[PPT] - Algorithmic Coalitional Game Theory Lecture 4: Shapley value Oskar PowerPoint Presentation

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Oskar Skibski

University of Warsaw

Algorithmic Coalitional Game Theory

Lecture 4: Shapley value

17.03.2019

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Payoff Division

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Assume all players in game !, # cooperate. Define a payoff vector $ ∈ ℝ'. Payoff Division In other words: how to split a joint payoff? What we want to achieve?

Stability?
Fairness?

TODAY

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Value of the Game

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

A value of the game is a function ! that for a given game associates a vector of " real numbers, one per each player. Thus, ! #, % ∈ ℝ( and !) #, % denotes the value of player * in game #, % Value of the Game During this (and the next) lecture we will associate ! with the function that given a game returns the payoff division.

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Shapley Value

! " " ! # # ! " # # " ! # # ! " " !

1 6 1 6 1 6 1 6 1 6 1 6

&'

( ), + =

1 ) ! .

/∈1(3)

+ &/

( ∪ 6

− + &/

(

,

where:

Π ) is the set of all possible permutations of players ',

i.e., functions 9: ) → 1, … , )

&/

( is the set of players that preceed 6 in permutation 9:

&/

( = {> ∈ ) ∶ 9 > < 9 6 }

Shapley Value

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Shapley Value

! " " ! # # ! " # # " ! # # ! " " !

1 6 1 6 1 6 1 6 1 6 1 6

&'

( ), + =

1 ) ! .

/∈1(3)

+ &/

( ∪ 6

− + &/

(

,

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Shapley Value

!"

# $, & =

1 $ ! *

+∈-(/)

& !+

# ∪ 2

− & !+

#

,

average marginal contribution in any permutation

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

6 4 5 5 4 6 6 4 5 6 6 6 6 5 4 4 5 5 4

1 6 1 6 1 6 1 6 1 6 1 6

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Shapley Value

!"

# $, & = ( )*+ |-| 1

|$| (

/∈1 - ∶/ # *)

& !#

/ ∪ 4

− & !#

/

$ − 1 ! ,

7 8 8 7 9 9 7 8 9 9 9 9

1 3 1 3 1 3

average marginal contribution in permutations in which 4 is at position ; position of 4

8 7 7 8 8 7

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Shapley Value

!"

# $, & = ( )*+ |-|./

1 |$| (

1⊆-∖{#}∶ 1 *)

& ! ∪ 8 − & !

./

)

average marginal contribution to coalitions of size : coalition size

; < ; < = = < ; = = = = ; < < ; ; <

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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1 3 1 3 1 3

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Shapley Value

! " # ! ! ! " #

$%

& ', ) = + ,-. |0|12

1 |'| +

4⊆0∖{&}∶ 4 -,

) $ ∪ ; − ) $

0 12 ,

average marginal contribution to coalitions of size = coalition size

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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1 3 1 3 1 3

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Shapley Value

! " # ! ! ! " #

$%

& ', ) =

+

,⊆.∖ &

1 '

. 12 ,

) $ ∪ 4 − ) $

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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1 3 1 3 1 3

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Shapley Value

! " # ! ! ! " # $%

& ', ) =

+

,⊆.∖ &

$ ! ' − $ − 1 ! ' ! ) $ ∪ 4 − ) $

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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1 3 1 3 1 3

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Shapley’s Axiomatization

The Shapley value is the unique value ! that satisfies:

Efficiency: ∑#∈% !# &, ( = ( & .
Symmetry: !# &, ( = !* # (&, , ( ) for every bijection ,: & → &.
Additivity: !# &, (0 + (2 = !# &, (0 + !#(&, (2).
Null-player: ∀4⊆% (( 6 = ( 6 ∪ 8

) ⇒ !# &, ( = 0 .

Shapley’s Axiomatization [Shapley 1953]

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Notation: games , ( and (0 + (2 are defined as follows: , ( 6 = ( , 8 ∶ 8 ∈ 6 and (0 + (2 6 = (0 6 + (2(6). Proof: On the blackboard.

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Shapley’s Axiomatization

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Sketch of the proof: Satisfiability: Easy to check. Uniqueness: Consider a class of games !" "⊆$ defined as follows: !" % = '1 )* + ⊆ %, ./ℎ123)41. Step 1: For every game it holds: 6 = ∑"⊆$(Δ" 6 ⋅ !") for some Δ" 6 . Step 2: If < satisfies Efficiency, Symmetry and Null-player, then <= >, !" = 1/|+| if ) ∈ + and <= >, !" = 0, otherwise. Step 3: From Step 1, Step 2 and Additivity: <= >, 6 = ∑"⊆$∶=∈"

CD E "

. Btw, it can be shown that Δ" 6 = ∑F⊆" −1

" H F ⋅ 6(%) (these are

called Harsanyi dividens).

