An Introduction to Cooperative Game Theorymjserna/docencia/agt-miri/slides/AGT13-coop-GT.pdf · De nitions Stability notions Other solution concepts Subclasses An Introduction to

DefinitionsStability notions

Other solution conceptsSubclasses

An Introduction to Cooperative Game Theory

Maria Serna

Fall 2016

AGT-MIRI Cooperative Game Theory



References

G. Chalkiadakis, E. Elkind, M. WooldridgeComputational Aspects of Cooperative Game TheoryMorgan & Claypool, 2012 Wikipedia.

G. OwenGame Theory3rd edition, Academic Press, 1995




TU characteristic function gamesOutcomesImputations

1 Definitions

2 Stability notions

3 Other solution concepts

4 Subclasses





Non-Cooperative versus cooperative Games

Non-cooperative game theory model scenarios where playerscannot make binding agreements.

Cooperative game theory model scenarios, where

agents can benefit by cooperating, andbinding agreements are possible.

In cooperative games, actions are taken by groups of agents,coalitions, and payoffs are given to

the group, that has to divided it among its members:Transferable utility games.individuals: Non-transferable utility games.

For the moment we focus on TU games

Notation: N, set of players, C ,S ,X ⊆ N are coalitions.














































the group, that has to divided it among its members:Transferable utility games.

individuals: Non-transferable utility games.






































Notation: N, set of players, C , S ,X ⊆ N are coalitions.





Characteristic Function Games

A characteristic function game is a pair (N, v), where:

N = {1, ..., n} is the set of players andv : 2N → R is the characteristic function.

for each subset of players C ⊆ N, v(C ) is the amount that themembers of C can earn by working together

usually it is assumed that v is

normalized: v(∅) = 0,non-negative: v(C ) ≥ 0, for any C ⊆ N, andmonotone: v(C ) ≤ v(D), for any C , D such that C ⊆ D

A coalition is any subset of NN itself is called the grand coalition.







N = {1, ..., n} is the set of players andv : 2N → R is the characteristic function.for each subset of players C ⊆ N, v(C ) is the amount that themembers of C can earn by working together

































A coalition is any subset of N

N itself is called the grand coalition.















Characteristic Function Games vs.Partition Function Games

In general TU games, the payoff obtained by a coalitiondepends on the actions chosen by other coalitionsthese games are also known as partition function games (PFG)

Characteristic function games (CFG): the payoff of eachcoalition only depends on the action of that coalitionin such games, each coalition can be identified with the profitit obtains by choosing its best action

We focus on characteristic function games, and use the termTU games to refer to such games





























Buying Ice-Cream Game

We have a group of n children, each has some amount ofmoney the i-th child has bi dollars.

There are three types of ice-cream tubs for sale:

Type 1 costs $7, contains 500gType 2 costs $9, contains 750gType 3 costs $11, contains 1kg

The children have utility for ice-cream but do not care aboutmoney.

The payoff of each group is the maximum quantity ofice-cream the members of the group can buy by pooling theirmoney

The ice-cream can be shared arbitrarily within the group.


















































































Ice-Cream Game: Characteristic Function

Charlie: $6 Marcie: $4 Pattie: $3

w = 500 w = 750 w = 100

p = $7 p = $9 p = $11

v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 750, v({C ,P}) = 750, v({M,P}) = 500

v({C ,M,P}) = 1000







w = 500 w = 750 w = 100

p = $7 p = $9 p = $11

v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 750, v({C ,P}) = 750, v({M,P}) = 500

v({C ,M,P}) = 1000







w = 500 w = 750 w = 100

p = $7 p = $9 p = $11

v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 750, v({C ,P}) = 750, v({M,P}) = 500

v({C ,M,P}) = 1000







w = 500 w = 750 w = 100

p = $7 p = $9 p = $11

v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 750, v({C ,P}) = 750, v({M,P}) = 500

v({C ,M,P}) = 1000





Outcomes

An outcome of a game Γ = (N, v) is a pair (CS , x), where:

CS = (C1, ...,Ck) is a coalition structure, i.e., partition of Ninto coalitions:

