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Direct PNAS a is article This interest. of conflict no wrote declare A.M. authors and The I.M.R., A.R., F.K.D., and data; paper. analyzed the I.M.R. and A.R., F.K.D., research; htpriiaei ie rp n h oe h sighting the lower the and trip, hunters given more a the is. higher is, in probability is season animal participate harvest given remaining that a the shooting show shorter of We meteo- the likelihood the situations. with the choice and 181,989 indeed data exploit that temperature, can probabil- these we precipitation, conditional These Matching phase, moon on shot. the animal. information not an construct recorded, but rological shooting directly were seen, of to not animal been ity us where have hunted allow 1), animals of (Fig. data which loca- type 2010 256 also and and covers but number 1999 panel the Our between only data. 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SUSTAINABILITY SCIENCE The hunter incurs a fixed cost ci per period (representing, e.g., his or her opportunity cost of labor, access fees to the hunting area, etc., measured in terms of utility), regardless of whether an animal is actually shot or not. Each period, the hunter observes one animal whose value is ex ante a random variable X . Clearly, the hunter goes hunting only if E[X ] > ci . The optimal policy in the basic setup is characterized by a “reservation value” that equals the expected benefit from follow- ing the optimal rule. Eq. 1 describes the “optimal shooting rule,” where ξi is the reservation value:

shoot animal if x ≥ ξi [1] do not shoot animal if x < ξi .

The reservation value ξi thus defines the selection pressure exerted by this hunter i: All animals whose trait value is at or above the threshold are subject to being hunted, whereas all ani- mals with a lower trait value are safe from being shot. The expected net return from policy Eq. 1 is given by E[max{ξi , X }] − ci and the reservation value can be expressed R ξi R ∞ as ξi = ξi dFi (x) + xdFi (x) − ci , which implies 0 ξi

Fig. 1. Map of Western Norway to indicate our study area. The graph shows R ∞ ci = (x − ξi )dFi (x). [2] the numbers of shot animals in Norway. ξi

Eq. 2 has an intuitive interpretation: The individual thresh- Theoretical Results old value ξi is chosen so that the marginal cost incurred by We develop a behavioral model to predict how constraints on not shooting in the current period and awaiting another period hunter decisions affect selectivity. The centerpieces of a model just equals the expected marginal return from one more obser- that describes the hunting decision are (i) that one shot can vation. The right-hand side of Eq. 2 defines a function Hi kill at most one animal, (ii) that the animal the hunter actually that maps a given trait value x into the utility gain that the observes is essentially an independent and identically distributed hunter i can expect by waiting for the next observation. That is, H (x) ≡ R ∞(y − x)dF (y) draw from the population, and (iii) that the hunter is constrained i x i . on how many animals can be shot. The function Hi is convex and strictly decreasing, and it sat- The first feature distinguishes hunting from harvesting, where isfies limx→∞Hi (x) = 0 and limx→0Hi (x) = E[X ]. This result many animals are removed from the population in one instant implies that whether a hunter chooses to shoot an animal (there are, for example, thousands of fish in a trawl net). The depends not only on characteristics of the animal (given by the second feature relates to the fact that animals move and that the trait value x), but also on characteristics of the hunter. Indeed, hunter cannot be sure to see a given animal again when he or she hunters may differ in terms of opportunity costs and also how lets it pass. The third feature stems from the fact that most hunt- they value a sighted animal. Fig. 2 illustrates how different costs ing is actively managed by quotas. The constraint could also more and different valuations affect the reservation value ξi and, generally result from specific investments that have to be made hence, the probability to shoot an animal. before hunting. Without loss of generality, we set this constraint so that the hunter is allowed to shoot exactly one animal. These three features make it very natural to frame the hunt- AB(A) Different costs (B) Different valuations ing decision as an optimal stopping problem: At each sighting, the individual hunter has to choose whether to shoot the cur- rent animal and stop hunting or to continue and wait for the next Hi(x) opportunity. Hi(x) = Hj(x) Hunting as an Optimal Stopping Problem. The animal population is Utility heterogeneous with respect to a given trait (for example, antler cost ci

