Meteorologische Treiber Modul Luftchemie 27.11.2009 Sabine Banzhaf.
ATGK 5 Banzhaf etc - mimuw.edu.pl
Transcript of 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
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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
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!"# $, & = 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
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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.
Probabilistic Values
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]
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Notation: game ,% and %& + %( are defined as follows:,% 9 = , ⋅ % 9 and %& + %( 9 = %& 9 + %((9).
Proof: On the blackboard.
Axiomatization of Prob. Values
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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}.
Probabilistic Values
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]
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Notation: game ,% and %& + %( are defined as follows:,% 9 = , ⋅ % 9 and %& + %( 9 = %& 9 + %((9).
Probabilistic ValuesProbabilistic
Values
Linearity + Dummy-player+ Positivity
+ Symmetry
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Semivalues
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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]
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Notation: game ) % is defined as follows: ) % - =% ) . ∶ . ∈ - .
Proof: On the blackboard.
Axiomatization of Semivalues
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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 * = |$|• 87 * = 8C(*) for every * ⊆ < ∖ {(, @}
• 87 < ∖ {(} = 8C < ∖ @
In result, there exists D: 1,… , G → [0,1] such that 87 * =D( * ) for every ( ∈ <, * ⊆ < ∖ {(}.Finally, from Dummy-player, ∑LMNOPQD(R) OPQ
L = 1.
SemivaluesProbabilistic
Values
Semivalues
BanzhafValue
Linearity + Dummy-player+ Positivity
+ Symmetry
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+ Efficiency
ShapleyValue
Random-Order Values
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A value is a random-order value if it is of the form:
!" #, % = '(∈*(,)
. / % 0(" ∪ 2 − % 0(" ,
for some .: Π # → [0,1] such that ∑(∈*(,) .(/) = 1.
Random-Order Values [Weber 1988]
Shapley value is a random-order value for . / = <, !.
Banzhaf value is a NOT a random-order value.
Random-Order Values
A value is a random-order value if and only if it is a probabilistic value that satisfies:• Efficiency: ∑"∈$%" &, ( = ( & .
Axiomatization of Random-Order Values [Weber 1988]
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Proof: On the blackboard.
Axiomatization of Random-Order
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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∈$∖/ %1(0) for every 0 ⊊ '.Now, we define % 3 for 3 = (45, 47, … , 49) as follows:
% 3 =:;<5
9 %=>({45, … , 4;?5})∑"@; %=A({45, … , 4;?5})
.
Details are left for the tutorials.
Weighted Shapley Value
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!"#$% &, ( = *
+⊆-
.$
.+⋅ 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
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!"#$ %, ' = 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
Linearity + Dummy-player+ Positivity
+ Efficiency + Symmetry
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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.
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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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