Refining measures of group mutual coherence

Abstract

The Condorcet efficiencies of plurality rule (PR), negative plurality rule (NPR), Borda rule (BR), plurality elimination rule (PER) and negative plurality elimination rule (NPER) were evaluated over parameters associated with six models of group mutual coherence in Gehrlein and Lepelley (Voting paradoxes and group coherence: the Condorcet efficiency of voting rules, 2010) It was found that BR was not always the most efficient voting rule, but it always performed quite well; while each of the other voting rules had identifiable regions of parameters in which they performed very poorly. By refining these parameters so that attention is focused on the particular model of group coherence that most closely reflects the voters’ preferences in a given voting situation, these conclusions are modified. The comparison of BR to PER and NPER changes significantly. The comparison of BR to PR and NPR remains similar, but the differences in the relative comparisons of efficiencies are somewhat reduced.

Appendices

Appendix: Condorcet efficiency of PR

$$ \begin{aligned} CE_{SD}^{PR} \left( {\infty ,IAC\left( {\alpha_{Mt} } \right)} \right) & = \frac{{137\alpha_{Mt}^{2} - 94\alpha_{Mt} + 16}}{{4\left( {29\alpha_{Mt}^{2} - 22\alpha_{Mt} + 4} \right)}},\quad {\text{for}}\;0 \le \alpha_{Mt} \le 1/4 \\ & = \frac{17}{20},\quad {\text{for}}\;1/4 \le \alpha_{Mt} \le 1/3. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{CP}^{PR} \left( {\infty ,IAC\left( {\alpha_{Mc} } \right)} \right) & = \frac{{2\left( {1782\alpha_{Mc}^{3} - 1800\alpha_{Mc}^{2} + 585\alpha_{Mc} - 62} \right)}}{{9\left( {457\alpha_{Mc}^{3} - 460\alpha_{Mc}^{2} + 150\alpha_{Mc} - 16} \right)}},\quad {\text{for}}\;1/4 \le \alpha_{Mt} \le 1/3 \\ & = \frac{{54,432\alpha_{Mc}^{4} - 51,840\alpha_{Mc}^{3} + 16,416\alpha_{Mc}^{2} - 1776\alpha_{Mc} + 13}}{{108\alpha_{Mc} \left( {457\alpha_{Mc}^{3} - 460\alpha_{Mc}^{2} + 150\alpha_{Mc} - 16} \right)}},\quad {\text{for}}\;1/6 \le \alpha_{Mc} \le 1/4 \\ & = \frac{152}{189},\quad {\text{for}}\;1/4 \le \alpha_{Mc} \le 1/3. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{NU}^{PR} \left( {\infty ,IAC\left( {\alpha_{Mb^{*}} } \right)} \right) & = \frac{20}{27},\quad {\text{for}}\; 1/ 3\le \alpha_{Mb^{*}} \le 3/8 \\ & = \frac{{220,032\alpha_{Mb^{*}}^{4} - 373,248\alpha_{Mb^{*}}^{3} + 231,552\alpha_{Mb^{*}}^{2} - 62,688\alpha_{Mb^{*}} + 6277}}{{54\left( {5808 \alpha_{Mb^{*}}^{4} - 9792\alpha_{Mb^{*}}^{3} + 6048\alpha_{Mb^{*}}^{2} - 1632\alpha_{Mb^{*}} + 163} \right)}},\quad {\text{for}}\; 5/ 8\le \alpha_{Mb^{*}} \le 2/3 \\ & = \frac{{14,400\alpha_{Mb^{*}}^{4} - 23,040\alpha_{Mb^{*}}^{3} + 6912\alpha_{Mb^{*}}^{2} + 4032\alpha_{Mb^{*}} - 1681}}{{1728(1 - \alpha_{{Mb^{*} }} )^{3} (7\alpha_{{Mb^{*} }} - 3)}},\quad {\text{for}}\; 1/ 2\le \alpha_{Mb^{*}} \le 5/8 \\ & = \frac{{112,322\alpha_{Mb^{*}}^{4} - 345,602\alpha_{Mb^{*}}^{3} + 397,442\alpha_{Mb^{*}}^{2} - 20,016\alpha_{Mb^{*}} + 3653}}{{864(\alpha_{Mb^{*}} - 1)^{3} (7\alpha_{Mb^{*}} - 3)}},\quad {\text{for}}\; 5/ 8\le \alpha_{Mb^{*}} \le 2/3 \\ & = \frac{{162\alpha_{Mb^{*}}^{4} + 642\alpha_{Mb^{*}}^{3} - 1922\alpha_{Mb^{*}}^{2} + 144\alpha_{Mb^{*}} - 31}}{{32(1 - \alpha_{Mb^{*}} )^{3} (7\alpha_{Mb^{*}} - 3)}},\quad {\text{for}}\; 2/ 3\le \alpha_{Mb^{*}} \le 3/4 \\ & = \frac{{15\alpha_{Mb^{*}} - 7}}{{2(7\alpha_{Mb^{*}} - 3)}},\quad {\text{for}}\; 3/ 4\le \alpha_{Mb^{*}} \le 1. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{PU}^{PR} \left( {\infty ,IAC\left( {\alpha_{Mt^{*}} } \right)} \right) & = \frac{4}{5},\quad {\text{for}}\; 1 / 3\le \alpha_{Mt^{*}} \le 3/8 \\ & = \frac{{2(1504\alpha_{Mt^{*}}^{4} - 2688\alpha_{Mt^{*}}^{3} + 1728\alpha_{Mt^{*}}^{2} - 480\alpha_{Mt^{*}} + 49)}}{{5808\alpha_{Mt^{*}}^{4} - 9792\alpha_{Mt^{*}}^{3} + 6048\alpha_{Mt^{*}}^{2} - 1632\alpha_{Mt^{*}} + 163}},\quad {\text{for}}\; 3 / 8\le \alpha_{Mt^{*}} \le 1/2 \\ & = 1,\quad {\text{for}}\; 1 / 2\le \alpha_{Mt^{*}} \le 1 \\ \end{aligned} $$

