Chapter 4 Voting procedures

Abstract Voting procedures focus on the aggregation of individuals' preferences to produce collective decisions. In practice, a voting procedure is characterized by ballot responses and the way ballots are tallied to determine winners. Voters are assumed to have clear preferences over candidates and attempt to maximize satisfaction with the election outcome by their ballot responses. Such responses can include strategic misrepresentation of preferences. Voting procedures are formalized by social choice functions, which map ballot response profiles into election outcomes. We discuss broad classes of social choice functions as well as special cases such as plurality rule, approval voting, and Borda's point-count method. The simplest class is voting procedures for two-candidate elections. Conditions for social choice functions are presented for simple majority rule, the class of weighted majority rules, and for what are referred to as hierarchical representative systems. The second main class, which predominates in the literature, embraces all procedures for electing one candidate from three or more contenders. The multicandidate elect-one social choice functions in this broad class are divided into nonranked one-stage procedures, nonranked multistage procedures, ranked voting methods, and positional scoring rules. Nonranked methods include plurality check-one voting and approval voting, where each voter casts either no vote or a full vote for each candidate. On ballots for positional scoring methods, voters rank candidates from most preferred to least preferred. Topics for multicandidate methods include axiomatic characterizations, susceptibility to strategic manipulation, and voting paradoxes that expose questionable aspects of particular procedures. Other social choice functions are designed to elect two or more candidates for committee memberships from a slate of contenders. Proportional representation methods, including systems that elect members sequentially from a single ranked ballot with vote transfers in successive counting stages, are primary examples of this class.

[1]  Steven J. Brams,et al.  The 3/2's Rule in Presidential Campaigning , 1974, American Political Science Review.

[2]  Donald G. Saari.  Millions of election outcomes from a single profile , 1992 .

[3]  Dolors Berga,et al.  Strategy-proofness and single-plateaued preferences , 1998 .

[4]  Peter J. Coughlin,et al.  Probabilistic Voting Theory , 1992 .

[5]  Peter C. Fishburn,et al.  Coincidence probabilities for simple majority and positional voting rules , 1978 .

[6]  Marco Scarsini.  A strong paradox of multiple elections , 1998 .

[7]  Gideon Doron,et al.  Single Transferrable Vote: An Example of a Perverse Social Choice Function , 1977 .

[8]  Steven J. Brams,et al.  CONSTRAINED APPROVAL VOTING: A VOTING SYSTEM TO ELECT A GOVERNING BOARD , 1990 .

[9]  Iain McLean.  E. J. Nanson, Social Choice and Electoral Reform , 1996 .

[10]  R. Niemi.  The Problem of Strategic Behavior under Approval Voting , 1984, American Political Science Review.

[11]  Samuel Merrill,et al.  Approval Voting: A ‘Best Buy’ Method for Multi-candidate Elections? , 1979 .

[12]  Dag Anckar.  Presidential elections in Finland: A plea for approval voting☆ , 1984 .

[13]  Gary W. Cox,et al.  The Efficient Secret: The Cabinet and the Development of Political Parties in Victorian England , 1987 .

[14]  Peter C. Fishburn,et al.  Approval voting, Condorcet's principle, and runoff elections , 1981 .

[15]  Donald G. Saari,et al.  The ultimate of chaos resulting from weighted voting systems , 1984 .

[16]  Peter C. Fishburn,et al.  A Strategic Analysis of Nonranked Voting Systems , 1978 .

[17]  S J Brams,et al.  The Science of Elections , 2001, Science.

[18]  Ron Holzman,et al.  To vote or not to vote: What is the quota? , 1989, Discret. Appl. Math..

[19]  Nicholas R. Miller.  A New Solution Set for Tournaments and Majority Voting: Further Graph- Theoretical Approaches to the Theory of Voting , 1980 .

[20]  William V. Gehrlein.  Condorcet efficiency and constant scoring rules , 1982, Math. Soc. Sci..

[21]  Martin Shubik,et al.  A Method for Evaluating the Distribution of Power in a Committee System , 1954, American Political Science Review.

[22]  Lani Guinier,et al.  The tyranny of the majority : fundamental fairness in representative democracy , 1994 .

[23]  S. Brams,et al.  Voting on Referenda: The Separability Problem and Possible Solutions , 1997 .

