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

Apr 18, 2023 372 likes •931 views

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

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Akaki Mamageishvili Paolo Penna

Tighter Bounds on the Inefficiency Ratio of Stable Equilibria in Load Balancing Games

ETH Zurich

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Outline

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

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Load Balancing (Games)

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Load Balancing (Games)

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Load Balancing (Games)

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Load Balancing (Games)

no improvement

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Load Balancing (Games)

OPT Nash Equilibrium

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Quality of Equilibria

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

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Quality of Equilibria

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

Opt

Nash

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Quality of Equilibria

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

Opt

PoS = best Nash

Opt

Nash

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Load Balancing (Games)

1 1 x x x x 1 1

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Load Balancing (Games)

1 1 x x x x 1 1 PoA =

2x 1+x → 4 3

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Load Balancing (Games)

1 1 x x x x 1 1 PoA =

2x 1+x → 4 3

PoA = 2

  • 1 −

1 m+1

  • In general:

PoS = 1

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Inefficiency Ratio of Stable Equilibria

PoA = worst Nash

Opt

PoS = best Nash

Opt

Nash

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Inefficiency Ratio of Stable Equilibria

PoA = worst Nash

Opt

PoS = best Nash

Opt

Nash Stable

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Inefficiency Ratio of Stable Equilibria

PoA = worst Nash

Opt

PoS = best Nash

Opt

Nash Stable noisy best response min potential

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Bounded Rationality

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Bounded Rationality

Noisy best response €10 €1 prob ∝ e10/noise prob ∝ e1/noise Strategies with higher payoff chosen with higher probability

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Inefficiency Ratio of Stable Equilibria

PoA = worst Nash

Opt

PoS = best Nash

Opt

Nash Stable noisy best response min potential

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Inefficiency Ratio of Stable Equilibria

PoA = worst Nash

Opt

PoS = best Nash

Opt

Nash Stable noisy best response min potential IRSE= worst Stable Nash

Opt (Asadpour, Saberi, 2009)

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Inefficiency Ratio of Stable Equilibria

PoA = worst Nash

Opt

PoS = best Nash

Opt

Nash Stable noisy best response min potential IRSE= worst Stable Nash

Opt

L2-norm L∞-norm

(Asadpour, Saberi, 2009)

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Inefficiency Ratio of Stable Equilibria

PoA = worst Nash

Opt

PoS = best Nash

Opt

Nash Minimize L2-norm ⇒ also good for L∞-norm (makespan)? Stable noisy best response min potential IRSE= worst Stable Nash

Opt (Asadpour, Saberi, 2009)

(Alon, Azar, Woeginger, Yadid, 1997)

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Our Contribution

7/6 ≤ IRSE ≤ 4/3

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Our Contribution

7/6 ≤ IRSE ≤ 4/3

mininimize L2-norm automatically 4/3-APX for L∞-norm

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Our Contribution

7/6 ≤ IRSE ≤ 4/3

mininimize L2-norm sometimes at least 7/6-APX of L∞-norm

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Our Contribution

7/6 ≤ IRSE ≤ 4/3 19/18 ≤ IRSE ≤ 3/2

Previous bunds:

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Our Contribution

7/6 ≤ IRSE ≤ 4/3 19/18 ≤ IRSE ≤ 3/2

Alon - Azar - Woeginger - Yadid, SODA 1997 Asadpour - Saberi, WINE 2009 Previous bunds:

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Our Contribution

7/6 ≤ IRSE ≤ 4/3

mininimize L2-norm sometimes at least 7/6-APX of L∞-norm

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Lower Bound (IRSE ≥ 7/6)

2

9 2

2 3 3 2

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Lower Bound (IRSE ≥ 7/6)

2

9 2

2 3 3 2 min potential

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Lower Bound (IRSE ≥ 7/6)

2

9 2

2 3 3 2 min potential

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Lower Bound (IRSE ≥ 7/6)

