Monday, February 23, 2009

The King's Salary.

After the revolution, each of the 66 citizens of a certain country, including the king, has a salary of $1. The king can no longer vote, but he does retain the power to suggest changes - namely, redistribution of salaries. Each person's salary must be a whole number of dollars, and the salaries must sum to $66. Each suggestion is voted on, and carried if there are more votes for than against. Each voter can be counted on to vote "yes" if his salary is to be increased, "no" if decreased, and otherwise not to bother voting.

The king is both, selfish and clever. What is the maximum salary he can obtain for himself, and how long does it take him to get it? (from P.Winkler, loosely inspired by real historical events in Sweden)


Unknown said...

$58 on the 7th vote.

Maria said...

Wayne, you are very close.
But the king can grab even more.
Please explain the answer in order for it to be counted. Many people are very curious to see it.
Here is one hint:
king must temporarily give up his own salary to get things started.

Maria said...

Here is how king can appropriate $63:
To start with there are 66 citizens (including the king) with a salary of $1.
King first proposes that 33 citizens have their salaries doubled to $2, at the expense of the remaining 33 (himself included). 33 citizens whose salaries are being doubled are voting "for" and king is also voting "for" while giving away his $1. So, we have 34 "for" votes and 32 "against" votes. Proposition passed. We now have 33 citizens earning $2 and 33 citizens (including the king) without the salary.
Next, king increases the salaries of 17 of the 33 salaried workers to $4. 17 votes "for", "16" against, others do not care.
In the same manner king slowly reduced the number of salary-receiving citizens to 9, 5, 3, 2. At this point there are two citizens earning $33.
As a last trick, king bribes three paupers with $1 each to help him turn over the two big salaries to himself, thus finishing with a royal salary of $63.

Maria said...

Wayne correctly pointed to me that there is a mistake in the above explanation:
instead of "king increases the salaries of 17 of the 33 salaried workers to $4 "(that would result in $68) it should say "king increases the salaries of 17 of the 33 salaried workers". Increase would be uneven with 15 getting $2 more (now they earn $4 each) and 2 get $1 more (earning $3 each).

He also proposed his alternative similar yet slightly different solution, where king gets rich gradually that is usually more logical but may be more prone to revolt:

1) 33 get $2 (and vote yes);32 get $0 (vote no); K = $0 (doesn’t vote).

2) 17 get raised from $2 to $3 (and vote yes); 16 reduced from $2 to $0 (and vote no); rest stay at $0. K = $15.

3) 9 get raised from $3 to $4; 8 get reduced from $3 to $0; rest stay at $0. K = $30.

4) 5 get raised from $4 to $5; 4 get reduced from $4 to 0; rest stay at $0. K = $46.

5) 3 get raised from $5 to $6; 2 get reduced from $5 to 0; rest stay at $0, K = $48.

6) 2 get raised from $6 to $7, 1 gets reduced from $6 to 0; rest stay at $0. K = $52.

7) 2 get reduced from $7 to $0. 3 get raised from $0 to $1; rest stay at $0. K = $63.

Pratik Poddar said...

I have written a problem in the solution provided above in this blog post... Please help me...

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