The Weighted Proportional Allocation Mechanism

1. The Weighted Proportional Allocation Mechanism Milan Vojnović Microsoft Research Joint work with Thành Nguyen Harvard University, Nov 3, 2009

2. Resource allocation problem provider users • Provider wants large revenue • User wants large surplus (utility – cost) • Resource with general constraints • Ex. network service, data centre, sponsored search Resource

3. Resource allocation problem (cont’d) providers users • Oligopoly – multiple providers competing to provide service to users • Each provider wants a large revenue 1 2 m

4. Desiderata • Simple auction mechanism • Small amount of information signalled to users • Easy to explain / understand by users • Accommodate resources with general constraints • High revenue and social welfare • Under strategic providers and strategic users

5. Outline • The mechanism • Applications • Game-theory framework and related work • Revenue and social welfare • Monopoly under linear utility functions • Generalization to an oligopoly and more general utility functions • Conclusion

6. The mechanism • Provider announces discrimination weights • Each user i submits a bid wi Payment by user i = wi Allocation to user i: • Discrimination weights so that allocation is feasible

7. Resource constraints • An allocation said to be feasible if where P is a polyhedron, i.e. for some matrix A and vector • Accommodates complex resources such as network of links, data centres, sponsored search Ex. n = 2 P

8. Ex 1: Network service provider users

9. Ex 1: Network service (cont’d) provider users

10. Ex 1: Network service (cont’d)

11. Ex 2: data centre resource allocation • Multi-job task scheduling • xi = 1 / (finish time for job i) • si,m = processing speed for job i at machine m • di,m = workload for job i at machine m task jobs

12. Ex 3. Sponsored search • Generalized Second Price Auction • Discrimination weights = click-through-rates • Assumes click-through-rates independent of which ads appear together

13. Ex 3: Sponsored search (cont’d) • xi = click-through-rate at slot i • Say $1 per click, so Ui(x) = x • GSP revenue: • Max weighted prop. revenue: (0,14) (4,5) (5,4) (6,0) (0,0)

14. Ex. 3: Sponsored Search (cont’d) • Revenue of weighted proportional allocation

15. Outline • The mechanism • Applications • Game-theory framework and related work • Revenue and social welfare • Monopoly under linear utility functions • Generalization to an oligopoly and more general utility functions • Conclusion

16. User’s objective • Price-taking – given price pi, user i solves: • Price-anticipating – given Ci and , user i solves:

17. Provider’s objective • Choose discrimination weights to maximize the revenue

18. Provider’s objective (cont’d) • Maximizing revenue also objective of some pricing schemes • Ex. well-known third-degree price discrimination • Assumes price taking users = price per unit resource for user i

19. Social optimum • Social optimum allocation is a solution to

20. Equilibrium: price-taking users • Revenue • Provider chooses discrimination weights where maximizes over • Equilibrium bids • Same revenue as under third-degree price discrimination

21. Equilibrium: price-anticipating users • Revenue R given by: • Provider chooses discrimination weights where maximizes over • Equilibrium bids

22. Related work • Proportional resource sharing – ex. generalized proportional sharing (Parkeh & Gallager, 1993) • Proportional allocation for network resources (Kelly, 1997) where for each infinitely-divisible resource of capacity C • No price discrimination • Charging market-clearing prices

23. Related work (cont’d) • Theorem (Kelly, 1997) For price-taking users with concave, utility functions, efficiency is 100%. • Assumes “scalar bids” = each user submits a single bid for a subset of resources (ex. single bid per path)

24. Related work (cont’d) • Theorem (Johari & Tsitsiklis, 2004) For price-anticipating users with concave, non-negative utility functions and vector bids, efficiency is at least 75%: • The worst-case achieved for linear utility functions. • Vector bids = each user submits individual bid per each resource (ex. single bid for each link of a path) (Nash eq. utility) (socially OPT utility)

25. Related work (cont’d) • Theorem (Hajek & Yang, 2004) For price-anticipating users with concave, non-negative utility functions and scalar bids, worst-case efficiency is 0.

26. Related work (cont’d) • Worst-case: serial network of unit capacity links

27. Outline • The mechanism • Applications • Game-theory framework and related work • Revenue and social welfare • Monopoly under linear utility functions • Generalization to an oligopoly and more general utility functions • Conclusion

28. Revenue • Theorem For price-anticipating users, if for every user i, is a concave function, thenwhere R-k is the revenue under third-degree price discrimination with a set of k users excluded, i.e.In particular:

29. Example • Unit-capacity resource: • Symmetric users with utility function U(x) • U(x) concave, and U’(x)x concave increasing on [0,1] revenue under third-degree price discrimination Ex.

30. Social welfare • Theorem For price-anticipating users with linear utility functions, efficiency > 46.41%:This bound is tight. • Worst-case: many users with one dominant user. (Nash eq. utility) (socially OPT utility)

31. Worst-case • Utilities: • Nash eq. allocation:

32. Proof key ideas • Utilities: P

33. Summary of properties • Competitive revenue and social welfare under linear utility functions and monopoly of a single provider • Revenue at least k/(k+1) times the revenue under third-degree price discrimination with a set of k users excluded • Efficiency at least 46.41%; tight worst case • Unlike to market-clearing where worst-case efficiency is 0

34. Outline • The mechanism • Applications • Game-theory framework and related work • Revenue and social welfare • Monopoly under linear utility functions • Generalization to an oligopoly and more general utility functions • Conclusion

35. Oligopoly: multiple competing providers 1 2 m providers users

36. Oligopoly (cont’d) • User i problem: choose bids that solve • Provider k problem: choose that maximize the revenue Rk over Pk where

37. d-utility functions • Def. U(x) a d-utility function: • Non-negative, non-decreasing, concave • U’(x)x concave over [0,x0]; U’(x)x maximum at x0 • For every :

38. Examples of d-utility functions “a-fair”

39. Social welfare • Theorem For price-anticipating users with d-utility functions and oligopoly of competing providers: • The worst-case achieved for linear utility functions. • The bound holds for any number of users n and any number of providers m. • Ex. for d = 1, 2, worst-case efficiency at least 31, 24% (Nash eq. utility) (socially OPT utility)

40. Proof key ideas • Bounding social welfare by an affine function separates to optimizations for individual providers • For provider k consider linear utility functions where

41. Conclusion • Proposed weighted proportional allocation mechanism • Simple; applies to general polyhedron constraints • Offers competitive revenue and social welfare • The revenue at least k/(k+1) times that under third-degree price discrimination with a set of k users excluded • Under linear utility functions, efficiency at least 46.41%; tight worst case • Efficiency lower bound generalized to an oligopoly of multiple competing providers and a general class of utility functions

42. To Probe Further • The Weighted Proportional Allocation Mechanism, Microsoft Research Technical Report, MSR-TR-2009-123