Much of our thinking is flawed because it is based on faulty intuition. But by using the framework and tools of probability and statistics, we can overcome this to provide solutions to many real-world problems and paradoxes. Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.
When it comes to situations like waiting for a bus, our intuition is often wrong.
Imagine, there’s a bus that arrives every 30 minutes on average and you arrive at the bus stop with no idea when the last bus left. How long can you expect to wait for the next bus? Intuitively, half of 30 minutes sounds right, but you’d be very lucky to wait only 15 minutes.
Say, for example, that half the time the buses arrive at a 20-minute interval and half the time at a 40-minute interval. The overall average is now 30 minutes. From your point of view, however, it is twice as likely that you’ll turn up during the 40 minutes interval than during the 20 minutes interval.
This is true in every case except when the buses arrive at exact 30-minute intervals. As the dispersion around the average increases, so does the amount by which the expected wait time exceeds the average wait. This is the Inspection Paradox, which states that whenever you “inspect” a process, you are likely to find that things take (or last) longer than their “uninspected” average. What seems like the persistence of bad luck is simply the laws of probability and statistics playing out their natural course.
Once made aware of the paradox, it seems to appear all over the place.
For example, let’s say you want to take a survey of the average class size at a college. Say that the college has class sizes of either 10 or 50, and there are equal numbers of each. So the overall average class size is 30. But in selecting a random student, it is five times more likely that he or she will come from a class of 50 students than of 10 students. So for every one student who replies “10” to your enquiry about their class size, there will be five who answer “50”. The average class size thrown up by your survey is nearer 50, therefore, than 30. So the act of inspecting the class sizes significantly increases the average obtained compared to the true, uninspected average. The only circumstance in which the inspected and uninspected average coincides is when every class size is equal.
We can examine the same paradox within the context of what is known as length-based sampling. For example, when digging up potatoes, why does the fork go through the very large one? Why does the network connection break down during download of the largest file? It is not because you were born unlucky but because these outcomes occur for a greater extension of space or time than the average extension of space or time.
Once you know about the Inspection Paradox, the world and our perception of our place in it are never quite the same again.
Another day you line up at the medical practice to be tested for a virus. The test is 99% accurate and you test positive. Now, what is the chance that you have the virus? The intuitive answer is 99%. But is that right? The information we are given relates to the probability of testing positive given that you have the virus. What we want to know, however, is the probability of having the virus given that you test positive. Common intuition conflates these two probabilities, but they are very different. This is an instance of the Inverse or Prosecutor’s Fallacy.
The significance of the test result depends on the probability that you have the virus before taking the test. This is known as the prior probability. Essentially, we have a competition between how rare the virus is (the base rate) and how rarely the test is wrong. Let’s say there is a 1 in 100 chance, based on local prevalence rates, that you have the virus before taking the test. Now, recall that the test is wrong one time in 100. These two probabilities are equal, so the chance that you have the virus when testing positive is 1 in 2, despite the test being 99% accurate. But what if you are showing symptoms of the virus before being tested? In this case, we should update the prior probability to something higher than the prevalence rate in the tested population. The chance you have the virus when you test positive rises accordingly. We can use Bayes’ Theorem to perform the calculations.
In summary, intuition often lets us down. Still, by applying the methods of probability and statistics, we can defy intuition. We can even resolve what might seem to many the greatest mystery of them all – why we seem so often to find ourselves stuck in the slower lane or queue. Intuitively, we were born unlucky. The logical answer to the Slower Lane Puzzle is that it’s exactly where we should expect to be!
When intuition fails, we can always use probability and statistics to look for the real answers.
Leighton Vaughan Williams, Professor of Economics and Finance at Nottingham Business School. Read more in Leighton’s new publication Probability, Choice and Reason.
Exploring the Will Rogers Phenomenon
A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press).
CLARIFYING THE CONCEPT
The Will Rogers Phenomenon occurs when moving an element from one group to another increases the average of both groups. It’s named after the comedian Will Rogers, who joked that people moving from Oklahoma to California raised the intelligence of both states.
Imagine moving a jigsaw piece from one box to another and somehow making both jigsaw puzzles look more complete. That’s the kind of surprising outcome the Will Rogers Phenomenon describes in the world of statistics.
I call it the Free Lunch Paradox.
ILLUSTRATIVE EXAMPLE
Let’s say we’re looking at two groups based on a medical condition. Group A has the condition, while Group B doesn’t.
Initially, Group A has a lower average life expectancy than Group B. But when one individual from Group B, who has a higher expectancy than Group A’s average but lower than Group B’s, is correctly diagnosed with the condition and moved to Group A, both groups’ averages increase.
This might seem odd because we haven’t changed anyone’s life expectancy; we’ve just reclassified one person. Yet, the averages for both groups go up.
Consider, for example, a study analysing the life expectancies of six individuals. We assess their life expectancies one by one and find that the first two individuals have a life expectancy of 5 and 15 years, respectively. They have been diagnosed with a specific medical condition. The remaining four individuals have life expectancies of 25, 35, 45, and 55 years, but they do not have the diagnosed condition. Consequently, the average life expectancy for those with the condition is 10 years, while for those without the condition, it is 40 years.
THE IMPACT OF IMPROVED DIAGNOSIS
Now, let’s suppose that advances in diagnostic medical science allow us to identify one of the individuals with a life expectancy of 25 years as actually having the medical condition, which was previously missed. This discovery prompts us to move this person from the undiagnosed group to the diagnosed group.
As a result, the average life expectancy within the group diagnosed with the condition increases from 10 to 15 years. The calculation for the new average is (5 + 15 + 25) divided by 3. Simultaneously, the average life expectancy of those not diagnosed with the condition also rises by five years, from 40 to 45 years. The calculation for this new average is (35 + 45 + 55) divided by 3.
THE ILLUSION OF CHANGE
Upon observing this scenario, we might be puzzled as to how moving a single data point can cause both groups’ averages to increase. The Will Rogers Phenomenon provides an explanation.
In this case, the data point being moved (the individual with a life expectancy of 25 years) is below the average of the group it is leaving, which is 40 years. Yet, it is above the average of the group it is joining, which is 10 years. This creates the illusion of improvement in both groups’ averages, despite there being no change in the actual values.
ADDITIONAL EXAMPLES
This phenomenon isn’t limited to medical data. For example, in education, say we move a student who’s not doing well in a class of high achievers to a class with lower overall scores. Suddenly, both classes seem to do better on average, even though the student’s performance hasn’t changed.
Suppose again there are two schools, School A and School B, with average test scores of 70% and 80%, respectively. School B now decides to send some of its lower-performing students (scores of below 80% but above 70%), while retaining its higher-performing students.
As a result, both schools’ average scores increase. This occurs because the students transferred from School B to School A have scores below School B’s average but above School A’s average.
This example highlights how the Will Rogers Phenomenon can manifest itself in different domains, influencing various statistical analyses and interpretations.
THE ROLE OF CONTEXT
Understanding the Will Rogers Phenomenon is crucial for individuals working with statistics and data analysis. It emphasises the significance of considering context and carefully interpreting statistical results, particularly when dealing with group comparisons.
By being aware of this phenomenon, we can avoid misconstruing statistical changes as genuine improvements or deteriorations in the underlying data. It reminds us that when data points move between groups, the resulting changes in averages may not reflect true progress but rather the consequences of shifting data categorisations.
APPLICATION TO REAL-WORLD SCENARIOS
Public Health Policy
In public health, the Will Rogers Phenomenon can have profound implications, particularly in the reporting and interpretation of disease rates and the effectiveness of interventions. For instance, if a new diagnostic technique becomes available that identifies milder cases of a disease previously undetected, the overall survival rate of the diagnosed population may increase. This is not necessarily because the treatment has improved but because the cohort now includes less severe cases. This can lead to the false conclusion that a new drug or treatment is more effective than it actually is, potentially influencing funding allocations, treatment protocols, and patient care strategies without genuine improvements in treatment efficacy.
Education Reforms
In the education sector, policy decisions are often influenced by the performance metrics of schools and universities. If educational standards change, causing students with lower grades to be reclassified from one performance category to another, it may appear that both the higher and lower performing groups have improved their average scores. This could lead to misleading conclusions about the success of new educational policies or teaching methods. For example, if a new grading policy causes borderline students to be classified into a lower-performing group, it might artificially inflate the average performance of both the higher and lower groups, leading to misguided policy decisions based on perceived improvements.
Economic Analysis
In economics, the phenomenon can impact the analysis of income data and the evaluation of economic policies. For example, if a government implements a new tax bracket that reclassifies some of the lower earners from the middle-income bracket to the lower-income bracket, it could appear that the average income in both brackets has increased. This could be misinterpreted as economic improvement resulting from the policy, leading to skewed analyses and subsequent policy decisions that do not accurately address the underlying economic conditions.
Environmental Policy
Consider the assessment of air quality in different regions. If new, more sensitive measuring techniques are employed that classify moderately polluted areas as highly polluted, it may appear that the average pollution levels in both the moderately and highly polluted categories have decreased. This could lead to incorrect conclusions about the effectiveness of environmental regulations and misdirected resources, impacting public health and environmental protection efforts.
Crime Statistics
Changes in the classification of crimes can lead to misunderstandings of crime trends. If, for example, certain types of thefts are reclassified, it might appear that both low-level and high-level crime rates have decreased, when in reality, only the classification criteria have changed. This can affect public perception, policy formulation, and resource allocation in law enforcement.
By providing these expanded examples, we can see how the Will Rogers Phenomenon extends far beyond statistical curiosity, affecting a wide range of important decisions in public health, education, economics, environmental policy, and criminal justice. Understanding this phenomenon is crucial for policymakers, educators, economists, and the public to avoid misinterpretations that can lead to significant real-world consequences.
CONCLUSION: SHEDDING LIGHT ON DATA PRESENTATION
Understanding the Will Rogers Phenomenon is crucial in fields like medicine, education, and any area where data is grouped and compared. It shows us that moving data around can create misleading impressions and thus highlights the need for careful data analysis and interpretation. By understanding this effect, we learn to interpret changes in averages within the appropriate context and to equip ourselves to better navigate the intricacies of data-driven knowledge in our increasingly data-centric world. In this way, it is the Will Rogers Phenomenon itself that provides us with the real free lunch!
The Ardross Heist
How the “Rational Deceivers” Stole the Series 4 Pot
If you’ve been following my Twisted Logic guides over the past year, you’ll know I’ve described The Traitors as a live-action laboratory for information asymmetry, a game not about spotting liars but about influencing how people see the truth. Last night’s Series 4 finale at Ardross Castle offered the cleanest empirical proof of that idea yet, in a classic display of incentives, signalling, and cognitive limits.
1. Beyond “Cheap Talk”: Rachel and the Power of Costly Signals
One of the recurring themes in my earlier guides is the futility of what economists call cheap talk, those costless, unverifiable claims like “I swear I’m Faithful” or “I’d never lie about this”. In a game like The Traitors, words are cheap. Everyone can say them. They convey almost no information. Rachel understood this.
Rather than relying solely on verbal assurances, she pivoted to costly signalling, behaviour that looks so socially risky that a rational liar shouldn’t dare attempt it.
Her strategy was what I’d call strategic outrageousness. Those breakfast confrontations weren’t lapses in control; they were signals. The implicit message was simple: a Traitor would be too frightened of the Round Table to behave like this. In contrast, the Faithfuls mistook social cost for authenticity. Rachel’s willingness to bruise were read as proof she couldn’t possibly be a Traitor. By embracing this character, she made herself cognitively awkward to suspect. Her behaviour was constantly explained away as “just Rachel”, allowing her to hide in plain sight.
2. Stephen and the “Stealth King” Strategy
If Rachel provided the noise, Stephen provided the vacuum.
Stephen’s win is a masterclass in exploiting what is called “bounded rationality”, the fact that humans, under stress, fatigue, and emotional pressure, rely on simplified mental shortcuts rather than exhaustive reasoning. He understood from the beginning that in a diverse group of people, the person who speaks just a little less than average is dramatically less likely to become a focal suspect.
Stephen never led a witch hunt. He rarely initiated accusations. Instead, he positioned himself in the middle of the pack, as reliable, agreeable, and essential to multiple alliances without ever becoming central to any of them.
This wasn’t passivity. It was optimisation. While others burned cognitive energy interpreting tone, eye contact, and breakfast silences, Stephen let the group exhaust itself. By the time the numbers favoured the hidden minority, the Faithful majority had spent their attention elsewhere. In a game obsessed with “reading people”, he won by giving almost nothing to read.
3. The Endgame: Self-Interest vs. Collective Success
The finale also illustrated the tension between individual self-interest and collective success in the endgame. As the prize pot grew, the Faithfuls continued to banish their own allies while coordination began to collapse. Stephen and Rachel didn’t need to push this dynamic. They simply let it run.
By the time the final fire pit arrived, decisions were no longer grounded in evidence. They were shaped by emotional investment. This is where the sunk-cost fallacy quietly did its damage.
Admitting the truth would have meant admitting that much of their judgment, and many of their earlier votes, had been wrong. Psychologically, that price was too high. Even when the logic pointed directly at the turret, they couldn’t afford to see it, except for one brief round of enlightenment which Rachel was lucky to escape. When the smoke finally cleared, the devastation wasn’t just about the money. It was about the collapse of a story they had paid too much to abandon.
4. The Verdict
This season felt different because the Traitors stopped playing defence. They moved the game beyond simply lying, into actively poisoning the group’s reasoning environment, shaping not just what people believed, but how they decided. Actions still speak louder than words, but in the hands of a master Traitor even your actions can be a lie.
Essentially, what happened was that Stephen and Rachel hacked the social software of the castle. As a result, Series 4 wasn’t a hunt. It was a harvest.
Life, Death, and Statistical Literacy
A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press).
Understanding complex statistical phenomena can be a daunting task, especially when they seem to defy common sense. One such concept is Simpson’s Paradox, a surprising phenomenon that occurs when a trend observed within several different groups of data disappears or reverses when these groups are combined. Think of it like a recipe. Individual ingredients might have distinct flavours, but when mixed together, the overall taste can be quite different. Similarly, separate sets of data may tell one story, but when combined, they can tell a completely different one.
Let’s look at some examples.
