Author: IMC

Credit: Unsplash/CC0 Public Domain

In just a few months, voters across America will head to the polls to decide who will be the next U.S. president. A new study draws on mathematics to break down how humans make decisions like this one.

The researchers, including Zachary Kilpatrick, an applied mathematician at CU Boulder, developed mathematical tools known as models to simulate the deliberation process of groups of people with various biases. They found that decision-makers with strong, initial biases were typically the first ones to make a choice.

“If I want good quality feedback, maybe I should look to people who are a little bit more deliberate in their decision making,” said Kilpatrick, a co-author of the new study and associate professor in the Department of Applied Mathematics. “I know they’ve taken their due diligence in deciding.”

The researchers, led by Samatha Linn of the University of Utah, published their findings August 12 in the journal Physical Review E.

In the team’s models, mathematical decision-makers, or “agents,” gather information from the outside world until, ultimately, they make a choice between two options. That might include getting pizza or Thai food for dinner or coloring in the bubble for one candidate versus the other.

The team discovered that when agents started off with a big bias (say, they really wanted pizza), they also made their decisions really quickly—even if those decisions turned out to run contrary to the available evidence (the Thai restaurant got much better reviews). Those with smaller biases, in contrast, often took so long to deliberate that their initial preconceptions were washed away entirely.

The results are perhaps not surprising, depending on your thoughts about human nature. But they can help to reveal the mathematics behind how the brain works when it needs to make a quick choice in the heat of the moment—and maybe even more complicated decisions like who to vote for.

“It’s like standing on a street corner and deciding in a split second whether you should cross,” he said. “Simulating decision making gets a little harder when it’s something like, ‘Which college should I go to?'”

Pouring water

To understand how the team’s mathematical agents work, it helps to picture buckets. Kilpatrick and his colleagues typically begin their decision-making experiments by feeding their agents information over time, a bit like pouring water into a mop pail. In some cases, that evidence favours one decision (getting pizza for dinner), and in others, the opposite choice (Thai food). When the buckets fill to the brim, they tip over, and the agent makes its decision.

In their experiment, the researchers added a twist to that set up: They filled some of their buckets part way before the simulations began. Those agents, like many humans, were biased.

The team ran millions of simulations including anywhere from 10 to thousands of agents. The researchers were also able to predict the behaviour of the most and least biased agents by hand using pen, paper and some clever approximations.

A pattern began to emerge: The agents that started off with the biggest bias, or were mostly full of water to begin with, were the first to tip over—even when the preponderance of evidence suggested they should have chosen differently. Those agents who began with only small biases, in contrast, seemed to take time to weigh all of the available evidence, then make the best decision available.

“The slowest agent to make a decision tended to make decisions in a way very similar to a completely unbiased agent,” Kilpatrick said. “They pretty much behaved as if they started from scratch.”

Neighbourhood choices

He noted that the study had some limitations. In the team’s experiments, for example, none of the agents knew what the others were doing. Kilpatrick compared it to neighbours staying inside their homes during an election year, not talking about their choices or putting up yard signs. In reality, humans often change their decisions based on the actions of their friends and neighbours.

Kilpatrick hopes to run a similar set of experiences in which the agents can influence each other’s behaviours.

“You might speculate that if you had a large group coupled together, the first agent to make a decision could kick off a cascade of potentially wrong decisions,” he said.

Still, political pollsters may want to take note of the team’s results.

“The study could also be applied to group decision making in human organizations where there’s democratic voting, or even when people give their input in surveys,” Kilpatrick said. “You might want to look at folks carefully if they give fast responses.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Daniel Strain, University of Colorado at Boulder

Australian students’ performance and engagement in mathematics is an ongoing issue.

International studies show Australian students’ mean performance in maths has steadily declined since 2003. The latest Program for International Student Assessment (PISA) in 2018 showed only 10% of Australian teenagers scored in the top two levels, compared to 44% in China and 37% in Singapore.

Despite attempts to reform how we teach maths, it is unlikely students’ performance will improve if they are not engaging with their lessons.

What teachers, parents, and policymakers may not be aware of is research shows students are using “non-thinking behaviours” to avoid engaging with maths.

That is, when your child says they didn’t do anything in maths today, our research shows they’re probably right.

What are non-thinking behaviours?

There are four main non-thinking behaviours. These are:

• slacking: where there is no attempt at a task. The student may talk or do nothing
• stalling: where there is no real attempt at a task. This may involve legitimate off-task behaviours, such as sharpening a pencil
• faking: where a student pretends to do a task, but achieves nothing. This may involve legitimate on-task behaviours such as drawing pictures or writing numbers
• mimicking: this includes attempts to complete a task and can often involve completing it. It involves referring to others or previous examples.

