Abstract
Estimating discrete choices under uncertainty typically rely on assumptions of expected utility theory. We build on the dynamic choice modeling literature by using a nonlinear case-based reasoning approach based on cognitive processes and forms expectations by comparing the similarity between past problems and the current problem faced by a decision maker. This study provides a proof of concept of a behavioral model of location choice applied to recreational fishers’ location choice behavior in Connecticut. We find the case-based decision model does well in explaining the observed data and provides value in explaining the dynamic value of attributes.
1. Introduction
The random utility model (RUM) is the workhorse of discrete choice analysis in economics, which includes but is not limited to location choice modeling, travel cost analysis, choice experiments, and contingent valuation. RUM spans both revealed-choice and stated-choice research across most disciplines of economics to explain choice behavior. The explosion in modeling discrete choice behavior and estimating demand from these choices can be traced to the early 1970s, when luminaries such as Daniel McFadden pioneered work in discrete choice modeling and economic choice (McFadden 1974, 2001; Manski 1977). The stochastic utility models underlying this literature in practice usually make strong assumptions about rationality. When applying these approaches to empirical data, the general practice is to choose models that exhibit high levels of rationality and are linear combinations of explanatory factors or a reduced-form specification.1 These assumptions are justified in the sense that estimation is easy to compute, the model is consistent with neoclassical theory, and the model is easy to interpret. However, the functional form of the utility and primitives of decision making is ideally based on the assumptions about how people make a choice (Koppelman 1981). The present work builds on the dynamic choice framework by introducing a method of estimation based on case-based decision theory (CBDT) to environmental and natural resource economics.
Expected utility theory (EUT) posits that decision makers conceptualize states of the world and assign probabilities to those states, updating with Bayes’s rule. CBDT takes a different approach, where knowledge or beliefs of all states of the world are not necessary. Instead, the primitives of CBDT are the problems, actions, and results, which as a triplet constitutes a case in CBDT. Information in CBDT enters through the memory of the decision maker, which includes past cases. Problems in CBDT are the choice situations faced by the decision maker. CBDT posits that decision makers use the psychological concept of similarity between past problems in memory and the current decision problem to maximize utility. CBDT sums over cases to find the utility of different actions for a given problem, while EUT sums over all state spaces for each action.
In many decision problems for environmental and natural resource economics, the primitives of EUT may be unnatural to define and difficult to formulate. For example, take a recreational outing to a beach. In EUT, the states of the world include all possible experiences from the trip to a particular beach is a daunting task to define. Asking a beach-goer what their priors were over each possible outcome and relative states of the world is likely to be unsuccessful. This does not seem like a plausible cognitive description of the decision problem. Furthermore, memory may be defined as the entire cross-sectional choice set, which may not be appropriate. CBDT instead supposes that decision makers under uncertainty ask how similar the trip is to a reference trip, perhaps to past trips to beaches. When faced with a new decision, the decision maker’s memory is used to form expectations about the utility of a beach trip. We can think about many environmental and natural resource decision problems that are similar in complexity to this example: location choice for recreation (surfing, birdwatching, hiking, fishing) or extractive natural resource decisions (irrigation, forestry, fisheries). For example, we can use the CBDT framework to model crop and irrigation technology choice decisions for farmers where the potential states of the world are difficult to define, yet history and experience may be available to guide the researcher on defining memory.
Recreational fishers face a location problem of where to go fishing. The EUT relevant state space would need to define all possible distributions of different species of fish and probabilities of catching the target species. This includes all theoretical possibilities as EUT fishers are never surprised by the existence of a state space; rather, they update priors about the likelihood of them occurring. Then fishers would need to assign utility outcomes to each state space, a considerable task. CBDT simplifies this task considerably and is a more natural way to formulate the decision problem. The fisher draws on experiences to form expectations and uses the similarity between decision “problems” to assess the utility of each location under the conditions of the current decision problem.
Several studies rely on nonlinear utility models to explain behavior in discrete choice literature (Martínez-Espiñeira 2006; Kim, Rasouli, and Timmermans 2014). When choice data exhibit repeated choices, studies have introduced state dependence (SD) as a control variable (Smith 2005; Cantillo, Dios Ortuzar, and Williams 2007; Abbott and Wilen 2010). SD is a specific way to address serial correlation (Heckman 1981). Like SD, CBDT is a dynamic choice framework that maps how past choices influence current choice behavior. The difference between conventional methods used in SD literature and our study is that CBDT is context-specific and based on the agent’s experience of past trips, the attributes of those trips, the trips’ success, and how similar those trips are to the current choice problem.
There are required assumptions when modeling discrete choice with linear additive parameters. First, individual choice is informed by all observations; in other words, a person’s memory is complete with all observable instances of the data. This implicitly relies on the idea that state spaces are known and calculated by individuals. Second, people use rule-based reasoning to make decisions based on the functional form of utility, namely, that it is linear and additive in components. People use rules that average the effect of dependent variables on the choice variable across observations. Case-based reasoning posits that people take cases from memory and compare the similarity of past problems with the current decision problem to form an expected utility of choices. In other words, individuals reason through analogies to make choices, rather than reason through rules. Based on this notion, people would expect similar problems to have similar outcomes (Gilboa and Schmeidler 1995). There is support in psychology and economics for case-based reasoning where people weigh their own experiences more than other available information. This suggests that there are apparent bounds to what is contained in a person’s memory while they make decisions. In short, there is a reasonable constraint on human choice (Shepard 1987; Pape and Kurtz 2013; Bleichrodt et al. 2017). Individuals may use rule-based reasoning or case-based reasoning or a combination in practice. Only careful inspection of observed choice can illuminate the decision process.
