Condorcet's Paradox (Theory and Decision Library C) by William V. Gehrlein

By William V. Gehrlein

The booklet compiles study on Condorcet's Paradox over a few centuries. It starts with a ancient assessment of the invention of Condorcet's Paradox within the 18th Century, studies quite a few stories carried out to discover real occurrences of the anomaly, and compiles examine that has been performed to increase mathematical representations for the chance that the ambiguity might be saw. Combines all ways which have been used to review this very attention-grabbing phenomenon.

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1. It is impossible to determine how the Border Democrats and Border Whigs split their votes between the two possible rankings for their respective parties. As a result, we initially ignore the eleven members of these two groups and consider 34 Condorcet’s Paradox the results of the comparisons on pairs of alternatives for the remaining 161 representatives. A = Approve the appropriation without the Proviso B = Approve the appropriation with the Proviso C = Take no action on either the appropriation or the Proviso Number Political Group of Voters Ranking Northern Administration Democrats 7 Northern Free Soil Democrats 51 Border Democrats 8 Southern Democrats 46 Northern Pro-War Whigs 2 Northern Anti-War Whigs 39 Border Whigs 3 Southern and Border Whigs 16 Preference ABC BAC ABC or ACB ACB CAB CBA BAC or BCA ACB Fig.

If we ignore the representatives in Group 6, since they are completely indifferent between the alternatives, a total of 184 votes are required for a candidate to win by majority rule. The rankings produce the result EMS (200-156) and SMI (194-162). A PMR cycle exists if we have IME, but the known preference relations for representatives only give a vote of 178-107. Blydenburgh goes on to produce strong evidence from other sources to induce the relative preferences on I and E for some of the representatives in Group 5, to obtain the required number of votes for I to obtain support from a majority of all voters.

7) Score B 3* 39  2* 31  1*11 190. Here, we have BBA when A is the PMRW, to show again that Borda Rule does not always elect the PMRW. Condorcet (1785c) then goes farther with the example voting situation in Fig. 6 to show a phenomenon that Fishburn (1974a) refers to as Condorcet’s Other Paradox. This argument involves analyzing this voting situation with a general weighted scoring rule with weights 3, O and 1, as described in earlier discussion. Condorcet computes Score A and Score B for this general weighted scoring rule: Score A 3*31  Ȝ* 39  1*11 Score B 3*39  Ȝ*31  1*11.

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