[24]  Donald G. Saari,et al.  Mathematical Structure of Voting Paradoxes: II. Positional Voting , 1999 .

[25]  C. Plott,et al.  The Probability of a Cyclical Majority , 1970 .

[26]  Bhaskar Dutta.  Covering sets and a new condorcet choice correspondence , 1988 .

[27]  Ran Raz.  VC-Dimension of Sets of Permutations , 2000, Comb..

[28]  Peter C. Fishburn,et al.  Some Logical Defects of the Single Transferable Vote , 1984 .

[29]  Samuel Merrill,et al.  The Effect of Approval Balloting on Strategic Voting under Alternative Decision Rules , 1987, American Political Science Review.

[30]  R. Niemi,et al.  A mathematical solution for the probability of the paradox of voting. , 1968, Behavioral science.

[31]  W. Riker,et al.  The Paradox of Voting and Congressional Rules for Voting on Amendments , 1958, American Political Science Review.

[32]  Samuel Merrill,et al.  A statistical model for Condorcet efficiency based on simulation under spatial model assumptions , 1985 .

[33]  Donald G. Saari,et al.  Mathematical structure of voting paradoxes , 2000 .

[34]  Barry R. Weingast,et al.  Uncovered Sets and Sophisticated Voting Outcomes with Implications for Agenda Institutions , 1984 .

[35]  Dominique Henriet,et al.  The Copeland choice function an axiomatic characterization , 1985 .

[36]  Donald G. Saari,et al.  The Borda dictionary , 1990 .

[37]  Kit Fine.  Some Necessary and Sufficient Conditions for Representative Decision on Two Alternatives , 1972 .

[38]  Paradoxes of voting in list systems of proportional representation , 1993 .

[39]  Donald G. Saari,et al.  A chaotic Exploration of Aggregation Paradoxes , 1995, SIAM Rev..

[40]  Robert F. Bordley,et al.  A Pragmatic Method for Evaluating Election Schemes through Simulation , 1983, American Political Science Review.

[41]  Steven J. Brams,et al.  Optimal resource allocation in presidential primaries , 1982, Math. Soc. Sci..

[42]  Donald G. Saari,et al.  Is approval voting an ‘unmitigated evil’?: A response to Brams, Fishburn, and Merrill , 1987 .

[43]  M. Satterthwaite.  Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions , 1975 .

[44]  William V. Gehrlein,et al.  Single-stage election procedures for large electorates , 1981 .

[45]  S. Merrill,et al.  Strategic Voting in Multicandidate Elections under Uncertainty and under Risk , 1981 .

[46]  Jerry S. Kelly,et al.  Voting Anomalies, the Number of Voters, and the Number of Alternatives , 1974 .

[47]  Hannu Nurmi,et al.  Computer simulations of approval and plurality voting: The frequency of weak pareto violations and condorcet loser choices in impartial cultures , 1986 .

[48]  William V. Gehrlein,et al.  The Condorcet criterion and committee selection , 1985 .

[49]  Salvador Barberà,et al.  Voting by Committees , 1991 .

[50]  I. Good.  A note on condorcet sets , 1971 .

[51]  H. Young.  Social Choice Scoring Functions , 1975 .

[52]  Dale T. Hoffman,et al.  A MODEL FOR STRATEGIC VOTING , 1982 .

[53]  H. Young.  Extending Condorcet's rule , 1977 .

[54]  W. Gehrlein.  Condorcet's paradox , 1983 .

[55]  William S. Zwicker.  The voters' paradox, spin, and the Borda count , 1991 .

[56]  J. Kelly.  Arrow Impossibility Theorems , 1978 .

[57]  D. J Hartfiel,et al.  A characterization result for the plurality rule , 1978 .

[58]  S. Brams,et al.  RESOURCE‐ALLOCATION MODELS IN PRESIDENTIAL CAMPAIGNING: IMPLICATIONS FOR DEMOCRATIC REPRESENTATION * , 1973, Annals of the New York Academy of Sciences.

[59]  Peter C. Fishburn,et al.  Monotonicity paradoxes in the theory of elections , 1982, Discret. Appl. Math..

[60]  J. Snyder.  Election Goals and the Allocation of Campaign Resources , 1989 .

[61]  Yasusuke Murakami.  Formal Structure of Majority Decision , 1966 .

