This lecture explores the computational complexity of finding mixed Nash equilibria. While polynomial-time algorithms exist for correlated equilibria, finding mixed Nash equilibria, even in two-player games, is computationally hard. The problem isn't NP-complete, but it's complete for the complexity class PPAD, suggesting inherent intractability and questioning the predictive power of Nash equilibria in general settings. This segment summarizes previously established results on computing correlated equilibria and course correlated equilibria in polynomial time, highlighting the contrast with the more limited positive results for pure and mixed Nash equilibria. This segment formally defines the computational problem of finding a mixed strategy Nash equilibrium in bimatrix games, emphasizing that the goal is to find any one equilibrium, making it the easiest equilibrium computation problem. The speaker establishes that no known polynomial-time algorithm exists for solving the problem, motivating the need to develop a suitable complexity theory to explain the intrinsic hardness of the problem. The segment introduces the concept of completeness to show the problem's hardness relative to other problems in a large class. This segment explains why NP-completeness is not the right notion for computing mixed strategy Nash equilibria, arguing that it's too strong a statement to prove. The speaker justifies the development of a weaker analog, PLS completeness, introduced in a previous lecture. This segment explores the possibility of FNP completeness for the problem, presenting a theorem that shows if computing mixed Nash equilibria is FNP-complete, it would imply NP=coNP, a highly unexpected result. The speaker explains the implications of this equivalence, relating it to the existence of short certificates for unsatisfiable formulas. This segment explains the concept of PLS (Polynomial Local Search) problems, emphasizing that these problems always possess a witness (locally optimal solution), unlike problems like SAT. The speaker contrasts PLS problems with those that may not have a solution, highlighting the significance of this guaranteed existence of a witness in the context of computational complexity. The speaker discusses the limitations of TFMP (Total FMP) as a complexity class for proving hardness results due to the lack of complete problems. The segment introduces the need for subclasses of TFMP that are both interesting (containing problems not known to be in P) and syntactically defined (allowing for complete problems), setting the stage for the introduction of PLS and PPAD. This segment uses the analogy of local search as traversing a directed acyclic graph (DAG) to illustrate the concept of PLS. The speaker explains how local search always terminates in a DAG due to the absence of cycles and the finite number of nodes, leading to the introduction of PPAD (Polynomial Parity Argument Directed) as a similar but distinct complexity class. This segment introduces Sperner's Lemma as a canonical PPAD problem. The speaker describes the problem of coloring a subdivided simplex (triangle) with three colors under specific boundary conditions, explaining that there always exists a trichromatic triangle (a triangle with all three colors). This problem's connection to PPAD is hinted at, setting the stage for the following explanation of its relationship to finding a solution. This segment details the intricate connection between Sperner's Lemma, a combinatorial result, Brouwer's Fixed-Point Theorem, a topological result, and Nash's Theorem, establishing the existence of mixed Nash equilibria in games. The explanation clarifies how Sperner's Lemma, through a limiting argument, implies Brouwer's theorem, which in turn implies the existence of Nash equilibria. This connection reveals a fundamental link between seemingly disparate mathematical fields and provides a constructive approach to approximating mixed Nash equilibria.This segment explains the reduction of finding Nash equilibria to Brouwer's Fixed-Point Theorem. It introduces the concept of a regularized best response function to ensure continuity, bridging the gap between fixed points and Nash equilibria. The discussion culminates in the explanation of how computing mixed Nash equilibria falls within the PPAD complexity class, characterized by path-following algorithms.This segment introduces the PPAD complexity class, a subclass of TFNP, and positions the problem of computing mixed Nash equilibria within it. The discussion clarifies that the PPAD class is defined by path-following problems and that the computation of approximate mixed Nash equilibria, for any number of players, belongs to this class. The segment also touches upon the implications of this classification for the computational complexity of finding Nash equilibria. The speaker defines PPAD problems, emphasizing their similarity to PLS problems in terms of three algorithms (start, next move, progress verification) but highlighting the key difference: PPAD involves directed graphs with cycles, unlike the DAGs in PLS. The speaker explains that PPAD problems are guaranteed to terminate at a witness, even with potentially exponentially many nodes. This segment presents the significant result that computing mixed Nash equilibria in two-player, nonzero-sum games is PPAD-complete. It highlights the historical context of this discovery, mentioning key papers and researchers involved in establishing this result. The explanation emphasizes the implications of this finding for the computational hardness of finding Nash equilibria and its significance in the field of computational game theory. This segment explains how finding a trichromatic triangle in Sperner's Lemma can be reduced to following paths in a directed graph. The speaker describes constructing a graph where nodes represent faces and edges connect faces based on color changes, demonstrating that following these paths leads to a trichromatic triangle. The speaker concludes by connecting this to the broader concept of PPAD problems and their relevance to computational complexity. This segment explores the implications of the PPAD-completeness of computing Nash equilibria. It discusses the potential limitations of Nash equilibria as a predictive model for real-world behavior, arguing that the intractability of finding these equilibria suggests that players may not be able to find them in a reasonable amount of time. The segment also considers alternative equilibrium concepts that might be more tractable and practically useful. This segment delves into the computational complexity of equilibrium problems, particularly focusing on the limitations in solving bi-matrix games and Nash equilibria using polynomial-time algorithms. The speaker introduces concepts like MP completeness, PLS completeness, and PPAD completeness to explain why certain equilibrium computations remain computationally challenging, setting the stage for a more advanced course focusing on cutting-edge mechanism design research.