1. Efficiency and the Redistribution of Welfare Milan Vojnovic Microsoft Research Cambridge, UK Joint work with Vasilis Syrgkanis and Yoram Bachrach

2. Contribution Incentives Rewards for contributions Credits Social gratitude Monetary incentives Online services Ex. Quora, Stackoverflow, Yahoo! Answers Other Scientific authorship Projects in firms 2

3. QuestionTopicSite 3

4. Another Example: Scientific Co-Authorship 4

5. Some Observations User contributions create value Ex. quality of the content, popularity of the generated content Value is redistributed across users Ex. Credits, attention, monetary payments Implicit and explicit signalling of individual contributions Ex. User profile page, rating scores, etc Ex. Wikipedia – not in an article, but by side means [Forte and Bruckman] Ex. Author order on a scientific publication 5

6. How efficient are simple local value sharing schemes with respect to social welfare of the society as a whole? 6

7. Outline Game Theoretic Framework Efficiency of Monotone Games under a Vickery Condition Efficiency of Equal and Proportional Sharing Production Costs Conclusion 7

8. Utility Sharing Game 8

9. Project Contribution Games 1 2 i n 1 2 j m Share of value 9

10. Monotone Games 10

11. Importance of Monotonicity 0 1 1 11

12. Vickery Condition Rewarded at least one’s marginal contribution 12

13. Local Value Sharing 13

14. DBLP database 2,132,763 papers 1,231,667 distinct authors 7,147,474 authors Scientific Co-Authorships 14

15. Scientific Co-Authorship (cont’d) 15 o random

16. Solution Concepts & Efficiency Nash Equilibrium (NE) Unilateral deviations Strong Nash Equilibrium (SNE) All possible coalitional deviations Bayes Nash Equilibrium (BNE) Incomplete information game Efficiency Worst case ratio of social welfare in an equilibrium and optimal social welfare 16

17. Outline Game Theoretic Framework Efficiency of Monotone Games under a Vickery Condition Efficiency of Equal and Proportional Sharing Production Costs Conclusion 17

18. Efficiency in Strong Nash Equilibrium 18

19. Efficiency in Nash Equilibrium 19

20. Local Vickery Condition } degree of substitutability 20

21. Degree of Substitutability 21

22. Degree of Substitutability (cont’d) 1 2 1 2 Budget 1 } 22

23. Outline Game Theoretic Framework Efficiency of Monotone Games under a Vickery Condition Efficiency of Equal and Proportional Sharing Production Costs Conclusion 23

24. Equal Sharing 24

25. Proportional Sharing 25

26. Proof Sketch 26

27. Local Smoothness 27

28. Efficiency of Smooth Games 28

29. Sufficient Condition for Smoothness 29

30. Efficiency by Smoothness: Fractional Exponent Functions 30

31. Efficiency by Smoothness: Exponential Value Functions 31

32. Tight Example 1 2 1 2 } 32

33. Tight Example (cont’d) 1 2 1 2 } 33

34. Efficiency and Incomplete Information 34

35. Universal Smoothness 35

36. Efficiency under Universal Smoothness 36

37. Outline Game Theoretic Framework Efficiency of Monotone Games under a Vickery Condition Efficiency of Equal and Proportional Sharing Production Costs Conclusion 37

38. Production Costs production cost Budget constraint (earlier slides) Constant marginal cost A convex increasing function Examples 38

39. Elasticity 39

40. Efficiency 40

41. Efficiency (cont’d) 41

42. Conclusion When the wealth is redistributed so that each contributor gets at least his marginal contribution locally at each project, the efficiency is at least ½ The degree of complementarity of player’s contributions plays a key role: the more complementary the worse Simple local value sharing Equal sharing: the efficiency is at least 1/k, where k is the maximum number of participants in a project Proportional sharing: guarantees the efficiency of at least ½ for any concave project value functions of the total contribution Production costs play a major function: the case of linear production costs is a special case for which the inefficiency can be arbitrarily small; at least a positive constant for any convex cost function of strictly positive elasticity 42