Understanding reinvestment in a miner growth scenario

Question

I got the following question which I can't figure out:

Alice, Bob, Carol, David and Eve are the five major bitcoin miners. They have each invested a million dollars in specialized equipment for bitcoin mining and have driven every other miner to look for other opportunities, i.e. every block in the bitcoin block chain is mined by one of them. The first four are happy just making money off of mining and save their money in bitcoin banks. Eve is willing to invest every bitcoin she makes in buying more mining equipment. Recall that the probability that a miner succeeds in mining a block is the number of hashes that the miner computes divided by the total number of hashes that the whole network computes. Assume that on average a dollar buys enough mining equipment to compute 2²⁵ SHA-256 hashes per second, that the mining fee for each block is 12.5 bitcoins and that each bitcoin is worth 500 dollars. What is the expected time required for Eve to be able to decide for every bitcoin transaction whether it is entered in the block chain, and therefore accepted, or not.

Can anyone explain how do I need to calculate this?

Answer 1

I will not give you the straight out answer, but i will guide you there.
Basically the question is asking when will Eve have 51% of the network hashing power.
Please note that in this question we dont really need to deal with hashing power, this question can be answered by calculating when Eve will have more than $4M+ worth of hashing hardware.
There is an easy approximation, and then there is a more difficult, but more precise way to calculate this.
Currently everyone has $1,000,000 of hashing power. So everybody has a 1 in 5 chance of finding a block.
Lets assume that Eve will get every 5th block (as she accumulates more hashing power, she will have a higher probability of finding new blocks - incorporating this into the answer is the more difficult but precise way).
So Eve will get 1/5 of all blocks, In a day she will get 144/5 = 28.8 blocks.
28.8 blocks @12.5 BTC @ $500 = $180,000
Now we need eve to have 51% of all hashing power. Or more than the others combined, so more than $4M So Eve needs to spend at least $3M on equipment.
$3M / $180,000 = 16.67 days So the approximation is 16.67 days, but Eve will get there faster because as she ads more equipment, her chances of finding new blocks grow.
For the more precise calculation we need to settle on the interval of Eve getting new hardware (hourly/daily/weekly/monthly etc.)
You can map it out and eventually you will arrive at a formula.
Hope this helps.