I gave the following answer which I’m sharing with you in order to elicit broader commentary (ideally from people who actually know something about social choice theory):

Conceptually, defining “power” should be straightforward.

Borrowing from standard social choice terminology, under any Social Welfare Function, which maps from the set of all preference profiles (list of policy preference rankings, one for every member of the society) to a unique social preference ranking, if my preferences correlate more with the social preferences than yours (where correlation is defined in an appropriate way), I am more powerful.

In a dictatorship, the correlation is 1 if I’m the dictator.

In a democracy, the correlation is high if I’m the median voter, low if I’m a member of the political fringe.

Patriarchy is then a system in which men’s policy preferences are more highly correlated with the social preferences than women’s.

David raised the following problem with my definition:

Suppose one percent of the population prefer outcome A to outcome B, ninety-nine percent the other way around. The social preference function, in situations where it has to choose between the two, chooses A two percent of the time.

The group of people who prefer B have more power than the group who prefer A, but does it make sense to say that an individual member of the group has more power? Might it make more sense to use a definition in which the question is not whether the social choice function correlates with my preferences but whether a change in my preferences produces a change in the social choice function in the same direction?

I think that’s a very good point. So here is my updated definition of power:

*If changes in A’s preference ranking are more highly correlated with changes in the social preference ranking than changes in B’s preference ranking are, A is more powerful than B.*

Is this how people in social choice have always defined power? If not, is there a deep problem with this definition which didn’t occur to me?

]]>

Well, researchers write “papers” and these are published in journals. There are more than 1000 journals (in economics) with probably more than 50 papers per year each, so at least 50.000 papers a year, clearly much more than any single person can read. There is a lot of specialization in economic research and there are also huge quality differences between papers and also between journals. Publishing is not easy. You write a paper and then you submit it to a journal. An editor looks at your paper and either rejects it outright or sends it on to referees (other researchers). These referees evaluate your paper along the following criteria: correctness (of facts and of the logic / math / stats used), originality (you cannot publish something that is not new), importance, plausibility, readability, and whether the paper changes our view of something. If any of these criteria are not met, the paper will typically be rejected. Of course referees also make mistakes (it is not easy, for instance, to judge the importance of things that have never been done before). Different journals have different standards and there is a relatively clear ranking of journals (still often debated). A simple ranking (only of better journals) that I like is given in this blog post, but there are other, conceptually perhaps even more appropriate, ranking methodologies. Of course there are bad papers in good journals and good papers in bad journals, but if I am interested in a certain topic I would usually start by looking up (using google scholar) what I can find on this topic in the top 5 journals (American Economic Review, Econometrica, Journal of Political Economy, Review of Economic Studies, Quarterly Journal of Economics) in recent years, or ideally recent survey papers in the Journal of Economic Perspective and the Journal of Economic Literature, and then I would start reading in zig-zag fashion from there: If you don’t understand something in this paper read the papers that are cited in this paper and that are supposed to explain these things better and then come back to the original paper. You may also want to read some of the papers that cite this paper (use google scholar for this). These papers will include things that you don’t understand, so read those papers that help you to understand these and then come back again.

In **my opinion**, What should you do if you want to become an economics researcher? You should study economics (mostly to see what questions economists have studied and how they have attempted to answer them). You should study mathematics, because this is the language of economic theory. You should study statistics (which requires mathematics), because this is the toolset for doing empirical research. You should learn how to write (academically) and how to give talks and presentations, because this is how science is communicated to other scientists and the rest of the world (and all this in English). All of this, you learn to some extent in the courses in economics, but in order to become a researcher, this is probably not sufficient. You should take up courses from the math and stats departments as well. Some of my colleagues even do improv theater (just to improve their science communication skills). Basically you want to have a large toolkit so that you can analyse any economics problem with the appropriate tools and not only with the tools you happen to know.

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? And I am probably wrong about this anyway.

]]>Als Professor an unserer Universität hat Heinz D. Kurz Generationen von Studierenden begleitet und geprägt. Der Economics Club Graz ist stolz ihn als Mitglied führen zu dürfen. Seine aufschlussreichen und oft humorvollen Ausführungen zu den verschiedensten Theorien großer Ökonomen sind legendär – mein persönlicher Favorit: das Wildlachs-Zuchtlachs-Modell aus der Theory of Production.