SIMPSON’S PARADOX IN MEDICINE TRIALS
Suppose you’re testing the effectiveness of two different types of medicine: a new drug and an old drug. Your goal is to determine which one is more effective at treating a certain condition. You administer these drugs to different groups of patients and then analyse how well each drug performs.
Let’s look at a two-day medical trial comparing two drugs.
On Day 1, the new drug showed a 70% success rate in a large group, while the old drug showed an 80% success rate in a much smaller group. This makes it seem like the old drug is better.
On Day 2, the new drug, applied to a small group, was less effective than on Day 1, while the old drug, applied to a larger group, was also less effective than on Day 1. Even so, once again the old drug seems to perform better than the new drug.
However, when we combine both days’ data, the new drug comes out ahead. This shift is a classic example of Simpson’s Paradox.
Day 1: Initial Observations
On the first day, you test the new drug on 90 patients, and it works for 63 of them, giving a success rate of 70%. In contrast, you administer the old drug to a smaller group of ten patients, and it works for eight of them, resulting in an 80% success rate. At this point, it seems like the old drug outperforms the new one. But let’s continue.
New drug: 90 patients; 63 successes. Success rate = 70%.
Old drug: 10 patients, 8 successes. Success rate = 80%.
Conclusion: Old Drug Outperforms New Drug
Day 2: More Data, More Surprises
The following day, the new drug is given to a different group of ten patients. This time, it only works for four of them, resulting in a decreased success rate of 40%. The old drug, on the other hand, is given to a larger group of 90 patients, and it works for 45 of them, indicating a 50% success rate. Once again, the old drug seems to outshine the new one.
New drug: 10 patients; 4 successes. Success rate = 40%.
Old drug: 90 patients, 45 successes. Success rate = 50%.
Conclusion: Old drug outperforms new drug.
COMBINING THE RESULTS: SIMPSON’S PARADOX AT WORK
When you merge the results from both days, however, an interesting thing happens. The new drug, which seemed less effective on each individual day, ended up working for 67 of the total 100 patients who took it, bringing the total success rate to 67%. The old drug, conversely, worked for only 53 out of 100 of its patients, resulting in a 53% success rate overall. This is contrary to what was observed on individual days and seems paradoxical. This flip is a classic example of Simpson’s Paradox.
New drug: 100 patients; 67 successes. Success rate = 67%.
Old drug: 100 patients, 53 successes. Success rate = 53%.
Conclusion: New drug outperforms old drug.
EXPLAINING THE PARADOX
The paradox in our medical trial example is heavily influenced by the size of the groups tested each day.
If we combine the results, larger group sizes on different days skew the overall success rate, revealing the paradox. The success rates are important, but the size of the groups being compared is crucial to understanding why the paradox occurs.
EXPLORING SIMPSON’S PARADOX IN DRUG EFFICACY TRIALS
Let’s delve deeper into Simpson’s Paradox using another example. Suppose this time you’re comparing a real drug to a placebo, a sugar pill, to see if the real drug can help patients recover from a specific illness.
You arrange the patients into four distinct age groups: elderly adults (Group A), middle-aged adults (Group B), young adults (Group C), and children (Group D).
The drug’s effectiveness is measured by the proportion of patients in each group who recover from their illness within two days of taking the medication.
THE SUGAR PILL EXPERIMENT
First, let’s take a look at the sugar pill group.
You distribute the sugar pill to different proportions of the four age groups:
Group A has 20 elderly adults, Group B has 40 middle-aged adults, Group C has 120 young adults, and Group D has 60 children.
The sugar pill helps 10% of the elderly (Group A), 20% of the middle-aged adults (Group B), 40% of the young adults (Group C), and 30% of the children (Group D).
To calculate the overall success rate, you add up the number of successful recoveries across all the groups (2 from Group A, 8 from Group B, 48 from Group C, and 18 from Group D) and divide by the total number of patients (240). The result is 76 successful recoveries out of 240 trials, giving an overall success rate of approximately 31.7%.
Group A: 20 elderly adults; 2 successes. Success rate = 10%.
Group B: 40 middle-aged adults; 8 successes. Success rate = 20%.
Group C: 120 young adults; 48 successes. Success rate = 40%.
Group D: 60 children; 18 successes. Success rate = 30%.
Total: 240 trials; 76 successes. Success rate = 31.7%.
THE REAL DRUG EXPERIMENT
Next, let’s look at the group given the real drug. This time, the group sizes are different: Group A has 120 elderly adults, Group B has 60 middle-aged adults, Group C has 20 young adults, and Group D has 40 children.
The real drug helps 15% of the elderly (Group A), 30% of the middle-aged adults (Group B), 90% of the young adults (Group C), and 45% of the children (Group D).
Again, to get the overall success rate, you add up the number of successful recoveries (18 from each group) and divide by the total number of patients (240). This time, the result is 72 successful recoveries out of 240 trials, resulting in an overall success rate of approximately 30%.
Group A: 120 elderly adults; 18 successes. Success rate = 15%.
Group B: 60 middle-aged adults; 18 successes. Success rate = 30%.
Group C: 20 young adults; 18 successes. Success rate = 90%.
Group D: 40 children; 18 successes. Success rate = 45%.
Total: 240 trials; 72 successes. Success rate = 30%.
A PARADOX EMERGES
At first glance, it seems that the sugar pill outperformed the real drug. After all, the overall success rate was higher for the sugar pill (31.7%) than for the real drug (30%). But if we examine the data more closely, we find that the real drug had a higher success rate within each age group.
So, why does the overall success rate favour the sugar pill, even though the real drug was more effective in every age category? The paradox again arises due to the different group sizes and composition.
For example, the group that took the sugar pill had a disproportionately large number of young adults (Group C). This demographic typically has higher natural recovery rates, skewing the overall success rate of the sugar pill upwards. On the contrary, the group that took the real drug had a higher proportion of elderly adults (Group A), who typically have lower recovery rates, leading to a lower overall success rate for the real drug.
A MATTER OF LIFE AND DEATH
A real-world example of Simpson’s Paradox in action can be seen in the context of the COVID-19 pandemic, specifically relating to a report published in November 2021 by the Office for National Statistics (ONS). It was titled ‘Deaths involving COVID-19 by vaccination status, England: deaths occurring between 2 January and 24 September 2021’.
The raw statistics showed death rates in England for people aged 10–59, listing vaccination status separately. Counterintuitively, the statistics showed that the death rates for the vaccinated in this age grouping were greater than those for the unvaccinated. These numbers were heavily promoted and highlighted on social media by anti-vaccine advocates, who used them to argue that vaccination increases the risk of death.
This claim was contrary, though, to efficacy and effectiveness studies showing that COVID-19 vaccines offered strong protection.
A CLOSER INSPECTION
Closer inspection of the ONS report reveals that over the period of the study, from January to September 2021, the age-adjusted risk of death involving COVID-19 was 32 times greater among unvaccinated people compared to fully vaccinated people. So how can we square this with the raw data? This is where Simpson’s Paradox comes in.
The paradox in the ONS statistics arises specifically because death rates increase dramatically with age, so that at the very top end of this age band, for example, mortality rates are about 80 times higher than at the very bottom end. A similar pattern is observed between vaccination rates and age. For example, in the 10–59 data set, more than half of those vaccinated are over the age of 40.