∪ki=1Ci = NCi ∩ Cj = ∅, for i 6= j

x = (x1, ..., xn) is a payoff vector, which distributes the valueof each coalition in CS:

xi ≥ 0, for all i ∈ N∑i∈C xi = v(C ), for each C ∈ CS , feasibility





Outcomes










Outcomes





xi ≥ 0, for all i ∈ N∑i∈C xi = v(C ), for each C ∈ CS ,

feasibility





Outcomes










Outcome:example

Suppose v({1, 2, 3}) = 9 and v({4, 5}) = 4

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is an outcome

(({1, 2, 3}, {4, 5}), (2, 3, 2, 3, 3)) is NOT an outcome astransfers between coalitions are not allowed





Outcome:example

Suppose v({1, 2, 3}) = 9 and v({4, 5}) = 4

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is an outcome






Outcome:example

Suppose v({1, 2, 3}) = 9 and v({4, 5}) = 4

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is an outcome






Imputations

An outcome (CS , x) is called an imputation if it satisfies individualrationality:

xi ≥ v({i}),

for all i ∈ N.

Notation: we denote∑

i∈C xi by x(C )





Imputations


xi ≥ v({i}),

for all i ∈ N.


i∈C xi by x(C )





Imputations


xi ≥ v({i}),

for all i ∈ N.


i∈C xi by x(C )




Core and variationsFairness: Shapley valueComputational Issues

1 Definitions

2 Stability notions


4 Subclasses





What Is a Good Outcome?

The solutions of a game should provide good outcomes.

Let us present some stability notions related to outcomes orimputations.

To simplify the presentation we consider superadditive games.





























Superadditive Games

A game G = (N, v) is called superadditive if

v(C ∪ D) ≥ v(C ) + v(D),

for any two disjoint coalitions C and D

Example: v(C ) = |C |2

v(C ∪ D) = (|C |+ |D|)2 ≥ |C |2 + |D|2 = v(C ) + v(D)





Superadditive Games

A game G = (N, v) is called superadditive if

v(C ∪ D) ≥ v(C ) + v(D),

for any two disjoint coalitions C and D

Example: v(C ) = |C |2

v(C ∪ D) = (|C |+ |D|)2 ≥ |C |2 + |D|2 = v(C ) + v(D)





Superadditive Games

In superadditive games, two coalitions can always mergewithout losing money; hence, we can assume that players formthe grand coalition

Convention: in superadditive games, we identify outcomeswith payoff vectors for the grand coalition

i.e., an outcome is a vector x = (x1, ..., xn) with x(N) = v(N)





Superadditive Games








Superadditive Games








Superadditive Games









Charlie: $4 Marcie: $3 Pattie: $3Ice-cream pots: w = (500, 750, 100) and p = ($7, $9, $11)

v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750

This is a superadditive game, so outcomes are payoff vectors!How should the players share the ice-cream?(200, 200, 350)?Charlie and Marcie can get more ice-cream by buying a 500g tubon their own, and splitting it equally(200, 200, 350) is not stable!







v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750








v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750

This is a superadditive game, so outcomes are payoff vectors!

How should the players share the ice-cream?(200, 200, 350)?Charlie and Marcie can get more ice-cream by buying a 500g tubon their own, and splitting it equally(200, 200, 350) is not stable!







v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750

This is a superadditive game, so outcomes are payoff vectors!How should the players share the ice-cream?

(200, 200, 350)?Charlie and Marcie can get more ice-cream by buying a 500g tubon their own, and splitting it equally(200, 200, 350) is not stable!







v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750

This is a superadditive game, so outcomes are payoff vectors!How should the players share the ice-cream?(200, 200, 350)?

Charlie and Marcie can get more ice-cream by buying a 500g tubon their own, and splitting it equally(200, 200, 350) is not stable!







v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750

This is a superadditive game, so outcomes are payoff vectors!How should the players share the ice-cream?(200, 200, 350)?Charlie and Marcie can get more ice-cream by buying a 500g tubon their own, and splitting it equally

(200, 200, 350) is not stable!







v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750






The core

The core of a game Γ is the set of all stable outcomes, i.e.,outcomes that no coalition wants to deviate from

core(Γ) = {(CS , x)|x(C ) ≥ v(C ) for any C ⊆ N}

each coalition earns at least as much as it can make on its own.

