I think Bamafam's reasoning is correct.  You can guarantee a win with 833,334 shares.  This is 9,166,674 votes, which is 4,583,337 per candidate.  The least the other shareholders would need to beat your candidates would be 4,583,338 x 10 = 45,833,380 votes.  (Possibly one fewer, since I don't know what would happen if the last three positions wound up in a 3-way tie.)  But the others have only 55,000,000 - 9,166,674 = 45,833,326.  So they can't beat you.

NEAS, can you check this reasoning? 

[NEAS: This is correct.]

 

  -- Mathochist

Cumulative Voting

Jacob: Can you give a simple illustration of the cumulative voting procedure?

Rachel: Suppose a firm has 101 shares and two positions on its Board of Directors. How many shares must one have to get a person on the Board of Directors?

Jacob: Let’s call the two positions on the Board of Directors as Position A and Position B.

The firm has 101 shares. A candidate needs 51 votes to be assured of a majority. If one investor has 50 shares and another investor has 51 shares, the investor with 51 shares wins each seat on the Board of Directors.

Rachel: That is true for individual elections, as we have in the United States. Your example shows the inequity it would create on the Board of Directors. The two investors have about the same number of shares (50 vs 51), so they should have about equal say in running the firm. But the investor with 51 shares wins every seat on the Board of Directors and has absolute control of the firm.

Jacob: Is this realistic? Publicly traded firms have thousands of shareholders, none of whom has more than 3% or 4% of the shares.

Rachel: It is very realistic because of proxy voting. The current management sends a letter to all shareholders explaining its perspective on running the firm and asking them to vote for its slate of candidates. Most shareholders buy this stock because they believe its management is reasonably good, so many of the shareholders agree.

Suppose another investor buys 10% of the firm’s stock and is the largest shareholder. Sending a different letter to other shareholders saying that the current management is not competent elicits the response: "Who are you? If you don’t like the firm, invest elsewhere."

Jacob: How do we solve this problem?

Rachel: We use a cumulative voting procedure. Let us continue with our example. Each share has two votes, since the Board of Directors has two seats. An investor with one share can vote for two people or give two votes to one person. The two candidates with the most votes get elected.

Suppose the investor with 51 shares wants to win both seats and gives 51 votes each to two candidates. The investor with 50 shares gives all 100 votes to one candidate and wins a seat on the Board of Directors.

Jacob: How many shares does an investor need to elect a candidate?

Rachel: Consider two scenarios:

Scenario #1: One investor has 33 shares and another investor has 68 shares, for a total of 101 shares. The investor with 68 shares chooses two candidates and gives each one 68 votes. The investor with 33 shares has a total of 66 votes. Even if the investor casts all votes for one candidate, this candidate comes in third and doesn’t get a seat.

Scenario #2: One investor has 34 shares and another investor has 67 shares, for a total of 101 shares. The investor with 34 shares gives all the 68 votes to one candidate. If the investor with 67 shares chooses two candidates and gives each one 67 votes, they comes in ties for second and third place.

The textbook shows the formula for getting a candidate on the Board of Directors.

OK, I'm going to throw in my two cents and give my intuitive understanding of this...

So there are 5 million shares, 11 directors to be elected, so 11 votes per share for a total of 55 million votes.

The way I see it, cumulative voting works like this: a bunch of candidates each get a number of votes from shareholders, and the 11 people with the most votes get the positions. Consider the following scenario: Candidate A gets 5.125M votes, Candidate B gets 4.875M votes, C gets 4.625M, D gets 4.375M, ... , T gets 0.375M, for a total of 20 candidates and 55M votes - and the first 11 candidates, A to K, get elected.

