 Okay, thank you, Dave. So we are going to start with our second research seminar. So Axel is doing a round two on the eising game and talking about critical phenomenon and non-inquilibrium dynamics. Thank you, Axel, and I will pass the mic to you. Thanks. Yeah, so this is kind of part two of the first talk I gave. So it can be useful to refresh that, so I'll go very quickly over some of the concepts from there. But yeah, you can go back to that. Right, so in the first talk I was kind of introducing these concepts of what would a phase transition even mean in a kind of game theory context and the common statistical physics methods don't really directly apply to it. And it was about introducing these ideas, but now this round is like, let's actually look at a simple toy model and probe for phase transitions and this kind of thing. And yeah, so this is a very simple model, but actually you'll see a lot of rich phenomena comes out of it. Right, which by the name is the model is just inspired by the eising model of physics, I'll talk about a little bit. Yeah, so this is a review of last time, so what are the common tools of statistical physics or statistical physics? So here, for instance, we'll talk about the eising model, which is basically a model for something like this, like you have some lattice and in each lattice side there is some little arrow. So these models, like the magnetic spin of some atom there, if all the spins are aligned, then your whole thing is like a magnet with an overall magnetic field and so on. So there's a model of these little arrows in a lattice and in the eising model, it's even simpler that these arrows could either point up or point down. And so they can't go like this, but the arrows have the option of just pointing up or pointing down. And basically, you have an energy function which tells you if it takes lower energy if the spins are aligned. So that's like if I remind the spins want to align. And you could also have some external field or something. So you say there's this external field that wants the spins to align in this particular direction, depending on the sign of J here. So in physics, you have an energy function. So you have some configuration of the spins up and down, however, and that has some energy. Then what you do in statistical physics is like you want so you can talk about a partition function, which is basically a weighted sum of all the possible configurations of spins up and down. And these are weighted by the temperature here. So this is a parameter that you put here that basically tells you how much weight the higher energy configurations have. So obviously the most important contribution here would be like the lowest energy here, so it's a theory of a ground state. We call it the lowest energy configuration plus fluctuations of higher energy around it. Then we have actually physical observable things, which are expectation values. So you say, so what's the expected value I will see of a particular spin here? Do I expect to see it up or down or what's the rest of us expected value? And how you do that is like you have the thing you want to observe and you basically average it with this distribution, which is weighted by that. So this is actually the definition of temperature. So temperature is just that. It's like how much do you weight other states in the partition function? No, so that's physics. So this is what I introduced last time. So what can we look at similar to this in game theory? So an important point here is like an important point in physics when you talk about critical phenomena is locality. So this energy function has some notion of locality with it. The idea is that the given spin interacts more strongly with the spins that are closest to it. And in this Ising model in particular, this is just an interaction between nearest neighbors. And so the direct interaction is between nearest neighbors. And it's assumed that there's no direct interaction with a spin that is very far away. So we're going to look at some like many agents or many many player game theory where there's some sense of locality where like my utility is more dependent, more strongly with players that are in some notion of distance closer to me. And then the analogy of this thermodynamic noise here given by the temperature will be like deviations from rationality. So you have like the stable configuration is like you have everyone being perfectly rational and you're in a Nash equilibrium. And then you add a little noise which can be parameterized to deviate around this equilibrium. So what you have is like a set of strategies for all the given forces. Is it the strategy for the height player? So one shot here means that the game will be only played once. So you don't collaborate with anyone and you have to come up with your strategy alone and you place your strategy. And then every player has their own utility. But their utility depends more strongly on what the strategies of the players around the given player was. So this is what this mutation is here. And then you have like when everyone is rational, you would have a Nash equilibrium where it's a particular configuration of spins that is basically the best that non-cooperating players can do. Now you introduce this idea of noise introspection which is kind of the analogy of statistical physics here. Okay, so let's assume I am less than rational. So what's the expected strategy that I would have? So if I say, if I'm not doing exactly the Nash equilibrium strategy but I have some noise around it and this can be modeled by some temperature parameter here. So this should be actually one over two here. But anyway. Right, so the idea is, so this is done by many levels. So at the first level I say, okay, so let's assume all my neighbors are doing the Nash strategy and I will weigh my strategy right with something similar to a statistical physics weight here. Right, so the highest contribution will come from the Nash equilibrium strategy. The highest contribution will come from the Nash equilibrium strategy here. But I accept fluctuations coming from other strategies. And then I sum over all my possible strategies weighted like this. And this would be