# Public vs Private Networks

## Public Networks Are Strictly Dominant Strategies in Infinitely Long Games

This is a proof demonstrating that there are no private/restricted networks that can't be made a strict topological subset of a public/decentralized one. Such that, when a network is defined as a set, Si, in a shared topological space on and "above" a spheroid (or in other words, bounded by a finite, earth shaped, 3-dimensional hole) with restricted networks denoted, Sp, and unrestricted networks denoted, Sd, there are no infinitely long games in which Sp ⊂ Sd does not occur and remain in perpetuity. Making Sd a strictly dominant strategy in infinitely long games.

### Caveat

This isn't to say that private Grassland networks are bad, necessarily. But as we'll show, they only make sense in circumstances where Si contains its limit points, like inside a building with walls on all sides and a roof. But if Si does not contain its limit points, there are no feasible economies of scale for an Si upon and above the surface of a spheroid (e.g. the earth).

As well, private networks should only be used on a limited scale since the act of creating a large, "dark" Grassland "stack", a set of pointers to 4D (space+time) coordinates on a "heap" used by a separate Grassland stack, would cause the sequestering of information from those parties and remove the mutual knowledge required to maintain peaceful Nash Equilibrium.

### Proof As Embeddings In Topological Spaces:

It is sensible to define Grassland networks as sets in a topological space (Topological space). That is to say, a set of points along with a set of neighbourhoods (Topological Neighbourhood) for each point.

1) We'll define each network as a set, S, in a shared, 3-dimensional topological space that's only bounded in the same finite region by an earth/spheroid shaped, 3-dimensional hole.

2) The spheroid is opaque and it is not possible in any way to place cameras at its center pointing outwards to a surface (precluding any economies of scale).

3) All the points of S are made up of 1) each camera's raycasted principal point and 2) the neighbourhood of raycasted points around it within the frame's field-of-view. The raycasted principal point (RPP) being the initial point at which an imaginary line (the "ray" in "raycasted") drawn outwards from the center (principal point) of the camera's aperture intersects the external, real-world object (an unprojection from the camera to the world).

4) We'll denote any private (restricted) network as Sp, and any decentralized (public) network as Sd.

5) And we'll define the "domination" of one set, S, by another set, S', to be whenever the points in S can be approximated (substituted) to a higher degree by points in S' than the other way around. To put it another way, the probability that any given x ∈ S is arbitrarily close to some y ∈ S' is higher than the probability that any given y ∈ S' is arbitrarily close to some x ∈ S. Therefore, the number of points in S' is greater than the number of points in S (even if it's just one more camera/node)

Since both Sp and Sd are on the surface of the earth, a spheroid having no edges, no set, S, can contain ALL its limit points (Limit Point) to the effect of keeping points of another set, S', from approximating points of S. And it goes without saying, that this is true along the vertical axis. So even continent's aren't topologically compact. They don't contain their limit points in regards to camera placement since along vertical axis the "sky's the limit". Which means there's always some way for points in an unrestricted Grassland network, Sd, to approximate more points in Sp. Topologically speaking, a greater number of points in Sp can be shown to be merely a point in a neighbourhood of some point in Sd.

### Proof As Embeddings In Infinite Sets:

We can also demonstrate this by finding an embedding of S into the infinite sets. And showing any differences in cardinality between them. In the sense that some infinities are larger than others.

The points of a private Grassland network would find an embedding in the set of all Integers. It can theoretically stretch infinitely in both directions, [-infinity, ..., -3, -2, -1, 0, 1, 2, 3, ..., infinity], with an infinite amount of points from a countably infinite amount of cameras. But it's still bound to certain values; Sp has more restrictions than Sd by definition. And there exists points in Sd which fall "outside" and can't be defined using the operation(s) that govern the elements of Sp. Since the domain and range of this operation(s) are, by design, restricted to the elements of Sp.

Therefore, because a decentralized (public) Grassland network doesn't have those restrictions it finds a complementary embedding into the Reals, [-infinity, ..., -Pi, ..., -1.001, ..., -0.001, ..., 0, ..., 0.001, ..., 1.001, ..., Pi, ..., infinity]. It's an infinity too but it's a much larger, uncountably infinite, infinity because it can theoretically stretch to infinity even between the Integers.

And it therefore approximates the Integers. Sp's point at "1" can be approximated by an infinite number of points in Sd. Moreover, "1" in Sp can be approximated to an arbitrary degree of accuracy by points in Sd from both sides, "0.9999...9" and from "1.0000...1". This is because y ∈ Sd has the freedom to approach any x ∈ Sp along every dimension of the space they're embedded into.

### Conclusion:

So since more points of Sp can be approximated to an arbitrary degree of accuracy by points in Sd than the other way around, and since Sd therefore contains the limit points of Sp, we can say that the set, Spn, comprised of the neighbourhoods of all points in Sp is merely a smaller subset of the set, Sdn, a set comprised of the neighbourhoods of all points in Sd. And therefore Sp is dominated by Sd according to point 5 above.