Monte Carlo Methods for Calculating Shapley-Shubik Power Index in Weighted Majority Games
MMonte Carlo Methodsfor Calculating Shapley-Shubik Power Indexin Weighted Majority Games ∗ Tokyo Institute of Technology Yuto UshiodaTokyo Institute of Technology Masato TanakaTokyo Institute of Technology Tomomi MatsuiJanuary 11, 2021
Abstract
This paper addresses Monte Carlo algorithms for calculating theShapley-Shubik power index in weighted majority games. First, weanalyze a naive Monte Carlo algorithm and discuss the required num-ber of samples. We then propose an efficient Monte Carlo algorithmand show that our algorithm reduces the required number of samplesas compared to the naive algorithm.keywords: Games/Voting, Probability/Applications, Statistics/Sampling,Monte Carlo algorithm
The analysis of power is a central issue in political science. In general, it isdifficult to define the idea of power even in restricted classes of the votingrules commonly considered by political scientists. The use of game theory tostudy the distribution of power in voting systems can be traced back to theinvention of “simple games” by von Neumann and Oskar Morgenstern [30].A simple game is an abstraction of the constitutional political machineryfor voting. ∗ A preliminary version of this paper was presented at the 21st Japan-Korea JointWorkshop on Algorithms and Computation (WAAC), August 26-27, Fukuoka, Japan,2018.This work was supported by JSPS KAKENHI Grant Numbers 26285045, 26242027. a r X i v : . [ c s . G T ] J a n n 1954, Shapley and Shubik [27] proposed the specialization of the Shap-ley value [26] to assess the a priori measure of power of each player in a simplegame. Since then, the Shapley-Shubik power index (S-S index) has becomewidely known as a mathematical tools for measuring the relative power ofthe players in a simple game.In this paper, we consider a special class of simple games, called weightedmajority games , which constitute a familiar example of voting systems. Let N be a set of players. Each player i ∈ N has a positive integer voting weight w i as the number of votes or weight of the player. The quota needed fora coalition to win is a positive integer q . A coalition N (cid:48) ⊆ N is a winningcoalition , if (cid:80) i ∈ N (cid:48) w i ≥ q holds; otherwise, it is a losing coalition .The difficulty involved in calculating the S-S index in weighted major-ity games is described in [13] without proof (see p. 280, problem [MS8]).Deng and Papadimitriou [9] showed the problem of computing the S-S in-dex in weighted majority games to be ϕ , ϕ , . . . , ϕ n ) denotes the S-S index and( ϕ A , ϕ A , . . . , ϕ A n ) denotes the estimator obtained by Algorithm A1 or A2.2able 1: Required Number of Samples.Required number of samplesProperty Algorithm A1 Algorithm A2(naive algorithm) (our algorithm)Pr (cid:104)(cid:12)(cid:12)(cid:12) ϕ A i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − δ ln 2 + ln(1 /δ )2 ε ln 2 + ln(1 /δ )2 ε (cid:18) i (cid:19) (Bachrach et al. [1]) (assume w ≥ · · · ≥ w n )Pr (cid:104) ∀ i ∈ N, (cid:12)(cid:12)(cid:12) ϕ A i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − δ ln 2 + ln(1 /δ ) + ln n ε ln 2 + ln(1 /δ ) + ln 1 . ε Pr (cid:34) (cid:88) i ∈ N (cid:12)(cid:12)(cid:12) ϕ A i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:35) ≥ − δ. n ln 2 + ln(1 /δ )2 ε n (cid:48)(cid:48) ln 2 + ln(1 /δ )2 ε An integer n (cid:48)(cid:48) denotes the size of a maximal player subset with mutuallydifferent weights. In this paper, we consider a special class of cooperative games called weightedmajority games . Let N = { , , . . . , n } be a set of players . A subset ofplayers is called a coalition . A weighted majority game G is defined by asequence of positive integers G = [ q ; w , w , . . . , w n ], where we may thinkof w i as the number of votes or the weight of player i and q as the quotaneeded for a coalition to win. In this paper, we assume that 0 < q ≤ w + w + · · · + w n .A coalition S ⊆ N is called a winning coalition when the inequality q ≤ (cid:80) i ∈ S w i holds. The inequality q ≤ w + w + · · · + w n implies that N is a winning coalition. A coalition S is called a losing coalition if S is notwinning. We define that an empty set is a losing coalition.Let π : { , , . . . , n } → N be a permutation defined on the set of players N , which provides a sequence of players ( π (1) , π (2) , . . . , π ( n )). We denotethe set of all the permutations by Π N . We say that the player π ( i ) ∈ N isthe pivot of the permutation π ∈ Π N , if { π (1) , π (2) , . . . , π ( i − } is a losingcoalition and { π (1) , π (2) , . . . , π ( i − , π ( i ) } is a winning coalition. For anypermutation π ∈ Π N , piv( π ) ∈ N denotes the pivot of π . For each player i ∈ N , we define Π i = { π ∈ Π N | piv( π ) = i } . Obviously, { Π , Π , . . . , Π n } becomes a partition of Π N . The S-S index of player i , denoted by ϕ i ,3s defined by | Π i | /n !. Clearly, we have that 0 ≤ ϕ i ≤ ∀ i ∈ N ) and (cid:80) i ∈ N ϕ i = 1. Assumption 1.
The set of players is arranged to satisfy w ≥ w ≥ · · · ≥ w n . Clearly, this assumption implies that ϕ ≥ ϕ ≥ · · · ≥ ϕ n . In this section, we describe a naive Monte Carlo algorithm and analyze itstheoretical performance.
Algorithm A1Step 0:
Set m := 1, ϕ (cid:48) i := 0 ( ∀ i ∈ N ). Step 1:
Choose π ∈ Π N uniformly at random.Put (the random variable) I ( m ) := piv( π ). Update ϕ (cid:48) I ( m ) := ϕ (cid:48) I ( m ) + 1. Step 2: If m = M , then output ϕ (cid:48) i /M ( ∀ i ∈ N ) and stop.Else, update m := m + 1 and go to Step 1.For each permutation π ∈ Π N , we can find the pivot piv( π ) ∈ N inO( n ) time. Thus, the time complexity of Algorithm A1 is bounded byO( M ( τ ( n ) + n )) where τ ( n ) denotes the computational effort required forrandom generation of a permutation.We denote the vector (of random variables) obtained by Algorithm A1by ( ϕ A1 , ϕ A1 , . . . , ϕ A1 n ). The following theorem is obvious. Theorem 1.
For each player i ∈ N , E (cid:104) ϕ A1 i (cid:105) = ϕ i . The following theorem provides the number of samples required in Al-gorithm A1.
Theorem 2.