my expected strategy at the first level of introspection. But then if I'm more rational like that, I assume also my neighbors are doing the same. So this is the first level that they're doing Nash, but then those neighbors are also not perfectly rational and doing a noisy introspection around their neighbors doing Nash strategy and so on. So the second round is like assuming my neighbors are also doing this kind of statistical approach and then I do a statistical approach on top of that. And then these ones are doing a statistical approach. So you can do n levels of introspection and then like kind of your actual expected value would be the limit of like infinite levels of introspection. This in principle could involve like many parameters here like they could have different temperatures at different levels, but for most of the top we'll just put the same parameter here to like describe that everyone at every level has the same amount of noise. All right, so this is the kind of thing that we're looking at. So instead of looking at energy functions, we're looking at utilities for individual players and instead of thermal noise, we're looking at this n levels of noise introspection around Nash equilibrium. Right, so this is the kind of things that we know about phase transitions in physical systems that we ask like we're gonna see something like this in Game Theory or something. So one thing is like so this is a thing that is well understood only in equilibrium statistical physics, which means so this is like when you're modeling statistical physics like I described this is something that is the system is not changing your modeling at this particular temperature and there's no other like forces or changing things on it. So something that happens in statistical physics is that you have like divergent correlation lengths, right, so close to a critical point, more and more and more distance spins in this easing model will be correlated with each other. So the correlation length there starts to diverge close to a critical point and that means it's anyway you can describe it with some critical exponents as well as most. And that's generally what there's like a phase transition happens between an ordered and a disordered phase like for instance in the easing model here this means that if you look at the expected value of a given spin so this will be the order parameter here which means that the order parameter is something that in one phase has expected value of zero and in another phase has a non-zero expected value. So if you are in a low temperature phase here so you expect that the spins actually can get aligned so the expected value for a given spin will be non-zero but if fluctuations are very high and you're in a disordered phase then expected value for any given spin will be zero because this can be up and down and they're not aligning the spins. So that's kind of the magnetization, demagnetization transition. So if you take a magnet and you hear it hot enough you will lose the magnet because the spins will not be aligned anymore. Right, yeah so something interesting is that there's an order of a phase transition so you can ask like what is the what is the of what order is my phase transition and because the point is there is some discontinuity happen at that transition point. So in physics this usually happens like you can talk about the behavior of this order parameter for for instance as I increase the temperature. And the idea is so if I'm so if I'm increasing the temperature and I look at how this order parameter behaves some derivative of this order parameter with respect to that temperature will be discontinuous. So in a first order phase transition the derivative of this guy is discontinuous as I cross that the thing. Now this is a simple case we physicists really love which is the pacing model because it's like simple to play with and see ideas but we don't really love the 1D easing model that much because it's kind of too simple so this is so what you can do here is okay so let's let's look at the 1D easing model which is I take these spins and just put them on a line so I have a line of spins and the nearest neighbors interact so this spin interacts with one ahead and one behind. Here in this case I am removing the external magnetic field doesn't matter much but just looking at the interaction between spins and something you can you can ask is I'd say you are at some you're looking at this model at some given temperature are you going to be in an ordered phase or a disordered phase so for a certain temperature will the spins be able to align or the thermal fluctuations be too high and you won't be able to align the spins despite they wanting to be aligned. So this model is nice in that you can do this exactly right so this is your energy function you calculate your partition function here and basically you can just sum over all possible states so you take these spins and yeah they are plus and minus one you can do all these sums and do this carefully and by the end this is your partition function here you can write it like this and you can also write a compute exactly things like correlation functions as a let me look at the correlation function of a given spin versus a spin that is our lattice sides away right and then you get this correlation function and you can so it looks like this like this exponential and since you can easily write it in terms of some correlation length here so this lets you kind of easily look for a phase transition if this correlation length diverges at some point and the one piece in model is nice in that you can calculate it exactly like this but it's not nice in that it's not very rich so you calculate like this and what you see is that there isn't a phase transition so there isn't a change of behavior in this guy here and so the only really phase transition happens only at exactly zero temperature so what this means is that in the 1d is in model for any finite temperature that you have you will not be able to align the spins they can only align at exactly zero temperature and this is a feature of the 1d this is different in 2d but we're not going to get there but in 1d as soon as you introduce