Risk Aversion and The Grand Gesture

(Alternate Title: Why doesn’t anyone ever show up at my door with roses?)

(Other Alternate Title: Sometimes I can’t sleep. This is the outcome.)

One of the things I have discovered during the early parts of my education in finance is that I am actually learning about human behavior. This was not something I’d ever considered, but it becomes apparent when you realize that all an economist does when they formulate a model is make an attempt to create a mathematical expression that will simulate an interaction that takes place naturally among people. Finance, being in essence a special form of applied economics, often involves making assumptions about human behavior in the face of risk. As I lay somewhere in-between waking and dreaming the other night, it dawned on me that I could easily model a simple form of the choices a man faces when considering asking a woman out, and use this to show, mathematically, why we are so timid.

On the show “How I Met Your Mother”, this scene stands out as an example.

Robin: Remember that guy Ted that I was telling you about? Well, I came home and I found him waiting in my apartment with the string quartet and roses and chocolates and…
The vet: Oh, that’s so sweet.
Robin: Ok, yes, it’s sweet in theory but isn’t it also kind of crazy? I mean a string quartet in my living room, who does that?
The vet: Nobody does that.
Robin: Exactly.
The vet: No honey, nobody does that.

Why? Why are men so scared to go the extra mile and really put themselves out there? We see it in movies and on TV all the time. Some man sings in public (Ten Things I Hate…), he breaks up a wedding (Wayne’s World 2), or he fights for her honor (Back to the Future, and yes, I chose stupid examples on purpose).  The point being, the guy makes a fool of himself, but ends up with the girl, because the latent cultural belief is that the “Grand Gesture” is the pinnacle of romantic overtures. So then, why is it so rare that we ever observe it in the wild?

I think, obviously, that the answer is well known to everyone. Risk. It’s a high risk, high reward situation. There’s a huge payoff to be had, but the cost of failure is punitive.  One my well ask though, isn’t the reward worth the risk? Well, let’s try to find out.

Allow me to introduce you to a little concept called the “utility” function. I use quotes around utility because the words has a specific definition in our context. We will simple refer to “utility” as the value one derives from the consumption of some good. One of the basic rules of a utility function is that it is a strictly increasing function. In other words, more is better than less. Think of money: you’d rather have two dollars as one dollar, and you’d prefer three to two, and you’d rather have one million and one dollars as opposed to just one million. For our case, the “good” we will consider is emotional well being. We will assign emotional well being a numeric value. Let’s assume that this is a linear mapping, so that an emotional rating of 100 is twice as good as 50.  So, we will define our utility function as u(x). You can read that as “you-of-ex”, with x being the e-score we mentioned above. This just says that our utility is a function of our e-score. The higher the e-score, the higher our utility, because more is preferred to less.

We will also make one little extra constraint on the utility function. It will be the property that represents the “risk aversion” of the individual. The way we will describe the situation is this: even though more is better than less, it gets better at a slower rate the more of something you already have. So, to say that a different way, $50 is more useful to someone who has $2,000 than $50 is to someone who has $200,000. We call this “decreasing marginal utility”. Mathematically, we express this as u”(x)<0. This is a statement about the shape of the function, but it’s basically just the “math” way of saying the same thing, i.e. we gain utility at a decreasing rate.

So, let’s assign our utility function an actual mathematical form. The natural logarithmic function suits our purposes. You won’t need to know much about it, but trust me that it matches the assumptions that we set out earlier. So, let u(x)=ln(x). To demonstrate that it is strictly increasing, consider ln(2.0)=.693, ln(3.0)=1.099, and ln(4.0)=1.386. So, it’s obviously increasing, but now look at the change from unit to unit. From 2 to 3, the function’s output rose by .406, but from 3 to 4, it only increased by .287. So, like I said, it’s always increasing, but it does so at a decreasing rate. This will become really important later.

