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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

- 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

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)

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)

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 value where
- 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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