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$$ \begin{aligned} CE_{PC}^{PR} \left( {\infty ,IAC\left( {\alpha_{Mc^{*}} } \right)} \right) & = \frac{104}{189},\quad {\text{for}}\; 1 / 3\le \alpha_{Mc^{*}} \le 3/8 \\ & = \frac{{74,112\alpha_{Mc^{*}}^{4} - 133,632\alpha_{Mc^{*}}^{3} + 86,400\alpha_{Mc^{*}}^{2} - 24,096\alpha_{Mc^{*}} + 2467}}{{108(1828\alpha_{Mc^{*}}^{2} - 3120\alpha_{Mc^{*}}^{2} + 1944\alpha_{Mc^{*}}^{2} - 528\alpha_{Mc^{*}} + 53)}},\quad {\text{for}}\; 3 / 8\le \alpha_{Mc^{*}} \le 1/2 \\ & = \frac{{19,008\alpha_{Mc^{*}}^{4} - 52,992\alpha_{Mc^{*}}^{3} + 57,024\alpha_{Mc^{*}}^{2} - 27,648\alpha_{Mc^{*}} + 4951}}{{432(\alpha_{Mc^{*}} - 1)^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 1 / 2\le \alpha_{Mc^{*}} \le 5/8 \\ & = \frac{{15,648\alpha_{{Mc^{^{*}} }}^{4} - 41,856\alpha_{Mc^{*}}^{3} + 42,912\alpha_{Mc^{*}}^{2} - 19,824\alpha_{Mc^{*}} + 3413}}{{216(\alpha_{Mc^{*}} - 1)^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 5 / 8\le \alpha_{Mc^{*}} \le 2/3 \\ & = \frac{{32\alpha_{Mc^{*}}^{4} - 1280\alpha_{Mc^{*}}^{3} + 2784\alpha_{Mc^{*}}^{2} - 2000\alpha_{Mc^{*}} + 455}}{{72(\alpha_{Mc^{*}} - 1)^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 2 / 3\le \alpha_{Mc^{*}} \le 3/4 \\ & = \frac{{4(73\alpha_{Mc^{*}} - 37)}}{{9(29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 3 / 4\le \alpha_{Mc^{*}} \le 1. \\ \end{aligned} $$

