Cooperativecoalitional game theory

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Cooperative/coalitional game theory

• A composite of slides taken from Vincent Conitzer
  
• and Giovanni Neglia
• (Modified by Vicki Allan)

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coalitional game theory

• There is a set of agents N
• Each subset (or coalition) S of agents can work
  together in various ways, leading to various
  utilities for the agents
• Cooperative/coalitional game theory studies which
  outcome will/should materialize
• Key criteria
• Stability No coalition of agents should want to
  deviate from the solution and go their own way
• Fairness Agents should be rewarded for what
  they contribute to the group
• (Cooperative game theory is the standard name
  (distinguishing it from noncooperative game
  theory, which is what we have studied in two
  player games). However this is somewhat of a
  misnomer because agents still pursue their own
  interests. Hence some people prefer coalitional
  game theory.)

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Example

• Three agents 1, 2, 3 can go out for Indian,
  Chinese, or Japanese food
• u1(I) u2(C) u3(J) 4
• u1(C) u2(J) u3(I) 2
• u1(J) u2(I) u3(C) 0
• So the base utility agent 1 gets for Indian food
  is 4.
• Each agent gets an additional unit of utility for
  each other agent that joins her. HOWEVER, going
  out to eat alone is NOT allowed.
• If all agents go for Indian together, they get
  utilities (6, 2, 4)
• All going to Chinese gives (4, 6, 2), all going
  to Japanese gives (2, 4, 6)
• Hence, the utility possibility set for 1, 2, 3
  is (6, 2, 4), (4, 6, 2), (2, 4, 6)
• For the coalition 1, 2, the utility possibility
  set is (5, 1), (3, 5), (1, 3) (why?)

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Stability the core

• u1(I) u2(C) u3(J) 4
• u1(C) u2(J) u3(I) 2
• u1(J) u2(I) u3(C) 0
• V(1, 2, 3) (6, 2, 4), (4, 6, 2), (2, 4, 6)
• V(1, 2) (5, 1), (3, 5), (1, 3)
• Suppose the agents decide to all go for Japanese
  together, so they get (2, 4, 6)
• 1 and 2 would both prefer to break off and get
  Chinese together for (3, 5) we say (2, 4, 6) is
  blocked by 1, 2
• Blocking only occurs if there is a way of
  breaking off that would make all members of the
  blocking coalition happier
• The core Gillies 53 is the set of all outcomes
  (for the grand coalition N of all agents) that
  are blocked by no coalition
• In this example, the core is empty (why?)
• In a sense, there is no stable (meaning people
  wont change) outcome. There is no way to form
  coalitions.

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Transferable utility

• Now suppose that utility is transferable you can
  give some of your utility to another agent in
  your coalition (e.g. by making a side payment)
• Then, all that we need to specify is a value for
  each coalition, which is the maximum total
  utility for the coalition
• Value function also known as characteristic
  function
• Def. A game in characteristic function form is a
  set N of players together with a function v()
  which for any subset S of N (a coalition) gives a
  number v(S) (the value of the coalition)
• Any vector of utilities that sums to the value is
  possible
• Hence, the total for utility possibility set for
  1, 2, 3
• (6, 2, 4)12, (4, 6, 2)12, (2, 4, 6)12
  Notice they totals wouldnt all have to be equal
  in other examples.

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Transferable utility

• Outcome is in the core if and only if every
  coalition receives a total utility that is at
  least its original value
• For every coalition C, v(C) Si in Cu(i)
• In above example,
• v(1, 2, 3) 12,
• v(1, 2) v(1, 3) v(2, 3) 8,
• v(1) v(2) v(3) 0
• Now the outcome (4, 4, 4) is possible it is also
  in the core (why?) and is the only outcome in the
  core.

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Emptiness multiplicity

• Example 2 Let us modify the above example so
  that agents receive no utility from being
  together (except being alone still gives 0)
• v(1, 2, 3) 6,
• v(1, 2) v(1, 3) v(2, 3) 6,
• v(1) v(2) v(3) 0
• Now the core is empty! Notice, the core must
  involve the grand coalition (giving payoff for
  each).
• Conversely, suppose agents receive 2 units of
  utility for each other agent that joins
• v(1, 2, 3) 18,
• v(1, 2) v(1, 3) v(2, 3) 10,
• v(1) v(2) v(3) 0
• Now lots of outcomes are in the core (6, 6, 6),
  (5, 5, 8),

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Issues with the core

• When is the core guaranteed to be nonempty?
• What about uniqueness?
• What do we do if there are no solutions in the
  core? What if many?

