Modeling Users’ Trust in Drought Forecasts

Majid Shafiee-Jood aVen Te Chow Hydrosystems Laboratory, Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, Urbana, Illinois
bDepartment of Engineering Systems and Environment, University of Virginia, Charlottesville, Virginia

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Tatyana Deryugina cDepartment of Finance, Gies College of Business, University of Illinois at Urbana–Champaign, Champaign, Illinois

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Ximing Cai aVen Te Chow Hydrosystems Laboratory, Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, Urbana, Illinois
dDOE Center for Advanced Bioenergy and Bioproducts Innovation, University of Illinois at Urbana–Champaign, Urbana, Illinois

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Abstract

Forecast valuation studies play a key role in understanding the determinants of the value of weather and climate forecasts. Such understanding provides opportunities to increase the value that users can obtain from forecasts, which can in turn increase the use of forecasts. One of the most important factors that influences how users process forecast information and incorporate forecasts into their decision-making is trust in forecasts. Despite the evidence from empirical and field-based studies, modeling users’ trust in forecasts has not received much attention in the literature and is therefore the focus of our study. We propose a theoretical model of trust in information, built into a forecast valuation framework, to better understand 1) the role of trust in users’ processing of drought forecast information and 2) the dynamic process of users’ trust formation and evolution. Using a numerical experiment, we show that considering the dynamic nature of trust is critical in more realistic assessment of forecast value. We find that users may not perceive a potentially valuable forecast as such until they trust it enough, implying that exposure to even highly accurate forecasts may not immediately translate into forecast use. Ignoring this dynamic aspect could overestimate the economic gains from forecasts. Furthermore, the model offers hypotheses with regard to targeting strategies that can be tested with empirical and field-based studies and used to guide policy interventions.

Significance Statement

A key factor that determines how users respond to forecast information is the extent to which they trust the information. We propose a model of trust in drought forecast information that captures how users’ trust forms and evolves over time and shows how trust influences users’ decisions. We find that even if a user is exposed to relatively accurate forecasts, he or she may not use them immediately because a minimum trust level must be developed before forecasts are perceived to be valuable. We also show that encouraging users to rely on poor forecasts can make them worse off, potentially deterring them from using forecasts in the future. Our findings highlight the importance of credible communication of forecast accuracy.

Shafiee-Jood’s ORCID: 0000-0002-5808-3393.

Deryugina’s ORCID: 0000-0003-0870-8655.

Cai’s ORCID: 0000-0002-7342-4512.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ximing Cai, xmcai@illinois.edu

Abstract

Forecast valuation studies play a key role in understanding the determinants of the value of weather and climate forecasts. Such understanding provides opportunities to increase the value that users can obtain from forecasts, which can in turn increase the use of forecasts. One of the most important factors that influences how users process forecast information and incorporate forecasts into their decision-making is trust in forecasts. Despite the evidence from empirical and field-based studies, modeling users’ trust in forecasts has not received much attention in the literature and is therefore the focus of our study. We propose a theoretical model of trust in information, built into a forecast valuation framework, to better understand 1) the role of trust in users’ processing of drought forecast information and 2) the dynamic process of users’ trust formation and evolution. Using a numerical experiment, we show that considering the dynamic nature of trust is critical in more realistic assessment of forecast value. We find that users may not perceive a potentially valuable forecast as such until they trust it enough, implying that exposure to even highly accurate forecasts may not immediately translate into forecast use. Ignoring this dynamic aspect could overestimate the economic gains from forecasts. Furthermore, the model offers hypotheses with regard to targeting strategies that can be tested with empirical and field-based studies and used to guide policy interventions.

Significance Statement

A key factor that determines how users respond to forecast information is the extent to which they trust the information. We propose a model of trust in drought forecast information that captures how users’ trust forms and evolves over time and shows how trust influences users’ decisions. We find that even if a user is exposed to relatively accurate forecasts, he or she may not use them immediately because a minimum trust level must be developed before forecasts are perceived to be valuable. We also show that encouraging users to rely on poor forecasts can make them worse off, potentially deterring them from using forecasts in the future. Our findings highlight the importance of credible communication of forecast accuracy.

Shafiee-Jood’s ORCID: 0000-0002-5808-3393.

Deryugina’s ORCID: 0000-0003-0870-8655.

Cai’s ORCID: 0000-0002-7342-4512.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ximing Cai, xmcai@illinois.edu

1. Introduction

Forecast valuation studies play a major role in the provision of climate information and services to user communities (Bruno Soares et al. 2018; Anderson et al. 2015). One of the main purposes of forecast valuation studies is to better understand forecast users’ attitudes toward forecasts and their perceptions about forecast value (Millner and Washington 2011; Bruno Soares et al. 2018). This understanding allows forecast producers and providers to employ more effective communication and targeting strategies that increase the likelihood of forecast use (PytlikZillig et al. 2010; Clements et al. 2013). Users’ attitudes toward forecasts play an important mediating role in explaining the relationship between forecast properties (e.g., accuracy) and forecast use (Klobas 1995; Kelton et al. 2008). Specifically, trust is a key intervening factor that directly affects users’ interpretation and use of forecast information, particularly in the context of weather-related hazards (Ziervogel 2004; Ripberger et al. 2015; Morss et al. 2016). Nevertheless, user trust is rarely considered in modeling studies of forecast valuation.

Our goal in this study is to model trust as an integral part of forecast valuation. To better understand how imperfect forecasts influence users’ trust and how that in turn influences the user-perceived forecast value, we propose a theoretical model of trust in information that is built into a forecast valuation framework. This model includes two main features: 1) information processing, which captures the role of trust as a mediating factor to describe how a user integrates forecast information with their prior knowledge, and 2) trust development, which captures the dynamic process of a user’s trust formation and evolution. We adopt the belief-updating model developed in Millner (2008) to formulate a user’s processing of forecast information and extend it by adding a learning component to capture how the user’s trust in the information evolves over time. We apply the proposed framework to a stylized crop-allocation decision problem in which the user receives a probabilistic drought forecast of the crop season and, considering the weather uncertainty, decides the optimal allocation between two crops.

In our model, we operationalize trust by perceived accuracy. The concept of trust has been studied within many disciplines, and these studies often define and measure trust differently (Kelton et al. 2008; Siegrist et al. 2012). It is widely acknowledged that trust is a complex, multidimensional construct (Mayer et al. 1995; Kelton et al. 2008), and as such, operationalizing and modeling trust is a challenging task (Love et al. 2013). Users’ perception about the accuracy of the information (i.e., perceived accuracy) is a known critical factor that determines the trustworthiness or credibility of information (Hertzum et al. 2002; Kelton et al. 2008). Even though perceived accuracy and trust are not necessarily the same, previous empirical studies have shown that trust in forecasts and perceived accuracy are positively and significantly correlated (Ripberger et al. 2015; Morss et al. 2016).

We model trust as a dynamic learning process. Considering the dynamic nature of trust underscores the important role of users interacting with forecasts and learning from their experiences (Teacy et al. 2006; Khodyakov 2007). Trust (or distrust) in forecasts is often built up over time by accumulating personal experiences with forecasts (Patt and Gwata 2002; Teacy et al. 2006; Kelton et al. 2008). Positive experiences with forecasts will result in higher levels of trust, while inaccurate or erroneous forecasts will result in lower levels of trust (Kelton et al. 2008; Pennesi 2013; Ripberger et al. 2015). While conceptual models of trust consider the trust evolution as a major component (see, e.g., Mayer et al. 1995; Kelton et al. 2008), the dynamic nature of trust creates the biggest challenge in measuring and modeling it. As such, despite the apparent importance of users’ interactions with forecasts, our conceptual understanding about how forecast accuracy might influence users’ trust and how trust evolution might affect future responses to forecasts is still limited. While different trust development mechanisms could exist (Kelton et al. 2008), we model trust dynamics based on the premise that users gather knowledge about forecasts through inferences drawn from their previous experiences with forecasts.

This study makes several contributions to the literature. Theoretical and modeling studies of forecast valuation traditionally establish a direct relationship between objective forecast accuracy and forecast value (see, e.g., Katz and Murphy 1997), implying that users find forecasts valuable once they reach a certain accuracy (Katz and Murphy 1997). Empirical studies, however, find that forecast information may not influence users’ beliefs about an uncertain event (e.g., below-normal rainfall in a season) if users do not trust the information, regardless of the accuracy of forecasts (Luseno et al. 2003). Furthermore, there is empirical evidence that users with greater trust in forecasts are more likely to take protective actions in response to warnings of weather-related hazards (Ripberger et al. 2015; Morss et al. 2016). Inspired by these findings, our model postulates that the relationship between objective forecast accuracy and forecast value—and by extension forecast use—is mediated through trust.

While modeling studies of forecast valuation largely ignore the role of trust, there are a number of noteworthy exceptions (Lee and Lee 2007; Millner 2008; Ziervogel et al. 2005).1 Lee and Lee (2007) introduce subjective forecast accuracy as a behavioral parameter and use a decision function to determine its impact on the decision-maker’s actions in a modified cost–loss problem. In the model developed in Millner (2008), a behavioral parameter is incorporated into the Bayesian belief-updating component to describe how a user’s perception about forecast accuracy influences their perceived forecast value. In both of these models, perceived accuracy is treated as a constant parameter, independent of forecast accuracy or forecast performance.

On the other hand, Ziervogel et al. (2005) consider the dynamic nature of users’ trust in a deterministic categorical forecast by using a simple heuristic developed based in part on empirical evidence from a role-playing exercise. In that model, however, farmers directly incorporate forecast information into their decision-making once the trust level exceeds a cutoff value, ignoring the role of users’ information processing. The model that we propose provides a more complete understanding of the role of trust in forecast valuation by simultaneously considering both information processing and trust dynamics. Our model also allows us to explore the dynamic relationship between forecast accuracy and forecast value. Last, using Bayesian inference to represent the dynamic nature of trust, we provide a flexible tool to explore the role of users’ prior knowledge about uncertain events, which is a critically important, but typically overlooked, factor in assessing forecast value (Sherrick et al. 2000).

