A group of people with assorted eye colors live on an island. There are 100 of them, and they are all perfect logicians. 95 of them have blue eyes, and 5 of them have red eyes. They can see the colors of everyone’s except for themselves. Once they realize that they have red eyes, they must commit suicide in the town center at midnight.
One day, a traveler comes and he tells the islanders, “I can see someone has red eyes,” before he leaves.
What is going to happen?
On the 5th night, all the red-eyers will commit suicide because they start to realize that they are the ones that have red eyes.
When the traveler announces publicly that at least one person with red eye is present, even though it was thought to be a mutual knowledge, that fact becomes common knowledge.
Common knowledge is a knowledge that a group of agents knows, and each agent also knows that the other agent knows the fact that they know about it.
Application:
- You and Jack were each given a card. The color of your card is red. The color of Jack’s card is unknown to you. The host told both of you that at least one of the cards is red. Then, the host asks if Jack knows the color of your card. Jack said no. Now, you can be sure that Jack’s card is red.
- Before the host speaks, you have obtained the following information:
- You have a red card
- Jack doesn’t know if your card is red
- After the host speaks and Jack replied, you have obtained the following information:
- You have a red card
- Jack doesn’t know if your card is red
- At least one of you have a red card
- Jack knows that at least one of you have a red card
- Knowing that the other person knows a fact is crucial.
- Before the host speaks, you have obtained the following information:
- A, B, C, D, E are friends, but E always late for 30 mins. A, B, C, and D spoke among each other that next time they will let E be 30 mins earlier.
- Without discussing, A, B, C, D’s plan would not have worked even though they have a mutual knowledge.
Benc Tan 