Wir gratulieren herzlich!

]]>

Nochmal zur Wiederholung: Das Rätsel ist, warum trotz einer Senkung der Körperschaftssteuer von 34 auf 25% des Unternehmensgewinns die Einnahmen aus dieser Steuer konstant geblieben bzw. sogar leicht gestiegen sind.

Rudis Theorie lautet, dass der Effekt durch Umgründungen von Personengesellschaften in Kapitalgesellschaften entstanden ist. Die Steuerreform bietet einen Anreiz dafür. Um das zu verstehen betrachten wir eine Unternehmerin mit 100.000 Euro Jahresgewinn. Sie hat vereinfacht gesagt zwei Möglichkeiten diesen zu versteuern. Entweder sie versteuert ihn als Einzelunternehmerin gemäß ihrem persönlichen Einkommenssteuertarif. Damit zahlt sie ca. 45.000 Euro***** an Einkommenssteuern (der genaue Betrag hängt von vielen Details ab), also 45% von ihrem Bruttoeinkommen. Die andere Möglichkeit wäre eine Kapitalgesellschaft zu gründen. Dann würde sie 100.000 Euro als Unternehmensgewinn deklarieren, davon zunächst die Körperschaftssteuer und dann bei Auszahlung des Nettogewinns nochmal Kapitalertragssteuer abführen. Nachstehende Tabelle zeigt wie viel sie vor und nach der Steuerreform von 2005 zahlen müsste:

Bei einer Umgründung hätte unsere Unternehmerin vor der Reform etwa 5,500 Euro jährlich mehr an Steuern gezahlt. Nach der Reform erspart sie sich durch Umgründung ca. 1,250 Euro pro Jahr.

Tatsächlich kam es nach 2005 zu einer solchen Umgründungswelle. Laut diesem Artikel in der „Presse” (ich danke Timon für den Link!) stieg die Zahl der GmbHs im relevanten Zeitraum um 5,2% während die Zahl der Personengesellschaften nur um 1,4% gestiegen ist. Im Zuge dieser Umgründungswelle stiegen die Einnahmen aus der Körperschaftssteuer, aber gleichzeitig sanken die Einnahmen aus der veranlagen Einkommenssteuer.

Also ist das Rätsel damit gelöst? Ich glaube nicht. Hier ist mein empirischer Einwand gegen Rudis Theorie.

Wenn der Umgründungseffekt der einzige Effekt der Steuerreform gewesen wäre, hätten die Gesamteinnahmen aus Körperschafts-, Kapitalertrags- und Einkommenssteuer sinken müssen. Jede Umgründung bewirkt ja, dass in Summe weniger Steuern gezahlt werden, andernfalls würde sich eine Umgründung nicht auszahlen. Außerdem sinken die Einnahmen aus der KöSt der schon bestehenden Kapitalgesellschaften.

Die nachfolgende Grafik kann diese Hypothese nicht wirklich bestätigen. Sie zeigt die inflationsbereinigten Einnahmen aus den drei relevanten Steuerarten vor und nach der Reform. (Datenquellen: Steuerstatistik der “Statistik Austria” und Konsumentenpreisindex von Eurostat.)

Man sieht: die grünen Balken (Einkommenssteuer) werden nach 2005 etwas kürzer, die roten (Körperschaftssteuer) und blauen (Kapitalertragssteuer) etwas länger, aber die Gesamtlänge der Balken verändert sich kaum. Vergleicht man den Durchschnitt der Einnahmen aus den drei Jahren vor der Reform (2002-2004) mit dem Durchschnitt der drei Jahre nach der Reform (2005-2007) ergibt sich folgendes Bild: Zwar sanken die realen Einnahmen aus der ESt um 443 Mio., jedoch wurde diese Abnahme mehr als kompensiert durch den Anstieg der Einnahmen aus KöSt (+585 Mio.) und KESt (+483 Mio.). In Summe stiegen die Gesamteinnahmen real um 625 Mio. also um ca. 6%.

