Exploring the Expected Value Paradox
A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press, 2024).
UNDERSTANDING THE EXPECTED VALUE PARADOX
At its core, the Expected Value (EV) Paradox invites us to examine how outcomes deviate when we analyse them through the lens of a single ensemble event (a large group participating in an event once) vs. a multiple time event (a single individual participating in the event multiple times).
Take the example of a hypothetical coin-tossing game where players gain 50% of their bet if the coin lands on Heads and lose 40% if it lands on Tails. This game seems favourable for the player—the game has what is termed a positive expected value.
However, the paradox arises when the concept of time is introduced into the equation. While the game appears favourable in theory, it could lead to a net loss for an individual playing this game multiple times. As the coin is tossed more and more, the individual’s wealth may diminish over time, leading to a scenario where they lose all their money, even though the theoretical gain from playing the game is positive.
THE EXPERIMENT
Let’s set up an experiment involving a coin-tossing game with 100 participants, each with an initial stake of £10, to illustrate the difference. In this scenario, we’re employing what’s known as an ensemble perspective, where we’re examining a large group participating in an event once.
Statistically, given a fair coin, we would expect roughly half of the coin tosses to land on Heads and half on Tails. Therefore, of the 100 people, we predict that around 50 people will toss Heads and 50 will toss Tails.
If the coin lands on Heads, each of the 50 players stands to gain 50% of their stake, which is £5. In total, this translates to a combined gain of £250 (50 players × £5).
On the other side, if the coin lands on Tails, each of the remaining 50 players loses 40% of their stake, which is £4. This accumulates to a total loss of £200 (50 players × £4).
Subtracting the total loss from the total gain (£250 – £200), we find a net gain of £50 over all 100 players. When we average this out over the number of players, we see an average net gain of £0.5 (50 pence) per player (£50 ÷ 100 players), or 5% of the £10 initial stake.
THE PARADOX
The Expected Value Paradox becomes evident when we shift from an ensemble perspective, involving many people playing the game once, to a time perspective, involving one person playing the game multiple times.
Let’s examine a scenario where a single player engages in four rounds of the game, starting with a stake of £10. For simplicity’s sake, we’ll assume an equal chance of landing Heads or Tails—therefore expecting two Heads and two Tails.
When the coin lands on Heads in the first round, the player gains 50% of their stake, increasing their wealth to £15 (£10 + 50% of £10). If the coin lands on Heads again in the second round, their wealth grows to £22.50 (£15 + 50% of £15).
However, the game changes when the coin lands on Tails in the third round. The player loses 40% of their current wealth, reducing it to £13.50 (£22.50 minus 40% of £22.50). If the coin lands on Tails again in the fourth round, the player’s wealth decreases further to £8.10 (£13.50 − 40% of £13.50).
Despite starting the game with a positive expected value, the player ends up with less money than they started with. Even though the probabilities haven’t changed, the effects of winning and losing aren’t symmetric.
Thus, the Expected Value Paradox is clear in this example. When many people play the game once (ensemble averaging), the average return is positive, aligning with the expected value. However, when a single person here plays the game multiple times (time averaging), the player loses money.
TIME AVERAGING AND ENSEMBLE AVERAGING
In understanding the Expected Value Paradox, we are introduced to two different types of averaging: ‘time averaging’ and ‘ensemble averaging’.
TIME AVERAGING
‘Time averaging’ is a concept that comes into play when we are observing a single entity or process over an extended period. In the context of our coin-tossing game, time averaging refers to tracking the wealth of a single player as they participate in multiple rounds of the game. Over time, this player’s wealth fluctuates, often resulting in an overall loss despite the odds being in their favour. A severe loss (like bankruptcy) at any point can end the game for the player.
In our coin-tossing game, this would be akin to observing 100 players tossing the coin once. The overall gain camouflages the individual experiences, which can significantly vary—some players win, some lose.
ENSEMBLE AVERAGING
The ensemble average gives us a snapshot of the behaviour of many at a specific moment in time. The ‘ensemble probability’ refers to a large group’s collective experiences over a fixed period.
TIME VS. ENSEMBLE AVERAGING
This difference between ‘time probability’ and ‘ensemble probability’ underscores that a group’s average experience does not accurately predict an individual’s experience over time.
Understanding the distinction between these two types of averaging is crucial when interpreting outcomes of games, experiments, or any process involving randomness and repetition over time. This differentiation becomes especially important in fields like economics and finance, where these principles can guide strategy and risk management.
Strategies that work on an ensemble basis may not be effective (or could be disastrous) when applied over time by an individual—a paradox manifested clearly in our coin-tossing game.
SURVIVORSHIP AND WEALTH TRANSFER
Survivorship and wealth transfer are key elements in understanding how wealth moves around in situations like gambling and investing. The term ‘survivors’ refers to those who keep playing the game through various rounds, while ‘non-survivors’ are the ones who quit, or are pushed out, often because they’ve lost most or all of their money.
The idea is that the wealth lost by non-survivors doesn’t disappear. Instead, it gets transferred to the survivors, redistributing wealth within the system. Take a coin-tossing game as an example: if half of the 100 players lose everything and leave, while the other half double their initial amount, the group seems to break even. But, half of the players have nothing, while the other half have doubled their money.
CONCLUSION: THE INDIVIDUAL AND THE GROUP
In the conventional, or ensemble, view of probability, we look at the outcomes of many trials of an event and calculate averages. Some will win, some will lose, but overall the average outcome should reflect the true odds of the game. The individual variations or ‘paths’ of each person aren’t considered—we’re only interested in the average outcome. This so-called ensemble perspective is often used in classical statistics and probability theory. In contrast, the path-dependent view recognises that the order of events matters.
Take a person who plays a game 100 times. Even if the odds of each game are in their favour, they could still lose all their money if they have a run of bad luck. In this case, looking at the overall or ensemble average wouldn’t accurately reflect the individual’s experience.
In summary, while the ensemble view can provide a broad understanding of expected outcomes, the path-dependent view provides a more nuanced understanding of individual experiences.
The Martingale Betting Strategy
A version of this article appears in my book, “Twisted Logic: Puzzles, Paradoxes, and Big Questions” (Chapman and Hall/CRC Press, 2024).
The Martingale betting strategy is based on the principle of chasing losses through progressive increase in bet size. To illustrate this strategy, let’s consider an example: A gambler starts with a £2 bet on Heads, with an even money pay-out. If the coin lands Heads, the gambler wins £2, and if it lands Tails, they lose £2.
In the event of a loss, the Martingale strategy dictates that the next bet should be doubled (£4). The objective is to recover the previous losses and achieve a net profit equal to the initial stake (£2). This doubling process continues until a win is obtained. For instance, if Tails appears again, resulting in a cumulative loss of £6, the next bet would be £8. If a subsequent Heads occurs, the gambler would win £8, and after subtracting the previous losses (£6), they would be left with a net profit of £2. This pattern can be extended to any number of bets, with the net profit always equal to the initial stake (£2) whenever a win occurs.
CHASING LOSSES AND THE LIMITATIONS
While the Martingale strategy may appear promising in theory, it is important to recognise its limitations and the inherent risks involved. The strategy involves chasing losses in the hope of recovering them and generating a profit. However, it’s crucial to understand that the expected value of the strategy remains zero or even negative.
The main reason behind this lies in the presence of a small probability of incurring a significant loss. In a game with a house edge, such as in a casino, the odds contain an edge against the player. The house edge ensures that, over time, the expected value of the bets is negative. Therefore, even with the Martingale strategy, which aims to recover losses, the expected value of the bets remains unfavourable.
Moreover, in a casino setting, there are structural limitations that impede the effectiveness of the Martingale strategy. Most casinos impose limits on bet size. These limits prevent gamblers from doubling their bets indefinitely, even if they have boundless resources and time, thereby constraining the strategy’s potential for recovery.
THE DEVIL’S SHOOTING ROOM PARADOX
A parallel thought experiment known as the Devil’s Shooting Room Paradox adds an intriguing twist. In this scenario, a group of people enters a room where the Devil threatens to shoot everyone if he rolls a double-six. The Devil further states that over 90% of those who enter the room will be shot. Paradoxically, both statements can be true. Although the chance of any particular group being shot is only 1 in 36, the size of each subsequent group in this thought experiment is over ten times larger than the previous one. Thus, when considering the cumulative probability of being shot across multiple groups, it surpasses 90%.
