Introduction to Game Theory Lirong Xia Fall, 2016 Homework 1 2 - PowerPoint PPT Presentation

Introduction to Game Theory Lirong Xia Fall, 2016

Homework 1 2

Announcements Ø We will use LMS for submission and grading Ø Please just submit one copy Ø Please acknowledge your team mates 3

Remarks Ø Show the math and formal proof • No math/steps, no points (esp. in midterm) • Especially Problem 1, 4, 5 Ø Problem 1 • Must use u(1M) etc. • Must hold for all utility function Ø Problem 2 • must show your calculation • For Schulze, if you have already found one strict winner, no need to check other alternatives • Kemeny outputs a single winner, unless otherwise mentioned Ø Problem 3.2 • b winning itself is not a paradox • people can change the outcome by not voting is not a paradox 4

Last class Ø Mallows’ model Ø MLE and MAP Ø P = {a>b>c, 2@c>b>a} Ø Likelihood Ø Prior distribution • Pr(a>b>c)=Pr(a>c>b)=0.3 • all other linear orders have prior 0.1 Ø Posterior distribution • proportional to Likelihood*prior 5

Last class Ø Plackett-Luce model • Example • alternatives {a,b,c} • parameter space {(4,3,3), (3,4,3), (3,3,4)} Ø MLE and MAP Ø P = {a>b>c, 2@c>b>a} Ø Likelihood Ø Prior distribution • Pr(4,3,3)=0.8 • all others have prior 0.1 Ø Posterior distribution • proportional to Likelihood*prior 6

Review: manipulation (ties are broken alphabetically) > > YOU > > Plurality rule Bob > > Carol > >

What if everyone is incentivized to lie? > > YOU Plurality rule > > Bob > > Carol

Today’s schedule: game theory Ø What? • Agents may have incentives to lie Ø Why? • Hard to predict the outcome when agents lie Ø How? • A general framework for games • Solution concept: Nash equilibrium • Modeling preferences and behavior: utility theory • Special games • Normal form games: mixed Nash equilibrium • Extensive form games: subgame-perfect equilibrium 9

A game Strategy Profile D Mechanism R 1 * s 1 s 2 R 2 * Outcome … … R n * s n • Players: N ={ 1 ,…,n } • Strategies (actions): - S j for agent j, s j ∈ S j - ( s 1 ,…, s n ) is called a strategy profile. • Outcomes: O • Preferences: total preorders (full rankings with ties) over O • often represented by a utility function u i : Π j S j → R • Mechanism f : Π j S j → O 10

A game of plurality elections > > Plurality rule YOU > > Bob > > Carol • Players: { YOU, Bob, Carol } • Outcomes: O = { , , } • Strategies: S j = Rankings( O ) • Preferences: See above • Mechanism: the plurality rule 11

A game of two prisoners Column player Cooperate Defect ( -1 , -1 ) ( -3 , 0 ) Cooperate Row player ( 0 , -3 ) ( -2 , -2 ) Defect Ø Players: Ø Strategies: { Cooperate, Defect } Ø Outcomes: {( -2 , -2 ), ( -3 , 0 ) , ( 0 , -3 ), ( -1 , -1 )} Ø Preferences: self-interested 0 > -1 > -2 > -3 : ( 0 , -3 ) > ( -1 , -1 ) > ( -2 , -2 ) > ( -3 , 0 ) • : ( -3 , 0 ) > ( -1 , -1 ) > ( -2 , -2 ) > ( 0 , -3 ) • 12 Ø Mechanism: the table

Solving the game Ø Suppose • every player wants to make the outcome as preferable (to her) as possible by controlling her own strategy (but not the other players’) Ø What is the outcome? • No one knows for sure • A “stable” situation seems reasonable Ø A Nash Equilibrium (NE) is a strategy profile ( s 1 ,…, s n ) such that • For every player j and every s j ' ∈ S j , f ( s j , s - j ) ≥ j f ( s j ' , s - j ) or u j ( s j , s - j ) ≥ u j ( s j ' , s - j ) • s - j = ( s 1 ,…, s j- 1 , s j +1 ,…, s n ) • no single player can be better off by deviating 13

Prisoner’s dilemma Column player Cooperate Defect ( -1 , -1 ) ( -3 , 0 ) Cooperate Row player ( 0 , -3 ) ( -2 , -2 ) Defect 14

A beautiful mind Ø “If everyone competes for the blond, we block each other and no one gets her. So then we all go for her friends. But they give us the cold shoulder, because no one likes to be second choice. Again, no winner. But what if none of us go for the blond. We don’t get in each other’s way, we don’t insult the other girls. That’s the only way we win. That’s the only way we all get [a girl.]” 15

A beautiful mind: the bar game Hansen Column player Blond Another girl ( 0 , 0 ) ( 5 , 1 ) Blond Nash Row player ( 1 , 5 ) ( 2 , 2 ) Another girl Ø Players: { Nash, Hansen } Ø Strategies: { Blond, another girl } Ø Outcomes: {( 0 , 0 ), ( 5 , 1 ) , ( 1 , 5 ), ( 2 , 2 )} Ø Preferences: self-interested 16 Ø Mechanism: the table

Does an NE always exists? Ø Not always Column player L R ( -1 , 1 ) ( 1 , -1 ) U Row player ( 1 , -1 ) ( -1 , 1 ) D Ø But an NE exists when every player has a dominant strategy • s j is a dominant strategy for player j, if for every s j ' ∈ S j , for every s - j , f ( s j , s - j ) ≥ j f ( s j ' , s - j ) 1. 17 2. the preference is strict for some s - j