a little bit of temperature you lose the alignment now let's talk about game theory so we can write down something inspired by the ising models this is the guy we're going to be playing with and this is what i call the ising game so the idea is similar thing so you have n number of players in one dimension right so they are aligned in a line each player has possible strategies of just plus and minus one this is the the play they can make they can either choose plus one or minus one and their utility for that game depends only on what their nearest neighbors do right so if my strategy aligns with my neighbor to the right that's good and if it aligns with the neighbor to the left that's also good and when this number of utility i can also put what i call here an external incentive which means that i say if this is positive this will favor using the strategy plus one right so so player wins by aligning with their neighbors and also by aligning with external incentive and in particular if this is j is positive then there's a clear Nash equilibrium where the best strategy for everyone to do is everyone just choose plus one and this is the best they can do in a very important way so everyone chooses plus so if everyone is rational everyone chooses plus one with this external incentive uh all right so a couple of questions we want to look at here so let's say everyone is rational everyone chooses plus one what if people started being non-rational right so we see in the in the easing model as soon as you introduce a little bit of noise this order is destroyed so by restoring symmetry here i mean like symmetry between plus and minus one right so this order is destroyed then players can equally do plus or minus one so so so how much noise is required to kind of destroy this order and another question is like okay so we have this Nash equilibrium kind of incentivized there by this external incentive that pushed everyone to choose plus one what if i then remove this incentive will this be stable or like results of will they uh keeping this uh all aligned or when i remove that will i lose that alignment as is the case in the one-d easing model if i remove the field so i can align of this all the spins if i introduce an external field but if i remove it then i lose the right kind of this one organization all right so this this is we can calculate this right so we can start doing noise insuspection of this uh easing game so for the first level this is pretty easy so what i do is i'm looking at the ith player and i assume the two nearest neighbors are doing the Nash strategy which is they're doing plus one and then i sum overall my possibilities around this Nash equilibrium so i have a contribution from plus one and plus minus one i can do this and this is the this kind of partition function here and the same way i can calculate a correlation function but in this game so this is something i discussed in the previous top that these two guys are actually uncorrelated here so this is the same as kind of calculating the expected value squared of one anyway this is stuff that i can do and at the first level of introspection it's easy to just i just kind of have to sum over plus and minus one and get this time and attention so this is what uh what i get right now it's not to this so this is something you you can kind of keep doing now the so i assume this is what now for the second round on it i assume this is what my neighbors do and then i do noise insuspection around that and and then i do that n times and what that result looks like is something like this where my nth level of my expected value is looking like like this where i have similar looking to this so tangent of this guy but here i kind of do more i hide the complexity here and calling this guy g for so also kind of an effective uh number here that is based on the previous one on the previous round and so on so what this actually looks like is like a tangent inside inside a tangent inside a tangent inside a tangent it's kind of a very nested expression with n levels of tange nested in there yeah so this is what the easing model with some given temperature n uh a temperature t at the nth level of introspection looks like and now i can so i have this expression i can calculate it now i can start probing and like okay so are there different phases do i see order and then so i know i'm starting from the order of this Nash equilibrium i've been pointing up well i break that at a specific amount of temperature or something like this uh so first kind of we can gain gain some intuition by kind of looking at different limits so first we can look at like let's say i am very close to rational so let that means let me put a very small amount of temperature let's say if i have so if this t is very small i am only slightly deviating from rational let's see what that looks like and so this is kind of a low temperature expansion and this can be done because this tange has a simple approximation where this t is very small then i can write it in terms of this exponential and then kind of i put this inside the tange and the next nested expression and by the end for small t's this expression for for for this looks like this and this is simple to understand so this is the the rational my expected value for strategy is the rational strategy which would have been one playing one and start deviating away from it by a small amount so the more irrational i am the less that i am likely to actually the more irrational everyone is so not just me but right so so it looks like so i prefer the rational i have my strategies one and then i start deviating away from it and that's a very sensible low temperature expansion now the other end is let's look at a high temperature expansion right so it's a similar thing so i have this tange expressions here this also have an approximation for very large values of temperature which is basically i can approximate the tange by its argument and when the argument is very small so i get this guy and now when i have n levels of introspection of that i have tange and then inside tange inside tange i'll get something that is proportional to one to the t to the n now i am assuming here that