some extensions to mechanism design -...

Introduction to Mechanism Design and AuctionsState of the Art

An Optimal Multi Unit Combinatorial AuctionConclusion and Future Work

Some Extensions to Mechanism Design

Sujit Prakash Gujar

Thesis Supervisor : Y Narahari

[email protected] Lab

Department of Computer Science and AutomationIndian Institute of Science

Bangalore-12

January 19, 2008

Sujit Prakash Gujar (CSA, IISc) Some Extensions to Mechanism Design January 19, 2008 1 / 31



Thesis Overview

Extending mechanism design to general situations motivated by realworld problems

Mechanism design with heterogeneous objects (multi dimensionalprivate information) is formidable challenge

Reducing budget imbalance1

Optimal multi unit combinatorial auction

1H. Moulin. “Efficient, strategy-proof and almost budget-balanced assignment”.Technical report, 2007.




Publications Based On Work

“Almost Budget Balanced Mechanism Design”, Sujit Gujar and YNarahari, Working Paper.

“Foundations of Mechanism Design: A Tutorial, Part 1: KeyConcepts and Classical Results”, Dinesh Garg, Y Narahari and SujitGujar. Sadhana - Indian Academy Proceedings in EngineeringSciences, to appear 2008.

“Foundations of Mechanism Design: A Tutorial, Part 2: AdvancedConcepts and Results”, Dinesh Garg, Y Narahari and Sujit Gujar.Sadhana - Indian Academy Proceedings in Engineering Sciences, toappear 2008.

“An Optimal Multi-Unit Combinatorial Procurement Auction withSingle Minded Bidders”, Sujit Gujar and Y Narahari, Submitted toManaging Complexity in Distributed World, MCDES 2008.




Agenda

1 Introduction to Mechanism Design and AuctionsMechanism DesignAuctions

2 State of the ArtMyerson’s WorkOptimal Auctions Beyond Myerson

3 An Optimal Multi Unit Combinatorial AuctionAssumptionsNotationNecessary and Sufficient Conditions for Bayesian IncentiveCompatibility and Individual RationalityAn Optimal AuctionAn Optimal Auction : Under Regularity Assumption

4 Conclusion and Future Work




Mechanism DesignAuctions

Agenda









Mechanism Design

Mechanism Design is the art of designing rules of a game to achievea specific outcome in presence of multiple self-interested agents,each with private information about their preferences.





Agenda









Auctions

First Price Auction (FPA) for selling a single item.

The bidder with the highest bid wins.She pays as much as she has bid for.

Second Price Auction (SPA) for selling a single item.

The bidder with the highest bid wins.She pays as much as the second highest bid.

Vickrey 2 showed : The truth revelation is dominant strategy insecond price auction.

2W. Vickrey. Counter speculation, auctions, and competitive sealed tenders.Journal of Finance, 16(1):8-37, March 1961.





Auctions


The bidder with the highest bid wins.

She pays as much as she has bid for.









Auctions











Auctions




The bidder with the highest bid wins.

She pays as much as the second highest bid.







Auctions











Space of Mechanisms




Myerson’s WorkOptimal Auctions Beyond Myerson

Agenda









Optimal Auction

Myerson3: Introduced the notion of “Optimal auction”

Maximizes revenue to the sellerSatisfies interim individual rationalityBayesian incentive compatibility

3R. B. Myerson. Optimal auction design. Mathematics of Operations Research,6(1):58-73, February 1981





Optimal Auction


Maximizes revenue to the seller

Satisfies interim individual rationalityBayesian incentive compatibility






Optimal Auction


Maximizes revenue to the sellerSatisfies interim individual rationality

Bayesian incentive compatibility






Optimal Auction


Maximizes revenue to the sellerSatisfies interim individual rationalityBayesian incentive compatibility






Agenda









Optimal Auctions Beyond Myerson

Armstrong, M. Optimal multi-object auctions. Review of EconomicStudies 67, 3 (July 2000), 455-81.

Kumar, A., and Iyengar, G. Optimal procurement auctions fordivisible goods with capacitated suppliers. Tech. rep., ColumbiaUniversity, 2006. Technical Report TR-2006-01.

Gautam, R. K., Hemachandra, N., Narahari, Y., and Prakash, H.Optimal auctions for multi-unit procurement with volume discountbids. Proceedings of IEEE Conference on E-Commerce Technology,CEC-2007, Tokyo, Japan (2007), 21-28.

Ledyard, J. O. Optimal combinatoric auctions with single-mindedbidders. In EC’07: Proceedings of the 8th ACM conference onElectronic commerce (New York, NY, USA, 2007), ACM Press, pp.237-242.