Ok, so, enter now our cast. We shall call our young suitor Westley. The object of his undisclosed affections is the beautiful young Buttercup. As I have just insinuated, Westley has not told Buttercup of his intentions, nor does he know her affections with regard to him. Westley starts this scenario with a baseline of 5000 emotional points. We will restrict his options to a binary choice. That means he only gets to pick from two different actions. Our setting: the annual fall ball – “Mega Autumnal Harvest Freakathon – MCMLXXX!!!”

Choice A: Westley decides that the better part of valor is discretion.  He chooses to make an attempt to single out Buttercup from amidst the crowd and confess his feelings privately in a straightforward undemanding way (perhaps during a dance, using the white noise of the crowd to cover his attempt). This allows Buttercup to let him down gently should it turn out that he is not exactly her choice of masked bandit. This spares him any public humiliation in that case, but it also limits his upside. But, let’s face it, if she says yes, he’s not worried about could-have-been.

Choice A has two possible outcomes, which we will list now.

Outcome 1A: Sweet success!! Buttercup smiles, looks away for just a moment to compose herself (after all, she’s been waiting for this moment for quite some time herself), and whispers, “Of course. What’s taken you so long!”

Westley is, at this point, fairly elated. He revels in his own private triumph, but does wonder how awesome it would have been if everyone would have been watching at the time. That would have put it over the top.

Success bonus: +1000 e-points.

Outcome 2A: Bitter defeat. Double sad face. ( Buttercup, falters for just a moment, but puts on a sweet, warm smile, and declines his proffer. Because he was so nice about it, however, she at least compliments him on his bravery. She’s somewhat amazed he had the guts to ask, since it was obvious to everyone that he had no shot.

Westley, dear, sweet Westley, is crestfallen. All his planning, all the months of pining away, have come to naught. But, on the bright side, she didn’t humiliate him, and other than a few close, chosen confidants, no one knows about what just happened, so his street cred has suffered minimal damage. He can hold a stiff upper lip, and walk out with dignity.

Failure penalty: -1000 e-points.

Now, we will proceed to his second option.

Choice B: Westley, being but a young, common day laborer, has aspirations. As a boy, he sat on his father’s knee and listened to the stories of days gone by. Days when the public outcry against the tyranny of illegitimate rulers would no longer be restrained, and the concentrated passions of good men wrought a new nation built upon individual liberty and respect for basic human rights bestowed upon us by an almighty righteous Creator. Days when men battled untold nightmares across the oceans. Days when men shaved their faces with nothing but a wedge of raw hand sharpened steel, iron willed self confidence, and the bowel churning fear of being the only guy who ever actually did cut his own throat while trimming his mustache.

Our protagonist has decided that he will be one of those men. Damn the consequences, the world will know how he feels. He will look back on this day as his proof of life, evidence of a spine, and “the day I grew a pair.” He will lie on his death bed with the bosom warming knowledge that he was not a coward in this all too short life. No sir (or ma’am, I suppose), Westley is going to step up to the plate and grab hold of this chance with both hands, and tightly. He may well strike out, but it will not be quietly to a slow curve over the outside corner. No, it will be on three Herculean swings that appear to be an attempt to screw himself into the ground feet-first.

Let there be lights. Westley jumps on stage. He sings an Elton John song. He does an interpretive dance. He tosses toffee and chocolate like it’s the freaking Macy’s Thanksgiving Day Parade. Men in penguin suits filter in through the back with row upon row of roses in a variety of colors.  He jumps off the stage, and in front of his beloved’s rather shocked face, he states clearly into the microphone, “Buttercup, I love you. I have always loved you. I will always love you. Would you like to go get some ice cream or something?” (Ed note: We never said he was smart.)

Choice B also has two possible outcomes.

