optimal mechanism design for multi unit combinatorial...

IntroductionOptimal Mechanism Design

Conclusion

Optimal Mechanism Design forMulti Unit Combinatorial Auctions

Sujit Prakash Gujar

Supervisor : Y [email protected]

E-Commerce LabDepartment of Computer Science and Automation

Indian Institute of Science, Bangalore-12

May 29, 2008

Sujit Prakash Gujar (CSA, IISc) Optimal Combinatorial Auctions May 29, 2008 1 / 23


Conclusion

Agenda

Introduction to Mechanism Design and Auctions

Optimal Auctions

An Optimal Multi-Unit Auction with Single Minded Bidders

Summary and Future work



Conclusion

Mechanism DesignAuctionsMotivation

Mechanism Design

Game Theory: Analysis of strategic interaction among players

Mechanism Design: Reverse engineering of game theory

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.



Conclusion


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

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



Conclusion


Auctions


The bidder with the highest bid wins.

She pays as much as she has bid for.







Conclusion


Auctions









Conclusion


Auctions




The bidder with the highest bid wins.

She pays as much as the second highest bid.





Conclusion


Auctions









Conclusion




Conclusion


Motivation

general situations motivated by the real world problems

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

optimal multi unit combinatorial auction



Conclusion

State of the ArtAn Optimal Multi Unit Combinatorial AuctionAn Optimal Auction

Optimal Auction

Myerson2: Introduced the notion of “Optimal auction”

Maximizes revenue to the sellerSatisfies interim individual rationalityBayesian incentive compatibility

2R. B. Myerson. Optimal auction design. Mathematics of Operations Research,6(1):58-73, February 1981



Conclusion


Optimal Auction


Maximizes revenue to the seller

Satisfies interim individual rationalityBayesian incentive compatibility




Conclusion


Optimal Auction


Maximizes revenue to the sellerSatisfies interim individual rationality

Bayesian incentive compatibility




Conclusion


Optimal Auction


Maximizes revenue to the sellerSatisfies interim individual rationalityBayesian incentive compatibility




Conclusion


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.



Conclusion


Optimal Multi Unit CombinatorialAuction in the Presence of Single

Minded, Capacitated Bidders



Conclusion


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



Conclusion


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

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



Conclusion


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 fromthe seller i when 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



Conclusion


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 )



Conclusion


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 )



Conclusion


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 ∀ i ∈ N

1 ρi (bi ) = ρi (ci ,qi ) +∫ ci

ciXi (t, qi )dt

2 ρi (bi ) ≥ 0, 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 ].



Conclusion


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 Theorem 1 holds true.

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

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.



Conclusion


Regularity Assumption

Regularity Assumption

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

fi (ci |qi )

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



Conclusion


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.



Conclusion

SummaryDirections for Future WorkReference

Summary

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



Conclusion


From here

Directions for future work,

relax the assumption of single minded bidders

multi-unit extensions

volume discounts

efficient auctions which are DSIC extension to Krishna’s work [1]



Conclusion


Literature Review

2007 Economics Noble laureates: L. Hurwicz, E. Maskin andMyerson [2]

L. Hurwicz, “Optimality and informational efficiency in resourceallocation processes”, in Mathematical Methods in the SocialSciences. Arrow, Karlin and Suppes(eds.). Stanford UniversityPress, 1960. [3]Introduced notion of incentive compatibility [4]

E. Clarke, “Multi-part pricing of public goods”, Public Choice, vol.11, pp. 17-23, 1971.

T. Groves, “Incentives in teams”, Econometrica, vol. 41, pp.617-631, 1973.

Groves mechanism are the only DSIC mechanisms which areallocatively efficient. [5].



Conclusion


Literature Review

R. B. Myerson, “Optimal auction design,” Mathematics ofOperations Research, vol. 6, no. 1, pp. 58-73, February 1981.

E. Maskin [6] Pioneer of Implementation theory

On Optimal Auctions: [7, 1, 8, 9, 10, 11, 12]

For more about mechanism design [13, 14, 15, 16, 17]



Conclusion


V. Krishna and M. Perry.Efficient mechanism design.SSRN eLibrary, 1998.

The Nobel Foundation.The Sveriges Riksbank Prize in Economic Sciences in Memory ofAlfred Nobel 2007: Scientific Background.Technical report, The Nobel Foundation, Stockholm, Sweden,December 2007.

L. Hurwicz.Optimality and informational efficiency in resource allocationprocesses.In Mathematical Methods in the Social Sciences. Arrow, Karlin andSuppes(eds.). Stanford University Press, 1960.

L. Hurwicz.On informationally decentralized systems.In Decision and Organization. Radner and McGuire. North-Holland,Amsterdam, 1972.

J. R. Green and J. J. Laffont.Sujit Prakash Gujar (CSA, IISc) Optimal Combinatorial Auctions May 29, 2008 21 / 23


Conclusion


Incentives in Public Decision Making.North-Holland Publishing Company, Amsterdam, 1979.

E. Maskin.Nash equilibrium and welfare optimality.Review of Economic Studies, 66:23–38, 1999.

J. G. Riley and Samuelson W.F.Optimal auctions.American Economic Review, 71(3):383–92, June 1981.

M. Armstrong.Optimal multi-object auctions.Review of Economic Studies, 67(3):455–81, July 2000.

A. Malakhov and R. V. Vohra.Single and multi-dimensional optimal auctions - a networkperspective.Discussion papers, Kellogg School of Management, NorthwesternUniversity, December 2004.

A. Kumar and G. Iyengar.



Conclusion


Optimal procurement auctions for divisible goods with capacitatedsuppliers.Technical report, Columbia University, 2006.

Raghav Gautam, N. Hemachandra, Y. Narahari, and HastagiriPrakash.Optimal auctions for multi-unit procurement with volume discountbidders.In Proceedings of IEEE Conference on Electronic Commerce (IEEECEC-2007), pages 21–28, 2007.

J. O. Ledyard.Optimal combinatoric auctions with single-minded bidders.In EC ’07: Proceedings of the 8th ACM conference on Electroniccommerce, pages 237–242, New York, NY, USA, 2007. ACM Press.

R. Myerson.Mechanism design.In J. Eatwell, M. Milgate, and P. Newman, editors, The NewPalgrave: Allocation, Information, and Markets, pages 191–206.Norton, New York, 1989.



Conclusion


N. Nisan, T. Roughgarden, E. Tardos, and V. Vazerani.Algorithmic Game Theory.Cambridge University Press, 2007.

Dinesh Garg, Y Narahari, and Sujit Gujar.Foundations of mechanism design: A tutorial - part 1: Key conceptsand classical results.Sadhana - Indian Academy Proceedings in Engineering Sciences,33(Part 2):83–130, April 2008.

Dinesh Garg, Y Narahari, and Sujit Gujar.Foundations of mechanism design: A tutorial - part 2: Advancedconcepts and results.Sadhana - Indian Academy Proceedings in Engineering Sciences,33(Part 2):131–174, April 2008.

Y. Narahari, D. Garg, N. Rama Suri, and H. Prakash.Game Theoretic Problems in Network Economics and MechanismDesign Solutions.Advanced Information and Knowledge Processing Series, Springer,London, 2008 (To appear).



Conclusion


Questions?



Conclusion


Thank You!!!


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