Computing Shapley Values, Manipulating Value Distribution Schemes, and Checking Core Membership in Multi-Issue Domains

1
Computing Shapley Values, Manipulating Value
Distribution Schemes, and Checking Core
Membership in Multi-Issue Domains

• Vincent Conitzer and Tuomas Sandholm

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Agenda

• Introduction to coalitional games
• Multi-Issue Domains
• How to distribute the gains
• The Core
• Shapley Value
• Other marginal contribution schemes
• Computing the Shapley value
• Manipulating contribution schemes
• Checking core membership

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Coalitional Games

• Coalition formation is a key part of automated
  negotiation between self-interested agents
• Several of companies can unite into a virtual
  organization to take more diverse orders and gain
  more profit
• Truck delivery companies can share truck space,
  as the cost is mostly dependant on the distance
  rather than on the weight carried
• Coalition formation has been studied extensively
  in game theory, and solution concepts were
  adopted in multi agent systems

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Coalitional Games Solutions

• Given a coalitional game we want to find the
  distribution of the gains of the coalition
  between the agents
• Different solution concepts have different
  objectives
• The Core promotes stability
• The Shapley value promotes fairness
• Game theory has studied these solution concepts
  for quite some time, but the computational aspect
  has received little attention

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Some Questions to Keep in Mind

• How much should each of the employees of the
  company be paid to make sure a group of them
  wont be bought away by another company?
• Get a value division in the core
• A few truck delivery companies unite to carry a
  high load of deliveries. How can the profits be
  divided fairly?
• The Shapley value division

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Coalitional Games With Side Payments

• The game is presented as a characteristic
  function
• Let A be the set of agents (players)
• Each potential coalition S has a value v(S)
• The value is independent of what the non members
  of the coalition do
• The characteristic function
• Typically it is increasing

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Super additivity

• The characteristic function is super additive if
  for all disjoint sets of a S,T we have
• This means every two subsets can do better if
  they unite
• Finally we would get the grand coalition of all
  the agents
• This does not always hold
• Hard optimization problem to decide what to do
  united
• Anti trust laws

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Multi Issue Domains

• The characteristic function is the sum of values
  of independent issues
• We have sub-games
• The characteristic function (for every subset S
  of A) is
• Every coalition gets the sum of what it gets in
  all the sub games
• If a game is a decomposition to increasing (super
  additive) sub games, it is also increasing (super
  additive)

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Games Concerning a Subset of the Agents

• We say only concerns a subset of the agents
  if
• Assuming that each of the sub games concerns only
  a small subset of the agents we can improve our
  calculations
• Our representation of the characteristic function
  now only requires a small fraction of the space
  it once took

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Solution Concepts

• On a super additive game, the grand coalition is
  likely to form, and the coalition gets v(A)
• How much does each agent gets?
• We want a value division
• We want to divide all the gains

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The Core

• The best known solution concept
• Proposed by Gillies (1953) and von Neumann
  Morgenstein (1947)
• A value division is in the core if no sub
  coalition has an incentive to break away
• A value division d is blocked by a sub coalition
  S if
• If d is blocked by S, it is not in the core
• Some coalitional games have an empty core

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Player Types

• Dummy players add nothing to all coalitions
• Equivalent players add the same to any coalition
  that does not contain any of the two players

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The Shapley Value (Cont.)

• A well know value division scheme
• Aims to distribute the gains in a fair manner
• A value division that conforms to the set of the
  following axioms
• Dummy players get nothing
• Equivalent players get the same
• If a game v can be decomposed into two sub games,
  an agent gets the sum of values in the two games
• Only one such value division scheme exists

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The Shapley Value

• Given an ordering of the agents in A, we
  define to be the set of
  agents of A that appear before a in
• The Shapley value is defined as the marginal
  contribution of an agent to its set of
  predecessors, averaged on all possible
  permutations of the agents

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A Simple Way to Compute The Shapley Value

• Simply go over all the possible permutations of
  the agents and get the marginal contribution of
  the agent, sum these up, and divide by A!
• Extremely slow
• Can we use the fact that a game may be decomposed
  to sub games, each concerning only a few of the
  agents?

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Computing the Shapley Value

• If v can be decomposed to several sub games, we
  know (from the axioms) that
• If only concerns then for any player
  a, we have

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Computing the Shapley Value

• We do not really need to sum over all possible
  orderings, but rather on all possible subsets of
  agents that arrive before player a
• For each such sub set we get the same marginal
  contribution of player a.
• If the sub set S has n agents, there are n!
  ordering on the players inside. There are then
  (A-n-1)! ways to complete this ordering to an
  ordering on all agents. We get

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Computing the Shapley Value Quickly in Multi
Issue Domains

• Compute the Shapley value for each sub game,
  using the previous formula, only taking into
  account the concerning agents, then sum these up
• If we assume computation of factorials,
  multiplication and addition in constant time we
  get an time complexity of or less
  precisely

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Marginal Contribution Based Value Division Schemes

• A marginal contribution scheme is a scheme that
  chooses some ordering of the players, and
  distributes the gains to the players according to
  their marginal contribution
• If on the chosen orderings you add much to the
  value of the coalition of the players before you
  on the ordering, you deserve a nice share of the
  profits

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Marginal Contribution Based Value Division Schemes

• For the Shapley value we have considered an
  average on all possible orders
• If we consider just one of the possible
  orderings, the value an agent gets depends on it
  location in the ordering
• Obviously, the agent has a specific location it
  wants to be in
• If the game is convex (you add to a coalition at
  least as much as you add to any of its subsets),
  you want to be last in the ordering

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Marginal Contribution Based Value Division
Schemes (Cont.)

