1 / 34

Beyond selfish routing: Network Games

Beyond selfish routing: Network Games. Network Games. NGs model the various ways in which selfish agents strategically interact in using a network They aim to capture two competing issues for agents: to minimize the cost they incur in creating/using the network

gage-hall
Télécharger la présentation

Beyond selfish routing: Network Games

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Beyond selfish routing:Network Games

  2. Network Games • NGs model the various ways in which selfish agents strategically interact in using a network • They aim to capture two competing issues for agents: • to minimize the cost they incur in creating/using the network • to ensure that the network provides them with a high quality of service

  3. Motivations • NGs can be used to model: • social network formation (edge represent social relations) • how subnetworks connect in computer networks • formation of P2P networks connecting users to each other for downloading files (local connection games) • how users try to share costs in using an existing network (global connection games)

  4. Setting • What is a stable network? • we use a NE as the solution concept • we refer to networks corresponding to Nash Equilibria as being stable • How to evaluate the overall quality of a network? • we consider the social cost: the sum of players’ costs • Our goal: to bound the efficiency loss resulting from selfishness

  5. A warming-up network game: The ISPs dilemma C, S: peering points s1 two Internet Service Providers (ISP): ISP1 e ISP2 t2 C S ISP1 wants to send traffic from s1 to t1 t1 ISP2 wants to send traffic from s2 to t2 s2 Long links incur a per-user cost of 1 to the ISP owning the link Each ISP can use two paths: either passing through C or not

  6. A (bad) DSE C, S: peering points s1 t2 Cost Matrix C S t1 ISP2 s2 ISP1  PoA is (4+4)/(2+2)=2 Dominant Strategy Equilibrium

  7. Our case study:Global Connection Games

  8. The model • G=(V,E): directed graph, k players • ce: non-negative cost of e E • Player i has a source node si and a sink node ti • Strategy for player i: a path Pi from si to ti • The cost of Pi for player i in S=(P1,…,Pk) is shared with all the other players using (part of) it: costi(S) =  ce/ke ePi this cost-sharing scheme is called fair or Shapley cost-sharing mechanism

  9. The model • Given a strategy vector S, the constructed network will be N(S)= i Pi • The cost of the constructed network will be shared among all players as follows: • Notice that each user has a favorable effect on the performance of other users (so-called cross monotonicity), as opposed to the congestion model of selfish routing cost(S)= costi(S) =  ce/ke=  ce ePi i i eN(S)

  10. Goals • Remind that given a strategy vector S, N(S) is stable if S is a NE • To evaluate the overall quality of a network, we consider its social cost, i.e., the sum of all players’ costs • A network is optimal or socially efficient if it minimizes the social cost

  11. Be optimist!: The price of stability (PoS) • Definition (Schulz & Moses, 2003): Given a game G and a social-choice minimization (resp., maximization) function f (i.e., the sum of all players’ payoffs), let S be the set of NE, and let OPT be the outcome of G optimizing f. Then, the Price of Stability (PoS) of G w.r.t. f is:

  12. Some remarks • PoA and PoS are (for positive s.c.f. f) •  1 for minimization problems •  1 for maximization problems • PoA and PoS are small when they are close to 1 • PoS is at least as close to 1 than PoA • In a game with a unique NE, PoA=PoS • Why to study the PoS? • sometimes a nontrivial bound is possible only for PoS • PoS quantifies the necessary degradation in quality under the game-theoretic constraint of stability

  13. An example 3 s2 1 3 1 1 3 1 2 s1 t2 t1 4 5.5

  14. An example 3 s2 1 3 1 1 3 1 2 s1 t2 t1 4 5.5 optimal network has cost 12 cost1=7 cost2=5 is it stable?

  15. An example 3 s2 1 3 1 1 3 1 2 s1 t2 t1 4 5.5 …no!, player 1 can decrease its cost cost1=5 cost2=8 …yes, and has cost 13! is it stable?  PoA  13/12, PoS ≤ 13/12

  16. An example 3 s2 1 3 1 1 3 1 2 s1 t2 t1 4 5.5 …a best possible NE: cost1=5 cost2=7.5 the social cost is 12.5  PoS = 12.5/12

  17. Addressed issues in GCG • Does a stable network always exist? • Does the repeated version of the game always converge to a stable network? • How long does it take to converge to a stable network? • Can we bound the price of anarchy (PoA)? • Can we bound the price of stability (PoS)?

  18. Theorem 1 Any instance of the global connection game has a pure Nash equilibrium, and best response dynamic always converges. Theorem 2 The price of anarchy in the global connection game with k players is at most k. Theorem 3 The price of stability in the global connection game with k players is at most Hk, the k-th harmonic number.

