The Game Theory Behind Miner Centralization | Bitcoin 2026

Yeah, thank you uh thank you for coming to my talk. Um I'm a mathematician and I this is the first time I've ever given a talk to a non-academic audience. Um so I didn't I didn't realize I'd be competing with like a member of the uh presidential family. So um thank you for coming. Um I'm going to try to do uh I'm a I I usually study like elliptic PTE but um I've been in Bitcoin for a little while and I started looking at the game theory at some point. Um it's worth writing it down. it's worth writing like figuring out what the assumptions are. Um, we all know that it works in practice, but yes, does it work in theory? Um, while that's kind of a uh, you know, a silly way to phrase the question, it actually is a good question to phrase because um, if we understand why the theory works, we know why the theory can break. So um game theory uh game theorists studies something uh called solution concepts. Um a solution concept is a it's an outcome that we expect games to take. Um these are going to be mathematically easy to write down or not necessarily easy but you know we can do it. Um and they have to make sense economically. Um, so the one that you've probably heard of is the Nash equilibrium. That's where uh all the all the players in the game are utility maxing. Um, and it makes a lot of sense economically. Everyone's trying to maximize their utility. Um, it's a good solution concept um because it it makes a lot of sense and it's easy to describe. Uh, it's not always available. Lots of games don't have a Nash equilibrium. Some of games have zero. Um, some have infinitely many. Um, and you know, sometimes it's not even something you would you would you would describe as being part of the game. Um, other games like this like a bargaining game, they don't have you don't have a Nash equilibrium. You have you have bargaining solution. Now, John Nash also described how he thought that would play out and it usually does. Um, you know, when you have like a go or, you know, tic-tac-toe or something like that. There's also subgame equilibria. There there's lots of different different ways to uh I mean different different uh solution concepts which you expect to see in different types of games like you might have a basian equilibrium when you're trying to guess about other people's strategies that kind of thing. Um coalitional games that that's what I'm going to talk about. Uh those are uh as it sounds like it's it's a a a game with say end players. Each of the end players has to decide amongst the other end players like who they want to form a coalition with. And the way this is described mathematically is via some sort of payoff function. Um this payoff function new which describes how much that coalition gets if they win if if they form. So they they get it if they form. Um they can't uh they can't join multiple coalitions. Um and you know obviously everybody in the coalition has to decide uh or has to agree that others are also in the coalition. So the game is specified by this this payout function new um but then we assume transferable utility which allows the players to you know the negotiate how the payout's going to come out in the end and this allows them to kind of play off one coalition against other ones. Um, so this is like the first like the non-interesting coalitional game and you know we we could do it. We you know if if I if somebody donated to Bitcoin I could do this. Um I could take like three people from the audience. I could say give me your Bitcoin address and I could send it to you the three of you in a two of three multi-IG transaction. Um and I would have just created like a a a very a very standard textbook uh coalitional game. um called uh Odd Man Out. Um and I can do it I with Bitcoin, you know, you can do it without violence. Uh usually the game involves like bank robbers in the forest or something, but you know, with Bitcoin, we can we can get rid of the violence. Um so, you know, how how I how you would spin that. The three of you, I don't know what you would do. You would get together um split it three ways. Two of you could just sort of meet up and and take it. Um, here's how it's specified. Uh, in terms of of the payout, it's it's, you know, you have the coalition of three players, they went the whole they get the whole bitcoin. They get to spend it however they want. Any coalition of two players also gets to spin it however they want. Um, and then any coalition of one player cannot move it unilaterally. Um, so in coalition games there's there's no Nash equilibrium. There's something called a core which is kind of feels like a Nash equilibrium um in the sense that it's it's it's an arrangement in which um every every coalition that could form um and to do better for all the members of the coalition um has formed. Sorry, it it there there's no temptation to form and and you know defect from a current arrangement and and get any any better. And this previous game, um, you can see no matter how the payout works, I mean, no matter how I I split up, you you decide to split up your your Bitcoin in 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's kind of maximally happy. Um, so there there is this is a it's kind of a standard solution concept. A lot of times it doesn't apply um like it like for example in this game. Um, so Games Theorist had to come up with a little bit more sophisticated ways to look for solution concepts which are not just, you know, um, everybody's happy. Um, so there's this idea of of objection and counter objection. And like an an objection is some proposal by some coalition that says, well, we could do better for