Monday, October 1, 2012

The Logic of Showing up Late

I've been pondering about the logic of showing up on time or late to casual meetings. In my own experience, people tag others as "shows up late" vs "shows up on time" and try to mirror this behavior. This might seem a little odd since it would perhaps be both simpler and better for everybody just to show up in time. But is it odd?

Consider the following situation (in the sense of game theory, it is a "game"):

We have two people that have decided to meet at a certain time. Let us call them A and B. They can each choose to arrive on time, or come late.

If both come on time, nobody has to wait for the other, so there is no "suffering" involved. Hence they both receive 0 "happiness points" each. This is usually called "utility" in more formal contexts.

If A shows up in time but B is late, A will have to wait for B, which is less pleasant. However, A will be in a worse mood, which will also make B enjoy their time together less. So A, because he has to wait, will suffer a penalty of -5. B, because A will be in a worse mood, will receive a utility of -2.
The exact numbers don't really matter for now: what's relevant is the relative order they have. What we are effectively stating is that a person is better off when not having to wait (0) compared to when he has to wait (-5). Being in worse company, due to having dragged down his or her mood by making them wait, falls in between (-2). Because the situation (game) is symmetric, the roles are reversed when A is late and B on time.

If A and B are both late, nobody has to wait for the other, so nobody suffers. They both receive a utility of 0. (Note that this is simplified: it's more realistic to assume that they will not be late by equal amounts, so that one may suffer more than the other. What we're saying is that as both show up late, nobody has to wait for the other for a "significant" amount of time, making it equivalent to both showing up in time for all practical purposes.)

To summarize:
StrategyB In TimeB Late
A In TimeA: 0, B: 0A: -5, B: -2
A LateA: -2, B: -5A: 0, B: 0

The assumption in how they choose strategies is as follows: they are both aware of all the information contained in the table above (the utilities that follow from different combinations of the choices both make). Furthermore, they both try to maximize their own utility, they both know the other is, they both know that the other knows this, etc. In other words, the assumptions are the obvious ones: if I come late and you don't, you're not gonna be happy. I know this. You know this. I know you know it. You know I know it. And so on. (This is called "common knowledge".)

What we want to know is the following: which strategy do we select (in time or late) to maximize our own happiness? Looking at the table, it should be clear to you that if I come late, you will want to come late too. If you come in time, you'll have to wait for me, receiving -5. If you come late too, you'll receive 0, which is better. On the other hand, if I come in time, you will want to come in time too, so as not to upset me by making me wait. In other words, this is a coordination game: all we need to do is make sure we both do the same thing.

Indeed, if we are both used to come in time when we meet, there is no benefit for any of us to start deviating from this behavior (without informing the other). If you show up late, we'll both be worse off, since you'll ruin my mood and that's gonna bite you back. The same goes for me. If we're both used to showing up late, nobody benefits from coming in time (again, without informing the other), since that just means waiting for the other one to show up. This means that both coming in time, and both coming late, are Nash equilibrium - it is the best each of us can do given the circumstances. Note that the same cannot be said of the two other situations: if I tend to come late and you tend to come in time, we can both be better off by either you starting to come late too, or me starting to show up in time. Thus the asymmetric options are not Nash equilibria.

In fact, the equilibria are both strict, meaning switching is always worse. Hence this is not only an equilibria in the sense that it requires rational foresight: any insect could "figure it out" by virtue of having the behavior (strategy) hardwired into its nervous system. (Game theorists call this an Evolutionarily Stable Strategy, or ESS.)

This answers the question I posed in the beginning of this post: is it odd that people "remember" which friends tend to come in time and not, only to mirror the behavior? The answer is: no. The reason is: because it doesn't matter which is it - in time or late - as long as both do the same thing, both are better off.

You might think that this is an artifact of me having chosen the same utility for both coming in time and both coming late. It is not so. We could change the utility of both showing up late to say -1 for both, reflecting the fact that there is less time to enjoy the meeting compared to both arriving in time. We could even assign a utility of 1 to both arriving late, reflecting less stress in getting ready (or whatever other reason you might find). The point is that the utility of both coming late or in time does not matter to our conclusion: it is identical as long as both being in time or both being late is better than the asymmetric cases.

