Games, Auctions, Learning, and the Price of Anarchy. Éva Tardos Cornell University

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1 Games, Auctions, Learning, and the Price of Anarchy Éva Tardos Cornell University

2 Games and Quality of Solutions Rational selfish action can lead to outcome bad for everyone Tragedy of the Commons Model: Value for each cow decreasing function of # of cows Too many cows: no value left

3 Example: Routing Games Traffic subject to congestion delays cars and packets follow shortest path Congestion game =cost (delay) depends only on congestion on edges

4 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

5 Results for non-atomic games Theorem 3 (Roughgarden-Tardos 02): In any network with continuous, nondecreasing latency functions cost of Nash with rates r i for all i cost of opt with rates 2r i for all i

6 Today: Online Markets Advertisement Effort Information Online markets use simple auctions for allocations How good is the resulting allocation?

7 Ideal Auction Basic Auction: single item Vickrey Auction $2 $5 $7 $3 $4 Pays $5 Player utility v i p i item value price paid Vickrey Auction (second price) Truthful Efficient Simple

8 Some Simple Auctions First/Second price GSP, etc Advertisement All Pay, War of attrition Effort Most not truthful

9 Truthful or Simple Vickrey, Clarke, Groves (VCG) truthful, but not simple Assignment efficient (maximizing social welfare) Payment welfare loss to others Online ads (Display/Search) customized by information about user: Search term, History of user, Time of the day, Geographic Data, Cookies, Budget Millions of ads each minute and all different! Needs a simple and intuitive scheme

10 Should Work in Composition Goods Effort Information Advertisement

11 Multiple Simple Auctions? Second price auction truthful and simple, but Two simultaneous second price auctions? Not How about sequentially? Not repeat offer to same or different seller

12 Goal [Syrkanis-Tardos 2013]: Design mechanisms such that a market composed of such mechanisms is approximately efficient? Local: local mechanism efficient Global mechanism efficient

13 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

14 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 utility i b i, b i v i S i p x x S i Players p 1 + S i p 1 Pure Nash sets market clearing prices (Bikhchandani 96) b i p 2 + p 2 p 3

15 What are Good Prices? Market clearing: utility i b v i S i p x x S i S i and i Theorem: Market clearing prices guarantee socially optimal outcome: maximizing i v i (S i ) of all allocations (S 1,S 2, ) Proof: Each player has value favorite items i (v i S i p x ) ( v i S i p x ) x S i i x S i

16 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

17 What is Selfish Outcome (2)? Do players find Nash? But.. if solution stable that is Nash 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

18 Learning outcome b 1 1 b 1 2 b n 1 b 2 1 b 2 2 b n 2 b 3 1 b 3 2 b n 3 b t 1 b t 2 b n t time Maybe here they don t know how to bid, who are the other players, Run Auction on ( b 1 1, b 1 2,, b 1 n ) Run Auction on ( b 1 t, b 2 t,, b n t ) By here they have a better idea 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)

19 Bayesian Beliefs F 1 v 1 Bayes-Nash Equilibrium: E v i utility i b v E v i utility i b i, b i v i F i v i b 1 (v 1 ) b i (v i ) F n v n b n (v n )

20 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

21 Market Clearing Prices Market clearing: Player i has a bid b i that guarantees utility i b i, b i v i S i p x x S i Robust: bid b i should not depend on b -i and other players. S i Do robust bids ever exists? p 1 p 1 + b i p 2 + p 2 p 3

22 Example: Approximately market clearing mechanism Claim: First price auction for a single item is ( 1, 1) smooth 2 User of value v i bid b i = 1 v 2 i, utility Claim: utility i b i, b i 1 v 2 i 1p i Proof Either wins and has utility v i p i = 1 v 2 i Or looses and hence price was p i 1 v 2 i

23 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 - bandwidth allocation (Johari-Tsitsiklis), - etc Public Projects Bandwidth Allocation

24 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 we get utility i b i, b i λ v i S i μ p x x S i Or just i utility i, b i λopt μ x p x b i, should not depend on b -i smooth games Roughgarden 09 Theorem Smooth mechanism at Nash has socially approximately optimal outcome (off by at most factor of λ/max 1, μ ) Proof: at Nash utility i b i, b i ) utility i (b

25 Robust Price of Anarchy Theorem(Syrkganis-T 13) Auction game (, )- smooth, 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) See in a bit Also true for games with uncertainty, assuming player types are independent

26 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 b 1, b 2,, b t, sequence of bids t ) t utility i b t t utility i (b i, b i (no regret) utility i b t t i t i, b i λt Opt μ t x p x (smooth)

27 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

28 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

29 No complements Submodular M 1 M 2 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 v S + x v S v S + x v(s ) Across Mechanisms (fractionally subadditive)

30 Simultaneous Composition Theorem (Syrkganis-T 13): simultaneous item auctions where each is (, )-smooth and players have fractionally subadditive valuations, then composition is also (, )-smooth Proof: valuation v S is fractionally subadditive maximum of j linear functions: v S = max j x S v x e.g., submodular optimal v j allocation j j x = v S x S 1 v, SS j 2, values x : marginal v i Svalue i = for ordering v i x S i x j, where S j jj User i bids in x auction is prefix x till for x, value and vs i x prefix without x. Real value v i S v x j i x S

31 Conclusion: Designing for Selfish Users Selfish users can ruin social welfare (Tragedy of the Commons) With care, we can design for selfish use and mitigate the welfare loss Guarantees very robust (Nash, learning, uncertainty)