Cohesive Games and Lessons Learned from the Field Theory of Games

My name is Jerry Thomas. I'm an adjunct professor at Milwaukee School of Engineering. And my talk today will be on cohesive games and what I've learned about teaching the field theory of games to MSOE students. Hello. Good afternoon, everybody. My name is Jerry Thomas. I am from the Milwaukee School of Engineering. I'm an adjunct professor there. My background is theoretical physics. I spent my postdoc at cern and I talked to a lot of guys that were working on string theory and field theory. And so that's kind of where I got my initial start about thinking about field theory of games. Although I wasn't thinking about games at the time, I spent some time in industry over 20 years and got involved with a lot of strategic thinking. And it occurred to me that I could maybe apply my knowledge of mathematics, not physics, because I wasn't going to try to make an analog of physics to games, but I thought I could use the mathematics that I had learned and apply that to game theory. I started working on the field theory of games and got help from Wolfram and published a couple of books on it. But I thought to make the next step, what I wanted to do was teach a course to undergraduates. And the challenge was the undergraduates that I could teach the course to were engineering students. They were not math students or physics students. And their background in math was somewhat dodgy and their background in physics was certainly nil. And so that was one challenge. And the other challenge is that schools like Milwaukee School of Engineering did not use Wolfram at all. And so my first challenge was to get the license there. So I've taught the course a number of times there now and, and I would like to tell you a couple of lessons that I've learned from teaching the course and also some lessons about what it's taught me about where I want to take the field theory of games next and what some of the challenges are. So the goal of the project, which is a multi year project, which I started at motorola in like 1995 or something, when I had nothing to do when I was sitting in airports at different parts of the world, was I wanted a theory of games that was deterministic, not stochastic. When I ran this idea behind people that were game theorists, they thought I was stupid because everybody knows that games are stochastic. And I said, no, no, no, that's like saying ph is Brownian motion. That's nice, but that's not the best we can do. And they said, well, tell me something we can do. And that was the problem. So I thought about that for a long time. So I made a lot of progress since then. I'd like to provide students or general public with tools for how to do games with a deterministic process, but somehow gloss over some of the mathematical complexities. And for those people that want to do the mathematical complexities, that's fine. And I can certainly point them to the books and the research for them to do that. The approach that I took was to create a toolkit that I could give students. I would teach them elements of the toolkit as we went along and learn the course. But if they didn't learn the toolkit or were they asleep when I gave that lecture, that's okay. The toolkit would guide them through. I learned this from industry that you don't have to. Everybody on the team doesn't have to understand. Understand everything. Somebody on the team has to understand something. So if you want to follow along later, and then you have to download the toolkit, which is up on Wolfram, and there's some initializations that have to go through. And my first challenge with students was simply this step. What is Wolfram? What is initializations? How to do the language. So I just introduce them to Wolfram's little book on Mathematica. And I would just start working through what's. How do I do one plus one is two, and little simple exercises like that. So by the time I got into the second or third week, I could walk through this stuff and they'd go, okay, that's all right. I don't understand it, but good. And I can get them to, okay, what is a game? And why do we talk about games in the first place? Because understand engineering students have no idea what game theory is. So I had to start with games. And I took an example out of a really nice little book that I found that was by an author that worked at the Rand Corporation. The model is very simple, but understand mathematics usually takes some very complicated thing and simplifies it. And here's a game that's really pretty simple but actually applied in the real world in the 50s, you'll understand that there were blues and reds, and you can understand who they were. And so here's a game where blue is going to defend two installations, and red is capable of attacking one or the other of the installations, but not both the installations have different values. So I can get the students thinking about that and get them to. This is a part where they can, whatever their background, they can attend this part of the subject and they can come up with their own examples. I make them come up with their own projects around what they happen to like or know. Including Pokemon. I got a couple of Pokemon games. Don't ask me how they did that. The more common game would be games on the Internet where they could say that's interesting. So at this point they've understood some Mathematica. I can talk them about game theory because they don't know game theory. So I have to teach them stochastic game theory. And since I'm not going to go through the mathematics of game theory with them, we're talking about linear programming or how to solve the things. It's all going to be done in Mathematica. They don't have to worry about