Date post: | 17-Dec-2015 |
Category: |
Documents |
Upload: | elijah-cole |
View: | 218 times |
Download: | 1 times |
How to divide prize money?
Milan VojnovićMicrosoft Research
Lecture to Trinity Mathematical Society, February 2nd, 2015
2
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
⋯
1 2 𝑛
𝑏1 𝑏2 𝑏𝑛
individuals
production outputs
prize purse
Order statistics:
6
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
7
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
8
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)
9
Standard All-Pay Contest (cont’d)
• Private valuations: independent identically distributed valuation with prior distribution on [0,1]
There is a unique BNE
, for
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 conditional on his of her valuation being of value :
11
Proof sketch
• an increasing symmetric BNE strategy
• It must hold , i.e.
12
Total Effort
• The expected total effort in symmetric BNE is equal to the expected value of the second largest valuation
Example: uniform prior distribution
13
Rank Order Allocation of Prizes
⋯
1 2 𝑛
𝑏1 𝑏2 𝑏𝑛
𝑤1 𝑤2
⋯𝑤𝑛
14
Symmetric Bayes-Nash Equilibrium
• Symmetric BNE given by, for
= distribution of the -th largest value from independent samples from
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
17
Proof Sketch (cont’d)• is single crossing : there exists :
for and for ,1]
h1(𝑥 )
h 𝑗(𝑥)
𝑥10
18
Proof Sketch (Cont’d)
+
+
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
non negative and non decreasing
22
Optimal Auction Design
• independent valuations with distributions • increasing with continuous density function on []
• Direct revelation mechanism
Allocation
Payment
23
Notation
• Expected allocation
• Expected payment
• Expected payoff -
• Welfare
• Revenue
24
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 ,
25
Necessary and Sufficient Conditions
is feasible if, and only if,
(M) is non decreasing for
(P)
(IR’) for
26
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
27
Welfare Optimal Auction (Cont’d)
• Second Prize Auction with a Reserve Price
• Allocation:
• Payment:
• For identical prior distributions:
Vickrey (1961)
28
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)
29
Regular Case
• Regular: all virtual valuation functions are increasing
• and
• For identical prior distributions: {}
• Example: uniform prior distribution
30
Maximum Individual Effort
• -virtual valuation function:
said to be regular if increasing
• is said to be regular if is regular for every integer
31
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
32
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)
33
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
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 of the optimal expected total effort
At least half of optimum
35
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)
36
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:
37
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)
38
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
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