The notion of efficiency

Efficiency is measured as the ratio of useful output to total input, which can be expressed with the mathematical formula r=P/C, where r is the efficiency, P is the amount of useful output and C is the amount of resources consumed.

The standard efficiency of a value transfer

In case of a value transfer we define the amount of useful output P as the total output of the transaction made available to its recipient (or recipients, if the transaction happens to have several recipients).

The amount of resources consumed C we define as the total amount related to the transfer, including the amount transferred, related transaction processing fees and rewards.

The efficiency of a value transfer - discussion

In case of bitcoin, there are attempts to define the efficiency of a value transfer using the total processing expenses per transaction. So, let's try to compare the total processing expenses per transaction to the standard efficiency notion.

The first example is a transfer of $1,000,000,000 in cash. The standard efficiency formula would give us that a transfer using an armoured car and a group of security guards may be efficient, since the useful transaction output is big.

In contrast to that, the standard efficiency formula gives us that using an armoured car and a group of security guards to transfer an amount of $0.50 would be inefficient.

What happens if we use the total processing expenses per transaction? Using an armoured car and a group of security guards to transfer the amount of $0.50 would have the same total processing expenses per transaction as using an armoured car and a group of security guards to transfer $1,000,000,000!

Another real-life example is an attempt to lower the transaction fees in the bitcoin network by merging several transactions into one with several outputs. This is a widely recognized way how to increase the efficiency of the transactions.

Using the standard efficiency formula merging the transactions makes sense, because the total useful output does not change, while the total processing expenses decrease, leading to a greater standard efficiency of the transfer.

On the other hand, the processing expenses per transaction will increase, because the merged transaction will be more expensive than any of the original transactions!

The above examples demonstrate that the total processing expenses per transaction are unrelated to the efficiency of the transfer.

Visa credit card network efficiency

In the Visa credit card network, the processing fees are characterized as a percentage of the payment sent. The values that can be found on the Internet are in the range of 1.43% to 2.4%.

Supposing that one intends to send an amount C to a merchant and using the lower bound of the payment processing percentage found on the Internet equal to 1.43%, one obtains that P=(100% - 1.43%) C=98.57% C, which means that the efficiency of the Visa credit card network in the given example is r=98.57% C/C=98.57%.

Mean efficiency of the bitcoin network

For the bitcoin network, we define P as the total output value, and C as the total output value plus the total mining rewards including the transaction fees.

When calculating the efficiency of the network, we are more interested in trend than in seasonal fluctuations. That is why, instead of using the daily total output and daily total mining rewards, we use running four-year geometric means of those values.

For example, on 31 August 2021, the running four-year geometric mean of the daily output value in the bitcoin network is $14,210,628,488 while the four-year geometric mean of the daily total mining rewards is $16,278,056, giving us the four-year mean efficiency of the bitcoin network of 99.8856%.

Note that there is a clear trend observable in the efficiency plot: the mean efficiency of BTC is increasing.

Similarly as in case of the Visa credit card network, we can define the mean payment processing percentage of the bitcoin network. For the reasons obvious from the Visa credit card example, we define it as p = 100% - r, where r is the mean efficiency of the network.

Conclusion

The efficiency of the BTC network is greater than the efficiency of the Visa credit card network.

When compared to BCH (see Selected Bitcoin Cash statistics), which has got the 31 August 2021 running four-year mean efficiency of 99.9203%, we see that BTC is not the most efficient commodity book entry coin.

Moreover, we used the total rewards for our BTC efficiency calculations above. These include, as its main constituent, also the supply increase of BTC. In contrast to that, the Visa network efficiency calculation does not include the supply increase of the fiat money and uses the smallest payment processing percentage found on the Internet. A more realistic Visa network efficiency calculation would shift the balance in favour of BTC even further.

On the other hand, in bitcoin network, all input UTXO (unspent transaction outputs) entering the transaction as its inputs, must be spent completely. That is why common transactions also have so-called change outputs, returning the value exceeding the intended transfer value plus transaction fees, back to the sender. It is quite complicated to find out which outputs are just change outputs, not contributing to the transferred value. The Can Bitcoin be Trusted? Quantifying the economic value of blockchain transactions article discusses this, finding out that the transferred value is about 8 times smaller than the sum of all transaction outputs. Dividing the sum of transaction outputs by 8 to estimate the transferred value, we obtain a smaller efficiency, equal to 99.09%, which is still greater than the calculated efficiency of the Visa credit card network.