Introduction
In a now classic study, psychiatric patients suffering from delusions observed a researcher draw coloured beads from one of two hidden jars: a mostly pink jar (85 pink beads, 15 green beads) or a mostly green jar (85 green, 15 pink). The patients had to decide which jar the beads were being drawn from and to indicate when they were ready to make this decision. Relative to control participants, deluded individuals required fewer draws before making a decision (Huq et al. Reference Huq, Garety and Hemsley1988). This effect, referred to as the ‘jumping-to-conclusions’ (JTC) bias, has been replicated numerous times with both deluded and delusion-prone participants (Colbert & Peters, Reference Colbert and Peters2002; Moritz & Woodward, Reference Moritz and Woodward2005). As a result, a tendency to gather insufficient evidence when forming beliefs and making decisions is thought to be a core cognitive component of delusion formation (Fine et al. Reference Fine, Gardner, Craigie and Gold2007; Garety & Freeman, Reference Garety and Freeman2013).
Although the JTC effect is well replicated and robust to many modifications of the basic ‘beads task’ paradigm, there are some fundamental limitations with the way this task is typically administered that call the above interpretation into question. The key problem is that the term ‘jumping to conclusions’ implies gathering insufficient evidence and reaching decisions prematurely, yet the standard JTC effect is a relative effect. The notions of premature and late decisions are only meaningful if there is some optimal point at which a rational individual should decide, and for such a point to exist there needs to be both an (opportunity) cost of incorrect decisions and a cost associated with collecting extra information. Investigations using the beads task paradigm typically make no attempt to incentivize participants explicitly (cf. Woodward et al. Reference Woodward, Munz, LeClerc and Lecomte2009; Lincoln et al. Reference Lincoln, Ziegler, Mehl and Rief2010), and no previous study has incorporated both of these elements. Without these aspects there is no objective basis for the decision to stop collecting data. As a result, although deluded and delusion-prone participants may ‘reach conclusions’ on this task more quickly than control participants (gathering less evidence), the standard unincentivized paradigm cannot justify the suggestion that they ‘jump to conclusions’ (objectively suboptimal evidence gathering).
To illustrate this point using a different example from the clinical literature, consider the relationship between depression and rational belief. It is well known that, relative to healthy individuals, depressed individuals have negative thoughts and expectations about the future (Beck, Reference Beck1976). It might be assumed that, whereas healthy individuals are essentially rational, depressed individuals have beliefs that are unwarrantedly pessimistic; perhaps they update their beliefs about future outcomes in a distorted fashion. However, recent research has clarified that healthy individuals have distorted beliefs about the future, being relatively disinclined to revise their beliefs in response to undesirable information (e.g. Sharot et al. Reference Sharot, Korn and Dolan2011; Eil & Rao, Reference Eil and Rao2012). Indeed, there is even evidence that people with mild depression show no bias when predicting future events (Sharot, Reference Sharot2011; Garrett et al. Reference Garrett, Sharot, Faulkner, Korn, Roiser and Dolan2014). This research demonstrates the perils of using relative comparisons to substantiate claims about deviations from rationality (Shah, Reference Shah2012). To show that a particular group of people is irrational, their beliefs and decisions must be compared to a rational standard, not simply to those of a different group.
In the present study we incorporated rigorous procedures from experimental economics (e.g. monetary incentives, decision theoretic analysis, detailed written instructions and comprehension checks, absence of deception) to address these issues. We set both the cost associated with gathering information and the reward for correct decisions, so these were equal for all participants. To make the task more relatable and engaging we used the ‘lakes and fish’ adaptation of the beads task (Speechley et al. Reference Speechley, Whitman and Woodward2010). On each trial, participants were shown a fisherman who had caught a series of fish from one of two lakes (Fig. 1). After seeing a fish from his catch, participants had the opportunity to see another fish or to decide which lake was the source of all fish in the series. Choosing the correct lake was rewarded but a small cost was incurred for each additional fish seen before making this decision. This combination of rewards and costs generated optimal decision points, that is points at which the expected payoff was maximalFootnote 1 Footnote †. In line with the standard, relative JTC effect, we hypothesized that delusion proneness would significantly predict decisions in this task (controlling for intelligence and risk aversion), such that more delusion-prone participants would decide after seeing fewer fish. Importantly, we also hypothesized that delusion-prone individuals would make objectively premature decisions, that is decisions made in advance of the point at which the expected payoff would be maximal. Only by testing the latter hypothesis could we confirm whether delusion-prone individuals do in fact jump to conclusions.
