Crowdsourcing and All-Pay Auctions

Milan VojnovicMicrosoft Research

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

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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

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Competition-based Crowdsourcing: An ExampleCrowdFlower

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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

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Example Prizes: TopCoder

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Example Participation: Tackcn

• A month in year 2010players

contests

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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

1 2 𝑛

⋮

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Strategic Equilibria

• A pure-strategy Nash equilibrium does not exist

• In general there exists a continuum of mixed-strategy Nash equilibrium

Moulin (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

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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

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Quantities of Interest

• Expected total effort:

• Expected maximum individual effort:

• Social efficiency:

Order statistics: (valuations sorted in decreasing order)

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Quantities of Interest (cont’d)

• In the symmetric Bayes-Nash equilibrium:

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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)

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Contests that Award Several Prizes: Examples

Kaggle TopCoder

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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)

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Rank Order Allocation of Prizes (cont’d)

• Expected total effort:

• Expected maximum individual effort:

V. – Contest Theory (2014)

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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)

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

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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 players

Values of prizes Valuations of players

Mixed-strategy Nash equilibrium in the limit of large :

V. - Contest Theory (2014)

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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 maximizes

payments

Myerson (1981)

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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

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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)

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Simultaneous All-Pay Contests

players

contests

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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

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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)

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Quantities of Interest

• Expected total effort is at least of the benchmark value

where

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

V. – Contest Theory (2014)

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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

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Example: Two Contests

Class 1 equilibrium strategy Class 2 equilibrium strategy

V. – Contest Theory (2014)

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Participation vs. Prize Value

• Taskcn 2009 – logo design tasks

any rate once a month every fourth day every second day

model

DiPalantino and V. (2009)

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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

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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

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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

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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