The Two-Thirds Game

Let’s start with the following puzzle.

There are a hundred people, you being one of them. You are all asked to choose a number between 0 and 100, both inclusive. The person who chooses a number that is closest to two-thirds the average wins the game.

In the above scenario, what number would you choose? Choose a number, and only then continue reading to understand the different strategies for this game.

Let’s just say that I assume that the numbers that other people choose are completely random. In such a scenario, the average of all choices is going to be close to 50. Hence, it seems rational for us to choose 33 as our own number.

However, when posed with the question, here is what I thought- I started off with the above assumption of the choices being random. According to this, my choice was going to be 33. But then I wondered if other people in the game would also have the same thought process that I did. If they did, they were all going to choose 33, bringing my answer down to 22 (since 2/3 of 33 is 22).

Next, I thought, if I had reduced my answer to 22, so would everyone else, and this lowered my answer even further. After numerous iterations, my answer terminated at 0, the lowest possible choice. I was assuming that everyone would choose 0, and there would be no winner.

Zero is the most stable answer to the above game. No matter how many times you run this game, nobody would want to change their answer once they arrive at zero. For any other set of choices, there would definitely be a winner, and everyone else would want to change their answers. Even if there wasn’t a winner, for example if everyone chose 33 in the first simulation of the game, there would definitely be someone who decides to change the choice to 22, destabilizing the game.

This game is a popular game in game theory, a field of study that has applications almost everywhere. Zero is the answer that one arrives at using common knowledge of rationality. Common knowledge of rationality means that I know or assume that everyone is rational and chooses 33, and then everyone assumes that everyone is rational and lowers their answer to 22, but then again everyone assumes that everyone would choose 22 and so on. This is an infinite loop of everyone assuming everyone would assume that everyone would assume…..that everyone is rational!

However, even if one person breaks the chain of common knowledge, it tips off the scales of the game. In such a scenario, it makes much more sense to choose a number greater than 0. And this is what happens in real-life scenarios, where the winning answer is close to 22.

Let’s look at what game theory is and how it has applications in various fields. In a nutshell, game theory is the science of decision-making. It governs how we should make our choices, especially when playing against a competitor. If we assume that our opponent acts rationally, meaning that he/she is looking at maximizing his/her gain from the game, we can come up with our own strategy.

A simple example of a game is the prisoner’s dilemma. Assume you are one of two prisoners. You and your accomplice are taken to two different rooms, and are asked to confess to your crimes. If both of you deny, you would both go to jail for one year. If both of you confess, you would both go to jail for 5 years each. However, if one of you confesses and the other denies, you can get a deal to testify against your accomplice and get off with zero jail time, while your partner goes to jail for 20 years!

Although common sense tells us that prisoners should deny, game theory tells us otherwise. What if my partner sells me off? I would go to jail for 20 years! Here is how game theory evaluates this situation:

If I confess, what are the two possibilities?
1. My partner also confesses, and we both get 5 years of jail.
2. My partner denies, and I get 0 years of jail time.
Here, my average jail time is 2.5 years.

If I deny, what are the two possibilities?
1. My partner also denies, in which case we both get 1 year of jail time.
2. My partner confesses, and I get 20 years of jail.
My average jail time in this case is 10.5 years!

It makes much more sense to confess, since the average jail time is significantly less. If both prisoners know game theory, they would both confess and spend 5 years in jail.

Game theory has applications in wars, since game theory is all about strategies. It has applications in stock markets and trading, since it tells us how to look at our expected gain, taking different possibilities in consideration. Even in nature, it has examples of how predators and prey should behave. In fact, the predator-prey game is another popular game in this field.

There are numerous categories of game theory. There are zero-sum games, wherein a gain for one player is a loss of equal magnitude for the other player(s), thus ensuring that the total gain-loss sum is zero, hence the name.

Another popular application of game theory tells us how attempts to maximize individual gain can be detrimental as compared to cooperative play. In fact, the prisoner’s dilemma that we looked at is an example. If we think beyond ourselves and about the two of us together, we realize that cooperation would allow us to both choose denial, which is exactly why we were taken to different rooms. There are numerous other formulations of games which tell us how cooperation leads to greater gains.

Game theory also successfully explains why we often find groups of restaurants (or shops, gas stations, etc) together, instead of them being spread evenly around a region. If you’re interested, do take a look at this video.

Hope you had fun reading this article. If you have any thoughts or comments, something that you would like to tell me as well as other readers, do drop a comment- I’m always eager to learn more! And if you think this article would be an interesting read for someone you know, share it with them!

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