Now, say that you have 416,666 shares, or just under 1/12th of the firm (5M/12=416,666.67). You're probably thinking, "oh yeah, I can definitely elect at least one member if I have this many shares..." but alas, the rest of the firm is owned by a single shareholder, Cecil P. Sneer, and he'd like to see each of his 11 buddies appointed to the board. Cecil has 5,000,000-416,666=4,583,334 shares and so 4,583,334 votes for each of his 11 buddies (4,583,334*11 total votes, divided among 11 candidates). You only have 416,666*11=4,583,326 votes, not quite enough to surpass the number of votes for any of the other candidates. However, if you owned just 1 more share, then you could be guaranteed to be in the top 11, with 416,667*11=4,583,337 votes (in fact, the rest would then have only 5,000,000-416,667=4,583,333 each). Even if the votes for the other 11 candidates were not divided evenly and there were some with more votes than you, there would still be at least one with less, so that you would make it to the top 11. If there were more than 11 candidates, there's still no way that more than 10 could get ahead of you (after your vote there are 55M-4,583,337=50,416,663 votes remaining - in order for 11 candidates to get ahead of you, there would have to be 4,583,338*11=50,416,718 votes available).

Hence the formula for the number of shares needed to guarantee the election of one director: S/(D+1) + 1  (rounded down to the next integer), and remember that S is the total number of shares and D is the total number of directors

More precisely, I think that it would be (S*D/(D+1) + 1)/D, where S*D/(D+1) + 1 would be the number of votes needed, which would then be divided by D and rounded up to get the minimum number of shares - but due to rounding this should probably give the same answer as the simpler formula above.

For N directors, the formula must then be N*S/(D+1) + 1 

...or by the "precise" formula (N*S*D/(D+1) + 1)/D rounded up to the next integer (both formulas give me the same answer for this homework problem, after rounding)

Somewhat of a derivation:

Say we have D director positions open – we want to elect at least N. Then there are D-N positions we “don’t care” about. Then the voting requirement is:

 

[our votes] / N > [remaining votes] / ( [remaining openings] + 1 )

 

where, [remaining votes] / ( [remaining openings] + 1 ) = ( [total votes] – [our votes] ) / ( ( D – N ) + 1 ) **Explained Below

 

 

We know that [our votes] = [our shares] * [openings] = X * D

And [total votes] = [total shares] * [openings] = S * D

 

Plugging in variables:

 

X * D / N > ( S * D – X * D ) / ( D – N + 1 )

 

After simplifying, we get:

 

X > S * N / ( D + 1 )

 

Since the right side will often not be an integer, it is generally written with an equal sign and a plus 1 [X = S * N / ( D + 1 ) + 1] and you take the integer part.

 

 

**Explanation:

 

The total votes we can place per individual we want to elect is: [our votes] / N

 

We need to be sure that there cannot be enough remaining votes so they can place a larger amount of votes PER INDIVIDUAL on MORE THAN the D – N remaining candidates that we “don’t care” about. Hence, the + 1.

 

For instance, say there are 3 openings and we want to elect 2. Assume we have enough votes to cast 40 to each of them.  If there are 79 votes remaining, then the remaining votes can cast 40 and 39 for another (i.e. enough to elect only one more, keeping your 2 safe). However, if there are 82 votes remaining, they can cast 41 and 41 and you lose one of your elections... So you see we really don’t care how many votes they can cast for the 1 remaining position. What we care about is how many they can cast for the remaining position AND another one (D – N + 1) = 2.

For anyone who has had some discrete mathematics, this is a good problem on which to use the pigeonhole principal.  Look at it this way:

Let X be the number of shares you hold.

You will have 11X votes.

Remaining votes:  V = 11*(5M – X)

 

In order for candidate C to win, at most 10 people must get more votes.

 

If 11 people get more votes than C, then each of them can get at most V/11 = 5M-X votes.  We must ensure this does not happen.

 

So the 11X votes for C must be more than the 5M-X votes for any other candidate.  We can then set up the inequality posted by jdcox1999 and solve for X.  For 2 candidates we simply need twice as many.

 

Hope this helps someone!  I just think there’s nothing quite as cool as some good pigeonhole principaling.