Fig. 1. A still from the lakes and fish task. The fisherman displays a series of six fish from his catch: B-W-W-B-W-B.
Method
Participants
Participants (n = 112, 59 females, 53 males; mean age = 19.94 years, s.d. = 2.92) were students at Royal Holloway, University of London, recruited online (ORSEE; Greiner, Reference Greiner2004). Participants received £6 for participation and a decision-based bonus. The Psychology Department Ethics Committee of Royal Holloway, University of London approved the study.
Materials
Lakes and fish task
Before the start of the experiment, participants were issued written instructions providing information about the ratios of black to white fish in the two lakes (i.e. 25:75 or 75:25), the incentive structure (rewards and costs) and the means of responding. It was emphasized that, on a given fishing trip (corresponding to a trial), the fisherman visited one of the lakes only and was equally likely to visit either lake. These instructions and the comprehension checks that followed were incorporated to minimize poor task comprehension, which has been shown to influence the JTC bias (Moritz & Woodward, Reference Moritz and Woodward2005; Balzan et al. Reference Balzan, Delfabbro and Galletly2012a ,Reference Balzan, Delfabbro, Galletly and Woodward b ). Visual stimuli were based on those used by Speechley et al. (Reference Speechley, Whitman and Woodward2010). Moreover, we aimed to reduce cognitive load through visual stimuli, which included explicit reminders of the ratios in the lakes, points to be earned and paid, and a string of the previously seen fish to avoid the confound of potentially impaired working memory (Dudley et al. Reference Dudley, John, Young and Over1997a ; Moritz et al. Reference Moritz, Woodward and Lambert2007; Garety et al. Reference Garety, Joyce, Jolley, Emsley, Waller, Kuipers, Bebbington, Fowler, Dunn and Freeman2013). We used the ratio of 25:75, rather than the 15:85 ratio used by Huq et al. (Reference Huq, Garety and Hemsley1988), to create optimal decision points later in the sequence.
After participants were shown the first fish from the fisherman's catch, they had the choice between deciding on a lake or seeing another fish for a small price (2 points; 1 point = £0.05). They could request to see additional fish until they either decided on a lake or all fish in the catch were shown (maximum 17), after which they were forced to decide on a lake. Choosing the correct lake earned them a reward (100 points). This procedure was repeated for six series of fish (trials). The first trial consisted of a randomly drawn series of fishFootnote 2 and was used for payout (participants were informed that the series of fish on one trial would be drawn at random and that they would be paid based on their performance on this particular trial; they did not, however, know which of the six trials was the random trialFootnote 3 ). The next five trials were tailored to have varying optimal decision points. The optimal decision point was always reached at a difference of three fish of the majority colour over the minority colour (see the Appendix for the calculation of the optimal decision point). After this point, unbeknown to the participants, all subsequent fish in the trial were of the majority colour to strengthen the stopping rule and avoid introducing contradictory evidence once the optimal point was reached. The sequences of fish in each non-random trial, including the optimal decision points, are shown in Table 1.
Table 1. The sequences of fish in the various trials (b = black fish; w = white fish); optimal decision points are shown in bold. Also shown are the number of participants who decided on the White lake (W) and on the Black lake (B), after each fish. Numbers in italics represent equivocal errors; numbers in bold italics represent inconsistent errors (see Jolley et al. Reference Jolley, Thompson, Hurley, Medin, Butler, Bebbington, Dunn, Freeman, Fowler, Kuipers and Garety 2014 )
Delusion proneness
We used the 21-item Peters et al. Delusions Inventory (PDI; Peters et al. Reference Peters, Joseph, Day and Garety2004) to measure delusional ideation. Sample items include ‘Do you ever feel as if there is a conspiracy against you?’ and ‘Do your thoughts ever feel alien to you in some way?’ For each yes/no item endorsed, participants are required to indicate their levels of distress, preoccupation and conviction for that item on five-point scales. The yes/no endorsement summed scores range from 0 to 21; the separate dimensions summed scores range from 0 to 105; and the total summed scores range from 0 to 336.
Intelligence
We used the 12-item version of Raven's Advanced Progressive Matrices (12-APM) to measure intelligence (Arthur & Day, Reference Arthur and Day1994). A time limit of 15 min was set for these 12 items. Intelligence was defined as the total number of correctly answered items (APM scores).
