voting power and target-based site prioritization
TRANSCRIPT
Biological Conservation 143 (2010) 1989–1997
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Biological Conservation
journal homepage: www.elsevier .com/ locate /biocon
Voting power and target-based site prioritization
Steven J. Phillips a,*, Aaron Archer a, Robert L. Pressey b, Desmond Torkornoo c,1, David Applegate a,David Johnson a, Matthew E. Watts d
a AT&T Labs-Research, 180 Park Avenue, Florham Park, NJ 07932, USAb Australian Research Council Centre of Excellence for Coral Reef Studies, James Cook University, Townsville, QLD 4811, Australiac Department of Industrial Engineering and Operations Research (IEOR), 4141 Etcheverry Hall, University of California, Berkeley, CA 94720-1777, USAd The Ecology Centre, University of Queensland, St. Lucia, QLD 4072, Australia
a r t i c l e i n f o
Article history:Received 31 December 2009Received in revised form 20 April 2010Accepted 28 April 2010Available online 26 June 2010
Keywords:Target-basedSite prioritizationIrreplaceabilityVoting powerSelection frequencyMarxanC-Plan
0006-3207/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.biocon.2010.04.051
* Corresponding author. Tel.: +1 212 787 3478; faxE-mail addresses: [email protected] (S.J. Ph
com (A. Archer), [email protected] (R.L. P(D. Torkornoo), [email protected] (D. Apple(D. Johnson), [email protected] (M.E. Watts).
1 This research was done while working at AT&T Lab
a b s t r a c t
Indices for site prioritization are widely used to address the question: which sites are most important forconservation of biodiversity? We investigate the theoretical underpinnings of target-based prioritization,which measures sites’ contribution to achieving predetermined conservation targets. We show a strongconnection between site prioritization and the mathematical theory of voting power. Current site prior-itization indices are afflicted by well-known paradoxes of voting power: a site can have zero prioritydespite having non-zero habitat (the paradox of dummies) and discovery of habitat in a new site can raisethe priority of existing sites (the paradox of new members). These paradoxes arise because of the razor’sedge nature of voting, and therefore we seek a new index that is not strictly based on voting. By negatingsuch paradoxes, we develop a set of intuitive axioms that an index should obey. We introduce a simplenew index, ‘‘fraction-of-spare,” that satisfies all the axioms. For single-species site prioritization, the frac-tion-of-spare(s) of a site s equals zero if s has no habitat for the species and one if s is essential for meetingthe target area for the species. In-between those limits it is linearly interpolated, and equals area(s)/(totalarea – target). In an evaluation involving multi-year scheduling of site acquisitions for conservation offorest types in New South Wales under specified clearing rates, fraction-of-spare outperforms 58 existingprioritization indices. We also compute the optimal schedule of acquisitions for each of three evaluationmeasures (under the assumed clearing rates) using integer programming, which indicates that there isstill potential for improvement in site prioritization for conservation scheduling.
� 2010 Elsevier Ltd. All rights reserved.
1. Introduction
A fundamental question in conservation biology is: how shouldwe allocate sparse resources to conserve as much biodiversity aspossible? In particular, which sites are most important and shouldtherefore be targeted for acquisition or other conservation actions?A wide variety of approaches have been used to address this ques-tion, ranging from global-scale identification of areas of extraordi-nary endemism with coincident threats (Myers et al., 2000; Olsonand Dinerstein, 2002; Eken et al., 2004) to optimization methods toidentify arrangements of sites that collectively meet conservationgoals (Possingham et al., 2006; Sarkar et al., 2006).
Here we focus on target-based site prioritization, where theconservation goal consists of quantitative targets for the natural
ll rights reserved.
: +1 973 360 8871.illips), [email protected].
ressey), [email protected]), [email protected]
s-Research.
features (species, habitat types, etc.) that we wish to conserve(Margules and Pressey, 2000). A site prioritization index assigns apriority to each site that measures how much it contributestowards meeting conservation targets and, in some cases, howurgently it must be protected to avoid compromising those targetsby threatening processes such as land use changes. While site pri-oritization indices can be used to propose reserve designs by iter-atively choosing the highest priority site, their main uses are forvisualization of conservation options and for interactive decision-support for conservation. They give conservation planners analternative to reserve design software that produces a single solu-tion, instead allowing exploration and understanding of the spaceof available conservation actions (Margules and Pressey, 2000;Pressey et al., 2009).
We note that there are alternative approaches to site prioritiza-tion that are not based on specific conservation targets. Zonation(Moilanen et al., 2005) starts from a full landscape and iterativelydiscards sites of lowest value from the edge of the remaining area.Another is the method of Davis et al. (2006) which orders conser-vation actions by their marginal utility. Related to these methods,continuous benefit functions (Arponen et al., 2005) can combine
1990 S.J. Phillips et al. / Biological Conservation 143 (2010) 1989–1997
the notions of utility functions and quantitative conservation tar-gets by specifying conservation benefit as a continuous increasingfunction of species representation relative to given targets.
There are often many alternative reserve systems that achieve agiven set of targets. The most widely used site prioritization indi-ces are based on the concept of irreplaceability, which measureseither the proportion of representative (target-achieving) reservesystems in which a particular site occurs, or the consequencesfor target achievement of a particular site becoming unavailablefor conservation management (Segan et al., 2010). We focus ontwo such indices:
1. The fraction of representative reserve systems of a given combi-nation size in which the site is both present and necessary forachieving the conservation targets (Ferrier et al., 2000). To dif-ferentiate this definition from other variants, we call thisapproach strict irreplaceability. For small problems, it can bemeasured exhaustively (Pressey et al., 1994); for larger prob-lems, it is approximated by various methods (Ferrier et al.,2000; Tsuji and Tsubaki, 2004; Pressey et al., 2009).
2. Selection frequency, whereby a simulated annealing algorithmsuch as Marxan (Ball and Possingham, 2000) is used to producemultiple reserve systems, and the irreplaceability of each siteis calculated as the proportion of solutions that contain it(Carwardine et al., 2007). This definition implicitly assumes thatthe algorithm produces sensible reserve systems; we will callthe algorithm ‘‘locally-optimal” if no site in its output can sim-ply be deleted without affecting the conservation targets.Marxan is locally-optimal (Ball and Possingham, 2000: page 18).