Here's my reasoning on this, presented as a dialog:

John:
I'm tired of dealing with the stooges on StagnantTech's board. How many shares do I need to buy to elect my best friend Chuckie to the board?

Kate:
Well, how many shares does StagnantTech have outstanding? Any how many seats are on their board?

John:
They have 5 million shares and 11 board members. The shares all have equal voting rights. So I need one 11-th, or 454,546 shares, right?

Kate:
No, you can get away with less. Imagine there are 10 other people, just like you, and each one is trying to elect a friend to the board. Between all 11 of you, you need to control enough shares to prevent a 12-th person from getting in. How many shares do you need to control?

John:
Well then I would need to control at least 416,667 shares. If 11 people control this many shares each, that would leave 416,663 shares for everyone else.

Kate:
Good, but you have to check for all possible upsets. For each of your shares, you get 11 votes, so that's 4,583,337 votes for your candidate. After you cast all your votes for Chuckie, how many votes are left for everyone else?

John:
Out of 55,000,000 total votes, that would leave 50,416,663 votes for everyone else.

Kate:
Now suppose there's a bloc of shareholders who want to keep Chuckie off the board. They have a list of 11 candidates, and they want to elect them all in order to shut you out. For each candidate, how many votes do they need to beat you?

John:
Let's see. My candidate has 4,583,337 votes. In order to beat me and keep Chuckie off the board, their 11 candidates would need 4,583,338 votes each. That's a total of 50,416,718 votes. They're 55 votes short!

Kate:
Very good. So that means you would need exactly 416,667 shares to elect Chuckie to StagnantTech's board.

John:
But I thought someone said I would need 416,668 shares. Whose right?

Kate:
We just proved that 416,667 shares will guarantee Chuckie's seat, even against an adversarial bloc of shareholders who are conspiring against you. One of their candidates would have to lose.

John:
Now wait a moment, Kate. If 50,416,663 votes are divided evenly among 11 candidates, then each one gets 4,583,333 votes each. That's an 11-way tie.

Kate:
Yes, but it's an 11-way tie for Second Place. Chuckie has 4,583,337 votes, all cast by you; that's a 4 vote lead for First Place. Chuckie still wins a seat, even if the remaining 10 seats are still contested.

John:
And that means I don't need to add an extra share. So we're right! But what if I also want to elect Hannibal. I want to elect both Chuckie and Hannibal. So do I need 833,334 shares?

Kate:
See if an adversarial bloc can shut you out.

John:
Well, I am casting 4,583,337 votes each for Chuckie and Hannibal. Out of 55 million votes, that would leave 45,833,326 votes for everyone else. In order to shut out just ONE of my two friends, an adversarial bloc would need 4,583,338 votes for each of 10 candidates, for a total of 45,833,380 votes. They're 12 votes short!

I cast all 11X of my votes to my candidate. In order for my candidate to lose, 11 candidates each must receive (11X+1) votes. Thus, I need X such that (11X+1)*11 > (55million - 11X). This simplifies to (11X+1)>(5millon-X) or X>(5million -1)/(11+1), which is slighly different than the formula given in the homework. Where did I go wrong?

Thanks!

This is how I derived the given formula.
‌If we want to elect one candidate, S shares must go to him, where S is greater than all other shares divided by 11 (which is the least number of candidates vying for a position). That is,
S > (5M - S)/11‌
From ‌‌‌this, we get S > 5M / (11+1). Thus S must be at least [5M / (11+1) ] + 1 = 416,667.667 . Rounding off we get the 416,668.

Same reasoning if we want to elect 2 members of the board. We should have T shares, so that for each of our 2 bets:
T/2 > [5M - (T/2)]‌‌/11‌
=> T > 2*(5M) / (11+1)
Therefore we need T‌‌‌  to be at least [2*(5M) / (11+1)] +1 = 833,334 shares.
 ‌‌