Optimal dynamic mechanisms with ex-post IR via bank accounts Vahab - - PowerPoint PPT Presentation
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
Vahab Mirrokni1 Renato Paes Leme1 Pingzhong Tang2 Song Zuo2
1Google Research 2Tsinghua University
AdAuction, 2016, Maastricht
Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 1 / 20
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 sale1. 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.
1All our results apply to the multi-item per stage case. Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 2 / 20
Google sells ad impressions to advertisers. Impressions may come from users’ searches on search engines. (Arrive
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 dynamic contract-based real-time real-time + bundling higher revenue higher revenue lack of competition complicated + high entering cost lower revenue commitment power not real-time 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
Seller: select stage mechanism Mt Buyer: observe vt and report it to Mt
Stage Mech Select Report Outcome: zt(vt), qt(vt)
Next stage
Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 5 / 20
Seller: select stage mechanism Mt Buyer: observe vt and report it to Mt
Stage Mech Select Report Outcome: zt(vt), qt(vt)
Next stage
Affect History
Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 6 / 20
Seller: select stage mechanism Mt Seller: spend st Buyer: deposit dt Buyer: observe vt and report it to Mt
Stage Mech Select Report Outcome: zt(vt), qt(vt) Add money Take money Affect
Go to next stage
Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 7 / 20
Seller: select stage mechanism Mt Seller: spend st Buyer: deposit dt Buyer: observe vt and report it to Mt
Stage Mech Select Report Outcome: zt(vt), qt(vt) Add money Take money Affect
Go to next stage
REVENUE Buyer’s Money
Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 8 / 20
Example
One buyer, two stages, i.i.d. valuations: v1, v2 ∼ F. F: Pr[v = 1] = Pr[v = 2] = 1/2.
Dynamic Mechanism
v1 = 1: posted-price p1 = 1, p2 = 2 v1 = 2: posted-price p1 = 1.5, p2 = 1 Revenue = 2.25 BIC & ex-post IR
Bank Account Mechanism
Seller sets M1 = posted-price at 1; Buyer reports its value v1, and deposits 0.5, if v1 = 2; if balance = 0.5, Seller spends 0.5, and sets M2 = posted-price at 1;
Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 9 / 20
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
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
This paper, ex-post IR
1 Reduction: dynamic to BAM 2 Closed-form 3-approx BAM 3 Deterministic 5-approx BAM 4 Computation of optimal BAM
FPTAS for discrete types
5 more ...
IJCAI 2016, interim IR
1 Reduction: dynamic to BAM 2 Trade-offs: deficits vs revenue 3 Practical subset: DRA 4 Computation of opt DRA 5 Empirical: optimal vs heuristic
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
Seller: select stage mechanism Mt Seller: spend st Buyer: deposit dt Buyer: observe vt and report it to Mt
Stage Mech Select Report Outcome: zt(vt), qt(vt) Add money Take money Affect
Go to next stage
IC①: Mt is IC IC②: ∆st = ∆utility of Mt+1 IR: dt ≤ utility of Mt
Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 13 / 20
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
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
Revenue =
+ st
payment spend
For each stage: Payment: qt ≤ Myerson revenue Spend: two upper bounds
st ≤ current balance — no over spend st ≤ range of expected utility of Mt+1 — IC 2 : ∆st = ∆expected utility of Mt+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
3-approx BAM = 1 3 (mech 1 + mech 2 + mech 3) Mech 1, payment: Myerson mechanism Mech 2, deposit: give-for-free mechanism dg
t = vt
Mech 3, utility range: mechanism with given utility expected utility of Mp
t+1 = st
Closed form construction for each
Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 17 / 20
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
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
Mirrokni, Paes Leme, Tang, Zuo Bank Account Mechanism AdAuction, 2016, Maastricht 20 / 20