1. How to divide prize money?
Milan Vojnović
Microsoft Research
Lecture to Trinity Mathematical Society, February 2nd, 2015

3. 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

4. Prizes, Prizes, Prizes

5. prize purse
𝑏 1
𝑏 2
𝑏 𝑛
production outputs ⋯
individuals 1 2 𝑛
Order statistics: 𝑏 (𝑛,1) ≥ 𝑏 𝑛,2 ≥⋯≥ 𝑏 (𝑛,𝑛)

6. Francis Galton’s Difference Problem (1902)
Split a unit prize budget between two placement prizes 𝑤 1 ,1− 𝑤 𝑤 1 = 𝑏 (𝑛,1) − 𝑏 (𝑛,3) 𝑏 (𝑛,1) − 𝑏 𝑛,3 + 𝑏 (𝑛,2) − 𝑏 (𝑛,3)
Assumption: 𝑏 1 , 𝑏 2 ,…, 𝑏 𝑛 independent and identically distributed random variables with distribution 𝐹
If 𝐹 has the domain of maximal attraction of type 3: lim 𝑛→∞ 𝐏𝐫 𝑤 1 ≤𝑥 =2𝑥−1 for 𝑥∈ 1/2, 𝐄 𝑤 1 = 3 4

7. Economist’s Approach
Assumption: individuals are strategic players that selfishly maximize their individual payoffs
Normal form game:
Players 𝑁={1,2,…,𝑛}
Strategies 𝒃= 𝑏 1 , 𝑏 2 ,…, 𝑏 𝑛 ∈ 𝐑 + 𝑛 (efforts)
Payoff functions 𝑠 𝑖 𝑣 𝑖 , 𝒃 = 𝑣 𝑖 𝑥 𝑖 𝒃 − 𝑐 𝑖 𝑏 𝑖
valuation
winning probability
production cost

8. Standard All-Pay Contest
Highest effort player wins with random time break
Linear production cost functions
Payoff functions: 𝑠 𝑖 𝑣 𝑖 , 𝒃 = 𝑣 𝑖 𝑥 𝑖 𝒃 − 𝑏 𝑖 , for 𝑖∈𝑁
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)

9. Standard All-Pay Contest (cont’d)
Private valuations: independent identically distributed valuation with prior distribution 𝐹 on [0,1]
There is a unique BNE 𝛽 𝑣 = 0 𝑣 𝑥𝑑 𝐹 𝑛−1 (𝑥) , for 𝑣∈[0,1]

10. 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 𝑖 conditional on his of her valuation being of value 𝑣 𝑖 : 𝑐 𝑖 𝑣 𝑖 = 0 𝑣 𝑖 𝑥𝑑 𝐹 𝑛−1 (𝑥)

11. Proof sketch 𝛽: 0,1 → 𝐑 + an increasing symmetric BNE strategy
𝑠 𝑖 𝑣 𝑖 , 𝑣 = 𝑣 𝑖 𝐹 𝑣 𝑛−1 − 𝑐 𝑖 𝑣
𝜕 𝜕𝑣 𝑠 𝑖 𝑣 𝑖 , 𝑣 = 𝑣 𝑖 𝐹 𝑣 𝑛−1 ′ − 𝑐 𝑖 ′ 𝑣
It must hold 𝜕 𝑠 𝑖 𝑣 𝑖 , 𝑣 𝑖 𝜕𝑣 =0, i.e. 𝑑 𝑑𝑣 𝑐 𝑖 𝑣 =𝑣 𝐹 𝑣 𝑛−1 ′
𝑐 𝑖 𝑣 𝑖 = 𝑐 𝑖 𝑣 𝑖 𝑥𝑑 𝐹 𝑛−1 (𝑥) = 0 𝑣 𝑖 𝑥𝑑 𝐹 𝑛−1 (𝑥)

12. Total Effort
The expected total effort in symmetric BNE is equal to the expected value of the second largest valuation
𝑅= 𝐄[ 𝑣 (𝑛,2) ]
Example: uniform prior distribution
𝑅=1− 2 𝑛+1

13. Rank Order Allocation of Prizes⋯
𝑤 1
𝑤 2
𝑤 𝑛
𝑏 1
𝑏 2
𝑏 𝑛 ⋯ 1 2 𝑛

14. Symmetric Bayes-Nash Equilibrium
Symmetric BNE given by
𝛽 𝑣 = 𝑗=1 𝑛−1 ( 𝑤 𝑗 − 𝑤 𝑗+1 ) 0 𝑣 𝑥𝑑 𝐹 𝑛−1,𝑗 (𝑥) , for 𝑣∈[0,1]
𝐹 𝑛−1,𝑗 = distribution of the 𝑗-th largest value from 𝑛−1 independent samples from 𝐹

