computational game theory amos fiat modified slides prepared for yishay mansour’s class

Computational Game TheoryAmos Fiat

Modified Slides prepared for Yishay Mansour’s class

Lecture 1 - Introduction

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Agenda Introduction to Game Theory Examples Matrix form Games Utility Solution concepts

Dominant Strategies Nash Equilibria

Complexity Mechanism Design: reverse game theory

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The study of Game Theory in the context of Computer Science, in order to reason about problems from the perspective of computability and algorithm design.

Computational Game Theory

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Computing involves many different selfish entities. Thus involves game theory.

The Internet, Intranet, etc.◦ Many players (end-users, ISVs, Infrastructure

Providers)◦ Players wish to maximize their own benefit and

act accordingly◦ The trick is to design a system where it’s

beneficial for the player to follow the rules

CGT in Computer Science

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Theory◦ Algorithm design◦ Complexity◦ Quality of game states (Equilibrium states in

particular)◦ Study of dynamics

Industry◦ Sponsored search ◦ Other auctions

CGT in Computer Science

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Rational Player◦ Prioritizes possible actions according to utility or

cost◦ Strives to maximize utility or to minimize cost

Competitive Environment◦ More than one player at the same time

Game Theory analyzes how rational players behave in competitive

environments

Game Theory

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Matrix representation of the game

The Prisoner’s Dilema

Thieves honor

Defect

Thieves honor

3,3 6,2

Defect 2,6 5,5

Row Player Column Player

2 < 3

5 < 6

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It is a dominant strategy to confess A dominant strategy is a “solution concept”

The Prisoner’s Dilema

Thieves honor

Defect

Thieves honor

3,3 6,2

Defect 2,6 5,5

6,10

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Internet Service Providers (ISP) often share their physical networks for free

In some cases an ISP can either choose to route traffic in its own network or via a partner network

ISP Routing

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ISP 1 needs to route traffic from s1 to t1

ISP 2 needs to route traffic from s2 to t2

The cost of routing along each edge is one

ISP Routing

A B

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ISP1 routes via B:◦Cost for ISP1: 1◦Cost for ISP2: 4

ISP Routing

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Cost matrix for the game:

ISP Routing

A B

A 3,3 6,2

B 2,6 5,5ISP 1

ISP 2

B,A: s1 to t1B,A: s2 to t2

Prisoners Dilemma Again

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The game consists of only one ‘turn’

All the players play simultaneously and are unaware of what the other players do

Players are selfish, seek to maximize their own benefit

Strategic Games

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N = {1,…,n} players Player i has actions We will say “action” or “strategy” The space of all possible action vectors is

A joint action is the vector a∈A Player i has a utility function If utility is negative we may call it cost

Strategic Games – Formal Model

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A strategic game:

Strategic Games – Formal Model

Players

Actions of each player

Utility of each player

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Action ai of player i is a weakly dominant strategy if:

Dominant Strategies

Action ai of player i is a strongly dominant strategy if:

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An outcome a of a game is Pareto optimal if for every other outcome b, some player will lose by changing to b

Pareto Optimality

Vilfredo Pareto

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St. Petersburg Paradox:◦ Toss a coin until tails, I pay

you

◦ What will you pay me to play?

Bernulli Utility

“Utility of Money”, “Bernulli Utility”

Completeness:Transitivity:Continuity:

Independence:

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Von Neumann–Morgenstern Rationality Axioms (1944)Preferences over lotteries

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Rationality Axioms

Utility function overlotteries, real valued,

expected utility maximization

Gamble A: 100% € 1,000,000Gamble B: 10% € 5,000,000

89% € 1,000,000 1% Nothing

Gamble C: 11% € 1,000,000 89% NothingGamble D: 10% € 5,000,000 90% Nothing

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Allias Paradox (1953)

Gamble A or B?

Gamble C or D?

