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

Milan Vojnovic

Microsoft Research

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


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

CrowdFlower


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 Participation: Tackcn

contests

  • A month in year 2010

players


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

  • 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

  • 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

  • Expected total effort:

  • Expected maximum individual effort:

  • Social efficiency:

    Order statistics: (valuations sorted in decreasing order)


Quantities of Interest (cont’d)

  • In the symmetric Bayes-Nash equilibrium:


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

Kaggle

TopCoder


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)

  • Expected total effort:

  • Expected maximum individual effort:

V. – Contest Theory (2014)


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?

  • 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

  • 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

  • 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

  • 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

  • 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

contests

players


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

  • 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 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

  • 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

Class 1 equilibrium strategy

Class 2 equilibrium strategy

V. – Contest Theory (2014)


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

  • 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

  • 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

  • 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)

  • 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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