Previous volumes of “Making Decisions” have alluded to voting processes, but were necessarily lacking in detail on this component of group decision-making. This volume remedies that deficiency, discussing some common voting systems in use for group decision-making. Some applications and issues that plague these systems are also considered.
Although “voting” is more often associated with political elections than decision-making, the two are perfectly compatible. An election, after all, is simply a group (constituency) voting to decide (elect) which alternative (candidate) to implement (inaugurate). Many descriptions of voting systems are given in the context of political elections; substituting key words, as shown above, often provides sufficient understanding to employ them for organizational decision-making.
Although it has not been stressed in this series, it is important to remember that decision-making is not always focused on selecting a single alternative. It can also be used to select multiple alternatives or to rank all acceptable or desirable alternatives. As such, voting systems can be employed to identify a single “winner” or multiple “winners” or to prioritize several alternatives.
Ties are possible in many voting systems; tie-breaking provisions should be specified before voting begins. Ties can be broken in a number of ways, including repeating a vote or combinations of methods. Tie-breakers will not be discussed in detail; basic knowledge of voting systems will guide a group’s choice of method.
There are many more systems – and variations on them – available than will be discussed here. An online search for “electoral systems” provides ample resources for further inquiry. Discussions of voting system types, jurisdictions or organizations that employ certain systems, potential misuse (e.g. manipulation), and other related topics can also be found.
While most of these topics will be left to the reader to explore independently, we will discuss a topic that most people consider central to a discussion of voting: “fairness.”
Voting System Fairness
There are many fairness criteria that can be considered for any voting system. Here, we will limit the discussion to four commonly-cited criteria. These four fairness criteria are summarized below.
(1) Condorcet criterion: A candidate that is preferred to all others in pairwise, or “head-to-head,” comparisons is a Condorcet winner. A voting system that always selects the Condorcet winner satisfies the Condorcet criterion.
(2) Monotonicity: Increasing preference for an alternative should not reduce its prospects for selection. Likewise, reducing preference for an alternative should not improve its prospects for selection. A voting system is monotonic if it impossible to prevent selection of an alternative by raising its ranking while the relative rankings of all other alternatives remain unchanged. Attempts to alter the outcome by manipulating rankings of a single alternative will not succeed in a monotonic voting system.
(3) Majority criterion: If an alternative garners the highest preference of a majority of voters, that alternative should be selected. A voting system that allows selection of an alternative with only minority support, when majority support exists, violates the majority criterion.
(4) Independence of Irrelevant Alternatives (IIA): The introduction or removal of a non-winning candidate should not change the rankings of other candidates or the outcome of an election. If a final selection is changed due to the removal of a non-winning alternative, the voting system has violated the IIA criterion. (For discussion of a similar issue, see rank reversal in “Vol. III: Analytic Hierarchy Process.”)
Arrow’s Impossibility Theorem states that no voting system will satisfy all fairness criteria in all circumstances. The preceding limited presentation is intended only to introduce the reader to the relevance of fairness criteria to the selection of a voting system to be used in an organization. These and others should be considered in the context of a specific decision-making group before defining a voting system to be used.
Voting System Examples
A number of voting systems are summarized below. The process followed to reach a decision is described. For comparison, several of the examples will use the same sample voting data. A simplified “theoretical” data set is provided in Exhibit 1 for this purpose. The example presented includes four alternatives (W, X, Y, Z); voters are divided among four voting blocs with unique preference profiles (A, B, C, D). The percentage (or number) of voters expressing each set of preferences is also shown.
Before we begin describing voting systems, it is important to note that many are known by multiple names. Also, sources vary on the definition of some terms and specific details of methods. These differences need not be alarming, but they highlight the need to fully define the voting system in use. Relying on a name alone may cause confusion and disagreement within a group.
Plurality may be the simplest and most common voting system in use. Choosing from a list of three or more alternatives, each group member casts a single vote for his/her “favorite.” The alternative with the highest number of votes “wins.” Plurality votes are susceptible to “spoiler effect;” a spoiler is a candidate that splits the vote with another, resulting in the selection of an alternative that could not otherwise win.
In the example presented in Exhibit 1, only the 1st-choice votes are considered (ranked votes are not submitted); alternative W wins with 42% of the vote. If alternative Y were not included, voting bloc C would have voted for Z (bloc C’s 2nd choice), giving Z 43% of the vote and a narrow victory. Thus, supporters of alternative Z may consider Y a spoiler.
Plurality was introduced in “Vol. IV: Fundamentals of Group Decision-Making” as a decision rule.
Instant Run-Off Voting (IRV)
For an instant run-off election, voters submit ranked-preference ballots, such as those compiled in Exhibit 1. If no candidate receives a majority of 1st-choice votes, the candidate with the fewest 1st-choice votes is eliminated. Votes cast for the eliminated candidate are redistributed to the remaining candidates according to those voters’ preference profiles. This process is repeated until a majority winner emerges. The result is “instant” because no additional votes need be cast; the original ballot contains all information necessary to make a final determination. The additional “rounds” of voting eliminate the spoiler effect inherent to plurality voting.
