A truthful Multi Item-Type Double-Auction Mechanism. Erel Segal-Halevi with Avinatan Hassidim Yonatan Aumann

Transcription

1 A truthful Multi Item-Type Double-Auction Mechanism Erel Segal-Halevi with Avinatan Hassidim Yonatan Aumann

2 Intro: one item-type, one unit Buyers: Value Sellers: Erel Segal-Halevi et al 3 Multi Item Double Auction

3 Intro: one item-type, one unit Buyers: k=5 efficient deals Value Sellers: Erel Segal-Halevi et al 4 Multi Item Double Auction

4 Intro: one item-type, one unit Buyers: k=5 efficient deals Value Gain from trade: Sellers: Erel Segal-Halevi et al 5 Multi Item Double Auction

5 Price Equilibrium k=5 efficient deals Price Erel Segal-Halevi et al 6 Multi Item Double Auction

6 Price Equilibrium Maximum gain k=5 efficient deals Price Erel Segal-Halevi et al 7 Multi Item Double Auction

7 Price Equilibrium Maximum gain Handles traders with many itemtypes if they are Gross-Substitutes (= no complementarities) k=5 efficient deals Price Erel Segal-Halevi et al 8 Multi Item Double Auction

8 Price Equilibrium Maximum gain Handles traders with many itemtypes if they are Gross-Substitutes (= no complementarities) Not truthful k=5 efficient deals Price Erel Segal-Halevi et al 9 Multi Item Double Auction

9 Some related work Bayesian prior: Single-sided auction: Myerson [1981], Blumrosen and Holenstein [2008], Segal [2003], Chawla et al. [ ], Yan [2011]. Double auction: Xu et al. [2010], Loertscher et al. [2014], Blumrosen and Dobzinski [2014], Colini-Baldeschi et al. [2016]. Prior-independent: Single-sided auction: Cole and Roughgarden [2014], Dhangwatnotai et al. [2015], Huang et al. [2015], Morgenstern and Roughgarden [2015], Devanur et al. 2011], Hsu et al. [2016]. Double auction: Baliga and Vohra [2003] single-parametric agents. Prior-free: Single-sided auction: Goldberg et al. [ ], Devanur et al. [2015], Balcan et al. [ ] Double auction: McAfee [1992] Erel Segal-Halevi et al 10 Multi Item Double Auction

10 McAfee (1992) (simplified) k=5 efficient deals Buyer price Seller price Erel Segal-Halevi et al 11 Multi Item Double Auction

11 McAfee (1992) Truthful (simplified) k=5 efficient deals Buyer price Seller price Erel Segal-Halevi et al 12 Multi Item Double Auction

12 McAfee (1992) Truthful Gain: (1-1/k) of maximum (simplified) k=5 efficient deals Buyer price Seller price Erel Segal-Halevi et al 13 Multi Item Double Auction

13 McAfee (1992) Truthful Gain: (1-1/k) of maximum Only single itemtype, single-unit (simplified) k=5 efficient deals Buyer price Seller price Erel Segal-Halevi et al 14 Multi Item Double Auction

14 McAfee (1992) Truthful Gain: (1-1/k) of maximum Only single itemtype, single-unit Extensions: Babaioff et al. [ ], Gonen et al. [2007], Duetting et al. [2014] Single-parametric agents. Blumrosen & Dobzinsky [2014] - Single item-type, Gain ~ 1/48. (simplified) k=5 efficient deals Erel Segal-Halevi et al 15 Multi Item Double Auction Buyer price Seller price

15 Prior-Free Double-Auctions Tru Gain Agents Equilibrium No 1 Multi-parametric (Gross-substitute) McAfee family Yes 1-o(1) Single-parametric / Single-item-type Our goal Yes 1-o(1) Multi-parametric, multi-item-type Erel Segal-Halevi et al 16 Multi Item Double Auction

16 Prior-Free Double-Auctions Tru Gain Agents Our goal Yes 1-o(1) Multi-item-type Our current assumptions: 1. Buyers at most g item-types, gross-substitute. Sellers 1 item-type, decreasing marginal gain. 2. Large market for each item-type x, k x ; at most m units per seller; 3. Bounded variability k max / k min c 4. Generic valuations no ties. Erel Segal-Halevi et al 17 Multi Item Double Auction

17 MIDA: Multi Item Double-Auction a. Random halving. b. Equilibrium calculation. c. Posted pricing. d. Random serial dictatorship. Erel Segal-Halevi et al 18 Multi Item Double Auction

18 MIDA step a: Random Halving Erel Segal-Halevi et al 19 Multi Item Double Auction

19 MIDA step a: Random Halving Right Left Erel Segal-Halevi et al 20 Multi Item Double Auction

20 MIDA step b: Equilibrium Calculation Gross-substitute traders price-equilibrium exists. Right p L Left p R Erel Segal-Halevi et al 21 Multi Item Double Auction

