### Table 6: When Will a Condorcet Winner be Chosen?

1995

"... In PAGE 33: ... A referendum may not select a Condorcet winner, and in many cases a referendum will select a Condorcet loser or a unanimous loser. Table6 summarizes the results in this paper by clarifying the conditions under which Condorcet winners will be chosen and Condorcet losers or universally Pareto dominated outcomes might be chosen. Solutions to the problem with referendums that retain the important charac- teristic of referendums | mass participation | bring with them a host of other problems.... ..."

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### Table 5: Sophisticated Voting Fails to Choose a Condorcet Winner

1995

"... In PAGE 23: ... While voter sophistication avoids many common voting paradoxes, it is insu#0Ecienttoavoid the problem of nonseparable preferences in referendums. In particular, sophisticated voting is unable to guarantee selection of a Condorcet winner when one exists, as the example in Table5 illustrates. Table 5: Sophisticated Voting Fails to Choose a Condorcet Winner... In PAGE 23: ... Voters with nonseparable preferences may not vote their #0Crst preference since their preference on each issue depends on the expected outcome of the other issue. In Table5 , voter 1 votes NY and voter 2 votes YN. Voter 3 is pivotal for each issue, and her best response to the others apos; voting strategies is... In PAGE 31: ...Table5 , the only stable vote trade produces YY, the Con- dorcet winner #28Schwartz 1977#29. When some voters have nonseparable preferences, then vote-trading is not simply notabad thing; it can be agood thing.... ..."

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### Table 1.2 Pair-wise Condorcet Tally Pairs F(10) H(4) Z(0) I(8) A(7) Total Winner

### Table 1: Probability that the quot;correct quot; option is the unique plurality winner

"... In PAGE 12: ... The great boast of the Condorcet jury theorem in its traditional form is that the probability of the correct option being the majority winner grows quite quickly with increases in the size of the electorate. To what extent can the extended plurality jury theorem make the same boast? To address that question, we present in Table1 some illustrative calculations. (All these calculations are based on Proposition 1 in Appendix 1.... In PAGE 12: ... (All these calculations are based on Proposition 1 in Appendix 1.) The first thing to note in Table1 is this. Where each voter has a probability of more than 0.... In PAGE 14: ...14 [ Table1 about here] The second thing to note from Table 1 is how plurality rule performs where voters are just slightly more likely to choose the correct option than incorrect ones. Where each voter apos;s probability of choosing the correct option from among k options is just over 1/k and the probability of choosing incorrect ones just under that, the probability of the correct option being the plurality winner increases much more slowly with increases in the size of the electorate.... In PAGE 14: ...14 [Table 1 about here] The second thing to note from Table1 is how plurality rule performs where voters are just slightly more likely to choose the correct option than incorrect ones. Where each voter apos;s probability of choosing the correct option from among k options is just over 1/k and the probability of choosing incorrect ones just under that, the probability of the correct option being the plurality winner increases much more slowly with increases in the size of the electorate.... In PAGE 21: ...51, but the number of options increases from k=2 to k=3, the probability of the correct outcome being chosen is greatly increased over that of the correct outcome being chosen in the two-option case. That has already been noted in connection with plurality voting, in our discussion of Table1 . What we see from Table 2, when comparing Scenarios 1 and 3, is that that is true (indeed, even more true) of all of the other standard social decision rules as well.... ..."

### Table 3. Confidence scores and corresponding vote patterns for three neural networks. An instance of the Condorcet paradox.

2000

"... In PAGE 7: ... This is an illustration of the so-called Borda voting paradox, named after the eighteenth century scientist who discovered it. Table3 demonstrates another classic voting paradox, due 4http://moneycentral.msn.... ..."

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