In our example, votes cast for alternative X are transferred to Y; Exhibit 2 shows the results of the first run-off. With X out of consideration, voting blocs C and D share a preference profile that gives Y 34% of the vote.
In the second run-off, alternative Z is eliminated, resulting in a two-way race between W and Y. Alternative W captures the votes originally cast for Z and the victory, with 66% of the vote. Results of the second and final run-off are shown in Exhibit 3.
Interestingly, limiting the process to a single run-off would have yielded a very different result. Alternatives Y and X would have been eliminated, transferring the votes from voting blocs C and D to alternative Z. This would have given Z a 58% to 42% victory over W. Defining a process before is critical to ensuring support for a decision.
A multi-voting process can be used to narrow the field of alternatives when a large number are under consideration. The number of votes and number of alternatives to be eliminated in each round can be adjusted to suit the group’s needs, but the “1/3 rule” is a common starting point. To apply the 1/3 rule, each group member votes for a number of alternatives equal to (or approximately) one third of the number under consideration. When the votes are tallied, a number of alternatives equal to (or approximately) one third of the number under consideration with the lowest vote counts are eliminated. Only one vote per alternative per person is allowed; there is no weighting of votes.
The process is repeated, adjusting the number of votes and eliminations as needed, until a manageable set of alternatives is attained. A final voting process is then initiated to determine the “overall winner.” The final vote can be conducted with the group’s choice of method.
The theoretical ballot data shown in Exhibit 1 could have resulted from a multi-voting process. Let’s say there were originally 12 alternatives that had not been disqualified. Two rounds of voting, with four votes cast by each group member and four alternatives eliminated each round leaves four viable alternatives in the running. We have already subjected this reduced set to plurality and instant run-off votes; demonstrations of other options are forthcoming.
Rather than picking a single winner, this process could be used to identify the four best projects, for example, all of which will be executed. The final vote count could also be used to prioritize four initiatives, defining the sequence in which they are to be pursued.
When multiple selections are to be made from a list of alternatives, block voting is one simple option. Each voter casts a number of votes equal to the number of alternatives to be accepted. The alternative with the highest vote count is accepted until all openings are filled.
If two openings are to be filled using the ballot data in Exhibit 1, simply count the 1st- and 2nd-choice votes. Alternative X is the 1st or 2nd choice of 81% of voters, while Z is the 1st or 2nd choice of 43% of voters. In this example, no tie-breaker is necessary, but it is possible for alternatives to receive equal numbers of votes.
Approval voting is a simple system that can be used to select a single winner or multiple winners. Approval votes can also be used to rank alternatives. Each voter casts one vote for each alternative that s/he “approves” or deems acceptable. The alternative(s) with the most “approvals” is/are selected. Ties could occur; groups using approval voting should be prepared with a defined tie-breaking process, whether selecting “winners” or ranking alternatives.
Approval voting does not reveal voters’ preferences among alternatives deemed acceptable. Preferences may need to be accounted for as a tie-breaking factor.
Returning to the sample ballot data of Exhibit 1, let’s assume that each voter’s first three choices are acceptable to that voter, while the fourth is not. Let’s also assume that we are accepting two alternatives. Summing the approval votes for each alternative, W and X are accepted with the approval of 85% and 81% of voters, respectively.
A proportional voting system, such as cumulative voting, is often used to elect multiple members to a board or committee. It can also be useful for various other decision-making or prioritization tasks. Each voter is allowed to cast a number of votes according to the following formula:
For corporate board elections, the weighting factor is the number of shares held by the voter. In other contexts, the weighting factor must be defined according to the impact of the decision on each voter. For example, those that will be required to invest the most to implement a decision – in terms of money, time, or other resources – should have the most influence over that decision. There are no strict rules for setting weighting factors in unregulated environments; there is only the ethical requirement that weighting factors be equitable and defensible.
Cumulative voting can support a variety of voter strategies. Votes can be distributed among multiple alternatives to elevate a subset above “the pack,” or others deemed less worthy of support. Alternately, support can be concentrated on one alternative to ensure its selection, leaving the remaining choices to the preferences of other voters.
To demonstrate cumulative voting, we again return to Exhibit 1 and make the following assumptions for compatibility and simplification:
The Borda Count is a weighted preferential voting method. If there are (n) alternatives, a voter’s 1st choice is awarded (n-1) points, his/her 2nd choice is awarded (n-2) points, and so on. When all ballots have been submitted, the points awarded to each alternative are summed to determine the winner.
Our sample preference ballot offers four choices. Therefore, 1st-choice alternatives are awarded three points. Points awarded are reduced by one for each step down in ranking. For our theoretical election, according to Exhibit 1, the alternatives accrue the following point totals:
W: (42 * 3) + (24 * 1) + (19 * 1) + (15 * 0) = 169
X: (15 * 3) + (42 * 2) + (24 * 2) + (19 * 0) = 177
Y: (19 * 3) + (15 * 2) + (42 * 1) + (24 * 0) = 129
Z: (24 * 3) + (19 * 2) + (15 * 1) + (42 * 0) = 125
Per Borda Count rules, alternative X is declared the winner. If multiple selections or prioritization is the goal, the point totals can also be used to rank the alternatives for these purposes.