The approach we take here is to step back from the definitionsand estimation of existing indices for site prioritization to addressthe following four questions:
1. What properties are desirable in a prioritization index?2. Do the existing indices have these properties?3. Is there a simple index that satisfies these properties?4. Do these properties capture the important aspects of site prior-
itization, so that an index designed simply to have these prop-erties will be useful?
In addressing these questions, we demonstrate and exploit theintimate connection between site prioritization and the mathe-matical theory of voting. We give simple examples showing thatexisting indices have some paradoxical properties. To avoid suchparadoxes, we define a set of axioms that we would prefer a siteprioritization index to obey. We introduce a new index, ‘‘frac-tion-of-spare,” which is the simplest index we could construct thatobeys all the axioms. We present a preliminary evaluation of thenew index by applying it to the same data set used in a previouscomparison of 52 site prioritization indices, involving schedulingof site purchases under annual budget constraints and anticipatedbiodiversity loss rates (Pressey et al., 2004). We also applied themarginal utility approach (Davis et al., 2006) to this dataset, yield-ing six new indices, so in total we compared fraction-of-spare to 58other prioritization approaches.
2. Mathematical theory of voting
We now outline how target-based site prioritization fits in theframework of mathematical voting theory, a topic in political sci-ence and game theory first explored in the 1950s (for a simpleintroduction, see Taylor, 1995). We focus on yes–no voting sys-tems: a collection of voters is deciding on the outcome of a singleproposition, with each voter voting ‘‘yea” or ‘‘nay”. A coalition is
any set of voters, and a yes–no voting system simply specifieswhich coalitions are winning: passage of the proposition is guaran-teed by ‘‘yea” votes from exactly the voters in winning coalitions.So, for example, if the proposition wins with majority support, thenthe winning coalitions are all sets containing a majority of thevoters.
To apply voting theory to conservation planning, we treat thesites as voters, and the proposition being decided is whether theconservation targets will be met. A winning coalition thus corre-sponds to a representative reserve system. As is the case with mostyes–no voting systems, conservation planning is ‘‘monotone”,which means that adding extra sites to a representative reservesystem keeps it representative.
A primary focus of political science is the concept of politicalpower. While political power takes many forms, in yes–no votingsystems it measures the importance of each voter for obtainingpassage of the proposition. In conservation planning language,political power measures the importance of each site for meetingthe conservation targets. Site prioritization can therefore be seenas political power in a yes–no voting system. In fact, strict irre-placeability turns out to be a variant of a well-known index ofpolitical power, the Banzhaf index (Banzhaf, 1965). The total Ban-zhaf power of a voter v is defined as the number of coalitions C thatsatisfy:
1. C contains v;2. C is a winning coalition;3. C is no longer a winning coalition if v is deleted from it.
The total Banzhaf power of a voter is often divided by the to-tal Banzhaf power of all voters, yielding the Banzhaf index of rel-ative voting power. In the simple majority rule examplementioned above, voters are interchangeable and therefore haveequal Banzhaf index, equaling one over the number of voters.Strict irreplaceability differs by counting only coalitions of a pre-determined size and by dividing by the total number of winningcoalitions of that size. In the C-Plan conservation software, thecoalition size is set to give a spread of estimated irreplaceabilityvalues between 0 and 1, but can be varied as needed (Ferrieret al., 2000). Selection frequency departs from the Banzhaf indexby counting coalitions weighted by the probability of theirbeing produced by the underlying simulated annealing software(Carwardine et al., 2007).
The Banzhaf index is only one of several well-studied indices ofpolitical power. The Shapley–Shubik index (Shapley and Shubik,1954) is better known in political science and game theory, andarguably suffers from less problematic paradoxes (Felsenthal andMachover, 1995). Although the Shapley–Shubik index offers analternative to existing irreplaceability variants in conservationplanning, we do not pursue it further here; rather, we argue againstbasing site prioritization on voting power.
2.1. Paradoxes of voting power
A body of literature in political science and game theorycompares the properties of various measures of voting power(Felsenthal and Machover, 1995; Taylor, 1995). Some propertiesare paradoxes – propositions that seem contrary to common sense,but are nevertheless true. Negating a paradox yields a compellingproposition, often termed an axiom, that we would like measuresof voting power to obey. Here we describe two paradoxes whichapply to any ‘‘reasonable” index of voting power, including theBanzhaf and Shapley–Shubik indices. (We use the term ‘‘reason-able” to mean the index assigns non-zero power to any voterwho can sometimes influence the outcome of the vote. In contrast,the index that simply assigns all voters zero power does not suffer
Table 1Voting system for the original European Economic Community, established by the Treaty of Rome in 1958. Passage of any proposition required at least 12 of the 17 total votes (i.e.,a two-thirds majority). Note that each winning coalition containing Luxembourg has at least 13 votes, as all other countries have even numbers of votes, so removing Luxembourgfrom a winning coalition cannot reduce the number of votes below 12. Therefore, Luxembourg could never affect the outcome of the vote. Presumably Luxembourg’srepresentative at the signing of the Treaty of Rome did not notice this paradox. The second line gives strict irreplaceability values for a combination size of four: there are sixrepresentative combinations (groups of four countries with at least 12 votes), with France, Germany and Italy contained and essential in five each, and Belgium and theNetherlands contained and essential in four each. The last line gives values of the fraction-of-spare index: the spare is five, which is the total number of votes (17) minus the target(12).
France Germany Italy Belgium Netherlands Luxembourg
Votes 4 4 4 2 2 1Irreplaceability 0.833 0.833 0.833 0.666 0.666 0Fraction-of-spare 0.8 0.8 0.8 0.4 0.4 0.2
S.J. Phillips et al. / Biological Conservation 143 (2010) 1989–1997 1991
from the paradoxes below, but is not reasonable.) Both paradoxesinvolve weighted voting systems: each voter has a weight, andwinning coalitions are those whose total weight exceeds a givenquota. In conservation planning, this corresponds to designing a re-serve system to achieve a single conservation target, for example aminimum area target for a single species; the weight of a voter cor-responds to the extent of habitat in a site.
The first paradox is that a voter may have strictly positiveweight but no voting power. A classic example is the EuropeanEconomic Community (EEC) established by the Treaty of Rome in1958 (Taylor, 1995). Although all member countries ostensiblyhad a say in running the Community, the voting system was suchthat Luxembourg could never affect the outcome of a vote (Table 1).Luxembourg is called a dummy voter, and had zero voting poweraccording to any reasonable definition.
The second paradox is the paradox of new members: if a newvoter is added to a voting system, the voting power of existing vot-ers may increase. This is counter-intuitive, since new voters shoulddilute the power of existing voters. The paradox can be demon-strated by considering adding a seventh country to the EEC, witha single vote: Luxembourg would no longer be a dummy voter,so would have non-zero power.
Because they afflict all reasonable indices of voting power, theseparadoxes are not flaws of particular indices. Instead, they areinherent in the razor’s-edge nature of voting: the vote is either suc-cessful or not, with no grey area in-between. Beyond theseinherent paradoxes of voting systems themselves, there are alsoparadoxes that apply to individual indices (Felsenthal andMachover, 1995).
3. Axioms for single-species site prioritization
We now consider the conservation goal of protecting a givenminimum extent of ‘‘habitat” (defined perhaps by suitable vegeta-tion types or occupied area) for a single species. We wish to giveeach site a priority based on the extent of habitat that it contains,henceforth simply called its extent.
Of course, conservation planning is generally more complex,potentially involving multiple species, varying site acquisitioncosts, acquisition scheduling, connectivity constraints, varying oruncertain habitat quality, climate change and so on; we addresssome of these factors later. Nevertheless, we argue that planningtools used for complex tasks should at least produce reasonable re-sults for the simplest cases.
As we have seen, site priorities can be defined in terms of votingpower in an equivalent weighted voting system, as done by irre-placeability. However, measures of voting power exhibit paradox-ical behaviour that is undesirable for conservation planning. It hasbeen argued that a prioritization index should be zero only for siteswith no elements of biodiversity (Wilhere et al., 2008), but irre-placeability suffers from the paradox of dummy voters. Indeed,
the European Economic Community example can be rephrased assingle-species site prioritization: the conservation target is 12units of habitat, and there are six sites containing 4, 4, 4, 2, 2,and 1 units of habitat, respectively (Table 1). Strict irreplaceabilityassigns Luxembourg zero priority because it is never necessary toachieve the conservation target, and similarly for selection fre-quency (assuming that the simulated annealing software is lo-cally-optimal). The paradox of new members applies todiscoveries of additional habitat: if an additional unit of habitatis found in each of 100 additional sites, Luxembourg’s irreplace-ability increases above zero. This contrasts with our intuitive senseof importance for conservation, as the additional habitat greatlyimproves the species’ survival prospects, and should reduce exist-ing sites’ priority. Note that considering the relative sizes or acqui-sition costs of sites might change selected sets of sites (e.g. Andoet al., 1998) and irreplaceability values relative to these examples,but will not necessarily avoid these paradoxes.
3.1. The axioms
To avoid these and other paradoxes, we follow the example ofgame theory and develop a set of axioms (negated paradoxes) thatwe would like a site prioritization index to obey (Table 2). The firsttwo negate the paradox of the existence of dummies and the par-adox of new members. Since these paradoxes are exhibited by allreasonable measures of voting power, negating them makes usconsider indices that are not based on voting power. The maxi-mum-values axiom requires the highest priority sites to be essen-tial for achieving conservation targets. Two monotonicity axiomsensure that priority increases with increasing extent of a site andthat making the conservation goal harder by raising the targetmakes all existing habitat more valuable. Lastly, the continuity ax-iom prevents razor’s-edge behaviour, requiring that a tiny changein the target or the estimated extent of a site should not greatlychange conservation priorities.
Of the six axioms, only the maximum-values and monotonicity-in-extent axioms are satisfied by strict irreplaceability. Selectionfrequency does not obey the continuity axiom, since the decisionsof the underlying simulated annealing software change at discretevalues of the inputs, resulting in jumps in selection frequency. Re-spect for the other axioms depends on the details of the simulatedannealing software. If the software is locally-optimal, then selec-tion frequency suffers from the paradox of the existence of dum-mies, and does not obey the minimum-values, cross-consistencyand monotonicity-in-target axioms.
Further axioms could be created, and examples abound in vot-ing theory (Felsenthal and Machover, 1995). However, we feel ourset of axioms encodes useful intuition about site prioritization. Thecase study below investigates whether these properties are suffi-cient for a prioritization index to have useful application inconservation.
Table 2Axioms for single-species site prioritization. Strict irreplaceability does not obey the monotonicity-in-target axiom because in the European Economic Community example (seetext), decreasing the target from 12 to 11 would stop Luxembourg being a dummy, and raise its irreplaceability from zero to a strictly positive value. Similarly, it does not obey thecontinuity axiom: if the extent of habitat in Luxembourg increases from 1 to 1.99 units its strict irreplaceability remains unchanged at 0, but it jumps above 0 if Luxembourg’sextent of habitat reaches 2.0.
Axiom Satisfied by strictirreplaceability
Notes
Minimum-values: sites should have priority = 0 if and only if they have zero habitat for the speciesof interest
No Negates paradox of dummy voters
Cross-consistency: adding a site or increasing the extent of a site should not increase the priority ofany other site. Conversely, deleting a site or reducing the extent of site should not decrease thepriority of any other site
No Negates paradox of new members
Maximum-values: sites should have priority equal to 1 if and only if they are essential, that is, if thetarget cannot be achieved without them
Yes Specifies range of priorities
Monotonicity-in-extent: increasing the extent of a site should not decrease its priority, anddecreasing its extent should not increase its priority
Yes
Monotonicity-in-target: increasing the target should not decrease the priority of any site.Conversely, decreasing the target should not increase the priority of any site
No
Continuity: priority should be a continuous function of all its arguments (site extents and target) No Prevents razor’s-edge behaviour
1992 S.J. Phillips et al. / Biological Conservation 143 (2010) 1989–1997
3.2. The fine print
Two exceptions are needed to make the axioms simultaneouslyachievable. First, if the target is more than the total extent of allsites, then even sites with a tiny (but non-zero) extent should havehigh priority, but sites with zero extent still should have priorityequal to zero. We therefore revise the continuity axiom to allowa discontinuity in this case, allowing a ‘‘jump” at zero extent. Inbiological terms, the site of a newly discovered population of a crit-ically endangered species instantly becomes valuable, no matterhow small the new population.
