routing games with progressive filling
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Routing Games with Progressive Filling
Kevin Schewior
TU Berlin, COGA Group
Presentation of my Masters Thesis
Advisor: Martin Hoefer (now MPII Saarbrucken)
Computer Science 1, RWTH Aachen
May 23, 2013
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Motivation
Consider a network modelling e.g. a road or computer network.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Motivation
Consider a network modelling e.g. a road or computer network.
Players (logistics companies, computer users) want to send
traffic from a specific source to a specific sink.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Motivation
Consider a network modelling e.g. a road or computer network.
Players (logistics companies, computer users) want to send
traffic from a specific source to a specific sink.
In order to achieve that, they choose paths from their source totheir sink nodes (as actions in a game).
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C C
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Motivation
Consider a network modelling e.g. a road or computer network.
Players (logistics companies, computer users) want to send
traffic from a specific source to a specific sink.
In order to achieve that, they choose paths from their source totheir sink nodes (as actions in a game).
We are considering a fair way to distribute bandwidth conform to
capacity constraints.
max-min fairness
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I t d ti P i Filli G Th ti A h C t ti l C l it Effi i f E ilib i F t W k
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Motivation
Consider a network modelling e.g. a road or computer network.
Players (logistics companies, computer users) want to send
traffic from a specific source to a specific sink.
In order to achieve that, they choose paths from their source totheir sink nodes (as actions in a game).
We are considering a fair way to distribute bandwidth conform to
capacity constraints.
max-min fairnessPlayers may have different priorities.
progressive filling as a generalization of max-min fairness
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Related Work
analysis of routing games with max-min fair allocations
Yang, Xue and Fang at ICNP 10
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Related Work
analysis of routing games with max-min fair allocations
Yang, Xue and Fang at ICNP 10
analysis of bottleneck congestion games
Harks, Hoefer, Klimm and Skopalik at ESA 10
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Related Work
analysis of routing games with max-min fair allocations
Yang, Xue and Fang at ICNP 10
analysis of bottleneck congestion games
Harks, Hoefer, Klimm and Skopalik at ESA 10
analysis of the maximum k-splittable flow (MkSF) problem
Bailer, Kohler and Skutella at ESA 02
Koch, Skutella and Spenke at WAOA 06
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Goals
We want to...
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g g pp p p y y q
Goals
We want to...
efficiently find equilibrium states (with a preferably high
throughput or other properties) in special cases or in general.
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g g pp p p y y q
Goals
We want to...
efficiently find equilibrium states (with a preferably high
throughput or other properties) in special cases or in general.efficiently compute or approximate optimal states in special
cases or in general.
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Goals
We want to...
efficiently find equilibrium states (with a preferably high
throughput or other properties) in special cases or in general.efficiently compute or approximate optimal states in special
cases or in general.
describe equilibrium states and optimal states in terms of
throughput (PoA, PoS).
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Overview
1 Introduction
2 Progressive Filling
3 Game Theoretic Approach
4 Computational Complexity
5 Efficiency of Equilibria
6 Future Work
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Max-Min Fairness
We are given a set of paths and want to determine the capacity sent
along the paths. Idea of Max-Min Fairness:
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Max-Min Fairness
We are given a set of paths and want to determine the capacity sent
along the paths. Idea of Max-Min Fairness:
Firstly, maximize the minimum bandwith.
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Max-Min Fairness
We are given a set of paths and want to determine the capacity sent
along the paths. Idea of Max-Min Fairness:
Firstly, maximize the minimum bandwith.
Secondly, maximize the second minimum bandwith.
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Max-Min Fairness
We are given a set of paths and want to determine the capacity sent
along the paths. Idea of Max-Min Fairness:
Firstly, maximize the minimum bandwith.
Secondly, maximize the second minimum bandwith.
...and so on...
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Progressive Filling for Max-Min Fairness
Idea:
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Progressive Filling for Max-Min Fairness
Idea:
Set the bandwidth of all players initially to 0.
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Progressive Filling for Max-Min Fairness
Idea:
Set the bandwidth of all players initially to 0.
Let the bandwidths of all players uniformally rise.
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Progressive Filling for Max-Min Fairness
Idea:
Set the bandwidth of all players initially to 0.
Let the bandwidths of all players uniformally rise.
When a link gets saturated, fix the bandwidths of the players on
this link and continue with the other players.
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Progressive Filling for Max-Min Fairness
Idea:
Set the bandwidth of all players initially to 0.
Let the bandwidths of all players uniformally rise.
When a link gets saturated, fix the bandwidths of the players on
this link and continue with the other players.
Theorem
This algorithm computes the max-min fair bandwidth allocation.
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General Progressive Filling
Players may have different priorities.
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General Progressive Filling
Players may have different priorities.