size). Assume that a given hunter i has a monotone preference cost c = c cost c i j j H (x) ordering over these trait values, where we define x ∈ [0, ∞) as j the value that this hunter assigns to a given animal. Let each ani- ξi ξj ξi ξj mal have a unique value so that the hunter’s valuation of the population can be described by a cumulative distribution func- Quality of observed animal (x)

tion Fi (x). Here, the population is defined with respect to the Fig. 2. The reservation value ξi is chosen so that the expected utility gain particular type of quota that the hunter holds. Assume that the from not shooting the current animal (with value x) equals the cost of wait- distribution Fi (x) is stationary and known to the hunter. This is ing for the next observation. When hunters differ in terms of cost but value unlikely to be literally true, but it is plausible that the hunter has animals in the same way (A), we can see that the hunter with the lower a general idea about the distribution of trait values in the pop- hunting cost cj implies a higher reservation value ξj and a longer duration of ulation and the opportunities to update his or her belief are so search. Hunters may face the same costs, but differ in terms of valuation (B). For hunter i we use a standard log-normal distribution [F = ln N (0, 1)]. limited that they can be neglected. The assumption that the dis- i For hunter j, we use a distribution Fj = ln N (−0.6, 2), which implies that tribution is stationary is warranted when the relevant population the valuation function is skewed to the right. This may represent a trophy is large with respect to the scale at which the decision is made. hunter who preferably shoots animals with large trait values, whereas many Note that a stationary distribution also implies that the vulnera- animals in the population are of low value to him. In this example, the tro- bility of animals does not change over time (but see Discussion). phy hunter has a higher reservation value and a longer duration of search.