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Young’s Axiomatization

The Shapley value is the unique value ! that satisfies:

Efficiency: ∑#∈% !# &, ( = ( & .
Symmetry: !# &, ( = !* # (&, , ( ) for every bijection ,: & → &.
Marginality: ∀1⊆%((3 4 ∪ 6

− (3 4 = (8 4 ∪ 6 − (8 4 ) ⇒ !# (3 = !# (8 .

Young’s Axiomatization [Young 1985]

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Notation: Marginal contribution vector :;# ( is defined as follows: :;# ( 4 = ( 4 ∪ 6 − ((4). Proof: On the blackboard.

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Young’s Axiomatization

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Sketch of the proof: Satisfiability: Easy to check. Uniqueness: We know that for every game: ! = ∑$⊆&(($ ⋅ *$) for some coefficients ($. Consider index of a game ,(!) defined as the minimum number of non- zero values ($ in the expression above. Step 1: If , ! = 0, then from Symmetry and Efficiency: ./ ! = 0. Step 2: Fix 0 ≥ 1. Assume that if , ! < 0, then ./ ! = 45

/(!).

Let , ! = 0 and 46, 48, … , 4: be coalitions with non-zero ($. Step 3: If ; ∉ 46 ∩ 48 ∩ ⋯ ∩ 4:, then for ? = ∑$⊆&∶/∈$(($ ⋅ *$) we have , ? < 0 and B(/ ! = B(/(?). So, from Marginality: ./ ! = 45

/(!).

Step 4: If ; ∈ 46 ∩ 48 ∩ ⋯ ∩ 4:, then from Symmetry and Efficiency: ./ ! = 45

/ ! .

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Myerson’s Axiomatization

The Shapley value is the unique value ! that satisfies:

Efficiency: ∑#∈% !# &, ( = ( & .
Balanced Contributions: !# &, ( − !# & ∖ , , ( = !- &, ( −

!-(& ∖ / , () for every /, , ∈ &.

Myerson’s Axiomatization [Myerson 1980]

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Notation: In game & ∖ / , ( the value of every coalition 1 ⊆ & ∖ {/} is the same as in game &, ( , and other coalitions do not exist. Proof: Tutorials.

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Hart and Mas-Colell’s Axiomatiz.

The Shapley value is the unique value ! that satisfies:

Efficiency: ∑#∈% !# &, ( = ( & .
Potential: there exists a function *: , → ℝ satisfying * ∅, ( = 0 s.t.

!# &, ( = * &, ( − *(& ∖ 4 , () for every &, ( and 4 ∈ &.

Hart and Mas-Colell’s Axiomatization [Hart & Mas-Colell 1989]

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Notation: , is the set of all possible games. Proof: On the blackboard.

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Hart and Mas-Colell’s Axiomatiz.

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Sketch of the proof: Satisfiability: For ! ", $ = ∑'⊆)

' *+ ! ) * ' ! ) !

$(.) it holds that .0

1 ", $ = ! ", $ − !(" ∖ 4 , $).

Uniqueness: Assume there exists a potential function !: 6 → ℝ. If " = {4}, then clearly ! 4 , $ − 0 = <1 4 , $ = $ 4 . Assume " > 1. From the definition: ? ⋅ ! ", $ = A

1∈)

! " ∖ 4 , $ + A

1∈)

<1 ", $ . From Efficiency, the second sum equals $ " . So, we get the recursive formula for !.

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Conclusions

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Shapley value is a fair payoff division in coalitional games.
It turns out to be a unique payoff division that satisfies

many desirable properties:

Efficiency, Symmetry, Additivity and Null-player
Efficiency, Symmetry and Marginality
Efficiency and Balanced Contributions
Efficiency and Potential

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References

[Hart & Mas-Colell 1989] S. Hart, A. Mas-Colell.

Potential, value and consistency. Econometrica 57, 589-614, 1989.

[Myerson 1980] R. Myerson.

Conference structures and fair allocation rules. International Journal of Game Theory 9, 169-82, 1980.

[Shapley 1953] L.S. Shapley.

A value for n-person games. Contributions to the Theory of Games II, 307-317, 1953.

[Young 1985] H.P. Young.

Monotonic solutions of cooperative games. International Journal of Game Theory 14, 65-72, 1985.

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory