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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.

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Crowdsourcing and All-Pay Auctions

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

  2. 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

  3. Competition-based Crowdsourcing: An Example CrowdFlower

  4. 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

  5. Example Prizes: TopCoder

  6. Example Participation: Tackcn contests • A month in year 2010 players

  7. 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

  8. 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

  9. 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

  10. Quantities of Interest • Expected total effort: • Expected maximum individual effort: • Social efficiency: Order statistics: (valuations sorted in decreasing order)

  11. Quantities of Interest (cont’d) • In the symmetric Bayes-Nash equilibrium:

  12. 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)

  13. Contests that Award Several Prizes: Examples Kaggle TopCoder

  14. 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)

  15. Rank Order Allocation of Prizes (cont’d) • Expected total effort: • Expected maximum individual effort: V. – Contest Theory (2014)

  16. 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)

  17. 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

  18. 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)

  19. 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)

  20. 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

  21. 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)

  22. Simultaneous All-Pay Contests contests players

  23. 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

  24. 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)

  25. 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)

  26. 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

  27. Example: Two Contests Class 1 equilibrium strategy Class 2 equilibrium strategy V. – Contest Theory (2014)

  28. Participation vs. Prize Value • Taskcn 2009 – logo design tasks any rate once a month every fourth day every second day DiPalantinoand V. (2009) model

  29. 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

  30. 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

  31. 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

  32. 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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