21 MIDA step c: Posted Pricing Right p L Left p R Erel Segal-Halevi et al 22 Multi Item Double Auction

22 MIDA step d: Random Dictatorship In case of over-demand/supply randomize. Right p L Left p R Erel Segal-Halevi et al 23 Multi Item Double Auction

23 MIDA step d: Random Dictatorship - Order buyers randomly; - Order sellers randomly; - First buyer buys from first sellers and goes home. - Seller goes home when marginal gain < 0. Buyers Sellers Erel Segal-Halevi et al 26 Multi Item Double Auction

24 MIDA step d: Random Dictatorship - Order buyers randomly; - Order sellers randomly; - First buyer buys from first sellers and goes home. - Seller goes home when marginal gain < 0. Buyers Sellers Erel Segal-Halevi et al 27 Multi Item Double Auction

25 MIDA step d: Random Dictatorship - Order buyers randomly; - Order sellers randomly; - First buyer buys from first sellers and goes home. - Seller goes home when marginal gain < 0. Buyers Sellers Erel Segal-Halevi et al 28 Multi Item Double Auction

26 MIDA step d: Random Dictatorship - Order buyers randomly; - Order sellers randomly; - First buyer buys from first sellers and goes home. - Seller goes home when marginal gain < 0. Buyers Sellers Erel Segal-Halevi et al 29 Multi Item Double Auction

27 MIDA step d: Random Dictatorship - Order buyers randomly; - Order sellers randomly; - First buyer buys from first sellers and goes home. - Seller goes home when marginal gain < 0. Buyers Sellers Erel Segal-Halevi et al 30 Multi Item Double Auction

28 MIDA step d: Random Dictatorship - Order buyers randomly; - Order sellers randomly; - First buyer buys from first sellers and goes home. - Seller goes home when marginal gain < 0. Theorem: If each seller sells one item-type and has decreasing-marginal-gains, then MIDA is truthful. Buyers Sellers Erel Segal-Halevi et al 31 Multi Item Double Auction

29 MIDA: Estimating the gain-from-trade

30 Four ways to lose gain-from-trade p L p OPT Right Left p R Erel Segal-Halevi et al 43 Multi Item Double Auction

31 Four ways to lose gain-from-trade Efficient sellers quitting: loss for buyers p L p OPT Right Left p R Erel Segal-Halevi et al 44 Multi Item Double Auction

32 Four ways to lose gain-from-trade Efficient sellers quitting: loss for buyers p L p OPT Right Left p R Erel Segal-Halevi et al 45 Multi Item Double Auction

33 Four ways to lose gain-from-trade Efficient sellers quitting: loss for buyers Efficient buyers quitting p L p OPT Right Left p R Erel Segal-Halevi et al 46 Multi Item Double Auction

34 Four ways to lose gain-from-trade Efficient sellers quitting: loss for buyers Efficient buyers quitting Inefficient sellers competing p L p OPT Right Left p R Erel Segal-Halevi et al 47 Multi Item Double Auction

35 Four ways to lose gain (left market) For every item-type x, define: B x* buyers who want x in p OPT B x- buyers who want x in p OPT but not in p R B x+ buyers who want x in p R but not in p OPT S x* sellers who offer x in p OPT S x- sellers who offer x in p OPT but not in p R S x+ sellers who offer x in p R but not in p OPT We lose B x- + S x+ random sellers and S x- + S x+ random buyers. So: E[Loss x ] ( B x- + B x+ + S x- + S x+ ) / B x* Erel Segal-Halevi et al 48 Multi Item Double Auction

36 Bounding the loss E[Loss x ] ( B x- + B x+ + S x- + S x+ ) / k x Price-equilibrium equations: for every x: Global population: B x* = S x* = k x Right market ( R = the subset sampled to Right): B x*r + B x+r - B x-r = S x*r + S x+r - S x-r Erel Segal-Halevi et al 49 Multi Item Double Auction

37 Bounding the loss E[Loss x ] ( B x- + B x+ + S x- + S x+ ) / k x Price-equilibrium equations: for every x: Global population: B x* = S x* = k x Right market ( R = the subset sampled to Right): B x*r + B x+r - B x-r = S x*r + S x+r - S x-r Concentration bounds: w.h.p: B x*r - B x* / 2 < err x S x*r - S x* / 2 < err x Erel Segal-Halevi et al 50 Multi Item Double Auction

38 Bounding the loss E[Loss x ] ( B x- + B x+ + S x- + S x+ ) / k x Price-equilibrium + Concentration bounds: With high probability: B x- R B x+r < 2 err x S x-r - S x+r < 2 err x Erel Segal-Halevi et al 51 Multi Item Double Auction