The implications for using the case-based reasoning framework on discrete choice questions are twofold. First, if the data-generating process that creates choice data is different from the model used, we will be more likely to underperform in out-of-sample prediction. A model consistent with what we know about choice behavior should be better at predicting out-of-sample, which are of particular importance to policy (i.e., climate change scenarios, hypothetical scenarios, location closures).2 Second, using the wrong model for inference on choices will impair our estimates for welfare. Therefore, a model that incorporates what we know about the psychology of choice and explains the data well is likely to provide a better measure of demand. Other researchers make the argument that welfare analysis should be based on our understanding of the behavioral processes that generate the data (Rubinstein and Salant 2012; Manzini and Mariotti 2014; Cerigioni 2021).
Several studies in the economics literature show that CBDT performs well in explaining empirical data. Ossadnik, Wilmsmann, and Niemann (2013) conduct a repeated choice experiment where individuals’ choice behavior was assessed based on an urn ball experiment using maximin decision criteria, a reinforcement learning model, and CBDT. The results revealed that CBDT explains the experimental data better than the other models. Guilfoos and Pape (2016, 2020) find that CBDT explains experimental game theory behavior well in prisoner’s dilemma and mixed strategy equilibria games. Pape and Kurtz (2013) and Kinjo and Sugawara (2016) show that CBDT explains data well regarding human classification learning and viewing decisions of Japanese TV dramas, respectively. CBDT predicts decisions well in several empirical settings. However, this theory has never been applied in nonmarket valuation studies or location choice modeling, and welfare implications have not yet been explored. Furthermore, CBDT has not been adapted to empirical applications in dynamic choice environments, except Pape and Kurtz (2013) and Guilfoos and Pape (2020). Our study builds on the estimation methods presented in Guilfoos and Pape (2020) and applies CBDT to a dynamic empirical application on observed choices outside a laboratory setting.
Location choice behavior is important for environmental policy and management. It reveals preferences for attributes and can illuminate important policy choices for nonmarket goods. We apply CBDT to recreational fishers’ location choice data set. Recreational fishers, unlike commercial ones, have varying motivations, such as spending time with friends and family, catching a trophy fish, deriving aesthetic pleasure, and catching a target species (Rubio, Brinson, and Wallmo 2014). Research on choice behavior of recreational fishers is essential, as this activity contributes a value addition of $50 billion, generating more than 553,000 employment opportunities, and providing $89 billion as annual income in the United States as of 2019.3 As a result, conserving fishing locations and maintaining an adequate level of fish populations to sustain recreational fishing are essential economic incentives for the nation. Fisheries management strives to conserve fishing areas, protect marine life, avoid fish stock depletion, and administer policy changes that may cause unintended consequences, especially in the behavior and distribution of recreational fishers (Pauly, Watson, and Alder 2005). However, choices made by fishers are dependent on numerous factors, some of which are uncertain and unobservable to the researcher (Holland 2008). Therefore, a clear understanding of site selection behavior enables us to design effective regulatory measures and understand how fishers respond to management policies (Cinti et al. 2010).
This article provides a proof of concept for CBDT in location choice modeling. We find that CBDT and the SD model appear to fit the recreational fishing location choice data well. In-sample goodness of fit favors the SD model, and out-of-sample goodness of fit favors CBDT. Sometimes, the goodness of fit is close between the two models. When investigating model differences by species, it appears that heterogeneity of choice behavior between species may be responsible for the good performance of the SD model using in-sample goodness of fit. Goodness of fit is important in identifying likely patterns of behavior and understanding the data-generating process (Atkinson 1981).
Using simulations, we show that there may be significant deviations in measures of willingness to pay (WTP) when applying a linear model if the underlying data-generating process is consistent with CBDT. A linear model consistently overstates WTP for the catch of a preferred species, which is statistically different from the “true” parameters and can be overstated by up to 35%. Furthermore, a CBDT model is consistent with the psychological mechanisms for choice, which may be why the data may appear to fit the CBDT framework slightly better. We contend that models that can match known mechanisms for choice may provide a better basis for making inferences about welfare; this appears to be true in other behavioral economic models, such as loss aversion or present-biased choices (Thaler 2016)
2. Rule-Based and Case-Based Reasoning in Location Choice
To clarify the differences between rule-based reasoning and case-based reasoning (reasoning by analogy), we provide examples of both. Suppose a person is interested in purchasing a boat and is deciding which boat satisfies certain attributes (size, color, style) while constrained by a budget. A rule-based decision would reason “I want to buy a boat, and boats cost $1,000 per additional foot of length,” while reasoning by analogy would state “My friend’s boat cost $20,000, and I want to buy a boat of the same size and characteristics, so it should cost a similar amount.” The predictions of rule-based reasoning and case-based reasoning could be similar, but the processes differ in the decision-making mechanisms. In location choice modeling, reasoning by analogy is intuitive, as agents choose to visit locations similar to past locations that generated high levels of utility. The same reasoning might present itself negatively as well: “We had a horrible time at beach A, and beach Z is very similar to beach A, so we will not visit beach Z.” Case-based reasoning can also fit into the RUM, although CBDT suggests a specific functional form and draws its inference through the concept of memory (Guilfoos and Pape 2020). CBDT is a close relative of reinforcement learning, which draws on similar psychological support (Shepard 1987; Gayer et al. 2007; Guilfoos and Pape 2016, 2020).
