The King Of Diamonds: A Dissected Gameplay of ‘Balance Scales’

– Manasvi Kshirasagar

FY BSc

Image Credits : Pinterest 

Imagine that you have been transported to a dystopian world of the Japanese show, Alice in the Borderland, a world where you must play games and compete with other players to survive. Each game has been allotted a playing card to signify their difficulty and the type of the game. The one you enter is a King of Diamonds, the diamond signifies the game is an Intelligence and Logic-based game. And the King? The King signifies that the game is of maximum difficulty. But don’t worry, for your convenience and assurance of survival, let’s make you the main character. With your main character plot armour, you won’t be dying off anytime soon. 

You enter the arena, and see the King of Diamonds, along with three other players, and you must beat the King to survive. You are shackled to a seat and you observe a balance is placed above your head. The game starts with 5 players and you hear the rules announcement:

“The game is called ‘Balance Scale’. You must choose a number between 0 to 100. The average of the number chosen by all players will be multiplied by 0.8, and this product will be the winning number. The player who guesses the winning number or the closest number wins the round. There will be only one winner in each round and every player except the winner will receive a negative point. When a player reaches ten negative points, the game will be over for them. The game starts with every player on zero. Each time a player receives a ‘GAME OVER’, a new rule will be added.”

 What number would be your choice? Well, let’s analyse with the help of game theory.

First of all, why the multiplication by 0.8? You imagine it would be to tip the balance of the average. A 0.8 would change the average so much that the players will be forced to think not just about their own guess but also about what the average guess of everyone else might be. It would successfully trap the players in a loop, where the winning number would change every round. Hence the name of the game; ‘Balance scale.’ Further, with every point deducted, a strong mixture of Sulphuric acid is added to the balance above your head. Once your score turns -10, it tips the balance, and the acid consumes you whole, killing you instantly. 

You assume that between a group of 5 people, choosing any number would round up the average to 50, which multiplied by 0.8 would give 40. But what if everyone else also thought this way and went with 40? Then you would have to go with 32 which is 40 multiplied by 0.8. But then again, what if everyone thought this way as well? You would have to go one step further and multiply it by 0.8 and now you’re stuck in an endless loop of assuming that the number you pick has a chance of getting picked by others, which means you would have to further multiply the number by 0.8. 

And so, a rational strategy would be to predict the number that others would pick, and multiply it by 0.8–a strategy which is very similar to a concept introduced by John Maynard Keynes in his work, The General Theory of Employment, Interest and Money. Keynes explains how investing in a stock market works with the example of an experimental beauty contest. In the Beauty Contest game, the entrants must choose 6 faces from 100 women. The entrants picking the most picked or most popular faces would win a prize. An amateur would go with the faces that they thought looked the most beautiful, but a seasoned professional would rationally think about the average opinion of people and pick the woman that would be the most beautiful for a majority of people to maximise their chances of winning. Similar to our game, where we mustn’t go with what we assume the average to be but predict what others will assume the average to be.

According to Keynes, a similar concept unfolds the mysteries of the stock market. Keynes, a macroeconomist extraordinaire, struggled immensely to get more than average returns in the stock market. He would predict fluctuations in the market based on policy changes and macroeconomic variables but that would never work. That is until he saw the advertisement for the beauty contest. He says we mustn’t even assume what the average opinion would be, but assume what the average opinion of the average opinion would be. 

Now, coming back to our game, the king has a four-point lead on you and sits on a 0, while you and the other players are at a -4. The average for the last round was 1.14 and with every round, you notice that with each consecutive multiplication by 0.8, the value of the average decreased and the players would use the winning number in the last round as their base for the prediction. Any rational player would now go with 0. But since you are the main character, with your almighty main character shield, you choose to go with 100 so the average will be 20 and the average multiplied by 0.8 would give 16. You hope that even if one player chooses 1, he would naturally win, given that 16 is closer to 1 than 0 and 100. To defeat the king, you take that risk. Your risk pays off and the King now has -1. In the next few rounds, you start choosing seemingly absurd numbers and even manage to get two wins. At the end of round 10, two players now have -10. As the acid starts to fall drop by drop on the shrieking players, you can only think of how you do not want to die this way.

