Monotonicity paradoxes in three-candidate elections using scoring elimination rules

Abstract

Scoring elimination rules (SER), that give points to candidates according to their rank in voters’ preference orders and eliminate the candidate(s) with the lowest number of points, constitute an important class of voting rules. This class of rules, that includes some famous voting methods such as Plurality Runoff or Coombs Rule, suffers from a severe pathology known as monotonicity paradox or monotonicity failure, that is, getting more points from voters can make a candidate a loser and getting fewer points can make a candidate a winner. In this paper, we study three-candidate elections and we identify, under various conditions, which SER minimizes the probability that a monotonicity paradox occurs. We also analyze some strategic aspects of these monotonicity failures. The probability model on which our results are based is the impartial anonymous culture condition, often used in this kind of study.

Appendices

Appendix 1: Analytical representations for the vulnerability of scoring elimination rules ($\mathbf {F}_{\mathbf {\lambda }})$ to monotonicity paradoxes with 3 candidates and large electorate

R1. For $\lambda \in \left[ 0,\frac{1}{2} \right]$,

$$\begin{aligned}&Pr\left( MLP,{{F}_{\lambda }},\infty \right) \\&\quad =\frac{1152{{\lambda }^{15}}-8232{{\lambda }^{14}}+34154{{\lambda }^{13}}-140310{{\lambda }^{12}}+364162{{\lambda }^{11}}-192564{{\lambda }^{10}}-1235457{{\lambda }^{9}}+2958975{{\lambda }^{8}}}{1728\left( 2+\lambda \right) \left( 1+\lambda \right) {{\left( 1-\lambda \right) }^{3}}{{\left( 2-\lambda \right) }^{4}}{{\left( 2-3\lambda \right) }^{2}}\left( 2+\lambda -4{{\lambda }^{2}} \right) }\\&\qquad +\frac{-2347767{{\lambda }^{7}}-422931{{\lambda }^{6}}+2009862{{\lambda }^{5}}-1213672{{\lambda }^{4}}-1032{{\lambda }^{3}}+310960{{\lambda }^{2}}-137280\lambda +19968}{1728\left( 2+\lambda \right) \left( 1+\lambda \right) {{\left( 1-\lambda \right) }^{3}}{{\left( 2-\lambda \right) }^{4}}{{\left( 2-3\lambda \right) }^{2}}\left( 2+\lambda -4{{\lambda }^{2}} \right) }. \end{aligned}$$

For $\lambda \in \left[ \frac{1}{2},1 \right]$ ,

$$\begin{aligned}&Pr\left( MLP,{{F}_{\lambda }},\infty \right) \\&\quad =\frac{640{{\lambda }^{13}}-10976{{\lambda }^{12}}+73691{{\lambda }^{11}}-270204{{\lambda }^{10}}+619225{{\lambda }^{9}}-949717{{\lambda }^{8}}+1006822{{\lambda }^{7}}-740377{{\lambda }^{6}}}{432{{\lambda }^{3}}{{\left( 2-\lambda \right) }^{3}}\left( 5\lambda -2 \right) \left( 4{{\lambda }^{2}}-3\lambda +2 \right) \left( 2{{\lambda }^{2}}-4\lambda +3 \right) }\\&\qquad +\frac{363979{{\lambda }^{5}}-102657{{\lambda }^{4}}+3872{{\lambda }^{3}}+8470{{\lambda }^{2}}-2864\lambda +312}{432{{\lambda }^{3}}{{\left( 2-\lambda \right) }^{3}}\left( 5\lambda -2 \right) \left( 4{{\lambda }^{2}}-3\lambda +2 \right) \left( 2{{\lambda }^{2}}-4\lambda +3 \right) }. \end{aligned}$$

R2. For $\lambda \in \left[ 0,\frac{1}{2} \right]$,

$$\begin{aligned}&Pr\left( LMP,F_{\lambda },\infty \right) \nonumber \\&\quad =\frac{64\lambda ^{7}+128\lambda ^{6}-92\lambda ^{5}-148\lambda ^{4}-652\lambda ^{3}+1411\lambda ^{2}-911\lambda +204}{5184(2-\lambda )\left( 1-\lambda \right) ^{4}}. \end{aligned}$$

