For HW13.2, I was using (5M shares /11)x2 but I am not sure if this is correct?

[NEAS: The denominator is too low, and you forgot to add 1 (to break ties).]

Shouldn't we use 55M/11*2?

5M/11*2 is less than 1M votes needed to elect 2 members.  Then there would be more than 54M votes left, which doesn't guarantee our members would be elected.

If you use 55M/11*2, then you get how many votes you need to elect at least 2 members.  But the question asks for how many shares are needed.  Since you get 11 votes per share, you would use (55M/11*2)/11 to get the answer, which is the same as 5M/11*2.

[NEAS: Be sure to distinguish between shares and votes.  But the solution is not quite correct.]

 In a different finance course I took they gave us this formula

Shares required = ((number of directors desired * total number of shares outstanding) / (total number of directors to be elected + 1)) + 1

Using this formula I got

((2*5,000,000) / (11+1))+1 so you need over 833,334 shares to elect atleast 2 members

[NEAS: This formula is correct.]

MC: I am not saying your formula is wrong.

extreme example:  1 share in the whole universe, 1 member to be select, and you desired that seat.

using your formula: desired share = (1)(1)/(1+2)+1 = 1.5 shares

But there is only 1 shares in the whole universe.  Contradicted.

Unless your formula imposed some conditions that you didn't listed.

[NEAS: This is rounding.  A more precise formula says: "The integer part of . . ."  The integer part of 1.5 is 1, which is the proper solution.]

I got the formula directly from S.B. Block & G.A Hirt, Foundations of Financial Management, 11th Edition, McGraw-Hill Irwin, 2005.

I also checked online and the New York Law Journal lists the same formula.  See page 3 in the link: http://www.stroock.com/SiteFiles/Pub341.pdf 

I think the formula makes sense if we look at what it's actually doing. 

We don't know how many people are going to by vying for the director positions.  However, with more people you need less votes to be elected.  So we need to figure out how many votes we would need in the smallest possible field (total number of directors to be elected + 1).  Then we're dividing the total number of shares equally among all the candidates and giving ourselves one more to guarantee our candidates win.

So we have 5M shares, and 11 votes per share(one per seat).  This is obvious.  So we would need 2500001 in votes per seat in order to win the election.  So (2,500,001 / 11) * 2 = 454,546 .  This number of shares will ensure that you have 5,000,002 votes to use on two specific seats and guarantee victory.

[NEAS: There are no specific seats.  This is cumulative voting.]

How is it guaranteed?

Per your argument, 5,000,002 votes.

There are 5M shares = 5M*11 = 55,000,000 votes.

That means there are 55,000,000 - 5,000,002 = 49,999,998 votes from others.

Even if they equally divide 49,999,998 votes among 11 seats.

They can put 4,545,545 votes per seat (for 11 seats)

How would you like to cast your vote to sure win 2 seats?

( you have 5,000,002 votes)