Choosing a Partner according to Decision Theory
A friend recently was telling me about a situation he had been in: choosing between two girls who were both keen to be his girlfriend. Revising for some upcoming exams (in the Theory of Individual and Strategic Decisions) gave me the boredom needed to think about how I could apply the theory of decisions to shed some light on this problem.
Disclaimer: To follow the situation in which I first considered the problem for my friend, I talk about two girls, feel free to insert guys / other as appropriate! The theory works just the same :)
[Ed: I became (but am no longer) girl C. ;) ]
Firstly, do we know our preferences?
Being able to make a choice first requires that we have preferences over our choice set, namely the would-be girlfriends. People have multiple attributes and the choices may be most preferred on one attribute so this is tough but persevere — we’re only at the first step.
If we are truly indifferent what should we do?
There’s just not a whisker to tell them apart — girl A is fantastically funny and has a great belly laugh; girl B has stunning intellect and challenges the way you think about even the tiniest things. Well, according to Prospect Theory, losses loom larger than gains — in other words rejecting girl A will dampen our spirits much more than saying ‘let’s do it!’ to girl B will gladden them, even though we were initially indifferent. Most importantly, we’d feel the same gnawing loss if we chose girl A and rejected girl B.
Solution Concept 1: Minimise Regret
The regret is likely to be worse if we get feedback on our decision. Seeing girl A getting cosy with someone else is likely enhance the feeling of regret — we could have been that guy! Girl A being in our close circle of friends, where gossip gets back to us, rather than someone at the periphery of our network increases the likelihood of this feedback. As does Facebook — does girl A post lots of pictures of what she’s doing and who with? Great — more regret. One solution concept for generating an optimal decision is minimising regret. The best girl to reject could indeed be the one that we won’t constantly be fondly reminded of.
Solution Concept 2: Subjective Expected Utility
Although generally not acceptable to talk about the utility of a girlfriend, the SEU is useful in helping us look into the future and thinking about how well we think it’ll work out with each girl. According to the matching hypothesis, partners well matched on physical attractiveness report being happier and deeper in love. (Of course, for the Hugh Hefners of the world, wealth can compensate for the odd stray hair and wrinkle). Are girls A or B substantially more or less attractive than ourselves? Are we well matched on other characteristics such as class background and level of education? Sad, but it makes a difference to our calculation of the subjective probability of it working out long term.
Solution Concept 3: Maxmin Expected Utility
My favourite way to think about this problem is to choose the best worst-case scenario. Think about when you’ve seen girl A and girl B really really grumpy — and go with the one who is the least worst!
Introducing the Preferences of the Girls — Solution Concept 4: Nash Equilibria
Of course, it would be nice if we could just decide who to go out with and they would just agree and we could amble off into the sunset. Unfortunately, we also have to consider what girl A and girl B want. They may have other choices, or we may be their least preferred choice with being alone ranking above us! :( Interestingly, whether we get to go out with girl B(whom we discover we preferred all along) may depend on individuals outside of us, girl A and girl B. If boy X chooses to go out with girl D instead of girl C, this frees up girl C to date girl B (to break the pretty sexist and heteronormative conventions of standard decision theory problems) meaning that girl B is no longer within our choice set! That the decision has been taken out of our hands may or may not be a relief but the lesson stands that it’s worth looking around your wider network to see who you can match-make. That way you’re more likely to push the Nash Equilibria in your favour. (If you need any tips on mutiple couple matching-making see pretty much any Shakespeare comedy).
Widening the choice set — Solution Concept 5: Optimal Stopping Theory
So far we have only considered two choices: girl A and girl B. Broadening our choice set completely changes the rules of the game. After all, girl A and girl B are not the only girls in the world with whom we could get along alright. So we have the question of whether we settle (irrespective of whether we choose girl A or girl B) or whether we continue to search? The catch is that we don’t know the subjective expected utility of the girls we are yet to meet. If we reject girl A and girl B, we may meet a girl who vastly surpasses them in all attributes — intelligent, attractive, great personality — take your pick of your dream girl. However, on the other hand, we may consider everyone we meet after rejecting girls A and B to be utterly disappointing.
