When you enter a lift, a bus, a doctor’s waiting room, or any other smallish place in which you and others are just waiting for something to happen, one of the key decisions you face is to choose where to stand or sit. How do we do this? What are the key factors (motives) behind our decisions? What are the consequences of this? What are the testable implications?

I found this in Chapter 2 “Territories of the self” part I “Preserves” in Goffman’s Relations in Public (recall my objectives):

“All of this may be seen in miniature in elevator behavior. Passengers have two problems: to allocate the space equably, and to maintain a defensible position, which in this context means orientation to the door and center with the back up against the wall if possible. The first few individuals can enter without anyone present having to rearrange himself, but very shortly each new entrant – up to a certain number – causes all those present to shift position and reorient themselves in sequence. Leave-taking introduces a tendency to reverse the cycle, but this is tempered by the countervailing resistance to appearing uncomfortable in an established distance from another. Thus, as the car empties, passengers acquire a measure of uneasiness, caught between two opposing inclinations – to obtain maximum distance from others and to inhibit avoidance behavior that might give offense.”

I have decided to write this blog post partly (especially when it comes to writing down the model) in the grand style of state of the art theory research papers. You will see what this means.

While I do not know of any specific game theory model that addresses this particular problem I am pretty confident that there is such a model out there in the literature. Please let me know if you know of one and I am happy to refer to it. If there is such a model out there I am pretty confident that it will be very similar to the one I am going to put forward here.

My first imitation of grand theory papers is to narrow down my vocabulary. While I would like the reader to think of any room in which people wait, such as lifts, busses, doctor’s waiting rooms, etc., I will refer to all of these as lifts.

Goffman then talks about three concerns individuals may have while on a lift. First, individuals care about the physical distance from other lift passengers; second, they care about “maintain[ing] a defensible position”; and third they do not want to offend others (unduly). I have decided to focus my blog post on one of these motivations, the first (and main one I think). Future blog posts (I am not planning any though) could tackle the other additional motivations. So the people in my lift will care only about the distance between them and their fellow lift passengers.

So what would a grand theory paper on this topic look like? Recall that a game has to have players, strategies, and outcomes (in terms of payoffs or “utils”). I don’t think we need incomplete information (information that is not shared by everyone in the game) here, so the game is one of complete information. While, of course, in any real-life lift individual passengers will have private information about many things, all this does not seem germane to the issue of where to place yourself. I am also ignoring that different people may have different desires about how close they would like to stand or sit to specific other fellow passengers. Again, an interested reader can modify the basic model to include a stalker or whatever other motive they would like to address for whatever situation they have in mind.

So we have reduced the problem to lifts and the single motivation of keeping one’s distance from other lift passengers. For a formal model we still have to make this even more concrete. Do we care about the average distance from all other fellow passengers or some other function of all these distances? My feeling here is that most likely we all care about the minimal distance from all other passengers. Suppose that all but one of the other passengers are bunched together in the far end of the lift, but the one remaining passenger has his or her nose almost touching yours (recall that this person is an uninteresting stranger to you). Contrast this with a situation in which all of your fellow passengers are evenly distributed in the lift with nobody standing super-close to you. You would probably prefer the second situation over the first. If you do, modelling your preference as caring about the minimal distance to your fellow passengers is probably not such a bad approximation to your real preferences.

Now let me finally express all this in the grand style of grand theory papers.

A lift, denoted by , is a closed and bounded subset of two-dimensional Euclidean space, i.e., . There are players. Each player’s strategy space is . Let denote player i’s choice of spot in the lift. [Note that we are assuming here that people have zero width, another assumption we could modify if we felt this would change things in an interesting way –  which I doubt. We are also assuming that people can stand right on the boundary of the lift. This is for technical reasons that M. G. can explain to you if you insist.] For any pair of points let be the Euclidian distance between these two points. Let  denote the vector of player placement in the lift. Each player i’s utility function is then given by .

It feels good to write this sort of thing.

So we have a model – a fictitious world with hypothesized people. What do we now expect to happen in this fictitious world we just created? As I argued before the most reasonable expectation in my view in situations like this (which we face over and over with always different opponents) is that we will get an evolutionary stable equilibrium of this game.

Now let’s play with our model and let’s find (evolutionary stable) equilibria of this game. Let’s do this by going through some real-life lift inspired examples. Suppose our lift is a square of some arbitrary size. Suppose first we have only two people. Where will they stand in our fictitious world? As far as I can see all equilibria are permutations of the two people standing in opposite corners of the lift. Why? Let me first suppose that one player places herself somewhere that is not on the boundary of the lift. But then she can always increase the distance from her fellow passengers by moving away from her fellow passenger. As she is not on the boundary she is able to do this. So both players (in equilibrium) must be on the boundary.  Suppose at least one of them is not in a corner of the lift. Call her person one. But then no matter where person two is located person one can again increase her distance from the other person by moving along the boundary one way or the other (or both if person two is standing on a perpendicular to the boundary side of the lift that person one is standing on). Finally, suppose the two people are placed in adjacent corners. Then one of them could increase the distance by moving towards the next corner away from the other person. This proves what I claimed. In this case the any equilibrium has the two people standing in opposite corners (as at the beginning of a boxing match).