that. This simplifies game theory down to about a lecture. So in one lecture they're able to identify a game matrix. So in this game it's a pretty simple matrix. Blue has four, 1, 3, 4, red has four, 1, 3, four. And game theory says there's a strategy for figuring out which. There's a way to figure out what is the best strategy for each player. So they get to the point where they can do that. And the strategies for blue are one time they'll do the minor target 1 to 3 over the major, the attack will be 3 to 1. The other way around it's a little anti intuitive which is you attack the target that's least valuable, but that's because you understand that the enemy is understanding that you're going to attack the most valuable. So you compensate. This is understanding games. And I explained to them how even this simple idea was not understood by the US military up until in the 90s. And they finally figured out don't talk about what an enemy you think they're going to do. Talk about what an enemy is capable of doing and set your strategies that way. That was a revolution in thinking, not a revolution in thinking worldwide. I think it goes back to China and some say in the art of War this is how you think about it. I'm not trying to teach them the theory of games. So I said, well there's a transition from the theory of games to broaden the perspective a little bit and think of a two player game as each player can play their strategies and the other player's strategies. That's a dumb idea, right? In order for that to work, each player you have to introduce a hedge strategy. This very simple idea was used by von Neumann in the early 50s as a way of reformulating a game so that rather than having to compute two separate games and figuring out what's happening, you can compute this composite game. Now the observation that I made and von Neumann made is there's a relationship between this form of the game, which is an anti symmetric matrix, and some other theories in physics, like the 3 of electromagnetism. Theory of electromagnetism has a magnetic field and an electric field, which is hard to understand, but you can combine them into one super matrix, which Einstein did, and hey, voila, you have an anti symmetric field, which is a nice way to solve the problem once, as opposed to two separate problems of computing electric fields and magnetic fields and trying to look at their interaction. So in this case, the 4134 shows up as the sort of the magnetic field part and the electric field part shows up at the edge. That's the hedge. Values are chosen just so that you get the original game back. So it's very clever. But you can make a theory of this and forget game theory. And this could come out of a field theory of games, where this is a natural tensor that comes out from the theory. So of course I don't tell them that. Well, I do tell them that, but I understand that that's the part where they're asleep. So once you have this, you can talk about not static behavior, but you can talk about the way in which behaviors change over time. They flow. And I can have a magnetic field and particles spiral around in the magnetic field, right? Well, games you can think of as points in space that spiral around that matrix. It's not a magnetic field, has nothing to do with magnets, but from the mathematics in geometry, that's a thing. It can spiral around. That was already understood by von Neumann. He just looked at it differently and said the way to talk about where the stable point is is to do linear algebra in convex spaces. Very complicated way to say it, but exactly the same. Once I can talk about them spiraling around, we can talk about how they behave in time. This is a part which I had a lot of time struggle with my students. They want to think about this spiraling around in time as simply an approximation to get to the answer of nothing's happening, it's stable. The analogy is if I was looking at building a bridge and I thought about the only property of a bridge is the fact that it doesn't move. I'm missing a lot of the physics. If an army marches across the bridge in step, they can break the bridge. In other words, the oscillations can generate harmonic motion. And that's very important, right? So dynamics is very, very important in game three as well. This is the marching across the bridge in step. And if you had the right forces, this could generate harmonics and maybe do some damage in behaviors that you would see in real world problems. Imagine that you have forces which are harmonic in the market. Big problems that occur over and over again. You can see that that can generate forces in society which get out of control. This is a theory that would allow you to talk about that. It's something that the students could wrap their heads around. So at this point, I say, okay, all we have to do is compute this. This is where it all falls apart. Because what's the dimension of space I'm working in? Well, I had for the attack defense, I had two dimensions for blue, two dimensions for red. That's four. One dimension for time. And then there's a couple of other dimensions which are associated with each player. Each player essentially has a worldview that's like a strategy. So I'm up to seven dimensions and oh yeah, I have time, right? I added time, so that's okay. So seven dimensions. I talked to my friends Wolfram, and I said, can you solve me a nonlinear nd solve equation in seven dimensions? And the first time I asked this, I asked it, can you do it in three? And the answer was no. But now we can do it in three. I'm perfectly confident then in a couple of years I can do it in seven. But what do I do in the meantime? Well, I go back and ask in other areas, are there similar problems? Well, in fluids, which