The value of information to a user depends on when the evaluation is made (Antonovitz and Roe 1988). Our valuation framework differs from most studies in the literature (see, e.g., Johnson and Holt 1997; Millner 2008) in that we assess the forecast value after it is received by the user but before the realization of the uncertain event. In the economics of information literature, this is referred to as the quasi ex ante value of information (Antonovitz and Roe 1988; Lawrence 1999). We focus on quasi ex ante valuation because our goal is to investigate the extent to which a publicly available forecast influences a user’s decisions. Quasi ex ante valuation is of particular relevance to our study because many farmers, especially in the United States, already have access to free forecast information. We also assess the forecast value after the uncertainty is resolved and the actual outcomes are observed (i.e., ex post value) to determine users’ realized benefits of using the forecasts. The ex post estimation of forecast value is specifically important as it provides insights into the accuracy of the ex ante measures of information value (Antonovitz and Roe 1986).

The remainder of this article is organized as follows. Section 2 introduces the components of the proposed forecast valuation framework. In section 3, we design a numerical experiment that we use to demonstrate the dynamics of trust formation and forecast evaluation. Section 4 presents the results, and section 5 concludes with a discussion of our findings and future research.

2. Method

The forecast valuation framework we propose consists of four components: 1) a decision-making component, 2) an information-processing component, 3) a learning component, and 4) a value assessment component. While the framework can be extended to consider any type of forecast, we focus on probabilistic forecasts of a dichotomous event (i.e., drought or no drought). Figure 1 shows the model’s time line and different components, and we provide a list of notations used in the article in Table 1.

Fig. 1.
Fig. 1.

The proposed forecast valuation framework: Boxes show (a) the information-processing component, (b) the value assessment component, (c) the decision-making component, and (d) the learning component.

Citation: Weather, Climate, and Society 13, 3; 10.1175/WCAS-D-20-0081.1

Table 1.

List of main symbols and notations.

Table 1.

At the beginning of each time step, a user receives a probabilistic drought forecast informing them about the weather during the upcoming crop season. The user integrates this information with their prior knowledge in a Bayesian manner and forms a new belief about the likelihood of the drought state during the crop season. They then assess the forecast value and decide on a crop allocation. If the forecast value is positive, the user makes the crop allocation decision based on their new belief about drought occurrence; otherwise, they make the decision based on their prior knowledge about drought occurrence. To represent the user’s decision-making under uncertainty, the model assumes that the user is an expected utility maximizer whose preferences are represented by a utility function with constant relative risk aversion (Gollier 2001). Once the uncertainty is resolved and the actual state of the weather is observed, the user updates their trust in the forecast and assesses the ex post value of information (Fig. 1). We provide a detailed explanation of the model’s components in the remainder of this section.

a. Decision-making problem

Consider a crop allocation problem involving two crops, A and B, where a forecast user must decide what proportion of the land to allocate to each crop given the uncertainty associated with the future weather during the upcoming crop season. Let θ ∈ Θ be a random variable representing the state of the weather during the crop season. For simplicity, we assume the weather has two states: normal (θ = 0) and drought (θ = 1). We adopt subjective or Bayesian interpretation of probability and define p˜(θ)2 as the user’s subjective belief that state θ occurs during the upcoming crop season; p˜(θ) embodies the user’s knowledge about the weather during the crop season (Lawrence 1999), which could be based on historical probabilities (also called climatological information) and the user’s experience (Johnson and Holt 1997; Sherrick et al. 2000). Suppose the crop yield (per unit area of land) for crops A and B is a function of the weather alone, denoted by yA(θ) and yB(θ), respectively. We assume that crop A is more drought tolerant and has lower yield variability and that crop B is a high-yield variety whose yield decreases significantly in drought conditions [i.e., yB(1) < yA(1) < yA(0) < yB(0)].

We assume that the user is a utility maximizer whose risk preferences are characterized by an increasing von Neumann–Morgenstern utility function, U = U[π(x, θ)]:
U=U[π(x,θ)]={π1r1rr1lnπr=1.
This utility function belongs to a class of utility functions with constant relative risk aversion that exhibit the property that wealthier users are more willing to take absolute risks (Gollier 2001); r ≥ 0 is the Arrow–Pratt coefficient of relative risk aversion, and π(x, θ) =π is the normalized payoff defined as
π(x,θ)=ω+xyA(θ)+(1x)yB(θ),
where x ∈ [0, 1] is the fraction of land allocated to crop A prior to the realization of θ, 1 − x is the land fraction allocated to crop B, and ω is the user’s normalized wealth. We assume that any direct and operation costs are embedded in ω. We normalize ω by land area and crop price, respectively, and assume the prices of the two crops are equal and do not depend on the occurrence of drought. Thus, π and ω are both expressed in the same unit as y (yield per unit area), denoted by u. The user’s optimization problem is given as
maxxEθ[U]=θ=01p˜(θ)U[ω+xyA(θ)+(1x)yB(θ)],
where Eθ[] is the expectation operator taken with respect to p˜(θ).3

b. Information processing: Updating beliefs about drought

This section outlines how a user updates their belief about drought occurrence based on a probabilistic drought forecast. Let θ^={0,1} be a deterministic forecast corresponding to state θ of the weather. We assume that θ^ is generated from a forecasting model with accuracy κ, that is, κ=p(θ^=θ|θ). We use Bayes’s theorem to represent how the user integrates the forecast with their prior belief about state θ [i.e., p˜(θ)]. Once the user receives the deterministic forecast, their updated belief (or posterior probability) about θ can be written as
p˜(θ|θ^)=p˜(θ^|θ)p˜(θ)p˜(θ^),
where p˜(θ^) is the normalization factor and p˜(θ^|θ) is the likelihood function (or simply likelihood). Now, suppose that the user receives a probabilistic forecast, q(θ^)[0,1], which is a probability mass function of θ^. Following Millner (2008), the user’s updated belief about θ given the probabilistic forecast q(θ^) can be written as
p˜[θ|q(θ^)]=θ^{0,1}q(θ^)p˜(θ|θ^)=θ^{0,1}q(θ^)p˜(θ^|θ)p˜(θ)p˜(θ^).
The likelihood term in Eqs. (4) and (5) expresses the probability of receiving a deterministic forecast θ^ if the user believes the state of the weather is θ. The likelihood, therefore, can be interpreted as a subjective measure of forecast accuracy (Adams et al. 1995; Millner 2008), as opposed to κ, which is an objective measure of forecast accuracy and a characteristic of the forecasts. We follow the assumptions of inter and intramodel democracy4 described in Millner (2008) and assume that the likelihood can be characterized by a single parameter τ ∈ [0, 1]; that is, τ=p˜(θ^=1|θ=1), 1τ=p˜(θ^=0|θ=1), 1τ=p˜(θ^=1|θ=0), and τ=p˜(θ^=0|θ=0). τ represents the user’s subjective belief about forecast accuracy (or perceived accuracy), and we use it in our model to operationalize the user’s trust in forecasts.5 Using this definition and expanding the normalization factor using the law of total probability, we can derive an expression for the user’s updated belief about the occurrence of the drought state:
p˜[θ=1|q(θ^)]=q(0)(1τ)p˜(θ=1)τp˜(θ=0)+(1τ)p˜(θ=1)+q(1)τp˜(θ=1)(1τ)p˜(θ=0)+τp˜(θ=1).

We explain Eq. (6) in the following (also see Fig. 2 for numerical examples). Because Θ = {0, 1} in the decision-making problem considered here, we characterize p˜(θ) by a single parameter p1:=p˜(θ=1), defined as the user’s subjective belief about the occurrence of the drought state during the crop season (or simply, the user’s belief about drought). As a result, p˜(θ=0)=1p1. Similarly, we can represent q(θ^) by a single parameter, pd: = q(1), defined as the probabilistic drought forecast.6 Last, we represent the user’s updated belief about drought [i.e., p˜[θ=1|q(θ^)]] by p1|pd. Based on Eq. (6), two factors contribute to how a user processes drought forecast information (i.e., pd): their trust in the information (i.e., τ) and their prior belief about drought (i.e., p1). Trust determines the extent to which the user is willing to abandon their prior belief about drought in favor of the forecast. When they have full trust in the forecast (i.e., τ = 1), their updated belief about drought is equal to the forecast (p1|pd=pd). In this case, the forecast is directly incorporated into decision-making. This is an implicit assumption in many numerical forecast valuation studies, as they directly use forecasts as inputs to the simulation or to optimization models (e.g., Cabrera et al. 2007; Letson et al. 2009; Asseng et al. 2012).

Fig. 2.
Fig. 2.

Updated belief about drought as a function of trust: (a) The sensitivity analysis of pd when p1 = 0.3, and (b) the sensitivity analysis of p1 when pd = 0.9.

Citation: Weather, Climate, and Society 13, 3; 10.1175/WCAS-D-20-0081.1

When trust is τ = 0.5, on the other hand, the user believes that the forecast is random and does not provide any information; hence p1|pd=p1. For τ > 0.5 (i.e., the user has a positive perception of the forecast), as trust increases from 0.5 to 1, the user’s updated belief about drought approaches the forecast (i.e., p1|pdpd). When τ < 0.5, however, they believe the forecast is wrong; in this case, forecasts still provide nontrivial information to the user (Murphy and Winkler 1987). When the user believes that the forecast is completely wrong (i.e., τ = 0), their updated belief about drought is exactly the opposite of what the forecast suggests: p1|pd=1pd.

