Vasilis Syrgkanis Microsoft Research NYC

Transcription

1 Vasilis Syrgkanis Microsoft Research NYC 1

2 Large Scale Decentralized-Distributed Systems Multitude of Diverse Users with Different Objectives

3 Large Scale Decentralized-Distributed Systems Multitude of Diverse Users with Different Objectives Sharing Computing Resources 3

4 Large Scale Decentralized-Distributed Systems Multitude of Diverse Users with Different Objectives Network Routing 4

5 Large Scale Decentralized-Distributed Systems Multitude of Diverse Users with Different Objectives Online Advertising 5

6 Centralized, engineered systems with clear objectives Platforms for interaction of diverse users 6

7 Centralized, engineered systems with clear objectives Platforms for interaction of diverse users Each optimizes their own objective Strategic user behavior can cause inefficiencies. 7

8 Centralized, engineered systems with clear objectives Platforms for interaction of diverse users Analyze efficiency of systems taking into account strategic behavior of participants Design systems for strategic users 8

9 Ad Impressions Goods Information Crowdsourcing 9

10 Ad Impressions Goods Information Crowdsourcing 10

11 Thousands of mechanisms run at the same time Players participate in many of them simultaneously or sequentially Environment too complex for optimal decision making Repeated game and learning behavior Incomplete information about environment (e.g. opponents) 11

12 Ad Impressions Goods How should we design efficient mechanisms for such markets? How efficient are existing mechanisms? Information Crowdsourcing 1

13 How should we design mechanisms such that a market composed of such mechanisms is approximately efficient? 13

14 We define the notion of a Smooth Mechanism. A market composed of Smooth Mechanisms is globally approximately efficient at equilibrium even under learning behavior and incomplete information. 14

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16 5 4 b 1 Pay 4 Vickrey (second-price) Auction Solicit bids Award to highest bidder Charge second highest bid b b 3 Classic Result. Dominant strategy equilibrium is efficient. Highest value wins 16

17 v 1 A = v 1 A B = 1 Buyers want one camera b 1 A b 1 B b A 1 1 A b B B 17

18 Buyers want one camera A 0 B 18

19 Buyers want one camera Allocation is inefficient A 0 B 19

20 0 A framework for robust and composable efficiency guarantees

21 Truthfulness doesn t compose No coordinator to run a centralized global truthful mechanism Centralized mechanism too complex or costly to implement rules with fast implementation 1

22 v 1 v i b 1 Utility = Value-Payment: u i b = (v i b i ) x i b b i Efficiency= Welfare v n SW b = v i x i b b n i

23 Classic Economics Approach 1. Characterize equilibrium. Analyze equilibrium properties 3

24 v 1 Characterize equilibrium strategy: b(v) Analyze equilibrium properties b 1 Example. v i U[0,1] then b v i = v i v i v n b i b n u i b = v i b Pr win = v i b b Set derivative w.r.t. b equal to 0: b = v i Marvelous theory! Revenue equivalence, Myerson s BNE characterization etc. 4

25 v 1 Characterize equilibrium strategy: b(v) Analyze equilibrium properties v b b 1 Example. v 1 U[0,1] and v U[0,] then? b 1 v = 3v 4 3v b Not v = scalable v 3v One uniform-one deterministic(vickrey 61) U a, b 1, U[a, b ]- Greismer et al 67 U a 1, b 1, U a, b - Kaplan, Samier 1 5

26 v 1 Characterize equilibrium strategy: b(v) Analyze equilibrium properties v b b 1 Example. v 1 U[0,1] and v U[0,] then? b 1 v = 3v 4 3v b Not v = scalable v 3v One uniform-one deterministic(vickrey 61) U a, b 1, U[a, b ]- Greismer et al 67 U a 1, b 1, U a, b - Kaplan, Samier 1 6

27 v 1 Characterize equilibrium strategy: b(v) Analyze equilibrium properties v b b 1 Example. v 1 U[0,1] and v U[0,] then? b 1 v = 3v 4 3v b Not v = scalable v 3v One uniform-one deterministic(vickrey 61) U a, b 1, U[a, b ]- Greismer et al 67 U a 1, b 1, U a, b - Kaplan, Samier 1 7

28 The Price of Anarchy Approach 8

29 v 1 v i v n b i b 1 = p + b i = 0 b n b n = 0 p = max i b i Pure Nash Equilibrium: b i maximizes utility u i b u i b i, b i Theorem. Any PNE is efficient. Proof. Highest value player can deviate to p + i i v i x i b u 1 b u i b p v 1 p u 1 p +, b i = v 1 p + u i b u i 0, b i = 0 i u i b i, b i = v 1 p 9

