Posted in Science & Nature

## Airplane Game

You are cordially invited to a game that lets you earn money very easily. The game works like this:

1. You pay \$1000 to be recruited as a passenger to a plane.
2. There are 8 passengers, managed by 4 crew members, who have 2 co-pilots above them, co-ordinated by a captain at the top.
3. Everytime the “plane” is filled with 8 passengers, the captain retires and is paid out \$8000.
4. When the captain retires, the plane is split into two planes and everyone else is promoted one step higher (co-pilots each become a captain, crew become co-pilots, passengers become co-pilots).
5. When each plane fills with 8 new patients, the captain of each plane gets paid out \$8000 and retires.

This seems like a very easy way to earn money. Where else could you invest money and guarantee a 700% return, only needing to recruit 7 new people into the game?

The problem with the airplane game is that it is a classic example of a pyramid scheme. At first glance, it seems that the payout of \$8000 is guaranteed because it seems that the promotions will keep coming.

But if you look at the mathematics, 8 people need to participate before the first player wins. 16 people have to participate for the second player to win. 80 people have to participate for the tenth person to win. If you are the one-thousandth person to join the game, you need a total number of 8000 people to be playing the game before you are paid out. At the end of the game, 87.5% of people playing will have lost money because they will never be paid out.

This is how simple exponential growth can result in a very real fraud, resulting in thousands of people losing their hard-earned money.

Posted in Philosophy

## Zero-Sum Game

Game theory is the study of using mathematical models to understand how rational decision-makers would strategically act in a given environment. One concept from game theory is that of the zero-sum game, where there is a finite amount of utility shared between players, meaning that if one person gains something, another must lose something to balance it out.

A classic example is a game of competitive sports, where there can be only one winner. For you to win, someone else must lose. A zero-sum game can have as few as two players (such as a singles tennis match) or many players (such as a game of poker, where every dollar you win is a dollar taken away from the other players).

From a young age, we see many examples of zero-sum games. We play sports and board games where there is a clear winner. We are marked on curve and compared to our classmates in exams. We compete for jobs and romantic partners. Competitiveness is driven into us and is sold as a survival skill.

This leads us to be prone to zero-sum thinking which can lead to many biases. Some studies show students acting more competitively and less inclined to help their peers if they were graded on a curve (e.g. percentiles), rather than grade categories (e.g. A, B, C). We think that if someone is a jack of all trades, they are masters of none, because surely no one can “have it all”. Many people oppose immigration because they believe that immigrants will take the finite number of jobs and houses. Some people negotiate aggressively in a deal, thinking that “your loss is my gain”. In severe cases, people may even sabotage others to increase their gains.

However, life is not always a zero-sum game. Game theory also describes non-zero-sum games, where the net balance of utility between all participants can be higher (or lesser) than zero. Simply put, in a non-zero-sum game, there can be more than one winner and sometimes, everyone can be a winner.

The best example of this is the mutual benefit born from cooperation. Zero-sum thinking may dictate that you must conquer your neighbouring tribe because they are your competition, but throughout history, cooperation, peace and harmony have prevailed as the winning strategy, because it results in greater net gain.

Happiness is also a non-zero-sum game, where just because someone else is happy, it does not take away from your happiness. But for some reason, some people cannot stand to watch others happy, or feel they must be happier than those around them. These people constantly try to “one-up” others, not recognising others’ happiness, or even sabotaging others and making them feel bad because they can’t stand to see other people be happier than them. This is an extremely toxic, unnecessary behaviour, that should be unacceptable in any kind of relationship, particularly between friends or family.

The far healthier behaviour is to be happy for others’ happiness, regardless of your life situation. This is why compassion is one of the keys for happiness. Realising that we can all find our own joy and contentness and help each other find happiness is a key step in being sustainably happy.

1 + 1 = 3

Posted in Psychology & Medicine

## Tit For Tat

In human society, there are many ways for a person to interact with others when in a group setting. Some may choose to be selfish and only be out for their best interests, while others may choose altruism and cooperate with each other. The mathematical model that tries to predict human behaviour and outcome in these settings is the Prisoner’s Dilemma – the core of game theory. Tit for tat is one strategy that can be employed in such a setting.

The basis of tit for tat is equivalent exchange. A tit for tat player always chooses to cooperate unless provoked. As seen in the Prisoner’s Dilemma, if both players cooperate, both benefit (let us say 3 points each); if one player defects, that person gains more than from cooperation (5 points) while the tit for tat player gains 0 points.
If a tit for tat player is provoked, that player will retaliate. However, the player is also quick to forgive. Ergo, if the other player chose to cooperate, the tit for tat player (following the principle of equivalent exchange), will also cooperate. If the other player defected, the tit for tat player loses the first round and then chooses to defect from then on.
Note that tit for tat strategy only works when there is more than one game so that the player has a chance to retaliate.

