Sincerity and Strategy-Proofness: Which System Is Most Honest?
In this chapter we shall introduce some basic ideas about voting that will be helpful in distinguishing better from worse voting strategies. By “better” we mean roughly those strategies that a voter would seriously consider in deciding for whom to vote. Crucial to this determination will be eliminating those “worse” strategies that a voter would never consider because they are “dominated”—under no circumstances would they yield a better outcome than some other strategies, and sometimes they would give unequivocally worse outcomes. Those strategies which are not dominated will be called “admissible.”
KeywordsVote System Preference Order Strategic Vote Approval Vote Feasible Strategy
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Footnotes to Chapter 2
- 2.Steven J. Brams, The Presidential Election Game (New Haven: Yale University Press, 1978), pp. 199–202.Google Scholar
- 3.We assume that if a voter votes for k candidates (k in s), then each candidate he votes for receives one full vote in the aggregation process (see Section 3.3). Cases in which votes receive different weights, depending on the number of candidates a voter votes for, are analyzed in Peter C. Fishburn, “Symmetric and Consistent Aggregation with Dichotomous Voting,” in Aggregation and Revelation of Preference, edited by Jean-Jacques Laffont (Amsterdam: North-Holland, 1979), pp. 201–208. This has been extended to ranked voting in P. C. Fishburn, “Dominant Strategies and Restricted Ballots with Variable Electorate,” Mathematical Social Sciences 2 4 (June 1982), 383–395. Among all systems examined in these articles, approval voting does very well on the basis of the criteria used for comparing systems.Google Scholar
- 4.Steven J. Brams, “One Man, N Votes,” Module in Applied Mathematics, Mathematical Association of America (Ithaca, NY: Cornell University, 1976); S. J. Brams, Comparison Voting, Innovative Instructional Unit (Washington, DC: American Political Science Association, 1978), which will appear in slightly revised form in Modules in Applied Mathematics: Political and Related Models, Vol. 2, edited by Steven J. Brams, William F. Lucas, and Philip D. Straffin, Jr. (New York: Springer-Verlag, 1982); and S. J. Brams, Presidential Election Game, Ch. 6.Google Scholar
- 5.George A. W. Boehm, “One Fervent Vote against Wintergreen” (mimeographed, 1976); and Steven J. Brams, “When Is It Advantageous to Cast a Negative Vote?” in Mathematical Economics and Game Theory: Essays in Honor of Oskar Morgenstern, edited by R. Henn and O. Moeschlin, Lecture Notes in Economics and Mathematical Systems, Vol. 141 (Berlin: Springer-Verlag, 1977), pp. 564–572.Google Scholar
- 6.It can happen that more complex voting systems than those discussed here are able to induce the same outcome from “strategic voting” as would obtain if all voters voted sincerely. See Bezalel Peleg, “Consistent Voting Systems,” Econometrica 46,1 (January 1978), 153–162; and Bhaskar Dutta and Prasanta K. Pattanaik, “On Nicely Consistent Voting Systems,” Econometrica 46 1 (January 1978), 163–170.MATHCrossRefMathSciNetGoogle Scholar
- 8.Allan Gibbard, “Manipulation of Voting Schemes: A General Result,” Econometrica 41,3 (May 1973), 587–601; and Mark Allen Satterthwaite, “Strategy-Proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions,” Journal of Economic Theory 10 2 (April 1975), 187–217. Recent results on manipulability relevant to approval voting are given in P. C. Fishburn, “Dominant Strategies and Restricted Ballots with Variable Electorate.”MATHCrossRefMathSciNetGoogle Scholar
- 9.P. C. Fishburn, “A Strategic Analysis of Nonranked Voting Systems.” Other properties that approval voting satisfies are delineated in different axiomatizations of approval voting in P. C. Fishburn, “Axioms for Approval Voting: Direct Proof,” Journal of Economic Theory 19 1 (October 1978), 180–185; and P. C. Fishburn, “Symmetric and Consistent Aggregation with Dichotomous Voting.”Google Scholar
- 10.Lying may also refer to the false announcement of preferences by a deceiver in a game of incomplete information that induces the deceived player (s) to change strategies that lead to better outcomes for the deceiver. This concept connotes more than “mere” insincerity and is formally analyzed in Steven J. Brams, “Deception in 2 × 2 Games,” Journal of Peace Science 2,2 (Spring 1977), 171–203; Steven J. Brams and Frank C. Zagare, “Deception in Simple Voting Games,” Social Science Research 6 3 (September 1977), 257–272; and S. J. Brams and F. C. Zagare, “Double Deception: Two against One in Three-Person Games,” Theory and Decision 13 1 (March 1981), 81–90. For a more informal development of deception in voting games, see S. J. Brams, Paradoxes in Politics: An Introduction to the Nonobvious in Political Science (New York: Free Press, 1976), pp. 172–176.Google Scholar