Home – The Game Theory of Everyday Life

A simple model of community enforcement

Here is the model for my previous blog post on (anonymous) community enfor cement. I would call it a simplified symmetric (single-population) version of the model in the paper by Michihiro Kandori entitled “Social norms and community enforcement” in the Review of Economic Studies 59.1 (1992): 63-80. The point of this blog post is to demonstrate that what I have claimed in the previous post can be made logically coherent. I can provide a reasonable and simple artificial world in which we obtain cooperative behavior under the fear of triggering a tipping point as a subgame perfect Nash equilibrium, meaning a self-enforcing situation that is even self-enforcing when the tipping point has been triggered and there is no way back.

There are $n$ people involved. These are those who are interested in going up the mountain, the people on the train, or the users of the communal kitchen. Time is discrete and runs from time points $0, 1, 2,$ until infinity. At every point in time $t$ one person is randomly drawn to undertake the activity (go up the mountain, use the bathroom, or the kitchen) so that each person has a $\frac{1}{n}$ probability of being drawn. The drawn person first observes (and only that) the state $x \in \{0,1,2,...\}$ of the resource (the amount of rubbish on the mountain, or the state of uncleanliness of the bathroom or kitchen). Then this person (after using the resource) decides whether to clean up ( $C$ – for cooperate or clean up) after themselves or not ( $D$ – for defect to use the language of the well-known prisoners’ dilemma game).

The instantaneous utility that the drawn person then receives shall be given by $\frac{1}{\lambda^x}$ , where $x$ is the state of the resource – let us assume here that it is simply equal to the number of people who have been drawn before this person and who have chosen action $D$ , that is not to clean up after themselves. This is the instantaneous utility this person receives when this person chooses $C$ . When this person chooses $D$ they get the same utility, plus a small but positive term $d$ for not having to clean up. Let us assume that $\lambda > 1$ , so this person receives less payoff the worse, that is the higher, the state of the resource $x$ is. As a function of $x$ , the function $\frac{1}{\lambda^x}$ starts at $1$ for $x=0$ and then exponentially decays to the limit value of $0$ when $x$ tends to infinity. When a person is not drawn to use the resource at some point in time this person receives an instantaneous utility of $0$ . Every person discounts the future exponentially with a discount rate $\delta \in [0,1)$ . This means that they evaluate streams of utils $u_t$ with the net present value $(1-\delta) \sum_{t=0}^{\infty} u_t \delta^t$ , where the $(1-\delta)$ term is a convenient normalization.

For a well-defined game theoretic model, we need to identify players, their information, their strategies, and their payoffs. We have players and what they know and we have their payoffs. We have not quite yet defined their possible strategies, but we have specified their actions. To conclude the model we, thus, only have to define players’ possible strategies. These are all possible functions from the set of possible values of $x$ , that is the set $\{0,1,2,...\}$ , to the set of actions $\{C,D\}$ . In principle, we should allow a bit more, as our players should probably remember what the state was at previous times when they were using the resource and also what they themselves did at these points in time, but this does not add or change anything of interest in our present analysis.

My claim then was that, at least for certain parameter ranges (for $\lambda$ , $\delta$ , $n$ , and $d$ ), the following strategy is a subgame perfect Nash equilibrium: Play $C$ if $x=0$ and play $D$ otherwise. This kind of strategy is often referred to, in the repeated game literature, as a “grim trigger” strategy. In order to see this, we need to check two things. First, suppose everyone uses this strategy, which means that the play path has everyone cooperating (keeping the resource clean), is it best for a drawn player to also do so? Second, suppose the “trigger” has been released by someone playing $D$ , that is by someone not cleaning up after themselves, is it best for a drawn person to then also play $D$ (to also not cleaning up after themselves)?

So, suppose first that everyone uses this strategy. Then a randomly drawn player at some point in time, that we can, without loss of generality, call time $0$ , finds the following payoff consequences for their two possible choices and for all time periods from then on:
$\begin{array}{ccccc} \mbox{time} & 0 & 1 & 2 & ... \\ C & 1 & \frac{1}{n} 1 & \frac{1}{n} 1 & ... \\ D & 1 + d & \frac{1}{n} \left(\frac{1}{\lambda} + d\right) & \frac{1}{n} \left(\frac{1}{\lambda^2} + d\right) & ... \end{array}$

The net present value for choosing $C$ at this point in time is then
$(1-\delta) \left( 1 + \frac{1}{n} \sum_{t=1}^{\infty} \delta^t \right)$ .
For choosing $D$ at this point in time it is
$(1-\delta) \left( 1 + d + \frac{1}{n} \sum_{t=1}^{\infty} \left(\frac{1}{\lambda^t} + d\right) \delta^t \right)$ .
The grim trigger strategy is a Nash equilibrium if and only if such a player would prefer
$C$ over $D$ ,
that is if and only if
$(1-\delta) \left( 1 + \frac{1}{n} \sum_{t=1}^{\infty} \delta^t \right) \ge (1-\delta) \left( 1 + d + \frac{1}{n} \sum_{t=1}^{\infty} \left(\frac{1}{\lambda^t} + d\right) \delta^t \right)$ .
According to my calculations, this is equivalent to
$\delta (\lambda-\delta) (1-d) \ge \left( d n (\lambda - \delta) \right) (1-\delta)$ .

It is not straightforward to derive nice bounds for $\delta$ (as a function of the other parameters) so that this inequality is satisfied. But we can at least say that, if people are sufficiently patient, that is for $\delta$ close to $1$ , the inequality is satisfied provided $d < 1$ as well, which I assumed anyway – I stated that I wanted $d$ positive and small.

For the second part of the argument, suppose that the “trigger” has been released and that everyone is playing $D$ . Suppose the drawn person at some given point in time faces a state of uncleanliness of $x>0$ . We can again reset the clock to zero without loss of generality. Then, the consequences of the two possible actions for this person are:
$\begin{array}{ccccc} \mbox{time} & 0 & 1 & 2 & ... \\ C & \frac{1}{\lambda^x} & \frac{1}{n} \left(\frac{1}{\lambda^x} + d\right) & \frac{1}{n} \left(\frac{1}{\lambda^{x+1}} + d \right) & ... \\ D & \frac{1}{\lambda^x} + d & \frac{1}{n} \left(\frac{1}{\lambda^{x+1}} + d\right) & \frac{1}{n} \left(\frac{1}{\lambda^{x+2}} + d \right) & ... \end{array}$

The net present value for choosing $C$ at this point in time is then
$(1-\delta) \left( \frac{1}{\lambda^x} + \frac{1}{n} \sum_{t=1}^{\infty} \left(\frac{1}{\lambda^{x+t-1}} + d\right) \delta^t \right)$ .
For choosing $D$ at this point in time it is
$(1-\delta) \left( \frac{1}{\lambda^x} + d + \frac{1}{n} \sum_{t=1}^{\infty} \left(\frac{1}{\lambda^{x+t}} + d\right) \delta^t \right)$ .

