Computation of Nash Equilibrium Jugal Garg Georgios Piliouras.

Slides:



Advertisements
Similar presentations
Lower Bounds for Local Search by Quantum Arguments Scott Aaronson (UC Berkeley) August 14, 2003.
Advertisements

Preprocessing Techniques for Computing Nash Equilibria Vincent Conitzer Duke University Based on: Conitzer and Sandholm. A Generalized Strategy Eliminability.
Complexity Theory Lecture 6
Shortest Vector In A Lattice is NP-Hard to approximate
1 A Graph-Theoretic Network Security Game M. Mavronicolas , V. Papadopoulou , A. Philippou  and P. Spirakis § University of Cyprus, Cyprus  University.
Introduction to Kernel Lower Bounds Daniel Lokshtanov.
Bilinear Games: Polynomial Time Algorithms for Rank Based Subclasses Ruta Mehta Indian Institute of Technology, Bombay Joint work with Jugal Garg and Albert.
6.896: Topics in Algorithmic Game Theory Lecture 8 Constantinos Daskalakis.
COMP 553: Algorithmic Game Theory Fall 2014 Yang Cai Lecture 21.
6.896: Topics in Algorithmic Game Theory Lecture 11 Constantinos Daskalakis.
6.853: Topics in Algorithmic Game Theory Fall 2011 Constantinos Daskalakis Lecture 13.
Towards a Constructive Theory of Networked Interactions Constantinos Daskalakis CSAIL, MIT Based on joint work with Christos H. Papadimitriou.
The Theory of NP-Completeness
Complexity 15-1 Complexity Andrei Bulatov Hierarchy Theorem.
Computability and Complexity 13-1 Computability and Complexity Andrei Bulatov The Class NP.
1 Computing Nash Equilibrium Presenter: Yishay Mansour.
Analysis of Algorithms CS 477/677
Computational Complexity, Physical Mapping III + Perl CIS 667 March 4, 2004.
Near-Optimal Network Design with Selfish Agents By Elliot Anshelevich, Anirban Dasgupta, Eva Tardos, Tom Wexler STOC’03 Presented by Mustafa Suleyman CIFTCI.
Backtracking Reading Material: Chapter 13, Sections 1, 2, 4, and 5.
Congestion Games (Review and more definitions), Potential Games, Network Congestion games, Total Search Problems, PPAD, PLS completeness, easy congestion.
1 Algorithms for Computing Approximate Nash Equilibria Vangelis Markakis Athens University of Economics and Business.
Network Formation Games. Netwok Formation Games NFGs model distinct ways in which selfish agents might create and evaluate networks We’ll see two models:
The Computational Complexity of Finding a Nash Equilibrium Edith Elkind, U. of Warwick.
Computing Equilibria Christos H. Papadimitriou UC Berkeley “christos”
Minimax strategies, Nash equilibria, correlated equilibria Vincent Conitzer
Theory of Computing Lecture 19 MAS 714 Hartmut Klauck.
6.896: Topics in Algorithmic Game Theory Lecture 7 Constantinos Daskalakis.
Computing Equilibria Christos H. Papadimitriou UC Berkeley “christos”
MCS312: NP-completeness and Approximation Algorithms
Chapter 11 Limitations of Algorithm Power. Lower Bounds Lower bound: an estimate on a minimum amount of work needed to solve a given problem Examples:
Computational Complexity Polynomial time O(n k ) input size n, k constant Tractable problems solvable in polynomial time(Opposite Intractable) Ex: sorting,
1 The Theory of NP-Completeness 2012/11/6 P: the class of problems which can be solved by a deterministic polynomial algorithm. NP : the class of decision.
Complexity Classes (Ch. 34) The class P: class of problems that can be solved in time that is polynomial in the size of the input, n. if input size is.
The Computational Complexity of Finding Nash Equilibria Edith Elkind Intelligence, Agents, Multimedia group (IAM) School of Electronics and CS U. of Southampton.
6.853: Topics in Algorithmic Game Theory Fall 2011 Constantinos Daskalakis Lecture 11.
Approximation Algorithms
Week 10Complexity of Algorithms1 Hard Computational Problems Some computational problems are hard Despite a numerous attempts we do not know any efficient.
1 The Theory of NP-Completeness 2 Cook ’ s Theorem (1971) Prof. Cook Toronto U. Receiving Turing Award (1982) Discussing difficult problems: worst case.
Ásbjörn H Kristbjörnsson1 The complexity of Finding Nash Equilibria Ásbjörn H Kristbjörnsson Algorithms, Logic and Complexity.
On the Complexity of Search Problems George Pierrakos Mostly based on: On the Complexity of the Parity Argument and Other Insufficient Proofs of Existence.
NP-Complete Problems. Running Time v.s. Input Size Concern with problems whose complexity may be described by exponential functions. Tractable problems.
Algorithms for solving two-player normal form games
Vasilis Syrgkanis Cornell University
1 Algorithms for Computing Approximate Nash Equilibria Vangelis Markakis Athens University of Economics and Business.
6.853: Topics in Algorithmic Game Theory Fall 2011 Constantinos Daskalakis Lecture 8.
Chapter 15 P, NP, and Cook’s Theorem. 2 Computability Theory n Establishes whether decision problems are (only) theoretically decidable, i.e., decides.
Parameterized Two-Player Nash Equilibrium Danny Hermelin, Chien-Chung Huang, Stefan Kratsch, and Magnus Wahlstrom..
6.853: Topics in Algorithmic Game Theory Fall 2011 Constantinos Daskalakis Lecture 9.
MCS 312: NP Completeness and Approximation Algorithms Instructor Neelima Gupta
1 Computability Tractable, Intractable and Non-computable functions.
The NP class. NP-completeness Lecture2. The NP-class The NP class is a class that contains all the problems that can be decided by a Non-Deterministic.
Comp/Math 553: Algorithmic Game Theory
The NP class. NP-completeness
Nash Equilibrium: P or NP?
P & NP.
Market Equilibrium Ruta Mehta.
Richard Anderson Lecture 26 NP-Completeness
COMP/MATH 553: Algorithmic Game Theory
Historical Note von Neumann’s original proof (1928) used Brouwer’s fixed point theorem. Together with Danzig in 1947 they realized the above connection.
Intro to Theory of Computation
Chapter 34: NP-Completeness
Chapter 11 Limitations of Algorithm Power
NP-Complete Problems.
Enumerating All Nash Equilibria for Two-person Extensive Games
Computing a Nash Equilibrium of a Congestion Game: PLS-completeness
15th Scandinavian Workshop on Algorithm Theory
2-Nash and Leontief Economy
Normal Form (Matrix) Games
Complexity Theory: Foundations
Presentation transcript:

