In a numerical profile, the value in each region represents the number or percentage of votes with the corresponding candidate preferences where the candidate closest to the region is the voters' first choice, the next closest candidate is their second choice and the most distant candidate is their last choice.  Thus, in the example below, there are 8 voters whose preferences are B > C > A; that is, have candidate B as their first choice, candidate C as their second and candidate A as their third.   Likewise, 10 voters have preferences A > C > B.  In what follows, we will represent these values as "bca" and "acb" respectively.

The results for a number of voting systems can be easily read from this triangle.  In plurality voting, each voter casts a ballot for his or her first choice only.  To calculate this, the two numbers closest to each candidate's vertex are added together and so we have:

10 + 20 = 30 points for Candidate A,

45 + 8 = 53 points for Candidate B and

25 + 35 = 60 points for Candidate C, making C the plurality winner.

In an anti-plurality election, each voter casts a vote for every candidate except his or her last choice.  Therefore, to decide the winner of the anti-plurality vote from the triangle, we add up the four numbers closest to each candidate's vertex.  These correspond to the candidate's first and second place votes.  

10 + 20 + 35 + 45 = 110 points for Candidate A,

45 + 8 + 20 + 25 = 98 points for Candidate B, and

35 + 25 + 10 + 8 = 78 points for Candidate C, making A the anti-Plurality winner.

Notice that anti-plurality voting is equivalent to each voter casting a ballot against his or her least favorite candidate, with the winner being the candidate who has the fewest voters in opposition.  The corresponding action in the numerical triangle is to add the numbers in the two regions farthest from a candidate's vertex and declare the candidate with the lowest score the winner.

In the Borda count, in a k-candidate election, an n-th place ranking earns (k-n) points.  Therefore, when there are three candidates, a first place vote receives two points and a second place vote receives one point.  This, to compute the Borda count results from the numerical representation triangle; we add twice the number of votes in the two regions closest to the candidate's vertex and then add the numbers in the regions representing that candidate's second place votes.

2(10 + 20) + 35 + 45 = 140 points for Candidate A,

2(45 + 8) + 20 + 25 = 151 points for Candidate B, and

2(35 + 25) + 10 + 8 = 138 points for Candidate C, making B the Borda count winner.

Plurality voting, anti-plurality voting and the Borda count are all examples of positional voting systems.  In any positional voting system, the voter provides a ranked list of all the candidates.  Each candidate is awarded a certain number of points based on his or her position on the ballot; naturally, a higher ranked candidate cannot receive fewer points than a lower ranked candidate.  In a three-candidate election, the plurality vote can be characterized by the weights <1, 0, 0>; that is, a voter's top choice receives one point and the second and third choices each receive zero.  Similarly, an anti-plurality vote is characterized by the weights <1, 1, 0> and a Borda count vote has weights <2, 1, 0>.  In general, a vote in a three-candidate positional system has weights <> with  and .  These weights are treated as points and summed; the candidate with the most points wins the election.

In a Condorcet election, candidates are compared pairwise to determine the societal preference between any two candidates.  These results are displayed on the outside of the triangle. Looking at the numbers on the bottom of the triangle, we see that 65 voters preferred A to B while 78 voters ranked B higher than A, and so, B would win a head-to-head election with A. Looking at the right-hand side of the triangle, we see B would also win a head to head election against C 73 votes to 70.  Since B wins head-to-head elections against every other candidate, he or she is the Condorcet winner in this example.  Similarly, C is the Condorcet loser in the example above because C loses head-to-head with both A and B.

Many consider the Condorcet winner, when one exists, to be the best possible societal choice.  Unfortunately, this method frequently gives indeterminate results.  Notice that if we had four more voters in the above example and they each ranked the candidates C > A > B, then our result would be a Condorcet paradox. The voters would prefer A to C, C to B, and B to A in the head-to-head elections; there is no Condorcet winner because no candidate is preferred to all of the others.

In a geometric profile our triangle is subdivided into the same ranking regions as in the numerical profile.  The data is depicted using graphical representations that are normalized to lie on the plane .  In the profile below, the locations of the plurality, anti-plurality and Borda count results are each plotted as distinct points.  The line segment connecting the plurality result (where the second-place votes are not counted) to the anti-plurality result (where the second-place votes count the same as the first-place votes) is called the procedure line.  Since the relative weight of the second place vote can range from 0 to 1, the result of the election under any positional voting system corresponds to a point on this line segment.

Approval voting is a non-positional voting system that has received a great deal of attention since Robert Weber introduced it in 1971.  In approval voting, voters are free to vote for as many candidates as they find acceptable. It is equivalent to allowing the electors to vote "yes" or "no" for each candidate, with the winner being the candidate who has the most "yes" votes. In a positional system, a given profile will have one outcome because the weight of each position on the ballot is fixed. In approval voting, each profile corresponds to many possible outcomes as voters are free to choose whether they vote for each candidate or not. In practice, it is reasonable to assume that voters always vote for their favorite candidate and never vote for their least favorite.  

The region seen enclosing the procedure line is the approval hull.  It includes all possible approval voting results where voters choose to approve of some combination of their top two choices.  There are eight points that can be found by either maximizing or minimizing the approval vote of each candidate.  The anti-plurality point maximizes each candidate's approval total by counting all first and second place votes.  Similarly, the plurality point minimizes every candidate's approval vote by only awarding first place votes.  This is done for every possible combination of candidates; for example, the topmost vertex of the approval hull shown below represents the election result where candidate C receives all of his or her first and second place votes, while candidates A and B only receive first place votes.  The Approval Hull is the convex closure of these 8 points.

The following picture uses the same data as the above numerical profile. The plurality winner is C, with a small margin over B and a large margin over A. The Borda count winner is B with small margins over A and B. And the anti-plurality winner is A with a small margin of victory over B and a larger disparity over C. The approval hull is largely split between each candidate with B receiving a little more than either C or A.

These representation triangles allow for a quick analysis of an election. The numerical profile provides a quick summary of the election results and easily compares each candidate head-to-head and allows for quick calculation of the election results under a variety of different voting systems.  The geometric profile provides a much simpler look at the election with a picture of the results. This picture provides us with the approval hull, the procedure line and the the plurality, anti-plurality, and Borda count points.

creates a numerical profile for the variables given. The profile is split into the six ranking regions (the six smaller triangles) and the amount of electors with each preference is in the corresponding region. Also, this command creates a chart with the counts for the plurality, anti-plurality, and Borda count results for each candidate where the greatest number is the winner. The candidates are color coded.  Because candidates' names can be different lengths the chart lists the color of the candidate rather than the name.

 will provide a geometric profile for the given election. This profile is split into the same six ranking regions as the  command. It includes a procedure line, plurality, anti-plurality, and Borda count points, the approval voting hull, and a handy legend. The area within the dashed lines is the anti-plurality region, the region where the anti-plurality vote must lie.

 combines the geometric and numerical profiles. It includes everything listed in  and .