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

Algorithmic Coalitional Game Theory Lecture 4: Shapley value Oskar Skibski University of Warsaw 17.03.2019

Payoff Division Payoff Division Assume all players in game !, # cooperate. Define a payoff vector $ ∈ ℝ ' . In other words: how to split a joint payoff? What we want to achieve? • Stability? TODAY • Fairness? 2 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Value of the Game During this (and the next) lecture we will associate ! with the function that given a game returns the payoff division. Value of the Game 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 #, % 3 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Shapley Value Shapley Value 1 ( ∪ 6 ( &' ( ), + = . + & / − + & / , ) ! /∈1(3) 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 } & / 1 1 1 ! # " ! " # # ! " 6 6 6 1 1 1 # " ! " # ! " ! # 6 6 6 4 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Shapley Value 1 ( ∪ 6 ( &' ( ), + = . + & / − + & / , ) ! /∈1(3) 1 1 1 ! # " ! " # # ! " 6 6 6 1 1 1 # " ! " # ! " ! # 6 6 6 5 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Shapley Value average marginal contribution in any permutation 1 # ∪ 2 # !" # $, & = * & ! + − & ! + , $ ! +∈-(/) 1 1 1 4 6 5 4 5 6 6 4 5 6 6 6 1 1 1 6 5 4 5 6 4 5 4 6 6 6 6 6 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Shapley Value |-| 1 / ∪ 4 / & ! # − & ! # !" # $, & = ( ( , |$| $ − 1 ! )*+ /∈1 - ∶/ # *) average marginal contribution in permutations position of 4 in which 4 is at position ; 1 1 1 3 3 3 7 9 8 7 8 9 9 7 8 9 8 7 8 9 7 8 7 9 7 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Shapley Value |-|./ 1 & ! ∪ 8 − & ! !" # $, & = ( ( |$| - ./ ) )*+ 1⊆-∖{#}∶ 1 *) average marginal contribution to coalitions of size : coalition size 1 1 1 3 3 3 < = ; < ; = = < ; = ; < ; = < ; < = 8 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Shapley Value |0|12 1 ) $ ∪ ; − ) $ $% & ', ) = + + |'| 0 12 , ,-. 4⊆0∖{&}∶ 4 -, average marginal contribution to coalitions of size = coalition size 1 1 1 3 3 3 " ! ! " # ! # ! 9 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Shapley Value 1 $% & ', ) = + ) $ ∪ 4 − ) $ . 12 ' ,⊆.∖ & , 1 1 1 3 3 3 " ! ! " # ! # ! 10 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Shapley Value $ ! ' − $ − 1 ! $% & ', ) = + ) $ ∪ 4 − ) $ ' ! ,⊆.∖ & 1 1 1 3 3 3 " ! ! " # ! # ! 11 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Shapley’s Axiomatization Shapley’s Axiomatization [Shapley 1953] 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 . • 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. 12 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Shapley’s Axiomatization Sketch of the proof: Satisfiability: Easy to check. Uniqueness: Consider a class of games ! " "⊆$ defined as follows: ! " % = '1 )* + ⊆ %, 0 ./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. C D E Step 3: From Step 1, Step 2 and Additivity: < = >, 6 = ∑ "⊆$∶=∈" . " " H F ⋅ 6(%) (these are Btw, it can be shown that Δ " 6 = ∑ F⊆" −1 called Harsanyi dividens ). 13 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Young’s Axiomatization Young’s Axiomatization [Young 1985] 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 . Notation: Marginal contribution vector :; # ( is defined as follows: :; # ( 4 = ( 4 ∪ 6 − ((4) . Proof: On the blackboard. 14 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Young’s Axiomatization 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 4 6 , 4 8 , … , 4 : be coalitions with non-zero ( $ . Step 3: If ; ∉ 4 6 ∩ 4 8 ∩ ⋯ ∩ 4 : , then for ? = ∑ $⊆&∶/∈$ (( $ ⋅ * $ ) we have , ? < 0 and B( / ! = B( / (?) . So, from Marginality: . / ! = 45 / (!) . Step 4: If ; ∈ 4 6 ∩ 4 8 ∩ ⋯ ∩ 4 : , then from Symmetry and Efficiency: . / ! = 45 / ! . 15 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Myerson’s Axiomatization Myerson’s Axiomatization [Myerson 1980] The Shapley value is the unique value ! that satisfies: Efficiency: ∑ #∈% ! # &, ( = ( & . • Balanced Contributions: ! # &, ( − ! # & ∖ , , ( = ! - &, ( − • ! - (& ∖ / , () for every /, , ∈ & . Notation: In game & ∖ / , ( the value of every coalition 1 ⊆ & ∖ {/} is the same as in game &, ( , and other coalitions do not exist. Proof: Tutorials. 16 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Hart and Mas-Colell’s Axiomatiz. Hart and Mas-Colell’s Axiomatization [Hart & Mas-Colell 1989] 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 ∈ & . Notation: , is the set of all possible games. Proof: On the blackboard. 18 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Hart and Mas-Colell’s Axiomatiz. 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 ! " ∖ 4 , $ + A < 1 ", $ . 1∈) 1∈) From Efficiency, the second sum equals $ " . So, we get the recursive formula for ! . 19 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Conclusions • 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 20 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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. 21 Oskar Skibski (UW) Algorithmic Coalitional Game Theory