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Stretch in Bottleneck Games
Costas Busch Rajgopal Kannan
Division of Computer Science and Eng., School of EECS Louisiana State University
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Stretch in Bottleneck Games Costas Busch Rajgopal Kannan Division - - PowerPoint PPT Presentation
Stretch in Bottleneck Games Costas Busch Rajgopal Kannan Division - - PowerPoint PPT Presentation
Stretch in Bottleneck Games Costas Busch Rajgopal Kannan Division of Computer Science and Eng., School of EECS Louisiana State University 1 Outline of Talk Introduction General games Linear Games Conclusions 2 Network Routing Each
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Network Routing
Each player corresponds to a pair of source-destination Objective is to select paths with small cost
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Main objective of each player is to minimize congestion: minimize maximum utilized edge
3 congestion C i player
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A player may selfishly choose an alternative path that minimizes congestion
C C 3 1 congestion
Congestion Games:
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Player cost function for routing :
i
i i
C p pc ) ( p
Congestion
- f selected path
Social cost function for routing :
C p SC ) (
p
Largest player cost
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We are interested in Nash Equilibriums where every player is locally optimal Metrics of equilibrium quality:
p
Price of Stability
) ( ) ( min
*
p SC p SC
p
Price of Anarchy
) ( ) ( max
*
p SC p SC
p
*
p
is optimal coordinated routing with smallest social cost
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Previous Result:
- Price of Stability is 1
- Price of Anarchy is
) log ( n L O
Maximum allowed path length
Costas Busch and Malik Magdon-Ismail, “Atomic Routing Games on Maximum Congestion,” Theoretical Computer Science, August 2009.
- Price of Anarchy is
) log (
2 2
n k O
Maximum cycle length
) 1 ( k
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Length of chosen path Length of shortest path
u v
Stretch=
5 . 1 8 12 stretch
shortest path
chosen path
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Our Results:
General bottleneck games: Linear bottleneck games:
- Price of Anarchy is
) (sm O
- Price of Anarchy is
) ( sm O
stretch Resources (edges)
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Introduction General games Linear Games Outline of Talk Conclusions
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Every player can have an arbitrary weight
- n each resource
Stretch :
s
Strategy weight: The sum of weights of utilized resources Maximum ratio of strategy weights
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- Price of Anarchy is
) (sm O ) (sm
- Price of Anarchy is at least
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1 1 1 1 1 1 1 resources
sm m
weight
Strategy 1 Strategy 2
Player 1 1
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1 resources
sm m
weight
Strategy 1 Strategy 2
Player 2 1 1 1 1 1 1 1
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1 resources
m
Player 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 resources
m
Player 1
1 Optimal solution
1
*
C
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sm
weight
player 2
sm
weight
player 1
Equilibrium
sm C
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Price of anarchy:
sm sm C C 1
*
Lower bound Can easily show matching upper bound
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Introduction General games Linear Games Outline of Talk Conclusions
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A player uses the same weight
- n each resource
Congestion on resource is linear function
- n total weight on resource
i
w
weight of player i:
i
S r i i
w a r C
:
) (
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- Price of Anarchy is
) ( sm O ) ( sm
- Price of Anarchy is at least
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1 2 n All players have same weight The linear coefficient is
1 a 1
i
w
Players
s n
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1 2 n Players Number of resources
s n n s n m
2
sm n
s n
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1 2 n
Strategy 1
Player i strategies
s n
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1 2 n
Strategy 2
Player i stategies i stretch
2 s
s n
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1 2 n
Optimal solution
i
1
*
C s n
All players use their second strategy
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1 2 n
s n
Equilibrium
n C
All players use their first strategy
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Price of anarchy:
1 1
*
sm n C C
Lower bound
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Upper Bound Analysis Lemma: for any set of resources it holds
sm Q Q C C
avg
| | ) (
*
Q
We apply the lemma to a special set
- f resources Q
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C
1 C 1 C
Support set of a resource r with maximum congestion Optimal strategy
- f player using r
r
Q
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Properties of support set :
Q
*
1 ) ( C C C Q Cavg | |
*
Q C C
sm Q Q C C
avg
| | ) (
*
+ Lemma:
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Therefore:
| | 1 |, | min
*
Q sm Q C C
Which gives: Price of Anarchy
sm O sm 1
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Proof of Lemma: for any set of resources it holds
sm Q Q C C
avg
| | ) (
*
Q
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) ( * ) (
| | | | ) (
Q P i i Q P i i Q r r avg
S s S C Q Q C ) ( ) (
) ( *
Q W Q W S
Q P i i
) (
) ( *
Q W S
Q P i i
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s Q C Q Q W C
avg
) ( ) (
*
| | | | ) ( ) 1 ( ) (
*
Q s Q Q C Q Q W C
avg
sm Q Q C C
avg
| | ) (
*
| | | | Q Q m
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Introduction General games Linear Games Outline of Talk Conclusions
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For non-uniform games: Price of anarchy:
) ( sm O
Max ratio of resource linear factors
j i
r r j i
a a
,max
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