Some Maths of use in the Computer Systems World

1. Some Maths of use in the Computer Systems World Milan Vojnović Microsoft ResearchCambridge, UK CCA Industrial Seminar, University of Cambridge, October 25, 2012

2. Outline • Online contests • Distributed systems • Algorithms for big data

10. TCO Contest Design

11. Some Questions • How do we best design an online contest? • How to allocate prizes? • How to infer skills of players? • There is, in fact, quite some maths behind it !

12. Standard All-Pay Contest ) Prize allocation

13. Game Model Payoff: Valuation or ability or skill Winning probability Production cost

14. Game Model (cont’d) • Strategically equivalent towhere = marginal production cost

15. Types of Games • Complete information: assumed to be common knowledge • Incomplete information (or “private values”): assumed to be a sample from a distribution , where is a common knowledge • Special case: i. i. d. valuations

16. Nash Equilibrium • A vector of efforts is said to be a pure-strategy Nash equilibrium if for every player : • Mixed-strategy: players use randomized strategies

17. Complete Information Game • There exists no pure-strategy Nash equilibrium • There exist a mixed-strategy Nash equilibrium, not necessarily unique • Fully characterized by Baye, Kovenock and de Vries (1996)

18. Example of Two Players • Equilibrium bid distributions 0 0

19. Example of Two Players (cont’d) • Expected total effort: • Expected maximum individual effort:

20. Incomplete Information Game • Assume that valuations are i.i.d. according to distribution F on [0,1] • There exists a symmetric pure-strategy Nash equilibrium • Total expected effort:

21. Optimality of the Winner-Take-All • Suppose a contest owner wants to maximize the expected total effort, or maximum individual effort • Theorem: For the contest among players with i.i.d. valuations, it is optimal to allocate the whole prize to the best performing player[Moldovanu and Sela 2001, Chawla, Hartline and Sivan 2012]

22. Optimal Contest • For a contest among players with i. i. d. valuations according to distribution the expected maximum individual effort is maximized by that maximizes where[Chawla, Hartline and Sivan, 2012] • The result follows from celebrated revenue equivalence theorem [Myerson, 81] • Replacing with corresponds to maximizing the expected total effort virtual valuation

23. Optimal Contest (cont’d) • Suppose is a monotone non-decreasing function for every positive integer • The optimal all-pay contest uses the winner-take-all prize allocation rule with minimum valuation • Ex. : the minimum required effort

24. Several Job Offers

25. Parallel Contests [DiPalantino and V., 2009]

26. Parallel Contests • There exists a symmetric Bayes-Nash equilibrium • Segmentation into skill levels

27. Reward vs. Participation • In a many auctions limit, the expected number of participants per contest: • = expected number per auction • = fraction of auctions of class • Diminishing returns of participation with reward

28. Does this Make Any Practical Sense ? • Rewards vs. participation observed at Taskcn any rate once a month every fourth day every second day

30. Statistical Inference of Skill • Probabilistic model: skill to performance • Ex. where represents skill of player • Ex 1. probit • Ex 2. logit

31. Maximum Likelihood Approach • Ex. Bradley-Terry Model [Zermelo, 1029] • Iterative solver: given an irreducible matrix of outcomes , and initial value :Guaranteed convergence to unique limit point (up to a multiplicative constant) [Ford, 1957]

32. Online Algorithms • Elo Rating System: • More complicated variant • Arbitrary number of participants in a contest

33. Bayesian Approach • Skill assumed to be a random variable • Ex. • Posterior distribution adjusted based on observed match outcomes • Ex. Graphical model • Examples: • Glicko rating system • TrueSkill™ (Xbox Halo game)

34. Outline • Online contests • Distributed systems • Algorithms for big data

35. A Catalogue of Problems • Ranking of information items • Information diffusion • Opinion formation • Distributed hypothesis testing

36. Distributed Mode Selection • Majority selection: given two alternatives, find the majority winner (a.k.a. consensus, quantile) • Plurality selection: given alternatives, find a plurality winner • Distributed preference data across nodes in a system • How do simple algorithms with bounded memory and communication perform ?

37. 0 1 1 0 0 1 0 1 1 • Each node to correctly decide whether 0 or 1 is preferred by majority of nodes

38. 0 1 0 0 0 1 0 1 1

39. Questions • Design an efficient algorithm • Probability of correctness • Convergence speed

40. Related WorkClassical Voter Model • Note copies the state of the contacted node • Binary memory and communication • Probability of incorrect outcome: [More general result: Hassin and Peleg, 2001] 0 1 1 0 0 1 0 1

41. Related Workm-ary Hypothesis Testing • Q: How many states does S need to decide the correct hypothesis w. p. going to 1 with the number of observations ? • m+1 necessary and sufficient (Koplowitz, 1975) 000110111110100011 Hi S i. i. d. mean

42. Ternary Algorithm • Three states: 0, e, 1 0 0 e 1 e 1 0 0 e e 0 1 [Perron, Vasudevanand V., 2009]

43. Ternary Dynamics • Assume complete graph • Markov process • U = number of nodes in state 0 • V = number of nodes in state 1 • n = total number of nodes

44. Probability of Error • Theorem: Assume ,

45. Probability of Error (Cont’d) • Lemma: is a solution ofwith boundary conditions for , and , for • Equivalence to the probability of hitting

46. Ballot Theorem Argument Number of paths from to that do not intersect the line

47. The Error Exponent • Theorem: For , , large • Ob.: Exponential decay for large

48. Plurality Selection • m alternatives • 2m states: weak strong … m 2 1 s s s’ s’ s’ s’ s’ s’ s s s s s’ s’ observer [Jung, Kim and V., 2012]

49. The Limit ODE

50. Convergence Time • Suppose • Given is -convergence time: