 All right Let us continue and the second talk is given by Sebastian van Strien from the Imperial College London And he will speak on fictitious games a class of peaceways smooth dynamical systems with very interesting properties So the reason I've started looking at a dynamics of games because I think there is a It's a need to look at the wider class of systems in dynamical systems This particular class of systems has been around for many many years since the 50s But I believe has not been studied significantly From a geometric point of view the many many papers on this Systems maybe in a hundreds of papers in the economic literature and the class of systems is very simple So your state space is the set of probability vectors in Say rr and rs. So you have actually two players in the game theoretic session setting and the state space Of this position of these things is a probability vector in these state spaces So you have a factor in This thing and so for example if I take the probability vectors in R3 These will be just a simplex and I'm going to represent that here in the triangle and If you take a point in These in this pair of state spaces. So this is the state space in this case It's four-dimensional. So you have two times two-dimensional state space Then the differential equation you're going to consider is that at every given point You're going to walk to some best response vector And actually as it happens this best response vector will always be sort of one of the corners of your simplex so if I'm here and here I will Go to somewhere and where I will go here will depend on this position So this player will look for this position is and we'll go to one of those corners And I may decide to go to this corner So that's the direction you're going to walk it towards and then this player will look where this one is and may say No, I'm not going to that corner, but I'm going to go that to that corner So that's the best response and I'm going to the next slide to give a precise definition of that But it's a very simple thing you're walking in the straight line And then at some point when you're you're told to go somewhere else you go somewhere else and I'll give you examples in a moment So it's just easy to write PA as a row vector and PB as a column vector And then you have payoff matrices a and b and you have payoff The payoff that you get at any moment is simply the this product is matrix products So that's the response that the payoff you get and the best response is simply saying well, I look at the vector That player B is in that's the position you're in and you're looking at the answer And what you're doing is you're looking at the the action that will give you the best response. So player Player will for example do this And Now what you will do here is to say well, this is a vector Consisting over a number of components and what vector here will be the best response Well, if this one is larger than all the others then you say well actually my best response is taking the first Component and that will be done that you will move to the first component And that's the best direction you're moving towards. So this is the The vector P That makes this the largest And of course this could be set values because for example, if you have here something like point three Point four point four point two then of course, it's ambiguous which those two would be the biggest So that's actually in simple terms the differential equation. So you have this differential equation with this Expression, I will give you an examples in a minute and the Nash Equilibrium is a sort of a notion that comes up a lot in In economics, but the short definition is simply that's a Position that's a best response to the position the other players in Advice for this one is a best response to the position. You're the one is in That's just the definition of best offer Nash Equilibrium and in the economic literature It's sort of written in words, but let's skip that and So in the 50s This differential equation was proposed and the reason it was proposed because it would be a natural way to find The Nash Equilibrium so two players would then naturally sort of converge to an end to a Nash Equilibrium if they followed this rule and that is not actually always the case but Anyway, it's sort of like going to a zero of this differential equation now, of course I mentioned already this could be a set So this is the right-hand set is set valued and of course, it's not even continuous So this of course means that it's kind of Not even clear that you have existence in uniqueness. In fact, you don't but in fact This right-hand side is lower similar continues. That means the solution exists But maybe you don't have a flow property and so on so they're all kind of Issues but in many cases actually or in the cases that we will study You will still have uniqueness and continuous of the flow in some cases and as I mentioned is a very large literature on this class of systems and There's also a strong relation between this and other systems, which I studied a lot like replicated dynamics, which I studied a lot in biology for example and So just to give you a flavor suppose you have two matrices a and b and Let's see what this means then So first of all, what is player B? They're going to do he's going to Look at the largest components of player a Because what is this here? P a And then you have this matrix P a B and of course B was chosen speed identity. So this is PB and What's the best response you can be a P a what's the best choice you can make here? Well, you just choose the largest so player B is trying to copy what the player one is trying to do That's the definition of this best response and so it also uniquely defines what the equation of the game is of the What the dynamics is so once you give this matrix B and a the dynamics is determined So player B is going to copy what player a is doing So if the first component of a is the largest then it's that's where you