Three of the five fastest runners were wearing our shoes: using the rules of speech to squeeze more information out of fewer words

Many years ago I saw an ad for a running shoe (maybe it was Reebok?) that said something like “At the New York Marathon, three of the five fastest runners were wearing our shoes.” I’m sure I’m not the first or last person to have realized that there’s more information there than it seems at first. For one thing, you can be sure that one of those three runners finished fifth: otherwise the ad would have said “three of the four fastest.” Also, it seems almost certain that the two fastest runners were not wearing the shoes, and indeed it probably wasn’t 1-3 or 2-3 either: “The two fastest” and “two of the three fastest” both seem better than “three of the top five.” The principle here is that if you’re trying to make the result sound as impressive as possible, an unintended consequence is that you’re revealing the upper limit.

The quote above, which came from Phil Price, a guest poster on Andrew Gelman’s must-read blog for statistics wonks, is an application of the pragmatics of speech.  You can read more about pragmatics in How to be a Good Professor–or Professional, but the basic idea is a simple one: we expect speakers to follow rules, and then we use those rules to interpret what was said.  The rules are specific to a particular conversation, so they are very local–I call them a LAAP, with the L for Local, rather than the Generally Accepted rules of GAAP.

Some conversations are cooperative–everyone is trying to reach the same goal.  So we can assume that if someone says something, they are being relevant, faithful, intelligible, efficient, diplomatic, and so on.  These rules, together with the assumption of cooperation, make conversations far more informative than they would otherwise be, because we can draw a lot of inferences from what people do and don’t say.  For example, if a friend says something offensive, we can infer that it must be especially relevant (“you should brush your teeth”).  If we ask them “where’s the closest grocery store” and they say “at the corner of 5th and Main”, we can also infer that the store is open for business during the usual hours, that it isn’t restricted only to military personnel, or that it’s closest in distance, but you have to drive 5 miles to cross a bridge to get there.

All of these inferences are called implicature, and we use it all of the time without realizing it.  If you ask someone if they are going to a big party tonight, and they answer “I have a test tomorrow”, you can assume they are also saying the words “no” and “because”.  If you are driving with someone else navigating (whether a person or an app), you can interpret silence as a clear statement that you don’t need to make a turn soon.

The example at the top of the post is somewhat different, because an advertisement is a rather non-cooperative conversation–someone is trying to sell us shoes.  This still allows some implicature, but of a different sort.  There are still rules, like “don’t lie”, but we assume that speakers step as close to the line as possible.  So “3 of the fastest 5 wore our shoes” is true, but we can also infer that “3 of the fastest 4” is false, or they would have said it, since it makes them look better.

In non-cooperative settings, silence is often interpreted as bad news.  Imagine you’re talking with someone who’s touting their shoes, or their stock, or their candidacy for a job.  You ask them about something that’s problematic, and the deflect the question to talk about something else.  If it were a cooperative conversation, you’d be reasonable to assume that the answer to your question isn’t that relevant.  But in a non-cooperative setting, you’re probably better off assuming an answer would make them look bad.

Very bad, in fact, because the situation unravels like a lemons problem. Imagine you can rank the quality of whatever’s in question on a scale of 0 to 100.  If you take silence to mean “30 or lower”, and you both know it, they will tell you the truth if it’s higher than 15.  But you’ll know that too, so you really need to think silence means “15 or lower”, and so on.  This only works if you’re sure they know the answer, and they don’t have other reasons to tell you–maybe it will impose some totally unrelated costs on them, like they’re so good they don’t want competitors to know.

Financial reporting is often like advertising:  firms will say the most beneficial thing they can, often quite non-cooperatively.  Sometimes that means pushing up to the limits of GAAP through measure management.  Sometimes it means making good news really easy to find, writing it clearly right up front in a press release or annual report, but burying bad news by writing it with big words and long sentences, in the middle of tedious boilerplate text everyone ignores.

Pragmatic analysis shows that regulations mandating disclosure often backfire–if firms are required to disclose everything, you can’t infer that they say anything because its relevant–they may be just following orders.  I have a paper on that topic, which recommends ‘elevated disclosure‘–if you are going to make firms disclose everything, also force them to elevate some topics to highlight their relevance.  Versions of this are done in other contexts, like credit card disclosures:  there’s a mass of mandated text, but certain facts (like interest rates and penalties) must be very easy to find.