Those who are in the upper ranges of the wide 10–59 age band are, therefore, both more likely to have been vaccinated and also more likely to die if infected with COVID-19 or for any other reason, and vice versa. Age is acting, in the terminology of statistics, as a confounding variable, as it is positively related to both vaccination rates and death rates. To put it another way, if you are older, you are more likely to die in a given period, and you are also more likely to be vaccinated. It is age that is driving up death rates not vaccinations. Without vaccinations, deaths would have been hugely greater from COVID-19.
STATISTICAL LITERACY
If we break down the band into narrower age ranges, such as 10–19, 20–29, 30–39, 40–49, and 50–59, the counterintuitive headline finding immediately disappears. In each age band, the death rates of the vaccinated are very much lower than those of the unvaccinated. This also applies in the higher age bands—60–69, 70–79, and 80 plus. The key point is that age is a crucial factor that must be considered when analysing the risk of death and the impact of vaccinations.
In this way, misrepresentation of statistics can have potentially devastating consequences for the lives of millions around the world. Statistical literacy is a real superpower in the global quest to protect and save these lives.
GUIDELINES AND STRATEGIES
Disaggregate the Data
Break Down Data into Subgroups: Disaggregating data by relevant subgroups (e.g. age, gender, region) can reveal underlying trends that the aggregated data might mask.
Question Initial Assumptions
Challenge Averages: Averages can be misleading. Always question what an average is concealing. Is it masking a wide distribution or skewing because of outliers?
Seek Out Hidden Variables (Confounding Variables)
Identify Potential Confounders: Simpson’s Paradox often arises due to the presence of hidden variables that influence both the predictor and outcome variables.
Use Visual Data Exploration
Plot Your Data: Visualising your data can help identify patterns, trends, and anomalies. Graphs can help spot where the trend within subgroups differs from the aggregated trend, potentially signalling Simpson’s Paradox.
CONCLUSION: RESOLVING SIMPSON’S PARADOX
Understanding the factors behind Simpson’s Paradox allows us to make much better sense of our data. Whether in stylised examples or in the real world of a global pandemic, the paradox underscores the importance of careful data analysis, particularly when dealing with grouped data. By taking account of the sizes and characteristics of different groups, we can navigate the potential pitfalls of Simpson’s Paradox and learn how to draw more informed conclusions. In a very real sense, millions of lives could depend on an understanding of this statistical reality.
Or Are You Real?
Pause for a moment and ask yourself a quietly subversive question.
How do you know that any of this is real?
Your life. Your memories. Your loved ones. The screen before you. Everything and everyone you know. Could it all be a brief hallucination, a flicker of awareness flashing and fading in a vast, indifferent void?
This isn’t a sceptical game from a philosophy seminar. It’s a live problem in modern physics and philosophy, known as the Boltzmann Brain paradox. And once you take it seriously, something surprising happens: the sheer coherence of your experience, its continuing stability, starts to look not like an obvious fact, but like a profound mystery.
Two Ways to Make a Mind
Imagine a universe without intent; just matter, energy, and time. How could a conscious observer arise in such a world?
1. The Hard Way
The universe begins in an awesomely ordered state. Physicists call this Low Entropy. Think of a brand-new deck of cards, perfectly sorted by suit and number. This extreme organisation allows for a complex, long-lasting “game” to follow.
Laws remain stable. Stars form, galaxies evolve, and life stirs on a small planet. Over billions of years, chemistry dreams itself awake, until one day someone sits reading an article about cosmology and doubt. This route demands what philosophers call Fine-Tuning: the idea that the fundamental constants of physics (like gravity) are dialled into a precise, “life-permitting” range.
It yields what we call a normal observer, a mind rooted in an ongoing, stable physical world.
2. The Easy Way
Now picture the opposite extreme: High Entropy. This is the universe after the deck has been shuffled for billions of years. In this world of pure chance, even the shuffled deck will produce a “straight flush” and then another and so on, just by accident. Given enough time, particles randomly arrange themselves into something along the lines of a functioning brain.
That brain would:
• Possess conscious thoughts
• Carry vivid but false memories
• Mistake illusion for reality
And almost instantly, it would dissolve again into the shuffle. That is a Boltzmann Brain.
The Unsettling Arithmetic: A Universe of Ghosts
Here’s the disquieting part: statistically, the easy way is far more common.
It is vastly “cheaper” for nature to generate hallucinating minds than to build a 14-billion-year-old stage for that mind to stand on. If the universe is a blind accident, for every one “real” brain that evolved over billions of years, there should be countless Boltzmann Brains popping into existence.
In a random, unguided cosmos, the “Normal Observers” are a vanishingly small minority. The “Blips” are the rule.
Which leads to a seemingly absurd, yet mathematically very serious, conclusion: If this picture of the universe is right, then you are almost certainly a Boltzmann Brain. You are statistically more likely to be a fleeting ghost than a real human. If that’s so, your trust in reality is a mathematical mistake.
When Fine-Tuning Stops Being Enough
The Fine-Tuning problem already puzzles physicists: why the laws of nature appear calibrated for life. The “Multiverse” is sometimes offered as a tidy answer; that with multiple universes, one like ours eventually appears.
But the Boltzmann Brain paradox in this case suddenly presses harder. Random multiverses produce exponentially more chaotic fluctuations, more opportunities for Boltzmann brains. Now the very scenario meant to explain our existence actually makes our persistent sanity a statistical nonsense. In this way, a universe that explains everything explains nothing. It turns truth itself into a mirage.
A Cosmos Biased Towards Coherence
So perhaps this assumption is wrong, and the universe does not drift aimlessly through probability space. Instead its laws are weighted, not arbitrarily, but towards order, endurance, and intelligibility. Such a universe is not hostile to reason, but profoundly hospitable to it. On such a view:
• The universe begins in Low Entropy for a reason.
• Its laws do not merely allow life but make understanding possible.
• Conscious minds are expected, not accidental.
By “rigging” the start of the universe to be so highly ordered, the “Normal Observers” become the majority, and the fleeting ghosts become the exception. Reality itself is tilted in favour of truth over illusion.
Why Reality Holds Together
Fine-tuning explains the conditions for life. The Boltzmann Brain problem asks why rational life can trust what it perceives. A universe committed to coherence answers both.
• When you lift your hand, it responds.
• When you remember yesterday, it existed.
• When you think, your thoughts are real.
That is not what a cosmic accident would predict. In a cosmic accident, your world should have vanished three sentences ago.
So… What Are We to Make of This?
If being itself tilts towards truth, then the fact that we can reason about the world without watching it collapse is more than the product of blind chance. Instead, it is woven into the structure of reality, minds not merely existing but with the capacity to know. Indeed, the remarkable thing is not just that consciousness occurs. It’s that it endures, anchored to a world intelligible enough to be shared. And if endurance, coherence, and truth are written this deeply into the grain of things, then perhaps the most rational response to the Boltzmann Brain paradox isn’t despair. It is wonder. Not least, it is wonder that we are here at all.
A Tennis Tale with a Twist
A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman & Hall/CRC Press).
THE WORLD OF BOBBY SMITH, A BUDDING TENNIS PRODIGY
Bobby Smith, a young tennis player, faces daunting odds in his journey to professional status. In his world, 1 in 1,000 schoolboy tennis players make it to the professional ranks.