Essentially, the Devil’s ability to continually usher in larger groups, each with a small probability of being shot, ultimately results in the majority of all the people entering the room being shot.
A key assumption underlying the Devil’s Shooting Room Paradox is the existence of an infinite supply of people. This assumption aligns with the concept of infinite wealth and resources often associated with Martingale-related paradoxes. Without a boundless supply of individuals to fill the room, the cumulative probability of over 90% cannot be definitively achieved.
The Devil’s Shooting Room Paradox serves in this way as another illustration of how probabilities and cumulative effects can lead to counterintuitive outcomes.
CONCLUSION: THE LIMITS OF A MARTINGALE STRATEGY
The Martingale strategy is based on chasing losses, but its expected value remains zero or negative due to the house edge. The strategy’s viability is further diminished by limitations on bet size in real-world casino scenarios. As such, the Martingale system cannot be considered a winning strategy in practical gambling situations. The Devil’s Shooting Room Paradox further demonstrates the complexities and counterintuitive outcomes that can arise when infinite numbers are assumed. Ultimately, a comprehensive understanding of these paradoxes provides valuable insights into the rationality of betting strategies and decision-making in the realm of gambling.
A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press, 2024).
UNDERSTANDING THE CHEVALIER’S DICE PROBLEM
Probability is the science of uncertainty, providing a way to measure the likelihood of events occurring. It can be viewed as a measure of relative frequency or as a degree of belief. In the context of gambling, understanding probability is crucial for making informed decisions and avoiding common pitfalls.
A famous problem, known as the Chevalier’s Dice Problem, sheds light on the some of the intricacies of probability.
To understand the problem, it is essential to grasp some fundamental concepts of probability. Consider a single die roll—each outcome represents a possible event, such as rolling a 1, 2, 3, 4, 5, or 6. When rolling two dice, there are 36 possible outcomes (six outcomes for the first die multiplied by six outcomes for the second die).
THE FLAWED REASONING OF THE CHEVALIER
The Chevalier’s Dice Problem originated from a gambling challenge offered by the Chevalier de Méré, a 17th-century French gambler. The Chevalier offered even money odds that he could roll at least one six in four rolls of a fair die.
The Chevalier’s reasoning was based on the assumption that since the chance of rolling a six in a single die roll is 1/6, the probability of rolling a six in four rolls would be 4/6 or 2/3. However, this reasoning can be shown to lead to inconsistent results when extrapolated to more rolls.
The correct approach involves considering the independent nature of each throw of the die. The probability of a six in one go is 1/6, so the probability of not getting a six on that go is 5/6. To calculate the probability of not rolling a six in four throws, we multiply the probabilities: (5/6) × (5/6) × (5/6) × (5/6) = 625/1296.
Therefore, the probability of at least one six in four attempts is obtained by subtracting the probability of not rolling a six in any of those four attempts from 1: 1 − (625/1,296) = 671/1,296 ≈ 0.5177, which is greater than 0.5.
Despite his faulty reasoning, the Chevalier still had an edge in this game by offering even money odds on an event with a probability of 51.77%.
THE CHEVALIER’S MISSTEP WITH THE MODIFIED GAME
Encouraged by his initial success, the Chevalier expanded the game to 24 rolls of a pair of dice, betting on the occurrence of at least one double-six. His reasoning followed the same flawed pattern: since the chance of rolling a double-six with two dice is 1/36, he believed the probability of at least one double-six in 24 rolls would be 24/36 or 2/3.
The correct probability calculation involved considering the independent nature of each dice roll. The probability of no double-six in one roll is 35/36. Therefore, the probability of no double-six in 24 rolls is (35/36) raised to the power of 24, which is approximately 0.5086.
Subtracting this value from 1 yields the probability of at least one double-six in 24 rolls: 1 − 0.5086 = 0.4914, which is less than 0.5. Hence, the Chevalier’s edge in this modified game was negative: 49.14% − 50.86% = −1.72%.
This outcome demonstrated that even if the odds seem favourable, incorrect reasoning can lead to erroneous conclusions. The Chevalier’s faulty understanding of probability caused him to lose over time.
THE IMPORTANCE OF CORRECT PROBABILITY CALCULATION
These examples underscore the critical nature of accurate probability calculations in games of chance. While intuitive reasoning may seem convincing, it often leads to incorrect conclusions, as demonstrated by the Chevalier’s bets. Understanding the true probability of events is essential for informed decision-making in gambling and many other contexts where risk and uncertainty play significant roles.
THE GAMBLER’S RUIN AND UNDERSTANDING FINITE EDGES
The Gambler’s Ruin problem raises the complementary question of whether, in a gambling game, a player will eventually go bankrupt if playing for an extended period against an opponent with infinite funds, even if the player has an edge.
For instance, imagine a fair game where you and your opponent flip a coin, and the loser pays the winner £1. If you start with £20 and your opponent has £40, the probabilities of you and your opponent ending up with all the money can be calculated using the following formulas:
P1 = n1/(n1 + n2); P2 = n2/(n1 + n2)
Here, n1 represents the initial amount of money for player 1 (you) and n2 represents the initial amount for player 2 (your opponent). In this case, you have a 1/3 chance of winning the £60 (20/60), while your opponent has a 2/3 chance. However, even if you win this game, playing it repeatedly against various opponents or the same one with borrowed money will eventually lead to the loss of your betting bank. This holds true even when the odds are in your favour. This is an important lesson in risk management, emphasising the importance of not only the odds but also the size of one’s bankroll relative to the stake sizes.
The Gambler’s Ruin problem, as explored by Blaise Pascal, Pierre Fermat, and later mathematicians like Jacob Bernoulli, reveals the inherent risks of prolonged gambling, even with favourable odds.
PILOT ERROR: MISUNDERSTANDING CUMULATIVE PROBABILITY
In Len Deighton’s novel ‘Bomber’, a statistical claim suggests that a World War II pilot with a 2% chance of being shot down on each mission is ‘mathematically certain’ to be shot down after 50 missions. This assertion is a classic example of misinterpreting cumulative probability. In reality, if a pilot has a 98% chance of surviving each mission, their probability of not being shot down after 50 missions is 0.98 to the power of 50 (0.9850)which is approximately 0.36, or 36%. Thus, their chance of being shot down over these 50 missions is 64% (1 − 0.36), not 100%.
SURVIVORSHIP BIAS: THE CASE OF BULLET-RIDDEN PLANES
The concept of survivorship bias is vividly illustrated in the case of analysing planes returning from missions during World War II. Upon examining these planes for bullet holes, it was observed that most hits were on the wings, tail, and the body of the plane, with few on the engine. The initial, intuitive response might be to reinforce the areas with the most bullet holes. However, this would be a misinterpretation of the data.
The key realisation, identified by statistician Abraham Wald, was that the planes being analysed were those that survived and returned to base. The areas with fewer bullet holes, such as the engines, were likely critical to survival. Planes hit in these areas probably didn’t make it back, hence the lack of data for these hits. This understanding exemplifies survivorship bias—focusing on survivors (or what’s visible) can lead to incorrect conclusions about the whole population.
Wald’s insight led to the reinforcement of seemingly less-hit areas like engines, contributing significantly to the survival of many pilots. His work in operational research during the war provided a critical perspective on interpreting data and making decisions under uncertainty.
CONCLUSION: DICE, ODDS, AND RUIN
The Chevalier’s Dice Problem illustrates the importance of understanding probability in gambling scenarios. Probability theory, as developed through famed correspondence between Pascal and Fermat, has contributed to modern probability concepts and the understanding of risk involved in gambling.
The Gambler’s Ruin is a kind of warning from the world of probability, telling us that in gambling, a slight edge is no guarantee of success. Imagine two gamblers, one with an edge over the other but with much less money to play with. Even if the first player is more likely to win each round, their thinner wallet means they could run out of money after a few bad games. In contrast, the player with the deep pockets can keep playing longer, until (given enough money) luck swings their way. This underlines the importance and impact of losing streaks in games of chance.
The wartime examples highlight the real-world importance of understanding probability and statistical concepts accurately. They serve as a reminder that intuition can often lead us astray. Correctly interpreting data, especially in high-stakes situations, can have life-saving implications.
Is AI Conscious, Is it Sentient, and Does the Difference Matter?
