routing games with progressive filling

7/30/2019 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

Kevin Schewior Routing Games with Progressive Filling 1/29
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Motivation

Consider a network modelling e.g. a road or computer network.

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Motivation


Players (logistics companies, computer users) want to send

traffic from a specific source to a specific sink.

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Motivation




In order to achieve that, they choose paths from their source totheir sink nodes (as actions in a game).


C C
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Motivation





We are considering a fair way to distribute bandwidth conform to

capacity constraints.

max-min fairness


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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Motivation





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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Related Work

analysis of routing games with max-min fair allocations

Yang, Xue and Fang at ICNP 10


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Related Work



analysis of bottleneck congestion games

Harks, Hoefer, Klimm and Skopalik at ESA 10


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Related Work



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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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...


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...


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



Firstly, maximize the minimum bandwith.


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Max-Min Fairness




Secondly, maximize the second minimum bandwith.


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Max-Min Fairness




Secondly, maximize the second minimum bandwith.

...and so on...




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Progressive Filling for Max-Min Fairness

Idea:


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Idea:

Set the bandwidth of all players initially to 0.


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Idea:


Let the bandwidths of all players uniformally rise.




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Idea:



When a link gets saturated, fix the bandwidths of the players on

this link and continue with the other players.


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Idea:



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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Thus, allow (Riemann) integrable functions vi : R` R`

assigning an allocation rate to each point in time.


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Further requirement:

80

vi ptqdt 8.


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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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v1 ptq

tv2 ptq

tv3 ptq

t




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v1 ptq

tv2 ptq

tv3 ptq

t


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v1 ptq

tv2 ptq

tv3 ptq

t

t




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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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N t1, ..., nu is the set of players,


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R t1, ..., mu is the set of resources,



F l D i i (1)
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cr P R is the capacity of resource r, for each r P R,



F l D i ti (1)
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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.



F l D i ti (2)
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Introduce a corresponding progressive filling game (PFG) to the

allocation model M and allocation rate functions pviqiPN:



F l D i ti (2)
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Strategic game GpM, pviqiPNq pN, pSiqiPN , pbiqiPNq where



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the players and strategies are kept,



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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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If the allocation is calculated with uniform allocation rate functions, we

call the corresponding game max-min fair game (MMFG).



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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.



Considered Subclasses
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According to the structure of allocation models, we distinguish

different subclasses of PFGs.



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A PFG G pN, pSiqiPN , pbiqiPNq is called



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symmetric game if we have Si Sj for all i,j P N (otherwise it is

called asymmetric),



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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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called asymmetric),




single-commodity network game if G is a symmetric network

game and





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called asymmetric),





game and

multi-commodity network game if G is an asymmetric network

game.





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called asymmetric),





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.



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.



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)



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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Now consider the sorted vector of finishing times of all players N, in

ascending order.





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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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ascending order.


this vector lexicographically increases. Reason:

let i be the player from C with the minimum finishing time in S



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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.


ascending order.




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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ascending order.






all the other players still get fixed after ti pSq



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ascending order.







PFGs have a lexicographical potential function



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ascending order.







PFGs have a lexicographical potential function

PFGs possess SE



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.



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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Theorem

In this class, the computation of SE is NP-hard.



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Theorem


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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Theorem



Digraph D from the


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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Digraph D from the


1`

`

1`

`

s

s1

s2

t

t1

t2


solvable or not:





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Digraph D from the


1`

`

1`

`

s

s1

s2

t

t1

t2


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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Digraph D from the


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



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



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



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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bottleneck congestion games the following way:Introduce upper bounds for players on each edge.





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Iteratively reduce the bounds on undesirable edges as long as

the bounds still allow a feasible state.



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Fix the players who prevent the bounds from being reduced andcontinue with the other ones.



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Theorem

Dual Greedy can be modified to calculate a SE in MMFGs.



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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.



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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Theorem

Dual Greedy computes a (2 1n

)-approximation to the social optimum.



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Theorem



Theorem

If we fix n 2, computing any better approximation is NP-hard.



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Theorem



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.