For $\lambda \in \left[ \frac{1}{2},1 \right]$,

$$\begin{aligned} Pr\left( LMP,F_{\lambda } ,\infty \right) =\frac{24\lambda ^{7}-180\lambda ^{6}+450\lambda ^{5}-658\lambda ^{4}+748\lambda ^{3}-363\lambda ^{2}+62\lambda +1}{1296(2-\lambda )\lambda ^{3}}. \end{aligned}$$

R3. For $\lambda \in \left[ 0,\frac{1}{2} \right]$,

$$\begin{aligned}&\Pr \left( MLP+LMP,{{F}_{\lambda }},\infty ~ \right) \\&\quad =\frac{21924{{\lambda }^{17}}-190404{{\lambda }^{16}}+574470{{\lambda }^{15}}-137734{{\lambda }^{14}}-3612736{{\lambda }^{13}}+10010609{{\lambda }^{12}}-9414159{{\lambda }^{11}}-9253752{{\lambda }^{10}}+39905950{{\lambda }^{9}}}{5184\left( 2+\lambda \right) \left( 1+\lambda \right) {{\left( 1-\lambda \right) }^{4}}{{\left( 2-\lambda \right) }^{4}}{{\left( 2-3\lambda \right) }^{3}}\left( 2-5\lambda +2{{\lambda }^{2}}+2{{\lambda }^{3}} \right) }\\&\qquad +\frac{-58786139{{\lambda }^{8}}+50542143{{\lambda }^{7}}-25453976{{\lambda }^{6}}+4420724{{\lambda }^{5}}+3523456{{\lambda }^{4}}-3107872{{\lambda }^{3}}+1168928{{\lambda }^{2}}-231040\lambda +19584}{5184\left( 2+\lambda \right) \left( 1+\lambda \right) {{\left( 1-\lambda \right) }^{4}}{{\left( 2-\lambda \right) }^{4}}{{\left( 2-3\lambda \right) }^{3}}\left( 2-5\lambda +2{{\lambda }^{2}}+2{{\lambda }^{3}} \right) }. \end{aligned}$$

For $\lambda \in \left[ \frac{1}{2},1 \right] ,$

$$\begin{aligned}&\Pr \left( MLP+LMP,{{F}_{\lambda }},\infty ~ \right) \\&\quad =\frac{97678{{\lambda }^{15}}-943184{{\lambda }^{14}}+4303838{{\lambda }^{13}}-12343296{{\lambda }^{12}}+24959502{{\lambda }^{11}}-37820113{{\lambda }^{10}}+44503368{{\lambda }^{9}}}{2592{{\lambda }^{3}}{{\left( 2-\lambda \right) }^{3}}\left( 2-5\lambda \right) \left( 2-3\lambda +4{{\lambda }^{2}} \right) \left( 3-4\lambda +2{{\lambda }^{2}} \right) {{\left( 1-2\lambda +3{{\lambda }^{2}} \right) }^{2}}}\\&\qquad +\frac{-41522415{{\lambda }^{8}}+31040508{{\lambda }^{7}}-18630170{{\lambda }^{6}}+8920492{{\lambda }^{5}}-3352717{{\lambda }^{4}}+959169{{\lambda }^{3}}-197544{{\lambda }^{2}}+26204\lambda -1680}{2592{{\lambda }^{3}}{{\left( 2-\lambda \right) }^{3}}\left( 2-5\lambda \right) \left( 2-3\lambda +4{{\lambda }^{2}} \right) \left( 3-4\lambda +2{{\lambda }^{2}} \right) {{\left( 1-2\lambda +3{{\lambda }^{2}} \right) }^{2}}}. \end{aligned}$$

