The Game Theory Behind Miner Centralization

Thank you. Thank you for coming to my talk. I'm a mathematician, and this is the first time I've ever given a talk to a non-academic audience. So I didn't realize I'd be competing with a member of the president's family. Thank you for coming.

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I usually study elliptic PDEs, but I've been in Bitcoin for a little while, and I started looking at the game theory at some point. It's worth writing it down. It's worth figuring out what the assumptions are. We all know that it works in practice, but does it work in theory? That's kind of a silly way to phrase the question, but it actually is a good question, because if we understand why the theory works, we know why the theory can break.

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Game theorists study something called solution concepts. A solution concept is an outcome that we expect games to take. These are going to be mathematically possible to write down, and they have to make sense economically.

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The one that you've probably heard of is the Nash equilibrium. That's where all the players in the game are utility maximizing. It makes a lot of sense economically. Everyone is trying to maximize their utility. It's a good solution concept because it makes sense, and it's easy to describe. It's not always available. Lots of games don't have a Nash equilibrium. Some games have zero. Some have infinitely many. And sometimes it's not even something that you would describe as being part of the game.

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Other games, like a bargaining game, don't have a Nash equilibrium. You have a bargaining solution. John Nash also described how he thought that would play out, and it usually does. When you have Go, or tic-tac-toe, or something like that, there are also subgame equilibria. There are lots of different solution concepts you would expect to see in different types of games. You might have Bayesian equilibrium when you're trying to guess other people's strategies, that kind of thing.

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Coalitional games are what I'm going to talk about. As it sounds, it's a game with, say, n players. Each of the n players has to decide among the other n players who they want to form a coalition with. The way this is described mathematically is via some sort of payoff function, nu, which describes how much that coalition gets if it forms. They can't join multiple coalitions. And obviously, everybody in the coalition has to agree that the others are also in the coalition.

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So the game is specified by this payoff function, nu, but then we assume transferable utilities, which allows the players to negotiate how the payout is going to come out in the end. This allows them to play one coalition against other ones.

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This is the first, non-interesting coalitional game. If somebody donated a Bitcoin, I could take three people from the audience, say, give me your Bitcoin address, and send it to the three of you in a two-of-three multisig transaction. I would have just created a very standard textbook coalitional game called Odd Man Out. I can do it with Bitcoin. You can do it without violence. Usually the game involves bank robbers in the forest or something, but with Bitcoin, we can get rid of the violence.

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How would you spend that, the three of you? I don't know what you would do. You could get together and split it three ways. Two of you could just meet up and take it.

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Here's how it's specified in terms of the payout. You have the coalition of three players. They get the whole Bitcoin. They get to spend it however they want. Any coalition of two players also gets to spend it however they want. Any coalition of one player cannot move it unilaterally.

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In coalitional games, there's no natural equilibrium. There's something called a core, which feels like a natural equilibrium in the sense that it's an arrangement in which every coalition that could form and do better for all of its members has formed. There is no temptation to form, defect from a current arrangement, and get any better.

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In this previous game, you can see that no matter how the payout works, no matter how you decide to split up your Bitcoin three ways, two people can always take whatever another person has and redirect it to themselves. So there's never going to be a core where everybody is kind of maximally happy. It is a standard concept. A lot of times it doesn't apply, like in this game.

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Game theorists had to come up with more sophisticated ways to look for solution concepts, not just everybody's happy. There's this idea of objection and counter-objection. An objection is some proposal by some coalition that says, well, we could do better for ourselves. If you take this Bitcoin and want to split it three ways, two people can always tell the third person, we can do better for ourselves if we take it and split it in half.

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But the counter-objection to that is that once you start this game, things might follow. Other people are going to make objections to the objection, and that's called a counter-objection. You can define these all mathematically in really nice ways.

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The idea here is that the more stable coalitions, the ones that we expect to see, are those where anytime you defect from it, you're going to end up with something that might, not at that step but maybe a step down the road, be a little bit worse.

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There are a lot of solution concepts for coalitional games: stable sets, kernel, the core, the bargaining set, nucleolus, and Shapley value. Mathematicians like the nucleolus because it's unique and it exists.

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For this particular game I was talking about, there's a stable set, which is the set of three-way splits. Without giving you the exact definition, I can tell you why it works. No matter what payout I suggest, say 30, 30, 40 or something, there are always two people who could form a coalition and get something better in this set of three payout vectors. Also, there is no payout vector that is strictly preferred by the full coalition among this set.

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Interestingly, the nucleolus, and I won't give you the definition, suggests that the three-way split is actually the most stable.

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This again highlights the fact that game theory is only a little bit predictive. We don't know exactly what's going to happen. We can show you some outcomes that we think are likely. We'd be surprised if it were 60, 40, zero. But a one-third, one-third, one-third split is reasonable, or a one-half, one-half split. Those are reasonable.

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To get back to Bitcoin, if you're looking at miners, you could ask if this is a weighted majority game. You can probably figure out what the weighted majority game is. You have a bunch of players, and they each have weights. If a coalition forms for a group of players and it is a majority, then they win the prize, whatever that prize is. This is very classical theory. It goes back to von Neumann in the 1940s. There is lots of theory behind it, lots of theory developed in the 1950s, 1960s, and 1970s. So I thought, can I apply this to mining?

