Beyond Routing Games Network Formation Games Network Games

Beyond Routing Games: Network (Formation) Games

Network Games (NG) n n NG model the various ways in which selfish users (i. e. , players) strategically interact in using a (either communication, computer, social, etc. ) network The Internet routing game is a particular type of network congestion game Other examples of NG: social network games, network design games, network creation games, graphical games, etc. Notice that each of these games is actually a class of games, where any element of the class is an instance of the game

A warming-up NG: The ISPs dilemma C, S: peering points s 1 Two Internet Service Providers (ISP) owning part of a network: ISP 1 e ISP 2 ISP 1 wants to send traffic from s 1 to t 1 t 2 C ISP 2 wants to send traffic from s 2 to t 2 s 2 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 S t 1

A (bad) DSE C, S: peering points s 1 t 2 Cost Matrix C ISP 2 ISP 1 avoid C through C avoid C 2, 2 5, 1 through C 1, 5 4, 4 S t 1 s 2 Po. A is (4+4)/(2+2)=2 Dominant Strategy Equilibrium

Network Formation Games (NFG) n NFG are NG that aim to capture two competing issues for players when using a network for communication purposes: n n n to minimize the afforded cost to be provided with a high quality of service Two big categories of NFG: n n Network Design Games (a. k. a. Global Connection Games): Users autonomously design a communication subnetwork embedded in an already existing network with the selfish goal of sharing costs in using it for a point-to-point communication Network Creation Games (a. k. a. Local Connection Games): Users autonomously form ex-novo a network that connects them for reciprocal communication (e. g. , downloading files in P 2 P networks, exchanging messages in social networks, etc. )

First case study: Network Design Games (a. k. a. Global Connection Games)

The model n n n 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 Let ke denote the load of edge e, i. e. , the number of players using e. The cost of Pi for player i in S=(P 1, …, Pk) is shared with all the other players using (part of) it, namely: costi(S) = ce/ke e P i this cost-sharing scheme is called fair or Shapley cost-sharing mechanism

The model n n Given a strategy vector S, the constructed network N(S) is given by the union of all paths Pi Then, the social-choice function (i. e. , the total cost of the constructed network) is the following: C(S)= costi(S) = ce/ke= ce i n i e Pi e N(S) 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

Setting n n n What is a stable network? We use NE as the solution concept How to evaluate the overall quality of a stable network? We compare its cost to that of an optimal (in general, unstable) network Our goal: to bound the efficiency loss resulting from selfishness

Bounding the loss of efficiency n n n Remind that a network is optimal or socially efficient if it minimizes the social cost (i. e. , it minimizes the social-choice function) We use the Po. A 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 cost of a best stable network and the cost of an optimal network?

The price of stability (Po. S) n Definition (Schulz & Moses, 2003): Given a (single-instance) game G and a social-choice function C, let S be the set of NE, and let OPT be the outcome of G either minimizing or maximizing C, depending on whether the payoff for a player is either a cost or a utility. Then, the Price of Stability (Po. S) of G w. r. t. C is: Po. SG(C) = n Remark: If the game consists of several instances, then its Po. S is the maximum/minimum among the Po. S of all the instances, depending on whether the payoff for a player is either a cost or a utility.

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

An example 3 s 2 3 1 s 1 3 2 4 1 1 1 t 1 5. 5 t 2

An example 3 s 2 s 1 1 3 3 2 4 optimal network has cost 12 cost 1=7 cost 2=5 is it stable? 1 1 1 t 1 5. 5 t 2

An example 3 s 2 s 1 3 2 1 1 1 4 t 1 t 2 5. 5 …no!, player 1 can decrease its cost 1=5 cost 2=8 is it stable? …yes, and has cost 13! Po. A 13/12, Po. S ≤ 13/12

An example 3 s 2 s 1 3 2 4 …a best possible NE: 1 1 1 t 2 5. 5 cost 1=5 cost 2=7. 5 the social cost is 12. 5 Po. S = 12. 5/12 Homework: find a worst possible NE

Our concerned issues in GCG Does a stable network always exist? Does the repeated version of the game always converge to a stable network? Can we bound the price of anarchy (Po. A)? Can we bound the price of stability (Po. S)?

