Tighter Bounds on the Inefficiency Ratio of Stable Equilibria in - PowerPoint PPT Presentation

Tighter Bounds on the Inefficiency Ratio of Stable Equilibria in Load Balancing Games Akaki Mamageishvili Paolo Penna ETH Zurich

Outline Load Balancing Games Inefficiency Ratio of Stable Equilibria Tighter Bounds for IRSE (Our Contribution)

Load Balancing (Games)

Load Balancing (Games)

Load Balancing (Games)

Load Balancing (Games) no improvement

Load Balancing (Games) Nash Equilibrium OPT

Quality of Equilibria Price of Anarchy: Let the players choose some equilibrium by themselves. How bad this can be? Nash

Quality of Equilibria Price of Anarchy: Let the players choose some equilibrium by themselves. How bad this can be? PoA = worst Nash Opt Nash

Quality of Equilibria Price of Anarchy: Let the players choose some equilibrium by themselves. How bad this can be? PoA = worst Nash Opt Nash PoS = best Nash Opt

Load Balancing (Games) x 1 1 1 x x x 1

Load Balancing (Games) x 1 1 1 x x x 1 1+ x → 4 2 x PoA = 3

Load Balancing (Games) x 1 1 1 x x x 1 1+ x → 4 2 x PoA = 3 � � 1 In general: PoA = 2 1 − m +1 PoS = 1

Inefficiency Ratio of Stable Equilibria PoA = worst Nash Opt Nash PoS = best Nash Opt

Inefficiency Ratio of Stable Equilibria PoA = worst Nash Opt Nash Stable PoS = best Nash Opt

Inefficiency Ratio of Stable Equilibria PoA = worst Nash Opt noisy best response Nash Stable min potential PoS = best Nash Opt

Bounded Rationality

Bounded Rationality Noisy best response prob ∝ e 10 / noise € 10 prob ∝ e 1 / noise € 1 Strategies with higher payoff chosen with higher probability

Inefficiency Ratio of Stable Equilibria PoA = worst Nash Opt noisy best response Nash Stable min potential PoS = best Nash Opt

Inefficiency Ratio of Stable Equilibria PoA = worst Nash Opt noisy best response Nash IRSE= worst Stable Nash Opt Stable min potential PoS = best Nash Opt (Asadpour, Saberi, 2009)

Inefficiency Ratio of Stable Equilibria PoA = worst Nash Opt noisy best response Nash IRSE= worst Stable Nash Opt Stable min potential PoS = best Nash Opt L 2 -norm L ∞ -norm (Asadpour, Saberi, 2009)

Inefficiency Ratio of Stable Equilibria PoA = worst Nash Opt noisy best response Nash IRSE= worst Stable Nash Opt Stable min potential PoS = best Nash Opt Minimize L 2 -norm ⇒ also good for L ∞ -norm (makespan)? (Alon, Azar, Woeginger, Yadid, 1997) (Asadpour, Saberi, 2009)

Our Contribution 7 / 6 ≤ IRSE ≤ 4 / 3

Our Contribution 7 / 6 ≤ IRSE ≤ 4 / 3 mininimize L 2 -norm automatically 4 / 3 -APX for L ∞ -norm

Our Contribution 7 / 6 ≤ IRSE ≤ 4 / 3 mininimize L 2 -norm sometimes at least 7 / 6 -APX of L ∞ -norm

Our Contribution 7 / 6 ≤ IRSE ≤ 4 / 3 Previous bunds: 19 / 18 ≤ IRSE ≤ 3 / 2

Our Contribution 7 / 6 ≤ IRSE ≤ 4 / 3 Previous bunds: 19 / 18 ≤ IRSE ≤ 3 / 2 Asadpour - Saberi, WINE 2009 Alon - Azar - Woeginger - Yadid, SODA 1997

Our Contribution 7 / 6 ≤ IRSE ≤ 4 / 3 mininimize L 2 -norm sometimes at least 7 / 6 -APX of L ∞ -norm

