In his memorable song, “Everybody Knows“, Leonard Cohen sings:
Everybody knows that you love me baby
Everybody knows that you really do
Everybody knows that you’ve been faithful
Ah give or take a night or two
Everybody knows you’ve been discreet
But there were so many people you just had to meet
Without your clothes
And everybody knows
What happens when “mutual knowledge” between a select few (say, the senior members of the ruling political party of India and those of the main opposition party) becomes “common knowledge” for all?
All hell breaks loose!
Allow me to tell you how, with the help of a story.
There is a unique village in Haryana, in which many married couples live. Each woman in this village immediately knows when another woman’s husband has been unfaithful but not when her own has been (“mutual knowledge”). Moreover, very strict rules of the village require that if a woman can prove her husband has been unfaithful, she must kill him that very day.
Assume that the women never inform other women of their cheating husbands. As it turns out, twenty of the men have been cheating on their wives, but since no woman can prove her husband has been cheating on her, the village life proceeds merrily along.
One morning, a wise old man with a long, white beard comes to the village. His mystical powers and honesty are acknowledged by all and his word is taken as the gospel truth.
He asks all villagers to gather together in the village compound and then makes this announcement:
“At least one of the men in this village has been unfaithful to his wife.”
Once this fact, already known to everyone, becomes “common knowledge,” what happens next?
The wise old man’s announcement will be followed by nineteen peaceful days and then, on the twentieth day, by a massive slaughter, twenty women will kill their husbands.
Case 1: One Cheating Husband
Let’s start with the assumption that there is only one cheating husband in the village, Mr. A. Everyone except Mrs. A already knows about him. So when the wise old man makes his announcement, only Mrs B learns something new from it. She thinks:
If any other husband was the cheater, I would have known about it. I don’t know about it. So, the cheater must be my husband. I have to kill him now!
Case 2: Two Cheating Husbands
Now, let’s assume there are two cheating husbands, Mr. A and Mr. B. Every woman except Mrs. A and Mrs. B knows about both these cases of infidelity. Mrs. A knows only of Mr. B’s, and Mrs. B knows only of Mr. A’s.
One day 1, how would Mrs A think? Imagine you are Mrs A. What would you think immediately after the wise old man has spoken?
“Aha! So there is at least one cheater. But I already know Mr. B is a cheater. Mrs B will soon figure it out and kill him! I’ll wait.”
Now imagine you are Mrs B? How would you think, Mrs B?
“Aha! So there is at least one cheater. But I already know Mr. A is a cheater. Mrs A will soon figure it out and kill him! I’ll wait.”
And so Mrs A and Mrs B wait for each other to kill their husbands on day 1 but neither of them does. This puzzles both of them. Mrs A thinks:
“OMG! Mrs B did not kill her husband on day 1. Why? Oh! I know why! There must be two cheating husbands. I already know that Mr. A is a cheater. So, who’s the other one? OMG! If it had been anyone else other than my own husband, I would have known! This means my husband has been cheating on me! I have to kill him now!
As it happens, Mrs B is also thinking exactly as Mrs A. And so, on day two, both Mrs A and Mrs B will kill their husbands.
Case 20: Twenty Cheating Husbands
By a process of mathematical induction involving proof by contradiction, we can conclude that if twenty husbands have been cheating on their wives, then their intelligent wives would finally be able to prove this on the twentieth day, the day of the righteous bloodbath.
The Naked Emperor
175 years ago, Hans Christian Andersen told the story of an emperor who had no clothes. In that story, you’ll remember
The naked emperor marched in the procession under the beautiful canopy, and all who saw him in the street and out of the windows exclaimed: “Indeed, the emperor’s new suit is incomparable! What a long train he has! How well it fits him!” Nobody wished to let others know he saw nothing…
But then suddenly, said a little child at last
“But he has nothing on at all.”
“Good heavens! listen to the voice of an innocent child,” said the father, and one whispered to the other what the child had said. “But he has nothing on at all,” cried at last the whole people.
In our present circumstances, there are some amongst the whole people who believe that
Sunlight is the best disinfectant.
God, I hope they are right…
This post was inspired by an excerpt from John Allen Paulos’ “Once Upon a Number”