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