1 Farmers, experts and students’ subjective probability distributions on methane emission 1 reductions in livestock farming: An experimental comparison across elicitation methods 2 Claudia Magnaperaa,c,1, Austeja Kazemekaitytea, Roberta Raffaellia, Simone Cerronia,b 3 aDepartment of Economics and Management, University of Trento, Trento, Italy 4 bCentre Agriculture Food Environment (C3A), University of Trento, Trento, Italy 5 cInstitute of Natural Resource Sciences (IUNR), Zurich University of Applied Sciences 6 (ZHAW), Wädenswil, Switzerland 7 1 Corresponding author, e-mail address: claudia.magnapera@unitn.it 8 This article has been accepted for publication and undergone full peer review but has not been 9 through the copyediting, typesetting, pagination and proofreading process, which may lead to 10 differences between this version and the Version of Record. 11 Please cite this article as: 12 Magnapera C., Kazemekaityte A., Raffaelli R., Cerroni S. (2025). Farmers, experts and 13 students’ subjective probability distributions on methane emission reductions in livestock 14 farming: An experimental comparison across elicitation methods. Bio-Based and Applied 15 Economics, Just Accepted. DOI:10.36253/bae-17310 16 17 Abstract 18 Subjective probabilities are important determinants of economic choice behaviour, and their 19 elicitation is not trivial. Different methods are available in the literature. This paper compares 20 three of them using an economic experiment with farmers, other experts (animal nutrition and 21 production scientists, vets, as well as animal feed, dairy, and meat company representatives), 22 and agricultural students: the frequency method (FM), the interval method (IM), and the 23 quadratic scoring rule method (QSR). These methods vary in the degree of complexity and 24 saliency of the incentive scheme. Elicited subjective probability distributions refer to methane 25 emissions reductions that can be achieved by changing animal diets in livestock farming. The 26 study investigates whether these methods produce consistent results across methods and 27 mailto:claudia.magnapera@unitn.it 2 participant groups. Subjective probability distributions do not significantly differ across 28 methods and participant groups. Overall, these results support that the use of less complex 29 methods such as the FM and IM may be preferable. 30 31 Introduction 32 33 Subjective beliefs help in understanding economic agents’ behaviour, particularly in 34 contexts characterised by risk and uncertainty (Manski, 2004; 2018), and the agri-food sector 35 is a fitting example (Cerroni and Rippo, 2023). Eliciting stakeholders’ beliefs along the agri-36 food supply chain is essential to disentangle the role these beliefs play in their decision-making 37 processes (Hardaker and Lien, 2010). There is a vast literature showing that farmers’ beliefs 38 are correlated with farmers’ decisions in different contexts: contract farming (e.g., Cerroni, 39 2020; Cerroni et al., 2023), insurance uptake (e.g., Arata et al., 2023; Čop et al., 2023), risk 40 management (e.g., Meraner and Finger, 2014; van Winsen et al., 2014), technology adoption 41 (e.g., Purvis et al., 1995; Tjernström et al., 2021), and adoption of sustainability practices (e.g., 42 Trujillo-Barrera et al., 2016; Dessart et al., 2019). Therefore, knowing farmers’ beliefs can help 43 policymakers designing information-based interventions that create awareness in the farming 44 community and facilitate a shift towards production patterns that maximize societal welfare 45 (Manski, 2018; Cerroni and Rippo, 2023). 46 However, eliciting beliefs is not a trivial task. A wide array of methods has been used 47 in the literature, and there is no consensus on the methods that provide the most reliable 48 estimates in mainstream economics (e.g., Trautmann and Van de Kuilen, 2015; Charness et al., 49 2021). The literature eliciting beliefs in agricultural economics, and more specifically farmers’ 50 and experts’ beliefs, is limited and very few elicitation methods have been employed to retrieve 51 expectations (Cerroni and Rippo, 2023). 52 3 This paper contributes to this literature by comparing farmers’ and experts’ beliefs 53 elicited via three different methods, the frequency method (FM) (Hurley and Shogren, 2005), 54 the interval method (IM) (Dufwenberg and Gneezy, 2000), and the quadratic scoring rule 55 (QSR) (Brier, 1950) using a framed field experiment (Harrison and List, 2004). A similar 56 exercise has been proposed on farmers’ risk preferences by Reynaud and Couture (2012). 57 Most attempts to elicit farmers’ beliefs use Likert scales that elicit qualitative judgments 58 regarding the probability and/or the magnitude of an event (e.g., Meraner et al., 2019; Feyisa 59 et al., 2023). Despite the Likert scales requiring a low cognitive effort from participants, 60 qualitative judgments cannot be incorporated into economic models of decision-making under 61 risk and uncertainty like the subjective expected utility model (SEUT, Savage, 1954). More 62 sophisticated methods can be used to retrieve farmers’ beliefs expressed in a probabilistic 63 fashion. These methods can be paired with the use of monetary incentives (see Cerroni et al., 64 2023 for a review), and, if properly designed, they allow eliciting subjective probability 65 distributions that can be incorporated in standard models of decision making under uncertainty 66 (e.g., Manski 2004; Cerroni et al., 2012). The literature on the elicitation of farmers' subjective 67 probability distributions is relatively limited, especially regarding direct comparison of 68 different incentivised methods (Cerroni and Rippo, 2023). 