For any ε > and < δ < , we have the following. (1) [1] If we set M ≥ ln 2 + ln(1 /δ )2 ε , then each player i ∈ N satisfiesthat Pr (cid:104)(cid:12)(cid:12)(cid:12) ϕ A1 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − δ. (2) If we set M ≥ ln 2 + ln(1 /δ ) + ln n ε , then Pr (cid:104) ∀ i ∈ N, (cid:12)(cid:12)(cid:12) ϕ A1 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − δ. If we set M ≥ n ln 2 + ln(1 /δ )2 ε , then Pr (cid:34) (cid:88) i ∈ N (cid:12)(cid:12)(cid:12) ϕ A1 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:35) ≥ − δ. The distance measure (cid:80) i ∈ N (cid:12)(cid:12)(cid:12) ϕ A1 i − ϕ i (cid:12)(cid:12)(cid:12) appearing in (3) is called the totalvariation distance .Proof. Let us introduce random variables X ( m ) i ( ∀ m ∈ { , , . . . , M } , ∀ i ∈ N ) in Step 1 of Algorithm A1 defined by X ( m ) i = (cid:26) i = I ( m ) ) , . It is obvious that for each player i ∈ N , { X (1) i , X (2) i , . . . , X ( M ) i } is a Bernoulliprocess satisfying ϕ A1 i = (cid:80) Mm =1 X ( m ) i /M , E (cid:104) ϕ A1 i (cid:105) = E (cid:104) X ( m ) i (cid:105) = ϕ i ( ∀ m ∈{ , , . . . , M } ). Hoeffding’s inequality [14] implies that each player i ∈ N satisfiesPr (cid:104)(cid:12)(cid:12)(cid:12) ϕ A1 i − ϕ i (cid:12)(cid:12)(cid:12) ≥ ε (cid:105) ≤ (cid:32) − M ε (cid:80) Mm =1 (1 − (cid:33) = 2 exp( − M ε ) . (1) If we set M ≥ ln(2 /δ )2 ε , thenPr (cid:104)(cid:12)(cid:12)(cid:12) ϕ A1 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − (cid:18) − /δ )2 ε ε (cid:19) = 1 − δ. (2) If we set M ≥ ln(2 n/δ )2 ε , then we have thatPr (cid:104) ∀ i ∈ N, (cid:12)(cid:12)(cid:12) ϕ A1 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) = 1 − Pr (cid:104) ∃ i ∈ N, (cid:12)(cid:12)(cid:12) ϕ A1 i − ϕ i (cid:12)(cid:12)(cid:12) ≥ ε (cid:105) ≥ − (cid:88) i ∈ N Pr (cid:104)(cid:12)(cid:12)(cid:12) ϕ A1 i − ϕ i (cid:12)(cid:12)(cid:12) ≥ ε (cid:105) ≥ − n (cid:88) i =1 − M ε ) ≥ − n (cid:88) i =1 (cid:18) − n/δ )2 ε ε (cid:19) = 1 − n (cid:88) i =1 δn = 1 − δ. (3) Obviously, the vector of random variables( M ϕ A1 , M ϕ A1 , · · · , M ϕ A1 n ) = (cid:32) M (cid:88) m =1 X ( m )1 , M (cid:88) m =1 X ( m )2 , · · · , M (cid:88) m =1 X ( m ) n (cid:33)
5s multinomially distributed with parameters M and ( ϕ , ϕ , · · · , ϕ n ). Then,the Bretagnolle-Huber-Carol inequality [29] (Theorem 8 in Appendix) im-plies thatPr (cid:34) (cid:88) i ∈ N (cid:12)(cid:12)(cid:12) ϕ A1 i − ϕ i (cid:12)(cid:12)(cid:12) ≥ ε (cid:35) = Pr (cid:34)(cid:88) i ∈ N (cid:12)(cid:12)(cid:12) M ϕ A1 i − M ϕ i (cid:12)(cid:12)(cid:12) ≥ M ε (cid:35) ≤ n exp (cid:0) − M ε (cid:1) ≤ n exp (cid:18) − (cid:18) ln(2 n /δ )2 ε (cid:19) ε (cid:19) = δ, and thus, we have the desired result. In this section, we propose a new algorithm based on the hierarchical struc-ture of the partition { Π , Π , . . . , Π n } . First, we introduce a map f i : Π i → Π N for each i ∈ N \ { } . For any π ∈ Π i , f i ( π ) denotes a permutationobtained by swapping the positions of players i and i − π (1) , π (2) , . . . , π ( n )). Because w i − ≥ w i (Assumption 1), it is easy to showthat the pivot of f i ( π ) becomes the player i −
1. The definition of f i directlyimplies that ∀{ π, π (cid:48) } ⊆ Π i , if π (cid:54) = π (cid:48) , then f i ( π ) (cid:54) = f i ( π (cid:48) ). Thus, we havethe following. Lemma 3.
