Adapted from: Game Theoretic Approach in Computer Science CS3150, Fall 2002 Introduction to Game Theory

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Adapted fromGame Theoretic Approach in Computer
ScienceCS3150, Fall 2002Introduction to Game
Theory

• Patchrawat Uthaisombut
• University of Pittsburgh

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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

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Restaurant Game
Malcolm
Julia
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Payoffs
Julia Julia
Wendys Dustys
Malcolm Wendys 2,1 0,0
Malcolm Dustys 0,0 1,2
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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

• 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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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, whats 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 1ltiltn, a set Ai of actions available for
  player i.
• For 1ltiltn, a payoff function uiA1??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
Wendy's Dusty's
Wendy's 2,1 0,0
Dusty's 0,0 1,2

• 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

• 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

• 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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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) ? A1?A2??An A
• a profile of actions one action from each player
• (a) (X,G,H,L,S)
• (a-k) (a) \ ak ? A1??Ak-1?Ak1??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) gt uk(a-k,y) for all y in Ak.

Wendy's Dusty's
Wendy's 2,1 0,0
Dusty's 0,0 1,2
Confess Deny
Confess -5,-5 0,-10
Deny -10,0 -1,-1
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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) gt
  uk(a-k,y) for all y in Ak

Wendy's Dusty's
Wendy's 2,1 0,0
Dusty's 0,0 1,2
Confess Deny
Confess -5,-5 0,-10
Deny -10,0 -1,-1
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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) gt 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)
  lt 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) lt 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) lt 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 maxx?Ak uk(a-k, x)
• Cross out all outcomes (a-k,x) such that uk(a-k,
  x) lt 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
74
52
37
68
55
34
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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
Stand Walk Run
Float X
Swim X X
Dive
Stand Walk Run
Float X
Swim X
Dive X
Row players best-response table
Column players 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

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The Prisoners Dilemma

• 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.

Confess Deny
Confess -5,-5 0,-10
Deny -10,0 -1,-1
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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

• Malcolm and Julia go to a restaurant.

Julia Julia
Wendy's Dusty's
Malcolm Wendy's 2,1 0,0
Malcolm Dusty's 0,0 1,2
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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 Julia
Swerve Straight
Malcolm Swerve 0,0 -1,1
Malcolm 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

• 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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Randomness in Payoff Functions

• 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

Venus Williams Venus Williams
DL CC
Serena Williams DL 50,50 80,20
Serena Williams CC 90,10 20,80
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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 EX 50.5 60.3 00.2
• 2.5 1.8 0 4.3
• In general, if X vi with probability pi
• Then EX S 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 Julia
Swerve Straight
Malcolm Swerve 0, 0 -1, 1
Malcolm Straight 1, -1 -2, -2
Malcolm p-mix
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Mixed Strategies in the Chicken Games
Julia Julia Julia
Swerve Straight q-mix
Malcolm Swerve 0, 0 -1, 1
Malcolm Straight 1, -1 -2, -2
Malcolm p-mix
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Finding Mixed Strategy Nash Equilibrium

1. Compute Rows payoffs as a function of q.
2. Find q that make Rows payoffs indifferent no
   matter what pure strategy she chooses.
3. Plot Rows best-response curve.
4. Do steps 1-3 for the Column player and p.
5. Plot Rows and Columns 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 cant change his strategy to do better
  and
• Julia cant change her strategy to do better
• Why cant Malcolm do better?
• Julia chooses a mix such that it doesnt matter
  what Malcolm does.
• Why cant Julia do better?
• Malcolm chooses a mix such that it doesnt matter
  what Julia does.

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Exercise

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

Venus Venus
DL CC
Serena DL 50,50 80,20
Serena CC 90,10 20,80
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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

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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?

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Simple Example 1
L(x) 1
s
t
L(x) x
(s,t) demand is 1 unit What is optimal flow to
minimize total latency? What is Nash
equilibrium? Price of Anarchy in this example?
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Simple Example 2
L(x) 1
s
t
L(x) xp for some integer p gt 0
(s,t) demand is 1 unit What is optimal flow to
minimize total latency? What is Nash
equilibrium? Price of Anarchy in this example?
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Braess Paradox
v
L(x) x
L(x) 1
s
t
L(x) x
L(x) 1
w
(s,t) demand is 1 unit What is optimal flow to
minimize total latency? What is Nash
equilibrium? Price of Anarchy in this example?
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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