can see that it would be stable by considering a man who prefers someone else to their current partner - since he proposed to his highest choice in each turn, he must have proposed to her before. And since he is no longer paired with her, that means he was either dropped or he was rejected. Since the woman’s tentative match can only improve, that means he would still be rejected now. As such, this is a stable match.

Thus we can see that in a world of perfect information and no matching costs, we can get to a Pareto optimal world via this Gale-Shapley algorithm, even if not everyone is completely satisfied. In fact, Hinge uses a variation of this to ensure that people are paired to those who are likely to mutually like each other, and Ariely, Hitsch and Hortaçsu (2010) have found that online dating is able to produce matches which approximate the optimal ones predicted by the algorithm. And if you found this question of market design intriguing, this is the field in which Al Roth applied these algorithms to real world problems of hospital intern allocations and public school admissions, garnering him and Lloyd Shapley the 2012 Nobel Prize in Economics.

All’s well that ends well to end up with you

Is any of this terribly helpful or insightful? Well, I’m posting this on Valentine’s Day instead of doing something with a boyfriend/girlfriend, so perhaps not. And the idea that we can infer something from how much time people spend with us, or that finding a partner is easier with a dating app and a clear profile seems pretty obvious. But if nothing else, this is a nice demonstration of the idea that Nobel-winning economic theories and models can be applied in surprising and unconventional places. Indeed, there’s plenty more than I haven’t touched on which applies in the realm of relationships: all of the work that got Gary Becker his 1992 Nobel. the idea of Nash equilibria as pioneered by the 1994 Nobel Laureate John Nash and the notion of adverse selection that got George Akerlof the prize in 2001.

So to close, I hope everyone has a happy Valentine’s Day! And to the people who read this blog (h/t @unofficialecon):