For any i ∈ N \ { } , the map f i : Π i → Π i − is injective. Figure 1 shows injective maps f , f , f induced by G = [50; 40 , , , π, π (cid:48) ) satisfies the conditionsthat π ∈ Π i , π (cid:48) ∈ Π j , i ≤ j , and π = f i − ◦ · · · ◦ f j − ◦ f j ( π (cid:48) ), we say that π (cid:48) is an ancestor of π . Here, we note that π is always an ancestor of π itself.Lemma 3 implies that every permutation π ∈ Π N has a unique ancestor,called the originator , π (cid:48) ∈ Π j satisfying that either j = n or its inverseimage f − j +1 ( π (cid:48) ) = ∅ . For each permutation π ∈ Π N , org( π ) ∈ N denotes thepivot of the originator of π ; i.e., Π org( π ) includes the originator of π .Now, we describe our algorithm. Algorithm A2Step 0:
Set m := 1, ϕ (cid:48) i := 0 ( ∀ i ∈ N ). Step 1:
Choose π ∈ Π N uniformly at random. Put the random variable L ( m ) := org( π ).Update ϕ (cid:48) i := (cid:26) ϕ (cid:48) i + 1 /L ( m ) (if 1 ≤ i ≤ L ( m ) ) ,ϕ (cid:48) i (if L ( m ) < i ) . f Π f Π f Π (2 , ○ , , ← (1 , ○ , , ← (1 , ○ , , ← (1 , ○ , , , ○ , , ← (1 , ○ , , ← (1 , ○ , , ← (1 , ○ , , , , ○ , ← (4 , , ○ , ← (4 , , ○ , , , ○ , ← (3 , , ○ , ← (2 , , ○ , , ○ , , ← (3 , ○ , , ← (2 , ○ , , , ○ , , ← (3 , ○ , , ← (2 , ○ , , , ○ , , ○ , , , , ○ , , , ○ , Figure 1: Injective maps f , f , f induced by G = [50; 40 , , , Step 2: If m = M , then output ϕ (cid:48) i /M ( ∀ i ∈ N ) and stop.Else, update m := m + 1 and go to Step 1.In the example shown in Figure 1, if we choose π = (3 , ○ , ,
1) at Step 1 ofAlgorithm A2, then org( π ) = 3 and Algorithm A2 updates( ϕ (cid:48) , ϕ (cid:48) , ϕ (cid:48) , ϕ (cid:48) ) := ( ϕ (cid:48) + (1 / , ϕ (cid:48) + (1 / , ϕ (cid:48) + (1 / , ϕ (cid:48) ) . For each permutation π ∈ Π N , we can find the originator org( π ) ∈ N inO( n ) time. Thus, the time complexity of Algorithm A2 is also bounded byO( M ( τ ( n ) + n )) where τ ( n ) denotes the computational effort required forrandom generation of a permutation.We denote the vector (of random variables) obtained by Algorithm A2by ( ϕ A2 , ϕ A2 , . . . , ϕ A2 n ). The following theorem is obvious. Theorem 4. (1)
For each player i ∈ N , E (cid:104) ϕ A2 i (cid:105) = ϕ i . (2) For each pair of players { i, j } ⊆ N , if ϕ i > ϕ j , then ϕ A2 i ≥ ϕ A2 j , (3) For each pair of players { i, j } ⊆ N , if ϕ i = ϕ j , then ϕ A2 i = ϕ A2 j . The following theorem provides the number of samples required in Al-gorithm A2. 7 heorem 5.