15. 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

16. Proof Sketch 𝑅= 𝑗=1 𝑛 𝑤 𝑗 𝑎 𝑗 𝑎 𝑗 = 0 1 𝐹 −1 𝑥 ℎ 𝑗 𝑥 𝑑𝑥
𝑅= 𝑗=1 𝑛 𝑤 𝑗 𝑎 𝑗
𝑎 𝑗 = 0 1 𝐹 −1 𝑥 ℎ 𝑗 𝑥 𝑑𝑥
ℎ 𝑗 𝑥 = 1−𝑥 𝐺 𝑗 ′ 𝑥
𝐺 𝑗 𝑥 = 𝑛−1 𝑗−1 𝑥 𝑛−𝑗 1−𝑥 𝑗−1

17. Proof Sketch (cont’d)
ℎ 1 is single crossing ℎ 𝑗 : there exists 𝑥 ∗ ∈[0,1]: ℎ 1 𝑥 ≤ ℎ 𝑗 (𝑥) for 𝑥∈ 0, 𝑥 ∗ and ℎ 1 𝑥 > ℎ 𝑗 (𝑥) for 𝑥∈( 𝑥 ∗ ,1]
ℎ 1 (𝑥) 1 𝑥
ℎ 𝑗 (𝑥)

18. Proof Sketch (Cont’d) 𝑎 1 − 𝑎 𝑗 = 0 1 𝐹 −1 𝑥 ℎ 1 𝑥 − ℎ 𝑗 𝑥 𝑑𝑥
𝑎 1 − 𝑎 𝑗 = 0 1 𝐹 −1 𝑥 ℎ 1 𝑥 − ℎ 𝑗 𝑥 𝑑𝑥
= 0 𝑥 ∗ 𝐹 −1 𝑥 ℎ 1 𝑥 − ℎ 𝑗 𝑥 𝑑𝑥 + 𝑥 ∗ 1 𝐹 −1 𝑥 ℎ 1 𝑥 − ℎ 𝑗 𝑥 𝑑𝑥
≥ 0 𝑥 ∗ 𝐹 −1 𝑥 ∗ ℎ 1 𝑥 − ℎ 𝑗 𝑥 𝑑𝑥 + 𝑥 ∗ 1 𝐹 −1 𝑥 ∗ ℎ 1 𝑥 − ℎ 𝑗 𝑥 𝑑𝑥
= 𝐹 −1 𝑥 ∗ ℎ 1 𝑥 − ℎ 𝑗 𝑥 𝑑𝑥 >0

19. 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

20. 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)

21. Proof Sketch 2 𝑅 1 −𝑅=2 0 1 𝛽 𝑥 𝑑 𝐹 𝑛 𝑥 −𝑛 0 1 𝛽 𝑥 𝑑𝐹(𝑥)
2 𝑅 1 −𝑅=2 0 1 𝛽 𝑥 𝑑 𝐹 𝑛 𝑥 −𝑛 0 1 𝛽 𝑥 𝑑𝐹(𝑥)
=𝑛 0 1 𝛽 𝑥 2 𝐹 𝑛−1 𝑥 −1 𝑑𝐹(𝑥)
=𝑛 0 1 𝛾 𝑦 𝑦 𝑛−1 2 𝑦 𝑛−1 −1 𝑑𝑦
≥𝑛𝛾 𝑦 ∗ 𝑦 𝑛−1 2 𝑦 𝑛−1 −1 𝑑𝑦
=𝑛𝛾 𝑦 ∗ 1 2𝑛−1 ≥0
𝛾 𝑦 =𝛽( 𝐹 −1 𝑦 )/ 𝑦 𝑛−1 non negative and non decreasing
2 ( 𝑦 ∗ ) 𝑛−1 −1=0

22. Optimal Auction Design
𝑣 1 , 𝑣 2 ,…, 𝑣 𝑛 independent valuations with distributions 𝐹 1 , 𝐹 2 ,…, 𝐹 𝑛
𝐹 𝑖 increasing with continuous density function 𝑓 𝑖 on [ 𝑏 𝑖 , 𝑏 𝑖 ]
Direct revelation mechanism (𝑥,𝑝)
Allocation 𝑥 𝒗 =( 𝑥 1 𝒗 , 𝑥 2 𝒗 ,…, 𝑥 𝑛 𝒗 ) Payment 𝑝 𝒗 =( 𝑝 1 𝒗 , 𝑝 2 𝒗 ,…, 𝑝 𝑛 𝒗 )