Peter Liljedahl studied Canadian maths lessons in all years of school, over 15 years. This research found up to 80% of students exhibit non-thinking behaviours for 100% of the time in a typical hour-long lesson.

The most common behaviour was mimicking (53%), reflecting a trend of the teacher doing all the thinking, rather than the students.

It also found when students were given “now you try one” tasks (a teacher demonstrates something, then asks students to try it), the majority of students engaged in non-thinking behaviours.

Australian students are ‘non-thinking’ too

Tracey Muir conducted a smaller-scale study in 2021 with a Year ¾ class.

Some 63% of students were observed engaged in non-thinking behaviours, with slacking and stalling (54%) being the most common. These behaviours included rubbing out, sharpening pencils, and playing with counters, and were especially prevalent in unsupervised small groups.

One explanation for students slacking and stalling is teachers are doing most of the talking and directing, and not providing enough opportunities for students to think.

How can we build “thinking” maths classrooms and reduce the prevalence of non-thinking behaviours?

Here are two research-based ideas.

Form random groups

Often students are placed in groups to work through new skills or lessons. Sometimes these are arranged by the teacher or by the students themselves.

Students know why they have been placed in groups with certain individuals (even if this is not explicitly stated). Here they tend to “live down” to expectations.

If they are with their friends they also tend to distract each other.

Our studies found random groupings improved students’ willingness to collaborate, reduced social stress often caused by self-selecting groups, and increased enthusiasm for mathematics learning.

As one student told us:

“I’m starting to like maths now, and working with random people is better for me so I don’t get off track.”

Get kids to stand up

Classroom learning is often done at desks or sitting on the floor. This encourages passive behaviour and we know from physiology that standing is better than sitting. But we found groups of about three students standing together and working on a whiteboard can promote thinking behaviours. Just the physical act of standing can eliminate slacking, stalling, and faking behaviours. As one student said, “Standing helps me concentrate more because if I’m sitting down I’m just fiddling with stuff, but if I’m standing up, the only thing you can do is write and do maths. ”

The additional strategy of only allowing the student with the pen to record others’ thinking and not their own, has shown to be especially beneficial. As one teacher told us: “The people that don’t have the pen have to do the thinking […] so it’s a real group effort and they don’t have the ability to slack off as much.”

Simple changes can work

While our studies were conducted in maths classrooms, our strategies would be transferable to other discipline areas.

So, while parents and educators may feel concerned about Australia’s declining mathsresults, by introducing simple changes to the classroom, we can ensure students are not only learning and thinking deeply about mathematics, but hopefully, enjoying it, too.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Tracey Muir and Peter Liljedahl, The Conversation

Around the world, political violence increased by 27% last year, affecting 1.7 billion people. The numbers come from the Armed Conflict Location & Event Data Project (ACLED), which collects real-time data on conflict events worldwide.

Some armed conflicts occur between states, such as Russia’s invasion of Ukraine. There are, however, many more that take place within the borders of a single state. In Nigeria, violence, particularly from Boko Haram, has escalated in the past few years. In Somalia, populations remain at risk amidst conflict and attacks perpetrated by armed groups, particularly Al-Shabaab.

To address the challenge of understanding how violent events spread, a team at the Complexity Science Hub (CSH) created a mathematical method that transforms raw data on armed conflicts into meaningful clusters by detecting causal links.

“Our main question was: what is a conflict? How can we define it?,” says CSH scientist Niraj Kushwaha, one of the co-authors of the study published in the latest issue of PNAS Nexus. “It was important for us to find a quantitative and bias-free way to see if there were any correlations between different violent events, just by looking at the data.”

“We often tell multiple narratives about a single conflict, which depend on whether we zoom in on it as an example of local tension or zoom out from it and consider it as part of a geopolitical plot; these are not necessarily incompatible,” explains co-author Eddie Lee, a postdoctoral fellow at CSH. “Our technique allows us to titrate between them and fill out a multiscale portrait of conflict.”

In order to investigate the many scales of political violence, the researchers turned to physics and biophysics for inspiration. The approach they developed is inspired by studies of stress propagation in collapsing materials and of neural cascades in the brain.

Kushwaha and Lee used data on violent battles in Africa between 1997 and 2019 from ACLED. In their analysis, they divided the geographic area into a grid of cells and time into sequential slices. The authors predicted when and where new battles would emerge by analysing the presence or absence of battles in each cell over time.

“If there’s a link between two cells, it means a conflict at one location can predict a conflict at another location,” explains Kushwaha. “By using this causal network, we can cluster different conflict events.”