39 Bounding the loss E[Loss x ] ( B x- + B x+ + S x- + S x+ ) / k x Price-equilibrium + Concentration bounds: With high probability: B x- R B x+r < 2 err x S x-r - S x+r < 2 err x Let s focus on the buyers. We have bounds on: B x- R B x+r We need bounds on: B x-, B x+ Erel Segal-Halevi et al 52 Multi Item Double Auction

40 Bounding the loss: step A We have bounds: B R x- B x+r < 2 err x B R 1- B 1+R < 2 err 1 B R 2- B 2+R < 2 err 2... B R g- B g+r < 2 err g We derive bounds on: B R x-, B x+r Erel Segal-Halevi et al 53 Multi Item Double Auction

41 Bounding the loss: step A We have bounds: B R x- B x+r < 2 err x We derive bounds on: B R x-, B x+r p R item g p R item 4 p R item 3 p R item 2 p R item 1 p OPT Erel Segal-Halevi et al 54 Multi Item Double Auction

42 Bounding the loss: step A We have bounds: B R x- B x+r < 2 err x We derive bounds on: B R x-, B x+r p R item g p R item 4 p R item 1 p R item 2 p R item 3 Erel Segal-Halevi et al 55 Multi Item Double Auction p OPT Theorem: The demand of gross-substitute agents moves only downwards (Segal-Halevi et al, 2016).

43 Bounding the loss: step A We have bounds: B R x- B x+r < 2 err x We derive bounds on: B R x-, B x+r p R item g p R item 4 p R item 1 p R item 2 p R item 3 Erel Segal-Halevi et al 56 Multi Item Double Auction p OPT Theorem: The demand of gross-substitute agents moves only downwards (Segal-Halevi et al, 2016).

44 Bounding the loss: step A We have bounds: B R x- B x+r < 2 err x We derive bounds on: B x-r, B x+r For every item x that became cheaper: B R 1- B 1+R < 2 err max B R 2- B 2+R < 2 err max... B R g- B g+r < 2 err max B R 1- = 0 B R 1+ < 2 err max B R 2- < 2 err max B R 2+ < 4 err max... B R g- < 2 g err max, B R g+ < 2 g err max Erel Segal-Halevi et al 58 Multi Item Double Auction

45 Bounding the loss: step B We have a bound: B R x-, B x+r < 2 g err max We need a bound on: B x-, B x+ When T is a deterministic set (like Bx* ) determined before randomization B x- and B x+ are random sets - depend on price determined after randomization! Our solution: bound the UI dimension of B x-, B x+ Erel Segal-Halevi et al 59 Multi Item Double Auction

46 UI Dimension of Random Sets UI Dimension property of a random-set. If UIDim(T) d then (Segal-Halevi et al, 2017): 1. Containment-Order Rule: If the support of T is ordered by containment, then UIDim(T) Union Rule: UIDim(T 1 U T 2 ) UIDim(T 1 ) + UIDim(T 2 ) 3. Intersection Rule: If T 1 < t then: UIDim(T 1 T 2 ) log(t)*(uidim(t 1 ) + UIDim(T 2 )) Erel Segal-Halevi et al 60 Multi Item Double Auction

47 Bounding the loss: step B We have a bound: B R x-, B x+r < 2 g err max We derive a bound on: B x-, B x+ Lemma: For every item-type x: Corollary: When k max >> 2 3g, w.h.p: B x-, B x+ < 3 * (2 g err max ) Erel Segal-Halevi et al 61 Multi Item Double Auction

48 Bounding the loss: step C We have a bound: B x-, B x+ < 3*2 g *err max Similarly: S x-, S x+ < 3*2 g *err max Lost deals in item x: < 12*(2 g err max ) Lost gain in item x < 12*(2 g err max ) / k x Lost gain overall < 12*(2 g err max ) / k min Lost gain overall < Const * o(k max ) / k min Theorem: Under large-market assumptions, gain-from-trade of MIDA approaches maximum. Erel Segal-Halevi et al 62 Multi Item Double Auction

49 Prior-Free Double-Auctions Tru Gain Agents Equilibrium No 1 Multi-parametric (Gross-substitute) McAfee family Yes 1-o(1) Single-parametric / Single-item-type MIDA Yes 1-o(1) Multi-parametric (Sellers: 1 type, Buyers: g types, Gross-substitute). Erel Segal-Halevi et al 63 Multi Item Double Auction

50 Acknowledments Game theory seminar in BIU Ron Peretz Simcha Haber Tom van der Zanden Assaf Romm Economic theory seminar in HUJI Econ.&Comp. seminar in HUJI Algorithms seminar in TAU Thank you! Erel Segal-Halevi et al 64 Multi Item Double Auction