an introduction to auction and mechanism design

An Introduction to Auctionand Mechanism Design

Yitian Chen, IPC [email protected]

November 21, 2014

Yitian Auction and Mechanism Design November 21, 2014 1 / 24

Outline

1 Warm-up For Mechanism Design

2 Mechanism Design in Advertising

3 Apply to Supply Chain Management?

4 Question & AnswerSusan Atheyhttp://www.gsb.stanford.edu/faculty-research/faculty/susan-athey

Warming-up

Outline

1 Warm-up For Mechanism DesignNegotiation vs Pricing vs AuctionWhen and why to choose AuctionEquation-based Model vs Agent-based Model

4 Question & Answer

Warming-up Negotiation vs Pricing vs Auction

Methods to sell an Item(Items)

Negotiation

Pricing

Auction

Warming-up When and why to choose Auction

When and Why to Auction

When to Auction [1]A seller has multiple potential buyersBuyers are symmetric and have independent valueBoth seller and buyers are not certain about the true value

Why to Auction [2]More profitsLess labor cost(Compare to Negotiation)

Auction and Mechanism DesignA seller has a Objective Function(Revenue, Social Value,Fairness...)A seller has a selling methods list(Pricing, Negotiation, First Price,Second Price, VCG)Mechanism Design: How to choose(create) selling method tomaximize Objective Function

Warming-up Equation-based Model vs Agent-based Model

Model thinking: from the point of Model

Equation-Based ModelMaximize the outcome(Objective function)Eg.. Generalized Linear Model; Decision Tree...SupervisedLearning

Agent-Based Modelindividuals: Agents interact with each other(cooperate, compete)behaviours: Eg..Every agent is try to maximise to maximize itsprofitoutcomes: Eg.. Seller is trying to maximize its revenue(Paretooptimization)

Advertising Mechanism

Outline

2 Mechanism Design in AdvertisingGeneralized First PriceGeneralized Second Price

Locally envy-free Nash EquilibriumThe beautiful VCG Auction

4 Question & Answer

Advertising Mechanism First price

A laptop example

Generalized first price auction

Bidder Bid Price OrderDell 6 6 1stJD 4 4 2ndLeno 2 2 3rdMRP 0.5

The problem of GFP

Advertising Mechanism Second price

Generalized second price

Bidder Bid Price OrderDell 6 4 1stJD 4 2 2ndLeno 2 0.5 3rdMRP 0.5

GSP Auction rules

Most search engines use Ecpm bid ∗ CTRinstead of BidEach advertiser i submits bid Bidi

Positions assigned in order of Bidi ∗ CTRi

Advertiser i ’s price Pi isBidi+1 ∗ CTRi+1/CTRi

Advertising Mechanism Equilibrium of GSP

Truthful?–Bidding the value

Bidder Value Bid Price Clicks Payoff OrderDell 6 6 4 200 (6− 4) ∗ 200 = 400 1stJD 4 4 2 120 (4− 2) ∗ 120 = 240 2ndLeno 2 2 0.5 50 (2− 0.5) ∗ 50 = 75 3rdMRP 0.5

Truthful?–If Dell Bidding 3...

Bidder Value Bid Price Clicks Payoff OrderJD 4 4 3 200 (4− 3) ∗ 200 = 200 1stDell 6 3 2 120 (6− 2) ∗ 120 = 480 > 400 2ndLeno 2 2 0.5 50 (2− 0.5) ∗ 50 = 75 3rdMRP 0.5

Nobody could improve its profit by exchanging its positionWe call this kind of equilibrium as Locally Envy-free NE

Locally Envy-free NE

Definition: An equilibrium is locallyenvy-free if no player can improve hispayoff by exchanging positions

Motivation: squeezing – if an equilibriumis not locally Envy-free NE, there might bean incentive to squeeze.

Inequality of LEF-NE(Vk − Pk ) ∗ Xk ≥ (Vk − Pk−1 ∗ Xk−1)No incentive to move below(Vk − Pk ) ∗ Xk ≥ (Vk − Pk+1 ∗ Xk+1)No incentive to move above

Hal Varianhttp://people.ischool.berkeley.edu/~hal/

Advertising Mechanism The beautiful VCG Auction

Think Second price with social value

Bidder Value Clicks social valueDell 6 200 ?JD 4 120 ?

Assume there is only one ad slot for sales.Dell participate in the auction, Dell win the slots and the totalvalue = 6 ∗ 200 = 1200, others’s social value is 0If Dell does not participate in the auction, JD win the slots and thetotal social value = 4 ∗ 200 = 800the hurt to others’s 800− 0 = 800, the price is the average hurt800/200 = 4

The VCG Auction

Definition:Wv = total value received byagents, where v is the vector of values

Payment rule: VCG payment of is W−i(0,X−i) - W−i (x) = value to others when ibids 0(does not bid), minus that when ibids truthfully(externality to others).Important: VCG is socially optimal-maximize the social welfare.

Vickreyhttp://en.wikipedia.org/wiki/William_Vickrey

Apply to Supply Chain Management?

Outline

3 Apply to Supply Chain Management?Procurement ProblemDynamic graph

4 Question & Answer

Apply to Supply Chain Management? Procurement Problem

ICP team is going to purchase 100 cupsSupplier Supply Num Unit price Sold partlyAda 80 2 YesBob 100 3 NoCarol 60 5 Yes

The outcome is determine by the bid:Supplier Ada and Carol win the auctionAda gets 80 cups supply and Carol gets 20cups supply (80 ∗ 2 + 5 ∗ 20) ≤ (3 ∗ 100)Price of Ada: If Ada does not participate,Bob wins the auction, the total social value is100 ∗ 3 = 300if Ada participate, others’ social value: Carol20*5 = 100, the social value hurt to others is(300-200) price = (300-100)/80=2.5Price of Carol : (3*100-80*2)/20=7

Apply to Supply Chain Management? Dynamic graph

Buying a path in a network

Many problem can be modelled as a directedgraph G = (V ,G).Each link E 3 e has a cost Ce ≥ 0Each edge owner can submit a bid .We want to buy a path between twovertices(s, t).Each edge owner havs incentive to changethe cost.

Buy a Path from a to e

Solution with VCG Auction

Find the shortest path(a− b − d − e)(Dijkstra Algorithm)If a− b does not participate, choose(a− d − e)The social value when (a− b) doesnot participate 7+4=11The others’ social value when a− bparticipate (b − d) and (d − e) 2+4=6the price of edge (a− b) 11-6=5The solution maximise the socialvalue(buyers and sellers’ total profit)

Question & Answer

Outline

4 Question & Answer

Question & Answer

References

[Auction Theory, 2002] Vijay Krishna

[Auction Versus Negotitations] Jeremy Bulow and Paul KlempererThe American Economic Review, Vol. 86, No. 1. (Mar., 1996), pp.180-194.

an introduction to auction and mechanism design

Economy & Finance