Return to the hypothetical example outlined above in which a group of stakeholders is asked for their preferences for 3 mutually exclusive environmental strategies, each of which has a different focus: A1 maximize harvest, A2 protect threatened species, and A3 maximize public amenity. The preferences expressed by 2 groups of 3 individuals (I) are arranged in Table 1.
Removing Dominated Alternatives
The simplest defensible action a decision maker can make is removing “dominated” alternatives. In general, dominated alternatives are not the favorite of any individual. Therefore, in Table 1a, the alternative set is reduced to A1 and A2. However, this does not provide a complete ranking of the remaining alternatives, A1 and A2, which are Pareto efficient alternatives (i.e., they are preferred by at least one individual).
This approach (removing dominated alternatives) is used in multicriteria decision analysis to simplify problems (e.g., Gregory & Keeney 2002). Hämäläinen et al. (2001) used it to analyze 5 water level management plans for a lake-river system in Finland involving 10 groups of stakeholders, including power producers, farmers, environmentalists, recreational users, and fishers and 10 ecological, economic, and sociological criteria.
Simple Majority (Plurality) Voting
Under simple majority voting, each voter has one vote, which may be cast for a single alternative. Thus, stakeholders cannot express their preferences for all of the alternatives; they merely identify their best option. Under simple majority voting, the aggregation rule is to sum the votes for each alternative. The alternative with the most votes is selected. There is no requirement that an alternative acquire a majority (>50%) of votes.
In antiplurality voting, individuals apply a single veto. The alternative with the largest number of vetoes is removed. The set of alternatives may be reduced to a desired size through successive vetoes (Armbruster & Boge 1983).
Consider the preferences of 9 individuals for 3 alternatives (Table 2). Under simple majority voting, A1 would be selected, even though 5 of the 9 individuals prefer either A2 or A3 to A1. Thus, the preference ordering suggested by the count of votes in Table 1 may not capture the full set of individual preferences and may result in unsatisfactory choices from the perspective of the majority. Below, we discuss how this property may be exploited to deliver outcomes at odds with the preferences of the majority of voters.
Table 2. Hypothetical ordinal preferences of 9 individuals (I) for 3 alternatives (A)
Above we noted that transitivity is a desirable property of a voting system. Unfortunately, if more than 2 alternatives exist, simple majority voting may not be transitive (Arrow 1951; Dodgson 1876; Gehrlein 1983). For example, consider the case in Table 1b, where 3 individuals specify their preferences for the 3 alternatives.
For A1 and A2, Oi(A1) > Oi(A2) holds for 2 individuals and Oi(A2) > Oi(A1) holds for one individual. Thus, R = (A1 > A2).
For A2 and A3, Oi(A2) > Oi(A3) holds for 2 individuals and Oi(A3) > Oi(A2) holds for one individual. Thus, R = (A2 > A3).
For A1 and A3, Oi(A1) > Oi(A3) holds for one individual and Oi(A3) > Oi(A1) holds for 2 individuals. Thus, R = (A3 > A1).
Thus, for this set, R is not transitive.
Another drawback of simple majority voting is that the preference of the group for one of 2 alternatives can be influenced by the inclusion or exclusion of other alternatives. For example, if we were to remove A3 from Table 2 and reassign the preferences, then A1 would have 4 votes and A2 would have 5. The votes for A2 have been effectively split between A2 and A3. If every choice involved only 2 alternatives, then the outcome could be determined equitably and consistently through simple majority voting. If there are many alternatives, the winner may attract only a small percentage of total votes cast, which may be unsatisfying even if it does not violate any of the other desirable qualities listed above.
Given the potential frailties of simple majority voting, it is perhaps surprising how widely it is used to make environmental management decisions. Examples include nomenclatural decisions in taxonomy (e.g., McNeill & Turland 2010). (Although, interestingly, for the naming of Acacia, a 60% majority threshold was required.) In environmental management it has been used to identify a preferred management strategy for threatened species (Marcot et al. 2006) and to identify preferred forest management options (Kangas et al. 2006; Hiltunen et al. 2008). Tisdell et al. (2006) used it to assess preferences for conservation objectives for Australian vertebrates. In Chile, members of the National Fisheries Council vote on proposals for total allowable catch by a simple majority vote (Leal et al. 2010).
In approval voting (Brams 2004) individuals select all alternatives of which they approve. The alternative(s) with the most votes is (are) selected. Generally, the option with the greatest overall support attracts the largest number of votes (Brams 2004). To illustrate its properties, we use Table 2 and specify that the 9 individuals who expressed preferences would approve of their first and second choices but not their third. Other configurations of approvals would lead to different results. Their approval votes would be as in Table 3.