Natürlich gab es im betrachteten Zeitraum jede Menge anderer Ereignisse, die die Steuereinnahmen beeinflusst haben könnten. Unter anderem gab es einen allgemeinen Konjunkturaufschwung, der besonders in den Jahren 2007 und 2008 die Unternehmensgewinne sprudeln ließ. Aber auch wenn man die Zeitreihe weiterzieht, sieht man keinen langfristigen Abwärtstrend in den Unternehmenssteuern.

Ich schließe daraus, dass die Senkung der KöSt sehr wohl einen positiven Effekt über die Umgründungen hinaus gehabt haben muss. Wie genau dieser Effekt ausgesehen haben könnte, würde eine genauere Untersuchung verlangen. Vielleicht möchte ja eine oder einer unserer emsigen VWL-Studierenden eine Bachelor-Arbeit dazu schreiben.

Bis dahin sage ich mal ein vorsichtiges Hurra für die Laffer-Kurve!

***) Berichtigung**: Der sachkundige Martin Kremsner schreibt, dass meine Berechnung der Einkommenssteuerlast fehlerhaft ist. Bei einem Einkommen von 100,000 Euro wäre die Steuerlast laut damaligem ESt-Tarif ca. 40,000 Euro. D.h. der steuerliche Anreiz zur Umgründung würde hier nicht bestehen. Ich habe das Zahlenbeispiel so gelassen um das zugrundeliegende Prinzip von Rudis Theorie zu erläutern. Ein besser gewähltes Beispiel wäre ein Unternehmen mit 200,000 Euro Gewinn, deren ESt-Last ca. 90,000 Euro betragen würde. Für so ein Unternehmen betrug die Steuerlast aus KöSt und KESt vor der Reform 101.000 Euro und danach 87.500 Euro. https://www.bruttonettorechner.at/einkommensteuer

]]>

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.

]]>Die folgende Grafik zeigt drei breite Steueraggregate im Verhältnis zum Bruttoinlandsprodukt: Einkommenssteuern (inkludiert Lohn- und Einkommenssteuer von Haushalten sowie Steuern auf das Einkommen von Kapitalgesellschaften), Gütersteuern (inkl. Mehrwertsteuern, Mineralölsteuer, Tabaksteuer, u. dgl.) und die Gesamtsumme aus Steuern und Sozialabgaben.

Was ich so erstaunlich daran finde? Wie flach alle diese Linien sind!

Das Gesamtsteueraufkommen bewegt sich immer um die 43% des BIP. Je 12% davon kommen von Einkommenssteuern und Gütersteuern, weiter 15% kommen aus Sozialbeiträgen und der Rest auf vermögensbezogene Steuern.

Und an dieser Struktur hat sich in den letzten zwei Jahrzehnten nichts verändert. Gar nichts. Weder die schwarz-blaue „Wende-Regierung“ zwischen 2000 und 2006, noch die „Große Koalition“ zwischen 2007 und 2016 lässt sich an dieser Grafik wirklich erkennen. In jeder Nationalratswahl wurde mindestens eine „Große Steuerreform“ angekündigt. Es gab drei Konjunkturzyklen und eine schwere Rezession. Nichts ist davon zu sehen.

Noch erstaunlicher: Wenn wir uns unsere europäischen Nachbarn anschauen, stellen wir fest, dass die fast genau dieselbe Steuerstruktur haben. Deutschland, Italien, Frankreich besteuern ihre Bürger auf ziemlich genau dieselbe Art und Weise – trotz erheblicher Unterschiede in der politischen Verfassung und der Struktur der Volkswirtschaft. (Die Schweiz stellt insofern eine Ausnahme dar, als sie wesentlich geringere Sozialabgaben haben, aber die Aufteilung von Einkommens- und Gütersteuern ist fast ident zu Österreich.)

Eine weitere erstaunliche Geschichte erzählen die Daten der Körperschaftssteuern:

Sehen Sie was 2005 passiert ist? Nein? Keine Sorge, ich auch nicht. Und das, obwohl in diesem Jahr der Körperschaftssteuersatz von 34% auf 25% gesenkt wurde! Der Steuersatz wurde um 9%-Punkte gesenkt, aber das Steueraufkommen blieb danach wie davor bei 2,3% des BIP. Die Laffer-Kurve lässt grüßen!

