The Green Eyed Logic Puzzle
Common knowledge has always been an intuition that people rarely delve deeper into. It is almost a mutual consensus that nobody talks about; an assumption that some knowledge should be understood by everyone. This is formed by the telling of information to everyone in this group at some time when they were growing up, cultivated by experience and cultural background. But what if we told everyone a piece of information simultaneously?
The Green Eyed Logic Puzzle by Ted-ed discussed the significance and power of common knowledge. The audience are asked to free 100 prisoners from an island using only one sentence that does not contain any new information. All prisoners grew up on the island, cannot communicate with each other, cannot see their reflections, and all had green eyes. The island had a peculiar rule: a prisoner can ask to leave everynight, but only prisoners with green eyes will be permitted to escape, whilst all others will be tossed in the volcano. All the prisoners want to leave, but will never take action unless they are certain that they have green eyes.The answer is to tell them: “at least one of you have green eyes”. The key to this answer is that now, all 100 prisoners will start keeping track of everyone’s eye color, and knows at others are doing the same. Imagin 2 prisoners, A and B. If A had green eyes and B had non-green eyes, A would know that she was the only green eyed prisoner, and would leave on the 1st night. But if both had green eyes, then they couldn’t be sure if the other was the only one, and each would stay put on the first night. But on the next day, the fact that the other did not leave, meant that they each saw a green eyed person, meaning both had green eyes. They would then both leave on the 2nd night. This logic applies to all 100 prisoners, and they would all leave on the 100th night.
This problem is very similar to the common knowledge problem in Chapter 19, where A, B and C will only take collective action if they are certain that the number of people willing to go exceeds their threshold. The only way to be sure, is to also know the threshold number for their neighbors, and see if the number of neighbors they each have equals the threshold number -1 (excluding themselves). Without this precondition fulfilled, participants are not certain enough that their collective action will be backed up and effective, and thus they will not go. The interesting thing about common knowledge is that it is intuitive: everyone is watching each other and everyone knows this without needing collaborative agreement.
Source: https://www.youtube.com/watch?v=98TQv5IAtY8