Consider the decision problem: "Given a two-player game, does there exist a Nash equilibrium?" It is in NP, since given as a certificate a candidate equilibrium we can check that the answer is yes in polynomial time. But we don't actually need to do this, since by Nash's theorem the answer is always yes!

It is therefore more helpful to rephrase the question as a search problem: NASH: "Given a two-player game, either find a Nash equilibrium or output "no" if none exists." This problem can be solved in polynomial time by a nondeterministic Turing machine by examining all possible supports for the equilibrium strategies, so it is a problem in NP. It is not known whether or not NASH is NP-complete. However, many other problems are NP-complete, such as 2NASH: "Given a game and a Nash equilibrium, find another one, or output “no” if none exist."

Notice that the Lemke-Howson algorithm described today is not a polynomial time algorithm. Just as the the Klee-Minty example for the simplex algorithm, there are examples in which the length of the Lemke-Howson path grows exponentially in the size of the game. I will say more about the algorithm in the next lecture.

There is a good lecture by Papadimitriou on the question of: Complexity of Finding a Nash Equilibrium. He defines PPAD as the class of all search problems which always have a solution and whose proof is based on the parity argument for directed graphs. This includes the problem of finding a Nash equilibrium, where the proof is based on the Lemke-Howson algorithm. In fact, the other proof, based on the Brouwer fixed-point theorem, is also in this class, since a proof of the Brouwer fixed-point theorem uses Sperner's lemma, which is itself based on a parity argument in a directed graph.

It is therefore more helpful to rephrase the question as a search problem: NASH: "Given a two-player game, either find a Nash equilibrium or output "no" if none exists." This problem can be solved in polynomial time by a nondeterministic Turing machine by examining all possible supports for the equilibrium strategies, so it is a problem in NP. It is not known whether or not NASH is NP-complete. However, many other problems are NP-complete, such as 2NASH: "Given a game and a Nash equilibrium, find another one, or output “no” if none exist."

Notice that the Lemke-Howson algorithm described today is not a polynomial time algorithm. Just as the the Klee-Minty example for the simplex algorithm, there are examples in which the length of the Lemke-Howson path grows exponentially in the size of the game. I will say more about the algorithm in the next lecture.

There is a good lecture by Papadimitriou on the question of: Complexity of Finding a Nash Equilibrium. He defines PPAD as the class of all search problems which always have a solution and whose proof is based on the parity argument for directed graphs. This includes the problem of finding a Nash equilibrium, where the proof is based on the Lemke-Howson algorithm. In fact, the other proof, based on the Brouwer fixed-point theorem, is also in this class, since a proof of the Brouwer fixed-point theorem uses Sperner's lemma, which is itself based on a parity argument in a directed graph.