Der österreichische Lohn- und Einkommenssteuertarif wurde mehrmals verändert. Die Progressionsstufen wurden angepasst, die Grenzsteuersätze neu bestimmt… und am Ende kamen immer 10% vom BIP raus.

Die Sozialversicherungssyteme in Italien, Spanien und Griechenland sind in ihren Details sicher sehr unterschiedlich. Im Aggregat nehmen sie alle genau dieselben 13% vom BIP in Anspruch – nur 2%-Punkte weniger als Deutschland und Österreich.

Ich habe keine Ahnung was das alles bedeutet. Aber ich finde es faszinierend.

]]>

I am continuing with my game theory inspired by Goffman’s work. 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 , and type B, then obviously . 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) . 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):

If this is the game and there is nothing else, and if , 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.

]]>

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

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

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

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

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

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

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

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

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

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

It feels good to write this sort of thing.

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

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

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

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

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

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

]]>I then looked at the expected return to a one Euro bet on the various countries winning the world cup under the assumption that gracenote’s estimates are completely correct. So if you believe in gracenote’s estimates as the abolute truth, what should you bet on?

Well, Brazil is the favorite according to gracenote but also in the betting odds. Gracenote gives them a 21% chance of winning the world cup, and at current best odds of 9:2 you would win 4,50 Euros if you put 1 Euro on Brazil. This means you would expect to get 4,5 * 0,21 = 0,945 Euros back. So if you are risk averse or risk neutral you should not bet on Brazil at these odds, but if you had to you could put a Euro on Brazil. Germany has similar best odds of 5:1, but gracenote does not rate them so highly, giving Germany only an 8% chance of winning. So you would only expect to win back 5*0,08=0,40 Euros for every Euro you place on Germany. This means that, if you could, you should “short sell” Germany to make money in expectation. This is not so easy to do in sports betting markets so let’s not pursue this here. It turns out that most of the better teams are not rated as highly by gracenote as they are in the betting odds.

So, again, what should you bet on if you believe in gracenote’s estimates? According to gracenote Peru has a 5% chance of winning the world cup. At current odds of 325:1 you would get an expected payout of 325*0,05=16,25 Euros for every Euro you put on them. This is an expected return better than anything you can get on the stock market I would guess. Mexico, Switzerland, Colombia (with expected payout of 3,75, 3,50, and 2,60 Euros for every Euro you put on them) are also high return bets.

I am afraid, though, that I believe in the efficiency of sport betting markets much more than in one sports data company’s estimates, so I will not follow these suggestions myself. If you want to know more about the efficiency or inefficiency of betting markets a good starting point would be a 1988 survey by Thaler and Ziemba.

One day later, on the 8th of June, I noticed that Peru’s odds have gone down to 200:1. Perhaps this was a reaction to the new information provided by gracenote (although I am not quite sure when their estimates were posted). You would, however, still make an expected winning of 200*0,05 = 10 Euros for every Euro you put on Peru if you believe gracenote’s estimates.

]]>

Is it because the Italian government was fiscally irresponsible, spending too much and taxing too litte? Or is it because investors demand such high interest rates on Italian government bonds? Or is it a consequence of Italy’s dismal economic performance in recent years?

To answer this question, we can take a simple decomposition of the debt-to-GDP ratio. First, remember the government budget constraint:

where B is public debt, G is spending, T is revenue and r is the interest rate. Second, take the time derivative of the debt-to-GDP ratio

Combine the two equations and denote the GDP growth rate dY/Y by g:

This equation allows us to decompose the total change in the debt-to-GDP ratio into a primary deficit component, an interest component and a growth component. The graph below shows this composition for Italy during the pre-crisis period (2000-2008) and the post-crisis period (2009-now).

In the years between the introduction of the euro and the financial crisis, Italy’s debt ratio decreased slightly by about 2 percent of GDP. During the years after the crisis, it increased by almost 30 percent of GDP.

What changed? As you can see by looking at the yellow and blue areas in the graph, it wasn’t interest payments or the primary surplus. Interest payments were around 5 percent of GDP both before and after the crisis and the Italian actually ran a primary surplus in both periods. What changed was the green area: the recent rise in the debt ratio is almost entirely due to Italy’s shrinking economy.

]]>