## Monday, August 15, 2005

### Fat Chance

A woman asks her mate, “Do I look fat?” What sort of answer is she hoping for? Co-blogger Tom’s wife d broaches this vexing issue in her most recent post (while revealing an aspect of Tom’s character of which I was previously unaware). She says the question is not about truth-seeking: “everyone knows that [truth] is not what one is looking for when they posit such questions.” Problem is, lots of guys don’t know this at all. Says commenter Gil:
“Do you want a convincing lie? Or a convincing expression of what you'd like to believe, whether it's true or not (and hope that we don't even consider whether or not it's true so we don't have to knowingly lie)? … What should a man who hasn't cultivated the skill of lying convincingly to his wife do when asked this type of question? If he answers only when the truth is pleasant, then avoiding answering will be taken as evidence that the truth is unpleasant. Would a consistent “No Comment” policy with respect to that type of question be acceptable?
Excellent questions all. Probability theory to the rescue! Suppose a woman thinks there’s an A chance she looks fat and a (1 - A) chance she doesn’t. If she poses the fateful query and she doesn’t actually look fat, then her mate will naturally say “no” (why not?). But if she does look fat, he faces a choice. Let B be the probability of his saying “no” (a lie) and (1 - B) the probability of his saying “yes” (the truth). We can construct the following probability table (where yes/no without quotation marks is the truth, and “yes”/“no” with quotation marks is what he said):After asking the question and getting an answer, the woman can update her beliefs. Upon hearing a “yes,” she can be sure she’s fat, because her mate would never have said so otherwise. But if she hears a “no,” there are two possibilities: either she’s not fat, or she’s fat but he lied. So what’s the probability that she’s actually not fat, given that her mate said she’s not? Using Bayes’ Rule, we can calculate as follows:
Pr(no | “no”) = (1 - A) / [(1 - A) + AB]
This is her updated probability that she doesn’t look fat. How does it compare to her prior assessment, (1 - A)? It turns out that it’s always larger (prove this algebraically by showing that [(1 - A) + AB] < 1), meaning her mate’s “no” makes her more confident that she’s not fat. But there’s one crucial exception! If B = 1, meaning her mate will never say she looks fat, then her updated probability reduces to (1 - A), the same as if she hadn’t asked. In other words, if her mate will never say she looks fat, then his answer provides no information whatsoever (about how she looks, that is). The question’s image-improving potential is dependent on the mate’s propensity to tell the truth. (Says Carina in d’s comments: “I'm more likely to appreciate compliments if I can expect the occasional honest criticism.”)

The above might seem a good argument for asking the question, if the woman’s mate does have some propensity for truth-telling (B < 1). But that’s not actually so. While a “no” will indeed increase her confidence ex post, from an ex ante perspective the anticipated boost in confidence is exactly offset by the less-than-100% probability she’ll actually get that “no.” Asking the question is a kind of gamble: a “no” will make her feel better, but a “yes” will confirm her fears. It turns out, under plausible assumptions, that on average these effects will cancel out regardless of her mate’s truthfulness.

So why do women ask this question? Two reasons. First, sometimes they actually do want the information. This seems especially likely when the question relates to a particular garment; the woman who asks “do these jeans make me look fat?” might in fact be seeking an assessment of the jeans, not herself. Second, they are seeking affirmation; the question is code for “do you still love me?” or “give me a compliment.” Now, both of these are acceptable goals. The problem is they’ve gotten lumped into a single question, which means the man has to adopt a single policy for both. As a result, each goal contaminates the other. The truth-seeking woman gets information tainted by her mate’s desire to please her, while the affirmation-seeking woman gets compliments weakened by his hesitation.

Steven Horwitz said...

And this, my friends, explains Glen's current marital status. ;)

W. said...

Great response, Steve. I gave a "huh?" to it at first. Apparently, my economics is more theory and common sense. Will have to work on that "higher-level math."

Glen Whitman said...

Steve -- admit it, you know what I'm saying is right! Are you saying my willingness to speak the mathematical truth dooms me to bachelorhood?

Tom W. Bell said...

You relate, Glen, that d's story revealed "an aspect of Tom’s character of which I was previously unaware." Am I to suppose that you refer to the character aspect of honesty? Or was it perhaps idiocy?

More charitably, I'll suppose you meant to say, "I hadn't realized before how much Tom knew about romance management!" Think about the effect my honest answer on d's subsequent queries in the "how do I look?" line: She stopped making them unless she really wanted and was prepared for an honest answer. I thus saved us both a lot of trouble in subsequent years.

What you have no doubt found true of teaching holds true in love, too: You can always start tough and ease up later. You cannot, however, effectively do the reverse.

Glen Whitman said...

Tom -- the character trait in question is "honesty under circumstances where honesty is typically not expected and often actively discouraged." In other words, an aversion to white lies. It's an aversion I share, by the way.