CBDT was introduced by Gilboa and Schmeidler (1995). This decision theory captures the thinking process of a decision maker based on the similarity of circumstances. The CBDT framework could be useful for exploring environmental and natural resource economic issues because it provides a framework to estimate welfare for new hypothetical location choices without complete knowledge and assignment of probabilities for all state spaces. For example, a new public park, the restoration of fishing ground, or other conservation initiatives can change the set of possible locations. This theory hypothesizes that decision makers rely on stored memory, experience, and reasoning by analogy to choose whether to visit locations and how they derive value from that choice.
How a resource user chooses a location to visit is difficult to know and construct (Hess, Daly, and Batley 2018). For example, fishers seem to qualitatively assess alternative locations to visit based on intuition and experience. Ethnographic interviews conducted by Holland (2008) show that fishers’ choice behavior often does not conform to the assumptions of expected utility. Some of the anecdotal findings include that safe and consistent returns were preferred over maximum fishing catch. However, as with other location choice modeling, fishing location research has relied on linear additive (LA) models. Bockstael and Opaluch (1983) were one of the first to incorporate uncertainty in the fisher’s choice model via RUM. Mistiaen and Strand (2000) use a mixed multinomial logistic model to understand the short-run heterogeneous risk preferences in fishing choice behavior. Similarly, several other studies also use LA models to examine fisher behavior when it comes to location choice preferences (Mistiaen and Strand 2000; Smith 2005; Ran, Keithly, and Kazmierczak 2011).
The recreational fishing literature focuses on collecting all attributes that could influence behavior, such as cost to travel to the fishing site, fishing quality, water quality, congestion in the site, expected catch, and site history (Morey, Shaw, and Rowe 1991; Train 1998; Rubio, Brinson, and Wallmo 2014). We propose to characterize the same attributes with similarity from past experiences generating expectations and form utility, much like reinforcement learning, where individuals choose locations based on expectations formed through case-based reasoning.
3. Methods
Random Utility Theory
In random utility theory, an individual decision maker faced with a finite choice set K assigns a utility value to each choice (U1,U2,…,UK), depending on a vector of individual-specific, time-specific, and alternative-specific characteristics denoted as X. The decision rule behind this framework hypothesizes that the decision maker would choose an alternative j ∈ K, where the utility derived from j is the maximum possible utility from the given choice set (McFadden and Train 2000; McFadden 2001). The probability of choosing the alternative j is given in
1
The random utility function Uj is the utility attained by the decision maker, given the vector of attributes influencing the decision. This utility is a combination of both deterministic and stochastic components, Uj((X;θ), εj). The deterministic component contains the observed vector of attributes, X, and θ is the parameter vector. j is the random component of utility. The unobserved portion is assumed to be independently and identically distributed (iid). Utility is then expressed as
2
where Unj is the utility function for the nth individual choosing the alternative j.
The functional form of utility could take many forms. The LA version takes information about the decision maker and site characteristics and uses equation [3] to model location choice. We refer to this model as the LA RUM:
3
CBDT
CBDT is a behavioral model of decision making that we incorporate into the RUM approach. This theory measures utility by incorporating the similarity between the current scenario and scenarios in memory, which are called “cases.” According to CBDT, every individual has a memory (M), which stores a set of cases (C). Each case is a combination of a set of problems (P), a set of actions (A) taken to resolve the problem, and the subsequent set of outcomes or results (R) from applying the action to the problem. CBDT assumes that people refer to their memory of cases and form expectations based on the weighted similarity of past cases and the current problem. A similarity function weighs the similarity between the current problem (p) and past problems (q). Past problems, q, need not be drawn from the decision maker’s own experience. These memories could be drawn from outside observations, or they could be hypothetical constructs. The expected utility is a combination of the cases in memory and the results of those cases, weighed by the similarity function. Another component considered in CBDT is the aspiration level (H). Aspiration denotes the satisficing amount of utility the person pursues. A combination of these components—(similarity function, utility function, and aspiration level)—provides an individual’s case-based utility (Gilboa and Schmeidler 1995).
In the recreational fisheries context, M is the set of fishing trips stored in the fisher’s memory. The problem, P, is defined as each fishing trip’s attributes, such as weather conditions, travel cost, or day of the week of the trip. The action, A, is the chosen fishing location. The result, R, is a binary indicator variable equal to one when the fisher catches the target species and zero otherwise.4 The aspiration level, H, for the fisher is the satisficing level of utility derived from the fishing trip. According to this model, the weighted similarity index between past (q) and current problems (p) of the fisher and the results of past trips will form their expectations of utility.
The psychology literature provides surprisingly specific guidance on the form of a similarity function and measures of distance between information in the definition of the problem. Shepard (1987) argues that a specific psychological function that generalizes distances in conditions that can be invariant to monotonic transformations is desirable. He further argues that this generalization is consistent with a general law of how any similarity between stimuli can be experienced by people. His work suggests an exponential decay similarity function, with Euclidean distance as an approximation of the general invariant monotonic function that generalizes between stimuli. Shepard (1987) further argues that these measures have an evolutionary basis and are found to be consistent with the learning data. While any similarity function that is decreasing in distance measures in practice could be applied to the data, we choose one that psychology has suggested emerges from the generalized learning responses from stimuli. This function establishes the resemblance between past problems and the decision maker’s current problem. As per CBDT, each fisher will have a set of cases stored in memory that can be referred to when making current decisions. The similarity function is
4
where w is the estimated weight between a vector of information from the current case (p) and past case (q), and where d(.) denotes a distance metric between elements of p and q. The greater the resemblance between information in the two cases, the greater the estimated weight given to the past case.