Thus, allow (Riemann) integrable functions vi : R` R`
assigning an allocation rate to each point in time.
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General Progressive Filling
Players may have different priorities.
Thus, allow (Riemann) integrable functions vi : R` R`
assigning an allocation rate to each point in time.
Further requirement:
80
vi ptqdt 8.
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General Progressive Filling
Players may have different priorities.
Thus, allow (Riemann) integrable functions vi : R` R`
assigning an allocation rate to each point in time.
Further requirement:
80
vi ptqdt 8.
A formalization will not be given here; we focus on moreimportant parts.
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Example for Progressive Filling
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Example for Progressive Filling
v1 ptq
tv2 ptq
tv3 ptq
t
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Example for Progressive Filling
v1 ptq
tv2 ptq
tv3 ptq
t
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Example for Progressive Filling
v1 ptq
tv2 ptq
tv3 ptq
t
t
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Example for Progressive Filling
v1 ptq
tv2 ptq
tv3 ptq
t
t
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Formal Description (1)
Introduce an allocation model M pN, R, pcrqrPR , pSiqiPNq where
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Formal Description (1)
Introduce an allocation model M pN, R, pcrqrPR , pSiqiPNq where
N t1, ..., nu is the set of players,
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Formal Description (1)
Introduce an allocation model M pN, R, pcrqrPR , pSiqiPNq where
N t1, ..., nu is the set of players,
R t1, ..., mu is the set of resources,
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F l D i i (1)
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Formal Description (1)
Introduce an allocation model M pN, R, pcrqrPR , pSiqiPNq where
N t1, ..., nu is the set of players,
R t1, ..., mu is the set of resources,
cr P R is the capacity of resource r, for each r P R,
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F l D i ti (1)
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Formal Description (1)
Introduce an allocation model M pN, R, pcrqrPR , pSiqiPNq where
N t1, ..., nu is the set of players,
R t1, ..., mu is the set of resources,
cr P R is the capacity of resource r, for each r P R,
Si PpRq is the set of strategies of player i, for each i P N.
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F l D i ti (2)
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Formal Description (2)
Introduce a corresponding progressive filling game (PFG) to the
allocation model M and allocation rate functions pviqiPN:
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F l D i ti (2)
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Formal Description (2)
Introduce a corresponding progressive filling game (PFG) to the
allocation model M and allocation rate functions pviqiPN:
Strategic game GpM, pviqiPNq pN, pSiqiPN , pbiqiPNq where
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Formal Description (2)
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Formal Description (2)
Introduce a corresponding progressive filling game (PFG) to the
allocation model M and allocation rate functions pviqiPN:
Strategic game GpM, pviqiPNq pN, pSiqiPN , pbiqiPNq where
the players and strategies are kept,
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Formal Description (2)
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Formal Description (2)
Introduce a corresponding progressive filling game (PFG) to the
allocation model M and allocation rate functions pviqiPN:
Strategic game GpM, pviqiPNq pN, pSiqiPN , pbiqiPNq where
the players and strategies are kept,
bi : S R` is the bandwidth function of Player i, for each i P N,
calculated by progressive filling using the functions pviqiPN.
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Formal Description (2)
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Formal Description (2)
Introduce a corresponding progressive filling game (PFG) to the
allocation model M and allocation rate functions pviqiPN:
Strategic game GpM, pviqiPNq pN, pSiqiPN , pbiqiPNq where
the players and strategies are kept,
bi : S R` is the bandwidth function of Player i, for each i P N,
calculated by progressive filling using the functions pviqiPN.
If the allocation is calculated with uniform allocation rate functions, we
call the corresponding game max-min fair game (MMFG).
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Formal Description (2)
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Formal Description (2)
Introduce a corresponding progressive filling game (PFG) to the
allocation model M and allocation rate functions pviqiPN:
Strategic game GpM, pviqiPNq pN, pSiqiPN , pbiqiPNq where
the players and strategies are kept,
bi : S R` is the bandwidth function of Player i, for each i P N,
calculated by progressive filling using the functions pviqiPN.
If the allocation is calculated with uniform allocation rate functions, we
call the corresponding game max-min fair game (MMFG).
The social welfare in SP Swill be defined to be
SW pSq :
iPN bi pSq.
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Considered Subclasses
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Considered Subclasses
According to the structure of allocation models, we distinguish
different subclasses of PFGs.
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Considered Subclasses
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Considered Subclasses
According to the structure of allocation models, we distinguish
different subclasses of PFGs.
A PFG G pN, pSiqiPN , pbiqiPNq is called
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Considered Subclasses
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Considered Subclasses
According to the structure of allocation models, we distinguish
different subclasses of PFGs.
A PFG G pN, pSiqiPN , pbiqiPNq is called
symmetric game if we have Si Sj for all i,j P N (otherwise it is
called asymmetric),
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Considered Subclasses
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Considered Subclasses
According to the structure of allocation models, we distinguish
different subclasses of PFGs.
A PFG G pN, pSiqiPN , pbiqiPNq is called
symmetric game if we have Si Sj for all i,j P N (otherwise it is
called asymmetric),
network game if it is played on the edges of a graph as resources
and the strategies of player i are the paths between certain
source and sink nodes si and ti, for each i P N,
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Considered Subclasses
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Considered Subclasses
According to the structure of allocation models, we distinguish
different subclasses of PFGs.
A PFG G pN, pSiqiPN , pbiqiPNq is called
symmetric game if we have Si Sj for all i,j P N (otherwise it is
called asymmetric),
network game if it is played on the edges of a graph as resources
and the strategies of player i are the paths between certain
source and sink nodes si and ti, for each i P N,
single-commodity network game if G is a symmetric network
game and
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Considered Subclasses
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Considered Subclasses
According to the structure of allocation models, we distinguish
different subclasses of PFGs.
A PFG G pN, pSiqiPN , pbiqiPNq is called
symmetric game if we have Si Sj for all i,j P N (otherwise it is
called asymmetric),
network game if it is played on the edges of a graph as resources
and the strategies of player i are the paths between certain
source and sink nodes si and ti, for each i P N,
single-commodity network game if G is a symmetric network
game and
multi-commodity network game if G is an asymmetric network
game.
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Considered Subclasses
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Considered Subclasses
According to the structure of allocation models, we distinguish
different subclasses of PFGs.
A PFG G pN, pSiqiPN , pbiqiPNq is called
symmetric game if we have Si Sj for all i,j P N (otherwise it is
called asymmetric),
network game if it is played on the edges of a graph as resources
and the strategies of player i are the paths between certain
source and sink nodes si and ti, for each i P N,
single-commodity network game if G is a symmetric network
game and
multi-commodity network game if G is an asymmetric network
game.
Sometimes we will w.l.o.g. consider multigraphs instead of graphs.
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k-Strong Equilibria
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g q
k-Strong Equilibrium (k-SE)
SP S is a k-SE
No C N with |C| k and S1C P
iPCSi exist such that
bi`
S1C, S C bi pSq, for all i P C.
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k-Strong Equilibria
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g q
k-Strong Equilibrium (k-SE)
SP S is a k-SE
No C N with |C| k and S1C P
iPCSi exist such that
bi`
S1C, S C bi pSq, for all i P C.
Special cases:
k 1: Nash Equilibrium (NE)
k n: Strong Equilibrium (SE)
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Existence of SE in PFGs
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Let S be a state in a PFG. For a player i, we call the point in timewhere his bandwidth gets fixed his finishing time ti pSq.
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Existence of SE in PFGs
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Let S be a state in a PFG. For a player i, we call the point in timewhere his bandwidth gets fixed his finishing time ti pSq.
Now consider the sorted vector of finishing times of all players N, in
ascending order.
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Existence of SE in PFGs
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Let S be a state in a PFG. For a player i, we call the point in timewhere his bandwidth gets fixed his finishing time ti pSq.
Now consider the sorted vector of finishing times of all players N, in
ascending order.
With each improvement step of a coalition C N from S to a state T,
this vector lexicographically increases.
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Existence of SE in PFGs
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Let S be a state in a PFG. For a player i, we call the point in timewhere his bandwidth gets fixed his finishing time ti pSq.
Now consider the sorted vector of finishing times of all players N, in
ascending order.
With each improvement step of a coalition C N from S to a state T,
this vector lexicographically increases. Reason:
let i be the player from C with the minimum finishing time in S
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Existence of SE in PFGs
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Let S be a state in a PFG. For a player i, we call the point in time
where his bandwidth gets fixed his finishing time ti pSq.
Now consider the sorted vector of finishing times of all players N, in
ascending order.
With each improvement step of a coalition C N from S to a state T,
this vector lexicographically increases. Reason:
let i be the player from C with the minimum finishing time in S
all the players who are fixed earlier than or at the same time as i
in S cannot be negatively affected by the improvement step
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Existence of SE in PFGs
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Let S be a state in a PFG. For a player i, we call the point in time
where his bandwidth gets fixed his finishing time ti pSq.
Now consider the sorted vector of finishing times of all players N, in
ascending order.
With each improvement step of a coalition C N from S to a state T,
this vector lexicographically increases. Reason:
let i be the player from C with the minimum finishing time in S
all the players who are fixed earlier than or at the same time as i
in S cannot be negatively affected by the improvement step
all the other players still get fixed after ti pSq
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Existence of SE in PFGs
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Let S be a state in a PFG. For a player i, we call the point in time
where his bandwidth gets fixed his finishing time ti pSq.
