Deep Learning for Revenue-Optimal Auctions with Budgets

Transcription

1 Deep Learning for Revenue-Optimal Auctions with Budgets Zhe Feng Harvard SEAS Based on joint work with Harikrishna Narasimhan (Harvard) and David C. Parkes (Harvard) 7/11/2018 AAMAS'18, Stockholm, Sweden 1

2 E. H. Clarke, 1971 T. Groves, 1973 Manelli &Vincent, 2006 Pavlov,2011 Hart & Nisan, 2012 Cai et al. 2012a; b Daskalakis et al Yao, 2017 History of Auction Design W. Vickrey Second Price Auction R. B. Myerson Optimal Single-item Auction present VCG Auction

3 Auction with budgets For bidders Willingness to pay: Valuation Ability to pay: financial (budget) constraints Utility is, when payment is larger than budget E.g. Rent an apartment near Harvard Valuation: Amenities, Distance from Harvard, Nearby environment Budget (affordable) Multiple items 7/11/2018 AAMAS'18, Stockholm, Sweden 3

4 Auction with budgets For bidders Willingness to pay: Valuation Ability to pay: financial (budget) constraints Utility is, when payment is larger than budget For auctioneer Maximize Social Welfare Maximize Revenue 7/11/2018 AAMAS'18, Stockholm, Sweden 4

5 Main question Design revenue-optimal auctions for budget constrained bidders 7/11/2018 AAMAS'18, Stockholm, Sweden 5

6 Optimal Auction Design with (private) Budgets (one item one bidder) Che and Gale, 2000 (one item two bidders) Malakhov and Vohra, 2008 Only one budget constrained bidder (one item multiple bidders) Pai and Vohra, 2014 BIC mechanism Discrete valuation and budget Optimal DSIC single-item auction with private budgets is still not fully understood! There are no results for multi-item auctions! 7/11/2018 AAMAS'18, Stockholm, Sweden 6

7 Results Summary Apply Deep Learning to design revenue-optimal auctions with private budgets. Extend RegretNet framework in [Dutting, ሷ Feng, Narasimhan, Parkes, 2017] to handle multi-item, DSIC & BIC setting. For settings with analytical solutions, RegretNet recovers almost-optimal auctions For settings where optimal solution is unknown, RegretNet outperforms well-known baselines. First automated mechanism design for budget constrained auction design. 7/11/2018 AAMAS'18, Stockholm, Sweden 7

8 Auctions with Private Budgets n bidders, m items Auction contains allocation rule g and payment rule p Each bidder has a type t i = v i, b i F = (F 1, F n ) is common knowledge Multi-dimensional setting Utility function: F i u i t i, t i, t i = ቊ v i g i t i, t i p i t i, t i, p i t i, t i b_i, o. w. Non-quasilinear: VCG fails Denote u i t i, t i : = u i (t i, (t i, t i )) 7/11/2018 AAMAS'18, Stockholm, Sweden 8

9 Auctions with Private Budgets Individual Rationality: u i t i, t i 0 Budget Constraints: p i t i, t i b i Incentive Compatibility Dominate Strategy IC (Strategy Proof): Under budget constraints, no matter what the other bidders report, truth-telling is always the weakly dominate strategy for each bidder. i, t, t i, u i t i, t i u i t i, t i. Bayesian IC, a weaker version of IC. Expected Revenue E t F p i (t) i 7/11/2018 AAMAS'18, Stockholm, Sweden 9

10 Challenges Multi-dimensional even for single-item auctions Bidders can misreport both value and budgets Non-quasilinear utility function VCG-based auctions fail! 7/11/2018 AAMAS'18, Stockholm, Sweden 10

11 VCG-based auction fails Run VCG Auction with truncated valuation (tv): min(value, budget) Not Strategy Proof! 7/11/2018 AAMAS'18, Stockholm, Sweden 11

12 VCG-based auction fails Value=$1, Budget = $10 Value =$10, Budget = $10 If win 1 chair, tv = $1 If win 2 chairs, tv = $2 If win 1 chair, tv = $10 If win 2 chairs, tv = $10 7/11/2018 AAMAS'18, Stockholm, Sweden 12

13 VCG-based auction fails Value=$1, Budget = $10 Value =$10, Budget = $10 If win 1 chair, tv = $1 If win 2 chairs, tv = $2 If win 1 chair, tv = $10 If win 2 chairs, tv = $10 Win 1 chair Pay $0 = $10 - $10 Win 1 chair Pay $1 = $2 - $1 Utility: $9 7/11/2018 AAMAS'18, Stockholm, Sweden 13

14 VCG-based auction fails Value =$1, Budget = $10 Value =$5, Budget = $10 If win 1 chair, tv = $1 If win 2 chairs, tv = $2 If win 1 chair, tv = $5 If win 2 chairs, tv = $10 Win 2 chairs Pay $2 = $2 - $0 Utility: $18 7/11/2018 AAMAS'18, Stockholm, Sweden 14

15 Data-driven Auction Design ($5, $9, $10) ($15, $2, $7).... ($9, $12, $6) Assume probability distribution for buyer s type Generate sample of type from distribution Use machine learning to discover the optimal auction 7/11/2018 AAMAS'18, Stockholm, Sweden 15