30 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 30

31 b 1 1 b 1 b 1 3 b 1 4 b 1 t b 1 T b i 1 b n 1 b i b n b i 3 b n 3 b i 4 b n 4 b i t b n t b i T b n T time Auction A 1 on (b 1 1,, b i 1,, b n 1 ) Vanishingly small regret for any fixed strategy x: T t=1 u i (b t ) T t=1 Auction A t on (b 1 t,, b i t,, b n t ) u i t x, b i o(t) Many simple rules: MWU (Hedge), Regret Matching etc. 31

32 Bid 3

33 Bid Days 33

34 F 1 v 1 Bayes-Nash Equilibrium: Mapping from values to bids Maximize utility in expectation F i v i b 1 (v 1 ) E v i u i b v E v i u i b i, b i v i b i (v i ) F n v n b n (v n ) Expected equilibrium welfare vs. Expected ex-post optimal welfare 34

35 What if conclusions for PNE of complete information directly extended to these more robust concepts Obviously: full efficiency doesn t carry over Possible, but we need to restrict the type of analysis 35

36 v 1 v i v n bp + 1 p = max i b i Recall. PNE is efficient because highest value player doesn t want to deviate to p + Challenge. Don t know p or v i in incomplete information Idea. Price oblivious deviation analysis Restrict deviation to not depend on p 36

37 v 1 v i v n b 1 = v 1 b i = 0 b i b n p b b n = 0 = max i b i Player 1 can deviate to b 1 = v 1 Either p b v 1 v Or u 1 1, b 1 = v 1 In any case: v 1 u 1, b 1 + p b v 1 Others can deviated to b i = 0: u i 0, b i 0 37

38 v 1 v i v n b 1 = v 1 b i = 0 p b b n = 0 = max i b i Player 1 can deviate to b 1 = v 1 This u i bguarantee u i b i, bextends i 1 v 1 to p(b) learning outcomes and to Bayesian beliefs. i i i u i (b) 1 v 1 p b SW(b) 1 OPT(v) 38

39 b 1 1 b 1 b 1 3 b 1 4 b 1 t b 1 T b i 1 b n 1 b i b n b i 3 b n 3 b i 4 b n 4 b i t b n t b i T b n T time Vanishingly small regret for fixed strategy b i : 1 T T t=1 i u i b t 1 T T t=1 T 1 T t=1 i u i b t i, b i o 1 SW b t 1 OPT v o 1 T 1 T t=1 v 1 pt o(1) 39

40 F 1 v 1 Deviation depends on opponent values F i v i b i v i, v i Need to construct feasible BNE deviations F n v n b i v i = b i v i, w i Each player random samples the others values and deviates as if that was the true values of his opponents Above works, due to independence of value distributions 40

41 Core Property v 1 v i v n b 1 = v 1 b i = 0 p b Player 1 can deviate to b 1 = v 1 Exists b i don t depend on current b = max i b i i u i b ii uu i ub i, b i i b i, b i v 1 i b i, b i 1 OPT 1 p(b) vv REV p(b) b i i u i (b) 1 v 1 p b b n = 0 SW(b) 1 OPT(v) 41

42 b 1 Utility = Value - Payment: v i X i b P i b b i X b = X1 b,, X n b M P b = P 1 b,, P n b b n Efficiency Measure: Social Welfare SW x = i v i (x i ) 4

43 b 1 b i Utility = Value - Payment: v i X i b P i b Combinatorial Auctions b n 43

44 b 1 b i Utility = Value - Payment: v i X i b P i b Public Projects b n 44

45 b 1 b i Utility = Value - Payment: v i X i b P i b Bandwidth Allocation b n 45

46 b 1 b i Utility = Value - Payment: v i X i b P i b b n Resource Sharing 46

47 Definition (S.-Tardos 13) λ, μ Smooth Mechanism Exist b i that don t depend on current b For any b i u i b i, b i λ OPT v μ REV b Closely related to smooth games [Roughgarden STOC 09], giving an intuitive market interpretation of smoothness 47

48 Theorem (S-Tardos 13) Mechanism is (λ, μ)-smooth, then every Nash Equilibrium achieves at least Extends no-regret learning outcomes of repeated game Extends to Bayesian Setting, assuming independent value distributions and even to no-regret under incomplete information. Extending Roughgarden EC 1 and S. 1 that used stricter universal smoothness property λ max{1, μ} of OPT. 48