R4. For $\lambda \in \left[ {0,}\frac{{1}}{{2}} \right]$,

$$\begin{aligned}&Pr\left( GMP,{{F}_{\lambda }},\infty \right) \\&\quad =\frac{6912{{\lambda }^{19}}-36288{{\lambda }^{18}}+9216{{\lambda }^{17}}-340448{{\lambda }^{16}}+3419456{{\lambda }^{15}}-4955750{{\lambda }^{14}}-19346650{{\lambda }^{13}}}{5184\left( 2+\lambda \right) \left( 1+\lambda \right) {{\left( 1-\lambda \right) }^{4}}{{\left( 2-\lambda \right) }^{4}}{{\left( 2-3\lambda \right) }^{3}}\left( 2-5\lambda +2{{\lambda }^{2}}+2{{\lambda }^{3}} \right) }\\&\qquad +\frac{66083799{{\lambda }^{12}}-39105246{{\lambda }^{11}}-117106696{{\lambda }^{10}}+232985335{{\lambda }^{9}}-121302386{{\lambda }^{8}}-98661200{{\lambda }^{7}}+175869928{{\lambda }^{6}}}{5184\left( 2+\lambda \right) \left( 1+\lambda \right) {{\left( 1-\lambda \right) }^{4}}{{\left( 2-\lambda \right) }^{4}}{{\left( 2-3\lambda \right) }^{3}}\left( 2-5\lambda +2{{\lambda }^{2}}+2{{\lambda }^{3}} \right) }\\&\qquad +\frac{-89966904{{\lambda }^{5}}-3185872{{\lambda }^{4}}+26547904{{\lambda }^{3}}-13852928{{\lambda }^{2}}+3243392\lambda -304896}{5184\left( 2+\lambda \right) \left( 1+\lambda \right) {{\left( 1-\lambda \right) }^{3}}{{\left( 2-\lambda \right) }^{4}}{{\left( 2-3\lambda \right) }^{3}}\left( 4{{\lambda }^{2}}-\lambda -2 \right) \left( 2-5\lambda +2{{\lambda }^{2}}+2{{\lambda }^{3}} \right) }. \end{aligned}$$

For $\lambda \in \left[ \frac{1}{2},1 \right]$,

$$\begin{aligned}&Pr\left( GMP, {{F}_{\lambda }},\infty \right) \\&\quad =\frac{432{{\lambda }^{13}}-4680{{\lambda }^{12}}+16644{{\lambda }^{11}}-16364{{\lambda }^{10}}-34424{{\lambda }^{9}}+109559{{\lambda }^{8}}-132472{{\lambda }^{7}}+92421{{\lambda }^{6}}-36452{{\lambda }^{5}}}{2592{{\lambda }^{3}}{{\left( 2-\lambda \right) }^{3}}{{\left( 1-2\lambda +3{{\lambda }^{2}} \right) }^{2}}} \\&\qquad +\frac{5392{{\lambda }^{4}}+2196{{\lambda }^{3}}-1279{{\lambda }^{2}}+243\lambda -8}{2592{{\lambda }^{3}}{{\left( 2-\lambda \right) }^{3}}{{\left( 1-2\lambda +3{{\lambda }^{2}} \right) }^{2}}}. \\ \end{aligned}$$

R5. For $0\le \lambda \le \frac{1}{2},$

$$\begin{aligned}&Pr\left( MLP+CW,F_{\lambda },\infty \right) \\&\quad =\frac{\left( 1-2\lambda \right) ^{3}(1296\lambda ^{12}+1404\lambda ^{11}-10368\lambda ^{10}-39093\lambda ^{9}+92536\lambda ^{8}+68065\lambda ^{7}-244480\lambda ^{6}+71028\lambda ^{5}+169312\lambda ^{4}-121504\lambda ^{3}-13552\lambda ^{2}+33216\lambda -7872)}{1728\left( 1+\lambda \right) \left( 2+\lambda \right) \left( 1-\lambda \right) ^{3}\left( 2-\lambda \right) ^{2}\left( 2-3\lambda \right) ^{4}(4\lambda ^{2}-\lambda -2)}. \end{aligned}$$

For $\frac{{1}}{{2}}{\le \lambda \le 1,}$

$$\begin{aligned} Pr\left( MLP+CW, F_{\lambda },\infty \right) =\frac{\left( 1-2\lambda \right) ^{3}(2\lambda ^{2}-13\lambda -1)}{432\lambda ^{3}}. \end{aligned}$$