Second, consider a site that is on the verge of being essential forachieving the target, because the total extent of all other sitesequals the target. We revise the maximum-values axiom to allowthis site to have priority equal to 1. This is required for consistencywith the continuity axiom, since an arbitrarily small increase in thetarget or decrease in the extent of another site would make thissite essential. For example, consider two sites A and B with99.99 ha and 10 ha of habitat respectively, and a conservation tar-get of 100 ha. Strict irreplaceability gives B priority 1, but reducesits priority to 0 if another 0.01 ha of habitat is discovered in A. Incontrast, the revised maximum-values axiom keeps the priorityof B at 1 in both cases.
4. The fraction-of-spare index
A simple approach to satisfying the axioms in Table 2 is to setthe priority of a site to 0 if its extent is zero (obeying the mini-mum-values axiom), 1 if its extent is large enough to be essentialfor the conservation target (obeying the maximum-values axiom),
Table 3Fraction-of-spare and irreplaceability values for three test cases, all with the same total extextent between sites. For each site extent, the table shows strict irreplaceability (the fracFerrier et al., 2000) for a site of that size (numbers in black). Irreplaceability values calculatcombinations containing the site (Pressey et al., 1994), appear in parentheses whenever theconsidering all possible representative combinations, rather than using the statistical estformulas (Appendix A of Ferrier et al., 2000) involve division by zero. ‘‘Combination size” reffor relevant combination sizes. As an example, manual calculations for Case 2, 8 ha site a
Test case Site extent (ha) Number of sites Fraction of spare Combi
1
1 100 5 0.52 100 1 0.5
8 50 0.04
3 100 1 0.5 0400 1 1.0 1
and to interpolate linearly in-between. This yields the fraction-of-spare index:
Fraction-of-spareðsÞ ¼
0 if areaðsÞ ¼ 01 if areaðsÞ > 0 & areaðsÞ
P total area� targetareaðsÞ
total area�target otherwise
8>>>><>>>>:
Note that the index is linearly related to extent, but capped at one.Intuitively, the linear scaling serves to make the index comparablebetween species, while the cap arises because we are defining theconservation goal in terms of reaching a specified target, with noadditional benefit thereafter.
For example, consider a region with 500 ha of habitat for thespecies of concern, with a conservation target of protecting300 ha. A site with 100 ha will get a priority of 0.5. The prioritydoes not depend on the distribution of extents among the othersites, but only on their combined area. To make this explicit, con-sider three different situations (Table 3):
� Case 1: four other sites have non-zero extent, all with 100 ha.� Case 2: fifty other sites have non-zero extent, all with 8 ha.� Case 3: only one other site has non-zero extent, namely 400 ha.
With fraction-of-spare, the priority of the 100 ha site is thesame (0.5) in all three cases. However, the priority of the othersites varies in each case: 0.5, 0.04, and 1.0, respectively. We believethese values fairly represent the conservation importance of theother sites relative to the 100 ha site.
ent (500 ha) and a target of 300 ha. The three cases differ in the breakdown of the totaltion of representative combinations in which the site is both present and necessary;ed using an earlier definition of irreplaceability, namely the fraction of representativetwo definitions result in different numbers. We did the exact calculations manually byimate of irreplaceability, since the estimate is undefined in many cases because theers to the number of sites in the combination being considered. Values are shown only
re presented in an Appendix A.
nation size (i)
2 3 4 5 26 27–37 38 39–51
0.6 0 (0.8) 01 1 0.745 0 (i/51)0.5 0 ði�1
50 Þ 0.194 (0.745) 0 (i/51)
01
S.J. Phillips et al. / Biological Conservation 143 (2010) 1989–1997 1993
For comparison, strict irreplaceability varies with changes in thearrangement of the other sites (Ferrier et al., 2000), and is highlydependent on combination size (Table 3). For the 8 ha sites, strictirreplaceability is not even unimodal as a function of combinationsize. If we do not know the exact number of sites that will be pro-tected or managed for conservation, it is not clear how to use theirreplaceabilities to guide conservation actions. For comparison,we also include calculations for an earlier variant of irreplaceabil-ity, namely the fraction of representative combinations containingthe site (Pressey et al., 1994). The latter definition drops therequirement that the site be necessary for achieving conservationtargets. This difference has a large and varied effect (Table 3),and for many combination sizes, it gives the 8 ha and 100 ha sitesequal priority, in contrast to fraction-of-spare.
5. Preliminary evaluation of fraction-of-spare
A useful way to compare indices is to simulate a pervasive real-ity of conservation planning: the simultaneous, incremental pro-tection and loss of native vegetation. Data from the forests ofupper north-eastern New South Wales have previously been usedin such a study comparing the effectiveness of a broad range ofindices for site prioritization (Pressey et al., 2004). The planningproblem was to stem the ongoing loss of biodiversity in the regionby scheduling the acquisition of sites for conservation action, in thecontext of limited annual conservation budgets and ongoing loss ofnative vegetation. We chose to use the same data set and planningproblem to compare the fraction-of-spare index against the 52indices compared by Pressey et al. (2004), while adding six indicesbased on marginal utility (Davis et al., 2006).
5.1. Data
The data set consists of a matrix listing the 1996 extent of eachof 107 forest types in each of 7948 sites defined by 400 ha gridcells. The conservation target for each forest type was a baselineof 10% of its estimated pre-European extent plus additional termsthat increased the target for particularly rare or vulnerable foresttypes. Where necessary, the resulting targets were trimmed to ex-tant areas. When converted to percentages of 1996 extant area, thetargets ranged from 13% to 100% (median 33%). The vulnerability ofeach forest type was defined as its annual clearing rate (in hectaresper year) estimated from analysis of satellite imagery between1972 and 1994. During the same period, the rate of reservationwas 7600 ha or 19 sites per year.