16 Deep Learning Rich Representation (Nonlinearity, Deep vs. Shallow) Fast Training Method (SGD, Adam) Robust tool chains (e.g. TensorFlow, Pytorch, GPUs) 7/11/2018 AAMAS'18, Stockholm, Sweden 16

17 Data-driven Auction Design (t 1 ) (t 2 ).... (t N ) Training Tune weights to maximize revenue s.t. incentive constraints and budget constraints Generalize RegretNet Framework [Dutting, ሷ Feng, Narasimhan, Parkes, 2017] 7/11/2018 AAMAS'18, Stockholm, Sweden 17

18 RegretNet: Architecture m identical items, n additive bidders, the type of bidder i is (v i, b i ). Parameters w. Allocation Net Payment Net Allocation: g w : R 2n Δ 1 Δ m Payment: p w : R 2n n R 0 ReLU units: relu x = max{0, x} 7/11/2018 AAMAS'18, Stockholm, Sweden 18

19 RegretNet: Metrics Deviation from IC (expected ex post regret) rgt i = E t max t i I(p i b i ) [u i t i, t i u i (t i, t i )] Ignoring measure zero events, IC iff rgt i = 0, for each bidder i Expected ex post IR-penalty: irp i = E t max 0, u i t i, t i Expected BC-penalty bcp i = E t max 0, p i (t) b i 7/11/2018 AAMAS'18, Stockholm, Sweden 19

20 RegretNet: Learning Problem g w, p w where w are parameters in Neural Nets Learning problem: min w L gw, p w = E t [ i p i w t ] s.t. i n, rgt i w = 0, irp i w = 0, bcp i w = 0 7/11/2018 AAMAS'18, Stockholm, Sweden 20

21 RegretNet: Learning Problem g w, p w where w are parameters in Neural Nets Learning problem: min w L gw, p w = E t [ i p i w t ] s.t. i n, rgt i w = 0, irp i w = 0, bcp i w = 0 Train via augmented Lagrangian Method to handle the IR, BC and regret constraints w t+1 argmin[l g w, p w + λ i rgt w i + ρ w 2 w rgt 2 i + ] i i (inner optimization) i n, λ t+1 i λ t w i + ρ rgt t+1 i Adaptively tune Lagrange multiplier. In our case, λ i always increases and we fix ρ > 0. 7/11/2018 AAMAS'18, Stockholm, Sweden 21

22 RegretNet: Empirical Optimization Randomly sample L type profiles from distribution, S = {t 1, t (L) }. Empirically estimate መL g w, p w = 1 σ L l p w t l, similarly for irp and bcp. 7/11/2018 AAMAS'18, Stockholm, Sweden 22

23 RegretNet: Empirical Optimization Randomly sample L type profiles from distribution, S = {t 1, t (L) }. Empirically estimate መL g w, p w = 1 σ L l p w t l, similarly for irp and bcp. To estimate regret, rgt i w = 1 L l [max t i I p i b i [u i t i, t i l Run SGD for inner optimization problem u i (t l )]] Max over additional fixed samples for each type profile, generated from uniform distribution. 7/11/2018 AAMAS'18, Stockholm, Sweden 23

24 Experiments: Can RegretNet recover known auction designs? 7/11/2018 AAMAS'18, Stockholm, Sweden 24

25 DSIC mechanism: 1-item, 1-bidder v U 0,1, b U 0,1 [Che & Gale, 2000] 7/11/2018 AAMAS'18, Stockholm, Sweden 25

26 DSIC mechanism: 1-item, 1-bidder v U 0,1, b U 0,1 [Che & Gale, 2000] RegretNet Optimal 7/11/2018 AAMAS'18, Stockholm, Sweden 26

27 DSIC mechanism: 1-item, 2-bidders v i Unif 1,2,, 10, one is unconstrained, other: b 2 = 4 [Malakhov & Vohra, 2008] 7/11/2018 AAMAS'18, Stockholm, Sweden 27

28 BIC mechanism: 1-item, 2-bidders v i Unif 1,2,, 10, one is unconstrained, other: b 2 = 4 [Malakhov & Vohra, 2008] 7/11/2018 AAMAS'18, Stockholm, Sweden 28

29 BIC mechanism: 1-item, 2-bidders 1-item, 2-bidders, v i U 0,1, b i U{0.22, 0.42} [Pai & Vohra, 2014] 7/11/2018 AAMAS'18, Stockholm, Sweden 29

30 Experiments: Can RegretNet discover new auction designs? 7/11/2018 AAMAS'18, Stockholm, Sweden 30

31 DSIC mechanism: 4-units, 2-bidders v i U 0,1, b i U[0,1] 7/11/2018 AAMAS'18, Stockholm, Sweden 31

32 DSIC mechanism: 2-item, 2-bidders v i Unif{1,2,, 10}, b i Unif{1,2,, 4}, unitdemand 7/11/2018 AAMAS'18, Stockholm, Sweden 32

33 Conclusion Extend RegretNet framework in [Dutting, ሷ Feng, Narasimhan, Parkes, 2017] to revenue-optimal auctions with private budgets (multi-item setting). Generalize RegretNet to BIC setting. Almost recover the optimal auctions where analytical result exists. For the settings where there is no theoretical analysis, our RegretNet outperforms well-known baselines. 7/11/2018 AAMAS'18, Stockholm, Sweden 33

34 Thanks! 7/11/2018 AAMAS'18, Stockholm, Sweden 34