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

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

Prizes, Prizes, Prizes

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)

Francis Galton’s Difference Problem (1902) Split a unit prize budget between two placement prizes 𝑤 1 ,1− 𝑤 1 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∈ 1/2,1 𝐄 𝑤 1 = 3 4

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

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)

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]

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)

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 0 + 0 𝑣 𝑖 𝑥𝑑 𝐹 𝑛−1 (𝑥) = 0 𝑣 𝑖 𝑥𝑑 𝐹 𝑛−1 (𝑥)

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

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

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

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

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

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)

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 ∗ 0 1 ℎ 1 𝑥 − ℎ 𝑗 𝑥 𝑑𝑥 >0

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

Max Individual vs. Total Effort In every BNE of the game that models standard all-pay contest, the expected maximum individual is at least 1 2 of the expected total effort Chawla, Hartline, Sivan (2012)

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 ∗ 0 1 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

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 )

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 ])

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

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

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

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)

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)

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

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

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 2 1− 1 𝑛+1

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)

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

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

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)

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 2 1− 1 𝑛 𝑅 ∗ (𝑛)

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 2 = 1

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

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

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 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

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