Abstract
Public lotteries for non-market goods provide a unique field counterpart to experimental lotteries with monetary payoffs for investigating risk and time preferences. In this paper random expected utility and rank dependent expected utility models of public lottery choice are estimated under constant relative risk aversion preferences and hyperbolic time preferences with a balanced panel of 7,924 participants in the 127 lotteries for elk hunting licenses in New Mexico in the year before and after a set of natural policy experiments. Consistent with experimental findings, significant heterogeneity in risk and time preferences is found in applicant choices, and applicants are found to significantly place a nonlinear weight on both the probability of being drawn and of harvesting an elk. Mean discount rate estimates are also on a par with experimental results documented in the growing joint risk-time preferences literature.
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Barseghyan et al. (2018) detail three steps for estimating risk preferences in the field, including identifying an appropriate setting, translating it into a set of lotteries from which choice models yield predictions of risk preferences, and accounting for preference heterogeneity in the process. Field settings investigated to date include homeowner choice of property insurance policy (Cohen & Einav, 2007), household choice of health-insurance coverage (Handel, 2013), and contestant choice in television game shows (Andersen et al., 2008b).
Returning to the Grand Canyon example, in 2018 the choice set contained 468 lotteries for launch dates between January 1 and December 31 of 2019. Payment of a non-refundable deposit ranging from $200 to $400 was required of successful applicants within five days of the random draws.
In the years evaluated the New Mexico lottery allowed applicants to identify their first and second most preferred lotteries. If licenses happened to remain for a hunt after the first round of random draws, then the licenses were randomly assigned in the second round if the number of unsuccessful first-round applicants who ranked the hunt as second most preferred exceeded the number of remaining licenses. If instead the number of unsuccessful first-round applicants was less than the number of remaining licenses, then the applicants would receive one of the licenses. Applicant first choice lotteries are used in the analysis below.
About 40% of the 127 lotteries had the lower of the two license fees, and a non-refundable $6 application fee was required in both years. For comparison, non-resident license fees in both years ranged from $275 to $750. The lotteries, rules, regulations, and application materials are documented in the annual New Mexico Big-Game Proclamation (NMDGF, 1996, 1997).
In the pre-payment period, resident and non-resident applicants were drawn from the same pool, and residents received about 78% of the licenses. In the post-payment period, applicants were drawn from separate pools, and the quota allotted residents 78% of the licenses in each lottery. Twelve percent were allotted to non-residents who contracted with a New Mexico guide and outfitting service.
In addition to the licenses allocated by lottery, the NMDGF had 8 elk hunts with an unlimited number of licenses, and these are also reported in Table 2.
Ninety-seven percent of the awarded licenses were purchased by applicants before the payment deadline, and remaining licenses were sold on a first-come first-serve basis.
In both years the number of licenses to be awarded ranged from 5 to 350, and the average lottery awarded 93.2 licenses. In allocating licenses to resident applicants under the 78% license quota, the NMDGF rounded to the number of whole licenses.
Because the hunts have varying attributes over which applicants have preferences, and the probability of being awarded a license varies over lotteries, expected utility will differ over the lotteries, and if two or more lotteries were to have the same expected utility then the level of risk associated with the lotteries will differ. Considering the vector of hunt attributes (x) and parameter vector β, let z = xʹβ reference the composite argument of the utility function (i.e., u(z)). Risk aversion is then reflected through convexity of u(z). For a hunt attribute in x that is continuous, convexity of u(z) and risk aversion may then be reflected through diminishing marginal utility in the attribute, all else constant.
Though the dominant characterization, findings that do not support hyperbolic time preferences have also been reported. For example, results from field experiments conducted by Attema et al. (2018) involving temporal decisions with health-related and monetary outcomes support constant rate discounting over hyperbolic discounting. Blavatskyy and Maafi (2018) also find limited evidence supporting hyperbolic discounting in repeated discrete choice experiments involving temporal decisions with monetary outcomes.
Reeling et al. (2020) also use multiple years of data in their lottery choice analysis. Given the dynamic structure of their choice model, they assume the choice environment is in a steady state, and the state variables in the model are determined by individual behavior in prior periods, which is sufficient for non-parametric identification of the choice probabilities as shown by Verdier and Reeling (2022). Identification of the utility parameters in their model follows by appealing to results from Hotz and Miller (1993).
Wakker (2010) provides a comprehensive survey of the evolution and empirical analysis of RDEU models and discusses several probability weighting functions evaluated in the economics and psychology literature.
The choice models were also specified according to the model proposed by Rouwendal (1989) in order to account for possible realization uncertainty associated with lottery choices. This entails including a scaling term g(.) that is an increasing function of p(k) as a hunt attribute. Resulting estimates included discount rates not differing significantly from zero or which are negative and risk aversion of a high degree. Akabua et al. (1999) compare results from Rouwendal’s model to those from Boxall (1995), which weights indirect utility by p(k) in a manner similar to the model evaluated here. And Reeling et al. (2020) propose an approach to account for realization uncertainty that is a variant of the heteroscedastic extreme value model proposed by Bhat (1995) and also compare results against those from Rouwendal’s model. Both report differences in welfare estimates between the various model specifications under assumed risk neutral preferences.