R6. For $0\le \lambda \le \frac{1}{2}$,

$$\begin{aligned} Pr\left( LMP+CW, F_{\lambda },\infty \right) = \frac{\left( 1-2\lambda \right) ^{2}(48\lambda ^{5}+160\lambda ^{4}+151\lambda ^{3}-123\lambda ^{2}-3216\lambda +204)}{5184\left( 2-\lambda \right) \left( 1-\lambda \right) ^{3}\left( 2-3\lambda \right) }; \end{aligned}$$

For $\frac{{1}}{{2}}{\le \lambda \le 1,}$

$$\begin{aligned} Pr\left( LMP+CW, F_{\lambda },\infty \right) = \frac{\left( 1-2\lambda \right) ^{2}(1+5\lambda +54\lambda ^{2}-18\lambda ^{3}+27\lambda ^{4}-6\lambda ^{5})}{1296\lambda ^{4}}. \end{aligned}$$

R7. For $0\le \lambda \le \frac{1}{2},$

$$\begin{aligned}&Pr\left( GMP+CW, F_{\lambda },\infty \right) \\&\quad =\frac{\left( 1-2\lambda \right) ^{2}(2592\lambda ^{13}-6264\lambda ^{12}-35640\lambda ^{11}-119577\lambda ^{10}+637043\lambda ^{9}-131110\lambda ^{8}-1661042\lambda ^{7})}{5184\left( 1+\lambda \right) \left( 2+\lambda \right) \left( 1-\lambda \right) ^{3}\left( 2-\lambda \right) ^{2}\left( 2-3\lambda \right) ^{4}(2+\lambda -4\lambda ^{2})}\\&\qquad +\frac{\left( 1-2\lambda \right) ^{2}(1637340\lambda ^{6}+633512\lambda ^{5}-1569488\lambda ^{4}+512368\lambda ^{3}+270864\lambda ^{2}-206592\lambda +36672)}{5184\left( 1+\lambda \right) \left( 2+\lambda \right) \left( 1-\lambda \right) ^{3}\left( 2-\lambda \right) ^{2}\left( 2-3\lambda \right) ^{4}(2+\lambda -4\lambda ^{2})}. \end{aligned}$$

For $\frac{{1}}{{2}}{\le \lambda \le 1,}$

$$\begin{aligned} Pr\left( GMP+CW, F_{\lambda },\infty \right) =\frac{\left( 1-2\lambda \right) ^{2}\left( 1+2\lambda +21\lambda ^{2}+66\lambda ^{3}+15\lambda ^{4}-6\lambda ^{5} \right) }{1296\lambda ^{4}}. \end{aligned}$$

R8. For $0\le \lambda \le \frac{1}{2},$

$$\begin{aligned}&Pr\left( MLP+B,F_{\lambda },\infty \right) \\&\quad =\frac{192\lambda ^{15}-1864\lambda ^{14}+6970\lambda ^{13}-16218\lambda ^{12}+36786\lambda ^{11}-46938\lambda ^{10}-94200\lambda ^{9}+422061\lambda ^{8}-516717\lambda ^{7}+96282\lambda ^{6}}{864\left( 1+\lambda \right) \left( 2+\lambda \right) \left( 3-\lambda \right) \left( 1-\lambda \right) ^{3}\left( 2-\lambda \right) ^{3}(4\lambda ^{2}-\lambda -2)(3\lambda ^{2}-7\lambda +3)}\\&\qquad +\frac{331613\lambda ^{5}-291197\lambda ^{4}+41232\lambda ^{3}+59820\lambda ^{2}-33318\lambda +5508}{864\left( 1+\lambda \right) \left( 2+\lambda \right) \left( 3-\lambda \right) \left( 1-\lambda \right) ^{3}\left( 2-\lambda \right) ^{3}(4\lambda ^{2}-\lambda -2)(3\lambda ^{2}-7\lambda +3)}. \end{aligned}$$

