Optimal dynamic mechanisms with ex-post IR via bank accounts Vahab Mirrokni 1 Renato Paes Leme 1 Pingzhong Tang 2 Song Zuo 2 1 Google Research 2 Tsinghua University AdAuction, 2016, Maastricht Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 1 / 20
Dynamic Environment: An Informal Description The seller has a sequence of items to sell to a single buyer. The items arrive over time. At each stage, there is one item for sale 1 . The item will be destroyed at the end of this stage, if not sold. Nobody knows the actual value of the t -th item until the beginning of the t -th stage. Stage-wise independent valuations, commonly known priors. Linear utility functions. The seller’s allocation rule and payment rule could depend on past stages. 1 All our results apply to the multi-item per stage case. Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 2 / 20
An Application: Ads Google sells ad impressions to advertisers. Impressions may come from users’ searches on search engines. (Arrive over time, destroyed immediately if not sold.) The value of each impression varies with (at least) the user’s information (location, time, age, gender, cookies, etc.). Currently, the auctions are rarely conducted dynamically. Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 3 / 20
Auction-based and Contract-based Advertising auction-based contract-based dynamic bundling real-time + real-time higher revenue higher revenue lack of competition complicated + high entering cost lower revenue not real-time commitment power We introduce a family of simple dynamic mechanisms — coined bank account mechanisms — to get around these two issues. Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 4 / 20
Static vs Dynamic vs Bank Account Select Seller: select stage mechanism M t Next stage Report Buyer: observe v t Stage and report it to M t Mech Outcome: z t (v t ), q t (v t ) Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 5 / 20
Static vs Dynamic vs Bank Account Select Seller: select stage mechanism M t Next stage Affect History Report Buyer: observe v t Stage and report it to M t Mech Outcome: z t (v t ), q t (v t ) Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 6 / 20
Static vs Dynamic vs Bank Account Select Seller: select stage mechanism M t Affect Bank Account Balance Report Buyer: observe v t Stage and report it to M t Mech Go to next stage Outcome: z t (v t ), q t (v t ) Buyer: deposit d t Add money Take money Seller: spend s t Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 7 / 20
Static vs Dynamic vs Bank Account Select Seller: select stage mechanism M t Affect Bank Account Balance Report Buyer: observe v t Stage and report it to M t Mech Go to next stage Outcome: z t (v t ), q t (v t ) Buyer’s Money Buyer: deposit d t Add money REVENUE Take money Seller: spend s t Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 8 / 20
A Toy Example Example One buyer, two stages, i.i.d. valuations: v 1 , v 2 ∼ F . F : Pr[ v = 1] = Pr[ v = 2] = 1 / 2. Bank Account Mechanism Dynamic Mechanism Seller sets M 1 = posted-price at 1; v 1 = 1: posted-price p 1 = 1, p 2 = 2 Buyer reports its value v 1 , v 1 = 2: posted-price and deposits 0 . 5, if v 1 = 2; p 1 = 1 . 5, p 2 = 1 if balance = 0 . 5, Seller spends 0 . 5, and sets M 2 = posted-price at 1; Revenue = 2 . 25 otherwise, M 2 = posted-price at 2. BIC & ex-post IR Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 9 / 20
Dynamic Bundle Revenue Comparison auction-based dynamic contract-based 2 2 . 25 3 Bank Account Mechanism interpretation: upon Buyer reporting, Seller sells a “dynamic bundle”, item + item(s) item + future benefits static bundle dynamic bundle “future benefits” sold via spends, implemented as discounts in the next stage mechanism. Selling “future benefits” brings the uncertainty of deficits . Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 10 / 20
Constraints on Dynamic Bundles item + item(s) item + future benefits static bundle dynamic bundle The “future benefits” must satisfy certain properties to ensure the mechanism being incentive compatible (IC) and individually rational (IR). Dominant Strategy IC: no difference with the static environment. [Mirrokni, et al. IJCAI’16]: Bayesian IC + interim IR. This paper: Bayesian IC + ex-post IR. [Papadimitriou, et al. SODA’16]: first paper on this setting, discrete and correlated types, focus on complexity. [Ashlagi, et al. EC’16]: the same setting, independent work. Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 11 / 20
Results overview This paper, ex-post IR IJCAI 2016, interim IR 1 Reduction: dynamic to BAM 1 Reduction: dynamic to BAM 2 Closed-form 3-approx BAM 2 Trade-offs: deficits vs revenue 3 Deterministic 5-approx BAM 3 Practical subset: DRA 4 Computation of optimal BAM 4 Computation of opt DRA FPTAS for discrete types 5 Empirical: optimal vs heuristic 5 more ... Rest of this talk IC IR constraints, BAM: simple and optimal Closed-form 3-approx BAM Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 12 / 20
Bayesian IC and ex-post IR Select Seller: select stage mechanism M t Affect IC ① : M t is IC Bank Account Balance Report Buyer: observe v t Stage and report it to M t Mech Go to next stage Outcome: z t (v t ), q t (v t ) IR: d t ≤ utility of M t Buyer: deposit d t Add money Take money Seller: spend s t IC ② : ∆s t = ∆utility of M t+1 Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 13 / 20
Bank Accounts — Simple yet Powerful Overcome the issues of general dynamic mechanisms Complicated − → simple structure (static mechanism + bank account) Commitment power − → ex-post IR without loss of generality Theorem (Optimal revenue is achievable) For any dynamic mechanism (full history) M, there is a (constructive) bank account mechanism B that is as good as M for the buyer Utl ( B ) = Utl ( M ) , and is (weakly) better than M for the seller Rev ( B ) ≥ Rev ( M ) . Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 14 / 20
3-approx BAM: simple & closed-form Theorem (3-approximation BAM) For any type distribution, there is a simple and closed-form BAM that 3 -approximates the optimal revenue. Theorem (Deterministic 5-approximation BAM) For any type distribution, there is a simple and deterministic BAM that 5 -approximates the optimal revenue. Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 15 / 20
Proof: 3-approximation BAM � � � Revenue = q t + s t t each stage: payment spend For each stage: Payment: q t ≤ Myerson revenue Spend: two upper bounds s t ≤ current balance — no over spend s t ≤ range of expected utility of M t +1 — IC 2 � : ∆ s t = ∆expected utility of M t +1 3-approx BAM = 1 3 (mech 1 + mech 2 + mech 3) mech 1: payment mech 2: deposit mech 3: utility range Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 16 / 20
Proof: 3-approximation BAM Cont’d 3-approx BAM = 1 3 (mech 1 + mech 2 + mech 3) Mech 1, payment: Myerson mechanism Mech 2, deposit: give-for-free mechanism d g t = v t Mech 3, utility range: mechanism with given utility expected utility of M p t +1 = s t Closed form construction for each Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 17 / 20
Summary Bank Account Mechanism: Simple and without loss of generality Efficient computation of optimal Closed-form/deterministic constant approximately optimal More insights ... Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 18 / 20
References Mirrokni, Vahab, Renato Paes Leme, Pingzhong Tang, and Song Zuo. “Dynamic auctions with bank accounts.” Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI). (2016). Papadimitriou, Christos, George Pierrakosm, Christos-Alexandros Psomas, and Aviad Rubinstein. “On the complexity of dynamic mechanism design.” Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms . SIAM, 2016. Ashlagi, Itai, Constantinos Daskalakis, and Nima Haghpanah. “Sequential Mechanisms with ex-post Participation Guarantees.” arXiv preprint arXiv:1603.07229 (2016). Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 19 / 20
Thanks for your attention! Thanks! & Questions? Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 20 / 20
Recommend
More recommend
