Voting Method Simulations for Investigating Voter Satisfaction Maximization
===========================================================================
John Huang, 5 February 2021
Executive Summary

These reports document election simulation results using the **votesim**
Python package. Source code can be `found on Github `_.
The objective of these simulations is to determine an ideal voting method
to replace firstpastthepost (FPTP), or plurality, voting as practiced in
the United States. These simulations attempt to
measure the ability of a voting method to choose a candidate
that maximally satisfies the electorate, through preference distance
minimization. Although computer simulations are limited in their ability to
realistically simulate human voting behavior, they are still useful
tools in determining the theoretical ideal efficiency and reliability
of a system when assuming rational agents.
Several different voting systems were assessed and compared to traditional plurality, firstpastthepost (FPTP) voting. The assessed voting systems include
 `Approval voting `_
 `Instant runoff voting (IRV) `_
 Toptwo Runoff
 Firstpastthepost (FPTP), or plurality voting
 `Scored voting `_
 `Majority judgment `_
 Condorcetcompliant `Smithminimax `_
 Condorcetcomplaint `Ranked Pairs `_
 `Smith score `_
 `STAR voting (Score then Automatic Runoff) `_
Voting method performance is measured using
`"Voter Satisfaction Efficiency (VSE)" `_ ,
a metric devised to measure
the utilitarian performance of a voting method. In this metric, 100% VSE symbolizes
the election of an ideal, maximally satisfactory candidate. 0% VSE symbolizes
the election of an average, middleofpack candidate that when averaged over
many elections, is
equivalent to random candidate selection.
These results show simulations for honest voting as well as strategic voting.
A simple metric which averaged the performance of honest, 1sided strategy,
and 2sided strategy was used to devise performance scores in the table below:
=============== ============
Election Method Average VSE
=============== ============
plurality 0.515
top_two 0.683
irv 0.725
score 0.762
approval50 0.772
maj_judge 0.812
ranked_pairs 0.849
smith_minimax 0.850
star5 0.858
smith_score 0.870
=============== ============
VSE results are also plotted, breaking down three strategy
assumptions made in the simulations  honest strategy,
onesided strategy, and twosided strategy. Results are plotted for
the 5th percentile of worst VSE, the mean VSE, and the 95th percentile
of best VSE.
.. figure:: tactical/TacticalVSE.png
Figure 1: Voter satisfaction efficiency for honest, onesided, and twosided strategy
This report ultimately finds that several methods have excellent
performance in their ability to choose candidates which satisfy
a greater number of voters than other methods. These best methods
include Condorcetcompliant methods, STAR voting, and SmithScore.
Among the highest performing methods, STAR voting is the simplest
to implement and compute.
Election Methods

In the United States, FPTP plurality voting is commonly practiced for local, state,
Congressional, and presidential elections. However several alternatives have been
proposed which will be described here.
* **Approval Voting**  In approval voting, voters may select any number of
candidates but may only select two options – approve or disapprove.
An example ballot is shown in Figure 1.
.. figure:: assets/approval.png
Figure 2: Approval voting ballot
* **Instant runoff voting (IRV)** – Also known as “Ranked Choice Voting”,
instant runoff allows voters to rank candidates from first to last.
To select the winner, instant runoff eliminates candidates onebyone based
on who receives the fewest topchoice votes. The eliminated candidate's
votes are then added to the total of their next choice.
An example ranked ballot is shown in Figure 2.
In this report the name IRV is used rather than ranked choice
to distinguish IRV from other ranked voting methods that have been assessed.
.. figure:: assets/ranking.png
Figure 3: Ranked voting ballot used for IRV and Condorcet methods
* **Condorcet** – Condorcet methods are a family of voting systems that
allow voters to rank candidates from first to last.
To select the winner, Condorcet methods simulate multiple headtohead elections.
The Condorcet criterion proposes that the best candidate is the
candidate which can win every single 1 vs 1 election combination.
Take for example a threeway election with candidates
Washington, Jefferson, and Monroe. For Washington to win, he must beat
Jefferson in a oneonone election, and then defeat Monroe in another
oneonone election. Condorcet methods compile ranking ballot data to
instantly perform these oneonone matchups.
However, it is possible that no candidate is able to beat all other candidates.
This scenario is called a Condorcet Paradox.
The many variants of Condorcet methods use various algorithms to resolve Condorcet paradoxes.
This report includes two ranked variants – **Rankedpairs** and **Smithminimax**.

