Weighted Voting Systems - W.H. Freeman

422 PART III Voting and Social Choicenumber being represented is the sum of the termsb n 2 n1 . The binary number 11001101 has b 1 1, b 2 0, b 3 b 4 1, b 5 b 6 0, and b 7 b 8 1. Hence it can be converted to decimal form as2 0 2 2 2 3 2 6 2 7 205Blocking coalition A coalition in opposition toa measure that can prevent the measure frompassing.C n k The number of voting combinations in a votingsystem with n voters, in which k voters say“yes” and n k voters say “no.” This number,referred to as “n-choose-k,” is given by theformulaC k n n!k! (n k)!Coalition The set of participants in a voting systemwho favor, or who oppose, a given motion. Acoalition may be empty (if, for example, the votingbody unanimously favors a motion, the opposingcoalition is empty), it may contain some but notall voters, or it may consist of all the voters.Critical voter A member of a winning coalitionwhose vote is essential for the coalition to win, ora member of a blocking coalition whose vote is essentialfor the coalition to block.Dictator A participant in a voting system who canpass any issue even if all other voters oppose itand block any issue even if all other voters approveit.Duality formula C n k C n nkDummy A participant who has no power in a votingsystem. A dummy is never a critical voter inany winning or blocking coaliton and is never thepivotal voter in any permutation.Equivalent voting systems Two voting systems areequivalent if there is a way for all the voters of thefirst system to exchange places with the voters ofthe second system and preserve all winning coalitions.Extra votes The number of votes that a winningcoalition has in excess of the quota.Extra-votes principle The critical voters in thecoalition are those whose weights are more thanthe extra votes of the coalition. For example, if acoalition has 12 votes and the quota is 9, there are3 extra votes. The critical voters in the coalitionare those with more than 3 votes.Factorial The number of permutations of n voters(or n distinct objects) is called n-factorial, or insymbols, n!. Because the empty coalition can beordered in only one way, 0! 1. When n is a positivewhole number, n! is equal to the product ofall the integers from 1 up to n. If n has more thanone digit, then n! is a pretty big number: 10! ismore than three million, and 1000! has 2568digits.Losing coalition A coalition that does not havethe voting power to get its way.Minimal winning coalition A winning coalitionthat will become losing if any member defects.Each member is a critical voter.Pascal’s triangle A triangular pattern of integers, inwhich each entry on the left and right edges is 1,and each interior entry is equal to the sum of thetwo entries above it. The entry that is located kunits from the left edge, on the row n units belowthe vertex, is C n k .Permutation A specific ordering from first to lastof the elements of a set; for example, an orderingof the participants in a voting system.Pivotal voter The first voter in a permutationwho, with his or her predecessors in the permutation,will form a winning coalition. Each permutationhas one and only one pivotal voter.Power index A numerical measure of an individualvoter’s ability to influence a decision, the individual’svoting power.Quota The minimum number of votes necessaryto pass a measure in a weighted voting system.Shapley–Shubik power index A numerical measureof power for participants in a voting system. Aparticipant’s Shapley–Shubik index is the numberof permutations of the voters in which he or she isthe pivotal voter, divided by the number of permutations(n! if there are n participants).Veto power A voter has veto power if no issuecan pass without his or her vote. A voter with vetopower is a one-person blocking coalition.Voting combination A list of voters indicating thevote of each on an issue. There is a total of 2 n