Fictional depictions of extra-legal bookies

In Lucky Number Slevin, there's a bit where a guy who's life is a disaster gets a second hand tip and makes a bet on a horse race with an (illegal) bookie he can't afford, and unsurprisingly it goes horrible wrong and they try to kill him.

The main moral is "prohibition makes for good films and disastrous government policy".

But then I got to thinking about the mechanics of running a bookie without access to law enforcement and banking infrastructure, and I didn't actually understand it.

I assumed, illegal bookies would exist on a spectrum. The more honest implementation being like a legal bookie: accept bets with cash up-front, or from people you're pretty sure are a good credit risk. Pay out if they win. That's it.

The other end of the spectrum being like a loan shark: extend credit to as many people as possible, let people get in over their heads, and then milk them for as long as possible before their life falls apart in ruins. If anyone decides to just not pay, force them or make an example out of them with physical violence.

But in Slevin, it seems like, the organised crime people knew in advance the mug was broke and could never really pay. So why do they accept the bet at all? As soon as the horse loses, they make a move on him. So they never expected to get *any* money from him whatever the race outcome. Even if you're *willing* to messily kill people, what do they gain by getting into that in the first place?

Is it just to get a splashy example so other people pay up? But don't you want them to dig themselves in FIRST? If you START by scaring everyone, maybe they just won't borrow money from you?

Topology / complex analysis / what

For a plot bunny (yes, really :)):

You have a multivalued function from a sphere onto "some surface", continuous everywhere except two points. (Or, equivalently, a function from "some surface" to the sphere, I guess?)

If you look at points on the surface which map onto the same point on the sphere, and connections between them of "paths" on the sphere (up to continuous deformation), I feel like they end up acting like the integers, where "+1" and "-1" correspond to a clockwise of anticlockwise circumnavigation. Or possibly some subset, a cyclic group of some finite order, if there are repeats. Is that right?

If you have *three* points, what can the relationship between the points look like? What about more?

I remember doing something like that but not what it's called.

I'm trying to put something like the shadows of amber onto a more concrete mathematical footing :)