69 Several incentive-compatible methods can elicit truthful subjective probability 70 distributions. Some studies have explored whether different elicitation methods produce 71 different subjective probability distributions (Trautmann and van de Kuilen, 2015), and we are 72 not aware of any study that has compared farmers’ subjective probability distributions elicited 73 via different incentivized and incentive-compatible mechanisms while linking potential 74 differences to complexity and saliency of the incentive scheme. Complexity refers to the 75 cognitive cost of the elicitation mechanism for participants (Charness et al., 2021). Saliency of 76 the incentive scheme refer to the potential influence that experimental payoffs have on subjects’ 77 4 actions in the experiment (Charness et al., 2016; Kleinlercher and Stöckl, 2017). This study 78 aims to fill this methodological gap by specifically comparing the FM (Hurley and Shogren, 79 2005), IM (Dufwenberg and Gneezy, 2000), and QSR (Brier, 1950). QSR has been already 80 used to elicit farmers’ subjective probabilities distributions in the literature (e.g., Cerroni, 2020; 81 Cerroni et al., 2023; Čop et al., 2023). While the FM and IM were used as well (e.g., Menapace 82 et al., 2013; Čop et al., 2023), these methods were never paired with a proper monetary 83 incentive scheme. The FM asks participants to report how frequently different outcomes of a 84 random variable occur. If the reported frequency matches exactly the empirical frequency 85 (known a priori), participants receive a fixed prize x. The IM is equivalent to the FM, except 86 that, in the IM participants are paid a fixed prize x if the reported frequency approximately 87 matches the empirical frequency. This implies that the saliency of pay-off is higher in the IM 88 than in the FM. According to Charness et al. (2021) FM and IM are less complex than QSR. 89 In particular, our study focuses on farmers and other experts’ beliefs about the potential 90 impact of adopting a new agricultural practice to improve the sustainability of food production. 91 Specifically, it elicits dairy farmers and other experts’ beliefs about the methane reductions that 92 can be achieved when introducing specific feed additives (i.e., essential oils) to bovine diets. 93 Other experts involved in our study are animal nutrition and production scientists, vets, as well 94 as animal feed, dairy, and meat company representatives. We also included a group of 95 postgraduate agricultural students (i.e. master-level), both as future sector participants and to 96 contribute to the ongoing debate on the use of student proxies in experiments. Including 97 agricultural students provides an additional test of whether farmer and agricultural students 98 behaviour are consistent in economic experiments (e.g., Grüner et al., 2022; Höhler et al., 99 2024). Some evidence finds that students and farmers can behave similarly in generic risk tasks 100 (e.g., Grüner et al., 2022), whereas other work warns that lack of domain expertise may lead to 101 divergent responses (e.g., Höhler et al., 2024). Students offer practical advantages for 102 5 experiments due to their accessibility and learning capacity (Grüner, 2022), but their 103 differences from farmers in age, income, and sector-specific experience may limit how well 104 their responses generalize to the farming population (e.g, Belot et al., 2015). Testing whether 105 student-elicited distributions align with those of practitioners therefore provides valuable 106 evidence on the external validity of experimental results in the agricultural context. 107 The emphasis on zootechnical feed additives, specifically essential oils, and their 108 impact on reducing methane emissions from livestock production stems from the fact that 109 livestock contributes 14.5% of the total greenhouse gas emissions (GHGEs) from 110 anthropogenic activities (FAO, 2017). Most livestock-related methane emissions are due to 111 enteric fermentation, and the use of essential oils in animal diets can help reduce the sector's 112 carbon footprint (Hristov et al., 2013; Honan et al., 2021). This aligns with EU policies, 113 including the EU Green Deal, the Farm-to-Fork strategy, and the planned revision of the 114 Industrial Emission Directive, which are pushing livestock farmers to lower their carbon 115 footprint. Consequently, farmers are encouraged to adopt innovations that support this goal 116 (Block et al., 2024). The introduction of feed additives such as essential oils is a readily 117 available and cost-effective option to reduce GHGEs at the farm level (Belanche et al., 2020). 118 119 In our experiment, each participant provides a subjective probability distribution three 120 times, each time facing a different elicitation mechanism. The order of exposure is randomized. 121 Results show that farmers and other experts’ subjective probability distributions do not differ 122 across methods, suggesting that it is preferable to implement simpler methods over complex 123 ones. Farmers, other experts, and students’ subjective probability distributions are similar. 124 This paper is structured as follows: we start with a review of the elicitation of beliefs in 125 the agricultural sector and a description of the three methods employed in this study. Next, we 126 6 present the empirical application, describe our sample, the experimental design, and the main 127 findings. Finally, we discuss the implications of these findings. 