We can also make three useful observations based on Eq. (6). First, if the user trusts the forecast (i.e., τ > 0.5), then the more surprising the forecast is relative to the user’s prior belief, the greater is the marginal impact of trust on the user’s updated belief about drought (e.g., see the greater slope associated with pd = 0.9 as compared with pd = 0.5 in Fig. 2a). Second, the same forecast could be interpreted differently depending not only on the user’s trust but also on their prior belief about drought (Fig. 2b). Third, as the difference between the forecast and the user’s prior belief increases, a higher level of trust will be needed to form the same perception about the likelihood of drought occurrence (see, e.g., the trust levels for the three cases of prior belief corresponding to the updated belief of p1|pd=0.7 in Fig. 2b).

c. Learning: Modeling trust formation and evolution

In this section, we propose a probabilistic model of trust to describe how a user’s trust in drought forecasts changes over time as they learn about the accuracy of the forecasts. We assume that the user is a Bayesian learner. The model considers the time line of information reception, belief updating, and decision-making presented in Fig. 1. Suppose that a user’s belief about drought is p1 and their trust in the forecast at time step t is represented by ft(τ) (i.e., a PDF over different values of τ ∈ [0, 1]).7 Once a forecast is received (i.e., pdt, indicated by point A in Fig. 1), the user’s updated belief about drought is the integral of Eq. (6) over all possible values of τ:
p1|pdt=01p˜[θ=1|q(θ^),τ]ft(τ)dτ,
where p˜[θ=1|q(θ^),τ] is calculated using Eq. (6) for any given value of τ ∈ [0, 1].
After the decision is made (point B in Fig. 1) and the uncertainty is resolved, the user observes the true state of the weather, denoted by φt ∈ Φ = {0, 1}, where φ = 1 indicates that a drought event has occurred (point C in Fig. 1). This observation, together with pdt, is used as new evidence or information to update the user’s trust in the forecast. Let ft(τ)=ft[τ|q(θ^),φt] be this updated or posterior belief. Using Bayes’s theorem and the law of total probability, ft(τ) can be derived for φt = 1 and φt = 0, respectively:8
ft[τ|q(θ^),φt=1]=pdtτft(τ)μτ+(1pdt)(1τ)ft(τ)1μτand
ft[τ|q(θ^),φt=0]=pdt(1τ)ft(τ)1μτ+(1pdt)τft(τ)μτ,
where the posterior distribution derived at time step t, ft(τ), will be used as the prior distribution in the next time step: ft+1(τ)=ft(τ). An implicit, but critical, assumption in our model of learning is that the characteristics of the forecasting system, including forecast accuracy, remain constant throughout the learning process.9

d. Value assessment

A user may assess the forecast value at three different stages depending on when the forecast information is available and when the uncertainty is resolved (Antonovitz and Roe 1988; Lawrence 1999): 1) before obtaining the information (“ex ante”), 2) after obtaining and processing the information but before the realization of the event (“quasi ex ante”), and 3) after the realization of the event (“ex post”). In our model, we assume that the user observes a publicly available, probabilistic drought forecast. Therefore, we use the quasi ex ante definition of forecast value (Antonovitz and Roe 1986). We also use the ex post valuation to measure the realized benefit of using the forecast.

Let xp1* and xpd* be the user’s optimal crop allocation decisions made without and after receiving the drought forecast information, respectively, formally defined as
xp1*=argmaxxθ=01p˜(θ)U[ω+xyA(θ)+(1x)yB(θ)]and
xpd*=argmaxxθ=01p˜[θ|q(θ^)]U[ω+xyA(θ)+(1x)yB(θ)],
where p˜[θ|q(θ^)] is calculated according to Eq. (7). Therefore, we use the full distribution of trust to calculate xpd*, unless stated otherwise. We calculate the quasi ex ante forecast value V by solving the following equation (Antonovitz and Roe 1986, 1988):
Eθ|pd{U[π(xp1*,θ)+V]}=Eθ|pd{U[π(xpd*,θ)]},
where Eθ|pd{} is the expectation operator taken over p˜[θ|q(θ^)] and V is expressed in the baseline unit u and is always nonnegative.10 This evaluation occurs before the uncertainty associated with the weather is resolved. We use Eq. (13) to calculate the ex post forecast value:
Vexp=π(xpd*,φ)π(xp1*,φ).
The first term on the right-hand side is the ex post payoff when the decision is made with the use of the forecast, and the second term on the right-hand side is the ex post payoff when the decision is made based on the user’s prior belief about drought. Therefore, the ex post forecast value determines the value that the user would realize from using the forecast; Vexp = 0 implies that xpd*=xp1* (i.e., forecast information did not change the user’s decision). The user would have been better off using the forecast if Vexp > 0 or worse off using the forecast if Vexp < 0.

3. Numerical experiment

Throughout the rest of this article, we assume that the climatological probability of a drought event is 30% and the user’s prior belief about the possibility that a drought event will occur is equal to the climatological probability of drought (i.e., p1 = 0.3). We also assume that the time series of the dichotomous drought events (φt) is given. We use the decision-making problem introduced in section 2a with yA(0) = 0.6, yA(1) = 0.4, yB(0) = 0.9, and yB(1) = 0.1, all expressed in the baseline unit u.

To demonstrate how a user’s trust in forecasts evolves over time, we evaluate a hypothetical scenario in which the accuracy of the forecasts is set as κ = 0.8. Given the time series of drought events and the forecast accuracy, we use an approach similar to ensemble forecasting to generate a time series of probabilistic drought forecasts. Specifically, we assume that the forecasting system generates N deterministic forecasts of the dichotomous event at each time step t at accuracy κ. Each deterministic forecast θ^it is referred to as an ensemble member, where i ∈ [1, N]. We assume that θ^it is a Bernoulli process, defined as follows:
θ^it~Be(1,κ)θ^it~Be(1,1κ)ifφt=1φt=0,
where Be(1, κ) indicates a binomial distribution with one trial and probability κ of success. To eliminate the possibility of ever obtaining an estimated probability of zero or one, once N ensemble members are produced, we use Eq. (15) to calculate the probabilistic drought forecast at time t (i.e., pdt) (Roulston and Smith 2002; Katz and Ehrendorfer 2006):
pdt=(i=1NI{θ^it=1})+0.5N+1,
where
I{θ^it=1}=1I{θ^it=1}=0ifθ^it=1θ^it=0.

We assess the value of drought forecasts under three scenarios (Table 2):

  1. Imperfect forecast/evolving trust scenario: the forecast is imperfect and the user’s trust evolves according to the model proposed in Eqs. (8) and (9) with the assumption that the user initially has no specific beliefs about the accuracy of the forecasts, represented by a uniform prior at t = 1;11

  2. imperfect forecast/full trust scenario: the forecast is imperfect but the user fully trusts it (i.e., τ = 1); and

  3. perfect forecast/full trust scenario: the forecast is perfect (i.e., κ = 1 or t:pdt=φt) and the user fully trusts it.12

Comparing the first two scenarios allows us to understand the importance of incorporating trust into our model. The third scenario serves a dual purpose by providing 1) an upper bound for the ex post forecast value and 2) insights into the validity of the ex ante forecast value (Antonovitz and Roe 1986). The imperfect forecast/evolving trust, imperfect forecast/full trust, and perfect forecast/full trust scenarios are indexed by s1, s2, and s3 subscripts, respectively.
Table 2.

Details of evaluation scenarios.

Table 2.

4. Results

We first use the numerical experiment introduced in the previous section to demonstrate the user’s trust evolution over time. We then investigate how trust and its evolution affect the user’s information processing and thus their perception of forecast value. At the end of this section, we show how the model is able to capture the impact of false-alarm events on the user’s trust.

a. Trust evolution

Figure 3b shows the evolution of the user’s trust in drought forecasts, represented by the mean of trust μτ, and one standard deviation intervals στ.13 Because the forecasts have a high accuracy (κ = 0.8), they provide significantly different information from the user’s prior knowledge about the forecasts in the beginning of the simulation (t < 10). In this period, the user’s trust has not yet formed [i.e., f(τ)is still close to the uniform prior and στ is still large], and therefore, new evidence (i.e., forecast–observation pairs) improves the user’s trust in the forecasts in most cases, moving subjective trust closer to the objective forecast accuracy. This learning period, however, causes a delay in how the user’s trust conforms to forecast accuracy (notice that the first time μτ = 0.8 is at t = 19). As the user forms a higher level of trust in forecasts (t ≥ 10) and their trust level approaches μτ = 0.8, which is equal to the objective forecast accuracy, new evidence leads to only small fluctuations of μτ around 0.8. When the forecasts are perfect, Fig. 3b shows that the user’s trust increases at a much higher pace during the learning period, surpassing the trust level of 0.8 only after 4 time steps, and quickly converging to τ = 1.

Fig. 3.
Fig. 3.

On the evolution of user’s trust and updated belief about drought: (a) The time series of drought forecasts and drought events. (b) The mean (μτ) and 1-std-dev intervals (μτ ± στ) of trust and the user’s trust evolution when the forecast is perfect (i.e., pdt=φt). (c) The user’s updated belief about drought based on imperfect and perfect forecasts.

Citation: Weather, Climate, and Society 13, 3; 10.1175/WCAS-D-20-0081.1

Figure 3c shows that the forecasts do not significantly change the user’s belief about drought (p1 = 0.3) in the beginning of the simulation (t ≤ 5). This is because the user has not yet formed a high level of trust in forecasts. As the user’s trust increases, however, their belief about drought occurrence shifts toward the forecasts. Nevertheless, even when the user forms a high level of trust in forecasts (with μτ ≈ 0.8), there is still a gap between their belief about drought and the drought forecasts. This gap is particularly significant for larger values of forecasts. For instance, at t = 51, after receiving a drought forecast of pd = 0.87, the user’s updated belief about drought is at 0.6. Figure 3c also shows that the user’s belief about drought based on imperfect drought forecasts (black line) significantly differs from their belief about drought based on perfect forecasts (dashed blue line). We discuss how these gaps translate into differences in the perceived forecast value in section 4c.