It is in the best interest of this person to choose $D$ rather than $C$ if and only if the latter is greater than or equal to the former, and this is the case, according to my calculations, if and only if
$\delta \le \frac{d n \lambda}{\lambda^{-x} (\lambda -1) + d n}.$
The right hand side is lowest for $x=1$ (among all integer values for $x > 0$ ), when it is
$\delta \le \frac{d n \lambda}{\lambda^{-1} (\lambda -1) + d n}.$
Rearranging, one can see that the right hand side of this inequality is greater than or equal to one, meaning that there is in fact no restriction on $\delta$ , if and only if
$\lambda \ge \frac{1}{d n}.$
Recall that $\lambda > 1$ . Then finally, this last inequality is satisfied if, for instance, $n$ is sufficiently large or $d$ is positive but very small.

All this together proves that, in the model given here, the strategy of cleaning up after yourself provided the resource is clean before you used it, and not cleaning up if the resource is not clean before you used it, is a subgame perfect Nash equilibrium: It is self-enforcing and the implicit threat of a tipping point in behavior is also self-enforcing. The model is very specific and many other versions would work just as well to make the same point.

August 17, 2021

Game Theory

Repeated Games, Subgame Perfect Equilibrium
Please leave the bathroom as you would like to find it

Many actions that we take affect other people that are not involved in the decision-making process. In economics, these effects are commonly referred to as “externalities” and the presence of externalities is one of the main concerns that may render free markets inefficient. “Inefficient” means that the ultimate outcome of people ignoring the externalities that they cause on other people is such that there is an alternative outcome that would be better (or at least as good) for all people! The presence of externalities is the main problem behind climate change and also at least one reason why we still have a problem with Covid-19. People, when making their holiday planning, car driving, air conditioning, car purchasing, et cetera decisions often ignore the effect their actions have on the environment and, thus, on all others. People make vaccination decisions weighing their own subjective assessment of the risks for themselves, without necessarily considering that with a vaccination they would also increase the protection from Covid-19 for everyone around them.

There are a variety of measures that governments or other organized groups of people can take to reduce the harm caused by people ignoring these externalities. One can, for instance, debate the (higher) taxation of fossil fuels or laws to force everyone to vaccinate. Often, such measures are probably necessary. There are (possibly rare) cases, however, in which even in fairly anonymous societies, the problem sorts itself out. It can do so through a mechanism of community enforcement. In this post I will describe an argument derived from a 1992 paper by Michihiro Kandori entitled “Social norms and community enforcement” in the Review of Economic Studies 59.1 (1992): 63-80. I will use the examples of mountain tops on which people do not leave their rubbish, bathrooms on trains that remain reasonably clean despite heavy usage, and communal kitchens (such as the one in my department) that despite a lack of regular professional cleaning service and despite a fair number of people using them, remain reasonably clean and usable.

I would first like to stress that in all three examples people are unlikely to observe your actions, so no one could punish you for bad behavior directly. When you are having your well-earned lunch or snack at the mountain top you are quite possibly alone at that moment. If you decide to leave some rubbish behind no one would see you doing it. I also hope that nobody can see what you do in the train bathrooms. And yes, occasionally you may not be alone in the department kitchen when you are making your coffee or warming up your lunch, but you are also often by yourself and unobserved. [It is, by the way, also not immediately clear what would happen if someone did observe your lack of adherence to the social norm of what is thought of as decent behavior (be it on the mountain top, the bathroom on the train, or the departmental kitchen). I often find that misbehavior in public may perhaps induce a fair bit of stern staring, but nothing much more than that. Nobody seems to want to engage in an altercation. An interesting phenomenon in its own right.] So why do people not leave rubbish on the mountain top, why do they clean up after themselves after using a bathroom, why do people wash, dry, and put away their dishes in the communal kitchen?

First, you might say, why wouldn’t you? Well, I guess the idea is that you would derive some benefit from not having to carry rubbish back down the mountain (after all it weighs something, also you might not have a good bag for your rubbish and it might soil all the other stuff you have in your backpack). You probably have to undertake some slightly unpleasant cleaning effort to keep the bathroom or the kitchen in a reasonable state. In fact, in your kitchen at home you might leave dirty dishes in the kitchen for quite some time, cleaning them later, while in a communal kitchen you probably do it (if at all) right away.

Then you might say, that ok, yes, it is a bit annoying having to do these things, but it is not too bad and anyway, you are a moral person. Maybe. I also would like to think that I am a moral person, but perhaps there is a more tangible reason behind our supposedly moral stance. [I generally don’t believe that people make all these decisions always so consciously. They may simply follow some more or less automatized protocols (perhaps as part of how they were raised as a child and now not often questioned). Then I will here provide a possible reason why such behavior might in fact be in your own self-interest despite the effort that is involved.]

Then you might say, and now you are on to something, that you have an interest in keeping the place clean, because you might want to use it yourself again. True, if it were your own mountain you would probably keep it clean. Perhaps you would not tidy up your kitchen immediately, but you would probably tidy it up at some point every day. But then you do not own the mountain and you are just one of many users. Wouldn’t this fact dilute your incentives to keep the place clean? Well, yes and no.

In fact, it seems quite plausible (and I have often observed this) that people do not clean up (much) after themselves if, for instance, the bathroom on the train is already in a bad state, even if they think that they might need to use it again. But they might well do so if the bathroom was clean when they started to use it. How can this be rationalized?

Let me sketch the model here (I will try and describe it in full detail in another post). Imagine you have a largish number of users of a place (such as the mountain top, the train bathroom, or the communal kitchen). Imagine that everyone uses this place infrequently but recurrently at random points in time. So, everyone always thinks that they may use this place again at some point in the future (I guess this works less well for the example of train bathrooms towards the end of the train ride – but then at that stage these bathrooms often are quite dirty). When they use it people can either be very clean (or clean up after themselves) or they can litter or soil the place. The instantaneous payoffs are such that people would (at that moment) prefer to litter or soil rather than clean. Finally, assume that nobody observes any actions of any other people, but everybody observes the state of the place when they use it (how much rubbish there is on the mountain, how clean the bathroom or kitchen is).

Then the following strategy, if employed by all people, can be made to be a subgame perfect Nash equilibrium of this symmetric stochastic repeated game, at least under some plausible conditions. [Subgame perfect Nash equilibrium means that this strategy is self-enforcing (everyone finds it in their interest to adhere to it when everyone else does) and does not involve a non-credible threat (any threats that are used to incentivize people to adhere to the strategy are also self-enforcing when they are supposed to be employed).] As long as the place is perfectly clean, keep the place clean (just as in the often-advertised statement “please leave the bathroom as you would like to find it”). If the bathroom is not perfectly clean, regardless of how bad it is, do not clean up after yourself.

If everyone follows this strategy, then the place would stay perfectly clean throughout. If, for some reason, somebody does not follow this strategy and does not clean up after themselves, this is a “tipping point” and the avalanche of dirt starts rolling: the place will just get dirtier and dirtier from then on. In reality, it may need more than one piece of rubbish on the mountain or more than just one sheet of loo paper on the floor to trigger the “tipping point”, and one could probably adapt the model (that you can find here) so that this would be the case. In any case we do get that people will behave very differently when the place is already a mess and when it is clean and it is, at least partly, the fear of triggering the tipping point that incentivizes people to behave and to internalize the externality that bad behavior would impose on others.