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Condorcet efficiency of NPR

$$ \begin{aligned} CE_{SD}^{NPR} \left( {\infty ,IAC\left( {\alpha_{Mt} } \right)} \right) & = \frac{{\left( {3\alpha_{Mt} - 1} \right)\left( {414\alpha_{Mt}^{2} - 282\alpha_{Mt} + 47} \right)}}{{9\left( {7\alpha_{Mt} - 2} \right)\left( {29\alpha_{Mt}^{2} - 22\alpha_{Mt} + 4} \right)}},\quad {\text{for}}\;0 \le \alpha_{Mt} \le 1/6 \\ & = \frac{{18,144\alpha_{Mt}^{4} - 20,736\alpha_{Mt}^{3} + 7344\alpha_{Mt}^{2} - 768\alpha_{Mt} - 17}}{{216\alpha_{Mt} \left( {7\alpha_{Mt} - 2} \right)\left( {29\alpha_{Mt}^{2} - 22\alpha_{Mt} + 4} \right)}},\quad {\text{for}}\;1/6 \le \alpha_{Mt} \le 1/4 \\ & = \frac{20}{27},\quad {\text{for}}\;1/4 \le \alpha_{Mt} \le 1/3. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{CP}^{NPR} \left( {\infty ,IAC\left( {\alpha_{Mc} } \right)} \right) & = \frac{{\left( {3\alpha_{Mc} - 1} \right)\left( {522\alpha_{Mc}^{2} - 366\alpha_{Mc} + 61} \right)}}{{9\left( {457\alpha_{Mc}^{3} - 460\alpha_{Mc}^{2} + 150\alpha_{Mc} - 16} \right)}},\quad {\text{for}}\;0 \le \alpha_{Mc} \le 1/6 \\ & = \frac{{5184\alpha_{Mc}^{4} + 6912\alpha_{Mc}^{3} - 4320\alpha_{Mc}^{2} + 384\alpha_{Mc} + 49}}{{216\alpha_{Mc} \left( {16 - 457\alpha_{Mc}^{3} + 460\alpha_{Mc}^{2} - 150\alpha_{Mc} } \right)}},\quad {\text{for}}\;1/6 \le \alpha_{Mc} \le 1/4 \\ & = \frac{104}{189},\quad {\text{for}}\;1/4 \le \alpha_{Mc} \le 1/3. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{NU}^{NPR} \left( {\infty ,IAC\left( {\alpha_{Mb^{*}} } \right)} \right) & = \frac{17}{20},\quad {\text{for}}\; 1 / 3\le \alpha_{Mb^{*}} \le 3/8 \\ & = \frac{{6780\alpha_{Mb^{*}}^{4} - 11,088\alpha_{Mb^{*}}^{3} + 6696\alpha_{Mb^{*}}^{2} - 1776\alpha_{Mb^{*}} + 175}}{{5808\alpha_{Mb^{*}}^{4} - 9792\alpha_{Mb^{*}}^{3} + 6048\alpha_{Mb^{*}}^{2} - 1632\alpha_{Mb^{*}} + 163}},\quad {\text{for}}\; 3 / 8\le \alpha_{Mb^{*}} \le 2/5 \\ & = \frac{{4280\alpha_{Mb^{*}}^{4} - 7088\alpha_{Mb^{*}}^{3} + 4296\alpha_{Mb^{*}}^{2} - 1136\alpha_{Mb^{*}} + 111}}{{5808\alpha_{Mb^{*}}^{4} - 9792\alpha_{Mb^{*}}^{3} + 6048\alpha_{Mb^{*}}^{2} - 1632\alpha_{Mb^{*}} + 163}},\quad {\text{for}}\; 2 / 5\le \alpha_{Mb^{*}} \le 1/2 \\ & = \frac{{344\alpha_{Mb^{*}}^{4} - 1056\alpha_{Mb^{*}}^{3} + 1200\alpha_{Mb^{*}}^{2} - 592\alpha_{Mb^{*}} + 105}}{{32(\alpha_{Mb^{*}} - 1)^{3} (7\alpha_{Mb^{*}} - 3)}},\quad {\text{for}}\; 1 / 2\le \alpha_{Mb^{*}} \le 3/4 \\ & = \frac{{11\alpha_{Mb^{*}} - 3}}{{4(7\alpha_{Mb^{*}} - 3)}},\quad {\text{for}}\; 3 / 4\le \alpha_{Mb^{*}} \le 1. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{PU}^{NPR} \left( {\infty ,IAC\left( {\alpha_{Mt^{*}} } \right)} \right) & = \frac{43}{54},\quad {\text{for}}\; 1 / 3\le \alpha_{{Mt^{^{*}} }} \le 3/8 \\ & = \frac{{192,456\alpha_{Mt^{*}}^{4} - 311,904\alpha_{Mt^{*}}^{3} + 187,056\alpha_{Mt^{*}}^{2} - 49,344\alpha_{Mt^{*}} + 4841}}{{27\left( {5808\alpha_{Mt^{*}}^{4} - 9792\alpha_{Mt^{*}}^{3} + 6048\alpha_{Mt^{*}}^{2} - 1632\alpha_{Mt^{*}} + 163} \right)}},\quad {\text{for}}\; 3 / 8\le \alpha_{Mt^{*}} \le 2/5 \\ & = \frac{{102,456\alpha_{Mt^{*}}^{4} - 167,904\alpha_{Mt^{*}}^{3} + 100,656\alpha_{Mt^{*}}^{2} - 26,304\alpha_{Mt^{*}} + 2537}}{{27(5808\alpha_{Mt^{*}}^{4} - 9792\alpha_{Mt^{*}}^{3} + 6048\alpha_{Mt^{*}}^{2} - 1632\alpha_{Mt^{*}} + 163)}},\quad {\text{for}}\; 2 / 5\le \alpha_{Mt^{*}} \le 1/2 \\ & = \frac{{135\alpha_{Mt^{*}}^{4} - 108\alpha_{Mt^{*}}^{3} - 270\alpha_{Mt^{*}}^{2} + 324\alpha_{Mt^{*}} - 85}}{{108(1 - \alpha_{Mt^{*}} )^{3} (7\alpha_{Mt^{*}} - 3)}},\quad {\text{for}}\; 1 / 2\le \alpha_{Mt^{*}} \le 2/3 \\ & = \frac{{31\alpha_{Mt^{*}} - 15}}{{4(7\alpha_{Mt^{*}} - 3)}},\quad {\text{for}}\; 3 / 3\le \alpha_{Mt^{*}} \le 1. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{PC}^{NPR} \left( {\infty ,IAC\left( {\alpha_{Mc^{*}} } \right)} \right) & = \frac{152}{189},\quad {\text{for}}\; 1 / 3\le \alpha_{Mc^{*}} \le 3/8 \\ & = \frac{{61,344\alpha_{Mc^{*}}^{4} - 100,224\alpha_{Mc^{*}}^{3} + 60,480\alpha_{Mc^{*}}^{2} - 16,032\alpha_{Mc^{*}} + 1579}}{{27\left( {1828\alpha_{Mc^{*}}^{4} - 3120\alpha_{Mc^{*}}^{3} + 1944\alpha_{Mc^{*}}^{2} - 528\alpha_{Mc^{*}} + 53} \right)}},\quad {\text{for}}\; 3 / 8\le \alpha_{Mc^{*}} \le 2/5 \\ & = \frac{{46,344\alpha_{Mc^{*}}^{4} - 76,224\alpha_{Mc^{*}}^{3} + 46,080\alpha_{Mc^{*}}^{2} - 12,192\alpha_{Mc^{*}} + 1195}}{{27\left( {1828\alpha_{Mc^{*}}^{4} - 3120\alpha_{Mc^{*}}^{3} + 1944\alpha_{Mc^{*}}^{2} - 528\alpha_{Mc^{*}} + 53} \right)}},\quad {\text{for}}\; 2 / 5\le \alpha_{Mc^{*}} \le 1/2 \\ & = \frac{{2376\alpha_{Mc^{*}}^{4} - 8640\alpha_{Mc^{*}}^{3} + 10,800\alpha_{Mc^{*}}^{2} - 5616\alpha_{Mc^{*}} + 1027}}{{108(\alpha_{Mc^{*}} - 1)^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 1 / 2\le \alpha_{Mc^{*}} \le 2/3 \\ & = \frac{{232\alpha_{Mc^{*}}^{4} - 672\alpha_{Mc^{*}}^{3} + 720\alpha_{Mc^{*}}^{2} - 336\alpha_{Mc^{*}} + 57}}{{4(\alpha_{Mc^{*}} - 1)^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 2 / 3\le \alpha_{Mc^{*}} \le 3/4 \\ & = \frac{{6(1 - \alpha_{Mc^{*}} )}}{{29\alpha_{Mc^{*}} - 13}},\quad {\text{for}}\; 3 / 4\le \alpha_{Mc^{*}} \le 1. \\ \end{aligned} $$