Before we illustrate how the user’s trust and its evolution influence their perception about forecast value, it is worth highlighting the complex relationship between the user’s trust and drought forecasts. We refer to a new evidence (i.e., forecast–observation pairs) that leads to a positive change in trust (Δμτ=μτt+1μτt>0) as trust strengthening, whereas trust-weakening evidence leads to a negative change in trust (Δμτ<0). According to Eqs. (8) and (9), the user’s trust is not only influenced by the new evidence but also depends on the user’s current trust level. In other words, as Fig. 4 illustrates, there are forecast–observation pairs that could be trust strengthening or trust weakening depending on the current level of trust. For example, notice that a forecast of pd ~ 0.3 that is followed by a no-drought event (φ = 0) is trust strengthening when trust level is relatively low (i.e., μτ ~ 0.66), but the same forecast–observation pair is trust weakening if trust level is relatively high (i.e., μτ ~ 0.77). This is because a forecast needs to be highly accurate to positively influence a user who already has a high level of trust, whereas a relatively less accurate forecast could be still trust strengthening if the user does not expect forecasts to be accurate. As the user’s trust converges to the forecast accuracy (i.e., μτκ = 0.8), the trust-strengthening events will be limited to forecast–observation pairs with pd > 0.8 and φ = 1 or pd < 0.2 and φ = 0. The positive impact that these events have on the user’s trust, however, is diminished when forecasts are pd < 0.8 or pd > 0.2, leading to the fluctuating pattern of trust around μτ = 0.8 as observed in Fig. 3b.

Fig. 4.
Fig. 4.

Change in the user’s trust (Δμτ) as a function of new information (i.e., a forecast–observation pair) and the user’s current trust level (μτ). The size of the symbols is proportional to the absolute value of Δμτ.

Citation: Weather, Climate, and Society 13, 3; 10.1175/WCAS-D-20-0081.1

b. Value of drought forecasts and threshold trust

Figure 5 demonstrates how a user’s trust in forecasts influences their perception of forecast value (i.e., quasi ex ante value V) for two cases of drought forecasts (pd = 0.5 and pd = 0.9). First, we find that the same forecast could be valued differently depending on how much the user trusts it. We also find that a user does not perceive a forecast to be valuable when their trust is at τ = 0.5. This is because the forecast does not alter their updated belief about drought, and as a result, the optimal decisions made with and without forecasts remain the same. The relationship between the quasi ex ante forecast value and the user’s trust in forecasts is not necessarily monotonic, implying that users with the same characteristics (i.e., r and ω) but different trust levels could have the same perception about forecast value. When the user trusts the forecasts (i.e., τ > 0.5), the forecast value monotonically increases with trust for both cases of pd = 0.5 and pd = 0.9. This holds true for different levels of risk aversion and wealth. Therefore, one way to increase the user-perceived forecast value is to increase the user’s trust in forecasts. When the user does not trust the forecast (i.e., τ < 0.5), the forecast could still provide nontrivial information, as the user can take advantage of the knowledge that forecasts are often wrong (Millner and Washington 2011).

Fig. 5.
Fig. 5.

Value of drought forecast information (specified with colors and contours) as a function of trust and risk aversion for (a),(b) ω = 0.5; (c),(d) ω = 1.5; and (e),(f) ω = 5 and pd = (left) 0.5 and (right) 0.9. Results are for p1 = 0.3.

Citation: Weather, Climate, and Society 13, 3; 10.1175/WCAS-D-20-0081.1

Even though intuition may suggest that information is more valuable to more risk-averse users, Fig. 5 shows that the relationship between the forecast value and the risk attitude is not necessarily monotonic. Consider Fig. 5d as an example. For a trust level τ > 0.6, we see that the forecast value first increases, and then decreases, as risk aversion increases. In other words, there is a specific value of risk aversion at which the forecast value is maximized. In Fig. 5d, this forecast-value-maximizing value of risk aversion corresponds to r = 2.1. For lower values of risk aversion, the optimal decision for an uninformed user who relies only on their prior belief about drought (i.e., p1 = 0.3) is to only plant crop B.14 Therefore, the forecast value comes from changing the user’s decision to include crop A in the crop mixture. As risk aversion increases beyond r = 2.1, however, the share of crop A in the uninformed user’s optimal crop mixture increases.15 Therefore, the decisions are less affected by the forecast and, consequently, the forecast value decreases. This nonmonotonic relationship fades away as the user’s wealth increases (see Fig. 5f). This occurs because the impact of risk aversion diminishes when wealth increases, a feature of utility functions with constant relative risk aversion.

Because of the uncertainty associated with forecasts and the complexities in real-world decision-making, it may not be realistic to assume that a user would rely on a forecast if they do not have trust in it (i.e., τ < 0.5) (Millner 2008). Focusing on τ ≥ 0.5 in Fig. 5, for any given combination of risk aversion and wealth, we can identify a minimum level of trust, referred to as threshold trust, beyond which the user perceives forecasts to be valuable (i.e., V > 0). Figure 6 shows how the threshold trust changes with risk aversion and wealth for two cases of drought forecasts (pd = 0.5 and pd = 0.9). The most important insight from this figure is that the threshold trust not only depends on user-related factors (e.g., risk aversion, wealth) but also varies with the forecast itself. In other words, a user may have different thresholds for different forecasts.

Fig. 6.
Fig. 6.

Threshold trust as a function of wealth and risk aversion under different levels of prior belief about drought (p1) and drought forecast (pd).

Citation: Weather, Climate, and Society 13, 3; 10.1175/WCAS-D-20-0081.1

We find that the more surprising a forecast is relative to the user’s prior belief about drought, the lower the threshold trust is, which may seem counterintuitive. However, a forecast that is closer to the prior belief is not as informative as a surprising forecast. As we show in Fig. 2a, for a user with p1 = 0.3, updated belief based on the forecast pd = 0.5 stays close to the prior belief. Therefore, the decisions made with and without forecast are identical (hence V = 0) except for large values of trust. For pd = 0.9, however, the decisions made with and without forecast are different, even for values of τ close to 0.5. As such, the user finds the forecast pd = 0.9 valuable at a smaller trust level.

c. Dynamic valuation of drought forecasts

In this section, we use the numerical experiment previously described to demonstrate how a user’s perception of drought forecast value changes over time as the user’s trust in forecast increases. Consider a user with r = 0.5, ω = 1.5, and p1 = 0.3. For this user, the optimal decision made without a forecast is to only plant crop B; hence, the forecast is valuable only if it would imply that crop A should be included in the crop allocation mixture. This requires the user to believe that there is at least a 45% chance that the drought would occur.16 Figure 7b shows that the quasi ex ante forecast value significantly differs across the three evaluation scenarios, particularly in the learning period (i.e., t < 10). In the imperfect forecast/evolving trust scenario, the quasi ex ante forecast value (i.e., Vs1) is zero for t < 10. This is because the user has not yet built sufficient trust in forecasts; hence, the updated belief about drought remains close to the prior belief (see Fig. 3c). During this stage, forecasts do not change the user’s actions and are therefore not perceived to be valuable. As the user’s trust increases and approaches the equilibrium level (i.e., μτ = 0.8), drought forecasts of pd > 0.5 are more likely to push the user’s belief about drought beyond 45%, and therefore, are perceived to be valuable.

Fig. 7.
Fig. 7.

(b) Quasi ex ante and (c)–(d) ex post forecast value. The Vs1, Vs2, and Vs3 are quasi ex ante values in imperfect forecast/evolving trust, imperfect forecast/full trust, and perfect forecast/full trust scenarios, respectively. The Vs1exp, Vs2exp, and Vs3exp are the corresponding ex post values. (a) The time series of drought forecasts and drought events. Note that trust evolution follows Fig. 3b.

Citation: Weather, Climate, and Society 13, 3; 10.1175/WCAS-D-20-0081.1

When a user has full trust in forecasts, their updated belief about drought is equal to the drought forecast (see Fig. 2). Hence, the quasi ex ante forecast value in the imperfect forecast/full trust scenario (Vs2) only changes with pd, regardless of how accurate the forecast is, and any forecast that predicts drought with more than a 45% chance (i.e., pd ≥ 0.45) is perceived to be valuable. As a result, a user with full trust in forecasts perceives them to be valuable (Vs2 > 0) at t = 2, 4, 5 while the same forecasts are not valuable for another user with a lower trust level (Vs1 = 0). Hence, these two users make different decisions (and will possibly experience different outcomes) only because they have different levels of trust in the same forecast. In the perfect forecast/full trust scenario, the user (with r = 0.5 and ω = 1.5) receives no value from the forecast when pdt=0 because the forecast does not change their decision (i.e., xpd*=xp1*=0). When pdt=1, however, knowing that a drought event will occur would change the user’s action (xpd*=1), resulting in a positive value of Vs2 = 0.3.

The ex post forecast value determines the value that the user would realize from using the drought forecast, and under the perfect forecast scenario (Vs3exp), it determines the maximum value that can be achieved by using a forecast at each time step.17 Our results show that the ex post forecast value in the imperfect forecast/evolving trust and imperfect forecast/full trust scenarios are identical for t > 5 (Figs. 7c,d). They are identical because the trust level under the imperfect forecast/evolving trust scenario is relatively high in this period, and therefore, the optimal decisions under both scenarios are similar. In contrast, Vs2exp differs from Vs1exp for t ≤ 5. Recall that Vs1 is always zero in this period because the user has not yet built up enough trust in the forecasts. As a result, Vs1exp is also zero. For a user who fully trusts the forecasts, however, the quasi ex ante forecast value (Vs2) is positive at t = 2, 4, 5 (see Fig. 7b), leading to positive values for the ex post forecast value (Vs2exp>0) in those time steps.