For joint estimation of risk and time preferences it is important to note that conditional logit choice probabilities may be non-monotonic if individuals are sufficiently risk averse. Apesteguia and Ballester (2018) show that monotonicity is ensured if the choice probabilities are specified as mixed (or random parameters) logit. Notwithstanding, the conditional logit model is adopted here due to the large number of parameters in the lottery choice models, nonlinearities across the CRRA utility function, hyperbolic discount factors, and probability weighting function, and the sizes of the lottery choice sets and applicant samples.
The application book probabilities are therefore assumed to be applicant subjective probabilities. However, lottery choice data could be augmented with data collected from belief elicitation tasks in order to estimate applicant subjective probabilities. Andersen et al. (2014b) and Harrison et al. (2017) evaluate such belief elicitation tasks with linear and quadratic scoring rules.
The range of mileage costs is similar to that evaluated in Hagerty and Moeltner (2005). They propose adjustments to mileage costs reported by the American Automobile Association to reflect vehicle type and towing load and consider average mileage costs between $0.30 and $0.61. As the New Mexico elk hunts occur in high elevation, mountainous regions across the state, and mature elk weigh between 500 and 700 pounds, actual travel costs may exceed those evaluated here. The average travel cost measured at $0.50 per mile is about $227 in both years, and the average travel cost of the chosen lottery is about $152.
The ‘quality’ hunt designation was the result of state legislation in 1995. Quality hunts “… are designed to provide a hunter with an increased opportunity to achieve a successful harvest, a harvest from a wider selection of mature deer or elk, and a pleasurable experience.” NMDGF (1996, p. 7; 1997, p. 13). Considering the inclusion of the number of licenses to be awarded as an attribute, as the number is the result of herd management objectives, it reflects the maximum number of elk the NMDGF will allow to be harvested in each hunt, which in general will be positively related to herd size. Larger herd sizes as reflected by hunts with larger numbers of licenses to be awarded will allow a license holder to select an elk to harvest from a larger number and wider variety of elk. The underlying hypothesis is that hunts with larger numbers of licenses will be more desirable to hunters, all else constant.
The full set of estimation results is reported in Appendix Table 12. For comparison, EU estimates of r range from 0.16 to 0.25 with per mile costs of $0.30 and from -0.01 to 0.12 with per miles costs of $0.70. RDEU estimates range from 0.27 to 0.30 with per mile costs of $0.30 and from 0.05 to 0.14 with per miles costs of $0.70. Estimation results with mileage costs of $0.30 and $0.70 are available in an appendix.
Mixed evidence of probability weighting is documented in the literature. Tversky and Kahneman (1992) estimated γ = 0.61 (p < 0.01), comparable to the estimates reported in Table 3. Harrison et al. (2020) also found evidence of probability weighting using a version of ω(p) proposed by Prelec (1998) and report ω(p) ≈ p for p ≈ 0.368. Alternatively, Laury et al. (2012) evaluated the Tversky and Kahneman weighting function used here and found no significant evidence of probability weighting.
For comparison, EU discount rate estimates range from 4.3% to 8.0% with per mile costs of $0.30 and from 17.8% to 28.7% with per miles costs of $0.70. RDEU estimates range from 3.4% to 4.9% with per mile costs of $0.30 and from 15.6% to 20.7% with per miles costs of $0.70. Results are available in an appendix.
Male equals 1 (0) if the applicant is male (female). Age is calculated as the number of years between applicant date of birth and the application deadline and scaled by 100 for estimation. Drawn is assigned a value of 1 to applicants in the post-payment period who were drawn in the pre-payment period and a value of 0 otherwise, and Not Drawn is assigned a value of 1 to applicants in the post-payment period who were not drawn in the pre-payment period and a value of 0 otherwise. In the pre-payment period Drawn and Not Drawn are both equal to 0.
The first and second stage outcomes are therefore discrete random variables. Alternatively, in the Grand Canyon river-rafting lotteries discussed earlier, individuals apply for permits a year in advance of launch dates, yet river flows and thus the recreational experience will depend upon Rocky Mountain snowfall in the following year. The first and second stage outcomes in this case are discrete and continuous, respectively.
Heterogenous preference estimates are reported in Appendix Table 15.
Haab and McConnell (2002) provide a detailed overview of the derivation and estimation of compensating and equivalent welfare measures from discrete choice models.
∆F0 and ∆F1 with the RDEU models are obtained by replacing the lottery probabilities in (23) and (26) with probability weights. ∆F0 and ∆F1 were also derived from the compound lottery models. The difference in ∆F0 and ∆F1 between the simple and compound lottery models is that u(x*) in (23) and (26) is replaced by p(h*)u(z*,1) + (1—p(h*))u(z*,0), where p(h*) is the average harvest rate. And with the RDEU compound lottery choice models, p(h*) is replaced by the respective probability weight.
It should be noted that because the lottery choice models are estimated with data that includes applicants who participated in both years versus those who participated in both years and those who participated in one year but not the other, the welfare estimates do not reflect the option for applicants to respond to a price change by not participating.
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Scrogin, D. Estimating risk and time preferences over public lotteries: Findings from the field and stream. J Risk Uncertain 67, 73–106 (2023). https://doi.org/10.1007/s11166-023-09404-4
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DOI: https://doi.org/10.1007/s11166-023-09404-4