* **Plurality** – Plurality voting, or Firstpastthepost (FPTP),
is the current voting method used in most American elections.
This voting rule is simple. You can only select one candidate,
and the candidate with the most votes wins.

* **Score voting** – Scored voting, or Range voting,
is a simple system based on rating or grading candidates.
For example, voters may grade each candidate from a scale of 0 to 5.
To calculate the winner, the candidate with the greatest sum of scores wins.
An example scored ballot is shown in Figure 3.
.. figure:: assets/scoring.png
Figure 4: Ranked voting ballot used for IRV and Condorcet methods
* **STAR voting** – STAR voting, or “Score Then Automatic Runoff”,
is a variant of score voting with an extra runoff round.
Score voting has been criticized by some voting theorists
to be vulnerable to tactical voting.
STAR voting was conceived in order to mitigate tactical voting concerns.
As with score voting,
two runoff candidates are chosen based on the sum of candidate scores.
However, during the runoff phase, the final winner is selected based
on the most preferred candidate. This runoff serves to encourage voters
to express the full range of ballot ratings.

* **TopTwo Runoff** – In top two runoff, the winner is determined from tworounds of voting.
The first round eliminates all but two candidates. The second round then determines
the winner from the final two winners. For this report’s simulations,
an automatic toptwo method is employed using ranked ballots.

* **Majority Judgment** – Majority judgment is another system based on
rating or grading candidates. However instead of determining the winner
from the greatest sum of scores (which is equivalent to the average score
for each candidate), the winner is instead determined from the median score.

* **SmithScore** – Smith Score is a hybrid combination of scored voting and
Condorcet voting systems. Smith score chooses the winner using a Condorcetstyle
selection as well as rated ballots. However, if a Condorcet Paradox is encountered,
SmithScore uses scored voting to resolve the paradox.
Election Model

Spatial Preference Model
++++++++++++++++++++++++
A "spatial model" is used as the base of the simulation's election model. The
spatial model is a simplified mathematical representation of an election process.
This model abstracts the choices of voters into "preference dimensions". Spatial
models have been used by many voting theorists, for example in Tideman and Plassman
[2] [3].
An example of a spatial election model is shown in Figure 1. In this model, voters and candidates are represented as points in space in two dimensions. These dimensions could be any arbitrary preference. For example, the preference could be the amount of money the voter wants to spend on the company party. Or, the preference could be a traditional political preference in the LiberalConservative spectrum. Or, the preference could be the desired rate of taxation.
Spatial models can be visualized for example in Figure 2. In this figure,
a 2dimensional spatial model is used. This model assumes that its voters
care about two difference preference categories which are independent from
one another. Alternatively, in a 1dimensional spatial model, voters only care
about a single preference category. In this model, voters prefer candidates
who lie closer in distance to themselves.
.. figure:: tactical/samplefptphonest.png
Figure 5: Voters and candidate preferences in example spatial model
Model Parameters
""""""""""""""""
Additional parameters of the model assessed are described in this section.
 6000 total voter/candidate distribution combinations were simulated.
 There are 51 voters for each election.
 Spatial preference dimensions of 1, 2, and 3 were used.
 Voter preferences are normally (Gaussian) distributed.
 Candidate preferences are uniformly distributed +/1.5 std deviations from the voter preference mean.
 There are either 3 or 5 candidates in each election.
Voter Satisfaction
""""""""""""""""""
This assessment asserts that the candidate which minimizes the preference
distance maximally satisfies the preferences all voters and therefore ought to be the
winner of an election. In other words, the candidate which maximizes utility
should win the election. In terms of geometry, this best candidate
is the one closest to the centroid of the voter preference distribution.
To calculate the net satisfaction, a metric called `"Voter Satisfaction Efficiency (VSE)" `_
is used, which was devised by statistician Jameso Quinn.
Voter Behavior