49 Simultaneous Second Price Single-Item Auctions Christodoulou, Kovacs, Schapira ICALP 08, Bhawalkar, Roughgarden SODA 11 Auctions based on Greedy Allocation Algorithms Lucier, Borodin SODA 10 AdAuctions (GSP, GFP) Paes-Leme Tardos FOCS 10, Lucier, Paes-Leme + CKKK EC 11 Simultaneous First Price Auctions Single-Item Auctions Bikhchandani GEB 96, Hassidim, Kaplan, Mansour, Nisan EC 11, Fu et al. STOC 13 Sequential First/Second Price Auctions Paes Leme, S, Tardos SODA 1, S, Tardos EC 1 All above can be thought as smooth mechanisms and some are even compositions of smooth mechanisms. 49

50 First price auction: 1 1, 1 -smooth (Improves Hassidim et al. EC 1) e First price combinatorial auction based on a a-approximate greedy algorithm is 1 e a, 1 -smooth (Improves Lucier-Borodin SODA 10) Marginal pricing multi-unit auctions is 1 1, 1 -smooth (Improves De Keijzer et al. e ESA 13) All-pay auction: 1, 1 -smooth (New result) First price position auction is 1, 1 -smooth Extends Paes Leme et al. FOCS 10 to more general valuations) Proportional bandwidth allocation mechanism is 1, 1 -smooth Extends Johari-Tsitsiklis 05, to incomplete information and learning outcomes 50

51 51

52 1 v i 1 i Unit-Demand Valuation v i S = max v j j S i v i v i 3 n 5

53 1 b i 1 i Unit-Demand Valuation v i S = max v j j S i b i b i 3 n 53

54 Can we derive global efficiency guarantees from local 1, 1 smoothness of each first price auction? APPROACH: Prove smoothness of the global mechanism GOAL: Construct global deviation IDEA: Pick your item in the optimal allocation and perform the smoothness deviation for your local value v i j, i.e. v i j v i j 0 0 j i p 1 54

55 Smoothness locally: Summing over players: u i b i, b i v j i i p j i b i u i b i, b i 1 OPTv v REV(b) b Implying 1, 1 smoothness property globally. 55

56 X 1 b 1 X j b j = X j 1 b j,, X j n b j X m b m P 1 b 1 P j b j = P j 1 b j,, P j n b j P m b m M 1 M j M m b i 1 b i j b i m Complex valuation over outcomes v i X i 1 b 1,, X i m b m 56

57 Theorem (S.-Tardos 13) Simultaneous composition of m mechanisms, each (λ, μ)-smooth and players have no complements* across mechanisms, then composition is also λ, μ -smooth. 57

58 Across Mechanisms M 1 M Marginal value for any allocation from some mechanism can only decrease, as I get non-empty allocations from more mechanisms No assumption about allocation structure and valuation within mechanism 59

59 Global Efficiency Theorem. A market composed of λ, μ -Smooth Mechanisms achieves λ of optimal welfare at no-regret learning outcomes and max{1,μ} under incomplete information, when players have nocomplement valuations across mechanisms. 60

60 Sequential Composition Smooth mechanisms compose sequentially when values are unit-demand*. Tight via: Feldman, Lucier, S. Limits of Efficiency in Sequential Auctions Hard Budget Constraints on Payments Same efficiency guarantees with respect to new welfare benchmark: Optimal welfare achievable after capping a player s value by his budget Limited complementarities Global efficiency degrades smoothly with size of complementarities Feige, Feldman, Immorlica, Izsak, Lucier, S., A Unifying Hierarchy of Valuations with Complements and Substitues 61

61 Revenue of non-truthful mechanisms via price of anarchy in multi-dimensional settings Price of anarchy for auction revenue : Hartline, Hoy, Taggart Other models of non-fully rational behavior: level-k, fictitious play Level-0 Meta-Models for Predicting Human Behavior in Games : J. Wright, K. Leyton- Brown Simple auctions with simple strategies: good mechanisms with small strategy spaces (single knob to turn, simple to optimize over) Utility target mechanisms : Hoy, Jain, Wilkens Simple auctions with simple strategies : Devanur, Morgenstern, S., Weinberg Algorithmic characterization of smoothness in multi-dimensional environments (similar to cyclic monotonicity) Uncertainty about own valuation, information asymmetry Auctions, Adverse Selection, and Internet Display Advertising, Arnosti, Beck, Milgrom Coalitional dynamics analogues of no-regret dynamics with good welfare properties Strong Price of Anarchy and Coalitional Dynamics : Bachrach, S., Tardos, Vojnovic 6

62 Many simple mechanisms are smooth Smooth mechanisms compose well Robust efficiency guarantees Useful design and analysis tool for efficiency in electronic markets/distributed resource allocation systems Thank you! 63