Algorithmic Coalitional Game Theory Lecture 7: Weighted Voting Games Oskar Skibski University of Warsaw 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 Simple Games Game (", $) is simple if $ & ∈ {0,1} such that $ " = 1 (unanimity) and $ & ≤ $ . for every & ⊆ . ⊆ " (monotonicity) . 3 Oskar Skibski (UW) Algorithmic Coalitional Game Theory
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 of 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 E ) = {1,2,3,4} X A , - = .1 if - ≥ 3 or - = 2 and 1 ∈ -, M P 0 otherwise. L E • ! = 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
Weighted Voting Games Weighted Voting Games A simple game is a weighted voting game if there exists a list & and quota + ∈ ℝ )* s.t.: of weights ! " , ! $ , … , ! & ∈ ℝ )* , - = /1 if 3 ! 4 ≥ +, 4∈5 0 otherwise. We will denote such a game by +; ! " , ! $ , … , ! & . We will write ! - instead of ∑ 4∈B ! 4 . 7 Oskar Skibski (UW) Algorithmic Coalitional Game Theory
Weighted Voting Games Not every simple game is a weighted voting game! E X / = {1,2,3,4} A 6 7 = M 1,2 , 3,4 P L E 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
Measuring Power in WVG ? How to measure power in Weighted Voting Games? 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 , otherwise. 9 Oskar Skibski (UW) Algorithmic Coalitional Game Theory
Measuring Power in WVG Shapley-Shubik Index [Shapley & Shubik 1954] ! ! $ − ! − 1 ! !!" # $, & = ( [3 is pivotal in !] $ ! )⊆+∖{#} Banzhaf Index [Penrose 1946] 1 >" # $, & = 2 + @A ( [3 is pivotal in !] )⊆+∖{#} 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
Measuring Power in WVG Deegan-Packel Index [Deegan & Packel 1978] 1 1 /0 7 2, 3 = ! " 8 = 9∈! ; ∶7∈9 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: /0 1 2, 3 = 1 14 < 5 21 = /0 5 2, 3 . So, a player with a smaller weight has a higher power! 11 Oskar Skibski (UW) Algorithmic Coalitional Game Theory
Measuring Power in WVG Let us concentrate on the Shapley-Shubik Index. Paradox of new members 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 size 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) . 12 Oskar Skibski (UW) Algorithmic Coalitional Game Theory
Measuring Power in WVG 13 Oskar Skibski (UW) Algorithmic Coalitional Game Theory
WVG Characterization ? Which simple games are weighted voting games? 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 $ , … , 6 7 of coalition sequences such that 6 +8" arises from 6 + through one-transfer. 14 Oskar Skibski (UW) Algorithmic Coalitional Game Theory
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 ! " # " {3,4,7} {3,4,6} {5,6} {3,5,7} {3,5,6} {4,5} {4,7} {6,7} Oskar Skibski (UW) Algorithmic Coalitional Game Theory
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 ! + ≥ ) . Characterization of WVG [Taylor & Zwicker 1995] A „simple game” is a weighted voting game if and only if it is trade robust. Proof: On the blackboard. 18 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. (1) ⇒ (2): A weighted voting game [+; - . , - ' , … , - 0 ] is trade robust, because for every trading transform 2 . , … , 2 0 , 3 . , … , 3 0 s.t. 2 . , … , 2 0 are winning coalitions: (- 3 . + ⋯ + - 3 0 )/7 = (- 2 . + ⋯ + - 2 0 )/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): T RADE R OBUST ( FOR A SET ) We will say that !: # → ℝ is trade robust for # ⊆ ' , if for every ( ≤ 2 + , -|/| 01 there are no sequences : ∪ 6 : s.t.: 9 , 2 : = 4 1 : ∪ 6 : , … , 4 9 2 = 4 1 ∪ 6 1 , … , 4 9 ∪ 6 1 9 : ∩ 6 : = ∅ for every > ∈ {1, … , (} 1) 4 ; ∩ 6 ; = ∅ and 4 ; ; : , … , 4 9 : ) is a trading transform 2) (4 1 , … , 4 9 , 4 1 : ⊆ # for every > ∈ {1, … , (} 3) 6 ; ⊆ # and 6 ; : + ⋯ ! 6 : 4) ! 6 1 + ⋯ ! 6 9 ≤ ! 6 1 9 and all coalitions from 2 all winning and all coalitions from 2 : are losing. 22 Oskar Skibski (UW) Algorithmic Coalitional Game Theory
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