will head That's what I'm going to explain now. So let's see. So if you have in the player B So play B if the player B is here on the right If it's here on the right then the player one So if player a is on the right Then player B will try to go to the right Well, here you have a simplex. It's a left and the right. So this is in this case that your simplex is simply probability factors in R2 and I can identify that with this interval zero one and So in that sense, it's just saying well if I identify the sigma a and sigma B in the same way it's simply saying well if the second component of The vector is the largest For player a that that's where player B will move and Player A will do exactly the opposite because he's saying well I don't like what the first player is the first component is the largest Minus of that will be the smallest and so he's going to do the opposite So we could sort of draw it in this way But in this particular case, it's also quite convenient to draw it in this case because here we have a state spec sigma a and Here's state space sigma B You're indifferent if your Vectors have equal components. So if this fact is half-half and here's half-half then you're indifferent And if one of the components larger than it will move here So for example on this part sigma B has this part as a larger component and sigma A will say well I'm the first player will sort of move to the opposite side This part is sort of doing first component most so this one will go in the opposite direction And if you put that together what will happen is that you move to the corners in This kind of alternating way and then you move to this corner, and then you move here And so this is the dynamics you would get So and that's just the definition that you will will obtain from this and of course here left and right It's always you know it takes a while to say what's left and was right But more generally if you take a three by three game then you have certain lines where a certain component Is the largest here for example if player B is in this part player A will move to what's the first component He will move to the third and here to the second component So player A is just looking where the player B is and depending on which part of B in this triangle Positions Play a will move to that particular corner. So if player B is here player A will say I'm going to move to this corner That's really the the situation here. So So they're divided in three parts and that's here in three parts and Depending on which part is the largest So for example here, this will determine that maybe The player will go to this one and this play will say well I'm moving towards here and then look at the labeling here and there's play one and so it move also to play Actually, if I draw this particular copy of drawing wrong because here should go to corner three Because here there was a label three To one so this tells you that player the first player the player a will move to corner one and Go to three and corner three was this one. So that's the So the dynamics is very simple. It's just going in straight lines it's not devalued when you're hitting these things and So here's a bit of the orbit here You can see that the starting point is where the star is and you start to move here Here moves the first the second player will move to the third corner and This one says well, I'm here in also moving to third corner But as soon across this line player a will be in this part and so player B will move towards this so the orbits are just piece by straight lines and you would imagine nothing simpler than that and That of course was the starting point because when I was you know So an economist told me this class of games is he said well It should be quite simple to analyze this and then there was this large literature But I'll tell you a little bit more about this but anyway when Yeah, there's you some game that means that a plus B is zero so the payoffs of the players are Complementary then in fact it turns out that the dynamics goes to this Nash equilibrium So if a plus B happened to be equal and then this you can see in this particular case also that it was also spiled towards the But actually the dynamics in general would be much more complicated turns out now and also Mark is that in fact if a and B and a tilde B tilde will have the same best response Then of course the dynamics is the same so even if a and B are not equal to zero if somehow the best responses for a comma B and a tilde B tilde will result in the same best response maps Then the dynamics will be exactly the same and in that case you also call them linear equivalents and A few non-zero some games which is known that the game converges But then the Nash equilibrium always turns out to be in a corner of a simplex And I'm going to study the case where the Nash equilibrium is in the theory So the Nash equilibrium This is clearly a Nash equilibrium because here you see that All three strategies are indifferent and So any vector will be in in that and here the same And that's the only Nash equilibrium of this particular game and there's a famous example from the 60s Due to Nobel Prize winner sharply Which shows that in general evolution does not converge to the equilibrium of the game, but you can have periodic behavior so What may talk will consist of a number of parts So first I will give a description of the dynamics and show that it can be quite more a lot more Simple than what you would know from a course on the French equations And also that the zero some games turn out to have a strong relationship with Hamiltonian systems and And also then I will ask a question Are there any non-zero games which still converge to an interior Nash equilibrium? And if I have time I will discuss also that actually most games seem to do better Players will do better if they use this dynamics than if they respond by Nash Nash is a very disc defensive response So if players would choose to both play this very simple