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The price of anarchy of finite congestion games
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1. The price of anarchy of finite congestion games Kapelushnik Lior Based on the articles: “The price of anarchy of finite congestion games” by Christodoulou & Koutsoupias “the price of routing unsplittable flow” by Awerbuch, Azar & Epstein

2. Agenda • Congestion games description • Price of anarchy definitions • Linear latency functions PoA upper and lower bounds • Polynomial latency functions PoA upper and lower bounds

3. Network congestion game • A directed graph G=(V,E) • For each edge exists a latency function • n users, user j have request • Request j assigned to path which is the strategy of player j • Users are none cooperative

4. Network game example s1 t1 t3 t2 s2 s3

5. Network game exampleagent paths (strategies) Agent 2 path 2 s1 t1 t3 t2 s2 Agent 2 path 1 s3

6. Network congestion game • Consider a strategy profile • Denote Ne(A) as the load on e in A • Mark as the possible paths for user i • Player j cost is the total cost of it’s strategy path • Strategy A is a NE if no player has reason to deviate from his strategy

7. Congestion game • More generalized than a network congestion game • N players, a set of facilities E • Each player i has a strategy chosen from several sets of facilities • Facility j have cost (latency) function

8. Congestion game definitions • Symmetric game (single-commodity) – all players choose from the same strategies • Asymmetric game (multi-commodity) – different players may have different strategy options • Mixed strategy for player i – a probability distribution over

9. Congestion game example f1 f2 f3 f4 F5 F6

10. Congestion game example f1 f2 f3 f4 F5 F6

11. Congestion game example f1 f2 f3 f4 F5 F6

12. Congestion game example f1 f2 f3 f4 F5 F6

13. Congestion games • Every network congestion game is a congestion game • Each edge will be represented by a facility • Each strategy path of a player will be replaced by the set of edges in the path

14. Network to congestion game e1 e2 fe1 fe2 fe3 s t e3 Instead of possible path strategies (e1->e2) and (e3) The congestion game strategies are ({fe1,fe2}, {fe3}}

15. Social cost • Two possible definitions • Definition 1: • Definition 2: or for weighted requests • When considering PoA the social cost definition of sum is equivalent to average (just divide by n)

16. Price of anarchy • The worst-case ratio between the social cost of a NE and the optimal social cost • Definition 1: • Definition 2:

17. Linear latency function • If an equivalent problem can be described with function • Duplicate an edge times

18. Avg. social cost PoA • Asymmetric case • Unweighted requests • pure strategy • Mixed strategy • Weighted requests • Pure + mixed strategies • Symmetric case • Unweighed pure strategy

19. Upper PoA bounds • Sketch of proof • Compare agent’s delay to the delay that would be encountered at the optimal path • Combine the bounds and transform to a relation between a total NE delay and the total optimal delay

20. Upper bound unweighted requests, fe(x)=x • Lemma: for a pair of nonnegative integers a,b

21. Upper bound unweighted requests, fe(x)=x • In a NE A and an optimal P allocation • The inequality holds since moving from a NE does not decrease the cost • Summing for all players we get

22. Upper bound unweighted requests, fe(x)=x cont’ • Using the lemma we get • And thus • And the upper bound is proven

23. Upper bound weighted requests • Notations: • J(e) – set of agents using e • P – NE strategies profile • Qj – request j path in P • X* - value X in optimal state • l – load vector of a system

24. Upper bound weighted requests Lemma 1: (follows from Cauchy-Schwartz inequality) Lemma 2: for any

25. Upper bound weighted requests since deviation from NE doesn’t decrease cost Multiplying by Wj we get

26. Upper bound weighted requests Summing for all agents we get Changing summation order we get Using lemma 2 we get

27. Upper bound weighted requests Using lemma 1 in previous expression

28. Lower bounds, unweighted requests, congestion game • Assume N≥3 agents, 2N facilities fe(x)=x • Facilities • Agent i strategies • Optimal allocation: each agent i chooses • Worst NE agent i choose • The cost for each agent is 2 in the optimal allocation and 5 in the NE, PoA is 5/2