In fact, these coordination situations (games of two players) always have another solution: a randomized strategy ("mixed strategy") of sometimes being on time, and sometimes being late. Not surprisingly, however, such a strategy is not a very good idea, because in this situation the parties do not coordinate their arrivals (if they did, it would not be a randomized strategy, but the cases we just looked at). In game theory jargon, there is a mixed Nash equilibrium, but it is not an ESS. The latter is obvious, since those who randomize are worse off than those who always manage to coordinate their arrivals.

Finally, there's the question of which is the better solution: for both to come late, or both to come in time? This depends on the utilities received by the two equilibria. In my table above, both receive 0 when in time, and 0 when late. So both solutions are equally good. However, if both would receive 1 when in time (and still 0 when both are late), it would be better for both to arrive in time. If both were to arrive late, realizing they would both benefit from switching to arriving in time (which is the Pareto optimal solution), they could simply both agree on showing up in time, knowing that the other one would for the simple fact that it benefits him/her more too. For example, when two people dislike to stress, it may be preferable for both to arrive late. For people that have scarcity of time, it may be preferable to arrive in time.

However, if the game is not symmetric, there need not be such a Pareto optimal solution: if by both being in time, A receives 1 and B 0, whereas when both are late, it is the opposite: A receives 0 and B 1, well, then A would prefer the solution of both arriving in time, and B the solution of both arriving late. Note that both are still worst off by not coordinating, so the Nash equilibria are still the same: both in time, or both late. All that has changed is that it is no longer ubiquitous which of the two is the "better one", simply because what's better for A is worse for B, and vice versa. However, having an asymmetric situation means that there is a fundamental difference between A and B, each player now knows whether it is "A" or "B" (they can't just be swapped arbitrarily), so it does not represent a situation where "everything is the same for each person". For example, A may prefer to arrive late because she is sensitive to stress. But B may prefer to arrive on time, because he has very little time for procrastination. What Nash equilibria can tell us, is that they will (well, should) coordinate so that they both arrive in time or late. However, which of the two they pick, in this situation (of asymmetry), is a complicated issue not answers by game theory (as far as I know). Nobody said agreeing would always be easy.

UPDATE: My friend Mads discussed some (psychological) possibilities for an equilibrium in the asymmetric cases (one is one time, the other is late). We can analyze such situations (of psychological preferences) simply by changing the person's utilities. Let's say that B really hates being on time. The utilities are like before, except that B receives an additional punishment of -3 for coming on time:

StrategyB In TimeB Late
A In TimeA: 0, B: -3A: -5, B: -2
A LateA: -2, B: -8A: 0, B: 0

One can immediately note that B's strategy of being late is better than being in time, no matter what A does: if A is on time, B receives -2, which is better than -3. If A is late, B receives 0 instead of -8 (this is in fact solely due to the punishment of B being in time when A is in time, no additional punishment would have been needed on top of B having to wait, i.e. receiving -5).

This means that B's strategy of being late strictly dominates (it is always better), and hence, We can reduce the problem to:

StrategyB Late
A In TimeA: -5, B: -2
A LateA: 0, B: 0

Now, by the same logic, A's strategy of being late strictly dominates being on time, so the only Nash equilibrium will be that both show up late. Intuitively, this is because B hates being on time, and A doesn't mind accommodating for B's need to show up on time.

It is presumably harder to be on time than showing up late, so that as soon as one person is a time optimist (meaning he shows up late) both will settle for "an implicit 10 more minutes on top of the decided time".

Nevertheless, let's assume that, except for B's dislike of being on time, A actually dislikes being late. We thus have a conflict of interest, which we represent by punishing A with -X more for being late:

StrategyB In TimeB Late
A In TimeA: 0, B: -3A: -5, B: -2
A LateA: -2-X, B: -8A: -X, B: 0

As before, B's strategy of being late dominates being on time:

StrategyB Late
A In TimeA: -5, B: -2
A LateA: -X, B: 0

We now see that what it all boils down to is just how much A dislikes being late (quantified by the value of -X). If he dislikes being late more than to be in time and still wait for B, he will (obviously) actually go there and wait for B. We would then arrive at the asymmetric solution where A arrives on time (knowing B will be late) and B showing up late.

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