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Advanced Topics in Algorithmic Game Theory

Jugal Garg

Lecture 1January 9, 2013

What is Game Theory?

I Modeling of the interaction among selfish agentsI Prediction of rational behavior in situation of conflicts

I Solution concepts

Wide applications

What is Game Theory?

I Modeling of the interaction among selfish agentsI Prediction of rational behavior in situation of conflicts

I Solution concepts

Wide applications

Markets

Online Advertisement

Voting

Traffic Routing

x/100 hours

x/100 hours

1 hour

1 hour

ST

A

B

Evolution

Game Theory

Wide Applications

I Planning, markets, auctions, networks, online advertisement,elections, evolution, ...

TypesI Simultaneous Move Game

I E.g. Rock Paper Scissors

I Sequential (one after another)I E.g. Chess

More Types

I One shot game/Repeated game

I Complete/Incomplete information game

Game Theory

Wide Applications

I Planning, markets, auctions, networks, online advertisement,elections, evolution, ...

TypesI Simultaneous Move Game

I E.g. Rock Paper Scissors

I Sequential (one after another)I E.g. Chess

More Types

I One shot game/Repeated game

I Complete/Incomplete information game

What is a game?

I Consists of agents, each having a set of strategies/play

I Every agent wants to maximize her payoff

Rock Paper Scissors

Rock Paper Scissor

Rock 0,0 -1,1 1,-1

Paper 1,-1 0,0 -1,1

Scissor -1,1 1,-1 0,0

Strategy: ?

Pure: every strategy in the strategy set of a playerStable: No player wants to deviate unilaterally

What is a game?

I Consists of agents, each having a set of strategies/play

I Every agent wants to maximize her payoff

Rock Paper Scissors

Rock Paper Scissor

Rock 0,0 -1,1 1,-1

Paper 1,-1 0,0 -1,1

Scissor -1,1 1,-1 0,0

Strategy: No stable pure strategies!

Pure: every strategy in the strategy set of a playerStable: No player wants to deviate unilaterally

Prisoner’s Dilemma

Story?

Cooperate Defect

Cooperate 5,5 0,6

Defect 6,0 1,1

Dominant Strategy: (Defect, Defect)regardless of what other player does!

I Not a social optimal solution!

Prisoner’s Dilemma

Story?

Cooperate Defect

Cooperate 5,5 0,6

Defect 6,0 1,1

Dominant Strategy: (Defect, Defect)regardless of what other player does!

I Not a social optimal solution!

Why study Game Theroy?

I To understand behavior of rational agents

I Prediction of the likely outcome of a situation of conflict

I Such predictions are called solution concepts, e.g. NashEquilibrium.

I Design Mechanism (“rules of the game”) to achieve socialoptimal/desirable outcome

Solution Concepts - Stable State (Equilibrium)

A state from which no player wants to deviate unilaterally.

In a game Γ = 〈N,Si , ui 〉 (normal form game)

I N: A set of players, {1, 2, . . . , n}I Si : set of actions/pure strategies of player i

I ui : payoff function of player i , i.e., ui : S1 × S2 × · · · × Sn → R

I S−i : set of action profiles of players except i ,

I S1 × . . . Si−1 × Si+1 × · · · × Sn

I (s1, s2, . . . , sn) ∈ S1 × S2 × · · · × Sn is an action profile

I s−i : an action profile of players except i , (s1, . . . si−1, si+1, . . . , sn)

I (si , s−i ): notation for an action profile (s1, . . . , sn)

Pure Strategy Nash Equilibrium

An action profile (s∗1 , . . . , s∗n) is said to be a pure strategy Nash

equilibrium iff ui (s∗i , s∗−i ) ≥ ui (si , s

∗−i ), ∀si ∈ Si ,∀i ∈ N

I (Defect, Defect) is a pure strategy Nash equilibrium inPrisoner’s Dilemma.

It may not always exist - Rock-Paper-Scissor?

I Probability distribution on strategies - Mixed strategies

Mixed Strategies

Recall the game Γ = 〈N, Si , ui 〉I A mixed strategy of player i is σi : Si → [0, 1] s.t.

∑si∈Si

σi (si ) = 1

I Note that every pure strategy is a “degenerate” mixed strategy.

I Si = (si1, . . . , sim), player i has m pure strategies.

I σij = σi (sij), probability of playing pure strategy sij .

I Set of mixed strategies of i : ∆(Si ) = {σij ≥ 0 |∑m

j=1 σij = 1}

The modified game Γ = 〈N,∆(Si ),Ui 〉I Ui : ∆(S1)× · · · ×∆(Sn)→ RI σi ∈ ∆(Si )

I σ(s1, . . . , sn) = Πi∈Nσi (si )

I Ui (σ1, . . . , σn) =∑

(s1,...,sn)∈S σ(s1, . . . , sn)ui (s1, . . . , sn)

Mixed Strategy Nash Equilibrium

An action profile (σ∗1, . . . , σ∗n) is said to be a mixed strategy Nash

equilibrium iff Ui (σ∗i , σ∗−i ) ≥ Ui (σi , σ

∗−i ), ∀σi ∈ ∆(Si ),∀i ∈ N

Back to RPS

Rock Paper Scissor

Rock 0,0 -1,1 1,-1

Paper 1,-1 0,0 -1,1

Scissor -1,1 1,-1 0,0

I (1/3, 1/3, 1/3) is the only Nash equilibrium - Prove (Homework)

Battle of Sexes

Story?

Movie Football

Movie 4,1 0,0

Football 0,0 1,4

Strategy: ?

2 Pure and 1 Mixed strategy equilibria - Homework

I Number of equilibria can be more than one!

Battle of Sexes

Story?

Movie Football

Movie 4,1 0,0

Football 0,0 1,4

Strategy: 2 Pure and 1 Mixed strategy equilibria - Homework

I Number of equilibria can be more than one!

Battle of Sexes

Story?

Movie Football

Movie 4,1 0,0

Football 0,0 1,4

Strategy: 2 Pure and 1 Mixed strategy equilibria - Homework

I Number of equilibria can be more than one!

Existence - Nash Theorem

I Finite game: finite number of players, each having finitelymany pure strategies.

I It is not at all obvious that why mixed strategy Nashequilibrium should exist in every finite game?

Theorem:Every finite game has a stable state (equilibrium) from which noplayer wants to deviate unilaterally.

I Proof through Brouwer fixed point theorem

Zero-sum Games

I∑

i ui (s) = 0, ∀s = (s1, . . . , sn) ∈ S1 × · · · × Sn

Two player case

I One player loss is another player gain

I Rock-Paper-Scissor

How to find an equilibrium state?

Matching Pennies

Head Tail

Head 1,-1 -1,1

Tail -1,1 1,-1

Strategy: (1/2, 1/2) is the only stable state!

History

I von Neumann (1928) - existence of equilibrium in two personzero sum games

I Minimax theorem

I Dantzig (1947) - MiniMax Theorem (Related to LP Duality)I Zero-sum games ↔ LP

I Nash (1951) - existence of equilibrium in finite non-zero sumgames

I through Brouwer’s fixed point theorem

MiniMax Theorem

I Maximize the minimum payoff: maxτ1∈∆(S1)

minτ2∈∆(S2)

u1(τ1, τ2)

- Guaranteed payoff for player 1

I Minimize the maximum loss: minτ2∈∆(S2)

maxτ1∈∆(S2)

u1(τ1, τ2)

- Guaranteed minimum loss for player 2

Theorem maxτ1∈∆(S1)

minτ2∈∆(S2)

u1(τ1, τ2) = minτ2∈∆(S2)

maxτ1∈∆(S2)

u1(τ1, τ2)

- A simple proof using LP duality

Homework Prove that (σ1, σ2) is a Nash equilibrium in a twoplayer zero sum game iff it is a minimax strategy, i.e.,σ1 = arg max

τ1∈∆(S1)min

τ2∈∆(S2)u1(τ1, τ2), σ2 = arg min

τ2∈∆(S2)max

τ1∈∆(S2)u1(τ1, τ2).