Now consider the sorted vector of finishing times of all players N, in
ascending order.
With each improvement step of a coalition C N from S to a state T,
this vector lexicographically increases. Reason:
let i be the player from C with the minimum finishing time in S
all the players who are fixed earlier than or at the same time as i
in S cannot be negatively affected by the improvement step
all the other players still get fixed after ti pSq
PFGs have a lexicographical potential function
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Existence of SE in PFGs
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Let S be a state in a PFG. For a player i, we call the point in time
where his bandwidth gets fixed his finishing time ti pSq.
Now consider the sorted vector of finishing times of all players N, in
ascending order.
With each improvement step of a coalition C N from S to a state T,
this vector lexicographically increases. Reason:
let i be the player from C with the minimum finishing time in S
all the players who are fixed earlier than or at the same time as i
in S cannot be negatively affected by the improvement step
all the other players still get fixed after ti pSq
PFGs have a lexicographical potential function
PFGs possess SE
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An Important Decision Problem
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The following problem is known to be NP-hard:
2 Directed Arc-Disjoint Paths Problem (2DADP Problem)
Input: A directed graph D pV, Aq and source-sink pairsps1, t1q , ps2, t2q P V2.
Output: The information whether there exist arc-disjoint paths from
s1 to t1 and from s2 to t2.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Computation of SE in single-commodity network PFGs
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Let v1 v2 be two constant allocation rate functions and consider the
class of single-commodity network PFGs with 2 players and v1, v2 as
allocation rate functions.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Computation of SE in single-commodity network PFGs
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Let v1 v2 be two constant allocation rate functions and consider the
class of single-commodity network PFGs with 2 players and v1, v2 as
allocation rate functions.
Theorem
In this class, the computation of SE is NP-hard.
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Computation of SE in single-commodity network PFGs
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Let v1 v2 be two constant allocation rate functions and consider the
class of single-commodity network PFGs with 2 players and v1, v2 as
allocation rate functions.
Theorem
In this class, the computation of SE is NP-hard.
We reduce from 2DADP. W.l.o.g., v1 1 and v2 1.
Digraph D from the
2DADP instance;capacity 1` each
1`
`
1`
`
s
s1
s2
t
t1
t2
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Computation of SE in single-commodity network PFGs
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Let v1 v2 be two constant allocation rate functions and consider the
class of single-commodity network PFGs with 2 players and v1, v2 as
allocation rate functions.
Theorem
In this class, the computation of SE is NP-hard.
We reduce from 2DADP. W.l.o.g., v1 1 and v2 1.
Digraph D from the
2DADP instance;capacity 1` each
1`
`
1`
`
s
s1
s2
t
t1
t2
We show that each SE certifies whether the 2DADP-instance is
solvable or not:
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Computation of SE in single-commodity network PFGs
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We reduce from 2DADP. W.l.o.g., v1 1 and v2 1.
Digraph D from the
2DADP instance;capacity 1` each
1`
`
1`
`
s
s1
s2
t
t1
t2
We show that each SE certifies whether the 2DADP-instance is
solvable or not:
Kevin Schewior Routing Games with Progressive Filling 16/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Computation of SE in single-commodity network PFGs
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We reduce from 2DADP. W.l.o.g., v1 1 and v2 1.
Digraph D from the
2DADP instance;capacity 1` each
1`
`
1`
`
s
s1
s2
t
t1
t2
We show that each SE certifies whether the 2DADP-instance is
solvable or not:
a) Each SE with two arc-disjoint paths from s to t certifies that the
instance is solvable:
in any SE, player 1 always uses a path of the form ps, s1, . . . , t1, tq
since player 2 uses an arc-disjoint path, he must indeed connect
s2 and t2
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Computation of SE in single-commodity network PFGs
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We reduce from 2DADP. W.l.o.g., v1 1 and v2 1.
Digraph D from the
2DADP instance;capacity 1` each
1`
`
1`
`
s
s1
s2
t
t1
t2
We show that each SE certifies whether the 2DADP-instance issolvable or not:
b) Each SE without two arc-disjoint paths from s to t certifies that the
instance is not solvable:
if the players share a common edge in a SE, player 1 getsbandwidth 1 and player 2 gets bandwidth
if there were two arc-disjoint paths in D, the players could switch
to these paths and strictly improve
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Complexity Results for PFGs
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computing SE computing optima
general PFGs NP-hard NP-hard
network PFGs NP-hard NP-hardsymmetric PFGs NP-hard NP-hard
single-commodity
network PFGsNP-hard NP-hard
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Complexity Results for MMFGs
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computing SE computing optima
general MMFGs NP-hard NP-hard
network MMFGs NP-hard NP-hardsymmetric MMFGs NP-hard NP-hard
single-commodity
network MMFGs
polynomial time
via Dual GreedyNP-hard
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Dual Greedy Algorithm
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Dual Greedy (Harks et al., ESA10) can be used to compute a SE in
bottleneck congestion games the following way:
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Dual Greedy Algorithm
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Dual Greedy (Harks et al., ESA10) can be used to compute a SE in
bottleneck congestion games the following way:Introduce upper bounds for players on each edge.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Dual Greedy Algorithm
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Dual Greedy (Harks et al., ESA10) can be used to compute a SE in
bottleneck congestion games the following way:Introduce upper bounds for players on each edge.
