1 Adapted from: Game Theoretic Approach in Computer Science CS3150, Fall 2002 Introduction to Game Theory Patchrawat Uthaisombut University of Pittsburgh.

1

Adapted from:Game Theoretic Approach in

Computer ScienceCS3150, Fall 2002

Introduction to Game TheoryPatchrawat Uthaisombut

University of Pittsburgh

2

Outline

• Example: Restaurant Game

• Formal Definition of Games

• Goal: Computing outcome of a game

• Examples: Computing game outcomes

• Mixed Strategies

• Selfish Routing and Price of Anarchy

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Restaurant Game

Wendy’sor

Dusty’s

Wendy’sor

Dusty’s

MalcolmJulia

4

Julia

Wendy’s Dusty’s

MalcolmWendy’s 2,1 0,0

Dusty’s 0,0 1,2

Payoffs

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A Play of the Restaurant Game

• The play• Row player chooses Dusty's.• Column player chooses Dusty's.

• The Outcome• They meet at Dusty's

• The Payoff• Row player gets 1.• Column player gets 2.

Wendy's Dusty's

Wendy's 2,1 0,0

Dusty's 0,0 1,2

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Outline







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Components of a Strategic Game

• Players• Who is involved?

• Rules• Who moves when?• What does a player know when he/she moves?• What moves are available?

• Outcomes• For each possible combination of actions by the players,

what’s the outcome of the game.

• Payoffs• What are the players’ preferences over the possible

outcomes?

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Key Assumptions

• Common knowledge• Everyone is aware of all player choices and

payoff functions

• Rationality of Players• Player will move to optimize individual payoff• All utility is expressed in the payoff function

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Formal Definition of Strategic Game

• A strategic game is a 3-tuple (n,A,u)• The number of players n.

• For 1<i<n, a set Ai of actions available for player i.

• For 1<i<n, a payoff function ui:A1…An R for player i.

Wendy's Dusty's

Wendy's 2,1 0,0

Dusty's 0,0 1,2

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Restaurant Game as a Strategic Game

• Players: n = 2• Player 1 = Malcolm• Player 2 = Julia

• Actions:• A1 = {Wendy's, Dusty's }• A2 = {Wendy's, Dusty's }

• Payoffs:• u1(Wendy's,Wendy's ) = 2• u1(Wendy's,Dusty's ) = 0• u1(Dusty's,Wendy's ) = 0• u1(Dusty's,Dusty's ) = 1

Wendy's Dusty's

Wendy's 2,1 0,0

Dusty's 0,0 1,2

• u2(Wendy's,Wendy's ) = 1• u2(Wendy's,Dusty's ) = 0• u2(Dusty's,Wendy's ) = 0• u2(Dusty's,Dusty's ) = 2

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Outline







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Goal: Compute Outcome

• Given a game, compute what the outcome should be• Key assumption: Rationality of players

• Ideas• Best response • Nash equilibrium• Dominant action or strategy• Dominated action or strategy

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Notations

• x Ak

• x is an action or a strategy of player k• Ak is a set of available actions for player k

• (a) = (a1, a2,…, an) A1A2…An = A• a profile of actions; one action from each player• (a) = (X,G,H,L,S)

• (a-k) = (a) \ ak A1…Ak-1Ak+1…An = A-k • actions of everybody except player k• (a-2) = (X,_,H,L,S)

• (a-k,y) = (a-k) y• (a-2,M) = (X,M,H,L,S)• (a-k,ak) = (a)

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Best Response Action

• An action x of player k is a best response to an action profile (a-k) if

• uk(a-k,x) > uk(a-k,y) for all y in Ak.

Confess Deny

Confess -5,-5 0,-10

Deny -10,0 -1,-1

Wendy's Dusty's

Wendy's 2,1 0,0

Dusty's 0,0 1,2

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Nash Equilibrium (local optimum)

• An action profile (a) is a Nash equilibrium if• for every player k, ak is a best response to (a-k)

• that is, for every player k, uk(a-k,ak) > uk(a-k,y) for all y in Ak

Confess Deny

Confess -5,-5 0,-10

Deny -10,0 -1,-1

Wendy's Dusty's

Wendy's 2,1 0,0

Dusty's 0,0 1,2

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Dominant Action or Strategy

• An action x of player k is a dominant action if • x is a best response to all (a-k) in A-k.

• That is, uk(a-k,x) > uk(a-k,y) for all y in Ak and any action profile (a-k) in A-k.

