S/he wants the Margaritaville, so what’s a girl to do?

So, your significant other has been lusting after a flat screen TV, espresso machine, or other prohibitively expensive luxury good for the past umpteen weeks, and you just saw an amazing deal for one on Amazon. Regardless of whether he actually buys the gadget, it would be nice to let him know you’re thinking of him. Do you say anything?

The best case scenario might be that he sees the ad, you get brownie points for having thought of him, and in the end he decides not to buy the thing. After all, you would rather he save the money for a rainy day, a nice vacation, or that expensive gift you’ve been eyeing… The worst case is, you don’t show him the ad and he goes and buys the thing anyway. If you show him the ad, at least you can put an upper bound on how much he spends (though, if the thing is a game console or smartphone, your payoff might still take a hit in terms of time away from you.)

If this sounds very familiar, this is an extensive form game you’ve played before. It would be easy if you knew exactly what he would do in each circumstance, and you could simply use backward induction to determine your best strategy.

But you don’t perfectly know what his payoffs are going to be. The good news is, you can still make an optimal strategic decision! You probably know, for instance, the approximate probability that if you do nothing, he’ll buy the thing anyway. We’ll call this p, and let’s assume it’s smallish because it’s expensive. Let p = 0.15. So, your expectation is -1.5.

If you show him the ad, what would the probability of buying the Margaritaville become?

-1.5 = -4q + 1(1-q) = -5q + 1
q = 0.5

His likelihood of buying has to become at least 50% before showing him the ad becomes a bad idea! If you don’t think he’ll become so much more likely to buy, you might be better off just showing him the ad.

What if p = 0.05, and q = 0.30? Your expected utility would be the same whether or not you show him the ad. Of course, if you are risk neutral like many of these games assume you are, just flip a coin. It doesn’t matter. If you’re a bit risk averse like most people are, you’ll find that the variance of the upper branch (~4.5) is higher than that of the lower branch (3.85), so it’s a mean preserving spread and you still prefer telling.

If you’re anything like me (not a given, because I’m kinda weird) you’ve found yourself in a similar situation at least once, and you’ve done a little bit of backwards induction to figure out what to do. Game theorists are most often criticized for assuming players are “strategic and perfectly rational” — the latest by Cosma Shalizi, according to Jordan Ellenberg at Quomodocumque:

What game theorists somewhat disturbingly call rationality is assumed throughout—in other words, game players are assumed to be hedonistic yet infinitely calculating sociopaths endowed with supernatural computing abilities.

Although you probably don’t draw a tree or calculate variances, a similar process of estimation happens in your head and you do indeed behave strategically, so it’s rational choice at work. And, as for sociopathy, economists are at worst amoral or mildly paternalistic (trust me, it’s for your own good 😉 ). It’s far from manipulating people for fun — cheap shot, Cosma.

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