Table 3. Hypothetical approval votes cast by the 9 individuals (I) in Table 2 for 3 alternatives (A)
In this example, A2 would be selected. Approval voting has the advantage over simple majority voting in that its results are transitive and it captures more information about the preference set P. However, it can generate counter-intuitive outcomes from the perspective of the choice of a single, most-preferred alternative. In this example, A2 was selected even though the fewest voters preferred it (Table 2). If a second alternative were selected based on the approval vote, it would be A3, even though most voters preferred A1. Nevertheless, most voters considered A2 to be good enough. Approval voting tends to promote moderate alternatives that may be satisfactory but not ideal from any one perspective (Kangas & Kangas 2002).
Approval voting is used widely to identify group preferences in ecology and conservation biology. Sutherland et al. (2009) (see also Morton et al. 2009) used Approval voting to reduce a long list of global conservation questions to a more manageable list. In Brown et al. (2010), each participant allocated 10 votes among 94 water research questions, and the questions with the most votes were selected, creating a priority list of 15 questions. Vignola et al. (2012) used it to identify acceptable alternatives for the management of a catchment, reconciling the priorities of upstream farmers with downstream hydropower producers. Kangas et al. (2006) reviewed its application to choices in forest management.
Preferential voting and the Borda count (following) use the entire preference order to determine R. Preferential voting (also called alternative vote or instant run-off) is designed to give equitable consideration to the full range of each voter's preferences. Under this system, if an alternative receives sufficient first preferences to achieve a quota (i.e., a minimum number of votes), it is selected. If no alternative achieves sufficient votes, the votes allocated to the least popular alternative are redistributed (transferred) to the remaining alternatives with the following procedure. Remove the alternative with the fewest primary votes. If there is a tie, remove the alternative with the fewest second preferences among those that tie. If there is a second tie, remove the alternative with the fewest third preferences among those that tie, and so on.
When there are m voters and more than one alternative is to be selected, transfers are made sequentially until n alternatives attain the quota needed for election. The quota, Q, is usually defined as (Fishburn & Brams 1983),
where the brackets indicate the integer part of the argument.
In Table 4, all 3 alternatives are scored first once, but A3 is never scored second, so it is removed. Of the remaining alternatives, A1 and A2, A2 has the fewest first preferences so it is removed. Thus, R = (A1 > A2 > A3).
Table 4. Hypothetical ordinal preferences of 3 individuals (I) for 3 alternatives (A) after the first round of voting and the second round of voting when preferences delineated in the first round are applied
|First round||Second round|
In preferential voting, an alternative that would attract the most votes in simple majority voting or an alternative that can defeat every other alternative in direct-comparison (one-on-one) voting (i.e., the Condorcet winner; see below) might not be selected (Fishburn & Brams 1983). Preferential voting may violate the monotonicity criterion, such that an increase in support for an alternative may turn it from a winner into a loser. In Table 5 decision 1, the fewest people prefer option A2. It so happens that all of the voters who prefer A2 also prefer A3 over A1. Their votes in round 2 transfer to option A3, which is selected.
Table 5. Example of preferential voting showing how an increase in support for an alternative can turn it from a winner into a loser. Numbers in the table are the number of votes received for each alternative
|Decision 1|| || || |
|Round 2||11|| ||18|
|Decision 2|| || || |
|Round 2|| ||15||14|
In decision 2, the only change is that support for A3 increases, with 4 votes being transferred from A1 to A3. As a result, A1 is removed first. It so happens that all of the voters who prefer A1 also prefer A2 over A3. The preferences associated with A1 transfer to A2, which is selected over A3. The outcome is that A3 is no longer selected even though support for it increased. Although this counter-intuitive outcome is possible, it required quite an unusual distribution of preferences (namely, all voters who prefer A2 also prefer A3 over A1 and all voters who prefer A1 also prefer A2 over A3). This raises the question of whether such arrangements of preferences are likely to arise in practice. We explore this question below.
Despite the appeal of systems that incorporate each individual's full set of preferences, we could find no examples of preferential voting in environmental science or conservation decision making. It is implemented in many jurisdictions in Australia to choose political representatives and has affected the development of public environmental policy there (Williams 2006).
The Borda Count
Like preferential voting, the Borda count uses the entire preference order of participants, but instead of reallocating the votes as is done under preferential voting, the preferences are tallied. When there are n alternatives, the best alternative gets n – 1 points, the second n – 2 points, the third n – 3 points, and so on. The least preferred option scores zero. The Borda counts for preferences shown in Table 4 are given in Table 6. The Borda tally aggregates individual preferences to generate the group's preferences. The tallies in Table 6 suggest the ranks R = (A1 > A2 > A3).