[62]  R. Niemi,et al.  The Choice of Voting Systems , 1976 .

[63]  H. Nurmi.  Comparing Voting Systems , 1987 .

[64]  Peter C. Fishburn,et al.  Expected utility and approval voting , 1981 .

[65]  Hans-Hermann Bock,et al.  Consensus rules for committee elections , 1998 .

[66]  John Craven.  Majority-consistent preference orderings , 1996 .

[67]  K. Arrow.  Social Choice and Individual Values , 1951 .

[68]  Robin Farquharson,et al.  Theory of voting , 1969 .

[69]  David C. Mcgarvey.  A THEOREMI ON THE CONSTRUCTION OF VOTING PARADOXES , 1953 .

[70]  Peter C. Fishburn,et al.  Robustness of positional scoring over subsets of alternatives , 1980 .

[71]  Kenneth O. May,et al.  A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision , 1952 .

[72]  Irène Charon,et al.  Note: A 16-vertex Tournament for Which Banks Set and Slater Set Are Disjoint , 1997, Discret. Appl. Math..

[73]  J. Banks.  Sophisticated voting outcomes and agenda control , 1984 .

[74]  Michael Suk-Young Chwe,et al.  Minority Voting Rights Can Maximize Majority Welfare , 1999, American Political Science Review.

[75]  Burt L. Monroe,et al.  Fully Proportional Representation , 1995, American Political Science Review.

[76]  Bengt Hansson,et al.  A proof technique for social choice with variable electorate , 1976 .

[77]  John R. Chamberlin,et al.  Selecting a Voting System , 1986, The Journal of Politics.

[78]  A. Tabarrok,et al.  President Perot or Fundamentals of Voting Theory Illustrated with the 1992 Election , 2001 .

[79]  Hannu Nurmi,et al.  Discrepancies in the outcomes resulting from different voting schemes , 1988 .

[80]  Philip D. Straffin,et al.  Topics in the theory of voting , 1980 .

[81]  Jeffrey T. Richelson,et al.  A comparative-analysis of social choice functions, I, II, III: A summary , 1979 .

[82]  H. Peyton Young,et al.  Fair Representation: Meeting the Ideal of One Man, One Vote , 1982 .

[83]  F. Roberts.  Characterizations of the plurality function , 1991 .

[84]  S. Merrill.  A Comparison of Efficiency of Multicandidate Electoral Systems , 1984 .

[85]  Peter C. Fishburn,et al.  An analysis of simple voting systems for electing committees , 1980 .

[86]  Gordon Tullock,et al.  Toward A Mathematics Of Politics , 1972 .

[87]  William G. Ludwin,et al.  Strategic voting and the borda method , 1978 .

[88]  Jill Van Newenhizen.  The Borda method is most likely to respect the Condorcet principle , 1992 .

[89]  Leigh Tesfatsion,et al.  Fair division with uncertain needs and tastes , 1985 .

[90]  Roger R. Davidson,et al.  Some Inconsistencies in Judging Problems , 1972, J. Comb. Theory, Ser. A.

[91]  Peter C. Fishburn,et al.  The responsiveness of approval voting: Comments on Saari and Van Newenhizen , 1988 .

[92]  A. Gibbard.  Manipulation of Voting Schemes: A General Result , 1973 .

[93]  H. Young,et al.  A Consistent Extension of Condorcet’s Election Principle , 1978 .

[94]  E. S. Staveley,et al.  Greek and Roman Voting and Elections , 1972 .

[95]  Gary W. Cox,et al.  Electoral Equilibrium under Alternative Voting Institutions , 1987 .

[96]  M. Truchon,et al.  A Borda Measure for Social Choice Functions , 1997 .

[97]  Pradeep Dubey,et al.  Mathematical Properties of the Banzhaf Power Index , 1979, Math. Oper. Res..

[98]  Douglas J. Amy,et al.  Behind the Ballot Box: A Citizen's Guide to Voting Systems , 2000 .

[99]  Paul R. Kleindorfer,et al.  Optimizer: IBM's Multi-Echelon Inventory System for Managing Service Logistics , 1990 .

[100]  P. Fishburn,et al.  Manipulability of voting by sincere truncation of preferences , 1984 .

[101]  Bernard Debord,et al.  An axiomatic characterization of Borda's k-choice function , 1992 .