The conservation planning task is to derive an annual scheduleof site acquisitions, with the annual budget constrained to equalthe prior rate of reservation, under the assumption that all foresttypes will continue to decline in extent on private tenure at theirprior rates of loss. Non-private tenure, including established re-serves, was assumed immune from loss of native vegetation.
Table 4Primary conservation criteria for a site s, used for deriving prioritization indices compacomplementarity (recalculation after selecting each site for reservation), without a weightindices. Two measures of vulnerability were added with and without complementarity fo
Primary criterion Definition
Richness The number of forest types in sMaximum rarity The maximum rarity of any forest type present in
which the forest type occurredSummed rarity The sum of rarity values for all forest types presenWeighted % target The sum over the forest types in s of their rarity vIrreplaceability A statistical estimate (Ferrier et al., 2000) of strictSummed irreplaceability The sum of statistical estimates of single-feature s
et al., 2000)
5.2. Combining vulnerability and fraction-of-spare
To incorporate the varying vulnerability (as defined above) ofdifferent forest types, we defined the aggregate summed-spare ofa site s as:
Summed-spareðsÞ¼
Xf :protectedðf Þ<targetðf Þ
Fraction-of-spareðs; f Þ � vulnerabilityðf Þ
Here the sum is over forest types (represented by the variable f) forwhich the extent protected has not yet reached the target. An alter-native formulation reduces the contribution of forest types that arealready close to their targets:
Xf :protectedðf Þ<targetðf Þ
Fraction-of-spareðs; f Þ�vulnerabilityðf Þ�closenessðf Þ
where closeness(f) is 1 minus the fraction of the target area for fthat is already protected. We refer to the two formulations assummed-spare with or without closeness factor. They represent dif-ferent interpretations of the concept of complementarity, i.e., thatprotected sites should be complementary in the biodiversity fea-tures they contain. Without closeness, features whose targets havenot been achieved contribute equally to the priority of a site, whilecloseness reduces their contribution as they approach their targets.The closeness factor will therefore tend to increase the priority ofsites that are more complementary to sites that have already beenprotected.
5.3. Simulations to compare outcomes for alternative indices
For each of 52 indices (Table 4), Pressey et al. (2004) ran a 96-year simulation. Each annual step involved protection of a speci-fied number of sites selected according to the index being tested,followed by deterministic loss of forest types in unprotected siteswith private tenure based on their prior rates of attrition. We sim-ulated the use of the summed-spare indices using the same taskparameters, using complementarity (recalculating our indices afterselecting each site for reservation).
We did additional simulations to test the new index under sev-eral different combinations of parameters. Our aim here was not toundertake exhaustive sensitivity trials but to determine whetherrelative performance of alternative indices was consistent acrossa few parameter combinations. We therefore followed the sensitiv-ity analyses of Pressey et al. (2004) (their Fig. 4), by performing 42-year simulations with clearing rates set at 0.5, 1.0 and 2.0 timestheir default values, and 21-year simulations with all targets dou-bled (but constrained not to exceed extant areas) and reservationrates set at 0.5, 1.0 and 2.0 times the defaults. As an additional sen-sitivity study, we performed 96-year simulations with the budgetvarying from 5 to 25 sites per year, with other parameters at theirdefault values.
red by Pressey et al. (2004). Each primary criterion was applied with and withouting for vulnerability and with three weighting methods for vulnerability, yielding 48r a total of 52 indices.
s, with rarity defined as 100/n, where n was the number of unreserved sites in
t in salues times the percent of the remaining target area present in sirreplaceability (see text for definition)
trict irreplaceability values, summed over the forest types contained in s (Ferrier
1994 S.J. Phillips et al. / Biological Conservation 143 (2010) 1989–1997
5.4. Performance evaluation of indices
Pressey et al. (2004) used two metrics to measure the perfor-mance of site prioritization indices. Both summarize the distribu-tion of forest type extents remaining at the end of the simulationperiod. Remaining extents refer not just to those within actual orsimulated reserves, but those outside reserves that had not beenlost in the course of the simulations. This reflects the key objectiveof retaining, not just formally protecting, biodiversity. Minimumretention measures the worst-case outcome across all forest types,i.e., the smallest remaining percentage of the respective targetamong forest types. Area compromised measures the total sum ofextents of forest types by which targets are made unachievableby loss during the simulation period. The rarest forest types are un-likely to contribute to this second metric because their targets aresmall, so we added a third metric, fractional area compromised:the sum across forest types of the fraction of target not achieved.
5.5. Marginal utility heuristics
We implemented six heuristics based on the marginal utilityapproach (Davis et al., 2006). Three were strictly based on marginalutility: at each step, the selected site is the one that most improvesthe utility of the set S of sites that have been protected so far. Theremaining three factored in vulnerability (which does not easily fit
Fig. 1. Performance of the fraction-of-spare site prioritization index with and withoutindices. Simulation parameters (targets, clearing rates, reservation rates) are the defaults(b) area compromised; and (c) fractional area compromised (not measured in the previ
in the utility framework) in a way that is consistent with our treat-ment of fraction-of-spare.
There are many ways that the utility of a schedule could be de-fined; we chose three that we believe are natural for this casestudy, corresponding to our three performance metrics. The first,based on the minimum retention metric, defines the utility of Sto be the minimum (over forest types) of the fraction of target pro-tected by sites in S. The second is based on the area compromisedmetric, and defines the utility of S as the sum (over forest types) ofthe total area protected in sites in S, with each term capped at thetarget for the respective forest type. Lastly, the third measure ofutility is based on the fractional area compromised metric, and de-fines the utility of S as the sum (over forest types) of the fraction ofthe target (capped at 1) protected by sites in S. To factor in vulner-ability, we simply weighted the area or fraction of target protectedfor each forest type by its vulnerability.
5.6. Optimal scheduling solutions for comparison with indices
For each of the three metrics, we expressed the task of findingan optimal acquisition schedule for that metric as an integer pro-gram (Nemhauser and Wolsey, 1988). Binary variables representthe decisions of whether each site is purchased each year, andthe amount of each forest type remaining at the end of thesimulation is expressed using linear constraints. We solved the
closeness factor, compared to optimal and (where applicable) the best of 58 otherfrom Pressey et al. (2004). Performance is measured using: (a) minimum retention;
ous study).
