Statistical Mechanics of Voting

David A. Meyer; Thad A. Brown

doi:10.1103/PhysRevLett.81.1718

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Statistical Mechanics of Voting

David A. Meyer^1,2,* and Thad A. Brown^1,3,†

¹Center for Social Computation/Institute for Physical Sciences, Los Alamos, New Mexico
²Project in Geometry and Physics, Department of Mathematics, University of California/San Diego, La Jolla, California 92093-0112
³Department of Political Science, 113 Professional Building, University of Missouri/Columbia, Columbia, Missouri 65211

^*Email address: dmeyer@chonji.ucsd.edu
^†Email address: polstab@showme.missouri.edu

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Phys. Rev. Lett. 81, 1718 – Published 24 August, 1998

DOI: https://doi.org/10.1103/PhysRevLett.81.1718

Abstract

Decision procedures aggregating the preferences of multiple agents can produce cycles and hence outcomes which have been described heuristically as “chaotic.” We make this description precise by constructing an explicit dynamical system from the agents' preferences and a voting rule. The dynamics form a one-dimensional statistical mechanics model; this suggests the use of the topological entropy to quantify the complexity of the system. We compute the expected complexity of a voting rule and the degree of cohesion/diversity among agents using random matrix models—ensembles of statistical mechanics models—in some representative cases.

References (22)

M. J. A. N. de Caritat, Marquis de Condorcet, Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix (l'Imprimèrie Royale, Paris,1785).
K. J. Arrow, Social Choice and Individual Values (Wiley, New York,1951); G. Chichilnisky, J. Math. Econom. 9, 223 (1982); Y. M. Baryshnikov, Adv. Appl. Math. 14, 404 (1993).
D. G. Saari, Adv. Appl. Math. 5, 286 (1984).
B. Hasslacher and M. W. Tilden, Robot. Auton. Syst. 15, 143 (1995).
The adjective in “weak tournament” allows, in analogy with the “weak order” formalization of preference, the possibility that $f_{p}$ contains both $a \leftarrow b$ and $b \leftarrow a$ . That is, both the individual preference orders in $p$ and their aggregate $f_{p}$ can include indifference between alternatives. For an early use of directed graphs/tournaments to analyze voting, see D. C. McGarvey, Econometrica 21, 608 (1953).
D. Black, J. Pol. Econom. 56, 23 (1948); R. Farquharson, Theory of Voting (Yale Univ. Press, New Haven, CT,1969).
This condition was earlier called the postulate of relevancy in E. V. Huntington, Science 88, 287 (1938).
W. H. Gottshalk and G. A. Hedlund, Topological Dynamics (AMS Colloquium Publications, Providence, RI,1955), Vol. 36.
D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics (Addison-Wesley, Reading, MA,1978).
C. E. Shannon, Bell System Tech. J. 27, 379 (1948); ibid.27, 623 (1948); W. Parry, Trans. Am. Math. Soc. 112, 55 (1964); R. L. Adler, A. G. Konheim, and M. H. McAndrew, 114, 309 (1965).
T. Li and J. Yorke, Am. Math. Monthly 82, 985 (1975); M. Misiurewicz, in Dynamical Systems, Proceedings of the CIME session, Bressanone, Italy, 1978, C. Marchioro (Liguori Editore, Napoli, Italy, 1980), p. 125.
B. Ward, J. Conflict Resolution 5, 379 (1961); R. D. McKelvey, J. Econom. Theory 12, 472 (1976); N. Schofield, Rev. Econom. Stud. 40, 575 (1978); L. Cohen, J. Econom. Theory 20, 1 (1979); R. D. McKelvey, Econometrica 47, 1085 (1979); E. C. Browne, P. A. James, and M. A. Miller, Behav. Sci. 36, 148 (1991); D. Richards, Soc. Choice Welf. 11, 109 (1994).
We must also note, however, that the entropy does not distinguish between the cycle in $f_{p_{3}}$ being a top cycle and being a bottom cycle, as it would be were all the edges from $a$ reversed, making $a$ the Condorcet winner (a pairwise unbeatable alternative).
E. P. Wigner, Ann. Math. 62, 548 (1955); M. L. Mehta and M. Gaudin, Nucl. Phys. 18, 420 (1960); F. J. Dyson, J. Math. Phys. 3, 140 (1962).
G. Th. Guilbaud, Économie Appliquée 5, 501 (1952).
R. M. May, Behav. Sci. 16, 143 (1971); P. C. Fishburn, 18, 212 (1973).
W. V. Gehrlein and P. C. Fishburn, J. Econom. Theory 19, 38 (1978).
J.-C. de Borda, Mem. Acad. R. Sci. 657 (1781).
The Borda count is not an IIA voting rule, so Arrow's theorem [[2]] does not apply. Nevertheless, alternatives may be equally ranked and, if we define the dynamics by choosing an equally ranked new alternative over the status quo, the system will then contain cycles.
A. H. Copeland, “A `reasonable' social welfare function,” mimeographed notes, University of Michigan seminar on applications of mathematics to the social sciences (November 1951); referred to in L. A. Goodman, in Decision Processes, R. M. Thrall, C. H. Coombs, and R. L. Davis (Wiley, New York, 1954), p. 39.
G. Chichilnisky, Am. Econom. Rev. 84, 427 (1994).
D. A. Meyer, chao-dyn/9710005; InterJournal Complex Systems, Article [123] http://dynamics.bu.edu/InterJournal/

[c1] M. J. A. N. de Caritat, Marquis de Condorcet, Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix (l'Imprimèrie Royale, Paris,1785).