For any ε > and < δ < , we have the following. (1) For each player i ∈ N = { , , . . . , n } , if we set M ≥ ln 2 + ln(1 /δ )2 ε i ,then Pr (cid:104)(cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − δ. (2) If we set M ≥ ln 2 + ln(1 /δ )2 ε , then Pr (cid:104) ∀ i ∈ N, (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − n (cid:88) i =1 (cid:18) δ (cid:19) i = 1 − (cid:32)(cid:18) δ (cid:19) + (cid:18) δ (cid:19) + (cid:18) δ (cid:19) + · · · + (cid:18) δ (cid:19) n (cid:33) . (3) If we set M ≥ | N ∗ | ln 2 + ln(1 /δ )2 ε , then Pr (cid:34) (cid:88) i ∈ N (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:35) ≥ − δ, where N ∗ = { i ∈ N \ { n } | ϕ i > ϕ i +1 } ∪ { n } , i.e., | N ∗ | is equal to the size ofthe maximal player subset, the S-S indices of which are mutually different. Proof. Let us introduce random variables X ( m ) i ( ∀ m ∈ { , , . . . , M } , ∀ i ∈ N ) in Step 2 of Algorithm A2 defined by X ( m ) i = (cid:26) /L ( m ) (if 1 ≤ i ≤ L ( m ) ) , L ( m ) < i ) . It is obvious that for each player i ∈ N , { X (1) i , X (2) i , . . . , X ( M ) i } is a collec-tion of independent and identically distributed random variables satisfying ϕ A2 i = (cid:80) Mm =1 X ( m ) i /M , E (cid:104) ϕ A2 i (cid:105) = E (cid:104) X ( m ) i (cid:105) = ϕ i , and 1 /i ≥ X ( m ) i ≥ /n ( ∀ m ∈ { , , . . . , M } ). Hoeffding’s inequality [14] implies that each player i ∈ N satisfiesPr (cid:104)(cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) ≥ ε (cid:105) ≤ (cid:32) − M ε (cid:80) Mm =1 (1 /i − (cid:33) = 2 exp( − M ε i ) . (1) If we set M ≥ ln(2 /δ )2 ε i , thenPr (cid:104)(cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − (cid:18) − /δ )2 ε i ε i (cid:19) = 1 − δ.
82) If we set M ≥ ln(2 /δ )2 ε , then we have thatPr (cid:104) ∀ i ∈ N, (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) = 1 − Pr (cid:104) ∃ i ∈ N, (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) ≥ ε (cid:105) ≥ − (cid:88) i ∈ N Pr (cid:104)(cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) ≥ ε (cid:105) ≥ − n (cid:88) i =1 − M ε i ) ≥ − n (cid:88) i =1 exp (cid:18) − /δ )2 ε ε i (cid:19) = 1 − n (cid:88) i =1 (cid:18) δ (cid:19) i . (3) We introduce random variables Y ( m ) (cid:96) ( ∀ m ∈ { , , . . . , M } , ∀ (cid:96) ∈ N ) inStep 2 of Algorithm A2 defined by Y ( m ) (cid:96) = (cid:26) (cid:96) = L ( m ) ) , . Because (cid:80) n(cid:96) =1 Y ( m ) (cid:96) = 1 ( ∀ m ), the above definition directly implies that X ( m ) i = 1 i Y ( m ) i + 1 i + 1 Y ( m ) i +1 + · · · + 1 n Y ( m ) n . For each player i ∈ N and i ≤ ∀ (cid:96) ≤ n , we define Π i(cid:96) = { π ∈ Π i | org( π ) = (cid:96) } .It is easy to show that | Π (cid:96) | = | Π (cid:96) | = · · · = | Π (cid:96)(cid:96) | for each (cid:96) ∈ { , , . . . , n } .The above definitions imply that12 (cid:88) i ∈ N (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) = 12 M (cid:88) i ∈ N (cid:12)(cid:12)(cid:12) M ϕ A2 i − M ϕ i (cid:12)(cid:12)(cid:12) = 12 M (cid:88) i ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 X ( m ) i − M | Π i | n ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 M (cid:88) i ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 n (cid:88) (cid:96) = i (cid:96) Y ( m ) (cid:96) − Mn ! n (cid:88) (cid:96) = i | Π i(cid:96) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 M (cid:88) i ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) (cid:96) = i (cid:32) M (cid:88) m =1 (cid:96) Y ( m ) (cid:96) − Mn ! | Π i(cid:96) | (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M n (cid:88) i =1 n (cid:88) (cid:96) = i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 (cid:96) Y ( m ) (cid:96) − Mn ! | Π i(cid:96) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 M n (cid:88) (cid:96) =1 (cid:96) (cid:88) i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 (cid:96) Y ( m ) (cid:96) − Mn ! | Π i(cid:96) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 M n (cid:88) (cid:96) =1 (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 (cid:96) Y ( m ) (cid:96) − Mn ! | Π (cid:96) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (since | Π (cid:96) | = | Π (cid:96) | = · · · = | Π (cid:96)(cid:96) | )= 12 M n (cid:88) (cid:96) =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 Y ( m ) (cid:96) − M (cid:96)n ! | Π (cid:96) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (cid:96) (cid:54)∈ N ∗ , we have the equalities | Π (cid:96) | = n ! ϕ (cid:96) = n ! ϕ (cid:96) +1 = | Π (cid:96) +1 | , which yields that f (cid:96) +1 : Π (cid:96) +1 → Π (cid:96) is a bijection and thus Π (cid:96) doesnot include any originator. From the above, it is obvious that, if (cid:96) (cid:54)∈ N ∗ , thenΠ (cid:96) = Π (cid:96) = · · · = Π (cid:96)(cid:96) = ∅ . For each (cid:96) ∈ { , , . . . , n } , { Y (1) (cid:96) , Y (2) (cid:96) , . . . , Y ( M ) (cid:96) } is a Bernoulli process satisfying E[ Y ( m ) (cid:96) ] = n ! (cid:80) (cid:96)i =1 | Π i(cid:96) | = (cid:96)n ! | Π (cid:96) | ( ∀ m ).Thus, (cid:96) (cid:54)∈ N ∗ implies that Y ( m ) (cid:96) = 0 for any m ∈ { , , . . . , M } . To summa-rize the above, we have shown thatif (cid:96) (cid:54)∈ N ∗ then M (cid:88) m =1 Y ( m ) (cid:96) − M (cid:96)n ! | Π (cid:96) | = M (cid:88) m =1 − M (cid:96)n ! 0 = 0 . Now, we have an upper bound of the total variation distance12 (cid:88) i ∈ N (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) ≤ M n (cid:88) (cid:96) =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 Y ( m ) (cid:96) − M (cid:96)n ! | Π (cid:96) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 M (cid:88) (cid:96) ∈ N ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 Y ( m ) (cid:96) − M (cid:88) m =1 E[ Y ( m ) (cid:96) ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Obviously, the vector of random variables (cid:16)(cid:80) Mm =1 Y ( m ) (cid:96) (cid:17) (cid:96) ∈ N ∗ is multino-mially distributed and satisfies that the total sum is equal to M . Then, theBretagnolle-Huber-Carol inequality [29] (Theorem 8 in Appendix) impliesthatPr (cid:34) (cid:88) i ∈ N (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) ≥ ε (cid:35) ≤ Pr (cid:34) M (cid:88) (cid:96) ∈ N ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 Y ( m ) (cid:96) − M (cid:88) m =1 E[ Y ( m ) (cid:96) ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ε (cid:35) = Pr (cid:34) (cid:88) (cid:96) ∈ N ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 Y ( m ) (cid:96) − M (cid:88) m =1 E[ Y ( m ) (cid:96) ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ M ε (cid:35) ≤ | N ∗ | exp (cid:0) − M ε (cid:1) ≤ | N ∗ | exp (cid:32) − (cid:0) | N ∗ | /δ (cid:1) ε ε (cid:33) = δ and thus, we have the desired result.The following corollary provides an approximate version of Theorem 5 (2).Surprisingly, it says that the required number of samples is irrelevant to n (number of players). 