The tally now looks like this:

Since two players succumbed to this round, two new rules were added. 

Rule 1: If two or more players choose the same number, it will become invalid even if the number was the winning number

Rule 2:  If a player manages to get the exact winning number, the loser’s penalty will be doubled (-2).

With these interesting rules, you resume the game. A 0 was a strong choice for you, but with the addition of Rule 1, you fear that there would be another player who would choose 0. This new rule has eliminated the possibility of a Nash equilibrium by negatively incentivising and eliminating the possibility that more than one player chooses the same number. Nash Equilibrium is a concept in game theory in which producers decide to continue with their pre-existing strategy to maximise profits. With the invalidity rule, two players choosing the Nash Equilibrium, 0, will cause their score to go down. To which you assume that even if one player chooses 0, and another player chooses any number from 2 to 100 (keeping the invalidity rule in mind), you still have a shot at winning if you go with 1. You could also go with 2, but that is a number that would only help you win if the other 2 players choose the same number or choose 0 and 100 respectively. So rationally, trying to keep a fallback option, you go with one. With your fingers trembling, you press 1. However, it seems that the King and Player 2 also had the same idea, and went with 1. There is no winner for this round since three players chose the same number. Now, you sit at -9 and cannot afford to lose even one more round. If you do, it is game over for you. The scores are now the following:

It’s starting to look like your main character plot armour isn’t really helping.

You calm yourself down and think harder. You try to go for the major win using Rule 2. You predict that Player 2 still has two more points to lose, so they will try to eliminate you, and even if they lose, they still have one more chance. They will assume that you and the King will invalidate each other, so choosing a random number is the only winning option. Supposing this theory, Player 2 will try to go for a number greater than 50, assuming if the King and us do end up invalidating each other, the numbers we choose will be below 50. Even so, 90-100 is far-fetched. Doubles are too obvious so 55, 66, 77, 88 are removed. Numbers ending with 0 are too easy so those are removed. Numbers with digits 3, 5, and 8 are numbers that people choose on a whim, but Player 2 will try to make an informed choice, so those are out too. Now, we have narrowed most of the numbers. Now, we eliminate numbers that we see in everyday life, movie names, game consoles, and commonly seen numbers. And we are left with 62 and 74. And it seems difficult to predict further. So once again, you take a risk and choose to go with 62. This is a game of life and death so don’t forget to multiply 0.8. if luck is by your side, you just might win. So you predict the winning number to be 23. 

The arena is now deadly silent, barring the voice of the announcer. The results are displayed and you see the following results; King:1, Player 2: 62, You:23. The winning number is 22.93. With your trembling feet, you have a wave of relief. You predicted the exact value! Congratulations. Player 2 gets 2 points deducted. The scores are now this way.

 Since a player is eliminated, a new rule is added.

Rule 3: If a player chooses 0, the player choosing 100 will be the winner of the round. 

Upon further thinking, you assume this rule is put in place to ensure a fair game so that the player in the lead doesn’t win by choosing 0. If both of you kept choosing 0, then according to the invalidity rule, the choice would get invalidated. Both players would lose a point but the player in the lead would automatically win since you would reach -10 much faster than him.

But with the introduction of this rule, you just might have a chance at winning. It is most rational for you, who is losing, to choose 100, and the King to choose any number between 0, 1 and 100 depending upon what he thinks your choice is. The outcome can be represented by the following decision matrix:

                          Table credits: Original content

It is rational for the King to pick 0 because the probability that he loses is only 1/3rd and so is the way for 1 and 100 as well.

So the probability that you die is 2/3rd.

Now, this is where your main character plot armour comes in and saves you. You somehow manage to manipulate the King into  0 for the next 3 consecutive rounds. If you hadn’t had your shield, you would have most probably lost. The King dies and order is restored. You walk out of the arena, relieved to see another day and having learned a few important lessons;

Firstly, Keynes had some great ideas. Secondly, learning and applying Economics does come in handy. And lastly, it’s of paramount importance to be the main character.

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