August 13, 2021

Game Theory

Proverb, Repeated Games, Subgame Perfect Equilibrium
Allocating fruit among household members

This story is loosely based on real-life events. In the summer, members of my wife’s family all come together to stay with the “grandparents” in one large household. There are typically around 12 of us and we have many small problems. One of these is the allocation of fruit (and other snack items – but I will concentrate on the fruit problem as it is the easiest to describe) among household members. Mind you, this is not a problem that we talk about (much) in the household. Many of us perhaps do not even recognize it as a problem, but it is a problem and one that we solve inefficiently.

Consider peaches. Almost all household members like peaches, but we all vary in our exact preferences. Some of us would happily eat many peaches at various points in time, some would rather eat just one at a specific time of day. Some prefer peaches when they are still quite hard, some prefer them when they are essentially mush. [Some even like them slightly moldy, at least that’s what I infer from fruit choice observations that I made.] So, we all have different bliss points as to the optimal stage in the ripening process when a peach should be eaten. I could probably show you a graph with peach ripening stage on the x-axis and derived pleasure from eating a peach on the y-axis and you would see a nice concave (for the mathematical economist) initially upward curve that reaches a peak at some point, the bliss point, and then bends down again. Each family member would have a somewhat different curve with different bliss points and different up- and downward curves before and after the bliss point, respectively.

Having drawn such a picture, one could go on to mark the level of derived pleasure from eating a peach, at which you would rather eat a different fruit, an apple, say, or even just nothing (rather than a moldy peach, for instance). Again, all this is different for all household members and would also depend on the ever-varying quality of apples (and that bit about the quality of apples that matters to whichever household member we are talking about). To make the fruit allocation problem even more complicated, we all seem to have variable preferences. Some days, when the weather is bad for instance, just don’t seem to be peach days, at least for some of us. Or you may have eaten or drunk something else today that does not go well with peaches. On top of that, peach quality seems to vary from batch to batch and so our preferences change accordingly as well.

Why am I talking so much about preferences? There are two reasons. One, we all want to be kind and let someone else eat a peach instead of us if they also want it. Second, if we think of the problem from the social (household) planner’s point of view, that person would like to allocate peaches to individuals in some sort of efficient way, give more peaches to people who like them more, perhaps also keep fairness in mind, etc. The problem now is that none of us perfectly know other household members’ personal peach preferences on any given day. This is both a problem for the social planner as well as for each of us when we consider eating a peach. [There is also the additional problem sometimes that the number of available peaches is not even always clear. Sometimes extra peaches are “hidden” in the larder, so there are more than you think. Sometimes some peaches are actually reserved to be made into a cake – and this fact may not be known to everyone – although sometimes the social planner labels peaches accordingly.]

So how do we solve this allocation problem, why is it inefficient, and what would other mechanisms look like? For us, it all begins with the purchasing decision (with nowadays online delivery – this also impacts peach batch quality, by the way) made once or twice a week by the head of household, in her capacity as household chief procurement officer (or CPO). The CPO makes these decisions, considering an amazing number of factors based on an incredibly high degree of empathy towards all household members. Yet, even the CPO is not fully aware of all aspects of the daily changing peach preference profile in the household. At the end of the day the CPO settles on some quantity of peaches and this is where our problem now begins.

There are many mechanisms that we could use to tackle our fruit allocation problem. Let me first unashamedly tell you that we do not use a market mechanism. What would a market mechanism look like? Well, we would all be given a share of all fruit and snack items that were purchased. We would all have something called “money” that we all accept in exchange for the various fruit and snack items. We would meet regularly in a “market”, the kitchen for instance, where we all set up shop and trade among us. Market prices would, we would hope, adapt on a daily and perhaps even hourly basis so that supply meets demand, so that no peach is left uneaten and no additional peach would be wanted to be eaten at these prices. As we can assume that there are no worrying externalities when it comes to fruit and snack choices (except perhaps that parents sometimes worry about the kids’ sweet choices) such a market mechanism is expected to deliver a, so-called, Pareto optimal allocation, an allocation of fruit and snack items that is such that if we wanted to improve the material (fruit and snack) well-being of one individual by adjusting the allocation, we would have to reduce the material well-being of another. We could also aim for a reasonable degree of fairness by adjusting the initial allocation (before trade happens) of fruits and snacks.

But, surprisingly, this is not what we do. Our system instead is as follows. All the peaches (with some of the qualifications pointed out above) are simply displayed in the kitchen and anyone is free to take one at any time. I am pretty sure that this leads to an inefficient (not Pareto-efficient) allocation, at least in our case. Not for the reason you might think of at first. True, in a world full of people who are interested only in their immediate material (fruit and snack) well-being, like a world of small children perhaps, we might find that all the peaches are eaten by the first person who spots them. You might say that, while this may not be fair, this is ok from an efficiency point of view. But not necessarily. Imagine one person eating all the peaches, because he or she spotted them first, and another eating all the chocolates, because he or she spotted them first. They might have both been better off had they traded some of their peaches against chocolates and vice versa. But, in any case, this is not the problem we have. Our problem is that people are too altruistic, I believe, and too careful not to eat a peach that somebody else might also like, perhaps at a later stage in the ripening process. The sad result is that many peaches simply remain un-eaten (and thrown away) at the end of the day. Well, sometimes, to be fair, they are rescued and baked into a cake (at a point where I am happy not to know how advanced those peaches already were in their ripening process). But even in this latter case, some of us, perhaps all of us, might have preferred a peak peach over a post-peak peach cake.

You might say that communication should solve the problem. Maybe. However, we have many other problems (and not only problems) to discuss and the peach problem does not seem high up there on the list of problems. Also, even if we did, most of us would probably find it hard to articulate our exact peach preferences (we often don’t seem to have a good prediction of our future preferences ourselves). Also, we only come together for about a month in the summer every year, and for such a short time, it may be inefficient to spend hours solving the inefficiency in our peach allocation. So, as much as it pains me, as a trained economist, to accept inefficiencies, I guess I will just have to accept it.

August 7, 2021

Economics, Mechanism Design

Market Making, Markets
Estimating the proportion of Corona cases

This short note makes one simple point. If you are interested in estimating the proportion of Corona infected people in some country or region, there is a simple and better (more precise) estimate than the one you obtain by computing the sample proportion. You can also read this in German here (and here).

Setup

Consider taking a (completely) random sample of $n$ individuals in some population in order to estimate the proportion of people in this population who have the Corona virus. Let $p$ denote this true proportion. I here assume that we already know, through potentially non-random medical testing, that there is a certain fraction $q$ of the population who definitely have the virus (or have had it). I will refer to these people as those that were declared to have the virus. I assume that whatever medical test was used to obtain this number was perfect, at least in one direction: anyone who has been declared to have the virus this way also actually has it. As, thus, necessarily $q \le p,$ we can write $p=\mu q,$ where we interpret $\mu \ge 1$ as the multiplier or ratio of actual virus cases relative to the declared virus cases. I am here interested in estimating $\mu$ from the random sample knowing $q.$ If we have an estimate for $\mu$ we get one for $p$ by multiplying the $\mu$ -estimate with $q$ .