Snow and sandpile avalanches

Observing the dynamics of the clusters, the scientists found that armed clashes spread like avalanches. “In a way evocative of snow or sandpile avalanches, a conflict originates in one place and cascades from there. There is a similar cascading effect in armed conflicts,” explains Kushwaha.

The team also identified a “mesoscale” for political violence —a time scale of a few days to months and a spatial scale of tens to hundreds of kilometers. Violence seems to propagate on these scales, according to Kushwaha and Lee.

Additionally, they found that their conflict statistics matched those from field studies such as in Eastern Nigeria, Somalia, and Sierra Leone. “We connected Fulani militia violence with Boko Haram battles in Nigeria, suggesting that these conflicts are related to one another,” details Kushwaha. The Fulani are an ethnic group living mainly in the Sahel and West Africa.

Policymakers and international agencies could benefit from the approach, according to the authors. The model could help uncover unseen causal links in violent conflicts. Additionally, it could one day help forecast the development of a war at an early stage. “By using this approach, policy decisions could be made more effectively, such as where resources should be allocated,” notes Kushwaha.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Complexity Science Hub Vienna

We’ve all been there: staring at a math test with a problem that seems impossible to solve. What if finding the solution to a problem took almost a century? For mathematicians who dabble in Ramsey theory, this is very much the case. In fact, little progress had been made in solving Ramsey problems since the 1930s.

Now, University of California San Diego researchers Jacques Verstraete and Sam Mattheus have found the answer to r(4,t), a longstanding Ramsey problem that has perplexed the math world for decades.

What was Ramsey’s problem, anyway?

In mathematical parlance, a graph is a series of points and the lines in between those points. Ramsey theory suggests that if the graph is large enough, you’re guaranteed to find some kind of order within it—either a set of points with no lines between them or a set of points with all possible lines between them (these sets are called “cliques”). This is written as r(s,t) where s are the points with lines and t are the points without lines.

To those of us who don’t deal in graph theory, the most well-known Ramsey problem, r(3,3), is sometimes called “the theorem on friends and strangers” and is explained by way of a party: in a group of six people, you will find at least three people who all know each other or three people who all don’t know each other. The answer to r(3,3) is six.

“It’s a fact of nature, an absolute truth,” Verstraete states. “It doesn’t matter what the situation is or which six people you pick—you will find three people who all know each other or three people who all don’t know each other. You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other.”

What happened after mathematicians found that r(3,3) = 6? Naturally, they wanted to know r(4,4), r(5,5), and r(4,t) where the number of points that are not connected is variable. The solution to r(4,4) is 18 and is proved using a theorem created by Paul Erdös and George Szekeres in the 1930s.

Currently r(5,5) is still unknown.

A good problem fights back

Why is something so simple to state so hard to solve? It turns out to be more complicated than it appears. Let’s say you knew the solution to r(5,5) was somewhere between 40–50. If you started with 45 points, there would be more than 10²³⁴ graphs to consider.

“Because these numbers are so notoriously difficult to find, mathematicians look for estimations,” Verstraete explained. “This is what Sam and I have achieved in our recent work. How do we find not the exact answer, but the best estimates for what these Ramsey numbers might be?”

Math students learn about Ramsey problems early on, so r(4,t) has been on Verstraete’s radar for most of his professional career. In fact, he first saw the problem in print in Erdös on Graphs: His Legacy of Unsolved Problems, written by two UC San Diego professors, Fan Chung and the late Ron Graham. The problem is a conjecture from Erdös, who offered $250 to the first person who could solve it.

“Many people have thought about r(4,t)—it’s been an open problem for over 90 years,” Verstraete said. “But it wasn’t something that was at the forefront of my research. Everybody knows it’s hard and everyone’s tried to figure it out, so unless you have a new idea, you’re not likely to get anywhere.”

Then about four years ago, Verstraete was working on a different Ramsey problem with a mathematician at the University of Illinois-Chicago, Dhruv Mubayi. Together they discovered that pseudorandom graphs could advance the current knowledge on these old problems.

In 1937, Erdös discovered that using random graphs could give good lower bounds on Ramsey problems. What Verstraete and Mubayi discovered was that sampling from pseudorandom graphs frequently gives better bounds on Ramsey numbers than random graphs. These bounds—upper and lower limits on the possible answer—tightened the range of estimations they could make. In other words, they were getting closer to the truth.

In 2019, to the delight of the math world, Verstraete and Mubayi used pseudorandom graphs to solve r(3,t). However, Verstraete struggled to build a pseudorandom graph that could help solve r(4,t).

He began pulling in different areas of math outside of combinatorics, including finite geometry, algebra and probability. Eventually he joined forces with Mattheus, a postdoctoral scholar in his group whose background was in finite geometry.

