Download - Crowdsourding and all-pay contests
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 TriposPart III, University of Cambridge - forthcoming book
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Competition-based Crowdsourcing: An Example
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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 20134
Example Prizes: TopCoder
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Example Participation: Tackcn
• A month in year 20106
players
contests
Game: Standard All-Pay Contest• 𝑛 players, 𝑣1, 𝑣2, … , 𝑣𝑛 valuations, linear production costs
• Quasi-linear payoff functions: 𝑠𝑖 𝒃 = 𝑣𝑖𝑥𝑖 𝒃 − 𝑏𝑖
• Simultaneous effort investments: 𝒃 = 𝑏1, 𝑏2, … , 𝑏𝑛 ,𝑏𝑖 = effort investment of player 𝑖
• Winning probability of player 𝑖: 𝑥𝑖(𝒃)highest-effort player wins with uniform random tie break
1 2 𝑛
⋮7
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 𝑣1, 𝑣2, … , 𝑣𝑛 are assumed to be private information of players, and independent samples from a prior distribution 𝐹 on [0,1]
• A strategy 𝛽: 0,1 → 0,1 is a symmetric Bayes-Nash equilibrium if it is a best response for every player 𝑖 conditional on that all other players play strategy 𝛽, i.e.
𝐄 𝑠𝑖 𝒃 𝑣𝑖 = 𝑣, 𝑏𝑖 = 𝛽 𝑣 ≥ 𝐄 𝑠𝑖 𝒃 𝑣𝑖 = 𝑣, 𝑏𝑖′ , for every 𝑣 and 𝑏𝑖′
𝛽 𝑣 = 0
𝑣
𝑥𝑑𝐹𝑛−1(𝑥)
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Quantities of Interest
• Expected total effort: 𝑅 = 𝐄 𝑖=1𝑛 𝑏𝑖
• Expected maximum individual effort: 𝑅 = 𝐄[max{𝑏1, 𝑏2, … , 𝑏𝑛}]
• Social efficiency: 𝐄 𝑖=1
𝑛 𝑣𝑖𝑥𝑖 𝒃
𝐄[𝑣(𝑛,1)]
Order statistics: 𝑣(𝑛,1) ≥ 𝑣 𝑛,2 ≥ ⋯ ≥ 𝑣 𝑛,𝑛 (valuations sorted in decreasing order)
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Quantities of Interest (cont’d)
• In the symmetric Bayes-Nash equilibrium:
𝑅 = 𝐄 𝑣 𝑛,2
𝑅1 = 𝐄 𝑣 𝑛−1,1 −𝑛 − 1
2𝑛 − 1𝐄[𝑣(2𝑛−1,1)]
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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
𝑅1 ≥ 𝑅/2
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 𝑤1 ≥ 𝑤2 ≥ ⋯ ≥ 𝑤𝑛 ≥ 0 are allocated to
players in decreasing order of individual efforts
• There exists a symmetric Bayes-Nash equilibrium given by
𝛽 𝑣 = 𝑗=1𝑛−1(𝑤𝑗 − 𝑤𝑗+1) 0
𝑣𝑥𝑑𝐹𝑛−1,𝑗(𝑥)
• 𝐹𝑚,𝑗 = distribution of the value of 𝑗-th largest valuation from 𝑚 independent samples from distribution 𝐹
• Special case: single unit-valued prize 1 = 𝑤1 > 𝑤2 = ⋯ = 𝑤𝑛 = 0 boils down to symmetric Bayes-Nash equilibrium in slide 9
14V. – Contest Theory (2014)
Rank Order Allocation of Prizes (cont’d)
• Expected total effort:
𝑅 = 𝑗=1𝑛−1 𝑤𝑗 − 𝑤𝑗+1 𝐄[𝑣(𝑛,𝑗+1)]
• Expected maximum individual effort:
𝑅1 = 𝑗=1𝑛−1 𝑤𝑗 −𝑤𝑗+1 𝐄 𝑣 𝑛−1,𝑗 −
𝑛−1 ! 2𝑛−1−𝑗 !
