I eat lunch with colleagues occasionally. Over the years I’ve learned that a 12:30 lunch, in fact, means 12:40. And 12:40 means 12:50. And so on. Time and again, otherwise smart and responsible people say one thing and do another. Why is this?

The answer is surprisingly simple: I hate waiting, but don’t hate *you* waiting.

Here’s the logic: You don’t know when I’ll show up, and waiting is costly. If you’re early you’ll have to wait. If you’re late you won’t. So you come late. And I do the same thing. Presto, we’re both late.

Seems obvious enough. But when I make this argument to non-economists they’re unpersuaded, preferring an explanation for lateness based on lazyness and stupidity rather than as the equilibrium of a social coordination game. So for their benefit, I’ll work it through with some simple game theory.

**The Lunch Model**

Let’s meet for lunch at 1:00pm. You try to guess my arrival time. Let’s say you think it’s normally distributed with a mean of 1:00pm and a standard deviation of, say, 10 minutes. That variation is plausible since random delays (think late buses) happen to the best of us.

Let’s assume “on time” means showing up within a standard deviation of 1:00pm. Also assume “early” and “late” mean showing up more than a standard deviation from 1:00pm — before 12:50pm and after 1:10pm respectively. Here’s a picture of this distribution of my arrival times:

Since my times are distributed normally we know the chances of me being on time are P(on time) = 0.68. Also we know the chances of being early and late are P(early) = P(late) = 0.16.

Assume people’s time is valuable, so whoever is early bears a time cost. We both have three options — show up early, late or on time — so let’s draw a 3×3 payoff matrix of the lunch game with waiting costs as the payoffs. Here it is:

Now, calculate the expected value of each option and pick the best. Do this by multiplying each payoff by its probability and summing for each option. Here’s the math:

E(early) = (-5)*(0.68) + (-10)*0.16) = -5

E(on time) = (-5)*(0.16) = -0.8

E(late) = 0

So your best — that is, least negative — option is to show up late. The math’s the same for me. So I show up late too. And our 1:00pm lunch becomes a 1:10 lunch. Q.E.D.

I leave it as an exercise to show we’ll both complain about each others’ lateness afterward.

**Update:** Several astute readers pointed out a flaw in the above “lunch model”. I’m always surprised by how seriously people take these tongue-in-cheek models. Lighten up people, this isn’t the American Economic Review!

In any case, for the econ nerds, here’s the problem and fix:

The distributions above represent my expectation of when you’ll arrive. In this stripped-down model where people naively assume other people will show up around noon, there’s technically no equilibrium.

Here’s why. If I assume you’ll show up at noon, I should show up at 12:10. But I know you’re rational and will make the same calculation and will show up at 12:10 also. So the distribution shifts right, and I repeat the calculation and decide to show up at 12:20. And you do the same thing, and so on. These iterations continue and we never show up.

That’s obviously nonsense. So how do we complete the model? We make choices of arrival times to aim at continuous, and add in a “social penalty” for late arrival.

A more rigorous model wouldn’t have a simple payoff matrix with 3 choices of “early, on-time, late.” Instead players would choose arrival times to aim at on a continuous scale (see Glen Whitman for more on this here). That’s realistic, although it complicates the math considerably because payoffs will be continuous functions rather than a simple 3×3 matrix.

Next, we have to add in a “social penalty” for being late — a cost incurred the later players arrive compared to the agreed upon time — something I mentioned briefly in a note at the end of the original post.

Think of the social penalty as the expected value of the lost income due to other’s reduced willingness to deal with you due to their irritation at your lateness. I won’t speculate on philosophically why people impose a social cost on latecomers — for the model we just need to know it’s there, not why — and take it as exogenous based on culture. The social penalty would be a simple function of minutes late compared to the agreed-upon time (not the other player’s arrival time!) with y’ > 0 and y’‘ > 0.

In this model we’ll get a very realistic equilibrium: Showing up early is costly due to time costs of waiting, and showing up very late is also costly because of the rising social penalty.

That implies there exists an equilibrium that minimizes costs between the two. Which is exactly what we see.

**Update 2:** Here’s an insightful March 2004 post from *Marginal Revolution* on the game theoretic approach to lateness and punctuality. Does this confirm or refute the “lunch model”?

Posted by andrewdavidchamber on Wednesday January 19, 2005 | Feedback?