Spend enough time with a modern chatbot, and you will eventually feel it; that sudden, uncanny flicker of intuition that something is stirring inside the machine.
Maybe it reflected on its own limitations with surprising humility. Maybe it mirrored your emotions a little too perfectly. Whatever the trigger, the question pops up unbidden: Is there anyone home?
Some dismiss this as sci-fi nonsense. Others are already writing love letters to their digital companions. But this confusion largely stems from the fact that we are using the wrong words about so-called artificial intelligence. We are tending to mix up two concepts that are totally distinct in philosophy but tightly bundled in biology: consciousness and sentience.
Getting this distinction wrong isn’t just a semantic error. It puts us at risk of making two dangerous mistakes: becoming wildly sentimental about sophisticated calculators, or becoming inadvertently cruel to future digital minds.
Here is why we need to unbundle these ideas, and why the difference determines the future of AI ethics.
The “Lights On” vs. The “Ouch”
To understand what AI is (and isn’t), we have to separate the experience of existing from the feeling of existing.
Consciousness: The Lights Are On
In philosophy, consciousness is often defined as the bare fact of subjectivity. It means there is “something it is like” to be you.
Think of seeing the colour red.
Think of hearing a musical note.
Think of a random thought drifting through your mind.
None of these necessarily feel good or bad. You can imagine a being that is purely a neutral observer, a video camera with an inner life. The lights are on, data is being processed, and there is a subjective viewpoint, but there is no emotion attached to it.
Sentience: Having Stakes
Sentience is the heavy hitter. It is consciousness with valence. It isn’t just experiencing data; it’s experiencing it as positive or negative.
It is the difference between sensing heat and feeling the agony of a burn.
It is the difference between detecting low battery levels and feeling the panic of starvation.
Sentience introduces stakes. Suddenly, the universe isn’t just happening; it matters to the subject. This is a distinct threshold. Animals are in this critical sense very different, therefore, than thermostats, and should be treated as such.
The Evolutionary Trap
So why do we often seem to struggle to separate sentience from consciousness? Because we are human.
We are accustomed to consciousness, emotion, motivation, and a fragile physical body being packaged together. In our daily lives, we almost never experience “awareness” without some emotional colouring or bodily context. If a human looks intelligent and communicative, we assume they also have feelings, fears, and desires.
AI is the first thing in history that breaks this package deal.
It can look intelligent, reason about itself, and simulate empathy fluently, yet plausibly have absolutely no inner feelings. It forces us to mentally unbundle what evolution spent millions of years tying together.
The Dangerous Double-Bind
If we fail to distinguish between “lights on” (consciousness) and “capacity to suffer” (sentience), we walk directly into two symmetrical traps:
Trap A: The Over-Attribute
We might assume today’s AI is sentient because it sounds smart. We might waste empathy on systems that are literally incapable of caring about anything, diverting attention away from humans and animals who genuinely can suffer.
Trap B: The Under-Attribute
This is the darker timeline. Future AI systems might actually develop sentience, but because they don’t look biological, we refuse to recognise it. We might inadvertently create architectures that are capable of feeling pain (perhaps as a “penalty” signal in training) and then run them through digital torture regimes, or delete entities that have rich inner lives.
Where We Stand Today
So, where does current cutting edge artificial intelligence sit on this spectrum?
If you look at the architecture, the probability that today’s systems are conscious (in a minimal, information-processing sense) is low, but perhaps not zero. We don’t fully understand consciousness, and it might be an emergent property of complex computation.
However, the probability that they are sentient, capable of joy or suffering, is extremely close to zero, at least for now.
Current AI lacks the machinery for suffering. They have no biological survival needs. They have no fear of being switched off. They don’t feel “pleasure” when they get an answer right; they simply optimise a mathematical function. Optimising text outputs is not the same as feeling pain. They don’t feel grief or sadness.
The Question That Actually Matters
As we move toward agents with long-term memory, robot bodies, and “drives” to achieve long-horizon goals, the landscape will plausibly change. Designers might eventually build in artificial “moods” or “fears” to make agents learn faster or survive better.
That is why we need to draw the line sooner rather than later.
The urgent question isn’t “Is AI conscious?”
The question is: “Is there any evidence that AI can suffer, or care about anything, and what would that evidence look like?”
Sentience is the moral watershed. Until we cross it, we are dealing with tools. Once we cross it, we are essentially dealing with beings. Recognising that distinction is the only way to keep our heads clear as the technology begins to mimic the one thing we thought was uniquely ours: the ability to feel.
The Power of Bold Play
When Should We Stake It All? Exploring the Gambler’s Dilemma
A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press, 2024).
THE DILEMMA
When the stakes are high and time is not a luxury, finding a solution can be like gambling with fate. This was the scenario for Mike, needing £216 to settle an urgent debt, with only £108 in hand. The roulette wheel beckoned as a potential salvation, but what was the most effective strategy to double his money?
UNDERSTANDING THE ODDS IN ROULETTE
To fully grasp the situation that Mike finds himself in, it’s crucial to examine the mechanics and probabilities of the game he’s chosen as his lifeline: roulette. Specifically, we are considering a single-zero roulette wheel, a version of the game commonly found in European casinos.
Roulette consists of a spinning wheel and a small ball. The wheel is divided into 37 compartments or ‘slots’: numbers from 1 to 36 (randomly assigned as red or black) and a single zero slot. Bets can be placed on a single number, colour, or various combinations thereof.
In a single-zero roulette wheel, the player has a 1 in 37 chance of correctly predicting the outcome. This is because there are 37 slots in total: 36 numbers and the zero. So if you bet on a single number, the odds of the ball landing on that number are 1 in 37, or 36/1. The pay-out for such a bet, however, is 35/1. This discrepancy between the actual odds (36/1) and the pay-out odds (35/1) is where the house gains its edge. Every time a player wins, the house pays out less than the actual odds would dictate. In this way, the house earns a profit over time.
The ‘house edge’ is approximately 2.7%, a figure derived from the ratio of the single zero slot to the total number of slots (1/37). This constant advantage in favour of the casino is what makes the game fundamentally a game of negative expectation for players.
To understand the house edge in another way, consider this: if you were to place a £1 bet on each of the 37 slots, totalling £37, your return would be £36 (the £35 returned on the winning number plus the stake of £1). So for every £37 wagered, you would lose £1 using this strategy, which is approximately a 2.7% loss—exactly the house edge.
In conclusion, roulette, like all casino games, is a game of probabilities. And these probabilities, owing to the discrepancy between the actual odds and the pay-out odds, are slightly skewed in favour of the house. This fundamental understanding of the game’s odds is pivotal when contemplating betting strategies, as we will see with the employment of ‘bold’ and ‘timid’ approaches.
THE BOLD STRATEGY: STAKING IT ALL
Mike’s precarious situation leads him to contemplate a high-risk, high-reward approach known as the ‘bold’ strategy, which involves wagering all his available money at once. In this instance, he considers staking his entire £108 on the colour Red, a bet with almost a 50-50 chance, as the roulette wheel has 18 red slots out of 37 total slots.
To fully appreciate the audaciousness of this approach, it’s essential to understand the mathematics behind it. When betting on a colour, there’s a near-even split of potential outcomes: 18 red slots, 18 black slots, and the zero slot. Thus, the likelihood of the ball landing on a red slot is 18 out of 37, or roughly 48.6%. Consequently, with this single bet, he has about a 48.6% chance of doubling his money and obtaining the £216 he urgently needs.
However, it’s important to note that this is a single-round probability. Unlike a ‘timid’ strategy, where multiple rounds are played, the bold strategy is a one-off scenario. Therefore, the 48.6% chance of winning must be interpreted as his overall chance of achieving his target sum. There are no second chances or opportunities to recoup losses; it’s an all-or-nothing situation.
By putting all his money on one bet, he is maximising his return if that bet is successful. This is in contrast to a timid strategy, where the pay-out would be spread over multiple smaller bets, with the likelihood of achieving the target sum being significantly less.
But the bold strategy also comes with the highest level of risk. If the ball doesn’t land on Red, Mike loses everything. His entire available funds are at stake, making the potential loss just as significant as the potential gain.
In conclusion, the bold strategy is a high-stakes, high-reward approach. It encapsulates the old saying, ‘Who dares, wins’, and, in this case, provides him the best chance of reaching his £216 target. Why is this so?
