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Robust Price of Anarchy Bounds via Smoothness Arguments Tim Roughgarden Stanford University 1 Price of Anarchy Definition: price of anarchy (POA) of a game (w.r.t. some objective function, eq concept): equilibrium objective fn value optimal obj fn value the closer to 1 the better 2 Price of Anarchy Definition: price of anarchy (POA) of a game (w.r.t. some objective function, eq concept): the closer to 1 the better equilibrium objective fn value optimal obj fn value 2x s 0 5 12 5x cost = 14+14 = 28 2x t 12 0 s 5 t 5x cost = 14+10 = 24 3 The Need for Robustness Meaning of a POA bound: if the game is at an equilibrium, then outcome is near-optimal. 4 The Need for Robustness Meaning of a POA bound: if the game is at an equilibrium, then outcome is near-optimal. Problem: what if can’t reach equilibrium? • (pure) equilibrium might not exist • might be hard to compute, even centrally – [Fabrikant/Papadimitriou/Talwar], [Daskalakis/ Goldbeg/Papadimitriou], [Chen/Deng/Teng], etc. • might be hard to learn in distributed way Worry: are our POA bounds “meaningless”? 5 Fix #1: Provable Convergence One Good Research Agenda: prove that “natural learning dynamics” converge (quickly) to equilibrium. • [Hart/Mas-Collell], [Even-Dar/Kesselman/Mansour], [Fisher/ Racke/Voecking], [Chien/Sinclair], [Awerbuch et al], [EvenDar/Mansour/Nadav], [Kleinberg/Piliouras/Tardos], etc. Problem: this “best-case scenario” has limited reach • non-existence/complexity issues • lower bounds on convergence time of natural dynamics (e.g., [Skopalik/Voecking STOC 08]) 6 Fix #2: Robust POA Bounds High-Level Goal: worst-case bounds that apply even to non-equilibrium outcomes! • best-response dynamics, pre-convergence – [Mirrokni/Vetta 04], [Goemans/Mirrokni/Vetta 05], [Awerbuch/Azar/Epstein/Mirrokni/Skopalik 08] • correlated equilibria – [Christodoulou/Koutsoupias 05] • coarse correlated equilibria aka “price of total anarchy” aka “no-regret players” – [Blum/Even-Dar/Ligett 06], [Blum/Hajiaghayi/Ligett/Roth 08] 7 Abstract Setup • n players, each picks a strategy si • player i incurs a cost Ci(s) Important Assumption: objective function is cost(s) := i Ci(s) Key Definition: A game is (λ,μ)-smooth if, for every pair s,s* outcomes (λ > 0; μ < 1): i Ci(s*i,s-i) ≤ λ●cost(s*) + μ●cost(s) [(*)] 8 Smooth => POA Bound Next: “canonical” way to upper bound POA (via a smoothness argument). • notation: s = a Nash eq; s* = optimal Assuming (λ,μ)-smooth: cost(s) = i Ci(s) [defn of cost] ≤ i Ci(s*i,s-i) [s a Nash eq] ≤ λ●cost(s*) + μ●cost(s) [(*)] Then: POA (of pure Nash eq) ≤ λ/(1-μ). 9 Why Is Smoothness Stronger? Key point: to derive POA bound, only needed i Ci(s*i,s-i) ≤ λ●cost(s*) + μ●cost(s) [(*)] to hold in special case where s = a Nash eq and s* = optimal. Smoothness: requires (*) for every pair s,s* outcomes. – even if s is not a pure Nash equilibrium 10 Some Smoothness Bounds • atomic (unweighted) selfish routing [Awerbuch/Azar/Epstein 05], [Christodoulou/Koutsoupias 05], [Aland/Dumrauf/Gairing/Monien/Schoppmann 06], [Roughgarden 09] • nonatomic selfish routing [Roughgarden/Tardos 00],[Perakis 04] [Correa/Schulz/Stier Moses 05] • weighted congestion games [Aland/Dumrauf/Gairing/Monien/Schoppmann 06], [Bhawalkar/Gairing/Roughgarden 10] • submodular maximization games [Vetta 02], [Marden/Roughgarden 10] • coordination mechanisms [Cole/Gkatzelis/Mirrokni 10] 11 Application: Out-of-Equilibria Definition: a sequence s1,s2,...,sT of outcomes is no-regret if: • for each player i, each fixed action qi: – average cost player i incurs over sequence no worse than playing action qi every time – simple hedging strategies can be used by players to enforce this (for suff large T) Theorem: [Roughgarden STOC 09] in a (λ,μ)smooth game, average cost of every noregret sequence at most [λ/(1-μ)] x cost of optimal outcome. 