dynamics their payoff on average will be much better than the Nash And so it's actually For both players a better response So let's look at the class of example. So let's look at this class of matrices, so you have this matrices a and b and Again the state space is just two triangles in R3. So it's a ball in our form and the Nash equilibrium is just this you know this intersection of these places where the two strategies are equivalent and Beta zero is actually the sharply example So when you take this parameter one place is copying and the other place always trying to play one the head so if I have a red shirt and The other one has a blue shirt The other one will try to copy my red shirt, but the other one says no he has a red shirt I prefer to go to the next thing so he will switch to the next Color shirt It turns out that from Peter's sigma In this game is equivalent to a zero sum game And so it's known then from this classical result that you converge to an interior Nash equilibrium and You know just to sort of look ahead a bit if you have go beyond this parameter sigma this called a mean and up to some other parameters You will have infimany horseshoes and and chaos But I'm going to show that it's more complicated this but let's sort of remember that you have this kind of dynamics and Effective role of beta is just that the the lines will change So here for beta zero you will get just These sort of division of lines will beat as positive They tilt in a certain way and we chose sort of a one parameter family simply just to make life a bit easier And of course you can analyze orbits very simply by just some kind of coding and for example here we we can call it because you have these three three Regions in your triangle for each of the place. We have nine regions So this is the region of our player a moves to the first second and third corner And he about player B will move to the second third and first corner and then there are some transitions that are loud and Some that will definitely not be allowed But this diagram doesn't indicate that any diagram you could choose will necessarily exist This says this is a possible path But it doesn't mean that all the paths that you could sort of possibly choose within this this diagram would actually exist And so this could actually be something like you know So there's some kind of coding in the kind of way that we're used to in dynamics And so for example if you have your one two three four five six That's a particular orbit and then you have here one two Three four five six and actually one of them corresponds to the sharply orbit which I will also draw in there But anyway the This is the first no basic result is that if you look at this family Then if you let beta between minus one and sigma Then in fact the exist period of orbits with this particular Itinerary and this is what this corresponds with the sharply orbit and which attracts an open set of initial conditions and For Peter beyond that say you have something like another orbit and Then you have this kind of anti sharply orbit which comes attracting and and Then effect when you have it beat between sigma one there's also an orbit which consists of mixed play So that means that actually in this kind of situation You have a pair of orbits Well, in fact in in the fourth space it consists of something like a triangle of six pieces of straight line one two three four five six but if I draw them in the in the Simplices They would correspond to something like you start here and maybe you start there. Well, this happens you move along this line and this moves here This one actually moves back on Itself and this one moves here and so on so you're only in different lines but of course these are projections of this figure in our form and so It kind of looks kind of hard to draw but it's just an orbit of is on these in different lines And he is similar so that orbit was playing an important role in our discussion and This notation simply means that you don't play one so you could be in different between so you're indifferent between two and three and And now of course you say well, okay That's it's great You have an attractor and then you probably have an umbrella also Well, actually the situation is already immediately much more complicated on that for example if you look at the stable set of the national equilibrium So what is that now? It turns out that this set is certainly not open but it's a counter union of counter reunion of of Of cones, so you have these points and It turns out that you have sort of orbits, which I draw schematically like this and you have a cone and The orbit will spiral along that cone to the Nash equilibrium and then you will have another region here and the cone and Or we will also spell around this So in sense the stable set is already something You know, we're usually more complicated than we used to and I had wallet theory. That's always a manifold He is not a manifold, but it's probably you know, I this is the only thing We've actually done but it's probably if you take the closure to some sort of counter set Object so already that's already a kind of Creating effect. We don't know but I think that should be doable that the stable set of this Nash equilibrium is an empty has empty interior although actually this conjecture later on I will sort of question that for other families So I'll come back to this conjecture, but within this family. I think this the only parameter where you would have this kind of So now note, of course, this diffraction is not smooth and not even continuous But actually it turns out that the flow for this particular family is actually continuous Except of the Nash equilibrium and I'll come back to it. You actually move there and find the time So it's already a lot more complicated So let me kind of explain that it's you know, what else you get So for example, if you take beat between zero and one you take some fixed n Then there are infinitely many periodic orbits of