# Game Theory—Second Week

As I keep on watching lectures on Open Yale Courses, I find that these lectures are fun to watch, because professor Ben Polak played games with students throughout the class and drew lessons from those games along the way. Even though each video is around seventy minutes, it is not tedious at all. I was initially drawn to this area of study by its name, because I was fond of playing games that require logical thinking, such as the prisoner’s dilemma. This course perfectly fulfills my expectation.

Currently, I have not seen too many mathematical components in this course. Most of the games played required purely logical thinking ability and rationality. This excites me, for it teaches me how to think in a rational way in order to reach a conclusion instead of learning how to solve math equations systematically.

Another interesting game was played in class.

‘Number Choosing Game:’ Select a random number between one and one hundred without consulting your classmates. The student who chooses the number that is closest to the 2/3 of the class average number is the winner of the game. Assume everyone in the class is rational.

Please ponder for five minutes and think about what number you would choose. (Hint: Be rational and think about dominated strategy I discussed in my first blog.)

The reasoning that leads to the answer of the game is as follows:

First, we eliminate number set (a collection of distinct numbers) [68, 100], because only when everyone in the class chooses 100, the average would be 100*(2/3)=66.67. As a result, any number larger than 67 is unlikely to be a winning strategy. To use economics jargon to characterize this situation, number set [68, 100] is dominated by number set [1, 67]. As the professor concluded in the last class—we should never play a dominated strategy, we should avoid choosing number larger than 67.

Because [68,100] is eliminated as this number set is dominated, we can rule out the possibility of choosing [67*(2/3), 67] which is [45, 67]. This reasoning holds true because rational players know that people will not play a dominated strategy and [45, 67] is a dominated strategy when we delete the dominated strategy [67, 100], which no rational people will choose. In other words, when we put ourselves in other people’s shoes and use our rationality by realizing others are rational, we rule out the possibility of choosing in number set [45, 67].

The same reasoning repeats again: [45*(2/3), 45] which is [30, 45] can be eliminated. This holds true, because we are rational and have knowledge of knowledge of knowledge of rationality.

This sequence repeats itself until it gets down to 1.

The following picture is a flowchart of the professor’s logic:

The professor concluded: “if we have the common knowledge of rationality in this class, the optimal choice would have been 1.”

In the class, interestingly, when no student realized this reasoning, the average of the whole class is around 13 which leads to a winning answer 13(2/3)=9 instead of 1. This phenomenon indicates that not everyone is rational. When professor asked them to play this game again after he explained his reasoning, almost all students chose 1. This fact shows not only that every student know how to play this game, but those students also know that the people around them know how to play. By showing the reasoning, the professor raised everyone’s sophistication at the game.

The professor eventually reached this conclusion: “Not only did it matter to put yourselves in other people’s shoes and think about their payoffs are, but you also need to put yourself into other people’s shoes and think about how sophisticated they are at playing games. And you need to think about how sophisticated they think you are at playing games. And you need to think about how sophisticated they think that you think that they are at playing games and so on.”

In a real world society, we have to consider a lot of factors when we are competing. For example, when we are competing against a company, we have to take into account the high sophistication of that company. If a company, however, is competing against a customer, the company probably does not need to worry too much because an individual is not likely to have the sophistication that a company has.

Overall, this game and this class in general enhanced my understanding of rationality and introduced me to the idea that always put myself into others’ shoes. I became growingly interested in this class, because it is not simply expanding my math knowledge but it is also showing me how I should react when I encounter different situations in my life.

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