ourselves. you know, if if you you take this Bitcoin, you want to split it three ways, two people could always tell the tell the third person like we're we we can do better for ourselves if we take it and split it in half. Um, but then the the counter objection to that is once you start this game, there's going to be things that might follow. So there's going to be a other people are going to make objections to that objection and that's called a counter objection. And so you can define these all mathematically in really nice ways. The the idea here is that like the more stable coalitions, the ones that we expect to to uh see um are those where anytime you defect from it, you're going to end up with something that might end up not that step, but maybe steps down the road a little bit worse. Um so there's a lot there's a lot of uh con solution concepts for coalitional games. stable sets kernel I mentioned the core um the bargaining set nucleololis mathematicians like the nucleololis because it's unique and it exists um sapply value um so for this particular game I was talking about there's a stable set which which is the set of three-way splits and I can without giving you the exact definition I can tell you that why why it works no matter what payout I I suggest you know you could you know 30 30 40 or something there's always going to be two people that could form a coalition and get something better in this set of three payout vectors. Um, also there's no there's no two or there's no payout vector which is preferred um by by uh strictly by full coalition amongst the set. Interestingly, the nucleololis, this is a I I will give you the definition, but um the coalition game theory behind the nucleololis suggests that the three-way split is actually the most stable. So, and this again, it it sort of highlights the fact that uh game theory, it's not it can it's only a little bit predictive. We don't know exactly what's going to happen. We can kind of show you some outcomes which we think are likely. Um um we you know we'd be surprised if it were like 6040 but a 1/3 split is reasonable or 1/2 split um th those are reasonable. So to get back to Bitcoin um if you're looking at miners uh you you could ask if this is a weighted majority game. So there's a you know you could probably just figure out what the the weighted majority game is. It's you have a bunch of players and they each have weights and if the the coalition which forms for a group of players is the majority then they win the prize whatever that prize is and this is very classical theory. It goes back to like vonoman in the 40s. Um lots of theory behind it. Lots of theory developed in the 50s and 60s and 70s. Um so you know I I thought well can I can I apply this to uh mining? Um there's a little bit of problem. It's like the theory doesn't quite go through. this this giant huge thing right in the middle of it. Um, and that's that the the decentralized coalition is I mean it's it sort of it is like a coalition in the sense that like it's there and everybody wins that if they participated but in it by not um colluding. So like if if a majority chooses not to collute, I mean they don't even have to collude to do this. Um they've sort of just created this grand decentralized coalition. And so what I've tried to do is like shoehorn that in the game theory and see how this fits with all the other the the classical solutions concepts. So here we go. Um here's here's the game. Um I'm I'm taking the miners which have hash rate add up to one. It's a fraction. And then um I'm I'm I'm taking the I'm I'm thinking of these as like mining companies that want some profits. So I'm taking that as a flow value. And I say if if you get if if the the network is decentralized, everybody gets their hash rate fraction times the the profits which are going out to the entire network. So I'm using this value D. This is the value of profits to the full network. And um you know there's a little bit of a simplification obviously. Um but then I'm also putting this this other option which says okay if if the uh some coalition a 51% group decides to collude and and take over the network um they get C and C is going to be the value and and obviously we know these are different numbers right the value of Bitcoin decentralized is different from the value of Bitcoin centralized the profits are going to be different. There's a whole there's a I mean so I'm leaving these numbers and this is the kind of assumption I think is important. Um we don't know what these numbers are. Um but we uh we're going to write down a model in terms of these numbers. Welcome to predict. The world is a market. Everything is a market. Get a 100% cash back up to $100 on your first predict bet if it loses. predict where everything is a market. Um, obviously there are some things which you know I this is the energy stage. I wouldn't suggest that hash cost is uniform across you know everybody has different hash costs. So it's not a perfect it's you know it's a toy model. Um, also the centralized payout, you know, if if miners in Texas colluded to take over the network and miners in China, I mean, it would do different things to the network. There would be different ramifications. Um, I'm not even going to touch selfish mining makes it too complicated. So, um, the big question is what's what's the difference between C and D? Like what what determines C versus what determines D? And like I said, I want to like write down the assumptions. It's going into everybody's model. We know that it works in practice, but but what what is the theory? Um, and I think that, you know, it's in the white paper. He ought to find he ought to find it more