Let's suppose that a group of 10 graduate students regularly go out to a pub for beer, and the tab for the 10 comes to $100 total. If they pay for their bill the way Americans pay for our taxes (based on our so-called "progressive" tax system), the breakout would be like this:
The first 4 people (the poorest) pay nothing. They get to drink for free.
The fifth pays $1
The sixth pays $3
The seventh pays $7
The eighth pays $12
The ninth pays $18
The tenth person (the richest) pays $59.
Being good friends and liberal progressives, that's what they all agree to do. It seems only fair that each person should pay what they can afford to pay, remembering the old adage they learned in school: "from each according to his ability, to each according to his need" (Karl Marx).
Every few days, the 10 good friends would meet up in the pub and would pay up as agreed upon.
Then one day, the proprietor gave them a deal. "Since you are such good customers, from now on", he said, "I'm going to reduce the cost of your tab by $20. You can just pay me $80!"
Everyone wanted to continue to pay their bill just the same way as they had before. So the first four people (the poorest) are unaffected. They continue to get to drink for free.
But what about the remaining 6 people? How should they split up the unexpected $20 savings "windfall" so that everyone would get "his fair share"? They figured that $20 shared by 6 comes out to $3.33 each. But if they simply subtracted that amount from each of the 6 paying friends, then person #5 and person #6 would actually be paid to have their beers since person #5 only paid $1 anyway and person #6 only paid $3!
What to do?
The pub owner came to their rescue. He suggested that each person's bill should be reduced by roughly the same amount, and he used his calculator to work out what that should be:
Persons 1-4 continue to get to drink for free
The fifth person, like the first four, now pays nothing and drinks for free (100% savings!)
The sixth pays just $2 instead of the original $3 (33% savings!)
The seventh pays just $5 instead of the original $7 (28% savings!)
The eighth pays just $9 instead of the original $12 (25% savings!)
The ninth pays just $15 instead of the original $18 (17% savings!)
The tenth pays just $49 instead of the original $59 (16% savings!)
All 6 friends were better off then before. And their first four buddies continued to drink for free, because they didn't have a lot of money.
They all felt pretty good about it.
After they thanked the pub owner and left to walk back to campus, they began to compare their savings under this new deal.
The sixth person was very quiet, though. Finally he blurted out. "You know, splitting up the bill that way wasn't fair! I only got a dollar out of that $20 we all saved, and yet (he pointed to the tenth person) he got $10!"
"Hey, you're right", shouted the seventh person. "I got cheated too. I only saved 2 dollars. It's unfair that he got back 5 times more than me!"
"Damn it! I've been ripped off too", yelled the eighth. "Why should he get back $10 when I got back only $3. The wealthy get all the breaks!"
"Wait a minute", screamed friends one through four. "We didn't get anything at all! The system exploits the poor!"
The first nine people surrounded the tenth person and beat him up.
The next day, tempers had cooled down and the nine friends showed back up at the pub. They were sorry for what they had done and they wanted to apologize to their tenth friend.
But the tenth person didn't show up for drinks. So the nine proceeded to drink without him.
When it came time to pay the tab, they discovered that they had a problem. They didn't have enough money among all nine of them to pay for even half of the bill!
Superbly put!