• That is, no matter what the other players do, x is a strategy for player k that is no worse than any other.

Confess Deny

Confess -5,-5 0,-10

Deny -10,0 -1,-1

Titanic Shrek

Titanic 3,2 1,3

Shrek 2,1 2,2

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Two Cases

• Dominant actions dictate the resulting Nash Equilibrium

• Dominant actions do not exist which means we need other methods

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Strictly Dominated Actions

• An action x of player k is a never-best response or a strictly dominated action if• x is not a best response to any action profile (a-k) in A-k

• That is, for any action profile (a-k) in A-k there exist an

action y in Ak such that uk(a-k,x) < uk(a-k,y)

• That is, no matter what the other players do, x is a strategy for player k that she should never use.

Titanic Shrek Sleep

Titanic 3,1 1,3 1,2

Shrek 2,3 2,1 2,2

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Iterated Elimination of Dominated Actions

• Procedure• Successively remove a strictly dominated action of a player

from the game table until there are no more strictly dominated actions

• Removing a dominated action• Reduce the size of the game• May make another action dominated• May make another action dominant

• If there is only 1 outcome remaining,• the game is said to be dominant solvable.• that outcome is the unique Nash equilibrium of the game

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Weakly Dominated Actions

• An action x of player k is a weakly dominated action if• for any action profile (a-k) in A-k there exists an

action y in Ak such that uk(a-k,x) < uk(a-k,y) and

• there exists an action profile (a-k) in A-k and an action y in Ak such that uk(a-k,x) < uk(a-k,y).

Titanic Shrek Sleep

Titanic 3,1 1,4 1,4

Shrek 2,3 2,2 2,1

Sleep 1,3 3,1 2,2

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Iterated Elimination of Weakly Dominated Actions

• Procedure• Same as before except• Remove weakly dominated actions instead of

strictly dominated actions

• Undesirable properties• The remaining cells may depend on the order that

the actions are removed. • May not yield all Nash equilibria.

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Best-Response Function

• A set-valued function Bk

• Bk(a-k) = {x Ak | x is a best response to (a-k) }• called the best-response function of player k.

• An action profile (ai) is a Nash equilibrium if• ak Bk(a-k) for all players k.

• An action x of player k is a dominant action if • x Bk(a-k) for all action profiles (a-k).

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Exhaustive Method

• Begin with a game table.

• We will incrementally cross out outcomes that are not Nash equilibria as follows:

• For each player k = 1..n• For each profile (a-k) in A-k

• Compute v = maxxAk uk(a-k, x)

• Cross out all outcomes (a-k,x) such that uk(a-k, x) < v

• The remaining outcomes are Nash equilibria.

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Example

Stand Walk Run

Float 62,65 38,74 34,32

Swim 68,38 55,52 31,36

Dive 33,37 32,30 22,28

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Solution

Stand Walk Run

Float 62,65 38,74 34,32

Swim 68,38 55,52 31,36

Dive 33,37 32,30 22,28

68 55 34

74

52

37

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Best-Response Table

Stand Walk Run

Float 62,65 38,74 34,32

Swim 68,38 55,52 31,36

Dive 33,37 32,30 22,28

Row player’s best-response table

Stand Walk Run

Float X

Swim X X

Dive

Stand Walk Run

Float X

Swim X

Dive X

Column player’s best-response table

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Outline







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The Prisoners’ Dilemma

Confess Deny

Confess -5,-5 0,-10

Deny -10,0 -1,-1

• The confession of a suspect will be used against the other.

• If both confess, get a reduced sentence.

• If neither confesses, face only minimum charge.

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Movie Game

• Two people go to a movie theatre.

Titanic Shrek

Titanic 3,2 1,3

Shrek 2,1 2,2

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Restaurant Game

Julia

Wendy's Dusty's

MalcolmWendy's 2,1 0,0

Dusty's 0,0 1,2

• Malcolm and Julia go to a restaurant.

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Concert Game

• Suppose both Malcolm and Julia are going to a concert instead of a dinner.

• Both like Mozart better than Mahler.

Mozart Mahler

Mozart 2,2 0,0

Mahler 0,0 1,1

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Chicken Game

• Malcolm and Julia dare one another to drive their cars straight into one another.

