if you re-did the maths with the same guy buying 1,000,000 shares at $10 a share then they'd still end up with a controlling share in the business (and the business would indeed be worth $20M since it was worth $10M before and now has an extra $10M in cash, meaning that shares would still be worth $10 each).

so i'm not sure why the original example needed to place an odd value on the newly issued shares. as far as i can tell, it just muddies things.

i wrote this on rights issues, which is kind-of related (works through a similar kind of argument, except this time framed so people keep the same fractional shares) - http://acooke.org/cute/EnersisEnd0.html [edit: just fixed url] (i'm no expert, i was just interested in that particular case in the chilean press recently).

in short: you don't need to screw anyone over. it can just be that the person putting most money in ends up with most of the ownership.

the two key points are: (1) people need to agree on the company's value before; (2) the company gets the cash (and so its value goes up). of course, that cash won't sit in the bank - but it's the company's responsibility (and the investor's hope) that it is spent in a way that increases the value of the company.

Edit: yes the value goes down of the shares, but the calculation 'post money' is $10M + $10M / 3m shares so 6.66 $/share.

After the new $10M in cash it is $20M/3,000,000, so they are worth $6.66/share