Speaker Arpita Biswas PhD Student (Google Fellow) Lab, Dept. of CSA, Indian Institute of Science, Bangalore

Email address: [email protected] 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 rational and intelligent agents”

GAME THEORY

“Mathematical framework for rigorous study of conflict and cooperation among rational and intelligent 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

GAME THEORY

“Mathematical framework for rigorous study of conflict and cooperation among rational and intelligent agents”

the agent would always choose an action that maximizes her/his (expected) utility.

preferences of the players expressed in terms of real numbers

• 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

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 other prisoner.

PRISONER’S DILEMMA

Bubly Two Players: Confess Not Confess • Bunty Bunty • Bubly

Two Actions: Confess • Confess • Not Confess

Not Confess

The utility matrix models the strategic conflict when two players have to choose their priorities PRISONER’S DILEMMA

Bubly Two Players: Confess Not Confess • Bunty Bunty • Bubly

Two Actions: Confess • Confess • Not Confess

< 푁, (퐴 ) , (푈 ) > Not 푖 푖∈푁 푖 푖∈푁 Confess 푁 ∶ set of players 퐴푖 ∶ set of actions for player 푖 푈 ∶ 퐴 × ⋯ × 퐴 → ℝ 푖 1 |푁| The utility matrix models the strategic conflict when Action profile or two players have to choose their priorities Strategy profile PRISONER’S DILEMMA

Bubly Two Players: Confess Not Confess • Bunty Bunty • Bubly

Two Actions: Confess • Confess • Not Confess

Not Confess PRISONER’S DILEMMA

Bubly Two Players: Confess Not Confess • Bunty Bunty • Bubly

Two Actions: Confess • Confess • Not Confess

Not Confess PRISONER’S DILEMMA

Bubly Two Players: Confess Not Confess • Bunty Bunty • Bubly

Two Actions: Confess • Confess • Not Confess

Not Confess PRISONER’S DILEMMA

Bubly Two Players: Confess Not Confess • Bunty Bunty • Bubly

Two Actions: Confess • Confess • Not Confess

Not Confess PRISONER’S DILEMMA

Bubly Two Players: Confess Not Confess • Bunty Bunty • Bubly

Two Actions: Confess • Confess • Not Confess

Not Confess PRISONER’S DILEMMA

Bubly Two Players: Confess Not Confess • Bunty Bunty • Bubly

Two Actions: Confess • Confess • Not Confess

Not Confess Nash Equilibrium

A strategy profile in which no player gains by changing only his/her own strategy (assuming no one else changes their strategy) PROJECT

Bob Two Players: Deep Learning Website Designing • Alice Alice • Bob

Two Actions: Deep • Deep Learning Learning Project • Website Designing Project Website Designing PROJECT COORDINATION GAME

Bob Two Players: Deep Learning Website Designing • Alice Alice • Bob

Two Actions: Deep • Deep Learning Learning Project • Website Designing Project Website Designing PROJECT COORDINATION GAME

Bob Two Players: Deep Learning Website Designing • Alice Alice • Bob

Two Actions: Deep • Deep Learning Learning Project • Website Designing Project Website Designing

Nash Equilibria PROJECT COORDINATION GAME

Bob Two Players: Deep Learning Website Designing • Alice Alice • Bob

Two Actions: Deep • Deep Learning Learning Project • Website Designing Project Website Designing

Nash Equilibria

Does there exist there any other Nash Equilibrium in this game? PROJECT COORDINATION GAME

Bob Two Players: Deep Learning Website Designing • Alice Alice • Bob

Two Actions: Deep • Deep Learning Learning Project (DL) • Website Designing Project (WD) Website Designing

Nash Equilibria

Does there exist there any other Nash Equilibrium in this game? 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

MIXED STRATEGY NASH EQUILIBRIUM EXISTENCE OF NASH EQUILIBRIA IN GAMES

 Does Nash Equilibrium always exist? EXISTENCE OF NASH EQUILIBRIA IN GAMES

 Does Nash Equilibrium always exist?