Julia

Swerve Straight

MalcolmSwerve 0,0 -1,1

Straight 1,-1 -3,-3

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Matching Pennies

Head Tail

Head 1,-1 -1,1

Tail -1,1 1,-1

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Outline







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Randomness in Payoff Functions

Venus Williams

DL CC

Serena Williams

DL 50,50 80,20

CC 90,10 20,80

• 2002 US open Final match.

• Serena is about to return the ball.

• She can either hit the ball down the line (DL) or crosscourt (CC)

• Venus must prepare to cover one side or the other

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Mixed Strategies

• What is a mixed strategy?• Suppose Ak is the set of pure strategies for

player k.• A mixed strategy for player k is a probability

distribution over Ak.• An actual move is chosen randomly according

to the probability distribution.

• Example:• Ak = { DL, CC }• “DL 60%, CC 40%” is a mixed strategy for k.

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Need for Mixed Strategies

• Multiple pure-strategy Nash equilibria• No pure-strategy Nash equilibria• Games where players prefer opposite outcomes

• Matching Pennies• Chicken• Sports• Attack and Defense

• Each player does very badly if her action is revealed to the other, because the other can respond accordingly.

• Want to keep the other guessing.• Mixed strategy Nash equilibrium always exists.

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Expectation

• Suppose X is a random variable.• Suppose X = 5 with probability 0.5• Suppose X = 6 with probability 0.3• Suppose X = 0 with probability 0.2• Then E[X] = 5*0.5 + 6*0.3 + 0*0.2• = 2.5 + 1.8 + 0 = 4.3• In general, if X = vi with probability pi

• Then E[X] = Σ vi pi

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Mixed Strategies in the Chicken Games

• Mixing 2 pure strategies• Swerve with probability p and Straight with

probability (1-p)• A continuous range of mixed strategies.

Julia

Swerve Straight

Malcolm

Swerve 0, 0 -1, 1

Straight 1, -1 -2, -2

p-mix

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Mixed Strategies in the Chicken Games

Julia

Swerve Straight q-mix

Malcolm

Swerve 0, 0 -1, 1

Straight 1, -1 -2, -2

p-mix

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Finding Mixed Strategy Nash Equilibrium

1. Compute Row’s payoffs as a function of q.2. Find q that make Row’s payoffs indifferent

no matter what pure strategy she chooses.3. Plot Row’s best-response curve.4. Do steps 1-3 for the Column player and p.5. Plot Row’s and Column’s best-response

curves together.6. Points where the 2 curves meet are Nash

equilibria.

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Why it is an equilibrium?

• It is a Nash equilibrium because• Malcolm can’t change his strategy to do better and

• Julia can’t change her strategy to do better

• Why can’t Malcolm do better?• Julia chooses a mix such that it doesn’t matter what

Malcolm does.

• Why can’t Julia do better?• Malcolm chooses a mix such that it doesn’t matter what

Julia does.

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Exercise

• Find mixed strategy Nash equilibrium in the following game.• Tennis match

Venus

DL CC

SerenaDL 50,50 80,20

CC 90,10 20,80

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Outline







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Selfish Routing

• Input• A directed graph G = (V,E)• Set of source-destination pairs {(si,ti)} where ri units of flow must be

transmitted from si to ti

• Each infinitesimal unit of flow is controlled by a selfish agent seeking to minimize its own latency.

• Latency functions L on each edge e• Le(x) is latency of edge e given load x on e

• Questions:• Identify the Nash Equilibria of the system• Price of Anarchy: How bad can the total latency of a Nash

Equilibrium be compared to that of a socially optimal solution?

s t

L(x) = x

L(x) = 1

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Simple Example 1

(s,t) demand is 1 unitWhat is optimal flow to minimize total latency?What is Nash equilibrium?Price of Anarchy in this example?

s t

L(x) = x

L(x) = 1

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Simple Example 2

s t

L(x) = xp for some integer p > 0

L(x) = 1


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Braess’ Paradox

s

v

t

w

L(x) = x

L(x) = x

L(x) = 1

L(x) = 1

0


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Price of Anarchy

• Approximation Algorithms• Lack of unbounded computing power leads to

loss of optimality

• Online Algorithms• Lack of complete information leads to loss of

optimality

• Noncooperative Games• Lack of coordination leads to loss of optimality

1 Adapted from: Game Theoretic Approach in Computer Science CS3150, Fall 2002 Introduction to Game Theory Patchrawat Uthaisombut University of Pittsburgh.

Documents

action x of player

action y

action profile

best response action

game examples

strategic game players

column player

player choices