Is it possible for a company to split into a parent company and child company? The child is publicly traded but the parent own 51% of child company's shares so continue to have decision making power.

What are the pros and cons of this approach?

Yes, and you can IPO but retain majority of the shares for yourself so you don’t lose majority vote. Carlyle group bought Booz Allen Hamilton and IPO’d it, but only sold 25% of the shares to the public. (Then they took out a billion dollar loan, spent it all on dividends which means Carlyle took a loan to pay themselves a bonus... and then immediately cut all employee benefits.. that would be when I quit /rant)

You don't need to do this, you can just make multiple classes of shares and only put one class on the market. Zuckerberg personally owns a majority share of facebook, and yet it is publicly traded. Facebook public shareholders have no voting power, because zuckerberg has two thirds of the votes.

There are other reasons to do such splits though; a popular is tax evasion (the parent company holds the intellectual property, and charges the child company enough fees that the child company doesn't make any profits), and of course private companies don't have the same disclosure obligations as public companies.

It's usually not seen very positively by the market unless the holder is viewed as extraordinarily visionary. Outside of tech, it's much harder to spin the visionary angle, so they often trade at a significant discount to their peers. It's the removal of a form of checks and balances. If the direction is good, shareholders would go along anyway, but if it's bad, there is nothing to do to exert influence except jump ship.

Is there a limit to a recursive structure of 51% holding companies? If not, one entity could hold all the decision making power without basically owning equity (in the limit)

That problem exists. For example, Israel enacted a law to prevent that after such problems arose:

https://en.m.wikipedia.org/wiki/Israel%27s_Anti-Concentratio...

Nice. You could even have a fractal structure. I dare a billionaire to make a Mandelbrot set with a bunch of shell companies.

Something like this was a minor plot point in Accellerando (book by Charles Stross). IP being rapidly bought & sold between a network of related inter-owned companies, to avoid ever holding it long enough to be sued or something.

Each company C_n would own 51% of the company C_{n-1} (with C_1 being the actual company). At every point each company owns a sizable chunk of equity.

You would have to file periodic public disclosures about your ownership stake and transactions in the company. Owning or controlling (directly, or indirectly, e.g., through a private company) 5% or more of the outstanding shares of any one share class makes you a "beneficial owner" under Section 13 of the Exchange Act and you become an "insider" under Section 16 of the Exchange Act if you own more than 10% of a class of a public company's registered securities. Section 16 also allows the company to recover "short swing" profits from insiders within a six-month transaction period so insiders need to plan transactions in advance to avoid the appearance of impropriety.

I'm also not sure what the benefit would be? The compliance/reporting/disclosure obligations would be more or less the same as creating a "founders class" of stock with juiced voting rights.

Yes, it is possible to jave such a system. Semapa, a portuguese company traded in the lisbon stock exchange has such a structure

But if there's a paper problem, if you take over one company in the chain and sneak through the company above it, suddenly you own the whole company? Especially one of the small ones?

It's extremely easy for the parent company to abuse the minority shareholders in the child company even if there are laws against it, because it can be hard to prove malicious intent.

It's outside of tech, but a recent case I have followed all the way through is Katanga Mining (TSX:KAT), now delisted. It was a company engaged in mining in the DRC, owned 88% at the time by mining conglomerate Glencore with the rest mostly by retail shareholders. The mine was still being built, so there were no or minimal revenues, but lots of debt from financing the construction. All that debt was contracted from Glencore directly and put on Katanga's book. The rates were pretty insane (12+%) considering the risk of the venture.

Now this is where the obvious conflict of interest arises. First, it's obvious that Glencore benefited from raising the interest rate as high as possible. The debt was becoming so large, and with interest rates so high, any future earnings from the mine were mostly going to go to repayment of that debt. If you're a holder of Katanga directly, that means you get to see no earnings, but if you're Glencore, you still receive all of them from the debt payments. Funnily enough, the DRC was also part of this venture, with a stake in the mine under deployment, thus sharing some of the debt. They complained about the scheme, as it was plainly obvious to them as well, and asked Glencore to eliminate part of this debt or risk seeing the mine nationalized. Glencore complied and moved some of that debt to Katanga's books. It still had an impact to them, but there was an even more disproportionate one to retail holders of Katanga.

There is nothing shareholders could have done because Glencore called all the shots due to their 88% ownership. The stock obviously tanked over time and Glencore eventually swept in to buy all of the remaining shares at extremely depressed valuations. Shortly before that happened, they had even done a stock offering for "debt recapitalization", which attracted no interest but tanked the share price even further.

In less extreme examples, you can still see it with royalty payments to the parent company, or other dubious transactions like that between the two.

Did glencore short or use derivatives on the retail shares as well?

Don’t know actually, but wouldn’t put it past them to do so under the justification of hedging.

Cargill is the largest private company in the world.