Crowdsourcing and all pay auctions
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Crowdsourcing and All-Pay Auctions. Milan Vojnovic Microsoft Research. Lecture series – Contemporary Economic Issues – University of East Anglia, Norwich, UK, November 10, 2014. This Talk.

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Crowdsourcing and all pay auctions

Crowdsourcing and All-Pay Auctions

Milan Vojnovic

Microsoft Research

Lecture series – Contemporary Economic Issues – University of East Anglia, Norwich, UK, November 10, 2014


This talk

This Talk

  • An overview of results of a model of competition-based crowdsourcing services based on all-pay auctions

  • Based on lecture notes Contest Theory, V., course, Mathematical Tripos Part III, University of Cambridge - forthcoming book


Competition based crowdsourcing an example

Competition-based Crowdsourcing: An Example

CrowdFlower


Statistics

Statistics

  • TopCoder data covering a ten-year period from early 2003 until early 2013

  • Taskcn data covering approximately a seven-year period from mid 2006 until early 2013


Example prizes topcoder

Example Prizes: TopCoder


Example participation tackcn

Example Participation: Tackcn

contests

  • A month in year 2010

players


Game standard all pay contest

Game: Standard All-Pay Contest

  • players, valuations, linear production costs

  • Quasi-linear payoff functions:

  • Simultaneous effort investments: = effort investment of player

  • Winning probability of player : highest-effort player wins with uniform random tie break


Strategic equilibria

Strategic Equilibria

  • A pure-strategy Nash equilibrium does not exist

  • In general there exists a continuum of mixed-strategy Nash equilibriumMoulin (1986), Dasgupta (1986), Hillman and Samet (1987), Hillman and Riley (1989), Ellingsen (1991), Baye et al (1993), Baye et al (1996)

  • There exists a unique symmetric Bayes-Nash equilibrium


Symmetric bayes nash equilibrium

Symmetric Bayes-Nash Equilibrium

  • Valuations are assumed to be private information of players, and independent samples from a prior distribution on [0,1]

  • A strategy is a symmetric Bayes-Nash equilibrium if it is a best response for every player conditional on that all other players play strategy , i.e., for every and


Quantities of interest

Quantities of Interest

  • Expected total effort:

  • Expected maximum individual effort:

  • Social efficiency:

    Order statistics: (valuations sorted in decreasing order)


Quantities of interest cont d

Quantities of Interest (cont’d)

  • In the symmetric Bayes-Nash equilibrium:


Total vs max individual effort

Total vs. Max Individual Effort

  • In any symmetric Bayes-Nash equilibrium, the expected maximum individual effort is at least half of the expected total effort

Chawla, Hartline, Sivan (2012)


Contests that award several prizes examples

Contests that Award Several Prizes: Examples

Kaggle

TopCoder


Rank order allocation of prizes

Rank Order Allocation of Prizes

  • Suppose that the prizes of values are allocated to players in decreasing order of individual efforts

  • There exists a symmetric Bayes-Nash equilibrium given by

  • = distribution of the value of -th largest valuation from independent samples from distribution

  • Special case: single unit-valued prize boils down to symmetric Bayes-Nash equilibrium in slide 9

V. – Contest Theory (2014)


Rank order allocation of prizes cont d

Rank Order Allocation of Prizes (cont’d)

  • Expected total effort:

  • Expected maximum individual effort:

V. – Contest Theory (2014)


The limit of many players

The Limit of Many Players

  • Suppose that for a fixed integer :

  • Expected individual efforts:

  • Expected total effort:

  • In particular, for the case of a single unit-valued prize (:

Archak and Sudarajan (2009)


When is it optimal to award only the first prize

When is it Optimal to Award only the First Prize?

  • In symmetric Bayes-Nash equilibrium both expected total effort and expected maximum individual effort achieve largest values by allocating the entire prize budget to the first prize.

  • Holds more generally for increasing concave production cost functions

Moldovanu and Sela (2001) – total effort

Chawla, Hartline, Sivan (2012) – maximum individual effort


Importance of symmetric prior beliefs

Importance of Symmetric Prior Beliefs

  • If the prior beliefs are asymmetric then it can be beneficial to offer more than one prize with respect to the expected total effort

  • Example: two prizes and three playersValues of prizes Valuations of players

Mixed-strategy Nash equilibrium

in the limit of large :

V. - Contest Theory (2014)


Optimal auction

Optimal Auction

  • Virtual valuation function:

  • said to be regular if it has increasing virtual valuation function

  • Optimal auction w.r.t. profit to the auctioneer: Allocation maximizespayments

Myerson (1981)


Optimal all pay contest w r t total effort

Optimal All-Pay Contest w.r.t. Total Effort

  • Suppose is regular. Optimal all-pay contest allocates the prize to a player who invests the largest effort subject to a minimum required effort of value .

