# The game theory of everyday life – gallantry

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.

$\begin{tabular}{c|cc} & Go & Wait \\ \hline Go & 0,0 & 2,1 \\ Wait & 1,2 & 0,0 \\ \end{tabular}$

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.