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Some Maths of use in the Computer Systems World

Some Maths of use in the Computer Systems World. Milan Vojnović Microsoft Research Cambridge, UK. CCA Industrial Seminar, University of Cambridge, October 25, 2012. Outline. Online contests Distributed systems Algorithms for big data. TCO Contest Design. Some Questions.

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Some Maths of use in the Computer Systems World

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  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

  3. TCO Contest Design

  4. 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 !

  5. Standard All-Pay Contest ) Prize allocation

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

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

  8. 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

  9. 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

  10. 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)

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

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

  13. 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:

  14. 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]

  15. 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

  16. 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

  17. Several Job Offers

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

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

  20. 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

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

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

  23. 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]

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

  25. 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)

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

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

  28. 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 ?

  29. 0 1 1 0 0 1 0 1 1 • Each node to correctly decide whether 0 or 1 is preferred by majority of nodes

  30. 0 1 0 0 0 1 0 1 1

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

  32. 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

  33. 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

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

  35. 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

  36. Probability of Error • Theorem: Assume ,

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

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

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

  40. 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]

  41. The Limit ODE

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

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