Let us use an example to illustrate why tit for tat strategy works. In this scenario, two tit for tat players and two defectors all play six games each, using the above point system (if both defect, they each receive 1 point). The results are as follows:
• Tit for tat vs defector: Tit for tat loses first round, both defect for next 5 rounds (5 vs 10)
• Tit for tat vs tit for tat: Both cooperate on every round (18 vs 18)
• Defector vs defector: Both defect on every round (6 vs 6)

When the points are added up, a tit for tat player gains 28 points (5 + 5 + 18) while a defector only gains 26 points (6 + 10 + 10). This is a surprising turn of events, as the defectors never lost a round and tit for tat players never “won” a round. This goes to show how cooperation leads to better long-term results while selfishness prevails.

There are shortcomings of this strategy. If there is a failure in communication and one tit for tat player mistakes the other’s actions as an “attack”, they will retaliate. The other player then retaliates to this and a vicious cycle is formed. This is the basis of many conflicts ranging from schoolyard fights to wars (although interestingly, tit for tat strategy is also found during wars in the form of “live and let live”). One way to prevent this is tit for tat with forgiveness, where one player randomly cooperates to try break the cycle (a defector would respond negatively while a tit for tat player will accept the cooperation), or the tit for two tats, where the tit for tat player waits a turn before retaliating, giving the opponent a chance to “make up for their mistake”.

Computer simulations have all proven that tit for tat strategy (especially the other two types mentioned just before) are extremely effective in games. In fact, it is considered one of the most optimal strategies in overcoming the Prisoner’s Dilemma.

In human societies, there is usually a mix of “nice people” and “selfish people”. By cooperating and trusting each other, we can produce a much greater gain over time compared to being selfish. And since society still unfortunately has “defectors”, you can retaliate to those who refuse to cooperate by defecting on them also. Ergo, a good approach to life is to initially reach out your hand to whoever you meet and treat them from there on according to how they respond. If they take your hand and want to cooperate, treat them with altruism and help them out. If they swat your hand away and try to use you for their selfish gain, it is fine to shun them and not help them out.

Through cooperation, understanding and connection, we can build a far more productive and efficient society, just like the ants.

Posted in Life & Happiness

## Slap Bet

If you ever have a disagreement with a friend and would like to bet over who is right, make the ultimate wager: the slap bet. Basically, whoever is right gets to slap the other person in the face as hard as they possibly can. On the surface it appears to be simple and harmless. But in reality, it is a deadly and formidable wager. For example, if one ever makes the unfortunate mistake of making a slap bet with the condition that the slap can occur at any place at any time, then they must live in fear of a slap appearing out of the blue and leaving a glowing, red and rather painful hand print on your face.

Being such a pricey bet, it is always useful to appoint a Slap Bet Commissioner. The Commissioner is responsible for resolving any problems that may arise regarding the bet, such as making a ruling. They must remain completely unbiased and hold the integrity of the slap bet above all else. They must also enforce the sacred rules of the slap bet, such as no premature slapulation. If the rules are disobeyed, the Commissioner has the power to endow one player the right to slap the other player (with completely subjective judgement of how many slaps they can get).

The slap bet is also highly customisable, where the players can settle on the number of slaps and the manner in which they will be delivered. Will the loser receive ten slaps in a row? Or will they get five slaps that can occur from the moment they lose to infinity?

A slap bet is the ultimate bet that is so satisfying and cathartic for the winner, but for the loser it is… well, let’s just say it is a real slap in the face.

Posted in Science & Nature

## Rock-Paper-Scissors

Rock-paper-scissors is a game with a long history. The earliest example of the game is a Chinese game called huoquan, which follows a cyclic rule where the frog eats the slug, the slug dissolves the snake and the snake eats the frog. The reason why rock-paper-scissors has been saved throughout history is because of the uncertainty it contains. Any hand you choose, the chance of winning is the same. Ergo, there is no single best choice and there is no move that will always win. But this is still a game played by people. It is not a game played by emotionless machines, meaning that you can use human psychology, the surfacing of emotion and specific signs and movements to help deduce your opponent’s hand. Mentalist Derren Brown can read tiny flickering of muscles in the opponent and microexpressions to pull off his “undefeatable rock-paper-scissors trick”, but this is near impossible for a normal person to try. However, you can use the following strategies to improve your odds.