Three assumptions of voter strategy were assessed in this report 
Honest behavior, 1sided strategy, and 2sided strategy.
Honest Strategy
+++++++++++++++
Honest strategy is defined to be a voter behavior where scores or ranks are
constructed monotonically and proportionately to voter preference distance from a candidate. Honest strategy produces an "honest winner" for the election.
For scored or rated ballots, honest voters in this simulation will normalize
to give their most preferred candidate maximum score, and their least preferred
candidate zero score.
Strategic Voters
++++++++++++++++
Strategic voters are voters who predict the election outcome and then
coordinate voting tactics on two **frontrunner** candidates.
The "topdog" frontrunner is the honest winner of the election.
The "underdog" frontrunner is an arbitrarily and iteratively chosen
candidate in which a voting coalition is formed in an effort to defeat
the topdog frontrunner.
Strategic voters will always coordinate their votes to express
maximum support for their preferred frontrunner.
Tactics
"""""""
`Tactics `_
are ballot manipulation routines voters may use to boost
support for their preferred candidate. The assessed tactics include:
 Bury  Rate or rank a frontrunner the worst rank or rating equal to a voter's true most hated candidate.
 Deep Bury  Rank a frontrunner the worst rank, below a voter's true most hated candidate.
 Compromise  Give a frontrunner the best rank or rating.
 Truncate Hated  All candidates equal or worse than the voter's hated frontrunner are given the lowest ranking or score.
 Truncate Preferred  All candidates worse than the voter's favorite frontrunner are given the lowest ranking or score.
 Bullet Preferred  The preferred frontrunner is given the best score or rank. All other candidates are given the lowest rank/score.
 Minmax Hated  Give worst rating or ranking to candidates equal or worse than a voter's hated frontrunner.
 Minmax Preferred  Give worst rating or ranking to candidates worse than a voter's favorite frontrunner.
OneSided Strategy
""""""""""""""""""
One sided strategy is defined to be a strategy where only supporters
of an underdog candidate use tactics, while topdog supporters use honest
strategy.
The goal of the assessment is calculate potential worstcase tactical scenarios, assuming rational underdog voters
with perfect information.
Given the enormous space of possible voter behavior, simulation postprocessing
chooses a onesided tactic that must increase the satisfaction of underdog strategic voters
but also minimize average voter satisfaction. Tactics are iterated over all possible
underdog candidates and test against all possible underdogagainsttopdog voting coalitions.
Every listed tactic is iteratively tested. All underdog coalition members use the same tactic,
though their ballot markings may still be different from oneanother.
If all tactics backfire against the underdog (ie, result in reduced satisfaction for all underdog coalitions),
then the honest election results are used. In other words, this analysis assumes
that onesided strategic voters are rational enough to avoid backfire.
TwoSided Strategy
""""""""""""""""""
In twosided strategy, underdog voters use the same strategy as in onesided strategy.
Topdog coalition members attempt to counter underdog strategy by bullet voting in favor of the honest winner.
In contrast to onesided strategy, twosided simulations do not filter out backfired twosided tactics.
Twosided strategy tests a voting method's ability to resist any sort of underdog manipulation.
For the scope of this assessment, no other topdog counter tactics were considered.
Comments on the Voter Strategy
""""""""""""""""""""""""""""""
Honest behavior represents an ideal where wellmeaning voters do not try to take advantage of tactics.
Onesided strategy represents the worst possible betrayal, where a candidate's voters conspire against honest
voters to maximize their coalition's electoral impact.
It is my opinion that onesided strategic resistance is a very important trait for an election method to
possess. If strategy is found to be highly effective, it would lead to greater and greater use of strategy.
It is also important to note that all tactics revolve around voter perception of who are "viable candidates",
which may or may not be based on realistic polling data. Obviously it is in the interest of campaigners
to present false information so that a candidate is perceived to be "viable" and therefore
worthy of applying tactical voting. The greater the tactical susceptibility of a method, the easier it is for
campaigners to manipulate the electoral result.
Example Election