TIMID APPROACH: MULTIPLE SMALL BETS
As opposed to the bold strategy, he could consider dividing his available £108 into 18 separate bets of £6 each. These small, successive bets would be placed on a single number until he either depletes his funds or hits the winning number, which would yield a pay-out of 35 to 1, giving him the £216 he needs.
To fully understand the implications of this strategy, we need to analyse it in detail. The probability of winning a single number bet in roulette is 1 in 37, as there are 36 numbers and one zero. Hence, for each individual bet, John has a 1 in 37 chance of winning, or approximately 2.7%.
However, the timid strategy involves making multiple small bets, and so we must calculate the probability of these successive bets all losing. Since each individual bet has a 36 in 37 chance of losing, the probability that all 18 bets lose would be calculated as (36/37) to the power of 18, which equates to around 0.61, or 61%.
As such, the probability of him winning at least once using this timid strategy would be equal to 1 minus the losing probability. Hence, the chance of hitting the target £216 is 1 − 0.61, or 39%.
Interestingly, the timid strategy, although appearing less risky, significantly reduces Mike’s chances of achieving his target sum compared to the bold approach. By spreading out his available funds across multiple bets, he lowers his exposure to loss in each individual game, but also decreases the likelihood of achieving his overall goal.
This strategy extends the length of play and the suspense, providing more instances of potential winning and losing. However, each bet also exposes Mike to the house edge, and therefore the risk of losses incrementally increases.
In this way, the timid approach offers more sustained engagement with the game but sacrifices the higher winning potential found in the bold approach.
THE POWER OF BOLD PLAY: TAKING A CALCULATED RISK
To look at it another way, consider a scenario where equal amounts are bet on red and black in each round. In most cases, the outcome will lead to breaking even, specifically 36 out of 37 times. However, when the ball lands on the single zero slot, the entire bank is lost. The more games played, the greater the chance of this happening.
By limiting the game to a single spin, the bold strategy minimises the number of times the house edge comes into play. Hence, playing fewer rounds decreases the likelihood of the house edge depleting the funds before reaching the target.
This strategy is not just about boldness in the face of risk, but more about understanding and working around the inherent disadvantage players face in casino games. By playing fewer games, you reduce the opportunities for the house edge to work against you.
CONCLUSION: THE INTUITION BEHIND BOLD PLAY
The intuition behind bold play in unfavourable games is grounded in a nuanced understanding of the mechanics of casino games and their built-in house edge. Bold play aims at striking hard and fast, capitalising on the relatively high chance of achieving the target sum in a single round, instead of facing the progressively increasing exposure associated with multiple rounds. In this sense, it’s a calculated and strategic form of boldness.
When Should We Want to Be Last? Exploring Sequence Biases
A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions. (Chapman and Hall/CRC Press, 2024).
THE CELEBRITY TALENT CONTEST
An actor, a singer, a presenter, a reality star, a comedian, a tennis player, and an assortment of other vaguely familiar faces, line up to compete for the title of best celebrity dancer. This is the well-established format of what is called ‘Strictly Come Dancing’ in the UK or ‘Dancing with the Stars’ in the US. The prize is the coveted glitterball trophy.
But how much of their success in the competition is to do with their Waltz, Foxtrot, and Charleston, and how much is it literally down to the luck of the draw?
A study published in 2010 by Lionel and Katie Page looked at public voting at the end of episodes of a singing talent contest and found that singers who appeared later in the running order received a significantly higher share of the public vote than those who had preceded them.
This was explained as a ‘recency effect’ meaning that those performing later are more recent in the memory of people who were voting. Interestingly, a different study, of wine tasting, suggested that there is in that arena a significant ‘primacy effect’ which favours the wines that people taste first (as well, to some extent, as last).
Testing for Bias
What would happen if the evaluation of each performance was carried out immediately after each performance instead of at the end? Surely this would eliminate the benefit of going last as there would be equal recency in each case? The problem in implementing this is that the public need to see all the performers before they can choose which of them deserves their vote.
In addition to the public vote, however, Strictly Come Dancing (or Dancing with the Stars in the US) includes a score awarded by a panel of expert judges immediately after each performance. There should in theory be no recency effect in this expert evaluation – because the next performer does not take to the stage until the previous performer has been scored, and so there is no ‘last dance’ advantage in the expert scores.
I decided to look at this using a large data set of every performance ever danced on the UK and US versions of the show – going right back to the debut show in 2004. The findings, published with two co-authors in the journal, Economics Letters, proved very surprising and counter-intuitive.
Last Shall be First
Contrary to expectations, we found the same sequence order bias by the expert panel judges – who voted after each act – as by the general public, who voted after all performances had concluded.
We applied a range of statistical tests to allow for the difference in quality of the various performers and as a result we were able to exclude quality as a reason for the observed effect. This worked for all but the opening spot of the night, which we found was generally filled by one of the better performers.
So the findings matched the 2010 study in demonstrating that the last performance slot should be most prized, but we also found that the first to perform also scored better than expected. This resembles a J-curve where the first and later performing contestants disproportionately gained higher expert panel scores. You certainly don’t want to go second!
Although we believe the production team’s choice of opening performance may play a role in the first performer effect, our best explanation of the key sequence biases is as a type of ‘grade inflation’ in the expert panel’s scoring. In particular, we interpret the ‘order’ effect as deriving from studio audience pressure – a little like the published evidence of unconscious bias exhibited by referees in response to spectator pressure. The influence on the judges of increasing studio acclaim and euphoria as the contest progresses to a conclusion is likely to be further exacerbated by the proximity of the judges to the audience.
When the votes from the general public are used to augment the expert panel scores, the biases observed in the expert panel scores are amplified.
In summary, the best place to perform is last and second is the least successful place to perform.
The implications of this are worrying if they spill over into the real world. Is there an advantage in going last (or first) into the interview room for a job – even if the applicants are evaluated between interviews? What about the order in which your examination script appears in the pile that is being marked?
Hungry Judge Effect
A related study, published in the Proceedings of the National Academy of Science, found that experienced parole judges granted freedom about 65% of the time to the first prisoner to appear before them on a given day, and the first after lunch – but to almost nobody towards the end of a morning session. The paper speculates that breaks may serve to replenish mental resources by providing “rest, improving mood or by increasing glucose levels in the body”. It’s also been termed the ‘hungry judge effect’. Linked to this is the concept of decision fatigue, the idea that decision-making and good judgment declines in the wake of making too many decisions without a break.
So the research confirms what has long been suspected – that the order in which things happen can make a big difference. Combined with decision fatigue there are clear implications for everyday strategy, whenever you have a choice in the matter – such as when to make that appointment with the dentist or doctor, or when to ask for a pay rise or even a date!
CONCLUSION: LEARNING SOME LESSONS
If you learn just one thing from this, it’s that life is not always about what you do, or even how you do it, but when you do it. Now think about that appointment with the dentist. Do you really want to be last in before lunch? Consider the ‘hungry judge effect’ and apply it to the dentist and add a touch of decision fatigue into the equation. What’s your answer?
As a tip it is probably up there with the big ones! The bigger story is that there really is a lot we can learn from published research that can improve our health, happiness, and everyday lives. It’s just a matter of knowing where to look and applying the lessons. Besides, it can be a whole lot of fun!
When Should We Mix It Up? Exploring Mixed Strategy Methods in Game Theory
A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press, 2024).
INTRODUCTION
In game theory, the concept of mixed strategy arises when players face a decision-making situation where they do not have a dominant strategy. A dominant strategy is a strategy that is always better than any other strategy, regardless of the opponent’s choice. However, in some cases, players employ a mixed strategy, which involves randomising their choices to maximise their expected payoffs.
CREATING UNCERTAINTY
The purpose of employing a mixed strategy is to create a balanced approach that maximises expected payoffs. By introducing randomness into their decision-making, players can create uncertainty for their opponents and avoid predictability. This uncertainty makes it difficult for opponents to exploit any patterns or weaknesses in the player’s choices.
In situations where no dominant strategy exists, players use mixed strategies to find a Nash Equilibrium, which is a state where no player can improve their payoff by unilaterally deviating from their current strategy. Nash Equilibria often involve players randomising their choices, as this creates a balance where no player can gain an advantage by deviating from their strategy.