1. Use paper on a beginner: Statistically, people prefer using rock. Males especially have a strong tendency to play rock.
2. Use scissors on an experienced player: People who know the first trick can be defeated by going one step further.
3. Use a hand that loses to the hand your opponent played: This uses the psychology of the opponent wanting to mix up hands and wanting to beat the hand you last played (which is the same as theirs as you drew).
4. Say what you will play and play that hand: In a competitive situation like rock-paper-scissors, people tend not to trust others. Thus, if you say you will play a certain hand, they will think is a trap and not play the hand that defeats that hand. For example, if you said you will play scissors, the opponent will play paper or scissors and you will either win or draw.
5. Do not give the opponent a chance to think: People have a subconscious tendency to play a hand that beats the hand that they played before. Without time to think, the subconscious takes action meaning that you can predict their move. If you do the same as strategy 3 and play a hand that loses against the opponent’s previous hand, you will win.
6. Suggest a certain hand: This is a form of hypnosis where you suggest something to the opponent’s subconscious. To use this trick, pretend to go over the rules by saying “rock, paper, scissors” then play a certain hand. The opponent will likely play the hand that the subconscious last saw.
7. If you keep drawing, use paper: This is the same as strategy 1.

Unfortunately, rock-paper-scissors has an equal probability of a win and a draw, meaning draws are rather common. Thus, a computer engineer called Samuel Kass devised a game where two additional hands are added: rock-paper-scissors-lizard-Spock. Lizard is played by making your hand into the shape of an animal’s head, while Spock is played using the Vulcan Salute from the science fiction show Star Trek, where you make a V-shape with two fingers on each side. The rules are as follows.

Scissors cut paper. Paper covers rock. Rock crushes lizard. Lizard poisons Spock. Spock smashes scissors. Scissors decapitate lizard. Lizard eats paper. Paper disproves Spock. Spock vaporizes rock. Rock crushes scissors.

As each hand has two ways of winning, the odds of winning is 10/25, or 2/5 and the odds of drawing is 5/25, or 1/5. As you can see, you have double the chance of winning compared to drawing, making the game much faster to play than the original game.

Posted in Science & Nature

## Monty Hall Problem

Imagine that you are on a game show and you are given the choice of three doors, where you will win what is behind the chosen door. Behind one door is a car; behind the others are goats, which you do not want. The car and the goats were placed randomly behind the doors before the show.

The rules of the game show are as follows:

• After you have chosen a door, the door remains closed for the time being.
• The game show host, Monty Hall, who knows what is behind the doors, opens one of the two remaining doors and the door he opens must have a goat behind it.
• If both remaining doors have goats behind them, he chooses one at random.
• After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door.

Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you: “Do you want to switch to Door 2?”

Most people believe that as an incorrect option (goat) is ruled out, their odds of winning the car go up from 1/3 to ½ even by staying on the same Door 1 and there is no benefit to switching. However, it is better to switch doors as this will double your odds of winning the car. To illustrate this point, the following three scenarios (with the car being behind Door 1, 2 or 3) can be imagined, using the above rules of the game:

In Scenario 1, you have already chosen the car (Door 1) so Monty Hall will randomly open Door 2 or 3. Switching will obviously lead you to losing the car. The chance of you losing after switching, therefore, is 1/6 + 1/6 = 1/3 (as either Door 2 or 3 could be opened)

In Scenario 2 and 3, because you chose the wrong door (goat) and Monty Hall will open the door with the goat behind it, switching will lead you to choosing the car (no other choices). As the odd of either scenario happening is 1/3 each, your odds of winning after a switch is 2/3 – double the odds of winning after not switching (1/3, the odd of your first guess being right).

Of course, this is only under the assumption that the rules of the game were followed and that Monty Hall will always open a door with a goat behind it. This problem and the answer suggested was extremely controversial as tens of thousands of readers refused to believe that switching could be a better choice. However, as the above illustration shows, the Monty Hall problem is a veridical paradox – a problem with a solution that appears ludicrous but is actually proven true by induction.

Posted in Science & Nature

## Duel

Three gunslingers called Good, Bad and Ugly duel to the death. They each stand an equal distance from each other and shoot at the same time. Good’s accuracy is 30%, Ugly’s accuracy is 70% and Bad’s accuracy is 100%. Who has the highest chance of survival?

Common sense dictates that Bad, with the highest accuracy, will have the highest survival rate. However, when the duel begins, the following scenario will occur.