,푵풂풔풉 푻풉풆풐풓풆풎, ퟏퟗퟓퟎ-. 푬풗풆풓풚 풇풊풏풊풕풆 풔풕풓풂풕풆품풊풄 풇풐풓풎 품풂풎풆 풉풂풔 풂풕 풍풆풂풔풕 풐풏풆 풎풊풙풆풅 풔풕풓풂풕풆품풚 푵풂풔풉 푬풒풖풊풍풊풃풓풊풖풎. EXISTENCE OF NASH EQUILIBRIA IN GAMES

 Does Nash Equilibrium always exist?

,푵풂풔풉 푻풉풆풐풓풆풎, ퟏퟗퟓퟎ-. 푬풗풆풓풚 풇풊풏풊풕풆 풔풕풓풂풕풆품풊풄 풇풐풓풎 품풂풎풆 풉풂풔 풂풕 풍풆풂풔풕 풐풏풆 풎풊풙풆풅 풔풕풓풂풕풆품풚 푵풂풔풉 푬풒풖풊풍풊풃풓풊풖풎.

 Is there an efficient algorithm for computing a mixed Nash equilibrium? EXISTENCE OF NASH EQUILIBRIA IN GAMES

 Does Nash Equilibrium always exist?

,푵풂풔풉 푻풉풆풐풓풆풎, ퟏퟗퟓퟎ-. 푬풗풆풓풚 풇풊풏풊풕풆 풔풕풓풂풕풆품풊풄 풇풐풓풎 품풂풎풆 풉풂풔 풂풕 풍풆풂풔풕 풐풏풆 풎풊풙풆풅 풔풕풓풂풕풆품풚 푵풂풔풉 푬풒풖풊풍풊풃풓풊풖풎.

 Is there an efficient algorithm for computing a mixed Nash equilibrium?

,푫풂풔풌풂풍풂풌풊풔 풆풕 풂풍. , ퟐퟎퟎퟔ-. 푭풊풏풅풊풏품 풎풊풙풆풅 풔풕풓풂풕풆품풚 푵풂풔풉 푬풒풖풊풍풊풃풓풊풖풎 풊풔 푷푷푨푫 − 풄풐풎풑풍풆풕풆 OTHER TYPES OF EQUILIBRIA

 Strongly Dominant Strategy Equilibrium (SDSE):

퐴푛 푎푐푡푖표푛 푝푟표푓푖푙푒 푎1, ⋯ , 푎푛 푖푠 푐푎푙푙푒푑 풔풕풓풐풏품풍풚 풅풐풎풊풏풂풏풕 풔풕풓풂풕풆품풚

풆풒풖풊풍풊풃풓풊풖풎 푓표푟 푎 푔푎푚푒 < 푁, 퐴푖 , 푈푖 >, 푖푓 ∀푖 ∈ 푁 푎푛푑 ∀푏푖 ∈ 퐴푖 ∖ *푎푖+,

푈푖 푎푖, 푏−푖 > 푈푖 푏푖, 푏−푖 ∀푏−푖 ∈ 퐴−푖 .

Bubly Confess Not Confess

Bunty

Confess

Not Confess OTHER TYPES OF EQUILIBRIA

 Weakly Dominant Strategy Equilibrium (WDSE):

퐴푛 푎푐푡푖표푛 푝푟표푓푖푙푒 푎1, ⋯ , 푎푛 푖푠 푐푎푙푙푒푑 풘풆풂풌풍풚 풅풐풎풊풏풂풏풕 풔풕풓풂풕풆품풚

풆풒풖풊풍풊풃풓풊풖풎 푓표푟 푎 푔푎푚푒 < 푁, 퐴푖 , 푈푖 >, 푖푓 ∀푖 ∈ 푁 푎푛푑 ∀푏푖 ∈ 퐴푖,

푈 푎 , 푏 ≥ 푈 푏 , 푏 ∀푏 ∈ 퐴 푎푛푑 푈 푎 , 푏 > 푈 푏 , 푏 푓표푟 푠표푚푒 푏 ∈ 퐴 . 푖 푖 −푖 푖 푖 −푖 −푖 −푖 푖 푖 −푖 푖 푖 −푖 −푖 −푖