The consequent case-based utility (CBU) function is
5
In equation [5], the CBU for individual i for location choice j, includes the similarity function s(w,p,q), the utility function, U(r), which denotes the utility derived from the result, r, and H which is the aspiration level. We have constrained the aspiration level to be zero because identification is confounded when estimating the initial attractions to locations and the aspiration level jointly.5 M denotes the level of memory the individual has that includes all the cases involved with the chosen alternative j. The CBU is then measured by taking the summation of the similarity function, weighted by the difference between U(r) and H (Guilfoos and Pape 2016). The maximum likelihood estimation procedure estimates the most probable parameters to obtain the observed data.
Distance Measures
The approximation of similarity and distance measures that Shepard (1987) suggests is the functional form we have used in equations [4] and [6], which uses a Euclidean distance measure (Shepard 1987; Nosofsky 1992). The suggested approximation also works with a “city block” distance function (Shepard 1964, 1987; Aulet and Lourenco 2021), which we also estimate. The distance function that follows the Euclidean distance metric (d(w,p,q)E) is
6
In equation [6], v denotes the explanatory variables used in the model. This similar functional form was used in Pape and Kurtz (2013) to describe data from a human classification learning problem experiment. Guilfoos and Pape (2020) used the same functional form in mixed strategy equilibria games and found that it performed well in describing the data from those experiments. The other measure of distance used in psychology is called the “city-block distance”:
7
SD Model
The conceptual understanding of SD, following Heckman (1981), is the influence of a person’s past experience on current decisions. The typical method to incorporate SD is to linearly add proxy variables that capture individual past experience to the utility function. To evaluate the general performance of CBDT, we estimate a model often used in the SD literature. Similar to Guadagni and Little (1983) and Keane (1997), variables that are serially correlated measures of individual past choices are included as controls in this model.
Apart from CBDT, we estimate two other models for comparison. The first model uses the cross-sectional data to make predictions with a LA combination of controls. The second model includes the additional SD variables, the weighted average of past location choices nested in the LA model. Equation [8] defines the SD variable xijt−1:
8
The SD variable is a dynamic measure of past location choices, where the variable α determines the weight of past choices. The variable y equals one for a visit to site j at time t for individual i and zero otherwise. The parameter α acts like a discount factor on past choices, similar to a recency parameter in CBDT, which measures the distance between current choice and past choices in the choice problem. The SD variable requires an initial value, which we set to zero. The combination of the LA model with all the controls, including the SD variables, is similar to models from Smith (2005) and Smith and Wilen (2002) in the fisheries literature.
The choice of α could be optimized by the researcher, adding another degree of freedom for model choice. In Appendix A, we show a range of α and the resulting model goodness-of-fit measures. We note that the SD model fit does not vary much based on the choice of α. The performance of the model is slightly affected by α when using out-of-sample measures. We use α = 0.50 in the main text because it performs well in the out-of-sample tests. Our findings are robust to different values of α.
Stochastic Choice Rule
A common stochastic choice rule applied in discrete choice modeling literature is the logit response model. The multinomial logistic model is used when the choice set faced by a person has multiple discrete alternatives (Sellar, Chavas, and Stoll 1986; Parsons, Jakus, and Tomasi 1999). For instance, recreational fishers have multiple fishing sites in their choice set. The choice probability that a decision maker chooses one of the alternatives, j ∈ K is
9
where Unj(β,xnj) is the utility of alternative j for individual n, which is a linear additive function of attributes (x) in the LA model and a summation of utility weighted similarity functions for CBDT. The sensitivity parameter, γ, which is assumed to be one in LA models, is estimated in CBDT. γ is important to the estimation of learning models on laboratory data of discrete choice and is considered in Guilfoos and Pape (2020). As γ approaches zero, the data appear to be completely random to the model predictions; as it approaches, the model appears to be more deterministic in fitting the data. The choice rule implies that the probability of a fisher choosing site j from choice set K is the exponential of the utility from site j divided by the sum of all of the exponentiated utilities (McFadden 1974; Ben-Akiva and Lerman, 1985).
An important critique against the multinomial logit choice model is the assumption of independence of irrelevant alternatives (IIA). This assumption implies that the utility of one alternative is solely influenced by individual-specific characteristics, which are constant across alternatives (Train 1998). To counter this limitation and account for other considerations, such as heterogeneity and taste variations, other models are used, such as nested logit, latent class, and mixed multinomial logit. Such models rely on location-specific variation that involves additional conditions to the multinomial choice probabilities (Ben-Akiva et al. 1997; McFadden 2001). We apply the original multinomial logit choice process to compare the LA rule-based model with the case-based reasoning model, CBDT. It is important to note that both models would suffer equally with such limitations.
4. Welfare Analysis with CBDT
Welfare estimation is essential for policy evaluation; therefore, we need to understand how CBDT choice affects our estimates of WTP for goods. An important assumption when measuring welfare in discrete choice models is the interpretation of the cost coefficient as the marginal utility from income. This monetary value is then used to compute fishers’ WTP estimates for a change in site attribute, holding all else constant (Hanemann 1983; McConnell 1995).
The theory of welfare valuation is unaffected by the CBDT assumption of a functional form of utility, but there are practical considerations to confront when implementing CBDT. For instance, based on the assumptions we make regarding memory, we need to construct a history of experiences, which resembles a representative individual from the data to understand how the payoffs from choices are incorporated into the choice set.
