Cooperative/coalitional 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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Collaboration

• Our focus this semester is on how agents
  collaborate.
• One view of collaboration is coalition formation.
• A coalition is simply a group of agents that work
  together to accomplish the task. Formation
  focuses on the process of forming such a
  coalition.

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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

• A set of twelve people all drive to work.
• They would like to form car pools.
• Some can pick up others on their way to work.
  Others have to go out of their way to pick up
  others.
• A car can only hold 5 people.
• Who should carpool together? How often should
  they each drive?

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Transferable utility

• Suppose that utility is transferable If you get
  more intrinsic benefit from a coalition, 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
• In the case of the carpool, how would we
  determine utility?
• Value function is 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)
• The payout is a vector of utilities that sums to
  the value is possible
• Can you think of other examples where a coalition
  has an associated utility?

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Suppose we have the following characteristic
function
Coalition Value of coalition
a 2
b 3
c 4
ab 6
ac 6
bc 7
abc 12
Agents are labeled a, b, c
How do we decide how to pay each member for
their contribution? We are assuming the final
(grand) coalition is the one that is formed.
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Some things we know
Coalition Value of coalition Possible payout
a 2 2
b 3 3
c 4 4
ab 6
ac 6
bc 7
abc 12
No one should do worse
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Other things we know
Coalition Value of coalition Possible payout Bad as doesnt use all value What about?
a 2 2 2
b 3 3 3
c 4 4 7
ab 6
ac 6
bc 7
abc 12
Somebody should get all the utility for the
grand coalition
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Other things we know
Coalition Value of coalition Possible payout Bad as doesnt use all value What about?
a 2 2 2
b 3 3 3
c 4 4 7
ab 6
ac 6
bc 7
abc 12
unstable as ab can do better
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Lets try a different example
Coalition Value of coalition
a 2
b 3
c 4
ab 10
ac 10
bc 7
abc 14
How do we decide how to pay each member for
their contribution?
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Suppose we have the following characteristic
function
Coalition Value of coalition Possibility
a 2 4
b 3 4
c 4 6
ab 10
ac 10
bc 7
abc 14
Who would pull out?
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Suppose we have the following characteristic
function
Coalition Value of coalition Possibility
a 2 6
b 3 4
c 4 4
ab 10
ac 10
bc 7
abc 14
Is this any better?
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Core

• Outcome is in the core if and only if every
  coalition receives a total utility that is at
  least its original value and no smaller
  coalition is tempted to pull out.
• For every coalition C, v(C) Si in Cu(i)
• For example,
• v(a,b,c) 12,
• v(a,b) v(a,c) v(b,c) 8,
• v(a) v(b) v(c) 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 multiplicityNo solution? many
solutions?

• Example 2
• v(a,b,c) 11,
• v(a,b) v(a,c) v(b,c) 8,
• v(a) v(b) v(c) 0
• Now the core is empty! Notice, the core must
  involve the grand coalition (giving payoff for
  each).
• Example 3
• v(a,b,c) 18,
• v(a,b) v(a,c) v(b,c) 8,
• v(a) v(b) v(c) 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
• Informally if agents did worse working
  together, there would be no motivation for
  coalitions.
• 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)

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Convexity

• a value system 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).
• For example, Does this have a core solution?
• v(a,b,c) 12,
• v(a,b) v(a,c) v(b,c) 8,
• v(a) v(b) v(c) 0
• Let A a,b and Bb,c
• v(AUB)-v(B) va,b,c vb,c 12 8 4
• v(A)-v(AnB) va,b vb 8 - 0 8
• Problem 4 lt 8

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Convexity

• 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 adds to the union.
• Works for any ordering of the agents

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The Shapley value motivation

• So, if there is no core, what do we do? It would
  be nice to say, This is what people should be
  paid
• Example no core
• v(a,b,c) 2,
• v(a,b) v(a,c) v(b,c) 2,
• v(a) v(b) v(c) 0
• If a,b decided to be a team, c is left out and
  may opt to work for less to attract a partner
• c says, Hey b, how about working with me? You
  can have 1.1. Ill take .9
• a says, Hey b, come back. You can have 1.2.
  Ill take .8
• c says to a, Hey a, work with me. We can each
  have 1.
• The solution isnt stable

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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)
• Example no core
• v(a,b,c) 2,
• v(a,b) v(a,c) v(b,c) 2,
• v(a) v(b) v(c) 0
• So, b could say, Hey a, work with me. I should
  get 2 as you were worth nothing without me. I
  should get the valued added (marginal value).

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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 simple 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.