Example: v({1, 2, 3}) = 9, v({4, 5}) = 4, v({2, 4}) = 7

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is NOT in the core

as x({2, 4}) = 6 and v({2, 4}) = 7

no subgroup of players can deviate so that each member ofthe subgroup gets more.





The core




Example: v({1, 2, 3}) = 9, v({4, 5}) = 4, v({2, 4}) = 7

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is NOT in the core

as x({2, 4}) = 6 and v({2, 4}) = 7






The core




Example: v({1, 2, 3}) = 9, v({4, 5}) = 4, v({2, 4}) = 7

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is NOT in the core

as x({2, 4}) = 6 and v({2, 4}) = 7






The core




Example: v({1, 2, 3}) = 9, v({4, 5}) = 4, v({2, 4}) = 7

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is NOT in the core

as x({2, 4}) = 6 and v({2, 4}) = 7






The core




Example: v({1, 2, 3}) = 9, v({4, 5}) = 4, v({2, 4}) = 7

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is NOT in the core

as x({2, 4}) = 6 and v({2, 4}) = 7






The core




Example: v({1, 2, 3}) = 9, v({4, 5}) = 4, v({2, 4}) = 7

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is NOT in the core

as x({2, 4}) = 6 and v({2, 4}) = 7






The core




Example: v({1, 2, 3}) = 9, v({4, 5}) = 4, v({2, 4}) = 7

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is NOT in the core

as x({2, 4}) = 6 and v({2, 4}) = 7






The core




Example: v({1, 2, 3}) = 9, v({4, 5}) = 4, v({2, 4}) = 7

(({1, 2, 3}, {4, 5}), (3, 3, 3, 3, 1)) is NOT in the core

as x({2, 4}) = 6 and v({2, 4}) = 7






Ice-cream game: Core


v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750

(200, 200, 350) is not in the core: v({C ,M}) > x({C ,M})(250, 250, 250) is in the core: alone or in pairs do not getmore.

(750, 0, 0) is also in the core:Marcie and Pattie cannot get more on their own!







v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750

(200, 200, 350)

is not in the core: v({C ,M}) > x({C ,M})(250, 250, 250) is in the core: alone or in pairs do not getmore.








v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750

(200, 200, 350) is not in the core: v({C ,M}) > x({C ,M})

(250, 250, 250) is in the core: alone or in pairs do not getmore.








v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750

(200, 200, 350) is not in the core: v({C ,M}) > x({C ,M})(250, 250, 250)

is in the core: alone or in pairs do not getmore.








v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750


(750, 0, 0)

is also in the core:Marcie and Pattie cannot get more on their own!







v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750


(750, 0, 0) is also in the core:

Marcie and Pattie cannot get more on their own!







v(∅) = v({C}) = v({M}) = v({P}) = 0

v({C ,M}) = 500, v({C ,P}) = 500, v({M,P}) = 0

v({C ,M,P}) = 750







Games with empty core?

Let Γ = (N, v), where N = {1, 2, 3} andv(C ) = 1 if |C | > 1 and v(C ) = 0 otherwise.

Consider an outcome (CS , x).

We have x1, x2, x3 ≥ 0, x1 + x2 + x3 = 1, and xi + xj = 1, fori 6= jAs, x1 + x2 + x3 ≥ 1, for some i ∈ {1, 2, 3}, xi ≥ 1/3.Assume that i = 1, we have x2 + x3 = 1− x1 ≤ 1− 1/3 ≤ 1!

Thus the core of Γ is empty.
































Core on payoff vectors

Suppose the game is not necessarily superadditive, but theoutcomes are defined as payoff vectors for the grand coalition.

Then the core may be empty, even if according to thestandard definition it is not.