When we take the random sample, we collect two pieces of information from each person. One, we check (again, for the sake of simplicity, with a perfect medical test) whether or not they have the virus. Two, we ask them (and the subject answers truthfully) whether they have already been declared as having the virus. I will call $X$ the total number of virus cases in the sample and $Y \le X$ the total number of already declared virus cases in the sample.

Estimator

Many people would probably be tempted to use $\hat{p}=\frac{X}{n}$ as the standard estimator for $p,$ and, thus, indirectly $\hat{\mu}_S=\frac{X}{qn}$ as the standard estimator for $\mu$ . It turns out that there is a better estimator that uses all available information. Let me call it the alternative estimator $\hat{\mu}_A$ . It is given by
$\hat{\mu}_A=1+\frac{X-Y}{qn}.$

In the Appendix below I derive (in a few simple steps) this estimator as an approximation of the maximum-likelihood estimator for the present problem. It, therefore, does have all the nice properties that maximum likelihood estimators have. But even if you are a maximum likelihood skeptic, we can actually just directly compare the precision (for all sample sizes) of the two estimators, by looking at their variances.

First note that, like the standard estimator, the alternative estimator is unbiased as
$\mathbb{E}\left[\hat{\mu}_A\right]=1+\frac{\mu qn - qn}{qn} = \mu.$

The variance of the two estimators are
$\mathbb{V}\left[\hat{\mu}_S\right] = \frac{\mu(1-\mu q)}{qn} \approx \frac{\mu}{qn},$
and, as $X-Y$ is binomially distributed with number of trials $n$ and success probability $\mu q \left(1-\frac{1}{\mu}\right),$
$\mathbb{V}\left[\hat{\mu}_A\right] = \frac{(\mu-1)(1-q(\mu-1))}{qn} \approx \frac{\mu-1}{qn},$
where the approximation is good when $q$ and $\mu$ are sufficiently small.

In this case the ratio of the two variances is given by
$\frac{\mathbb{V}\left[\hat{\mu}_A\right]}{\mathbb{V}\left[\hat{\mu}_S\right]} = \frac{\mu-1}{\mu} < 1.$

Thus, especially, if $\mu$ is not much larger than 1, the alternative estimator is quite a bit more precise. Note also, that the alternative estimator can never be below 1.

Austrian Corona cases

In Austria, from 1st to 6th of April, a random sample of $n=1544$ was checked for the Corona virus. I will here ignore the disturbing sample selection problem that actually 2000 people were supposed to participate and 456 did not participate. Of those who participated the number of cases found, $X,$ was 5 and the number of already declared cases among them, $Y,$ was either 2 or 3. There was some weighting in these numbers which I am not fully informed about. I will ignore these issues here, but at least will look at both cases for $Y.$ At the same day the proportion $q=1/758$ (11383 declared cases among 8,636.364 people in Austria).

Using the, here also easily applicable, Clopper-Pearson method to compute 95\% confidence bounds, we get the following estimates and bounds derived from the two different estimators.
$\begin{array}{c|ccc} & \hat{\mu}_S & \hat{\mu}_A (Y=3) & \hat{\mu}_A (Y=2) \\ \hline \mbox{estimate } \mu & 2,46 & 1,98 & 2,47 \\ \mbox{lower bound } \mu & 0,87 & 1,12 & 1,30 \\ \mbox{upper bound } \mu & 5,72 & 4,54 & 5,30 \\ \mbox{lower bound cases } & 9866 & 12738 & 14845 \\ \mbox{estimated cases } & 27.968 & 22570 & 28164 \\ \mbox{upper bound cases } & 65126 & 51726 & 60331 \\ \end{array}$
As you can see, the confidence bounds are much narrower for the alternative estimator than for the standard estimator.

A Thought

If we could assume, which sadly we often probably cannot, that the proportionality factor $\mu$ is the same in all regions of interest, while $q$ is observably not, then one could take a specific random sample that would even be much better than a random sample of all people. In Austria, for instance, the $q$ for Landeck in Tirol is about $q_L=1/50,$ while in Neusiedl am See in Burgenland it is about $q_N=1/1000.$

Then a random sample of people in Landeck would produce a much more precise estimate for $\mu$ than a random sample of people in Neusiedl. The variance for the Neusiedl estimator would be 20 (the ratio of $q_L/q_N$ ) times as large as that for Landeck.

Another Thought

Of course, there is nothing specific about the setup here that makes it only applicable to counting virus cases. This estimator could be used in all cases in which we are interested in the true proportion of some attribute A in some population, when we know that only A’s can also have attribute B and we know how many B’s there are. Looking at it like that I am sure this estimator is known. So I am here just reminding you all about it.

Appendix

We here derive the alternative estimator as an approximation to the maximum likelihood estimator. Taking a truly random sample, we know that $X$ is binomially distributed with number of trials $n$ and success probability $\mu q.$ Conditional on $X$ we know that $Y$ is binomially distributed with number of trials $X$ and success probability $\frac{1}{\mu}.$ The likelihood function is, therefore, given by
$\mathcal{L}(\mu;X,Y) = {n \choose X} (\mu q)^X\left(1-\mu q\right)^{(n-X)} {X \choose Y} \left(\frac{1}{\mu}\right)^Y \left(1-\frac{1}{\mu}\right)^{(X-Y)}.$

The log-likelihood function is then proportional to
$\ell(\mu;X,Y)=X \ln(\mu q) + (n-X) \ln\left(1-\mu q\right) - Y\ln(\mu) + (X-Y) \ln\left(1-\frac{1}{\mu}\right).$

The maximum likelihood estimator, thus, has to satisfy
$\begin{array}{lll} \frac{X}{\mu q}q + \frac{n-X}{1-\mu q} (-q) - \frac{Y}{\mu} + \frac{X-Y}{1-\frac{1}{\mu}} \frac{1}{\mu^2} & = & 0 \\ \frac{X-Y}{\mu} - \frac{q(n-X)}{1-\mu q} + \frac{X-Y}{\mu(\mu-1)} & = & 0. \end{array}$

If $\mu q$ is small, we can approximate $1-\mu q$ by 1. We then get
$\mu=1+\frac{X-Y}{q(n-X)}.$
If $X$ is, in expectation, much smaller than $n,$ we can approximate this further to get
$\hat{\mu}_A=1+\frac{X-Y}{qn}.$

April 22, 2020

Statistics

Estimator, Maximum Likelihood, Standard Error
A joke about economic methodology

This is a joke that I heard many times and once on a big stage at the 2014 annual meeting of the Verein für Socialpolitik where some supposedly important person from a supposedly important central bank (if I recall correctly) used it as a criticism of current economic methodology (as this person understood it) and generalizing it to mean it as a criticism of any economic methodology that uses math (if I understood this person correctly).

