After reading your response I realized my error in logic.

So we must look at our shares X vs everyone elses 5M - X

I will uses the assumption everyone else votes exactly the same, one vote per seat. 

So in order to insure success in one seat we need X shares where

11*X > 5M - X

so X = 416,666.67

so to insure two seats we need 833,333.3333 or 833,334 shares.  this gives us 4583337 votes per seat.  If there is only one other holder(or a bunch of holders voting the same way) they will have 4166666 votes per seat.  If the combine votes to insure a specific seat that wil only further insure our success in two seats.

[NEAS: Correct.]

Jercox came to the same conclusion as MC's quoted formula.  (I had a hard time understanding the formula.  BTW, the second equation of the article is not quite right, there was typo in the formula, if anyone care)

Jercox's reasoning is much better.

The total votes from the 11 people elected doesn't need to add to 55 million. It just needs to add to enough that the amount left is less than the 11th person has (if the 11th person gets a million votes, then the total of the 11 has to add to 54 million so that no one can knock him out.)

Jacob: Can you explain cumulative voting?

Rachel: Let us compare voting in the United States vs Europe (and most countries of the world). The elections in Iraq, for instance, are European style Parliamentary elections, not U.S. style Congressional elections.

Suppose a State gets two Senators, and each citizen in the State gets to vote for each Senator. The citizens are loyal party members and they vote along party lines. The State has 55% Democrats and 45% Republicans.

In the U.S., we have separate ballots for each Senate seat. Both Senate seats are won by the Democratic candidate.

In Europe, each party submits a list of two names. Citizens vote for a party, not for a Senator. The voting result is 55% Democratic and 45% Republican. The closest division is 1 Senator for each party.

Each system has its advantages. In the U.S., we never vote for parties. In theory, every election is between individuals, not between parties. A person may vote for the Republican candidate in one Senate seat and the Democratic candidate in the other Senate seat.

The problem with the U.S. system is that a slight edge in citizens may give an enormous disparity in Congress. Suppose every election district in the U.S. has a 55% to 45% split between Democrats and Republicans. If people vote along party lines, the Democrats pick up every seat in Congress.

The European system says: "This makes no sense. In practice, people vote along party lines. The Congress should end up 55% Democrats and 45% Republicans.

Jacob: How does this relate to cumulative voting?

Rachel: Suppose a Board of Directors has 20 members. The managing officers of the firm own 10% of the stock. Most small stockholders will vote for whomever the management wants, and they give their votes to management to vote by proxy. These small stockholders have 45% of the stock, so management controls 55% of the votes.

Minority shareholders, such as a large pension fund, own 45% of the stock. They want a different direction for the firm, and they want to elect members of the Board of Directors.

With voting for individual members, the management would control all 20 seats on the Board. With cumulative voting, management controls 11 seats, and the minority shareholders control 9 seats.

Jacob: Are you saying that the European system is better? Are there advantages to the U.S. system?

Rachel: In the European system, no one runs for office himself or herself. In the U.S., if a person can persuade voters that he or she is the best candidate, that person can win election. One doesn’t need a party for support.

Jacob: Does this happen a lot in the U.S.?

Rachel: It occurs all the time. To get on the ballot, a person needs signatures. To run a campaign, a person may use personal funds or find wealthy supporters. Candidates generally work through parties, since they get the party’s campaign structure. But it is the individuals who promote themselves and get the party nomination; the party elders have little say. This occurs in every election we have. Even for Presidential elections, Bill Clinton and Jimmy Carter were outsiders, not the first choice of the Democratic Party, when they first ran for President.

We have 55 million votes and 11 members. To guarantee a majority for one membe we have to cast 27.5 million + 1 vote. That way no one can outvote this candidate. We have to cast 18,333,334 votes for each candidate to guarantee that no one can vote a majority for any other candidates. Specifically, there are a total of 36,666,668 votes cast for our desired candidates which leaves only 18,333,332 votes for any other candidate which cannot be a majority. This tactic requires 333,333,333.4545 shares, or at least 3,333,334 shares.