The conditional indirect case-based utility function (CBV), is
10
where y denotes the income for individual i, Qj is the attribute for choice j, and x denotes other explanatory variables affecting utility. Equation [11] demonstrates how a change in policy that alters the site attribute from Q0 to Q1 can be measured:
11
To compute the value of a change of compensating surplus (CS) in site attributes, we need to make assumptions about all site attributes. Similar to the LA form of utility models when variables are held at their means in the numerator of equation [11], in CBDT, we need to make assumptions on the values of variables in the similarity function. When valuing a change in result, such as catching a target species of fish, the similarity function is held at some assumed value. On the other hand, when valuing a change in the attribute, Q, in the similarity function, we must consider if the attribute affects the result, (r), as well as the similarity function. The indirect CBU as a function of a particular Q is
12
In CBV, we make assumptions about the past problems in memory, qQ, by taking the average distribution of past attributes (QA) or by another measure of a representative past. Other possible choices for welfare calculations are calculating welfare for specific populations of memory or calculating the distribution of welfare for the sample population. For instance, it may be preferable to assume a specific memory distribution if trying to obtain the welfare gain or loss for a specific subpopulation or type of fisher. Assumptions are also required regarding how 
affects the result, r. To measure how attributes affect results, we need to establish a functional form, which measures the effect of the attributes on the results:
13
We use the predictions from equation [13] to construct the average result, 
, conditional on attribute 
for a particular site j and estimate the location choice model using CBDT as outlined already. Last, we need a measure of the marginal utility of income, y, to interpret the effect of a change in the attribute on utility in dollar terms. We hypothesize that the marginal utility of income could be rule-based or case-based. If rule-based, we would typically recover a constant marginal utility of income. If case-based, the derivative of CBU regarding the cost (or measure of income) would potentially affect both the result, r, and the comparison to past cases through the estimated weights in the similarity function. The estimates from the location choice model and the predictions from equation [13] are used as inputs into equation [12].
5. Data
We use data from Connecticut recreational fishers to test the empirical fit of each model. The data used in this study were obtained from the Volunteer Angler Survey (VAS) program provided by the Connecticut Department of Energy and Environmental Protection.6 Fishing trip and catch information is voluntarily recorded in survey logbooks by anglers, who are encouraged to return-mail them when completed. Weather data are obtained from the National Oceanic and Atmospheric Administration (NOAA) National Centers for Environmental Information and joined to the trip data by day of the trip.7
After accounting for missing values, the VAS data received have a total of 3,182 day trip records taken by 51 survey participants from 2013 to 2016. A concern with location choice data is the potential for a selection problem of who goes fishing or which fishers choose to report trips. This would need to incorporate a first-stage estimate to model the selection process and use a CBDT framework to model the data-generating process of site selection as a second stage. Since the selection of our data is based on voluntary participation, it may have a selection bias, which could interact with a travel cost coefficient; yet the model comparisons of the second stage are still valid as all our models use the same sample.
The area assigned to recreational anglers in Connecticut is appropriated into six area codes. Each three-digit area code denotes an area of the Long Island Sound defined by NOAA.8 Figure 1 presents the areas used in this study as recorded in the “Fishing Vessel Trip Report.” The trip report also contains the species caught, the number of fish caught, and each catch’s weight and size. Observations recorded from the Long Island Sound but not noted on the map have been grouped into a sixth area denoted as “other.” The smallest unit of observation for location is the area codes used in our definition of location. Fishing sites may be aggregated, which could create aggregation bias (Parsons and Needelman 1992). However, all models would suffer from aggregation bias, so it should not affect model selection criteria.
Map of Long Island Sound, Denoting Study Areas 141–146
The variables in this data set are workday, if a trip is taken on a weekday, the month and year of the trip, daily average wind speed, daily average temperature, daily average precipitation, fisher ID, fishing hours on a trip, and trip mode (type of boat or if the shore). The variables described are used to derive the key variables commonly used in fisheries literature (McConnell, Strand, and Blake-Hedges 1995; Hunt 2005; Timmins and Murdock 2007). The derived key variables of interest are site congestion, expected catch rate, site history, and period. The summary statistics of the same are included in Table 1.
Summary Statistics of Key Variables
Expected Catch Rate
The expected catch represents the expected payout received in terms of fish caught per unit effort from each site. We construct this variable based on multiple attributes of the trip. It is estimated using the number of anglers, fishing hours, area, trip mode, weather variables, workday, year, and month. This predicted measure for catch rate is estimated using a Poisson process model, an approach popularized by McConnell, Strand, and Blake-Hedges (1995). Further details and estimated results of the Poisson process model are reported in Appendix B.
Weather is an important aspect of recreation behavior (Chan and Wichman 2020; Dundas and von Haefen 2020). Weather can affect the decision to go fishing on a particular day and which location to visit through the expected catch. We use weather as an input to expected catch. Aspects of weather and climate (wind speed, temperature, precipitation) are used to define the expected catch for an area. We define trace precipitation for weather when the average daily precipitation is less than 0.005 in. Weather can also shift behavior owing to climate change, though we do not model all of these behaviors in this article. Other adaptations that fishers make to adjust to climate change may include shifting the time of day to fish to adjust to extreme temperatures (Dundas and von Haefen 2020).
Indexing Memory
In CBDT, each case in the decision maker’s memory, which are previous fishing trips in this study, is chronologically ordered and indexed using a variable we call period. This is a constructed variable that equals the accumulated number of trips taken by a fisher. Period is a measure of recency in CBDT. A relatively recent case may have a more considerable influence in the decision-making process than an older case. To account for this, we include period in the model as an attribute. In the LA model, this variable acts as a proxy for individual fishing experience in our sample size.