Iteratively reduce the bounds on undesirable edges as long as
the bounds still allow a feasible state.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Dual Greedy Algorithm
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Dual Greedy (Harks et al., ESA10) can be used to compute a SE in
bottleneck congestion games the following way:Introduce upper bounds for players on each edge.
Iteratively reduce the bounds on undesirable edges as long as
the bounds still allow a feasible state.
Fix the players who prevent the bounds from being reduced andcontinue with the other ones.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Dual Greedy Algorithm
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Dual Greedy (Harks et al., ESA10) can be used to compute a SE in
bottleneck congestion games the following way:Introduce upper bounds for players on each edge.
Iteratively reduce the bounds on undesirable edges as long as
the bounds still allow a feasible state.
Fix the players who prevent the bounds from being reduced andcontinue with the other ones.
Theorem
Dual Greedy can be modified to calculate a SE in MMFGs.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Dual Greedy Algorithm
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Dual Greedy (Harks et al., ESA10) can be used to compute a SE in
bottleneck congestion games the following way:Introduce upper bounds for players on each edge.
Iteratively reduce the bounds on undesirable edges as long as
the bounds still allow a feasible state.
Fix the players who prevent the bounds from being reduced andcontinue with the other ones.
Theorem
Dual Greedy can be modified to calculate a SE in MMFGs.
Theorem
Dual Greedy can be implemented to run in polynomial time for
single-commodity network MMFGs.
Kevin Schewior Routing Games with Progressive Filling 19/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Approximation Guarantee of Dual Greedy
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Consider symmetric MMFGs. We utilize some ideas and
constructions from work on the MkSF problem.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Approximation Guarantee of Dual Greedy
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Consider symmetric MMFGs. We utilize some ideas and
constructions from work on the MkSF problem.
Theorem
Dual Greedy computes a (2 1n
)-approximation to the social optimum.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Approximation Guarantee of Dual Greedy
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Consider symmetric MMFGs. We utilize some ideas and
constructions from work on the MkSF problem.
Theorem
Dual Greedy computes a (2 1n
)-approximation to the social optimum.
Theorem
If we fix n 2, computing any better approximation is NP-hard.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Approximation Guarantee of Dual Greedy
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Consider symmetric MMFGs. We utilize some ideas and
constructions from work on the MkSF problem.
Theorem
Dual Greedy computes a (2 1n
)-approximation to the social optimum.
Theorem
If we fix n 2, computing any better approximation is NP-hard.
Theorem
Asymptotically, computing any approximation with a guarantee
smaller than 65
for fixed n is NP-hard.
Kevin Schewior Routing Games with Progressive Filling 20/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
k-Strong Price of Anarchy/Stability
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k-Strong Price of Anarchy (k-SPoA)k-SPoApGq : maxSPS SWpSq
min SPS:S is k-SE SWpSq
the optimal social welfare
the worst social welfare possible in a k-SE
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k-Strong Price of Anarchy/Stability
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k-Strong Price of Anarchy (k-SPoA)k-SPoApGq : maxSPS SWpSq
min SPS:S is k-SE SWpSq
the optimal social welfare
the worst social welfare possible in a k-SE
k-Strong Price of Stability (k-SPoS)
k-SPoSpGq : maxSPS SWpSqmax SPS:S is k-SE SWpSq the optimal social welfarethe best social welfare possible in a k-SE
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
k-Strong Price of Anarchy/Stability
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k-Strong Price of Anarchy (k-SPoA)
k-SPoApGq : maxSPS SWpSqmin SPS:S is k-SE SWpSq
the optimal social welfare
the worst social welfare possible in a k-SE
k-Strong Price of Stability (k-SPoS)
k-SPoSpGq : maxSPS SWpSqmax SPS:S is k-SE SWpSq the optimal social welfarethe best social welfare possible in a k-SE
Special cases:
k 1: Price of Anarchy/Stability (PoA/PoS)
k n: Strong Price of Anarchy/Stability (SPoA/SPoS)
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k-Strong Price of Anarchy/Stability
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k-Strong Price of Anarchy (k-SPoA)
k-SPoApGq : maxSPS SWpSqmin SPS:S is k-SE SWpSq
the optimal social welfare
the worst social welfare possible in a k-SE
k-Strong Price of Stability (k-SPoS)
k-SPoSpGq : maxSPS SWpSqmax SPS:S is k-SE SWpSq the optimal social welfarethe best social welfare possible in a k-SE
Special cases:
k 1: Price of Anarchy/Stability (PoA/PoS)
k n: Strong Price of Anarchy/Stability (SPoA/SPoS)
Extension to classes of games: The k-SPoA/S is the supremum of
the individual k-SPoA/S.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in single-commodity network PFGs
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Theorem
The PoS in single-commodity network PFGs is at least n.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in single-commodity network PFGs
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Theorem
The PoS in single-commodity network PFGs is at least n.
