Statisticians Count Euros and Find More Than Money
By OTTO POHL
BERLIN When the euro was introduced in 12 countries last Jan. 1, politicians
called it the herald of a new era in European integration the advent of
a common currency that would help break down political and cultural barriers
as well as economic ones. But for mathematicians, the euro has turned
those 12 countries (plus the three tiny states of the Vatican, Monaco and
San Marino) into an almost ideal laboratory, providing a eurosandcents
way to measure the extent of European integration and perhaps to understand
other phenomena as well, from the course of epidemics to the movement of
stock prices.
While the euro is worth the same in every country that uses it, each
one mints its own euro coins, with a distinctive design on the reverse. Every
time a euro coin from Finland appears in Greece, for example, it provides
a tiny but precise data point about the relationship between the countries.
So to find out more about their continent, thousands of Europeans are digging
through their pockets and keeping track of their coins, and mathematicians
are tabulating the results in order to figure out just how fast the coins
are spreading. "This is a historically unique opportunity," said Dr.
Dietrich Stoyan, a statistics professor at the University of Freiberg in
Germany, who is in charge of one of the coincounting projects. But the goal
is not just to learn more about Europe. "I hope that studying this process
will help people studying epidemics," Dr. Stoyan said. What makes this special
is that the precise launching date of the coins is known. "We know when this
`epidemic' broke out," he noted. Moreover, the total amount minted
was roughly proportional to each country's economy compared with the overall
European economy. Germany, an economic powerhouse, minted 32.6 percent of
all coins, France about 15 percent, and tiny Luxembourg (which used the face
of its Grand Duke Henri) just twotenths of 1 percent. (All of the euro notes
look the same.) Each country introduced its own coins exclusively within
its borders. On New Year's Day, the coins, all of them valid across
the entire euro zone, began to spread across national borders. Now, statisticians
are looking forward to the summer travel season, when German tourists can
simply pay for their espresso in Paris with the coins they received in change
that morning at the baker's, back home in Munich. But how does one
model this mingling? Is it simple diffusion, like opening 12 cans of gas
in a giant room and watching the molecules mix? Or does the relationship
between each country depend on a complex set of equations between each country
that considers the distance between the countries and the number of commuters,
travelers and bank trucks going back and forth? Dr. Stoyan has gone
the complex route. His model is composed of 144 interdependent differential
equations that take as many of the known variables into account as possible,
including traveler volume data from travel bureau associations. Too
many variables, says a group from the University of Amsterdam, which has
chosen simply to take a highlevel look at the coin flow. Its model, based
on a branch of probability theory called Markov chains, assumes that a relatively
constant percentage of Dutch coins will leave the Netherlands each month,
and that a different, smaller percentage of Dutch coins that have already
left the country will return. "It's just a guess," said Misja Nuyens, a doctoral
candidate in probability theory at the University of Amsterdam, who is a
member of the group. "We'll see if we were right, or sadly mistaken."
Each experiment has a Web site for participants to post the contents of their
wallets. Data are tabulated and posted as the experiments go on. So
far, Dr. Stoyan and the Amsterdam group have been surprised to learn that
largedenomination coins for one and two euros move much faster than smaller
ones. Neither knows exactly why that is. Dr. Stoyan hypothesizes that people
tend to dump their small coins out of their pockets at the end of the day
and are therefore less likely to take them traveling. Mr. Nuyens believes
that the coins are used much more often. And when will the coins reach
statistical equilibrium? The models have yielded different results. The Dutch
group believes that half of all coins in Holland will be of foreign origin
a year from now and that statistical equilibrium across Europe will be reached
in five to seven years. "I don't think it will go that fast," said Dr. Stoyan, whose differential equation model predicts it will take several decades.
Dr. Ger Koole, a mathematics professor at the Free University in Amsterdam
who is involved with the Dutch project, hopes the findings will help expand
understanding of applied mathematics. Phenomena like the "on hold"
times at call centers and stock price movements, he said, are similar to
the movements of euro coins: systems whose development depends largely or
entirely on the previous state the system was in. For example, to
predict the number of calls that will be on hold, you need to know how many
calls were on hold in the previous minute and then calculate the probability
that calls will have been added (new calls coming in) or have been removed
(calls answered or hung up). These simple calculations, repeated for every
time period in question, quickly can predict the development of complex systems.
Dr. Koole says it is too early to know what will be learned from the euro
experiments. The only way to improve understanding of these phenomena is
"the constant application of theory to see what you end up with," he said.
The groups agree that the experiment has so many unknowns that it is difficult
to gain scientific clarity, at least at this point. "I have the distinct
impression that a real mania has broken out among coin collectors," Dr. Stoyan
said, which would take rare coins disproportionately more often out of circulation.
In addition, the selfselected group of participants who log their coins
each month is not necessarily a statistically representative sample.
The best data sources have turned out to be classrooms. The two groups report
that science and math teachers have latched onto the projects as a way to
illustrate their subjects. They either ask all of the students to examine
the change in their pockets or buy rolls of coins at the bank and count them
in class. Each team has a different model for the effect of the summer
travel season. The Dutch expect that July and August, the two busiest months,
will each count as two or three offseason months. Dr. Stoyan's model integrates
the increased number of travelers. The two teams are hoping for a spike in
interest in the projects, which will get even more coins counted and improve
the results. Dr. Stoyan, though, is already dreaming of one change
that would really make a difference. "Now if I could just get them to send
me the actual money," Dr. Stoyan said with a laugh.