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Condorcet efficiency of BR

$$ CE_{SD}^{BR} \left( {\infty ,IAC\left( {\alpha_{Mt} } \right)} \right) = CE_{SP}^{BR} \left( {\infty ,IAC\left( {\alpha_{Mb} } \right)} \right) $$

(48)

$$ \begin{aligned} CE_{CP}^{BR} \left( {\infty ,IAC\left( {\alpha_{Mc} } \right)} \right) & = \frac{{2107\alpha_{Mc}^{3} - 2372\alpha_{Mc}^{2} + 810\alpha_{Mc} - 88}}{{6(457\alpha_{Mc}^{3} - 460\alpha_{Mc}^{2} + 150\alpha_{Mc} - 16)}},\quad {\text{for}}\; 0\le \alpha_{Mc} \le 1/6 \\ & = \frac{{3403\alpha_{Mc}^{4} - 3236\alpha_{Mc}^{3} + 1026\alpha_{Mc}^{2} - 112\alpha_{Mc} + 1}}{{6\alpha_{Mc} (457\alpha_{Mc}^{3} - 460\alpha_{Mc}^{2} + 150\alpha_{Mc} - 16)}},\quad {\text{for}}\; 1 / 6\le \alpha_{Mc} \le 1/4 \\ & = \frac{37}{42},\quad {\text{for}}\; 1 / 4\le \alpha_{Mc} \le 1/3. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{NU}^{BR} \left( {\infty ,IAC\left( {\alpha_{Mb^{*}} } \right)} \right) & = \frac{13}{15},\quad {\text{for}}\; 1/ 3\le \alpha_{Mb^{*}} \le 3/8 \\ & = \frac{{20,016\alpha_{Mb^{*}}^{4} - 32,832\alpha_{Mb^{*}}^{3} + 19,872\alpha_{Mb^{*}}^{2} - 5280\alpha_{Mb^{*}} + 521}}{{3(5808\alpha_{Mb^{*}}^{4} - 9792\alpha_{Mb^{*}}^{3} + 6048\alpha_{Mb^{*}}^{2} - 1632\alpha_{Mb^{*}} + 163)}},\quad {\text{for}}\; 3/ 8\le \alpha_{Mb^{*}} \le 2/5 \\ & = \frac{{10,016\alpha_{Mb^{*}}^{4} - 16,832\alpha_{Mb^{*}}^{3} + 10,272\alpha_{Mb^{*}}^{2} - 2720\alpha_{Mb^{*}} + 265}}{{3(5808\alpha_{Mb^{*}}^{4} - 9792\alpha_{Mb^{*}}^{3} + 6048\alpha_{Mb^{*}}^{2} - 1632\alpha_{Mb^{*}} + 163)}},\quad {\text{for}}\; 2/ 5\le \alpha_{Mb^{*}} \le 5/12 \\ & = \frac{{191,712\alpha_{Mb^{*}}^{4} - 321,600\alpha_{Mb^{*}}^{3} + 197,280\alpha_{{Mb^{^{*}} }}^{2} - 52,800\alpha_{Mb^{*}} + 5225}}{{45(5808\alpha_{Mb^{*}}^{4} - 9792\alpha_{Mb^{*}}^{3} + 6048\alpha_{Mb^{*}}^{2} - 1632\alpha_{Mb^{*}} + 163)}},\quad {\text{for}}\; 5/ 1 2\le \alpha_{Mb^{*}} \le 1/2 \\ & = \frac{{8832\alpha_{Mb^{*}}^{4} - 29376\alpha_{Mb^{*}}^{3} + 36,288\alpha_{Mb^{*}}^{2} - 19344\alpha_{Mb^{*}} + 3647}}{{1440(\alpha_{Mb^{*}} - 1)^{3} (7\alpha_{Mb^{*}} - 3)}},\quad {\text{for}}\; 1/ 2\le \alpha_{Mb^{*}} \le 2/3 \\ & = \frac{{784\alpha_{Mb^{*}}^{4} - 4032\alpha_{Mb^{*}}^{3} + 6336\alpha_{Mb^{*}}^{2} - 3888\alpha_{Mb^{*}} + 789}}{{480(\alpha_{Mb^{*}} - 1)^{3} (7\alpha_{Mb^{*}} - 3)}},\quad {\text{for}}\; 2/ 3\le \alpha_{Mb^{*}} \le 3/4 \\ & = \frac{{15\alpha_{Mb^{*}} - 7}}{{2(7\alpha_{Mb^{*}} - 3)}},\quad {\text{for}}\; 3/ 4\le \alpha_{Mb^{*}} \le 1. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{PU}^{BR} \left( {\infty ,IAC\left( {\alpha_{{Mt^{*} }} } \right)} \right) & = \frac{13}{15},\quad {\text{for}}\; 1 / 3\le \alpha_{{Mt^{*} }} \le 3/8 \\ & = \frac{{20016\alpha_{{Mt^{*} }}^{4} - 32832\alpha_{{Mt^{*} }}^{3} + 19872\alpha_{{Mt^{*} }}^{2} - 5280\alpha_{{Mt^{*} }} + 521}}{{3(5808\alpha_{{Mt^{*} }}^{4} - 9792\alpha_{{Mt^{*} }}^{3} + 6048\alpha_{{Mt^{*} }}^{2} - 1632\alpha_{{Mt^{*} }} + 163)}},\quad {\text{for}}\; 3 / 8\le \alpha_{{Mt^{*} }} \le 2/5 \\ & = \frac{{2516\alpha_{{Mt^{*} }}^{4} - 4832\alpha_{{Mt^{*} }}^{3} + 3072\alpha_{{Mt^{*} }}^{2} - 800\alpha_{{Mt^{*} }} + 73}}{{3(5808\alpha_{{Mt^{*} }}^{4} - 9792\alpha_{{Mt^{*} }}^{3} + 6048\alpha_{{Mt^{*} }}^{2} - 1632\alpha_{{Mt^{*} }} + 163)}},\quad {\text{for}}\; 2 / 5\le \alpha_{{Mt^{*} }} \le 5/12 \\ & = \frac{{4(7071\alpha_{{Mt^{*} }}^{4} - 12,264\alpha_{{Mt^{*} }}^{3} + 7704\alpha_{{Mt^{*} }}^{2} - 2100\alpha_{{Mt^{*} }} + 211}}{{9(5808\alpha_{{Mt^{*} }}^{4} - 9792\alpha_{{Mt^{*} }}^{3} + 6048\alpha_{{Mt^{*} }}^{2} - 1632\alpha_{{Mt^{*} }} + 163)}},\quad {\text{for}}\; 5 / 1 2\le \alpha_{{Mt^{*} }} \le 1/2 \\ & = \frac{{3825\alpha_{{Mt^{*} }}^{4} - 8316\alpha_{{Mt^{*} }}^{3} + 6534\alpha_{{Mt^{*} }}^{2} - 2172\alpha_{{Mt^{*} }} + 257}}{{72(1 - \alpha_{{Mt^{*} }} )^{3} (7\alpha_{{Mt^{*} }} - 3)}},\quad {\text{for}}\; 1 / 2\le \alpha_{{Mt^{*} }} \le 3/5 \\ & = \frac{{225\alpha_{{Mt^{*} }}^{4} - 648\alpha_{{Mt^{*} }}^{3} + 702\alpha_{{Mt^{*} }}^{2} - 336\alpha_{{Mt^{*} }} + 59}}{{9(\alpha_{{Mt^{*} }} - 1)^{3} (7\alpha_{{Mt^{*} }} - 3)}},\quad {\text{for}}\; 3 / 5\le \alpha_{{Mt^{*} }} \le 2/3 \\ & = 1,\quad {\text{for}}\; 2 / 3\le \alpha_{{Mt^{*} }} \le 1. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{PC}^{BR} \left( {\infty ,IAC\left( {\alpha_{Mc^{*}} } \right)} \right) & = \frac{37}{42},\quad {\text{for}}\; 1 / 3\le \alpha_{Mc^{*}} \le 3/8 \\ & = \frac{{6294\alpha_{Mc^{*}}^{4} - 10,440\alpha_{Mc^{*}}^{3} + 6372\alpha_{Mc^{*}}^{2} - 1704\alpha_{Mc^{*}} + 169}}{{3\left( {1828\alpha_{Mc^{*}}^{4} - 3120\alpha_{Mc^{*}}^{3} + 1944\alpha_{Mc^{*}}^{2} - 528\alpha_{Mc^{*}} + 53} \right)}},\quad {\text{for}}\; 3 / 8\le \alpha_{Mc^{*}} \le 2/5 \\ & = \frac{{\left( {3\alpha_{Mc^{*}} - 1} \right)(1473\alpha_{Mc^{*}}^{3} - 1989\alpha_{Mc^{*}}^{2} + 861\alpha_{Mc^{*}} - 121)}}{{3\left( {1828\alpha_{Mc^{*}}^{4} - 3120\alpha_{Mc^{*}}^{3} + 1944\alpha_{Mc^{*}}^{2} - 528\alpha_{Mc^{*}} + 53} \right)}},\quad {\text{for}}\; 2 / 5\le \alpha_{Mc^{*}} \le 5/12 \\ & = \frac{{111,834\alpha_{Mc^{*}}^{4} - 188,640\alpha_{Mc^{*}}^{3} + 115,560\alpha_{Mc^{*}}^{2} - 30,720\alpha_{Mc^{*}} + 3005}}{{90\left( {1828\alpha_{Mc^{*}}^{4} - 3120\alpha_{Mc^{*}}^{3} + 1944\alpha_{Mc^{*}}^{2} - 528\alpha_{Mc^{*}} + 53} \right)}},\quad {\text{for}}\; 5 / 1 2\le \alpha_{Mc^{*}} \le 1/2 \\ & = \frac{{18,966\alpha_{Mc^{*}}^{4} - 36,816\alpha_{Mc^{*}}^{3} + 22,500\alpha_{Mc^{*}}^{2} - 3648\alpha_{Mc^{*}} - 329}}{{360(1 - \alpha_{Mc^{*}} )^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 1 / 2\le \alpha_{Mc^{*}} \le 3/5 \\ & = \frac{{9159\alpha_{Mc^{*}}^{4} - 30,684\alpha_{Mc^{*}}^{3} + 38,250\alpha_{Mc^{*}}^{2} - 20,652\alpha_{Mc^{*}} + 3974}}{{360(\alpha_{Mc^{*}} - 1)^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 3 / 5\le \alpha_{Mc^{*}} \le 2/3 \\ & = \frac{{893\alpha_{Mc^{*}}^{4} - 4468\alpha_{Mc^{*}}^{3} + 6990\alpha_{Mc^{*}}^{2} - 4324\alpha_{Mc^{*}} + 898}}{{120(\alpha_{Mc^{*}} - 1)^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 2 / 3\le \alpha_{Mc^{*}} \le 3/4 \\ & = \frac{{3709\alpha_{Mc^{*}} - 1789}}{{120(29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 3 / 4\le \alpha_{Mc^{*}} \le 1. \\ \end{aligned} $$

