Price of Anarchy Smoothness Price of Stability. Price of Anarchy. Algorithmic Game Theory

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1 Smoothness Price of Stability Algorithmic Game Theory

2 Smoothness Price of Stability Recall Recall for Nash equilibria: Strategic game Γ, social cost cost(s) for every state s of Γ Consider Σ PNE as the set of pure Nash equilibria of Γ is a ratio: PoA = ma s Σ PNE cost(s ) min s Σ cost(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 N c i(s) throughout. Is there a technique to bound the price of anarchy in many games?

3 Smoothness Price of Stability Eample: Congestion Games with Linear Delay Functions PoA in CGs with linear delays d r() = a r +b r, for a r,b r > 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 verte) strategy: w 0 u 0 v

4 Smoothness Price of Stability Eample: Congestion Games with Linear Delay Functions Optimum s A bad PNE s cost(s ) = = 4 0 cost(s) = = 10 PoA in this game at least 2.5. Is this the worst-case?

5 Smoothness Price of Stability 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 c i (s i,s i ) λ cost(s )+µ cost(s) (1) i 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.

6 Smoothness Price of Stability Proof PoA for PNE Proof: Let s be the worst PNE and s = s be an optimum solution. Then: cost(s) = i N c i (s) i N c i (s i,s i ) (as s is NE) 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) cost(s ) λ 1 µ. (Theorem)

7 Smoothness Price of Stability Smoothness Eamples Theorem Every congestion game with affine delay functions is ( 5, 1 3 3) -smooth. Thus, the PoA is upper bounded by 5/2 = 2.5.

8 Smoothness Price of Stability 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 3 y z2. Recall that delays are d r(n r) = a rn r +b r and consider the numbers a r,b r 0. We multiply the above inequality by a r 0 and then add b ry to the left and 5/3 b ry +1/3 b rz to the right-hand side. This implies a ry(z +1)+b ry 5 3 (ary2 +b ry)+ 1 3 (arz2 +b rz). Thus with y = n r and z = n r the above inequality can be used to show (a r(n r +1)+b r)nr 5 3 (an r +b)nr + 1 (anr +b)nr. 3

9 Smoothness Price of Stability Tightness in Affine Congestion Games Summing up these inequalities for all resources r R, we get (a r(n r +1)+b r)nr 5 (a rnr +b r)nr + 1 (a rn r +b r)n r 3 3 r R r R r R = 5 3 cost(s )+ 1 3 cost(s). (5/3, 1/3)-smoothness is shown by observing that c i (Si,S i ) r R(a r(n r +1)+b r)nr, i N 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 d r(n r +1) upon switching to Si unilaterally. (Theorem)

10 Smoothness Price of Stability 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 eample). 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 eample for affine delays above). Thus, we have tightness and universal worst-case network structures in large classes of Wardrop and congestion games.

11 Smoothness Price of Stability Decreasing Delays: Fair Cost Sharing Games Fair Cost Sharing Game Set N of n players, set R of m resources Player i allocates some resources, i.e., strategy set Σ i 2 R Resource r R has fied cost c r 0. Cost c r is assigned in equal shares to the players allocating r (if any). Fair cost sharing games are congestion games with delays d r() = c r/. Social cost turns out to be the sum of costs of resources allocated by at least one player: cost(s) = c i (S) = d r(n r) = n r c r/n r = c r. i N i N r S i r R r R n r 1 n r 1

12 Smoothness Price of Stability Price of Stability Theorem Every fair cost sharing game is (n,0)-smooth. Thus, the PoA is upper bounded by n.the class of fair cost sharing games is tight, i.e., there are games in which the PoA for pure Nash equilibria is eactly n. PoA is large, but pure Nash equilibrium is not necessarily unique. What do other pure NE cost, what about the best one? Price of Stability for Nash equilibria: Consider Σ PNE as the set of pure Nash equilibria of a game Γ Price of Stability is the ratio: PoS = min s Σ PNE cost(s ) cost(s ) PoS is a best-case ratio and measures how much the best PNE costs in comparison to an optimal state of the game.

13 Smoothness Price of Stability Price of Stability in Cost Sharing Games Theorem The Price of Stability for pure Nash equilibria in fair cost sharing games is at most H n = n = O(logn). Proof: Rosenthal s potential function for cost sharing delays is Φ(S) = r R n r i=1 c r/i = r R n r 1 c r H n r R n r 1 c r ( ) nr = cost(s) H n.

14 Smoothness Price of Stability Price of Stability in Cost Sharing Games In Φ(S) we account for each player allocating resource r a contribution of c r/i for some i = 1,...,n r, whereas in his cost c i (S) we account only c r/n r. Hence, for every state S of a cost sharing game we have cost(s) Φ(S) cost(s) H n. Now suppose we start at the optimum state S and iteratively perform improvement steps for single players. This eventually leads to a pure Nash equilibrium. Every such move decreases the potential function. For the resulting Nash equilibrium S we thus have Φ(S) Φ(S ) and cost(s) Φ(S) Φ(S ) cost(s ) H n. This proves that there is a pure Nash equilibrium that is only a factor of H n more costly than S. (Theorem)

15 Smoothness Price of Stability Recommended Literature Chapters 18 and 19.3 in the AGT book. (PoA and PoS bounds) B. Awerbuch, Y. Azar, A. Epstein. The Price of Routing Unsplittable Flow. STOC (PoA for pure NE in congestion games) G. Christodoulou, E. Koutsoupias. The of finite Congestion Games. STOC (PoA for pure NE in congestion games) T. Roughgarden. Intrinsic Robustness of the. STOC (Smoothness Framework and Unification of Previous Results)