this particular period. I Mean, I will explain it in a different language in the moment or maybe I will do that now already So in fact this six n where I've drawn you one two three four five six So the six comes from there and what actually happens if I take a section here in This section if I take a point You may come back after after and steps So this may have under this return map for this section Here's this period of orbit if I take a section here, then this may be a pair to orbit of p. It's n and And so I draw a six n but that's just because of notation here before tell me have maybe two That turns out that you don't have isolated number of orbits appear to Have a whole sequence of them another pair another pair, so in fact the number of fixed points is Countable a number of points appear to is countable. So it's really It's not their periodic orbits of a certain period and not isolated So I'll come back to this picture in moments the other thing is that if you look at the parameter between sigma and zero and sigma Then there are many orbits that lie in this kind of vision. It turns out that these orbits actually reach a Nashic level in finite time They they're really just like in a differential equation can go a finite time to infinity here they will go in a finite time to zero and And For beta between sigma one this period of orbits that I mentioned there actually part of some horseshoes You really have some horseshoes that appear. So if I look at the first term of this On this section. There is actually this this will have horseshoes. You can construct explicitly that this has Horseshoes or in fact I will come to another thing that this also has some kind of random walk phenomena Which I will discuss And I believe that actually in this whole region you don't have periodic attractors So you have true chaos in the sense that there's no no We know that horseshoes, but we don't know that necessarily there are no attractors of high peers, but we believe there aren't any and So here's sort of this diagram again with the itineraries So for example these non-essential period of orbits to the following they follow the Shapley orbit and then they go a few times round This kind of decoration and they go here in six and so they move so they can do some extra stuff and Well in chaos in the sense of strategies. I mean, this is what of course economists really are thinking here You move towards three then you move towards one then you move to what's two This is really what you kind of aiming towards and these sequence are totally You know random in the sense that this is just Some sort of a shift space and a substance of finite type which are in silence and Here for example is here the orbits are draw here's for example the Shapley orbit here start at this point here and immediately move You know you start as it happens here You move towards here and as soon as this point reaches this line This one will Change direction and as soon as you don't hear this line this one will be here and the move towards that line So you just move in and then the Shapley orbit is this kind of densely drawn blue thing which is kind of where you converge to and If you take a parameter beyond this golden mean then the orbit will be sort of chaotic and you can Let you see also pictures in a different display So I'm going to move directly to another setting Just to Zero some games because that was for me there the Canada first surprising thing that actually there's a connection between zero some and Hamiltonian systems which actually nobody had noticed before and Well as noticed before each orbit will converge to the equilibrium That was known already, but now you could also ask how do they converge to equilibrium? So what you could do is Actually the following so I'm going to draw this chromatically have this Nash equilibrium, and I'm going to draw a sphere Around it and I'm going to ask, you know, there is this orbit and it will converge to this But actually as it happens this dynamics has a well-defined Projected dynamics on the sphere So you can actually project the dynamics just using spherical dynamics and that way you get dynamics on on the boundary of your space And what's the benefit of looking now that it just tells you how you're going to go, you know, with it That's the spherical direction. So how you may still switch a lot of Directions while you're moving towards us So for example in a two by two case which I've drawn here if I draw the boundary here, you would just get Circular motion you would go like this and so yeah, it's just like rotation in High dimensions it turns out that something similar happens. In fact This dynamics turns out to be Hamiltonian Provide you take a non-standard symplectic structure. So let me kind of explain a little bit more What I do so for example, if you take the Hamiltonian of this form So this is just something related to these matrices. So that actually is just the the level set of your Hamiltonian and this is sort of like that kind of like the energy or something and This is as before then if you take this as your level sets, then you get actually Projected on this level set the orbits will go actual orbit will converge to Nash, but there is actually if you identify this With your level set and on this set. There's a projected dynamics, which turns out to be Hamiltonian and So let me say a little bit more about that because well, I mentioned already this is a sphere made of hyper planes and In fact the Hamiltonian flow Turns out to be just a continuous flow. In fact, not just a continuous flow a translation flow. It just is translation So it's actually in that sense also it seems to my mind also something that should be studied Translation flows and higher dimensions In this case, so it's a piece vice a translation flow and it has no stationing solutions and Well, as I mentioned it's sort of Hamiltonian in the sense if you take this under the standard Supplective structure would get this but you have