Well, the point is that paying a stock dividend is equivalent to doing nothing. All that happens is that 1 share turns into for example 1.05 shares. Price per share should go down by ~ 5%, but total market cap stays the same. A rational shareholder should completely indifferent as to whether the company gives out a stock dividend or not.

For a cash dividend that's not true. Something real happens: the company has less cash afterwards (and the shareholder more).

There's a school of thought that says that the value of share should be the discounted sum of all future _cash_ dividends. According to it a company that only does share dividends should be worth nothing.

The dilution does not affect everyone equally. Consider the extreme case of a company with only two outstanding shares and 100$ in assets, so each share trades at 50$. Now, the company emits an extreme dividend of 50$ per share. The first shareholder choses cash andthe second one stocks. Now, the first one ends up with 50$ in cash an 1/3 of a company worth 50$, while the second shareholder ends up owning 2/3 of the firm, which is worth only 33$.

You're missing one important event. In the moment that the dividends are issued (regardless of whether they are stock or cash) a part of the company's value transfers into those dividends.

Your example cannot work because a company worth $100, cannot create new shares worth $100 ($50 per share). (It could theoretically distribute a cash dividend of $50 per share, but then the value of the outstanding shares would be $0).

If it were to distribute say 50% per share. Then it would create two new shares (in addition to the existing two). This would mean the new value per share would be $25.

So investor A has one share worth $25 and a cash dividend of $25 (he owns 1/4 of the company) Investor B has one share worth $25 and one (dividend) share worth $25 (in total 1/2 of the company).

This is also an example of a repeated game, so over time if they maintained their strategies, investor B would own the whole company.

Also depending on how the company operates, there is another element of game theory because the controlling shareholder could elect to cancel all future dividends. if you're the newly minted majority shareholder, it's probably in your favor to do this immediately. In that case, the Nash equilibrium seems to be both parties choosing the stock dividend (and effectively getting nothing).

While the 2 shareholder example is an oversimplification in this example, after enough plays of the game (i.e. dividends) a large shareholder could take control of the company if the other shareholders always choose cash.