Online Mechanisms

René van TwistJaap Landheer

1

Lecture 2 (part 1/2)

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Overview

• Adaptive Limited-supply Market

• Efficiency and revenue

• Ex-Post IC

• Simple price based auction

2

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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

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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

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N

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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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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⎢⎣

⎥⎦

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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⎢

⎣⎥⎦

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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.

8

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)

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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

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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

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Revenue • An online market is c-competitive for

revenue if:

: set of inputs available for the adversary : second-highest value in type profile

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minz∈Z

ΕRev(p(θz ))R*(θz )

⎧⎨⎩

⎫⎬⎭≥1c

Rev(p(θ)) = pi (θ)i∑

ZR*(θ) θ

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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 .

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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)

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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

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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:

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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 )

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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:

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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

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