Γ = (N, v) with N = {1, 2, 3, 4} andv(C ) = 1 if |C | > 1 and v(C ) = 0 otherwise

not superadditive: v({1, 2}) + v({3, 4}) = 2 > v({1, 2, 3, 4})no payoff vector for the grand coalition is in the core:either {1, 2} or {3, 4} get less than 1, so can deviateBut (({1, 2}, {3, 4}), (1/2, 1/2, 1/2, 1/2)) is in the core




































not superadditive: v({1, 2}) + v({3, 4}) = 2 > v({1, 2, 3, 4})

no payoff vector for the grand coalition is in the core:either {1, 2} or {3, 4} get less than 1, so can deviateBut (({1, 2}, {3, 4}), (1/2, 1/2, 1/2, 1/2)) is in the core









not superadditive: v({1, 2}) + v({3, 4}) = 2 > v({1, 2, 3, 4})no payoff vector for the grand coalition is in the core:either {1, 2} or {3, 4} get less than 1, so can deviate

But (({1, 2}, {3, 4}), (1/2, 1/2, 1/2, 1/2)) is in the core














ε-Core

When the core is empty, we may want to find approximatelystable outcomes.

We need to relax the notion of the core:core: (CS , x) : x(C ) ≥ v(C ), for all C ⊆ N

ε-core: {(CS , x) : x(C ) ≥ v(C )− ε, for all C ⊆ N}Γ = (N, v), N = {1, 2, 3} andv(C ) = 1 if |C | > 1 and v(C ) = 0 otherwise

1/3-core is non-empty: (1/3, 1/3, 1/3) ∈ 1/3-coreε-core is empty for any ε < 1/3:xi ≥ 1/3, for some i = 1, 2, 3, so x(N \ {i}) ≤ 2/3,v(N \ {i}) = 1





ε-Core









ε-Core


We need to relax the notion of the core:

core: (CS , x) : x(C ) ≥ v(C ), for all C ⊆ N







ε-Core









ε-Core



ε-core: {(CS , x) : x(C ) ≥ v(C )− ε, for all C ⊆ N}

Γ = (N, v), N = {1, 2, 3} andv(C ) = 1 if |C | > 1 and v(C ) = 0 otherwise






ε-Core









ε-Core









Least Core

If an outcome (CS , x) is in the ε-core, the deficit v(C )− x(C )of any coalition is at most ε

We are interested in outcomes that minimize the worst-casedeficit

Let ε∗(Γ) = inf{ε|ε-core of Γ is not empty}.It can be shown that, for all Γ, the ε∗(Γ)-core is not empty.

The ε∗(Γ)-core is called the least core of Γ andε∗(Γ) is called the value of the least core

Γ = (N, v), N = {1, 2, 3}, v(C ) = (|C | > 1)

1/3-core is non-empty: (1/3, 1/3, 1/3) ∈ 1/3-coreε-core is empty for any ε < 1/3The least core is the 1/3-core.





Least Core





Γ = (N, v), N = {1, 2, 3}, v(C ) = (|C | > 1)






Least Core





Γ = (N, v), N = {1, 2, 3}, v(C ) = (|C | > 1)






Least Core



Let ε∗(Γ) = inf{ε|ε-core of Γ is not empty}.

It can be shown that, for all Γ, the ε∗(Γ)-core is not empty.


Γ = (N, v), N = {1, 2, 3}, v(C ) = (|C | > 1)






Least Core





Γ = (N, v), N = {1, 2, 3}, v(C ) = (|C | > 1)






Least Core





Γ = (N, v), N = {1, 2, 3}, v(C ) = (|C | > 1)






Least Core





Γ = (N, v), N = {1, 2, 3}, v(C ) = (|C | > 1)

1/3-core is non-empty: (1/3, 1/3, 1/3) ∈ 1/3-coreε-core is empty for any ε < 1/3

The least core is the 1/3-core.





Least Core





Γ = (N, v), N = {1, 2, 3}, v(C ) = (|C | > 1)






Stability vs. Fairness

Outcomes in the core may be unfair.

Γ = ({1, 2}, v) withv(∅) = 0, v({1}) = v({2}) = 5, v({1, 2}) = 20

(15, 5) is in the core: player 2 cannot benefit by deviating.However, this is unfair since 1 and 2 are symmetric

How do we divide payoffs in a fair way?







Γ = ({1, 2}, v) withv(∅) = 0, v({1}) = v({2}) = 5, v({1, 2}) = 20

(15, 5) is in the core: player 2 cannot benefit by deviating.