cap. 1
(top edge:
1 ` )cap.
1 `
cap. 1
(bot. edge:
1 ` )
...
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in single-commodity network PFGs
Th
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Theorem
The PoS in single-commodity network PFGs is at least n.
cap. 1
(top edge:
1 ` )cap.
1 `
cap. 1
(bot. edge:
1 ` )
...
Players:
one player () with constant
allocation rate function v1 1
n 1 players (,,,...) with
constant allocation rate
functions vi
n
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PoS in single-commodity network PFGs
Th
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Theorem
The PoS in single-commodity network PFGs is at least n.
cap. 1
(top edge:
1 ` )cap.
1 `
cap. 1
(bot. edge:
1 ` )
...
any NE:
player takes the path with
capacity 1 `
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PoS in single-commodity network PFGs
Th
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Theorem
The PoS in single-commodity network PFGs is at least n.
cap. 1
(top edge:
1 ` )cap.
1 `
cap. 1
(bot. edge:
1 ` )
...
any NE:
player takes the path with
capacity 1 `
all the other players ,,,...
must hence share an edge
with player
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PoS in single-commodity network PFGs
Theorem
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Theorem
The PoS in single-commodity network PFGs is at least n.
cap. 1
(top edge:
1 ` )cap.
1 `
cap. 1
(bot. edge:
1 ` )
...
optimal state:
All the players choose parallel
paths with capacity 1 each through
the network.
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PoS in single-commodity network PFGs
Theorem
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Theorem
The PoS in single-commodity network PFGs is at least n.
cap. 1
(top edge:
1 ` )cap.
1 `
cap. 1
(bot. edge:
1 ` )
...
Price of Stability:
social welfare in a NE:
1 ` 2 if 1
social welfare in optimal state:
n
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PoS in single-commodity network PFGs
Theorem
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Theorem
The PoS in single-commodity network PFGs is at least n.
cap. 1
(top edge:
1 ` )cap.
1 `
cap. 1
(bot. edge:
1 ` )
...
Price of Stability:
social welfare in a NE:
1 ` 2 if 1
social welfare in optimal state:
n
sup!
n1`2 | 1 0
) n
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PoA in PFGs
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Theorem
The PoA in PFGs is at most n.
Kevin Schewior Routing Games with Progressive Filling 23/29 Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in symmetric MMFGs
By the approximation guarantee of Dual Greedy, we know that the
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SPoS in this case is at most 2 1n
.
Kevin Schewior Routing Games with Progressive Filling 24/29 Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in symmetric MMFGs
By the approximation guarantee of Dual Greedy, we know that the
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SPoS in this case is at most 2 1
n
.
We find the same lower bound on the PoS.
Theorem
The PoS in single-commodity MMFGs isat least2 1n
.
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PoS in symmetric MMFGs
By the approximation guarantee of Dual Greedy, we know that the
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SPoS in this case is at most 2 1
n
.
We find the same lower bound on the PoS.
Theorem
The PoS in single-commodity MMFGs isat least2 1n
.
consider a network consisting of source and sink nodes s and t
and n parallel edges
Kevin Schewior Routing Games with Progressive Filling 24/29 Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in symmetric MMFGs
By the approximation guarantee of Dual Greedy, we know that the
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SPoS in this case is at most 2 1
n
.
We find the same lower bound on the PoS.
Theorem
The PoS in single-commodity MMFGs isat least2 1n
.
consider a network consisting of source and sink nodes s and t
and n parallel edges
one of these edges has capacity n and the other n 1 edges
have capacity 1
Kevin Schewior Routing Games with Progressive Filling 24/29 Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in symmetric MMFGs
By the approximation guarantee of Dual Greedy, we know that the1
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SPoS in this case is at most 2 1
n
.
We find the same lower bound on the PoS.
Theorem
The PoS in single-commodity MMFGs isat least2 1n
.
consider a network consisting of source and sink nodes s and t
and n parallel edges
one of these edges has capacity n and the other n 1 edges
have capacity 1
any NE: everyone uses the n-edge, social welfare n
Kevin Schewior Routing Games with Progressive Filling 24/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in symmetric MMFGs
By the approximation guarantee of Dual Greedy, we know that the1
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SPoS in this case is at most 2 1
n
.