  • Example:uniform distribution: minimum required effort

  • If is not regular, then an “ironing” procedure can be used


Optimal all pay contest w r t max individual effort

Optimal All-Pay Contest w.r.t. Max Individual Effort

  • Virtual valuation:

  • is said to be regular if is an increasing function

  • Suppose is regular. Optimal all-pay contest allocates the prize to a player who invests the largest effort subject to a minimum required effort of value

  • Example:uniform distribution: minimum required effort =

Chawla, Hartline, Sivan (2012)


Simultaneous all pay contests

Simultaneous All-Pay Contests

contests

players


Game simultaneous all pay contests

Game: Simultaneous All-Pay Contests

  • Suppose players have symmetric valuations (for now)

  • Each player participates in one contest

  • Contests are simultaneously selected by the players

  • Strategy of player = contest selected by player = amount of effort invested by player


Mixed strategy nash equilibrium

Mixed-Strategy Nash Equilibrium

  • There exists a symmetric mixed-strategy Nash equilibrium in which each player selects the contest to participate according to distribution given by

V. – Contest Theory (2014)


Quantities of interest1

Quantities of Interest

  • Expected total effort is at least of the benchmark valuewhere

  • Expected social welfare is at least of the optimum social welfare

V. – Contest Theory (2014)


Bayes nash equilibrium

Bayes Nash Equilibrium

  • Contests partitioned into classes based on values of prizes: contests of class 1 offer the highest prize value, contests of class 2 offer the second highest prize value, …

  • Suppose valuations are private information and are independent samples from a prior distribution

  • In symmetric Bayes Nash equilibrium, players are partitioned into classes such that a player of class selects a contest of class with probability

DiPalantino and V. (2009)

number of contests of class through


Example two contests

Example: Two Contests

Class 1 equilibrium strategy

Class 2 equilibrium strategy

V. – Contest Theory (2014)


Participation vs prize value

Participation vs. Prize Value

  • Taskcn 2009 – logo design tasks

any rate

once a month

every fourth day

every second day

DiPalantinoand V. (2009)

model


Conclusion

Conclusion

  • A model is presented that is a game of all-pay contests

  • An overview of known equilibrium characterization results is presented for the case of the game with incomplete information, for both single contest and a system of simultaneous contests

  • The model provides several insights into the properties of equilibrium outcomes and suggests several hypotheses to test in practice


Not in this slide deck

Not in this Slide Deck

  • Characterization of mixed-strategy Nash equilibria for standard all-pay contests

  • Consideration of non-linear production costs, e.g. players endowed with effort budgets (Colonel Blotto games)

  • Other prize allocation mechanisms – e.g. smooth allocation of prizes according to the ratio-form contest success function (Tullock) and the special case of proportional allocation

  • Productive efforts – sharing of a utility of production that is a function of the invested efforts

  • Sequential effort investments


References

References

  • Myerson, Optimal Auction Design, Mathematics of Operations Research, 1981

  • Moulin, Game Theory for the Social Sciences, 1986

  • Dasgupta, The Theory of Technological Competition, 1986

  • Hillman and Riley, Politically Contestable Rents and Transfers, Economics and Politics, 1989

  • Hillman and Samet, Dissipation of Contestable Rents by Small Number of Contestants, Public Choice, 1987

  • Glazer and Ma, Optimal Contests, Economic Inquiry, 1988

  • Ellingsen, Strategic Buyers and the Social Cost of Monopoly, American Economic Review, 1991

  • Baye, Kovenock, de Vries, The All-Pay Auction with Complete Information, Economic Theory 1996


References cont d

References (cont’d)

  • Moldovanu and Sela, The Optimal Allocation of Prizes in Contests, American Economic Review, 2001

  • DiPalantino and V., Crowdsourcing and All-Pay Auctions, ACM EC 2009

  • Archak and Sundarajan, Optimal Design of Crowdsourcing Contests, Int’l Conf. on Information Systems, 2009

  • Archak, Money, Glory and Cheap Talk: Analyzing Strategic Behavior of Contestants in Simultaneous Crowsourcing Contests on TopCoder.com, WWW 2010

  • Chawla, Hartline, Sivan, Optimal Crowdsourcing Contests, SODA 2012

  • Chawla and Hartline, Auctions with Unique Equilibrium, ACM EC 2013

  • V., Contest Theory, lecture notes, University of Cambridge, 2014


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