However, this is unfair since 1 and 2 are symmetric








Γ = ({1, 2}, v) withv(∅) = 0, v({1}) = v({2}) = 5, v({1, 2}) = 20









Γ = ({1, 2}, v) withv(∅) = 0, v({1}) = v({2}) = 5, v({1, 2}) = 20







Marginal Contribution

A fair payment scheme rewards each agent according to hiscontribution.

Attempt: given a game Γ = (N, v), set

xi = v({1, ..., i − 1, i})− v({1, ..., i − 1}).

The payoff to each player is his marginal contribution to thecoalition of his predecessors

We have x1 + ...+ xn = v(N) thus x is a payoff vector

However, payoff to each player depends on the order

Γ = ({1, 2}, v) withv(∅) = 0, v({1}) = v({2}) = 5, v({1, 2}) = 20x1 = v({1})− v(∅) = 5, x2 = v({1, 2})− v({1}) = 15







Attempt:

given a game Γ = (N, v), set

xi = v({1, ..., i − 1, i})− v({1, ..., i − 1}).












xi = v({1, ..., i − 1, i})− v({1, ..., i − 1}).












xi = v({1, ..., i − 1, i})− v({1, ..., i − 1}).












xi = v({1, ..., i − 1, i})− v({1, ..., i − 1}).












xi = v({1, ..., i − 1, i})− v({1, ..., i − 1}).












xi = v({1, ..., i − 1, i})− v({1, ..., i − 1}).









Average Marginal Contribution

Idea: Remove the dependence on ordering taking the averageover all possible orderings.

Γ = ({1, 2}, v) withv(∅) = 0, v({1}) = v({2}) = 5, v({1, 2}) = 20

1, 2: x1 = v({1})− v(∅) = 5, x2 = v({1, 2})− v({1}) = 152, 1: y2 = v({2})− v(∅) = 5, y1 = v({1, 2})− v({2}) = 15

z1 = (x1 + y1)/2 = 10, z2 = (x2 + y2)/2 = 10the resulting outcome is fair!

Can we generalize this idea?







Γ = ({1, 2}, v) withv(∅) = 0, v({1}) = v({2}) = 5, v({1, 2}) = 20

1, 2: x1 = v({1})− v(∅) = 5, x2 = v({1, 2})− v({1}) = 152, 1: y2 = v({2})− v(∅) = 5, y1 = v({1, 2})− v({2}) = 15









Γ = ({1, 2}, v) withv(∅) = 0, v({1}) = v({2}) = 5, v({1, 2}) = 20

1, 2: x1 = v({1})− v(∅) = 5, x2 = v({1, 2})− v({1}) = 152, 1: y2 = v({2})− v(∅) = 5, y1 = v({1, 2})− v({2}) = 15









Γ = ({1, 2}, v) withv(∅) = 0, v({1}) = v({2}) = 5, v({1, 2}) = 20

1, 2: x1 = v({1})− v(∅) = 5, x2 = v({1, 2})− v({1}) = 152, 1: y2 = v({2})− v(∅) = 5, y1 = v({1, 2})− v({2}) = 15









Γ = ({1, 2}, v) withv(∅) = 0, v({1}) = v({2}) = 5, v({1, 2}) = 20

1, 2: x1 = v({1})− v(∅) = 5, x2 = v({1, 2})− v({1}) = 152, 1: y2 = v({2})− v(∅) = 5, y1 = v({1, 2})− v({2}) = 15







Shapley Value

A permutation of {1, ..., n} is a one-to-one mapping from{1, ..., n} to itselfΠ(N) denote the set of all permutations of N

Let Sπ(i) denote the set of predecessors of i in π ∈ Π(N)

For C ⊆ N, let δi (C ) = v(C ∪ {i})− v(C )

The Shapley value of player i in a game Γ = (N, v) with nplayers is

Φi (Γ) =1

n!

∑π∈Π(N)

δi (Sπ(i))

In the previous slide we have Φ1 = Φ2 = 10





Shapley Value





Φi (Γ) =1

n!

∑π∈Π(N)

δi (Sπ(i))






Shapley Value





Φi (Γ) =1

n!

∑π∈Π(N)

δi (Sπ(i))






Shapley Value





Φi (Γ) =1

n!

∑π∈Π(N)

δi (Sπ(i))






Shapley Value





Φi (Γ) =1

n!