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SUSTAINABILITY SCIENCE hunters participate in a given trip, and (iii) the lower the sight- season, the less likely it is to shoot a given seen male deer. In ing probability. Fig. 3 illustrates how these effects play out for a other words, the shorter the remaining time horizon is, the less given hunter i with cost ci and valuation Fi . selective the hunters are. In contrast to team size and the number of remaining days in Empirical Evidence from Red Deer Hunting in Norway the season, there is no single covariate that captures our third the- In this section, we test our theory by analyzing a unique dataset oretical prediction. However, if we consider the total number of on red deer hunting incidents in Norway. Red deer are a sexually observed animals in a given trip as a proxy for the probability of dimorphic species, with large differences in body weight between seeing an animal, we see that these coefficients are negative and males and females (14). Red deer prefer habitats that offer both statistically significant for all categories. This result suggests that forage and cover within close range. Mating takes place in Octo- the higher the probability to see an animal is, the less likely it is that ber, with a distinct rutting period where males defend smaller a given male deer is shot, exactly as the theoretical model predicts. harems of females or land occupied by females. Males mature Additionally, we control for weather, day of the week, and the around 1–3 y of age, but do not allocate much energy into the moon phase. We find no effect of precipitation, but we find a rut until they reach 3–4 y and females mature around 1.5 to 2.5 y lower probability of shooting a deer during weekends than during (15). A single calf is born in June. weekdays. Moreover, we find that a larger visible fraction of the It is rare that red deer reach old age. Young individuals experi- moon is positively correlated with the probability of shooting. If ence particularly high hunting pressure, and older males experi- a larger fraction of the moon implies better visibility, this may— ence higher hunting pressure than older females (3). For males, at first sight—seem at odds with our third prediction. However, the chance of survival from 1.5 y to 2.5 y is only 52%, and it is whereas better current visibility certainly increases the chances 55% from 2.5 y to 3.5 y. The corresponding survival rates for to take a clear targeted shot, it does not affect the hunter’s gen- females are much higher (81% and 82%, respectively, based on eral estimate of the probability of seeing an animal at future trips, capture–mark–recapture analyses in ref. 3). which is the aspect that our third theoretical prediction relates to. In Norway, the hunting rights belong to the landowners. The In addition to the three key predictions on season length, team land of one or more landowners constitutes the lowest level of size, and sighting probability, our model of hunters’ shooting the local management units (vald). Quotas are area specific to decisions implies that the probability of shooting a given ani- this level and set by the municipalities. Quotas are based on sex mal increases nonlinearly as the season approaches its end. We and age (calves, yearling males and females, adult males, and test for nonlinearities by estimating the model with the variable adult females). It is also allowed to shoot younger animals on remaining days in cubic form. The estimated relationships are adult quotas. Here, we concentrate on the older males as the plotted in Fig. 4, whereas the regression coefficients are shown main category of interest. Results for females, calves, and year- in SI Appendix, Table S2, column 4. We find striking similari- lings are shown in SI Appendix, Table S3. ties with our theoretical predictions and our empirical results (compare Figs. 3 and 4). In particular, the probability to shoot Results increases nonlinearly as the season comes to an end. This effect is The reservation value in the mind of the hunter is unobserv- stronger the more hunters are in a team, also in line with theory. able. However, we can test the implications of the theoretical We do not find any evidence that hunters are guided purely by model. We estimate the probability of shooting a male deer con- self-interest, which would imply that the presence of one fellow ditional on seeing it with a probit binary outcome model that hunter would be sufficient to shoot every animal upon sighting. includes location-specific random effects and year-specific fixed Our empirical results are, however, consistent with hunters being effects; see Materials and Methods for details. The results, shown guided by social norms. Typically such norms are stronger in in Table 1, strongly support our theoretical predictions. smaller groups, which is again supported by the empirical results. First, the coefficient on team size is positive and highly statis- We have extensively tested the robustness of our results and tically significant, confirming our first theoretical prediction that the restrictiveness of the implied linearity. First, we run several more hunters in the team increase the probability to shoot an specifications of the estimation model (SI Appendix, section 2). animal upon sighting. Second, the coefficient on the number of Second, we discuss hunter heterogeneity (SI Appendix, section remaining days in the season is negative and highly statistically 3). Although there are no data on hunter preferences or hunter significant. This means that the more days that are left in the types (the data on hunting effort are anonymous), a first step is to distinguish trips taking place during the weekend from week- Table 1. Probability to shoot an animal upon sighting: Results from day trips. Despite the probability to shoot being lower during the a binary outcome model using probit regression with location- weekend, we find that our theoretical predictions hold equally specific random effects and year-specific fixed effects for weekend and weekday hunters. Third, animal behavior, not only hunter behavior, can also influence harvesting vulnerabil- Coefficient SE ity [occurring through, e.g., differential use of open habitat (16) Team size 0.048∗∗∗ 0.006 or behavioral responses to prevailing weather and moon phase Remaining days −0.003∗∗∗ 0.001 (17)]. In addition, hunting early in the season will always change Seen male −0.079∗∗∗ 0.021 the size and composition of the population later in the season. Seen yearling −0.028∗∗ 0.011 If these effects are large and statistically significant, it will intro- Seen female −0.016∗∗ 0.007 duce bias. In SI Appendix, section 4, we show that a “popula- Seen calf −0.030∗∗∗ 0.009 tion depletion” effect is not important. Finally, we address the Weekend −0.084∗∗∗ 0.023 possible concern of reverse causality that may arise if the num- Precipitation 0.000 0.001 ber of hunters is not exogenous. We use an instrumental variable Moon fraction 0.090∗∗∗ 0.033 strategy to take such reverse causality into account. SI Appendix, Constant −0.478∗∗∗ 0.137 Table S8 presents the results, confirming the robustness of our Observations 22,705 findings. Log likelihood −10,049.1 Clusters 245 Discussion SEs are clustered at the management unit (vald). Hunting has a crucial impact on many wildlife populations, and ∗P<0.10, ∗∗P<0.05, ∗∗∗P<0.01. this is particularly the case for ungulates. For a large part of

