Characterizing distribution rules for cost sharing games

1. Characterizing distribution rules for cost sharing games Thesis defense May 28, 2013 by Raga GopalakrishnanComputing and Mathematical Sciences, Caltech

2. Distributed resource allocation problems • Allocate scarce resources to distributed agents, such that some objective function is optimized. • Several examples throughout this talk…

3. Example: Network formation [ Jackson 2003 ] D1 6 S1 1 1 6 1 1 S2 6 D2 • Problems: • NP-hard, not scalable, not reliable, … • Agents are people; not everyone may be happy with the outcome. • Centralized optimization: • Build the optimal network (cost 10). • Recover this cost from S1 and S2.

4. Example: Network formation [ Jackson 2003 ] D1 6 S1 1 1 6 1 1 S2 6 D2 • Distributed solution: • Let sources play a noncooperative game by choosing the edges they want and pay for them. • Q: How to share the cost of the common edge? Outcome depends on how this cost is shared!

5. Example: Network formation game [ Anshelevich et al. 2004 ] • Q: How to share the cost of the common edge? • Option 1: S1 pays 5 • S2 pays 1 • Option 2: S1 pays 3 • S2 pays 3 D1 D1 6 6 S1 S1 1 1 1 1 3+3 5+1 1 1 1 1 S2 S2 D2 6 6 D2 sub-optimal Nash equilibrium optimal network is a Nash equilibrium

6. Cost sharing games D1 ? S1 ? ? ? ? ? ? ? S2 D2 Multi-project management Network formation [ Anshelevich et al. 2004 ] Common Feature: The distribution rules used to share the global cost/revenue determine the players’ local utility functions, ? ? Facility location [ Vetta 2002 ]

7. Cost sharing games D1 ? S1 ? ? ? ? ? ? ? S2 D2 Multi-project management Network formation [ Anshelevich et al. 2004 ] Common Feature: The distribution rules used to share the global “welfare” determine the players’ local utility functions, and hence the outcome! ? ? Facility location [ Vetta 2002 ]

8. Cost sharing games D1 ? S1 ? ? ? ? ? ? ? S2 D2 Multi-project management Network formation [ Anshelevich et al. 2004 ] Goal: Design distribution rules that result in a “desirable” outcome. ? ? Facility location [ Vetta 2002 ]

9. Goal of the thesis: Characterize the space of all distribution rules that result in a “desirable” outcome for a broad class of cost sharing games. D1 ? S1 ? ? ? ? ? [ G., Marden, Wierman. NetGCooP 2011. ] [ G., Marden, Wierman. EC 2013. ] [ full version under submission to MOR. ] ? ? S2 D2 ? ? Multi-project management Network formation Facility location

10. Goal of the thesis: Characterize the space of all distribution rules that result in a stable outcome for a broad class of cost sharing games. D1 ? S1 ? ? ? ? ? [ G., Marden, Wierman. NetGCooP 2011. ] [ G., Marden, Wierman. EC 2013. ] [ full version under submission to MOR. ] ? ? S2 D2 “equilibrium” +efficiency, +tractability, … ? ? Multi-project management Network formation Facility location

11. Variations – distributed resource allocation 1 Scheduling games in service systems Focus: Designing dispatch policies [ G., Doroudi, Wierman. MAMA 2011. ] 2 Staffing games in service systems Focus: Investigating square-root staffing policy for large systems [ Working paper, in preparation for submission to OR. ] Joint work with Ward and Wierman. Scenario: Heterogeneous multi-server system m1 FCFS l m2 scheduling ? staffing ? mm

12. Variations – distributed resource allocation Content caching and replication Focus: Investigating for and computing pure Nash equilibria [ G., Kanoulas, Karuturi, Rangan, Rajaraman, Sundaram. LATIN 2012. ] Scenario: Content distribution network ? ? ? ? ? ? ? ? ?

