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 :doc:`for 3-way elections <../simple3way/simple3way>`.

 
Assessed systems include:

 - `Approval voting <https://en.wikipedia.org/wiki/Approval_voting>`_
 - `Instant runoff voting (irv) <https://en.wikipedia.org/wiki/Instant-runoff_voting>`_
 - `Scored voting <https://en.wikipedia.org/wiki/Score_voting>`_
 - `STAR voting <https://en.wikipedia.org/wiki/STAR_voting>`_
 - `Ranked pairs <https://en.wikipedia.org/wiki/Ranked_pairs>`_
 - `Smith minimax <https://electowiki.org/wiki/Smith//Minimax>`_
 - `Smith score <https://electowiki.org/wiki/Smith//Score>`_

These new systems are compared to the traditional 
`plurality voting <https://en.wikipedia.org/wiki/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 <https://en.wikipedia.org/wiki/Condorcet_paradox>`_
   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:: scenario-categories-1.png
    :scale: 60 %
    :align: center   
    
    Figure 1: Scenario category probabilities for 1-Dimensional Spatial Model
   
.. figure:: scenario-categories-2.png
    :scale: 60 %
    :align: center   
    
    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 
:doc:`the previous report <../simple3way/simple3way>`, 
are plotted below in Figure 3 and 4 
for 1 and 2 dimensional models.

.. figure:: regrets-1.png
    :scale: 75 %
    :align: center   
   
    Figure 3: Voter regrets for 1-Dimensional Spatial Model
   
.. figure:: regrets-2.png
    :scale: 75 %
    :align: center   
       
    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:: scenarios-vs-dimension.png
    
    Figure 5: Occurrences of Scenarios vs Model Preference Dimensions
   
   
.. figure:: regrets-3.png
    :scale: 50 %
    :align: center   
       
    Figure 4: Voter regrets for 3-Dimensional Spatial Model
   
   
.. figure:: regrets-4.png
    :scale: 50 %
    :align: center   
       
    Figure 4: Voter regrets for 4-Dimensional Spatial Model
   

.. figure:: regrets-5.png
    :scale: 50 %
    :align: center   
       
    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:: scenarios-vs-candidates.png

    Figure 6: Occurrences of Scenarios vs # of Candidates


Conclusions
------------
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.