Abstract
This essay measures and analyzes for a special class of point-voting schemes (the Borda method, plurality rule and the unrestricted point-voting scheme) sensitivity to preference variation (a simple change in the socially winning alternative resulting from alteration of a single voter's preferences) and vulnerability to individual strategic manipulation (a change in the winning alternative that benefits the voter whose preferences are altered). Assuming that society (n voters with linear preference orders on a finite set of m alternatives) satisfies the impartial-culture assumption, that is, each randomly selected voter is equally likely to hold any one of the randomly picked possible preference orders on the alternatives, we demonstrate:
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(i)
for a given rule and a fixed number of voters, the sensitivity to individual preference variation and the vulnerability to individual strategic manipulation are greater, the larger the total number of alternatives.
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(ii)
For a given rule and a fixed number of alternatives, the vulnerability to individual strategic manipulation, in general, is not greater the smaller the total number of voters. Such a relationship does hold, however, if n is sufficiently large.
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(iii)
For any given combination of number of voters and number of alternatives, the unrestricted point-voting scheme is more sensitive to preference variation than the Borda method, which, in turn, is more exposed to such variation relative to the plurality rule. A similar conclusion does not hold with respect to vulnerability to individual strategic manipulation, unless the number of voters is sufficiently small.
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References
Arrow, K. (1963). Social choice and individual values, 2nd ed. New York: Wiley.
Barbera, S. (1977). The manipulation of social choice mechanisms that do not leave ‘too much’ to chance. Econometrica 45: 1573–1588.
Chamberlin, J. (1980). A mathematical programming approach to assessing the manipulability of social choice functions. A paper presented at the 1980 Public Choice Society Meetings.
De Meyer, F., and Plott, C.R. (1970). The probability of a cyclical majority. Econometrica 38: 345–354.
Fishburn, P. (1973). The theory of social choice. Princeton, N.J.: Princeton University Press.
Garman, M.B., and Kamien, M. (1968). The paradox of voting: Probability calculations. Behavioral Science 13:306–316.
Gärdenfors, P. (1973). Positionalist voting functions. Theory and Decision 4:1–24.
Gärdenfors, P. (1976). Manipulation of social choice functions. Journal of Economic Theory 13:217–228.
Gärdenfors, P. (1979). On definitions of manipulation of social choice functions. In J.-J. Laffont (Ed.), Aggregation and revelation of preferences. Amsterdam: North Holland.
Gehrlein, W.V., and Fishburn, P.C. (1976). The probability of the paradox of voting: A computable solution. Journal of Economic Theory 13:14–25.
Gibbard, A. (1973). Manipulations of voting schemes: A general result. Econometrica 41:587–601.
Gibbard, A. (1977). Manipulation of schemes that mix voting with chance. Econometrica 45:665–681.
Kannai, Y., and Peleg, B. (1984). A note on the extension of an order on a set to the power set. Journal of Economic Theory 32:172–175.
Kelly, J.S. (1974). Voting anomalies, the number of voters and the number of alternatives. Econometrica 42:239–251.
Kelly, J. (1977). Strategy-proofness and social choice functions without single-valuedness. Econometrica 45: 439–446.
Klahr, D. (1966). Computer simulation of the paradox of voting. American Political Science Review 60:384–390.
Niemi, R.G., and Weisberg, H.F. (1968). A mathematical solution for the probability of the paradox of voting. Behavioral Science 13:317–323.
Nitzan, S., Paroush, J., and Lampert, S. (1980). Preference expression and misrepresentation in point voting schemes. Public Choice 35:421–436.
Packard, D.J. (1979). Preference relations. Journal of Mathematical Psychology.
Pattanaik, P.K. (1973). On the stability of sincere voting situations. Journal of Economic Theory 6:558–574.
Pattanaik, P.K. (1974). Stability of sincere voting under some classes of non-binary group decision procedures. Journal of Economic Theory 8:206–224.
Pattanaik, P.K. (1975). Strategic voting without collusion under binary and democratic group decision rules. The Review of Economic Studies 42:93–103.
Peleg, B. (1979). A note on manipulation of large voting schemes. Theory and Decision. 401–413.
Satterthwaite, M. (1975). Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory 10:187–217.
Sen, A.K. (1970). Collective choice and social welfare. San Francisco: Holden-Day.
Sengupta, M. (1980). Monotonicity, independence of irrelevant alternatives and strategy-proofness of social decision functions. Review of Economic Studies 47:393–407.
Tullock, G., and Campbell, C.D. (1970). Computer simulation of a small voting system. Economic Journal 80:97–104.
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I am indebted to the anonymous referees of this journal and to P. Aranson, B. Peleg and A. Rubinstein for their very helpful comments and suggestions. I also wish to express my appreciation to Yoav Ben-Zvi for his ingenious programming skills which were crucial to the results contained in this paper.
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Nitzan, S. The vulnerability of point-voting schemes to preference variation and strategic manipulation. Public Choice 47, 349–370 (1985). https://doi.org/10.1007/BF00127531
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DOI: https://doi.org/10.1007/BF00127531