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Superadditivity

• v is superadditive if for all coalitions A, B
  with AnB Ø, v(AUB) v(A) v(B)
• Informally, the union of two coalitions can
  always act as if they were separate, so should be
  able to get at least what they would get if they
  were separate
• Usually makes sense
• Previous examples were all superadditive
• Given this, always efficient for grand coalition
  to form
• Without superadditivity, finding a core is not
  possible.

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Convexity

• v is convex if for all coalitions A, B,
  v(AUB)-v(B) v(A)-v(AnB)
• In other words, the amount A adds to B (in the
  union) is at least as much it adds to the
  intersection.
• One interpretation the marginal contribution of
  an agent is increasing in the size of the set
  that it is added to. The term marginal
  contribution means the additional contribution.
  Precisely, the marginal contribution of A to B is
  v(AUB)-v(B)
• Example, suppose we have three independent
  researchers. When we combine them at the same
  university, the value added is greater if the
  set is larger.

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Convexity

• v is convex if for all coalitions A, B,
  v(AUB)-v(B) v(A)-v(AnB)
• Previous examples were not convex (why?)
• v is convex if for all coalitions A, B,
  v(AUB)-v(B) v(A)-v(AnB). Let A 1,2 and
  B2,3
• v(AUB)-v(B) 12 8
• v(A)-v(AnB) 8 - 0
• In convex games, core is always nonempty. (Core
  doesnt require convexity, but convexity produces
  a core.)
• One easy-to-compute solution in the core agent i
  gets u(i) v(1, 2, , i) - v(1, 2, , i-1)
• Marginal contribution scheme- each agent is
  rewarded by what it ads to the union.
• Works for any ordering of the agents

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The Shapley value Shapley 1953

• In dividing the profit, sometimes agent is given
  its marginal contribution (how much better the
  group is by its addition)
• The marginal contribution scheme is unfair
  because it depends on the ordering of the agents
• One way to make it fair average over all
  possible orderings
• Let MC(i, p) be the marginal contribution of i in
  ordering p
• Then is Shapley value is SpMC(i, p)/(n!)
• The Shapley value is always in the core for
  convex games
• but not in general, even when core is nonempty,
  e.g.
• v(1, 2, 3) v(1, 2) v(1, 3) 1,
• v 0 everywhere else

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Example v(1, 2, 3) v(1, 2) v(1, 3)
1,v 0 everywhere else
Compute the Shapley value for each. Is the
solution in the core?
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Axiomatic characterization of the Shapley value

• The Shapley value is the unique solution concept
  that satisfies
• (Pareto) Efficiency the total utility is the
  value of the grand coalition, Si in Nu(i) v(N)
• Symmetry two symmetric players (add the same
  amount to coalitions they join) must receive the
  same utility
• Dummy if v(S? i) v(S) for all S, then i
  must get 0
• Additivity if we add two games defined by v and
  w by letting (vw)(S) v(S) w(S), then the
  utility for an agent in vw should be the sum of
  her utilities in v and w
• most controversial axiom (for example,
  participant is cost-share of a runway and
  terminal is its cost-share of the runway plus
  his cost-share of the terminal)

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Computing a solution in the core

• Can use linear programming
• Variables u(i)
• Distribution constraint Si in Nu(i) v(N)
• Non-blocking constraints for every S, Si in
  Su(i) v(S)
• Problem number of constraints exponential in
  number of players (as you have values for all
  possible subsets)
• but is this practical?

