Congestion Games with affine functions Maria Serna Fall 2016 - PowerPoint PPT Presentation

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Congestion Games with affine functions Maria Serna Fall 2016 AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References 1 Congestion games and variants 2 Affine Congestion games 3 Weighted Congestion Games 4 References AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Congestion games AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Congestion games A congestion game ( E , N , ( d e ) e ∈ E , ( c i ) i ∈ N ) is defined on a finite set E of resources and has n players using a delay function d e mapping N to the integers, for each resource e . The actions for each player are subsets of E . The cost functions are the following: � c i ( a 1 , . . . , a n ) = d e ( f e ( a 1 , . . . , a n )) e ∈ a i being f e ( a 1 , . . . , a n ) = |{ i | e ∈ a i }| . AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Weighted congestion games AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Weighted congestion games A weighted congestion game ( E , N , ( d e ) e ∈ E , ( c i ) i ∈ N , ( w i ) i ∈ N ) is defined on a finite set E of resources and has n players. Player i has an associated natural weight w i . Using a delay function d e mapping N to the integers, for each resource e . The actions for each player are subsets of E . The cost functions are the following: � c i ( a 1 , . . . , a n ) = d e ( f e ( a 1 , . . . , a n )) e ∈ a i being f e ( a 1 , . . . , a n ) = � { i | e ∈ a i } w i . AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References 1 Congestion games and variants 2 Affine Congestion games 3 Weighted Congestion Games 4 References AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References PoA for affine congestion games Consider unweighted congestion games such that for each resource e d e ( x ) = a e x + b e , for a e , b e > 0. AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Smoothness A game is called ( λ, µ )-smooth, for λ > 0 and µ � 1 if, for every pair of strategy profiles s and s , we have � c i ( s − i , s ′ i ) � λ C ( s ′ ) + µ C ( s ) . i ∈ N AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Smoothness A game is called ( λ, µ )-smooth, for λ > 0 and µ � 1 if, for every pair of strategy profiles s and s , we have � c i ( s − i , s ′ i ) � λ C ( s ′ ) + µ C ( s ) . i ∈ N Smoothness directly gives a bound for the PoA: AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Smoothness A game is called ( λ, µ )-smooth, for λ > 0 and µ � 1 if, for every pair of strategy profiles s and s , we have � c i ( s − i , s ′ i ) � λ C ( s ′ ) + µ C ( s ) . i ∈ N Smoothness directly gives a bound for the PoA: Theorem λ In a ( λ, µ ) -smooth game, the PoA for PNE is at most 1 − µ . AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Proof of smoothness bound on PoA Let s be the worst PNE and s ∗ be an optimum solution. � � c i ( s − i , s ∗ C ( s ) = c i ( s ) � i ) i ∈ N i ∈ N � λ C ( s ∗ ) + µ C ( s ) Substracting µ C ( s ) on both sides gives (1 − µ ) C ( s ) � λ C ( s ∗ ) . AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Theorem Every congestion game with affine delay functions is (5 / 3 , 1 / 3) -smooth. Thus, PoA � 5 / 2 . AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Theorem Every congestion game with affine delay functions is (5 / 3 , 1 / 3) -smooth. Thus, PoA � 5 / 2 . The proof uses a technical lemma: Lemma (Christodoulou, Koutsoupias, 2005) For all integers y , z we have y ( z + 1) � 5 3 y 2 + 1 3 z 2 . AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Proof of smoothness for affine functions Recall that d e ( x ) = a e x + b e . Note that using the Lemma a e y ( z +1)+ b e y � a e (5 3 y 2 +1 3 z 2 )+ b e y = 5 3( a e y 2 + b e y )+1 3( a e z 2 + b e z ) . AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Proof of smoothness for affine functions Recall that d e ( x ) = a e x + b e . Note that using the Lemma a e y ( z +1)+ b e y � a e (5 3 y 2 +1 3 z 2 )+ b e y = 5 3( a e y 2 + b e y )+1 3( a e z 2 + b e z ) . Taking y = f e ( s ∗ ) and z = f e ( s ) we get ( a e ( f e ( s )+1)+ b e ) f e ( s ∗ ) � 5 3( a e f e ( s ∗ )+ b e ) f e ( s ∗ ))+1 3( a e f e ( s )+ b e ) f e ( s )) . AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Proof of smoothness for affine functions Recall that d e ( x ) = a e x + b e . Note that using the Lemma a e y ( z +1)+ b e y � a e (5 3 y 2 +1 3 z 2 )+ b e y = 5 3( a e y 2 + b e y )+1 3( a e z 2 + b e z ) . Taking y = f e ( s ∗ ) and z = f e ( s ) we get ( a e ( f e ( s )+1)+ b e ) f e ( s ∗ ) � 5 3( a e f e ( s ∗ )+ b e ) f e ( s ∗ ))+1 3( a e f e ( s )+ b e ) f e ( s )) . Summing up all the inequalities ( a e ( f e ( s ) + 1) + b e ) f e ( s ∗ ) � 5 3 C ( s ∗ ) + 1 � 3 C ( s ) . e ∈ E AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Proof of smoothness for affine functions ( a e ( f e ( s ) + 1) + b e ) f e ( s ∗ ) � 5 3 C ( s ∗ ) + 1 � 3 C ( s ) . e ∈ E AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Proof of smoothness for affine functions ( a e ( f e ( s ) + 1) + b e ) f e ( s ∗ ) � 5 3 C ( s ∗ ) + 1 � 3 C ( s ) . e ∈ E But, � c i ( s − i , s ∗ � ( a e ( f e ( s ) + 1) + b e ) f e ( s ∗ ) i ) � i ∈ N e ∈ E as there are at most f e ( s ∗ ) players that might move to resource r . Each of them by unilaterally deviating incur a delay of ( a e ( f e ( s ) + 1) + b e . AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References Proof of smoothness for affine functions ( a e ( f e ( s ) + 1) + b e ) f e ( s ∗ ) � 5 3 C ( s ∗ ) + 1 � 3 C ( s ) . e ∈ E But, � c i ( s − i , s ∗ � ( a e ( f e ( s ) + 1) + b e ) f e ( s ∗ ) i ) � i ∈ N e ∈ E as there are at most f e ( s ∗ ) players that might move to resource r . Each of them by unilaterally deviating incur a delay of ( a e ( f e ( s ) + 1) + b e . This gives the (5 / 3 , 1 / 3)-smoothness. AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References 1 Congestion games and variants 2 Affine Congestion games 3 Weighted Congestion Games 4 References AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References PNE in Weighted Congestion Games There are weighted network congestion games without PNE AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References PNE in Weighted Congestion Games There are weighted network congestion games without PNE (see blackboard example) AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References PNE in Weighted Congestion Games There are weighted network congestion games without PNE (see blackboard example) For affine delay functions PNE always exist AGT-MIRI, FIB-UPC Congestion Games

Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References PNE in Weighted Congestion Games There are weighted network congestion games without PNE (see blackboard example) For affine delay functions PNE always exist Show that the following Φ( s ) is a weighted potential function AGT-MIRI, FIB-UPC Congestion Games