Price of Anarchy PDF
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PriceofAnarchy Smoothness PriceofStability Price of Anarchy Algorithmic Game Theory AlexanderSkopalik AlgorithmicGameTheory2012 PriceofAnarchy PriceofAnarchy Smoothness PriceofStability Recall Price of Anarchy Recall Price of Anarchy for Nash equilibria: ◮ Strategic game Γ, social cost cost(s) for every state s of Γ ◮ Consider ΣPNE as the set of pure Nash equilibria of Γ ◮ Price of Anarchy is a ratio: PoA= maxs′∈ΣPNE cost(s′) mins∈Σcost(s) PoA is a worst-case ratio and measures how much the worst PNE costs in comparison to an optimal state of the game. Assumption We here choose cost(s)= i∈Nci(s) throughout. P Is there a technique to bound the price of anarchy in many games? AlexanderSkopalik AlgorithmicGameTheory2012 PriceofAnarchy PriceofAnarchy Smoothness PriceofStability Example: Congestion Games with Linear Delay Functions PoA in CGs with linear delays dr(x)=ar ·x+br, for ar,br >0: In the following game, there are 4 players going from (1) u to w, (2) w to v, (3) v to w and (4) u to v. Essentially, each player has a short (direct edge) and a long (along the 3rd vertex) strategy: w x x 0 x x u v 0 AlexanderSkopalik AlgorithmicGameTheory2012 PriceofAnarchy PriceofAnarchy Smoothness PriceofStability Example: Congestion Games with Linear Delay Functions Optimum s∗ A bad PNE s x x x x 0 x 0 x x x 0 0 cost(s∗) = 1 + 1 + 1 + 1 = 4 cost(s) = 3 + 2 + 2 + 3 = 10 PoA in this game at least 2.5. Is this the worst-case? AlexanderSkopalik AlgorithmicGameTheory2012 PriceofAnarchy PriceofAnarchy Smoothness PriceofStability A general approach Definition A game is called (λ,µ)-smooth for λ>0 and µ≤1 if, for every pair of states s,s′ ∈Σ, we have ci(si′,s−i)≤λ·cost(s′)+µ·cost(s) (1) iX∈N Smoothness directly gives a bound for the PoA: Theorem In a (λ,µ)-smooth game, the PoA for pure Nash equilibria is at most λ . 1−µ AlexanderSkopalik AlgorithmicGameTheory2012 PriceofAnarchy PriceofAnarchy Smoothness PriceofStability Proof PoA for PNE Proof: Let s be the worst PNE and s′ =s∗ be an optimum solution. Then: cost(s) = ci(s) ≤ ci(si∗,s−i) (as s is NE) iX∈N iX∈N ≤ λ·cost(s∗)+µ·cost(s) (by smoothness) On both sides subtract µ·cost(s), this gives (1−µ)·cost(s)≤λ·cost(s∗) and rearranging yields cost(s) λ ≤ . cost(s∗) 1−µ (Theorem) AlexanderSkopalik AlgorithmicGameTheory2012 PriceofAnarchy PriceofAnarchy Smoothness PriceofStability Smoothness Examples Theorem Every congestion game with affine delay functions is 5,1 -smooth. Thus, the 3 3 PoA is upper bounded by 5/2 = 2.5. (cid:0) (cid:1) AlexanderSkopalik AlgorithmicGameTheory2012 PriceofAnarchy PriceofAnarchy Smoothness PriceofStability Tightness in Affine Congestion Games Proof of (5/3,1/3)-smoothness: We use the following lemma: Lemma (Christodoulou, Koutsoupias, 2005) For all integers y,z ∈Z we have y(z+1)≤ 5 ·y2+ 1 ·z2 . 3 3 Recall that delays are dr(nr)=arnr +br and consider the numbers ar,br ≥0. We multiply the above inequality by ar ≥0 and then add bry to the left and 5/3·bry +1/3·brz to the right-hand side. This implies ary(z+1)+bry ≤ 5(ary2+bry)+ 1(arz2+brz) . 3 3 Thus with y =nr∗ and z =nr the above inequality can be used to show 5 1 (ar(nr +1)+br)nr∗ ≤ (anr∗+b)nr∗+ (anr +b)nr . 3 3 AlexanderSkopalik AlgorithmicGameTheory2012 PriceofAnarchy PriceofAnarchy Smoothness PriceofStability Tightness in Affine Congestion Games Summing up these inequalities for all resources r ∈R, we get 5 1 (ar(nr +1)+br)nr∗ ≤ (arnr∗+br)nr∗+ (arnr +br)nr 3 3 rX∈R rX∈R rX∈R 5 1 = ·cost(S∗)+ ·cost(S) . 3 3 (5/3,1/3)-smoothness is shown by observing that ci(Si∗,S−i)≤ (ar(nr +1)+br)nr∗ , iX∈N rX∈R because there are at most nr∗ many players that might pick resource r upon switching to Si∗. Each of these players then sees a delay of at most dr(nr +1) upon switching to Si∗ unilaterally. (Theorem) AlexanderSkopalik AlgorithmicGameTheory2012 PriceofAnarchy PriceofAnarchy Smoothness PriceofStability Tightness in General Congestion Games Theorem (Roughgarden, 2003, Informal) For a large class of non-decreasing, non-negative latency functions, the PoA for pure NE in Wardrop games is λ/(1−µ), and it is achieved on a two-node, two-link network (like Pigou’s example). Theorem (Roughgarden, 2009, Informal) For a large class of non-decreasing, non-negative delay functions, the PoA for pure NE in congestion games is λ/(1−µ), and it is achieved on an instance consisting of two cycles with possibly many nodes (like the example for affine delays above). Thus, we have tightness and universal worst-case network structures in large classes of Wardrop and congestion games. AlexanderSkopalik AlgorithmicGameTheory2012 PriceofAnarchy
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