1. Potential games are necessary to ensure pure Nash equilibria in cost sharing games

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

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

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

5. cost sharing games 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 ] [ … ] [ 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 ] [ … ] [ Marden and Wierman 2008 ] [ Panagopoulou and Spirakis 2008 ] [ … ] [ Vetta 2002 ] [ Hoefer 2006 ] [ Dürr and Thang 2006 ] [ Chekuri et al. 2007 ] [ Hansen and Telelis 2008 ] [ … ] [ Chekuri et al. 2007 ] [ Cardinal and Hoefer 2010 ] [ Bilò et al. 2010 ] [ Buchbinder et al. 2010 ] [ … ] [ Johari and Tsitsiklis 2004 ] [ Panagopoulou and Spirakis 2008 ] [ Marden and Effros 2009 ] [ Harks and Miller 2011 ] [ von Falkenhausen and Harks 2013 ] [ … ] Key feature: Distribution rules outcome!

6. Most prior work studies two distribution rules Marginal Contribution (MC) [ Wolpert and Tumer 1999 ] average marginal contribution over player orderings Shapley Value (SV) [ Shapley 1953 ] externality experienced by all other players

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

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

9. set of players set of resources S1S1 S2S2 D1 D2 Example: Formal model “welfare” = revenue / negative cost

10. Recall:

11. “all games”

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

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

14. Tractability Consequences Efficiency Incentive compatibility Budget- balance Four other important properties: don‘t guarantee PNE in all games guarantee PNE in all games

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

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

17. “coalition” “contribution of the coalition” Proof sketch [ Shapley 1953 ]

18. Proof sketch no yes DECOMPOSITON 1

19. Proof sketch DECOMPOSITON Proof: Establish a “fairness” condition 1 CONSISTENCY 2

20. Potential games are necessary to ensure pure Nash equilibria in cost sharing games