• If we randomly choose a permutation the
  expectancy of the value distribution for an agent
  is its Shapley value
• This requires a trusted source of randomness /
  cryptography
• Another way is to show that even if an agent has
  total control on the ordering chosen, it would
  still be computationally intractable for that
  agent to find the optimal ordering for him
• The computational complexity is used as a barrier
  for manipulation

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Maximal Marginal Contribution

• Let v be a game decomposed as follows
• and the game only concerns
• We are given an agent a and a number k, and are
  asked if there is some such that
  the value
• We want to see if we can find a subset of the
  agents to which as marginal contribution is at
  least k
• These would be the agents before a in the
  ordering a would choose

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NP-Completeness of Max-Marginal-Contribution

• Conitzer and Sandholm have shown that
  Max-Marginal-Contribution is NP-Complete, even in
  the case that and all s take values
  in 0,1,2
• The problem is in NP since for a given subset of
  agents we can simply calculate the marginal
  contribution of a to this subset

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NP-Completeness of Max-Marginal-Contribution

• NP-hardness is proven by reducing an arbitrary
  MAX2SAT instance to a Max-Marginal-Contribution
  instance
• In MAX2SAT we are given a set V of Boolean
  variables and a set of clauses C, each with 2
  literals, and a target number r of satisfied
  clauses
• For each variable v in V there is an agent Av
• We also have an agent a, whose contribution we
  want to maximize
• For every clause c there is a sub game (issue)
  tc, that only concerns the agents a and the
  agents representing the variables in the clause c

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NP-Completeness of Max-Marginal-Contribution

• The sub game results are as follows
• 1 point for having a in the coalition
• 1 point for having all the agents representing
  the negative literals
• But, if you want to get 2 points, you also have
  to have one of the agents representing the
  positive literals
• The marginal contribution we want is kr

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NP-Completeness of Max-Marginal-Contribution

• If there is a solution to MAX2SAT with r
  satisfied clauses, take V - the variables set to
  true
• What is the marginal contribution of a to this
  subset?
• Hint you either satisfied the clause by setting
  one of the negative literals to false, or if you
  didnt, youve set one of the positive literals
  to true
• Given a solution S to max-marginal-contribution,
  look at the assignment of true to everything in
  S, false otherwise
• If a sub game tc has increased the value by 1 due
  to adding a, what can you say about the clause?
• Open question we have used increasing games
  here, so the problem is NP-Complete even if the
  game is known to be increasing. What is the
  complexity for super additive games?

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Checking Core Membership

• Let v be a game decomposed as follows
• and the game only concerns
• We are given a value division that may not even
  be feasible
• If it isnt we can increase only the value of the
  grand coalition to the point where it is (the
  help of an outside benefactor for the stability)
• We are asked if the division is in the core, or
  if there is no blocking sub coalition for it

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NP-Completeness of Checking Core Membership

• Conitzer and Sandholm have shown that checking
  core membership (CHECKE-IF-BLOCKED) is
  NP-Complete
• The problem is in NP since for a given subset of
  agents we can simply calculate the sum of their
  values in the division and see if it is less than
  v(S)

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NP-Completeness of Checking Core Membership

• NP-hardness is proven by reducing an arbitrary
  VERTEX-COVER instance to a core membership
  problem
• We have an agent for each vertex, av, and another
  special agent a
• We have a sub game for each edge, that only
  concerns agent a and the agents of the edges
  vertices
• The value of the sub game is 1 if the coalition
  contains agent a and at least one of the edges
  vertices (we have agent a, and the edge is
  covered)
• The value distribution to check

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NP-Completeness of Checking Core Membership

• If there is a vertex cover with W vertices
• What is the value of the coalition of these
  vertices and agent a?
• How much do they get according to the value
  distribution?
• If a set of agents is a blocking coalition
• It has to contain agent a (or they get nothing)
• Consider the set of vertices of the agents in the
  blocking coalition, W
• How much do they get according to the value
  distribution?
• Can the number of vertices in W be smaller than
  r?
• To block, v(S) must be greater than v(a), since a
  is in the blocking coalition
• But then we have to get 1 for every sub game, so
  we have covered all the edges, with r vertices or
  less

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Conclusions

• Coalitional games important for automated
  negotiation between agents
• Such games can be decomposed to sub games
  (issues) which only concern some of the agents
• We can quickly compute the Shapley value in some
  of these cases
• Other marginal contribution value distribution
  schemes can be manipulated, but the general case
  is hard (an NP-complete problem)
• So such distribution schemes are acceptable in
  some cases, even if some of the agents have
  control on the chosen ordering
• Checking if a value distribution is stable (in
  the core) is hard (and NP-Complete problem in the
  general case)

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Open Questions

• NTU games (no side payments)
• Finding value divisions the are even harder to
  manipulate (eg. PSPACE-hard)
• Finding stability concepts that take into account
  the complexity of finding a beneficial deviation
• The complexity of other (longer term) solution
  concepts