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There's a little bit of a problem. The theory doesn't quite go through. There's this giant thing right in the middle of it, and that's the decentralized coalition. It is like a coalition in the sense that it's there, and everybody wins if they participate in it by not colluding. If a majority chooses not to collude, they don't even have to collude to do this. They've sort of just created this grand decentralized coalition. So what I've tried to do is shoehorn that into game theory and see how this fits with all the other classical solution concepts.

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Here is the game. I'm taking the miners, which have hash rate that adds up to one. It's a fraction. I'm thinking of these as mining companies that want some profits, so I'm taking that as a flow value. If the network is decentralized, everybody gets their hash rate fraction times the profits that are going out into the entire network. I'm using this value D. This is the value of profit to the full network. There is a little bit of a simplification, obviously.

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Then I'm also putting in this other option, which says that if some coalition, a 51% group, decides to collude and take over the network, they get C. C is going to be the value. Obviously, we know these are different numbers. The value of Bitcoin decentralized is different from the value of Bitcoin centralized. The profits are going to be different. I'm leaving these numbers, and this is the kind of assumption I think is important. We don't know what these numbers are, but we're going to write down a model in terms of these numbers.

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Obviously, there are some things which, this being the Energy Stage, I wouldn't suggest that hash cost is uniform across the board. Everybody has different hash costs, so it's not perfect. It's a toy model. Also, the centralized payout, if miners in Texas colluded to take over the network versus miners in China, it would do different things to the network. There would be different ramifications. I'm not even going to touch selfish mining. It makes it too complicated.

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The big question is, what's the difference between C and D? What determines C versus what determines D? Like I said, I want to write down the assumptions that go into everybody's model. We know that it works in practice, but what is the theory?

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I think it's in the white paper: "He ought to find it more profitable to follow the rules." We've kind of echoed this, and it is part of the folklore. We all talk about, if somebody says, what if miners centralized, what would happen? You often hear this story about GHash.io in 2014. They got about 51%. It was just a pool, but they got above 50% or something, and then Bitcoin crashed from 630 to 580 or something. Peter Todd went on Reddit and said he was selling some coins.

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Everyone since then has had this assumption, this hypothesis, that if anyone were to ever do anything damaging to the decentralization of the network, everyone would just sell their Bitcoin. I don't think this is true anymore.

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In 2014, this was sort of like Old Testament times. Bitcoin was still an experiment. You did not have a Bitcoin conference. You did not have Michael Saylor. You did not have a lot of this stuff that is going on here. So I would suggest that we need to look at this number more closely.

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If miners did centralize, I think a lot of people wouldn't sell their Bitcoin because there are still 21 million. It still would work. Unless you're maybe Iran or something, it would still work for most people just doing most of their investment TradFi things. So it's an interesting question how much this holds in 2026, or how much it will hold in 2032 or 2038. It held in 2014, but it's a dynamic thing.

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Other things that might affect or give miners incentive to centralize: you get to pick the fees. You can charge some users, you can charge whales more if you want. They might not like it, but you can do it. You can also crank down the hash rate because you don't really need to compete anymore. You can crank it up if someone competes with you, but you can kind of wind it down until all the other miners have traded away. That's cheaper.

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Of course, there are a lot of complicated things that could happen here. One thing is, if you're in a jurisdiction where, and I'm not a lawyer, obviously, if the authorities are hostile, they might call you a money transmitter or something and make your life miserable. Or if they have reasons to want to control the blockchain for certain reasons, like stopping global adversaries from receiving payments, they might be very happy just to have you control the network. This can work both ways. You can think of it as a benefit or a cost.

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I've explained C and D in the model. I worked out the solution concepts. In the Satoshi days, a long time ago, when Bitcoin was sort of fragile, we thought of the ratio C over D as basically zero. If miners were to centralize in 2012, the experiment would have been over. C is zero, so C over D is significantly less than one.

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But we can look at what happens as that increases. It turns out it depends on your particular model. This is a simplification: I've got n miners. There is going to be a value. Think of M as maybe a little bit more than 51%, and then M star is a little bit less than one. As long as this ratio of C over D stays less than this M, the grand decentralized coalition is still the core. It's still the stable set. It's the kernel. It's the nucleolus. So it's all of the solution concepts. It has the checkmate grand slam. This explains why, in theory, it works now.

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Of course, the question is what happens later if this C gets to be a larger number. There is this transition phase where you go between this number, which is a little bit bigger than one-half, and this number, which is closer to one, where the grand decentralization is no longer the core. So it's not this super stable thing that everyone's going to be immediately attracted to. People will be tempted away from it. It is not in the stable set either. There's no stable set, but it is going to be in the nucleolus. It is the unique thing in the nucleolus.

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This suggests that we could get to this C over D, and decentralized mining is still going to be predicted by game theory, even though people will start to be tempted. It will look appetizing, maybe, to centralize. But for these objection and counter-objection reasons, people will probably stay decentralized.

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But then what happens later? Once C over D gets bigger than one, game theory predicts that miners will centralize and redistribute according to their relative power.

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The black-pill takeaway is that if we get too much TradFi in the system and nobody cares if mining is decentralized, it won't stay decentralized. Or at least that's what game theory would try to predict. Thank you.