View by Category

Loading...

PPT – Nash Equilibrium PowerPoint presentation | free to view - id: 1fc15e-ZDc1Z

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Nash Equilibrium

- A strategy profile s (s1, , sn) is a Nash

Equilibrium if, for all agents i, si is a best

response to s-i. - two strategies si and s-i are in Nash equilibrium

if - under the assumption that agent i plays si, agent

j can do no better than play s-i and - under the assumption that agent j plays s-i,

agent i can do no better than play si. - Each agent chooses a strategy that is a best

response to the other agents strategies.

Nash Equilibrium (NE)

- How hard is the Nash Equilibrium to compute?

Normal Form Games

- Finite, n-person normal form game (N, A, O, m,

u) - N is a finite set of n players, indexed by i
- A A1, . . . , An is a set of actions for each

player i - a A is an action profile
- O is a set of outcomes
- m A O
- u u1, ... , un, a utility function for each

player, where ui A

Game Definitions

- Common payoff games a game where all action

profiles a A1 x x An for any pair of agents i,

j, result in ui(a) uj(a). - Constant sum games A game in which a constant c

exists st for each strategy profile a A1 x A2 it

is the case that u1(a) u2(a) c.

Linear Programming

- A linear program is defined by
- A set of real-valued variables
- A linear objective function
- A weighted sum of the variables
- A set of linear constraints
- The requirement that a weighted sum of the

variables must be greater than or equal to some

constant

Linear programming

- maximize Si wixi
- subject to Si wcixi bc c C
- xi 0 xi X
- These problems can be solved in polynomial time

using interior point methods. - Interestingly, the (worst-case exponential)

simplex method is often faster in practice.

NE of two-player zero-sum games

- Easiest NE to find via linear programming in

polynomial time. - Let G (1,2, A1, A2, u1, u2) be a two-player

zero-sum game. - Ui is the unique expected utility for player i

in equilibrium. - We know that the players interests are opposing,

what does this tell us about U1 and U2? - U1 - U2

NE of two-player zero-sum games

- minimize U1
- subject to u1(aj1, ak2) sk2 U1

j A1 - sk2 1
- sk2 0 k A2
- Variables
- U1 is the expected utility for player 1
- sa22 is player 2s probability of playing action

a2 under his mixed strategy - each u1(a1, a2) is a constant.

NE of two-player zero-sum games

- minimize U1
- subject to u1(aj1, ak2) sk2 rj1 U1

j A1 - sa22 1
- sk2 0 k A2
- rj1 0 j A1
- Introduce slack variables, rj1.

NE of two-player general-sum games

- The NE of a two-player general-sum game cannot be

represented by a linear program. Why?

NE of two-player general-sum games

- Computation of the NE in this case is not

NP-Complete. Why? - However, it does appear to be in the PPAD

complexity class.

NE of two-player general-sum games

- PPAD Class of problems
- Define a family of directed graphs, G(n).
- Let each graph in G(n) contain a number of nodes

that is exponential in n. - Let Parent N N and Child N N
- Let there be one graph g G(n) for every such

pair of Parent and Child functions as long as - An edge exists from a node j to a node k iff

Parent(k) j and Child(j) k. - There must exist one distinguished node 0 N

with exactly zero parents.

NE of two-player general-sum games

- Reformulate previous solution to explicitly

consider both players. - u1(aj1, ak2) sk2 rj1 U1 j A1
- u2(aj1, ak2) sk1 rj2 U2 k A1
- sj1 1, sk2 1
- sj1 0, sk2 0 j A1, k A2
- rj1 0, rk2 0 j A1 , k A2
- rj1 sj1 0, rk2 sk2 0 j A1 , k A2

NE of two-player general-sum games

- The complementarity condition requires that when

an action is played by a particular player with

positive probability, the corresponding slack

variable must be zero. - Each slack variable represents the players

incentive to deviate from the corresponding

action. - Linear Complementarity Problem (LCP)

NE of two-player general-sum games

- Solve the LCP with the Lemke-Howson algorithm

a12

a22

How many pure strategies does agent a1 have?

a11

What about agent a2?

a21

a31

NE of two-player general-sum games

- Lemke-Howson algorithm

a12

a22

a11

a21

a31

How do we represent a2s Strategy space?

