Settling the Complexity of Two-Player Nash Equilibrium

@article{Chen2006SettlingTC,
  title={Settling the Complexity of Two-Player Nash Equilibrium},
  author={Xi Chen and Xiaotie Deng},
  journal={2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)},
  year={2006},
  pages={261-272}
}
  • Xi ChenXiaotie Deng
  • Published 21 October 2006
  • Economics
  • 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
Even though many people thought the problem of finding Nash equilibria is hard in general, and it has been proven so for games among three or more players recently, it's not clear whether the two-player case can be shown in the same class of PPAD-complete problems. We prove that the problem of finding a Nash equilibrium in a two-player game is PPAD-complete 
View on IEEE
cs.tau.ac.il
612 Citations
Highly Influential Citations
46
Background Citations
344
Methods Citations
49
Results Citations
26

Figures from this paper

612 Citations

The complexity of computing a Nash equilibrium

This proof uses ideas from the recently-established equivalence between polynomial time solvability of normal form games and graphical games, establishing that these kinds of games can simulate a PPAD-complete class of Brouwer functions.

Reducibility among equilibrium problems

The main result is that the problem of solving a game for any constant number of players, is reducible to solving a 4-player game.

The approximation complexity of win-lose games

The hard core of the complexity of Nash equilibria is distill, showing that even correctly computing a logarithmic number of bits of the equilibrium strategies of a two-player win-lose game is as hard as the general problem.

Complexity of Pure-Strategy Nash Equilibria in Non-Cooperative Games

Game theory in general and the concept of Nash equilibrium in particular have lately come under increased scrutiny by theoretical computer scientists. Computing a mixed Nash equilibrium is a case in…

Nash equilibria in random games

We consider Nash equilibria in 2-player random games and analyze a simple Las Vegas algorithm for finding an equilibrium. The algorithm is combinatorial and always finds a Nash equilibrium; on m /spl…

On the Approximation of Nash Equilibria in Sparse Win-Lose Games

The problem of finding an approximate Nash equilibrium with a polynomial precision is PPAD-hard even for two-player sparse win-lose games, which is mainly based on a new class of prototype games called Chasing Games.

Complexity of Verifying Game Equilibria

This work identifies a certain class of games where Nash or Bayesian Nash equilibria can be verified in polynomial time and shows that verifying a dominant strategy equilibrium is NP-complete even for normal form games.

Sparse Games Are Hard

It is shown that the problem of computing a Nash equilibrium remains PPAD-hard to approximate in fully polynomial time for sparse games, and a simple andPolynomial-time algorithm is given for finding exact Nash equilibria in a class of sparse win-lose games.

How discontinuous is Computing Nash Equilibria?

The degree of discontinuity of several solution concepts from non-cooperative game theory are investigated, including Nash equilibria and pure and correlated Equilibria.

Algorithmic Game Theory: The Complexity of Finding Nash Equilibria

The recent proof that finding a Nash equilibrium is complete for the complexity class PPAD, even in the case of two players, is outlined, evidence that the problem is intractable.
...

References

SHOWING 1-10 OF 29 REFERENCES

The complexity of computing a Nash equilibrium

This proof uses ideas from the recently-established equivalence between polynomial time solvability of normal form games and graphical games, establishing that these kinds of games can simulate a PPAD-complete class of Brouwer functions.

Reducibility among equilibrium problems

The main result is that the problem of solving a game for any constant number of players, is reducible to solving a 4-player game.

Computing Nash Equilibria: Approximation and Smoothed Complexity

A key geometric lemma for finding a discrete fixed-point is proved, a new concept defined on n + 1 vertices of a unit hypercube, which enables the authors to overcome the curse of dimensionality in reasoning about fixed-points in high dimensions.

On the Approximation and Smoothed Complexity of Leontief Market Equilibria

It is proved that the Leontief market exchange problem does not have a fully polynomial-time approximation scheme, unless PPAD ⊆ P.

Graphical Models for Game Theory

The main result is a provably correct and efficient algorithm for computing approximate Nash equilibria in one-stage games represented by trees or sparse graphs.

Equilibrium Points of Bimatrix Games

An algebraic proof is given of the existence of equilibrium points for bimatrix (or two-person, non-zero-sum) games. The proof is constructive, leading to an efficient scheme for computing an…

Equilibrium Points in Bimatrix Games

An algorithm for computing all equilibrium points (situations) for the case of bimatrix (i.e., finite two-person, non-cooperative, non-zero-sum) games is given.

Hard‐to‐Solve Bimatrix Games

The Lemke-Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in…

COMPLEMENTARY PIVOT THEORY OF MATHEMATICAL PROGRAMMING

On equilibrium points in bimatrix games

We discuss sensitivity of equilibrium points in bimatrix games depending on small variances (perturbations) of data. Applying implicit function theorem to a linear complementarity problem which is…