Let’s stay with the square lift, but let us now consider three and then four players. These cases are already much more complicated. Take three players. Let us first see what constellations are not equilibria. Even before that let me say that I think one can prove that all three people need to be on the boundary in any equilibrium. Suppose two individuals are in adjacent corners. Then the optimal placement of the third person is exactly in the middle of the lift side that is opposite to the two other passengers. But then the other two passengers finding this third person further away than the one standing in the adjacent corner next to them, can increase the minimal distance by moving along the boundary of the lift closer towards the third person. So this is not an equilibrium. If both corner people were to move in such a fashion we will not reach an equilibrium either, I think, as then at some point each of them will find it better to move away to their original corner again.

Now suppose all three people place themselves in distinct corners of the lift. Then one corner is unfilled. Then the two people in corners adjacent to the empty corner could each unilaterally increase the minimal distance to their fellow lift passengers by moving a little bit towards the empty corner. If they keep doing this at the same speed then they would at some point each hit a place where each finds that they are no equidistant from both of their fellow passengers. The third passenger can also not reduce the distance to the other two. We have reached an equilibrium. In fact one, with a non-trivial and empirically testable implication of placement in the largest equilateral that could fit in a square.

It is possible that the game with three people has multiple equilibria. I have not explored this further. The game with four people, however, definitely has multiple equilibria. In fact it has infinitely many equilibria (all with different equilibrium happiness – I mean payoff). I am not sure whether one can generally prove that in the four player case in equilibrium all people must be on the boundary of the lift. For five players this is definitely not true. But let me give you an infinity of equilibria in which all four players are placed in symmetric positions on the boundary of the lift. Take any side of the lift and place one player on any point on this side. Then put the other three players on the same position on the other three boundary sides in such a way that the four players form a square. Note that, no matter what initial point you chose for the first player, if any player now moves in any feasible direction she will only reduce the distance to one of her closest neighbors. This means that any such constellation is a Nash equilibrium. I do not know whether they are all evolutionary stable. I would certainly only expect the equilibrium in which all four people stand, respectively, in the four corners of the lift. If it is true that this is what happens empirically in real square lifts, then we would need to think about why it is that the people do not put themselves in one of the infinitely other equilibrium positions. I guess these may not be evolutionary stable, but maybe the reason lies somewhere else.

The more I think about it, the more I realize that my blog post has only scratched the surface of already undertaken or yet to be done lift placement research.

Chapter 1.II on “Vehicular Units” of Goffman’s Relations in Public has many more “nuggets” that are amenable to a game theoretic analysis in addition to the one I described in my previous post. In footnote 23 on page 17, for instance, he talks about what we would call “common knowledge” and that eye contact is perhaps the only way to establish it (referring here to the earlier work by Lewis 1969, Scheff 1967, and Schelling 1960). This could lead one to discuss Ariel Rubinstein’s “email game” (1989, ECMA) and some of the literature thereafter (and before). On page 14, Goffman talks about “gamesmanship” in whether or not we let others “catch our eye”. I would like to think here about pedestrians visibly (to all who do not do the same) refusing to “scan” their environment by looking at their smartphone while walking. This would lead me to discuss a paper of Hurkens and Schlag (2002, IJGT) and possibly beyond that. There is also Goffman’s discussion of the apparently commonly observed practice of the “interweaving” of cars when they have to go from two lanes into one. I have not yet seen a game theoretic treatment of this phenomenon and I am not quite sure (at the moment) how one would explain it.

But in this post I want to take up Goffman’s brief mention (on pages 14-15) of special circumstances that seem to necessarily lead to what he calls “gallantry”. This is when a path that pedestrians take in both directions at some point becomes too narrow for two people to pass simultaneously. Then one has to wait to let the other person pass. But who should wait and who should be first to pass?

There are indeed often “norms” in place that dictate a form of “gallantry”. For instance, men should give way to women, younger people should give way to older people, or people going down should give way to people coming up (if the path has a non-negligible slope).

Before I go deeper into this subject, let me quickly state that there are other forms of gallantry which I (and Goffman) do not talk about here (there). One could, for instance, go out of one’s way to open a door for someone else, perhaps even a door that one is not going through oneself, such as a car door. This is interesting also, but not the subject of this post. Here I only look at cases in which some form of “gallantry” is really almost required for these two people to (eventually) go on with their lives: in the end one person has to let the other person pass first. There is no other way.

A nice illustration of the potential problem in such cases is described by Lady Mary Montagu (thanks to my father for pointing this out to me!) in her letter XI to Mrs. J. 26 September 1716 of her collected correspondence (http://ota.ox.ac.uk/id/N31507), when she was travelling through Europe and writing about it. This is something she wrote about Vienna. See also footnote 20, page 15 in Goffman’s book for a similar story.

