Monday, August 11, 2014

Swiper No Swiping or: How I Learned to Stop Worrying and Love the Game Theoretic Analysis of Hookup Culture

"It is a truth universally acknowledged, that a single man in possession of a good fortune must be in want of a wife."

-J. Austen, Pride and Prejudice

I gather from the kids these days that mobile phone applications have replaced seedy dive bars as the go-to venue for taking a spin on the ol' genital mashing lottery¹. "Swipe right" is the "hey baby" of 2014. To credit technology for the apparent rise of casual sexual encounters is perhaps a bit premature. To illustrate why, let's swipe right on some elementary game theory.

Consider a single player. Call her Ann. Ann finds herself at the start of her sexual career and is faced with two options:
a) Participate in the casual sex market
b) Participate in the marriage market
Since this is ~elementary~ game theory, omit from consideration opting out altogether, or exotic choices like... well, use your imagination.

Before that choice, Ann was secretly assigned by Nature to one of two types.
Type L (for licentious). There is no penalty for choosing (a) after type is revealed. There is a penalty for choosing (b) after type is revealed.
Type D (for demure). There is a penalty for choosing (a) after type is revealed. There is no penalty for choosing (b) after type is revealed.

My conscience urges me to point out that what I'm about to do is methodologically suspect. I am going to assign numbers to utility. Doing this covertly drafts several stealth assumptions into this model, some of which I find terribly implausible. I excuse myself by noting that I'm doing a comparative statics problem here, rather than utility comparison. On the margin, you can (probably) toss out assumptions of well-ordered convex utility functions and still get similar results. Whether or not the model holds up when tested at extreme parameter values is questionable. So as far as a bit of insight into why some people might marginally choose regrettable mating decisions, this might be useful, if not entirely accurate.

My conscience temporarily sated, let's define a few structural parameters.

First, we need Ann's type assignment probability. Remember that Ann doesn't know what her type is as she makes her choice over (a) or (b). Only afterwards will she find out. Let's call this probability α. With Pr = α, Ann is Type L, and with Pr = (1-α), Ann is Type D.

Next, we'll want a parameter for a successful casual sex match. With Pr = β, Ann finds a partner for the evening. With Pr = (1-β), she spends the night alone.

Next, let's define γ as Ann's discount rate. For arithmetic purposes, this is just a multiplier that lets us compare payoffs earned in the future to payoffs earned now. For social science purposes, this is a contentious mess. Empirically decomposing the inputs to individual discount rates is a hassle, and even when aggregating, it's hard to measure, and harder yet to isolate specific treatment effects. Over a person's lifespan the discount rate can change wildly, some of it because of normal lifecycle reasons, some of it for cohort reasons, some cultural, some from exogenous shock, some for psychological reasons, some from conformity, some idiosyncratic. Beware stories that attempt to reduce discount rates to a monocause. I include this here for discussion, but I'll omit it from the model since I don't want to make things too intractable. Not yet anyway. Maybe in a follow-up post.

Finally, we'll need a parameter for the probability that Ann will participate successfully in the marriage market. Let's use ω. With Pr = ω, Ann marries. With Pr = (1-ω), Ann ends up with cats. Lots of cats.

At the risk of drifting into differential equations territory, β and ω are both partially functions of [potential partners'] α parameters. That is to say, the more people in the {temporary mating|marriage} market, the higher the probability of finding a match in that market. For the purposes of this exercise, assume that participants know which market they are in and that no participants attempt to participate in both markets at the same time. Modeling infidelity is another exercise entirely.

Let's say that a successful booty call yields and a successful marriage yields w units of pleasure. Failure of either one yields 0.

If Ann is Type L, she'll pay a penalty of -c for successfully participating in the marriage market, no penalties otherwise.
If Ann is Type D, she'll pay a penalty of -d for successfully participating in the casual sex market, no penalties otherwise.

With Pr = α, Ann earns a payoff of:
β*v + (1-β)*(0) for playing (a)
ω*w + (1-ω)*(0) + ω*(-c) for playing (b)
With Pr = (1-α), Ann earns a payoff of:
β*v + (1-β)*(0) + β*(-d) for playing (a)
ω*w + (1-ω)*(0) for playing (b)

Ann is therefore indifferent between playing (a) and (b) when
α(β*v + (1-β)*(0)) + (1-α)(β*v + (1-β)*(0) + β*(-d)) = α(ω*w + (1-ω)*(0) + ω*(-c)) + (1-α)(ω*w + (1-ω)*(0))

βv - βd + αβd = ωw - αωc

If βv - βd + αβd > ωw - αωc, Ann will play (a). Casual sex.
If βv - βd + αβd < ωw - αωc, Ann will play (b). Marriage market.

Casual sex is increasing in the probability of a successful right-swipe and costs of regrettable marriage; decreasing in social stigma against licentiousness and the probability of a happy marriage. Reverse that for marriage market participation.

And this gets interesting when we consider that the probability parameters are something of a commons. The more people that participate in the casual sex market, the greater the probability that Ann will find a match for any given attempt. If, ex post, Ann is a Type D attempting an (a) strategy, she would prefer to have had incentives encouraging her to pursue a (b) strategy instead. Similarly, if Ann is a Type L attempting a (b) strategy, she'd regret settling down and probably be a bit disgruntled that her society had provided her incentives to marry that schlub Jasper (no relation to the Peanut Butter Kid).

Think of "traditional" marriage and everything that goes along with it as a set of institutions that attempt to avert the regrets of licentiousness. The tradeoffs are the foregone opportunities of short-term associations. Knocking down that Chestertonian fence can reveal that if Ann is a Type D living the life of a Type L, the costs could be quite real and quite salient.

There's an interesting tragedy in there. If identifying type ex ante is either very costly or downright impossible, and if Ann's incentive structure is the product of human action, but not human design, a semi-stable equilibrium emerging from the joint and several negotiations between human minds, what is her optimal strategy? Which parameters should she try to strengthen? Which should she try to weaken? She can't choose her culture. Can she choose her discount rate? Can she select her cultural filters? By choosing the non-dominant strategy, does she run afoul of a concentrated costs-diffuse benefits dilemma? How can she overcome this problem? What would you do?

¹ Apologies for the horrific idiom. Blame it on my low blood sugar.

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Do you have suggestions on where we could find more examples of this phenomenon?