Coalitional Skill Games

1. Coalitional Skill Games Yoram Bachrach Jeffrey S. Rosenschein November 2007

2. Agenda • Skill based models of cooperation • Coalitional games and solution concepts • Payoff vectors • The Core • The Shapley value and Banzhaf power index • The CSG model • Restricted CSGs – TCSG, WTSG and thresholds • Overview of results • Veto and dummy players • Core representation and emptieness • The Shapley value and Banzhaf index • Conclusion

3. Skill Based Models of Cooperation • Cooperation in multiagent systems • Several selfish agents working together • Different subsets of the agents can achieve various goals • Focus on various skills agents have, which contribute to completing tasks • Study the complexity of computing game theoretic solution concepts

4. Coalitional Games With Transferable Utility • Agents obtain utility when cooperating • A characteristic function indicates how much utility any coalition achieves • The utility can be divided among the agents in any way • Game properties • Increasing: If then • Super-additive: for all A,B • Simple games: coalitions either win or loose

5. Payoffs • Define how the total utility is distributed • A payoff vector such that • Individual rationality • Otherwise, an agent can do better working alone • The payoff of a coalition C is • A coalition C is blocking if p(C) < v(C)

6. Solution Concepts • Reasonable payoffs • Stability: when agents behave rationally, which payoff vectors do not give them an incentive to split the coalition apart? • Fairness: which payoff vectors reflect the contribution of the agents to the coalition? • Power • Which agent has the most influence on the outcome?

7. The Core (Stability) • The set of all payment vectors that are not blocked by any coalition • For any coalition C, p(C) ≥ v(C) • No coalition has an incentive to split off from the grand coalition • Proposed by Gillies (1953) and von Neumann & Morgenstein (1947)

8. The Shapley Value (Fairness) • Given an ordering of the agents in I, we denote the set of agents that appear before i in • The Shapley value is defined as the marginal contribution of an agent to its set of predecessors, averaged on all permutations

9. The Banzhaf Index (Power) • Used for measuring “real power” in weighted voting systems • Suitable to any simple coalitional game • Counts the number of coalition when an agent is pivotal out of all wining coalitions containing that agent

10. The CSG Model • A simple domain • Agents , Skills , Tasks • Each agent owns a set of skills • Each task requires a set of skills • A coalition owns the skills • A coalition can achieve any task it has the required skills for

11. Coalitional Skill Games • The utility is determined by the set of the tasks a coalition can achieve • Very basic model of cooperation • No measure of capability for performing a task • Probability of success, quality of performance • No notion of skill quantity • Required amounts of resources • No plans for achieving a task • Direct representation is still exponential in the number of tasks

12. Restricted forms of CSGs • TCSG – Task Count Skill Games • Utility is the number of achieved tasks • WTSG – Weighted Task Skill Games • Each task has a weight • A subset of tasks has weight • Utility is the weight of achieved tasks • Polynomial representation • List of skills for each agent and for each task • List of task weights • Misses synergies between tasks

13. Simple Games and Threshold Versions • Coalitions can either win or loose • Require a threshold of utility to win • TCSG-T • Number of achieved tasks must exceed k • WCSG-T • Weight of achieved tasks must exceed k • STSG: Single Task Skill Game • Need to achieve all the skills to win • Can be characterized a single task, which requires all the skills

14. Problems in CSG • Coalition Value (CV) • Compute the value of a coalition • Veto (VET) • Test of an agent is veto (present in all wining coalitions) • Dummy (DUM) • Test if an agent is a dummy (contributes nothing to any coalition) • Core Not Empty (CNE) • Test if there is some payoff vector in the core • Core (COR) • Compute and return a representation of the core • There may be infinitely many payoff vectors in the core • Shapley (SH) • Compute the Shapley value of an agent • Banzhaf (BZ) • Compute the Banzhaf index of an agent

15. Overview of the Results

16. Characteristic Functions in CSGs • Polynomial to compute which tasks a coalition can achieve • Iterate through the required skills for the task, and check if any member of the coalition has them • Easy to compute the characteristic function • TCSG – count the number of achieved tasks • WTSG – sum the weights of achieved tasks • General CSG – requires access to an oracle for computing the characteristic function given the subset of achieved tasks