Outcome 1B: Perfect Victory!!!11One!! A slight mistiness creeps into the eyes of many. An older gentleman comments “I say, it seems to have gotten a bit dusty in here,” as he steps out so that no one will see him cry. Buttercup beams like the glory of the rising sun. She throws herself at young Westley in what appears to be an attempt to swallow his face whole. Women want him. Men want to be him. The florist refuses to accept his money, on account of this being the most beautiful thing she has ever seen. Buttercup’s father asks Westley to please marry his daughter so as to strengthen his own progeny’s genetic heritage. He even offers to pay for the honeymoon.

Success bonus: +500,000 e-points

“K-K-K-K-COMBO!!!”:  +100,000 e-points. All who witnessed will tell their grandchildren. The story persists through the millennia.

Made for TV bonus: +100,000 e-points. Funding to put their seven children (hey, they really like each other) through college is covered by the Hollywood exec who decides this is going to be next summer’s blockbuster.

Manliness bonus: +100,000 e-points. Overheard on the way out, “Dude. That guy has balls that clang.”

Twoo Wuv bonus: +100,000 e-points. Turns out, she wasn’t just caught up in the moment. They celebrate their 75th wedding anniversary next Tuesday.

Unquestionable Decision Making bonus: +95,000 e-points. Westley now has the ultimate card to play whenever anyone else has relationship problems. He and  Buttercup become “that couple” that always go on and on about how they got together at every opportunity. Everyone hates it, but they don’t care. Don’t question success.

Total: +995,000 e-points.

Outcome 2B: Epic Fail.  The crowd stands in stunned silence. Buttercup, horrified, shrieks, “Oh my God! What is *wrong* with you! You freak!” She then runs out of the room sobbing in humiliation. Westley, for his part, hasn’t moved. He does appear to have gone a slight shade of green. Internally, the reason for his immobility is not resolve. No, instead, it is the all too common reaction to fear. Simply, he’s doing his best to resist the sensation of impending explosive incontinence. A young child, seeing the tortured disfigurement in Westley’s face turns to his mother, who simply whispers, “Dear God, what is that thing?” Westley manages to creep out the back, attempting to ignore the growing laughter in the auditorium.

Failure penalty: -1,000 e-points.

“Finish Him!”: -1,000 e-points. As a follow up, Westley is served papers later that week. Buttercup is suing for emotional trauma. There’s also a restraining order included, just for kicks. Good times!

Humiliation penalty: -1,000 e-points. Overheard from another young lady on the way out: “Can you imagine?  I would have just thrown up on him then and there. I mean, what was he thinking? Everyone knows he had no shot with her. And ice cream… what was that? What a creep.”

Short changed penalty: -1,000 e-points. Umm… yeah. He only made it to the car before his “discomforts” overtook him. Now he needs to buy a new driver’s seat, and have the car fumigated. We won’t even mention the cost of replacing the tux he rented.

“Pay up sucker!”: -999 e-points. Do you have any idea how much those freaking roses cost?!

Total: -4,999 e-points.

Math note: I didn’t take the full 5,000 because I assume that if his e-index falls to zero, Westley commits suicide. I didn’t want to go there. It just so happens that ln(0) is negative infinity, but ln(1)=0. I think the intuition of that works well.

Ok, now let’s look at this mathematically. Westley has to make his decision now, before he knows how Buttercup will react. Because of this, we say that Westley is about to maximize his *expected* utility. If he knew what was going to happen, he’d just maximize his utility. The difference is, now Westley has to face some risk.

Let’s assume that Westley feels that either reaction is equally likely. She might love him, she might not. It’s 50/50.

So, what is Westley’s expectation about his potential utility with regards to each of his two options?

Choice A: 50% chance of receiving 1000 e-points, and a 50% chance of losing 1000 e-points. So, he expects to basically end up with a zero net change in his emotional well being.

Choice B: 50% chance of receiving 995k e-points, and a 50% chance of losing 4999 e-points. So, he is actually looking like he’d be much better off taking the risk here, right?