∑π∈Π(N)

δi (Sπ(i))






Shapley Value: Probabilistic Interpretation

Φi is i ’s average marginal contribution to the coalition of itspredecessors, over all permutations

Suppose that we choose a permutation of players uniformly atrandom, then Φi is the expected marginal contribution ofplayer i to the coalition of his predecessors



















Player’s properties

Given a game Γ = (N, v)

A player i is a dummy in Γ if

v(C ) = v(C ∪ {i}), for any C ⊆ N

Two players i and j are said to be symmetric in Γ if

v(C ∪ {i}) = v(C ∪ {j}), for any C ⊆ N \ {i , j}








v(C ) = v(C ∪ {i}), for any C ⊆ N










v(C ) = v(C ∪ {i}), for any C ⊆ N










v(C ) = v(C ∪ {i}), for any C ⊆ N







Shapley value: Axiomatic Characterization

Properties of the Shapley value:

Efficiency: Φ1 + ...+ Φn = v(N)

Dummy: if i is a dummy, Φi = 0

Symmetry: if i and j are symmetric, Φi = Φj

Additivity: Φi (Γ1 + Γ2) = Φi ((Γ1) + Φi (Γ2)

Theorem

The Shapley value is the only payoff distribution scheme that hasproperties (1) - (4)

Γ = Γ1 + Γ2 is the game (N, v) with v(C ) = v1(C ) + v2(C )





Shapley value: Axiomatic Characterization

Properties of the Shapley value:

Efficiency: Φ1 + ...+ Φn = v(N)

Dummy: if i is a dummy, Φi = 0

Symmetry: if i and j are symmetric, Φi = Φj

Additivity: Φi (Γ1 + Γ2) = Φi ((Γ1) + Φi (Γ2)

Theorem

The Shapley value is the only payoff distribution scheme that hasproperties (1) - (4)

Γ = Γ1 + Γ2 is the game (N, v) with v(C ) = v1(C ) + v2(C )





Computational Issues

We have defined some solution concepts

can we compute them efficiently?

We need to determine how to represent a coalitional gameΓ = (N, v)?

Extensive list values of all coalitionsexponential in the number of players nSuccinct a TM describing the function vsome undecidable questions might arise

We are usually interested in algorithms whose running time ispolynomial in n

So what can we do? subclasses?






































Checking Non-emptiness of the Core: Superadditive Games

An outcome in the core of a superadditive game satisfies thefollowing constraints:

xi ≥ 0 for all i ∈ N∑i∈N

xi = v(N)∑i∈C

xi ≥ v(C ), for any C ⊆ N

A linear feasibility program, with one constraint for eachcoalition: 2n + n + 1 constraints





Checking Non-emptiness of the Core: Superadditive Games

An outcome in the core of a superadditive game satisfies thefollowing constraints:


xi = v(N)∑i∈C

xi ≥ v(C ), for any C ⊆ N

A linear feasibility program, with one constraint for eachcoalition: 2n + n + 1 constraints





Superadditive Games: Computing the Least Core

Starting from the linear feasibility problem for the core

min ε


xi = v(N)∑i∈C

xi ≥ v(C )− ε, for any C ⊆ N

A minimization program, rather than a feasibility program





Superadditive Games: Computing the Least Core

Starting from the linear feasibility problem for the core

min ε


xi = v(N)∑i∈C

xi ≥ v(C )− ε, for any C ⊆ N

A minimization program, rather than a feasibility program





Computing Shapley Value

Φi (Γ) =∑

π∈Π(N) δi (Sπ(i))

Φi (Γ) is the expected marginal contribution of player i to thecoalition of his predecessors

Quick and dirty way:

Use Monte-Carlo method to compute Φi (Γ)

Convergence guaranteed by Law of Large Numbers






Φi (Γ) =∑











Φi (Γ) =∑











Φi (Γ) =∑









Banzhaf indexNucleolusKernelStable set

1 Definitions

2 Stability notions


4 Subclasses





Banzhaf index

The Banzhaf index of player i in game Γ = (N, v) is

βi (Γ) =1

2n−1

∑C⊆N

[v(C ∪ {i})− v(C )]

Dummy player, symmetry, additivity, but not efficiency.