To illustrate the mechanics of tactical voting, a simple firstpastthepost plurality election's
results are presented in this section. This election was performed with 201 voters and 5 candidates.
The results of an honest election are plotted in Figure 6. Candidate and voters are plotted in their
2dimensional preference space. Candidates are denoted as stars, and voters are denoted as circles.
Each of the 5 candidates is denoted Red, Blue, Green, Yellow, and Cyan.
In the honest election, the Blue candidate wins by plurality with 68 votes,
or 34% of total votes. An honest election also results in a VSE of 0.83.
.. figure:: tactical/samplefptphonest.png
Figure 6: Voters and candidate preferences, assuming honest behavior
Clearly it is possible to form a coalition to defeat blue, yet which candidate should lead the charge?
The next two Figures 7 and 8 propose a challenger coalition and observe the results.
.. figure:: tactical/samplefptptactical1.png
Figure 7: Voter tactical preferences assuming a Red or Green candidate underdog coalition
In Figure 7 on the top row, a onesided Red Coalition is capable of defeating Blue
by 79 to 68 votes.
In this strategy, the Red coalition decides to ignore Yellow, Green, and Cyan candidates.
Noncoalition members that otherwise would support blue have wasted 44 votes on Yellow, 6 votes on Green,
and 4 votes on Cyan. This election would result in 0.89 VSE which improves the results of an honest election.
However in a twosided struggle where a Blue Coalition is constructed, the Blue coalition can amass
122 to 79 votes, resulting in a Blue winner.
Figure 7 on the bottom row shows a potential coalition with Green candidate. In a one sided election,
Green is capable of amassing 105 votes vs 68 votes for Blue. Moreover, even if Blue constructs
their own coalition in a twosided strategy, Green still wins with 105 against 96 votes.
The Greenled coalition would result in a VSE of 1.00 which is the optimal result.
.. figure:: tactical/samplefptptactical2.png
Figure 8: Voter tactical preferences assuming a Yellow or Cyan candidate underdog coalition
Figure 8 presents the potential coalitions for Yellow or Cyan candidates. Notably, it is
possible for Yellow to defeat Blue 80 to 68 if a onesided strategy is used. Such a combination
results in the worst VSE of 3.09. It is also possible for Cyan to defeat Blue in a onesided
election by 77 to 68 votes with a resulting VSE of 0.36. However in both of these elections,
a twosided Blue strategy can resist these challenges to maintain a VSE of 0.83.
This simulator is interested in recording the worst case tactical results of an election.
In our example, all four underdog candidates are capable of challenging and defeating
the topdog honest winner in a onesided strategy. The worst case scenario is a Yellow victory;
therefore the onesided VSE recorded for this election is 3.09.
One underdog candidate is capable of defeating
the topdog honest winner in a twosided strategy. The worst case scenario for this election
however is coalition formation by the losing underdog candidates. Therefore a twosided
VSE of 0.83 is recorded for this election.
Results

The main VSE result is repeated here for 5th percentile, mean and 95th percentile VSE,
given three different strategy assumptions.
.. figure:: tactical/TacticalVSE.png
Figure 9: Voter satisfaction efficiency for honest, onesided, and twosided strategy
Plurality voting is the most susceptible to strategic onesided voting with a
VSE of 0.19. In other words the resulting winner for one sided elections are closer
to random selection than the VSE "Best winner". Plurality overall has the worst
VSE for honest, onesided, and twosided behavior.
Approval voting, score voting, and instant runoff also have mediocre strategy
resistance with VSE of 0.51, 0.42, and 0.53 respectively. However their performance
is substantially better than plurality.
High tier results are the Condorcet methods and STAR voting with onesided strategy VSE
ranging from 0.67 to 0.72. These methods in general also have high honest average VSE ranging
from 0.95 to 0.97.
Based on these results, I recommend the replacement of plurality voting with
any of the above tested voting methods, all of which are superior to plurality
in the scenarios tested. However for optimal results, I recommend
STAR voting or Condorcet methods.
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References

1. Quinn, J. “VSESIM’. Center for Election Science. http://electionscience.github.io/vsesim/VSE/. Accessed 6 June 2020.
2. Tideman, T. Plassmann, F. “Which voting rule is most likely to choose the best candidate?” Public Choice, March 2012.
3. Tideman, T. Plassmann, F. “The Source of Election Results: An Empirical Analysis of Statistical Models of Voter Behavior”. Journal of Economic Literature, June 2008.