For $\frac{{1}}{{2}}{\le \lambda \le 1,}$

$$\begin{aligned}&Pr\left( MLP+B, F_{\lambda },\infty \right) \\&\quad =\frac{32\lambda ^{9}-448\lambda ^{8}+2183\lambda ^{7}-5282\lambda ^{6}+7180\lambda ^{5}-5588\lambda ^{4}+2261\lambda ^{3}-301\lambda ^{2}-85\lambda +24}{432\lambda ^{3}\left( 2-\lambda \right) ^{2}(4\lambda -2\lambda ^{2}-3)}. \end{aligned}$$

R9. $\hbox { For } 0\le \lambda \le \frac{1}{2}$,

$$\begin{aligned}&Pr\left( LMP+B,F_{\lambda },\infty \right) \\&\quad =\frac{16\lambda ^{8}+8\lambda ^{7}-68\lambda ^{6}+16\lambda ^{5}-144\lambda ^{4}+655\lambda ^{3}-864\lambda ^{2}+481\lambda -102}{2592\left( 1-\lambda \right) ^{4}\left( \lambda -2 \right) }. \end{aligned}$$

For $\frac{{1}}{{2}}{\le \lambda \le }\frac{{2}}{{3}},$

$$\begin{aligned}&Pr\left( LMP+B, F_{\lambda },\infty \right) \\&\quad =\frac{\left( 1-\lambda \right) (8\lambda ^{8}-92\lambda ^{7}+446\lambda ^{6}-732\lambda ^{5}+347\lambda ^{4}-83\lambda ^{3}+13\lambda ^{2}-5\lambda +2)}{1296\lambda ^{5}\left( 2-\lambda \right) }. \end{aligned}$$

For $\frac{{2}}{{3}}{\le \lambda \le 1,}$

$$\begin{aligned}&Pr\left( LMP+B, F_{\lambda },\infty \right) \\&\quad = \frac{56\lambda ^{13}-804\lambda ^{12}+4786\lambda ^{11}-15522\lambda ^{10}+30388\lambda ^{9}-37701\lambda ^{8}+30137\lambda ^{7}-15144\lambda ^{6}+4388\lambda ^{5}-475\lambda ^{4}-128\lambda ^{3}+69\lambda ^{2}-17\lambda +2}{1296\lambda ^{5}\left( 2-\lambda \right) (1-3\lambda +\lambda ^{2})}. \end{aligned}$$

R10. For $\le \lambda \le \frac{1}{2},$

$$\begin{aligned}&Pr\left( MLP+LMP+B, F_{\lambda },\infty \right) \\&\quad =\frac{13392\lambda ^{17}-151218\lambda ^{16}+710178\lambda ^{15}-1635907\lambda ^{14}+981368\lambda ^{13}+5246294\lambda ^{12}-18571308\lambda ^{11}+32814574\lambda ^{10}-37581933\lambda ^{9}+29118925\lambda ^{8}}{1296(2\lambda ^{3}+2\lambda ^{2}-5\lambda +2)\left( 1+\lambda \right) \left( 1-\lambda \right) ^{4}\left( 2-\lambda \right) ^{3}(3\lambda ^{2}-7\lambda +3)\left( 2-3\lambda \right) ^{2}\left( 4\lambda -3 \right) }\\&\qquad +\frac{-14401003\lambda ^{7}+3048779\lambda ^{6}+1455765\lambda ^{5}-1604349\lambda ^{4}+717238\lambda ^{3}-186681\lambda ^{2}+27720\lambda -11836}{1296(2\lambda ^{3}+2\lambda ^{2}-5\lambda +2)\left( 1+\lambda \right) \left( 1-\lambda \right) ^{4}\left( 2-\lambda \right) ^{3}(3\lambda ^{2}-7\lambda +3)\left( 2-3\lambda \right) ^{2}\left( 4\lambda -3 \right) }. \end{aligned}$$

For $\frac{{1}}{{2}}{\le \lambda \le }\frac{{2}}{{3}},$

$$\begin{aligned}&Pr\left( MLP+LMP+B, F_{\lambda },\infty \right) \\&\quad =\frac{\left( 1-\lambda \right) \left( 3+38\lambda -219\lambda ^{2}+403\lambda ^{3}-391\lambda ^{4}+201\lambda ^{5}-47\lambda ^{6} \right) }{1296\lambda ^{4}(\lambda -2)(2\lambda ^{2}-4\lambda +3)}. \end{aligned}$$