We extend the ex post assessment of forecast value by calculating a value score, denoted by VS, which measures the long-term economic benefit that a user receives from using forecasts in the imperfect forecast/evolving trust (πs1tot) or imperfect forecast/full trust scenario (πs2tot) in T time steps relative to the case where the user relies on climatological information (πclimtot). The VS is normalized by the economic benefit that the user receives in the perfect forecast/full trust scenario (πs3tot) and is defined according to Eq. (16):
VSs1=πs1totπclimtotπs3totπclimtotand
VSs2=πs2totπclimtotπs3totπclimtot,
where VSs1 and VSs2 are the value score in the imperfect forecast/evolving trust and imperfect forecast/full trust scenarios, respectively, and πtot=t=1Tπt(xt*,φt). We interpret the value score as a percentage improvement in economic benefit in T time steps between using climatological information and the perfect forecast/full trust scenario; VS = 0 implies that the forecasts provide no additional value than does relying on climatological information, and negative values of VS imply that the user is better off if they ignore the forecasts.

In Fig. 8, we calculate VSs1 and VSs2 for two users with different coefficients of risk aversion (i.e., r = 0.5 and r = 5) and for two scenarios of forecast accuracy (i.e., κ = 0.6 and κ = 0.8). We keep the users’ wealth at ω = 1.5. To draw more robust conclusions, we use 20 time series of drought events, and, for each one, we generate 50 time series of drought forecasts. Therefore, we calculate VS based on 1000 scenarios in total. Figure 8 shows that a more risk-averse user benefits less from the forecasts regardless of forecast accuracy. This is because a more risk-averse user allocates a significant portion of their land to crop A (see Fig. B1a), given that crop A’s yields exhibit much less variation to the weather, helping the users to minimize their risk exposure; however, crop A is less profitable when droughts occur with 30% frequency, leading to less economic benefits for the user.

Fig. 8.
Fig. 8.

Value score for imperfect forecast/evolving trust (VSs1) and imperfect forecast/full trust (VSs2) scenarios under two scenarios of forecast accuracy [(a) κ = 0.8 and (b) κ = 0.6] for two different values of risk aversion (r = 0.5 and r = 5).

Citation: Weather, Climate, and Society 13, 3; 10.1175/WCAS-D-20-0081.1

When forecast accuracy is high (e.g., κ = 0.8), a user with full trust (τ = 1) in forecasts benefits more from using them than a user whose trust in forecasts evolves based on the model proposed in this study [i.e., ft(τ), ∀t]. As we show in Fig. 8a, for a user with a low level of risk aversion (r = 0.5), VSs2 = 1, implying that the user’s economic benefit under the imperfect forecast/full trust scenario is equal to the benefit they gain under the perfect forecast/full trust scenario. If the same user does not fully trust the forecasts, however, our results show that the trajectory of economic benefit (VSs1) they gain is similar to the trajectory of their trust (see Fig. 3b), implying that, as the user develops a higher level of trust in the forecast, they can gain higher benefits from it.

Fully trusting the forecasts is not always beneficial to the user if the forecast accuracy is low. As we show in Fig. 8b for κ = 0.6, even though a less risk-averse (r = 0.5) user with full trust in forecasts will benefit more from using them (i.e., VSs2 > VSs1), VSs2 for a more risk-averse user (r = 5) is negative and less than VSs1, suggesting that the user is worse off by trusting the forecasts.

d. Impact of false-alarm events

Recent empirical studies have found that errors (e.g., false alarms) in forecast and warning systems could influence a user’s response to the forecast information (Ripberger et al. 2015). As we show in Fig. 9, our model is able to capture the impact of a false-alarm event on the user’s trust. However, this experience does not necessarily deter the user from using forecasts in the future. Our results show that, after experiencing the false alarm, the user’s trust recovers, and, depending on the timing of the false-alarm event, the user’s trust in the forecast could pass the threshold trust, enabling the user to possibly benefit from an accurate drought forecast in the near future.

Fig. 9.
Fig. 9.

The impact of a false-alarm event on the user’s trust. The forecast time series in the top panel is similar to the one shown in Fig. 3. For implementing the false alarm at time t, we simply replace the forecast at time t in the top panel by the forecast of pdt=0.8.

Citation: Weather, Climate, and Society 13, 3; 10.1175/WCAS-D-20-0081.1

5. Discussion and conclusions

In this study, we propose a framework to model users’ trust in forecasts as an integral part of forecast valuation. This framework includes a dynamic learning component to simulate how users’ trust in drought forecasts evolves based on their experiences, allowing us to better understand the role of trust evolution in the users’ perception of forecast value.

Applying the framework to a stylized example of drought management, our results show that the users’ trust in drought forecasts significantly influences the way the information is integrated with their prior knowledge. We find that users tend to ignore a forecast when they have limited or no exposure to the forecast.18 We also find that users’ perceived value of forecasts monotonically increases with their trust level when they have confidence in forecasts (i.e., τ > 0.5). The perceived forecast value is the highest when the users have full trust in forecasts, implying that directly incorporating the forecast information into a decision-making model without considering the impact of the users’ trust will likely overestimate their perception of forecast value.

Unlike most theoretical forecast valuation studies in the literature, we do not assume a direct relationship between forecast quality and forecast value in our framework. We show that when users initially lack prior knowledge about the forecasts, there is a significant difference between their trust in the forecast and the actual forecast accuracy. We model how users build up trust as more information about the forecasting systems is accumulated over time. When the forecast accuracy is relatively high, we show that a user’s trust in drought forecasts converges to an equilibrium that is equal to the objective forecast accuracy, but with a time lag. We show that even though forecasts could potentially be valuable, the user may not perceive the forecasts as valuable until they can trust them enough.

We therefore hypothesize that a user must develop a minimum level of trust—which we refer to as threshold trust—before they start to integrate the forecast into their decision-making. In the numerical experiment we use here, the time lag is relatively short. However, even a short time lag that is required for a user to surpass this threshold could delay the effective use of potentially valuable forecasts, particularly in the critical first stage of exposure to a new technology. This delay highlights the role of agricultural extension services and boundary organizations and raises the question of what strategies should be employed to facilitate and accelerate the use of forecasts.

One potential solution is to design incentive mechanisms or to target opinion leaders to encourage the use of forecasts in user communities. This is, however, a delicate task. Our results show that encouraging users to form a high level of trust in forecasts that are not sufficiently accurate can make the users worse off, potentially discouraging them from using forecasts in the future. On the other hand, we find that when forecasts are highly accurate, users with higher trust level in forecasts will benefit more from them. Therefore, when attempts are made to increase the uptake of forecasts, it is important to properly communicate forecast accuracy so that users are encouraged to hold a belief about the forecast accuracy supported by objective measures.

We make two key assumptions in formulating the proposed trust model. First, admitting that trust is a complex, multifaceted construct, we operationalize trust by the user’s perceived accuracy. Perceived accuracy, however, is only one determinant of trust (Kelton et al. 2008; Ripberger et al. 2015). With more insights from quantitative empirical studies, it is possible for future studies to develop multivariable models of trust that capture other attributes such as objectivity, validity, and reputation (Teacy et al. 2006; Kelton et al. 2008). Second, because we use Bayesian inference in modeling trust development, our model cannot fully capture the asymmetric nature of trust dynamics—trust is typically created slowly but can be destroyed in an instant (Slovic 1999). For instance, as we show in Fig. 9, even though a false alarm significantly impacts the user’s trust in the forecasts, trust recovers rather quickly. In real world, however, trust-destroying events could have lasting impacts on the users’ perception toward the forecast (Slovic 1993, 1999; Glantz 1982). With justification from empirical research, future research can use non-Bayesian learning models to better capture the asymmetric nature of trust dynamics, which can improve our ability to better understand the impact of missed events or false alarms on users’ behavior (Ripberger et al. 2015; Lim et al. 2019).

Despite the simplified assumptions that we make in modeling trust and users’ decision-making, as well as in drought forecast representation, the framework presented here still provides a reasonable description of what farmers in rainfed agricultural systems, particularly in the midwestern United States, could potentially do in response to seasonal drought forecasts. Our framework is particularly applicable to the real-world settings where farmers receive free weather forecasts, that is, where the quasi ex ante value is the most appropriate value metric. Our framework, however, cannot be directly used to value forecast information systems, which requires the use of the full ex ante value. We leave it to future work to explore forecast valuation when a user does not know a priori what forecast will be received or has to incur a cost to access or use the forecasts. There are also other fruitful areas for future research. For instance, once the model’s assumptions are tested and validated for a case study, it can be used to develop testable hypotheses to better understand the complex relationships between forecast accuracy, trust evolution, and forecast use. Future research can also extend the framework presented here into a numerical setting that allows for more realistic decision-making problems and forecast representations and can use it to guide field studies and direct extension-targeting efforts.

Acknowledgments

We acknowledge funding support from the NOAA Climate Program Office’s Coping with Drought Initiative (Grant NA18OAR4310257). We declare that we have no conflict of interest. We also thank the three anonymous reviewers for their constructive comments that helped to improve our work.

Data availability statement

We did not use or analyze any datasets during this study. The time series of drought events and drought forecasts generated for this study and the sources codes for the model are publicly available in a GitHub repository (https://github.com/mshjood/Trust_WCAS_2021/).