Site Congestion
The variable site congestion refers to the number of other fishers encountered during the fishing trip. The effect of congestion as a site attribute is important when modeling location choice preferences. The standard hypothesis is that congestion initially acts as a proxy for the popularity of the site. However, to a certain degree, it is considered less desirable and acts as a disutility in the site choice model (Timmins and Murdock 2007). In this study, congestion is the individual share of total fishing trips taken the previous year in the same month in the same location. The share of a participant in proportion to the number of site visitors measures the likelihood of running into others on a trip to a particular location. In this method, we presume that fishers formed expectations regarding the congestion of a site while they were making the site choice decision. Therefore, we use the fisher’s previous visitation experience in that site to measure congestion. The share of each individual across total site visitors gives us an insight into how much site space they occupy as well as the frequency of encountering another fisher (Schuhmann and Schwabe 2004). A smaller individual share implies more congestion at the specific site. Considering the year before reduces the limitations of recall memory, and the same month is used to account for the seasonal nature of this recreational activity (Kolstoe, Cameron, and Wilsey 2018).
The construction of site congestion is based on the anglers in our data set, which acts as a proxy for actual congestion. This variable therefore may contain measurement error in the variation between the actual measure due to anglers who did not report their day trips.
Site History
The familiarity of the site is another attribute that may affect its utility. Site history is a binary indicator for whether the chosen site was visited in the previous period by the same person. This measure is a direct way to capture the incidence of repeat visitation and its importance in site selection.
CBDT Variables
In the CBDT model, we use the variables period, site congestion, site history, and expected catch rate to define the problem, (P). The result (R), or payout, is a binary indicator of whether the targeted species was caught on the trip. This result is a proxy for actual utility received on a trip as that is not observable. The set of actions (A) is the fishing area locations.
6. Model Goodness-of-Fit Comparison
The in-sample quantitative fit of LA, CBDT, and SD models is compared using the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). When comparing the information criteria for the estimated results, a relatively smaller AIC or BIC value means that the model has better goodness of fit (Atkinson 1981).
Out-of-sample predictions for all models are also conducted. For model selection, the out-of-sample procedure is preferred because the in-sample fit can be more easily manipulated by adding controls, which may mask how well the underlying model is performing. We use a method of roll-forward samples to estimate out-of-sample fit as measured by log-likelihood. In this approach, a percentage of the decision maker’s choice data, which comprises cases ordered chronologically, is used to predict the remaining hold-out sample. We conducted a rolling window selection for out-of-sample fit comparison using 15%, 25%, 50%, 75%, and 90% of choice data for all models.
Last, we use a non-nested model selection test developed by Vuong (1989) to evaluate if the models are statistically different from each other. The Vuong test determines whether the CBDT model is preferred using in-sample measures.
7. Results
Table 2 reports the measures of fitness for the LA, SD, and CBDT models. Using the in-sample goodness of fit, we find that the SD model is the best model by all criteria. CBDT models perform similarly with either a Euclidean distance or city-block distance metric. Despite having 30 parameters more than CBDT, which has 10 parameters, the BIC measure for both the SD and the CBDT models is similar. Therefore, the penalty for parameters from BIC makes the models similar in in-sample goodness of fit. Both dynamic models outperform the LA model, indicating that the dynamic history of behavior is important in this context. We find that model selection using the Vuong (1989) non-nested model selection test favors the SD model and is statistically significant at the 1% level (z-statistic = 2.73). We provide the details of the LA model in Appendix C.
Comparison of Model Selection Criteria
The SD and CBDT models fit the data well. To guard against overfitting, we compared the out-of-sample goodness of fit for all models. We compare the rolling out-of-sample goodness of fit based on fishers’ memories. Table 3 reports the log-likelihood value for all models. CBDT, with both distance metrics, performs better than all other models in general, except for the 50% memory hold-out sample. We take this as an indication that the SD model may be overfitting the data and is not as parsimonious as the CBDT model. Given the simpler and more intuitive model performs well in this task, the results suggest CBDT as a model that explains the observed data well.
Out-of-Sample Fit: Log-likelihood Comparison
Table 4 contains the coefficient estimates from the CBDT model. The coefficients for Area i are relative initial attractions to the stated location areas. These parameters are similar to attractions to strategies that learning rules accumulate in behavioral game theory (Guilfoos and Pape 2020). This initial attraction serves a role similar to fixed effects in a LA model. Initial attractions to locations and aspiration levels are not separable in estimation, and therefore aspiration is left out. In Table 4, columns (1)–(4), we incrementally add controls to the model of CBDT using the Euclidean distance metric. In column (5), we report the CBDT model using city-block distance metric.
Estimated Parameters Using Case-Based Decision Theory
The coefficients in the LA model are log odds ratios; with CBDT, the coefficients are the weights, wv, given to that parameter in the similarity function as specified in equation [6]. The weights from the CBDT model, if positive, indicate a degree of similarity between current and past cases. For instance, the estimated weight for site congestion is substantial, positive, and statistically significant. It indicates a high degree of similarity between the congestion level for the current and past locations. The weight for expected catch increases in statistical significance using the city-block distance metric. This means that although model goodness of fit is robust to different distance functions, the interpretation of individual pieces of the problem definition is sensitive to the functional form used to define distance. The estimated weight for the variable period accounts for recency. In other words, the similarity weight for period accounts for the temporal distance between the current choice problem and trips that occurred in the past. Trips further in the past are given less weight in forming expectations. All the parameter estimates are logical and intuitive in interpretation. To interpret the relative importance of each weight, we must standardize them, which we do in Appendix D using the estimates from Table 4, column (4). Recency and then congestion appear to play the most important roles in the weight of information of the problem set.
