Arpita rpita Bisw swas as PhD Stude udent nt (Google Fellow) - - PowerPoint PPT Presentation

Speaker

Arpita rpita Bisw swas as

PhD Stude udent nt (Google Fellow) Game Theory y Lab, Dept. . of CSA, Indi dian n Institut stitute of Science nce, , Bangalore re Email address ress: arpita

ta.bisw iswas@li as@live.in e.in

OUTLINE

Game Theory – Basic Concepts and Results Mechanism Design Cooperative Game Theory Real-World Applications

GAME THEORY

“Mathematical framework for rigorous study of conflict and cooperation among rati tiona nal and int ntellige gent nt agents”

GAME THEORY

“Mathematical framework for rigorous study of conflict and cooperation among rati tiona nal and int ntellige gent nt agents” the agent would always choose an action that maximizes her/his (expected) utility.

• competent enough to make any inferences about the

game that a game theorist can make.

• can carry out the required computations involved in

determining a best response strategy

GAME THEORY

“Mathematical framework for rigorous study of conflict and cooperation among rati tiona nal and int ntellige gent nt agents” the agent would always choose an action that maximizes her/his (expected) utility.

• competent enough to make any inferences about the

game that a game theorist can make.

• can carry out the required computations involved in

determining a best response strategy preferences of the players expressed in terms of real numbers

PRISONER’S DILEMMA

The problem is as follows:

• Two individuals arrested for a robbery (witnessed by several people).
• The police suspects that they were guilty of a similar crime earlier, but

were never caught.

• The prisoners are lodged in separate prisons and interrogation happens

separately

• The police tells each prisoner that:
• a. “If you are the only one to confess, you’ll get a light sentence of 1 year

while the other would be sentenced to 10 years in jail”.

• b. “If both of you confess, both of you would be sentenced for 5 years” .
• c. “If neither of you confess, then each of you would get 3 years in jail”.
• The police also informs each prisoner that the same has been told to the
• ther prisoner.

PRISONER’S DILEMMA

Bunty ty Bubly Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess Two Player ers: s:

• Bunty
• Bubly

Two Ac Action ions: s:

• Confess
• Not Confess

The utilit ity y matrix rix models the strategic conflict when two players have to choose their priorities

PRISONER’S DILEMMA

Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess The utilit ity y matrix rix models the strategic conflict when two players have to choose their priorities < 𝑂, (𝐵𝑗)𝑗∈𝑂 , (𝑉𝑗)𝑗∈𝑂 > 𝑂 ∶ set of players 𝐵𝑗 ∶ set of actions for player 𝑗 𝑉𝑗 ∶ 𝐵1 × ⋯ × 𝐵|𝑂| → ℝ Action profile or Strategy profile Bunty ty Bubly Two Player ers: s:

• Bunty
• Bubly

Two Ac Action ions: s:

• Confess
• Not Confess

PRISONER’S DILEMMA

Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess Bunty ty Bubly Two Player ers: s:

• Bunty
• Bubly

Two Ac Action ions: s:

• Confess
• Not Confess

PRISONER’S DILEMMA

Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess Bunty ty Bubly Two Player ers: s:

• Bunty
• Bubly

Two Ac Action ions: s:

• Confess
• Not Confess

PRISONER’S DILEMMA

Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess Bunty ty Bubly Two Player ers: s:

• Bunty
• Bubly

Two Ac Action ions: s:

• Confess
• Not Confess

PRISONER’S DILEMMA

Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess Bunty ty Bubly Two Player ers: s:

• Bunty
• Bubly

Two Ac Action ions: s:

• Confess
• Not Confess

PRISONER’S DILEMMA

Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess

Nash h Eq Equi uilibr briu ium

Bunty ty Bubly Two Player ers: s:

• Bunty
• Bubly

Two Ac Action ions: s:

• Confess
• Not Confess

PRISONER’S DILEMMA

Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess

Nash h Eq Equi uilibr briu ium

A strategy profile in which no player gains by changing only his/her own strategy (assuming no one else changes their strategy) Bunty ty Bubly Two Player ers: s:

• Bunty
• Bubly

Two Ac Action ions: s:

• Confess
• Not Confess

PROJECT COORDINATION GAME

Alice Bob Deep p Le Learn rning ng Deep Learning Webs bsite e Desi signi gning Websi site Designin gning Two Player ers: s:

• Alice
• Bob

Two Ac Action ions: s:

