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- Settling Complexity of NASH equilibrium
- Joint work with CHEN Xi
- and
- Extensions with CHEN Xi and Shanghua TENG

Importance of complexity issue of NASH

- Papadimitriou STOC 2001 invited talk
- Fortnow Computational Complexity Column
- We know only a few natural problems in NP that

are not known to be NP-complete or in P. The two

most often named are factoring and graph

isomorphism. Another that has come to forefront

is Nash Equilibrium. - http//weblog.fortnow.com/2005/02/complexity-of-na

sh-equilibrium.html

- Outline of Story
- Nash Equilibrium (Mathematicians)
- Algorithmic Solutions (OR and Economists)
- Complexity (Computer Scientists)

- Nash Equilibrium

Zero Sum Game

- N players
- Each chooses one strategy from a set of

strategies (which is in general different for

different players) - Once each player fixes its strategy, the game has

an outcome for each player (which again is in

general different for different players) which

sums up to zero.

Notations

- N players i1,2,,n
- Let Si be the set of strategies of player i
- Let si be the strategy chosen by player i
- Let ui(s1, s2,, sn) be the payoff to player i
- Zero sum game is one such that it always hold

that - Sumui(s1, s2,, sn)i1,2,,n0

An Example Matching Pennies

- Input
- Two players Evan and Odette
- The protocol
- Each flips a coin
- The winner will be decided by the outcome of the

flipped coins 0 for the number and 1 for the

picture. - If coins sum to odd, Oddete wins
- Otherwise, Evan wins.
- Loser pays winner M coins

Evans Payoff Matrix

Odettes Payoff Matrix

Joint Payoff Matrixnumbers in each cell sum to

zero

von Neuman min-max Theorem

- Pure strategy game may not always have an

equilibrium - Any two player zero sum game has a mixed strategy

equilibrium. - John von Neumann (1928), zur Theorie der

Gesellshaftsspiele, Mathematische Annalen 100,

pp.295-320.

Pure vs Mixed Strategies

- Pure strategy Each player fixes its decision

on its decided strategy such as always choosing

the picture for the outcome of the coin. - Mixed strategy equilibrium Each player chooses

each side of the coin with a certain probability. - Flipping it is usually regard as assign a

probability of ½ for each side of the coin.

pure strategy equilibirum

When both choose betrayal, none of them can do

better by shifting away from its strategy of

betrayal. Therefore, it is an equilibrium point.

Non-existence of pure strategy equilibrium

At any pair of strategies of the two Players, one

of them can do better By shifting away from its

strategy.

Mixed strategy equilibrium

Each chooses number or picture with probability ½.

Evans payoff when Both choose ½ and ½.

When Odette chooses half and half, the payoff to

Evan is always zero. Therefore, no matter what

other strategy Evan shifts into, it cannot do

better than the strategy of ½ and ½. The same

argument also hold for Odette.

Proof of von Neuman min-max Theorem via Linear

Programming

- Any two player zero sum game has a mixed strategy

equilibrium. - Let the payoff matrix of row player be A, an m by

n matrix. - The payoff matrix of the column player is A.
- Row player chooses its pure strategies with

probability x1, x2, , xn - Column player chooses its pure strategies with

probability y1, y2, , ym

Von Neumanns Maximin Approach

- The row player wants to choose a mixed strategy

to guarantee a payoff such that no matter

whatever pure strategy the column player choose - The column player wants to do the same thing.
- Luckily, those two strategies can be achieved

simultaneously, by linear program duality.

Non-zero-sum Game

Another Game

- Cournot game
- Between two firms.
- They produce the same good
- The price of good is a decreasing function of the

total quantity of goods they jointly produce - The function is known to both firms.
- Strategy The quantity one firm chooses to

produce. - Example 200 pieces will have price 5, 300 piece

price 3, 400 piece price 1

Joint Payoff Matrix

Reduction to zero sum game

- Reduce to three player zero sum game
- Introduce a third person
- With only one strategy
- The payoff is the negation of the sum of the

other two players. - Similar argument holds for games of more players.

Solutions?

- We can still define a maximin solution for every

player. - However, we dont have a minmax theorem as in the

two player case.

von Neumanns Proposal

- Study co-operative game behavior for multiple

players games. - The von Neumann-Morgenstern solution

Nashs approach of Equilibrium

- Nash proved that an equilibrium point exists for

any number of players in a non-cooperative

setting. - Nash, J. F. "Non-Cooperative Games." Ann. Math.

