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

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

Oct 17, 2023 957 likes •1.27k views

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

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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

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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

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

Congestion games

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

Congestion games

A congestion game (E, N, (de)e∈E, (ci)i∈N) is defined on a finite set E of resources and has n players using a delay function de mapping N to the integers, for each resource e. The actions for each player are subsets of E. The cost functions are the following: ci(a1, . . . , an) =

  • e∈ai

de(fe(a1, . . . , an)) being fe(a1, . . . , an) = |{i | e ∈ ai}|.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

Weighted congestion games

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

Weighted congestion games

A weighted congestion game (E, N, (de)e∈E, (ci)i∈N, (wi)i∈N) is defined on a finite set E of resources and has n players. Player i has an associated natural weight wi. Using a delay function de mapping N to the integers, for each resource e. The actions for each player are subsets of E. The cost functions are the following: ci(a1, . . . , an) =

  • e∈ai

de(fe(a1, . . . , an)) being fe(a1, . . . , an) =

{i|e∈ai} wi.

AGT-MIRI, FIB-UPC Congestion Games

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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

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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 de(x) = aex + be, for ae, be > 0.

AGT-MIRI, FIB-UPC Congestion Games

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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

  • i∈N

ci(s−i, s′

i) λC(s′) + µC(s).

AGT-MIRI, FIB-UPC Congestion Games

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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

  • i∈N

ci(s−i, s′

i) λC(s′) + µC(s).

Smoothness directly gives a bound for the PoA:

AGT-MIRI, FIB-UPC Congestion Games

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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

  • i∈N

ci(s−i, s′

i) λC(s′) + µC(s).

Smoothness directly gives a bound for the PoA: Theorem In a (λ, µ)-smooth game, the PoA for PNE is at most

λ 1−µ.

AGT-MIRI, FIB-UPC Congestion Games

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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(s) =

  • i∈N

ci(s)

  • i∈N

ci(s−i, s∗

i )

λC(s∗) + µC(s) Substracting µC(s) on both sides gives (1 − µ)C(s) λC(s∗).

AGT-MIRI, FIB-UPC Congestion Games

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Theorem Every congestion game with affine delay functions is (5/3, 1/3)-smooth. Thus, PoA 5/2.

AGT-MIRI, FIB-UPC Congestion Games

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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 3y2 + 1 3z2.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

Proof of smoothness for affine functions

Recall that de(x) = aex + be . Note that using the Lemma aey(z+1)+bey ae(5 3y2+1 3z2)+bey = 5 3(aey2+bey)+1 3(aez2+bez).

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

Proof of smoothness for affine functions

Recall that de(x) = aex + be . Note that using the Lemma aey(z+1)+bey ae(5 3y2+1 3z2)+bey = 5 3(aey2+bey)+1 3(aez2+bez). Taking y = fe(s∗) and z = fe(s) we get (ae(fe(s)+1)+be)fe(s∗) 5 3(aefe(s∗)+be)fe(s∗))+1 3(aefe(s)+be)fe(s)).

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

Proof of smoothness for affine functions

Recall that de(x) = aex + be . Note that using the Lemma aey(z+1)+bey ae(5 3y2+1 3z2)+bey = 5 3(aey2+bey)+1 3(aez2+bez). Taking y = fe(s∗) and z = fe(s) we get (ae(fe(s)+1)+be)fe(s∗) 5 3(aefe(s∗)+be)fe(s∗))+1 3(aefe(s)+be)fe(s)). Summing up all the inequalities

  • e∈E

(ae(fe(s) + 1) + be)fe(s∗) 5 3C(s∗) + 1 3C(s).

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

Proof of smoothness for affine functions

  • e∈E

(ae(fe(s) + 1) + be)fe(s∗) 5 3C(s∗) + 1 3C(s).

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

Proof of smoothness for affine functions

  • e∈E

(ae(fe(s) + 1) + be)fe(s∗) 5 3C(s∗) + 1 3C(s). But,

  • i∈N

ci(s−i, s∗

i )

  • e∈E

(ae(fe(s) + 1) + be)fe(s∗) as there are at most fe(s∗) players that might move to resource r. Each of them by unilaterally deviating incur a delay of (ae(fe(s) + 1) + be.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

Proof of smoothness for affine functions

  • e∈E

(ae(fe(s) + 1) + be)fe(s∗) 5 3C(s∗) + 1 3C(s). But,

  • i∈N

ci(s−i, s∗

i )

  • e∈E

(ae(fe(s) + 1) + be)fe(s∗) as there are at most fe(s∗) players that might move to resource r. Each of them by unilaterally deviating incur a delay of (ae(fe(s) + 1) + be. This gives the (5/3, 1/3)-smoothness.

AGT-MIRI, FIB-UPC Congestion Games

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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

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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

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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

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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

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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

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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 U(s) =

  • i∈N

wi

  • e∈si

(aewi + be) C(s) =

  • i∈N

wici(s) Φ(s) = (C(s) + U(s))/2.

AGT-MIRI, FIB-UPC Congestion Games

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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 U(s) =

  • i∈N

wi

  • e∈si

(aewi + be) C(s) =

  • i∈N

wici(s) Φ(s) = (C(s) + U(s))/2. It can be shown that Φ(s′) − Φ(s) = wi(ci(s′) − ci(s)).

AGT-MIRI, FIB-UPC Congestion Games

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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

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Contents Congestion games and variants Affine Congestion games Weighted Congestion Games References

References

Chapters 18 and 19.3 in the AGT book. (PoA and PoS bounds).

  • B. Awerbuch, Y. Azar, A. Epstein. The Price of Routing

Unsplittable Flow. STOC 2005. (PoA for pure NE in congestion games).

  • G. Christodoulou, E. Koutsoupias. The Price of Anarchy of

finite Congestion Games. STOC 2005. (PoA for pure NE in congestion games)

  • T. Roughgarden. Intrinsic Robustness of the Price of Anarchy.

STOC 2009. (Smoothness Framework and Unification of Previous Results)

  • D. Fotakis. A Selective Tour Through Congestion Games,

LNCS 2015.

AGT-MIRI, FIB-UPC Congestion Games