THE TEST: BOBBY’S GATEWAY TO THE ACADEMY
Bobby takes a crucial test to join the prestigious tennis academy, which serves as a breeding ground for future professionals. Though he passes the test and enters the academy, we must consider what this really indicates about his chances of turning pro.
THE FALLACY: MISINTERPRETING PROBABILITIES
It’s crucial not to confuse the probability of Bobby turning pro (given his academy entry) with the inverse, the probability of him entering the academy if destined to turn pro. While all professionals come from the academy, not all academy members become professionals.
THE GATEKEEPER: A SPECIAL TEST
Bobby is given a test designed to gauge the potential of young tennis players, which is used to determine who will have the privilege of becoming a member of the tennis academy, a training and nurturing ground for aspiring professionals. Bobby takes this test with the goal of securing membership.
THE CHALLENGE: OVERCOMING THE ODDS
The test is taken by a thousand of these budding tennis players, including Bobby, all of whom want to enter the academy. Just 5% of those tested will gain entry to the academy and then fail to become professional players.
Graduation from the academy is also a condition of entry to the professional tour in Bobby’s world. As such, we can rule out anyone who does not gain entry to the academy as a future professional player.
BOBBY’S TRIUMPH: ENTERING THE ACADEMY
Fortunately for Bobby, he aces the test and joins the academy. This is a crucial step for him. After all, every professional player in Bobby’s world, as we have noted, is a graduate of the academy.
It might now seem almost certain that he has a bright future ahead in the world of professional tennis. But is this a correct assessment?
Well, it is undeniable that without entrance to the academy there is no way for Bobby to achieve professional status, but he has aced the test and is now a member of the academy. Give the accuracy of the test in sifting talent, can we now look forward with some confidence to his future sporting career?
Well, determining the probability of Bobby becoming a professional tennis player if he scores well enough on the test to gain entry to the academy is a complex matter. It involves factors beyond just his entrance to the academy. Many other elements, such as his dedication, talent, and the competitive environment, play roles in determining his chances. Even so, it does look promising, or does it?
THE FALLACY: AN ILLUSION OF CERTAINTY
Back to the test result, we must be very careful not to confuse the probability of Bobby going on to a professional tennis career given his entrance to the academy with its inverse—the probability that he would enter the academy if he were to go on to attain the professional ranks.
In our example, the probability of his entrance to the academy given that Bobby will make it to professional circles is a sure thing. All future professional players will be graduates of the academy. What we seek to know, however, is something different—it is the probability that Bobby will become a professional player given that he enters the academy. This is a very different question.
Put another way, the fallacy arises from confusing two distinct probabilities:
The probability of a hypothesis being true (Bobby will become a professional tennis player) given some evidence (entrance to the academy).
The probability of the evidence (entrance to the academy) given the hypothesis is true (Bobby will become a professional player).
In simple terms, if we know that Bobby became a professional, he definitely went to the academy. But that’s not what we’re interested in. We want to know the odds of Bobby becoming a professional, given that he got into the academy.
So what is the actual chance that Bobby will become a professional tennis player if he scores well enough on the test to gain entry to the academy?
CALCULATING THE REAL PROBABILITY: BEWARE OF FALSE POSITIVES
When we dig deeper into the data, we uncover some revealing insights. Consider the 5% of students who pass the test and enter the academy but don’t go on to become professional players—they are the ‘false positives’ in our scenario. If we assume 1,000 students take the test, 50 such ‘false positives’ get into the academy.
Add to them the one student who does become a pro (from the original pool of 1,000), and you find that Bobby’s chances of turning pro, even after making it into the academy, are just 1 in 51. This translates to approximately 1.96%.
This will only change if we know some additional information about Bobby.
THE MEDICAL ANALOGY: VIRUS TESTING
Interestingly, this concept aligns with the ‘false positives’ problem in the medical field, particularly in regard to virus testing. Let’s take a group of 1,000 people getting tested for a certain virus. Even with a test accuracy of 95%, about 5% of those tested (50 individuals) will also test positive despite not carrying the virus. On top of these, one individual does have the virus. Thus, if you test positive, the probability of actually carrying the virus is again about 1.96%, unless there is some additional information we need to take into account.
A MATHEMATICAL ASSURANCE: BAYES’ THEOREM
Though we’ve already figured out Bobby’s chances of turning pro, there’s another way to confirm our findings. This alternative method involves Bayes’ theorem. This theorem helps us calculate the updated probability of a hypothesis (in our case, Bobby turning pro) after obtaining new evidence (Bobby entering the academy).
The formula is expressed as follows:
ab/[ab + c (1 − a)]
where
a is the prior probability, i.e. the probability that Bobby will turn pro before we know his test results (0.001, as Bobby is one among 1,000),
b is the probability of Bobby entering the academy if he will turn pro (which is 100%, as all pros in Bobby’s world are academy graduates), and
c is the probability of Bobby entering the academy if he won’t turn pro (which is 50 out of 999, as out of the 999 kids who won’t turn pro, 50 will enter the academy).
By plugging these values into Bayes’ theorem, we confirm that Bobby’s chances of becoming a professional, despite gaining entry to the academy, are not 95% as one might think, but around 1.96%.
CRUNCHING NUMBERS: THE HARD REALITY
To summarise, let’s analyse the situation numerically. Among the 1,000 kids applying for the academy, 50 will be accepted but won’t make it to professional status. One will eventually turn pro. So, out of the 51 kids admitted, only one will become a professional. Therefore, the chance of becoming a professional tennis player if you enter the tennis academy is 1 in 51, or roughly 1.96%, unless there is some additional information that we need to take into account.
THE TWIST: A SUCCESS STORY
Despite the low probability, Bobby turns out to be the exception. He defies the odds and ends up winning the Australian Open under a different name.
CONCLUSION: BEYOND THE NUMBERS
Bobby’s story highlights how statistical probabilities can mislead our intuition. Understanding these concepts is crucial, whether assessing the future of a tennis player or interpreting medical test results. Despite the odds, individuals like Bobby can defy statistics, reminding us that while numbers describe populations, they don’t predetermine individual destinies.
The Cosmological Argument Most People Miss
Most discussions surrounding the “cosmological argument” default to a simplified premise: “Everything has a cause, therefore the universe must have a cause, and that cause is God”. This is usually met with the equally familiar retort: “If everything needs a cause, what caused God?” But that exchange misses the strongest version of the argument entirely. The deepest form of the cosmological argument isn’t about what happened 13.8 billion years ago at the Big Bang. It asks something far more immediate: What is sustaining the universe in existence right now?
This is at the heart of the “Hierarchical Cosmological Argument”. It doesn’t care about the start of the clock; it cares about the “floor” beneath your feet.
1. Borrowed Being: A Universe on Suspenders
We usually picture causation as a linear chain through time, like falling dominoes. But the hierarchical argument concerns simultaneous dependence. Think of a chandelier. It hangs from a chain, the chain from a hook, the hook from a beam, and the beam from the foundation of the house. At no point does the chain support itself. Its “lifting power” is entirely borrowed. If you remove the hook, the chandelier falls immediately, not after a delay, but now.
The claim is that reality has this same vertical structure:
• A tree exists because its molecular structure is held together.
• Molecules depend on the bonding properties of atoms.
• Atoms depend on subatomic particles and quantum fields.