54, 286-295, 1951. - His trick was the use of best-response functions
- a recent theorem that had just emerged -

Kakutani's fixed point-theorem. - Won nobel prize in 1994

An intuition for a proof

- x is the best response to y
- Payoff1(x,y) maximizes Payoff1(x,y)
- y is the best response to x
- Payoff2(x,y) maximizes Payoff2(x,y)
- Best response function
- BR(x,y) (x', y') (maps unit square to itself)
- Its fixed point
- BR(x,y) (x, y)
- is NASH equilibrium by definition.

Previous Work of Nash

- As an undergraduate, he had inadvertently (and

independently) proved Brouwer's fixed point

theorem. - Later on, he went on to break one of Riemann's

most perplexing mathematical conundrums. - Source
- http//cepa.newschool.edu/het/profiles/nash.htm

Extensions

- General equilibrium in Economics proven by

Arrow-Debreu using Kakutanis fixed point

theorem. - Arrow, K.J., and G. Debreu (1954). Existence of

an equilibrium for a competitive economy.

Econometrica 22 (July) 26590. - Both are Nobel prize winners (1972, 1982).

Algorithmic Solutions

Relationships between the two models

- Two player zero sum game can be solved by linear

programming - The next easiest problem is two player non-zero

sum game. - N player non-zero sum game can solve n player

zero sum game (2 and more) - N1 player zero-sum game can solve n player

non-zero sum game (3 and more)

Linear Complementary Problem

- uMvb0
- v0
- ltu,vgt 0
- It is complementary since either u_i0 or v_i0

for each i.

Mathematical Programming of Players

- Let A and B be the payoff matrices to the two

players. - The players problems are thus
- Max xtAy xte1, x 0, y fixed
- MaxxtBy yte1, y0, x fixed

Dual LPs

- The dual of Max xtAy xe1, x 0, y fixed
- Min z etz Ay, z 0
- In LCP form xt(ez-Ay)0
- MaxxtBy ety 1, y0, x fixed
- Min w we xtB, w 0
- In LCP form (we-xtB)y0

Rescale the equations

- Divide by z on each side and set ylt-y/z
- Qy et Ay, y0
- xt(e-Ay)0
- Divide by w on each side and set xlt-x/w
- P x e xtB, x 0
- (e-xtB)y0

Nash Equilibrium as LCP

- v(xt,yt)t
- uMvb 0
- be
- The matrix M is gt
- The equations are
- vt (Mve)0
- v 0

LCP formulation of NASH

- Formulation in LCP
- Cottle, RW and Dantzig, GD (1962-3),

Complementary Pivot Theory of MathematicalProgram

ming, Linear Algebra and its Applications 1,

103-125, 1968. - Algorithmic Solution
- C.E. Lemke and J.T. Howson, Jr., Equilibrium

Points in Bimatrix Games, J. SIAM 12, pp.418-423,

1964 - Review Article
- http//www.informs.org/History/dantzig/LinearProgr

amming_article.pdf

More on Lemke and Howson

- The underlying logic, involving motions on the

edges of an appropriate polyhedron, has been the

starting point of the path-following methodology - More details on significance and relevance, see
- 1978 John von Neumann Theory Prize Winning

Citations for John F. Nash and Carlton E. Lemke - http//www.informs.org/Prizes/vonNeumannDetails.ht

ml

Main Idea of Lemke and Howson algorithm

- Start with v0
- Choose an index k to be dropped
- That is, vk0 no longer holds
- As the complementary condition
- vt (Mve)0
- Requires we set Mkv10
- vk increases
- until Miv10 for some i
- If ik, we have a non-trivial solution, i.e., a

Nash equilibrium. - Otherwise, vi can be increased afterward.
- Change both vk and vi simultaneously such that

Miv10 - while keep other vs zero.

Main Idea of Lemke and Howson algorithm

- NOTE move along edges of polytope with (n-1)

constraints. - There is only one choice in the non-degenerate

case.

Extensions

- Scarf developed a path-following approach to

solve the fixed point problem, approximately and

to solve the general equilibrium problem. - Equivalently an algorithm for n person games.
- von Neumann Award (1983).

Lower Bound on Lemke and Howson

- An example is constructed such that an

exponential number of steps is necessary no

matter which initial index is chosen - R. Savani and B. von Stengel (2005),

Hard-to-Solve Bimatrix Games. Econometrica, to

appear. - http//www.maths.lse.ac.uk/Personal/stengel/bvs-pu

bl.html

- Complexity

Combinatorial Nature of NASH

- Though the problem is defined on continuous

variables, its proof of existence depended on - Fixed point theorem
- Dependent on Sperners lemma
- Path following algorithm
- Relied on Combinatorial structures of polytopes.