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Example v(a,b,c) v(a,b) v(a,c)
v(b,c 2,v 0 everywhere else
a b c
abc 0 2 0
acb 0 0 2
bac 2 0 0
bca 0 0 2
cab 2 0 0
cba 0 2 0
avg 4/6 4/6 4/6
Compute the Shapley value for each. Is the
solution in the core?
Appealing as all appeared to have same
characteristics
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Example v(a,b,c) 15v(a,b) 3v(a,c)
4v(b,c 9,v 2 everywhere else
b c
abc
acb
bac
bca
cab
cba
avg
Compute the Shapley value for each. Is the
solution in the core?
Without doing the math, what would you guess?
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Example v(a,b,c) 15v(a,b) 3v(a,c)
4v(b,c 9,v 2 everywhere else
a b c
abc 2 1 12
acb 2 11 2
bac 1 2 12
bca 6 2 7
cab 2 11 2
cba 6 7 2
avg 19/6 34/6 37/6
Compute the Shapley value for each. Is the
solution in the core?
Is this reasonable?
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Briefly describe the characteristics of core
and shapley
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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 equivalent 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
  its utilities in v and w

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Computing a solution in the core

• Can use linear programming (a formal mathematical
  method)
• 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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Example Minimum Spanning Tree game

• For some games the characteristic function (value
  of each coalition) representation is immediate
  others it is more difficult
• Communities a,b c want to be connected to a
  nearby power source
• Possible transmission links costs as in figure
• Notice that distance does not determine cost
• Maybe the soil is rocky in certain directions

a
40
100
40
c
40
source
20
50
b
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Example Minimum Spanning Tree game

• What links should be built to get to power
  source?
• What should each community pay?
• Can c say, Hey b, Im getting my power through
  you. Dont expect any reimbursement because you
  already had to connect. Is that fair?

a
40
100
40
c
40
source
20
50
b
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Example Minimum Spanning Tree game

• Can a say, Hey guys, lets share the cost
  equally. The total cost of all of us connecting
  is 100. Lets each do a third.
• Can c say, No way. I should be the cheapest as I
  was the cheapest by myself. You are gaining
  more.

a
40
100
40
c
40
source
20
50
b
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Example Minimum Spanning Tree game

• Communities a,b,c want to be connected to a
  nearby power source

Red currently paying Black cost of new
arrangement Green Improvement We show what is
gained from the coalition. How to divide the
gain?
v(void) 0 v(a) 0 v(b) 0 v(c) 0 v(ab)
-90 100 50 60 v(ac) -80 100 40
60 v(bc) -60 50 40 30 v(abc) -100 100
50 40 90
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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(a) v(b) v(c)0
• v(ab) 60, v(ac) 60, v(bc) 30
• v(abc) 90
• Analitically, in getting to a group of three, you
  must make sure you do better than the group of 2
  cases
• abgt60, iff clt30
• a cgt60, iff blt30
• b cgt30, iff x1lt60

1
2
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The Core

• Lets divide up the 90 in savings. Lots of
  choices
• Lets choose a division of the profits in the
  core x(60,25,5)
• Is this fair compared to what their costs were
  without cooperation?
• The payoffs represent the savings, the costs
  (what they really pay) under x are
• cost(a)100-6040,
• cost(b)50-2525
• cost(c)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(a) v(b) v(c)0
• v(a,b) 60, v(a,c) 60, v(b,c) 30, v(a,b,c)
  90

Value added by
a b c
abc
acb
bac
bca
cab
cba
avg
Coalitions
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The Shapley value computation

• MST game
• v(void) v(a) v(b) v(c)0
• v(a,b) 60, v(a,c) 60, v(b,c) 30, v(a,b,c)
  90

Value added by
a b c
abc 0 60 30
acb 0 30 60
bac 60 0 30
bca 60 0 30
cab 60 30 0
cba 60 30 0
avg 40 25 25
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 a, b, and c require 1, 2, and 3 KM,
repectively to land. Thus, a runway of 3 must be
build, but what part of the total cost for a 3 KM
runway should each pay? Instead of looking at
utility given, look at how much increased cost
was required.
a b c
abc
acb
bac
bca
cab
cba
avg
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The Shapley value has been used for cost
sharing. Suppose three planes share a runway.
The planes a, b, and c require 1, 2, and 3 KM,
respectively to land. Thus, a runway of 3 must be
build, but what part of the total cost for a 3 KM
runway should each pay? Instead of looking at
utility given, look at how much increased cost
was required.
a b c
abc 1 1 1
acb 1 0 2
bac 0 2 1
bca 0 2 1
cab 0 0 3
cba 0 0 3
avg 2/6 5/6 11/6