temperature is large so this is a number that will vanish to zero as i take n to infinity right as i take more and more introspection so if temperature is high enough my expected value will go to exactly zero right so what i see is something that looks like an ordered phase and a disordered phase so at low temperature as expected i have uh uh uh so at low temperature i have order so i have something that is not exactly the perfectly rational strategy but close to it at some point in between that breaks down and my strategy is indifferent between one and minus one because everyone is on notion uh yes zero zero temperature is perfectly rational um yeah that's sensational so temperature here is just some way i am parametrizing the how irrational people are right so this is a way to to rational the try to parametrize the noise of like people are not perfectly rational but are kind of random around rationality right okay uh right uh okay so so we so we establish kind of the limits make sense so at high at low temperature we have something some non-zero expected value some ordered value that makes sense at at large temperatures this drops to zero so kind of this plus minus one symmetry is restored when does this change happen is there a point and we identify a point somewhere in between where these differencing behaviors happen and they actually can right so the idea here is that right so so remember we have this kind of nested expression where we have a tange inside a tange inside a tange so kind of i start with the initial parameter g here and put it in a tange and then put that in under the tange and so on so if it happens that when i take the initial when i take this thing and i put it inside a tange it becomes smaller every time and then i put it in another tange and then it keeps getting smaller and smaller and smaller then that thing will inevitably just go to zero when i do like a large n level of introspection so the question is is there is there a particular value of of temperature where this thing will start getting smaller and smaller such that it will get to zero and if it doesn't do that then so so there's that point where where g n is smaller than g n minus one when that happens you have zero expectation value so this is kind of when that phase transition happens and you can kind of solve this and anyway so this is the the critical temperature you identify with this expression here so if my if my temperature if my temperature is lower than the one given here i will have a finite a non-zero expected value for my strategy but if i introduce a temperature higher than this then my strategy is just equally plus and minus one so yeah so this is a phase transition and we identified the critical temperature which happens at this point now there's something kind of cool here yeah can i just do a quick question so if when you do infinite introspection the rational decision is to pick zero doesn't that mean that everyone will pick zero and so you get symmetry as well i think you phrased that wrong so so it's not that you that you did this introspection and the rational decision is zero i mean the rational decision is doing the nice strategy the the thing is that this is the best that you can do because you're not rational right so you can do so if if there's enough noise and if you are irrational and everyone else is irrational this is the you can't do better than just be random okay so you're so noisy that i don't know let me just choose whatever okay got it yeah no okay so right so so this is what the summary here so i have this expected values here that so at low temperature looks like something finite so small deviations from one from the rational thing then so it's smoothly decreasing from the rational thing then at some at some critical point after that critical point it drops to zero but this is the something that is peculiar here that is unlike physics when this happens usually in physics you look at okay so let me so let's say this expected value was decreasing from one and if it was physics it will decrease all the way to zero and then stay in zero so that is actually that would be a first order first phase transition so the expected value is not discontinuous but the derivative of it is yeah now what we see here is actually a zero toward the phase transition which is weird so so it's decreasing from one and then it reaches this critical point at that point its value you can calculate it it's actually one fourth and then after that it drops immediately to zero that's it so the expected value itself is discontinuous as i change the temperature right so it's a non-zero value non-zero value and then so then zero so so this is a kind of interesting thing that is on physics like and this is a feature of this weird noise introspection here it happens because of this nested structure like this is not going to happen in physics uh yeah so that's kind of cool all right uh so this this is about it for that so so this is kind of the so for the original questions we said to answer this this is it the the model is nice enough that it gave us a bit more which we will see now but uh yeah so these are the questions we we had from before and so we can touch again on the rest of the the questions that i asked before so let's say okay so i aligned everyone with with this external incentive and i can see now okay it's it's uh everything is stable until i go beyond this temperature right but if i do that and then i actually remove the external incentive this still works right so this is still a finite finite temperature so yeah so so the idea is that unlike the 1d easing model of physics this uh ordered phase is stable when i remove the incentive so that's kind of cool also yeah so um yeah uh so if i do this with the easing model i introduce an external field i kind of will get a critical temperature that depends on this external field but if i remove the external field that critical temperature goes to zero so so any amount of noise will destroy that order but here uh i can align everyone then remove the field and i can still have some order at some non-zero temperature okay so now we have that and there's something kind of cooler that we can start