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 player corresponds to a pair of source-destination Objective is to select paths with small cost 3

Main objective of each player is to minimize congestion: minimize maximum utilized edge player i  congestion 3 C 4

Congestion Games: A player may selfishly choose an alternative path that minimizes congestion     congestion 1 3 C C 5

Player cost function for routing : p i Congestion pc ( p ) C  i i of selected path Social cost function for routing : p SC ( p ) C  Largest player cost 6

We are interested in Nash Equilibriums p where every player is locally optimal Metrics of equilibrium quality: Price of Stability Price of Anarchy ( ) SC p ( ) SC p min max * ( ) * SC p p ( ) SC p p is optimal coordinated routing * p with smallest social cost

Previous Result: Costas Busch and Malik Magdon-Ismail, “Atomic Routing Games on Maximum Congestion,” Theoretical Computer Science, August 2009. • Price of Stability is 1  • Price of Anarchy is ( log ) O L n Maximum allowed path length  • Price of Anarchy is 2 2 ( log ) O k n  k (  Maximum cycle length 1 ) 8

Length of chosen path Stretch= Length of shortest path 12  stretch 1 . 5  8 shortest path u v chosen path 9

Our Results: General bottleneck games: • Price of Anarchy is ( sm ) O stretch Resources (edges) Linear bottleneck games: • Price of Anarchy is ( sm ) O 10

Outline of Talk Introduction General games Linear Games Conclusions 11

Every player can have an arbitrary weight on each resource Strategy weight: The sum of weights of utilized resources Stretch : s Maximum ratio of strategy weights 12

• Price of Anarchy is ( sm ) O  • Price of Anarchy is at least ( sm ) 13

weight Player 1 sm 1 Strategy 1 1 1 1 resources m 1 1 1 1 Strategy 2 14

weight Player 2 sm 1 Strategy 1 1 1 1 m 1 resources 1 1 1 Strategy 2 15

*  Optimal solution 1 C 1 1 1 1 1 1 1 1 resources m m 1 1 resources 1 1 1 1 1 1 Player 2 Player 1 16

weight weight sm sm player 2 player 1 Equilibrium C  sm 17

Price of anarchy: C sm  1  sm * C Lower bound Can easily show matching upper bound 18

Outline of Talk Introduction General games Linear Games Conclusions 19

A player uses the same weight on each resource weight of player i: w i Congestion on resource is linear function on total weight on resource   ( ) C r a w i  : i r S i 20

• Price of Anarchy is ( sm ) O • Price of Anarchy is at least  ( sm ) 21

Players 1 n 2 n s  All players have same weight 1 w i  The linear coefficient is 1 a 22

Players 1 n 2 n s Number of resources 2 n n n     sm m n s s 23

Strategy 1 1 n 2 n s Player i strategies 24

Strategy 2 1 n 2 i n s Player i stategies stretch  2 s 25

*  Optimal solution 1 C 1 n 2 i n s All players use their second strategy 26

C  Equilibrium n 1 n 2 n s All players use their first strategy 27

Price of anarchy: C n sm   * 1 1 C Lower bound 28

Upper Bound Analysis Lemma: for any set of resources Q it holds ( ) | | *  C Q Q avg C sm We apply the lemma to a special set of resources Q 29

Support set of a resource r with Q maximum congestion r C 1  C C 1  Optimal strategy of player using r 30

Properties of support set : Q     * ( ) 1 C avg Q C C C C C  * | | Q ( ) | | *  C Q Q + Lemma: avg C sm 31

Therefore:   C sm     min | |, 1 Q *   | | C Q Which gives:   Price of Anarchy  sm  1 O sm 32

Proof of Lemma: for any set of resources Q it holds ( ) | | *  C Q Q avg C sm 33

       * ( ) | | | | C Q Q C S s S avg r i i    ( ) ( ) r Q i P Q i P Q     * ( ) ( ) S W Q W Q i  ( ) i P Q    i  * ( ) S W Q  ( ) i P Q 34

  ( ) C Q ( ) W Q   avg * C Q s     ( 1 ) ( ) | | C Q Q ( ) W Q   * avg C   | | Q s Q    | | | | m Q Q ( ) | | *  C Q Q avg C sm 35

Outline of Talk Introduction General games Linear Games Conclusions 36

For non-uniform games: Price of anarchy:  ( ) O sm Max ratio of resource linear factors a   r , max i a i j r j 37

For non-uniform games:   8   Price of anarchy: O sm   3   We can remove the dependence on  38