10 orollary 6. For any ε > and < δ (cid:48) < , we have the following. If weset M ≥ ln 2 + ln(1 /δ (cid:48) ) + ln 1 . ε , then Pr (cid:104) ∀ i ∈ N, (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − δ (cid:48) . Proof. If we put δ = δ (cid:48) / . (cid:104) ∀ i ∈ N, (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − (cid:32)(cid:18) δ (cid:19) + (cid:18) δ (cid:19) + (cid:18) δ (cid:19) + · · · + (cid:18) δ (cid:19) n (cid:33) ≥ − δ (cid:32) (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + · · · + (cid:18) (cid:19) n − (cid:33) ≥ − δ (cid:32) (cid:18) (cid:19) + (cid:18) (cid:19) (cid:32) (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + · · · (cid:33)(cid:33) = 1 − δ (cid:32) (cid:18) (cid:19) + (cid:18) (cid:19) (cid:18) − (1 / (cid:19)(cid:33) ≥ − . δ = 1 − δ (cid:48) . Here, we note that ln 2 (cid:39) . . (cid:39) . N ∗ defined inTheorem 5 (3), since the problem of verifying the asymmetricity of a givenpair of players is NP-complete [21]. The following corollary is useful in somepractical situations. Corollary 7.
For any ε > and < δ < , we have the following. If weset M ≥ n (cid:48)(cid:48) ln 2 + ln(1 /δ )2 ε , then Pr (cid:34) (cid:88) i ∈ N (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:35) ≥ − δ, where n (cid:48)(cid:48) = |{ i ∈ N \ { n } | w i > w i +1 } ∪ { n }| , i.e., n (cid:48)(cid:48) is equal to the size ofa maximal player subset with mutually different weights. Proof. Since ϕ i > ϕ i +1 implies w i > w i +1 , it is obvious that | N ∗ | ≤ n (cid:48)(cid:48) andwe have the desired result.A game of the power of the countries in the EU Council is defined by[255; 29 , , , , , , , , , , , , , , , , , , , , , , , , , , n = 27 and n (cid:48)(cid:48) = 9. A weighted majority game definedby [23](Section 12.4) for a voting process in United States has a vector ofweights[270; 45 , , , , , , , , , , , , , , , , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) , , , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) , , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) , where n = 51 and n (cid:48)(cid:48) = 19 . This section reports the results of our preliminary numerical experiments.All the experiments were conducted on a windows machine, i7-7700 [email protected] (RAM) 16GB. Algorithms A1 and A2 are implemented by Python3.6.5.We tested the EU Council instance and the United States instance de-scribed in the previous section. In each instance, we set M in Algorithm A1and A2 (the number of generated permutations) to M ∈ { × , × , . . . , × } . For each value M , we executed Algorithms A1 and A2,100 times. Figures 2 and 3 show results of some players. For each value M ,we calculated the mean number of | ϕ i − ϕ A i | , denoted by (cid:98) ε i , in an averageof 100 trials. The horizontal axes of Figures 2 and 3 show the value 1 / (cid:98) ε i .Under the assumption