The complexity issue

- The complexity of Nash Equilibrium is therefore

well defined in the standard computational

complexity theory. - However, it has been an open problem whether

there is a polynomial time algorithm since the

early work of Lemke-Howson, of forty years ago.

PPAD

- Polynomial Parity Argument, Directed Version
- Characterize the proof technique employed in the

proof of many mathematical problems - Papadimitriou. On the complexity of the parity

argument and other inefficient proofs of

existence. JCSS 48, pp.498-532, 1994.

Define PPAD end of line

- A graph of exponential size
- 2n nodes
- Each node has at most one outgoing edge and at

most one incoming edge - A polynomial time Turing machine computing the

successor of every node. - Node 1 has no incoming edge and has one outgoing

edge. - Output Requirement find another node with

exactly one edge (incoming or outgoing).

Problems in PPAD

- Sperner Lemma, Fixed Point, NASH
- In addition, Sperner Lemma and Fixed Point are

PPAD-Complete - That means, if they can be solved in polynomial

time, any problem in PPAD can be solved in

polynomial time.

NASH remains open

- NASH is in PPAD by Lemke and Howson algorithm
- It is not known to be PPAD complete

A Big Breakthrough

- The complexity of computing a Nash equilibrium

- Constantinos Daskalakis, Paul W. Goldberg,

Christos H. Papadimitriou - which proves 4 player NASH is PPAD-Complete.
- October 10, 2005, in ECCC
- http//www.eccc.uni-trier.de/eccc-reports/2005/TR0

5-115/index.html - Conjecture
- 3 player game is hard
- 2 player game is polynomial solvable

The Next Step

- 3-NASH is PPAD-Complete
- Xi Chen and Xiaotie Deng
- ECCC, Nov 18, 2005
- Three-Player games are hard
- Constantinos Daskalakis, Christos H.

Papadimitriou - Nov 29, 2005, in ECCC

Importance of Complexity of 2NASH

- At the frontier of EASY problem Two player zero

sum game is easy (LP), 2NASH is the next easiest

problem. - At the frontier of HARD problem Lemke-Howson

algorithm, started as a solution for 2NASH, has

been the original of many related problems that

employed the path following methodology. Most of

the important ones are PPAD-hard, among them are - Fixed point problem
- Many person NASH Equilibrium
- Sperner problem

The result

- Xi Chen and Xiaotie Deng
- Settling the Complexity of Two Player Nash

Equilibrium - Dec 4, 2005 in ECCC

The main structure of proof

- Reduce end of line to
- 3D SPERNER, which is reduced to
- 3D BROUWER FIXED POINT, discrete version
- R player NASH Equilibrium
- Degree 3 graphic NASH equilibrium, approximate

version - 4 player NASH, approximate version

Our simplified structure of proof

- Reduce end of line to
- 2D SPERNER, which is reduced to
- 2D BROUWER FIXED POINT, discrete version
- 2 player NASH, exact version

2D discrete fixed point problem

Problems in PPAD

- Sperner Lemma, Fixed Point, NASH
- In addition, 3D Sperner Lemma and 3D Fixed Point

were PPAD-Complete - That means, if they can be solved in polynomial

time, any problem in PPAD can be solved in

polynomial time.

SPERNER Lemma

- Given a triangulated triangle,
- Given a proper labeling of its nodes,
- Labels 0,1,2
- A condition on labels on the boundaries.
- There must be a triangle with all three labels

appear in its three nodes.

Triangulation of a triangle

Ai is colored with i Any point on the boundary

is labeled with labels on the two

endpoints. Labels on the interior points Are not

restricted. SPERNER LEMMA There is a triangle

with all Three labels.

Other reasons why 2NASH could not be hard?

- 2D Sperner was open
- 2D fixed point was open

The results

- Xi Chen and Xiaotie Deng
- On Complexity of 2D fixed point
- ICALP 2006

Relationship of 2D Sperner with 2NASH

- Affirms the possibility that 2NASH could be

harder - Simplifies the proof structure

Differences from NP-hard Proofs

- Problems of exponential size search space
- Local properties of the input/output structure
- Which can be verified by a polynomial time

algorithm also as an input parameter. - Reductions should reduce the polynomial

algorithms for the local properties to each other

Define PPAD Another End

- A graph of exponential size
- 2n nodes
- Each node has at most one outgoing edge and at

most one incoming edge - A polynomial time Turing machine computing the

successor of every node. - Node 1 has no incoming edge and has one outgoing

edge. - Output Requirement find another node with

exactly one edge (incoming or outgoing).