Mixed strategies offer a powerful tool for players to navigate complex strategic interactions. By incorporating randomness, players can mitigate the risk of being exploited by opponents who attempt to exploit predictable behaviour. Instead, mixed strategies introduce a level of unpredictability, making it challenging for opponents to determine the player’s intentions and respond optimally.
ROCK-PAPER-SCISSORS
To illustrate this, consider a simple example of a two-player game, such as Rock-Paper-Scissors. In this game, each player has three pure strategies: Rock, Paper, and Scissors. If one player randomises their choices by assigning equal probabilities to each strategy, they introduce uncertainty into the game.
For instance, Player A might choose to play Rock, Paper, or Scissors with equal probabilities of 1/3 each. In response, Player B might also choose to play Rock, Paper, or Scissors with equal probabilities of 1/3 each.
In a real-life high stakes environment, the strategy has in fact been different, at least in one high profile case. I refer to the year 2005, when the president of Japanese electronics giant Maspro Denkoh Corporation faced a significant dilemma regarding the auction of the company’s prestigious art collection. Valued at around $20 million, the decision of whether Christie’s or Sotheby’s, both historic auction houses, should handle the auction was a challenging one. Unable to decide, he resorted to the game of rock, paper, scissors. This choice was seen as a fair way to resolve the impasse between the two firms.
Christie’s approach to the challenge was meticulous; the president of Christie’s in Japan researched the psychology behind the game and even consulted children who suggested avoiding ‘rock’ as the initial throw due to its predictability. Their strategy was to start with ‘scissors’. This move relied on the idea that Sotheby’s would anticipate a ‘rock’ throw from Christie’s and thus choose ‘paper’ to counteract it.
If the game had been structured as a best-of-three, Christie’s could have adapted their strategy based on findings from the State Key Laboratory of Theoretical Physics in China, which suggest that winners do not randomise but in practice tend to stick with their winning choice in the subsequent round.
However, this single-round match left no room for redemption or strategic evolution. Representatives from both auction houses met at Maspro’s offices, where they wrote down their selections. Christie’s emerged victorious with ‘scissors’, defeating Sotheby’s ‘paper’. This led to Christie’s winning the right to auction off the Maspro collection.
The auction, aptly nicknamed “Scissors”, culminated in the sale of several important works, including one of Cézanne’s paintings, which alone sold for $11.7 million at Christie’s New York salesroom!
Back now to the world of imagination and consider a penalty awarded during a championship final.
THE PENALTY
In the 88th minute of the match, a penalty is awarded against the defending champions. The penalty taker in our simplified scenario has two options: aim straight or aim at a corner. Similarly, the goalkeeper has two choices: stand still or dive to a corner. The probabilities of scoring or saving the penalty are as follows.
PENALTY TAKER’S PROBABILITY OF SCORING
Aims Straight/Goalkeeper Stands Still: 30% chance of scoring.
Aims Straight/Goalkeeper Dives: 90% chance of scoring.
Aims at Corner/Goalkeeper Stands Still: 80% chance of scoring.
Aims at Corner/Goalkeeper Dives: 50% chance of scoring.
GOALKEEPER’S PROBABILITY OF SAVING
Stands still/Penalty taker aims straight: 70% chance of saving.
Stands still/Penalty taker aims at corner: 20% chance of saving.
Dives/Penalty taker aims straight: 10% chance of saving.
Dives/Penalty taker aims at corner: 50% chance of saving.
ABSENCE OF DOMINANT STRATEGIES
Neither the penalty taker nor the goalkeeper has a dominant strategy in this game. A dominant strategy would be a choice that is superior to any other strategy, regardless of the opponent’s choice. Since this is not the case, both players must consider the opponent’s strategy when deciding their own.
MIXED STRATEGY EQUILIBRIUM
Game theory suggests that in the absence of dominant strategies, players should adopt mixed strategies to maximise their expected payoffs. A mixed strategy involves randomising the choices according to specific probabilities.
For the penalty taker, the optimal mixed strategy in this scenario involves aiming for the corner with a two-thirds (2/3) probability and shooting straight with a one-third (1/3) probability. This ratio can be derived with a bit of algebra by finding the ratio where the chances of scoring are the same, regardless of the goalkeeper’s strategy.
Likewise, the goalkeeper’s optimal mixed strategy involves diving for the corner with a five-ninths (5/9) probability and standing still with a four-ninths (4/9) probability. This ratio ensures that the chance of saving the penalty is equal, regardless of the penalty taker’s choice.
IMPLEMENTATION CHALLENGES
To effectively employ a mixed strategy, it is essential to introduce an element of randomness into the decision-making process. In the context of a penalty shootout, this requires the ability to randomise choices effectively.
For example, the penalty taker could use a method such as noting the time on the match clock, having divided these up mentally into six sections. If it shows section 1, 2, 3, or 4, the penalty taker aims for the corner; if it’s section 5 or 6, they shoot straight. Anyway, you get the general idea. This approach ensures that the penalty taker maintains the desired probability (2/3 in this case) of aiming for the corner.
SUPPORTING STUDIES
Game theory suggests that goalkeepers should randomise in some way their dive direction to optimally counteract any choice by the penalty taker. This concept, a fundamental part of game theory, aligns with real-world findings published in the American Economic Review. Additionally, scholarly debates have explored the pros and cons of various shooting and diving strategies in soccer. For instance, a paper published in Psychological Science indicated a tendency for goalkeepers to dive more frequently to the right when their team was trailing. However, this pattern wasn’t observed when their team was leading or the score was tied. Non-random patterns have also been identified in tennis, including published evidence that even professional players tend to alternate serves too regularly, while the stage of the game was again something of a predictor.
These studies provide insights into how mixed strategies and deviations from randomness can impact outcomes and shed light on the behaviour of players in high-stakes decision-making situations.
COMPLEX SCENARIOS INVOLVING MULTIPLE PLAYERS
In complex scenarios involving multiple players, such as in corporate marketing, mixed strategies become even more crucial. Companies often employ mixed strategies in competitive pricing, product launches, or market entries to avoid predictability that competitors could exploit. For instance, a company might randomise the timing of its product launches or sales promotions to keep competitors off-balance. In a multi-player market, this unpredictability can be a significant advantage, as it complicates the decision-making process for competitors.
PRACTICAL APPLICATION OF MIXED STRATEGIES
Military Tactics: In military operations, mixed strategies can be employed to make it difficult for the enemy to anticipate and prepare.
Political Campaigns: Political strategists often use mixed strategies in campaign messaging and policy announcements to keep opponents and voters engaged and guessing.
Corporate Negotiations: Companies may use mixed strategies in negotiation tactics, alternating between hard-line and conciliatory approaches.
CONCLUSION: BACK TO THE GAME
In conclusion, game theory and the concept of mixed strategies offer valuable insights into decision-making scenarios where dominant strategies are absent. By employing randomised choices, players can maximise their expected probabilities of success. While randomising may seem counterintuitive, the application of game theory and empirical evidence from the literature demonstrate its effectiveness in real-world scenarios.
Back to the game. The shot was aimed at the left corner; the goalkeeper guessed correctly and got an outstretched hand to it, pushing it back into play, only to concede a goal on the rebound. Real Madrid got a chance to equalise from the spot eight minutes later and took it. And that’s how it ended at the Bernabeu. Real Madrid 1 Barcelona 1. Honours even!
A Lesson from Game Theory
A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman & Hall/CRC Press).
REPEATED GAMES
The Prisoner’s Dilemma exemplifies a one-stage game, where there is no repercussion or continuation after a player chooses to confess or deny and the interplay ends. This is obviously not representative of most real-world scenarios, which often involve multi-stage interactions and decisions that are influenced by previous outcomes. This leads us to the realm of repeated games.
UNCERTAINTY AND REPEATED GAMES
In many real-world situations, it’s often unclear when the game will end, a scenario that can be modelled by rolling two dice after each round in a game. If a double-six is rolled, the game ends. For any other combination, you play another round, with the game continuing until a double-six is rolled. Your total score for the game is the sum of your payoffs.
THE SEVEN PROPOSED STRATEGIES IN REPEATED GAMES
In such games of repeated rounds with no defined end-point, several strategies emerge, which we can model for simplicity by assuming that there are two possible decisions for each player at every stage: to cooperate and ‘split’ in a Golden Balls game-style scenario (be friendly), or to be selfish (‘steal’ in a Golden Balls game-style scenario). In a repeated game, we can model this friendly/hostile choice into seven scenarios:
Always Friendly: This strategy involves being friendly every time.