Computation of Nash Equilibrium Jugal Garg Georgios Piliouras

Recap – Two player game 2

Nash Equilibrium 3

Best Response Polyhedron 4 Complementarity Conditions ( CC )

Characterization 5

Quadratic Programming Formulation 6

Zero-sum Game 7

Zero-sum Game 8

Support Enumeration Algorithm 9

Support Enumeration Algorithm How many possible support? 10

Support Enumeration Algorithm 11

Support Enumeration Algorithm 12

Corollaries 13

Lemke-Howson Algorithm (1964) 14

Preliminaries 15

Best Response Polyhedron (Recall) 16

Linear Complementarity Problem (LCP) Formulation 17 Complementarity Conditions ( CC )

LCP Characterization 18

Preliminaries 19

Fully-labeled Points 20

Fully-labeled Characterization 21

Lemke-Howson Algorithm 22 (0,0) This path is called k-almost fully-labeled – only label k is missing

Convergence No cycling Cannot comes back to starting vertex (0,0) Has to terminate in finite time Exponential in worst case ( Savani, von Stengel’04 ) 23

Structural Results 24

Break Q. How we get existence and oddness of the number of equilibria from Lemke-Howson algorithm? 25

PTIME for Special Cases Zero-sum games - LP formulation Both rank(A) and rank(B) are constant (Lipton, Markakis, Mehta’04) Either rank(A) or rank(B) is constant (G., Jiang, Mehta’11) – Proof on board Rank-based hierarchy (rank(A+B)) – Zero-sum games = rank-0 games – Rank-1 games ( Adsul, G., Mehta, Sohoni’ 11 ) 26

Rank-1 Games Number of Nash equilibrium can be exponential (von Stengel’12) For any c, rank-1 games can have c number of connected components (Kannan-Theobold’07) Rank-1 games are strictly general than zero- sum games and LP (Dantzig’47) In general rank-1 QP is NP-hard to solve – surprisingly those arising from rank-1 games are polynomial time solvable. 27

Approximate Nash Equilibrium 28

Approximate Nash Equilibrium 29

FPTAS for Constant Rank Games 30

Quasi-polynomial Time Algorithm (Lipton, Markakis, Mehta’03) 31

Complexity Considerations 32 Nash You Are here

Standard Complexity Classes P: The set of decision problems for which some algorithm can provide an answer in polynomial time NP: set of all decision problems for which for the instances where the answer is "yes“, we can verify in polynomial time that the answer is indeed yes. coNP: Same as above with yes->no 33

Equilibrium Computation The goal is to find a function that maps objects (games) to the mixed strategy profiles. This is NOT a decision problem (yes/no). Examples: Add two numbers and find the outcome Is the sum of two numbers odd? 34

Function Complexity Classes FP: The set of function problems for which some algorithm can provide an answer in polynomial time FNP: set of all function problems for which the validity of an (input, output) pair can be verified in polynomial time by some algorithm. 35

Function Complexity Classes FP: A binary relation P(x,y) is in FP if and only if there is a deterministic polynomial time algorithm that, given x, can find some y such that P(x,y) holds. FNP: A binary P(x,y), where y is at most polynomially longer than x, is in FNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(x,y) holds given both x and y. 36

Function Complexity Classes FP: The set of function problems for which some algorithm can provide an answer in polynomial time FNP: set of all function problems for which the validity of an (input, output) pair can be verified in polynomial time by some algorithm. TFNP: The subclass of FNP for which existence of solution is guaranteed for every input! 37

Non-constructive arguments Local Search: Every directed acyclic graph must have a sink. Pigeonhole Principle: If a function maps n elements to n-1 elements, then there is a collision. Handshaking lemma: If a graph has a node of odd degree, then it must have another. End of line: If a directed path has an unbalanced node, then it must have another. 38

From non-constructive arguments to complexity classes PLS: All problems in TFNP whose existence proof is implied by Local Search arg. PPP: All problems in TFNP whose existence proof is implied by the Pigeonhole Principle. PPA: All problems in TFNP whose existence proof is implied by the Handshaking lemma. PPAD: All problems in TFNP whose existence proof is implied by the End-of-line argument. 39

From non-constructive arguments to complexity classes PPA FNP PPAD PPP PLS P 40

PPAD [Papadimitriou 1994] Suppose that an exponentially large graph with vertex set {0,1} n is defined by two circuits: P N node id END OF THE LINE: Given P and N: If 0 n is an unbalanced node, find another unbalanced node. Otherwise say “yes”. PPAD = { Search problems in FNP reducible to END OF THE LINE} possible previous possible next P(v)=u and N(u)=v u v Finding Nash equilibria, even in 2-person games is PPAD –complete. [Daskalakis, Goldberg, Papadimitriou 05], [Chen, Deng 06’] 41