128 1. Literature Review 129 1.1. Elicitation of farmers’ subjective probability distributions 130 Farmers’ probabilistic beliefs can be elicited using many different approaches. Most 131 studies elicit probabilistic qualitative judgments about the occurrence of given events using 132 Likert scales. Among others, events refer to climatic adverse events, loss in production or 133 income, and efficacy of an innovative risk-reducing technology or insurance product in 134 reducing production or income losses (see Meraner et al., 2019; Feyisa et al., 2023; Villacis et 135 al., 2023). A typical question would be asking how likely an x% loss in production due to 136 climatic events is on a five-point scale from 1 (very unlikely) to 5 (very likely). Probabilistic 137 qualitative judgments are relatively easy to elicit as they require a relatively low design effort 138 from a researcher’s perspective and a relatively low cognitive effort from a participant’s 139 perspective (e.g., Delavande et al., 2011). However, these strengths are counterbalanced by 140 some limitations. Likert scales are not incentive-compatible, meaning that they do not induce 141 respondents to truthfully reveal their beliefs. In addition, probabilistic qualitative judgments do 142 not facilitate inter- and intra-personal comparisons (e.g. Manski, 2004). Finally, they cannot be 143 incorporated into decision making models of decision making under risk and uncertainty (e.g. 144 Cerroni et al., 2012). 145 A useful way to circumvent some of these disadvantages is the use of direct elicitation 146 methods (or direct introspection). Participants are directly asked to state the probability of an 147 event to occur. If several direct questions are asked about different states of the world, it is 148 possible to map the entire probability distribution of an outcome variable. This approach has 149 been widely used to elicit farmers’ subjective probability in the literature in both developing 150 and developed countries (see Hardaker and Lien, 2010; Cerroni, 2020; Cerroni and Rippo, 2023 151 7 for reviews). Direct methods are generally paired with the use of visual aids that facilitate 152 farmers’ understanding of the elicitation task. An example would be asking farmers to allocate 153 a given number of tokens (generally 100) among different states of the world that may 154 characterize a given outcome variable, suggesting that each token is equivalent to a probability 155 point. Čop et al. (2023) and Fezzi et al. (2021) used this approach to elicit farmers’ subjective 156 probability distributions regarding farm income losses in Croatia and regarding loss in 157 production due to extreme climatic events in Italy. Direct methods are widely used to elicit 158 subjective probability distributions in developing countries (Delavande et al., 2011; Cerroni 159 and Rippo, 2023). While direct introspection is relatively straightforward to implement for 160 researchers, it might be cognitively demanding and challenging for respondents who are not 161 always able and willing to express the likelihood of events in a probabilistic way (Manski, 162 2004). New web interfaces were recently created to facilitate farmers’ understanding of direct 163 elicitation methods (see Crosetto and de Haan, 2023). Direct methods are not generally 164 incentive-compatible; hence, they do not overcome the issue of the potential of misreported 165 probabilities (Trautmann and van de Kuilen, 2015; Cerroni, 2020; Charness et al., 2021). 166 To overcome the latter issue and elicit truthful beliefs, it is possible to use indirect 167 methods. If associated with a proper incentive scheme, these methods are incentive compatible. 168 These methods ask respondents to make choices between lotteries and allow retrieving 169 subjective probabilities distributions based on respondents’ gambling behaviour. For a 170 comprehensive review of these methods, we refer to Cerroni and Rippo (2023). To date, only 171 a very few studies have used these methods to elicit farmer’s subjective probability 172 distributions. All these studies have used proper scoring rules. Linear scoring rules have been 173 used by Grisley and Kellong (1983) and Smith and Mandac (1995) in developing countries. 174 Quadratic scoring rules have been used by Cerroni (2020) in Scotland, Cerroni et al. (2023) in 175 Zimbabwe, and Čop et al. (2023) in Croatia. These methods are not free from criticism. A 176 8 recent study by Danz et al. (2022) argued that individuals’ actual behaviour in response to 177 incentives may lead respondents to deviate from truth-telling when eliciting beliefs. 178 1.2. Comparing methods for eliciting subjective probabilities distributions 179 Reliable subjective probability distributions are often needed in economic modelling to 180 explain or predict human behaviour. Hence, it is essential to have reliable measures of 181 individuals’ subjective beliefs. Several methods are available and choosing the right one for 182 eliciting truthful beliefs is a relevant issue (Charness et al., 2021). The reliability of elicited 183 subjective probability distributions is contingent upon the methodology employed and the 184 participants’ comprehension of it. 185 Charness et al. (2021) ranked the methods available to elicit beliefs according to a scale 186 based on complexity, which is the subjective burden a rule places on a decision-maker (Oprea, 187 2020). In our study, complexity pertains to the extent of the understanding required to generate 188 utility-maximising belief reports. Some methods, such as the FM and the IM, have a low degree 189 of complexity. In contrast, some others, such as QSR, entail a higher degree of complexity and 190 may not be fully understood by participants at the cost of truthful belief reports. 