Dominant-strategy NE Ø For player j , strategy s j dominates strategy s j ’, if for every s - j , u j ( s j , s - j ) ≥ u j ( s j ' , s - j ) 1. 2. the preference is strict for some s - j Ø Recall that an NE exists when every player has a dominant strategy s j , if • s j dominates other strategies of the same agent Ø A dominant-strategy NE (DSNE) is an NE where • every player takes a dominant strategy • may not exists, but if exists, then must be unique 18

Prisoner’s dilemma Column player Cooperate Defect ( -1 , -1 ) ( -3 , 0 ) Cooperate Row player ( 0 , -3 ) ( -2 , -2 ) Defect Defect is the dominant strategy for both players 19

The Game of Chicken Ø Two drivers for a single-lane bridge from opposite directions and each can either (S)traight or (A)way. • If both choose S, then crash. • If one chooses A and the other chooses S, the latter “wins”. • If both choose A, both are survived Column player A S ( 0 , 0 ) ( 0 , 1 ) A Row player ( 1 , 0 ) ( -10 , -10 ) S NE 20

Rock Paper Scissors Ø Actions: {R, P, S} Ø Two-player zero sum game No pure NE Column player R P S ( 0 , 0 ) ( -1 , 1 ) ( 1 , - 1 ) R Row player ( 1 , - 1 ) ( 0 , 0 ) ( 1 , - 1 ) P ( 1 , - 1 ) ( 1 , - 1 ) ( 0 , 0 ) S 21

Rock Paper Scissors: Lirong vs. young Daughter Ø Actions • Lirong: {R, P, S} • Daughter: {mini R, mini P} Ø Two-player zero sum game Daughter mini R mini P No pure NE ( 0 , 0 ) ( -1 , 1 ) R Lirong ( 1 , - 1 ) ( 0 , 0 ) P ( 1 , - 1 ) ( 1 , - 1 ) S 22

Computing NE: Iterated Elimination Ø Eliminate dominated strategies sequentially Column player L M R Row player ( 1 , 0 ) ( 1 , 2 ) ( 0 , 1 ) U ( 0 , 3 ) ( 0 , 1 ) ( 2 , 0 ) D 23

Iterated Elimination: Lirong vs. young Daughter Ø Actions • Lirong: {R, P, S} • Daughter: {mini R, mini P} Ø Two-player zero sum game Daughter mini R mini P No pure NE ( 0 , 0 ) ( -1 , 1 ) R Lirong ( 1 , - 1 ) ( 0 , 0 ) P ( - 1 , 1 ) ( 1 , - 1 ) S 24

Normal form games Ø Given pure strategies: S j for agent j Normal form games Ø Players: N ={ 1 ,…,n } Ø Strategies: lotteries (distributions) over S j • L j ∈ Lot( S j ) is called a mixed strategy • ( L 1 ,…, L n ) is a mixed-strategy profile Ø Outcomes: Π j Lot( S j ) Column player Ø Mechanism: f ( L 1 ,…, L n ) = p L R • p ( s 1 ,…, s n ) = Π j L j ( s j ) Row ( 0 , 1 ) ( 1 , 0 ) U Ø Preferences: player ( 1 , 0 ) ( 0 , 1 ) D • Soon 25

Preferences over lotteries Ø Option 1 vs. Option 2 • Option 1: $0@50%+$30@50% • Option 2: $5 for sure Ø Option 3 vs. Option 4 • Option 3: $0@50%+$30M@50% • Option 4: $5M for sure 26

Lotteries Ø There are m objects. Obj={ o 1 ,…, o m } Ø Lot(Obj): all lotteries (distributions) over Obj Ø In general, an agent’s preferences can be modeled by a preorder (ranking with ties) over Lot(Obj) • But there are infinitely many outcomes 27

Utility theory • Utility function: u : Obj → ℝ Ø For any p ∈ Lot(Obj) • u ( p ) = Σ o ∈ Obj p ( o ) u ( o ) Ø u represents a total preorder over Lot(Obj) • p 1 > p 2 if and only if u ( p 1 )> u ( p 2 ) 28

Example utility Money Money 0 5 30 5M 30M Utility 1 3 10 100 150 Ø u (Option 1) = u ( 0 ) × 50% + u ( 30 ) × 50%=5.5 Ø u (Option 2) = u ( 5 ) × 100%=3 Ø u (Option 3) = u ( 0 ) × 50% + u ( 30M ) × 50%=75.5 Ø u (Option 4) = u ( 5M ) × 100%=100 29

Normal form games Ø Given pure strategies: S j for agent j Ø Players: N ={ 1 ,…,n } Ø Strategies: lotteries (distributions) over S j • L j ∈ Lot( S j ) is called a mixed strategy • ( L 1 ,…, L n ) is a mixed-strategy profile Ø Outcomes: Π j Lot( S j ) Ø Mechanism: f ( L 1 ,…, L n ) = p, such that • p ( s 1 ,…, s n ) = Π j L j ( s j ) Ø Preferences: represented by utility functions u 1 ,…, u n 30

Mixed-strategy NE Ø Mixed-strategy Nash Equilibrium is a mixed strategy profile ( L 1 ,…, L n ) s.t. for every j and every L j ' ∈ Lot( S j ) u j ( L j , L - j ) ≥ u j ( L j ' , L - j ) Ø Any normal form game has at least one mixed- strategy NE [Nash 1950] Ø Any L j with L j ( s j )=1 for some s j ∈ S j is called a pure strategy Ø Pure Nash Equilibrium • a special mixed-strategy NE ( L 1 ,…, L n ) where all strategies are pure strategy 31