 So let's speak a little bit about how hard is it to compute a Nash equilibrium in a normal form game. This is an involved topic, and I'll just give you a taste for it. Let me draw your attention to two specific algorithms for computing a sample Nash equilibrium in a game. These are two out of a long line of algorithms that have been investigated, and these are sort of two extreme. The one of them starts with a mathematical formulation of the problem called a linear complementarity problem, and once you set it up as a mathematical optimization problem, you can apply various algorithms to that, and the most famous one for two-player games is due to Lemke and Halsey. And this is an algorithm that really displays a deep understanding of the mathematical structure of what a game is and the nature of Nash equilibrium. Perhaps the other extreme is what is called the support enumeration method, a recent procedure that doesn't have as deep an insight into the structure of the problem. It says simply the following, it says if you fix the support of the strategies of the player and the support of the strategy players are those actions that are played with non-zero probability, if you fix that support then the problem becomes very easy. You can set it up as a linear program and solve it efficiently. And that will be the end of it if it weren't for the case that indeed there are an exponential number of supports to explore. And so the trick in this procedure is to explore them cleverly using clever heuristics, and that's called the support enumeration. There's a clever heuristic for how to enumerate those supports and check them one by one. Although the latter procedure is not as smart or as insightful as the Lemke-Housen, it turns out that in practice it tends to run very fast. So we've seen the algorithm, people have tried very hard to find algorithms computing a sample Nash equilibrium, and it does seem hard. The question is can we somehow capture that formally within the complexity hierarchy? And for that we need to introduce a new concept. The essential concept is that of the new class of problems called P P A D for polynomial parity arguments directed graph introduced by Christos Papadimitri in 1994. We won't go into detail, but just so you know the chronology, P P A D is a specialization of a class called T F N P, which is in turn was a specialization of a problem called F N P. Going in detail here is beyond the scope of what we want to speak about, but it does help us now position the complexity of finding a sample Nash equilibrium in the complexity hierarchy. And again, we have the class of polynomial time problems, of problems that can be verified in polynomial time, with these being the hardest among them. And given that P P A D turns out to reside somewhere within this class. Now again, we don't know whether this entire class does not collapse and all become one of the same. It's why do you believe that it does not, but proof doesn't exist. However, we do know that P P A D lies someplace in between P and N P. Now what does that have to do with the problem of computing a Nash equilibrium? Well, that's where the following theorems come in. Originally it was shown that the problem of computing a Nash equilibrium is complete for this class P P A D. The problem is, it's the hardest among all problems in that class. Initially proved for four players, then for all, for games with three or more players, and then finally in 06 for all, all, all class of games. And so we, why do you believe that the problem is not polynomial, cannot prove it, but we do know where it resides within the vector hierarchy that we are familiar with.