looking at which is okay so this was all uh like a one one round game where everyone just plays one so you choose now plus or minus one and that's it the game is over but what if we had let's say two rounds right so everyone plays once and then i can observe everyone's first round play and then i play again so something kind of i i kind of sneaked in here is that in this one round game if i want from the start to perfectly align everyone i have to introduce this external incentive so kind of once i have everyone aligned i can actually remove it but if if i if i'm just playing one round and i don't have this field the the players don't know if they should choose plus or minus one so aligning is better but i don't know what everyone else will do so i that choose randomly plus or minus one so i say if we don't have an external incentive and we're doing this this first this two round game in the first game the only thing rational players can do is choose whatever plus plus or minus one so with 50 i'll choose plus one 50 i'll choose minus one there's i mean there's no reason to do anything else than that and then so let's so let's say the first round everyone just draws from some random distribution and that's the first play then for the second round everyone observes what everyone else do and plays again so can the second round improve on this right so can they do better by the second round by by by looking on the first round so the answer is yes by doing this noisiness inspection but so what they would do is this are the the questions here so like by the second round with some with some correlation among spins among strategies emerge because at the first round they're just strategy they're uncorrelated and uh yeah and will they do better like getting closer towards some better equilibrium that favors them so what they need to do is so like so you know so in this case for the for the first round everyone will just do some random strategy but for the second round they can do a noisy introspection but where the starting point of that noisy introspection is the players the strategies that were done in the first round I'll show you more what that means exactly and for example we will look at this guy here so let's say let's calculate the correlation function between a given player and a player that is two sides away so two players away so in the first round these two players are just playing random so this is just zero they're uncorrelated but by right so by the second round what I do is so I can start doing introspection right so let me so this is the first round here the first correlation is just zero uh let's say for now that let's say what's the best that the players can do by the second round the best that they can do is that say everyone is perfectly Russian right so let's assume that everyone has temperature equal to zero and they start doing noisy introspection with temperature equal to zero so what it looks like is that I start so I'm gonna choose my next strategy doing noisy introspection based on what my neighbors did in the first round with a temperature that is going to zero so if I do that and if I do just one level the first level of introspection for that so I just look at my neighbors and base my strategy on my nearest neighbors I immediately increase the does the correlation length of this the correlation function for the second step to one half from zero to one half so so so by just doing one level of perfectly rational introspection by the next step uh this guys will be a lot more coordinated right and this is good because everyone's utility will go up if they're more likely to agree with each other and this actually keeps increasing with with every level of introspection I do okay so I do this for the nearest neighbors and then I do the next level which will take into account the next nearest neighbors and so this there's actually a nice simple formula that you can write for this so so let's say this is so the the second game for perfectly rational players will increase and increase the more levels of introspection everyone does so let's say this is the probability that the strategy I of the player I on the player two steps away uh for the second round of the game the probability that those two players will align in this strategy that they will be either both plus one or both minus one uh and this is like for the end level of introspection that they do so if you look for example at the at the first round of the game where they're just doing equally plus or minus one the probability that they align is just one half because that's just randomly they might align they might not align there's a one half chance chance that they are aligned but if they are doing this perfectly rational introspection level by level this probability of alignment increases with every level of introspection and this is a little formula that can be derived for it so this is the probability they had for their previous step and this is the actually this should be squared here anyway so you can calculate what is if I do one more round of introspection what would be my new probability of alignment and you can calculate and it keeps increasing increasing and increasing so the idea that is kind of cool here is that in principle perfectly rational players can reach perfect coordination by just the second round of the game by doing this noise introspection with zero temperature there's a caveat to this which is okay so they can do this but kind of this is a computation they have to do so to do more and more noisy introspection this is like kind of a calculation you have to do so and you kind of need infinite time to do this infinite introspection so it's kind of a the more you do the more computation you have to do but the more realistically you will talk also about okay so they can play twice and how much time do they have before they need to play again and that would limit how good they can get so that kind of how many levels of introspection they can do right so this is for perfectly rational so the kind of cool thing here is that in this game in principle with an infinite amount of time between plays they can perfectly coordinate by the by the next