At every level, these things do not exist by their own nature. They are “contingent”—they receive their existence from a deeper layer. This raises a sharp question: Can the entire structure be made of nothing but borrowers?
2. Why the Chain Can’t Be All Mirrors
One response is to suggest an infinite chain of dependence. But adding links to the chain doesn’t explain how the chain is staying in the air.
Imagine a room of mirrors. Each mirror reflects light but produces none. You can add a thousand mirrors, or an infinite sea of them, but without a light source, the room remains pitch black. An infinite chain of “borrowers” does not explain the gift of existence; it merely postpones the explanation forever.
If everything has existence only because something else “lends” it, then existence itself becomes a miracle without a source. At that point, one must either accept that reality is a “Brute Fact”, meaning it exists for no reason at all, or admit there is a foundation.
3. From Potentiality to Pure Act
To sharpen this, classical philosophy looks at the difference between Potentiality and Actuality. Everything we encounter is a mixture of both. A coffee bean is actually a bean but potentially a cup of espresso. It cannot “upgrade” itself; it requires something already actual (hot water and pressure) to flip the switch from potential to actual. If the universe is just a vast collection of these “switches,” there must be a First Mover that isn’t a mixture of potential and act. There must be something that is Pure Actuality:
• It has no “unrealised potential.”
• It doesn’t need to be “switched on” by anything else.
• It doesn’t just have existence; its very nature is existence.
4. Why call the First Cause “God”?
Why call this source “God” rather than a fundamental field of physics? Because anything described by physics is composite—it is a “LEGO set” of parts, laws, and properties. If a thing has parts, it depends on those parts to exist. It is still borrowing its being.
Whatever sits at the absolute bottom of the hierarchy must be:
1. Undivided (Non-composite): It cannot have parts, or it would depend on them. It is a single, seamless reality.
2. Necessary: It cannot not exist; it is the ground that allows everything else to be.
3. Unique: There can only be one “Pure Act,” as two would require a difference to distinguish them, and difference implies composition.
When classical theists speak of God, they aren’t talking about a “super-being” living inside the universe. They are talking about the substance of reality itself, the “Hook” from which the entire chain of physics hangs.
5. The Cost of Denial
Does this argument compel belief? Not like a mathematical trap. You can always walk away. But walking away has a price. To reject the foundation, you must accept that at the very bottom of reality, reason simply stops. You must accept that the “mirrors” are reflecting light that comes from nowhere. You must embrace ultimate arbitrariness as the source of all order.
Or you can ask: Why is there, right now, something rather than nothing? Why has the world not vanished into nothingness while you were reading this sentence?
The Hierarchical Cosmological Argument provides a real and meaningful answer to both questions. It is, for this reason, an argument that quite simply refuses to go away.
From Possibility to Necessity
The God Logic: Why Possibility Implies Necessity.
Many arguments for God’s existence ask us to look out: at the intricate dance of galaxies, the fine-tuning of physical constants, the sheer improbability of life, and far more. These arguments frame God as a hypothesis to be inferred from the evidence.
But there is also another, radically different argument, one that has perplexed and fascinated philosophers for nearly a millennium. It requires no telescope or microscope. It looks instead to the very nature of meaning, possibility, and reality itself.
This is the Ontological Argument. First formulated by St. Anselm of Canterbury in the 11th century, its claim is as startling as it is profound: if you can coherently conceive of God, then God must exist not only in your mind but in reality.
The Definition: The Unsurpassable Limit of Thought
Anselm begins not with a statement of faith, but with a definition rooted in logic:
God is “that than which nothing greater can be conceived”.
This isn’t a poetic flourish; it is a strict logical boundary. To grasp it, imagine ranking all conceivable things by their “greatness”, not in terms of size or moral character, but of their perfection or fullness of being.
Consider these examples of how we intuitively understand “greatness” in this philosophical context:
• A being that possesses consciousness is greater than one that does not.
• A being that is all-powerful is greater than one with limitations.
• A being that exists independently is greater than one that depends on something else for its existence.
Now, push this concept to its absolute limit. If you can think of a being, and then imagine a greater one, you have not yet conceived of God. By definition, God is the logical ceiling, the maximal and unsurpassable case.
The Pivot: Why Existence in Reality is a Perfection
Here we arrive at the argument’s crucial and most debated step. Ask yourself a simple question: which is greater, something that exists only as an idea, or something that exists in reality as well?
Imagine the most perfect island, complete with pristine beaches, ideal weather, and untold riches. Now, which is greater: the idea of this island in your mind, or this island existing in the actual world?
Everything screams that reality is superior. An imagined masterpiece lacks what a real one possesses: actuality.
Anselm applies this directly to his definition of God.
1. We have a coherent idea of God as the greatest conceivable being.
2. But if this being existed only in our minds, it would not be the greatest conceivable being. We could conceive of something greater: the same being, but with the added perfection of existing in reality.
3. This creates a logical contradiction: the greatest conceivable being would not be the greatest conceivable being.
Since contradictions are logically impossible, the initial premise, that God exists only as an idea, must be false. Therefore, if the concept of God is coherent, God must exist in reality.
The Modern Upgrade: From Possibility to Necessity
For centuries, many philosophers dismissed Anselm’s argument as a clever but flawed “proof by definition”. The most famous early objection came from the monk Gaunilo, who argued that if Anselm was right, one could define a “perfect island” into existence.
Philosophers have since noted a key distinction: an island has no intrinsic maximum. For any “perfect” island, we can always conceive of a “greater” one. God, as a being of maximal perfection, is not subject to this flaw.
In the 20th century, philosophers like Alvin Plantinga revived the argument using the tools of modal logic—the logic of possibility, necessity, and contingency. This modern version is widely considered more robust.
It proceeds as follows:
1. Possibility: It is at least logically possible that a “maximally great being” exists. A being is maximally great if it is all-powerful, all-knowing, and morally perfect in every possible world. To refute this, one must show the concept is logically incoherent, like a “married bachelor”.
2. Necessity: A key attribute of a maximally great being is that it would exist necessarily. It could not be a contingent being that just happens to exist, as depending on anything else for its existence would be an imperfection.
3. The Axiom: A foundational principle of modal logic states that “If it is possible that a necessary being exists, then that necessary being does exist”.
4. Conclusion: Therefore, if the existence of a maximally great being is even logically possible, even 0.0000000000000000001% possible, or far less, it must actually exist.
As such, the modern argument shifts the debate from whether existence is a “perfection” to a starker question:
Is the concept of a maximally great being logically coherent?
Where the Real Dispute Lies
This is what makes the ontological argument so intellectually potent. It bypasses debates over empirical evidence and forces the sceptic into a very specific corner.
To reject the argument, it is not enough to say, “I see no evidence for God”, even if you don’t. One must be prepared to argue that the very idea of a maximally great being is as logically self-contradictory as a four-sided triangle.
If the concept is even possibly coherent, however, the logic compels the conclusion that such a being exists necessarily and, therefore, actually. That is a far heavier burden of proof for the atheist than is commonly assumed.
The Takeaway
The ontological argument does not present God as another object to be found within the universe. It presents God as the very ground of reality, a being whose nature, if coherent, if even possible at all, does not permit non-existence.
And if God is possible, even 0.0000000000000000001% possible, then we are led to the inescapable logical conclusion that God exists.
