I spent part of the holidays poring over Eric Budish’s important paper, The Economic Limits of Bitcoin and the BlockChain. Using a few equilibrium conditions and some simulations, Budish shows that Bitcoin is vulnerable to a double spending attack.

In a double spending attack, the attacker sells say bitcoin for dollars. The bitcoin transfer is registered on the blockchain and then, perhaps after some escrow period, the dollars are received by the attacker. As soon as the bitcoin transfer is registered in a block–call this block 1–the attacker starts to mine his own blocks which do not include the bitcoin transfer. Suppose there is no escrow period then the best case for the attacker is that they mine two blocks 1′ and 2′ before the honest nodes mine block 2. In this case, the attacker’s chain–0,1′,2′–is the longest chain and so miners will add to this chain and not the 0,1… chain which becomes orphaned. The attacker’s chain does not include the bitcoin transfer so the attacker still has the bitcoins and they have the dollars! Also, remember, even though it is called a double-spend attack it’s actually an n-spend attack so the gains from attack could be very large. But what happens if the honest nodes mine a new block before the attacker mines 2′? Then the honest chain is 0,1,2 but the attacker still has block 1′ mined and after some time they will have 2′, then they have another chance. If the attacker can mine 3′ before the honest nodes mine block 3 then the new longest chain becomes 0,1′,2′,3′ and the honest nodes start mining on this chain rather than on 0,1,2. It can take time for the attacker to produce the longest chain but if the attacker has more computational power than the honest nodes, even just a little more, then with probability 1 the attacker will end up producing the longest chain.

As an example, Budish shows that if the attacker has just 5% more computational power than the honest nodes then on average it takes 26.5 blocks (a little over 4 hours) for the attacker to have the longest chain. (Most of the time it takes far fewer blocks but occasionally it takes hundreds of blocks for the attacker to produce the longest chain.) The attack will always be successful eventually, the key question is what is the cost of the attack?

The net cost of a double-spend attack is low because attackers also earn block rewards. For example, in the case above it might take 26 blocks for the attacker to substitute its longer chain for the honest chain but when it does so it earns 26 block rewards. The rewards were enough to cover the costs of the honest miners and so they are more or less enough to cover the costs of the attacker. The key point is that attacking is the same thing as mining. Budish assumes that attackers add to the computation power of the network which pushes returns down (for both the attacker and interestingly the honest nodes) but if we assume that the attacker starts out as honest–a Manchurian Candidate attack–then there is essentially zero cost to attacking.

It’s often said that Bitcoin creates security with math. That’s only partially true. The security behind avoiding the double spend attack is not cryptographic but economic, it’s really just the cost of coordinating to achieve a majority of the computational power. Satoshi assumed ‘one-CPU, one-vote’ which made it plausible that it would be costly to coordinate millions of miners. In the centralized ASIC world, coordination is much less costly. Consider, for example, that the top 4 mining pools today account for nearly 50% of the total computational power of the network. An attack would simply mean that these miners agree to mine slightly different blocks than they otherwise would.

Aside from the cost of coordination, a small group of large miners might not want to run a double spending attack because if Bitcoin is destroyed it will reduce the value of their capital investments in mining equipment (Budish analyzes several scenarios in this context). Call that the Too Big to Cheat argument. Sound familiar? The Too Big to Cheat argument, however, is a poor foundation for Bitcoin as a store of value because the more common it is to hold billions in Bitcoin the greater the value of an attack. Moreover, we are in especially dangerous territory today because bitcoin’s recent fall in price means that there is currently an overhang of computing power which has made some mining unprofitable, so miners may feel this a good time to get out.

The Too Big to Cheat argument suggests that coins are vulnerable to centralized computation power easily repurposed. The tricky part is that the efficiencies created by specialization–as for example in application-specific integrated circuits–tend to lead to centralization but by definition make repurposing more difficult. CPUs, in contrast, tend to lead to decentralization but are easily repurposed. It’s hard to know where safety lies. But what we can say is that any alt-coin that uses a proof of work algorithm that can be solved using ASICs is especially vulnerable because miners could run a double spend attack on that coin and then shift over to mining bitcoin if the value of that coin is destroyed.

What can help? Ironically, traditional law and governance might help. A double spend attack would be clear in the data and at least in general terms so would the attackers. An attack involving dollars and transfers from banks would be potentially prosecutable, greatly raising the cost of an attack. Governance might help as well. Would a majority of miners (not including the attacker) be willing to fork Bitcoin to avoid the attack, much as was done with The DAO? Even the possibility of a hardfork would reduce the expected value of an attack. More generally, all of these mechanisms are a way of enforcing some stake loss or capital loss on dishonest miners. In theory, therefore, proof of stake should be less vulnerable to 51% attacks but proof of stake is much more complicated to make incentive-compatible than proof of work.

All of this is a far cry from money without the state. Trust doesn’t have the solidity of math but we are learning that it is more robust.

Hat tip to Joshua Gans and especially to Eric Budish for extensive conversation on these issues.

Addendum: See here for more on the Ethereum Classic double spend attack.