to take a non-standard supplective structure to obtain the actual dynamics that you want to get So Yes, the flow is continuous on the on the effect is continuous in this particular case It's continuous everywhere because you're some did then the flow is the continuous. That's one of the things we show also and And so the Hamiltonian flow agrees with this geographic projection so from the Nash equilibrium on on this on this field and and this has actually quite So suppose you take for example the mages I took before with the parameter beta Picking this called a mean which is supposed to be one which was equivalent to zero sum Let's see what happens then well. I mentioned already you had this particular orbit gamma So gamma was this orbit here this Orbit where both players were indifferent and One of the funny things is that actually this orbit usually you take a section Transversal to your orbit Effective is also section which is spun by the orbit. So it's a totally different section and Let's call this B or something And it turns out that's a global section for your dynamics In your three-dimensional sphere orbits from will come here and then come out and come back here Now, of course, this is different from the usual setting because if you come close to here In fact orbits will spiral really close by It's not as you would expect from a from a differential situation where orbits are more or less Parallel because if you're close to this situation here orbits will zigzag zigzag here and the speed in the in the direction perpendicular to your You know to the line is as fast as here So you have this kind of direction you're disparling and the closer you are to this thing the finer this parallel is So the first term up to this will extend as I to the identity on the boundary So it's quite different from the usual thing if you have a Mechanical system and you have the what's called the the the cipher to flow or something, you know with all the Ventura, that's not the situation that you have here So what I'm claiming here is the following look at the stable manifold of this This is very of this periodic orbit and now look at the points that converge with a certain speed to your set So here you want to go to a certain speed So normally when you have manifold of a certain speed, that's just a kind of a particular manifold Now here you may go at this speed and here may go somewhere else at some speed And so that's the the hyperbolic picture here. It's quite different for every choice of this speed close to zero There will be actually a set which is not empty So these are actually kind of you know from you can go at any speed you want to this set gamma and In fact, if you think about it, it's not so surprising because in fact if I drew this Section S again, I could actually draw here also an ally or the fact they won't be an ally But something like an ally and in fact if I look at the this is my section S So that's the thing that again is transversal to your orbit and I look here at this first return map in this way Then in fact, it's like a random walk This point can move a number of steps further out and number further steps in and there is some kind of Allowable sequence that goes You know a few steps out a few steps in a few steps out few steps in and so if you think about this way It's actually not very surprising at all but from a you know abstract point of view of course, it's quite and well, this is another way of Representing this section that I've drawn here here. I've drawn this kind of nicely in the plane But of course in force space is how to draw But you know, this is roughly how these things this consists of a number of of planes and And the orbit sort of will come back in and so on and actually on the next one I draw it more like the shape of a Pac-Man, you know here you have this kind of The same thing with these two wings Remember here you have these two wings and I'm going to put them more flat and just a different presentation And so here's here you have your orbits so what you have here is Well, what do you have there? So let me draw that in this roughly this shape Well, what you have here, of course is it is a is a torus So you have a two-dimensional torus In this three-dimensional space and in fact this whole space this whole thing is foliated by to write So that's just an elliptic region where in fact the orbit is appeared to and it's coming back And I have still ten minutes, right? at least almost and So and then here that seems to be in a region where you have a completely a gothic Motion here, which is something that I haven't been able to prove but so you have an elliptic part and and a sort of a gothic part and So one question is can there be every elliptic orbit sort of inside here Or is it really sort of? fully And Well, I've drawn this already. So here is for example line the image of this line is actually sort of an infinite spiral So it's not of course a smooth map this first to turn map to this section which I draw here as a Z is actually is a sort of a piecewise smooth object and so it's sort of like a spiral and And that's actually the reason why also the certain lines when which the Accountable number of fixed points and there's other lines on which a countable number of points appeared to So now I'm coming to the next question. So so there was a question that actually Rows in the following way. So let me Go back to this for any matrices not only if you have two matrices that are Zero sum as a pair But also if you take any matrices, you can arrange it. So this has level sets, which are homomorphic to spheres You can find equivalent games with the same best response so that this actually this will change this As such but that so this will be homomorphic to spheres and Then of course you could say let's do that in general not just for a zero sum game But let's look at the dynamics in a spherical You know in spherical coordinates So you take the the distance to the Nash equilibrium, which you'd