“It turned out that the pseudorandom graph we needed could be found in finite geometry,” Verstraete stated. “Sam was the perfect person to come along and help build what we needed.”

Once they had the pseudorandom graph in place, they still had to puzzle out several pieces of math. It took almost a year, but eventually they realized they had a solution: r(4,t) is close to a cubic function of t. If you want a party where there will always be four people who all know each other or t people who all don’t know each other, you will need roughly t³ people present. There is a small asterisk (actually an o) because, remember, this is an estimate, not an exact answer. But t³ is very close to the exact answer.

The findings are currently under review with the Annals of Mathematics. A preprint can be viewed on arXiv.

“It really did take us years to solve,” Verstraete stated. “And there were many times where we were stuck and wondered if we’d be able to solve it at all. But one should never give up, no matter how long it takes.”

Verstraete emphasizes the importance of perseverance—something he reminds his students of often. “If you find that the problem is hard and you’re stuck, that means it’s a good problem. Fan Chung said a good problem fights back. You can’t expect it just to reveal itself.”

Verstraete knows such dogged determination is well-rewarded: “I got a call from Fan saying she owes me $250.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to University of California – San Diego

What influences the choices we make, and what role does the behaviour of others have on these choices? These questions underlie many aspects of human behaviour, including the products we buy, fashion trends, and even the breed of pet we choose as our companion.

Now, a new Stanford study that used population and statistical models to analyse the frequency of specific moves in 3.45 million chess games helps reveal the factors that influence chess players’ decisions. The researchers’ analysis of chess games revealed three types of biases described by the field of cultural evolution, which uses ideas from biology to explain how behaviours are passed from person to person. Specifically, they found evidence of players copying winning moves (success bias), choosing atypical moves (anti-conformity bias), and copying moves by celebrity players (prestige bias).

The study summarizing their results was published Nov. 15 in the Proceedings of the Royal Society B: Biological Sciences.

“We are all subject to biases,” said Marcus Feldman, the Burnet C. and Mildred Finley Wohlford Professor in the Stanford School of Humanities and Sciences and senior author. “Most biases are acquired from our parents or learned from our teachers, peers, or relatives.”

Feldman, a professor of biology, co-founded the field of cultural evolution 50 years ago with the late Luca Cavalli-Sforza, professor of genetics at Stanford School of Medicine, as a framework for studying changes in human behaviour that can be learned and transmitted between people. In the past, many studies of cultural evolution were theoretical because large datasets of cultural behaviour didn’t exist. But now they do.

The way chess is played has evolved over time too.

“Over the last several hundred years, paintings of chess playing show a change from crowded disorganized scenes to the quiet concentration we associate with the game today,” said Noah Rosenberg, the Stanford Professor in Population Genetics and Society in H&S.

“In the 18th century, players subscribed to a knightly sort of behaviour,” said Egor Lappo, lead author and a graduate student in Rosenberg’s lab. “Even if a move obviously led to a win, if it could be interpreted as cowardly, the player would reject it. Today, this is no longer the case.”

“The thesis of the paper is that when an expert player makes a move, many factors could influence move choice,” Rosenberg said. “The baseline is to choose a move randomly among the moves played recently by other expert players. Any deviations from this random choice are known in the field of cultural evolution as cultural biases.”

“In the mid-century players eschewed the Queen’s Gambit,” Feldman said. “There didn’t seem to be anything rational about this choice. In a large database of chess games by master-level players, the players’ biases can change over time, and that makes chess an ideal subject to use to explore cultural evolution.”

Playing the game

Chess is often called a game of perfect information because all pieces and their positions are clearly visible to both players. Yet simply knowing the present location of all pieces won’t win a chess game. Games are won by visualizing the future positions of pieces, and players develop this skill by studying the moves made by top chess players in different situations.

Fortunately for chess players (and researchers), the moves and game outcomes of top-level chess matches are recorded in books and, more recently, online chess databases.

In chess, two players take turns moving white (player 1) and black (player 2) pieces on a board checkered with 64 positions. The player with the white pieces makes the first move, each piece type (e.g., knight, pawn) moves a specific way, and (except for a special move called castling) each player moves one piece each turn.

There are few move options in the opening (beginning) of a chess game, and players often stick to tried-and-true sequences of moves, called lines, which are frequently given names like Ruy Lopez and the Frankenstein-Dracula Variation. The opening lines of master and grandmaster (top-level) players are often memorized by other players for use in their own games.

The researchers considered chess matches of master-level players between 1971 and 2019, millions of which have been digitized and are publicly available for analysis by enthusiasts.

“We used a population genetics model that treats all chess games played in a year as a population,” Lappo said. “The population of games in the following year is produced by players picking moves from the previous year to play in their own games.”