Experimental ”Fact”:

Experimental “Fact”:

Gamble A: 100% € 1,000,000Gamble B: 10% € 5,000,000

89% € 1,000,000 1% Nothing

Gamble C: 11% € 1,000,000 89% NothingGamble D: 10% € 5,000,000 90% Nothing

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Allias Paradox

“Fact”:

“Fact”:

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Expected Utility Theory

VNM Axioms Expected Utility MaximizationMixed Nash Equilibrium exists

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Assume there’s a shared resource (network bandwidth) and N players.

Each player “uses” the common resource, by choosing Xi from [0,1].

If

Otherwise,

Tragedy of the commons

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Tragedy of the commons

Given that the otherplayers are fixed, whatIs the best response?

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Tragedy of the commons

This is an equilibriumNo player can improve

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Tragedy of the commons

The case for Privatization or central control of commons

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A Nash Equilibrium is an outcome of the game in which no player can improve its utility alone:

Alternative definition: every player’s action is a best response:

Nash Equilibrium

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The payoff matrix:

Battle of the Sexes

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The payoff matrix:

Battle of the Sexes

Row player has no incentive to

move up

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The payoff matrix:

Battle of the Sexes

Column player has no

incentive to move left

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The payoff matrix:

Battle of the Sexes

So this is an Equilibrium state

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The payoff matrix:

Battle of the Sexes

Same thing here

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2 players need to send a packet from point O to the network.

They can send it via A (costs 1) or B (costs 2)

Routing Game

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The cost matrix:

Routing Game

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The cost matrix:

Routing Game

Equilibrium states

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2 players, each chooses Head or Tail Row player wins if they match the column

player wins if they don’t Utility matrix:

Matching Pennies

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2 players, each chooses Head or Tail Row player wins if they match the column

player wins if they don’t Utility matrix:

Matching Pennies

Row player is fine, but Column player wants to move left

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2 players, each chooses Head or Tail Row player wins if they match the column

player wins if they don’t Utility matrix:

Matching Pennies

Column player is fine, but Row player wants to move up

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2 players, each chooses Head or Tail Row player wins if they match the column

player wins if they don’t Utility matrix:

Matching Pennies

Row player is fine, but Column player

wants to move right

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2 players, each chooses Head or Tail Row player wins if they match the column

player wins if they don’t Utility matrix:

Matching Pennies

Column player is fine, but Row player wants

to move down

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2 players, each chooses Head or Tail Row player wins if they match the column

player wins if they don’t Utility matrix:

No equilibrium state!

Matching Pennies

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Players do not choose a pure strategy (one specific strategy)

Players choose a distribution over their possible pure strategies

For example: with probability p choose Heads, and with probability 1-p choose Tails

Mixed Strategies

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Row player chooses Heads with probability p and Tails with probability 1-p

Column player chooses Heads with probability q and Tails with probability 1-q

Row plays Heads: Row plays Tails:

Matching Pennies

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Each player selects where is the set of all possible distributions over Ai

An outcome of the game is the Joint Mixed Strategy

An outcome of the game is a Mixed Nash Equilibrium if for every player

Mixed Strategy

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2nd definition of Mixed Nash Equilibrium:

Definition:

Definition:

Property of Mixed Nash Equilibrium:

Mixed Strategy

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No pure strategy Nash Equilibrium, only Mixed Nash Equilibrium, for mixed strategy (1/3, 1/3, 1/3) .

Rock Paper Scissors

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N ice cream vendors are spread on the beach

Assume that the beach is the line [0,1] Each vendor chooses a location Xi, which

affects its utility (sales volume). The utility for player i :

X0 = 0, Xn+1 = 1

Location Game

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For N=2 we have a pure Nash Equilibrium:

No player wants to move since it will lose space

For N=3 no pure Nash Equilibrium:

The player in the middle always wants to move to improve its utility

Location Game

0 11/2

0 11/2

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If instead of a line we will assume a circle, we will always have a pure Nash Equilibrium where every player is evenly distanced from each other:

Location Game

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N companies are producing the same product

Company I needs to choose its production volume, xi ≥ 0

The price is determined based on the overall production volume,

Each company has a production cost: The utility of company i is:

Cournot Competition

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Case 1: Linear price, no production cost

◦ Utility:

◦ Pure Nash Equilibrium is reached at:

Cournot Competition

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Case 2: Harmonic price, no production cost

◦ Company i’s utility:

◦ Companies have incentive to produce as much as they can – no pure or mixed Nash Equilibrium

Cournot Competition

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n players wants to buy a single item which is on sale

Each player has a valuation for the product, Assume WLOG that Each player submits its bid, , all players

submit simultaneously.