APPENDIX A

Derivations

Here we provide a step-by-step derivation of Eqs. (8) and (9). Before receiving the forecast, the user’s trust is represented by f(τ). Once the user receives the probabilistic forecast q(θ^) and observes the state φ = 1 of weather, their updated trust, f(τ)=f[τ|q(θ^),φ=1], can be derived as follows:
f[τ|q(θ^),φ=1]=θ^q(θ^)f(τ|θ^,φ=1).
When θ^ is binary (i.e., θ^{0,1}), q(1) = pd and q(0) = 1−pd. Therefore, expanding the summation in the above equation,
f[τ|q(θ^),φ=1]=pdf(τ|θ^=1,φ=1)+(1pd)f(τ|θ^=0,φ=1).
Now we determine the two conditional terms in this equation. For f(τ|θ^=1,φ=1), we have
f(τ|θ^=1,φ=1)=p˜(θ^=1,φ=1|τ)f(τ)p˜(θ^=1,φ=1),
where p˜(θ^=1,φ=1|τ)=[p˜(θ^=1|φ=1,τ)p˜(φ=1,τ)]/f(τ). Note that p˜(θ^=1|φ=1,τ)=τ and p˜(φ=1,τ)=p˜(φ=1)f(τ)=p1f(τ), and we use the law of total probability in the denominator: p˜(θ^=1,φ=1)=01p˜(θ^=1,φ=1|τ)f(τ)dτ. Hence, Eq. (A3) now becomes
f(τ|θ^=1,φ=1)=τp1f(τ)01τp1dτ.
We can further simplify Eq. (A4): we can pull p1 out of the integral because it is a constant and replace 01τf(τ)dτ by μτ. Therefore, we have
f(τ||θ^=1,φ=1)=τf(τ)μτ.
We follow a similar procedure for the second conditional term in Eq. (A2):
f(τ|θ^=0,φ=1)=p˜(θ^=0,φ=1τ)f(τ)p˜(θ^=0,φ=1),
where p˜(θ^=0,φ=1|τ)=(1τ)p1; therefore
f(τ|θ^=0,φ=1)=(1τ)p1f(τ)01(1τ)p1f(τ)dτ.
We can pull p1 out of the integral and replace 01(1τ)f(τ)dτ by 1 − μτ (knowing that 01f(τ)dτ=1) to simplify Eq. (A7):
f(τ|θ^=0,φ=1)=(1τ)f(τ)1μτ.
Replacing the conditional terms in Eq. (A2) by Eqs. (A5) and (A8),
f[τ|q(θ^),φ=1]=pdτf(τ)μτ+(1pd)(1τ)f(τ)1μτ
We follow a similar procedure for the case that the user observes the state φ = 0 of the weather to derive their updated trust f(τ)=f[τ|q(θ^),φ=0]:
f[τ|q(θ^),φ=0]=pdf(τ|θ^=1,φ=0)+(1pd)f(τ|θ^=0,φ=0),
where
f(τ|θ^=1,φ=0)=(1τ)f(τ)1μτand
f(τ|θ^=0,φ=0)=τf(τ)μτ.
Replacing the conditional terms in Eq. (A10) by Eqs. (A11) and (A12), we have
f[τ|q(θ^),φ=0]=pd(1τ)f(τ)1μτ+(1pd)τf(τ)μτ

APPENDIX B

Crop Allocation Decision-Making

For the decision-making problem considered in the model, the optimal crop allocation decision x* depends on several factors, including crop yield distribution y, initial wealth ω, the coefficient of risk aversion coefficient r, and beliefs about drought p1. We make the following assumptions throughout our analysis: yA(0) = 0.6, yA(1) = 0.4, yB(0) = 0.9, and yB(1) = 0.1. Figure B1 shows how the optimal decision changes with r, p1, and ω. For a risk-neutral user (or decision-maker) (i.e., r = 0), the optimal decision is to plant only crop B (i.e., x* = 0) if p1 < 0.5. As risk aversion increases, for any given p1 or ω, a greater fraction of the land is allocated to crop A because yields of crop A exhibit much less weather-related variation, helping risk-averse users to minimize their risk exposure. Holding r constant, the fraction of land allocated to crop A increases as a user’s belief about drought occurrence (p1) increases. This occurs because crop A has a higher yield than crop B in drought conditions. More land is allocated to crop B as ω increases, because wealthier farmers’ treatment of uncertainty more closely resembles that of a risk-neutral user.

Fig. B1.
Fig. B1.

Optimal crop allocation decisions as a function of user’s risk aversion (r), prior belief about drought (p1), and wealth (ω): (a) The sensitivity analysis for p1 when ω = 1.5, and (b) the sensitivity analysis for ω when p1 = 0.3.

Citation: Weather, Climate, and Society 13, 3; 10.1175/WCAS-D-20-0081.1

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  • Khodyakov, D., 2007: Trust as a process: A three-dimensional approach. Sociology, 41, 115132, https://doi.org/10.1177/0038038507072285.

    • Search Google Scholar
    • Export Citation
  • Klobas, J. E., 1995: Beyond information quality: Fitness for purpose and electronic information resource use. J. Inf. Sci., 21, 95114, https://doi.org/10.1177/016555159502100204.

    • Search Google Scholar
    • Export Citation
  • Kox, T., and A. H. Thieken, 2016: To act or not to act? Factors influencing the general public’s decision about whether to take protective action against severe weather. Wea. Climate Soc., 9, 299315, https://doi.org/10.1175/WCAS-D-15-0078.1.

    • Search Google Scholar
    • Export Citation
  • Lawrence, D. B., 1999: The Economic Value of Information. Springer, 405 pp.

  • LeClerc, J., 2014: Communicating weather and climate uncertainty: Exploratory research in cognitive psychology. Ph.D. dissertation, University of Washington, 181 pp., https://digital.lib.washington.edu/researchworks/bitstream/handle/1773/26360/LeClerc_washington_0250E_13285.pdf?sequence=1&isAllowed=y.

  • Lee, K.-K., and J.-W. Lee, 2007: The economic value of weather forecasts for decision-making problems in the profit/loss situation. Meteor. Appl., 14, 455463, https://doi.org/10.1002/met.44.

    • Search Google Scholar
    • Export Citation
  • Letson, D., C. E. Laciana, F. E. Bert, E. U. Weber, R. W. Katz, X. I. Gonzalez, and G. P. Podestá, 2009: Value of perfect ENSO phase predictions for agriculture: Evaluating the impact of land tenure and decision objectives. Climatic Change, 97, 145170, https://doi.org/10.1007/s10584-009-9600-8.

    • Search Google Scholar
    • Export Citation
  • Lim, J. R., B. F. Liu, and M. Egnoto, 2019: Cry wolf effect? Evaluating the impact of false alarms on public responses to tornado alerts in the southeastern United States. Wea. Climate Soc., 11, 549563, https://doi.org/10.1175/WCAS-D-18-0080.1.

    • Search Google Scholar
    • Export Citation
  • Love, B., M. Mackert, and K. Silk, 2013: Consumer trust in information sources: Testing an interdisciplinary model. SAGE Open, 3, https://doi.org/10.1177/2158244013492782.

    • Search Google Scholar
    • Export Citation
  • Luseno, W. K., J. G. McPeak, C. B. Barrett, P. D. Little, and G. Gebru, 2003: Assessing the value of climate forecast information for pastoralists: Evidence from southern Ethiopia and northern Kenya. World Dev., 31, 14771494, https://doi.org/10.1016/S0305-750X(03)00113-X.

    • Search Google Scholar
    • Export Citation
  • MacKenzie, D. I., J. D. Nichols, J. A. Royle, K. H. Pollock, L. L. Bailey, and J. E. Hines, 2018: Fundamental principals [sic] of statistical inference. Occupancy Estimation and Modeling—Inferring Patterns and Dynamics of Species Occurrence, 2d ed., Elsevier, 71–111.

  • Mayer, R. C., J. H. Davis, and F. D. Schoorman, 1995: An integrative model of organizational trust. Acad. Manage. Rev., 20, 709734, https://doi.org/10.5465/amr.1995.9508080335.

    • Search Google Scholar
    • Export Citation
  • Millner, A., 2008: Getting the most out of ensemble forecasts: A valuation model based on user-forecast interactions. J. Appl. Meteor. Climatol., 47, 25612571, https://doi.org/10.1175/2008JAMC1899.1.

    • Search Google Scholar
    • Export Citation
  • Millner, A., 2009: What is the true value of forecasts? Wea. Climate Soc., 1, 2237, https://doi.org/10.1175/2009WCAS1001.1.

  • Millner, A., and R. Washington, 2011: What determines perceived value of seasonal climate forecasts? A theoretical analysis. Global Environ. Change, 21, 209218, https://doi.org/10.1016/j.gloenvcha.2010.08.001.

    • Search Google Scholar
    • Export Citation
  • Morss, R. E., and M. H. Hayden, 2010: Storm surge and “certain death”: Interviews with Texas coastal residents following Hurricane Ike. Wea. Climate Soc., 2, 174189, https://doi.org/10.1175/2010WCAS1041.1.

    • Search Google Scholar
    • Export Citation
  • Morss, R. E., K. J. Mulder, J. K. Lazo, and J. L. Demuth, 2016: How do people perceive, understand, and anticipate responding to flash flood risks and warnings? Results from a public survey in Boulder, Colorado, USA. J. Hydrol., 541, 649664, https://doi.org/10.1016/j.jhydrol.2015.11.047.

    • Search Google Scholar
    • Export Citation
  • Murphy, A. H., and R. L. Winkler, 1987: A general framework for forecast verification. Mon. Wea. Rev., 115, 13301338, https://doi.org/10.1175/1520-0493(1987)115<1330:AGFFFV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Patt, A., and C. Gwata, 2002: Effective seasonal climate forecast applications: Examining constraints for subsistence farmers in Zimbabwe. Global Environ. Change, 12, 185195, https://doi.org/10.1016/S0959-3780(02)00013-4.

    • Search Google Scholar
    • Export Citation
  • Pennesi, K., 2013: Predictions as lies in Ceará, Brazil: The intersection of two cultural models. Anthropol. Quart., 86, 759789, https://doi.org/10.1353/anq.2013.0038.

