Question: It seems counter intuitive that calculating more vertices instead of just reading more from vram would be faster. But if memory bandwidth is the issue that makes tessellation worth it, then why do things like displacement mapping exist? In the tessellation shader, if you read from a texture, you accessing vram more anyway. Are texture look ups less expensive than more original vertices? Why is tessellation fast? 

Say you had an vertex amplification of 32 with a very low polygon model. Would this be faster than say a higher polygon model with only a tessellation vertex amplification of 8 or something. Or in other words, do you linearly gain performance with the more you tessellate?

Answer: 
There is no single point that gives tessellation better performance in every possible instance. Different benefits and trade offs apply in every use case. Some things that might contribute to making tessellation faster than alternatives:

* **Memory Bandwidth:** Modern computers are largely limited by memory speeds. Even if you use textures, a single read could be as low as 4 bytes instead of the 32+ bytes typically required to store vertex data.
* **Level of Detail (LOD):** With tessellation shaders you can avoid having areas that too detailed while still insuring that the rest of the scene has sufficient detail.
* **Fewer Verticies:** Means fewer executions of the vertex shader and every stage of the pipeline before that.
* **Less CPU Overhead:** Potentially requires fewer draw calls, especially if you no longer have to do LOD on the CPU.

There are probably other factors that I have missed...
There's always a trade off between processor and memory. Tessellation is a way that you can save memory and bandwidth but at the cost of GPU performance. 

**Why you should use tessellation:**
Tessellation with displacement maps significantly reduces memory bandwidth for animated or multi-instance objects in the scene. But its not very useful for static single objects.

Lets say you have a sprite that is running across the screen. If the sprite is a high detail (1 million plus vertices) then each time the animation routine moves/deforms the mesh, all 1 million vertices are transformed and reloaded to the GPU each frame. 

However if you use a low detail model (50-100k vertices) with tessellation and displacement. Then you store the displacement map on the gpu once. you update the 50k mesh for animation and reload significantly less mesh each frame, then the GPU tessellates up to 4-5 million virts using the displacement map which is already loaded. 

The end result is that you get 2-4x the mesh detail with 1/20th the memory bandwidth. Now imagine you have 20-30 of these sprites on screen at a time.

**Why you should not use tessellation:** To add this detail on the fly the gpu has to burn up processing power in order to calculate the 3d position of each tessellated vertex before it starts running all the other shaders. 

The major distinction you need to watch out for is that this only helps you when you are instancing and/or animating geometry. 

If you have a high detail static mesh that never moves and only have one instance of it on the screen, then it is faster to upload the geometry in full detail. Tessellation would just add complexity and eat up cycles in the pipeline. 

**There is a trade-off:**
There is a slight memory bandwidth perk you get by using tessellation for static meshes. Because vertices need 3 coordinates of floating point to be understood by the gpu. But sampling a displacement map uses 1 coordinate of fixed point data to be useful. Because the displacement map is normalized against adjacent vertices. So it calculates the extra data on the fly. But this calculation is performed every frame for each tessellated vertex. This eats up shader time that would not be needed if you used static mesh. 

However if you turn the tessellation down or off for LOD purposes it SAVES shader time for objects that don't need detail, compared to a high detail static mesh. 

So tessellation is always a good idea for improving detail on dynamic/instanced meshes. 

But for static meshes or singleton meshes, its a trade off between LOD capability and pipeline complexity. A high detail mesh in the distance is burning up more compute time then a tessellated mesh with tessellation turned off. But a high detail mesh in the foreground takes up less compute time than a tessellated mesh with tessellation turned up. 

One big thing to consider however is that slowly turning up the tessellation as an object gets closer, looks way better than instantly replacing a low detail mesh with a high detail mesh. So when smooth LOD is a big concern then definitely go with tessellation. 
...or use tessellation to a certain point, then replace it with a high detail mesh. But that's only a good idea if you are not worried about running out of memory and keep both versions on the gpu at all times. Otherwise you will be burning up bandwidth again swapping them around.

Again, always a trade off between memory usage and processor usage.