round of the game but then if you introduce temperature right so if you assume that not everyone is perfectly rational then you can do this a similar kind of kind of procedure and you will see that this this probability of coordination will be slightly less than this premium formula that we have here and again a similar thing happens that if I reach a certain critical temperature then actually no coordination is possible at all by the second round right so it's a perfectly rational plays players will perfectly coordinate by the next round if they're less than rational they will less than perfectly coordinate but if I take this noisy enough then it's impossible to coordinate basically no matter how many rounds they keep playing they will just not coordinate uh so that's uh yeah I don't know about that. Axel on the on the initial case where the temperature is zero so the good strategy do you think would be after the first round just pick the majority so would that like if you do infinite introspection the majority doesn't really matter to you because the majority may be far away from you so there's n players and the majority were pointing where plus one but my neighbors were actually minus one then I should I should try to coordinate with my neighbors I don't care about the majority. But doesn't the infinite introspection mean that you are going like to the neighbors of the neighbors and the neighbors of the neighbors and so in the end the limit should be the majority. It's kind of it turns out agreeing with that so yeah kind of but it's uh I mean it comes about in a more local way so I mean so in the end like something has to break the symmetry so for the first round everyone is equally symmetrically plus minus one but uh but by the next round that symmetry must be broken right so everyone must be plus one or minus one that will I mean probably that will agree with what was the majority thing in the first round. So just to summarize here what are the main points here so guys I guess so the we're looking at the like the game theoretical analogy to statistical mechanics where you have thermo fluctuations and WP this n level noise introspection then we uh just look at so in this talk we looked at a particular simple model which where we can calculate things and it's actually rich enough that we can see a couple of nice phenomena and that would be the what we call the easing game. All right so then it's things are a bit interesting so like so unlike in the physical 1D easing model the the easing game actually has a phase transition at some non-zero temperature so that is a phenomenon that happens in game theory that is not happening in physics and even weird there is that this is a zero order transition phase transition that definitely is not happening in these physical models and so this is this is weird right so and then uh yeah so the other thing is right so if there's no external stimulus right and they just play the game once everyone is just playing randomly plus or minus one but perfectly rational players would be able to perfectly align by the next run of the game uh but yeah so uh less less than perfectly rational players will will align less or will have less probability of of aligning as well as if they don't have enough infinite time to do this infinite computation right right and yeah so this above is true that they kind of are getting closer to alignment unless they pass a critical amount of of noise and then they just will not align by the second run that's really cool it's so nice to see this i was going to ask you a basic question which i i didn't know to interrupt earlier but i'm just trying to think like how do i map this onto a situation in filecoin and the thing i'm kind of wondering about is here there's an there's an assumption of locality right so like what is like a mapping between like things that i care about in filecoin to i mean it's concept of locality okay let's reduce it a bit to let's reduce it to bitcoin maybe which is a bit less i mean in a sense that that i mean it's a simple thing i want to let's say i want to make transactions and pay for these transactions right so uh right so so what bitcoin is trying to accomplish with an external incentive is to get right so they introduce this incentive that is artificially motivating people to do this right so it's it's motivating validators to validate transactions and motivates people to use what but kind of this incentive is going away right so in in the end so what we would like is for it to be a stable point where it would be a good strategy for people to make to do this exchange is not really where people are making their transactions and paying validators transactions fees for for that so people were not doing this before i mean people could have done this with uh i mean so it's kind of ipfs versus filecoin so filecoin existed people could i'm sorry ipfs existed people could have started making kind of our deals for to store data and okay store my data and i'll give you a money for it and so on but they are not naturally doing it also can i first can i push them to do it so can i align so this is i mean this is of course a very very simple toy model where we're talking about plus and one but naturally those plus and ones are just being plus and minus one or not aligning i introduced this external incentive to make it make them align so let's say they could be doing those those transactions and paying for them or they could not be doing that right but i want to push them to do that so introduce this incentive and i pushed the people to make this transaction market to make this data storage market but i would like to for this to be stable if i remove that incentive also that's so and so this is kind of the question so i mean first thing is that yeah so people are not perfectly rational so so i mean there will be some level of noise so the question is i introduce these incentives in in bitcoin that make people make these transactions uh you introduce these incentives in filecoin that make people do uh data and so on uh will it be stable if i remove those incentives so can i use the incentives to kind of push people to do a desired activity and then remove