Table 6. Hypothetical Borda counts for 3 individuals (I) for 3 alternatives (A) applied to the preferences in Table 4
The Borda count has been used to aggregate preferences in many contexts (Black 1958; Hwang & Lin 1987). Kijazi and Kant (2010) used a Borda count to establish group preferences for alternatives for forest use on Mount Kilimanjaro. Laukkanen et al. (2005) and Hiltunen et al. (2008) evaluated group preferences for forest management plans in Finland, applying several voting methods including the Borda count. Laukkanen et al. (2005) asked individuals to provide an ordering of forest plans on each of several criteria with approval voting and then used the Borda count to combine individual preferences into a group ordering.
A Condorcet winner has the appealing property that it is preferred to any other alternative by a majority of the voters (see Elkind et al. 2011). Condorcet functions find the preferred alternative by examining all pairwise comparisons of alternatives for each individual (a total of n × (n – 1)/2 comparisons for n alternatives). Each individual votes on each comparison. The score for an alternative is given by the number of times it is ranked higher than another. That is, Oi(Aj, Ak) = 1 if and only if Aj ≻i Ak and O(Aj) is summed over n alternatives and m individuals:
For example, Table 7 gives the preferences of 4 individuals for 3 alternatives. The resulting table of ordinal preferences, O, and resulting Condorcet scores are given in Table 8. The O(A1) is the sum of columns O(A1, A2), and O(A1, A3) = 6, O(A2) = 2, and O(A3) = 3 so that R = (A1 > A3 > A2).
Table 7. Hypothetical ordinal preferences of 4 individuals (I) for 3 alternatives (A)
Table 8. Condorcet scores for pairwise comparisons of alternatives in Table 7
| ||(A1, A2)||(A1, A3)||(A2, A1)||(A2, A3)||(A3, A1)||(A3, A2)|
Ghanbarpour et al. (2005) and Zendehdel et al. (2010) used a Condorcet function to aggregate group opinions about management alternatives for a watershed in Iran. Pairwise procedures such as the Condorcet function ignore portions of the information that are used by other procedures such as the Borda count and preferential voting. Of course, simple majority voting ignores even more.
Many other voting systems have been developed. They have not been used widely for environmental decision making, but we document them so that readers may pursue them if they require a particular feature for a specific context.
Variants of the Condorcet function, including Copeland's function (Hwang & Lin 1987), select alternatives based on pairwise comparisons, their differences depending on the form of the comparisons (e.g., Craven 1992; Risse 2001). Dodgson's (1876) function orders alternatives according to the number of pairwise comparisons that would require alteration for an alternative to be preferred to all others (Hwang & Lin 1987; Elkind et al. 2011).
In the Hare system, sometimes called 2-stage plurality, individuals cast one vote for one alternative. These are tabulated and the alternative with the smallest tally is removed from A. This process is repeated until one alternative remains. Hare voting requires that votes be tallied and results disseminated before another round of voting can take place, significantly increasing the complexity of the voting process.
Nanson's (1883) function identifies preferences through iterative application of the Borda count (Hwang & Lin 1987). Alternatives with the lowest Borda count are sequentially eliminated until a set of alternatives with the same Borda count remains. Group preferences derived from Nanson's function may differ from those derived from simple Borda counts.
Fishburn's (1977) function gives a group preference A1 > A2 if and only if A1 outscores or ties A2 by simple majority and A1 outscores or ties at least one other alternative that outscores A2. Kemeny's (1959) function scores alternatives by the number of times each is ranked higher than another. Its goal is to maximize the agreement between final group preferences and individual voters’ preferences; matrix algebra is used to find a solution (Fishburn 1977; Hwang & Lin 1987). It is considered far too complex for many voters to understand.
Some systems allocate multiple votes to each individual. If individuals may allocate multiple votes to a single alternative if they strongly prefer it (e.g., Mendoza & Prabhu 2009), the method is equivalent to assigning weights rather than voting. This is sometimes called cumulative voting. New voting systems continue to be developed. The examples outlined here illustrate the richness of potential approaches.
Individual preferences can be expressed by the strength of preferences among alternatives (sometimes called cardinal data). Data on preference strengths can be collected via focus groups, deliberative multicriteria analysis, or remotely via surveys of stated preferences or willingness to pay (e.g., Saaty 1980; Gregory & Keeney 2002; Chee 2004). These methods often require travel and face-to-face workshops, extensive sampling protocols, or complex question formats. Cost and fatigue can make these systems impractical.
Do the Differences Matter?
As illustrated above, counter-intuitive and even unintended outcomes are possible when voting methods are applied unthinkingly. Riker (1982), Poundstone (2008), and others argue that because all voting procedures have at least one serious flaw and opportunities for strategic manipulation are ubiquitous, voting results cannot express collective opinion. The list of important properties of voting systems in Table 9 is incomplete, but even so, no method satisfies them all.