this is a lot in common with fluids as well, the flow is a complicated behavior. It has viscosity and things like that. Very complicated, right? So to solve the equations, if I'm not very strong, what do I do? Well, I watch streamlines. What is the streamline? You take your air and you put a bunch of dust in it and you watch the little dust bolts. That's a streamline. The path of a streamline is easy to calculate. I don't need in seven dimensions. It's a one dimensional equation. So I can write one dimensional equations, maybe two dimensional equations, and we can get answers. So I said, okay, you now know how to solve one dimensional equations. I give you the toolkit to do that. Let's figure out how to solve it. So, okay, we walk through some solving and I say, look, voila, we get some solutions. Here's a two dimensional projection of what's going on in the seven dimensional space. And one of the projections is essentially the streamline for player one, and the other is a streamline for player two, whatever that means. It's two dimensions. And what you're seeing is a plot of the flow vector value, which means the flow is stationary in time, but it's not zero. Have you ever seen a tornado? You see the tornado is whirling around. If you look at it, the little dust particles form a certain stability that's stationary. You'll get the spot on Jupiter, who just had an astronomical tuck. It doesn't move. What is it really doing? Very complicated turbulence. This is what's going on. Stationary. But it's actually giving you some time. Dynamic dependence. So good. And we can look at different things. Okay, so I have to go through and make different kinds of projections of what's going on. And I say, okay, very nice. And how do I take a step back and what do I want to teach students that's important about game theory and about the way decisions are made without getting involved too much in the mathematics. You notice I'm skimming over a lot of complicated dirt, right? So one of the things that I skimmed over in talking about games, when I talk about, talk to game theorists and they tell me I'm stupid, which is probably true, but not relevant. They say, where in this theory do you deal with the fact that you don't know what the hell is going on? You're talking about two players, assuming that that's all there is. But they're in a societal context. There's other forces going on. Your information is imperfect. And so they built up this entire superstructure of how to deal with games with imperfect information. They've spent, what, maybe 50 years doing this. They said, okay, physicists have spent probably since Newton dealing with the same problem. The way Newton deals with it is they say, yes, I have a car that's moving along the highway. I talk about one variable. It's moving in this direction. It's not doing anything. What about the 10 to the 26 other problems that I'm ignoring? The fact that it doesn't go through the floor or the asphalt. What businesses do is they reflect the other things that are going on as constraints. And they say constraints are still deterministic, but you have energy and momentum to conserve. So what we want to do in this theory is put those constraints in the theory. What are those constraints? I have two people that are playing the game, but it's in a context of lots of others. In the Cold war, in the 50s, red and blue was a subset of all the different things going on. There were other major countries that were being ignored, but they of course had some influence. They didn't have the major influence. So you want to take them into account in terms of the constraints. So how do you take them into account? Well, the theory says everything is the geometry. So there's a part of the geometry which you're ignoring. What are the characteristics of the geometry? Well, it has certain symmetry properties, certain eigenvalues, eigenvectors. If you can simply characterize those in a phenomenological way, you may be able to use those as an approximation and still get a very good result for the thing that you care about that the way which this theory will take those into account. Here's an example of taking into account one of those effects, the contextual scalars. If you think about what's missing, the tensor that you're trying to evaluate is the symmetric tensor. It has time is one. In space there are time eigenvalues and space eigenvalues. Time eigenvalues can talk, you can call in physics energy density. In geometry you call it just the curvature of space. Space eigenvalues. In physics you call the pressure or the stress. Here we'll just say they're the space eigenvalues. That's just the characteristic of the geometry, not of any, any analogy. So here's a case where we have some stresses with some gradients. And I like to sail sailboats, and so I know that whenever I have stresses, high forces move from high pressure to low. And therefore I created some stresses and that will add some forces that will change the behavior that we started with. This is the part that we didn't consider, the part that we don't know about. The fact that it's not a fully. The information is not fully determined, so we get different results. If you want to know more detail, my friends have helped me with a book, and in two volumes it goes through all the stuff and tells you where all the toolkit elements come from. But for the talk, the point is there are things that you can compute. So at this point, I've done what I can do with my students. The next step, of course, is for me, how do I deal with these stresses, how do I understand stresses in some more detail? And in fact, this leads me to some idea of thermodynamics, believe it or not. And why do I say thermodynamics? Because I don't have a