 

Under a cumulative voting system you can vote as much as you want for your favorite candidate, so to guarantee your choices are picked you have to have a majority.

 

[NEAS: Under cumulative voting, the seats are not specific.  We do not need a majority of the shares to elect a desired candidazte.]

Actuizzle, may I clarify your statement: "This tactic requires 333,333,333.4545 shares, or at least 3,333,334 shares." ?

I think you meant to say:

"This tactic requires 36,666,667 votes, or at least 3,333,334 shares." 

You realize, this method shows that the answer is simply 2/3 the number of outstanding shares (5,000,000), and it shows that the number of total members, 11, is irrelevant.

The variety of answers we’ve seen to this question on voting reflects the variety of ways to understand the question. For instance, do we understand it as: “How many shares are needed to guarantee the election of at least 2 members of the board?” or perhaps: “How many shares are needed to guarantee the election of at least 2 specific members of the board?” ?    

"He's right."

"No she's right"

"You're both right."

"Wait a sec! They can't both be right!"

"You're right, too!"

We have left the student postings to show the reasoning required. The NEAS postings (one is an addition to a student posting) show the correct answer. The question is not ambiguous and only one answer is correct.

What confuses me about this question is how someone with anything less than half the shares will be able to insure the election of even one candidate.  Hypothetically, it seems possible that for every vote you cast for candidate X a vote could be cast for candidate Y, thus resulting in your candidate losing the election. 

I guess this question boils down to asking: how can a cumulative voting procedure insure that 11 positions for Board of Directors will be filled?  What if all votes go towards only 5 candidates?

The NEAS analogy just doesn't make sense to me, in European elections you vote down party lines, but in this scenario there are no party lines.  If you consider each candidate to be a "party" then what if everyone voted for the same party?  In the analogy, the candidates from the party would take all the seats.  Who would fill the remaining positions in our scenario? 

I'm so confused; the book does a horrible job explaining this.

[NEAS: The analogy with European parlimentary elections relates to specific seats vs non-specific seats, not to parties.  The U.S. has parties just like Europe.  But in the U.S., we vote for a specific seat, such as the Senator from State X.  In Europe, we vote for a general seat.  If the parliment has 100 seats, the 100 candidates with the highest votes get seats.]

 

I think the problem with the question is that people are attacking the problem incorrectly.

I see the question as there's a board of directors that has X members, and some insurer with 5 million shares can (and will) elect 11 members of the X. So to elect two members of the X, one would need to have a similar proportion of votes for the X, namely:

(2 members of the board )*(5 million shares / 11 million members of the board) + 1 share in case of a tie = 909091 + 1 = 909092 votes.

I think in the bigger picture, the number of total seats in the board is insignificant, but with a share to board member ratio, the number of shares is easy to compute via dimensional analysis.

[NEAS: The number of seats is relevant.  You forgot to add 1 to the denominator.]

Formula should be

S/(D+1) + 1 for one guaranteed seat.

Formula S/D + 1 gives 5M/11 + 1 = 454,546 shares.

Suppose I have 450,000 shares = 4,950,000 votes. In order to be beaten by 11 others, they must have at least 4,950,001 votes each, i.e. 54, 450, 011 votes.  This gives a vote total of 59,400,011 votes, more than the total number of votes allowed (55M).


Yes, having 454,546 shares guarantees a win, but so does 450,000 shares.  A better perspective might be, "what is the maximum number of votes I can receive without winning?"

That answer would be S/(D+1), or in this case, 5M/12 = 416,666.  This number of shares leaves 50,416,674 votes, which allows 11 other people to receive 416,666 (or 416,667, accounting for whole numbers) votes.   

The correct answer to the question is 416,667 shares per director for a guaranteed victory.