MiniMax Theorem

I Maximize the minimum payoff: maxτ1∈∆(S1)

minτ2∈∆(S2)

u1(τ1, τ2)

- Guaranteed payoff for player 1

I Minimize the maximum loss: minτ2∈∆(S2)

maxτ1∈∆(S2)

u1(τ1, τ2)

- Guaranteed minimum loss for player 2

Theorem maxτ1∈∆(S1)

minτ2∈∆(S2)

u1(τ1, τ2) = minτ2∈∆(S2)

maxτ1∈∆(S2)

u1(τ1, τ2)

- A simple proof using LP duality

Homework Prove that (σ1, σ2) is a Nash equilibrium in a twoplayer zero sum game iff it is a minimax strategy, i.e.,σ1 = arg max

τ1∈∆(S1)min

τ2∈∆(S2)u1(τ1, τ2), σ2 = arg min

τ2∈∆(S2)max

τ1∈∆(S2)u1(τ1, τ2).

MiniMax Theorem

I Maximize the minimum payoff: maxτ1∈∆(S1)

minτ2∈∆(S2)

u1(τ1, τ2)

- Guaranteed payoff for player 1

I Minimize the maximum loss: minτ2∈∆(S2)

maxτ1∈∆(S2)

u1(τ1, τ2)

- Guaranteed minimum loss for player 2

Theorem maxτ1∈∆(S1)

minτ2∈∆(S2)

u1(τ1, τ2) = minτ2∈∆(S2)

maxτ1∈∆(S2)

u1(τ1, τ2)

- A simple proof using LP duality

Homework Prove that (σ1, σ2) is a Nash equilibrium in a twoplayer zero sum game iff it is a minimax strategy, i.e.,σ1 = arg max

τ1∈∆(S1)min

τ2∈∆(S2)u1(τ1, τ2), σ2 = arg min

τ2∈∆(S2)max

τ1∈∆(S2)u1(τ1, τ2).

Traffic Routing

x/100 hours

x/100 hours

1 hour

1 hour

ST

A

B

Traffic Routing

x/100 hours

x/100 hours 1 hour

1 hour

ST

A

B

50

50

Braess’s Paradox

0

x/100 hours

x/100 hours 1 hour

1 hour

ST

A

B

Adding extra resources is not always good

Braess’s Paradox

0

x/100 hours

x/100 hours 1 hour

1 hour

ST

A

B

100

Adding extra resources is not always good

Price of Anarchy

Price of Anarchy (PoA) = Performance of worst Nash equilibriumOptimal performance

I Measures the loss in performance due to free will

I In the traffic example, it is 4/3

Mechanism Design

I Design a system such that PoA is small

I Outcomes are social optimal

I What incentives given so that we get close to optimalperformance

Single Item Auction

I One item for sale

I n people are interested

I vi : valuation of i th person (private)

How to allocate this item to the person with highest valuation?

I We ask them to bid: suppose bi ’s are the bids

I pi : payment made by bidder i

The problem is to design a mechanism

I Truthful bids can be revealed (Incentive compatible)

I Item is given to a person with highest valuation

I How to design payments?

First price Auction

I Suppose b1 > b2 > · · · > bnI p1 = b1 and pi = 0, ∀i ≥ 2

what bidders will do?

I Suppose two bidders with v1 = 10 and v2 = 5

I Nash Equilibrium?

Second price Auction

I Give item to the highest bidder and charge second highest bid.

Show that

I Bidding your true value is dominant strategy - truthful

Vickrey Auction

I Generalized to Vickrey-Clarke-Groves mechanism (VCG)

Algorithmic Game Theory?

Why worry about computation?

I Need of efficient algorithms for computation - PredictionI Boom and scale in e-commerce

I Auctions, resource allocation, routing, ...

I Evolution, planning, designing, ...

We need practical and efficient algorithms!

Algorithmic Game Theory?

Why worry about computation?

I Need of efficient algorithms for computation - PredictionI Boom and scale in e-commerce

I Auctions, resource allocation, routing, ...

I Evolution, planning, designing, ...

We need practical and efficient algorithms!

Guess half the average

I A game with n players

I Everyone need to guess an integer between [0, 100]

I Winner - who guesses a number that is nearest to the half ofthe average guesses of all players

I Payoff of winner is 1 and others is 0I In case of tie, all winners get 1

What is your guess?

What is Nash Equilibrium guess?

Guess half the average

I A game with n players

I Everyone need to guess an integer between [0, 100]

I Winner - who guesses a number that is nearest to the half ofthe average guesses of all players

I Payoff of winner is 1 and others is 0I In case of tie, all winners get 1

What is your guess?

What is Nash Equilibrium guess?