Price of Anarchy Smoothness
Price of Anarchy
Algorithmic Game Theory
Alexander Skopalik Algorithmic Game Theory Price of Anarchy Price of Anarchy Smoothness
Recall Price of Anarchy
Recall Price of Anarchy for Nash equilibria: ◮ Strategic game Γ, social cost cost(s) for every state s of Γ PNE ◮ Consider Σ 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 Price of Anarchy Price of Anarchy Smoothness
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 Price of Anarchy Price of Anarchy Smoothness
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 Price of Anarchy Price of Anarchy Smoothness
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) i∈N X
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 Price of Anarchy Price of Anarchy Smoothness
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) i∈N i∈N X ∗ X ≤ 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 Price of Anarchy Price of Anarchy Smoothness
Beyond PNE?
In general games, players might not stabilize at a pure Nash equilibrium and only use no-regret algorithms to play the game.
What is the cost when all players play with no-regret algorithms?
Definition In a game, the price of anarchy for no-regret sequences or price of total anarchy T is the smallest ρ ≥ 1 such that, for every sequence of states s1, s2,..., s with the no-regret property for every player, we have
T 1 t ∗ o(T ) cost(s ) ≤ ρ · cost(s )+ . T t=1 T X For T →∞, the average social cost of a sequence becomes ρ times the optimal cost. If we set s1 = . . . = sT = s′ for a PNE s′, then no player experiences regret. Thus, the no-regret PoA can only be larger than the PoA for PNE.
Alexander Skopalik Algorithmic Game Theory Price of Anarchy Price of Anarchy Smoothness
Smoothness can do more...
In the previous theorem, we only used that condition (1) holds for every pure Nash equilibrium. According to the definition, however, it holds for every pair of states. This allows to prove Theorem In a (λ, µ)-smooth game, the PoA for no-regret sequences is at most λ . 1 − µ
Proof: Let s∗ denote the state of optimum social cost. We define t t ∗ t ∆i := ci (s ) − ci (si , s−i ). For a sequence with no-regret property for every player i, we have
T T T 1 t 1 t t ∆i ≤ ci (s ) − min ci (si , s−i ) = Ri (T )/T = o(1) . s ∈Σ T t=1 T t=1 i i t=1 ! X X X
Alexander Skopalik Algorithmic Game Theory Price of Anarchy Price of Anarchy Smoothness
Proof PoA for No-Regret
This allows us to bound
T T T t t ∗ t t cost(s ) = ci (s )= ci (si , s−i )+∆i t=1 t=1 i∈N t=1 i∈N X X X X X T T ∗ t t ≤ [λ · cost(s )+ µ · cost(s )] + ∆i , t=1 t=1 i∈N X X X leading to
T T 1 t 1 ∗ Ri (T ) (1 − µ) · cost(s ) ≤ λ · cost(s )+ T t=1 T t=1 i∈N T X X X and finally T 1 t λ ∗ cost(s ) ≤ · cost(s )+ o(1) , − T t=1 1 µ X as desired. (Theorem)
Alexander Skopalik Algorithmic Game Theory Price of Anarchy Price of Anarchy Smoothness
Smoothness Examples
Theorem 1 Every Wardrop routing game with linear delay functions is 1, 4 -smooth. Thus, the PoA is upper bounded by 4/3, even for no-regret sequences.
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 Price of Anarchy Price of Anarchy Smoothness
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 Price of Anarchy Price of Anarchy Smoothness
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 Price of Anarchy