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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?