S.J. Phillips et al. / Biological Conservation 143 (2010) 1989–1997 1995
integer programs to determine the best achievable value of eachmetric.
5.7. Results
For a budget of 19 sites per year, Pressey et al. (2004) reportedthe best performance across 52 indices as: area compro-mised = 86 ha and minimum retention = 96% (for summed irre-placeability and irreplaceability respectively, with different waysof incorporating vulnerability, both with complementarity). Thecorresponding numbers for summed-spare without closeness fac-tor were 48.3 ha and 97.9% (Fig. 1). For summed-spare with close-ness factor the numbers were 47.9 ha and 97.9%. Therefore,according to both minimum retention and area compromised,summed-spare outperforms the 52 previously considered indices,both with and without closeness factor. The heuristics based onmarginal utility all performed much worse than summed-spare:the best marginal utility version was based on fractional area com-promised, and achieved 87.2% minimum retention with 239 hacompromised; its fractional area compromised (0.40) was alsomuch worse than summed-spare (0.08). For budgets below 14sites/year, the closeness factor slightly improved summed-spare’sperformance by all metrics (Fig. 1), while above 14 sites/year re-sults were unchanged.
The sensitivity analyses showed slightly mixed results forsummed-spare when compared with previous indices (Fig. 2). Inone case, a previous index (summed irreplaceability with vulnera-bility Vs) achieved lower area compromised, namely 250 ha com-pared to 393 ha (summed-spare with closeness factor) and429 ha (without closeness factor) for doubled clearing rate. In theother five cases, summed-spare achieved better results, with andwithout closeness factor, than all previous indices.
The optimal solutions produced by the integer programachieved markedly better performance than all solutions basedon site prioritization indices whenever the conservation goals werehard to achieve, either because the annual budget for site acquisi-tion was lower (Fig. 1) or because the targets or clearing rate werehigher (Fig. 2). However, the optimal solutions for the three evalu-ation statistics might differ in terms of the selection and scheduleof sites protected, so there might not be a single solution thatsimultaneously achieves the optimal performance shown in allthree plots of Fig. 1.
Fig. 2. Performance (measured by sum of area compromised for each feature) of priosimulation (b) varying reservation budget with doubled targets and 21-year simulation.and complementarity (Pressey et al., 2004).
6. Discussion
Our work yields several insights regarding tradeoffs for conser-vation prioritization, while suggesting a number of directions forfuture research.
6.1. Simplicity versus optimality
The fraction-of-spare index is extremely simple, and thereforemuch quicker to compute than strict irreplaceability, its statisticalapproximations or multiple runs of Marxan for selection fre-quency. More importantly, it is simple to understand and explain,and therefore highly suitable for use in visualization and decision-support tools such as C-Plan (Pressey et al., 2009) that facilitatenegotiations between conservation biologists and other stakehold-ers affected by conservation decisions (Ferrier et al., 2000).
Simplicity comes at a price: for any conservation planning prob-lem, a heuristic solution will likely give sub-optimal results. In ourcase study, site acquisition schedules produced by fraction-of-spare were not as good at retaining forest types in a 96-yearsimulation as optimal schedules determined using integer pro-gramming. However, the optimal solutions are only optimal forthe assumed threats and loss rates over the whole simulation per-iod. These values are only estimates, and errors therein would dra-matically reduce the effectiveness of the ‘‘optimal” solutions. Inpractice, therefore, it may be preferable to use a simple adaptivescheduling approach, where purchases in each year are based onupdated information about threats, loss rates and the remainingdistribution of biodiversity features; we leave as a topic of futureresearch the relative performance of ‘‘optimal” and heuristic meth-ods in case of parameter errors. In our case study, fraction-of-spareoften produced near-optimal schedules (achieving at least 97% ofeach conservation target under default parameter values), demon-strating that it can be a useful conservation tool.
6.2. Razor’s edge behaviour: continuity versus uncertainty
Voting systems exhibit razor’s edge behaviour, where a singlevoter can reverse the outcome of the vote. Site prioritization indi-ces that implicitly rely on voting, such as popular variants of irre-placeability are therefore inherently discontinuous. A compellingjustification for preferring continuity is that the data available for
ritization indices under sensitivity analysis: (a) varying clearing rate for 42-yearBoth irreplaceability and summed irreplaceability were applied with vulnerability
1996 S.J. Phillips et al. / Biological Conservation 143 (2010) 1989–1997
calculating site priorities are always estimates; the inevitable smallerrors therein should not drastically change conservation priority.In our case study the historical clearing rates, conservation targets(based on model-derived measures of pre-European extents) andthe matrix of extents of forest types in each site are all estimates.The fraction-of-spare index, being continuous, guarantees a degreeof resilience to the inevitable errors in those estimates.
A second reason to prefer continuous site prioritization indicesis that conservation benefit is inherently continuous (Arponenet al., 2005). For example, if the target is to protect 100 ha of hab-itat for a species, protecting 99 ha is clearly a partial success. Inaccordance with that principle, the performance measures usedin our case study and the preceding study (Pressey et al., 2004)were continuous, punishing solutions more for targets missed bya wider margin.
While fraction-of-spare and continuous benefit functions(Arponen et al., 2005) both use continuity to avoid problems of ra-zor’s edge behaviour, we note that they are addressing differentconservation problems. Continuous benefit functions describe theutility of a whole reserve system, while fraction-of-spare is a prior-itization index, summarizing the immediate options for conserva-tion. As demonstrated by the poor performance of the marginalutility heuristics, a continuous benefit function by itself does notgive much insight into which sites are in most urgent need of pro-tection. Indeed, the marginal utility heuristic based on minimumretention performed badly because the utility remained at zerothroughout the entire simulation – no site ever had non-zero areafor all forest types that were as yet completely unprotected, so allsites had zero marginal utility, and the unprotected forest types re-mained unprotected.