    • Search Google Scholar
    • Export Citation
  • PytlikZillig, Q. Hu, K. G. Hubbard, G. D. Lynne, and R. H. Bruning, 2010: Improving farmers’ perception and use of climate predictions in farming decisions: A transition model. J. Appl. Meteor. Climatol., 49, 13331340, https://doi.org/10.1175/2010JAMC2270.1.

    • Search Google Scholar
    • Export Citation
  • Ripberger, J. T., C. L. Silva, H. C. Jenkins-Smith, D. E. Carlson, M. James, and K. G. Herron, 2015: False alarms and missed events: The impact and origins of perceived inaccuracy in tornado warning systems. Risk Anal., 35, 4456, https://doi.org/10.1111/risa.12262.

    • Search Google Scholar
    • Export Citation
  • Roulston, M. S., and L. A. Smith, 2002: Evaluating probabilistic forecasts using information theory. Mon. Wea. Rev., 130, 16531660, https://doi.org/10.1175/1520-0493(2002)130<1653:EPFUIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Sherrick, B. J., S. T. Sonka, P. J. Lamb, and M. A. Mazzocco, 2000: Decision-maker expectations and the value of climate prediction information: Conceptual considerations and preliminary evidence. Meteor. Appl., 7, 377386, https://doi.org/10.1017/S1350482700001584.

    • Search Google Scholar
    • Export Citation
  • Siegrist, M., M. Connor, and C. Keller, 2012: Trust, confidence, procedural fairness, outcome fairness, moral conviction, and the acceptance of GM field experiments. Risk Anal., 32, 13941403, https://doi.org/10.1111/j.1539-6924.2011.01739.x.

    • Search Google Scholar
    • Export Citation
  • Slovic, P., 1993: Perceived risk, trust, and democracy. Risk Anal., 13, 675682, https://doi.org/10.1111/j.1539-6924.1993.tb01329.x.

  • Slovic, P., 1999: Trust, emotion, sex, politics, and science: Surveying the risk-assessment battlefield. Risk Anal., 19, 689701, https://doi.org/10.1111/j.1539-6924.1999.tb00439.x.

    • Search Google Scholar
    • Export Citation
  • Teacy, W. T. L., J. Patel, N. R. Jennings, and M. Luck, 2006: TRAVOS: Trust and reputation in the context of inaccurate information sources. Auton. Agents Multi-Agent Syst., 12, 183198, https://doi.org/10.1007/s10458-006-5952-x.

    • Search Google Scholar
    • Export Citation
  • Turco, M., A. Ceglar, C. Prodhomme, A. Soret, A. Toreti, and J. Doblas-Reyes Francisco, 2017: Summer drought predictability over Europe: Empirical versus dynamical forecasts. Environ. Res. Lett., 12, 084006, https://doi.org/10.1088/1748-9326/aa7859.

    • Search Google Scholar
    • Export Citation
  • Tuyl, F., R. Gerlach, and K. Mengersen, 2008: A comparison of Bayes–Laplace, Jeffreys, and other priors: The case of zero events. Amer. Stat., 62, 4044, https://doi.org/10.1198/000313008X267839.

    • Search Google Scholar
    • Export Citation
  • Yuan, X., E. F. Wood, N. W. Chaney, J. Sheffield, J. Kam, M. Liang, and K. Guan, 2013: Probabilistic seasonal forecasting of African drought by dynamical models. J. Hydrometeor., 14, 17061720, https://doi.org/10.1175/JHM-D-13-054.1.

    • Search Google Scholar
    • Export Citation
  • Ziervogel, G., 2004: Targeting seasonal climate forecasts for integration into household level decisions: The case of smallholder farmers in Lesotho. Geogr. J., 170, 621, https://doi.org/10.1111/j.0016-7398.2004.05002.x.

    • Search Google Scholar
    • Export Citation
  • Ziervogel, G., M. Bithell, R. Washington, and T. Downing, 2005: Agent-based social simulation: A method for assessing the impact of seasonal climate forecast applications among smallholder farmers. Agric. Syst., 83, 126, https://doi.org/10.1016/j.agsy.2004.02.009.

    • Search Google Scholar
    • Export Citation
1

Another notable study in the literature is Millner (2009), which models the dynamics of the user’s decision to follow a forecast as a function of forecast performance, that is, its ex post value. However, the model does not include an explicit measure of the user’s confidence in the forecast and therefore overlooks the information-processing component that is considered in our model.

2

Throughout the paper, we use p() to represent objective probability and p˜() to represent subjective probability (i.e., the user’s belief).

3

In Fig. B1, we show how the optimal land allocation decision x* changes with the user’s risk aversion r, wealth ω, and prior belief p1.

4

From the user’s perspective, these assumptions imply that the ensemble members have the same accuracy and are weighted equally and that the errors are distributed equally between incorrect states. See Millner (2008, p. 2564) for more details.

5

Even though some studies in the literature consider confidence as an element of trust (Kelton et al. 2008; Kox and Thieken 2016), we use trust and confidence interchangeably in this study.

6

We interpret pd as probabilistic seasonal drought forecasts for the crop season, for example, based on the 6-month standardized precipitation index (e.g., Yuan et al. 2013) or standardized precipitation evapotranspiration index (e.g., Turco et al. 2017) for August made available to the user before the crop season begins.

7

We assume that the user’s prior belief about drought (p1) does not change from one time step to another. The justification is that p1 has been formed over many years of experience and is therefore not affected by new evidence at each time step. Also, an implicit assumption here is that the user’s belief updating and decision-making do not deviate from rationality tenets.

8

See appendix A for derivations.

9

Our proposed model can be applied to scenarios in which the forecast accuracy itself changes over time. In that case, however, our model would not represent a fully rational learner without a modification that, for example, incorporates a Bayesian method of inference for time-varying parameters, which is beyond the scope of our study.

10

See the proof in Antonovitz and Roe (1986, p. 106).

11

Uniform priors (also known as Bayes–Laplace prior) are widely used in Bayesian inference when there is not much prior information available. Sometimes they are referred to as noninformative priors, although this term may not be precise (MacKenzie et al. 2018, p. 89). Jeffreys prior and Haldane prior are also widely used as noninformative priors, but the best methods for choosing such priors are still an issue of considerable debate (Kass and Wasserman 1996; Tuyl et al. 2008).

12

We could also have another scenario with perfect forecast information but with variable trust, similar to the first scenario. However, when the forecast is perfect, f(τ) quickly converges to τ = 1 (Fig. 3b), which does not provide any additional insights.

13

We define the mean of trust as μτ=01τf(τ)dτ and the standard deviation of trust as στ=01(μττ)2f(τ)dτ.

14

We can derive the optimal solution of x* = 0 (i.e., planting only crop B) by solving the optimization problem presented in Eq. (3) given r = 0.5, ω = 1.5, and p1 = 0.3.

15

Refer to Fig. B1.

16

Using Eq. (3), we can see that, for a user with r = 0.5 and ω = 1.5, the optimal allocation of land to crop A becomes nonzero when the user’s belief about drought reaches 45%.

17

Note that the quasi ex ante and ex post values of forecasts are equal in the perfect forecast/full trust scenario (i.e., Vs3exp=Vs3) because there is no uncertainty involved in calculating Vs3. Hence Vs3exp=0.3 if there is a drought event (φt = 1), and Vs3exp=0 otherwise.

18

By “exposure to the forecast,” we are referring to decision-makers observing a forecast and considering whether to use the forecast in their decision-making. This terminology has been used in the literature—for instance, in Morss and Hayden (2010, p. 183) and LeClerc (2014, p. 129).

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  • Khodyakov, D., 2007: Trust as a process: A three-dimensional approach. Sociology, 41, 115132, https://doi.org/10.1177/0038038507072285.

    • Search Google Scholar
    • Export Citation
  • Klobas, J. E., 1995: Beyond information quality: Fitness for purpose and electronic information resource use. J. Inf. Sci., 21, 95114, https://doi.org/10.1177/016555159502100204.

    • Search Google Scholar
    • Export Citation
  • Kox, T., and A. H. Thieken, 2016: To act or not to act? Factors influencing the general public’s decision about whether to take protective action against severe weather. Wea. Climate Soc., 9, 299315, https://doi.org/10.1175/WCAS-D-15-0078.1.

    • Search Google Scholar
    • Export Citation
  • Lawrence, D. B., 1999: The Economic Value of Information. Springer, 405 pp.

  • LeClerc, J., 2014: Communicating weather and climate uncertainty: Exploratory research in cognitive psychology. Ph.D. dissertation, University of Washington, 181 pp., https://digital.lib.washington.edu/researchworks/bitstream/handle/1773/26360/LeClerc_washington_0250E_13285.pdf?sequence=1&isAllowed=y.

  • Lee, K.-K., and J.-W. Lee, 2007: The economic value of weather forecasts for decision-making problems in the profit/loss situation. Meteor. Appl., 14, 455463, https://doi.org/10.1002/met.44.

    • Search Google Scholar
    • Export Citation
  • Letson, D., C. E. Laciana, F. E. Bert, E. U. Weber, R. W. Katz, X. I. Gonzalez, and G. P. Podestá, 2009: Value of perfect ENSO phase predictions for agriculture: Evaluating the impact of land tenure and decision objectives. Climatic Change, 97, 145170, https://doi.org/10.1007/s10584-009-9600-8.

    • Search Google Scholar
    • Export Citation
  • Lim, J. R., B. F. Liu, and M. Egnoto, 2019: Cry wolf effect? Evaluating the impact of false alarms on public responses to tornado alerts in the southeastern United States. Wea. Climate Soc., 11, 549563, https://doi.org/10.1175/WCAS-D-18-0080.1.

    • Search Google Scholar
    • Export Citation
  • Love, B., M. Mackert, and K. Silk, 2013: Consumer trust in information sources: Testing an interdisciplinary model. SAGE Open, 3, https://doi.org/10.1177/2158244013492782.