• Deep Learning

Project

• Website Designing

Project

PROJECT COORDINATION GAME

Alice Bob Deep p Le Learn rning ng Deep Learning Webs bsite e Desi signi gning Websi site Designin gning Two Player ers: s:

• Alice
• Bob

Two Ac Action ions: s:

• Deep Learning

Project

• Website Designing

Project

PROJECT COORDINATION GAME

Alice Bob Deep p Le Learn rning ng Deep Learning Webs bsite e Desi signi gning Websi site Designin gning Two Player ers: s:

• Alice
• Bob

Two Ac Action ions: s:

• Deep Learning

Project

• Website Designing

Project Nash h Equil quilib ibria

PROJECT COORDINATION GAME

Alice Bob Deep p Le Learn rning ng Deep Learning Webs bsite e Desi signi gning Websi site Designin gning Two Player ers: s:

• Alice
• Bob

Two Ac Action ions: s:

• Deep Learning

Project

• Website Designing

Project Does s there exist st there e any ot

• ther

er Nash h Equi quilibr ibrium ium in this s game? e? Nash h Equil quilib ibria

PROJECT COORDINATION GAME

Alice Bob Deep p Le Learn rning ng Deep Learning Webs bsite e Desi signi gning Websi site Designin gning Two Player ers: s:

• Alice
• Bob

Two Ac Action ions: s:

• Deep Learning

Project (DL)

• Website Designing

Project (WD) Does s there exist st there e any ot

• ther

er Nash h Equi quilibr ibrium ium in this s game? e? Alice: With probability 2/3 choose DL and with probability 1/3 choose WD Bob: With probability 1/3 choose DL and with probability 2/3 choose WD Nash h Equil quilib ibria

MIXED STRATEGY NASH EQUILIBRIUM

EXISTENCE OF NASH EQUILIBRIA IN GAMES

• Does Nash Equilibrium always exist?

EXISTENCE OF NASH EQUILIBRIA IN GAMES

𝑭𝒘𝒇𝒔𝒛 𝒈𝒋𝒐𝒋𝒖𝒇 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒋𝒅 𝒈𝒑𝒔𝒏 𝒉𝒃𝒏𝒇 𝒊𝒃𝒕 𝒃𝒖 𝒎𝒇𝒃𝒕𝒖 𝒑𝒐𝒇 𝒏𝒋𝒚𝒇𝒆 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝑶𝒃𝒕𝒊 𝑭𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏. ,𝑶𝒃𝒕𝒊 𝑼𝒊𝒇𝒑𝒔𝒇𝒏, 𝟐𝟘𝟔𝟏-.

• Does Nash Equilibrium always exist?

EXISTENCE OF NASH EQUILIBRIA IN GAMES

𝑭𝒘𝒇𝒔𝒛 𝒈𝒋𝒐𝒋𝒖𝒇 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒋𝒅 𝒈𝒑𝒔𝒏 𝒉𝒃𝒏𝒇 𝒊𝒃𝒕 𝒃𝒖 𝒎𝒇𝒃𝒕𝒖 𝒑𝒐𝒇 𝒏𝒋𝒚𝒇𝒆 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝑶𝒃𝒕𝒊 𝑭𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏. ,𝑶𝒃𝒕𝒊 𝑼𝒊𝒇𝒑𝒔𝒇𝒏, 𝟐𝟘𝟔𝟏-.

• Does Nash Equilibrium always exist?
• Is there an efficient algorithm for computing a mixed Nash equilibrium?

EXISTENCE OF NASH EQUILIBRIA IN GAMES

𝑭𝒘𝒇𝒔𝒛 𝒈𝒋𝒐𝒋𝒖𝒇 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒋𝒅 𝒈𝒑𝒔𝒏 𝒉𝒃𝒏𝒇 𝒊𝒃𝒕 𝒃𝒖 𝒎𝒇𝒃𝒕𝒖 𝒑𝒐𝒇 𝒏𝒋𝒚𝒇𝒆 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝑶𝒃𝒕𝒊 𝑭𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏. ,𝑶𝒃𝒕𝒊 𝑼𝒊𝒇𝒑𝒔𝒇𝒏, 𝟐𝟘𝟔𝟏-.

• Does Nash Equilibrium always exist?
• Is there an efficient algorithm for computing a mixed Nash equilibrium?

