Games, Auctions, Learning, and the Price of Anarchy Éva Tardos Cornell University
Games and Quality of Solutions • Rational selfish action can lead to outcome bad for everyone Model: • Value for each cow decreasing function of Tragedy of the Commons # of cows • Too many cows: no value left
Example: Routing Games • Traffic subject to congestion delays • cars and packets follow shortest path Congestion game =cost (delay) depends only on congestion on edges
What is Selfish Outcome? We will use: Nash equilibrium – Current strategy “best response” for all players (no incentive to deviate) Theorem [Nash 1952]: – Always exists if we allow randomized strategies
Results for non-atomic games Theorem 3 (Roughgarden- Tardos’02): • In any network with continuous, nondecreasing latency functions cost of opt with rates cost of Nash with rates r i 2r i for all i for all i
Today: Online Markets Effort Advertisement Information Online markets use simple auctions for allocations How good is the resulting allocation?
Ideal Auction Basic Auction: single item Vickrey Auction $2 $5 $7 $3 $4 Pays $5 Player utility 𝑤 𝑗 − 𝑞 𝑗 item value – price paid Vickrey Auction – Truthful (second price) – Efficient – Simple
Some Simple Auctions Advertisement First/Second price GSP, etc Effort All Pay, War of attrition Most not truthful
Truthful or Simple Vickrey, Clarke, Groves Online ads (VCG) truthful, but not (Display/Search) simple customized by information about user: Search term, History of user, Time of the day, • Assignment efficient Geographic Data, Cookies, Budget (maximizing social welfare) Millions of ads each minute and all different! • Payment welfare loss to others Needs a simple and intuitive scheme
Should Work in Composition Effort Goods Advertisement Information
Multiple Simple Auctions? Second price auction truthful and simple, but… Two simultaneous second price auctions? Not Not How about sequentially? repeat offer to same or different seller
Goal [Syrkanis-Tardos 2013]: Design mechanisms such that a market composed of such mechanisms is approximately efficient? Local: local mechanism efficient Global mechanism efficient
Today Mixing Auction Types First/Second price/All pay/ etc. Each interested in one or more items, but has different values Key idea: Auctions that price
First Price Auction: Good prices Outcome of first price auction: Each player i has a bid b’ i , such that if current bids are b -i and prices are p j we get ∗ − ′ , 𝑐 −𝑗 ≥ 𝑤 𝑗 𝑇 𝑗 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 𝑐 𝑗 𝑞 𝑦 ∗ 𝑦∈𝑇 𝑗 ∗ 𝑇 𝑗 Players Pure Nash sets market clearing prices 𝑞 1 ( Bikhchandani’96 ) + 𝑞 1 ′ 𝑐 𝑗 𝑞 2 + 𝑞 2 𝑞 3
What are Good Prices? ∗ − Market clearing: 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 𝑐 ≥ 𝑤 𝑗 𝑇 𝑗 𝑞 𝑦 ∗ 𝑦∈S 𝑗 ∗ and 𝑗 𝑇 𝑗 Theorem: Market clearing prices guarantee socially optimal outcome: maximizing 𝑤 𝑗 (𝑇 𝑗 ) of all 𝑗 allocations (S 1 ,S 2 ,…) Proof: Each player has value favorite items ∗ − 𝑞 𝑦 ) (𝑤 𝑗 𝑇 𝑗 − 𝑞 𝑦 ) ≥ ( 𝑤 𝑗 𝑇 𝑗 𝒋 𝒋 ∗ 𝑦∈𝑇 𝑗 𝑦∈𝑇 𝑗
Robust solution concepts • Pure Nash of Complete Information is very brittle – Pure Nash might not always exist – Game might be played repeatedly, with players using learning algorithms (correlated behavior) – Players might not know other valuations – Players might have probabilistic beliefs about values of opponents
What is Selfish Outcome (2)? Do players find Nash? if solution stable that is Nash But….. Finding Nash is hard algorithmic problem (Daskalakis-Goldberg- Papadimitrou’06) No regret learning: do at least as well as any fixed strategy with hindsight. If converges: Nash equilibrium…
Learning outcome b 1 1 b 1 2 b 1 3 b 1 t b 2 1 b 2 2 b 2 3 b 2 t … … … … b n 1 b n 2 b n 3 b n t time Maybe here they don’t know By here they have a better how to bid, who are the other idea… Run Auction on Run Auction on players, … ( b 1 1 , b 2 1 , …, b n 1 ) ( b 1 t , b 2 t , …, b n t ) Vanishingly small regret for any fixed strat b’ : ∑ t utility i ( b i t , b -i t ) ≥ ∑ t utility i ( b’ , b -i t ) – o(T) Simple randomized strategies guarantee vanishing regret (regret matching, multiplicative weight)