[c2] K. J. Arrow, Social Choice and Individual Values (Wiley, New York,1951); G. Chichilnisky, J. Math. Econom. 9, 223 (1982); Y. M. Baryshnikov, Adv. Appl. Math. 14, 404 (1993).

[c3] D. G. Saari, Adv. Appl. Math. 5, 286 (1984).

[c4] B. Hasslacher and M. W. Tilden, Robot. Auton. Syst. 15, 143 (1995).

[c5] The adjective in “weak tournament” allows, in analogy with the “weak order” formalization of preference, the possibility that $f_{p}$ contains both $a \leftarrow b$ and $b \leftarrow a$ . That is, both the individual preference orders in $p$ and their aggregate $f_{p}$ can include indifference between alternatives. For an early use of directed graphs/tournaments to analyze voting, see D. C. McGarvey, Econometrica 21, 608 (1953).

[c6] D. Black, J. Pol. Econom. 56, 23 (1948); R. Farquharson, Theory of Voting (Yale Univ. Press, New Haven, CT,1969).

[c7] This condition was earlier called the postulate of relevancy in E. V. Huntington, Science 88, 287 (1938).

[c8] W. H. Gottshalk and G. A. Hedlund, Topological Dynamics (AMS Colloquium Publications, Providence, RI,1955), Vol. 36.

[c9] D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics (Addison-Wesley, Reading, MA,1978).

[c10] C. E. Shannon, Bell System Tech. J. 27, 379 (1948); ibid.27, 623 (1948); W. Parry, Trans. Am. Math. Soc. 112, 55 (1964); R. L. Adler, A. G. Konheim, and M. H. McAndrew, 114, 309 (1965).

[c11] T. Li and J. Yorke, Am. Math. Monthly 82, 985 (1975); M. Misiurewicz, in Dynamical Systems, Proceedings of the CIME session, Bressanone, Italy, 1978, C. Marchioro (Liguori Editore, Napoli, Italy, 1980), p. 125.

[c12] B. Ward, J. Conflict Resolution 5, 379 (1961); R. D. McKelvey, J. Econom. Theory 12, 472 (1976); N. Schofield, Rev. Econom. Stud. 40, 575 (1978); L. Cohen, J. Econom. Theory 20, 1 (1979); R. D. McKelvey, Econometrica 47, 1085 (1979); E. C. Browne, P. A. James, and M. A. Miller, Behav. Sci. 36, 148 (1991); D. Richards, Soc. Choice Welf. 11, 109 (1994).

[c13] We must also note, however, that the entropy does not distinguish between the cycle in $f_{p_{3}}$ being a top cycle and being a bottom cycle, as it would be were all the edges from $a$ reversed, making $a$ the Condorcet winner (a pairwise unbeatable alternative).

[c14] E. P. Wigner, Ann. Math. 62, 548 (1955); M. L. Mehta and M. Gaudin, Nucl. Phys. 18, 420 (1960); F. J. Dyson, J. Math. Phys. 3, 140 (1962).

[c15] G. Th. Guilbaud, Économie Appliquée 5, 501 (1952).

[c16] R. M. May, Behav. Sci. 16, 143 (1971); P. C. Fishburn, 18, 212 (1973).

[c17] W. V. Gehrlein and P. C. Fishburn, J. Econom. Theory 19, 38 (1978).

[c18] J.-C. de Borda, Mem. Acad. R. Sci. 657 (1781).

[c19] The Borda count is not an IIA voting rule, so Arrow's theorem [[2]] does not apply. Nevertheless, alternatives may be equally ranked and, if we define the dynamics by choosing an equally ranked new alternative over the status quo, the system will then contain cycles.

[c20] A. H. Copeland, “A `reasonable' social welfare function,” mimeographed notes, University of Michigan seminar on applications of mathematics to the social sciences (November 1951); referred to in L. A. Goodman, in Decision Processes, R. M. Thrall, C. H. Coombs, and R. L. Davis (Wiley, New York, 1954), p. 39.

[c21] G. Chichilnisky, Am. Econom. Rev. 84, 427 (1994).

[c22] D. A. Meyer, chao-dyn/9710005; InterJournal Complex Systems, Article [123] http://dynamics.bu.edu/InterJournal/

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