Yoram Bachrach

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) lt 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 -gt 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