    • Search Google Scholar
    • Export Citation
  • Luseno, W. K., J. G. McPeak, C. B. Barrett, P. D. Little, and G. Gebru, 2003: Assessing the value of climate forecast information for pastoralists: Evidence from southern Ethiopia and northern Kenya. World Dev., 31, 14771494, https://doi.org/10.1016/S0305-750X(03)00113-X.

    • Search Google Scholar
    • Export Citation
  • MacKenzie, D. I., J. D. Nichols, J. A. Royle, K. H. Pollock, L. L. Bailey, and J. E. Hines, 2018: Fundamental principals [sic] of statistical inference. Occupancy Estimation and Modeling—Inferring Patterns and Dynamics of Species Occurrence, 2d ed., Elsevier, 71–111.

  • Mayer, R. C., J. H. Davis, and F. D. Schoorman, 1995: An integrative model of organizational trust. Acad. Manage. Rev., 20, 709734, https://doi.org/10.5465/amr.1995.9508080335.

    • Search Google Scholar
    • Export Citation
  • Millner, A., 2008: Getting the most out of ensemble forecasts: A valuation model based on user-forecast interactions. J. Appl. Meteor. Climatol., 47, 25612571, https://doi.org/10.1175/2008JAMC1899.1.

    • Search Google Scholar
    • Export Citation
  • Millner, A., 2009: What is the true value of forecasts? Wea. Climate Soc., 1, 2237, https://doi.org/10.1175/2009WCAS1001.1.

  • Millner, A., and R. Washington, 2011: What determines perceived value of seasonal climate forecasts? A theoretical analysis. Global Environ. Change, 21, 209218, https://doi.org/10.1016/j.gloenvcha.2010.08.001.

    • Search Google Scholar
    • Export Citation
  • Morss, R. E., and M. H. Hayden, 2010: Storm surge and “certain death”: Interviews with Texas coastal residents following Hurricane Ike. Wea. Climate Soc., 2, 174189, https://doi.org/10.1175/2010WCAS1041.1.

    • Search Google Scholar
    • Export Citation
  • Morss, R. E., K. J. Mulder, J. K. Lazo, and J. L. Demuth, 2016: How do people perceive, understand, and anticipate responding to flash flood risks and warnings? Results from a public survey in Boulder, Colorado, USA. J. Hydrol., 541, 649664, https://doi.org/10.1016/j.jhydrol.2015.11.047.

    • Search Google Scholar
    • Export Citation
  • Murphy, A. H., and R. L. Winkler, 1987: A general framework for forecast verification. Mon. Wea. Rev., 115, 13301338, https://doi.org/10.1175/1520-0493(1987)115<1330:AGFFFV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Patt, A., and C. Gwata, 2002: Effective seasonal climate forecast applications: Examining constraints for subsistence farmers in Zimbabwe. Global Environ. Change, 12, 185195, https://doi.org/10.1016/S0959-3780(02)00013-4.

    • Search Google Scholar
    • Export Citation
  • Pennesi, K., 2013: Predictions as lies in Ceará, Brazil: The intersection of two cultural models. Anthropol. Quart., 86, 759789, https://doi.org/10.1353/anq.2013.0038.

    • Search Google Scholar
    • Export Citation
  • PytlikZillig, Q. Hu, K. G. Hubbard, G. D. Lynne, and R. H. Bruning, 2010: Improving farmers’ perception and use of climate predictions in farming decisions: A transition model. J. Appl. Meteor. Climatol., 49, 13331340, https://doi.org/10.1175/2010JAMC2270.1.

    • Search Google Scholar
    • Export Citation
  • Ripberger, J. T., C. L. Silva, H. C. Jenkins-Smith, D. E. Carlson, M. James, and K. G. Herron, 2015: False alarms and missed events: The impact and origins of perceived inaccuracy in tornado warning systems. Risk Anal., 35, 4456, https://doi.org/10.1111/risa.12262.

    • Search Google Scholar
    • Export Citation
  • Roulston, M. S., and L. A. Smith, 2002: Evaluating probabilistic forecasts using information theory. Mon. Wea. Rev., 130, 16531660, https://doi.org/10.1175/1520-0493(2002)130<1653:EPFUIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Sherrick, B. J., S. T. Sonka, P. J. Lamb, and M. A. Mazzocco, 2000: Decision-maker expectations and the value of climate prediction information: Conceptual considerations and preliminary evidence. Meteor. Appl., 7, 377386, https://doi.org/10.1017/S1350482700001584.

    • Search Google Scholar
    • Export Citation
  • Siegrist, M., M. Connor, and C. Keller, 2012: Trust, confidence, procedural fairness, outcome fairness, moral conviction, and the acceptance of GM field experiments. Risk Anal., 32, 13941403, https://doi.org/10.1111/j.1539-6924.2011.01739.x.

    • Search Google Scholar
    • Export Citation
  • Slovic, P., 1993: Perceived risk, trust, and democracy. Risk Anal., 13, 675682, https://doi.org/10.1111/j.1539-6924.1993.tb01329.x.

  • Slovic, P., 1999: Trust, emotion, sex, politics, and science: Surveying the risk-assessment battlefield. Risk Anal., 19, 689701, https://doi.org/10.1111/j.1539-6924.1999.tb00439.x.

    • Search Google Scholar
    • Export Citation
  • Teacy, W. T. L., J. Patel, N. R. Jennings, and M. Luck, 2006: TRAVOS: Trust and reputation in the context of inaccurate information sources. Auton. Agents Multi-Agent Syst., 12, 183198, https://doi.org/10.1007/s10458-006-5952-x.

    • Search Google Scholar
    • Export Citation
  • Turco, M., A. Ceglar, C. Prodhomme, A. Soret, A. Toreti, and J. Doblas-Reyes Francisco, 2017: Summer drought predictability over Europe: Empirical versus dynamical forecasts. Environ. Res. Lett., 12, 084006, https://doi.org/10.1088/1748-9326/aa7859.

    • Search Google Scholar
    • Export Citation
  • Tuyl, F., R. Gerlach, and K. Mengersen, 2008: A comparison of Bayes–Laplace, Jeffreys, and other priors: The case of zero events. Amer. Stat., 62, 4044, https://doi.org/10.1198/000313008X267839.

    • Search Google Scholar
    • Export Citation
  • Yuan, X., E. F. Wood, N. W. Chaney, J. Sheffield, J. Kam, M. Liang, and K. Guan, 2013: Probabilistic seasonal forecasting of African drought by dynamical models. J. Hydrometeor., 14, 17061720, https://doi.org/10.1175/JHM-D-13-054.1.

    • Search Google Scholar
    • Export Citation
  • Ziervogel, G., 2004: Targeting seasonal climate forecasts for integration into household level decisions: The case of smallholder farmers in Lesotho. Geogr. J., 170, 621, https://doi.org/10.1111/j.0016-7398.2004.05002.x.

    • Search Google Scholar
    • Export Citation
  • Ziervogel, G., M. Bithell, R. Washington, and T. Downing, 2005: Agent-based social simulation: A method for assessing the impact of seasonal climate forecast applications among smallholder farmers. Agric. Syst., 83, 126, https://doi.org/10.1016/j.agsy.2004.02.009.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    The proposed forecast valuation framework: Boxes show (a) the information-processing component, (b) the value assessment component, (c) the decision-making component, and (d) the learning component.

  • Fig. 2.

    Updated belief about drought as a function of trust: (a) The sensitivity analysis of pd when p1 = 0.3, and (b) the sensitivity analysis of p1 when pd = 0.9.

  • Fig. 3.

    On the evolution of user’s trust and updated belief about drought: (a) The time series of drought forecasts and drought events. (b) The mean (μτ) and 1-std-dev intervals (μτ ± στ) of trust and the user’s trust evolution when the forecast is perfect (i.e., pdt=φt). (c) The user’s updated belief about drought based on imperfect and perfect forecasts.

  • Fig. 4.

    Change in the user’s trust (Δμτ) as a function of new information (i.e., a forecast–observation pair) and the user’s current trust level (μτ). The size of the symbols is proportional to the absolute value of Δμτ.

  • Fig. 5.

    Value of drought forecast information (specified with colors and contours) as a function of trust and risk aversion for (a),(b) ω = 0.5; (c),(d) ω = 1.5; and (e),(f) ω = 5 and pd = (left) 0.5 and (right) 0.9. Results are for p1 = 0.3.

  • Fig. 6.

    Threshold trust as a function of wealth and risk aversion under different levels of prior belief about drought (p1) and drought forecast (pd).

  • Fig. 7.

    (b) Quasi ex ante and (c)–(d) ex post forecast value. The Vs1, Vs2, and Vs3 are quasi ex ante values in imperfect forecast/evolving trust, imperfect forecast/full trust, and perfect forecast/full trust scenarios, respectively. The Vs1exp, Vs2exp, and Vs3exp are the corresponding ex post values. (a) The time series of drought forecasts and drought events. Note that trust evolution follows Fig. 3b.

  • Fig. 8.

    Value score for imperfect forecast/evolving trust (VSs1) and imperfect forecast/full trust (VSs2) scenarios under two scenarios of forecast accuracy [(a) κ = 0.8 and (b) κ = 0.6] for two different values of risk aversion (r = 0.5 and r = 5).

  • Fig. 9.

    The impact of a false-alarm event on the user’s trust. The forecast time series in the top panel is similar to the one shown in Fig. 3. For implementing the false alarm at time t, we simply replace the forecast at time t in the top panel by the forecast of pdt=0.8.

  • Fig. B1.

    Optimal crop allocation decisions as a function of user’s risk aversion (r), prior belief about drought (p1), and wealth (ω): (a) The sensitivity analysis for p1 when ω = 1.5, and (b) the sensitivity analysis for ω when p1 = 0.3.

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