In real life, I do not believe that the options market is zero sum. The reason is that market makers hedge their options positions with actual stock. This means that when
1) Options traders buy more calls than they sell, and/or sell more puts than they buy, market makers on the other side of the trades buy and hold more BRK stock to hedge their long-calls and short-puts positions
2) Options traders sell more calls than they buy and/or buy more puts than they sell, market makers on the other side of the trades sell BRK stock short to hedge thir short-calls and long-puts positions.
Indeed, there are special rules for options market makers that allow them to more easily and with fewer restrictions make short sales to hedge their options positions...

The things you say are true, but it doesn't mean the options market isn't a zero sum game.

If you consider every security trade by people who are participants in the the options market (as you seem to), including their trades in shares, then no, it's not zero sum, because share ownership isn't a zero sum game. But I don't think that's a useful way to delineate "the options market"...you're including non-options in the discussion.

If you consider only the set of all standard traded option contracts, that is in aggregate a zero sum game. An option contract can't ever create or destroy the underlying shares, nor ever affect the value of the underlying shares in any way.

None of the ways that an option contract can be bought, or sold, or come into existence, or cease to exist, or expire, or be exercised, can change the number of shares in existence or their value. So there is an absolute perimeter that you can draw around the options market that is nothing but the option contracts themselves, and the money passing between the people writing them, buying them, selling them, or seeing them expire. An option assignment doesn't create a share, it just obliges someone to move some already existing shares from one place to another.

Ultimately, all of the money received from selling options exactly equals the amount of money paid to buy options. It's an identity, true in every sub-period and over time.

What is more interesting to me is that there is no reason to believe that subsets of option trading are zero sum. I suspect that there is a long run net transfer of funds from buyers of options to sellers of options, though it may be modest. The amount gained by one group would have to equal the amount lost by the other group, however, as the boundary around all option contracts is a zero sum game.

Jim