# 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} }

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

## 612 Citations

### The complexity of computing a Nash equilibrium

- EconomicsSTOC '06
- 2006

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

- Economics, MathematicsSTOC '06
- 2006

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

- EconomicsSODA '07
- 2007

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

- EconomicsOR
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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

- Economics46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
- 2005

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

- EconomicsAAAI
- 2018

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

- EconomicsCEEMAS
- 2007

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

- Computer Science, EconomicsWINE
- 2006

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?

- EconomicsArXiv
- 2009

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

- Economics, Computer Science
- 2007

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

- EconomicsSTOC '06
- 2006

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

- Economics, MathematicsSTOC '06
- 2006

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

- Computer Science2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
- 2006

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

- EconomicsFAW
- 2006

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

- Computer ScienceUAI
- 2001

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

- Economics, Mathematics
- 1962

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

- Economics
- 1958

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

- Economics
- 2006

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…

### On equilibrium points in bimatrix games

- Economics
- 1996

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…