Examples¶

These are some example scripts to demonstrate the various simulations that can be done, and to verify the simulator by reproducing results of already-published works.

Wikipedia¶

Likelihood of a Condorcet cycle¶

Wikipedia’s analytical example of likelihood of a Condorcet paradox in an impartial culture model

	3	101	201	301	401	501	601
WP	5.556	8.690	8.732	8.746	8.753	8.757	8.760
Sim	5.553	8.682	8.748	8.743	8.747	8.759	8.752
Diff	0.003	0.008	0.016	0.003	0.006	0.002	0.008

Source: wikipedia_condorcet_paradox_likelihood.py

Niemi and Weisberg 1968, Klahr 1966¶

Niemi, R. G., & Weisberg, H. F. (1968). A mathematical solution for the probability of the paradox of voting. Behavioral Science, 13(4), 322.

Reproduction of Condorcet paradox likelihood column and table:

Table 1: Probability That There Is No Majority Winner¶

	2	3	4	5	6	7	10	23	49
Niemi	0.000	0.088	0.175	0.251	0.315	0.369	0.489	0.712	0.841
Sim	0.002	0.095	0.179	0.253	0.321	0.372	0.488	0.728	0.843
Diff	0.002	0.008	0.004	0.002	0.006	0.003	0.001	0.015	0.003

Source: niemi_1968_table_1.py

Table 2: Probabilities of No Majority Winner, P(m, n), for Equally Likely Rank Orders¶

	3	5	7	9	11	13	15	17	19	21	23	25	27	29	59
3	0.0558	0.0698	0.0755	0.0779	0.0799	0.0812	0.0819	0.0826	0.0831	0.0830	0.0842	0.0839	0.0845	0.0840	0.0864
4	0.1110	0.1388	0.1506	0.1556	0.1599	0.1625	0.1641	0.1653	0.1662	0.1678	0.1675	0.1687	0.1697	0.1697	0.1728
5	0.1599	0.1996	0.2146	0.2230	0.2288	0.2331	0.2347	0.2367	0.2387	0.2395	0.2406	0.2416	0.2421	0.2437	0.2480
6	0.2023	0.2518	0.2707	0.2814	0.2884	0.2925	0.2948	0.2974	0.2993	0.3012	0.3030	0.3038	0.3036	0.3052	0.3110

(This also covers the same values as Figure 6. of Klahr, D. (1966). A Computer Simulation of the Paradox of Voting. American Political Science Review, 60(2), 388.)

Source: niemi_1968_table_2.py

Weber 1977/1978¶

Weber, R. J. (1978). Comparison of Public Choice Systems. Cowles Foundation Discussion Papers.

Reproduction of utility and effectiveness tables and formulas:

Table 4: The expected social utility of the elected candidate¶

Reproduce Table 4 from p. 17 of Comparison of Voting Systems: The expected social utility of the elected candidate, under three voting systems.

	Standard	Borda	Approval
2	1.2505	1.2922	1.2920
3	1.8330	1.8748	1.8644
4	2.3889	2.4236	2.4210
5	2.9171	2.9768	2.9727
10	5.5970	5.6701	5.6714
15	8.2248	8.3207	8.3246
20	10.8325	10.9470	10.9537
25	13.4320	13.5583	13.5659
30	16.0177	16.1577	16.1669

Source: weber_1977_table_4.py

The Effectiveness of Several Voting Systems¶

Reproduce table from p. 19 of Reproducing Voting Systems using Monte Carlo methods (incomplete)

	Standard	Vote-for-half	Borda
2	81.37	81.71	81.41
3	75.10	75.00	86.53
4	69.90	79.92	89.47
5	65.02	79.09	91.34
6	61.08	81.20	92.61
10	50.78	82.94	95.35
255	12.78	86.37	99.80

Source: weber_1977_effectiveness_table.py

Closed-form effectiveness expressions¶

Reproduce table from p. 19 of Reproducing Voting Systems using Weber’s closed-form expressions to verify them.

