# On the probabilities of winning the world cup

I just read an article on the bbc about sports data company gracenote’s estimates of countries’ winning probabilities for the upcoming soccer world cup. I then looked up the best current betting odds on oddschecker. These are, of course, subject to change. I looked at them on the morning (Pacific Time) of the 7th of June.

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.

# Italy’s public debt, decomposed

Why is Italy’s debt so high?

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: $\displaystyle dB = G - T + rB.$

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 $\displaystyle d\left(\frac{B}{Y}\right) = \frac{dB}{Y} - \frac{B}{Y}\frac{dY}{Y}.$

Combine the two equations and denote the GDP growth rate dY/Y by g: $\displaystyle d\left(\frac{B}{Y}\right) = \frac{G-T}{Y} + (r-g)\frac{B}{Y}.$

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.

# A mistake in probability theory in David Hume’s “Of Miracles”

When should a rational individual believe in a miracle?

David Hume, the great skeptical philosopher, answered: practically never. His argument ran as follows: Miracles are extremely rare events and thus have a very low prior probability. On the other hand,  people can be misled rather easily either by their own senses or by other people. Therefore, the rational reaction to hearing a miracle story is to reject it, except the evidence supporting it is overwhelming. “Extraordinary events require extraordinary evidence” became a popular summary of Hume’s point of view.

Here is a famous passage from Hume’s “Of Miracles” explaining the point:

When anyone tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that the fact, which he relates, should really have happened. I weigh the one miracle against the other; and according to the superiority, which I discover, I pronounce my decision, and always reject the greater miracle.

This argument sounds intuitively plausible and compelling, but it is mistaken. In fact Hume is committing an elementary error in probability theory, which shouldn’t be held against him since “Of Miracles” predates the writings of Bayes and Laplace.

In the language of modern probability theory, Hume asks us as to compare the prior probability that miracle X occurred, $\displaystyle Pr(X)$, to the probability of seeing the evidence Y supporting miracle X even though X did not in fact occur, i.e. the conditional probability of Y given the negation of X, $\displaystyle Pr(Y | \neg X).$ Econometricians would call the latter the likelihood of Y under the hypothesis not-X. If $\displaystyle Pr(X) < Pr(Y | \neg X),$ Hume says we should reject X in favor of not-X.

But this inference is unwarranted. What a rational observer ought to ask is: Given the evidence Y, is it more likely that X occurred or that it didn’t occur? We are looking for the posterior odds of X conditional on Y: $\displaystyle \frac{ Pr(X | Y)} { Pr(\neg X | Y) }.$

Bayes’ theorem immediately gives us what we are looking for: $\displaystyle \frac{ Pr(X | Y)} { Pr(\neg X | Y) } = \frac{ Pr(Y | X) }{Pr(Y | \neg X) } \frac{ Pr(X) }{Pr(\neg X)}$

This equation makes it clear that even if Hume’s inequality $\displaystyle Pr(X) < Pr(Y | \neg X),$ holds, it is possible that the posterior odds of X are greater than 1. All we need for such as result is that the likelihood of having evidence Y under the hypothesis that X occurred is sufficiently higher than the likelihood of Y under the alternative hypothesis that X did not occur. In econometric terms, the likelihood ratio must exceed a critical value which depends on the prior odds against the miracle: $\displaystyle \frac{ Pr(Y | X) }{ Pr(Y | \neg X) } > \frac{ Pr(\neg X) }{ Pr(X) }.$

To conclude: A rational observer is justified in believing a miracle if the evidence for it is sufficiently more likely under the hypothesis that the miracle really did occur than under the hypothesis that it didn’t so as to offset the low prior odds for the miracle. Just comparing the low prior probability of a miracle to the probability of receiving false evidence in favor of it is not enough and can be misleading.

# The game theory of everyday life – gallantry

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

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

# America vs. Europe: some personal observations

I’ve spent a lot of time in the US in recent years, and will spend much more time there next year. I get asked a lot about the differences between living in America and in Europe, which caused me to compile a list of the differences that I found most striking. Some of them may be clichés, others may be more surprising. All of them are true, I believe, but my confidence in their truth is not uniform. I have discussed many them with friends and colleagues on both sides of the Atlantic, and found them controversial to varying degrees. Some of these differences have an obvious explanation, others seem hard to understand or even puzzling. I’m looking for economic explanations of these differences and might blog about that in the future. Comments are highly welcome!

