atgk 5 banzhaf etc - mimuw.edu.pl

Oskar Skibski

University of Warsaw

Algorithmic Coalitional Game Theory

Lecture 5: Banzhaf Value and probabilistic values

24.03.2020

Payoff Division

2Oskar 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?

In other words: how important is each player?or what is player’s expected value?

TODAY

Banzhaf Value

Oskar Skibski (UW) Algorithmic Coalitional Game Theory 3

!"# $, & = 12 * +, -

.⊆*∖{#}& 3 ∪ 5 − & 3 .

Banzhaf Value [Banzhaf 1965, Penrose 1946]

„What is the point of taking this Shapley’s weighted average?”

Generalization of the Shapley value and Banzhaf value: probabilistic values.

Probabilistic Values


A value is a probabilistic value if it is of the form:

!" #, % = '(⊆*∖{"}

." / % / ∪ 1 − % / ,

for some .": 2*∖{"} → [0,1] such that ∑(⊆*∖{"} ."(/) = 1.

Probabilistic Values [Weber 1988]

Shapley value is a probabilistic value for ." / = |(|! * ? ( ?@ !* ! =

@|*| ⋅

@B CDE

.

Banzhaf value is a probabilistic value for ." / = @F B CD.


A value is a probalistic value if and only if it satisfies:• Linearity: !" #, %& + %( = !" #, %& + !"(#, %() and !" #, ,% = , ⋅ !"(#, %) for a constant , ∈ ℝ.

• Positivity: !" #, % ≥ 0 if % is monotone.• Dummy-player: ∀3⊆5∖{"} (% 9 ∪ {;} − % 9 = =) ⇒

!" #, % = = .

Axiomatization of Probabilistic Values [Weber 1988]


Notation: game ,% and %& + %( are defined as follows:,% 9 = , ⋅ % 9 and %& + %( 9 = %& 9 + %((9).

Proof: On the blackboard.

Axiomatization of Prob. Values


Sketch of the proof:

Assume ! satisfies Linearity, Positivity and Dummy-player.

• From Linearity: !" #, % = ∑(⊆* +(%(-).• From Dummy-player, we have +(∪{"} + +( = 0 for every- ⊆ # ∖ 5 , - ≠ ∅. So, we get that: !" #, % = ∑(⊆*∖{"} 8((% - ∪ 5 − %(-)) for some 8(.

• Also from Dummy-player we get ∑(⊆*∖{"} 8( = 1.• Finally, from Positivity: 8( ≥ 0 for every - ⊆ # ∖ {5}.


A value is a probalistic value if and only if it satisfies:• Linearity: !" #, %& + %( = !" #, %& + !"(#, %() and !" #, ,% = , ⋅ !"(#, %) for a constant , ∈ ℝ.

• Milnor Axiom: min3⊆5∖{"} % 9 ∪ ; − % 9 ≤ !" #, % ≤ max3⊆5∖{"} % 9 ∪ ; − % 9

Axiomatization of Probabilistic Values [Mondered 1988]


Notation: game ,% and %& + %( are defined as follows:,% 9 = , ⋅ % 9 and %& + %( 9 = %& 9 + %((9).

Probabilistic ValuesProbabilistic

Values

Linearity + Dummy-player+ Positivity

+ Symmetry


Semivalues


A value is a semivalue if it is of the form:

!" #, % = '(⊆*∖{"}

.( 0 ) % 0 ∪ 3 − % 0 ,

for .: 0,… , 8 − 1 → [0,1] such that ∑>?@ABC.(D) ABC> = 1.

Semivalues [Dubey, Neyman, Weber 1981]

Shapley value is a semivalue for .(D) = CA EFG

H.

Banzhaf value is a semivalue for .(D) = CIEFG.

Semivalues

A value is a semivalue if and only if it is a probabilistic valuethat satisfies:• Symmetry: !" #, % = !' " (#, ) % ) for every bijection):# → #.

Axiomatization of Semivalues [Dubey, Neyman, Weber 1981]


Notation: game ) % is defined as follows: ) % - =% ) . ∶ . ∈ - .


