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On the probability of a Condorcet winner among a large number of alternatives

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  • Lisa Sauermann

Abstract

Consider $2k-1$ voters, each of which has a preference ranking between $n$ given alternatives. An alternative $A$ is called a Condorcet winner, if it wins against every other alternative $B$ in majority voting (meaning that for every other alternative $B$ there are at least $k$ voters who prefer $A$ over $B$). The notion of Condorcet winners has been studied intensively for many decades, yet some basic questions remain open. In this paper, we consider a model where each voter chooses their ranking randomly according to some probability distribution among all rankings. One may then ask about the probability to have a Condorcet winner with these randomly chosen rankings (which, of course, depends on $n$ and $k$, and the underlying probability distribution on the set of rankings). In the case of the uniform probability distribution over all rankings, which has received a lot of attention and is often referred to as the setting of an "impartial culture", we asymptotically determine the probability of having a Condorcet winner for a fixed number $2k-1$ of voters and $n$ alternatives with $n\to \infty$. This question has been open for around fifty years. While some authors suggested that the impartial culture should exhibit the lowest possible probability of having a Condorcet winner, in fact the probability can be much smaller for other distributions. We determine, for all values of $n$ and $k$, the smallest possible probability of having a Condorcet winner (and give an example of a probability distribution over all rankings which achieves this minimum possible probability).

Suggested Citation

  • Lisa Sauermann, 2022. "On the probability of a Condorcet winner among a large number of alternatives," Papers 2203.13713, arXiv.org.
  • Handle: RePEc:arx:papers:2203.13713
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    References listed on IDEAS

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    1. William Gehrlein, 2002. "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences ," Theory and Decision, Springer, vol. 52(2), pages 171-199, March.
    2. Norman Schofield, 1978. "Instability of Simple Dynamic Games," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 45(3), pages 575-594.
    3. McKelvey, Richard D, 1979. "General Conditions for Global Intransitivities in Formal Voting Models," Econometrica, Econometric Society, vol. 47(5), pages 1085-1112, September.
    4. DeMeyer, Frank & Plott, Charles R, 1970. "The Probability of a Cyclical Majority," Econometrica, Econometric Society, vol. 38(2), pages 345-354, March.
    5. McKelvey, Richard D., 1976. "Intransitivities in multidimensional voting models and some implications for agenda control," Journal of Economic Theory, Elsevier, vol. 12(3), pages 472-482, June.
    6. William V. Gehrlein, 2006. "Condorcet’s Paradox," Theory and Decision Library C, Springer, number 978-3-540-33799-7, September.
    7. Ilia Tsetlin & Michel Regenwetter & Bernard Grofman, 2003. "The impartial culture maximizes the probability of majority cycles," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 21(3), pages 387-398, December.
    8. Bell, Colin E, 1981. "A Random Voting Graph Almost Surely Has a Hamiltonian Cycle When the Number of Alternatives Is Large," Econometrica, Econometric Society, vol. 49(6), pages 1597-1603, November.
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