[102]  H. Nurmi.  Voting paradoxes and referenda , 1998 .

[103]  Peter C. Fishburn,et al.  Proportions of profiles with a majority candidate , 1979 .

[104]  Peter C. Fishburn,et al.  Probabilities of election outcomes for large electorates , 1978 .

[105]  Dale T. Hoffman,et al.  Relative Efficiency of Voting Systems , 1983 .

[106]  Barry Nalebuff,et al.  An Introduction to Vote-Counting Schemes , 1995 .

[107]  Gideon Doron,et al.  He Hare Voting System is Inconsistent , 1979 .

[108]  H. Young.  Condorcet's Theory of Voting , 1988, American Political Science Review.

[109]  Peter C. Fishburn,et al.  The probability of the paradox of voting: A computable solution , 1976 .

[110]  Peter C. Fishburn,et al.  Comment on The Problem of Strategic Voting under Approval Voting (Vol. 78, December 1984, pp. 952-958) , 1985, American Political Science Review.

[111]  Peter C. Fishburn,et al.  Discrete Mathematics in Voting and Group Choice , 1984 .

[112]  E. M. Bolger.  MONOTONICITY AND OTHER PARADOXES IN SOME PROPORTIONAL REPRESENTATION SCHEMES , 1985 .

[113]  Peter C. Fishburn,et al.  Paradoxes of Preferential Voting , 1983 .

[114]  Lee Spector,et al.  Would the Borda Count Have Avoided the Civil War? , 1999 .

[115]  Steven J. Brams,et al.  The Presidential Election Game , 1978 .

[116]  Williams V. Gehrlein.  Condorcet efficiency of two stage constant scoring rules , 1993 .

[117]  D. Saari.  Basic Geometry of Voting , 1995 .

[118]  H. Moulin.  Condorcet's principle implies the no show paradox , 1988 .

[119]  P. Fishburn,et al.  Proportional representation in variable-size legislatures , 1984 .

[120]  Peter C. Fishburn.  The Theory of Representative Majority Decision , 1971 .

[121]  Robert M. May,et al.  Some mathematical remarks on the paradox of voting , 1971 .

[122]  P. Slater.  Inconsistencies in a schedule of paired comparisons , 1961 .

[123]  Nicholas R. Miller.  Committees, Agendas and Voting , 1995 .

[124]  D. Saari.  Geometry of voting , 1994 .

[125]  Jean-François Laslier,et al.  Slaters's winners of a tournament may not be in the Banks set , 1991 .

[126]  Steven J. Brams,et al.  Rational politics: Decisions, games, and strategy , 1988 .

[127]  James Abello.  The Weak Bruhat Order of SSigma, Consistent Sets, and Catalan Numbers , 1991, SIAM J. Discret. Math..

[128]  Peter C. Fishburn,et al.  Inverted orders for monotone scoring rules , 1981, Discret. Appl. Math..

[129]  Roger B. Myerson,et al.  Axiomatic derivation of scoring rules without the ordering assumption , 1993 .

[130]  Michael Dummett,et al.  The Borda count and agenda manipulation , 1998 .

[131]  T. Schwartz.  Rationality and the Myth of the Maximum , 1972 .

[132]  Ariel Rubinstein,et al.  A further characterization of Borda ranking method , 1981 .

[133]  Marji Lines,et al.  Approval voting and strategy analysis: A Venetian example , 1986 .

[134]  Peter C. Fishburn,et al.  Acyclic sets of linear orders , 1996 .

[135]  Hervé Raynaud.  The individual freedom allowed by the value restriction condition , 1982 .

[136]  Alfred De Grazia,et al.  Mathematical Derivation of an Election System , 1953 .

[137]  Peter C. Fishburn,et al.  A nail-biting election , 2001, Soc. Choice Welf..

[138]  J. H. Smith.  AGGREGATION OF PREFERENCES WITH VARIABLE ELECTORATE , 1973 .

[139]  Steven J. Brams,et al.  Approval voting in practice , 1991 .

[140]  W. Riker.  The art of political manipulation , 1987 .

[141]  Mike Staring.  Two Paradoxes of Committee Elections , 1986 .

[142]  Peter C. Fishburn,et al.  Utility theory for decision making , 1970 .

[143]  Dan S. Felsenthal,et al.  On combining approval with disapproval voting , 1989 .