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Theory of cooperative games with sidepayments

• It starts with von Neumann and Morgenstern (1944)
• Two main (related) questions
• which coalitions should form?
• how should a coalition which forms divide its
  winnings among its members?
• The specific strategy the coalition will follow
  is not of particular concern...
• Note there are also cooperative games without
  sidepayments

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Example Minimum Spanning Tree game

• For some games the characteristic form
  representation is immediate
• Communities 1,2 3 want to be connected to a
  nearby power source
• Possible transmission links costs as in figure

1
40
100
40
3
40
source
20
50
2
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Example Minimum Spanning Tree game

• Communities 1,2 3 want to be connected to a
  nearby power source

v(void) 0 v(1) 0 v(2) 0 v(3) 0 v(12)
-90 100 50 60 v(13) -80 100 40
60 v(23) -60 50 40 30 v(123) -100 100
50 40 90
A strategically equivalent game. We show what is
gained from the coalition. How to divide the
gain?
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The important questions

• Which coalitions should form?
• How should a coalition which forms divide its
  winnings among its members?
• Unfortunately there is no definitive answer
• Many concepts have been developed since 1944
• stable sets
• core
• Shapley value
• bargaining sets
• nucleolus
• Gately point

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The Core

• What about MST game? We use value to mean what
  is saved by going with a group.
• v(void) v(1) v(2) v(3)0
• v(12) 60, v(13) 60, v(23) 30
• v(123) 90
• Analitically, in getting to a group of three, you
  must make sure you do better than the group of 2
  cases
• x1x2gt60, iff x3lt30
• x1x3gt60, iff x2lt30
• x2x3gt30, iff x1lt60

1
2
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The Core

• Lets choose an imputation in the core
  x(60,25,5)
• The payoffs represent the savings, the costs
  under x are
• c(1)100-6040,
• c(2)50-2525
• c(3)40-535

FAIR?
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The Shapley value computation

• Consider the players forming the grand coalition
  step by step
• start from one player and add other players until
  N is formed
• As each player joins, award to that player the
  value he adds to the growing coalition
• The resulting awards give an value added
• Average the value added given by all the possible
  orders
• The average is the Shapley value k

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The Shapley value computation

• MST game
• v(void) v(1) v(2) v(3)0
• v(1,2) 60, v(1,3) 60, v(2,3) 30, v(1,2,3)
  90

Value added by
Coalitions
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The Shapley value computation

• MST game
• v(void) v(1) v(2) v(3)0
• v(12) 60, v(13) 60, v(23) 30, v(123) 90

Value added by
Coalitions
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The Shapley value computation

• MST game
• v(void) v(1) v(2) v(3)0
• v(12) 60, v(13) 60, v(23) 30, v(123) 90

Value added by
Coalitions
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The Shapley value computation

• A faster way
• The amount player i contributes to coalition S,
  of size s, is v(S)-v(S-i)
• This contribution occurs for those orderings in
  which i is preceded by the s-1 other players in
  S, and followed by the n-s players not in S
• ki 1/n! ?Si in S (s-1)! (n-s)! (v(S)-v(S-i))

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The Shapley value has been used for cost
sharing. Suppose three planes share a runway.
The planes require 1, 2, and 3 KM to land. Thus,
a runway of 3 must be build, but how much should
each pay? Instead of looking at utility given,
look at how much increased cost was required.
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The Shapley value has been used for cost
sharing. Suppose three planes share a runway.
The planes require 1, 2, and 3 KM to land. Thus,
a runway of 3 must be build, but how much should
each pay? Instead of looking at utility given,
look at how much increased cost was required.
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An application voting power

• A voting game is a pair (N,W) where N is the set
  of players (voters) and W is the collection of
  winning coalitions, s.t.
• the empty set is not in W (it is a losing
  coalition)
• N is in W (the coalition of all voters is
  winning)
• if S is in W and S is a subset of T then T is in
  W
• Also weighted voting game can be considered
• The Shapley value of a voting game is a measure
  of voting power (Shapley-Shubik power index)
• The winning coalitions have payoff 1
• The loser ones have payoff 0

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An application voting power

• The United Nations Security Council in 1954
• 5 permanent members (P)
• 6 non-permanent members (N)
• the winning coalitions had to have at least 7
  members,
• but the permanent members had veto power
• A winning coalition had to have at least seven
  members including all the permanent members
• The seventh member joining the coalition is the
  pivotal one he makes the coalition winning

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An application voting power

• 462 (11!/(5!6!)) possible orderings
• Power of non permanent members
• (PPPPPN)N(NNNN)
• 6 possible arrangements for (PPPPPN)
• 1 possible arrangements for (NNNN)
• The total number of arrangements in which an N is
  pivotal is 6
• The power of non permanent members is 6/462
• The power of permanent members is 456/462, the
  ratio of power of a P member to a N member is
  911
• In 1965
• 5 permanent members (P)
• 10 non-permanent members (N)
• the winning coalitions has to have at least 9
  members,
• the permanent members keep the veto power
• Similar calculations lead to a ratio of power of
  a P member to a N member equal to 1051