  19. The potential function method For any finite game, an exact potential function is a function that maps every strategy vector S to some real value and satisfies the following condition: • S=(s1,…,sk), s’isi, let S’=(s1,…,s’i,…,sk), then (S)-(S’) = costi(S)-costi(S’). A game that does possess an exact potential function is called potential game

  20. Lemma 1 Every potential game has at least one pure Nash equilibrium, namely the strategy vector S that minimizes(S). Proof: consider any move by a player i that results in a new strategy vector S’. Since (S) is minimum, we have: (S)-(S’) = costi(S)-costi(S’)  0 player i cannot decrease its cost, thus S is a NE. costi(S)  costi(S’)

  21. Convergence in potential games Observation: any state S with the property that (S) cannot be decreased by altering any one strategy in S is a NE by the same argument. This implies the following: Lemma 2 In any finite potential game, best response dynamic always converges to a Nash equilibrium Proof:best response dynamic simulates local search on .

  22. …turning our attention to the global connection game… Let  be the following function mapping any strategy vector S to a real value [Rosenthal 1973]: (S) = eN(S) e(S) where (recall that ke is the number of players using e) e(S) = ce · H = ce · (1+1/2+…+1/ke). ke

  23. Lemma 3 ( is a potential function) Let S=(P1,…,Pk), let P’i be an alternative path for some player i, and define a new strategy vector S’=(S-i,P’i). Then: (S) - (S’) = costi(S) – costi(S’). Proof: • It suffices to notice that: • If edge e is used one more time in S: (S+e)=(S)+ce/(ke+1) • If edge e is used one less time in S: (S-e)=(S) - ce/ke • (S) -(S’) = (S) -(S-Pi+P’i) = (S)– ((S) - ePi ce/ke + eP’ice/(ke+1))= costi(S) – costi(S’).

  24. Existence of a NE Theorem 1 Any instance of the global connection game has a pure Nash equilibrium, and best response dynamic always converges. Proof:From Lemma 3, a GCG is a potential game, and from Lemma 1 and 2 best response dynamic converges to a pure NE.

  25. Price of Anarchy: a lower bound k s1,…,sk t1,…,tk 1 optimal network has cost 1  best NE: all players use the lower edge PoS is 1  worst NE: all players use the upper edge PoA is k

  26. k k k k     i=1 i=1 i=1 i=1 Upper-bounding the PoA Theorem 2 The price of anarchy in the global connection game with k players is at most k. Proof: Let OPT=(P1*,…,Pk*) denote the optimal network, and let i be a shortest path in G between si and ti. Let w(i) be the length of such a path, and let S be any NE. Observe that costi(S)≤w(i) (otherwise the player i would change). Then: cost(S) = costi(S)≤ w(i)≤ w(Pi*)≤ k·costi(OPT) = k·cost(OPT).

  27. PoS for GCG: a lower bound >o: small value t1,…,tk 1/k 1/(k-1) 1/3 1 1/2 sk-1 . . . s1 s2 s3 sk 1+ 0 0 0 0 0

  28. PoS for GCG: a lower bound >o: small value t1,…,tk 1/k 1/(k-1) 1/3 1 1/2 sk-1 . . . s1 s2 s3 sk 1+ 0 0 0 0 0 The optimal solution has a cost of 1+ is it stable?

  29. PoS for GCG: a lower bound >o: small value t1,…,tk 1/k 1/(k-1) 1/3 1 1/2 sk-1 . . . s1 s2 s3 sk 1+ 0 0 0 0 0 …no! player k can decrease its cost… is it stable?

  30. PoS for GCG: a lower bound >o: small value t1,…,tk 1/k 1/(k-1) 1/3 1 1/2 sk-1 . . . s1 s2 s3 sk 1+ 0 0 0 0 0 …no! player k-1 can decrease its cost… is it stable?

  31. PoS for GCG: a lower bound >o: small value t1,…,tk 1/k 1/(k-1) 1/3 1 1/2 sk-1 . . . s1 s2 s3 sk 1+ 0 0 0 0 0 A stable network k social cost:  1/j =Hk  ln k + 1 k-th harmonic number j=1

  32. Lemma 4 Suppose that we have a potential game with potential function , and assume that for any outcome S we have cost(S)/A  (S)  B cost(S) for some A,B>0. Then the price of stability is at most AB. Proof: Let S’ be the strategy vector minimizing  (i.e., S’ is a NE) Let S* be the strategy vector minimizing the social cost we have: cost(S’)/A  (S’)  (S*)  B cost(S*)  PoS ≤ cost(S’)/cost(S*) ≤ A·B.

  33. Lemma 5 (Bounding  ) For any strategy vector S in the GCG, we have: cost(S)  (S)  Hk cost(S) Proof:Indeed: (S) = eN(S) e(S) = eN(S) ce· Hke  (S)  cost(S) = eN(S) ce and (S) ≤ Hk· cost(S) = eN(S) ce· Hk.

  34. Upper-bounding the PoS Theorem 3 The price of stability in the global connection game with k players is at most Hk, the k-th harmonic number Proof:From Lemma 3, a GCG is a potential game, and from Lemma 5 and Lemma 4 (with A=1 and B=Hk), its PoS is at most Hk.

More Related