To view this presentation, you'll need to enable Flash.

Show me how

After you enable Flash, refresh this webpage and the presentation should play.

Loading...

PPT – The Computational Complexity of Finding a Nash Equilibrium PowerPoint presentation | free to view - id: 20d556-ZDc1Z

The Adobe Flash plugin is needed to view this content

View by Category

Presentations

Products
Sold on our sister site CrystalGraphics.com

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

The Computational Complexityof Finding a Nash

Equilibrium

- Edith Elkind, U. of Warwick

Based On

- Reducibility Among Equilibrium Problems

(Goldberg, Papadimitriou) Aug 2005 - The Complexity of Computing a Nash Equilibrium

(Goldberg, Daskalakis, Papadimitriou) Sep 2005 - 3-NASH is PPAD-Complete

(Chen, Deng) Nov 2005 - Three-Player Games Are Hard

(Daskalakis, Papadimitriou) Nov 2005 - Settling the Complexity of 2-Player

Nash-Equilibrium (Chen, Deng) Dec 2005

Normal Form Games

- finite set of players 1, , n
- each player has k actions
- (pure strategies) 1, , k
- payoffs of the ith player Pi 1, , kn ? R

Row player

Column player

Nash Equilibrium

- Nash equilibrium a strategy profile such that
- noone wants to deviate given other players

strategies, i.e., each players strategy is a

best response to others strategies - (0, 0) and (1, 1) are both NE

Row player

Column player

Pure vs. Mixed Strategies

- NE in pure strategies may not exist!
- matching pennies
- Mixed strategy a probability distribution over

actions - 50 tail, 50 head

Row player

Column player

Existence of NE

- Theorem (Nash 1951)

any game in normal form has an equilibrium in

mixed strategies - 1 000 000 question
- how to find one?

Finding mixed NE in 2 x 2 Games

Suppose R plays 1 w.p. r EP(C) from playing

0 (1-r)1 EP(C) from playing 1 r3 1-r

3r iff r ¼

Suppose C plays 1 w.p. c EP(R) from playing 0

(1-c)2 EP(R) from playing 1 c1 (1-c)2 c

iff c 2/3

NE r1/4, c2/3

Row player

Column player

2 players, k actions

- Representation two k x k matrices
- Checking for pure NE easy
- at most k2 of them
- Checking for mixed NE
- all straightforward methods are exptime
- Lemke-Howson algorithm is exptime, too (previous

talk) - For 2 players all NE are rational
- but not for 3 and more players

n players, 2 actions

- Representation payoffs to each player for every

action profile (vector in 0, 1n) n2n numbers - graphical games
- players are associated with the vertices of a

graph - each players payoff depends on his own action

and actions of his neighbors - n players, max degree d gt n2d1 numbers

W

t0, u0, v0, w0 12 t1, u0, v0, w0 31

. t1, u1, v1, w1 -6

Ws payoffs (16 cases)

T

V

U

Algorithms for NE in Graphical

Games

- Bounded-degree trees
- Exp-time algorithm/poly-time approximation

algorithm to find all NE (Kearns, Littmann,

Singh, UAI 2001) - ??? poly-time algorithm to find a single NE

(Kearns, Littmann, Singh, NIPS2001) - shown to be incorrect in E., Goldberg, Goldberg,

ACM EC06 - Graphs of max degree 2
- poly-time algorithm (EGG06)

Is Finding NE NP-hard?

- Reminder a problem P is NP-hard if you can

reduce 3-SAT to it - yes-instance 3-SAT ? yes-instance of P
- no-instance 3-SAT ? no-instance of P
- Problem each instance of NASH is

a yes-instance! - every game has a NE
- need complexity theory for search problems
- Side note pure Nash for n players, NE of total

value gt K are NP-hard

Reducibility Among Search Problems

S X Y

T X Y

- S associates x in X with a solution set S(x)
- Total search problem for any x, S(x) is not empty

If T is easy, so is S

Equivalences GP05

deg d graphical game G NE of G

d2-player game G NE of G

d-Graphical Game GG ? d2-Player Game G

- Color the graph of GG

d(u,v) 2 ? color(u) ? color(v) - Each color is a player of G
- RED chooses a red vertex in GG

and an action for that vertex in GG - payoffpayoff1payoff2
- payoff1 BLUE tries to guess which vertex RED

chose RED pays a penalty if BLUE guesses

correctly - payoff2 if all neighbors of a chosen vertex are

also chosen, it gets same payoff as in GG, else 0

r-Player Game G ?

3-Graphical Game GG

- Si space of pure strategies of player i
- S- i S1 Si-1Si1 .. Sr
- xij the probability that ith player uses jth

strategy - xs x1s1 x2s2 xrsr (for s in S-i)
- uijs utility of the ith player when he plays j

and others play according to s

NE 0 xij 1 Sj xij 1 Ss in

S uijsxs gt Ss in S uijsxs implies xij 0

-p

-p

r-Player Game G ?

3-Graphical Game GG

- Vertex Vij for any pair (playeri, actionj)
- Want PrVij plays 1 Pr i plays j in Gxij
- Idea graphical games can do math!
- Enforce constraints from the previous slide

v1

v2

v3

Need gadgets for , , c, , min, max,

u

Set payoffs to u, v3 so that pv3pv1 pv2

Equivalences GP05

deg d graphical game G NE of G

d2-player game G NE of G

Combining Reductions GP05

Completeness Results?

- Can we prove that any total search problem is

reducible to r-NASH? - Not really the class T of all total search

problems is a semantic class - not known how to find complete problems for these
- Want to pick a large subclass S of T s.t.
- S includes some natural problems
- there are problems that are complete for S
- in particular, r-NASH is complete for S

END OF THE LINE

- Input Boolean circuits

S (Successor), P

(Predecessor) - n inputs, n outputs
- S(0n) ? 0n, P(0n) 0n
- Output x ? 0n s.t.
- S(P(x)) ? x or P(S(x)) ? x
- Intuition G(V, E)
- V Sn
- E (x,y) yS(x), xP(y)

PPAD

- PPAD Polynomial Parity Argument, Directed

version - PPAD is the class of all search problems that are

reducible to END OF THE LINE

search problem solution

g

f

circuits S, T end of the

line

r-NASH is in PPAD

- Proof on Nashs theorem
- existence of NE reduces to Brouwers fixpoint

theorem - Brouwers fixpoint theorem reduces to Sperners

lemma - Sperners lemma is proven by a parity argument

(similar to END OF THE LINE) - Reduction of r-NASH to END OF THE LINE can be

extracted from these proofs (Papadimitriou 94)

Brouwers Fixpoint Theorem

- Brouwers Theorem Any continuous mapping from

the simplex to itself has a fixpoint. - Nash ? Brouwer proof sketch
- set of all strategy profiles ? simplex
- mapping (s1, , sn) ? (s1d1, , sndn), where

di is a shift in the direction of best response

to (s1, , si-1, si1, , sn) - NE is a point where noone wants to deviate, i.e.,

a fixpoint

Sperners Lemma

- Proper coloring
- vertices on BC are not blue
- vertices on AC are not green
- vertices on AB are not yellow
- Sperners Lemma

there exists a trichromatic

triangle - Brouwers theorem ? Sperners Lemma
- x is blue if the grad(F) at x points away from A,

etc. - trichromatic triangle has no direction
- repeat at increased resolution

Opposite Direction 3D-BROUWER

- Input
- 3D unit cube divided into 23n cubelets
- cijk is the center of Kijk
- f(cijk)cijkdijk, dijk is in d0, d1, d2, d3,

where - d1(a, 0, 0), d2(0, a, 0)
- d3(0, 0, a), d0(-a, -a, -a)
- circuit C 0, 1 3n ? 0, 1, 2, 3 selects dijk
- Output
- a panchromatic cubelet, i.e., one that has all of

d0, d1, d2, d3 among its 8 neighbors

3D-BROUWER is PPAD-complete

- Papadimitriou (1994) shows that a more

complicated version of 3D-BROUWER is

PPAD-complete - This version was proven hard in DGP05
- Reduction from END OF THE LINE
- embed the line L into 3d cube
- protect L from color 0 using three other colors
- color the rest of inner cubelets with 0

r-NASH vs 3D BROUWER

- Existence of NE follows from Brouwers fixpoint

theorem - NE are special cases of Brouwers fixpoints
- just how special?
- Can any fixpoint be represented as a NE of a

game? - DGP05 YES! ? 4-NASH is PPAD complete
- Proof
- 4-NASH ? deg 3 Graphical Nash
- graphical games can compute fixpoints

4-NASH to 3-NASH

- Daskalakis, Papadimitriou modify arithmetic

gadgets so that the graph is 3-colorable - Chen, Deng same gadgets, but allow for small

error

2-NASH

- Chen, Deng
- avoid graphical games
- reduce directly from 3D-BROUWER to

2-NASH using arithmetic gadgets similar to

graphical game gadgets - Game over?

Graphical Games Open Problems

- Degree
- deg 3 PPAD-complete (DGP05b)
- deg 2 polynomial time solvable (EGG06)
- Pathwidth
- paths poly-time
- pathwidth 1 maybe algorithm from EGG06 still

works - pathwidth 2 any KLS-style algo is exptime

(EGG06) - pathwidth gt r, r constant PPAD-complete (EGG06)
- Finding NE on trees?

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Presentation Removal Request Contact Us Send Us Feedback

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "The Computational Complexity of Finding a Nash Equilibrium" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!