We find the same lower bound on the PoS.
Theorem
The PoS in single-commodity MMFGs isat least2 1n
.
consider a network consisting of source and sink nodes s and t
and n parallel edges
one of these edges has capacity n and the other n 1 edges
have capacity 1
any NE: everyone uses the n-edge, social welfare n
optimal state: everyone uses an individual edge, social welfare
asymptotically 2n 1
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in multi-commodity network MMFGs
Theorem
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The PoS in multi-commodity network MMFGs is in pnq.
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in multi-commodity network MMFGs
Theorem
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The PoS in multi-commodity network MMFGs is in pnq.
W.l.o.g. 2 | n.
s1 t1 s2 t2 s3 t3 s4 tn2 sn
2`1 sn tn
2`1 tn
. . .1 1 1 1 1
0
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in multi-commodity network MMFGs
Theorem
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The PoS in multi-commodity network MMFGs is inpnq.
W.l.o.g. 2 | n.
s1 t1 s2 t2 s3 t3 s4 tn2 sn
2`1 sn tn
2`1 tn
. . .1 1 1 1 1
0
any equilibrium state:
0-edge is not chosen at all
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in multi-commodity network MMFGs
Theorem
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The PoS in multi-commodity network MMFGs is in
pnq.
W.l.o.g. 2 | n.
s1 t1 s2 t2 s3 t3 s4 tn2 sn
2`1 sn tn
2`1 tn
. . .1 1 1 1 1
0
any equilibrium state:
0-edge is not chosen at all
hence, each 1-edge is used by n2` 1 players
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in multi-commodity network MMFGs
Theorem
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The PoS in multi-commodity network MMFGs is in
pnq.
W.l.o.g. 2 | n.
s1 t1 s2 t2 s3 t3 s4 tn2 sn
2`1 sn tn
2`1 tn
. . .1 1 1 1 1
0
any equilibrium state:
0-edge is not chosen at all
hence, each 1-edge is used by n2` 1 players
overall, the social welfare is n 1n2`1
2nn 2
O p1q
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Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in multi-commodity network MMFGs
Theorem
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The PoS in multi-commodity network MMFGs is inpnq.
W.l.o.g. 2 | n.
s1 t1 s2 t2 s3 t3 s4 tn2 sn
2`1 sn tn
2`1 tn
. . .1 1 1 1 1
0
optimum state:
0-edge is chosen by players n2` 1, . . . , n
Kevin Schewior Routing Games with Progressive Filling 25/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in multi-commodity network MMFGs
Theorem
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The PoS in multi-commodity network MMFGs is inpnq.
W.l.o.g. 2 | n.
s1 t1 s2 t2 s3 t3 s4 tn2 sn
2`1 sn tn
2`1 tn
. . .1 1 1 1 1
0
optimum state:
0-edge is chosen by players n2` 1, . . . , n
hence, each 1-edge is used by exactly one player
Kevin Schewior Routing Games with Progressive Filling 25/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in multi-commodity network MMFGs
Theorem
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The PoS in multi-commodity network MMFGs is inpnq.
W.l.o.g. 2 | n.
s1 t1 s2 t2 s3 t3 s4 tn2 sn
2`1 sn tn
2`1 tn
. . .1 1 1 1 1
0
optimum state:
0-edge is chosen by players n2` 1, . . . , n
hence, each 1-edge is used by exactly one player
overall, the social welfare is n2 pnq
Kevin Schewior Routing Games with Progressive Filling 25/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
PoS in multi-commodity network MMFGs
Theorem
Th P S i lti dit t k MMFG i i
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The PoS in multi-commodity network MMFGs is in p
nq
.
W.l.o.g. 2 | n.
s1 t1 s2 t2 s3 t3 s4 tn2 sn
2`1 sn tn
2`1 tn
. . .1 1 1 1 1
0
Price of Stability:
any NE: social welfare of O p1q
optimal state: social welfare of pnq
thus, PoS ispnqOp1q pnq
Kevin Schewior Routing Games with Progressive Filling 25/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
k-SPoA in single-commodity network MMFGs
First let k 1, i.e., consider the PoA and the following network:
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...
n players
in optimum social welfare of n
Kevin Schewior Routing Games with Progressive Filling 26/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
k-SPoA in single-commodity network MMFGs
First let k 1, i.e., consider the PoA and the following network:
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...
n players
in optimum social welfare of n; in a NE social welfare of 1
PoA=n
Kevin Schewior Routing Games with Progressive Filling 26/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
k-SPoA in single-commodity network MMFGs
Now let k arbitrary but fixed and consider the k-SPoA and the
following network:
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g
......
