Raga Gopalakrishnan Caltech CU-Boulder Adam Wierman (Caltech) Jason Marden (CU-Boulder)

1. Potential games are necessary to ensure pure Nash equilibria in cost sharing games Raga GopalakrishnanCaltech CU-Boulder Adam Wierman (Caltech) Jason Marden (CU-Boulder)

2. Network formation [ Anshelevich et al. 2004 ] D1 6 S1 1 1 6 ?+? 1 1 S2 6 D2 cost sharing games • Model for distributed resource allocation problems. • Self-interested agents make decisions and share the resulting cost.

3. Network formation [ Anshelevich et al. 2004 ] D1 6 S1 1 1 6 3+3 1+5 1 1 S2 6 D2 cost sharing games A Nash equilibrium Also optimal! Unique Nash equilibrium Suboptimal

4. Network formation [ Anshelevich et al. 2004 ] D1 6 S1 1 1 6 ?+? 1 1 S2 6 D2 cost sharing games Key feature: Distribution rules outcome! Can we understand this better?

5. Network formation games • Facility location games • Congestion games • Routing games • Multicast games • Coverage games • … [ Anshelevich et al. 2004 ] [ Corbo and Parkes 2005 ] [ Fiat et al. 2006 ] [ Albers 2009 ] [ Chen and Roughgarden 2009 ] [ Epstein, Feldman, and Mansour 2009 ] [ … ] [ Vetta 2002 ] [ Hoefer 2006 ] [ Dürr and Thang2006 ] [ Chekuri et al. 2007 ] [ Hansen and Telelis 2008 ] [ … ] [ Rosenthal 1973 ] [ Milchtaich 1996 ] [ Christodoulou and Koutsoupias 2005 ] [ Suri, Tóth, and Zhou 2007 ] [Bhawalkar, Gairing, and Roughgarden 2010 ] [ … ] [ Roughgarden and Tardos 2002 ] [ Kontogiannis and Spirakis 2005 ] [ Awerbuch, Azar, and Epstein 2005 ] [ Chen, Chen, and Hu 2010 ] [ … ] cost sharing games [ Chekuri et al. 2007 ] [ Cardinal and Hoefer 2010 ] [ Bilò et al. 2010 ] [ Buchbinder et al. 2010 ] [ … ] Key feature: Distribution rules outcome! [ Marden and Wierman 2008 ] [ Panagopoulou and Spirakis 2008 ] [ … ] [ Johari and Tsitsiklis 2004 ] [ Panagopoulou and Spirakis 2008 ] [ Marden and Effros 2009 ] [ Harks and Miller 2011 ] [ von Falkenhausen and Harks 2013 ] [ … ]

6. Most prior work studies two distribution rules Marginal Contribution (MC) [ Wolpertand Tumer 1999 ] Shapley Value (SV) [ Shapley 1953 ] externality experienced by all other players average marginal contribution over player orderings Extensions: weighted and generalized weighted versions parameterized by “weight system”

7. Most prior work studies two distribution rules Marginal Contributions (MC+) [ Wolpertand Tumer 1999 ] Shapley Values (SV+) [ Shapley 1953 ] externality experienced by all other players average marginal contribution over player orderings Extensions: weighted and generalized weighted versions parameterized by “weight system” Both guarantee PNE in all games! Question: Are there other such distribution rules? Short answer: NO!for any fixed cost functions

8. all distribution rules SV+ guarantee PNE in all games 1 guarantee PNE in all games guarantee PNE in all games don‘t guarantee PNE in all games don‘t guarantee PNE in all games don‘t guarantee PNE in all games ? MC+ SV+ 2 guarantee potential game MC+

9. Formal model “welfare” = revenue / negative cost local welfare function distribution rule set of resources action set of player set of players D1 Example: S1 player 1’s share of S2 D2

10. Distribution rules: Marginal Contributions (MC+) (parameterized by weight system ) Shapley Values (SV+) (parameterized by weight system ) D1 6 S1 1 1 6 1 1 S2 6 D2 1+5 3+3 0+0

11. “all games” D1 D2 6 1 6 1 1 S1 6 1 6 1 6 1 6 1 6 1 6 1 S2 6 1 D1 D2

12. The inspiration for our work • [ Chen, Roughgarden, and Valiant 2010 ] guarantee PNE in all games • There exists : don‘t guarantee PNE in all games ? Our characterization budget-balanced guarantee potential game • For any : guarantee PNE in all games don‘t guarantee PNE in all games ? actual welfare distributed

13. The inspiration for our work • [ Chen, Roughgarden, and Valiant 2010 ] guarantee PNE in all games • There exists : don‘t guarantee PNE in all games guarantee PNE in all games don‘t guarantee PNE in all games Our characterizations budget-balanced guarantee potential game • For any : guarantee PNE in all games don‘t guarantee PNE in all games

14. Consequences Four other important properties: Incentive compatibility Budget-balance Tractability Efficiency • If budget-balance is required, set . Just optimize over . • Theorem:If budget-balance is not required, weights () don’t matter! Just optimize over . guarantee PNE in all games don‘t guarantee PNE in all games

15. Consequences Four other important properties: Incentive compatibility Budget-balance Tractability Efficiency guarantee PNE in all games don‘t guarantee PNE in all games Easier to control budget-balance More tractable exponential time! guarantee PNE in all games don‘t guarantee PNE in all games “preprocessing”

16. Consequences Four other important properties: Incentive compatibility Budget-balance Tractability Efficiency private values Future work:Design incentive compatible cost sharing mechanisms for a noncooperativesetting

17. Proof sketch Restrict to the case where where is arbitrary Restrict to the case of characterizing only budget-balanced Basis representation:Any can be written as Inclusion functions: Special welfare functions : [ Shapley 1953 ] “contribution of the coalition” “coalition” is said to be a “contributing” coalition in if • Technique:Establish a series of necessary conditions on

18. Proof sketch DECOMPOSITON 1 Proof:Establish a “fairness” condition on How much of should get? no Does contain a contributing coalition of that contains ? should not depend on the “noncontributing” players yes

19. Proof sketch DECOMPOSITON 1 Proof:Establish a “fairness” condition CONSISTENCY 2 Proof: Construct a “universal” equivalent to all

20. Potential games are necessary to ensure pure Nash equilibria in cost sharing games Ragavendran GopalakrishnanCaltech CU-Boulder Adam Wierman (Caltech) Jason Marden (CU-Boulder)