12 Smooth => POTA Bound • notation: s1,s2,...,sT = no regret; s* = optimal Assuming (λ,μ)-smooth: t cost(st) = t i Ci(st) [defn of cost] 13 Smooth => POTA Bound • notation: s1,s2,...,sT = no regret; s* = optimal Assuming (λ,μ)-smooth: t cost(st) = t i Ci(st) = t i [Ci(s*i,st-i) + ∆i,t] [defn of cost] [∆i,t:= Ci(st)- Ci(s*i,st-i)] 14 Smooth => POTA Bound • notation: s1,s2,...,sT = no regret; s* = optimal Assuming (λ,μ)-smooth: t cost(st) = t i Ci(st) = t i [Ci(s*i,st-i) + ∆i,t] [defn of cost] [∆i,t:= Ci(st)- Ci(s*i,st-i)] ≤ t [λ●cost(s*) + μ●cost(st)] + i t ∆i,t [(*)] 15 Smooth => POTA Bound • notation: s1,s2,...,sT = no regret; s* = optimal Assuming (λ,μ)-smooth: t cost(st) = t i Ci(st) = t i [Ci(s*i,st-i) + ∆i,t] [defn of cost] [∆i,t:= Ci(st)- Ci(s*i,st-i)] ≤ t [λ●cost(s*) + μ●cost(st)] + i t ∆i,t [(*)] No regret: t ∆i,t ≤ 0 for each i. To finish proof: divide through by T. 16 Why Important? • bound on no-regret sequences implies bound on inefficiency of mixed and correlated equilibria • bound applies even to sequences that don’t converge in any sense no regret correlated eq mixed Nash pure Nash • no regret much weaker than reaching equilibrium • e.g., if every player uses “multiplicative weights” then get o(1) regret in poly-time 17 Tight Game Classes Theorem: [Roughgarden STOC 09] for every set C, unweighted congestion games with cost functions restricted to C are tight: maximum [pure POA] = minimum [λ/(1-μ)] congestion games w/cost functions in C (λ ,μ): all such games are (λ ,μ)-smooth 18 Tight Game Classes Theorem: [Roughgarden STOC 09] for every set C, unweighted congestion games with cost functions restricted to C are tight: maximum [pure POA] = minimum [λ/(1-μ)] congestion games w/cost functions in C (λ ,μ): all such games are (λ ,μ)-smooth • weighted congestion games [Bhawalkar/ Gairing/Roughgarden ESA 10] and submodular maximization games [Marden/Roughgarden CDC 10] are also tight in this sense 19 Further Applications no regret correlated eq mixed Nash approximate Nash pure Nash bestresponse dynamics Theorem: in a (λ,μ)-smooth game, everything in these sets costs (essentially) λ/(1-μ) x OPT. 20 More Applications? Theorem: [Nadav/Roughgarden 10] Consider a (λ,μ)-smooth game for optimal choices of λ,μ. Then precisely the “aggregate” coarse correlated equilibria have cost ≤ λ/(1-μ) x OPT. •essentially, bound holds if and only if the average (rather than per-player) regret is non-positive Proof: convex duality. 21 When Is a POA Bound a Smoothness Proof? Need to show: for every pair s,s* outcomes: i Ci(s*i,s-i) ≤ λ●cost(s*) + μ●cost(s) Generally sufficient: prove POA bound for pure Nash equilibria such that: • invoke best-response condition once/player • hypothetical deviation by i independent of s-i Non-example: network creation games [Fabrikant et al], [Albers et al], [Demaine et al], etc. 22 Application: POA of NonTruthful Mechanisms Mechanism Design: design protocol with desirable outcome despite selfish participants • example: Vickrey (second-price) auction • focus thus far on "truthful" mechanisms Non-truthful mechanisms: motivated by • simplicity (e.g., sponsored search auctions) • low communication complexity • low computational complexity 23 Application: POA of NonTruthful Mechanisms Fact: plausible outcomes of non-truthful mechanisms = Bayes-Nash equilibria • POA results for simple non-truthful auctions in [Christodoulou/Kovacs/Schapira ICALP 08] and [Borodin/Lucier SODA 10] • first bound POA of Nash equilibria, then "by magic" same bound holds for Bayes-Nash Fact: these are automatic consequences of smoothness. 24 POA of Non-Truthful Mechanisms (continued) [Paes Leme/Tardos FOCS 10] • • POA upper bounds for welfare of Generalized Sponsored Search auction different bounds for pure (1.618), mixed (4), and Bayes-Nash eq (8) (without smoothness) Open Questions: • compute best smoothness bound • prove a lower bound separating the bestpossible pure vs. Bayes-Nash POA 25 POA of Non-Truthful Mechanisms (continued) [Bhawalkar/Roughgarden 10] • POA upper bounds for welfare of subadditive combinatorial auctions with “item bidding” – • • generalizes [Christodoulou/Kovacs/Schapira 08] pure Nash: non-smooth upper bound of 2 Bayes-Nash: lower bound of 2.01, upper bound of 2 ln m [m = # goods] Open Questions: • is Bayes-Nash POA = O(1)? 26 Local Smoothness and Splittable Congestion Games Local smoothness: require smoothness condition only for outcomes that are “close” – • • • • assume strategy sets = convex subset of Rn can only decrease optimal value of λ/(1-μ) [Harks 08]: local smoothness gives improved POA bounds for splittable congestion games [Roughgarden/Schoppmann 10]: matching lower bounds => first tight bounds in this model [RS10]: local smoothness bounds extend to correlated eq but not to no-regret outcomes! 27 Splittable Congestion Games: Open Questions Open Question #1: determine optimal POA bounds for no-regret sequences. Open Question #2: is (the distribution of) every no-regret sequence a convex combination of pure Nash equilibria? Open Question #3: prove something non-trivial about the POA of symmetric splittable congestion games. 28