1. Arbitration, Fairness and Stability Revenue Division in Collaborative Settings Yair Zick Advisor: Prof. Edith Elkind

2. Acknowledgments Edith Elkind

3. Acknowledgments My research collaborators, past and present! Yoram Bachrach, Nina Balcan, Alejandro Carbonara, George Chalkiadakis, Amit Datta, Anupam Datta, Yuval Filmus, Kobi Gal, Nick Jennings, Ian Kash, Peter Key, Yoad Lewenberg, Vangelis Markakis, Moshik Mash, Reshef Meir, Svetlana Obraztsova, Joel Oren, Dima Pasechnik, Maria Polukarov, Ariel Procaccia, Jeffery S. Rosenschein, Shayak Sen, Nisarg Shah, Arunesh Sinha, Alexander Skopalik and Junxing Wang.

4. Thesis Outline Goal: find reasonable revenue divisions among collaborative agents. Features: agents may belong to more than one coalition; causes interdependencies. Key insights: -Agents’ reaction to deviation strongly governs stability of payoff division. -We develop a framework for handling reaction to deviation: arbitration functions -New insights about stability of various MAS, and their computational complexity.

5. Cooperative Games Players divide into coalitions to perform tasks Coalition members can freely divide profits. How should profits be divided?

6. Cooperative Games

7. Induced Subgraph Games We are given a weighted graph Players are nodes; value of a coalition is the value of the edges in the induced subgraph. Applications: markets, collaboration networks. 1 7 3 9 2 8 4 5 6 3 2 5 4 1 3 6 1 3 1 4 5 7 2 10 22

8. Network Flow Games We are given a weighted, directed graph Players are edges; value of a coalition is the value of the max. flow it can pass from s to t. Applications: computer networks, traffic flow, transport networks. s t 3 7 5 10 1 3 6 1 3 1 4 5 7 2

9. Cost Sharing Ride sharing: how to fairly split a taxi fare? Airport $50 $60 $70

10. Cost Sharing Ride sharing: how to fairly split a taxi fare?

11. Cost Sharing Ride sharing: how to fairly split a taxi fare?

12. Cooperative Games 125364

14. OCF Games (Chalkiadakis et al., JAIR 2010) Agents have divisible resources How should profits be divided? Overlapping Coalition Formation (OCF)

15. OCF Games

16. Each vector in [0,1] n describes how much each agent contributes; called a coalition. A coalition structure: a list of coalitions such that the sum of contributions from each agent is at most 1 (no agent contributes more than 100%).

17. OCF Games Features Simple model Applicable to many settings: Matching markets Network/multicommodity flows Agents completing tasks 15 62 4 37 8 s t s t 5 4 6 5 7 5 9 1 4 5 2 3 6 10 3 5 3 8 4 7 4 8 Red: 8 tons, worth 10$/kg Blue: 12 tons, worth 7$/kg 7 9

18. Outcome 17 8 6 9 14 10 8 5 3 5 3 4 4 6 6 0 2 4 5 3 2 7 4 1 3 2 5 4 4

19. Making them Offers they Can’t Refuse: The Arbitration Function Deviations and Reactions to Deviation in OCF Games

20. Deviation in OCF Games

21. 3 5 3 4 4 6 6 0 2 4 5 3 7 1 3 2 5 4 4 2 1 4 8 2 Total payoff: 11 Total payoff: 11 14 12 138 9 2

22. The Arbitration Function

23. Profitable Deviation Current outcomeDeviate; how much do we get from arbitration fn.? Form coalitions, divide revenue so that all profit!

24. 3 5 3 4 4 6 6 0 2 4 5 3 7 1 3 2 5 4 4 2 1 4 8 2 Optimistic Refined Sensitive Conservative ≺ ≺ ≺

25. “Agent Rights” vs. Stability The more generous the arbitration function, the more egalitarian the payoff division. However, freedom to deviate causes social instability.