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 (k-SPoA)k-SPoApGq : maxSPS SWpSq

min SPS:S is k-SE SWpSq



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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k-Strong Price of Anarchy (k-SPoA)

k-SPoApGq : maxSPS SWpSqmin SPS:S is k-SE SWpSq





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 (k-SPoA)

k-SPoApGq : maxSPS SWpSqmin SPS:S is k-SE SWpSq





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.



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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Theorem


cap. 1

(top edge:

1 ` )cap.

1 `

cap. 1

(bot. edge:

1 ` )

...




Th


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Theorem


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

Kevin Schewior Routing Games with Progressive Filling 22/29 Introduction Progressive Filling Game Theoretic Approach Computational Complexity Efficiency of Equilibria Future Work


Th
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Theorem


cap. 1

(top edge:

1 ` )cap.

1 `

cap. 1

(bot. edge:

1 ` )

...

any NE:

player takes the path with

capacity 1 `



Th
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Theorem


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



Theorem
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Theorem


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.



Theorem
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Theorem


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



Theorem
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Theorem


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


PoA in PFGs
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Theorem

The PoA in PFGs is at most n.


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

.



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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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n

.


Theorem


.

consider a network consisting of source and sink nodes s and t

and n parallel edges



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n

.


Theorem


.



one of these edges has capacity n and the other n 1 edges

have capacity 1



By the approximation guarantee of Dual Greedy, we know that the1
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n

.


Theorem


.




have capacity 1

any NE: everyone uses the n-edge, social welfare n




By the approximation guarantee of Dual Greedy, we know that the1
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n

.


Theorem


.




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



PoS in multi-commodity network MMFGs

Theorem
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The PoS in multi-commodity network MMFGs is in pnq.




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




Theorem
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The PoS in multi-commodity network MMFGs is inpnq.

W.l.o.g. 2 | n.


2`1 sn tn

2`1 tn

. . .1 1 1 1 1

0

any equilibrium state:

0-edge is not chosen at all




Theorem
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The PoS in multi-commodity network MMFGs is in

pnq.

W.l.o.g. 2 | n.


2`1 sn tn

2`1 tn

. . .1 1 1 1 1

0



hence, each 1-edge is used by n2` 1 players




Theorem


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The PoS in multi-commodity network MMFGs is in

pnq.

W.l.o.g. 2 | n.


2`1 sn tn

2`1 tn

. . .1 1 1 1 1

0



hence, each 1-edge is used by n2` 1 players

overall, the social welfare is n 1n2`1

2nn 2

O p1q




Theorem
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W.l.o.g. 2 | n.


2`1 sn tn

2`1 tn

. . .1 1 1 1 1

0

optimum state:

0-edge is chosen by players n2` 1, . . . , n




Theorem
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W.l.o.g. 2 | n.


2`1 sn tn

2`1 tn

. . .1 1 1 1 1

0

optimum state:


hence, each 1-edge is used by exactly one player




Theorem
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W.l.o.g. 2 | n.


2`1 sn tn

2`1 tn

. . .1 1 1 1 1

0

optimum state:


hence, each 1-edge is used by exactly one player

overall, the social welfare is n2 pnq




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.


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



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




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




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




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



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Theorem

The k-SPoA for single-commodity network MMFGs is in `

nk

.



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Theorem


nk

.

Conjecture


nk

.



SPoA in symmetric MMFGs
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Theorem

For n 2, it holds that SPoA 2 1n 3

2.



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Theorem


2.

Theorem

For arbitrary n, we have SPoA 4n 2n 1

(i.e. asymptotically 4).



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Theorem


2.

Theorem

For arbitrary n, we have SPoA 4n 2n 1

(i.e. asymptotically 4).

Conjecture

For arbitrary n, we conjecture SPoA 2 1n

.



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



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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





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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



Future Work


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For single-commodity network MMFGs:



Future Work


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find a tight bound on the SPoA for n 2



Future Work
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find a tight bound on the k-SPoA for fixed k



Future Work

F i l di k MMFG
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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)



Future Work

F i l dit t k MMFG


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2 1n

is not yet tight)

Other tasks:



Future Work

F i l dit t k MMFG


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2 1n

is not yet tight)

Other tasks:

consider other subclasses of PFGs such as matroid games


routing games with progressive filling

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