Multi-dimensional Spatial Voting Simulations

John Huang, 25 May 2020

Executive Summary

This report documents simulation results for multi-dimensional spatial election models. In these simulations:

  • The number of candidates was varied from 3, 4, 5, 7, and 9.

  • Candidates were uniformly generated within 1.5 standard deviations of voter population preferences.

  • The number of preference dimensions was varied from 1 to 5.

  • All elections had 101 voters generated from a normal distribution.

  • All voters were assumed to grade candidates relative to their best and worst candidate.

Other details about the voting model and assessment methodology are presented in the previous report for 3-way elections.

Assessed systems include:

These new systems are compared to the traditional plurality voting system.

From these simulations we can draw the following conclusions:

  1. For 1-dimensional models, Condorcet methods such as ranked pairs and smith-minimax have superior performance.

  2. Increasing preference dimensions from 1 to 2 increases the prevalence of Condorcet Paradoxes from 0% to 5.15%.

  3. Increasing preference dimensions reduces Condorcet method satisfaction.

  4. Cardinal methods have excellent multi-dimensional performance. At 5 dimensions, score is the best performing system.

  5. The best, well-rounded methods are hybrid methods such as STAR voting and Smith-Minimax for 1-dimensional, 2-dimensional and multi-dimensional performance.

1-Dimensional vs 2-Dimensional Models

Newly generated scenario categorization for 1 and 2 preference dimensions are shown below. The following observations can be made:

  • The transition to 2-dimensions increases the occurrence of Condorcet Paradoxes from 0% to 5.152%.

  • The transition to 2-dimensions also increases the occurrence of Condorcet failures, where the Condorcet winner is not coincident with the Utility winner (Scenarios C, CP, PU, and M) from 3.75% to 12.28%

  • The transition to 2-dimensions changes the most likely scenario from “CU” (49.824% to 34.526% of scenarios) to “CPU” (27.736% to 36.58% of scenarios).


Figure 1: Scenario category probabilities for 1-Dimensional Spatial Model


Figure 2: Scenario category probabilities for 2-Dimensional Spatial Model

Due to increasing Condorcet paradoxes and Condorcet failures, Cardinal and Hybrid voting systems have superior performance in multi-dimensional problems. Voter regrets, normalized as in the previous report, are plotted below in Figure 3 and 4 for 1 and 2 dimensional models.


Figure 3: Voter regrets for 1-Dimensional Spatial Model


Figure 4: Voter regrets for 2-Dimensional Spatial Model

Increasing Preference Dimensionality to 3, 4, and 5

Further increasing preference dimensions to 3, 4, and 5 dimensions increases the prevalence of the easiest-to-solve scenario category “CPU”, where the Condorcet, plurality, and utility winner are coincident. All other more complex scenario prevalences are reduced with each additional dimension.

Therefore for a “conservative” (in terms of engineering risk aversion) assessment of a voter method, it ought to be sufficient to only assess the 1-dimensional and 2-dimensional cases which contain the greatest occurrences of voting system failure scenarios.


Figure 5: Occurrences of Scenarios vs Model Preference Dimensions


Figure 4: Voter regrets for 3-Dimensional Spatial Model


Figure 4: Voter regrets for 4-Dimensional Spatial Model


Figure 4: Voter regrets for 5-Dimensional Spatial Model

The Effect of Greater Number of Candidates

As the number of candidates increases, the likelihood of Condorcet failures tend to increase. For 2-dimensional models, the likelihood that the Condorcet winner is not the utility winner increases from 8.47% to 10.93% from 3 to 9 candidates. The likelihood of a Condorcet Paradox increases from 1.09% to 5.37%.


Figure 6: Occurrences of Scenarios vs # of Candidates


STAR voting or a similar hybrid cardinal method is recommended for multi-dimensional problems with higher occurrences of Condorcet Cycles. Condorcet methods remain well suited for 1-dimensionally polarized elections. Score also has excellent performance, assuming no tactical voting. Under min-max strategy equivalent to approval25 or approval50 voting, regret may be significantly increased.