[144]  Hannu Nurmi,et al.  Voting paradoxes and how to deal with them , 1999 .

[145]  Peter C. Fishburn,et al.  Majority efficiencies for simple voting procedures: Summary and interpretation , 1982 .

[146]  Gary W. Cox,et al.  Strategic Electoral Choice in Multi-Member Districts: Approval Voting in Practice? , 1984 .

[147]  V Gehrlein William,et al.  CONDORCET'S PARADOX AND THE CONDORCET EFFICIENCY OF VOTING RULES. , 1997 .

[148]  H. P. Young,et al.  An axiomatization of Borda's rule , 1974 .

[149]  D. Saari.  Susceptibility to manipulation , 1990 .

[150]  Y. Murakami.  Logic and Social Choice , 1968 .

[151]  H. Moulin.  Axioms of Cooperative Decision Making , 1988 .

[152]  Steven J. Brams,et al.  When is it Advantageous to Cast a Negative Vote , 1977 .

[153]  R. Myerson.  Analysis of Democratic Institutions: Structure, Conduct and Performance , 1995 .

[154]  D. Saari.  The Source of Some Paradoxes from Social Choice and Probability , 1987 .

[155]  Peter C. Fishburn,et al.  Efficacy, power and equity under approval voting , 1981 .

[156]  D. Black.  The theory of committees and elections , 1959 .

[157]  M. Lake,et al.  A New Campaign Resource Allocation Model , 1979 .

[158]  Philip A. Schrodt,et al.  The Logic of Collective Choice. , 1986 .

[159]  P. Fishburn.  The Theory Of Social Choice , 1973 .

[160]  Peter C. Fishburn,et al.  An analysis of voting procedures with nonranked voting , 1977 .

[161]  Peter C. Fishburn,et al.  On the sum-of-ranks winner when losers are removed , 1974, Discret. Math..

[162]  Donald G. Saari,et al.  A dictionary for voting paradoxes , 1989 .

[163]  W. Riker,et al.  The Strategy of Rhetoric: Campaigning for the American Constitution , 1996 .

[164]  Peter C. Fishburn,et al.  An analysis of simple two‐stage voting systems , 1976 .

[165]  David Epstein,et al.  Uncovering some subtleties of the uncovered set: Social choice theory and distributive politics , 1997 .

[166]  H. Moulin.  On strategy-proofness and single peakedness , 1980 .

[167]  K. Arrow.  A Difficulty in the Concept of Social Welfare , 1950, Journal of Political Economy.

[168]  Depankar Ray,et al.  On the practical possibility of a ‘no show paradox’ under the single transferable vote , 1986 .

[169]  S. Shapiro,et al.  Mathematics without Numbers , 1993 .

[170]  Peter C. Fishburn,et al.  Yes-no voting , 1993 .

[171]  Peter C. Fishburn,et al.  Axioms for approval voting: Direct proof , 1978 .

[172]  John R. Chamberlin,et al.  Representative Deliberations and Representative Decisions: Proportional Representation and the Borda Rule , 1983, American Political Science Review.

[173]  Vincent Merlin,et al.  On the relationship of the Condorcet winner and positional voting rules , 1997 .

[174]  Peter C. Fishburn,et al.  Does Approval Voting Elect the Lowest Common Denominator , 1988 .

[175]  Shmuel Nitzan,et al.  The vulnerability of point-voting schemes to preference variation and strategic manipulation , 1985 .

[176]  Murat R. Sertel,et al.  Characterizing approval voting , 1988 .

[177]  H. Young.  Optimal Voting Rules , 1995 .

[178]  Peter C. Fishburn.  Heights of representative systems , 1979, Discret. Appl. Math..

[179]  Donald G. Saari,et al.  Inconsistencies of Weighted Summation Voting Systems , 1982, Math. Oper. Res..

[180]  Dan S. Felsenthal,et al.  The measurement of voting power , 1998 .

[181]  Murat R. Sertel,et al.  The majoritarian compromise is majoritarian-optimal and subgame-perfect implementable , 1999 .

[182]  P. Fishburn.  Paradoxes of Voting , 1974, American Political Science Review.

[183]  Shahriar Shahriari,et al.  On Uniform f-vectors of Cutsets in the Truncated Boolean Lattice , 2000, Comb..