To search for possible cultural biases in the dataset of chess moves and games, the researchers used mathematical models to describe patterns that correspond to each kind of bias. Then they used statistical methods to see if the data matched (“fit”) the patterns corresponding to those cultural biases.

A value consistent with players choosing randomly from the moves played the year before indicated there was no cultural bias. This was the average “baseline” strategy. Success bias (copying winning moves) corresponded to values that were played by winning players in the previous year. Prestige bias (copying celebrity moves) corresponded to values that matched the frequencies of lines and moves played by the top 50 players in the previous year. Anti-conformity bias (unpopular moves) corresponded to choosing moves played infrequently in the previous year.

In the paper, the researchers focused on three frequently played moves at different depths of the opening to explore possible biases in early game play—the Queen’s Pawn opening, the Caro-Kann opening, and the Najdorf Sicilian opening.

Before the Queen’s Gambit was cool

For a game that is synonymous with strategy, relatively little is known about the factors that could affect a player’s choice of strategy. This study revealed evidence of cultural biases in the openings of master-level games played between 1971 and 2019.

In the Queen’s Pawn opening, players sometimes choose outlandish moves to rattle their opponents (anti-conformity bias). In the Caro-Kann opening, the study found that players mimic moves associated with winning chess games more often than expected by chance (success bias). And in the Najdorf Sicilian, players copy moves played by top players in famous games (prestige bias).

“The way people get their information about chess games changed between 1971 and 2019,” Rosenberg said. “It is easier now for players to see recent games of master- and grandmaster-level players.”

“The data also show that over time it is increasingly hard for the player with white pieces to make use of their first-move advantage,” Lappo said.

Many of the results align with ideas that are common knowledge among chess players, such as the concept that playing well-known lines is generally preferable to in-the-moment strategies in the opening. The researchers suggest that their statistical approach could be applied to other games and cultural trends in areas where long-term data on choices exist.

“This dataset makes questions related to the theory of cultural evolution useful and applicable in a way that wasn’t possible before,” Feldman said. “The big questions are what behaviour is transmitted, how is it transmitted, and to whom is it transmitted. With respect to the moves we analysed, Egor has the answers, and that is very satisfying.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Holly Alyssa MacCormick, Stanford University

Math enthusiasts around the world, from college kids to rocket scientists, celebrate Pi Day on Thursday, which is March 14 or 3/14—the first three digits of an infinite number with many practical uses.

Around the world many people will mark the day with a slice of pie—sweet, savory or even pizza.

Simply put, pi is a mathematical constant that expresses the ratio of a circle’s circumference to its diameter. It is part of many formulas used in physics, astronomy, engineering and other fields, dating back thousands of years to ancient Egypt, Babylon and China.

Pi Day itself dates to 1988, when physicist Larry Shaw began celebrations at the Exploratorium science museum in San Francisco. The holiday didn’t really gain national recognition though until two decades later. In 2009, Congress designated every March 14 to be the big day—to hopefully spur more interest in math and science. Fittingly enough, the day is also Albert Einstein’s birthday.

Here’s a little more about the holiday’s origin and how it’s celebrated today.

WHAT IS PI?

Pi can calculate the circumference of a circle by measuring the diameter—the distance straight across the circle’s middle—and multiplying that by the 3.14-plus number.

It is considered a constant number and it is also infinite, meaning it is mathematically irrational. Long before computers, historic scientists such as Isaac Newton spent many hours calculating decimal places by hand. Today, using sophisticated computers, researchers have come up with trillions of digits for pi, but there is no end.

WHY IS IT CALLED PI?

It wasn’t given its name until 1706, when Welsh mathematician William Jones began using the Greek symbol for the number.

Why that letter? It’s the first Greek letter in the words “periphery” and “perimeter,” and pi is the ratio of a circle’s periphery—or circumference—to its diameter.

WHAT ARE SOME PRACTICAL USES?

The number is key to accurately pointing an antenna toward a satellite. It helps figure out everything from the size of a massive cylinder needed in refinery equipment to the size of paper rolls used in printers.

Pi is also useful in determining the necessary scale of a tank that serves heating and air conditioning systems in buildings of various sizes.

NASA uses pi on a daily basis. It’s key to calculating orbits, the positions of planets and other celestial bodies, elements of rocket propulsion, spacecraft communication and even the correct deployment of parachutes when a vehicle splashes down on Earth or lands on Mars.

Using just nine digits of pi, scientists say it can calculate the Earth’s circumference so accurately it only errs by about a quarter of an inch (0.6 centimeters) for every 25,000 miles (about 40,000 kilometers).

