Algorithmic Coalitional Game Theory Lecture 7: Weighted Voting Games - - PowerPoint PPT Presentation

Oskar Skibski

University of Warsaw

Algorithmic Coalitional Game Theory

Lecture 7: Weighted Voting Games

07.04.2020

Weighted Voting Games

Nassau County Board: § Hempstead #1: 9 votes § Hempstead #2: 9 votes § North Hempstead: 7 votes § Oyster Bay: 3 § Glen Cove: 1 § Long Beach: 1 What is the power of each part? 2

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Simple Games

3

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Game (", $) is simple if $ & ∈ {0,1} such that $ " = 1 (unanimity) and $ & ≤ $ . for every & ⊆ . ⊆ " (monotonicity). Simple Games

Simple Games

• A coalition ! is a winning coalition if " ! = 1. Otherwise, it

is a losing coalition.

• Simple games can be equivalently characterized by the set
• f winning coalitions:

% = ! ⊆ ' ∶ " ! = 1 .

• A winning coalition ! is minimal if removing any player from

it makes it a losing coalition.

• The set of minimal winning coalitions is defined as follows:

%* = ! ∈ % ∶ ! ∖ - ∉ % for every - ∈ ! . 4

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Simple Games

A player ! is a:

• veto player if it belongs to all winning coalitions, i.e., " ∈ $

implies ! ∈ ";

• dictator if it is a veto player and all coalitions containing !

are winning, i.e., " ∈ $ ⇔ ! ∈ ". 5

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Simple Games

• ! =

1,2 , 1,3 , 1,4 , 1,2,3 , 1,2,4 , 1,3,4 , 2,3,4 , 1,2,3,4

• !( =

1,2 , 1,3 , 1,4 , 2,3,4

• There is no veto player nor dictator.

This is the game called „My Aunt and I” that we have already considered. 6

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

) = {1,2,3,4} , - = .1 if - ≥ 3 or - = 2 and 1 ∈ -,

• therwise.

E X A M P L E

Weighted Voting Games

7

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

A simple game is a weighted voting game if there exists a list

• f weights !", !$, … , !& ∈ ℝ)*

& and quota + ∈ ℝ)* s.t.:

, - = /1 if 3

4∈5

!4 ≥ +,

• therwise.

We will denote such a game by +; !", !$, … , !& . Weighted Voting Games We will write ! - instead of ∑4∈B !4.

Weighted Voting Games

Not every simple game is a weighted voting game! Assume a weighted voting game !; #$, #&, #', #( is equivalent to the above game. We get:

• #$ + #& ≥ ! and #' + #( ≥ !
• #$ + #' < ! and #& + #( < !

Hence, 2! > #$ + #& + w' + w( ≥ 2! – contradiction! 8

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

/ = {1,2,3,4} 67 = 1,2 , 3,4

E X A M P L E

Measuring Power in WVG

In particular, for the game [50; 49, 49, 2] all players are symmetric! We say that player * is pivotal in coalition + if + ∉ - but + ∪ * ∈ -. We will calculate how often a player is pivotal. We will use Iverson brackets: 0 = 1 if 0 is true, 0 = 0,

• therwise.

9

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

How to measure power in Weighted Voting Games?

?

Measuring Power in WVG

These are just different names for the Shapley value and the Banzhaf value applied to weighted voting games. 10

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

!!"# $, & = (

)⊆+∖{#}

! ! $ − ! − 1 ! $ ! [3 is pivotal in !]

Shapley-Shubik Index [Shapley & Shubik 1954]

>"# $, & = 1 2 + @A (

)⊆+∖{#}

[3 is pivotal in !]

Banzhaf Index [Penrose 1946]

Measuring Power in WVG

Consider game: [26; 20, 6, 5, 2, 1, 1, 1, 1]. In this game we have:

!" = { 1,2 , 1,3,4 , 1,3,5 , 1,3,6 , 1,3,7 , 1,3,8 , 1,4,5,6,7,8 }

Calculating DP index for player 2 and 3 we get: /01 2, 3 = 1 14 < 5 21 = /05 2, 3 . So, a player with a smaller weight has a higher power! 11

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

/07 2, 3 = 1 !" 8

9∈!;∶7∈9

1 =

Deegan-Packel Index [Deegan & Packel 1978]