the incentive and they will keep doing that or if i remove the incentive just irrational noise will be too strong to destroy this and people will just stop using bitcoin or filecoin when the incentives go away so is the graph over which the sun runs is this between storage providers or is it between storage providers and their clients yeah i mean yeah i mean i guess uh that's also for this case is why it's like kind of bitcoin is easier problem because there's fewer things you can do right so you would have people that want to make transactions and people that put so different so you have two different kinds of players at one different thing so that so it's more complicated because you have two different types of utilities there's validators so validators would benefit from this artificial incentive plus from transaction fees and people who want to send transactions will benefit from the value that it has to them to send the transaction because they consider that value minus the transaction cost so they will but they will do that if if what they value that transaction is higher than what cost them right so kind of this like so block reward is so i have these two two types of utilities interacting and block reward is something that this is distorting what is the best thing for people to do so in a way so block reward distorts the validator utility function and that okay i'm gonna do this even if i if people aren't paying me because i'm just getting this external reward when they do that they are actually free to lower their transaction fees right because i'm getting block reward so i'll i'll just validate for nothing right so then that affects the the other users you know okay so so transaction costs are lower so i can yeah this i it's worth it to me to send this transaction but if i if i remove the block reward it does affect propagates everywhere and so so even if if if at that point that the effect propagates everywhere even if players were sorry even if we would if it would be the best thing to keep using this transaction system maybe just a little bit of irrationality will push it to not being the best system okay interesting thank you thank you i was just gonna ask one other question very quickly so like what would be a really interesting result seems to be if i understood correctly if you've got something at finite temperature and you buy some incentive fields to align people's behavior that's fine and then you take away that incentive field you're saying that the system will remain polarized is that that right yeah so that is so i mean for me like should we take away any like practical ideas for this i mean it's i mean the ideal thing is okay so it's a what the eventual thing of this okay so we'll formulate an actual model of bitcoin or whatever where we have these utilities of of users and these utilities of of validators and we'll try to formulate it in a way like this but i mean that's a more complicated model than plus and minus one but but there seems to be something saying there that if you apply incentives temporarily this can be effective in the long term yes over interpreting so yeah it's over interpreting that this okay a simple model says that but it's encouraging and that simple model says that so it's kind of cool that's the best i can say it's kind of cool and it's also i mean what is kind of cooler is that this is even stronger than in physics so because in the in the 1d physical is in model this is not stable upon removing the external field right but kind of this introspection thing is more this stable point this table face is more stable than in the statistical mechanics so that's that's encouraging for these simple models yeah so i mean it's nice i'm not going to say statements about what that means for falcon or bitcoin but but it's encouraging that it gives a hope that oh maybe maybe crypto economics can work that's kind of the outcome here yeah it's really cool thank you yeah and by the way that's crypto economics is j crypto economics is j to turn this into a crypto icon talk i mean by j i mean this external incentive crypto economics done this is crypto economics nice yeah so this to extend the analogy further this is like like applying a magnetic field but what about if we dope the system or like is there any other ways we can warp the field i mean how do we extend this yeah i mean there's yeah i mean there's as many things to modify this i mean what my criteria here was like was the simplest thing i can think of and then let's see if that's calculable and and hopefully that's rich enough to show me something cool and i was lucky that it was but of course we can keep complicating this and we can add next to nearest neighbors we can add in homogeneity right so like the values of this constant can be different side by side and things like this yeah many things we can add different types of utilities so we can add higher dimensions also so this this can also be done i align all the players in a line but you can also do to the it's actually yeah i i'd really like to see this for a lot of the token versions or if you insert some agents who have a fixed polarization and they're always always aligned in a certain way how does that affect the state globally of the whole system in this kind of the introspective recursive thing that would be like really interesting as well anyway that sounds probably doable yes so yes it's a very simple model but i mean there shouldn't be a phase transition between this and falcon so it's like uh i want to have a more complicated model accounting for that but but this is this good this is this is a formalism that you could use to understand the falcon thing without much more complicated model that you're not going to be able to solve exactly like this one but i mean the same is true in physics like this is really like this in model because you can play a lot with it but real systems you want to describe can be a lot more complicated than this in model but that doesn't stop people from first trying to understand this in one right it looks like we're about at time i can stop the recording unless anybody else has any further questions this stuff thanks so much excel very interesting