IT’S NOT JUST MATH, THOUGH

Every year the San Francisco museum that coined the holiday organizes events, including a parade around a circular plaque, called the Pi Shrine, 3.14 times—and then, of course, festivities with lots of pie.

Around the country, many events now take place on college campuses. For example, Nova Southeastern University in Florida will hold a series of activities, including a game called “Mental Math Bingo” and event with free pizza (pies)—and for dessert, the requisite pie.

“Every year Pi Day provides us with a way to celebrate math, have some fun and recognize how important math is in all our lives,” said Jason Gershman, chair of NSU’s math department.

At Michele’s Pies in Norwalk, Connecticut, manager Stephen Jarrett said it’s one of their biggest days of the year.

“We have hundreds of pies going out for orders (Thursday) to companies, schools and just individuals,” Jarrett said in an interview. “Pi Day is such a fun, silly holiday because it’s a mathematical number that people love to turn into something fun and something delicious. So people celebrate Pi Day with sweet pies, savory pies, and it’s just an excuse for a little treat.”

NASA has its annual “Pi Day Challenge” online, offering people plenty of games and puzzles, some of them directly from the space agency’s own playbook such as calculating the orbit of an asteroid or the distance a moon rover would need to travel each day to survey a certain lunar area.

WHAT ABOUT EINSTEIN?

Possibly the world’s best-known scientist, Einstein was born on March 14, 1879, in Germany. The infinite number of pi was used in many of his breakthrough theories and now Pi Day gives the world another reason to celebrate his achievements.

In a bit of math symmetry, famed physicist Stephen Hawking died on March 14, 2018, at age 76. Still, pi is not a perfect number. He once had this to say,

“One of the basic rules of the universe is that nothing is perfect. Perfection simply doesn’t exist. Without imperfection, neither you nor I would exist.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Curt Anderson

The market for collectible digital assets, or non-fungible tokens, is an interesting example of a physical system with a large scale of complexity, non-trivial dynamics, and an original logic of financial transactions. At the Institute of Nuclear Physics of the Polish Academy of Sciences (IFJ PAN) in Cracow, its global statistical features have been analysed more extensively.

In the past, the value of money was determined by the amount of precious metals it contained. Today, we attribute it to certain sequences of digital zeros and ones, simply agreeing that they correspond to coins or banknotes. Non-fungible tokens (NFTs) operate by a similar convention: their owners assign a measurable value to certain sets of ones and zeros, treating them as virtual equivalents of assets such as works of art or properties.

NFTs are closely linked to the cryptocurrency markets but change their holders in a different way to, for example, bitcoins. While each bitcoin is exactly the same and has the same value, each NFT is a unique entity with an individually determined value, integrally linked to information about its current owner.

“Trading in digital assets treated in this way is not guided by the logic of typical currency markets, but by the logic of markets trading in objects of a collector’s nature, such as paintings by famous painters,” explains Prof. Stanislaw Drozdz (IFJ PAN, Cracow University of Technology.)

“We have already become familiar with the statistical characteristics of cryptocurrency markets through previous analyses. The question of the characteristics of a new, very young and at the same time fundamentally different market, also built on blockchain technology, therefore arose very naturally.”

The market for NFTs was initiated in 2017 with the blockchain created for the Ethereum cryptocurrency. The popularization of the idea and the rapid growth of trading took place during the pandemic. At that time, a record-breaking transaction was made at an auction organized by the famous English auction house Christie’s, when the art token Everyday: The First 5000 Days, created by Mike Winkelmann, was sold for $69 million.

Tokens are generally grouped into collections of different sizes, and the less frequently certain characteristics of a token occur in a collection, the higher its value tends to be. Statisticians from IFJ PAN examined publicly available data from the CryptoSlam (cryptoslam.io) and Magic Eden (magiceden.io) portals on five popular collections running on the Solana cryptocurrency blockchain.

These were sets of images and animations known as Blocksmith Labs Smyths, Famous Fox Federation, Lifinity Flares, Okay Bears, and Solana Monkey Business, each containing several thousand tokens with an average transaction value of close to a thousand dollars.

“We focused on analysing changes in the financial parameters of a collection such as its capitalization, minimum price, the number of transactions executed on individual tokens per unit of time (hour), the time interval between successive transactions, or the value of transaction volume. The data covered the period from the launch date of a particular collection up to and including August 2023,” says Dr. Marcin Watorek (PK).

For stabilized financial markets, the presence of certain power laws is characteristic, signaling that the likelihood of large events occurring is greater than would result from a typical Gaussian probability distribution. It appears that the operation of such laws is already evident in the fluctuations of NFT market parameters, for example, in the distribution of times between individual trades or in volume fluctuations.