AssumptionsNotationNecessary and Sufficient Conditions for Bayesian Incentive Compatibility and Individual RationalityAn Optimal AuctionAn Optimal Auction : Under Regularity Assumption

Optimal Multi Unit CombinatorialAuction in the Presence of Single

Minded, Capacitated Bidders





Agenda









Assumptions

1 The sellers are single minded

2 The sellers can collectively fulfill the demands specified by the buyer

3 The sellers are capacitated

4 The seller will never inflate his capacity(This is an important assumption)

5 All the participants are rational and intelligent





Agenda









Notation

I Set of items the buyer is interested in buying, {1, 2, . . . , m}Dj Demand for item j , j = . . . m

N Set of sellers. {1, 2, . . . , n}ci True cost of production of one unit of bundle of interest to the seller i ,

ci ∈ [ci , ci ]

qi True capacity for bundle which seller i can supply, qi ∈ [qi , qi ]

ci Reported cost by the seller i

qi Reported capacity by the seller i

bi Bid of the seller i . bi = (ci , qi )

b Bid vector, (b1, b2, . . . , bn)

b−i Bid vector without the seller i , i.e. (b1, b2, . . . , bi−1, bi+1, . . . , bn)

ti (b) Payment to the seller i when submitted bid vector is b

Ti (bi ) Expected payment to the seller i when he submits bid bi .Expectation is taken over all possible values of b−i





Notation

xi = xi (b) Quantity of the bundle to be procured from the seller iwhen the bid vector is b

Xi (bi ) Expected quantity of the bundle to be procured from the seller iwhen he submits bid bi .Expectation is taken over all possible values of b−i

fi (ci , qi ) Joint probability density function of (ci , qi )

Fi (ci , qi ) Cumulative distribution function of fi (ci , qi )

fi (ci |qi ) Conditional probability density function of production costwhen it is given that the capacity of the seller i is qi

Fi (ci |qi ) Cumulative distribution function of fi (ci |qi )

Hi (ci , qi ) Virtual cost function for seller i ,

Hi (ci , qi ) = ci + Fi (ci |qi )fi (ci |qi )

ρi (bi ) Expected offered surplus to seller i , when his bid is bi

Table: Notation





Agenda









Incentive Compatibility

× Sellers may not be willing to reveal their true types.

√Offer them incentives for reporting true costs and capacities.

We propose the following incentive structure, ∀i ∈ N,

ρi (bi ) = Ti (bi )− ciXi (bi ), where bi = (ci , qi )





Incentive Compatibility

× Sellers may not be willing to reveal their true types.√Offer them incentives for reporting true costs and capacities.

We propose the following incentive structure, ∀i ∈ N,

ρi (bi ) = Ti (bi )− ciXi (bi ), where bi = (ci , qi )





Necessary and Sufficient Conditions for Bayesian IncentiveCompatibility and Individual Rationality

We proved,

Theorem 1

Any mechanism in the presence of single minded, capacitated sellers isBIC and IR iff

1 ρi (bi ) = ρi (ci ,qi ) +∫ ci

ciXi (t, qi )dt

2 ρi (bi ) non-negative, and non-decreasing in qi ∀ ci ∈ [ci , ci ]

3 The quantity which seller i is asked to supply, Xi (ci , qi ) isnon-increasing in ci ∀qi ∈ [qi , qi ].





Agenda









An Optimal Auction

The buyer’s problem is to solve,

min Eb

∑ni=1 ti (b) s.t.

1 ti (b) = ρi (b) + cixi (b)

2 All three conditions in Theorem 1 hold true.

3 She procures at least Dj units of each item j .

We have shown that an optimal auction for the buyer in the presence ofthe single minded sellers is,

Optimal Auction

min∫ q

q

∫ c

c

(∑ni=1 Hi (ci , qi )xi (ci , qi )

)f (c , q)dc dq s.t.

1. ∀; i , Xi (ci , qi ) is non-increasing in ci ,∀ qi .2. The Buyer’s minimum requirement of each item is satisfied.





Agenda









Regularity Assumption

Regularity Assumption

Hi (ci , qi ) = ci +Fi (ci |qi )

fi (ci |qi )

is non-increasing in qi and non-decreasing in ci .





An Optimal auction : Under regularity Assumption

The buyer’s optimal auction is,

minn∑

i=1

xiHi (ci , qi ) subject to

1 0 ≤ xi ≤ qi

2 Buyer’s demands are satisfied.

The buyer pays each seller i the amount

ti = cix∗i +

∫ ci

ci

xi (t, qi )dt (1)

where x∗i is what agent i has to supply after solving the above problem.Note: This auction enjoys Dominant Strategy Incentive Compatibility.




Summary So Far...

We have seen,

Necessary and sufficient conditions for BIC and individual rationality

Characterization of an optimal multi unit combinatorial procurementauction in the presence of single minded capacitated bidders

An optimal auction, for the same, which is dominant strategyincentive compatible if some regularity condition holds true




Directions for Future Work

Design of an optimal combinatorial auction for more general settingis a formidable challenge

Extending Moulin’s work for heterogeneous objects

Repeated/Stochastic mechanism design




Questions?




Thank You!!!


some extensions to mechanism design -...

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