An obvious caveat: My observations are biased due to the fact that I’ve only seen parts of America (California, part of the Northwest, part of the Midwest, and New England), and only parts of Europe (Central Europe, part of Scandinavia, part of France and England). One should realize that large cultural, political and economic differences exist both within the US and Europe, so all the statements below refer to averages with wide confidence bands around them.

Economic life:

1. Prices are normally stated net of sales taxes. (high confidence, uncontroversial)
2. Tips are higher and more common. (high confidence, uncontroversial)
3. Tap water is much more heavily chlorinated. (high confidence, uncontroversial)
4. Air-conditioning is vastly more common both in private homes and offices. (high confidence, uncontroversial)
5. The average quality of houses is much lower. (high confidence, somewhat controversial)
6. The proportion of people living in single-family houses as opposed to apartment buildings is much higher. (medium confidence, uncontroversial)
7. The price of gasoline is about 50% lower. (high confidence, uncontroversial)
8. There are both more cars per person and cars are much bigger on average. (medium confidence, uncontroversial)
9. Automatic cars are vastly more common. (high confidence, uncontroversial)
10. The price of necessities (food, clothing, personal hygiene) is lower, but not much, and the quality is generally lower. (low confidence, highly controversial)
11. Food is bought and sold in much larger quantities. For instance, the smallest available bottle size for milk is usually half a gallon (about 1.9 liters). (high confidence, uncontroversial)
12. Extreme obesity is vastly more common. (high confidence, uncontroversial)
13. The quality of infrastructure (roads, railways, electricity grid) is lower. (medium confidence, somewhat controversial)
14. Roads are more often built in strictly rectangular patterns, both in cities and on the countryside. (high confidence, uncontroversial)

Social life:

1. Racial diversity is immensely higher, especially in urban areas, but also in rural areas. (high confidence, uncontroversial)
2. So is religious diversity. (high confidence, uncontroversial)
3. Religion plays a more central part of public life, including in politics (high confidence, uncontroversial)
4. Interest in family history and genealogy is much higher. (high confidence, uncontroversial)
5. Bodily contact between people in everyday interactions is much less frequent and more often regarded as inappropriate. (medium confidence, somewhat controversial)
6. Conversations are much less formal both in professional and private contexts. (medium confidence, uncontroversial)
7. Small talk is a much more important part of everyday life both in professional and private contexts. (high confidence, uncontroversial)
8. Adolescents and young adults seem to be more mature both in terms of physical appearance and character development. (low confidence, highly controversial)
9. Elderly people seem to be more familiar with, and more adept at, using new technology as well as social media. (high confidence, somewhat controversial)
10. Knowledge about foreign countries (geography, history, politics) is generally much poorer. (high confidence, highly controversial)
11. Patriotism is more wide-spread, more frequently expressed and more strongly felt. (high confidence, somewhat controversial)

Political life:

1. There are more elected (as opposed to appointed) public officials and elections occur at higher frequencies. (high confidence, uncontroversial)
2. Personal political opinions are more frequently expressed in public. For instance, pumper stickers with political messages are a much more common sight. (medium confidence, somewhat controversial)
3. Political polarization is more profound. (medium confidence, somewhat controversial)
4. There are more prohibition and warning signs on the streets as well as in public and private buildings and facilities. (low confidence, highly controversial)
5. There are more local political initiatives  such as petitions, awareness campaigns, fund raising events etc. (medium confidence, somewhat controversial)

# The game theory of everyday life – 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.

# The game theory of everyday life inspired by the work of Erving Goffman – Introduction

In April 2018 I spent a week at the Research Center for Social Complexity (CICS in Spanish) at the Universidad del Desarrollo (UDD) teaching a PhD research course on game theoretic modelling. The idea of this course, developed together with Carlos Rodriguez-Sickert, was to make it an experiential course of model building from question to model. We would start by reading parts of chapters of two books by Erving Goffman that deal with how people interact in public places and then attempt to provide game theoretic models of what we read.