Table 9. Flaws of voting systems (modified from Nurmi 2012; see Richelson 1981)
| ||Simple|| || || || |
|Condorcet winner is chosen||0||0||0||0||1|
Arrow's theorem applies to open and secret ballots, even when voters are unaware they are voting, such as when they reveal preferences through their purchases (French 1986). Saari (2001) notes that the same problems arise when individuals make choices based on a number of independent and incommensurate criteria, as is commonplace in conservation management. Consider an example in which a manager is required to select a conservation reserve from among 4 areas that contain 4 threatened species, 3 threatened communities, and 4 important ecosystem services. A field survey results in the data in Table 10.
Table 10. Hypothetical measures for each of 11 criteria for 4 candidate conservation reserves (based on tables 1.1 and 1.2 in Saari 2001)
|Measure||Area A||Area B||Area C||Area D|
|Populations of threatened species 1||0||20||10||80|
|Populations of threatened species 2||4||0||3||2|
|Populations of threatened species 3||30||20||18||12|
|Populations of threatened species 4||2||9||8||15|
|Extent of threatened ecosystem 1||400||50||80||100|
|Extent of threatened ecosystem 2||7||0||2||3|
|Extent of threatened ecosystem 3||0||25||30||10|
|Index of water quality||3||7||8||5|
|Index of cultural value||4||8||6||10|
|Index of pollination services||44||80||60||100|
The scores for the 11 criteria generate a rank order for the manager's preference for each reserve. There is no clear winner over all criteria, and no clear loser, so the manager cannot eliminate any dominated alternatives. A natural approach is to select the ‘best of the best’ (Saari, 2001), where the manager finds the top-ranked alternative for each criterion and selects the option ranked best most often. This is equivalent to plurality voting.
The manager's objective may be to avoid making a bad choice on any of the criteria. One way of doing so awards votes to an option if it is either first or second on each criterion. This is equivalent to approval voting. Alternatively, the manager may wish to account for all attributes in a balanced fashion, awarding 3 votes to the best option, 2 to the second best option, and 1 to the third best option for each criterion. This is equivalent to Borda count. In this hypothetical but quite realistic example, area A is chosen under a plurality vote, area B is chosen under an approval vote, and area D is chosen under a Borda count or preferential vote.
There are several problems with Table 9 as a basis for choosing a voting system. First, it is not comprehensive about what is good in a system. Some of these other attributes have been outlined in the preceding discussion. Second, the criteria are not equally important or independent. For example, the Condorcet winner criterion is strictly incompatible with the consistency criterion (Young 1974 in Nurmi 2012). Perhaps most importantly, the potential for violations of the criteria says nothing about how likely these violations are to arise in particular circumstances in which the system is used.
Assessing the likelihood of violations can be difficult because it relies on assumptions about the distributions and independence of preferences (see Nurmi 2012). In general, though, given strong preferences (full ordering of preferences), a small numbers of voters (less than about 10) or many alternatives (more than about 10), as might happen in a small committee, it is likely the Condorcet system will not produce a preferred alternative. The probabilities of intransitive outcomes are even greater than failing to find a Condorcet winner (Klahr 1966; Jones et al. 1995). In contrast, such problems are unlikely to arise if there are many voters with weak preferences (such that individuals may be indifferent to some alternatives), few alternatives (Jones et al. 1995), a few relatively strong candidates (Tangian 2000), or coalitions of voters with similar preferences (Nurmi 2012).
All voting rules are susceptible to strategic manipulation to greater or lesser degrees (Satterthwaite 1975). People designing voting systems may manipulate outcomes by including alternatives so that votes for a preferred alternative are split among 2 or more similar alternatives (Gibbard 1973). It may be in the best interests of a voter to vote for an alternative other than the one they prefer most, or not to vote at all, to secure their desired result (Kangas & Kangas 2002). A major drawback of simple majority voting, for example, is that it is relatively easy to manipulate by introducing irrelevant alternatives (Lehtinen 2008). Consider the preferences in Table 11 (derived from Table 2). Under simple majority voting, A2 is selected (decision 1). A person who controls the design of the voting system may have a personal preference for A1. If they know the likely distribution of votes among the alternatives, they may add another alternative with attributes similar to A2, in which case A1 would be selected if votes were now distributed evenly across A2 and the new alternative (decision 2).
Table 11. An example of a strategic split of votes
|Decision 1||4||5|| |
Strategic voting and coalitions have been explored at length in political science (see Taylor 1995; Hodge & Klima 2005). Generally, manipulation requires the system designer or the voter to know about other voters’ preferences. Voters with more information have more power. For instance, approval voting is more difficult to manipulate than simple majority voting because it requires additional information about the distribution of approvals (Lepelley & Valognes 2003). Thomas et al. (2010) showed how strategic voting behavior may evolve over time into special interest-driven voting blocks that “capture” state agencies, leading to decisions about the environment by public regulatory agencies that accord with commercial or recreational interests.