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Approaches

• Stable sets (Core)
• sets of imputations J
• internally stable (no imputations in J is
  dominated by any other imputation in J)
• externally stable (every imputations not in J is
  dominated by an imputation in J)
• incorporate social norms
• Bargaining sets
• the coalition is not necessarily the grand
  coalition (no collective rationality)
• Nucleolus
• minimize the unhappiness of the most unhappy
  coalition
• it is located at the center of the core (if there
  is a core)
• Gately point
• similar to the nucleolus, but with a different
  measure of unhappiness

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Nucleolus Schmeidler 1969

• Always gives a solution in the core if there
  exists one
• Always uniquely determined
• A coalitions excess e(S) is v(S) - Si in Su(i)
  (There was more available that we didnt get. We
  assume v(S) is limited by what is actually
  available.)
• The excess value is what the coalition was worth
  that it wasnt rewarded. It is what they were
  shorted.
• For an outcome, list all coalitions excesses in
  decreasing order
• E.g. consider
• v(1, 2, 3) 6,
• v(1, 2) v(1, 3) v(2, 3) 6,
• v(1) v(2) v(3) 0
• For payoff (2, 2, 2), the list of excesses is 2,
  2, 2, 0, -2, -2, -2
• (for coalitions 1, 21, 32, 3 1, 2, 3
  1 2 3, respectively)
• For payoff (3, 3, 0), the list of excesses is 3,
  3, 0, 0, 0, -3, -3 (coalitions 1, 3, 2, 3
  1, 2, 1, 2, 3, 3 1, 2)
• The first is more fair that the second as the
  shorted amounts are less.

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• Nucleolus is the (unique) payoff that
  lexicographically minimizes the list of excesses
• Lexicographic minimization minimize the first
  entry first, then (fixing the first entry)
  minimize the second one, etc.
• It is like dictionary ordering.
• So for each possible outcome, you make a list of
  excesses for each coalition, and sort them in
  order.
• Then, the one that lexicographically minimizes
  the list is selected.
• The idea is that you are trying to be fair so
  that no group receives a lot less benefit than
  another.

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Marriage contract problem Babylonian Talmud,
0-500AD

• A man has three wives
• Their marriage contracts specify that they
  should, respectively, receive 100, 200, and 300
  in case of his death
• but there may not be that much money to go
  around
• Talmud recommends
• If 100 is available, each wife gets 33 1/3
• If 200 is available, wife 1 gets 50, other two
  get 75 each
• If 300 is available, wife 1 gets 50, wife 2 gets
  100, wife 3 gets 150
• What is going on?
• Define v(S) max0, money available - Si in N-S
  claim(i)
• Any coalition can walk away and obtain 0
• Any coalition can pay off agents outside the
  coalition and divide the remainder
• Talmud recommends the nucleolus! Aumann
  Maschler 85

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• Talmud recommends
• If 100 is available, each wife gets 33 1/3
• If 200 is available, wife 1 gets 50, other two
  get 75 each
• If 300 is available, wife 1 gets 50, wife 2 gets
  100, wife 3 gets 150
• for coalitions 1, 21, 32, 3 1, 2, 3 1
  2 3,
• Case 1 all have equal claim on the 100 (as they
  all get at least that)
• excess 33, 33, 33, 0, 67,67,67
• if you split 17, 34, 49
• excess 49, 34, 17, 0, 83, 66, 51 (worse)
• Case 2 Only two people have claim on the last
  100. The two divide that equally.
• excess 75, 75, 50, 0, 50, 125,125
• if you split the first equally and divide the
  rest 33, 83,83
• excess 84, 84, 33, 0, 67, 117,117 (would be
  better, Right??)
• Case 3
• excess 150, 100, 50, 0, 50, 100, 150
• If 100, 100, 100 excess 100, 100, 100, 0, 0,
  100, 200 (worse)
• If 33, 83, 116 excess 116, 83, 33, 0,67,117,
  184 (worse than first)