There are some screwy results in the economic game theory of infinitely iterated games which clearly don't apply in real life, but could make grist for the mill for fictional economics. Specifically, if you have an infinitely iterated game with rational players, in any game with some net benefit from cooperation, each player can compel each other player to take any finite-cost action they want by threatening to never cooperate again unless they do the requested action. There's a lot of reasons that this breaks down in practice, but its a weird quirk of the math applied blindly.

So lets say you want an anarcho-capitalist city that has no property rights but yet still manages to function. The way you do that is to make the entire economy revolve around futures, and have the major players in the economy be immortal (these don't have to be individuals - it could be immortal factions, companies, etc). Wealth is explicitly a collection of promises of future action or labor rather than tied to physical goods. The trick is, if you're the first to break your promise to provide those futures on demand, everyone else takes that into account and it affects the future value of your own currency. To put it another way, its an economic arena where every individual is basically a state and has the ability to print their own money at will, and the market decides what their money is worth.

It's likely that such a society would develop an outer ring of barter or threat based economics. If you can trade in future promises of labor, someone would figure out that they can receive property in exchange for not killing someone (since property is after all only a finite value, whereas death is an infinite penalty). So there'd have to be some other emergent thing acting as a counter-balance to that - again, I'm thinking that as long as the major participants whose value 'counts' are effectively immortal organizations rather than people, then the death of a particular member of the organization has only finite value and can be offset by the organization as a whole refusing to deal with the murderer's organization from that time forward.

This kind of thing would develop a lot of deep-seated feuds and would get quickly clogged with all of those agreements and ultimatums and threats. So in practice, navigating the market would be an insanely involved process, and it wouldn't look anything like an actual free market in the slightest (but could very well hold pretenses of being so in how it presents itself to outsiders).