1. Beyond selfish routing: Network Games

2. Network Games NGs model the various ways in which selfish users (i.e., players) strategically interact in using a network They aim to capture two competing issues for players: 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: formation of social network (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? the social cost is given by the sum of players’ costs Our goal: to bound the efficiency loss resulting from selfishness

5. Our case study: Global Connection Games

6. The model G=(V,E): directed graph, k players c e : non-negative cost of e  E Player i has a source node s i and a sink node t i Strategy for player i: a path P i from s i to t i Let k e denote the number of users using edge e. The cost of P i for player i in S=(P 1,…,P k ) is shared with all the other players using (part of) it: cost i (S) =  c e /k e ePiePi this cost-sharing scheme is called fair or Shapley cost-sharing mechanism

7. The model Given a strategy vector S, the constructed network N(S) is given by the union of all paths P i Then, the cost of the constructed network (social-choice function) is the following : 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 C(S)=  cost i (S) =   c e /k e =  c e ePiePi ii e  N(S)

8. Bounding the loss of efficiency Remind that a network is optimal or socially efficient if it minimizes the social cost We use the PoA to estimate the loss of efficiency we may have in the worst case, as given by the ratio between the cost of a worst stable network and the cost of an optimal network What about the ratio between the costs of a best stable network and of an optimal network?

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

10. Some remarks PoA and PoS are (for positive s.c.f. C)  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

11. An example 1 1 1 1 3 3 3 2 4 5.5 s1s1 s2s2 t1t1 t2t2

12. An example 1 1 1 1 3 3 3 2 45.5 s1s1 s2s2 t1t1 t2t2 optimal network has cost 12 cost 1 =7 cost 2 =5 is it stable?

13. An example 1 1 1 1 3 3 3 2 45.5 s1s1 s2s2 t1t1 t2t2  PoA  13/12, PoS ≤ 13/12 cost 1 =5 cost 2 =8 is it stable? …yes, and has cost 13! …no!, player 1 can decrease its cost

14. An example 1 1 1 1 3 3 3 2 45.5 s1s1 s2s2 t1t1 t2t2 the social cost is 12.5  PoS = 12.5/12 cost 1 =5 cost 2 =7.5 …a best possible NE: Homework: find a worst possible NE

15. 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)?

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

17. 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: The potential function method  S=(s 1,…,s k ), s’ i  s i, let S’=(s 1,…,s’ i,…,s k ), then  (S)-  (S’) = cost i (S)-cost i (S’). A game that does possess an exact potential function is called potential game

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

19. In any finite potential game, best response dynamic always converges to a Nash equilibrium Lemma 2 Proof: best response dynamic simulates local search on . 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: Convergence in potential games

20. …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 k e is the number of players using e)  e (S) = c e · H = c e · (1+1/2+…+1/k e ). keke

21. Let S=(P 1,…,P k ), let P’ i be an alternative path for some player i, and define a new strategy vector S’=(S -i,P’ i ). Then: Lemma 3 (  is a potential function)  (S) -  (S’) = cost i (S) – cost i (S’). Proof: It suffices to notice that: If edge e is used one more time in S:  (S+e)=  (S)+c e /(k e +1) If edge e is used one less time in S:  (S-e)=  (S) - c e /k e  (S) -  (S’) =  (S) -  (S-P i +P’ i ) =  (S) – (  (S) -  e  P i c e /k e +  e  P’ i c e /(k e +1))= cost i (S) – cost i (S’).

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

23. Price of Anarchy: a lower bound s 1,…,s k t 1,…,t k k 1 optimal network has cost 1 best NE: all players use the lower edge worst NE: all players use the upper edge PoS is 1 PoA is k 

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

25. PoS for GCG: a lower bound 1+ ... 1 1/2 1/(k-1) 1/k 1/3 0 000 0 s1s1 sksk s2s2 s3s3 s k-1 t 1,…,t k  >o: small value

26. 1+ ... 1 1/2 1/(k-1) 1/k 1/3 0 000 0 s1s1 sksk s2s2 s3s3 s k-1 t 1,…,t k  >o: small value The optimal solution has a cost of 1+  is it stable? PoS for GCG: a lower bound

27. 1+ ... 1 1/2 1/(k-1) 1/k 1/3 0 000 0 s1s1 sksk s2s2 s3s3 s k-1 t 1,…,t k  >o: small value …no! player k can decrease its cost… is it stable? PoS for GCG: a lower bound

28. 1+ ... 1 1/2 1/(k-1) 1/k 1/3 0 000 0 s1s1 sksk s2s2 s3s3 s k-1 t 1,…,t k  >o: small value …no! player k-1 can decrease its cost… is it stable? PoS for GCG: a lower bound

29. 1+ ... 1 1/2 1/(k-1) 1/k 1/3 0 000 0 s1s1 sksk s2s2 s3s3 s k-1 t 1,…,t k  >o: small value The only stable network social cost: C(S)=  1/j = H k  ln k + 1 k-th harmonic number j=1 k PoS for GCG: a lower bound

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

31. Lemma 5 (Bounding  ) C(S)   (S)  H k C(S). For any strategy vector S in the GCG, we have:  (S) =  e  N(S)  e (S) =  e  N(S) c e · H k e Proof: Indeed:   (S)  C(S) =  e  N(S) c e and  (S) ≤ H k · C(S) =  e  N(S) c e · H k.

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