“It is not from Austria that one can write with vivacity, and I am already infected with the phlegm of the country. Even their amours and their quarrels are carried on with a surprising temper, and they are never lively, but upon points of ceremony. There, I own, they shew all their passions, and ’tis not long since two coaches meeting in a narrow street at night, the ladies in them not being able to adjust the ceremonial of which should go back, sat there with equal gallantry till two in the morning, and were both so fully determined to die upon the spot rather than yield, in a point of that importance, that the street would never have been cleared till their deaths, if the Emperor had not sent his guards to part them, and even then they refused to stir, till the expedient could be found out, of taking them both our in chairs, exactly in the same moment. After the ladies were agreed, it was with some difficulty, that the pass was decided between the two coachmen, no less tenacious of their rank than the ladies.”

There are many ways one could model this situation as a game. One could emphasize the time dimension of the problem, which would lead us to call this a game of attrition, as perhaps the quote from Lady Mary Montagu suggests we should. I think, however, that this is typically not the most important issue in this situation. I want to model a situation where the two people would in principle not mind terribly if they are the one to wait, but in which they still, at least slightly, prefer to pass first if possible.

While in reality this game often has incomplete information as well – see for instance my blog post on this very problem that drivers face when navigating the narrow lanes in Cornwall – I will here model this as a game with complete information. This will suffice for my purposes here as you will see.

So the idea is this. Each person can decide between “Go” and “Wait”. If both Go or both Wait they have not yet solved the problem and I normalize these cases (equally) as giving them both zero payoffs. We could give them different payoffs in the two cases, but this does not affect the analysis as long as these payoffs are less than one. You can see that each person would prefer to be the one to Go while the other Waits.

The important thing for my discussion of this problem is that the game is symmetric. This means that, without any other information, the two players are in exactly the same position and will find it impossible to coordinate on an asymmetric outcome unless by luck.

In fact the theory suggests that play in such cases (that can be emulated in the artificial environment of a lab – see more about this below) eventually will converge to the unique symmetric evolutionary stable strategy. Here this would be that 2/3 of all people play Go and 1/3 play Wait.

Note that this is not a great outcome as in most cases (in 5 out of 9 cases) the two individuals will do the same thing (to which as you remember we attached payoffs of zero). In reality this means that the two individuals will now engage in some form of communication and additional maneuvering necessitating some delay with some loss of payoff (in the form of time) to both.

When, however, the two individuals have commonly understandable observable differences, such as one being a man and one a woman, or one being old and the other young, or one coming from a low place and the other a high one (this could also be in terms of status), norms can develop that take this possible information into account. In fact the theory suggests that this would be the case. The theory I am here referring to is developed by Selten (1980, “A Note on Evolutionary Stable Strategies in Asymmetric Animal Conflicts,” Journal of theoretical Biology, 84, 93-101) building on the ground-breaking work by Maynard-Smith and Price (who have invented the concept of evolutionary stability in a symmetric game very much like this one). There is a lovely lab experiment about this, very much confirming the theoretical findings in both cases (symmetric outcome without information and asymmetric outcomes with information about the characteristics of the opponents) by Oprea, Henwood, and Friedman (2011) “Separating the Hawks from the Doves: Evidence from continuous time laboratory games”, Journal of Economic Theory, 146 (6), 2206—2225).

Note that sometime these norms turn into laws, such as the car driving down has to give way to a car driving up when the street is too narrow for both cars at the same time. I believe this to be a law in Austria, for instance. It does not seem to be enforced much, perhaps exactly for the very reason that it is a norm, an evolutionary stable equilibrium. In some cases such norms have made it into books of etiquette. In these cases a violation of the norm is not only against your immediate interest but may also be severely frowned upon by others with possible social sanctions being imposed afterwards. Although this seems unnecessary, again, for the very reason that it is already in everyone’s best interest to adhere to this norm (if all others do).

What I would find interesting would be a study of which characteristics are more likely to be used in such a norm. It seems to me that a person’s height, for instance, would not be such a great choice, as there would be many cases where it would be unclear who of the two is actually the taller. I suppose age suffers from the same problem, though. A binary and very obvious characteristic, such as man and woman, seems a very natural first thing to condition on. But of course it also does not solve the problem fully. What if two women meet at this narrow path? Then they have to use a more refined subnorm.

I guess it would also be bad to use a characteristic that one could influence. Imagine that we use hair color as a characteristic and suppose we have a norm that dark hair goes first and light hair has to wait. Then there would be an incentive (albeit I admit a very small one – possibly overridden by other more important incentives about hair color) to die your hair dark so as to always have the right to go first. In any case, in principle one could condition on all sorts of characteristics such as eye color or size of ears or who approached the narrow bit of the path first (actually probably a norm that is often in place) or who saw the other first. No matter what the norm is, I guess it will never be so perfect as to allow individuals to solve the problem perfectly in all cases. For any norm there is probably still a positive likelihood of the two individuals being the same as far as the norm is concerned (such as both being women of roughly the same age who have arrived at this narrow bit of the path more or less at the same time) and if this does not happen too often, evolutionary pressure to put a subnorm in place in such cases is pretty small.