better word for it. What I do have is in economic systems, business systems, this system. There's a sense of what behavior is that stationary, that steady state when all the transients die out. There are certain attributes of steady state which you can articulate. Well, thermodynamics is one way of talking about what is a steady state. And so what are some properties of steady state? Well, typically, stresses like this are functions only of the curvature. They simplify. So if we make some assumptions like that and walk through, we can get some rather similar pictures, but now pictures where the stress and the and the curvature are now correlated with each other. At this point, I'm going, how am I ever possibly going to convince myself, much less anybody else, that this all hangs together? In the closing thing, I said, let's do the following. I'm going to skip this part. Some nice things you can read about. I'm going to talk about Navier Stokes. Navier Stokes is a well defined set of equations in three dimensions. And this is something we can all sort of relate to. Right? But I can write a Navier Stokes that essentially looks like game theory, where I put some. I tweak it a little bit so that I can make some good analogies. But this is essentially just fluid theories. Okay, so we can compute it. I learned that I have to have viscosity. I actually learned that in those other equations I wrote. I also needed viscosity. Why? I went to talk to Oliver and Oliver told me that my equations were going to crash if I didn't add this little term that was going to be proportional to the Laplacian. And I said, okay, what is that term? Well, that's the viscosity term. Now in Navier Stokes, I have to add it because that's the physics. I started thinking about viscosity and said, oh, wait a minute. Viscosity is a statement ultimately that stress is proportional to strain. If I strain a system, I create stress. Do I have that in games? Of course. If we have a country that is pushing another country that causes a certain movement that generates some stresses and they have behaviors which are no longer game theory. This is very important. Let's compute that theory and compute the same things. Go through all the dirty details. But we'll get some pictures now for the curvature quote. And in fact, some frame dependent things which I didn't talk about, but I said, this is geometry. And frame dependent things are gravity. And it says gravity's force depends upon where matter is. I put that in and I get that out. I put in cohesion, which is a statement of between, that the stresses are proportional to the energy density or the curvature, they become proportional. I see a little funny peak here. It's hard to see what's going on. But in three dimensions, no problem. In three dimensions I can look at the energy density. I found something which is really hard to find out of Navier Stokes. I got a donut. Okay, how did I get a donut? Because I have a market force. The game theory force is circulating in the middle. And I figure out just the right collection of parameters to create the donut. Now, I have a mathematician friend and I. We want to create two donuts that are like this. And if we can, then we'll have a model for the universe, for the early universe, which is a whole other topic. Okay, let's see if I can go to the next thing. Oh, there we go. And here's the flow. And you can see that the flow is that donut shape is in the middle. That's where things are flowing around. How do I connect with what I was doing in my streamlines? Well, now I can actually do my streamlines, assuming that things were on a. I looked at streamlines that go through points along the bottom and points along the top. That's essentially the streamlines I was looking at that I can do in seven dimensions. But here I can see what it means where all the dimensions are there. And I can see that my line generates essentially a surface that I'm really looking at. And if I compare the surface using the same parameterization of those other plots, I can see that that plot is reflective of what's going on in the three dimension. So I have to learn how to see from seven dimension projections into some two dimensional world and figure out how to interpret what I'm seeing. And then the final piece, I would say is I really want to follow the matter flow, not just velocity flow, which means I have to look at the density times the flow. And that will show me that my donut has nothing on the boundaries, of course. And then the flow itself is also not on the boundary. So it all starts to make sense. And I can also put together with what happens on a line. Then my concluding remark, because I think I'm out of town, right? My concluding remark is I've looked at these equations as stationary in time. And the way people normally analyze Navier Stokes as a time dependent and all sorts of interesting things with the time dependency, my question was, if I analyze in a static, where do I see all these interesting things and the way I think I see them? And this has to be proved, my conjecture is that what happens when I make the Reynolds number too big, my equations blow up. And I have an analogy. There's something called the logistics equation. Some of you know what that is, right? And it's a simple algebraic equation that says the population is equal to a constant times 1 minus the population. And if the constant gets too big, which is like the Reynolds number, the thing breaks up. But it doesn't break up crazily. It has two solutions that it oscillates back and forth, it bifurcates and then it trifurcates, and then it goes into chaotic solutions. My conjecture is the same thing will occur here, but in function space. Have to prove that. So that concludes my talk.