The joke goes like this. A police officer patrols the city at night and finds a perhaps slightly inebriated person apparently looking for something under the dim light of a street lamp. The police officer approaches said person and inquires: “Are you looking for something?” The perhaps slightly inebriated person responds: “Yes, I am looking for my keys.” “Where did you lose them?” the police officer asks. To which the perhaps slightly inebriated person answers: “Over there.” loosely waving at a bunch of bushes in the distance. “Why aren’t you looking for your keys over there then?” the police officer wonders out loud. “Well I only have light here” the slightly inebriated person replies.

How does this apply to research methodology in economics? Think of the slightly inebriated person in the joke as your economic researcher (now you see why this person had to be slightly inebriated). Think of the keys as the answer to a research question and think of the light as the research methodology that the economic researcher applies. The economic researcher is thus just as unlikely to find the right answer using their methodology as the slightly inebriated person is to find their keys.

I like this joke because it does ring true. I am sure it is a valid criticism of thousands of research articles in economics every year. But I do not come to the same conclusion that I believe the supposedly important person from a supposedly important central bank came to which is that, if I understood correctly, we should abandon serious mathematical modelling in economics (in favor, I believe this person indicated, of large scale simulation studies with intricately interwoven agents who all behave mechanically according to some simple heuristic). A brief aside: I do not mind if some people try simulation as a means of getting to an answer. I did not intend to here write a criticism of simulation as a methodology, although I am not overly optimistic of its usefulness. Simulation, as I see it, can only ever provide a fairly dim light. But it could on occasion be in just the right place to find the beginnings of an answer. But I believe that when we find a mathematical model to be unhelpful we should not abandon math completely but we should try and find or develop a more appropriate math to deal with the situation.

The more math we have at our disposal the more stats we have at our disposal the more light we have and the more likely will it be that we will find answers to our economic problems.

A final note, perhaps, and again very much only my opinion: the more brilliantly creative and intuitive you are the less you may need to know the tools. But then again who is brilliant?

October 29, 2018

Economics
On towels, parking spots, and job protection

You just arrived at your dream summer resort. You had a restful night almost entirely uninterrupted by mosquitoes. You just woke up and had a leisurely and plentiful breakfast. You are making your way to the swimming pool that looked so enticing on the webpage. And what do you find? You find towels. In fact you find towels on every single one of the lounge chairs that the resort has provided. While almost no lounge chair is actually occupied, not a single lounge chair is really available. Economics is supposedly (primarily?) about the allocation of scarce resources. So what about the scarce resource that is a lounge chair next to the pool in a holiday resort?

Let us first approach this problem from the viewpoint of the resort. How did they decide how many lounge chairs they would provide? Probably there is a bit of a space problem. Or let’s say it differently, there is probably a temptation for the resort management to divide the space between hotel rooms and lounge chair space in such a way that there are more rooms so that they can have more paying guests. It probably makes little sense to have more lounge chairs than beds (unless the resort was open to day visitors as well – which we shall not assume here). But should one have fewer lounge chairs than beds? I can imagine the conversation in management about this issue. Somebody will have pointed out that it’s probably ok to have more beds than lounge chairs because not all people who sleep in the resort will also need a lounge chair. Some guests may make day trips to other places. Also even if all guests will want a lounge chair, they probably do not all need one at the same time. The average guest might spend, let’s say four hours every day in a lounge chair (seems a long time to me). And not all guests will want to spend the same four hours in lounge chairs. There may be morning people and afternoon people, before lunch people and after lunch people, early lunch and late lunch people, et cetera.

In fact they are probably right (these are after all just the managers I imagine in my head). It is probably true for many resorts that at any given moment during the day the actual number of lounge chairs needed is smaller than the total number of lounge chairs in the resort. After all we often observe many lounge chairs with only a towel on it. So there is actually no real scarcity and yet we find that there are some people who cannot find a lounge chair when they want one. The problem is that this “game” between the resort guests can have two equilibria and it is easy to get stuck in the “bad” equilibrium.

To see this consider this. What do you do the next day do when you observe that all lounge chairs are reserved through the early placing of towels? Well, you have two options. Either you give up your hope of getting a lounge chair or you get up early and place a towel on a chair yourself.

What do you do if you find that there are always lounge chairs available (in nice locations around the pool)? You don’t even think about getting up early just to place a towel on a chair.

This means that both situations are self-enforcing. If no one places towels in the morning (and there is no real scarcity at any given moment in time) then no one will even consider reserving lounge chairs with towels in the morning. If however people do place towels in the morning and, if you do not you do not find a lounge chair when you need one, you will quite possibly get up early in the morning to do the same. In fact, there may be a race such that you have to get up earlier and earlier to find an empty lounge chair for your towel. An equilibrium is then found in such a way that exactly (in pure theory only) so many people get up early enough to place a towel on a lounge chair as there are lounge chairs. These people are those that care relatively less about sleeping in the morning. This, by the way, is called Harsanyi purification (of mixed Nash equilibria).

So how can you slip from the “good” equilibrium to the “bad” one and what could the resort do to prevent the “bad” equilibrium? I guess that most resorts have a variety of more or less attractive lounge chair locations. So I guess it is possible that some people start putting towels on the most attractive locations, which then starts a gradual chain reaction that eventually all lounge chairs get “toweled” if I am allowed to invent this word (it is not underlined in my editor, so I guess this word exists already). Another possibility is that some large enough group of tourists, perhaps with experience from other resorts and not knowing that in their current resort there is no real need for this, do get up and place towels and cover so many chairs that for the remaining chairs there now is a real scarcity at some point in time.

If the true reason for the lounge chair is that we are indeed simply in a bad equilibrium then the resort can introduce some simple and effective policy measures to restore the good equilibrium. They could, for instance, simply not allow the “toweling” of lounge chairs. They could remove towels after some time. A bit costly, this one, as someone has to monitor the pool area and enforce this rule. But they may not need to do it for too long as, once the good equilibrium has reestablished itself, they can stop enforcing the rule (as it is self-enforcing).

So what about the parking spots and job protection, the other two topics I mentioned in the title? Well, you can figure out for yourself how one could use the towels as an analogy for these two problems.

August 9, 2018

Economics, Evolutionary, Game Theory

Multiple Equilibria, Nash Equilibrium
Inspired by Goffman – the secret handshake

You are visiting another university and have arranged to meet someone from that university in the lobby of the hotel you are staying at. The hotel lobby is busy with many people and (for some strange reason) neither you nor the person you are supposed to meet have recognizable pictures on their webpages. How will you find each other? What is the mechanism behind it? How is this possible at all?

I am continuing with my game theory inspired by Goffman’s w ork. This is an excerpt of what Goffman says about this on page 95, Part Three “Focused Interaction”, Chapter 6 “Face Engagements” in his “Behavior in Public places”:

“As these various examples suggest, mutual glances ordinarily must be withheld if an encounter is to be avoided, for eye contact opens one up for face engagement. I would like to add, finally, that there is a relationship between the use of eye-to-eye glances as a means of communicating a request for initiation of an encounter, and other communication practices. The more clearly individuals are obliged to refrain from staring directly at others, the more effectively will they be able to attach special significance to a stare, in this case, a request for an encounter. The rule of civil inattention thus makes possible, and “fits” with, the clearance function given to looks into others’ eyes. The rule similarly makes possible the giving of a special function to “prolonged” holding of a stranger’s glance, as when unacquainted persons who had arranged to meet each other manage to discover one another in this way.”