that M = α/ (cid:98) ε i , we estimated α by the least squaresmethod. Table 2 shows the results and ratios of α of two algorithms.Table 2: Comparison of Algorithms A1 and A2.EU Council α Alg. A1 Alg. A2 ratioPlayer 1 0.0557 0.0022 25.318Player 13 0.0199 4 . × − . × − α Alg. A1 Alg. A2 ratioPlayer 1 0.0489 0.0181 2.7017Player 26 0.0088 1 . × − . × − n ). Here, we discuss the constantfactors of O( n ) computations. We tested the cases that weights w i are gen-erated uniformly at random from the intervals [1 ,
10] or [1 , / (cid:80) i ∈ N w i . For each n ∈ { , , . . . , } , we executed Al-gorithms A1 and A2 by setting M = 10 , an + b , we estimated a and b by the leastsquares method. Figure 4 shows that for each permutation, the computa-tional effort of Algorithm A2 increases about 5-fold comparing to AlgorithmA1. In this paper, we analyzed a naive Monte Carlo algorithm (Algorithm A1)for calculating the S-S index denoted by ( ϕ , ϕ , . . . , ϕ n ) in weighted ma-jority games. By employing the Bretagnolle-Huber-Carol inequality [29](Theorem 8 in Appendix), we estimated the required number of samplesthat gives an upper bound of the total variation distance.We also proposed an efficient Monte Carlo algorithm (Algorithm A2).13igure 3: United States.The time complexity of each iteration of our algorithm is equal to that ofthe naive algorithm (Algorithm A1). Our algorithm has the property thatthe obtained estimator ( ϕ A2 , ϕ A2 , . . . , ϕ A2 n ) satisfiesboth [ if ϕ i < ϕ j then ϕ A2 i ≤ ϕ A2 j ] and [ if ϕ i = ϕ j then ϕ A2 i = ϕ A2 j ] . We also proved that, even if we consider the propertyPr (cid:104) ∀ i ∈ N, (cid:12)(cid:12)(cid:12) ϕ A2 i − ϕ i (cid:12)(cid:12)(cid:12) < ε (cid:105) ≥ − δ, the required number of samples is irrelevant to n (the number of players). APPENDIX (Bretagnolle-Huber-Carol inequality)
Theorem 8. [29]
If the random vector ( Z , Z , . . . , Z n ) is multinomially distributed withparameters ( p , p , . . . , p n ) and satisfies Z + Z + · · · + Z n = M then Pr (cid:34) n (cid:88) i =1 | Z i − M p i | ≥ M ε (cid:35) ≤ n exp( − M ε ) . (cid:34) n (cid:88) i =1 | Z i − M p i | ≥ M ε (cid:35) = Pr (cid:34) S ⊆{ , ,...,n } (cid:88) i ∈ S ( Z i − M p i ) ≥ M ε (cid:35) = Pr (cid:34) ∃ S ⊆ { , , . . . , n } , (cid:88) i ∈ S ( Z i − M p i ) ≥ M ε (cid:35) ≤ (cid:88) S ⊆{ , ,...,n } Pr (cid:34)(cid:88) i ∈ S ( Z i − M p i ) ≥ M ε (cid:35) = (cid:88) S ⊆{ , ,...,n } Pr (cid:34)(cid:88) i ∈ S Z i − M (cid:88) i ∈ S p i ≥ M ε (cid:35)
For any subset S ⊆ { , , . . . , n } , there exists a Bernoulli process ( X (1) S , X (2) S , . . . , X ( M ) S )satisfying (cid:80) i ∈ S Z i = (cid:80) Mm =1 X ( m ) S and E[ X ( m ) S ] = (cid:80) i ∈ S p i ( ∀ m ∈ { , , . . . M } ).Hoeffding’s inequality [14] implies that (cid:88) S ⊆{ , ,...,n } Pr (cid:34)(cid:88) i ∈ S Z i − M (cid:88) i ∈ S p i ≥ M ε (cid:35) = (cid:88) S ⊆{ , ,...,n } Pr (cid:34) M (cid:88) m =1 X ( m ) S − E (cid:34) M (cid:88) m =1 X ( m ) S (cid:35) ≥ M ε (cid:35) = (cid:88) S ⊆{ , ,...,n } Pr (cid:34) M M (cid:88) m =1 X ( m ) S − M E (cid:34) M (cid:88) m =1 X ( m ) S (cid:35) ≥ ε (cid:35) ≤ (cid:88) S ⊆{ , ,...,n } exp( − M ε )= 2 n exp( − M ε ) . QED
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