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Fig. Probability to shoot sn h utn esna olt civ management achieve to tool a as season hunting the Using the quantify to developed well now are models Population .2 .3 .4 .5 .6 60 h rdciepoaiiyo o esnlnt n emsize team and length season how on probability predictive The 50 Remaining daysinthehuntingseason 40 Team size:1 Team size:2 Team size:10 30 20 10 0 u aaa h rplevel. trip the at data our 3,3 bevtosi oa) u eann apecnit f1199dis- 181,989 of situations. consists choice sample remaining tinct shot year Our been that total). has in in category observations remaining observations certain all (32,639 a identify remove of we and animal category Therefore, an specific shot. where that being filled. year for the are the are that in animals quotas know day as do last time long the we as what that filled at fact not the and is exploiting quota quota infor- by the no obstacle is of this There overcome size regulation. 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Methods and Materials pattern. selection insti- emerging underlying the pre- and shape constraints the tutions social for which account by manner sustain- should dictable achieve management wildlife To for future management. consequences ability, ecosystem unwanted pop- wider and and potentially past abundance wildlife on with of quota estimates structure annual biased ulation new to lead the will basing harvesting Simply case: hunting phy on indirectly operating (17). process hunters climate harvesting the how actual understanding the on affect put may been (30). has dynamics population attention for on little se focuses Very per quotas literature of management selectivity how and wildlife size the or qual- whereas (28) and (29), maturation weather ity plant winter affect of on precipitation severe and focused terms of temperature typically in effects is behavior direct literature explaining hunter effects climate through The operates selectivity. that climate of of applications future framework. fruitful to our potentially and the culture to of points aspects This important hunting (27). in selectivity differences affect that also known non- methods well also is when It com- weekends hunting. during are is residents common evening more whereas is work, and between hunting in drive morning hunters local in for weekdays Sit-and-wait during fields way mon groups. is agricultural the between this, at animal predictably to hunting differ an Linked may shoot population. hunting the hunter in of to differences the to probability of due presumably composition the weekend, (26). the that during hunters, selectivity lower show individual identify affect to results socioeconomic us likely allow our Such not well do will (25). data as our hunters Although generally (13), among more hunters differences hunters deer among red as Norwegian among tinguished as changes prey of approaches. choice migration their of timing whether quantify to interesting oevr ehv acltdtermiigdy ntesao teach at season the in days remaining the calculated have we Moreover, tro- the beyond far relevant policy highly are findings Our effect indirect the is analysis our of aspect interesting Another dis- be can “types” hunter different that evidence is There nNra,i smnaoyb a ofilotadt omta con- that form data a out fill to law by mandatory is it Norway, In al S1 Table Appendix, SI rsnssmaysaitc of statistics summary presents NSEryEdition Early PNAS | f6 of 5

SUSTAINABILITY SCIENCE Estimation Strategy. Knowing the number of both seen and shot animals of negative; theoretical Prediction 1), and N, the number of hunters in the each category allows us to construct the conditional probability of shooting team (we expect β2 to be positive; theoretical Prediction 2). Furthermore, 0 an animal. That is, we split each hunting incidence into several observa- X i,j,k,t includes selected control variables that attempt to capture the sight- tions and assign our dependent variable a value of 1 when an animal of ing probability (Prediction 3). Furthermore, we include a location-specific category i has been seen and shot and a value of 0 if it has been seen but random effect cj and a year-specific fixed effect yt . ui,j,k,t is the idiosyncratic not shot. We estimate the binary choice per category and provide all results 2 error [with ui,j,k,t ∼ N (0,σu)]. All regression results have been estimated category-specific (SI Appendix, section 2). The main specification then takes in Stata, using the command “xtprobit.” Fig. 4 has been produced with the the (general) form command “margins,” assuming that the random effect is zero.

Pr(Y = 1) = β + β R + β N + X0 β + c + y + u , 0 1 j,k,t 2 j,k,t i,j,k,t 3 j t i,j,k,t ACKNOWLEDGMENTS. We thank N. C. Stenseth for encouraging comments and acknowledge the Center for Advanced Study in Oslo, funding and host- where we estimate the probability to shoot animal Y of category i (condi- ing the project “Climate effects on harvested large mammal populations” tional on seeing it) at hunting site j during trip k, in year t. Our key vari- in 2015–2016 (A.M. and I.M.R.), as well as financial support from the Norwe- ables of interest are R, the number of remaining days (we expect β1 to be gian Research Council Grant 215831 (to A.R. and F.K.D.).

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