For $\frac{{2}}{{3}}{\le \lambda \le 1,}$

$$\begin{aligned}&Pr\left( MLP+LMP+B, F_{\lambda },\infty \right) \\&\quad =\frac{262145\lambda ^{15}-2806405\lambda ^{14}+14018622\lambda ^{13}-4332348{1\lambda }^{12}+92617090\lambda ^{11}-145109259\lambda ^{10}+172250955\lambda ^{9}-157940181\lambda ^{8}+112979163\lambda ^{7}}{1296\lambda ^{3}\left( 2-\lambda \right) ^{2}{(7\lambda ^{2}-7\lambda +2)}^{2}(5\lambda ^{2}-5\lambda +2)(2\lambda ^{2}-4\lambda +3)(5\lambda ^{2}-10\lambda +4)}\\&\qquad + \frac{-63187783\lambda ^{6}+27477878\lambda ^{5}-9151584\lambda ^{4}+2267216\lambda ^{3}-395792\lambda ^{2}+43680\lambda -2304}{1296\lambda ^{3}\left( 2-\lambda \right) ^{2}{(7\lambda ^{2}-7\lambda +2)}^{2}(5\lambda ^{2}-5\lambda +2)(2\lambda ^{2}-4\lambda +3)(5\lambda ^{2}-10\lambda +4)}. \end{aligned}$$

R11 For $0\le \lambda \le \frac{1}{2},$

$$\begin{aligned}&Pr\left( GMP+B, F_{\lambda },\infty \right) \\&\quad = \frac{13824\lambda ^{22}-43776\lambda ^{21}-365856\lambda ^{20}+1724416\lambda ^{19}-229432\lambda ^{18}-6007556\lambda ^{17}-6224444\lambda ^{16}+30150396\lambda ^{15}+80042172\lambda ^{14}}{2592\left( 3-4\lambda \right) \left( 2-3\lambda \right) ^{2}(2\lambda ^{3}+2\lambda ^{2}-5\lambda +2)\left( 2+\lambda \right) \left( 1-\lambda \right) ^{3}\left( 3-\lambda \right) \left( 1+\lambda \right) (3\lambda ^{2}-7\lambda +3)\left( 2-\lambda \right) ^{3}(4\lambda ^{2}-\lambda -2)}\\&\qquad + \frac{-342217609\lambda ^{13}+252096276\lambda ^{12}+573002396\lambda ^{11}-1338479620\lambda ^{10}+908077340\lambda ^{9}+375040562\lambda ^{8}-1105263361\lambda ^{7}}{2592\left( 3-4\lambda \right) \left( 2-3\lambda \right) ^{2}(2\lambda ^{3}+2\lambda ^{2}-5\lambda +2)\left( 2+\lambda \right) \left( 1-\lambda \right) ^{3}\left( 3-\lambda \right) \left( 1+\lambda \right) (3\lambda ^{2}-7\lambda +3)\left( 2-\lambda \right) ^{3}(4\lambda ^{2}-\lambda -2)}\\&\qquad + \frac{+791048112\lambda ^{6}-158072500\lambda ^{5}-141402888\lambda ^{4}+125915520\lambda ^{3}-46522080\lambda ^{2}+8823168\lambda -705024}{2592\left( 3-4\lambda \right) \left( 2-3\lambda \right) ^{2}(2\lambda ^{3}+2\lambda ^{2}-5\lambda +2)\left( 2+\lambda \right) \left( 1-\lambda \right) ^{3}\left( 3-\lambda \right) \left( 1+\lambda \right) (3\lambda ^{2}-7\lambda +3)\left( 2-\lambda \right) ^{3}(4\lambda ^{2}-\lambda -2)}. \end{aligned}$$

For $\frac{{1}}{{2}}{\le \lambda \le }\frac{{2}}{{3}},$

$$\begin{aligned} Pr\left( GMP+B, F_{\lambda },\infty \right) = \frac{-{8\lambda }^{10}+68\lambda ^{9}-162\lambda ^{8}+180\lambda ^{7}-353\lambda ^{6}+549\lambda ^{5}-286\lambda ^{4}+112\lambda ^{3}+42\lambda ^{2}+18\lambda -4}{1296\lambda ^{5}{(\lambda -2)}^{2}}. \end{aligned}$$