2D Sperner

- A triangle of exponential size
- 2n nodes in the base and in the height
- 22n triangles
- The nodes are properly labeled
- A polynomial time Turing machine computing the

label of each node - Output Requirement find a small triangle with

all three labels.

Outline of Proof

- Planar embedded version of Another End
- Even though Another End is planar, its direct

embedding is in general not planar and an

efficient embedding is not known. - A carefully designed labeling processing to

construct an instance of the Sperners problem.

The structure for a planar embedding

- Denote N2n
- Embed a complete graph to a planar grid of size

3NN, 6N with about NN/2 crossings - This can be done by a polynomial time algorithm

locally, i.e., report the edges enters and leaves

any node in the grid. - Use the edges in the planar embedding of the

complete graphs for the embedding of Another

End - Difficulty make sure each node has at most one

incoming and at most one outgoing edge.

The complete graph embedding

Four types of gadgets at crossing

The line graph embedding

- Denote N2n
- Use the edges in the planar embedding of the

complete graphs for the embedding of Another

End - Trick turn left or right on crossings to keep

the property of single incoming edge and single

outgoing edge. - It does not preserve the graph, but preserves all

the ends of lines.

Detours at Crossings

Properties

- Properties of the reduction
- The new graph is larger than the original by an

exponential factor. - At each node of the grid, it can be calculated in

polynomial time - Whether an embedded line goes through it, and in

which direction. - Whether two embedded lines are at it or close

enough to change its incoming/outgoing edges. - The set of the ends of the lines is exactly the

same as before the reduction

Embedding of a complete graph of three nodes

Embedding of a Subgraph (02,21)

Theorem

- Another End of Lines embedded on 2D is

PPAD-complete - Xi Chen, Xiaotie Deng, On the Complexity of 2D

Discrete Fixed Point Problem, ICALP 2006.

Outline of 2D SPERNER

- Start with Another End of Lines embedded on 2D
- Encode it so that
- We can triangulate a triangle and properly label

it. - Do it locally according to the coordinates.
- A completely labeled triangle correspond to an

end except the start. - 0 on path, 1 on left, 2 on right and everywhere
- The end of path will have a triangle of three

labels.

A direct line/cycle graph

Corresponding triangulation

2D Brower Fixed Point

- Input Exponential size grid
- A function of three values (0,1), (1,0),

(-1,-1). - Output A unit square on vertices of which the

function values have all three possibilities

2D Brower Fixed Point simplex version

- Input Exponential size grid
- A function of three values (0,1), (1,0),

(-1,-1). - Output A triangle on a unit square on vertices

of which the function values have all three

possibilities

Reduction to Bimatrix Game

Special Features as a Natural Problem

- Input two m by n matrices A and B
- Output two vectors x and y as Nash equilibrium
- The input size is no longer exponential
- The input data are explicitly given
- The solution is exact

Reduction from 2D Fixed Point

- Code logic operations (and arithmetic operations)

by Games of Two Players - Use them to encode polynomial algorithms to

calculate the function values from input variable

values - Make the output function value equal to the

unspecified input value (fixed point) - Nash equilibrium obtains the result which derive

the fixed point.

A two player game for addition

- Player one
- Input Nodes a, b and
- Output node c
- Player two intermediate node d
- Pure strategies s1lt(a, 1), (d, 1)gt s2lt(b, 1),

(d, 1)gt s3lt(c, 1), (d, 0)gts4lt(c, 1), (d,

1)gts5lt(c, 0), (d, 0)gt - Payoffs
- For Player one one for s4, s5 zero otherwise
- For Player two one for s1, s2, s3 zero

otherwise.

Proof addition gate

- Pure strategies s1lt(a, 1), (d, 1)gt s2lt(b, 1),

(d, 1)gt s3lt(c, 1), (d, 0)gts4lt(c, 1), (d,

1)gts5lt(c, 0), (d, 0)gt - Payoffs
- For Player one one for s4, s5 zero otherwise
- For Player two one for s1, s2, s3 zero

otherwise. - Equilibrium probability
- Player one x on a, y on b, z on c
- Player two w on d
- At equilibrium player two has the same utility

choosing zero or one - Player two xyz
- Player one w(1-w)

Other operations

- Can be implemented by a two player game in a

similar manner. - Leave them as homework for students.