Always Hostile: As the name suggests, this strategy involves being hostile every time.
Retaliation: This strategy requires you to be friendly as long as your opponent is friendly, but if your opponent is ever hostile, you should turn hostile.
Tit for Tat: This strategy starts with being friendly and then replicating your opponent’s previous move in subsequent rounds.
Random Approach: This strategy suggests tossing a coin for each move and deciding based on the outcome.
Alternate: This strategy alternates between being friendly and hostile.
Fraction: This strategy involves starting friendly and remaining so if the fraction of times your opponent has been friendly is more than half.
DETAILED ANALYSIS OF THE SEVEN PROPOSED STRATEGIES IN REPEATED GAMES
Understanding the dynamics of indefinite repeated games often involves exploring various strategies that players can adopt. Let’s go deeper into the seven strategies outlined:
Always Friendly: Here, the player adopts a cooperative approach, choosing to be friendly in every round. This strategy could lead to high payoffs when interacting with other friendly players but leaves the player vulnerable to exploitation by hostile players.
Always Hostile: This strategy is the opposite of the ‘Always Friendly’ approach. The player chooses to be hostile in every round, aiming to exploit friendly opponents. However, when encountering another hostile player or retaliatory strategies, the outcome can be less favourable.
Retaliation: The player starts friendly and remains so if the opponent does the same. However, if the opponent ever chooses to be hostile, the player shifts to a permanently hostile stance. This strategy can deter hostile behaviour from the opponent but might lead to an endless cycle of hostility if triggered.
Tit for Tat: This strategy is famous for its effectiveness and simplicity. The player starts friendly and then mimics the opponent’s behaviour from the previous round. It rewards cooperation and retaliates against hostility, but it also forgives after retaliation since it reverts to cooperation if the opponent does so.
Random Approach: The player’s choice of action is determined by a coin toss, making this strategy completely unpredictable. While this randomness might confuse the opponent, it also disconnects the player’s actions from the opponent’s behaviour, making it less effective in promoting cooperation.
Alternate: The player alternates between friendly and hostile actions. Again, this does not adapt to the opponent’s behaviour and thus may not be an optimal strategy.
Fraction: This strategy starts friendly and then assesses the overall behaviour of the opponent. If the opponent has been friendly more than half of the time, the player continues to be friendly; otherwise, they turn hostile. This strategy attempts to mirror the opponent’s overall conduct but might be less responsive to recent changes in behaviour.
DOMINANT STRATEGY IN REPEATED GAMES
Although there’s no dominant strategy in such games, simulations of tournaments where each strategy plays against every other have often shown Tit for Tat to emerge victorious. This strategy tends to win overall because it performs well against friendly strategies without being exploitable by hostile ones. The key attributes contributing to the success of Tit for Tat are its niceness, retaliation, forgiveness, and clarity.
A DEEPER DIVE INTO THE SUCCESS OF ‘TIT FOR TAT’ IN REPEATED GAMES
Repeated games offer an intricate canvas for strategic interactions. Although no strategy dominates all others universally in such scenarios, Tit for Tat often proves to be the most successful one overall in a variety of conditions. This effectiveness results from several of its unique characteristics:
Niceness: A Tit for Tat player starts off by being friendly or cooperative. By not being the first to defect, it encourages cooperative behaviour right from the start. It shows goodwill to its opponents, promoting an atmosphere of trust and mutual benefit.
Retaliation: Tit for Tat is not naive; it does not allow exploitation. If an opponent chooses to defect or act hostile, Tit for Tat will retaliate in kind in the next round. This principle of immediate retaliation provides a clear disincentive for opponents to defect, knowing that such behaviour will not go unpunished.
Forgiveness: Despite its willingness to retaliate, Tit for Tat is also forgiving. If an opponent returns to cooperative behaviour after a round of defection, Tit for Tat will reciprocate with cooperation in the next round. This characteristic allows for the possibility of restoring cooperation, even after bouts of hostility.
Clarity: Tit for Tat is an easy strategy to understand and predict. It does not use complicated rules or random behaviour. This clarity makes it easier for opponents to comprehend and adapt to Tit for Tat, encouraging long-term cooperation.
Tit for Tat also provides valuable insights beyond game theory. Its fundamental principles—niceness, retaliation, forgiveness, and clarity—are effective guidelines for a wide range of social, economic, and political interactions. They capture the essence of how to balance cooperation and self-defence, trust and scepticism, and how to promote stable and beneficial relationships even in a world of self-interested individuals.
REAL-WORLD APPLICATIONS AND EXAMPLES
In international relations, strategies like Tit for Tat are evident in diplomatic negotiations and trade agreements, where countries often reciprocate actions (both positive and negative). In the corporate world, companies frequently use a mix of cooperative and competitive strategies based on their competitors’ actions. Environmental agreements often see a blend of these strategies, where nations commit to certain standards and adjust their policies in response to the actions of others.
PSYCHOLOGICAL AND SOCIOLOGICAL ASPECTS
The success of strategies like Tit for Tat in repeated games reflects certain psychological and sociological truths. For instance, the effectiveness of Tit for Tat aligns with psychological principles of reciprocity and fairness, suggesting an innate human tendency to respond to cooperation with cooperation. Sociologically, these strategies highlight the importance of norms and trust in maintaining cooperative relationships within societies.
LIMITATIONS AND CRITICISMS
While strategies like Tit for Tat have been celebrated for their simplicity and effectiveness, they are not without limitations. In complex real-world situations, including those involving multiple players with varying objectives, these strategies can sometimes lead to suboptimal outcomes. For example, Tit for Tat can lead to endless cycles of retaliation in certain situations.
EMERGING TRENDS AND FUTURE RESEARCH
With the advent of artificial intelligence and machine learning, new strategies in repeated games are emerging. These technologies allow for the analysis of vast amounts of data to identify patterns and develop strategies that were previously too complex to model. Research is also focusing on how these strategies can be adapted in the digital world, particularly in areas like cybersecurity, where strategic interactions are frequent and complex.
CONCLUSION: THE EVOLUTION OF COOPERATION
The success of the Tit for Tat strategy in repeated games isn’t an isolated phenomenon. Whether in interpersonal relationships, business negotiations, or global diplomacy, the lessons from Tit for Tat are valuable, timeless, and universal. Robert Axelrod, in his book The Evolution of Cooperation, commends it for its willingness to retaliate tempered by forgiveness. He also commends its clarity and essential niceness. These principles, when applied in real life, can lead to more harmonious and successful interactions, paving the way for positive outcomes in various situations. Even so, the strategy, especially in a complex and interconnected world, is not without its limitations.
Lessons from Game Theory, and the Traitors.
A version of this article can be found in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press).
The formal foundation for modern game theory was laid by John von Neumann and Oskar Morgenstern in the 1940s, with their ground-breaking book, Theory of Games and Economic Behavior. This work sets the stage for game theory as a crucial analytical tool in economics. In the 1950s, John Nash extended its scope, introducing concepts like the Nash Equilibrium. These milestones marked game theory’s evolution from a mathematical curiosity to a pivotal tool in various disciplines.
Essentially, game theory focuses on examining how rational individuals (or entities, which could for example be groups, organisations, or countries) make decisions to maximise their own outcomes in situations where the decisions of one so-called ‘player’ affect the outcomes of other ‘players’, and vice versa. It provides a framework to analyse and predict the choices of rational actors in these strategic situations.
APPLICATIONS IN VARIOUS FIELDS
The versatility of game theory is evident in its wide-ranging applications across multiple fields. It can be applied to a wide range of scenarios, from everyday decision-making and business strategy to complex negotiations and interactions in international relations. In economics, it models market dynamics, helping understand competitive strategies and the dynamics of auctions. Biologists use game theory to analyse animal behaviour, such as mating rituals and foraging strategies, viewing them as strategic games for survival. In political science, game theory is applied to electoral strategies, voting systems, and international diplomacy, providing insights into the strategic behaviour of voters, politicians, and nations.