For $\frac{\mathrm {2}}{\mathrm {3}}\mathrm {\le \lambda \le 1,}$

$$\begin{aligned}&Pr\left( GMP+B, F_{\lambda },\infty \right) \\&\quad =\frac{68600\lambda ^{22}-1406300\lambda ^{21}+12829670\lambda ^{20}-69452725\lambda ^{19}+251061702\lambda ^{18}-647047199\lambda ^{17}+1240417731\lambda ^{16}-1822285490\lambda ^{15}}{1296\lambda ^{5}\left( 2-\lambda \right) ^{2}(5\lambda ^{2}-10\lambda +4)(5\lambda ^{2}-5\lambda +2){(7\lambda ^{2}-7\lambda +2)}^{2}{(\lambda ^{2}-3\lambda +1)}^{2}}\\&\qquad + \frac{2095718215\lambda ^{14}-1913651176\lambda ^{13}+1396641104{1\lambda }^{12}-811844407\lambda ^{11}+367749071\lambda ^{10}-121125045\lambda ^{9}+21583778\lambda ^{8}+4200519\lambda ^{7}}{1296\lambda ^{5}\left( 2-\lambda \right) ^{2}(5\lambda ^{2}-10\lambda +4)(5\lambda ^{2}-5\lambda +2){(7\lambda ^{2}-7\lambda +2)}^{2}{(\lambda ^{2}-3\lambda +1)}^{2}}\\&\qquad + \frac{-5309200\lambda ^{6}+2474472\lambda ^{5}-770232\lambda ^{4}+172176\lambda ^{3}-27008\lambda ^{2}+2688\lambda -128}{1296\lambda ^{5}\left( 2-\lambda \right) ^{2}(5\lambda ^{2}-10\lambda +4)(5\lambda ^{2}-5\lambda +2){(7\lambda ^{2}-7\lambda +2)}^{2}{(\lambda ^{2}-3\lambda +1)}^{2}}. \end{aligned}$$

Appendix 2: Proof of Proposition 4.2

Let x be a voting situation at which a wins against b in the runoff and loses against c after being moved up by some voters in their rankings. Then by Proposition 4.1, MLP occurs under $F_{\lambda }$ in favor of voters changing their preferences if and only if there exists $t\in \left[ 0,x_{4} \right]$ such that:

$$\begin{aligned} S_\lambda \left( x,a \right)>S_\lambda \left( x,\mathrm {c} \right) , S_\lambda \left( x,a \right) >S_\lambda \left( x,c \right) \hbox { and } aMAJ(x)b \end{aligned}$$

(MLP7)

$$\begin{aligned} S_\lambda \left( z,a \right)>S_\lambda \left( z,c \right) , S_\lambda \left( z,a \right) >S_\lambda \left( z,c \right) \hbox { and }cMAJ(z)a \end{aligned}$$

(MLP8)

with $z=(x_{1}+t,x_{2},x_{3},x_{4}-t,x_{5}+x_{6},0)$. Note that (MLP7) is equivalent to (MLP1), (MLP2) and (MLP3). Then we only have to prove that given (MLP1), (MLP2) and (MLP3), (MLP8) holds if and only if (MLP4) and (MLP5) hold for $\lambda \in \left[ 0,\frac{1}{2} \right]$; and that (MLP4) and (MLP6) hold for $\in \left[ \frac{1}{2},1 \right]$. Clearly, $S_\lambda \left( z,a \right) >S_\lambda \left( z,c \right)$ is a consequence of $S_\lambda \left( x,a \right) >S_\lambda \left( x,c \right)$ since some voters move a up from x to z. Therefore given (MLP1), (MLP2) and (MLP3), (MLP8) is now equivalent to $S_\lambda \left( z,a \right) >S_\lambda \left( z,c \right)$ and cM(z)a. That is

$$\begin{aligned} \left( 2\lambda -1 \right) t<-\lambda x_{1}+{\lambda x}_{2}-x_{3}-\left( 1-\lambda \right) x_{4}+x_{5}+x_{6}= & {} T_{1}\\ 2t <-x_{1}-x_{2}-x_{3}+x_{4}+x_{5}+x_{6}= & {} T_{2}. \end{aligned}$$

