Price of Stability for Network Design with Fair Cost Allocation [Anshellevich, Dasgupta, Kleinberg, Tardos, Wexler, Roughgarden : FOCS ’04] Presented by Mangesh Gupte

Network Design Games Graph G = (V,E) , cost function c : E ̶> R Users U : user i wants to connect terminals (si , ti) ( buy a si-ti path ) Cost for common edges is shared equally ci(e) = c(e) / xe c(e) = cost of edge e xe = number of users using edge e We want to find the Price of Stability (Cost of Best Nash / Optimal Cost) of this Game

Examples Game with high Price of Anarchy 1+ε s1, s2,.., sk t1, t2, .., tk k Game with high Price of Stability s1, s2,.., sk 1 1/n 1/2 1+ε t1 t2 tn

Price of Stability of the fair connection game is at most H(k) Construct a potential function Φ : (S1, S2, …, Sk) ̶> R Φ(S) – Φ(S’) = ui(S’) – ui(S) : S, S’ differ in strategy for player i Define : Φ(S) = Σe in E c(e) Hxe c(S) ≤ Φ(S) ≤ Hk c(S) SOPT : optimal solution. Start from SOPT and apply best response to get SNASH c(SNASH) ≤ Φ(SNASH) ≤ Φ(SNASH) ≤ Hk c(SOPT)

If the edge cost function is concave, non-decreasing then the PoS is Hk Define Φ(S) = Σe in E Σx=1..xe ce(x) / x Again Φ(S) - Φ(S’) = ui(S’) – ui(S) : S, S’ differ in strategy for player i Σx=1..xe ce(x) / x ≤ Hxe ce(xe) ce(k) / k ≤ ce(j) / j for j ≤ k Hence , c(S) = Σe in E ce(x) / xe ≤ Σe in E Σx=1..xe ce(x) / x = Φ(S) Using same arguments c(S) ≤ Φ(S) ≤ Hxe c(S)

More General cost function Cost of an edge : ce is arbitrary non-decreasing, satisfies conditions : Φ(S) = Σe in E Σx=1..xe ce(x) / x c(S) ≤ A Φ(S) Φ(S) ≤ B c(S) Then, the Price of Stability is AB These are called Potential Games. Example : fe(x) = ce(x) / x + de(x) , where d is a polynomial of degree l. Then de(1) + de(2) + … + de(t) = 1l + 2l + .. + tl = (l+1) tl+1 = (l+1) de(t+1)

It is NP-hard to know if there is a Nash Equilibrium of cost at most C 3D Matching : Given sets X, Y, Z and subsets of the form (xi, yj, zk) , can we cover all elements exactly. Given an instance of 3D matching we create an instance of the connection game Set C = |X|+|Y|+|Z| and create C users with a distinct source node for each one Connect as given by diagram, to the unique sink node t

t 3 3 Subsets X Y Z

It is easy to see that Every 3D matching is a Nash Equilibrium of the fair connection game If there is a 3D matching, then a Nash Equilibrium will be a matching Hence, finding Nash Equilibrium of cost at most C is equivalent to telling if there exists a 3D matching, which is NP-Hard

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