How would factors of this calculation change for the rate of doubling, scalability (the maximum amount at which it begins to become less efficient), risk and other factors you might bring up a reference to that I'm not thinking of? I'm a programmer and wondering the purchasing prices for machines of this calibre.
About the same amount you'd get for a perpetuum mobile - a machine that continuously runs without external energy. Because it is equally impossible.
The calculable value is infinite, as it can recover any price in a reasonable time.
One of the consequences of calculating with infinity is that it seems to result in unintuitive or even contradicting results (if you don't know how to handle it)
When you invest in the bot's trades up to the point of diminishing returns, you get some typical dollar return per year, minus the cost of capital (interest you could otherwise earn on the money invested). This net return is the "earnings" of the bot considered like any other business. Then, standard valuation techniques (accounting for risk) could estimate a P/E ratio and thus a price for the bot. It is very important to keep the algorithm a trade secret or the value will be diluted.
How the reliability/effectiveness of the bot would be demonstrated is another question. A consistent track record in real trading is the best evidence, but even then there is a risk that its "edge" unexpectedly stops working at some point, either because the market dynamics shift or because other traders independently discover (and arbitrage away) the inefficiency it's exploiting.
Your biggest problem will be proof that it can double my money each period of time without fail, and that it has been doing so for a long time. That proof would have to be independently collected, documented and verified; and that is the problem. Without that proof you are Bernie Madoff, or any other Ponzi scheme
In the bank example in the United States we have the FDIC insurance system to eliminate the risk, and in the Vanguard example decades long history of their low costs and the investing in the S&P 500 index as external proof. You charging a ton of money upfront would be a huge risk, and the unproven history would be like a Ponzi scheme.
Lets say it scales up to 1 Billion dollars, and doubles every month. That means if invest $1,000 using your robot then in 10 months I will have $1,000,000 and 10 months later I would have $1 Billion . I would want years of data to prove this. Of course if you had this data you would have no reason to sell. You seem to expect that I would pay you a ton of money up front, but that would not be possible, because I don't have a ton of money to risk. The only way I could pay would be as a percentage of my profits.
Another thing just occurred to me. Does that $1 Billion limit apply to me or collectively to all the other buyers of the robot. If The number of users gets to large we would be the market, and there would be no ability to double our money, which for the later buyers of the robot would sound exactly like a Ponzi scheme.
What i'm thinking is that at this point what you're selling wouldn't be worth money it would be worth time which is far more valuable.
If the hypothetical that you state is that it doubles the amount of value of something, then it exponentially grows value. A fixed price of the same type of value that it generates is worthless. If this can be agreed on then in this sense it's priceless. But what about time? You can't double time; you can only take away time, or reallocate time (change how you spend your time).
But you asked how much would it sell for. So for the individual selling this machine, the value of it would come down to how much money do they need in order to reallocate the time they have to do what they want. Say this machine took a week to double your money. The selling price would be a lot higher than if it took 10 or 20 years to double the money.