𝑛−1−𝑗 ! 2𝑛−1 !𝐄 𝑣 2𝑛−1,𝑗
15V. – Contest Theory (2014)
The Limit of Many Players
• Suppose that for a fixed integer 𝑘: 𝑤1 ≥ 𝑤2 ≥ ⋯ ≥ 𝑤𝑘 > 𝑤𝑘+1 = ⋯ = 𝑤𝑛 = 0
• Expected individual efforts:
lim𝑛→∞
𝐄 𝑏 𝑛,𝑖 = 𝑗=1𝑘 1
2𝑗+𝑖−1
𝑗 + 𝑖 − 2𝑗 − 1
𝑤𝑗
• Expected total effort:lim𝑛→∞
𝑹 = 𝑗=1𝑘 𝑤𝑗
• In particular, for the case of a single unit-valued prize (𝑘 = 1):
lim𝑛→∞
𝐄 𝑏 𝑛,𝑖 =1
2𝑖
16Archak 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 17
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 w1, w2 = 𝑤, 1 − 𝑤Valuations of players 𝑣 = 𝑣1 > 𝑣2 = 𝑣3 = 1
Mixed-strategy Nash equilibrium in the limit of large 𝑣:
18V. - Contest Theory (2014)
Optimal Auction
• Virtual valuation function: 𝜓 𝑣 = 𝑣 −1−𝐹(𝑣)
𝑓(𝑣)
• 𝐹 said to be regular if it has increasing virtual valuation function
• Optimal auction w.r.t. profit to the auctioneer: (𝑥, 𝑝)
Allocation 𝑥 maximizes
𝐄 𝑖=1𝑛 𝜓 𝑣𝑖 𝑥𝑖 𝒗
payments 𝑝𝑖 𝒗 = 𝑣𝑖𝑥𝑖 𝒗 − 0𝑣𝑖 𝑥𝑖 𝒗−𝑖 , 𝑣 𝑑𝑣
Myerson (1981)19
Optimal All-Pay Contest w.r.t. Total Effort
• Suppose 𝐹 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 𝜓−1 0 .
• Example: uniform distribution: minimum required effort = 1/2
• If 𝐹 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: 𝜓𝑛 𝑣 = 𝑣𝐹 𝑣 𝑛−1 −1−𝐹 𝑣 𝑛
𝑛𝑓(𝑣)
• 𝐹 is said to be regular if 𝜓𝑛(𝑣) is an increasing function
• Suppose 𝐹 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 𝜓𝑛
−1 0 𝐹𝑛−1 𝜓𝑛−1 0
• Example: uniform distribution: minimum required effort = 1/(𝑛 + 1)
Chawla, Hartline, Sivan (2012)21
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 𝑖: 𝑎𝑖 , 𝑏𝑖
𝑎𝑖 = contest selected by player 𝑖𝑏𝑖 = amount of effort invested by player 𝑖
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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 𝒑 given by
𝑝𝑗 = 1 − 1 −1
𝑚Φ 𝑚
1
𝑤𝑗
1/(𝑛−1)
𝑗 = 1,2,… , 𝑚
0 o.w.
• Φ𝑗 =1
1
𝑗 𝑙=1𝑗 1
𝑤𝑙
1/(𝑛−1) , 𝑗 = 1,2,… ,𝑚
• 𝑚 = max 𝑗|𝑤𝑗1/(𝑛−1)
> 1 −1
𝑗Φ𝑗
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V. – Contest Theory (2014)
Quantities of Interest
• Expected total effort is at least ¼ of the benchmark value
𝑅∗ = 𝑗=1𝑘 𝑤𝑗 where 𝑘 = min{𝑚, 𝑛}
• Expected social welfare is at least 1 − 1/𝑒 of the optimum social welfare
𝑊∗ = 𝑗=1𝑘 𝑤𝑗
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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 𝐹
• In symmetric Bayes Nash equilibrium, players are partitioned into classes such that a player of class 𝑙 selects a contest of class 𝑗 with probability
𝛼𝑗𝑙 =
Φ𝑙
𝑀𝑙
1
𝑤𝑗
1/(𝑛−1)
1 ≤ 𝑗 ≤ 𝑙
1 o.w.
DiPalantino and V. (2009)
26𝑀𝑙 = number of contests of class 1 through 𝑙
Example: Two Contests
• 𝑎 =𝑤2
𝑤1
1/(𝑛−1)
Class 1 equilibrium strategy Class 2 equilibrium strategy
27V. – Contest Theory (2014)
Participation vs. Prize Value
• Taskcn 2009 – logo design tasks
any rate once a month every fourth day every second day
28model
DiPalantino and V. (2009)
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
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