Yoram Bachrach Jeffrey S. Rosenschein November 2007

 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

 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

 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

 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)

 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?

 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)

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

 Simply need to return a list of the veto players

 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

 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

 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

 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

 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

 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

 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