Measuring Power in WVG

12

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

New members can increase the power of existing members. E.g., in [4; 2, 2, 1] the last player is a null player, but in 4; 2, 2, 1,1 it is not. Paradox of new members Let us concentrate on the Shapley-Shubik Index. Splitting into two different players may increase the power. E.g., in [( + 1; 2, 1, 1, … , 1] first player has the power 1/(, but if he splits, in [( + 1; 1, 1, 1, 1, … , 1] he gets 2/(( + 1). Paradox of size

Measuring Power in WVG

13

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

For two sequences of (possibly overlapping) coalitions !", !$, … , !& , (

", ( $, … , ( & the latter one arises from the

former one through one-transfer if there exist indices ), * s.t.: (

+ = !+ ∪ . , ( / = ! / ∖ {.} and ( 3 = !3, for 4 ∉ {), *}.

A sequential one-transfer is a sequence 6", 6$, … , 67 of coalition sequences such that 6+8" arises from 6+ through

• ne-transfer.

14

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Which simple games are weighted voting games?

?

WVG Characterization

A sequence !", !$, … , !&, '

", ' $, … , ' & is a trading transform

if there exists a sequential one-transfer (", ($, … , () such that (" = !", !$, … , !& and () = ('

", ' $, … , ' &).

Simpler: A sequence !", !$, … , !&, '

", ' $, … , ' & is a trading transform

if multiset of players in !", !$, … , !& and '

", ' $, … , ' & are

equal: ⊔. !. =⊔. '

..

15

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

!" #

"

{3,4,7} {5,6} {3,5,6} {4,7} {3,4,6} {3,5,7} {4,5} {6,7}

WVG Characterization

A simple game is called !-trade robust if there is no trading transform "#, "%, … , "', (

#, ( %, … , ( ' for ) ≤ ! such that

"#, "%, … , "' are winning coalitions and (

#, ( %, … , ( ' are losing

coalitions. If a game is !-trade robust for every ! ∈ ℕ, then it is trade robust. 17

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Let a „simple game” be a coalitional game in which characteristic function assigns only values 0 and 1 (no unanimity and monotonicity) Let a „simple game” be a weighted voting game if there exists a list of (possibly negative) weights !", !$, … , !& ∈ ℝ& and quota ) ∈ ℝ such that * + = 1 iff ! + ≥ ). 18

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

A „simple game” is a weighted voting game if and only if it is trade robust. Characterization of WVG [Taylor & Zwicker 1995] Proof: On the blackboard.

WVG Characterization

Sketch of the proof: We show the equivalence of the following three conditions for a „simple game” (", $):

• (1): (", $) is a weighted voting game
• (2): (", $) is trade robust
• (3): (", $) is 2'(-trade robust.

(1) ⇒ (2): A weighted voting game [+; -., -', … , -0] is trade robust, because for every trading transform 2., … , 20, 3

., … , 3 0 s.t. 2., … , 20 are winning coalitions:

(- 3

. + ⋯ + - 3 0 )/7 = (- 2. + ⋯ + - 20 )/7 ≥ +

which implies all coalitions 3

., … , 3 0 cannot be losing.

19

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof: We show the equivalence of the following three conditions for a „simple game” (", $):

• (1): (", $) is a weighted voting game
• (2): (", $) is trade robust
• (3): (", $) is 2'(-trade robust.

(2) ⇒ (3): From the definition. (3) ⇒ (1): The main challenge. 20

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof: Assume a simple game (", $) is trade robust. We need to define a weight function &: " → ℝ that will correspond to (", $). We will inductively construct the weight function &. 21

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof (continued): 22

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

We will say that !: # → ℝ is trade robust for # ⊆ ', if for every ( ≤ 2+ , -|/|01 there are no sequences 2 = 41 ∪ 6

1, … , 49 ∪ 6 9 , 2: = 41 : ∪ 6 1 :, … , 49 : ∪ 6 9 : s.t.:

1) 4; ∩ 6

; = ∅ and 4; : ∩ 6 ; : = ∅ for every > ∈ {1, … , (}

2) (41, … , 49, 41

:, … , 49 :) is a trading transform

3) 6

; ⊆ # and 6 ; : ⊆ # for every > ∈ {1, … , (}

4) ! 6

1 + ⋯ ! 6 9 ≤ ! 6 1 : + ⋯ ! 6 9 :

and all coalitions from 2 all winning and all coalitions from 2: are losing. TRADE ROBUST (FOR A SET)

WVG Characterization

Sketch of the proof (continued): 23

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

!" #

"

!"