[184]  Amnon Rapoport,et al.  Efficacy in small electorates under plurality and approval voting , 1990 .

[185]  Gary W. Cox,et al.  Making Votes Count: Strategic Coordination in the World's Electoral Systems , 1997 .

[186]  Steven J. Brams,et al.  Proportional Representation , 1998 .

[187]  F. Galton.  One Vote, One Value , 1907, Nature.

[188]  Hannu Nurmi,et al.  Mathematical models of elections and their relevance for institutional design , 1986 .

[189]  W. Riker,et al.  Liberalism Against Populism: A Confrontation Between the Theory of Democracy and the Theory of Social Choice , 1982 .

[190]  Donald G. Saari,et al.  Mathematical Structure of Voting Paradoxes: I. Pairwise Votes , 1999 .

[191]  Jeffrey C. Lagarias,et al.  Optimal pairs of score vectors for positional scoring rules , 1982 .

[192]  Gerald J. Glasser,et al.  Game Theory and Cumulative Voting for Corporate Directors , 1959 .

[193]  P. Pattanaik,et al.  Necessary and Sufficient Conditions for Rational Choice under Majority Decision , 1969 .

[194]  J. Kelly.  Social Choice Theory: An Introduction , 1988 .

[195]  Duff Spafford.  Does Canada Need a New Electoral System? William P. Irvine Kingston: Institute of Intergovernmental Relations, Queen's University, 1979, pp. xii, 99 , 1980, Canadian Journal of Political Science.

[196]  Jean-François Laslier,et al.  Tournament Solutions And Majority Voting , 1997 .

[197]  Peter C. Fishburn,et al.  Interprofile Conditions And Impossibility , 1987 .

[198]  Charles R. Johnson,et al.  How Large are Transitive Simple Majority Domains , 1984 .

[199]  Dan S. Felsenthal,et al.  A comparative analysis of sincere and sophisticated voting under the plurality and approval procedures , 1988 .

[200]  P. Fishburn.  Condorcet Social Choice Functions , 1977 .

[201]  Steven J. Brams,et al.  The AMS Nomination Procedure Is Vulnerable to ‘Truncation of Preferences’ , 1982 .

[202]  Donald G. Saari,et al.  Chaotic Elections! - A Mathematician Looks at Voting , 2001 .

[203]  Peter C. Fishburn,et al.  Condorcet proportions and Kelly's conjectures , 1979, Discret. Appl. Math..

[204]  Donald G. Saari,et al.  Analyzing a nail-biting election , 2001, Soc. Choice Welf..

[205]  T. Tideman,et al.  Independence of clones as a criterion for voting rules , 1987 .

[206]  Samuel Merrill,et al.  Strategic decisions under one-stage multi-candidate voting systems , 1981 .

[207]  B. Grofman,et al.  Approval Voting, Borda Winners, and Condorcet Winners: Evidence From Seven Elections , 1998 .

[208]  Shigehiro Serizawa,et al.  A Maximal Domain for the Existence of Strategy-Proof Rules☆ , 1998 .

[209]  J. Mill.  Considerations on Representative Government , 1861 .

[210]  Scott L. Feld,et al.  The structure of the Banks set , 1990 .

[211]  H. Keiding.  Heights of representative systems: A proof of Fishburn's conjecture , 1984 .

[212]  Richard Edwin Stearns,et al.  The Voting Problem , 1959 .

[213]  J. Nagel.  The Approval Ballot as a Possible Component of Electoral Reform in New Zealand , 1987 .

[214]  N. Tideman.  The Single Transferable Vote , 1995 .

[215]  Ken-ichi Inada,et al.  A Note on the Simple Majority Decision Rule , 1964 .

[216]  E. Maskin,et al.  The Implementation of Social Choice Rules: Some General Results on Incentive Compatibility , 1979 .

[217]  S. Brams,et al.  The paradox of multiple elections , 1998 .

[218]  Ki Hang Kim,et al.  Introduction to mathematical consensus theory , 1980 .

[219]  D. Saari,et al.  The problem of indeterminacy in approval, multiple, and truncated voting systems , 1988 .

[220]  S. Brams,et al.  Would Ross Perot Have Won the 1992 Presidential Election Under Approval Voting? , 1994, PS: Political Science & Politics.

[221]  Peter C. Fishburn,et al.  Coalition voting , 1992 .