Assume an investor takes all dividends and buys more shares.

Example:

Assume 10M shares at $1 each. With a 10 cent dividend ($1M total), an investor with 100k shares (1% of the company) receives $10k. The market cap drops to $9M (since cash holdings decreased) and the stock price drops to $0.90. They then buy ~11k (10k/.9) shares and have 111k shares, or 1.1%.

Now instead assume the company does a buyback of 1M shares. Now there are 9M shares, each still priced at $1 (cash holdings decreased). The investor has 1.11% of the company.

Lots of assumptions here (no fluctuation in price due to market reaction, cash is not discounted in market cap, etc) but the numbers check out in a perfect world.

The point is that with a Dividend investors can choose whether to reinvest, whereas with a buyback everyone essentially reinvests.

Well but that just confirms my whole point. In a buyback the company invests its cash into stock instead of distributing it as dividend. It therefore makes an investment decision for the investor, who never gets to see any cash.

The whole point of giving out dividends as a company is saying to an investor "here, you can invest this cash better than we can". If you're not giving out dividends you think you have better investment foresight than your investors, which may often be true.

What you as a company then do then with the cash, whether you're investing it in R&D, personnel, etc.. or buy stock doesn't change the mathematics of things for the investor, even if the stock you buy happens to be your own stock.

Yep. Except for the fact that without taxes the investor can just sell the stock if they disagree with the investments the company is making, resulting in the same cash return as a dividend.

In my example with the buyback, if the investor sold 0.11% of the company after the buyback, the result would look just like what would happen if they did not reinvest the dividend. 1% of the company and $10k cash.

Ok, but it's not the buy-back that increases the per-share value for the selling investor, it's the fact that the company has generated positive Free Cash Flow that it is free to invest.

True, though investors often discount cash on hand ("a bird in the hand is worth two in the bush") because of the uncertainty of ever seeing that money. YHOO is probably the easiest example as of late, with their massive goodwill write downs of acquisitions. So the act of either a dividend or a buyback removes that uncertainty and can result in an increase in per share value.

For simplicity, assume that the company buys back stock from all shareholders equally, and then the shareholders trade amongst themselves.

Dividends and buybacks are simply a way to get money from the company to the shareholder. The total number of outstanding shares is of no import---as you can see in stock splits.