The results estimated up to now include all fish species. We check the robustness of our results by analyzing each target species separately, using CBDT and the Euclidean distance metric. Table 5 reports the estimated weights for the top four target species considered by recreational fishers in our data set. The BIC model selection criteria show CBDT better fits the data than LA for all species. CBDT also has a better fit than the SD model for striped bass, bluefish, and black sea bass but not for fluke.
Model Selection and Estimated Weights for Different Target Species
In terms of interpretation, the coefficient for expected catch shows significance for striped bass and fluke. On the other hand, the estimated weight for site congestion is significant, positive, and substantial for all target species, especially black sea bass. This finding implies that choice location weighs cases in the past with similar congestion very high when constructing location preferences. This finding conforms to the existing literature regarding the importance of including congestion effects when modeling recreational choice behavior (Schuhmann and Schwabe 2004; Timmins and Murdock 2007; Bujosa et al. 2015).
Another point of interest is that the goodness of fit seems to favor CBDT in the samples for different species. These results suggest that heterogeneity plays an important role in modeling decisions. The parameters in CBDT are significantly different and CBDT performs better by in-sample fit compared with the other models. This shows the importance of heterogeneity by species. Including interaction terms in the model by species—or species-specific models—may be more appropriate when estimating location choice for anglers.
Simulation of Welfare Changes
We use simulated data to demonstrate the errors in estimating welfare when ignoring nonlinear aspects of dynamic choice when the data-generating process is from a case-based decision maker. We conducted this simulation for two reasons. First, it allows us to add a measure for marginal utility of money, which is lacking in our recreational fishing data. Second, we can run controlled experiments with simulated data varying the relationships between random variables.
The generated discrete choice data follows equation [14], where we index the current period (t) to reference past periods (q), in memory. Decision maker i considers attributes, k, for two locations j = [1,2] with a random variable for travel cost, C. We use specifications of travel cost data from Melstrom and Lupi (2013) to inform our simulated data. The site attributes (k) are expected catch rate, site congestion, and the index for time (period). In addition, we assume the error term, εijt, to be independent and identically distributed and from the logistic function. Following the premise behind CBDT, memory is constructed on the three previous periods, after which the fourth and subsequent periods are forgotten. Memory in this and other empirical work (Pape and Kurtz 2013; Guilfoos and Pape 2016, 2020) appears to be highly discounted. Using the assumption of only three periods in memory is similar to highly discounting further periods. The result (or reinforcement mechanism) is a binary indicator that equals one if the fisher caught their preferred species at location j, referenced as catch:
14
Descriptive statistics for the parameters and the distributions of random variables are provided in Table 6. Each simulation contains 1,000 fishers, over 20 time periods (40,000 observations) and is repeated 500 times. We have left aspiration levels out of our simulation since we did not estimate them in the empirical section. It is worth noting that if included, aspiration levels would shift welfare measures similar to intercept terms from a LA model.
Description of Simulated Data
In Table 6, the correlation parameter describes the level of correlation between ECR and C. After each simulation, we use the standard logistic model to estimate the coefficients from equation [15]. We use a Wald test to assess if the recovered coefficients are equal to the “real” coefficients that generated the data. The travel cost coefficient, β1, and the coefficient on a prior catch at location j, β4, are used to assess how the marginal WTP for a target species is estimated. Because we assumed LA cost structure, the “real” coefficient is equal to −0.065, which is the marginal utility of money. While the marginal increase in the previous period catch is one over the average similarity function from the previous period, 0.517, we can further accumulate the value of all past catches as far back as a person’s memory goes to assess the cumulative effects of catches at a particular location. WTP for a site is acquired in the same way, provided we assume a value for past catches or the expected value of catching the preferred species:
15
The error rate in identifying β1 is high (100%) with a p-value < 0.05, with a statistical difference in the real cost coefficient and the estimated cost coefficient at close to 100% of the time. The error rate in identifying the marginal value of a previous catch is also high (100%) with a p-value < 0.05. The mode and mean of point estimates for 
are systematically lower than the “true” parameter, which inflates the WTP of any attribute. As the correlation between a random variable in the similarity function and the LA part of the data-generating process increases, so do the issues with precision around the marginal utility of money and with the bias in the estimated 
.
The marginal WTP for a preferred species by construction is $7.96. The LA model retrieves between $9.36 and $10.73, with a larger bias with high correlations between random variables. This demonstrates a concern with the bias in welfare when the data-generating process is case-based and nonlinear in ways that a linear specification misspecifies.
This simulation demonstrates that in dynamic processes it is easy to misspecify the choice data-generating mechanism when omitting the dynamic aspect. Of course, this is a weak empirical test of the importance of CBDT, as many types of dynamic data-generating mechanisms would also produce results that are different from what the linear models would predict. Future work must investigate which theories are consistent with the data and how to structurally estimate known behavior and construct tests of validity for the behavioral theories.
Discussion and Limitations
We find some support to recommend case-based reasoning to empirical location choice data. First, our results show that CBDT does a good job explaining the data using out-of-sample goodness of fit. CBDT does well in reproducing the choice data across different cutoffs. Replicating the data-generating process is of particular concern for the external validity of estimates and welfare estimates when considering nonmarket valuation.