Taking into consideration the sign of the coefficient $2\lambda -1$ and the fact that $t\in ]0,x_{4}]$, it appears that t exists if and only if for $\lambda \in [ 0,\frac{1}{2} [$, max $(\frac{T_{1}}{2\lambda -1},0)<t<$ min $(\frac{T_{2}}{2},x_{4})$ and for $\lambda \in ] \frac{1}{2},1 ]$, $<t<$ min $(\frac{T_{1}}{2\lambda -1},\frac{T_{3}}{2},x_{4})$. Now, for $\lambda \in [ {0,}\frac{{1}}{{2}}[$, there exists t such that max($\frac{T_{1}}{2\lambda -1},0)<t\mathrm {<}$ min $(\frac{T_{2}}{2},x_{4})$ if and only if:

$$\begin{aligned} 0-\frac{T_{2}}{2}=\frac{q_{4}}{2}<0 \end{aligned}$$

(MLP9)

$$\begin{aligned} \frac{T_{1}}{2\lambda -1}-\frac{T_{2}}{2}=\frac{q_{5}}{2(1-2\lambda )}<0 \end{aligned}$$

(MLP10)

$$\begin{aligned} \frac{T_{1}}{2\lambda -1}-x_{4}=\frac{q_{7}}{1-2\lambda }<0 \end{aligned}$$

(MLP11)

$$\begin{aligned} \mathrm {0-}x_{\mathrm {4}}<0. \end{aligned}$$

(MLP12)

Since (MLP9) and (MLP10) are respectively equivalent to (MLP4) and (MLP5), to complete the proof for $\lambda \in [ 0,\frac{1}{2} [$, we have to prove that (MLP11) and (MLP12) can be discarded. We omit (MLP12) as it has no influence on the 5-dimensional volume computed with $0\le x_{\mathrm {4}}$. To see that (MLP12) is redundant, we simply rewrite $q_{7}$ as a sum of non positive terms:

$$\begin{aligned} q_{7}= & {} \frac{1-\lambda }{\lambda ^{2}-\lambda +1}\left( q_{1}+q_{2} \right) -\frac{\lambda ^{2}+\lambda }{\lambda ^{2}-\lambda +1}x_{2}-\frac{\lambda ^{2}+\lambda }{\lambda ^{2}-\lambda +1}x_{3}\\&+\frac{{5\lambda }^{2}-5\lambda +2}{{2(\lambda }^{2}-\lambda +1)}\frac{q_{5}}{1-2\lambda }+\frac{3\lambda (1-\lambda )}{\lambda ^{2}-\lambda +1}\frac{q_{6}}{2}<0. \end{aligned}$$

Similarly, for $\lambda \in ] \frac{1}{2},1 ]$, there exists t such that $<t\mathrm {<}min(\frac{T_{1}}{2\lambda -1},\frac{T_{2}}{2},x_{4})$ if and only if:

$$\begin{aligned} 0-\frac{T_{2}}{2}=\frac{q_{4}}{2}<0 \end{aligned}$$

(MLP13)

$$\begin{aligned} 0-\frac{T_{1}}{2\lambda -1}=\frac{q_{6}}{2\lambda -1}<0 \end{aligned}$$

(MLP14)

$$\begin{aligned} 0-x_{4}<0. \end{aligned}$$

(MLP15)

As mentioned above, (MLP15) has no influence on the evaluation of the 5-dimensional volume computed for $x_{j}\ge 0$, $j=1,\ldots , 6$. Finally (ML13) and (MLP14) are respectively equivalent to (MLP4) and (MLP6). For $\lambda =\frac{\mathrm {1}}{2}$ , (MLP1), (MLP2), (MLP3) and (MLP4) hold from (MLP7) and (MLP8). Moreover, (MLP5) and (MLP6) are now equivalent. $\square$