Theorem 1 Any instance of the global connection game has a pure Nash equilibrium, and best response dynamic always converges. Theorem 2 The Po. A in the global connection game with k players is at most k (i. e. , every instance of the game has Po. A ≤ k), and this is tight (i. e. , we can exhibit an instance of the game whose Po. A is k). Theorem 3 The Po. S in the global connection game with k players is at most Hk, the k-th harmonic number (i. e. , every instance of the game has Po. S ≤ Hk), and this is tight (i. e. , we can exhibit an instance of the game whose Po. S is Hk)

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=(s 1, …, sk), s’i si, let S’=(s 1, …, s’i, …, sk), then (S)- (S’) = costi(S)-costi(S’). A game that does possess an exact potential function is called potential game

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 costi(S) costi(S’) player i cannot decrease its cost, thus S is a NE.

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. Furthermore, improving moves for players decrease the value of the potential function. Together, these observations imply the following result. Lemma 2 In any finite potential game, best response dynamics always converges to a Nash equilibrium However, it can be shown that converging to a NE may take an exponential (in the number of players) number of steps!

…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 k = ce · (1+1/2+…+1/ke). e

Lemma 3 ( is a potential function) Let S=(P 1, …, 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: When player i switches from Pi to P’i, some edges of N(S) increase their load by 1, some others decrease it by 1, and the remaining do not change it. Then, it suffices to notice that: • If load of edge e increases by 1, its contribution to the potential function increases by ce/(ke+1) • If load of edge e decreases by 1, its contribution to the potential function decreases by ce/ke (S) - (S’) = (S) - (S-Pi+P’i) = (S) – ( (S) - e Pi ce/ke + e P’i ce/(ke+1))= costi(S) – costi(S’).

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. However, it can be shown that finding a best response for a player is NP-hard!

Price of Anarchy: a lower bound k s 1, …, sk t 1, …, tk 1 optimal network has cost 1 best NE: all players use the lower edge Po. S is 1 worst NE: all players use the upper edge Po. A is k

Upper-bounding the Po. A Theorem 2 The price of anarchy in the global connection game with k players is at most k. Proof: Let OPT=(P 1*, …, 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: k k k i=1 i=1 C(S) = costi(S) ≤ w( i) ≤ w(Pi*) ≤ k k·costi(OPT) = k· C(OPT). i=1 (since a path is used by at most k players)

Po. S for GCG: a lower bound >o: small value t 1, …, tk s 1 1 s 2 0 1/3 1/2 s 3 0 1/(k-1) 1/k sk-1 . . . 0 0 0 sk 1+

Po. S for GCG: a lower bound >o: small value t 1, …, tk s 1 1 s 2 0 1/3 1/2 s 3 0 1/(k-1) 1/k sk-1 . . . 0 0 The optimal solution has a cost of 1+ is it stable? 0 sk 1+

Po. S for GCG: a lower bound >o: small value t 1, …, tk s 1 1 s 2 0 1/3 1/2 s 3 0 1/(k-1) 1/k sk-1 . . . 0 0 …no! player k can decrease its cost… is it stable? 0 sk 1+

Po. S for GCG: a lower bound >o: small value t 1, …, tk s 1 1 s 2 0 1/3 1/2 s 3 0 1/(k-1) 1/k sk-1 . . . 0 0 …no! player k-1 can decrease its cost… is it stable? 0 sk 1+

Po. S for GCG: a lower bound >o: small value t 1, …, tk s 1 1 s 2 1/3 1/2 0 s 3 0 1/(k-1) 1/k sk-1 . . . 0 0 sk 1+ 0 The only stable network social cost: C(S)= k 1/j = Hk ln k + 1 j=1 k-th harmonic number

Lemma 4 Suppose that we have a potential game with potential function , and assume that for any outcome S we have C(S)/A (S) B C(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: C(S’)/A (S’) (S*) B C(S*) Po. S ≤ C(S’)/C(S*) ≤ A·B.

Lemma 5 (Bounding ) For any strategy vector S in the GCG, we have: C(S) Hk C(S). Proof: Indeed: (S) = e N(S) e(S) = e N(S) ce· Hke (S) C(S) = e N(S) ce and (S) ≤ Hk· C(S) = e N(S) ce· Hk.

Upper-bounding the Po. S 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 Po. S is at most Hk.