How do we represent a1s Strategy space?

NE of two-player general-sum games

- Lemke-Howson algorithm
- Label the strategies.
- A pair of strategies (s1, s2) is a NE iff it is

completely labeled (L(s1) L(s2) A1 A2).

s31

s22

Where are the multiply labeled points?

(0,0,1)

(0,1)

(0,1/3,2/3)

(1/3, 2/3)

(1,0,0)

s11

(2/3, 1/3)

s12

(2/3,1/3,0)

(1,0)

(0,1,0)

s21

NE of two-player general-sum games

- Lemke-Howson algorithm
- Identify the NE

(0,0,1)

a11 , a21 , a12

(0,1)

a11 , a12

(0,1/3,2/3)

(1/3, 2/3)

a11 , a12 , a22

a11 , a21

a21 , a31 , a12

(1,0,0)

(2/3, 1/3)

(0,0,0)

a21 , a31

a11 , a21 , a31

(2/3,1/3,0)

a31 , a12 , a22

(0,0)

(1,0)

a31 , a22

a11 , a22

(0,1,0)

a11 , a31 , a22

((0,0,1), (0,1)), ((0,1/3,2/3), (2/3,

1/3)), ((2/3,1/3,0), (1/3, 2/3))

NE of two-player general-sum games

- Lemke-Howson algorithm
- Initialize
- (s1, s2) (0, 0)
- Find an s1 G1 s.t. s1 is adjacent to 0
- x 1
- Repeat
- sx sx
- let aji be the label that occurs in both s1 and

s2 - Find an sx Gx s.t. sx is adjacent to sx and

aji L(sx) - x 3 x
- Until (s1, s2) is a completely labeled pair.

NE of two-player general-sum games

(0,0,1)

(0,0,1)

a11 , a21 , a12

a11 , a21 , a12

(0,1)

(0,1)

a11 , a12

a11 , a12

(0,1/3,2/3)

(0,1/3,2/3)

(1/3, 2/3)

(1/3, 2/3)

(1/3, 2/3)

a11 , a12 , a22

a11 , a12 , a22

a11 , a21

a11 , a21

a11 , a21

a21 , a31 , a12

(1,0,0)

(2/3, 1/3)

(2/3, 1/3)

(2/3, 1/3)

(0,0,0)

(0,0,0)

a21 , a31

a21 , a31

a21 , a31

a11 , a21 , a31

a11 , a21 , a31

(2/3,1/3,0)

(2/3,1/3,0)

a31 , a12 , a22

a31 , a12 , a22

(0,0)

(1,0)

(1,0)

(0,0)

(1,0)

a31 , a22

a31 , a22

a31 , a22

a11 , a22

a11 , a22

(0,1,0)

(0,1,0)

a11 , a31 , a22

a11 , a31 , a22

((2/3,1/3,0), (1/3,2/3)) ((0,0,0), (0,0))

((0,1,0), (0,1)) ((2/3,1/3,0), (0,1))

((2/3,1/3,0), (1/3,2/3))

((2/3,1/3,0), (1/3,2/3)) ((2/3,1/3,0),

(1/3,2/3)) ((0,1/3,2/3)), (1/3,2/3)

((0,1/3,2/3), (2/3, 1/3))

((0,0,1), (1,0)) ((0,0,0), (0,0)) ((0,0,1)),

(0,0) ((0,0,1), (1,0))

NE of two-player general-sum games

- Lemke-Howson algorithm
- Advantages
- Guaranteed to find a sample NE.
- Non-determinism is concentrated in the first

move. - Disadvantages
- Not guaranteed to find all NE.
- Does not provide guidance on choosing a good

first move.

NE of two-player general-sum games

- Heuristics and the Support-Enumeration Method can

be combined to provide an algorithm to find NE. - Search for NE can be reduced to searching the

space of supports. - Use a feasibility program that tests the

specified supports. - Complete
- Worst case performance exponential

Specific NE of General-sum Games

- Idea is to find an equilibrium with a specific

property. - Properties
- Uniqueness
- Pareto optimal
- Guaranteed payoff
- Guaranteed social welfare
- Subset inclusion
- Subset containment
- NP-Hard when applied to NE.

NE for n-player general-sum games

- Cannot formulate as a linear complementarity

problem. - Sequence of linear complementarity problems

(SLCP). - Each LCP is an approximation of the problem and

is used to developed the next approximation in

the sequence.

NE for n-player general-sum games

- Formulate the problem as a minimum of a function.
- Constrained optimization problem
- Unconstrained optimization problem
- Disadvantages
- Both have local minima that do not correspond to

the NE.

NE for n-player general-sum games

- Simplicial subdivision algorithms
- Consider
- the space of mixed strategies is a simlpex
- The players best response is a function from

points on the simplex to other points on the

simplex. - Scarfs algorithm locates the fixed points.
- Add a variable that expresses the accuracy of the

current iterations approximation. - Worst case complexity exponential in the number

of players and the number of digits of accuracy.

NE for n-player general-sum games

- Generalize SEM to the n-player case
- The feasibility program becomes non-linear.
- Algorithm must accommodate multiple variables in

the feasibility problem. - Use standard numerical techniques for non-linear

optimization. - Reverse the lexicographic ordering between size

and balance of supports.

All NE of General-sum Games

- Idea is to determine all equilibria of a game.
- Important when designing a game and need to know

all possible stable outcomes. - Worst case exponential in the number of actions

for each player.

Dominant Strategies

- A strategy dominates another when the first

strategy is always at least as good as the

second, independent of the other players

actions. - Iterative removal
- Strictly dominant strategies order does not

matter. - Very weakly and weakly dominant strategies

removal order can have an affect. - Potentially remove some equilibrium of the

original game. - Potentially remove a larger set of strategies and

result in a smaller game.

Domination by a Pure Strategy

- for all pure strategies ai Ai for player i where

ai ? si do - dom true
- for all pure strategy profiles a-i A-i for the

players other than i do - if ui(si, a-i) ui(ai, a-i) then
- dom false
- break
- end if
- end for
- if dom true then return true
- end for
- return false

Domination by a Mixed Strategy

- Recall that mixed strategies cannot be

enumerated. - Strict Domination
- Requires a linear program.
- Minimize
- Subject to

Domination by a Mixed Strategy

- Very weakly dominate

Domination by a Mixed Strategy

- Weak domination

Maximize

Subject to

Iterated Dominance

- Strategy Elimination Does there exist some

elimination path under which the strategy si is

eliminated? - Reduction Identity Given action subsets A-i Ai

for each player i, does there exist a maximally

reduced game where each player i has the actions

A-i? - Uniqueness Does every elimination path lead to

the same reduced game? - Reduction Size Given constants ki for each

player i, does there exist a maximally reduced

game where each player i has exactly ki actions? - Iterated strict dominance problems are all in P.
- Iterated weak or very weak dominance problems

are NP-complete.

Correlated Equilibrium n-player general-sum games

Variables p(a) constants ui(a)

Could find the social-welfare maximizing the

correlated equilibrium by adding an objective

function

Maximize

Correlated Equilibrium

- Complexity of P when applied to CE
- Uniqueness
- Pareto Optimal
- Guaranteed payoff
- Subset inclusion
- Subset containment

CE to NE Calculation

Intuitively, correlated equilibrium has only a

single randomization over outcomes, whereas in NE

this is constructed as a product of independent

probabilities.

Changing this program so that it finds NE

requires the first constraint to be

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 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "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!

Committed to assisting Vanderbilt University and other schools with their online training by sharing educational presentations for free