Performance, Putts, and Parables
A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press, 2024).
THE PROBLEM OF LOSS AVERSION
Loss aversion, a key concept in behavioural economics, explains our tendency to prefer avoiding losses over acquiring equivalent gains. This principle is illustrated in diverse settings, from biblical parables to professional sports.
LOSS AVERSION IN THE VINEYARD
The Gospel of Matthew in the New Testament of the Bible relates the Parable of the Workers in the Vineyard. Here’s a breakdown of the parable:
Setting: A landowner goes out early in the morning to hire workers for his vineyard.
Early Morning Hiring: The landowner hires the first group of workers at daybreak, agreeing to pay them a denarius for the day’s work, which was a typical day’s wage at the time.
Subsequent Hirings: Throughout the day, the landowner hires additional groups of workers. He goes out again at the third hour (around 9 a.m.), the sixth hour (noon), the ninth hour (3 p.m.), and even the 11th hour (5 p.m.) to hire more workers. With these groups, he doesn’t specify an exact wage, but promises to pay them ‘whatever is right’.
End of the Day—Payment Time: When evening comes, the workers line up to receive their wages, starting with those who were hired last. To everyone’s surprise, the workers hired at the 11th hour receive a full day’s wage (a denarius).
Discontent among the First Hired: Seeing this, the workers who were hired first expect to receive more than their agreed-upon denarius. However, they too receive the same wage. This leads to grumbling against the landowner. They feel it’s unfair that they worked the whole day, bearing the heat, and yet received the same as those who worked just an hour.
Landowner’s Response: The landowner points out that he did not cheat the first workers; he paid them the agreed-upon wage.
Reference Points and Relative Outcomes: The workers who commenced early had a clear reference point; a denarius for a day’s toil. Despite receiving the agreed wage, they perceived receiving the same pay as those who worked less as a loss. Observing others receiving equivalent wages for fewer hours of work framed their wages in this way. This episode underlines the significance of reference points in human decisions, emphasising the relational aspect of outcome evaluations, surpassing absolutes. The dissatisfaction emanating from the early workers is a classic example of loss aversion. It is a central feature of modern Prospect Theory, albeit some 2,000 years ahead of its time.
BENCHMARKS AND BEHAVIOUR
An intriguing study on the behaviour of New York City cab drivers focused on how they decided the duration of their work shifts in relation to their daily earnings. Contrary to the expectations set by traditional economic theories, which suggest workers would maximise their hours on days with higher demand (and thus potential earnings), the study discovered that cab drivers tended to work fewer hours on days where they were earning more per hour and worked longer hours on less lucrative days. This behaviour aligns more closely with the concept of setting earning targets—drivers tended to end their shifts once they reached a certain income goal, regardless of how quickly they achieved it. This finding, led by Colin Camerer and Linda Babcock, challenged the rational agent model in economics, suggesting that real-world decisions are influenced more by psychological factors and personal benchmarks than traditional economics would expect.
BENCHMARKS ON THE GOLF COURSE
In the game of professional golf, the reference point takes on a very different aspect. This time the problem does not arise because those playing for just a couple of holes earn the same as those who complete four rounds. Instead, the problem arises from the so-called ‘Par’ assigned to each hole, which is a benchmark indicating the number of strokes that a skilled golfer, typically a ‘scratch golfer’ – who plays without any handicap – is expected to take to complete that particular hole under standard playing conditions. While the total number of shots should be the player’s real concern in most competitions, regardless of the assigned ‘par’ scores, the fear of failing to achieve par on individual holes may trigger the influence of loss aversion. In this case, the aversion is to underperforming expectations.
FINDING EVIDENCE OF LOSS AVERSION
Contrary to what might seem rational, analysis of more than 2.5 million putts, with detailed measurements of initial and final ball placement, reveals that professional golfers are indeed significantly influenced by the artificial par reference point. Specifically, it can be shown from the data that golfers are less accurate when putting to score better than par on a hole than when aiming for par. The data suggests, as such, that professional golfers exert more effort to avoid missing par on a hole than in scoring better than par. But why? Their only true objective in most competitions should be to minimise the number of shots taken, regardless of the ‘par’ assigned to each hole.
A paper published by Devon Pope and Maurice Schweitzer in the American Economic Review in 2011 examines a range of possible explanations, systematically eliminating them one by one until the true cause becomes evident. They conclude that golfers view par as their ‘reference’ score. Therefore, a missed par putt is perceived by golfers, perhaps subconsciously, as a more significant loss than a missed ‘birdie’ putt, i.e. one shot better than par. The reality is that par is an artificial construct; all that really matters in strokeplay competitions is the total number of shots taken. This implicit mental bias, however, leads to irrational behaviour during the game, with golfers unable to adjust their strategy even when made aware of this bias.
Interestingly, the researchers also observed a tendency for equivalent ‘birdie’ putts to come up slightly short of the hole in comparison to par putts, further confirming the hypothesis that a fear of a loss to par impacts the players’ putting strategies.
EXPLORING ALTERNATIVE EXPLANATIONS
Despite the compelling evidence for loss aversion, alternative explanations were considered. The possibility that birdie putts might originate from more precarious positions, for example, was explored. However, comprehensive data and rigorous analysis ruled out competing theories.
DYNAMICS OF LOSS AVERSION ACROSS TOURNAMENT ROUNDS
Interestingly, the accuracy gap between par and birdie putts varied across the tournament rounds. It was largest during the initial round and decreased significantly by the fourth round. This fluctuation suggests that the influence of loss aversion and the salience of the par reference point are neither automatic nor immutable and may be affected by factors such as competitor scores later in the tournament.
IMPLICATIONS: BENEFICIAL KNOWLEDGE FOR FORECASTERS AND PSYCHOLOGISTS
This unique insight into professional golfer behaviour has profound implications. It provides valuable information for sports forecasters or even those betting on the match in-play. Moreover, it serves as critical knowledge for sports psychologists working with professional golfers. If these psychologists could find a way to subtly reframe a golfer’s perception of a birdie putt, they could significantly improve a golfer’s performance and earnings over time.
CONCLUSION: PERSISTENCE OF BENCHMARKS AND LOSS AVERSION
This analysis demonstrates that even in a high-stakes, competitive market setting, loss aversion persists among experienced agents. Even top-performing golfers in the study displayed signs of this enduring bias, highlighting its pervasive influence in decision-making scenarios.
The concept has, of course, much broader significance than in competitive sports or even at the workplace. It has been shown to exist in so many of our personal choices and perceptions. By considering how we respond in our own lives to artificial benchmarks and reference points, it has the potential to significantly improve our everyday decisions and actions for the better.
A spoiler-free guide; including the Secret Traitor twist
How to Win the New Traitors
At first glance, The Traitors looks like a game about spotting liars. Watch a little longer and you realise it’s something else entirely. It’s a game about how people react to uncertainty, who feels safe to keep around, and when being right actually makes you dangerous.
The latest UK format adds an elegant complication: the Secret Traitor. Alongside the familiar, visible (to viewers) Traitors, there is now one extra player who knows who the Traitors are, but is unknown to everyone else, including them. This Secret Traitor provides a shortlist of candidates for the visible Traitors to eliminate from. This twist doesn’t just add drama. It changes how the game should be played.