m	Standard	Vote-for-half	Best Vote-for-or-against-k	Borda
2	81.65%	81.65%	81.65%	81.65%
3	75.00%	75.00%	87.50%	86.60%
4	69.28%	80.00%	80.83%	89.44%
5	64.55%	79.06%	86.96%	91.29%
6	60.61%	81.32%	86.25%	92.58%
10	49.79%	82.99%	88.09%	95.35%
∞	0.00%	86.60%	92.25%	100.00%

Source: weber_1977_expressions.py

Vote-for-k effectiveness¶

Compare Monte Carlo method with closed-form expressions for Vote-for-k method

	1	2	3	4	half
2	81.6				81.6
3	75.0	75.2			75.0
4	69.4	80.1	69.2		80.1
5	64.4	79.1	79.1	64.9	79.1
6	60.8	76.6	81.4	76.6	81.4
7	57.1	74.0	81.1	81.1	81.1
8	54.4	71.3	79.7	82.3	82.3
9	51.8	68.5	77.8	82.0	82.0
10	49.7	66.4	75.9	81.5	83.0

Source: weber_1977_verify_vote_for_k.py

Merrill 1984¶

Merrill, S. (1984). A Comparison of Efficiency of Multicandidate Electoral Systems. American Journal of Political Science, 28(1), 23–48.

Reproduction of Condorcet efficiency (CE) and social utility efficiency (SUE) tables and figures:

Table 1 / Fig. 1: CE random society¶

Method	2	3	4	5	7	10
Plurality	100.0	79.1	68.4	61.8	51.4	41.1
Runoff	100.0	96.2	89.6	83.4	72.3	60.3
Hare	100.0	96.2	92.5	89.1	83.7	77.0
Approval	100.0	75.6	70.0	67.4	63.8	61.2
Borda	100.0	90.9	87.4	85.9	84.6	83.9
Coombs	100.0	96.9	93.4	91.0	86.4	81.7
Black	100.0	100.0	100.0	100.0	100.0	100.0
SU max	100.0	84.1	79.6	78.4	77.3	77.5
CW	100.0	91.7	83.1	75.6	64.3	52.9

Source: merrill_1984_table_1_fig_1.py

Table 2: CE spatial model¶

(Markdown can’t handle colspan or multiple header rows)

Disp	1.0	1.0	1.0	1.0	0.5	0.5	0.5	0.5
Corr	0.5	0.5	0.0	0.0	0.5	0.5	0.0	0.0
Dims	2	4	2	4	2	4	2	4
Plurality	57.5	65.8	62.2	78.4	21.7	24.4	27.2	41.3
Runoff	80.1	87.3	81.6	93.6	35.4	42.2	41.5	61.5
Hare	79.2	86.7	84.0	95.4	35.9	46.8	41.0	69.9
Approval	73.8	77.8	76.9	85.4	71.5	76.4	73.8	82.7
Borda	87.1	89.3	88.2	92.3	83.7	86.3	85.2	89.4
Coombs	97.8	97.3	97.9	98.2	93.5	92.3	93.8	94.5
Black	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0
SU max	82.9	85.8	85.3	90.8	78.1	81.5	80.8	87.1
CW	99.7	99.7	99.7	99.6	98.9	98.6	98.7	98.5

Source: merrill_1984_table_2.py

Fig 2a 2b: Spatial model scatter plots¶

Source: merrill_1984_fig_2a_2b.py

Fig 2c 2d: CE spatial model¶

2.c¶

Method	2	3	4	5	6	7
Black	100.0	100.0	100.0	100.0	100.0	100.0
Coombs	100.0	99.4	98.6	97.8	96.9	96.0
Borda	100.0	91.4	89.2	87.1	85.7	84.6
Approval	100.0	85.9	79.8	73.9	70.1	66.8
Hare	100.0	94.1	86.6	78.9	71.7	65.2
Runoff	100.0	94.1	87.1	79.7	72.8	66.1
Plurality	100.0	80.6	67.6	57.4	49.3	42.6