26. Stability in OCF Games The Arbitrated Core

29. … so, we can’t share coalitions where I get paid (could reduce unhappiness)! I can transfer money to you if we share a coalition where I get paid! We can deviate without those guys!

30. Some Arbitrated Cores Classic CGT assumes conservative reaction to deviation!

31. Is it all for Naught? Refined core: a coalition that is not changed by deviation will still pay deviators; otherwise it will pay nothing. Things get interesting here…

32. The Refined Core Interesting observation: it is possible that one optimal coalition structure can be stabilized w.r.t. the refined arbitration function, but another would not!

33. The Refined Core 123 10 9 Must get at least 9 from each coalition! Leaves me with at most 2!

34. The Refined Core 123 10 Must get at least 9! Must get at least 5! 20

35. The Refined Core

36. Computing Solutions to OCF Games Finding optimal coalition structures and payoffs in polynomial time

37. Computational Aspects

38. These problems become easy when: Agents can only form small coalitions Agent interactions are simple (interaction graph) Discrete, poly-bounded, agent resources. Arbitration function is simple (local).

39. 3 5 3 4 4 6 6 0 2 4 5 3 7 1 3 2 5 4 4 2 1 4 8 2 Local: the decision of how much to give depends only on the effect on the coalition. Global: may depend on the effect deviation had on other coalitions. Can be 0, all, some… but independent of how deviators behave outside the coalition!

40. Computational Aspects In order to find stable outcomes in polynomial time, stick to the issues!

41. Specific Classes of OCF Games We identify a class of OCF games for which the optimistic core is not empty, and for which stable outcomes can be efficiently computed.

42. s t s t 5 4 6 5 7 5 9 1 4 5 2 3 6 10 3 5 3 8 4 7 4 8 Red: 8 tons, worth 10$/kg Blue: 12 tons, worth 7$/kg 7 Construct LP Generate Dual

43. Summary We introduce a new concept to the study of OCF games: the arbitration function. We reexamine solution concepts in light of the arbitration function. We examine computational aspects arising in OCF games. We identify sufficient conditions for core non-emptiness in OCF games. Also in thesis: – Iterated revenue sharing (to appear in IJCAI 2015) – Nucleolus, bargaining set and Shapley value for OCF games (read my thesis!)

44. Future Work Conceptual: – how lenient can the arbitration function be without destabilizing the game? – what games can be stabilized by a given arbitration function? (partial answer in this thesis) Computational: – can we find better algorithms with some assumptions on the valuation function? – Can we find approximately optimal/approximately stable solutions? – Computing other solution concepts?

45. Future Work New Frameworks: – In application domains (some work on cellular networks) – In economic domains (many market domains already form overlapping coalitions; arbitration functions are natural) – Arbitrators as a benchmark of the desirability of social strictness/lenience. A new research paradigm for the study of strategic interaction (not just in CGT!) New Directions: – Handling uncertainty – Applying cooperative game theory (and OCF solution concepts) in ML.

46. Publications this thesis is based on: -Yair Zick and Edith Elkind. Arbitrators in Overlapping Coalition Formation Games (AAMAS’11) -Yair Zick, Georgios Chalkiadakis and Edith Elkind. Overlapping Coalition Formation Games: Charting the Tractability Frontier (AAMAS’12) (see newer version on ArXiv) -Yair Zick, Evangelos Markakis and Edith Elkind. Stability via Convexity and LP Duality in Games with Overlapping Coalitions (AAAI’12) -Yair Zick, Evangelos Markakis and Edith Elkind. Arbitration and Stability in Cooperative Games with Overlapping Coalitions (JAIR’14) -Yair Zick, Yoram Bachrach, Ian Kash and Peter Key. Non-Myopic Negotiators See What's Best (IJCAI’15) Thank you! Questions? My e-mail: yairzick@cmu.edu My website: http://www.cs.cmu.edu/~yairzick/