Price of Anarchy Smoothness Price of Stability
Price of Anarchy
Algorithmic Game Theory
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy Smoothness Price of Stability
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:
′ maxs′ ΣPNE cost(s ) PoA = ∈ mins∈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 ci (s) throughout. P
Is there a technique to bound the price of anarchy in many games?
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy Smoothness Price of Stability
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
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy Smoothness Price of Stability
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?
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy 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
′ ′ 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
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy 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) = 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)
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy Smoothness Price of Stability
Smoothness Examples
Theorem 5 1 Every congestion game with affine delay functions is 3 , 3 -smooth. Thus, the PoA is upper bounded by 5/2 = 2.5.
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy 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
5 2 1 2 y(z + 1) ≤ · y + · z . 3 3
Recall that delays are dr (nr ) = ar nr + br and consider the numbers ar , br ≥ 0. We multiply the above inequality by ar ≥ 0 and then add br y to the left and 5/3 · br y +1/3 · br z to the right-hand side. This implies
5 2 1 2 ar y(z + 1) + br y ≤ (ar y + br y) + (ar z + br z) . 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
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy Smoothness Price of Stability
Tightness in Affine Congestion Games
Summing up these inequalities for all resources r ∈R, we get
∗ 5 ∗ ∗ 1 (ar (nr + 1) + br )nr ≤ (ar nr + br )nr + (ar nr + 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)
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy 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 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.
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy Smoothness Price of Stability
Decreasing Delays: Fair Cost Sharing Games
Fair Cost Sharing Game ◮ Set N of n players, set R of m resources R ◮ Player i allocates some resources, i.e., strategy set Σi ⊆ 2 ◮ Resource r ∈R has fixed cost cr ≥ 0. ◮ Cost cr is assigned in equal shares to the players allocating r (if any).
Fair cost sharing games are congestion games with delays dr (x) = cr /x.
Social cost turns out to be the sum of costs of resources allocated by at least one player:
cost(S) = ci (S) = dr (nr ) = nr · cr /nr = cr . iX∈N iX∈N rX∈Si rX∈R rX∈R nr ≥1 nr ≥1
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy 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 exactly 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 ΣPNE as the set of pure Nash equilibria of a game Γ ◮ Price of Stability is the ratio:
′ mins′ ΣPNE cost(s ) PoS = ∈ 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.
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy 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 1 1 1 most Hn = 1 + 2 + 3 + ... + n = O(log n).
Proof: Rosenthal’s potential function for cost sharing delays is
nr 1 1 1 Φ(S) = cr /i = cr · 1 + + + ... + 2 3 nr rX∈R Xi=1 rX∈R nr ≥1
≤ cr ·Hn rX∈R nr ≥1 = cost(S) ·Hn .
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy 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 cr /i for some i =1,..., nr , whereas in his cost ci (S) we account only cr /nr . Hence, for every state S of a cost sharing game we have
cost(S) ≤ Φ(S) ≤ cost(S) ·Hn .
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 ) ·Hn .
This proves that there is a pure Nash equilibrium that is only a factor of Hn more costly than S ∗. (Theorem)
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy Price of Anarchy 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 2005. (PoA for pure NE in congestion games)
◮ G. Christodoulou, E. Koutsoupias. The Price of Anarchy of finite Congestion Games. STOC 2005. (PoA for pure NE in congestion games)
◮ T. Roughgarden. Intrinsic Robustness of the Price of Anarchy. STOC 2009. (Smoothness Framework and Unification of Previous Results)
Alexander Skopalik Algorithmic Game Theory 2012 Price of Anarchy