profitable to follow the rules. And we've always, you know, we we've we've kind of echoed this and this is part of the folklore and this is kind of like uh we all talk about like uh you know, if somebody says, you know, what what is the minor centralized? What would happen? Um, often you hear the story about gash.io in in 2014 they got like 51% it was just a pool but or they got above 50% or something and then uh you know Bitcoin crashed from 630 to 580 or something. Um, Peter Todd went on Reddit and said he was selling some coins. And and so everyone since then has has had this uh this assumption, this this hypothesis that if anyone were to ever do anything damaging to the decentralization of the network, everyone would just sell their Bitcoin. Um, I don't think this is true anymore, right? Like 2014, this was sort of like old testament times, right? I mean people Bitcoin was still an experiment. It was still used. I mean you did not have a Bitcoin conference. You did not have like Michael Sailor. Um you did not have um you know a lot of the stuff that's going on here. So I I would suggest that we need to look at this number more closely because you know if if miners did centralize I think a lot of people wouldn't sell their Bitcoin because it's still there's still 21 million. Um it still would work. um unless you're maybe like Iran or something. Uh it would still work for most people just doing most of their like investment credi things. So um it's you know it's an interesting question whether to how much this holds you know in 2026 or you know how how much it will hold in 20 32 or or 2038. It held in 2014 but but it's it's a dynamic thing. Um other other things that might affect or or give give uh miners incentive to centralize, you get to pick the fees, right? You know, you can you can charge some users, you can charge the whales more if you want. They might not like it, but you can you can do it. Um you can also kind of crank down the hash rate because you you don't really need to compete anymore. I mean, just you can crank it up if someone competes with you, but you can kind of wind it down until all the other miners have kind of traded away. Um, and so that's cheaper. Um, of course, you know, there's a lot a lot of complicated things that could that could happen here. Uh, you know, one thing is, you know, if if you're in a jurisdiction where it's where the I mean, I'm not a lawyer obviously, but like a if if they're hostile, if the if if if the authorities are hostile, they might call you a money transmitter or something, but um and and make your life miserable or, you know, if if they have reasons to uh want to control the blockchain for certain reasons, you know, um like stopping global adversaries from receiving payments, they might be very happy just to have you control the network. Um so you know and this can work both ways. You can think of this as a benefit or a cost. Okay. So there's a you know I've I've explained I've explained the C and the D and the and the and the model. Um there is a uh I I I worked out the uh the solution concepts. So in the Satoshi days, you know, back, you know, long time ago, um, when when Bitcoin was sort of fragile, um, we thought of the ratio of C to D is basically zero. Like if if if miners were to centralize in 2012, experiment would have been over. Um, C is zero. Um, so CD was C over D is significantly less than one. But we can look at what happens as that increases. And so turns out it depends on your particular I'm saying I'm kind of it's a simplification I got n minors um there's going to be a value think of m is like maybe a little bit more than 51% and then m star is a little bit less than one and as long as this this ratio of cd stays less than this m the grand decentral coalition is still the core it's still the stable set it's the kernel it's the nucleio So, it's all of the it's all of the solution concepts. It's kind of got the uh it's got the checkmate grand slam and that I mean this explains like why in theory it works. Now, um of course the question is what happens later if this C gets to be a larger number. So, there's this transition phase where you go between this number which is a little bit bigger than 1/2 and this number which is closer to one. Um where the grand decentralization is no longer the core. So, it's not like this this super stable thing that everyone's going to be immediately attracted to. People will be tempted away from it, but it's and it's not in the stable set either. There's no stable set. Um, but it is going to be in the nucleolus. So, uh or is the unique thing in the nuc in the nucleolist. So this suggests that like while you know we could get to this C over D um the decentralized mining is still going to be predicted by game theory even though people will start to be tempted. It it it will it will look look appetizing maybe to to centralize but for you know um these objection and counter objection reasons people will probably uh stay decentralized but then what happens later once once M gets bigger than or once C overd gets bigger than one the game theory predicts that miners will centralize and they will redistribute according to their relative power. So, um, the black pill takeaway is that, you know, if if we, uh, if we get too much trades if mining is decentralized, it won't stay decentralized. Or at least that's what Game Theory would try to predict. Thank you. Every year this community comes together to celebrate, to debate, to build what comes next. And every year the stage gets bigger. Sound money center stage. So, where do you go to celebrate the next chapter in Bitcoin history? You come home. Nashville, July 2027. Thank you very