Weighted Voting Systems - W.H. Freeman

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CHAPTER 11 Weighted Voting Systems 42935. Find the minimal winning coalitions of theweighted voting system [7 : 3, 3, 3, 1, 1, 1] anddetermine the Banzhaf index.36. A new weight-1 voter joins the system ofExercise 35. Again, describe the minimal winningcoalitions and determine the Banzhaf power index.Does the presence of this new voter increase ordecrease the share of power of each weight-1 voter?37. Compute the Shapley–Shubik power index forthe systems in Exercises 35 and 36. How does theaddition of the new voter affect the power of theother three weight-1 voters?✎WRITING PROJECTS1. The most important weighted voting system inthe United States is the Electoral College (seeSpotlight 11.1). Three alternate methods to electthe president of the United States have beenproposed:Direct election. The Electoral College would beabolished, and the candidate receiving aplurality of the votes would be elected. Mostversions of this system include a runoffelection or a vote in the House ofRepresentatives in cases where no candidatereceives more than 40% of the vote.District system. This system could be adoptedby individual states without amending theConstitution or passing a federal law. It isnow in use by two states, Maine andNebraska. In each congressional district, andin the District of Columbia, the candidatereceiving the plurality would select oneelector. Furthermore, in each state, includingthe District of Columbia, the candidatereceiving the plurality would receive twoelectors. In effect, the unit rule would beretained for the District of Columbia and forstates with a single congressional district.Larger states might have electors representingboth parties.◆ 38. An alumni committee consists of 3 richalumni and 12 recent graduates. To pass ameasure, a majority, including at least 2 of the richalumni, must approve. Is this equivalent to aweighted voting system? If so, find the weights anda quota; if not, explain why not.◆ 39. Show that if a state uses the district systemto choose its electors in a two-candidatepresidential election, as Maine and Nebraska do,then some electoral permutations are impossible.Give an example of an impossible permutation.How would this affect the calculation of theShapley–Shubik power index?Proportional system. Each state and the Districtof Columbia would have fractional electoralvotes assigned to each candidate in proportionto the number of popular votes received.Under this system, Governor Bush, whoreceived 4,567,429 popular votes out of10,965,822 cast in California in 2000, wouldhave received4,567,429 54 electoral votes1 0,965,822 22.4918 electoral votesVice President Gore would have received5,861,203 54 electoral votes1 0,965,822 28.8629 electoral votesThe Green Party received 418,707 votes inCalifornia and would be entitled to 2.06188electoral votes. There were four other partiesthat received between them less than oneelectoral vote. Obviously, no actual electorswould be chosen.Should the present electoral college, operatingunder the unit rule, be replaced by one of these
430 PART III Voting and Social Choicesystems? Reference: The Presidential Election Game,by Steven Brams, which contains useful referencesto Senate hearings on electoral college reform.2. Write an essay on weighted voting in theCouncil of Ministers of the EuropeanCommunity. Compute the Banzhaf and Shapley–Shubik indices for the system as it was in 1958. Inlater years, the number of member nationsincreased significantly, and you may want to usethe power index calculator, available on the Webat www.whfreeman.com/fapp. If they differsignificantly in their allocation of power, whichindex represents the true balance of power best?3. California has 10.22% of the votes in ElectoralCollege, but according to Spotlight 11.4 that statehas more than 11% of the power in the ElectoralCollege, as measured by either of our powerindices. Discuss the appropriateness of each powerindex as a measure of voting power in the ElectoralCollege. Is the disproportionate power of Californiain the Electoral College a problem that the UnitedStates should address? Assume that California hasacquired additional congressional seats as a resultof migration. Calculate the Banzhaf index whenCalifornia has 65, 75, and 100 electors. In eachcase take the electoral votes that are to be awardedto California from other states.What would happen if all states, exceptCalifornia, adopted the district system forchoosing electors? See Writing Project 1 for adiscussion of the district system.SUGGESTED READINGSBRAMS, STEVEN J. Game Theory and Politics, 2ndEd., Dover Publications, New York, 2004.Iannucci v. Board of Supervisors of Washington County20 N.Y. 2d 244, 251, 229 N.E. 2d 195, 198, 282N.Y.S. 2d 502, 507 (1967). This code will help alaw librarian find this case for you. It opened a“mathematical quagmire.”FELSENTHAL, DAN S., and MOSHÉMACHOVER, The Measurement of Voting Power:Theory and Practice, Problems and Paradoxes, EdwardElgar, Cheltenham, U.K., 1998. This book has adetailed analysis of the Council of Ministers of theEuropean Community, and a thorough treatmentof the power indices mentioned in this chapter.TAYLOR, ALAN D. Mathematics and Politics:Strategy, Voting Power, and Proof, Springer-Verlag,New York, 1995. Chapter 4 covers weighted votingsystems and their analysis using the Shapley–Shubik and Banzhaf power indices. It has nomathematical prerequisites, but it does includecarefully written logical arguments that must becarefully read.
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