23. Notation Expected allocation 𝑥 𝑖 𝑣 =𝐄[ 𝑥 𝑖 (𝒗)| 𝑣 𝑖 =𝑣]
Expected payment 𝑝 𝑖 𝑣 =𝐄 𝑝 𝑖 𝒗 𝑣 𝑖 =𝑣
Expected payoff 𝑠 𝑖 𝑣 𝑖 , 𝑣 = 𝑣 𝑖 𝑥 𝑖 𝑣 - 𝑝 𝑖 𝑣 ; 𝑠 𝑖 𝑣 𝑖 = 𝑠 𝑖 𝑣 𝑖 , 𝑣 𝑖
Welfare 𝑤 𝑟 =𝐄 𝑖=1 𝑛 𝑣 𝑖 𝑥 𝑖 𝒗 +𝑟(1−𝐄[ 𝑖=1 𝑛 𝑥 𝑖 𝒗 ])
Revenue 𝑠 0 𝑟 =𝐄 𝑖=1 𝑛 𝑝 𝑖 𝒗 +𝑟(1−𝐄[ 𝑖=1 𝑛 𝑥 𝑖 𝒗 ])

24. Feasible Auction Mechanism
An auction mechanism (𝑥,𝑝) is feasible if it satisfies the following conditions:
(RC) Resource Constraint: 𝑥 𝑖 𝒗 ≥0, 𝑖=1,2,…, 𝑛, 𝑖=1 𝑛 𝑥 𝑖 𝒗 ≤1
(IR) Individual Rationality: 𝑠 𝑖 𝑣 ≥0 for all 𝑣∈[ 𝑏 𝑖 , 𝑏 𝑖 ], 𝑖=1,2,…, 𝑛
(IC) Incentive Compatibility: 𝑠 𝑖 𝑣 𝑖 , 𝑣 𝑖 ≥ 𝑠 𝑖 ( 𝑣 𝑖 , 𝑣), for all 𝑣∈[ 𝑏 𝑖 , 𝑏 𝑖 ], 𝑖=1,2,…, 𝑛

25. Necessary and Sufficient Conditions
(𝑥,𝑝) is feasible if, and only if, (M) 𝑥 𝑖 (𝑣) is non decreasing for 𝑣∈[ 𝑏 𝑖 , 𝑏 𝑖 ] (P) 𝑝 𝑖 𝑣 =𝑣 𝑥 𝑖 𝑣 − 𝑏 𝑖 𝑣 𝑥 𝑖 𝑥 𝑑𝑥− 𝑎 𝑖 𝑥 𝑖 𝑎 𝑖 − 𝑝 𝑖 𝑎 𝑖 (IR’) 𝑠 𝑖 𝑎 𝑖 ≥0 for 𝑖=1,2,…,𝑛

26. Welfare Optimal Auction
Suppose that (𝑥,𝑝) is such that 𝑥 maximizes 𝐄 𝑖=1 𝑛 𝑣 𝑖 −𝑟 𝑥 𝑖 𝒗 subject to the constraints (M) and (RC) and that payment is given by 𝑝 𝑖 𝒗 = 𝑣 𝑖 𝑥 𝑖 𝒗 − 𝑏 𝑖 𝑣 𝑥 𝑖 𝑣, 𝒗 −𝑖 𝑑𝑥
Then, (𝑥,𝑝) is a welfare optimal auction

27. Welfare Optimal Auction (Cont’d)
Second Prize Auction with a Reserve Price
Allocation: 𝑥 𝑖 𝒗 =𝟏( 𝑣 𝑖 > 𝜃 𝑖 𝒗 −𝑖 )
Payment: 𝑝 𝑖 𝒗 = 𝜃 𝑖 𝒗 −𝑖 𝟏 𝑥 𝑖 𝒗 =1
For identical prior distributions: 𝜃 𝑖 𝒗 −𝑖 =max{ max 𝑗≠𝑖 𝑣 𝑗 ,𝑟}
Vickrey (1961)

28. Optimal Auction: Revenue
Suppose that (𝑥,𝑝) is such that 𝑥 maximizes 𝐄 𝑖=1 𝑛 𝜓 𝑖 𝑣 𝑖 −𝑟 𝑥 𝑖 𝒗 where 𝜓 𝑖 𝑣 =𝑣−(1− 𝐹 𝑖 𝑣 )/ 𝑓 𝑖 (𝑣)
subject to the constraints (M) and (RC) and that payment is given by 𝑝 𝑖 𝒗 = 𝑣 𝑖 𝑥 𝑖 𝒗 − 𝑏 𝑖 𝑣 𝑥 𝑖 𝑣, 𝒗 −𝑖 𝑑𝑥
Then, (𝑥,𝑝) is a revenue optimal auction
Myerson (1982)

