Hi everyone I'm a new user here, very interested in the potential of Bitcoin, and eager to see it succeed.

I'm curious about all aspects of Bitcoin security, and I was trying to figure out the cost of owning 50% of the network hashrate. I found some rough calculations by others online, but was not very satisfied, so I made some calculations of my own and I would like to share it with you.

The "cost of 50% ownership" is defined here as the cost C of buying N number of mining devices where N is the existing number of devices in the network. The idea is that, with time, N reaches an equilibrium level that is dependent on the total miner profit, and the minimum ROI that miners are prepared to accept. By making assumptions about these quantities, we can have a theoretical estimate of what C would be.

The graph below plots [C / BTC market cap] vs. [Average TX fee per BTC], the model being that C is proportional to both the market cap and the tx fees.

http://i.imgur.com/LeNa4mTl.pngIn terms of interpreting the graph (disregarding accuracy concerns for now) we see that in order for the cost C to be 10% of the market cap (22 million out of 220 million USD today), transaction fees need to be about 1.5% on average. In contrast, if transaction fees are 0.02 percent (roughly the level today), then the cost is 0.15% of the market cap (330k USD today).

The calculation details are as follows -

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Scenario assumptions:

1a. State-of-the-art mining hardware is accessible and is used by all miners.

1b. Miner income from block reward is negligible compared to transaction fees (true in long term)

1c. There exists an equilibrium in the amount of time taken for a mining device to break even on its fixed cost. At any level below this equilibrium, more miners will enter the market, and any higher, miners will leave.

Three values used in the calculation, which are somewhat arbitrary and open to dispute:

2a. Equilibrium time for miners to break even (BE): EQ_BE_TIME = 12 months.

This in practice is determined by the market and depends on the mining risk profile. I am just guessing a value here.

2b. Electricity-to-Hardware cost ratio: ELE_COST_RATIO = 0.35

This is (electricity cost of operating 1 device for duration EQ_BE_TIME) / (cost of 1 device), and I have estimated this using the Avalon ASIC data from

https://en.bitcoin.it/wiki/Mining_hardware_comparison:

ELE_COST_RATIO = 0.12 (USD per kWh) * 0.4 (kW for ASIC device) * 8640 (hours per year) / 1200 (cost of device) = 0.35

2c. Daily transaction volume ratio: DAILY_TX_RATIO = 0.025.

This is the ratio of transaction volume per day / number of coins in circulation, and I estimated this from

http://blockchain.info/charts/estimated-transaction-volume.

The equilibrium number N of mining devices in the network is related to the above three values as follows:

(TX_FEE_RATIO * DAILY_TX_RATIO * MARKET_CAP) * EQ_BE_TIME * 30 / N = DEVICE_COST * (1 + ELE_COST_RATIO)

where TX_FEE_RATIO is the average transaction fee per BTC, MARKET_CAP is the total BTC capitalization, and DEVICE_COST is the cost of 1 mining device.

The total value of all mining hardware in the network, C, relative to the MARKET_CAP, is

N*DEVICE_COST/MARKET_CAP and this is a function of TX_FEE_RATIO, after fixing the other parameters as explained above.

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Comments and criticisms on the methodology / assumptions / values are all welcome.

eyci