Lower Bound (IRSE ≥ 7/6) 2 2 3 9 3 2 2

Lower Bound (IRSE ≥ 7/6) 2 2 3 9 min potential 3 2 2

Lower Bound (IRSE ≥ 7/6) 2 2 3 9 min potential 3 2 2

Lower Bound (IRSE ≥ 7/6) 2 2 3 9 min potential 3 2 2 2 3 opt 2 9 3 2 2

Lower Bound (IRSE ≥ 7/6) 3 4 3 5 min potential 14 4 5 2 3 3 5 4 opt 3 14 4 5 2 3

Lower Bound (IRSE ≥ 7/6) 10 3 4 3 5 min potential 14 4 5 2 3 9 3 5 4 opt 3 14 4 5 2 3

Lower Bound (IRSE ≥ 7/6) m − 1 3 4 3 5 min potential 14 4 5 2 3 3 5 4 opt 3 14 4 5 2 3

Lower Bound (IRSE ≥ 7/6) m − 1 3 4 3 5 min potential 14 4 5 2 3 3 m − 3 3 5 4 opt 3 14 4 5 2 3

Lower Bound (IRSE ≥ 7/6) m − 1 3 4 3 5 min potential 14 4 5 2 3 2 m − 3 3 m − 3 3 5 4 opt 3 14 4 5 2 3

Lower Bound (IRSE ≥ 7/6) m − 1 3 4 3 5 min potential 14 5 m − 3 4 5 2 2 3 2 m − 3 3 m − 3 3 5 4 opt 3 14 4 5 2 3

Lower Bound (IRSE ≥ 7/6) 7 m − 4 3 2 4 3 5 min potential 14 5 m − 3 4 5 2 2 3 2 m − 3 3 m − 3 3 5 4 opt 3 14 4 5 2 3

Our Contribution 7 / 6 ≤ IRSE ≤ 4 / 3 mininimize L 2 -norm automatically 4 / 3 -APX for L ∞ -norm

Upper Bound (IRSE ≤ 4/3) α > 1 / 3 opt =1 3 min potential L 1 ≥ L 2 L m ≥ · · · · · · ≥

Upper Bound (IRSE ≤ 4/3) α > 1 / 3 smallest opt =1 3 min potential L 1 ≥ L 2 L m ≥ · · · · · · ≥

Upper Bound (IRSE ≤ 4/3) α > 1 / 3 smallest opt =1 3 min potential L 1 ≥ L 2 L m ≥ · · · · · · ≥

Upper Bound (IRSE ≤ 4/3) α > 1 / 3 smallest opt =1 3 β > 2 / 3 min potential L 1 ≥ L 2 L m ≥ · · · · · · ≥

Upper Bound (IRSE ≤ 4/3) α > 1 / 3 smallest opt =1 3 β > 2 / 3 min potential L 1 ≥ L 2 L m ≥ · · · · · · ≥ α > 1 / 3 β > 2 / 3 OR α > 1 / 3 IN EVERY MACHINE

Upper Bound (IRSE ≤ 4/3) α > 1 / 3 smallest opt =1 3 β > 2 / 3 min potential L 1 ≥ L 2 L m ≥ · · · · · · ≥ smallest α > 1 / 3 · · · β > 2 / 3 · · · α > 1 / 3

Upper Bound (IRSE ≤ 4/3) α > 1 / 3 smallest opt =1 3 β > 2 / 3 min potential L 1 ≥ L 2 L m ≥ · · · · · · ≥ α > 1 / 3 β > 2 / 3 OR α > 1 / 3 IN EVERY MACHINE

Upper Bound (IRSE ≤ 4/3) α > 1 / 3 smallest opt =1 3 β > 2 / 3 min potential L 1 ≥ L 2 L m ≥ · · · · · · ≥ x > x ′ y > y ′

Upper Bound (IRSE ≤ 4/3) α > 1 / 3 smallest opt =1 3 β > 2 / 3 min potential L 1 ≥ L 2 L m ≥ · · · · · · ≥ x > x ′ 3 y > y ′

Conclusions PoA ≈ 2 Nash Stable PoS = 1

Conclusions PoA ≈ 2 Nash 4 / 3 IRSE Stable 7 / 6 PoS = 1

Conclusions PoA ≈ 2 Nash ? 4 / 3 IRSE Stable 7 / 6 PoS = 1 Minimize L 2 -norm ⇒ also good for L ∞ -norm (makespan)?

Conclusions PoA ≈ 2 Nash ? 4 / 3 IRSE Stable 7 / 6 PoS = 1 Global properties? 3

Thank You!!