So let's speak a little bit about how hard it is 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, a long line of algorithms, algorithms that've been investigated and these are sort of too extreme. The one of them starts with a mathematical formulation of the problem called linear complementarity problem and you, once you set it up as a mathematical optimization problem, you can apply various algorithms to that and the most famous one for 2 players games is due to Lemke and Howson. 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. On, perhaps, the other exterme is is what is called the Support Enumeration Method, a 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 supports 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 would be the end of it. If it weren't for the case that 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 a Support Enumeration as a clever heuristic word for how to interwrite those supports and check them one by one. although, the, the ladder procedure is not as smart or as insightful as the Lemke-Howson it turns out that in practice tends it tends to run very fast. So, we've seen the algorithm, people have tried very hard to find algorithms as computing a sample Nash equilibrium and it does seem very hard. The question is, can we somehow capture that formally within the complexity hierarchy. And and and for that we need to introduce a new, new new concepts. the essential concept is that of the new class of problems called PPAD, for Polynomial Parity Arguments on Directed graph introduced by Christos Papadimitriou in 1994. we want to go into detail, but just so you know the chronology PPAD is a specialization of the class called TFNP which is in turn, was a specialization of a problem called FNP. go in detail here is, is, is beyond the scope of, 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 heirarchy. 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, PPAD turns out to reside somewhere within this class. Now, we can, we don't know, whether this entire class does not collapse, and all become one and the same. It's widely believed that it does not, but proof doesn't exist. However, we do know that PPAD lies someplace in between P and NP. Now, what does that have to do with the problem of computing a Nash equilibrium? Well, that's where the, the following theorems come in. originially, it was shown that the problem of computing a Nash equilibrium is compolete for this class PPAD. That is, it's, the hardest among all problem in that class. In this, we prove for 4 players, then for games for 3 or more players, and then finally in 06' for all class of games. And so, we widely believe that the problem is not polynomial, we cannot prove it, but we do know where it resides in, within the complex hierarchy that we are familiar with.