Good’s most rational decision is to shoot Bad rather than Ugly. Reason being, shooting the person with the higher accuracy improves your survival rate in the next round. Ugly also chooses to shoot Bad instead of Good as it is the best choice. Lastly, Bad shoots Ugly instead of Good. This scenario can be explained by the following diagram:

Thus, the probability of Bad being alive after the first round is (1-0.3)(1-0.7)=0.21, or 21%. This is because Ugly is killed by Bad on the first shot. On the second round, the probability of Good dying is the same as Bad’s survival rate of the first round, which is 21%. Therefore, Good’s survival rate is 79%. On the other hand, Bad’s survival rate becomes 0.21(1-0.3)=0.147, or 14.7%

Ultimately, the survival rate of each shooter is: Ugly 0%, Bad 14.7%, Good 21%, making Good the most likely winner. This illustrates the fundamental principles of game theory – an extremely useful theory that helps predict the many choices we make in life.

Posted in History & Literature

## Election

If you think of an election as a game, it is a rather fascinating game. Let us pretend that you are a player participating in a game called Elections. As a player, what is the most rational decision you can make?

The obvious answer is to not vote on the day of the election and do something else instead. As elections tend to be decided by a significant difference in the number of votes between parties, the probability that your single vote will make a difference is near 0. Ergo, instead of wasting your time submitting a vote that will have virtually no effect on the results, you are better off doing something more productive.

However, despite this, there are fools who vote in every election. The fools’ votes pile up and make the world keep on moving forward.

Posted in Psychology & Medicine

## Confidence

Two psychologists, Bob Josephs and Pran Mehta, performed an interesting experiment studying the how extroverted and introverted people react differently to a rigged game. They told a pair of participants to play a game where they had to draw lines to connect numbers in sequence as they popped up in a grid. They also told them that it was to study their spatial awareness and intelligence. The pair were given the game in a competitive setting at the same time so one could tell if they were “winning” or not.
The grid could be easily rigged to determine who would win. Josephs and Mehta posited that men and women with high testosterone levels would have high confidence in their spatial awareness, while those with low testosterone would be the opposite.
What they found was quite interesting.

When those with high confidence in their abilities lost a game, they were more distressed relative to when they won (as measure by their cortisol, a stress hormone, level). Those with low confidence were more distressed when they won a game.
Furthermore, after winning a game these participants would show a fall in their ability to reason and solve logic problems.

The reason behind this perplexing result is likely to be a cause of “mismatch”. It has been hypothesised that human beings are very protective of their self-identity and when this is challenged, they try stubbornly to rationalise their identity even if it means a negative outcome. For example, a person who believes they are not creative will dress and act to show this trait, even if it means others will see him in a negative light.
In the case of the game, the participants were confused as they won the game when they believed they would do badly.
This same effect has been found in studies looking at pay raises. Those with self-esteem issues are less likely to be satisfied with a raise as they feel that “they do not deserve it”. They are also more likely to quit after a raise rather than before. It is quite possible that this would also apply to students with low self-esteem, as they would expect lower grades and (subconsciously) actively achieve lower grades to satisfy their self-identity.

Posted in Life & Happiness

## Eleusis Game

The victory condition for this card game, named after an ancient Greek city, is quite simple: discover the pre-determined law via induction.
This game needs at least four people, with one person acting the position of God. God decides on a certain law (in the form of a single statement) and writes it down on a paper, thus creating the way of the universe
Next, the deck of cards is split evenly between the other players, then one person places a card in the centre. After “the world begins to exist”, God looks at the card and says “This card qualifies” or “This card fails”. The next player also places a card in the centre and the God judges whether it fits the way of the universe.

Players carefully study which cards qualify or fail to try discover the way of the universe. If someone thinks they figured the law out, he or she proclaims themself as the prophet, who begins to take over the role of God to announce whether the cards qualify or not. If at any point the prophet is wrong, he is dismissed. If the prophet correctly judges ten cards in a row, he states his hypothesised law and compares it to the piece of paper. The prophet wins if the two laws coincide, but is dismissed if it is not. When all 52 cards are played without a successful prophet, God becomes victorious.

However, as this is a game, the way of the universe cannot be too complex. To make it fun, the God player must devise a law that is simple yet difficult to discover. For example, the law “Alternate a card higher than 9 and a card lower than 9” is tricky as players tend to focus on picture cards or the colours of the cards. Also, laws such as “Only red cards qualify, except the tenth and thirtieth cards are disqualified” and “Accept all cards that are not the 7 of Hearts” are illegal, as they are too detailed. A God who comes up with such ways of the universe that cannot be found using logic and the scientific method loses his right to play the game. Ergo, the God must seek simplicity that is not easily conceived.

So what is the most successful strategy for this game? Even if there is the risk of being dismissed, proclaim yourself as the prophet as soon as possible for the best chance of winning.