Among the statistical parameters analysed by the researchers from the IFJ PAN was the Hurst exponent, which describes the reluctance of a system to change its trend. The value of this exponent falls below 0.5 when the system has a tendency to fluctuate: all rises increase the probability of a decrease (or vice versa).

In contrast, values above 0.5 indicate the existence of a certain long-term memory: after a rise, there is a higher probability of another rise; after a fall, there is a higher probability of another fall. For the token collections studied, the values of the Hurst exponent were between 0.6 and 0.8, thus at a level characteristic of highly reputable markets. In practice, this property means that the trading prices of tokens from a given collection fluctuate in a similar manner in many cases.

The existence of a certain long-term memory of the system, reaching up to two months in the NFT market, may indicate the presence of multifractality. When we start to magnify a fragment of an ordinary fractal, sooner or later, we see a structure resembling the initial object, always after using the same magnification. Meanwhile, in the case of multifractals, their different fragments have to be magnified at different speeds.

It is precisely this non-linear nature of self-similarity that has also been observed in the digital collectors’ market, among others, for minimum prices, numbers of transactions per unit of time, and intervals between transactions. However, this multifractality was not fully developed and was best revealed in those situations where the greatest fluctuations were observed in the system under study.

“Our research also shows that the price of the cryptocurrency for which collections are sold directly affects the volume they generate. This is an important observation, as cryptocurrency markets are already known to show many signs of statistical maturity,” notes Pawel Szydlo, first author of the article in Chaos: An Interdisciplinary Journal of Nonlinear Science.

The analyses carried out at IFJ PAN lead to the conclusion that, despite its young age and slightly different trading mechanisms, the NFT market is beginning to function in a manner that is statistically similar to established financial markets. This fact seems to indicate the existence of a kind of universalism among financial markets, even of a significantly different nature. However, its closer understanding will require further research.

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Credit of the article given to Polish Academy of Sciences

Example 5

A sequence of functions {fn:R→R}{fn:R→R} which converges to 0 pointwise but does not converge to 0 in L1L1.

This works because: The sequence tends to 0 pointwise since for a fixed x∈Rx∈R, you can always find N∈NN∈N so that fn(x)=0fn(x)=0 for all nn bigger than NN. (Just choose N>xN>x!)

The details: Let x∈Rx∈R and fix ϵ>0ϵ>0 and choose N∈NN∈N so that N>xN>x. Then whenever n>Nn>N, we have |fn(x)−0|=0<ϵ|fn(x)−0|=0<ϵ.

Of course, fn↛0fn↛0 in L1L1 since∫R|fn|=∫(n,n+1)fn=1⋅λ((n,n+1))=1.

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*Credit for article given to Tai-Danae Bradley*

Human coexistence depends on cooperation. Individuals have different motivations and reasons to collaborate, resulting in social dilemmas, such as the well-known prisoner’s dilemma. Scientists from the Chatterjee group at the Institute of Science and Technology Austria (ISTA) now present a new mathematical principle that helps to understand the cooperation of individuals with different characteristics. The results, published in PNAS, can be applied to economics or behavioural studies.

A group of neighbours shares a driveway. Following a heavy snowstorm, the entire driveway is covered in snow, requiring clearance for daily activities. The neighbours have to collaborate. If they all put on their down jackets, grab their snow shovels, and start digging, the road will be free in a very short amount of time. If only one or a few of them take the initiative, the task becomes more time-consuming and labor-intensive. Assuming nobody does it, the driveway will stay covered in snow. How can the neighbours overcome this dilemma and cooperate in their shared interests?

Scientists in the Chatterjee group at the Institute of Science and Technology Austria (ISTA) deal with cooperative questions like that on a regular basis. They use game theory to lay the mathematical foundation for decision-making in such social dilemmas.

The group’s latest publication delves into the interactions between different types of individuals in a public goods game. Their new model, published in PNAS, explores how resources should be allocated for the best overall well-being and how cooperation can be maintained.

The game of public goods

For decades, the public goods game has been a proven method to model social dilemmas. In this setting, participants decide how much of their own resources they wish to contribute for the benefit of the entire group. Most existing studies considered homogeneous individuals, assuming that they do not differ in their motivations and other characteristics.

“In the real world, that’s not always the case,” says Krishnendu Chatterjee. To account for this, Valentin Hübner, a Ph.D. student, Christian Hilbe, and Maria Kleshina, both former members of the Chatterjee group, started modeling settings with diverse individuals.

A recent analysis of social dilemmas among unequals, published in 2019, marked the foundation for their work, which now presents a more general model, even allowing multi-player interaction.

