Characterizing distribution rules for cost sharing games

1. Characterizing distribution rules for cost sharing games Raga GopalakrishnanCaltech Joint work with Jason R. Marden & Adam Wierman

2. Cost sharing games: Self-interested agentsmake decisions, and share the incurred cost among themselves. • Key Question: How should the cost be shared? Lots of examples: Network formation games Facility location games Profit sharing games

3. Cost sharing games: Self-interested agentsmake decisions, and share the incurred cost among themselves. • Key Question: How should the cost be shared? Lots of examples: Network formation games Facility location games Profit sharing games D1 S1 S2 D2

4. Cost sharing games: Self-interested agentsmake decisions, and share the incurred cost among themselves. • Key Question: How should the cost be shared? Lots of examples: Network formation games Facility location games Profit sharing games

5. Cost sharing games: Self-interested agentsmake decisions, and share the incurred cost among themselves. • Key Question: How should the cost be shared? Lots of examples: Network formation games Facility location games Profit sharing games

6. Cost sharing games: Self-interested agentsmake decisions, and share the incurred cost among themselves. • Key Question: How should the cost be shared? Lots of examples: Network formation games • [Jackson 2003][Anshelevich et al. 2004] Facility location games • [Goemans et al. 2000] [Chekuri et al. 2006] Profit sharing games • [Kalai et al. 1982] [Ju et al. 2003] Huge literature in Economics Growing literature in CS New application: Designing for distributed control [Gopalakrishnan et al. 2011][Ozdaglar et al. 2009][Alpcan et al. 2009]

7. Cost sharing games (more formally): utility function of agent welfare function set of resources action set of agent set of agents/players D1 S1 Example: S2 D2

8. Cost sharing games (more formally): utility function of agent welfare function set of resources action set of agent set of agents/players Assumption: is separable across resources set of agents choosing resource in allocation

9. Cost sharing games (more formally): welfare function at resource utility function of agent set of resources action set of agent set of agents/players Assumption: is scalable common base welfare function

10. Cost sharing games (more formally): utility function of agent resource-specific coefficients set of resources welfare function action set of agent set of agents/players

11. Cost sharing games (more formally): utility function of agent resource-specific coefficients set of resources welfare function action set of agent set of agents/players Assumption: Utility functions are also separable/scalable common base distribution rule (portion of welfare at to agent )

12. Cost sharing games (more formally): resource-specific coefficients distribution rule set of resources welfare function action set of agent set of agents/players Goal: Design the distribution rule

13. Requirements on the distribution rule The distribution rule should be: • Budget-balanced • “Stable” and/or “Fair” • “Efficient”

14. Requirements on the distribution rule The distribution rule should be: • Budget-balanced • “Stable” and/or “Fair” • “Efficient”

15. Requirements on the distribution rule The distribution rule should be: • Budget-balanced • “Stable” and/or “Fair” • “Efficient” Lots of work on characterizing “stability” and “fairness” Core Nash equilibrium [von Neumann et al. 1944] [Nash 1951] [Moulin 1992] [Albers et al. 2006] [Gillies 1959] [Devanur et al. 2003] [Chander et al. 2006]

16. Requirements on the distribution rule The distribution rule should be: • Budget-balanced • “Stable” and/or “Fair” • “Efficient” Lots of work on characterizing “stability” and “fairness” Core Nash equilibrium [von Neumann et al. 1944] [Nash 1951] [Moulin 1992] [Albers et al. 2006] [Gillies 1959] [Devanur et al. 2003] [Chander et al. 2006]

17. Requirements on the distribution rule The distribution rule should be: • Budget-balanced • “Stable” and/or “Fair” • “Efficient” Has good Price of Anarchy and Price of Stability properties

18. The Shapley value [Shapley 1953] A player’s share of the welfare should depend on their“average” marginal contribution Example: If players are homogeneous, Note: There is also a weighted Shapley value Players are assigned ‘weights’

19. Properties of the Shapley value + Guaranteed to be in the core for “balanced” games • [Shapley 1967] • + Results in a potential game • [Ui 2000] + Guarantees the existence of a Nash equilibrium • Often intractable to compute • [Conitzer et al. 2004] • Not “efficient” in terms of social welfare e.g. Price of Anarchy/Stability • [Marden et al. 2011] approximations are often tractable [Castro et al. 2009]

20. Research question: Are there distribution rules besides the (weighted) Shapley value that always guarantee a Nash equilibrium? If so: can designs be more efficient and/or more tractable? If not: we can optimize over to determine the best design!