29. Regular Case Regular: all virtual valuation functions are increasing
𝑥 𝑖 𝒗 =𝟏( 𝑣 𝑖 > 𝜃 𝑖 𝒗 −𝑖 ) and 𝑝 𝑖 𝒗 = 𝜃 𝑖 𝒗 −𝑖 𝟏 𝑥 𝑖 𝒗 =1
𝜃 𝑖 𝒗 −𝑖 = inf 𝑣∈ 0,1 : 𝜓 𝑖 𝑣 ≥𝑟 and 𝜓 𝑖 𝑣 ≥ 𝜓 𝑗 𝑣 𝑗 , 𝑗=1,2,…,𝑛
For identical prior distributions: 𝜃 𝑖 𝒗 −𝑖 =max{ max 𝑗≠𝑖 𝑣 𝑗 , 𝜓 −1 (𝑟)}
Example: uniform prior distribution 𝜓 −1 𝑟 =(𝑟+1)/2

30. Maximum Individual Effort
𝑅 1 =𝐄 𝑖=1 𝑛 𝑥 𝑖 𝒗 𝜓 𝑣 𝑖 ;𝑛
𝑛-virtual valuation function:
𝜓 𝑣;𝑛 =𝑣𝐹 𝑣 𝑛−1 − 1−𝐹 𝑣 𝑛 𝑛𝑓(𝑣)
said to be regular if increasing
𝐹 is said to be regular if 𝜓 𝑣;𝑛 is regular for every integer 𝑛≥2

31. Optimal All-Pay Contest
Suppose that valuations are i.i.d. with regular distribution 𝐹
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
𝜓 −1 0;𝑛 𝐹 𝑛−1 𝜓 −1 0;𝑛
Example: uniform prior distribution 𝜓 −1 0;𝑛 = 1 𝑛+1 1/𝑛
𝑅 1 = − 1 𝑛+1

32. Comparison with Standard All-Pay Contest
The expected total effort in symmetric BNE of the game that models standard all-pay contest with 𝑛+1 players is at least as large as that of the optimal expected total effort in the game with 𝑛 players Same holds for the expected maximum individual effort
Chawla, Hartline, Sivan (2012)

33. If the prize is allocated
Proof Sketch
If the prize is allocated
in Round 1
else ⋯ 1 2
𝑛+1
Standard All-Pay Contest ⋯ 1 2 𝑛
Round 1: Optimal All-Pay Contest
𝑛+1
Round 2

34. 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 1−1/𝑛 of the optimal expected total effort
⇒ At least half of optimum

35. Proof Sketch
𝑅(𝑛) = expected total effort in BNE in standard all-pay contest
𝑅 ∗ (𝑛) = optimal total effort in BNE of optimal all-pay contest
𝑅 𝑛 ≥ 𝑅 ∗ 𝑛−1 (1) (slide 32)
𝑅 ∗ (𝑛) 𝑛 = 1 𝑛 𝐄 𝜓 𝑣 𝑛,1 𝟏 𝜓 𝑣 𝑛,1 >0 = 𝜓 −1 (0) 1 𝜓 𝑣 𝐹 𝑛−1 𝑣 𝑑𝐹(𝑣) ↓ with 𝑛
⇒ 𝑅 ∗ (𝑛−1) 𝑛−1 ≥ 𝑅 ∗ (𝑛) 𝑛 (2)
(1) and (2) ⇒𝑅 𝑛 ≥ 1− 1 𝑛 𝑅 ∗ (𝑛)

36. Competitiveness of Standard All-Pay Contest
The expected maximum individual effort in symmetric BNE of the game that models the standard all-pay contest is at least (1−1/𝑛)/2 of the optimal expected total effort
Proof sketch:
𝑅 1 𝑛 ≥ 1 2 𝑅 𝑛 ≥ − 1 𝑛 𝑅 ∗ (𝑛)

37. The Importance of Symmetric Priors
If the prior distributions are asymmetric then it may be optimal to split a prize purse between two or more position prizes
(𝑤,1−𝑤)
𝑣= 𝑣 1
≥ 𝑣 2
= 𝑣 = 1

38. The Importance of Symmetric Priors (cont’d)
𝐵 1 (𝑥)
The large 𝑣 limit:
𝐄 𝑏 1 = 1 2 , 𝐄 𝑏 2 =𝐄 𝑏 3 = 1−𝑤 2
𝑅= 3 2 −𝑤
Ex winner-take-all: 𝑅= 1 2
Ex 2 :1 prize split: 𝑅= 5 6
1−𝑤 𝑤 𝑥
1 2
𝐵 2 (𝑥)
1−𝑤 𝑤
1 2 𝑥

39. 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

40. 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

41. 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

42. Topics not Covered in the Talk
Smooth allocation of prizes, e.g. proportional allocation
Simultaneous contests
Sequential contests
Productive efforts: utility sharing mechanisms