191 To the best of our knowledge, there is limited research that compares complex methods 192 to simpler ones, and this paper aims to fill the gap. Trautmann and Van de Kuilen (2015) 193 compared subjective probabilities related to whether a participant accepted or rejected an 194 allocation submitted by a proposer in a simple two-player ultimatum game using different 195 elicitation techniques. Specifically, they tested non-incentivised direct method (i.e., direct 196 introspection) against incentivised and incentive compatible methods such as outcome-197 matching, probability matching, and QSR with or without corrections for deviations from risk 198 neutrality. Internal validity was investigated by testing for additivity and consistency of players’ 199 beliefs with their allocation decisions during the game, while external validity was investigated 200 9 by testing whether elicited beliefs matched objective probability measured at the end of the 201 game (i.e. accuracy). Results suggest that incentivised and incentive-compatible methods 202 predict better respondents’ behaviour during the game than non-incentivised introspection 203 without necessarily having better performances in terms of additivity and accuracy. Schlag et 204 al. (2015) and Charness et al. (2021) discuss the strengths and limitations of different elicitation 205 mechanisms and conclude that more research in this direction is needed. For a comprehensive 206 discussion of the most suitable methods to elicit subjective probability distributions for 207 agricultural research, which particularly focuses on developing countries, we suggest reading 208 Cerroni and Rippo (2023). We contribute to this literature by comparing three subjective 209 probability distribution methods that differ in terms of complexity, and saliency of the incentive 210 scheme. 211 212 1.3. Frequency method (FM), Interval method (IM), and Quadratic scoring rules (QSR) 213 The FM is among the least complex methods. Participants are asked to guess the 214 empirical frequency of a random variable, either when the random variable has two possible 215 outcomes (Hurley and Shogren, 2005) or multiple ones (Schlag and Tremewan, 2021). In both 216 cases, participants are rewarded with a fixed prize x if and only if their reported frequencies 217 match exactly the empirical frequencies. This rewarding mechanism can be applied either when 218 empirical frequencies are observable or not yet observable. In the latter case, a few options to 219 match unobservable frequencies would be by asking experts about it, or by using forecasts 220 based on historical data. The main advantage of the method is that it is very easy to implement 221 and easy to understand, in fact, it does not require any mathematical calculations, and natural 222 frequencies are processed better by individuals than other formats (e.g., numbers, percentages) 223 (Schlag and Tremewan, 2021). These characteristics (i.e., better understanding, fast 224 responding, and less likelihood of choosing focal points) make the frequency method FM 225 10 theoretically robust to deviations from risk neutrality and therefore more likely to be 226 empirically incentive compatible (Schlag and Tremewan, 2021). Nevertheless, when faced with 227 situations where there are many possible outcomes and unfamiliar topics, people may feel that 228 their chances of achieving a reward are low because they must make correct predictions for 229 each potential outcome. As a result, they may tend to misreport their beliefs, as they perceive 230 the likelihood of being rewarded is tied to accurate predictions. 231 This major disadvantage of the FM can be alleviated by the IM, which is de facto a 232 variation of the latter. It is still ranked as a simple method. Yet, it mitigates the low likelihood 233 of being rewarded by allowing participants to guess correctly when their reported frequencies 234 are approximately equal to the underlying ones. The primary challenge is in establishing an 235 appropriate margin of error to not incentivize under- and over-reporting. While this approach 236 significantly reduces complexity by retaining similar characteristics to the previous method, 237 the literature reflects a scarcity of applications utilising the IM (e.g., Dufwenberg and Gneezy, 238 2000; Charness and Dufwenberg, 2006). A similar approach is eliciting beliefs using most-239 likely intervals (MLI) (Schlag and van der Weele, 2015). The interval method aims at providing 240 a probability distribution within a certain percentage range of the underlying distribution, while 241 MLI consists of specifying an interval that contains the most likely outcomes. This specification 242 changes how the reward is calculated. While in the IM, participants are rewarded based on how 243 well their entire probability distribution aligns with the underlying probability distribution 244 within a certain margin of error, in the MLI participants are typically rewarded based on the 245 width of the interval they provide, with narrower intervals being rewarded more favourably. 246 Moving to more complex methods, QSR is a frequently used incentive-compatible 247 method to elicit beliefs, which first appeared in Brier (1950). It is a specification of the family 248 of the proper scoring rules, which assign a numerical score to the individuals’ probabilistic 249 predictions regarding future outcomes, rewarding them for the accuracy of their reports, while 250 11 having severe punishments if they miss the true allocation in that interval (Harrison et al., 251 2017). The payoff score is defined as 252 𝑆 = (2 × 𝑟𝑘) − ∑ (𝑟𝑖) 2 𝑖=1,..,𝐾 (Eq .1) 253 where 𝐾 corresponds to the number of intervals, 𝑟𝑘 are the reports of the likelihood that the 254 event (i.e., methane reduction) falls in the interval k = 1, … , K. Each 𝑟𝑘 ≥ 0 , ∀𝑘 and 255 ∑ (𝑟𝑖)𝑖=1,..,𝐾 = 1 (Matheson and Winkler, 1976). The method is incentive-compatible under the 256 assumption of risk neutrality and expected utility maximisation (Kadane and Wrinkler, 1988; 257 Schlag and van der Weele, 2015; Winkler and Murphy, 1970). Under these assumptions, an 258 individual who is sufficiently risk averse