Bubly Confess Not Confess

Bunty

Confess

Not Confess OTHER TYPES OF EQUILIBRIA

 Very Weakly Dominant Strategy Equilibrium (VWDSE):

퐴푛 푎푐푡푖표푛 푝푟표푓푖푙푒 푎1, ⋯ , 푎푛 푖푠 푐푎푙푙푒푑 풗풆풓풚 풘풆풂풌풍풚 풅풐풎풊풏풂풏풕 풔풕풓풂풕풆품풚

풆풒풖풊풍풊풃풓풊풖풎 푓표푟 푎 푔푎푚푒 < 푁, 퐴푖 , 푈푖 >, 푖푓 ∀푖 ∈ 푁 푎푛푑 ∀푏푖 ∈ 퐴푖,

푈푖 푎푖, 푏−푖 ≥ 푈푖 푏푖, 푏−푖 ∀푏−푖 ∈ 퐴−푖 .

Bubly Confess Not Confess

Bunty

Confess

Not Confess NO DOMINANT STRATEGY EQUILIBRIA (PROJECT COORDINATION GAME)

Bob Deep Learning Website Designing Alice

Deep Learning

Website Designing 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 other agents will do, and (hence) how agents should act – Useful for taking a profitable action as well as predicting behavior of others.

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

MECHANISM DESIGN

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

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

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

Courtesy: Google images CAKE CUTTING

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

Courtesy: Google images CAKE CUTTING

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

Courtesy: Google images CAKE CUTTING

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

Courtesy: Google images CAKE CUTTING

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

Courtesy: Google images CAKE CUTTING

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

Courtesy: Google images CAKE CUTTING: MECHANISM DESIGN

Solution: Cut and Choose

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 chooser 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 own eyes.

CAKE CUTTING: MORE THAN TWO KIDS

I want the cake to I want one a piece be split into exactly with at least one 4 equal parts fourth of all the fruits

I want at least one- I want a piece eighth of strawberry with at least one- cream and at least one- fourth of all the eighth of kiwi cream kiwi pieces.

Courtesy: Google images

Courtesy: Google images FAIR DIVISION

Courtesy: Google images DESIRABLE PROPERTIES OF A MECHANISM

 Allocative Efficiency: Allocation should maximize the sum of value obtained by all the players.

 Individual Rationality: “Players do not loose anything by participating in the game” or “Voluntary Participation”

 Dominant Strategy :”Strategy-proofness”

 Non-Dictatorship: “There is no agent for whom all outcomes turn out to be favored outcomes.”

COOPERATIVE GAME THEORY

 There is an incentive to cooperate , 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 EXAMPLE: PROJECT CONTRACT

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.

EXAMPLE: PROJECT CONTRACT

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>? EXAMPLE: PROJECT CONTRACT

Recall: EXAMPLE: PROJECT CONTRACT

Recall: EXAMPLE: PROJECT CONTRACT

Recall: EXAMPLE: PROJECT CONTRACT

Recall:

Shapley value split: <50, 50, 200> OTHER SOLUTION CONCEPTS IN COOPERATIVE GAME THEORY

 Stable Sets   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

 Resource Allocation: Find a “fair” split of resources among agents  Procurement Auction: Design an auction that maximizes social utilities  Crowdsourcing: Design a mechanism to complete as many task as possible with maximum quality.  Online Education Platforms (MOOCs): Designing incentives to improve participation level of students and instructors.  Social Network Analysis: 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

Separate auction for every query: • Positions awarded in order of bid (more on this later). • Advertisers pay bid of the advertiser in the position 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

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.

Simple setting One “ad slot” and N competing advertisers Which ad to show and what should the advertiser pay?

Solving this requires a mechanism comprising an allocation rule and a payment rule. 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.

Simple setting One “ad slot” and N competing advertisers Which ad to show and what should the advertiser pay?

Solving this requires a mechanism comprising an allocation rule and a payment rule.

VCG (Vickrey-Clarke-Groves) mechanism :

* Allocation rule: Give the ad-slot to the advertiser with maximum valuation/bid * Payment 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.

https://www.youtube.com/watch?v=WhbIPNoJAec