21. Research question: Are there distribution rules besides the (weighted) Shapley value that always guarantee a Nash equilibrium? Our (surprising) answer: NO, for any submodular welfare function. “decreasing marginal returns” natural way to model many real-world problems

22. The inspiration for our work • Theorem (Chen, Roughgarden, Valiant): • There exists a welfare function , for which no distribution rules other than the weighted Shapley value guarantee a Nash equilibrium in all games. • [Chen et al. 2010] A game is specified by Our result • Theorem: • For any submodular welfare function , no distribution rules other than the weighted Shapley value guarantee a Nash equilibrium in all games.

23. The inspiration for our work • Theorem (Chen, Roughgarden, Valiant): • Given all games posses aNash equilibrium if and only if is a weighted Shapley value. • [Chen et al. 2010] Our result • Theorem: • Given and any submodularall games posses a Nash equilibrium if and only if is a weighted Shapley value.

24. Our result • Theorem: • For any submodular welfare function , no distribution rules other than the weighted Shapley value guarantee a Nash equilibrium in all games. Consequences • Can obtain the best distribution rule by optimizing the player weights, • Can always work within a potential game • Small, well-defined class of games • Several learning algorithms for Nash equilibrium • Fundamental limits on tractability and efficiency

25. Proof Sketch First step: Represent using a linear basis • Define a -welfare function: • Given any , there exists a set , and a sequence of weights indexed by , such that: Proof technique: Establish a series of necessary conditions on “magnitude of contribution” “contributing coalition”

26. Proof Sketch (A single T-Welfare Function) • Proof technique: Establish a series necessary conditions on is not formed in Don’t allocate welfare to any player What is required of Allocate welfare only to players in , independent of others • is formed in is completely specified by is a weighted Shapley value

27. Proof Sketch (General Welfare Functions) • Proof technique: Establish a series necessary conditions on no coalition from is formed in Don’t allocate welfare to any player What is required of Allocate welfare only to players in these formed coalitions, independent of others a coalition from is formed in is the basis weighted Shapley value corresponding to , with weights Key challenge: Each basis might use different !

28. Proof Sketch (General Welfare Functions) • Proof technique: Establish a series necessary conditions on no coalition from is formed in Don’t allocate welfare to any player What is required of Allocate welfare only to players in these formed coalitions, independent of others a coalition from is formed in Weights of common players in any two coalitions must be linearly dependent is a weighted Shapley value is submodular

29. Research question: Cost Sharing Games Are there distribution rules besides the (weighted) Shapley value that always guarantee a Nash equilibrium? Our answer: NO, for any submodular welfare function. what about for other welfare functions? Understand what causes this fundamental restriction – perhaps some structure of action sets?

30. Characterizing distribution rules for cost sharing games Raga GopalakrishnanCaltech Joint work with Jason R. Marden & Adam Wierman

31. References • [von Neumann et al. 1944] • [Nash 1951] • [Shapley 1953] • [Gillies 1959] • [Shapley 1967] • [Kalai et al. 1982] • [Moulin 1992] • [Goemans et al. 2000] • [Ui 2000] • [Devanur et al. 2003] • [Jackson 2003] • [Ju et al. 2003] • [Anshelevich et al. 2004] • [Conitzer et al. 2004] • [Albers et al. 2006] • [Chander et al. 2006] • [Chekuri et al. 2006] • [Alpcan et al. 2009] • [Ozdaglar et al. 2009] • [Chen et al. 2010] • [Gopalakrishnan et al. 2011] • [Marden et al. 2011]