Introduction

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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.

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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

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

particular)◦ Study of dynamics

Industry◦ Sponsored search ◦ Other auctions

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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

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Matrix representation of the game

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Thieves honor

Defect

Thieves honor

3,3 6,2

Defect 2,6 5,5

Row Player Column Player

2 < 3

5 < 6

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

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Thieves honor

Defect

Thieves honor

3,3 6,2

Defect 2,6 5,5

6,10

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

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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

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A B

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

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Cost matrix for the game:

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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

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

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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

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A strategic game:

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Players

Actions of each player

Utility of each player

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

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

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

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Vilfredo Pareto

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

you

◦ What will you pay me to play?

18“Utility of Money”, “Bernulli Utility”

Completeness:Transitivity:Continuity:

Independence:

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Preferences over lotteries

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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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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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“Fact”:

“Fact”:

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VNM Axioms Expected Utility MaximizationMixed Nash Equilibrium exists

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,

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

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This is an equilibriumNo player can improve

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The case for Privatization or central control of commons

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:

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The payoff matrix:

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The payoff matrix:

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Row player has no incentive to

move up

The payoff matrix:

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Column player has no

incentive to move left

The payoff matrix:

32

So this is an Equilibrium state

The payoff matrix:

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Same thing here

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)

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The cost matrix:

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The cost matrix:

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Equilibrium states

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

player wins if they don’t Utility matrix:

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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:

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

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

player wins if they don’t Utility matrix:

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Column player is fine, but Row

player wants to move up

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

player wins if they don’t Utility matrix:

40

Row player is fine, but Column player

wants to move right

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

player wins if they don’t Utility matrix:

41

Column player is fine, but Row player wants

to move down

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!

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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

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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:

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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

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2nd definition of Mixed Nash Equilibrium:

Definition:

Definition:

Property of Mixed Nash Equilibrium:

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

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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

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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

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0 11/2

0 11/2

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:

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http://martin.zinkevich.org/lemonade/ http://tech.groups.yahoo.com/group/

lemonadegame/

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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:

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Case 1: Linear price, no production cost

◦ Utility:

◦ Pure Nash Equilibrium is reached at:

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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

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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.

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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:

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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

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58

pureNash

mixed Nash

correlated eq

no regret

best-responsedynamics

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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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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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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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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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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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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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Note: 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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Initial Network:

s tx 1

½

x1½

½

½

cost = 1.5

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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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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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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