“The public good in our game can be anything, such as environmental protection or combating climate change, to which everybody can contribute,” Hübner explains. The players have different levels of skills. In public goods games, skills typically refer to productivity.

“It’s the ability to contribute to a particular task,” Hübner continues. Resources, technically called endowment or wealth, on the other hand, refer to the actual things that participants contribute to the common good.

In the snowy driveway scenario, the neighbours vary significantly in their available resources and in their abilities to use them. Solving the problem requires them to cooperate. But what role does their inequality play in such a dilemma?

The two sides of inequality

Hübner’s new model provides answers to this question. Intuitively, it proposes that for diverse individuals to sustain cooperation, a more equal distribution of resources is necessary. Surprisingly, more equality does not lead to maximum general welfare. To reach this, the resources should be allocated to more skilled individuals, resulting in a slightly uneven distribution.

“Efficiency benefits from unequal endowment, while robustness always benefits from equal endowment,” says Hübner. Put simply, for accomplishing a task, resources should be distributed almost evenly. Yet, if efficiency is the goal, resources should be in the hands of those more willing to participate—but only to a certain extent.

What is more important—cooperation efficiency or stability? The scientists’ further simulations of learning processes suggest that individuals balance the trade-off between these two things. Whether this is also the case in the real world remains to be seen. Numerous interpersonal nuances also contribute to these dynamics, including aspects like reciprocity, morality, and ethical issues, among others.

Hübner’s model solely focuses on cooperation from a mathematical standpoint. Yet, due to its generality, it can be applied to any social dilemma with diverse individuals, like climate change, for instance. Testing the model in the real world and applying it to society are very interesting experimental directions.

“I’m quite sure that there will be behavioural experiments benefiting from our work in the future,” says Chatterjee. The study could potentially also be interesting for economics, where the new model’s principles can help to better inform economic systems and policy recommendations.

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Credit of the article to be given Institute of Science and Technology Austria

This post is the fourth example in an ongoing list of various sequences of functions which converge to different things in different ways.

Also in this series:

Example 1: converges almost everywhere but not in L1L1
Example 2: converges uniformly but not in L1L1
Example 3: converges in L1L1 but not uniformly
Example 5: converges pointwise but not in L1L1
Example 6: converges in L1L1 but does not converge anywhere

Example 4

A sequence of (Lebesgue) integrable functions fn:R→[0,∞)fn:R→[0,∞) so that {fn}{fn} converges to f:R→[0,∞)f:R→[0,∞) uniformly,  yet ff is not (Lebesgue) integrable.

‍Our first observation is that “ff is not (Lebesgue) integrable” can mean one of two things: either ff is not measurable or ∫f=∞∫f=∞. The latter tends to be easier to think about, so we’ll do just that. Now what function do you know of such that when you “sum it up” you get infinity? How about something that behaves like the divergent geometric series? Say, its continuous cousin f(x)=1xf(x)=1x? That should work since we know∫R1x=∫∞11x=∞.∫R1x=∫1∞1x=∞.Now we need to construct a sequence of integrable functions {fn}{fn} whose uniform limit is 1x1x. Let’s think simple: think of drawring the graph of f(x)f(x) one “integral piece” at a time. In other words, define:

This works because: It makes sense to define the fnfn as  f(x)=1xf(x)=1x “chunk by chunk” since this way the convergence is guaranteed to be uniform. Why? Because how far out we need to go in the sequence so that the difference f(x)−fn(x)f(x)−fn(x) is less than ϵϵ only depends on how small (or large) ϵϵ is. The location of xx doesn’t matter!

Also notice we have to define fn(x)=0fn(x)=0 for all x<1x<1 to avoid the trouble spot ln(0)ln⁡(0) in the integral ∫fn∫fn. This also ensures that the area under each fnfn is finite, guaranteeing integrability.

The details: Each fnfn is integrable since for a fixed nn,∫Rfn=∫n11x=ln(n).∫Rfn=∫1n1x=ln⁡(n).To see fn→ffn→f uniformly, let ϵ>0ϵ>0 and choose NN so that N>1/ϵN>1/ϵ. Let x∈Rx∈R. If x≤1x≤1, any nn will do, so suppose x>1x>1 and let n>Nn>N. If 1<x≤n1<x≤n, then we have |fn(x)−f(x)|=0<ϵ|fn(x)−f(x)|=0<ϵ. And if x>nx>n, then∣∣1xχ[1,∞)(x)−1xχ[1,n](x)∣∣=∣∣1x−0∣∣=1x<1n<1N<ϵ.|1xχ[1,∞)(x)−1xχ[1,n](x)|=|1x−0|=1x<1n<1N<ϵ.

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*Credit for article given to Tai-Danae Bradley*