would be drawn to report a probability distribution 259 close to a uniform one, or report probabilities equal to 50% in the case of binary events 260 (Andersen et al., 2014). It is good practice to accompany the task with measurements of 261 individuals’ risk attitudes, to control for the initial assumption of risk neutrality (Andersen et 262 al., 2014; Offerman et al., 2009). However, Harrison et al. (2017) demonstrated that having a 263 large enough set of intervals (i.e. states of the world k) allows a reliable elicitation of subjective 264 probability distributions without undertaking calibration for risk attitudes. QSR is classified as 265 a complex elicitation method by Charness et al. (2021). Participants may have difficulties in 266 understanding the multiple layering of the QSR, which, accompanied by a substantial aversion 267 to complexity (Oprea, 2020), might impair truthful reports. 268 269 12 2. Empirical application 270 The Global Methane Pledge at a global level, the EU Green Deal, the Farm to Fork 271 Strategy, and the EU Methane Strategy at a European level, are among the variety of initiatives 272 currently underway to mitigate the significant climate impact of methane (CH4) from one of its 273 predominant sources, that is the agricultural sector. Mitigating methane emissions would allow 274 for short-term effects on the climate, due to the evidence that CH4 has about 82 times the 275 climate-changing impact of carbon dioxide (CO2) over 20 years (Smith et al., 2021). In the 276 agricultural sector, animal production is a significant contributor, particularly through 277 ruminants’ enteric fermentation, which alone accounted for 69% of CH4 emissions within the 278 sector at the European level (EEA, 2020). 279 There is a vast range of mitigation technologies available in agriculture, such as 280 production intensification, dietary and rumen manipulation, and selection of low-CH4-281 producing animals (Beauchemin et al., 2022). Some of these may be available at low cost and 282 bring additional benefits for farms, but barriers such as insufficient knowledge and expertise 283 and lack of financial incentives (Long et al., 2016) need to be addressed since their uptake is 284 uneven across the European Union. 285 Different methane-reducing technologies have different potential for mitigation (Arndt 286 et al., 2022). In our empirical application, we consider the wide class of CH4-reducing essential 287 oils that can be categorised as zootechnical feed additives for animal diets. Including essential 288 oils in bovine feed is a practice that is effective in the short term, and studies suggest it is 289 leading to a positive impact on feed efficiency and hence milk production (Wells, 2024). 290 Essential oils, derived through a steam distillation process of various parts of plant species, act 291 as rumen modifiers and can cause an inhibition of the methanogenesis process in ruminants 292 (Glasson, 2022). Many studies show that, even though there is a substantial reduction in enteric 293 CH4, the results are variable (Honan et al., 2021; Hristov et al., 2013). Arndt et al. (2022) built 294 13 a database reporting the effects of various mitigation strategies, including 115 studies on 295 additives. Feed additives, such as essential oils, have a mean value of the reduction in daily 296 CH4 emissions of -8.3%, with a 95% confidence interval (-9.8%, -6.8%). The variation depends 297 on several factors such as the environment, the animal’s diet and health, and others. 298 3. Methods 299 3.1. Sample characteristics and data collection 300 The sample consists of 60 individuals, evenly distributed between two groups: 20 dairy 301 farmers from various Italian regions, 10 livestock experts, and 30 master’s students from 302 various agri-food faculties.1 For simplicity, the group of farmers and experts is going to be 303 called the “dairy sector” throughout the paper. The sample was recruited using a snowball 304 sampling method, a non-probabilistic approach where initial participants were asked to refer 305 others who fit the study criteria. Data was gathered during the summer and the autumn of 2023 306 primarily through one-to-one with farmers and experts or group experimental sessions with 307 students on the Zoom platform.2 Participants were asked to report their probability on the web 308 interface generated via the software o-Tree (Chen et al., 2016). Ethical clearance was granted 309 by the Ethical Review Board at the University of Trento (Italy).3 310 311 1The 10 experts consulted are: 3 professionals from a public unit for forage resources and livestock production (including a technologist, a technician, and a department head), 2 veterinarians, 2 experts in animal feed development and commercialization, 1 head of a dairy consortium, 1 food processing technologist, and 1 agronomist. 210 sessions in total (4 participants in session 1, 3 participants in session 2, 3 participants in session 3, 3 participants in session 4, 1 participant in session 5, 6 participants in session 6, 1 participant in session 7, 4 participants in session 8, 4 participants in session 9, 1 participant in session 10). While the procedures and instructions were identical across settings, we acknowledge different settings might have introduced subtle differences, resulting in a minor limitation of the study design. 3We did not collect demographic data, as the primary aim was to compare elicitation methods. This was intended to reduce respondent burden and keep participants focused on the tasks. 