# The game theory of everyday life – where to stand in a lift

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 $L$, is a closed and bounded subset of two-dimensional Euclidean space, i.e., $L \subset \mathcal{R}^2$. There are $n \ge 2$ players. Each player’s strategy space is $L$. Let $x_i \in L$ 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 $x_i,x_j \in L$ let $d(x_i,x_j)$ be the Euclidian distance between these two points. Let $x = (x_1,...,x_n)$ denote the vector of player placement in the lift. Each player i’s utility function is then given by $u_i(x) = \min_{j \neq i} d(x_i,x_j)$.

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.

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