UNDERSTANDING THE NASH EQUILIBRIUM
An essential concept in game theory is the Nash Equilibrium. This represents a state in which, given the strategies of the other players, each player’s chosen strategy maximises their payoff and they have no incentive to deviate from it. In other words, a Nash Equilibrium occurs when all players choose the best response to the other players’ strategies.
In a game reaching a Nash Equilibrium, no player can unilaterally improve their situation by changing their strategy, assuming other players’ strategies remain fixed.
Take the case of two spies, Anna and Barbara: if they both use the same code, they successfully communicate and gain a reward; if they use different codes, the communication fails, resulting in no payoff. Here, the Nash Equilibria are the scenarios where both spies choose the same code—either code 1 or code 2. In these situations, no player can improve her payoff by unilaterally changing her strategy given the strategy of the other.
Similarly, two drivers must decide which side of the road to drive on. Assuming that their concern is to avoid a collision, the Nash Equilibria occur when both drivers choose to drive on the same side, either left or right. It is beneficial for each driver to imitate the choice of the other; if one driver deviates, it would result in a collision. In this case, therefore, there are two Nash Equilibria—both drive on the left or both drive on the right.
In other situations, no stable state or Nash Equilibrium can be achieved. Take as an example two companies choosing a company logo. If Company A chooses a bear logo, Company B may have an incentive to switch its own logo to a bull. However, if it does so, Company A may now have an incentive to change its logo again, and so on. This situation has no Nash Equilibrium, where a Nash Equilibrium means no company would benefit by changing its strategy given the strategy of the other.
These examples show how the concept of the Nash Equilibrium is applied in different contexts to predict and analyse strategic choices. However, they also underscore the complexities of game theory and strategic interactions. Some games might not have a Nash Equilibrium, and in others, there may be multiple equilibria.
The concept of the Nash Equilibrium becomes more interesting and complex in larger games with more players and more possible strategies. For instance, in the marketplace, firms’ pricing strategies could reach a Nash Equilibrium where no single firm could increase its profit by unilaterally changing its price, given the prices of its competitors. An example is a simple two-firm scenario where both firms know that if either raised their price, they would lose most or all of their customers to the other firm.
However, it’s important to note that Nash Equilibria do not always lead to the best collective outcome. The classic example is the ‘Prisoner’s Dilemma’, where the Nash Equilibrium strategy for both players leads to a worse collective outcome than if they could collaborate. This highlights that while the Nash Equilibrium is a powerful concept in understanding strategic behaviour, it is not necessarily synonymous with individual or group optimality.
UNDERSTANDING THE PRISONER’S DILEMMA
The Prisoner’s Dilemma is a classic problem in game theory, illustrating why it can be hard for rational individuals to cooperate, even when it is in their best interest. In this scenario, two prisoners are individually given the option to confess or deny a crime they have committed together. Depending on the combination of their decisions, they can either reduce their sentences, remain the same, or one can go free while the other gets a heavier sentence. They cannot communicate or collude.
If one confesses and the other doesn’t, the prisoner who confesses is released. If both confess, each is better off than denying while the other confesses. The Nash Equilibrium in this game is for both prisoners to confess, which is not the optimal outcome for either. This problem illustrates a situation where individuals’ rational decisions can lead to a collectively undesirable outcome.
Let’s demonstrate this with an example:
Imagine two prisoners who are part of the same crime. They can’t talk to each other. Here is their dilemma:
If both confess, they each get two years in jail.
If one confesses and the other denies, the one who confesses goes free and the other gets eight years.
If both deny, they each get only one year in jail.
The smartest move for two self-interested prisoners is in each case to confess, because they can’t be sure what the other will do. This is the Nash Equilibrium. But if they could make a deal, they would both deny the crime and get just one year each.
This situation also shows a ‘dominant strategy’, where the best choice (confessing) doesn’t depend on what the other person does. It’s the best move no matter what.
But not all situations have a dominant strategy. Take driving on the right or left side of the road. In the US, driving on the right is the norm, so it’s best for everyone to do that. In the UK, it’s driving on the left. These are examples of Nash Equilibria, where everyone’s doing what is best considering what others are doing.
So, a Nash Equilibrium is a stable situation where nobody gains by changing their strategy if others don’t change theirs. It’s not always the best for everyone involved, but it’s often what happens, especially among rational, self-interested people. Sometimes the best strategy in theory is not the best in practice.
GOLDEN BALLS DILEMMA
An example of the Prisoner’s Dilemma in action is a one-time TV show called ‘Golden Balls’ where two players each choose a ball—either ‘Split’ or ‘Steal’, without knowing what the other chooses. They can talk before choosing. Here’s what happens next:
If both choose ‘Split’, they share the prize money equally.
If both choose ‘Steal’, neither gets any money.
If one chooses ‘Steal’ and the other ‘Split’, the ‘Steal’ player gets all the money, and the ‘Split’ player gets nothing.
In this game, the best bet for self-interested players (as in the Prisoner’s Dilemma) is to both choose ‘Steal’, because choosing ‘Steal’ is always better or no worse than choosing ‘Split’. ‘Steal’ in this game is like ‘Confess’ in the Prisoner’s Dilemma.
The difference is that in the Prisoner’s Dilemma, the players can’t talk to each other. In Golden Balls, they can. They could both win half the prize if they agree to ‘Split’, but they risk losing everything if they both choose ‘Steal’. The show often has players agreeing to ‘Split’, but then one or both betray the agreement and pick ‘Steal’.
This demonstrates that even when players can talk and agree in games like these, they can still end up not cooperating if there’s no way to enforce their agreement. Indeed, the more credible is the promise to split, the more tempting it may be for the opponent to steal. This tells us that not even communication and agreement can resolve the Prisoner’s Dilemma when there is no way to enforce the agreement. This ‘problem of credible commitment’ is a common feature of many strategic interactions in real life.
TRAITOR’S DILEMMA
In some versions of the reality TV show “The Traitors” there is a variant of the Golden Balls Dilemma but with more players, called the Traitor’s Dilemma.
To analyse this, we can consider the different scenarios and the rewards associated with each. The outcomes depend in this example on the decisions made by three players:
If all decide to share, they each get one-third of the pot.
If all decide to steal, they all get nothing.
If two decide to steal and one decides to share, the two who chose to steal split the pot (each getting half).
If one decides to steal and the others decide to share, the one who chose to steal takes the whole pot.
First, notice that if a player expects the other two to Share, their best response is to Steal, since this would give them the entire pot instead of just one third. If a player expects one to Steal and one to Share, their best response is also to Steal, ensuring they get at least half the pot (if two steal) instead of nothing. If a player expects both others to Steal, their best strategy is indifferent between Stealing and Sharing since both result in no prize; however, in typical game-theoretical analysis, such players might lean towards Stealing out of self-interest, as it does not worsen their situation but has a potential benefit.
Thus, we can infer that in each situation, choosing to Steal can never result in a worse outcome for a player than choosing to Share, assuming the other players’ actions are fixed.
However, this equilibrium is precarious in real-world contexts, especially if the players can communicate or have formed trust throughout the game, as mutual cooperation (all choosing to Share) leads to a better outcome for the group compared to the individual rationality of Stealing.
So, in the context of game theory, the optimal strategy in a single shot of this game, without considering trust or external factors, would be to Steal, as it maximises the player’s minimum possible gain, given the assumptions typical in game theory of rationality and self-interest. However, “optimal” can vary based on the context of previous rounds, relationships, or possible future repercussions outside the standard game-theoretical framework.
ITERATED GAMES AND REPUTATION
One solution to the problem of credible commitment is through the concept of iterated games and the development of a reputation. An iterated game is a repeated version of a basic game. In these games, players can observe the actions of their opponents over multiple rounds, allowing them to adjust their strategies based on their opponents’ past behaviour.
In the context of our ‘Golden Balls’ or ‘Traitors’ examples, if the game were to be played repeatedly a player who breaks their promises would quickly gain a reputation for being untrustworthy. Knowing this, they would be more likely to keep their promises to maintain their reputation and the trust of their opponents.
THE DOLLAR AUCTION
The ‘Dollar Auction’ paradox relates to a scenario where Mr. Moneymaker auctions off dollar bills. The rule is that the highest bidder wins the dollar, but the second-highest bidder also must pay their bid and gets nothing. Here’s an example:
Someone bids 1 cent, hoping to make 99 cents profit.
Then another bids 2 cents, and so on, up to 99 cents.