29. Lower bounds, unweighted requests, network game S1 t1 S2 t2 S3 t3

30. Lower bounds, weighted requests, network game V fe(x)=0 U fe(x)=x W

31. Lower bounds, weighted requests, network game V fe(x)=0 Agent 1 U Agent 4 fe(x)=x Agent 3 Agent 2 W Optimal cost, player 1 use UV, player 2 use UW, Player 3 use VW, player 4 use WV Total cost:

32. Lower bounds, weighted requests, network game Agent 2 V fe(x)=0 Agent 3 U fe(x)=x Agent 1 Agent 4 W NE cost, player 1 use UWV, player 2 use UVW, Player 3 use VUW, player 4 use WUV Total cost:

33. Linear congestion symmetric games lower bound of PoA • The upper bound for asymmetric games with avg. social cost also holds for symmetric games • The lower bound both max and avg. social cost is (5N-2)/(2N+1) • Next is a game description which achieves this PoA for N players

34. Lower bound game construction for symmetric games • The facilities will be in N sets of the same size P1,P2,…,Pn • Each Pi is a pure strategy and in optimal allocation each player i plays Pi • Each Pi contains facilities • At NE player i plays alone facilites of each Pj • At NE each pair of players play together facilities of each Pj

35. Lower bound game construction for symmetric games cont’ • At NE A, • We want that at NE no players will switch to Pj • For NE we need • Which proves the PoA of (5N-2)/(2N+1)

36. Max social cost PoA • Unweighted pure strategy cases only • Symmetric case • Lower bound already shown • Asymmetric case

37. Asymmetric case upper bound • Let A be a NE, P optimal allocation, w.l.o.g Max(A)=c1(A), the NE imply • Denote the players in A that use facilities of P1 • The avg. social cost lower bound showed

38. Asymmetric case upper bound • Combining the last 2 inequalities • substitute in the first inequality

39. Asymmetric case lower bound k k k V1 V2 V3 Vk Vk+1 k • One player wants to go from V1 to Vk+1 • For each i k-1 players wants to go from Vi to Vi+1 • Between Vi and Vi+1 one path with length 1, k-1 paths with length k • Opt – all players use different paths, cost is k • NE – all players use the 1 length paths cost is PoA is

40. Symmetric case upper bound • Let A be a NE, P optimal allocation, w.l.o.g Max(A)=c1(A), the NE imply • Summing for all possible j in P and using the lemma • Using the avg. social cost lower bound

41. Polynomial latency function • The latency functions are polynomials of bounded degree p • The proofs for PoA of linear latency functions are quite similar to those of polynomial latencies

42. Polynomial latencies cost PoA • For polynomials of degree p, nonnegative coefficients • Avg. social cost weighted requests, unweighted requests, symmetric games, asymmetric games, pure strategies, mixed strategies • Max social cost • Pure strateties symmetric games • Pure strateties asymmetric games

43. Upper bound unweighted requests polynomial latencies • Instead of the lemma for linear functions for a pair of nonnegative integers a,b • A new lemma is used, if f(x) polynomial in x with nonnegative coefficients, of degree p, for nonegative x and y • Where

44. Upper bound unweighted requests polynomial latencies cont’ • In a NE A and an optimal P allocation • The inequality holds since moving from a NE does not decrease the cost • Summing for all players we get

45. Upper bound unweighted requests polynomial latencies cont’ • Using the lemma we get • And thus • And the upper bound is proven

46. Lower bound game construction for symmetric games • The facilities will be in N sets of the same size P1,P2,…,Pn • Each Pi is a pure strategy and in optimal allocation each player i plays Pi • Each Pj contains N facilities • At NE player i plays

47. Lower bound game construction for symmetric games cont’ • At NE A, • We want that at NE no players will switch to Pj • For NE we need to select N such that • For opt • The PoA is • Which proves the PoA of when choosing N that satisfies the equation

48. Asymmetric case lower boundalmost like in the linear case k k k V1 V2 V3 V(k^p) V(k^p+1) • One player wants to go from V1 to • For each i k-1 players wants to go from Vi to Vi+1 • Between Vi and Vi+1 a path with length 1, k-1 paths with length • Opt – all players use different paths, cost is • NE – all players use the 1 length paths cost is