𝒋𝒕 𝑸𝑸𝑩𝑬 − 𝒅𝒑𝒏𝒒𝒎𝒇𝒖𝒇 𝑮𝒋𝒐𝒆𝒋𝒐𝒉 𝒏𝒋𝒚𝒇𝒆 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝑶𝒃𝒕𝒊 𝑭𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏 ,𝑬𝒃𝒕𝒍𝒃𝒎𝒃𝒍𝒋𝒕 𝒇𝒖 𝒃𝒎. , 𝟑𝟏𝟏𝟕-.

OTHER TYPES OF EQUILIBRIA

• Strongly

gly Dominant inant Strategy egy Equi quilibr ibriu ium m (SDSE SE): ): 𝐵𝑜 𝑏𝑑𝑢𝑗𝑝𝑜 𝑞𝑠𝑝𝑔𝑗𝑚𝑓 𝑏1, ⋯ , 𝑏𝑜 𝑗𝑡 𝑑𝑏𝑚𝑚𝑓𝑒 𝒕𝒖𝒔𝒑𝒐𝒉𝒎𝒛 𝒆𝒑𝒏𝒋𝒐𝒃𝒐𝒖 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝒇𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏 𝑔𝑝𝑠 𝑏 𝑕𝑏𝑛𝑓 < 𝑂, 𝐵𝑗 , 𝑉𝑗 >, 𝑗𝑔 ∀𝑗 ∈ 𝑂 𝑏𝑜𝑒 ∀𝑐𝑗 ∈ 𝐵𝑗 ∖ *𝑏𝑗+, 𝑉𝑗 𝑏𝑗, 𝑐−𝑗 > 𝑉𝑗 𝑐𝑗, 𝑐−𝑗 ∀𝑐−𝑗 ∈ 𝐵−𝑗 . Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess Bunty ty Bubly

• Weakly

y Domin inant nt Strategy egy Equi quilibr ibriu ium m (WDS DSE): E):

OTHER TYPES OF EQUILIBRIA

𝐵𝑜 𝑏𝑑𝑢𝑗𝑝𝑜 𝑞𝑠𝑝𝑔𝑗𝑚𝑓 𝑏1, ⋯ , 𝑏𝑜 𝑗𝑡 𝑑𝑏𝑚𝑚𝑓𝑒 𝒙𝒇𝒃𝒍𝒎𝒛 𝒆𝒑𝒏𝒋𝒐𝒃𝒐𝒖 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝒇𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏 𝑔𝑝𝑠 𝑏 𝑕𝑏𝑛𝑓 < 𝑂, 𝐵𝑗 , 𝑉𝑗 >, 𝑗𝑔 ∀𝑗 ∈ 𝑂 𝑏𝑜𝑒 ∀𝑐𝑗 ∈ 𝐵𝑗, 𝑉𝑗 𝑏𝑗, 𝑐−𝑗 ≥ 𝑉𝑗 𝑐𝑗, 𝑐−𝑗 ∀𝑐−𝑗 ∈ 𝐵−𝑗 𝑏𝑜𝑒 𝑉𝑗 𝑏𝑗, 𝑐−𝑗 > 𝑉𝑗 𝑐𝑗, 𝑐−𝑗 𝑔𝑝𝑠 𝑡𝑝𝑛𝑓 𝑐−𝑗 ∈ 𝐵−𝑗 . Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess Bunty ty Bubly

• Very Weakly Dominant

inant Strategy egy Equi quilibr ibriu ium m (VWDSE DSE): ):

OTHER TYPES OF EQUILIBRIA

𝐵𝑜 𝑏𝑑𝑢𝑗𝑝𝑜 𝑞𝑠𝑝𝑔𝑗𝑚𝑓 𝑏1, ⋯ , 𝑏𝑜 𝑗𝑡 𝑑𝑏𝑚𝑚𝑓𝑒 𝒘𝒇𝒔𝒛 𝒙𝒇𝒃𝒍𝒎𝒛 𝒆𝒑𝒏𝒋𝒐𝒃𝒐𝒖 𝒕𝒖𝒔𝒃𝒖𝒇𝒉𝒛 𝒇𝒓𝒗𝒋𝒎𝒋𝒄𝒔𝒋𝒗𝒏 𝑔𝑝𝑠 𝑏 𝑕𝑏𝑛𝑓 < 𝑂, 𝐵𝑗 , 𝑉𝑗 >, 𝑗𝑔 ∀𝑗 ∈ 𝑂 𝑏𝑜𝑒 ∀𝑐𝑗 ∈ 𝐵𝑗, 𝑉𝑗 𝑏𝑗, 𝑐−𝑗 ≥ 𝑉𝑗 𝑐𝑗, 𝑐−𝑗 ∀𝑐−𝑗 ∈ 𝐵−𝑗 . Confess ess Confess ess Not

• t Confess

ess Not

• t

Confess ess Bunty ty Bubly

NO DOMINANT STRATEGY EQUILIBRIA (PROJECT COORDINATION GAME)

Alice Bob Deep p Learning ng Deep Learning Websi site e Desi signing gning Websi site Designing gning

OTHER CATEGORIES OF GAMES

• Repeated games
• Dynamic games
• Stochastic games
• Network games
• Multi-level games (Stackelberg games)
• Differential games

. . .