14 3.2. Experimental design 312 In this study, farmers, experts, and students were asked to report their subjective 313 probability distribution regarding the reduction (in percentage value) of enteric methane 314 emissions which could be achieved using essential oils in bovine diets in a dairy farm across 315 three consecutive tasks. Specifically, each participant completed the same task three times, each 316 time under different elicitation methods: FM, IM, and QSR. The only difference among the 317 three tasks was how the monetary reward was attributed. To control for order effects, the order 318 of the task was randomized. Participants were divided into two groups. Specifically, one group 319 was exposed to tasks in the following order: i) FM, ii) IM, and iii) QSR, and the other group as 320 follows: i) QSR, ii) FM, and iii) IM. The FM was always presented before the IM to facilitate 321 the understanding of the IM, which is a variation of the FM. 322 Before the elicitation of the subjective probability distributions, participants were 323 provided with the following standardised information: i) a description of policy initiatives 324 aiming at reducing methane emissions from the agricultural sector, ii) a description of strategies 325 to reduce methane emissions in dairy farming with a focus on essential oils, and iii) a 326 description of benefit and opportunities of using essential oils. Experimental instructions are 327 available in Appendix A. 328 Regardless of the elicitation mechanism, participants were informed that the reduction 329 in methane emissions can fall in any percentage value between -6.6% and -10% given the 330 information retrieved from the literature (Arndt et al., 2022). Using the mean and confidence 331 interval reported in Arndt et al. (2022), we approximated an empirical benchmark distribution 332 by fitting a normal distribution, against which participants’ subjective probabilities were 333 compared. The normal distribution can be a good approximation because it provides a 334 continuous, two‐parameter reference distribution for comparing participants’ subjective 335 probabilities without further assumptions. Participants were instructed to allocate 70 tokens 336 15 across seven predefined intervals each representing a range of methane emission reduction 337 percentages using the interface depicted in Figure 1. These intervals included the following 338 percentages of reduction: (-10%, -9.6%), (-9.5%, -9.1%), (-9%, -8.6%), (-8.5%, -8.1%), (-8%, 339 -7.6%), (-7.5%, -7.1%), and (-7%, -6.6%). They were informed that the number of tokens 340 allocated to each interval should reflect their perceived likelihood of the corresponding methane 341 emission reduction occurring in the future. A higher allocation of tokens to a particular interval 342 indicates a higher subjective likelihood of that reduction range according to the participant’s 343 beliefs. 344 345 346 Figure 1. Example of the interface used in the economic experiment. 347 348 The experiment was incentivised. Participants did not receive any participation fee, but 349 they could get a reward of up to €25, knowing that, at the end of the session, one of the three 350 tasks would have been randomly extracted and used to determine their final payoff. Each 351 elicitation method has a different incentivisation scheme. The FM pays €25 if and only if the 352 probability distribution matches exactly the empirical distribution, otherwise it pays €0. 353 Matching exactly meant that the number of tokens allocated by the participant perfectly 354 corresponded to the probabilities defined by the empirical distribution. The IM pays €25 if and 355 only if the subjective probability distribution matches the empirical distribution with an error 356 margin equal to 10% in each interval, otherwise it pays €0. The payment scheme of the QSR is 357 completely different as the QSR allows for a more continuous spectrum of payoffs as explained 358 in Eq. 1. 359 16 4. Results and discussion 360 4.1. Comparisons across elicitation methods 361 Figure 2 shows the subjective probability distributions across elicitation methods 362 when the sample is pooled (dairy sector plus students). 363 364 365 Figure 2 Within-group average subjective probability distributions per task 366 367 We test differences in subjective probability distributions across elicitation methods 368 using the Wilcoxon signed-rank test for paired samples.4 Results suggest no statistically 369 significant differences across methods when the sample is pooled (dairy sector plus students). 370 Results hold when we estimate a generalized linear model with an underlying binomial 371 distribution for our dependent variable and a logistic link function a la Papke and Woolridge 372 4Results from the non-parametric tests are available in Table 4 in the Appendix B. 17 (1996) (see Table 1). This modelling approach is particularly well suited when the state space 373 of the dependent variable ranges from 0 to 1 as is the case in our study (e.g. Cerroni et al., 374 2012). Our dependent variable 𝑝𝑖,𝑘 is the probability associated with each participant 𝑖 to each 375 interval 𝑘 (i.e. state of the world) shown in Figure 2.5 Our discrete independent variables are 376 associated with the elicitation mechanism: 𝐹𝑀𝑘 equals 1 if the probability was elicited using 377 the FM, and 𝐼𝑀𝑘 equals 1 if the probability was elicited using the IM. The variable 𝑄𝑆𝑅𝑘 is 378 used as a baseline and equals 1 if the probability was elicited using the QSR.6 Standard errors 379 are clustered at the individual level. The model is specified as follows: 380 381 𝑃𝑖,𝑘 = 𝛼 + 𝛽𝐹𝑀𝐹𝑀𝑖 ,𝑘 + 𝛽𝐼𝑀𝐼𝑀𝑖 ,𝑘+ 𝜀𝑖,𝑘 (Eq. 2) 382 383 Table 1. Effect of elicitation methods on subjective probabilities (for each interval) Dep. Var.: (𝑃𝑖,𝑘 ) 1 2 3 4 5 6 7 FM 0.255 0.157 0.012 -0.130 0.294 0.156 -0.355 (0.619) (0.554) (0.486) (0.499) (0.500) (0.531) (0.518) IM 0.053 0.130 0.144 0.083 0.061 -0.011 -0.477 (0.643) (0.557) (0.476) (0.481) (0.518) (0.516) (0.532) Constant -2.358*** -2.031*** -1.592*** -1.596*** -1.803*** -1.757*** -1.576*** (0.459) (0.403) (0.344) (0.345) (0.370) (0.364) (0.343) Observations 180 180 180 180 180 180 180 LL -33.676 -28.225 -39.625 -34.450 -37.143 -36.065 -51.568 Note: *p<0.1; **p<0.05; ***p<0.01 384 LL stands for Log-Likelihood 385 Robust standard error in brackets 386 387 388 389 5In our study, seven possible states of the world were introduced. 6QSR serves as the baseline category, and thus it is not explicitly included as a variable in the equation. 18 These results suggest that different methods elicit equivalent subjective probability 390 distributions. We can conclude that less complex (i.e. cognitively demanding) elicitation 391 methods (FM and IM) are performing equivalently to a complex elicitation method such as the 392 QSR. Subjective probability distributions are stable regardless of the use of completely 393 different incentive schemes (FM and IM vs QSR). In addition, the difference in the saliency of 394 payoffs between FM and IM does not produce any significant effect on the elicited subjective 395 probability distributions. Therefore, we conclude that the use of less cognitively demanding 396 techniques like IM and IF might be a viable approach for the elicitation of subjective probability 397 distributions. We acknowledge that failure to detect differences across methods within subjects 398 may be because subjects tend to confirm their beliefs over time, showing a sort of confirmatory 399 bias (e.g., Rabin and Schrag, 1999; Zappalà, 2023). We evaluated the presence of confirmatory 400 bias by taking advantage of the task randomization. More specifically, we are comparing 401 subjective probability distributions elicited via the FM and the QSR when these were the first 402 methods faced by our participants. This between-subject analysis rules out the presence of 403 confirmatory bias. Results from the Kolmogorov-Smirnov test suggest that there is no 404 difference in subjective probabilities across groups, indicating that subjective probability 405 distributions are similar across methods even when confirmatory bias does not play any role.7 406 Results hold when estimating generalised linear models (Eq. 3) (see Table 2), as above. In this 407 case 𝐹𝑀_𝑓𝑖𝑟𝑠𝑡𝑘 equals 1 if the probability was elicited using the FM as a first task, and 408 𝑄𝑆𝑅_𝑓𝑖𝑟𝑠𝑡𝑘 equals 1 if the probability was elicited using the QSR as the first task. The latter 409 is used as a baseline. 410 411 𝑃𝑖,𝑘 = 𝛼 + 𝛽𝐹𝑀𝑓𝑖𝑟𝑠𝑡𝑘 𝐹𝑀𝑓𝑖𝑟𝑠𝑡𝑘 + 𝜀𝑖,𝑘 (Eq. 3) 412 413 414 7Results from the non-parametric tests are available in the Appendix B in Table 5. 19 Table 2. Effect of confirmatory bias on subjective probabilities (for each interval) Dep. Var.: (𝑃𝑖,𝑘 ) 1 2 3 4 5 6 7 FM_first -0.308 -0.589 0.313 0.459 0.112 0.056 -0.286 (0.509) (0.451) (0.406) (0.423) (0.418) (0.437) (0.435) Constant -2.080*** -1.622*** -1.729*** -1.893*** -1.746*** -1.844*** -1.678*** (0.368) (0.311) (0.323) (0.342) (0.325) (0.336) (0.317) Observations 180 180 180 180 180 180 180 LL -33.843 -27.684 -39.304 -34.388 -37.762 -36.414 -51.831 Note: *p<0.1; **p<0.05; ***p<0.01 415 LL stand for Log-Likelihood 416 Robust standard error in brackets 417 418 419 420 4.2. Comparisons across groups 421 Figure 3 and Figure 4 show subjective probability distributions across elicitation 422 methods for the dairy sector (farmers plus experts) and students, respectively. These provide 423 insights into probabilistic beliefs of dairy farmers, experts, and students across different 424 outcomes. It is noticeable that, overall, the probability mass is concentrated in the three central 425 intervals, suggesting that these groups generally expect moderate to average outcomes of future 426 reduction in methane emissions in dairy farming. This is a typical stance when evidence or 427 confidence in the topic is neither very strong nor very weak. However, it is interesting to note 428 that students assign a moderate probability (around 10%) to the scenario with the lowest 429 reduction in methane emissions. This probability rises to approximately 15% and 20% when 430 assessed by farmers and experts, indicating a degree of scepticism about the efficacy of 431 essential oils in reducing methane emissions, and signalling a need for further research on the 432 topic. 433 434 20 435 436 437 Figure 3. Dairy sector’s average subjective probability distributions per task 438 439 440 441 Figure 4. Students’ average subjective probability distributions per task 442 443 444 21 Differences in subjective probability distributions across groups (dairy sector vs 445 students) were assessed using the Kolmogorov-Smirnov test for the between-subjects 446 comparison.8 Results indicate that there are no significant differences in token allocation across 447 groups. Equivalent results are obtained when we estimate a generalized linear model (binomial 448 distribution and logistic link function as above) where the dependent variable 𝑝𝑖,𝑘 is the 449 probability associated with each participant 𝑖 to each interval 𝑘 (i.e., state of the world) shown 450 in Figure 1 (see Table 3). Our discrete independent variable is 𝐷𝑎𝑖𝑟𝑦_𝑆𝑒𝑐𝑡𝑜𝑟𝑘 that equals 1 if 451 the probability was elicited from a farmer or an expert (otherwise = 0), while students were 452 used as a baseline. Standard errors are clustered at the individual level. The model is specified 453 as follows: 454 455 𝑃𝑖,𝑘 = 𝛼 + 𝛽𝐷𝑎𝑖𝑟𝑦_𝑆𝑒𝑐𝑡𝑜𝑟𝐷𝑎𝑖𝑟𝑦_𝑆𝑒𝑐𝑡𝑜𝑟𝑖,𝑘 + 𝜀𝑖,𝑘 (Eq. 4) 456 457 Table 3. Effect of being a farmer or expert (dairy sector) on subjective probabilities (for each interval) Dep. Var.: (𝑃𝑖,𝑘 ) 1 2 3 4 5 6 7 Dairy Sector -0.157 -0.339 0.210 0.106 -0.537 0.179 0.461 (0.509) (0.453) (0.392) (0.400) (0.419) (0.430) (0.441) Constant -2.175*** -1.775*** -1.648*** -1.663*** -1.436*** -1.904*** -2.088*** (0.348) (0.299) (0.286) (0.288) (0.268) (0.314) (0.336) Observations 