13. Variations – distributed resource allocation • No people involved! Objective function is not cost/revenue. • pretend that agents are people, model them as players in a “cost” sharing game. Game theoretic control Focus: Designing the entire game [ G., Marden, Wierman. HotMetrics 2010. ] Scenario: Distributed control frequency frequency ? F2 F2 F1 F3 F1 F3 ? ? Wireless access point assignment Wireless frequency selection Sensor coverage

14. Variations – distributed resource allocation Bandwidth allocation in multi-tenant datacenters Focus: Designing a robust bandwidth allocation scheme [ working paper ] Joint work with Attar, Jeyakumar, Narayana, Prabhakar. ? VM VM ? VM VM Scenario: Bandwidth allocation in datacenter networks ? ? VM VM ? VM VM

15. Goal of the thesis: Characterize the space of all distribution rules that result in a stable outcome for a broad class of cost sharing games. D1 ? S1 ? ? ? ? ? [ G., Marden, Wierman. NetGCooP 2011. ] [ G., Marden, Wierman. EC 2013. ] [ full version under submission to MOR. ] ? ? S2 D2 ? ? Multi-project management Network formation Facility location

16. Model players (researchers)

17. Model players (researchers) resources (projects)

18. Model

19. Model Global welfare: Denote the set of all (distinct) local welfare functions:

20. Model Scalability assumption: Denote by

21. Model Scalability assumption: Denote by

22. Model Scalability assumption: Denote by

23. Model An allocation Players’ overall utility is the sum of their shares across all the resources they choose Example:

24. Model Design! An allocation Players’ overall utility is the sum of their shares across all the resources they choose Example:

25. Model a broad model that includes: • Multi-project management • Network formation games • Facility location games • Multicast games • Congestion games • Routing games • Coverage games • … [ Anshelevich et al. 2004 ] [ Vetta 2002 ] [ Chekuri et al. 2007 ] [ Rosenthal 1973 ] [ Roughgarden and Tardos 2002 ] [ Marden and Wierman 2008,2013 ]

26. Goal:

27. Goal: Given the set of players

28. Goal: Given the set of players and the local welfare functions,

29. Goal: Given the set of players and the local welfare functions, design local distribution rules

30. Goal: • Given the set of players and the local welfare functions, design local distribution rules that result in a “desirable” outcome

31. Goal: • Given the set of players and the local welfare functions, design local distribution rules • that result in a “desirable” outcome regardless of the set of resources and players’ action sets.

32. Class of all games with set of players , set of local welfare functions , corresponding local distribution rules Goal: • Given the set of players and the local welfare functions, design local distribution rules • that result in a “desirable” outcome regardless of the set of resources and players’ action sets.

33. Class of all games with set of players , set of local welfare functions , corresponding local distribution rules Goal: Given any and , design • such that all games in are “desirable”.

34. “desirable” properties for a game • Static: • Stability • Equilibrium concepts • Efficiency • Price of anarchy • Price of stability • Budget-balance • Tractability • Poly-time computable • Locality • Separable • … • Dynamic: • Existence of distributed learning rules that converge to one of the stable outcomes • Existence of distributed learning rules that converge to the most efficient stable outcome • Good convergence rates • Simple, intuitive learning dynamics perform well, e.g., “best-response” • …

35. “desirable” properties for a game • Static: • Stability • Equilibrium concepts • Efficiency • Price of anarchy • Price of stability • Budget-balance • Tractability • Poly-time computable • Locality • Separable • … • Dynamic: • Existence of distributed learning rules that converge to one of the stable outcomes • Existence of distributed learning rules that converge to the most efficient stable outcome • Good convergence rates • Simple, intuitive learning dynamics perform well, e.g., “best-response” • …

36. “desirable” properties for a game • Static: • Stability • Equilibrium concepts

37. “desirable” properties for a game • Static: • Stability • pure Nash equilibrium • dominant strategy equilibrium • mixed Nash equilibrium • correlated equilibrium • coarse-correlated equilibrium • … An allocation/outcome that satisfies:

38. “desirable” properties for a game • Static: • Stability • pure Nash equilibrium • Efficiency An allocation/outcome that satisfies: • Price of Anarchy (PoA) • Ratio of the optimum welfare to the welfare of the worst Nash equilibrium. • Price of Stability (PoS) • Ratio of the optimum welfare to the welfare of the bestNash equilibrium. Goal: Given any and , design such that all games in are “desirable”.

39. “desirable” properties for a game • Static: • Stability • pure Nash equilibrium • Efficiency An allocation/outcome that satisfies: • Price of Anarchy (PoA) • Ratio of the optimum welfare to the welfare of the worst Nash equilibrium. • Price of Stability (PoS) • Ratio of the optimum welfare to the welfare of the bestNash equilibrium. Goal: Given any and , design such that all games in possess a pure Nash equilibrium.

