For 5,000 years we've come to rely on judgements of creditworthiness when it comes to deciding how much to trust each other in financial dealings. We made many inventions, everything from inventing physical money to central banks, each of which has embedded some idea a trust model.
The bitcoin paper wants us to believe the block chain is only broken (in the double spend sense) in theory if the bad user gets a hold of over half of the CPU power. The block chain is the implementation of the idea that truth is power. This argument is the essence of the architecture's claim to our trust. And the paper bases that argument on Gambler's ruin. This argument ends by suggesting that only when the attacker block chain creation likelihood is greater than the main network will their node become 'truth'. This bad node contains double spent transactions, enriching the bad node owner, giving them an incentive.
This game of two opponents, the good node pool and the bad, with probabilities $p$ and $q$, $p+q=1$ is set as a version of Gambler's ruin where the opponent has deep pockets. In that case, the likelihood of broken trust is certain if bad is momentarily more powerful than good, in a block chain sense. But becomes exponentially vanishingly small as time passes otherwise, $(q/p)^z$. But the bad node player isn't playing this version of Gambler's ruin (which mathematically is a Bernoulli Trial). They're playing Gambler's ruin with a free option to reset the game to all-square as many times as they like. This is a Bernoulli trial with a reflecting boundary.
The maths is different. Therefore the threshold point q=0.5 is different, lower. Therefore the main trust claim expressed in the paper is broken. Imagine gambler's ruin where you have a pot of capital as does your opponent. If your opponent falls behind, then he will need to recoup his losses if he's ever to go on to ruin you. A tough challenge. But the block chain bad pool can skip this step and always reset to all-square simply by taking another copy of the good block chain and having another attempt to double spend
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