Crowdsourcing and All-Pay Auctions
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Transcript of Crowdsourcing and All-Pay Auctions
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