While Borda Count violates the majority criterion, it is valuable when seeking a consensus decision. In our example, alternative X has much broader support than any other alternative. Consider the analysis summarized in Exhibit 4. Assume that voters whose 1st or 2nd choice is ultimately selected are satisfied with the outcome and those whose 3rd or 4th choice is ultimately selected are disappointed with the result. The goal is to maximize the former and minimize the latter. As can be seen in Exhibit 4, this is clearly achieved by selecting alternative X; a majority of voters would be disappointed if any other alternative were selected!
Score or range voting is similar in concept to the evaluations in Analytic Hierarchy Process (See Vol. III). Voters score each alternative on a scale of 0 to 9 (or other range of choice). Each is scored relative to the others to indicate the magnitude of preferences among the alternatives. Those that are unfamiliar can be scored “no opinion.” The alternative with the highest average score is declared the winner.
“No opinion” scoring causes great concern for some. A system of “fake votes” can be instituted to counteract its influence, but the very name of it could cause confusion and distrust of the system. Voters may also choose to counter it by giving minimum scores (i.e. 0) to unfamiliar candidates in order to avoid the “unknown lunatic wins” scenario. This occurs when a small but fervent group of supporters score an obscure candidate at the maximum while others offer no opinion of the unknown candidate. The result is a very high average score – and potential victory – for a candidate with very little popular support.
“No opinion” voting is unlikely to be relevant in organizational decision-making. Should an issue arise, however, decision-making groups must be prepared to resolve it. An alternate approach is offered here: score each “no opinion” vote at the median of the range. This approach offers the following advantages:
The Copeland method determines a winner by subjecting all candidates to pairwise comparisons with all others. To do this, the relative position of two candidates on a preference ballot is considered. Of the paired candidates, the preferred one receives one vote. The candidate that receives the most votes wins the pairwise “election” and earns one point. If there is a tie, each candidate earns ½ point. When all pairwise match-ups have been evaluated, the points for each candidate are tallied. The candidate with the most points is declared the winner.
The preference ballot data in Exhibit 1 requires six pairwise comparisons among the four alternatives. In the first match-up, W is preferred to X by voting blocs A and C; W receives 42 + 19 = 63 votes. Alternative X is preferred to W by blocs B and D, receiving 24 + 15 = 39 votes. Alternative W wins the pairwise match-up and earns one point. The other pairings are evaluated in the same way. Results of the pairwise match-ups are summarized below.
Summing the points earned in pairwise comparisons, we get the following totals:
W: 2 points; X: 2 points; Y: 1 point; Z: 1 point.
Alternatives W and X are tied with two pairwise victories each. An instant run-off is an obvious choice of tie-breaker; simply return to the W vs. X match-up. The 63 to 39 victory in the pairwise match-up earns W the overall win using this tie-breaking method.
As shown by applying various voting systems to a single set of data, the outcome is affected by the method employed. Trust in a voting system can also be affected by its satisfaction or violation of fairness criteria. Exhibit 5 summarizes the performance of the voting systems discussed. Satisfaction or violation of the four fairness criteria described and the winner(s) chosen by each are presented.
The choices of fairness criteria to consider depends upon the culture and philosophy of the organization within which voting and decision-making takes place. The availability of advanced voting technology simplifies the application of multiple voting systems to a single set of data, allowing organizations to evaluate the consistency of outcomes with varying levels of “fairness” as defined by their choices of criteria. Consistent, fair outcomes can increase confidence in decision-making processes and build a more cohesive team.
As mentioned previously, there are many voting systems and variations thereof available. Further modification to suit an organization’s unique needs is also acceptable. Whatever systems, criteria, or guidelines are adopted or developed should be documented in an organizational decision-making standard, as described in “Vol. IV: Fundamentals of Group Decision-Making.” This standard is the foundation of consistency and fairness that maximizes organizational support for decisions.
If your organization would like additional guidance on voting systems, voting technology, decision-making standards, or related topics, contact JayWink Solutions for a consultation.
For a directory of “Making Decisions” volumes on “The Third Degree,” see “Vol. I: Introduction and Terminology.”
[Link] “Voting Methods Overview.”
[Link] “All About Voting Methods.”
[Link] “Voting Research - Voting Theory.” Paul Cuff, Sanjeev Kulkarni, Mark Wang, John Sturm; Princeton University.
[Link] “Voting Theory.” David Lippman, The OpenTextBookStore, 2017.
[Link] “Fair Representation Voting Methods.”
[Link] “Range Voting.”
[Link] “Mathematical Foundations of AI, Lecture 6: Social Choice 1.” Ariel D. Procaccia; Carnegie Mellon University, 2008.
[Link] “Seven Methods for Effective Group Decision-Making.”
Jody W. Phelps, MSc, PMP®, MBA
JayWink Solutions, LLC
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