Yoram Bachrach
Jeffrey S. Rosenschein
November 2007
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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
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Cooperation in multiagent systems
◦ Several selfish agents working together
◦ Different subsets of the agents can achieve various goals
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Focus on various skills agents have, which
contribute to completing tasks
Study the complexity of computing game
theoretic solution concepts
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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
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Define how the total utility is distributed
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A payoff vector
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Individual rationality
such that
◦ Otherwise, an agent can do better working alone
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The payoff of a coalition C is
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A coalition C is blocking if p(C) < v(C)
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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?
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Power
◦ Which agent has the most influence on the
outcome?
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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)
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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
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Used for measuring “real power” in weighted
voting systems
◦ Suitable to any simple coalitional game
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Counts the number of coalition when an
agent is pivotal out of all wining coalitions
containing that agent
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A simple domain
◦ Agents
Tasks
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, Skills
,
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
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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
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Direct representation is still exponential in
the number of tasks
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TCSG – Task Count Skill Games
◦ Utility is the number of achieved tasks
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WTSG – Weighted Task Skill Games
◦ Each task has a weight
◦ A subset of tasks has weight
◦ Utility is the weight of achieved tasks
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Polynomial representation
◦ List of skills for each agent and for each task
◦ List of task weights
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Misses synergies between tasks
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Coalitions can either win or loose
◦ Require a threshold of utility to win
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TCSG-T
◦ Number of achieved tasks must exceed k
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WCSG-T
◦ Weight of achieved tasks must exceed k
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STSG: Single Task Skill Game
◦ Need to achieve all the skills to win
◦ Can be characterized a single task, which requires all
the skills
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Coalition Value (CV)
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Veto (VET)
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Dummy (DUM)
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Core Not Empty (CNE)
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Core (COR)
◦ Compute the value of a coalition
◦ Test of an agent is veto (present in all wining coalitions)
◦ Test if an agent is a dummy (contributes nothing to any coalition)
◦ Test if there is some payoff vector in the core
◦ Compute and return a representation of the core
 There may be infinitely many payoff vectors in the core
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Shapley (SH)
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Banzhaf (BZ)
◦ Compute the Shapley value of an agent
◦ Compute the Banzhaf index of an agent
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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
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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
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A Veto player
is present in all winning coalitions
◦ Or any coalition with a non zero value
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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
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Can simply check
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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
misses some
◦ Since
is a superset of C, it also covers
but
 Divide the game into sub-games for various tasks and test
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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
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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
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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
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Simply need to return a list of the veto players
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Unique skill agents
◦ Agents who have a certain skill no one else has
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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
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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
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Similarly to Shapley, we can show computing the
Banzhaf index is NP-hard
◦ Can we give a better computational characterization?
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#P – the counting version of NP
◦ The number of accepting paths of a non-deterministic TM
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A problem is #P-complete if we can polynomial
reduce any problem in #P to this problem
Computing the Banzhaf index in CSGs is #Pcomplete
◦ Even for the most restricted case of STSG
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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
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Compact representation of TU coalitional games
◦ Bilbao - Cooperative Games on Combinatorial Structures, 2000
◦ Conitzer & Sandholm
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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
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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
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Suggested a skill based model of cooperation
◦ A basic general model
◦ Restricted form games – TCSG and WTSG
◦ Restricted simple threshold versions
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Analyzed complexity of several problems and
game theoretic solution concepts
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Computing the value of a coalition
Testing for veto and dummy players
Computing the core
Computing the Shapley value and Banzhaf index
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Complexity of other game theoretic solution
concepts in CSGs:
◦ Least-core and epsilon-core
◦ Nucleolus
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Other restricted forms of CSGs
Richer models
◦ Allowing some synergies between tasks
◦ Composition of games