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Condorcet efficiency of PER

$$ CE_{SD}^{PER} \left( {\infty ,IAC\left( {\alpha_{Mt} } \right)} \right) = 1,\quad {\text{for}}\;0 \le \alpha_{Mt} \le 1/3. $$

(53)

$$ \begin{aligned} CE_{CP}^{PER} \left( {\infty ,IAC\left( {\alpha_{Mc} } \right)} \right) & = \frac{{3951\alpha_{Mc}^{3} - 3960\alpha_{Mc}^{2} + 1287\alpha_{Mc} - 137}}{{9\left( {457\alpha_{Mc}^{3} - 46\alpha_{Mc}^{2} 0 + 150\alpha_{Mc} - 16} \right)}},\quad {\text{for}}\;0 \le \alpha_{Mc} \le 1/6 \\ & = \frac{{83,160\alpha_{Mc}^{4} - 85,536\alpha_{Mc}^{3} + 28,080\alpha_{Mc}^{2} - 2928\alpha_{Mc} - 17}}{{216\alpha_{Mc} \left( {457\alpha_{Mc}^{3} - 460\alpha_{Mc}^{2} + 150\alpha_{Mc} - 16} \right)}},\quad {\text{for}}\;1/6 \le \alpha_{Mc} \le 1/4 \\ & = \frac{181}{189},\quad {\text{for}}\;1/4 \le \alpha_{Mc} \le 1/3. \\ \end{aligned} $$

(54)

$$ \begin{aligned} CE_{NU}^{PER} \left( {\infty ,IAC\left( {\alpha_{Mb^{*}} } \right)} \right) & = \frac{134}{135},\quad {\text{for}}\; 1 / 3\ge \alpha_{Mb^{*}} 3/8 \\ & = \frac{{158,112\alpha_{Mb^{*}}^{4} - 266,112\alpha_{Mb^{*}}^{3} + 164,160\alpha_{Mb^{*}}^{2} - 44,256\alpha_{Mb^{*}} + 4417}}{{27(5808\alpha_{Mb^{*}}^{4} - 9792\alpha_{Mb^{*}}^{3} + 6048\alpha_{Mb^{*}}^{2} - 1632\alpha_{Mb^{*}} + 163)}},\quad {\text{for}}\; 1 / 2\ge \alpha_{{Mb^{^{*}} }} 2/3 \\ & = \frac{{928\alpha_{Mb^{*}}^{4} - 3072\alpha_{Mb^{*}}^{3} + 3744\alpha_{Mb^{*}}^{2} - 1968\alpha_{Mb^{*}} + 369}}{{96(\alpha_{Mb^{*}} - 1)^{3} (7\alpha_{Mb^{*}} - 3)}},\quad {\text{for}}\; 2 / 3\ge \alpha_{Mb^{*}} 3/4 \\ & = 1,\quad {\text{for}}\; 3 / 4\ge \alpha_{Mb^{*}} 1. \\ \end{aligned} $$

(55)

$$ CE_{PU}^{PER} \left( {\infty ,IAC\left( {\alpha_{Mt^{*}} } \right)} \right) = 1,\quad {\text{for}}\;0 \le \alpha_{Mt^{*}} \le 1/3 $$

(56)

$$ \begin{aligned} CE_{PC}^{PER} \left( {\infty ,IAC\left( {\alpha_{Mc^{*}} } \right)} \right) & = \frac{184}{189},\quad {\text{for}}\; 1 / 3\le \alpha_{Mc^{*}} \le 3/8 \\ & = \frac{{50,976\alpha_{Mc^{*}}^{4} - 86,400\alpha_{Mc^{*}}^{3} + 53,568\alpha_{Mc^{*}}^{2} - 14496\alpha_{Mc^{*}} + 1451}}{{27\left( {1828\alpha_{Mc^{*}}^{4} - 3120\alpha_{Mc^{*}}^{3} + 1944\alpha_{Mc^{*}}^{2} - 528\alpha_{Mc^{*}} + 53} \right)}},\quad {\text{for}}\; 3 / 8\le \alpha_{Mc^{*}} \le 1/2 \\ & = \frac{{2136\alpha_{Mc^{*}}^{4} - 6912\alpha_{Mc^{*}}^{3} + 8208\alpha_{Mc^{*}}^{2} - 4212\alpha_{Mc^{*}} + 775}}{{54(\alpha_{Mc^{*}} - 1)^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 1 / 2\le \alpha_{Mc^{*}} \le 2/3 \\ & = \frac{{(8\alpha_{Mc^{*}}^{2} - 12\alpha_{Mc^{*}} + 3)(8\alpha_{Mc^{*}}^{2} - 24\alpha_{Mc^{*}} + 15)}}{{18(\alpha_{Mc^{*}} - 1)^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 2 / 3\le \alpha_{Mc^{*}} \le 3/4 \\ & = \frac{{16(2\alpha_{Mc^{*}} - 1)}}{{29\alpha_{Mc^{*}} - 13}},\quad {\text{for}}\; 3 / 4\le \alpha_{Mc^{*}} \le 1. \\ \end{aligned} $$