6.3. Spatial configuration
Our cross-consistency axiom negates the paradox of new mem-bers, requiring for example that the discovery of a new site with apopulation of a threatened species should not increase the priorityof existing sites. This makes sense for conservation targets that de-pend only on the extent of protected habitat for each species. How-ever, many species are sensitive to the spatial arrangement ofhabitat, so a contiguous protected cluster of sites may be morevaluable than the sum of its parts. Alternatively, avoidance ofshared threats may favor protecting well-separated sites, as nearbysites may be lost to the same cyclone or wildfire. In such cases, thevalue of a site depends on the spatial arrangement of nearby sites,and the cross-consistency axiom would be inappropriate. Somespatial structure is easily represented in Marxan (and thereforeselection frequency) through the use of boundary-length modifiers(McDonnell et al., 2002). An approach has been proposed for C-Planthat reweights irreplaceability values for compactness (Presseyet al., 2009); such retrospective weighting could also be used withfraction-of-spare.
6.4. Site costs
We have studied site prioritization for achieving conservationtargets, without regard for economic concerns. However, siteacquisition costs are important for most conservation planningtasks (Ando et al., 1998; Naidoo et al., 2006). The fraction-of-spareindex lends itself to incorporating costs, for example as a cost-ben-efit ratio (measuring the ratio of fraction-of-spare to cost of eachsite).
An alternative site prioritization index that incorporates costs isreplacement cost: the loss of biodiversity or extra economic costincurred when a site is forcibly excluded (or included) in the re-serve system (Cabeza and Moilanen, 2006). Calculating replace-ment cost formally requires optimal solution of multiple reserve
design problems, though in practice heuristic solutions would beused.
6.5. Unachievable goals
For single-species prioritization, if the conservation goal isunobtainable because the target exceeds the total extent, themonotonicity-in-target axiom and maximum-values axioms to-gether imply that all sites with non-zero area must have priorityequal to 1. An alternative would be to assume that an unobtainabletarget makes protecting the species pointless because it is doomedto extinction, so all priorities should be zero. We reject the latterapproach on the grounds that conservation targets are only esti-mates based on limited information about the prospects of species,so missed conservation targets do not mean certain extinction. In-stead, an unobtainable target suggests that all remaining areashould be protected to maximize the species’ probability of sur-vival. In practice, unobtainable conservation targets are typicallytrimmed to the present extent of species and other features ofinterest (e.g. Pressey et al., 2003). Otherwise strict irreplaceabilityand selection frequency are not defined, although some studieshave defined variants of irreplaceability for such situations (Kiesteret al., 1996; Jacobi et al., 2007).
6.6. Maximum values
The maximum value of the fraction-of-spare index is 1, as is thecase for other site prioritization indices such as irreplaceability andselection frequency. Indeed, we codified this as a requirement inthe maximum-values axiom. We feel that this requirementmatches people’s intuitive concept of priority, and for conservationplanning it has the benefit of allowing priorities for different spe-cies to be assessed and compared on a common 0–1 scale. By com-bining priorities for different species in an aggregated summed-spare, we have used the fact that a given priority value has thesame interpretation across species. Having all priorities boundedabove by 1 also limits the potential for a single species to dominatethe analysis.
6.7. Incorporating vulnerability
The summed-spare index combined target-based priority withvulnerability in a very simple way, and other combinations couldbe studied. Our method for incorporating vulnerability can alsobe applied to summed irreplaceability. We adapted summed irre-placeability to incorporate the vulnerability of individual features,in the same way as the summed-spare index. We found that thisimproved slightly over the best methods compared by Presseyet al. (2004) but did not match the performance of summed-spare.This established that the improvement of summed-spare over theprevious indices was due to the inherent nature of summed-spareand not to the method of incorporating feature-specific vulnerabil-ities. The vulnerability-weighted indices we have studied herecould also be compared against the ordering algorithm of Moilanenet al. (2009); we leave such comparisons for future work.
7. Conclusion
We have postulated a set of axioms that we would prefer siteprioritization indices to obey. They cannot be satisfied by indicesbased on voting power, such as the two most common definitionsof irreplaceability. We therefore introduced a simple new methodfor single-species site prioritization called fraction-of-spare whichdoes satisfy the axioms. We evaluated two indices based on frac-tion-of-spare in a case study involving the scheduling of site
S.J. Phillips et al. / Biological Conservation 143 (2010) 1989–1997 1997
acquisitions for conservation of multiple forest types in New SouthWales. The two indices differ in their interpretation of the conceptof complementarity, but yield similar results. We found that bothof our indices outperformed 52 indices compared in a previousstudy (Pressey et al., 2004), as well as six indices based on marginalutility (Davis et al., 2006).
Acknowledgements
RLP would like to thank the Australian Research Council forfinancial support.
Appendix A. Manual calculations for Table 3
Here we detail the calculations of strict irreplaceability for the8 ha site for Test Case 2 in Table 3; the calculations for other casesare similar. We use ‘‘n1:n2” to represent a combination with n1
100 ha sites and n2 8 ha sites. We use nk
� �to represent the num-
ber of ways to choose k objects from a collection of n objects, which
equals n!n!ðn�kÞ!. We use the identity n
k
� �þ n
kþ 1
� �¼ nþ 1
kþ 1
� �.
� Combination size 26. Representative combinations have type1:25. Each 8 ha site is in half such combinations, and all sitesare necessary in such combinations. Therefore, both the earlierand later definition of strict irreplaceability yield a value of 0.5.� Combination size i 2 {27. . .37}. Representative combinations
have type 1:(i � 1). Each 8 ha site is in a fraction ði�1Þ50 of such
combinations, but none is essential.� Combination size 38. Representative combination have type
1:37 or 0:38, totaling 5037
� �þ 50
38
� �¼ 51
38
� �representative
combinations. Each 8 ha site is present in4936
� �þ 49
37
� �¼ 50
37
� �of those, for a fraction of
5037
� ��5138
� �¼ 38=51 ¼ 0:745, but is only essential in
4937
� �of them (those of type 0:38) for a fraction of
4937
� ��5138
� �¼ 38�13
50�51 ¼ 0:194.
� Combination size i 2 {39. . .51}. Similar to combination size 38,except that 8 ha sites are never essential.
References
Ando, A., Camm, J., Polasky, S., Solow, A., 1998. Species distributions, land values,and efficient conservation. Science 279, 2126–2128.