2.d¶

Method	2	3	4	5	6	7
Black	100.0	100.0	100.0	100.0	100.0	100.0
Coombs	100.0	98.2	95.9	93.4	90.9	88.4
Borda	100.0	89.2	86.3	83.8	82.1	80.8
Approval	100.0	84.0	76.9	71.5	67.8	64.7
Hare	100.0	72.2	50.3	35.8	26.0	19.7
Runoff	100.0	72.2	50.6	35.3	24.4	16.9
Plurality	100.0	55.9	34.7	21.5	13.5	8.5

Source: merrill_1984_fig_2c_2d.py

Table 3 / Fig 3: SUE random society¶

Method	2	3	4	5	7	10
Plurality	100.0	83.3	75.1	69.5	62.5	54.9
Runoff	100.0	89.1	83.9	80.4	75.1	69.1
Hare	100.0	89.0	84.8	82.5	79.9	77.3
Approval	100.0	95.5	91.3	89.3	87.8	86.8
Borda	100.0	94.7	94.3	94.4	95.3	96.2
Coombs	100.0	90.2	86.8	85.2	84.0	82.9
Black	100.0	92.9	92.0	92.1	93.2	94.6

Source: merrill_1984_table_3_fig_3.py

Table 4: SUE spatial model¶

(Markdown can’t handle colspan or multiple header rows)

Disp	1.0	1.0	1.0	1.0	0.5	0.5	0.5	0.5
Corr	0.5	0.5	0.0	0.0	0.5	0.5	0.0	0.0
Dims	2	4	2	4	2	4	2	4
Plurality	72.1	79.1	80.4	92.4	4.0	6.3	25.2	52.9
Runoff	90.5	94.2	92.0	97.5	36.6	43.6	53.3	75.3
Hare	91.7	94.7	94.3	98.4	46.4	57.7	58.7	83.6
Approval	96.2	97.0	96.8	98.5	95.6	96.8	95.8	98.0
Borda	97.8	98.6	98.3	99.4	96.6	97.7	97.4	99.0
Coombs	97.0	97.5	97.7	98.7	94.0	94.3	95.0	96.7
Black	97.3	97.8	98.0	99.0	95.5	96.1	96.5	98.0

Source: merrill_1984_table_4.py

Fig 4a 4b: SUE spatial model¶

4.a¶

Method	2	3	4	5	7
Black	100.0	97.1	97.1	97.3	97.8
Coombs	100.0	97.0	96.8	97.0	97.4
Borda	100.0	98.7	98.2	97.9	97.7
Approval	100.0	98.7	97.4	96.2	95.3
Hare	100.0	94.1	92.7	91.7	90.2
Runoff	100.0	94.1	92.0	90.5	87.4
Plurality	100.0	84.4	77.3	72.0	64.8

4.b¶

Method	2	3	4	5	7
Black	100.0	95.4	95.2	95.5	96.1
Coombs	100.0	94.9	94.1	94.0	94.1
Borda	100.0	97.9	97.1	96.6	96.2
Approval	100.0	98.5	96.8	95.5	94.5
Hare	100.0	70.2	55.8	46.7	34.7
Runoff	100.0	70.1	51.4	36.8	13.6
Plurality	100.0	50.0	23.2	4.0	-24.7

Source: merrill_1984_fig_4a_4b.py

Winner distribution plots¶

Somewhat similar to:

Edith Elkind, Piotr Faliszewski, et al. (2019) What Do Multiwinner Voting Rules Do?
Kiran Tomlinson, Johan Ugander, Jon Kleinberg (2023) Moderation in instant runoff voting

Tomlinson 2023¶

Kiran Tomlinson, Johan Ugander, Jon Kleinberg (2023) Moderation in instant runoff voting

Figure 3¶

Source: tomlinson_2023_figure_3.py

Figure 3 comparison with other systems¶

Source: tomlinson_2023_figure_3_updated.py