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Transcript of 1 Adapted from: Game Theoretic Approach in Computer Science CS3150, Fall 2002 Introduction to Game...
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
5
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
6
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
7
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?
8
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
9
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
11
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
12
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
13
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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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
• 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
28
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.
30
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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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
35
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
36
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
39
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
41
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.
42
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.
43
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
44
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
45
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
46
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
47
Simple Example 2
s t
L(x) = xp for some integer p > 0
L(x) = 1
(s,t) demand is 1 unitWhat is optimal flow to minimize total latency?What is Nash equilibrium?Price of Anarchy in this example?
48
Braess’ Paradox
s
v
t
w
L(x) = x
L(x) = x
L(x) = 1
L(x) = 1
0
(s,t) demand is 1 unitWhat is optimal flow to minimize total latency?What is Nash equilibrium?Price of Anarchy in this example?