Arponen, A., Heikkinen, R., Thomas, C., Moilanen, A., 2005. The value of biodiversityin reserve selection: representation, species weighting, and benefit functions.Conservation Biology 19, 2009–2014.
Ball, I.R., Possingham, H.P., 2000. MARXAN (V1.8.2): Marine Reserve Design UsingSpatially Explicit Annealing, A Manual. 8. University of Queensland,Queensland, Australia.
Banzhaf, J.F., 1965. Weighted voting doesn’t work: a mathematical analysis. RutgersLaw Review 19 (2), 317–343.
Cabeza, M., Moilanen, A., 2006. Replacement cost: a practical measure of site valuefor cost-effective reserve planning. Biological Conservation 132, 336–342.
Carwardine, J., Rochester, W.A., Richardson, K.S., Williams, K.J., Pressey, R.L.,Possingham, H.P., 2007. Conservation planning with irreplaceability: does themethod matter? Biodiversity and Conservation 16, 245–258.
Davis, F., Costello, C., Stoms, D., 2006. Efficient conservation in a utility-maximization framework. Ecology and Society 11 (1), art. 33.
Eken, G., Bennun, L., Brooks, T., Darwall, W., Fishpool, L., Foster, M., Knox, D.,Langhammer, P., Matiku, P., Radford, E., Salaman, P., Sechrest, W., Smith, M.,Spector, S., Tordoff, A., 2004. Key biodiversity areas as site conservation targets.BioScience 54, 1110–1118.
Felsenthal, D.S., Machover, M., 1995. Postulates and paradoxes of relative votingpower – a critical re-appraisal. Theory and Decision 38 (2), 195–229.
Ferrier, S., Pressey, R.L., Barrett, T.W., 2000. A new predictor of the irreplaceability ofareas for achieving a conservation goal, its application to real-world planning,and a research agenda for further refinement. Biological Conservation 93, 303–325.
Jacobi, S., ReVelle, C., Pressey, R., Williams, J., 2007. Novel operations researchmethods for efficiently determining irreplaceable sites for conservation.Environmental Modeling and Assessment 12, 91–103.
Kiester, A., Scott, J., Csuti, B., Noss, R., Butterfield, B., Sahr, K., White, D., 1996.Conservation prioritization using GAP data. Conservation Biology 10, 1332–1342.
Margules, C., Pressey, R., 2000. Systematic conservation planning. Nature 405, 243–253.
McDonnell, M., Possingham, H., Ball, I., Cousins, E., 2002. Mathematical methods forspatially cohesive reserve design. Environmental Modeling and Assessment 7,107–114.
Moilanen, A., Arponen, A., Stokland, J., Cabeza, M., 2009. Assessing replacement costof conservation areas: how does habitat loss influence priorities? BiologicalConservation 142, 575–585.
Moilanen, A., Franco, A., Early, R., Fox, R., Wintle, B., Thomas, C., 2005. Prioritizingmultiple-use landscapes for conservation: methods for large multi-speciesplanning problems. Proceedings of the Royal Society of London B 272, 1885–1891.
Myers, N., Mittermeier, R., Mittermeier, C., da Fonseca, G., Kent, J., 2000. Biodiversityhotspots for conservation priorities. Nature 403, 853–858.
Naidoo, R., Balmford, A., Ferraro, P., Polasky, S., Ricketts, T., Rouget, M., 2006.Integrating economic costs into conservation planning. Trends in Ecology andEvolution 21, 681–687.
Nemhauser, G.L., Wolsey, L.A., 1988. Integer and Combinatorial Optimization.Wiley-Interscience, New York, NY, USA.
Olson, D., Dinerstein, E., 2002. The Global 200: priority ecoregions for globalconservation. Annals of the Missouri Botanical Garden 89, 199–224.
Possingham, H., Wilson, K., Andelman, S., Vynne, C., 2006. Protected areas: goals,limitations, and design. In: Groom, M., Meffe, G., Carroll, C.R. (Eds.), Principles ofConservation Biology, 3rd ed. Sinauer Associates, Sunderland, Massachusetts,pp. 507–533.
Pressey, R., Cowling, R., Rouget, M., 2003. Formulating conservation targets forbiodiversity pattern and process in the Cape Floristic Region, South Africa.Biological Conservation 112, 99–127.
Pressey, R., Johnson, I., Wilson, P., 1994. Shades of irreplaceability: towards ameasure of the contribution of sites to a reservation goal. Biodiversity andConservation 3, 242–262.
Pressey, R., Watts, M., Barrett, T., 2004. Is maximizing protection the same asminimizing loss? Efficiency and retention as alternative measures of theeffectiveness of proposed reserves. Ecology Letters 7, 1035–1046.
Pressey, R., Watts, M., Barrett, T., Ridges, M., 2009. The C-Plan conservation planningsystem: origins, applications, and possible futures. In: Moilanen, A., Wilson, K.,Possingham, H. (Eds.), Spatial Conservation Prioritization: QuantitativeMethods and Computational Tools. Oxford University Press, Oxford, pp. 211–234.
Sarkar, S., Pressey, R., Faith, D., Margules, C., Fuller, T., Stoms, D., Moffett, A., Wilson,K., Williams, K., Williams, P., Andelman, S., 2006. Biodiversity conservationplanning tools: present status and challenges for the future. Review ofEnvironment and Resources 31, 123–159.
Segan, D., Carwardine, J., Klein, C., Grantham, H., Pressey, R., 2010. Can wedetermine conservation priorities without clear objectives? BiologicalConservation 143, 2–4.
Shapley, L., Shubik, M., 1954. A method for evaluating the distribution of power in acommittee system. American Political Science Review 48, 787–792.
Taylor, A.D., 1995. Mathematics and Politics: Strategy, Voting, Power and Proof.Springer-Verlag, New York.
Tsuji, N., Tsubaki, Y., 2004. Three new algorithms to calculate the irreplaceabilityindex for presence/absence data. Biological Conservation 119, 487–494.
Wilhere, G., Goering, M., Wang, H., 2008. Average optimacity: an index to guide siteprioritization for biodiversity conservation. Biological Conservation 141, 770–781.