1 On the price of anarchy and stability of correlated equilibria of linear congestion games By George Christodoulou Elias Koutsoupias Presented by Efrat Naim Part of slides taken from George Christodoulou and Elias Koutsoupias web site

2 Agenda  Congestion Games  An example  Definitions  Bounds for correlated Price of stability of congestions games  Bounds for correlated price of anarchy of congestion games  Related Work  Results

3 Congestion Games  Introduced in [Rosenthal, 1973]  Each player has a source and destination. Pure strategies are the path from source to destination  The cost on each edge depends on the number of the players using it. dc ba

4 Congestion Games  N players  M facilities (edges)  Pure strategy (path) is a subset of facilities. Each player can select among a collection of pure strategies (pure strategy set)  Cost of facility depends on the number of players using it  The objective of each player is to minimize its own total cost  Pure strategy profile s = (s 1,……..s N )

5 An Example e d c a b

6 From a to c e d c a b

7 An Example From a to c e d c a b

8 An Example From a to c e d c a b

9 An Example From e to c e d c a b

10 An Example From e to c e d c a b

11 An Example Nash Equilibrium Player 1 has cost 1+1=2 Player 2 has cost 1+1 =2 e d c a b

12 An Example Another Nash Equilibrium Player 1 has cost 2+1+1=4 Player 2 has cost 2+1+1=4 e d c a b

13 Mixed Strategy A mixed strategy for a player is a probability distribution over its pure strategy set. Mixed strategy profile p = (p 1,…….p N ) e d c a b 1/2

14 Correlated Strategy A correlated strategy q for a set of players is any probability distribution over the set S = X i€N Si e d c a b e d c a b 1/2 0 0 LRLR L R 1 2

15 Correlated Equilibrium  Introduced in [Auman,1974]  Consider a mediator that makes a random experiment with a probability distribution q over the strategy space S.  q is common knowledge to the players  The mediator, with respect to the outcome s€S, announces privately the strategy s i to the player i.  Player i is free to obey or disobey to the mediator’s recommendation, with respect to his own profit.  Player i doesn’t know the outcome of the experiment If the best for every player is to follow mediator’s recommendation, then q is a correlated equilibrium.

16 Correlated Equilibrium Cost for player i for pure strategy A is n e(A) = number of players using e in A Given a correlated strategy q, the expected cost of a player i€N is A correlated strategy q is a correlated equilibrium if it satisfies the following

17 Price of Anarchy Price of anarchy A social cost (objective) of a pure strategy profile A is the sum of players costs in A: and

18 Price of Anarchy e d c a b Player 1 has cost 2+1+1=4 Player 2 has cost 2+1+1=4 PoA = (4+4)/ (2+2) = 2

19 Price of Stability Price of stability

20 Price of Stability e d c a b Player 1 has cost 1+1=2 Player 2 has cost 1+1=2 PoS = (2+2)/ (2+2) = 1

21 PoS and PoA of congestion games  By PoS(PoA) for a class of games, we mean the worst case Pos(PoA) over this class.  UpperBound: must hold for every congestion game  LowerBound: Find such a congestion game

22 Correlated PoS – Upper Bound We consider linear latencies f e (x) = a e x+b e Lemma 1: For every pair of non negative integers  it holds

23 Correlated PoS – Upper Bound Theorem 1: Let A be a pure Nash equilibrium and P be any pure strategy profile such that P(A)<= P(P), then SUM(A)<=8/5AUM(P) Where P is the potential of strategy profile This show that the correlated price of stability is at most 1.6

24 Correlated PoS – Upper Bound Proof: Let X be a pure strategy profile X=(X 1,………X N ) From the potential inequality

25 Correlated PoS – Upper Bound A is Nash equilibrium so Summing for all the players we get Adding the two inequalities and use Lemma 1

26 Price of Stability - Lower Bound Dominant Strategies: Each player prefers a particular strategy (dominant), no matter what the other players will choose. Dominant Strategies Nash Equilibrium Correlated Equilibrium

27 Price of Stability - Lower Bound Theorem 2: There are linear congestion games whose dominant equilibrium have price of stability of the SUM social cost approaching as the number of players N tends to infinity. So this holds for correlated equilibrium.

28 Price of Stability - Lower Bound A strategies type  P strategies (equilibrium) (optimal social cost) N-1 N N-1 N

29 Price of Stability - Lower Bound A strategies type  P strategies (equilibrium) (optimal social cost) N-1 N N-1 N

30 Price of Stability - Lower Bound A strategies type  P strategies (equilibrium) (optimal social cost) N-1 N N-1 N

31 Price of Stability - Lower Bound We will fix  and m such that in every allocation (A1,……A K,P K+1 ……P N ), players prefer their Ai strategies. In order to be dominant (A 1, ……. A N )

32 Price of Stability - Lower Bound And it is satisfied by For, the price of anarchy tends to as N tends to infinity.

33 Correlated PoA- Upper Bound Theorem 4: The correlated price of anarchy of the average social cost is 5/2. Lemma 2: For every non negative integers  :

34 Correlated PoA- Upper Bound Proof: Let q be a correlated equilibrium and P be an optimal allocation. Summing for all players The optimal cost is

35 Correlated price of anarchy

36 Correlated price of anarchy – cont. Sum over all players i We finally obtain

37 Asymmetric weighted games Theorem 6: For Linear weighted congestions games, the correlated price of anarchy of the total latency is at most Lemma 3: For every non negative real  : Build so satisfy Achieved by

38 Asymmetric weighted games Proof: Q – correlated equilibrium, P - optimal allocation  e(s) - total load on the facility e for allocation s multiply with  i Nash inequality

39 Asymmetric weighted games And for all players: PoA = C(q)/ C(P) = (3+√ 5)/2 ≈ Using Lemma 3

40 Related Work  Max social cost, parallel links [Mavronicolas, Spirakis,2001], [Czumaj, Vocking, 2002]  Max social cost, Single-Commodity Network [Fotakis, Kontogiannis,Spirakis,2005]  Sum social cost, parallel links [Lucking, Mavronicals, Monien, Rode, 2004]  Splittable, General Network [Roughgarden, Tardos, 2002]  Max, Sum social cost, General Network [Awebuch, Azar, Epstein, 2005] [Christodoulou, Koutsoupias, 2005]

41 Results For linear congestion games Correlaed PoS = [1.57,1.6] Correlatted PoA = 2.5 For weighted congestion games Correlated PoA = (3+√ 5)/2 ≈ 2.618

42 The End