...
. . .n disjointpaths per
gadget
k gadgets
nk players each
in optimum social welfare of n
Kevin Schewior Routing Games with Progressive Filling 26/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
k-SPoA in single-commodity network MMFGs
Now let k arbitrary but fixed and consider the k-SPoA and the
following network:
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g
......
...
. . .n disjointpaths per
gadget
k gadgets
nk players each
in optimum social welfare of n; in a k-SE social welfare of k
k-SPoA= nk
Kevin Schewior Routing Games with Progressive Filling 26/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
k-SPoA in single-commodity network MMFGs
http://find/ -
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Theorem
The k-SPoA for single-commodity network MMFGs is in `
nk
.
Kevin Schewior Routing Games with Progressive Filling 26/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
k-SPoA in single-commodity network MMFGs
http://find/ -
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Theorem
The k-SPoA for single-commodity network MMFGs is in `
nk
.
Conjecture
The k-SPoA for single-commodity network MMFGs is in `
nk
.
Kevin Schewior Routing Games with Progressive Filling 26/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
SPoA in symmetric MMFGs
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Theorem
For n 2, it holds that SPoA 2 1n 3
2.
Kevin Schewior Routing Games with Progressive Filling 27/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
SPoA in symmetric MMFGs
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Theorem
For n 2, it holds that SPoA 2 1n 3
2.
Theorem
For arbitrary n, we have SPoA 4n 2n 1
(i.e. asymptotically 4).
Kevin Schewior Routing Games with Progressive Filling 27/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
SPoA in symmetric MMFGs
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Theorem
For n 2, it holds that SPoA 2 1n 3
2.
Theorem
For arbitrary n, we have SPoA 4n 2n 1
(i.e. asymptotically 4).
Conjecture
For arbitrary n, we conjecture SPoA 2 1n
.
Kevin Schewior Routing Games with Progressive Filling 27/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Overview of results on PoS/PoA for MMFGs
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PoS SPoS SPoA PoA
general MMFGs
network MMFGs
symmetric MMFGs
single-commodity
network MMFGs
Kevin Schewior Routing Games with Progressive Filling 28/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Overview of results on PoS/PoA for MMFGs
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PoS SPoS SPoA PoA
general MMFGs pnq pnq pnq n
network MMFGs pnq pnq pnq n
symmetric MMFGs
single-commodity
network MMFGs
Kevin Schewior Routing Games with Progressive Filling 28/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Overview of results on PoS/PoA for MMFGs
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PoS SPoS SPoA PoA
general MMFGs pnq pnq pnq n
network MMFGs pnq pnq pnq n
symmetric MMFGs 2 1n 2 1n
4n 2n 1
n
single-commodity
network MMFGs2 1
n2 1
n 4n 2
n 1n
Kevin Schewior Routing Games with Progressive Filling 28/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Future Work
http://find/http://goback/ -
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For single-commodity network MMFGs:
Kevin Schewior Routing Games with Progressive Filling 29/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Future Work
http://find/http://goback/ -
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For single-commodity network MMFGs:
find a tight bound on the SPoA for n 2
Kevin Schewior Routing Games with Progressive Filling 29/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Future Work
http://find/ -
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123/126
For single-commodity network MMFGs:
find a tight bound on the SPoA for n 2
find a tight bound on the k-SPoA for fixed k
Kevin Schewior Routing Games with Progressive Filling 29/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Future Work
F i l di k MMFG
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For single-commodity network MMFGs:
find a tight bound on the SPoA for n 2
find a tight bound on the k-SPoA for fixed k
find a tight bound on the inapproximability of optimal states for
n 2 (and possibly a better approximation than Dual Greedy if
2 1n
is not yet tight)
Kevin Schewior Routing Games with Progressive Filling 29/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Future Work
F i l dit t k MMFG
http://find/http://goback/ -
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125/126
For single-commodity network MMFGs:
find a tight bound on the SPoA for n 2
find a tight bound on the k-SPoA for fixed k
find a tight bound on the inapproximability of optimal states for
n 2 (and possibly a better approximation than Dual Greedy if
2 1n
is not yet tight)
Other tasks:
Kevin Schewior Routing Games with Progressive Filling 29/29
Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work
Future Work
F i l dit t k MMFG
http://find/http://goback/ -
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126/126
For single-commodity network MMFGs:
find a tight bound on the SPoA for n 2
find a tight bound on the k-SPoA for fixed k
find a tight bound on the inapproximability of optimal states for
n 2 (and possibly a better approximation than Dual Greedy if
2 1n
is not yet tight)
Other tasks:
consider other subclasses of PFGs such as matroid games
Kevin Schewior Routing Games with Progressive Filling 29/29
http://find/http://goback/