This is wonderful stuff. I am here just going to explain in game theoretic terms what is going on in the background of the last sentence of the quote. So we have two people who would like to meet. But there are many other people there as well, who we cannot ex-ante distinguish from the person we would like to meet. So how should we model this? I would like to model this as a one-at-a-time two person game of two people potentially trying to engage with each other. I would say that there are two types of individuals, type A who would like to meet another type A and type B who would like to be left in peace. This is private information. Only I know whether I am type A (trying to find another type A) or type B (hoping to be left in peace). This means we have a game of incomplete information. So there are two players and each one could be of type A or type B. To close the model informationally, so that we can work with it, it makes sense here to assume that both players of all types have common knowledge of the likelihood of anyone being of type A, let’s say of $\mu$ , and type B, then obviously $1-\mu$ . This is, of course, an empirically completely implausible assumption, but you will see that it actually does not play a huge role. In fact we will find an equilibrium of this game that will be an equilibrium for any positive (and relatively small) $\mu (< 1/2)$ . The game has a first stage, which I will describe in a bit. Let me first describe the second (and last) stage. Eventually both players can unilaterally choose to verbally engage with the other player. So let us give both players two strategies each: engage (E) and not engage (N). So we have players (each of possibly different types) and we have strategies for each player. All that is left is to specify their payoffs. Well, what do we want? We want that B types do not want to engage. So let’s give B types a zero payoff whenever they choose E and let’s give B types a payoff of one if they choose N. Note that this gives B types a (strictly) dominant action of choosing N (not to engage).

What about A types? They want to engage with other A types but do not want to engage with B types (consider the embarrassment and need for a lengthy further explanation when you inadvertently try to engage with the wrong person). So they should get a payoff of zero if they engage with a B type and a one if they do not engage with a B type. They should get a zero if they fail to engage with an A type and get a one if they do engage with a B type. All that matters, of course, is that one is larger than zero. We could have chosen 100 and 6 and it would all be the same.

So let us summarize this in matrix form. There are four possible encounters, I am an A type and meet another A type, I am an A type and meet a B type, I am a B type and meet an A type, and I am an A type and meet a B type. Here are the payoffs we just chose for me in these for encounters (I choose row, my opponent chooses column):
$\begin{tabular}{ccc} & \mbox{opp is A type} & \mbox{opp is B type} \\ \\ \mbox{I am A type} & \begin{tabular}{c|cc} & E & N \\ \hline E & 1 & 1 \\ N & 0 & 0 \\ \end{tabular} & \begin{tabular}{c|cc} & E & N \\ \hline E & 0 & 0 \\ N & 1 & 1 \\ \end{tabular} \\ \\ \mbox{I am B type} & \begin{tabular}{c|cc} & E & N \\ \hline E & 0 & 0 \\ N & 1 & 1 \\ \end{tabular} & \begin{tabular}{c|cc} & E & N \\ \hline E & 0 & 0 \\ N & 1 & 1 \\ \end{tabular} \end{tabular}$
If this is the game and there is nothing else, and if $\mu < 1/2$ , then there is a unique equilibrium in this game in which both A and B types choose not to engage. B types do this because they find not to engage a dominant strategy (maybe they don’t even think about the possibility of engaging anyone) and the (in equilibrium sad) A types do this because it is (sufficiently) more likely that their opponent is a B type so that they are too worried about being embarrassed if they try to engage them.

So this is all very sad. But luckily the game people actually play is not fully described yet. We have not taken into account Goffman’s statement about the “special function to “prolonged” holding of a stranger’s glance”. Before the two players decide whether or not to engage verbally, they can first both send a (not very costly) “message” to their opponent by holding a prolonged stare. Now Goffman in his book in the pages leading up to the quote I provided above discusses at length why prolonged stares are typically not used by people as these are considered rude. You can read this for yourself. Whatever the reason, it seems a fact that people do not ordinarily treat other people to a prolonged stare. This means, that we can choose to deliberately employ this otherwise almost never used “message” in special encounters. This “message” can then act as a secret handshake. By the way I first encountered this secret handshake in a just slightly different context in a 1990 paper by Arthur Robson (Journal of Theoretical Biology 144, 379-396). It supposedly is very similar (but I again think the context is slightly different – actually quite different – one has to be careful to distinguish between cooperation and coordination I think) to the so-called green beard effect allegedly proposed by William Hamilton in 1964 and Richard Dawkins in his popular 1976 “selfish gene” book.

So how does this secret handshake work here and how do we model it? Before the two players play this game as described so far, they can first choose whether or not to send this message of the prolonged stare. So how can this generate new outcomes? Well, the two players can condition their engagement level on whether or not their opponent (and they themselves) employed this prolonged stare. In fact the following behavior is an equilibrium of this game. Every A type first uses a prolonged stare, B types do not. Then an A type decides to engage if and only if their opponent gave them a prolonged stare as well. In this equilibrium everyone is now as happy as they can be. B types get briefly stared at by some A types, but are then not engaged and they of course never engage their opponent themselves, while A types by means of the prolonged stare are able to identify other A types and engage exactly those. This is I think a fair description of how and why indeed two strangers can thus meet in a busy hotel lobby in such a way that disinterested others would not even notice that the two did not know each other before.

One of my PhD students at CICS (Research Center for Social Complexity at the Universidad del Desarrollo in Santiago, Chile) told me the following story. There used to be a bar in Santiago called the Club Amsterdam. You could order a beer, get a beer and then pay. You could also, allegedly, order a beer and at the same time put down a 5000 pesos note on the table and you would get a beer and a small portion of cocaine. No more communication was needed. I am not sure whether this situation should be modelled by the game I provided above – maybe one should have to think about the police here – but the problem seems quite similar and its solution employs a secret handshake as well.

July 22, 2018

Evolutionary, Game Theory, Incomplete Information

Bayes Nash Equilibrium, Evolutionary Stability, Multiple Equilibria
Inspired by Goffman – 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.

July 21, 2018

Evolutionary, Game Theory

Evolutionary Stability, Nash Equilibrium
Inspired by Goffman – 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.

May 30, 2018

Evolutionary, Game Theory

Evolutionary Stability, Nash Equilibrium
Inspired by Goffman – pedestrian traffic

Our starting point is Goffman’s Relations in Public Chapter 1.II on “Vehicular Units”. Goffman is here interested in the norms that regulate traffic, especially but not only pedestrian traffic. He first quotes Edward Alsworth Ross, Social Control, New York: The Macmillan Company (1908), page 1: “A condition of order at the junction of crowded city thoroughfares implies primarily an absence of collisions between men or vehicles that interfere one with another.”

