The Weighted Proportional Resource Allocation

1. The Weighted Proportional Resource Allocation Milan Vojnović Microsoft Research Joint work with Thành Nguyen University of Cambridge, Oct 18, 2010

2. Resource Allocation Problem provider users • Resource with general constraints • Ex. network service, data centre, sponsored search • Everyone is selfish: • Provider wants large revenue • Each user wants large surplus (utility – cost) Resource

3. Resource Allocation Problem (cont’d) providers users • Multiple providers competing to provide service to users • Everyone is selfish 1 2 m

4. Desiderata • Simple auction mechanisms • Small amount of information signalled to users • Easy to explain to users • Accommodate resources with general constraints • High revenue and social welfare • Under “everyone is selfish”

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

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

7. The Weighted Resource Allocation (cont’d) • Similar results hold also for “weighted payment” auction (Ma et al, 2010); an auction for specific resource constraints; results not presented in this slide deck • Weighted Payment Auction: • Provider announces discrimination weights • Each user i submits a bid wi Payment = Ciwi Allocation: • C = resource capacity

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

9. Ex 1: Network Service provider users

10. Ex 1: Network Service (cont’d) provider users

11. Ex 1: Network Service (cont’d)

12. Ex 2: Compute Instance Allocation • Multi-machine multi-job 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

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

14. Ex 3: Sponsored Search (cont’d) • xi = click-through-rate for 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)

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

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

17. User’s Objective • Price-taking: given price pi, user i solves: • Price-anticipating: given Ci and , user i solves:

18. Provider’s Objective • Choose discrimination weights to maximize own revenue

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

20. Social Optimum • Social optimum allocation is a solution to

21. Equilibrium: Price-Taking Users • Revenue • Provider chooses discrimination weights where maximizes over • Equilibrium bids • Same revenue as under third-degree price discrimination

22. Equilibrium: Price-Anticipating Users • Revenue R given by: • Provider chooses discrimination weights where maximizes over • Equilibrium bids

23. 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

24. 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)

25. 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)

26. 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.

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

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

29. 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 worst-case set of k users excluded, i.e.In particular:

30. Proof Key Idea • Sufficient condition: for every there exists and

31. 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)

32. Worst-Case • Utilities: • Nash eq. allocation:

33. Proof Key Ideas • Utilities: Q P

34. Summary of Results • 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 • In contrast to market-clearing where worst-case efficiency is 0

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

36. Oligopoly: Multiple Competing Providers 1 2 m providers users

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

38. 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 :

39. Examples of d-Utility Functions “a-fair”

40. Social Welfare • Theorem For price-anticipating users with d-utility functions and oligopoly of competing providers: (Nash eq. utility) (socially OPT utility) • 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%

41. Proof Key Ideas

42. Conclusion • Established revenue and social welfare properties of weighted proportional resource allocation in competitive settings where everyone is selfish • Identified cases with competitive revenue and social welfare • The revenue is at least k/(k+1) times the revenue under third-degree price discrimination with a set of k users excluded • Under linear utility functions, efficiency is at least 46.41%; tight worst case • Efficiency lower bound generalized to multiple competing providers and a general class of utility functions

43. To Probe Further • The Weighted Proportional Allocation Mechanism, Microsoft Research Technical Report, MSR-TR-2010-145