180 180 180 180 180 180 180 LL -33.884 -27.904 -39.335 -34.399 -37.208 -36.064 -51.289 Note: *p<0.1; **p<0.05; ***p<0.01 458 LL stands for Log-Likelihood 459 Robust standard error in brackets 460 461 462 8Results from the non-parametric tests are available in Table 6 in the Appendix B. 22 These results may be driven by the fact that agricultural students are very aware of 463 climate change mitigation strategies that can be implemented at the farm level and hence they 464 are as knowledgeable and aware as farmers and experts regarding the specific topic under 465 investigation. On the other hand, it might be possible that, given the novelty of the topic and 466 the high degree of uncertainty related to the efficacy of introducing feed additives to reduce 467 methane emissions from livestock production, both students, farmers and experts have little 468 knowledge and awareness of the problem. 469 470 5. Conclusion 471 This study compares three indirect methods —FM, IM, and QSR—to elicit farmers’, 472 other experts’ and students’ subjective probability distributions about a specific agricultural 473 outcome using a framed economic experiment. The outcome variable considered is the methane 474 emission reduction which can be achieved by the introduction of zootechnical feed additives to 475 bovine diets. Feed additives, more specifically essential oils, are readily available and cost-476 effective solutions to reduce the carbon footprint of animal production (Belanche et al., 2020). 477 As the EU is designing policies that require farmers to reduce their GHGEs (e.g., the Green 478 Deal, the Farm-to-fork Strategy, and the Industrial Emission Directive), understanding farmers’ 479 and other stakeholders’ beliefs regarding the effectiveness of GHGE-reducing innovations can 480 help anticipate adoption behaviour. 481 The three elicitation methods differ in two main features: the level of complexity (i.e., 482 the cognitive cost of the elicitation mechanism for participants) and the saliency of the incentive 483 scheme (i.e., the ability of the incentive scheme to alter behavior). Following Charness et al. 484 (2021), we classify the QSR as complex, and the FM and IM as simple. Similarly, the IM and 485 the QSR have more salient incentive schemes than the FM. The QSR has been recently used to 486 elicit farmers’ subjective probability distributions using proper incentive schemes (Cerroni et 487 23 al., 2020; Cerroni et al., 2023; Čop et al., 2023), while the FM and IM have been applied without 488 incentivization in earlier studies (e.g., Menapace et al., 2013; Čop et al., 2023). 489 In our study, each participant was asked to provide a subjective probability distribution 490 when exposed to these three different methods. The order of the tasks was randomized to 491 mitigate order effects. Our sample consists of farmers, other experts (i.e., animal nutrition and 492 production scientists, veterinaries as well as feed, dairy, and meat company representatives), 493 and postgraduate agricultural students. Our results indicate that, on average, the subjective 494 probability distributions elicited using the three methods do not differ significantly, implying a 495 consistent elicitation through tasks. This leads us to suggest that less complex elicitation 496 methods (FM and IM) may be preferable when feasible in experimental studies. No difference 497 has been highlighted between groups. In fact, subjective probability distributions were elicited 498 consistently across the dairy sector (i.e., farmers and experts), and agricultural students. Dairy 499 farmers, experts, and students generally expect moderate to average reductions in methane 500 emissions from essential oils. However, while students see a moderate chance of achieving the 501 lowest methane reduction possible, dairy farmers and experts are more sceptical, highlighting 502 the need for further research on the efficacy of essential oils. 503 Little variation across groups may suggest that agricultural students are as 504 knowledgeable as experts and farmers about reductions in methane emission that can be 505 achieved using essential oils. Nevertheless, an alternative explanation is that, given the high 506 degree of uncertainty surrounding the outcome variable, both the dairy sector (farmers and 507 experts) and students form similar uncertain beliefs. 508 In this regard, as the exercise focused on a highly uncertain outcome, that is methane 509 reductions achievable through the use of essential oils, about which even experts hold wide-510 ranging opinions, it remains an open question whether method performance would remain 511 24 equivalent when eliciting beliefs about more familiar or less ambiguous outcomes (for example, 512 crop yields or weather events). 513 Although our primary objective was methodological (comparing FM, IM, and QSR) 514 rather than producing broadly representative estimates, the modest sample size may still cast 515 doubt on external validity. To reinforce and extend these findings, future research should 516 replicate the experiment with larger, stratified samples spanning different regions, crop 517 systems, and farm structures. 518 In practical terms, our results offer clear guidance for researchers in agricultural and 519 environmental economics: simpler, low-burden elicitation methods can be as effective as 520 complex scoring rules, easing implementation without compromising data quality. To build on 521 this work, future research should (1) replicate the comparison across different domains of 522 uncertainty, (2) employ larger, stratified samples to bolster statistical power, and (3) explore 523 alternative interfaces or incentive schemes. 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