parametrized by this H So H is zero will be the Nash equilibrium. And if you Have here then it's kind of the distance to the Nash equilibrium and the spherical coordinates So you take coordinates on your on your on your level set And as it turns out if you take a time parametre reparameterization Where you take it? New time in this way so you so speed up depending on the distance to the origin Then you get this particular equation here. So this is the part That lives on these level sets So on these level sets, that's what the spherical dynamics is going to do You see the spherical dynamics has as is not aware of what is happening in the radial direction So it's the spherical direction, which is doing whatever it's doing and then there's the radial direction and Of course as you as you can see here also, you know, because this is a compact space The solutions will exist always for all time in this new time in this new parameter Precision tau so On the next slide I will copy this and explain what this alpha is but anyway, so the what I'm doing here I'm just looking at spherical This is the spherical part in spherical coordinates these dynamics and that's just rewritten in a new time parametre station Oops So here's my System again and where alpha is actually defined. Well, let's see what this is Alpha minus one is equal to something and this something means a plus b and This is looking at the best response of a and b So it's like the the payoff that they jointly get, you know the payoff of a plus the payoff of player B and n times the is the dimension of the sphere we're talking about and So in fact if you define this a of tau in this way So at a particular moment you look at this number alpha minus one, which is this object That's the divergence of the fear is spherical of the factor field But now only considered as effect field on the spherical parts So I just look at the spherical direction and there I look at the diversions of this factor field and So that's actually the spherical direction and then I can look here at the isotopic volume expansion along an orbit and So I'm going to define here laptop plus and laptop mine minus So these are something like exponents, but actually in terms of your volume expansion rather than in sort of So it's not really like the Lyapunov exponent with volume expansion And then you have a sort of a result that says if laptop minus is positive then you then that particular for one particular orbit, then you go with Roved tau will be of something like this and in terms of original time That will mean actually that this orbit will go in finite time to Nash if laptop minus is negative then this orbit will definitely not go to Nash and It will actually sort of keep it a positive distance and where n is here the dimension and and so this means also the solution exists for all time and If laptop is great equal plus plus plus is great equal zero then the limb in will be zero and If this is negative then solutions exist for all time, but they could also lay between a final value and and zero So in other words what happens in the radio whether or not an orbit will go to Nash equilibrium Can be read off for what happens on the spherical part? That's what this simply says and for example if the induced flow is volume preserving Then solution will necessarily go to the Nash equilibrium in infinite time now Why did we we look at this? Well, the reason was that actually there is a conjecture by one of the leading people in the subject half power and Who says this is one of the main open problems in in this emotionally dynamics area, and it says the following Consider the dynamics we have been looking at with a unique interior Nash equilibrium and Assume that this Nash equilibrium is stable So in a sense that all orbits near Nash equilibrium will go to the Nash equilibrium So assume that then the assertion in this conjection is that the game should be equivalent to a zero sum game and I will sort of explain why that may not be the case and so I'm hoping to construct a count example and One possible scenario for a count example and I've done some kind of numerics I mean if you don't know where to look Then of course, it would be very hard to find examples, but Think of the following scenario think of the scenario where you had this this Picture here and I'll draw it just like this and here you have these ellipses and Here is this indifference plane and so this this first term map is piecewise affine It's continuous and piecewise affine on the boundary its identity and It could easily imagine a situation where in this part the dynamics is area contracting So what would happen then so what would happen then is that well air contract means that You know points here, which is close by is still going to follow roughly this circular motion And so it will it will contract but for many times here it will actually then then Not contract area at all, but then it picks up a bit of air contraction But it takes more and more time because it's going to be slowly converge to this parent orbit And slower and slower it will pick up this air contraction and this is precisely the scenario that was kind of Need here if you can find something where you where you have This laptop plus or maybe love the minus is actually zero in perfect shape He have an attractor, but it's not hyperbolic And then what you need in addition is that there is no other attractors somewhere else Propelers are fine in terms of area because they will converge to this Nash equilibrium, but Attractors are definitely not fine So let me just You know sort of close this by saying well I feel this is a quite an exciting area There are not many people looking at it from a classical dynamical systems point of view or a dramatic point of view And so you know one of the reasons I give this talk and so why I enjoyed it because it's sort of although There are lots of papers on it. I think it's actually Unexplored area, so I hope some of you will join the fun