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Online Mechanisms
René van TwistJaap Landheer
1
Lecture 2 (part 1/2)
maandag 6 juni 2011
Overview
• Adaptive Limited-supply Market
• Efficiency and revenue
• Ex-Post IC
• Simple price based auction
2
maandag 6 juni 2011
Adaptive Limited-Supply Market
• One indivisible item
• agents with type
• assume: no early-arrival misreports
• goal: good revenue and efficiency properties
N θi = (ai ,di ,wi )∈ T × T × >0
3
maandag 6 juni 2011
Secretary problem• known number off applicants in turn.
• Each applicant have a quality.
• Interviewer needs to learn the relative rank of each applicant.
• Irrevocable decision to hire or not.
• Goal: hire the best applicant
4
N
maandag 6 juni 2011
Secretary problem
• Random ordering hypothesis:you can select a subset of applicants to inspect but you cannot control the assignment of quality to applicant.
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maandag 6 juni 2011
Secretary problem
• Method: - First interview the first t - 1 applicants- Then hire the next candidate with quality equal or better in the first part.
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1j −1j= t+1
N∑ ≤ 1 < 1j −1j= t
N∑t −1 = N
e⎢⎣
⎥⎦
maandag 6 juni 2011
Auction Problem
• Modify the adversarial model in the secretary problem.
• Look at the first bids (the learning phase)
• Sell the item to the agent that report a greater value then the greatest seen so far.
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Ne⎢
⎣⎥⎦
maandag 6 juni 2011
Truthful?• Six agents, transition after 2 bids.
If all agents are truthful wins in period 6 with payment 6. If reports then it wins in period 5 with payment 4.
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θ1 = (1, 7,6) θ2 = (3, 7,2) θ3 = (4,8, 4)θ4 = (6, 7,8) θ5 = (9,10,3) θ6 = (10,11,9)
θ4
θ1 (5, 7,6)
maandag 6 juni 2011
Simple Variation• Bid of an agent is a claim of its type.
• Let be the top two bits so far.
• Transition between phase happens after the th bid, call the period .
• If an agent bidding is still present sell it to him for price , otherwise sell it to the next agent bidding at least for price .
• Break ties at random.
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p ≥ q
p
pq
p
N e⎢⎣ ⎥⎦ τ
maandag 6 juni 2011
Strongly truthful• Theorem 16.21: The auction is strongly truthful in
the single unit, limited supply environment with no early-arrival misreports.
Proof by: case analysis: - if - if , wins, don’t trigger transition, no tie- if , wins, - if , wins triggers transition,
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di < τai ≤ τ ≤ diτ < ai p < wiai ≤ τ ≤ di q < wi
maandag 6 juni 2011
Revenue • An online market is c-competitive for
revenue if:
: set of inputs available for the adversary : second-highest value in type profile
11
minz∈Z
ΕRev(p(θz ))R*(θz )
⎧⎨⎩
⎫⎬⎭≥1c
Rev(p(θ)) = pi (θ)i∑
ZR*(θ) θ
maandag 6 juni 2011
Efficiency and Revenue • Theorem 16.23: The auction is -
competitive in efficiency and -competitive for revenue in the single-unit, limited supply environment in the limit as .
12
e + o(1)e2 + o(1)
N →∞
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Proof efficiency
• The competitive ratio is at least as great as the probability of selling to the highest value agent.
• The probability of selling to highest value agent at
the transition is at least:
• The probability of selling to highest value agent after the transition is at least
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N e⎢⎣ ⎥⎦N
= 1e − o(1)
1 e − o(1)
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Proof Revenue• Ratio at least as great as the probability of selling
to the highest value agent for the price of the second highest value agent.
• For selling at the transition the probability is: (Chance that they both bid before )
• For selling after the transition the probability is: which is at least
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1 e( )2 − o(1) τ
1 e( ) ⋅ 1−1 e( ) − o(1) 1 e( )2 − o(1)
maandag 6 juni 2011
Ex-Post IC
• A Mechanism is ex-post IC (EPIC) if truth revelation is a best-response contingent on other agents being truthful, and whatever the types of other agents are.
• Monotonicity provides EPIC but not neccesairly DSIC
15
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Simple price-based online auction
• Define an agent-independent price schedule: to agent i in period t.
• Interesting decision set
• Stochastic events
• Payment
• If agent i gets allocated by the decision policy in period t then the following must hold:
16
qit (L,θ− i ,ω )∈
L ∈Liω ∈Ω
p(ai ,di ,L i )(θ− i ,ω ) = mint∈[ai ,di ] qi
t (Li ,θ− i ,ω )
θi = (ai ,di ,(ri ,Li )) ri ≥ qit (Li ,θ− i ,ω )
maandag 6 juni 2011
Simple price-based online auction
• This approach isn’t completely general:critical-value: wins in period 2But or don’t win.
This policy would require:
17
v(1,2)c (θ− i ) = 20 v(1,1)
c (θ− i ) = v(2,2)c (θ− i ) = 30
θi = (1,2,25)θi ' = (1,1,28) θi ' = (2,2,28)
qi1(θ− i ) > 28 ∧ qi
2 (θ− i ) > 28 ∧min(qi
1(θ− i ),qi2 (θ− i )) ≤ 25
maandag 6 juni 2011
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