Good morning Dave, et al.,

> >      Myopic Miners: This bribery attack relies on all miners
> >
> >
> > being rational, hence considering their utility at game conclu-
> > sion instead of myopically optimizing for the next block. If
> > a portion of the miners are myopic and any of them gets to
> > create a block during the first T − 1 rounds, that miner would
> > include Alice’s transaction and Bob’s bribery attempt would
> > have failed.
> > In such scenarios the attack succeeds only with a certain
> > probability – only if a myopic miner does not create a block
> > in the first T − 1 rounds. The success probability therefore
> > decreases exponentially in T . Hence, to incentivize miners
> > to support the attack, Bob has to increase his offered bribe
> > exponentially in T .
> This is a good abstract description, but I think it might be useful for
> readers of this list who are wondering about the impact of this attack
> to put it in concrete terms. I'm bad at statistics, but I think the
> probability of bribery failing (even if Bob offers a bribe with an
> appropriately high feerate) is 1-exp(-b*h) where `b` is the number of
> blocks until timeout and `h` is a percentage of the hashrate controlled
> by so-called myopic miners. Given that, here's a table of attack
> failure probabilities:
> "Myopic" hashrate
> B 1% 10% 33% 50%
> l +---------------------------------
> o 6 | 5.82% 45.12% 86.19% 95.02%
> c 36 | 30.23% 97.27% 100.00% 100.00%
> k 144 | 76.31% 100.00% 100.00% 100.00%
> s 288 | 94.39% 100.00% 100.00% 100.00%
> So, if I understand correctly, even a small amount of "myopic" hashrate
> and long timeouts---or modest amounts of hashrate and short
> timeouts---makes this attack unlikely to succeed (and, even in the cases
> where it does succeed, Bob will have to offer a very large bribe to
> compensate "rational" miners for their high chance of losing out on
> gaining any transaction fees).
> Additionally, I think there's the problem of measuring the distribution
> of "myopic" hashrate versus "rational" hashrate. "Rational" miners need
> to do this in order to ensure they only accept Bob's timelocked bribe if
> it pays a sufficiently high fee. However, different miners who try to
> track what bribes were relayed versus what transactions got mined may
> come to different conclusions about the relative hashrate of "myopic"
> miners, leading some of them to require higher bribes, which may lead
> those those who estimated a lower relative hash rate to assume the rate
> of "myopic" mining in increasing, producing a feedback loop that makes
> other miners think the rate of "myopic" miners is increasing. (And that
> assumes none of the miners is deliberately juking the stats to mislead
> its competitors into leaving money on the table.)

A thought occurs to me, that we should not be so hasty to call non-myopic 
strategy "rational".
Let us consider instead "myopic" and "non-myopic" strategies in a population of 

I contend that in a mixed population of "myopic" and "non-myopic" miners, the 
myopic strategy is dominant in the game-theoretic sense, i.e. it might earn 
less if all miners were myopic, but if most miners were non-myopic and a small 
sub-population were myopic and there was no easy way for non-myopic miners to 
punish myopic miners, then the myopic miners will end up earning more (at the 
expense of the non-myopic miners) and dominate over non-myopic miners.
Such dominant result should prevent non-myopic miners from arising in the first 

The dominance results from the fact that by accepting the Alice transaction, 
myopic miners are effectively deducting the fees earned by non-myopic miners by 
preventing the Bob transaction from being confirmable.
On the other hand, even if the non-myopic miners successfully defer the Alice 
transaction, the myopic miner still has a chance equal to its hashrate of 
getting the Bob transaction and its attached fee.
Thus, myopic miners impose costs on their non-myopic competitors that 
non-myopic miners cannot impose their myopic competitors.
If even one myopic miner successfully gets the Alice transaction confirmed, all 
the non-myopic miners lose out on the Bob bribe fee.

So I think the myopic strategy will be dominant and non-myopic miners will not 
arise in the first place.

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