Banzhaf index

The Banzhaf index of player i in game Γ = (N, v) is

βi (Γ) =1

2n−1

∑C⊆N

[v(C ∪ {i})− v(C )]

Dummy player, symmetry, additivity, but not efficiency.





Nucleolus

The nucleolus is a solution concept that defines a uniqueoutcome for a superadditive game.

Consider Γ = (N, v), C ⊆ N and a payoff vector x .

The deficit of C with respect to x is defined asd(x ,C ) = v(C )− x(C ).Any payoff vector x defines a 2n deficit vectord(x) = (d(x ,C1), . . . , d(x ,C2n)).Let ≤lex denote the lexicographic order

The nucleolus N (Γ) is the setN (Γ) = {x | d(x) ≤lex d(y) for all imputation y}.Can be computed by solving a polynomial number ofexponentially large LPs.





Nucleolus



The deficit of C with respect to x is defined asd(x ,C ) = v(C )− x(C ).

Any payoff vector x defines a 2n deficit vectord(x) = (d(x ,C1), . . . , d(x ,C2n)).Let ≤lex denote the lexicographic order






Nucleolus



The deficit of C with respect to x is defined asd(x ,C ) = v(C )− x(C ).Any payoff vector x defines a 2n deficit vectord(x) = (d(x ,C1), . . . , d(x ,C2n)).

Let ≤lex denote the lexicographic order






Nucleolus









Nucleolus




The nucleolus N (Γ) is the setN (Γ) = {x | d(x) ≤lex d(y) for all imputation y}.

Can be computed by solving a polynomial number ofexponentially large LPs.





Nucleolus









Kernel

The kernel consists of all outcomes where no player cancredibly demand a fraction of another player’s payoff.

Consider Γ = (N, v), i ∈ N and a payoff vector x .the surplus of i over the player j with respect to x is

Si ,j(x) = max{v(C )− x(C ) | C ⊆ N, i ∈ C , j /∈ C}

The kernel of a superadditive game Γ, K(Γ) is the set of allimputations x such that, for any pair of players (i , j) either:

Si,j(x) = Sj,i (x), orSi,j(x) > Sj,i (x) and xj = v({j}), orSi,j(x) < Sj,i (x) and xi = v({i}).

The kernel always contains de nucleolus, thus it is non-empty.





Kernel











Kernel











Kernel











Kernel











Stable set

Consider Γ = (N, v) superadditive and two imputations y , z .

y dominates z via coalition C if y(C ) ≤ v(C ) and yi > zi , forany i ∈ Cy dominates z (y dom z) if there is a coalition C such that ydominates z via coalition CFor a set of imputations JDom(J) = {z | there exists y ∈ J, y dom z}.

A set of imputations J is a stable set of Γ if {J, Dom(J)} is apartition of the set of imputations.

Stable sets form the first solution proposed for cooperativegames [von Neuwmann, Morgensten, 1944].

There are games that have no stable sets [Lucas, 1968].





Stable set










Stable set


y dominates z via coalition C if y(C ) ≤ v(C ) and yi > zi , forany i ∈ C

y dominates z (y dom z) if there is a coalition C such that ydominates z via coalition CFor a set of imputations JDom(J) = {z | there exists y ∈ J, y dom z}.








Stable set


y dominates z via coalition C if y(C ) ≤ v(C ) and yi > zi , forany i ∈ Cy dominates z (y dom z) if there is a coalition C such that ydominates z via coalition C

For a set of imputations JDom(J) = {z | there exists y ∈ J, y dom z}.








Stable set










Stable set










Stable set










Stable set









1 Definitions

2 Stability notions


4 Subclasses




Some subclasses of cooperative games

Simple gamesv(C ) ∈ {0, 1} and monotone

Weighted voting gamesInfluence games

Combinatorial Optimization gamesv depends on some measure of a formed structure

Induced subgraph gamesNetwork flow gamesMinimum cost spanning tree gamesFacility location games


















Documents

An Introduction to Cooperative Game Theorymjserna/docencia/agt-miri/slides/AGT13-coop-GT.pdf · De nitions Stability notions Other solution concepts Subclasses An Introduction to