1 atomic routing games on maximum congestion costas busch department of computer science louisiana...
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1
Atomic Routing Games on Maximum Congestion
Costas BuschDepartment of Computer Science
Louisiana State University
Collaborators: Rajgopal Kannan, LSUMalik Magdon-Ismail, RPI
2
Introduction
Price of Stability
Price of Anarchy
Outline of Talk
Bicriteria Game
3
Network Routing
Each player corresponds to a pair of source-destination
Objective is to select paths with small cost
4
Main objective of each player is to minimize congestion: minimize maximum utilized edge
3 congestion C
iplayer
5
A player may selfishly choose an alternativepath that minimizes congestion
CC 31 congestion
Congestion Games:
6
Player cost function for routing :i
ii Cppc )(
p
Congestionof selected path
Social cost function for routing :
CpSC )(
p
Largest player cost
We are interested in Nash Equilibriumswhere every player is locally optimal
Metrics of equilibrium quality:
p
Price of Stability
)(
)(min
*pSC
pSCp
Price of Anarchy
)(
)(max
*pSC
pSCp
*p is optimal coordinated routing with smallest social cost
8
Results:
• Price of Stability is 1
• Price of Anarchy is )log( nLO
Maximum allowed path length
9
Introduction
Price of Stability
Price of Anarchy
Outline of Talk
Bicriteria Game
10
We show:
• QoR games have Nash Equilibriums
(we define a potential function)
• The price of stability is 1
11
],,,,,[)( 21 Nk mmmmpM
number of players with cost km k
Routing Vector
12
Routing Vectors are ordered lexicographically
],,,[)( 21 NmmmpM
],,,[)( 21 NmmmpM
= = = =
],,,,,[)( 11 Nkk mmmmpM
],,,,,[)( 11 Nkk mmmmpM
< < = =
)()( pMpM
)()( pMpM )( pp
)( pp
13
If player performs a greedy movetransforming routing to then:p p pp
i
Lemma:
Proof Idea:
Show that the greedy move gives a lower order routing vector
14
kk
ii Cppck )(
ii Cppck )(
Player Costi
Before greedy move:
After greedy move:
Since player cost decreases:
15
],,,,,,,[)( 11 Nkkk mmmmmpM
Before greedy move player was counted herei
],,,,,,,[)( 11 Nkkk mmmmmpM
After greedy moveplayer is counted herei
16
],,,,,,,[)( 11 rkkk mmmmmpM
],,,,,,,[)( 11 rkkk mmmmmpM
> ==No change
Definite Decrease
possibledecrease
possibleincreaseor decrease
Possible increase
>
END OF PROOF IDEA
17
Existence of Nash Equilibriums
Greedy moves give lower order routings
Eventually a local minimum for every playeris reached which is a Nash Equilibrium
18
minp
Price of Stability
Lowest order routing :
*min )( SCpSC
• Is a Nash Equilibrium
• Achieves optimal social cost
1)(
Stability of Price*min
SC
pSC
19
Introduction
Price of Stability
Price of Anarchy
Outline of Talk
Bicriteria Game
20
We show for any restricted QoR game:
Price of Anarchy = )log( nLO
21
Path of player
Consider an arbitrary Nash Equilibriump
i
iCedgemaximum congestionin path
22
must have an edge with congestion
Optimal path of player
In optimal routing :*p
i
iC
1 iCC
)(111*)( ppcCCCppc iiii
**)( CpSC
Since otherwise:
23
C
00
0
edges use that Paths:
Congestion of Edges :
E
CE
In Nash Equilibrium social cost is: CSC
0 0
24
C 1C1C
0 0
Edges in optimal paths of 0
25
C 1C1C
0 01 1
11
1
edges use that Players:
1 least at Congestion of Edges :
E
CE
26
C 1C1C 2C 2C2C2C
0 01 1
Edges in optimal paths of 1
27
C *SC *SC 2C 2C2C
0 01 1
2C
2 2
22
2
edges use that Players:
2 least at Congestion of Edges :
E
CE
28
In a similar way we can define:
jj
j
E
jCE
edges use that Players:
least at Congestion of Edges :
29
,,,,
,,,,
3210
3210
EEEEWe obtain sequences:
There exist subsequence:
110
110
,,,
,,,,
s
ss EEEE
||2|| 1 jj EEWhere: ||2|| 1 ss EEand1sj
ns log
30
||))1((|| 11 ss EsCL
|||| 1*
s
s
EC
Maximum edge utilization
Minimum edge utilization
LMaximum path length
)log(* nLOCC
ns log ||2|| 1 ss EE
Known relations
31
Worst Case Scenario:
32
Introduction
Price of Stability
Price of Anarchy
Outline of Talk
Bicriteria Game
33
We consider Quality of Routing (QoR) congestion games where the pathsare partitioned into routing classes:
QQQ ,,, 21
)()()( 21 QSQSQS
With service costs:
Only paths in same routing class can causecongestion to each other
34
An example:
•We can have routing classes)(lognO
•Each routing class contains paths with length in range
jQ
]2,2( 1jj
12)( jjQS•Service cost:
•Each routing class uses a different wireless frequency channel
35
Player cost function for routing :i
iii SCppc )(
p
Congestionof selected path
Cost of respectiverouting class
36
Social cost function for routing :
SCpSC )(
p
Largest player cost
37
Results:
• Price of Stability is 1
• Price of Anarchy is
)log),(min( ** nSCO
38
We consider restricted QoR games
For any path :p )(|| pSp
Path length Service Cost of path
39
We show for any restricted QoR game:
Price of Anarchy = )log),(min( ** nSCO
40
Path of player
Consider an arbitrary Nash Equilibriump
i
iCedgemaximum congestionin path
41
must have an edge with congestion
Optimal path of player
In optimal routing :*p
i
iC
*SCC i
)(111 *** ppcSCCSSCSCcp iiiiiiii
***)( SCpSC
Since otherwise:
42
C
00
0
edges use that Paths:
Congestion of Edges :
E
CE
In Nash Equilibrium: SCSC
0 0
43
C *SC *SC
0 0
Edges in optimal paths of 0
44
C *SC *SC
0 01 1
11
*1
edges use that Players:
least at Congestion of Edges :
E
SCE
45
C *SC *SC *2SC *2SC *2SC *2SC
0 01 1
Edges in optimal paths of 1
46
C *SC *SC *2SC *2SC *2SC
0 01 1
*2SC
2 2
22
*2
edges use that Players:
2least at Congestion of Edges :
E
SCE
47
In a similar way we can define:
jj
j
E
jSCE
edges use that Players:
least at Congestion of Edges : *
48
,,,,
,,,,
3210
3210
EEEEWe obtain sequences:
There exist subsequence:
110
110
,,,
,,,,
s
ss EEEE
||2|| 1 jj EEWhere: ||2|| 1 ss EEand1sj
ns log
49
||))1((|| 1*
1 ss ESsCL
||
|| 1*
s
s
EC
Maximum edge utilization
Minimum edge utilization
*SLMaximum path length
)log( **
nSOC
C
ns log ||2|| 1 ss EE
Known relations
50
)log( **
nSOC
C
)log),(min( Anarchy of Price ****
nSCOSC
SC
We have:
By considering class service costs, we obtain:
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