$

#

" $

{3,4,7} {5,6} {3,5,6} {4,7} {3,4,6} {3,5,7} {4,5} {6,7} {1} {2} {2} {1,2}

4 9 9 9 4

{/}

9

0(/)

WVG Characterization

Sketch of the proof (continued): For ! = ∅, we get that for every $ ≤ 2'()* there are no sequences + = ,* ∪ .

*, … , ,1 ∪ . 1 , +2 = (

) ,*

2 ∪ . * 2, … , ,1 2 ∪

.

1 2 such that:

1) ,5 ∩ .

5 = ∅ and ,5 2 ∩ . 5 2 = ∅ for every 7 ∈ {1, … , $}

2) (,*, … , ,1, ,*

2, … , ,1 2) is a trading transform

3) .

5 ⊆ ∅ and . 5 2 ⊆ ∅ for every 7 ∈ {1, … , $}

4) = .

* + ⋯ = . 1 ≤ = . * 2 + ⋯ = . 1 2

and all coalitions from + all winning and all coalitions from +2 are losing. Clearly, this follows from 2'(-trade robustness. 24

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof (continued): For ! = #, we get that for every $ ≤ 2'()* = 1 (i.e., $ = 1) there are no coalitions (-* ∪ /

*) and (-* 1 ∪ / * 1) such that:

1)

• * ∩ /

* = ∅ and -* 1 ∩ / * 1 = ∅

2) (-*, -*

1) is a trading transform (i.e., -* = -* 1)

3) /

* ⊆ # and / * 1 ⊆ #

4) 6 /

* ≤ 6(/ * 1)

and coalition (-* ∪ /

*) is winning and (-* 1 ∪ / * 1) is losing.

This means that every winning coalition has higher weight than every losing, i.e., game is weighted. 25

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof (continued): 26

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

For a „simple game” (", $), set & ⊆ ", weight function (: & → ℝ which is trade robust for & and , ∈ " ∖ & there exists / ∈ ℝ such that (0: & ∪ {,} → ℝ with (0 , = / and (0 5 = ( 5 for 5 ∈ & is trade robust for & ∪ {,}. MAIN LEMMA This means we can inductively construct the weight function.

WVG Characterization

Sketch of the proof (continued): Proof of MAIN LEMMA: We say what does it mean that value ! is „too low” or „too high” and show: STEP 1: Every wrong value is either too low or too high. STEP 2: If ! is too low, then !" < ! is also too low. Analogously, if ! is too high, then !" > ! is also too high. STEP 3: Value ! cannot be both too low and too high. STEP 4: Values which are too low are bounded above. Values which are too high are bounded below. STEP 5: The supremum of too low values is a too low value. The infimum of the high values is a too high value. 27

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

(−∞ too low ] [ too high +∞)

WVG Characterization

Sketch of the proof (continued): A real value ! is „too low” if there exist two sequences: "# ∪ %

#, … , "( ∪ % ( , "# ) ∪ % # ), … , "( ) ∪ % ( )

that show *) is not trade robust for + ∪ {-} and / ∈ {1, … , 2} ∶ - ∈ %

4

> / ∈ {1, … , 2} ∶ - ∈ %

4 ) .

A real value ! is „too high” if there exist two sequences: "# ∪ %

#, … , "( ∪ % ( , "# ) ∪ % # ), … , "( ) ∪ % ( )

that show *) is not trade robust for + ∪ {-} and / ∈ {1, … , 2} ∶ - ∈ %

4

< / ∈ {1, … , 2} ∶ - ∈ %

4 ) .

28

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof (continued): 29

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

!" #

"

!"

$

#

" $

{5,6} {5,7} {5,6} {5,7} {1, 3} {1, 2}

9 4

{3}

.

Assume 3 is more important than 2, but . < 9.

. 4

{2}

9

WVG Characterization

Sketch of the proof (continued): STEP 0: If ! is „too low”, then there exists two sequences: "# ∪ %

#, … , "( ∪ % ( , "# ) ∪ % # ), … , "( ) ∪ % ( )

that show *) is not trade robust for + ∪ {-} such that - does not appear in any %

/ ):

0 ∈ {1, … , 3} ∶ - ∈ %

/ )

= 0. If ! is „too high”, then there exist two sequences: "# ∪ %

#, … , "( ∪ % ( , "# ) ∪ % # ), … , "( ) ∪ % ( )

that show *) is not trade robust for + ∪ {-} such that - does not appear in any %

/:

0 ∈ {1, … , 3} ∶ - ∈ %

/

= 0. 30

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof (continued): 31

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

!" #

"

!"

