A little puzzle

[cjp]

Imagine a very small country with 66 citizens. Some time ago, a revolution occurred, and you have become the new president. Initially, every citizen (including you) earns the same amount of money, which is one coin per day.

Now, suppose you are a kind of an evil dictator. You managed that you cannot be voted away by elections, but you don't like the small amount of money you get. You can make laws to change that, but unfortunately, each law needs to pass a poll, and a law is rejected if it doesn't get a majority of votes (more than 50%). Also, the new constitution allows every citizen to vote, except for you, the president.

Luckily, the other citizens are as greedy as you, which makes them very predictable. On a poll, they vote like this:
If the new law increases someones income, he/she votes "yes"
If a new law decreases someones income, he/she votes "no"
If a new law doesn't change someone's income, he/she doesn't vote

You have to know that someone's income can only be an integer number, so it can be 1 coin/day, 2 coins/day etc, but not one coin for every two days. It can be zero.

Now the question is: what is the maximum amount of income you can give yourself by changing the law?

 

[silent_protagonist]

Well, this is as far as my logic takes me:

• First law:33 citizens, including yourself, have all income removed. The resulting 33 coins are distributed evenly among the remaining 33 citizens. The law passes 33/32. (Resulting incomes: 32-0$, 33-2$, You-0$)

• Second law: Of the 33 citizens now receiving 2 coins annually, 16 have all income removed. Of the resulting 32 coins, 17 are distributed evenly among the 17 remaining two-coiners, and the remaining 15 coins are given to you. The 32 citizens receiving no income are not affected. The law passes 17/16. (Resulting incomes: 17-3$, 48-0$, You-15$)

• Third law: Of the 17 citizens now receiving 3 coins annually, 8 have all income removed. Of the resulting 24 coins, 9 are distributed evenly among the 9 remaining thee-coiners, and the remaining 15 coins are given to you. The 48 citizens receiving no income are not affected. The law passes 9/8. (Resulting incomes: 56-0$, 9-4$, You-30$)

• Fourth law: Of the 9 citizens now receiving 4 coins annually, 4 have all income removed. Of the resulting 16 coins, 5 are distributed evenly among the 5 remaining four-coiners, and the remaining 11 coins are given to you. The 56 citizens receiving no income are not affected. The law passes 5/4. (Resulting incomes: 60-0$, 5-5$, You-41$).

• Fifth law: Of the 5 citizens now receiving 5 coins annually, 2 have all income removed. Of the resulting 10 coins, 3 are distributed evenly among the 3 remaining five-coiners, and the remaining 7 coins are given to you. The 60 citizens receiving no income are not affected. The law passes 3/2. (Resulting incomes: 62-0$, 3-6$, You-48$).

• Sixth law: Of the 3 citizens now receiving 6 coins annually, 1 has all income removed. Of the resulting 6 coins, 2 are distributed evenly among the 2 remaining six-coiners, and the remaining 4 coins are given to you. The 62 citizens receiving no income are not affected. The law passes 2/1. (Resulting incomes: 63-0$, 2-7$, You-52$).

• Seventh law: Both of the 2 citizens now receiving 7 coins annually have all income removed. Of the resulting 14 coins, 3 are divided between 3 of the no-income citizens, the remaining 11 coins are given to you. The other 60 citizens receiving no income are not affected. The law passes 3/2.

• Final Results: 62-0$, 3-1$, You-63$

Can't figure out a way get those last three coins. I tried one other "endgame," which ended with the three coins split between 2 citizens, but it took more steps.

It's be interesting to generate a function for this problem, and see if and how the number of steps required is dependent on the initial number of citizens, or if it will always take seven.

 

[Scarecrow]

Rampant inflation! Mint another 66 zillion coins, so everybody has an income of 66 zillion and one coins per day, then use logic similar to silent_protagonist's to give yourself all but 3 coins. Everybody's income is still an integer.

 

[cjp]

• Final Results: 62-0$, 3-1$, You-63$

Correct!

Isn't it interesting how people's own greediness in the end turns against them? I like this kind of game theory paradoxes.

 

[silent_protagonist]

Correct!

Isn't it interesting how people's own greediness in the end turns against them? I like this kind of game theory paradoxes.

:speakcool:
It's was also interesting to watch the voter turnout go down. Remind you of any other political systems we know?


BTW I like the way you think Scarecrow. When the world is mine, you will be my Treasury Secretary.