Auction

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Case 1: First price auction◦ The player with the highest bid wins◦ The price equals the bid◦ 1st Equilibrium is:

The first player needs to know the valuation of the second player – not practical

◦ 2nd Equilibrium is:

Auction

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Case 2: Second price auction: Vickrey Auction◦ The player with the highest bid wins◦ The price equals the second highest bid

No incentive to bid higher than one’s valuation - a player’s utility when it bids its valuation is at least as high than when it bids any other value

This mechanism encourages players to bid truthfully Mechanism Design: reverse game theory –

set up a game so that the equilibria has a desired property

Auction

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Equilibrium Concepts

pureNash

mixed Nash

correlated eq

no regret

best-responsedynamics

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Traffic Flow: the Mathematical Model a directed graph G = (V,E) k source-destination pairs (s1 ,t1), …,

(sk ,tk) a rate (amount) ri of traffic from si to ti

for each edge e, a cost function ce(•)◦ assumed nonnegative, continuous,

nondecreasing

s1 t1

c(x)=x Flow = ½

Flow = ½c(x)=1

Example: (k,r=1)

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Routings of Traffic

Traffic and Flows: fP = amount of traffic routed on si-ti path P flow vector f routing of traffic

Selfish routing: what are the equilibria?

s t

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Nash Flows

Some assumptions: agents small relative to network (nonatomic

game) want to minimize cost of their path

Def: A flow is at Nash equilibrium (or is a Nash flow) if all flow is routed on min-cost paths [given current edge congestion]

xs t

1Flow = .5

Flow = .5

s t1

Flow = 0

Flow = 1x

Example:

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History + Generalizations

model, defn of Nash flows by [Wardrop 52]

Nash flows exist, are (essentially) unique◦ due to [Beckmann et al. 56]◦ general nonatomic games: [Schmeidler 73]

congestion game (payoffs fn of # of players)◦ defined for atomic games by [Rosenthal 73]◦ previous focus: Nash eq in pure strategies exist

potential game (equilibria as optima)◦ defined by [Monderer/Shapley 96]

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The Cost of a Flow

Def: the cost C(f) of flow f = sum of all costs incurred by traffic (avg cost × traffic rate)

s t

x

1

½

½

Cost = ½•½ +½•1 = ¾

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The Cost of a Flow

Def: the cost C(f) of flow f = sum of all costs incurred by traffic (avg cost × traffic rate)

Formally: if cP(f) = sum of costs of edges of P (w.r.t. the flow f), then:

C(f) = P fP • cP(f)

s ts t

x

1

½

½

Cost = ½•½ +½•1 = ¾

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Inefficiency of Nash FlowsNote: Nash flows do not minimize the cost observed informally by [Pigou 1920]

Cost of Nash flow = 1•1 + 0•1 = 1 Cost of optimal (min-cost) flow = ½•½ +½•1 = ¾ Price of anarchy := Nash/OPT ratio = 4/3

s t

x

10

1 ½

½

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Braess’s Paradox

Initial Network:

s tx 1

½

x1½

½

½

cost = 1.5

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Braess’s Paradox

Initial Network: Augmented Network:

s tx 1

½

x1½

½

½

cost = 1.5

s tx 1

½

x1½

½

½0

Now what?

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Braess’s Paradox

Initial Network: Augmented Network:

s tx 1

½

x1½

½

½

cost = 1.5 cost = 2

s t

x 1

x10

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Braess’s Paradox

Initial Network: Augmented Network:

All traffic incurs more cost! [Braess 68]

see also [Cohen/Horowitz 91], [Roughgarden 01]

s tx 1

½

x1½

½

½

cost = 1.5 cost = 2

s t

x 1

x10