How to divide prize money?

Milan VojnovićMicrosoft Research

Lecture to Trinity Mathematical Society, February 2nd, 2015

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• 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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Prizes, Prizes, Prizes

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⋯

1 2 𝑛

𝑏1 𝑏2 𝑏𝑛

individuals

production outputs

prize purse

Order statistics:

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Francis Galton’s Difference Problem (1902)• Split a unit prize budget between two placement prizes

• Assumption: independent and identically distributed random variables with distribution

• If has the domain of maximal attraction of type 3:

for

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Economist’s Approach

Assumption: individuals are strategic players that selfishly maximize their individual payoffs

Normal form game:• Players • Strategies (efforts)• Payoff functions

valuation winning probability production cost

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Standard All-Pay Contest• Highest effort player wins with random time break

• Linear production cost functions

• Payoff functions: , for

• There exists no pure-strategy Nash equilibrium

• There exists a mixed-strategy Nash equilibrium• For three or more players, a continuum of mixed-strategy Nash equilibria• Moulin (1986), Dasgupta (1986), Hillman and Samet (1987), Hillman and Riley (1989),

Ellingsen (1991), Baye et al (1993, 1996)

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Standard All-Pay Contest (cont’d)

• Private valuations: independent identically distributed valuation with prior distribution on [0,1]

There is a unique BNE

, for

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Revenue Equivalence Theorem

Suppose:• The valuation parameters are i.i.d. with differentiable distribution F • Standard auction (item allocated to the highest bidder)• The expected payment by a player with valuation zero is zero

Then, every symmetric increasing equilibrium has the same expected payment

The expected payment by player conditional on his of her valuation being of value :

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Proof sketch

• an increasing symmetric BNE strategy

• It must hold , i.e.

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Total Effort

• The expected total effort in symmetric BNE is equal to the expected value of the second largest valuation

Example: uniform prior distribution

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Rank Order Allocation of Prizes

⋯

1 2 𝑛

𝑏1 𝑏2 𝑏𝑛

𝑤1 𝑤2

⋯𝑤𝑛

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Symmetric Bayes-Nash Equilibrium

• Symmetric BNE given by, for

= distribution of the -th largest value from independent samples from

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Total Effort: Winner-Take-All Optimality

• Suppose that the production cost functions are linear • The goal is to maximize the expected total effort in symmetric BNE

Then, it is optimal to allocate entire prize purse to the first place prize

• This holds more generally for increasing concave production cost functions

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Proof Sketch

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Proof Sketch (cont’d)• is single crossing : there exists :

for and for ,1]

h1(𝑥 )

h 𝑗(𝑥)

𝑥10

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Proof Sketch (Cont’d)

+

+

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Max Individual Effort: Winner-Take-All Optimality

• Suppose that the production cost functions are linear• The goal is to maximize the expected maximum individual effort in

symmetric BNE

Then, it is optimal to allocate entire prize purse to the first place prize

• This generalizes to increasing concave production cost functions

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Max Individual vs. Total Effort

In every BNE of the game that models standard all-pay contest, the expected maximum individual is at least of the expected total effort

Chawla, Hartline, Sivan (2012)

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Proof Sketch

non negative and non decreasing

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Optimal Auction Design

• independent valuations with distributions • increasing with continuous density function on []

• Direct revelation mechanism

Allocation

Payment

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Notation

• Expected allocation

• Expected payment

• Expected payoff -

• Welfare

• Revenue

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Feasible Auction Mechanism

An auction mechanism is feasible if it satisfies the following conditions:

(RC) Resource Constraint:

, ,

(IR) Individual Rationality:

for all ,

(IC) Incentive Compatibility:

, for all ,

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Necessary and Sufficient Conditions

is feasible if, and only if,

(M) is non decreasing for

(P)

(IR’) for

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Welfare Optimal Auction

Suppose that is such that maximizes

subject to the constraints (M) and (RC) and that payment is given by

Then, is a welfare optimal auction

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Welfare Optimal Auction (Cont’d)

• Second Prize Auction with a Reserve Price

• Allocation:

• Payment:

• For identical prior distributions:

Vickrey (1961)

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Optimal Auction: Revenue

Suppose that is such that maximizes

where

subject to the constraints (M) and (RC) and that payment is given by

Then, is a revenue optimal auction

Myerson (1982)

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Regular Case

• Regular: all virtual valuation functions are increasing

• and

• For identical prior distributions: {}

• Example: uniform prior distribution

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Maximum Individual Effort

• -virtual valuation function:

said to be regular if increasing

• is said to be regular if is regular for every integer

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Optimal All-Pay Contest

• Suppose that valuations are i.i.d. with regular distribution • Goal is to maximize the expected maximum individual effort in a BNE

Then, it is optimal to allocate entire prize purse to the first place prize subject to minimum required effort of value

Example: uniform prior distribution

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Comparison with Standard All-Pay Contest

The expected total effort in symmetric BNE of the game that models standard all-pay contest with players is at least as large as that of the optimal expected total effort in the game with players

Same holds for the expected maximum individual effort

Chawla, Hartline, Sivan (2012)

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Proof Sketch

⋯

1 2 𝑛+1

Standard All-Pay Contest

⋯

1 2 𝑛

Round 1: Optimal All-Pay Contest

If the prize is allocated in Round 1

else

𝑛+1

Round 2

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Competitiveness of Standard All-Pay Contest

The expected total effort in symmetric BNE of the game that models the standard all-pay contest is at least of the optimal expected total effort

At least half of optimum

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Proof Sketch

• = expected total effort in BNE in standard all-pay contest• = optimal total effort in BNE of optimal all-pay contest

(1) (slide 32)

with

(2)

(1) and (2)

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Competitiveness of Standard All-Pay ContestThe expected maximum individual effort in symmetric BNE of the game that models the standard all-pay contest is at least of the optimal expected total effort

Proof sketch:

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The Importance of Symmetric PriorsIf the prior distributions are asymmetric then it may be optimal to split a prize purse between two or more position prizes

𝑣=𝑣1 ≥𝑣2 = 1

(𝑤 ,1−𝑤)

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The Importance of Symmetric Priors (cont’d)The large limit:

• Ex winner-take-all: • Ex prize split:

𝐵1(𝑥)

𝐵2(𝑥)𝑥

𝑥

12

12

𝑤

𝑤

1−𝑤

1−𝑤

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Conclusion

• Optimality of winner-take-all prize allocation under symmetric prior distributions and concave production cost functions• Both for expected total and expected maximum individual effort

• The expected maximum individual effort is at least ½ of the expected total effort in a BNE for standard all-pay contest• The expected total effort in BNE of standard all-pay contest is at least

½ of that in the BNE under optimal all-pay contest• If the prior distributions are asymmetric, then it may be optimal to

split the prize purse over two or more placement prizes

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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 Crowdsourcing 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: Incentive Mechanisms and Ranking Methods, forthcoming

book, Cambridge University Press, 2015

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Topics not Covered in the Talk

• Smooth allocation of prizes, e.g. proportional allocation

• Simultaneous contests

• Sequential contests

• Productive efforts: utility sharing mechanisms