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 to ensure that the network provides them with a high quality of service

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)

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

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

A (bad) DSE s1 s2 t1 t2 C S C, S: peering points avoid C through C avoid C 2, 25, 1 through C 1, 54, 4 Cost Matrix ISP1 ISP2  PoA is (4+4)/(2+2)=2 Dominant Strategy Equilibrium

Our case study: Global Connection Games

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 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

The model Given a strategy vector S, the constructed network will be N(S)=  i P i 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)=  cost i (S) =   c e /k e =  c e ePiePi ii e  N(S)

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

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:

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

An example s1s1 s2s2 t1t1 t2t2

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

An example 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

An example 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:

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)?

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

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

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.

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

…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

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’).

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

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 

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: cost(S) = cost i (S) ≤ w(  i ) ≤ w(P i * ) ≤ k·cost i (OPT) = k· cost(OPT). i=1 k  k  k  k 

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

1+  /2 1/(k-1) 1/k 1/ 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

1+  /2 1/(k-1) 1/k 1/ 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

1+  /2 1/(k-1) 1/k 1/ 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

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

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*) cost(S)/A   (S)  B cost(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 cost(S*) cost(S’)/A   PoS ≤ cost(S’)/cost(S*) ≤ A·B.

Lemma 5 (Bounding  ) cost(S)   (S)  H k cost(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)  cost(S) =  e  N(S) c e and  (S) ≤ H k · cost(S) =  e  N(S) c e · H k.

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