Andrew Chamberlain has a clever model showing that arriving up late for a lunch date is an equilibrium outcome. However, I think Andrew’s calculations are incomplete. Read Andrew’s post first, or else this post will make no sense.
Early on, Andrew treats arrival time as randomly distributed: your mean arrival time is 1:00 pm, with some chance of arriving early and some chance of arriving late. But later, Andrew treats your arrival time as a matter of choice: you choose to arrive at 1:10 pm because that maximizes your expected utility. The problem is that if I choose my own arrival time, then I can also predict that you will choose your own arrival time. It’s therefore incorrect for me to base my expected utility calculations on the assumption that your arrival time is randomly distributed with a mean of 1:00 pm. And the same goes for your calculations.
To make the model consistent, we should assume that what each individual chooses is the time to aim at. The chosen time then becomes the mean of a random distribution. Thus, if you choose to aim for 1:10 pm, there’s some chance you’ll arrive “early” at 1:00 pm (the actual time we agreed upon), and some chance you’ll arrive late at 1:20 (which is especially late relative to the agreed upon time). Performing the very same calculations that Andrew used to show that my utility is maximized by arriving (technically, aiming to arrive) at 1:10 pm, I can show that my utility is actually maximized by aiming to arrive at 1:20 pm. Why do our calculations differ? In Andrew’s approach, each chooser naively assumes the other guy is aiming for 1:00 pm, with any deviation resulting only from randomness. In my approach, you assume the other guy is doing the same self-interested calculation you are.
But the process doesn’t stop there. The other guy, predicting that I will aim for 1:20 pm, will rationally aim for 1:30 pm. And I can predict this, so I will rationally aim for 1:40 pm… Where does the process stop? The answer depends on how many rounds of double think we are willing to permit in our model. If we allow infinitely many iterations, there is no equilibrium to this game. Neither of us will ever show up. On the other hand, if we stop at (say) three iterations (each of us does the calculation three times), then we both arrive at 1:30 pm.
It might seem logical to limit the number of iterations, since human beings have finite cognitive resources. But if you just read the previous two paragraphs and understood them, then you’ve effectively just done an infinite number of iterations in shorthand form. So we’re back to no equilibrium (except, perhaps, an equilibrium that involves deliberate randomization). Alternatively, we might suppose that the utility figures change as the time gets later and later – I get hungrier and hungrier, and care less and less about companionship. As a result, there might be some time late enough that both players will aim for it, since the gains from eating sooner just barely outweigh the expected gain from avoiding any possible wait time. (In Andrew’s payoff matrix, that would mean the payoffs on the main diagonal are not all the same.) Making this conclusion rigorous, however, would require a more complex game theoretic model than I’m willing to devise right now.