Bayesian Beliefs Bayes-Nash Equilibrium: ′ , 𝑐 −𝑗 𝑤 −𝑗 𝐹 𝑤 −𝑗 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 𝑐 𝑤 ≥ 𝐹 𝑤 −𝑗 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 𝑐 𝑗 𝐺 1 ∼ 𝑤 1 𝑐 1 (𝑤 1 ) 𝐺 𝑗 ∼ 𝑤 𝑗 𝑐 𝑗 (𝑤 𝑗 ) 𝑐 𝑜 (𝑤 𝑜 ) 𝑤 𝑜 𝐺 𝑜 ∼
Direct extensions • What if conclusions drawn for the Pure Nash equilibrium of the complete information setting could be directly extended to these robust notions? • Possible, but we need to restrict the type of analysis
Market Clearing Prices Market clearing: Player i has a bid b’ i that guarantees ∗ − ′ , 𝑐 −𝑗 ≥ 𝑤 𝑗 𝑇 𝑗 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 𝑐 𝑗 𝑞 𝑦 ∗ 𝑦∈𝑇 𝑗 Robust: bid b’ i should not depend on b -i and other players. ∗ 𝑇 𝑗 Do robust bids ever exists? 𝑞 1 + 𝑞 1 ′ 𝑐 𝑗 𝑞 2 + 𝑞 2 𝑞 3
Example: Approximately market clearing mechanism Claim: First price auction for a single item is ( 1 2 , 1) smooth User of value 𝑤 𝑗 bid 𝑐′ 𝑗 = 1 2 𝑤 𝑗 , utility ′ , 𝑐 −𝑗 ≥ 1 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 𝑐 𝑗 2 𝑤 𝑗 − 1𝑞 𝑗 Claim: Proof 1 • Either wins and has utility 𝑤 𝑗 − 𝑞 𝑗 = 2 𝑤 𝑗 1 • Or looses and hence price was 𝑞 𝑗 ≥ 2 𝑤 𝑗
Examples of approximately market clearing auction games • First price auction (1-1/e,1) approx – See also Hassidim et al EC’12, Syrkhanis’12 • All pay auction ( ½,1)-smooth • First position auction (GFP) is (½,1)-smooth • Second price auction is (½,0,1)-smooth (no overbidding) • Generalized second price (GSP) is (½,0,1)-smooth Other applications include: - public goods Public Projects - bandwidth allocation (Johari-Tsitsiklis), - etc Bandwidth Allocation
Auctions with OK Prices: Smooth Approximately market clearing: Player i has a bid b’ i , such that if current bids are b -i and item prices are p j ∗ − 𝜈 ′ , 𝑐 −𝑗 ≥ 𝜇 𝑤 𝑗 𝑇 𝑗 we get 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 𝑐 𝑗 𝑞 𝑦 ∗ 𝑦∈𝑇 𝑗 Or just 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 , 𝑐 −𝑗 ≥ 𝜇𝑃𝑄𝑈 − 𝜈 𝑞 𝑦 𝑦 𝑗 ′ , should not depend on b -i 𝑐 𝑗 smooth games Roughgarden’09 Theorem Smooth mechanism at Nash has socially approximately optimal outcome (off by at most factor of 𝜇 / 𝑛ax 1, 𝜈 ) ′ , 𝑐 −𝑗 ) ≤ 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 (𝑐 Proof: at Nash 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 𝑐 𝑗
Robust Price of Anarchy Theorem(Syrkganis- T’13 ) Auction game ( , )- smooth, then See in • Price of anarchy is at most max(1, )/ a bit • Preserved in Composition (assuming no complements) • Also true for mixed equilibria (and even correlated equilibria = learning outcomes) • Also true for games with uncertainty, assuming player types are independent
Robust Price of Anarchy Theorem (Syrkganis- T’13 ) Auction game ( , )-smooth game, then • Price of anarchy is at most max(1, )/ • Also true for learning outcomes (= coarse correlated equilibria) and mixed equilibria Proof: let 𝑐 1 , 𝑐 2 , … , 𝑐 𝑢 , … sequence of bids 𝑢 ) 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 𝑐 𝑢 ≥ 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 (𝑐 𝑗 ′ , 𝑐 −𝑗 (no regret) 𝑢 𝑢 𝑢 ≥ 𝜇𝑈 𝑃𝑞𝑢 − 𝜈 𝑞 𝑦 ′ , 𝑐 𝑗 𝑢 𝑣𝑢𝑗𝑚𝑗𝑢𝑧 𝑗 𝑐 𝑗 (smooth) 𝑢 𝑢 𝑦 𝑗
Robust Price of Anarchy Theorem (Syrkganis- T’13 ) Auction game ( , )- smooth game, then • Price of anarchy is at most max(1, )/ • Preserved in Composition (assuming no complements) • Also true for mixed equilibria (and even correlated equilibria = learning outcomes) • Also true for games with uncertainty, assuming player types are independent
Simultaneous Composition Corollary: Simultaneous first price auction has price of anarchy of e/(e-1) if player values have no complements • Simultaneous all-pay auction: price anarchy 2 • Mix of first price and all pay, price of anarchy ≤ 2
No complements Submodular … 𝑵 𝟐 𝑵 𝟑 … item auctions: Submodular: Marginal value for any allocation can only decrease, by getting more items: For all sets S S’, and item x S’ we have 𝑤 𝑇 + 𝑦 − 𝑤 𝑇 ≥ 𝑤 𝑇 ′ + 𝑦 − 𝑤(𝑇 ′ ) Across Mechanisms (fractionally subadditive)
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