1. Stop Playing Detective
The most common mistake, especially among Faithfuls, is treating the game like a puzzle to be solved. That instinct is understandable. It’s also usually fatal.
You don’t win The Traitors by proving who the Traitors are. You survive by not giving the group a reason to get rid of you. Those are very different skills. Players who sound sharp, confident, and decisive often leave early — not because they’re wrong, but because they look like future trouble. Being useful is safer than being impressive.
2. Confidence Is Riskier Than Being Wrong
Nothing raises eyebrows faster than certainty.
- Confident Faithfuls look like they might organise votes later.
- Confident Traitors look like they know more than they should.
- Confident Secret Traitors don’t stay secret for long.
The safest tone is calm uncertainty:
- Ask rather than accuse.
- Share doubts instead of conclusions.
- Be willing to change your mind out loud.
People rarely banish someone for being unsure. They often banish people who sound settled.
3. What the Secret Traitor Changes
The Secret Traitor quietly removes a comforting illusion: that the Traitors are a tight, informed unit acting with shared purpose. They aren’t, and this has three knock-on effects:
Strange behaviour is harder to interpret
Not every odd move is a “tell” anymore. Some confusion is baked into the structure.
Traitors have less control than they did
Knowing that someone else has influence, but not knowing who, makes bold play risky.
Blending in becomes even more powerful
Players who don’t dominate discussions or force narratives are harder to justify removing. The game shifts away from detection and towards social equilibrium.
4. How to Play the Secret Traitor
The Secret Traitor sits in the most powerful position, and the most precarious one.
The key is restraint. Don’t look clever. Your strength is invisibility. The moment others think you’re steering outcomes, you’re vulnerable. So make decisions that feel obvious. When influencing eliminations, aim for choices that don’t spark debate. If no one talks about your move, it’s probably a good one.
Let others absorb the drama. If tension follows a decision, allow louder players to carry it. Stay adjacent, not central. Think in weeks, not days. Short-term manoeuvring is tempting. Longevity comes from patience. The best Secret Traitor move is often the one nobody notices.
5. Likeability Outlasts Insight
The players who last tend to do small things well:
- They listen.
- They acknowledge others’ worries.
- They soften disagreements instead of sharpening them.
- They don’t insist on being heard.
Someone who’s “probably right but irritating” is far more at risk than someone who’s “possibly wrong but calming”. In this game, people vote out threats, not errors.
6. Why Players Really Get Banished
Most banishments aren’t about guilt or innocence. They happen because someone is:
- Too intense
- Too articulate
- Too eager
- Too early
There’s a real paradox here: good reasoning increases danger unless it’s carefully disguised as reflection. Often, the smartest thing to do is stop just before you land the point.
Final Thought: It’s Not a Mystery — It’s a Social Test
The Traitors isn’t about uncovering truth. It’s about managing fear, comfort, and trust in a group that never has enough information. The Secret Traitor twist simply makes that reality harder to ignore.
If you want to survive, and maybe win:
- Be understandable, not dazzling.
- Be flexible, not fixed.
- And remember that how you make people feel matters more than what you know.
That’s the real game, secret or otherwise.
Can prediction markets find missing MH370?
Confronting uncertainty.
For more than a decade, Malaysia Airlines Flight MH370 has occupied a strange place in the modern imagination: a wound that has never closed. A plane with 239 people on board does not simply vanish in the 21st century, or at least it shouldn’t. And yet it did!
Now, once again, the southern Indian Ocean is being searched.
A quiet, highly technical operation is underway, led by Ocean Infinity, under a renewed “no-find, no-fee” agreement with the Malaysian government. Its vessel, Armada 86 05, is deploying autonomous underwater vehicles capable of descending nearly 20,000 feet, scanning the seabed with sonar, magnetometers, and high-resolution 3D mapping. The target zone, around 5,800 square miles, has been refined using years of accumulated analysis.
There are no dramatic press conferences this time, no daily briefings. Just machines slipping silently into black water, searching terrain no human will ever see.
What we know, and what we still don’t
MH370 disappeared on 8 March 2014, forty minutes after take-off from Kuala Lumpur bound for Beijing. Military radar later showed the Boeing 777 deviating sharply from its planned route, flying south for hours into one of the most remote regions on Earth. Satellite data confirmed continued flight, but not where it ended, or why.
The largest multinational search in aviation history followed, at enormous cost. It failed to locate the main wreckage or flight recorders. And yet the evidence is no longer a blank page.
A flaperon, part of a wing control surface, was recovered on Réunion Island in 2015 and identified by investigators as almost certainly originating from MH370. Additional fragments, judged “very likely” to be from the aircraft, later washed up along the East African coast and Indian Ocean islands. Oceanographers refined drift models. Satellite analysts revisited the data again and again. The picture narrowed, even if it never snapped fully into focus.
This is where MH370 now sits: not in a fog of ignorance, but in a haze of probability.
Why this search feels different
Ocean Infinity has been here before. A 2018 seabed search came up empty. An earlier phase of this renewed effort was paused due to weather. Scepticism is not only understandable; it is rational. What has changed is not just technology, but synthesis.
The current search area reflects years of accumulated judgment across disciplines: aviation, satellite communications, oceanography, wreck recovery. It is, in effect, the best collective guess we can now make about where the aircraft lies.
And that brings me back to an idea I first explored in this context nearly ten years ago.
The problem of the “lone expert”
We like to imagine breakthroughs coming from a single decisive insight: the brilliant analyst, the overlooked data point, the final piece of the puzzle. But MH370 resists that narrative. No single expert, model, or dataset has been enough.
In problems like this, where uncertainty is vast and information fragmented, history suggests a different approach can work better: aggregating judgment.
In 1968, when the US Navy submarine USS Scorpion was lost, the search area was overwhelming. Instead of relying on one authoritative theory, experts were asked to make independent probabilistic assessments. When those assessments were combined, the wreck was found within a few hundred metres of the predicted location. The lesson is not mystical. It is practical. Groups of informed people, when aggregated properly, can outperform even the best individual experts.
In a sense, Ocean Infinity’s search already embodies this idea. But it does so informally, behind closed doors. A structured mechanism, such as a carefully designed prediction market restricted to qualified experts, can help surface neglected hypotheses, test assumptions, and dynamically re-weight search priorities as new information emerges.
This is not about “betting” in some probability exercise on tragedy. It is about recognising that uncertainty itself can be measured, and that human judgment, when pooled intelligently, is a tool rather than a weakness. The wisdom of the crowd is often greater than even its strongest member.
Confronting uncertainty
For the families and friends of the 239 people on board, this search is about being able to say, finally, this is where they are. It is about burial, mourning, and the end of limbo. For the rest of us, MH370 is a reminder of something deeply unsettling: that even in an age of satellites, big data, and constant connectivity, parts of the world, and parts of our systems, remain frighteningly opaque.
If Ocean Infinity succeeds, it will be a triumph of persistence and engineering. If it fails, the story should not end in resignation. The question then becomes not “why didn’t we look harder?”, but “did we think hard enough about how we look at all?”
The challenge is a great one – it is about how we should confront uncertainty, share knowledge, and search together when no one has the full answer.
And that question, unlike the aircraft, has never really disappeared.
Professor Leighton Vaughan Williams 