40. Existing distribution rules • The Shapley value[ Shapley 1953 ] • A player’s share of the welfare should depend on their“average” marginal contribution • Intuition: • Imagine the players in arriving one at a time to the resource, according to some order. • When player arrives, depending on when he arrived, he sees some subset of players already present. • Player can be thought of as contributing . • The Shapley value is his expected marginal contribution over all possible orders, assuming each order is equally likely. • Example: when all players are ‘identical’, .

41. Properties of the Shapley value • Static: • Stability • pure Nash equilibrium • Efficiency • Price of anarchy • Price of stability • Budget-balance • Tractability • Poly-time computable • Locality • Separable • … • Dynamic: • Existence of distributed learning rules that converge to one of the stable outcomes • Existence of distributed learning rules that converge to the most efficient stable outcome • Good convergence rates • Simple, intuitive learning dynamics perform well, e.g., “best-response” • …

42. Properties of the Shapley value • Static: • Stability • pure Nash equilibrium • Efficiency • Price of anarchy • Price of stability • Budget-balance • Tractability • Poly-time computable • Locality • Separable • … • Dynamic: • Existence of distributed learning rules that converge to one of the stable outcomes • Existence of distributed learning rules that converge to the most efficient stable outcome • Good convergence rates • Simple, intuitive learning dynamics perform well, e.g., “best-response” • …

43. Properties of the Shapley value • Static: • Stability • pure Nash equilibrium • Efficiency • Price of anarchy • Price of stability • Budget-balance • Tractability • Poly-time computable • Locality • Separable • … • Dynamic: • Existence of distributed learning rules that converge to one of the stable outcomes • Existence of distributed learning rules that converge to the most efficient stable outcome • Good convergence rates • Simple, intuitive learning dynamics perform well, e.g., “best-response” • …

44. “desirable” properties for a game • Static: • Stability • pure Nash equilibrium • Dynamic: • Existence of distributed learning rules that converge to one of the stable outcomes • Existence of distributed learning rules that converge to the most efficient stable outcome • Good convergence rates • Simple, intuitive learning dynamics perform well, e.g., “best-response” Potential game There exists a player-independent function , called the potential function, which encodes utility differences due to unilateral deviations by any player.

45. Properties of the Shapley value • Potential game • guarantees stability (pure Nash equilibrium), good dynamic properties • Efficiency • Price of anarchy • Price of stability • Budget-balance • Tractability • Poly-time computable • Locality • Separable • …

46. Properties of the Shapley value • Potential game • guarantees stability (pure Nash equilibrium), good dynamic properties • Efficiency • Price of anarchy • Price of stability • Budget-balance • Tractability • Poly-time computable • Locality • Separable • … • Optimal in many specific settings, e.g., network coding, network formation. [ Marden and Effros 2009 ] [ Roughgarden 2009] • Lower bound of 2 for submodular welfare functions. [ Marden and Wierman 2013 ]

47. Properties of the Shapley value • Potential game • guarantees stability (pure Nash equilibrium), good dynamic properties • Efficiency • Price of anarchy • Price of stability • Budget-balance • Tractability • Poly-time computable • Locality • Separable • …

48. Properties of the Shapley value • Potential game • guarantees stability (pure Nash equilibrium), good dynamic properties • Efficiency • Price of anarchy • Price of stability • Budget-balance • Tractability • Poly-time computable • Locality • Separable • … • Requires computing exponentially many marginal contributions • Intractable in general [ Matsui and Matsui 2000 ]

49. Existing distribution rules • Extensions of the Shapley value • Weighted Shapley value : : • Parameterized by a vector of strictly positive player weights • Expected marginal contribution according to a probability distribution (that depends on ) with full support on all orders. • Generalized weighted Shapley value : : • Parameterized by a weight system , where is a vector of strictly positive player weights, and is an ordered partition of the set of players. • Expected marginal contribution according to a probability distribution (that depends on ) with support only on those orders that respect : players in arrive before players in when . • They lead to weighted/generalized weighted potential games. Their properties are similar to those of the Shapley value.

50. Existing distribution rules • The marginal contribution • [ Wolpert and Tumer 1999 ] • A player’s share of the welfare is their • marginal contribution to the welfare • Similar extensions:weighted marginal contribution ( ) and generalized weighted marginal contribution ( ) can be defined.