The trouble is after we’ve rejected them, neither A nor B are going to be very amenable to getting back together. Mathematicians have helpfully given us an equation to help us estimate at what point in our search for a partner we should stop searching and pick the first partner who beats all the other potentials we met during our search period — apparently 37% of the way through our dating lives (i.e. if you start dating at the age of 18 and want to be in a committed relationship thinking about kids by the age of 32, find 37% of that interval). Applying this carries two risks: i) that we may never met a partner who beats all the potentials we came across in the search period; ii) that all the potentials we met in the search period were so rubbish that accepting the next best that comes along means we settle for someone rubbish. Nevertheless, it is apparently the optimal way to approach a search. (See Hannah Fry’s TED talk for an entertaining walk through the formula.)
[An aside — scarily for me 37% through is bang on my current age!]
Although I said at the beginning that this story would apply equally well to girls as boys, it is worth noting that optimal stopping strategy is likely to vary systematically between men and women if one considers kids to be on the horizon and the differential time constraints on reproduction. Although my biological clock is not yet ticking; I do slightly resent that time pressure enters the equation for us girls. It is strange to consider that new reproductive technologies such as IVF which push back starting a family could influence which partner we end up with by elongating a woman’s search period. (Just a thought experiment — I’m not aware of any research in this area.)
How wide can we widen the choice set? Applying Drake’s Equation to Dating
An economist at Warwick, trying to explain why he didn’t have a girlfriend, applied Drake’s equation to his love life — an equation to estimate the number of highly evolved civilisations that might exist in our galaxy. He concluded — on the basis of fairly broad suitability criteria such as age range, location range, intelligence, probability of meeting etc — that there were 26 potential girlfriends out there and he had a 1 in 285,000 chance of meeting one of them on an average night out in London.
Although this sounds rather depressing, 26 potential incredible girlfriends isn’t bad! And we were having trouble choosing between 2! [You see how I switched the framing.. ;) ] Moreover, our likelihood of bumping into them is much much more likely than the 1 in 285,000 chance quoted. Why? Because of networks. As with Granovetter’s famous demonstration of the importance of friends of friends helping you with the job search, there really is a strength in weak ties. Most people find their partners through friends. (Whether this is their optimal partner is, of course, another question.) Our friend network is much more important to our dating life than one would think!
Intertemporal Preferences
So we can widen the choice set — we can continue searching — albeit at the risk of loosing both girl A and girl B but saving ourselves the hassle of regretting choosing one and not the other. So we now have three options: girl A, girl B or a set of choices induced by ‘continue searching’. I now introduce girl C, an element of the set ‘continue searching’. To make it simple, we assume (unlike in the above paragraph) that we will meet her for certain at a specified point in the future and we assume that she’s brilliant, absolutely all you could ever ask for in a girlfriend. The question is how brilliant would she have to be to be preferred to girls A and B? Research shows that we heavily discount payoffs in the short-time. So, it is not good enough for girl C to be marginally better than girls A and B — she must be much much better — whether we meet her in a year or 5 years’ time. It is tempting to say, well, we have girl A or girl B now — we want someone to call now, someone to share breakfast-in-bed with at the weekend, someone to take as a ‘plus one’ next week’s dreaded wedding... Contrary to the advice of The Supremes, my anecdotal experience is that people try to hurry love all the time.
All this is getting very complicated…
Yes, and somehow people manage most of the time! A revolution in decision science occurred when psychologists such as Kahneman (of ‘Thinking Fast; Thinking Slow’ fame) and Tversky (among others) started looking at heuristics — rules of thumb we use to make decisions. And found them everywhere! And the expected utility calculations? Practically nowhere. I still think that classical decision theory (all the mathematical models) are useful for finding the optimal choice but descriptively they are terrible.
Although we’re interested in an optimal solution here, a quick rule of thumb (since your ‘rational thinking’ is going to be morphed by emotions and intuitions anyway): do you love her?
Great, get on with it then. :)