Main difficulties

- Comparison operator (lt or gt) cannot be

implemented by Nash equilibrium exactly nor

approximately. - Each operation can be done by two players but it

is hard to combine all the operations to a two

player game. - The dependence of an operation on its

input/output must not interference with other

operations - This part is the trickiest part of the Chen/Deng

paper.

Summary Discussion

Mile Stones

- Existence
- Von Neuman zero sum game 1928
- Nash non-zero sum game 1951
- Algorithms
- Cottle and Dantzig, LCP Model, 1962-3
- Lemke and Howson, path-following, 1964
- R. Savani and B. von Stengel, Exponential lower

bound of LH algorithm, 2004 - Complexity
- Papadimitriou, NASH is in PPAD,1991
- Daskalakis, Goldberg, Papadimitriou, 4-NASH is

PPAD-complete, Oct 10, 2005 - ChenDeng, DaskalakisPapadimitriou, 3-NASH is

PPAD-complete Nov18, 2005, Nov29 2005 - Chen and Deng, 2 player Nash Equilibrium is

PPAD-complete Dec 04, 2005

The end brings us back to the beginning

- Lemkes algorithm solves the two player nash

equilibrium developed a new combinatorial

algorithmic paradigm in computation of a large

body of problems in continuous variables. - In comparison, Dantzigs simplex algorithm.
- Problems include nash, fixed points, general

equilibrium - In the end, the solution structure of Lemke to

bimatrix game that started the history is proven

to be reduced to any solution to the bimatrix

game. - But still Lemke-Howsons algorithm is considered

the most practical one (with a proven convergence

theorem) for Nash.

A New Start

- Chen Deng Teng
- No FPTAS for two player nash equilibrium
- No polynomial time smoothed algorithm.
- and provides a first nontrivial lower bound in

smoothed analysis, for a central problem in

computing.

Past related work

- Michael D. Hirsch, Christos H. Papadimitriou,

Stephen A. Vavasis - Exponential lower bounds for finding Brouwer fix

points. J. Complexity 5(4) 379-416 (1989) - C. Papadimitriou
- The Complexity of the Parity Argument and Other

Inefficient proofs of Existence (1991)

Past related work

- Xi Chen, Xiaotie Deng
- On algorithms for discrete and approximate

brouwer fixed points. STOC 2005 323-330 - On the Complexity of 2D Discrete Fixed Point

Problem, ICALP 2006.

Concepts of Discrete Fixed Point

- IIMura (2002) Fixed point on lattice point for

direction preserving functions. - Chen and Deng (STOC 2005) Fixed point algorithms

on lattice for functions with 2n values in n

coordinates (, -) in each coordinate,

introducing the concept of bad cubes. - Chen and Deng (COCOON 2006) bad simplex.
- Daskalakis, Goldberg, Papadimitriou (STOC 2006)

A new discrete definition of the fixed point set

(on 238 points in 3D, closely related to bad

cubes), and consider functions with n1 values,

one is each coordinate and one diagonal ray, with

no other restrictions. - Chen/Deng/Teng (ECCC 2006) A new definition of

the discrete fixed point set (dependent on

function values on n1 points in an n-dimensional

space, related to bad simplex).

Related Progress

- IIMura (2002) A fixed point theorem on lattice

points for direction preserving functions. - Chen and Deng (STOC 2005) Fixed point algorithms

with oracle model, closing the gap by improving

both upper and lower bound. - Daskalakis, Goldberg, Papadimitriou (STOC 2006)

4NASH is PPAD complete - Chen and Deng (ECCC2005) PPAD-complete result for

bimatrix NASH - Chen and Deng (ICALP 2006) Hardness of 2D

SPERNER, 2D fixed points. - Chen/Deng/Teng (ECCC2006) A high dimension fixed

point hardness result, for a high dimension cube

with a constant side length, and NFPTS results

for NASH, and no polynomial time smoothed

algorithm for NASH.

Open problems

- Could PPAD still in P?
- Or it is NP-hard
- How would we place it with other complexity

classes?

Open problems

- Several other related problems discussed in PPAD

class by Papadimitriou. Could we improve those

results?

Open problems

- Oracle results
- Fixed point problem
- Sperners Lemma
- What about other problems?
- Is there an oracle version of NASH?

Open problems

- The negative result for the smoothed analysis

derived later with Chen and Teng is the first

lower bound for a non-trivial problem. Could the

idea be of use for other problems?

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