Analyzing these games show how agents can rationally form beliefs over what

• ther agents will do, and (hence) how agents should act – Useful for taking a

profitable action as well as predicting behavior of others.

MECHANISM DESIGN

How would you create the rules of a game to achieve a desired objective? Ans: Mechanism hanism Design ign

MECHANISM DESIGN

How would you create the rules of a game to achieve a desired objective? Ans: Mechanism hanism Design ign

• “Reverse Engineering of Games”
• “Art of designing the rules of a game to achieve a specific desired outcome”

Game Theory, along with Mechanism Design have emerged as an important tool to model, analyze, and solve decentralized design problems in engineering involving multiple autonomous agents that interact strategically in a rational and intelligent way.

CAKE CUTTING

Courtesy: Google images

CAKE CUTTING

I want no less than half the cake I want no less than half the cake

Courtesy: Google images

CAKE CUTTING

I want no less than half the cake I want no less than half the cake

Courtesy: Google images

CAKE CUTTING

I want no less than half the cake I want no less than half the cake

Courtesy: Google images

CAKE CUTTING

I want no less than half the cake I want no less than half the cake

Courtesy: Google images

CAKE CUTTING

I want no less than half the cake I want no less than half the cake

Courtesy: Google images

CAKE CUTTING

I want no less than half the cake I want no less than half the cake

Courtesy: Google images

CAKE CUTTING: MECHANISM DESIGN

Soluti tion

• n:

: Cut ut and d Choo

• ose

se

Mother makes one of the kids “cutter” and the other “chooser”

• Cutter : Cuts the cake into two halves
• Chooser: Gets to select one of the haves

The cutter can cut the cake to two pieces that she considers equal. Then, regardless of what the chooser does, she is left with a piece that is as valuable as the other piece. The ch choo

• oser

ser can select the piece which he considers more valuable. Then, even if the cutter divided the cake to pieces that are very unequal (in the chooser's eyes), the chooser still has no reason to complain because he chose the piece that is more valuable in his

• wn eyes.

CAKE CUTTING: MORE THAN TWO KIDS

I want one a piece with at least one fourth of all the fruits I want a piece with at least one- fourth of all the kiwi pieces. I want the cake to be split into exactly 4 equal parts I want at least one- eighth of strawberry cream and at least one- eighth of kiwi cream

Courtesy: Google images

FAIR DIVISION

Courtesy: Google images

FAIR DIVISION

Courtesy: Google images

DESIRABLE PROPERTIES OF A MECHANISM

• Allocat

cativ ive Effici iciency ncy: Allocation should maximize the sum of value

• btained by all the players.
• Individ

ividua ual Ration ionalit ity: “Players do not loose anything by participating in the game” or “Voluntary Participation”

• Domin

inant nt Strategy egy Incen enti tive e Compat atib ibilit ity:”Strategy-proofness”

• Non-Dict

Dictat atorshi ship: p: “There is no agent for whom all outcomes turn out to be favored outcomes.”

COOPERATIVE GAME THEORY

• There is an incentive to cooperate collusion, binding contract, side

payment

• Players can form a group and cheat the system to get a better pay-off

Questions of Interest

• What are the conditions for forming stable coalitions?
• When will a single coalition (grand coalition) be formed?
• What is a “fair” distribution of payoffs among players

SHA HAPLEY LEY VALUE UE

EXAMP AMPLE: LE: PROJECT OJECT CONT NTRA RACT CT

A single project worth Rs. 300. One contractor (C) Two laborers (A and B).

• The contractor alone cannot finish the project without the laborers.
• The laborers cannot get the project contract without the contractor.
• If the contractor gets the project, it can be completed with the help of at

least one laborer.

EXAMP AMPLE: LE: PROJECT OJECT CONT NTRA RACT CT

A single project worth Rs. 300. One contractor (C) Two laborers (A and B).