Goffman on page 6 then states the following: “Take, for example, techniques that pedestrians employ in order to avoid bumping into one another. These seem of little significance. However, there are an appreciable number of such devices; they are constantly in use and they cast a pattern on street behavior. Street traffic would be a shambles without them.”

In this post I want to take up this claim and provide a model that allows us to discuss how people avoid bumping into each other. I will use Goffman’s work to help me to identify the appropriate model for this issue.

Let me first identify the players. It seems that, while there are many people involved in street traffic, typically we encounter these people one by one. So I think for a first attempt it might be sufficient to study the situation of two people who are currently on course to bump into each other and who are trying to get past each other in order to avoid a collision. So we have two players in often fairly symmetric positions. Now here is one statement by Goffman (on page 8) about actions: “Pedestrians can twist, duck, bend, and turn sharply, and therefore, unlike motorists, can safely count on being able to extricate themselves in the last few milliseconds before impending impact.” Despite the fact that Goffman mentions so many possible actions I will for a first attempt consider only two. Try to pass on the left or try to pass on the right. But if we feel it may be useful we can go back and think more about the possible moves pedestrians can make. Now what about payoffs? Talking about cars or road traffic, Goffman, on page 8, states that “On the road, the overriding purpose is to get from one point to another.” For pedestrian or street traffic he states “On walks and in semi-public places such as stadiums and stores, getting from one point to another is not the only purpose and often not the main one”. He has more to say about payoffs on page 8: “Should pedestrians actually collide, damage is not likely to be significant, whereas between motorists collision is unlikely (given current costs of repair) to be insignificant.” All this strikes me as important to understand pedestrian traffic. Let me see why. Suppose we ignore these last few statements, especially the one about pedestrians often having more than one purpose. We might then be tempted to say, and perhaps this is a good model of car traffic, that the game is simple. We have two players (the drivers facing each other), each has two possible choices (pass on the left L or pass on the right R) and if they pass each other that’s great (they both get a payoff of say one) and if they bump into each other that’s awful (they both get a payoff of say zero). In other words the game can be written in matrix form as follows:
$\begin{tabular}{c|cc} & L & R \\ \hline L & 1,1 & 0,0 \\ R & 0,0 & 1,1 \\ \end{tabular}$
What are the evolutionary stable norms of behavior in this game? They must be a Nash equilibrium, which means no player should have an incentive to deviate from the norm. Could the norm be that everyone passes on the right? Yes! If everyone passes on the right, you would be foolish to pass on the left, because that would mean you bump into everyone and get a payoff of zero! If you instead also pass everyone on the right, you indeed do get past everyone and you enjoy your payoff of one. Completely analogously the norm could be that everyone passes everyone on the left. And indeed both of these norms exist for car traffic. In Japan people drive on the left, in Chile they drive on the right (most of the time). Recall that Goffman was well aware of the possibility of different norms being possible (in different societies or places) – see the previous post.

A quick aside: game theory experts will have noted that the game has a third Nash equilibrium, an equilibrium in so-called mixed strategies. Under a Harsanyi purification (Harsanyi, 1973) interpretation of this mixed equilibrium we could describe it like this. Half of all people pass on the left and the other half of all people pass on the right. This is an equilibrium, because if that’s indeed what the others are doing you are equally well of passing on the left and passing on the right: half of the time you avoid an accident and the other half of the time you are dead (have an accident) either way. This is an equilibrium, but not an evolutionary stable one. Why not? Suppose slightly more than half of all people pass on the left. After a while you might notice this and then you find it slightly better to also start passing people on the left. But then the more people pass on the left the better this strategy becomes and gradually we move towards the norm of everyone passing on the left. This is probably more or less how the whole thing evolved in the early days of cart traffic. You may want to read Peyton Young’s “Individual Strategy and Social Structure” Princeton University Press (2001).

This all seems fine for cars, but what about pedestrians who supposedly, according to Goffman, use many “devices” and “techniques […] in order to avoid bumping into one another”? There seems to be absolutely no need for this here. So I think something is missing from this game. We should recall that pedestrians, according to Goffman, have side interests in addition to getting from A to B as fast as possible. It does not seem to be the pedestrian’s only goal to get past the oncoming person, the pedestrian might have a slight preference for which side would be better for her. Think of a person that you face in a corridor and that person, after they pass you, would like to turn left to the bathroom for instance. This person probably has a slight preference, when possible, to pass you on her left. The problem with this now is that you don’t necessarily know that she wants to go to the bathroom, and thus, you don’t know that she prefers passing you on her left. That’s why she might want to use a “device” – a signal of some sort – that tells you that she wants to pass on the left. Ok, so hold on a moment. We need to proceed slowly. I first need to discuss this game without “devices” so that we can see why “devices” might be useful. So how do I take into account these possible side issue preferences that pedestrians might have? Well, I need to modify the payoffs people get and these payoffs are now different for different people and I need to make the information about these preferences private. What I mean is that I will assume that everyone knows their own preferences (or payoffs) – they know whether or not they want to go to the bathroom on the left after they pass you – but you, their “opponent” do not know. So how does this work? I will simply change the game as follows:
$\begin{tabular}{c|cc} & L & R \\ \hline L & 1-u,1-v & 0,0 \\ R & 0,0 & u,v \\ \end{tabular}$
What are these u’s and v’s? You should think of each u and v representing a possible person with a particular preference for passing left and right. A person with a u (or v) close to a half is a person who cares only about getting past their “opponent” and does not care in any way on which side this happens. A person with a u (or v) less than but close to one is a person who would much prefer to pass their “opponent” on the right. Say this person really urgently needs the bathroom just behind you on her right. A person with a u (or v) greater than but close to zero is similar but has a strong preference to pass on the left.

How do we do this? Well, this is one of Harsanyi’s great contributions to the body of game theory. We assume that both u and v are drawn from some distribution F on a subset of the real line that includes the interval from zero to one. Then every person learns their own u (or v) but learns nothing (as yet) about their opponent’s v (or u). Every person only knows that her opponent’s v (or u) is random and that the randomness is described by the cumulative distribution function F. In fact we here make a radical assumption and one that we should probably challenge later. We assume that not only does every person believe her opponent’s v (or u) is distributed according to F, but we also assume that everyone knows this fact, and that everyone knows that everyone knows this, and so on ad infinitum (as game theorists like to say). In short we assume that this distribution F that governs the likelihood of the various preference types you might encounter is common knowledge among the two players. Modern game theory also has ways of dealing with deviating from this assumption. But for the moment we shall assume it. Under an evolutionary interpretation this assumption is less worrisome than one might initially think, but we should probably come back to it.