2

9 2

2 3 3 2 2

9 2

2 3 3 2 min potential

  • pt
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Lower Bound (IRSE ≥ 7/6)

min potential 3

14 2

3 5 4 4 3

14 2

5 3 3 3 4 5 5 4

  • pt
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Lower Bound (IRSE ≥ 7/6)

min potential 3

14 2

3 5 4 4 3

14 2

5 3 3 3 4 5 5 4

  • pt

10 9

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Lower Bound (IRSE ≥ 7/6)

min potential 3

14 2

3 5 4 4 3

14 2

5 3 3 3 4 5 5 4

  • pt

m − 1

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Lower Bound (IRSE ≥ 7/6)

min potential 3

14 2

3 5 4 4 3

14 2

5 3 3 3 4 5 5 4

  • pt

3m − 3 m − 1

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Lower Bound (IRSE ≥ 7/6)

min potential 3

14 2

3 5 4 4 3

14 2

5 3 3 3 4 5 5 4

  • pt

3m − 3 2m − 3 m − 1

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Lower Bound (IRSE ≥ 7/6)

min potential 3

14 2

3 5 4 4 3

14 2

5 3 3 3 4 5 5 4

  • pt

3m − 3 2m − 3

5m−3 2

m − 1

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Lower Bound (IRSE ≥ 7/6)

min potential 3

14 2

3 5 4 4 3

14 2

5 3 3 3 4 5 5 4

  • pt

3m − 3 2m − 3

5m−3 2 7m−4 2

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Our Contribution

7/6 ≤ IRSE ≤ 4/3

mininimize L2-norm automatically 4/3-APX for L∞-norm

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Upper Bound (IRSE ≤ 4/3)

3

  • pt =1

L1 ≥ L2 Lm ≥ ≥ · · · · · · α > 1/3 min potential

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Upper Bound (IRSE ≤ 4/3)

3

  • pt =1

L1 ≥ L2 Lm ≥ ≥ · · · · · · α > 1/3 min potential smallest

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Upper Bound (IRSE ≤ 4/3)

3

  • pt =1

L1 ≥ L2 Lm ≥ ≥ · · · · · · α > 1/3 min potential smallest

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Upper Bound (IRSE ≤ 4/3)

3

  • pt =1

L1 ≥ L2 Lm ≥ ≥ · · · · · · α > 1/3 min potential smallest β > 2/3

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Upper Bound (IRSE ≤ 4/3)

3

  • pt =1

L1 ≥ L2 Lm ≥ ≥ · · · · · · α > 1/3 min potential smallest β > 2/3 α > 1/3 α > 1/3 β > 2/3 OR IN EVERY MACHINE

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Upper Bound (IRSE ≤ 4/3)

3

  • pt =1

L1 ≥ L2 Lm ≥ ≥ · · · · · · α > 1/3 min potential smallest β > 2/3 α > 1/3 α > 1/3 β > 2/3 · · · smallest · · ·

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Upper Bound (IRSE ≤ 4/3)

3

  • pt =1

L1 ≥ L2 Lm ≥ ≥ · · · · · · α > 1/3 min potential smallest β > 2/3 α > 1/3 α > 1/3 β > 2/3 OR IN EVERY MACHINE

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Upper Bound (IRSE ≤ 4/3)

3

  • pt =1

L1 ≥ L2 Lm ≥ ≥ · · · · · · α > 1/3 min potential smallest β > 2/3 x > x′ y > y ′

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Upper Bound (IRSE ≤ 4/3)

3

  • pt =1

L1 ≥ L2 Lm ≥ ≥ · · · · · · α > 1/3 min potential smallest β > 2/3 x > x′ y > y ′ 3

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Conclusions

PoA ≈ 2 PoS = 1 Nash Stable

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Conclusions

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

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Conclusions

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

?

Minimize L2-norm ⇒ also good for L∞-norm (makespan)?

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Conclusions

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

?

Global properties? 3

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Thank You!!