There are limitations to the CBDT approach. When applying models to empirical data, the researcher often does not know much about the choice data, such as the experiences that shaped preferences. Therefore, in constituting an individual’s memory using CBDT, we may leave out or misconstrue what is in memory or how a particular memory enters into utility. In panel data, where repeat observations are available, there is a natural definition of the memory to draw on: the past experience of the decision maker. Our framework naturally suggests itself to panel data, in which memory can be easily defined. Much of the work done on location choice has used surveys in the past, and often it is not a panel of data. Yet we imagine that other methods could be used to supplement survey data or cross-sectional data to define memory or history. One potential method is to join cell phone data that gives insight into location choice histories.9 Another is to develop repeat surveys with more extensive histories, ratings of past trips, or particularly salient experiences. However, domain-specific knowledge about information available to decision makers should be used in defining all primitives of the decision problem in empirical applications.
Exploration is another aspect that may be relevant for fisher behavior. In the deterministic formulation of CBDT, not reaching a satisficing level of utility leads to more exploration. Three elements affect this exploration behavior in our applied setting, which differ from this deterministic interpretation. First, we have a stochastic decision process, where the exploration may occur randomly. Second, we have few locations in the choice set and estimate initial attractions to each, which may confound preferences for exploration as preferences for the areas. Third, if the coefficient weights in the similarity function are estimated to be negative, that would lead to greater exploration or variety-seeking behavior in those attributes of the problem.
One difficulty in measuring how a location choice enters into utility is the result of a particular choice. In our case, we use the catch that a recreational fisher is rewarded for fishing in a particular location. This is a proxy for actual utility. An ideal data set would be a panel of choice observations where the information set and result are known to the researcher. The lack of a result is a limitation in most travel cost studies. CBDT suggests that this is a vital piece of information that would reinforce choices in a repeated choice setting. In our setting, recreational fishers may be motivated by the number of fish caught, type of fish, size of the fish caught, or spending quality time with friends and family. Information about the level of success attained due to a past choice made is an essential determining factor behind how people make future decisions. While catching a target species is a good proxy measure of the result, in other settings, a measure of success of a choice may be difficult or impossible to know or omitted from survey data.
Another limitation of this study is omitted variables. We lack information about fishers’ characteristics, such as income, education, and travel cost to the site. Although we contend that the omitted variables do not favor one model over the other, a complete set of variables is desirable. Fisheries economists have applied other models that incorporate alternative-specific characteristics into the model. A prospective future application in CBDT accounts for unobserved heterogeneity by allowing parameters to vary across observations. Such a model would be comparable to a latent class or mixed multinomial logistic model.
8. Conclusions
In this study, we find that CBDT explains location choice behavior well. In-sample goodness of fit favors a LA SD model, while out-of-sample goodness of fit favors CBDT. Using both models, or mixes of behavioral models, in investigating empirical choice can only offer more insight into the mechanisms for choice and the importance of information to decision makers. Building on the SD literature, our work confirms that dynamic elements in fishery location choice are extremely important.
The compact and parsimonious CBDT model is promising for behavioral modeling of discrete choice data. It may explain data better and adds an element of value to the dynamic importance of information. Further research is needed to better match and collect data for behavioral decision-making models, such as CBDT. However, we can imagine that future survey efforts may capture explicit measures of success of trips and aspiration values. Further work may also find when or if this type of behavioral modeling is needed to understand the observed choice.
Care needs to be taken when considering discrete choice modeling and nonmarket valuation work. Using simulation data, we demonstrate the reduced-form model’s potential bias, assuming that the data-generating process is case-based. This work and past empirical work on CBDT (Kahneman 2003; Gayer et al. 2007; Ossadnik, Wilmsmann, and Niemann 2013; Bleichrodt et al. 2017; Guilfoos and Pape 2020) suggest themselves to applications outside location choice modeling. Behavioral modeling is not limited to the functional form of choice but can involve cognition, rationalization, or other psychological aspects of choice. The extension of behavioral modeling, and specifically case-based reason modeling, to other choice settings may provide more accurate welfare estimates if the models better match our understanding of how people make decisions.
Acknowledgments
We thank the Connecticut Department of Energy and Environmental Protection for providing access to the VAS program data on recreational fishers. We are also grateful to Christopher Dumas and Andreas Pape, who provided valuable feedback and advice. Thanks to the participants at the 2019 Southern Economic Association conference and the 2019 Northeastern Agricultural and Resource Economics Association conference for their comments. This work is supported by the USDA Hatch Program and the University of Rhode Island’s College of Environmental and Life Sciences.
Footnotes
Appendix materials are freely available at http://le.uwpress.org and via the links in the electronic version of this article.
↵1 There is a robust literature on learning models and Markov decision models that do not use these assumptions. However, these are typically not used in nonmarket valuation or location choice modeling.
↵2 Behavioral anomalies can be important to model selection; e.g., if loss framing is important, a model based on prospect theory may be appropriate and performs better out-of-sample.
↵3 See https://www.fisheries.noaa.gov/resource/document/fisheries-economics-united-states-report-2019.
↵4 We explored many possible choices for the result. These could be the number of fish caught or the weight of the accumulated catch. The target species is a good proxy for the result in this setting.
↵5 This point is made in Guilfoos and Pape (2020).
↵6 Details about the VAS program in Connecticut are available at http://www.ct.gov/deep/cwp/view.asp?a=2696&q=322750.
↵7 Details about the National Centers for Environmental Information and how to obtain weather data and information are available at https://www.ncdc.noaa.gov/.
↵8 The NOAA “Fishing Vessel Trip Report (VTR) Reporting Instructions” for the Greater Atlantic region provides details about the areas appropriated into grid codes in the New England region; available at https://www.greateratlantic.fisheries.noaa.gov/public/nema/apsd/vtr_inst.pdf.
↵9 Safegraph or similar data companies could be used to mine data on visitations and potential representative memories for defining the information of the problem. We thank an anonymous reviewer for this suggestion.
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