# Thoughts on modelling

Initial plan is to handle just two operators.

## Environment

### Transition function

#### Debris
- Currently using a decay formulation for debris.
- Probably better to use a mass conversion from satellite -> debris
    - This would have to be a stochastic approach to handle satellite destruction.
    - I envision it would be better because it would represent how much debris 
    is created better and would easigly lend itself to a size based debris 
    approach.
    - It doesn't represent slow degradation well though. It does handle destructive collisions better.
- Maybe I can ignore satellite on satellite collisions in an initial model because it is the lowest.
    
#### Satellite Stocks

- Probably just fine as is.
    

### Market

- Current plan is to use a cournot market approach with inverse demand: \[ P = a-bQ \]
- Need to look up dynamic cournot market lit.

## Operator's Problem

- Restrict policy to integers (e.g. 1:10)
- Use a symmetric policy and value function for each operator.

### Operator State
- Initally I am going to track one type of debris, the operator's fleet size, and the number of satellites in all other fleets.

## Social Planner's Problem

I see two different classes of problems that the Social Planner could face, and both could take various forms.
Here are the two areas and some variations:

1. Maximizing long term expected utility/value
    a. Maximize expected value
    b. Preserve value x% of the time
2. Preserving the resource (manage debris levels)
    a. Keep debris below a threshold.
    b. Keep debris from growing faster than Y rate.
    c. Keep kessler syndrome from occuring in Z% over ZZ periods.
    
The first case "should" subsume the second in that minimal debris over a long period should permit more earnings. 
But, if the time-discounting factor is low enough, this might not be the case.
And this brings up a big issue in that the discount factor of the Social Planner and the Operators might differ 
a lot.
The question it poses is "Should we use this resource as we best see fit or preserve it for our descendants?"

## modelling state

I need to track all constellation sizes and debris size.