So how do we “solve” this game? There are two ways one can look at a game with incomplete information. One can either consider each possible person (with a specific u or v) separately – the so-called interim view – or one can consider the problem from the ex-ante point of view, where each person has a strategy for all possible u’s that this person could end up with. These two approaches are equivalent but sometimes one is easier than the other for the analyst. Here the second, the ex-ante, approach is easier. So consider a person who many times throughout her life has to navigate pedestrian traffic. In each situation she might have a different u. Sometimes she just wants to get past her opponent, her u is a half, sometimes she wants to turn left right after passing her opponent, her u is close to zero, sometimes she wants to turn right right after passing her opponent, her u is close to one. She develops a strategy as a function of her u. Now what would be a good strategy? Suppose there is some norm of behavior that people follow, a function from their u’s to passing left or right. For some such norm, what would be the best individual response to this norm? As you with your u do not know your opponent type, their v, knowing the norm that is in place only tells you with what probability (or frequency) your opponents will choose left or right. Suppose you know this probability of opponents going left (from your knowledge of the norm and the distribution function F) and call this probability $\alpha$ , then what is your implicit tradeoff between going left and going right? Recall that we are at the moment studying a situation where people do not communicate with each other (they do not use any “devices”). Well, if you go left you avoid bumping into each other with probability $\alpha$ and you do bump into each other with the remaining probability $1-\alpha$ . Your average (or expected) payoff from going left yourself is, thus, $\alpha (1-u)$ . Similarly, your average (or expected) payoff from going right is $(1-\alpha)u$ . When is left (strictly) better than right for you? Well if and only if $(1-u)\alpha > (1-\alpha)u$ . Calculating we get $u < \alpha$ .

This means that, whatever the norm is, your best response to this norm is to use a simple cut-off strategy. Basically what you do is this. You observe the frequency of people going left and right (induced, as we said, by the combination of the prevailing norm and the distribution of preference types F) and you choose left yourself if your u for this interaction is less than the observed frequency of left and choose right otherwise.

But if this is your best response to this norm, then it is everybody’s best response to this norm and it will become the norm itself! So everyone will be using the same cut-off strategy! But what will the cut-off be? Well if everyone uses a cut-off of say x, some real number between zero and one, then the probability that people use action left is the probability that their u is less than x, which is given by $F(x)$ . So if the cut-off people use is x, the probability of people going left is F(x) and this is the best response cut-off they will use. So we must have that x=F(x).

So any stable norm must at least satisfy that we are in equilibrium, meaning that x=F(x). But is this enough for a stable norm of behavior? Not quite. To discuss this it is best to consider two examples of possible distributions F that could be present in different places of human pedestrian traffic.

Suppose the preference u is, like so many things in life, normally distributed. Let’s say it is normally distributed with a mean of a half and a relatively low variance so that not too many people have a u less than zero or more than one. Please excuse the low tech (but I think sufficient) rendering of this example:

What are the Nash equilibria of this game with such a distribution F? There are three. First we have F(x)=x for a value of x that is positive but pretty close to zero. What does this mean? It means that the norm is such that almost everyone attempts to pass others on the right except for very few people who have a very strong interest to pass on the left. This is an equilibrium that is pretty close to the equilibrium of the car-driving game of always driving on the right. There is a similar equilibrium with x=F(x) where x is just less than but very close to one. Here almost everyone attempts to pass others on the left except for very few people who have a strong interest to pass on the right. There is another equilibrium, however, at x equal to one half, where we also have x=F(x). Here we have that everyone who has the slightest inclination for passing on the left attempts to pass on the left and everyone who has the slightest inclination for passing on the right attempts to pass on the right. This is a mayhem equilibrium. But is it stable? No. Why not? Suppose that people use a slightly larger cut-off than one half, call it y. Then we find that, as F is quite steep at one half, F(y) > y. This means that no people’s best response cut-off F(y) is higher than the prevailing cut-off of y. So we expect people to adjust their cut-off upwards. This will go on until we reach the other equilibrium with a cut-off close to one. Similarly, a cut-off of just less than one half will lead to lower and lower cut-offs and eventually to the equilibrium cut-off close to zero.

So what have we achieved? Not so much. The whole situation is very similar to the much simpler game without the u’s and v’s and all that. So, again, it seems that we would not need any “devices” and “techniques” of “scanning” and “intention display” (Relations in Public, pages 11 and 12) in this situation. Even without this we obtain a stable norm of behavior in which there are (almost) no collisions. I will come back to this after another example.

Suppose now that the place of pedestrian traffic that we are interested in has a very different F. Suppose that most people have a relatively strong preference for either left or right. For instance you can imagine a doorway that people need to get through before they then want to turn left or right pretty quickly after that. For these people, encountering each other in the doorway, the density f behind the cumulative distribution F is probably best described as being relatively high around low and high values of u and relatively low for medium values of u close to one half. Let us assume that F is symmetric around one half. Let us also assume that still there is almost no weight (in f) on values of u less than zero and larger than one. A picture of this situation:

Now what equilibria do we get here? Actually we get only one equilibrium and it is a mayhem equilibrium. It is cutoff equilibrium with cutoff x equal to one half, much as the mayhem equilibrium in the normal distribution case. But now the mayhem equilibrium is stable. Why? Because F is rather flat around the value of one half, if we consider a cut-off of y that is slightly larger than one half we have that F(y) < y and the best response cut-off is thus smaller than the y cut-off and we expect that the cut-off evolves back to a value of one half.

By the way, what I have described here is essentially the paper “Evolution in Bayesian Games II: Stability of Purified Equilibrium” by Bill Sandholm, Journal of Economic Theory, 136 (2007), 641-667.

Now you might say that we do not often observe such a stable mayhem equilibrium and you are probably right. In fact this is where we should finally introduce Goffman’s “devices” and “techniques” of “scanning” and “intention display” (Relations in Public, pages 11 and 12). The way I would model this (and this is now finally ongoing research I am currently undertaking with Yuval Heller at Bar Ilan University) is as follows. I would allow the players after they know their own u to send one from a set of possible messages to their opponent, to be understood as their “intention display”. I would assume both players are “scanning” for messages of their opponent and that the players can then condition where they try to pass their opponent on the two observed messages. You may want to think about this as players making a slight movement towards the left or right (this can be done a long time before the two actually meet) with the idea of signaling their intention as to where they would prefer to pass their opponent. What Yuval and I find so far is that for many distributions F (including the two I mentioned before) there is a universal and simple strategy (or norm) that is evolutionary stable. If you have a u less than one half you send the message to be read as “I intend to pass on the left” and if you have a u greater than one half you send the message to be read as “I intend to pass on the right”. If both send the same message they follow through with their displayed intentions. If they send different messages – that is, a slight conflict of interest is revealed – they fall back on a background norm of always passing on the left (or always passing on the right). We are not quite done yet with this project, but I hope you will be able to read about it very soon.

So how did game theory add to Goffman’s study? In many ways I think. First, we had to be very explicit about the various strategies (potential norms) people could be following in our model. Second, we can then explain why a specific norm among all the potential norms is expected (a stable equilibrium). Third, the formal analysis allows us to identify conditions under which different potential norms are stable or unstable. Fourth, we can now ask new questions. For instance, is a stable norm of behavior in pedestrian traffic efficient (maximizes the sum of utilities)? The answer, by the way, is typically no. And finally, the theory is so explicit in its predictions that it can be tested.

May 3, 2018

Evolutionary, Game Theory, Incomplete Information

Bayes Nash Equilibrium, Evolutionary Stability