Axiomatization of Semivalues


Sketch of the proof:

Consider a game !"# $ = &1 () * ⊊ $,

0 ./ℎ123(41.We know that if 67 !"# = 87(*) for every * ⊆ < ∖ {(}.From Symmetry we show that for every players (, @ ∈ <:• 87 * = 87($) for every *, $ ⊆ < ∖ {(} such that * = |$|• 87 * = 8C(*) for every * ⊆ < ∖ {(, @}

• 87 < ∖ {(} = 8C < ∖ @

In result, there exists D: 1,… , G → [0,1] such that 87 * =D( * ) for every ( ∈ <, * ⊆ < ∖ {(}.Finally, from Dummy-player, ∑LMNOPQD(R) OPQ

L = 1.

SemivaluesProbabilistic

Values

Semivalues

BanzhafValue


+ Symmetry


+ Efficiency

ShapleyValue

Random-Order Values


A value is a random-order value if it is of the form:

!" #, % = '(∈*(,)

. / % 0(" ∪ 2 − % 0(" ,

for some .: Π # → [0,1] such that ∑(∈*(,) .(/) = 1.

Random-Order Values [Weber 1988]

Shapley value is a random-order value for . / = <, !.

Banzhaf value is a NOT a random-order value.

Random-Order Values

A value is a random-order value if and only if it is a probabilistic value that satisfies:• Efficiency: ∑"∈$%" &, ( = ( & .

Axiomatization of Random-Order Values [Weber 1988]



Axiomatization of Random-Order


Sketch of the proof:Clearly, a random-order value is an efficient probabilisticvalue. We will prove that every efficient probabilistic value is a random-order value.

A probabilistic value satisfies Efficiency iff: ∑"∈$ %"(' ∖ {*}) =1 and ∑"∈/ %"(0 ∖ {*}) = ∑1∈$∖/ %1(0) for every 0 ⊊ '.Now, we define % 3 for 3 = (45, 47, … , 49) as follows:

% 3 =:;<5

9 %=>({45, … , 4;?5})∑"@; %=A({45, … , 4;?5})

.

Details are left for the tutorials.

Weighted Shapley Value


!"#$% &, ( = *

+⊆-

.$

.+⋅ Δ+ ( ,

for some .:& → ℝ4.

Weighted Shapley Value [Shapley 1953, Kalai and Samet 1987]

Notation: Δ+ ( are Harsanyi dividends, defined as follows:Δ+ ( = ∑7⊆+ −1 7 : + ⋅ ((<); they satisfy: ( = ∑+⊆- Δ+ ( ⋅ >+

Weighted Shapley value is a random-order value for ? @ = ∏BCD

E %FG%FH4%FI4⋯4%FG

, where @ = (KD, KL, … , KE).

Weighted Shapley value is NOT a semivalue.

Owen Value


!"#$ %, ' = 1|Π$ % | ,

-∈/0(2)' 4-# ∪ {7} − ' 4-# ,

for some partition : = {4;, … , 4=}, where Π$ % ⊆ Π(%) isa set of permutations in which players from differentcoalitions do not mix, i.e., forms separate blocks.

Owen Value [Owen 1972]

Owen value is a random-order value.

Owen value is NOT a semivalue.

ConclusionsProbabilistic

Values

Random-OrderValues

Semivalues

ShapleyValue

BanzhafValue

Owen Value

WeightedShapley

Value


+ Efficiency + Symmetry


References• [Banzhaf 1965] J.F. Banzhaf III.

Weighted voting doesn't work: A game theoretic approach. Rutgers Law Review 19, 343, 1965.

• [Dubey et al. 1981] P. Dubey, A. Neyman, R.J. Weber.Value theory without efficiency. Mathematics of Operations Research 6, 122-128, 1981.

• [Kalai & Samet 1987] E. Kalai, D. Samet.On weighted Shapley values. International Journal of Game Theory 16, 205-222, 1987.

• [Monderer 1988] D. Monderer.Values and semivalues on subspaces of finite games. International Journal of Game Theory 17, 301-310, 1988.

• [Owen 1972] G. Owen.Multilinear extensions of games. Management Science 18, 64-79, 1972.


References• [Penrose 1946] L.S. Penrose.

The elementary statistics of majority voting. Journal of the Royal Statistical Society, 53-57, 1946.

• [Shapley 1953] L.S. Shapley.A value for n-person games. Contributions to the Theory of Games II, 307-317, 1953.

• [Weber 1988] R.J. Weber.Probabilistic values for games. The Shapley Value. Essays in Honor of Lloyd S. Shapley, 101-119, 1988.


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