View by Category

Loading...

PPT – Computing Nash Equilibrium PowerPoint presentation | free to download - id: 9249b-NTU4Y

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Computing Nash Equilibrium

- Presenter Yishay Mansour

Outline

- Problem Definition
- Notation
- Last week Zero-Sum game
- This week
- Zero Sum Online algorithm
- General Sum Games
- Multiple players approximate Nash
- 2 players exact Nash

Model

- Multiple players N1, ... , n
- Strategy set
- Player i has m actions Si si1, ... , sim
- Si are pure actions of player i
- S ?i Si
- Payoff functions
- Player i ui S ? ?

Strategies

- Pure strategies actions
- Mixed strategy
- Player i pi distribution over Si
- Game P ?i pi
- Product distribution
- Modified distribution
- P-i probability P except for player i
- (q, P-i ) player i plays q other player pj

Notations

- Average Payoff
- Player i ui(P) EsPui(s) ? P(s)ui(s)
- P(s) ?i pi (si)
- Nash Equilibrium
- P is a Nash Eq. If for every player i
- For any distribution qi
- ui(qi,P-i) ? ui(P)
- Best Response

Two player games

- Payoff matrices (A,B)
- m rows and n columns
- player 1 has m action, player 2 has n actions
- strategies p and q
- Payoffs u1(pq)pAqt and u2(pq) pBqt
- Zero sum game
- A -B

Online learning

- Playing with unknown payoff matrix
- Online algorithm
- at each step selects an action.
- can be stochastic or fractional
- Observes all possible payoffs
- Updates its parameters
- Goal Achieve the value of the game
- Payoff matrix of the game define at the end

Online learning - Algorithm

- Notations
- Opponent distribution Qt
- Our distribution Pt
- Observed cost M(i, Qt)
- Should be MQt, and M(Pt,Qt) Pt M Qt
- cost on 0,1
- Goal minimize cost
- Algorithm Exponential weights
- Action i has weight proportional to bL(i,t)
- L(i,t) loss of action i until time t

Online algorithm Notations

- Formally
- Number of total steps T is known
- parameter b 0lt b lt 1
- wt1(i) wt(i) bM(i,Qt)
- Zt ? wt(i)
- Pt1(i) wt1(i) / Zt
- Initially, P1(i) gt 0 , for every i

Online algorithm Theorem

- Theorem
- For any matrix M with entries in 0,1
- Any sequence of dist. Q1 ... QT
- The algorithm generates P1, ... , PT
- RE(AB) ExA ln (A(x) / B(x) )

Relative Entropy

- For any two distributions A and B
- RE(AB) ExA ln (A(x) / B(x) )
- can be infinite
- B(x) 0 and A(x) ? 0
- Always non-negative
- log is concave
- ? ai log bi ? log ? ai bi
- ? A(x) ln B(x) / A(x) ? ln ? A(x) B(x) / A(x) 0

Online algorithm Analysis

- Lemma
- For any mixed strategy P
- Corollary

Online Algorithm Optimization

- b 1/(1 sqrt2 (ln n) / T)
- additional loss
- O(sqrt(ln n )/T)
- Zero sum game
- Average Loss v
- additional loss O(sqrt(ln n )/T)

Example Zero Sum

Two players General sum games

- Input matrices (A,B)
- No unique value
- Computational issues
- find some Nash,
- all Nash
- Can be exponentially many
- identity matrix
- Example 2xN

Computational Complexity

- Complexity of finding a sample equilibrium is

unknown - no proof of NP-completeness seems possible

(Papadimitriou, 94) - Equilibria with certain properties are NP-Hard
- e.g., max-payoff, max-support
- (Even) for symmetric 2-player games
- ? NE with expected social welfare at least k?
- ? NE with least payoff at least k?
- ? Pareto-optimal NE?
- ? NE with player 1 EU of at least k?
- ? multiple NE?
- ? NE where player 1 plays (or not) a particular

strategy?

Gilboa Zemel, Conitzer Sandholm

Two players General sum games

- player 1 best response
- Like for zero sum
- Fix strategy q of player 2
- maximize p (Aqt) such that ?j pj 1 and pj ?0
- dual LP minimize u such that u ? Aqt
- Strong Duality p(Aqt) u p u
- p( u Aq) 0
- complementary system
- Player 2 q(v- pB) 0

Nash Linear Complementary System

- Find distributions p and q and values u and v
- u ? Aqt
- v ? pB
- p( u Aq) 0
- q(v- pB) 0
- ?j pj 1 and pj ? 0
- ?j qj 1 and qj ? 0

Two players General sum games

- Assume the support of strategies known.
- p has support Sp and q has support Sq
- Can formulate the Nash as LP

Approximate Nash

- Assume we are given Nash
- strategies (p,q)
- Show that there exists
- small support
- epsilon-Nash
- Brute force search
- enumerate all small supports!
- Each one requires only poly. time
- Proof!

Nash Linear Complementary System

- Find distributions p and q and values u and v
- u ? Aqt
- v ? pB
- p( u Aq) 0
- q(v- pB) 0
- ?j pj 1 and pj ? 0
- ?j qj 1 and qj ? 0

Lemke Howson

- Define labeling
- For strategy p (player 1)
- Label i if (pi0) where i action of player 1
- Label j if action j (payer 2) is best response

to p - bj p ? bkp
- Similar for player 2
- Label j if (qj0) where j action of player 2
- Label i if action i (payer 1) is best response

to q - ai q ? ajq

LM algo

- strategy (p,q) is Nash if and only if
- Each label k is either a label of p or q (or

both) - Proof!
- Example

Lemke-Howson Example

G1

G2

a3

a5

(0,0,1)

(0,1)

1

2

4

(0,1/3,2/3)

4

2

(1/3,2/3)

1

a1

3

(2/3,1/3)

5

(1,0,0)

a4

(2/3,1/3,0)

(1,0)

5

3

(0,1,0)

a2

U2

U1

Lemke-Howson Example

G1

G2

a3

a5

(0,0,1)

(0,1)

1

2

4

(0,1/3,2/3)

4

2

(1/3,2/3)

1

a1

3

(2/3,1/3)

5

(1,0,0)

a4

(2/3,1/3,0)

(1,0)

5

3

(0,1,0)

a2

U2

U1

LM non-degenerate

- Two player game is non-degenerate if
- given a strategy (p or q)
- with support k
- At most k pure best responses
- Many equivalent definitions
- Theorem For a non-degenerate game
- finite number of p with m labels
- finite number of q with n labels

LM Graphs

- Consider distributions where
- player 1 has m labels
- player 2 has n labels
- Graph (per player)
- join nodes that share all but 1 label
- Product graph
- nodes are pair of nodes (p,q)
- edges if (p,p) an edge then (p,q)-(p,q) edge

LM

- completely labeled node
- node that has mn labels
- Nash!
- node k-almost completely labeled
- all labeling but label k.
- edge k-almost completely labeled
- all labels on both sides except label k
- artificial node (0,0)

LM Paths

- Any Nash Eq.
- connected to exactly one vertex which is
- k-almost completely labeled
- Any k-almost completely labeled node
- has two neighbors in the graph
- Follows from the non-degeneracy!

LM algo

- start at (0,0)
- drop label k
- follow a path
- end of the path is a Nash

Lemke-Howson Algorithm

a3

a5

(0,0,1)

G1

G2

(0,1)

1

2

4

(0,1/3,2/3)

4

2

(1/3,2/3)

1

a1

3

(2/3,1/3)

5

(1,0,0)

a4

(2/3,1/3,0)

(1,0)

5

3

(0,1,0)

a2

Lemke-Howson Algorithm

a3

a5

G2

(0,0,1)

G1

(0,1)

1

2

4

(0,1/3,2/3)

4

2

(1/3,2/3)

1

a1

3

(2/3,1/3)

5

(1,0,0)

a4

(2/3,1/3,0)

(1,0)

5

3

(0,1,0)

a2

Lemke-Howson Algorithm

a3

a5

(0,0,1)

G1

G2

(0,1)

1

2

4

(0,1/3,2/3)

4

2

1

(1/3,2/3)

a1

3

(2/3,1/3)

5

(1,0,0)

a4

(2/3,1/3,0)

(1,0)

5

3

(0,1,0)

a2

Lemke-Howson Other Equilibria

a3

a5

G1

(0,0,1)

G2

(0,1)

1

2

4

(0,1/3,2/3)

4

2

1

(1/3,2/3)

a1

3

(2/3,1/3)

5

(1,0,0)

a4

(2/3,1/3,0)

(1,0)

5

3

(0,1,0)

a2

LM Theorem

- Consider a non-degenerate game
- Graph consists of disjoint paths and cycles
- End points of paths are Nash
- or (0,0)
- Number of Nash is odd.

LM Sketch of Proof

- Deleting a label k
- making support larger
- making BR smaller
- Smaller BR
- solve for the smaller BR
- subtract from dist. until one component is zero
- Larger support
- unique solution (since non-degenerate)

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 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: "Computing 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!