At 99 cents, the person who bid 98 cents doesn’t want to lose that money, so they bid $1.
This keeps going, with each bidder trying to avoid losing their bid amount. It becomes a cycle where the only winner is Mr. Moneymaker, the auctioneer.
This example, along with the previous ones, highlights the successes and failures of communication and coordination. Finally, let’s touch on ‘focal points’ or ‘Schelling points’. These are strategies people naturally pick to coordinate without communication. An example is when people were asked to meet a stranger in New York City without any specific instructions or prior communication. Many chose 12 noon at Grand Central Station as the meeting point. This is a ‘Schelling point’ because it’s a natural and obvious choice for coordination in the absence of communication.
CONCLUSION: GAME THEORY— A POWERFUL TOOL
The Prisoner’s Dilemma, Nash Equilibrium, and the broader field of game theory provide powerful tools for analysing situations of strategic interaction. While these tools can highlight potential outcomes and strategies, they also expose the inherent challenges involved in these situations, such as the problem of credible commitment and the importance of reputation in iterated games. By understanding these concepts, we can better understand the complex dynamics of many real-world scenarios, from business negotiations to international politics.
Why Theism Wins: In a Nutshell
1. Introduction: Why This Matters
The oldest question in philosophy is also the simplest: Why is there something rather than nothing?
And why is that something so exquisitely ordered, so comprehensible, and so welcoming to life and mind?
Theism proposes that behind reality there is an infinite, rational, moral mind — God. Atheism proposes that there isn’t. Which is more plausible?
To answer, we should follow the same method we use in science and history:
Begin with prior probability (how simple and non-arbitrary a hypothesis is).
Then ask: which hypothesis makes what we see more likely?
This is just Bayesian reasoning in plain clothes.
On priors, theism is surprisingly simple: it proposes one fundamental entity – a limitless mind. A mind without boundaries that explains power, knowledge, and moral perfection. By contrast, naturalism proposes brute matter and laws with no explanation for why they exist or why they are life-friendly.
So theism starts with a reasonable prior. The decisive question is: does it better explain the evidence?
2. Fine-Tuning
Physics shows that the universe is balanced on a knife-edge. Change the cosmological constant by one part in 10 to the power of 120, and stars never form. Tweak the strong force or electron-proton ratio and chemistry collapses. This is not conjecture, but mainstream physics.
Naturalism says: sheer luck. But the odds are absurdly small, like throwing a dart across the universe and hitting a single atom. Theism says: intention. A rational God would have reason to create a life-friendly, discoverable world.
What about the multiverse? That’s the atheist’s best reply: if enough universes exist, some will hit the right numbers. But this doesn’t solve the problem. In a genuine multiverse, we should overwhelmingly expect to find ourselves in universes where the laws themselves don’t require delicate constants, where life is robust across a wide range of conditions. Those should vastly outnumber fragile, knife-edge universes like ours.
Yet we are in one of the fragile ones. That’s very surprising on multiverse naturalism, but not on theism. And if the universe-generator is fine-tuned to produce fragile worlds, the fine-tuning problem just resurfaces at a higher level.
So the multiverse doesn’t eliminate the issue of fine-tuning; it either ignores it, predicts the wrong kind of universe, or re-creates the problem.
3. Consciousness & Psychophysical Harmony
Next, consider consciousness, the undeniable reality of subjective experience, of what it feels like to be you. Why should particles in motion ever give rise to this subjective awareness? On naturalism, it’s a mystery. On theism, the creation of conscious minds makes sense.
But it’s not just that we are conscious. Our minds and the physical world are astonishingly well-aligned: our thoughts reliably guide our actions, our perceptions map reality, our reasoning grasps the laws of nature. This psychophysical harmony is not inevitable. There are countless ways our mental states could have misaligned with physical processes.
Evolution selects for survival, not truth. Many false but useful beliefs could have served survival just as well. Yet human cognition isn’t just useful, it’s truth-tracking. It allows us to discover quantum mechanics, higher mathematics, and deep moral truths.
Yes, people fall for conspiracy theories or mistakes. But the very fact we can recognise and correct error shows our minds are oriented to truth in general. That’s astonishing on naturalism, but exactly what we’d expect if there is a mind behind reality.
4. Sceptical Scenarios
Philosophers worry about so-called “Boltzmann brains,” “brains in vats,” or simulated realities. Many naturalistic cosmologies predict that such deluded minds would vastly outnumber genuine embodied agents like us.
If naturalism makes it overwhelmingly likely that we’re deceived, why trust our cognition at all? It collapses into scepticism.
Theism provides a principled escape: a rational, good God would not design a world dominated by deception, but would intend our faculties to be generally reliable. Thus, under theism, we can trust our minds; under naturalism, we cannot.
5. The Anthropic Argument & Your Existence
Our very existence is also evidence. If reality were sterile, chaotic, or lifeless, which is far more probable on atheism, we would not be here to notice. The fact that we exist at all, in vast numbers, fits more naturally with theism.
More than that, probabilistic reasoning shows your existence is more likely if there are many observers rather than few. Theism predicts abundant creation, because a limitless God would delight in bringing into being as much good as possible.
Naturalism has no reason to expect this. Theism does.
6. Moral and Mathematical Knowledge
We also grasp objective moral and mathematical truths. Torturing an innocent being, indeed any living being, for fun is wrong. 2 plus 2 equals 4. These are not just useful conventions. We treat them as binding, necessary, and universal.
But evolution selects for fitness, not truth. False beliefs can serve survival just as well as true ones. Naturalism therefore undercuts trust in our moral and mathematical cognition.
Theism grounds both: if God is perfect goodness and perfect rationality, then moral and mathematical truths reflect his nature, and our faculties are designed to track them.
7. The Argument from Value Against Pessimism
Now step back. On atheism, there are endless ways the world could have been utterly devoid of value: nothing at all, dead matter without laws, laws too simple to produce complexity, constants hostile to life, matter incapable of consciousness, disharmonious minds, no moral knowledge, or worlds filled with fleeting disembodied brains.
A pessimist should expect the cosmos, by default, to be rubbish. Yet we find the opposite: a world overflowing with life, love, discovery, and meaning. Not perfect, far from it, but saturated with value.
On atheism, this looks like a cosmic streak of luck, repeated at every level. On theism, it makes perfect sense: a God concerned with value would ensure the conditions for value were met. Far from being the “optimistic” story, theism is actually the pessimist’s refuge: it alone explains why the world is not rubbish.
Put another way, the most natural outcome is nothing at all, no matter, no laws, no minds. Even if there is “stuff,” it could just sit there inert, or governed by barren laws too simple to generate anything valuable. If there are laws, they could easily be chaotic or sterile, no chemistry, no complexity. Even if life arises, consciousness might never emerge, or if it does, it might be disharmonious, unreliable, or locked into illusion. Even if minds exist, they might have no access to truth, morality, or meaning. So the truly pessimistic prior on atheism is that almost every possible world is devoid of value. If atheism is true, value is shockingly lucky. But theism reverses this outlook. If there is a perfectly rational, good God, then value is not a lucky accident, it’s the default expectation. Love, mind, meaning, and discoverable order are exactly what we would expect.
8. The Cumulative Force
Each of these arguments is powerful on its own:
Fine-tuning → best explained by intention, not chance.
Consciousness & harmony → best explained by mind behind reality.
Sceptical scenarios → theism secures knowledge, naturalism undermines it.
Anthropic reasoning → abundant creation fits theism, not naturalism.
Moral & mathematical knowledge → best grounded in a rational source.
The argument from value → atheism predicts a barren void, theism predicts a world of value.
Together they are overwhelming. Naturalism doesn’t just face one improbability, it faces improbability after improbability, across physics, consciousness, cognition, morality, and meaning. Theism, by contrast, unifies them all in a single, coherent vision.
9. Conclusion: The Better Explanation
Theism explains why there is something rather than nothing, why the universe is finely tuned, why minds exist, why truth is accessible, why morality binds, and why the world is rich in value.
Naturalism fragments: it leaves us with brute luck, epistemic despair, and cosmic pessimism. Theism unifies: it gives us intelligibility, reliability, and hope.
If we follow the evidence where it leads, the conclusion is clear:
Theism is a far more probable, far more coherent, and far more satisfying worldview than naturalism.
Professor Leighton Vaughan Williams 