Well, now we have to go back to the constraints we set forth earlier. Remember, Westley operates out of his utility function, and it factors in his risk aversion. That’s why we say he is maximizing his expected utility, and not just his expected emotional point standing. Why? Because, like I said earlier: just like money, the more you have of something, the less it is worth incrementally.

So, for Choice A, Westley’s expected utility is: .5*ln(6000)+.5*ln(4000)

All we did is plug in his post-decision emotional well being into his utility function, and multiply them by their respective probability.

For Choice A, this expected utility is 8.497.

Doing the same for Choice B, his expected utility with that decision is:

.5*ln(1000000)+.5*ln(1)= 6.908

Whoa. Isn’t that weird? Even though Westley would expect to have a much better average outcome if he goes with Choice B, he won’t choose it. He will wuss out and go with Choice A. Why? Risk. Implicit in the “risk averse” concept is that people put more weight on the downside than they do the upside.

An example: Someone walks up to you, and offers you this gamble. They will flip a coin. If it’s heads, you win $50,000. If it’s tails, you lose $50,000. You have to pay it. There’s no way to weasel out. The coin is fair, verifiably. The money is real, and it’s right there in front of you. Do you take the gamble? Most people won’t, because they are risk averse. The downside, though technically “equal” in magnitude, outweighs the upside.  As an aside, there also exists the concept of “risk neutral” and “risk loving” as human behaviors. Risk loving is pretty rare, and only exists in a few situations.

So, what could we do to change this outcome? Well, what if Westley had a better idea that Buttercup might be interested? What if he was 65% sure that she liked him?

Choice A: .65*ln(6000)+.35*ln(4000)=8.558

Choice B: .65*ln(1000000)+.35*ln(1)=8.980

Hey! Now it looks like the guy will take the risk, which is what all of us romantic softies really wanted in the first place. So, this explains a little something else to us: we *do* see some big gestures, but they only really occur after a relationship has already started. A guy might get down on one knee in front of a thousand people, but he generally already knows what’s about to happen. It’s probably just as much (or more) a function of his expectation of her reaction than it is the actual value to him of that reaction.

I expect that no one reading this will really consider this revolutionary information. Most people favor the overly romantic “Grand Gesture” as the theoretical peak, but so few attain it because the cost of failing is just too much to overcome. Perhaps men are cowards, but it’s not because they prefer it. They just can’t afford not to be extremely caution. Why, consider dear Westley. Suppose after the failure, he decides to move on with his life and pursue some other young lady. If said young lady had witnessed the cataclysmic wreckage of his attempt with Buttercup, do you think she’s going to go out with the guy? I think not. So, by virtue of making the supreme effort, Westley sacrifices his other alternatives. Perhaps I should have listed that in the costs above.

Speaking of extending this analysis, some of you (being clever in your own right) may be asking the question, “Well, why does the guy *have* to ask her out?” Good question. I considered putting that in, as a third option, but I felt that it would best describe the situation to show a man who has made up his mind to go forward. If you prefer a more mathematical viewpoint, we could say that Choice C is take no action at all. But, alas, because of our protagonist’s sense of honor, he would suffer a cowardice penalty. I would say, let’s dock him 4001 e-points, because he regrets it for the rest of his life, and he is never really able to get over Buttercup. Without “knowing” one way or the other, his soul is tortured. He wilted in the face of the trial, and now he must live with the knowledge that he never pursued his one true love.

ln(999)=6.907

Thus, his utility from doing nothing is a worse option than Choice B. Please note that I had to chose 999 e-points as his final total, since ln(1000) is exactly equal to .5*ln(1000000). It’s just one of the interesting properties of the natural log function.

Fortunately for our two love starved heroes, their story is somewhat well known. He distinctly said, “to blave”, and as we all know, to blave means “to bluff”, and he died after being tortured in the Pit of Despair. It’s sad, but true.

Now that I have hand forged, lo, these 3000+ words; I hope that you are appreciative. Knowledge is power. Or pow’r, if you were the writer of many nineteenth century hymns.