17. Veto Players • A Veto player is present in all winning coalitions • Or any coalition with a non zero value • Non veto players have a certain winning coalition C that they are not a part of • CSGs are increasing • If C wins, so does • If looses, so does any subset of it, or any coalition that does not contain • Can simply check

18. Dummy Players • Dummy players contribute nothing to any coalition • Can be tested in polynomial time for TCSG and WTSG, but is co-NPC for threshold versions • Denote the set of agents who do not cover skill s as • Non dummies have a certain skill s that covers • They contribute to a coalition C, so C covers but misses some • Since is a superset of C, it also covers • Divide the game into sub-games for various tasks and test

19. Threshold Dummy is Hard • Found an polynomial algorithm for TCSG and WTSG • What about threshold versions? • Can still be a dummy even if your addition to a coalition makes it achieve more tasks • Maybe for all such coalition, this is not enough to make the coalition go over the threshold • Dummy is co-NPC for threshold versions • Reduction from 3SAT • Hard to test if there are coalitions which can achieve exactly k tasks • If you are an agent who always adds exactly one task, testing if you are a dummy for threshold k is really testing if there is a coalition that covers exactly k tasks

20. The Core • The Core can have infinitely many vectors in it • Cannot always find a polynomial representation for it • Can be done in simple games • No veto players -> the core is empty • Any agent has a winning coalition C that does not contain him • Give anything to that agent, and C blocks - it gets less than 1 • Otherwise, any payoff vector that gives all the gains to the veto player (in any way) is in the core • Only a winning coalition can bock • It must contain all the veto agents • If all the gains go to the veto agents, that coalition gets a total payoff of 1, which is exactly what it gains, so it cannot block

21. The Core in Simple CSGs • Simply need to return a list of the veto players

22. The Core in Non-Simple CSGs • Unique skill agents • Agents who have a certain skill no one else has • If there are not unique skill agents, the core is empty • Consider an agent • Coalition covers all the skills, and wins, so it blocks any payoff vector where gets anything • But this was any agent, so the core is empty

23. The Shapley Value • Only dummy agents have a Shapley value of 0 • Testing non-dummies in TCSG-T and WTSG-T is NPC • Computing the Shapley value is NP hard

24. The Banzhaf Index • Similarly to Shapley, we can show computing the Banzhaf index is NP-hard • Can we give a better computational characterization? • #P – the counting version of NP • The number of accepting paths of a non-deterministic TM • A problem is #P-complete if we can polynomial reduce any problem in #P to this problem • Computing the Banzhaf index in CSGs is #P-complete • Even for the most restricted case of STSG

25. #P-completeness of Banzhaf • Reduction from #SET-COVER • Counting the number of different set cover • #SC-K – counting the number of set covers with size of at most k • Known to be #P-complete • Solving #SC-k easily allows solving #SC • We need the other way around, which is harder but true • We add an agent with a new required skill • The Banzhaf index of this agent is proportional to the number of coalitions in which he is critical • This agent is critical exactly for a set of agents which cover all the other skills, so given the index we can get the #SC solution

26. Related Work • Compact representation of TU coalitional games • Bilbao - Cooperative Games on Combinatorial Structures, 2000 • Conitzer & Sandholm • Complexity of determining nonemptiness of the core, 2003 • Computing shapley values, manipulating value division schemes, and checking core membership in multi-issue domains, 2004 • Deng & Papadimitriou – on the complexity of cooperative solution concepts, 1994 • Power indices complexity • Matsui & Matsui – Banzhaf and Shapley in WVGs is NPC • Deng & Papadimitriou – Shapley in WVG is #P-C • Bachrach & Rosenschein –Banzhaf in network flow games is #P-C • Similar models • Wooldridge & Dunne - CRGs (Coalitional Resource Games) and QCG (Qualitative Coalitional Games • Yokoo, Conitzer, Sandholm, Ohta and Iwasaki - coalitional games in open anonymous environments

27. Conclusion • Suggested a skill based model of cooperation • A basic general model • Restricted form games – TCSG and WTSG • Restricted simple threshold versions • Analyzed complexity of several problems and game theoretic solution concepts • Computing the value of a coalition • Testing for veto and dummy players • Computing the core • Computing the Shapley value and Banzhaf index

28. Future Work • Complexity of other game theoretic solution concepts in CSGs: • Least-core and epsilon-core • Nucleolus • Other restricted forms of CSGs • Richer models • Allowing some synergies between tasks • Composition of games