The paper "The Complexity of Computing a Nash Equilibrium" by Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou resolves the computational complexity of finding Nash equilibria in games with four or more players. The authors show that the problem of computing a Nash equilibrium in such games is PPAD-complete, a complexity class known for problems with solutions guaranteed by a directed graph-theoretic argument. The proof uses reductions between normal-form games and graphical games, demonstrating that these games can implement arbitrary members of a PPAD-complete class of Brouwer functions. The paper also introduces "arithmetical gadgets" that allow for arithmetic operations in graphical games, such as addition, multiplication, and comparison of real numbers. The main result is a reduction from END OF THE LINE, a problem in PPAD, to 4-NASH, proving that 4-NASH is PPAD-complete. The authors discuss the challenges and solutions in their proof, including the use of "brittle comparators" to handle boundary issues and the construction of a graphical game that simulates a Brouwer function. The paper concludes with open problems, including the complexity of finding Nash equilibria in games with two or three players and the computation of approximate Nash equilibria.The paper "The Complexity of Computing a Nash Equilibrium" by Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou resolves the computational complexity of finding Nash equilibria in games with four or more players. The authors show that the problem of computing a Nash equilibrium in such games is PPAD-complete, a complexity class known for problems with solutions guaranteed by a directed graph-theoretic argument. The proof uses reductions between normal-form games and graphical games, demonstrating that these games can implement arbitrary members of a PPAD-complete class of Brouwer functions. The paper also introduces "arithmetical gadgets" that allow for arithmetic operations in graphical games, such as addition, multiplication, and comparison of real numbers. The main result is a reduction from END OF THE LINE, a problem in PPAD, to 4-NASH, proving that 4-NASH is PPAD-complete. The authors discuss the challenges and solutions in their proof, including the use of "brittle comparators" to handle boundary issues and the construction of a graphical game that simulates a Brouwer function. The paper concludes with open problems, including the complexity of finding Nash equilibria in games with two or three players and the computation of approximate Nash equilibria.