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Condorcet efficiency of NPER

$$ \begin{aligned} CE_{SD}^{NPER} \left( {\infty ,IAC\left( {\alpha_{Mt} } \right)} \right) & = \frac{{2\left( {927\alpha_{Mt}^{3} - 936\alpha_{Mt}^{2} + 315\alpha_{Mt} - 35} \right)}}{{9\left( {7\alpha_{Mt} - 2} \right)\left( {29\alpha_{Mt}^{2} - 22\alpha_{Mt} + 4} \right)}},\quad {\text{for}}\;0 \le \alpha_{Mt} \le 1/6 \\ & = \frac{{5562\alpha_{Mt}^{4} - 5832\alpha_{Mt}^{3} + 1998\alpha_{Mt}^{2} - 228\alpha_{Mt} + 1}}{{27\alpha_{Mt} \left( {7\alpha_{Mt} - 2} \right)\left( {29\alpha_{Mt}^{2} - 22\alpha_{Mt} + 4} \right)}},\quad {\text{for}}\;1/6 \le \alpha_{Mt} \le 1/4 \\ & = \frac{134}{135},\quad {\text{for}}\;1/4 \le \alpha_{Mt} \le 1/3. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{CP}^{NPER} \left( {\infty ,IAC\left( {\alpha_{Mc} } \right)} \right) & = \frac{{8\left( {369\alpha_{Mc}^{3} - 414\alpha_{Mc}^{2} + 144\alpha_{Mc} - 16} \right)}}{{9\left( {457\alpha_{Mc}^{3} - 460\alpha_{Mc}^{2} + 150\alpha_{Mc} - 16} \right)}},\quad {\text{for}}\;0 \le \alpha_{Mc} \le 1/6 \\ & = \frac{{12,744\alpha_{Mc}^{4} - 12,960\alpha_{Mc}^{3} + 4320\alpha_{Mc}^{2} - 492\alpha_{Mc} + 5}}{{27\alpha_{Mc} \left( {457\alpha_{Mc}^{3} - 460\alpha_{Mc}^{2} + 150\alpha_{Mc} - 16} \right)}},\quad {\text{for}}\;1/6 \le \alpha_{Mc} \le 1/4 \\ & = \frac{184}{189},\quad {\text{for}}\;1/4 \le \alpha_{Mc} \le 1/3. \\ \end{aligned} $$

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$$ CE_{NU}^{NPER} \left( {\infty ,IAC\left( {\alpha_{Mb^{*}} } \right)} \right) = 1,\quad 0 \le \alpha_{Mb^{*}} \le 1/3. $$

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$$ \begin{aligned} CE_{PU}^{NPER} \left( {\infty ,IAC\left( {\alpha_{{Mt^{*} }} } \right)} \right) & = \frac{131}{135},\quad {\text{for}}\; 1 / 3\le \alpha_{Mt*} \le 3/8 \\ & = \frac{{162,000\alpha_{Mt^{*}}^{4} - 271,296\alpha_{Mt^{*}}^{3} + 166,752\alpha_{Mt^{*}}^{2} - 44,832\alpha_{Mt^{*}} +\, 4465}}{{27\left( {5808\alpha_{Mt^{*}}^{4} - 9792\alpha_{Mt^{*}}^{3} + 6048\alpha_{Mt^{*}}^{2} - 1632 \propto + 163} \right)}},\quad {\text{for}}\; 3 / 8\le \alpha_{{Mt^{^{*}} }} \le 1/2 \\ & = \frac{{297\alpha_{Mt^{*}}^{4} - 540\alpha_{Mt^{*}}^{3} + 270\alpha_{Mt^{*}}^{2} - 17}}{{27\left( {1 - \alpha_{Mt^{*}} } \right)^{3} (7\alpha_{Mt^{*}} - 3)}},\quad {\text{for}}\; 1 / 2\le \alpha_{Mt^{*}} \le 2/3 \\ & = 1,\quad {\text{for}}\; 2 / 3\le \alpha_{Mt^{*}} \le 1. \\ \end{aligned} $$

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$$ \begin{aligned} CE_{PC}^{NPER} \left( {\infty ,IAC\left( {\alpha_{Mc^{*}} } \right)} \right) & = \frac{181}{189},\quad {\text{for}}\; 1 / 3\le \alpha_{Mc^{*}} \le 3/8 \\ & = \frac{{51,948\alpha_{Mc^{*}}^{4} - 87,696\alpha_{Mc^{*}}^{3} + 54,216\alpha_{Mc^{*}}^{2} - 14,640\alpha_{Mc^{*}} + 1463}}{{27\left( {1828\alpha_{Mc^{*}}^{4} - 3120\alpha_{Mc^{*}}^{3} + 1944\alpha_{Mc^{*}}^{2} - 528\alpha_{Mc^{*}} + 53} \right)}},\quad {\text{for}}\; 3 / 8\le \alpha_{Mc^{*}} \le 1/2 \\ & = \frac{{327\alpha_{Mc^{*}}^{4} - 1308\alpha_{Mc^{*}}^{3} + 1854\alpha_{Mc^{*}}^{2} - 1092\alpha_{Mc^{*}} + 221}}{{27(\alpha_{Mc^{*}} - 1)^{3} (29\alpha_{Mc^{*}} - 13)}},\quad {\text{for}}\; 1 / 2\le \alpha_{Mc^{*}} \le 2/3 \\ \end{aligned} $$

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