On the complexity of the parity argument and other inefficient proofs of existence
Journal of Computer and System Sciences - Special issue: 31st IEEE conference on foundations of computer science, Oct. 22–24, 1990
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
On the Complexity of Two-PlayerWin-Lose Games
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Settling the complexity of computing two-player Nash equilibria
Journal of the ACM (JACM)
Complexity results about Nash equilibria
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
The Complexity of Computing a Nash Equilibrium
SIAM Journal on Computing
On the computational complexity of Nash equilibria for (0,1) bimatrix games
Information Processing Letters
Weighted Boolean formula games
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
On the Complexity of Nash Equilibria and Other Fixed Points
SIAM Journal on Computing
When the players are not expectation maximizers
SAGT'10 Proceedings of the Third international conference on Algorithmic game theory
The complexity of decision problems about nash equilibria in win-lose games
SAGT'12 Proceedings of the 5th international conference on Algorithmic Game Theory
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We introduce two new decision problems, denoted as ∃ RATIONAL NASH and ∃ IRRATIONAL NASH, pertinent to the rationality and irrationality, respectively, of Nash equilibria for (finite) strategic games. These problems ask, given a strategic game, whether or not it admits (i) a rational Nash equilibrium where all probabilities are rational numbers, and (ii) an irrational Nash equilibrium where at least one probability is irrational, respectively. We are interested here in the complexities of ∃ RATIONAL NASH and ∃ IRRATIONAL NASH. Towards this end, we study two other decision problems, denoted as NASH-EQUIVALENCE and NASH-REDUCTION, pertinent to some mutual properties of the sets of Nash equilibria of two given strategic games with the same number of players. NASH-EQUIVALENCE asks whether the two sets of Nash equilibria coincide; we identify a restriction of its complementary problem that witnesses ∃ RATIONAL NASH. NASHREDUCTION asks whether or not there is a so called Nash reduction (a suitable map between corresponding strategy sets of players) that yields a Nash equilibrium of the former game from a Nash equilibrium of the latter game; we identify a restriction of it that witnesses ∃ IRRATIONAL NASH. As our main result, we provide two distinct reductions to simultaneously showthat (i)NASH-EQUIVALENCEis co-NP-hard and ∃ RATIONAL NASHis NP-hard, and (ii)NASH-REDUCTION and ∃ IRRATIONALNASH are NP-hard, respectively. The reductions significantly extend techniques previously employed by Conitzer and Sandholm [6, 7].