$

#

" $

{4,6} {5,6} {5,7} {4,5} {5,6} {5,7} {1} {1, 3} {1, 3} {1, 2}

4 4 9 4

{3}

/

Assume 3 is more important than 2, but / < 9.

/ 4

{2}

9 /

3 3

WVG Characterization

Sketch of the proof (continued): 32

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

!" #

"

!"

$

#

" $

{4,6} {3,5,6} {5,7} {3,4,5} {5,6} {5,7} {1} {1} {1} {1, 2}

4 4 9 4

{3}

/

Assume 3 is more important than 2, but / < 9.

4

{2}

9

WVG Characterization

Sketch of the proof (continued): STEP 1: Every wrong value is either too low or too high. In sequences in which ! appears the same number of times

• n both sides, we can move them from "

# to $#. In this way,

we get the contradiction with the inductive assumption. 33

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof (continued): STEP 2: If ! is too low, then !" < ! is also too low. Analogously, if ! is too high, then !" > ! is also too high. If we change the value from ! to !′ all conditions will still be

• satisfied. Hence, if two sequences were showing that & is not

trade robust for ' ∪ {*}, then it still shows that. 34

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof (continued): STEP 3: Value ! cannot be both too low and too high. Assume ! is „too low” and sequences ( ) $% ∪ '

%, … , $* ∪

'

* , $% + ∪ ' % +, … , $* + ∪ ' * + are the witness such that , does

not appear on the right side at all and on the left side - times. Assume ! is also „too high” and sequences ( ) .% ∪ /

%, … , .0 ∪

/

0 , .% + ∪ / % +, … , .0 + ∪ / +

are the witness such that , does not appear on the left side at all and on the right side 1 times. Now, we can take two sequences: 1 × $% ∪ '

%, … , $* ∪ ' * + - × .% ∪ / %, … , .0 ∪ /

1 × $%

+ ∪ ' % +, … , $* + ∪ ' * + + - × .% + ∪ / % +, … , .0 + ∪ / +

and move all appearances of , from '

4// 4 to $4/.4 which

contradicts the inductive assumption. 35

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof (continued): STEP 4: Values which are too low are bounded above. Values which are too high are bounded below. ! = 2 ⋅ 2% & ∑(∈* + , is the upper bound for too low values, because if - appears on the left hand side we get: .

/01 2

.

(∈34

+(,) > ! − 2% & .

(∈*

+ , ≥ ≥ 2% & .

(∈*

+ , ≥ .

/01 2

.

(∈3

4 :

+ , . Hence, condition 4) is violated. For the lower bound take – !. 36

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

WVG Characterization

Sketch of the proof (continued): STEP 5: The supremum of too low values is a too low value. The infimum of the high values is a too high value. Take a sequence !" < !$ < ⋯ that converge to the

• supremum. For each value there exists a pair of sequences

that shows that !& is too low. Since we have a finite number of possible pairs of sequences of coalitions, one pair must appear infinite number of times. Hence, by taking only the corresponding values !&' < !&( < ⋯ we get a sequence converging to the supremum for which there exists a single pair of sequences. From this, we get that this pair is also a witness that the supremum is too low (from condition (4)). 37

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Conclusions

Today, we discussed:

• class of weighted voting games,
• 3 power indices,
• paradoxes regarding power, and
• characterization using trading transfers.

38

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

References

• [Deegan & Packel 1978] J. Deegan, E.W. Packel.

A new index of power for simplen-person games. International Journal of Game Theory 7, 113-123, 1978.

• [Penrose 1946] L.S. Penrose.

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

• [Shapley & Shubik 1954] L.S. Shapley, M. Shubik.

A method for evaluating the distribution of power in a committee

• system. American Political Science Review 48, 787-792, 1954.
• [Taylor & Zwicker 1992] A. Taylor, W. Zwicker.

A characterization of weighted voting. Proceedings of the American Mathematical Society 115, 1089-1094, 1992.

39

Oskar Skibski (UW) Algorithmic Coalitional Game Theory