• The contractor alone cannot finish the project without the laborers.
• The laborers cannot get the project contract without the contractor.
• If the contractor gets the project, it can be completed with the help of at

least one laborer. How to split the cost among the contractor and the two laborers? <100,100,100>?

EXAMP AMPLE: LE: PROJECT OJECT CONT NTRA RACT CT

Recall:

EXAMP AMPLE: LE: PROJECT OJECT CONT NTRA RACT CT

Recall:

EXAMP AMPLE: LE: PROJECT OJECT CONT NTRA RACT CT

Recall:

EXAMP AMPLE: LE: PROJECT OJECT CONT NTRA RACT CT

Recall: Shapley value split: <50, 50, 200>

OTHER SOLUTION CONCEPTS IN COOPERATIVE GAME THEORY

• Stable Sets
• Core
• Kernel
• Nucleolus
• The Gately Point

. . .

OTHER SOLUTION CONCEPTS IN COOPERATIVE GAME THEORY

• Stable Sets
• Core
• Kernel
• Nucleolus
• The Gately Point

. . .

Non–Cooperative Game Theory Mechanism Design Cooperative Game Theory

REAL-WORLD APPLICATIONS

• Resour
• urce

ce Allocation:

• cation: Find a “fair” split of resources among agents
• Procuremen

ement t Aucti ction:

• n: Design an auction that maximizes social utilities
• Crowdsour

dsourcin cing: g: Design a mechanism to complete as many task as possible with maximum quality.

• Online

ne Educat cation

• n Platf

tfor

• rms

ms (MOO OOCs Cs): Designing incentives to improve participation level of students and instructors.

• Social

al Net etwor

• rk

k Analysis ysis: Discovering influential nodes, providing incentives to ensure maximum spread of information over a network.

. . .

REAL-WORLD APPLICATIONS

REAL-WORLD APPLICATIONS

REAL-WORLD APPLICATIONS

REAL-WORLD APPLICATIONS

REAL-WORLD APPLICATIONS

SPONSORED (KEYWORD) SEARCH AUCTION

Separat ate e aucti tion

• n for every que

query: y:

• Posit

itions

• ns awarded

ed in order of bid (more e on this s later). ).

• Advertis

tisers ers pay bid of the adver erti tise ser r in the posit ition

• n below.

.

SPONSORED (KEYWORD) SEARCH AUCTION

Separate auction for every query:

• Positions are assigned to ads in the order of their bids.
• The payment is typically a function of the bids of the advertisers.

SPONSORED (KEYWORD) SEARCH AUCTION

Simp mple e set etting ting One “ad slot” and N competing advertisers Which ad to show and what should the advertiser pay? Solving this requires a mechan hanism sm comprising an allocat cation ion rule and a payme ment nt rule. Separate auction for every query:

• Positions are assigned to ads in the order of their bids.
• The payment is typically a function of the bids of the advertisers.

SPONSORED (KEYWORD) SEARCH AUCTION

Separate auction for every query:

• Positions awarded in order of bid (more on this later).
• The payment is typically a function of the bids of the advertisers.

VCG (Vickrey-Clarke-Groves) mechanism : * * Allocat cation ion rule: Give the ad-slot to the advertiser with maximum valuation/bid Simp mple e set etting ting One “ad slot” and N competing advertisers Which ad to show and what should the advertiser pay? Solving this requires a mechan hanism sm comprising an allocat cation ion rule and a payme ment nt rule. * * Pa Paymen ent t rule: Take the second-highest bid value from the selected advertiser

PAY PER CLICK AUCTION

• Each advertiser pays a value only when an user clicks the ad
• Each “ad” is an arm and “click probabilities” are the stochastic rewards

(parameters to be learnt)

• Additionally, each advertiser bids a value that s/he is willing to pay when

an user click the ad (strategic parameter)

• Payment received is a function of both:
• 1. Click probabilities of the ads
• 2. Declared bids of the advertisers

GOAL: Design a mechanism (allocation rule and payment rule) that ensures truthful elicitation of bids (“strategy proof-ness”) as well as maximizes the total payment received from the advertisers within a limited number of trials.

USEFUL LINKS

http://www.gametheory.net http://www.gametheorysociety.org http://william-king.www.drexel.edu/top/eco/game/game.html http://levine.sscnet.ucla.edu/ http://plato.acadiau.ca/courses/educ/reid/games/General_Games_Links.htm The book followed for preparing this lecture: “Game Theory and Mechanism Design” by Prof. Y. Narahari.

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• utube.com
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