# [bitcoin-dev] Block size implementation using Game Theory

Anthony Towns aj at erisian.com.au
Fri Aug 7 02:16:50 UTC 2015

```On 7 August 2015 at 09:52, Wes Green via bitcoin-dev <
bitcoin-dev at lists.linuxfoundation.org> wrote:​

> Bitcoin is built on game theory. Somehow we seem to have forgotten that
> and are trying to fix our "block size issue" with magic numbers, projected
> percentage growth of bandwidth speeds, time limits, etc... There are
> instances where these types of solutions make sense, but this doesn't
> appear to be one of them. Lets return to game theory.
>
> Proposal: Allow any miner to, up to, double the block size at any given
> time - but penalize them. Using the normal block reward, whatev
> ​H​
> er percentage increase the miner makes over the previous limit is taken
> from both the normal reward and fees. The left over is rewarded to the next
> miner that finds a block.
> ​​
>
> What this would look like: The miners start seeing what looks like natural
> network growth, and make the decision (or program an algorithm, the beauty
> is it leaves the "how" up to the miners) to increase the blocksize. They
> think that, in the long run, having larger blocks will increase their
> revenue and its worth taking the hit now for more fees later. They increase
> the size to 1.25 MB. As a result, they reward would be 18.75 (75%). The
> miner fees were .5BTC. The miner fees are also reduced to .375BTC.
>

The equilibrium for that game is just keeping the same block size, isn't
it? Assume there's a huge backlog of fee-paying transactions, such that you
can trivially fill 1MB, and easily fill 1.25MB for the forseeable future;
fee per MB is roughly stable at "f". Then at any point in time, a miner has
the choice between receiving 25 + f btc, or increasing the blocksize to 1+p
MB and earning (25+f+pf) * (1-p) = f-ppf = 25(1-p) + f(1-pp) < 25+f. Even
if you think bigger blocks are good long term, wouldn't you reason that
other people will think so too, so why pay the price for it yourself,
instead of waiting for someone else to pay it and just reaping the benefit?

An idea I've been pondering is having the block size adjustable in
proportion to fees being paid. Something like "a block is invalid if
a(B-c)*B > F" where B is the block's size, F is the total fees for the
block, and a and c are scaling parameters -- either hardcoded in bitcoin,
or dynamically adjusted by miner votes. ATM, bitcoin's behavour is
effectively the same as setting c=1MB, a>=21M BTC/byte.

Having a more reasonable value for a would make it much easier to produce a
fee market for bitcoin transactions -- if the blocksize is currently around
some specific "B", then the average cost per byte of a transaction is just
"a(B-c)". If you pay more than that, then a miner will include your txn
sooner, increasing the blocksize beyond average if necessary; if you pay
less, you may have to wait for a lull in transactions so that the blocksize
ends up being smaller than average and miners can afford to include your
transaction (or someone might send an unnecessarily high fee paying txn
through, and yours might get swept along with it).

To provide some real numbers, you need to make some assumptions on fee
levels. If you're happy with:

- 1 MB blocks are fine, even if no one's paying any fees
- if people are paying 0.1 mBTC / kB (=0.1 BTC/MB) in fees on average then
8MB is okay

then a(1-c)=0, so c=1MB, and a(8-1)=0.1, so a=0.0143 and the scaling works
out like:

- 1MB blocks: free transactions, no fees
- 2MB blocks: 0.0143 mBTC/kB, 0.02 btc in fees/block
- 4MB blocks: 0.043 mBTC/kB, 0.17 btc in fees/block
- 8MB blocks: 0.1 mBTC/kB, 0.8 btc in fees/block
- 20MB blocks: 0.27 mBTC/kB, 5.4 btc in fees/block
- 40MB blocks: 0.56 mBTC/kB, 22 btc in fees/block

In the short term, miners can just maximise fees for a block -- ie, add the
highest fee/byte txns in order until adding the next one would invalidate
the block.

Over the long term, you'd still want to be able to adjust a and c -- as the
price of bitcoin in USD/real terms goes up, a should decrease
proportionally; as hardware/software improve, a should decrease and/or c
should increase.  Essentially miners would want to choose a,c such that the
market for block space clears at a price of some \$x/byte, where \$x is
determined by their costs -- ie, hardware/software constraints. If they set
a too high, or c too low, then they'll be unable to accept some
transactions offering \$x/byte, and thus lose out. If they set a too low or
c too high, they'll be mining bigger blocks for less reward, and lose out
that way too. At the moment, I think it's a bit of both problems -- c is
too low (meaning some transactions get dropped), but a is too high (meaning
fees are too low to pay for the effort of bigger blocks).

Note that, as described, miners could try cheating this plan by making a
high fee transaction to themselves but not publishing it -- they'll collect
the fee anyway, and now they can mine arbitrarily large blocks. You could
mitigate this by having a(B-c) set the /minimum/ fee/byte of every
transaction in a block, or alternatively by enforcing each miner pay a
significant %ge of collected fees to the miner of the next block(s).

​Cheers,
aj​

--
Anthony Towns <aj at erisian.com.au>
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