Risk aversion
We used Holt & Laury's (Reference Holt and Laury2002) measure to gauge how risk averse participants were. This measure involves 10 decisions between two gambles, for example a decision between option A ‘a 4/10 chance of winning £2.00, a 6/10 chance of winning £1.60’ and option B ‘a 4/10 chance of winning £3.85, a 6/10 chance of winning £0.10’ (Holt & Laury, Reference Holt and Laury2002; based on p. 1645). With each successive decision, the probability of the high payoff outcome in each option increases by 10%, such that the expected value of the ‘risky’ option B increases. Risk aversion was defined as the total number of times option A was chosen, so that the higher the scores, the higher the risk aversion.
Procedure
Participants were tested in groups ranging from 20 to 26 people. All sessions were conducted on a local computer network using z-Tree software (Fischbacher, Reference Fischbacher2007) in the EconLab at Royal Holloway, University of London. After correctly answering a series of comprehension questions based on detailed instructions, participants started the task. As mentioned, the first trial was always a truly random sequence. The next five trials were presented in counterbalanced order across sessions. Incomplete counterbalancing was accomplished using a Latin square procedure to maximize the difference between the trial orders across the five sessions. The first decision (i.e. see another fish or decide on a lake) in the first trial had a time limit of 60 s, to allow participants to get acquainted with the task (note that this trial was not analysed). To discourage participants from making formal calculations (e.g. using mobile phones), each subsequent decision (i.e. see another fish or decide on a lake) in that trial and all decisions in subsequent trials had a time limit of 20 s. Once all participants had completed the task, a different decision task was presented (van der Leer et al., unpublished data). Then, participants completed the risk aversion measure, next the PDI, followed by the 12-APM, and finally some demographic questions.
Statistical analyses
There were two types of decisions in our study (as in all beads task studies). First, after each fish was presented, participants had to decide whether or not to sample a further fish. Second, once they had decided not to sample any more fish (or when they had reached the end of the sequence), they had to decide which of the two lakes the fisherman was fishing from (‘lake decision’). The former decisions were of primary interest, as the number of pieces of information (fish, in this case) that are requested (the number of ‘draws to decision’) comprise the main dependent measure in studies of JTC. To investigate relative effects of delusion proneness we regressed draws to decision on total summed PDI scores (McKay et al. Reference McKay, Langdon and Coltheart2006). We ran sequential regression analyses to check whether effects remained when intelligence and risk aversion were accounted for.
Our paradigm, however, enables more than just relative comparisons. The incentive structure of our task (incorporating both rewards and costs) generates an optimal decision point for any given sequence of fish (i.e. an optimal number of draws to take before deciding which of the two lakes the fisherman was fishing from). To investigate objectively premature decisions (bona fide ‘jumps to conclusions’), we compared participants’ decision points to the optimal decision points across the five non-random trials, using one-sample t tests. We investigated whether objectively premature decisions were made by our sample as a whole and/or by low- and high-delusion-prone subsets of the sample.
Ethical considerations
All procedures contributing to this work comply with the ethical standards of the relevant national and institutional committees on human experimentation and with the Helsinki Declaration of 1975, as revised in 2008.
Results
Data screening
There was no multicollinearity between predictor variables and predictors were linearly related to the outcome variable. Residuals were normally distributed, and the assumption of homoscedasticity was met. One PDI score was an outlier according to the Mahalanobis distance criterion, but it was a valid score (i.e. well within the range of possible scores) so we included it and applied a square-root transformation. After this transformation no more outliers were detected through Cook's and Mahalanobis’ distances; the standardized residuals indicated that less than 5% of participants were outliers (Field, Reference Field2009).
Descriptive statistics
Descriptive statistics for risk aversion, PDI scores and APM scores are provided in Table 2. PDI scores were not significantly different between males (mean = 68.83, s.d. = 43.137) and females (mean = 80.66, s.d. = 37.691); t 110 = 1.816, p = 0.072. Table 1 shows the numbers of lake decisions made at each point of each sequence along with the associated errors. As in Jolley et al. (Reference Jolley, Thompson, Hurley, Medin, Butler, Bebbington, Dunn, Freeman, Fowler, Kuipers and Garety2014), we defined two types of lake decision errors: deciding when the evidence collected to that point favoured neither lake (‘equivocal errors’) and deciding on a lake that was disfavoured by the evidence collected to that point (‘inconsistent errors’)Footnote 4 . Table 3 shows the mean draws to decision in each sequence.
Table 2. Descriptive statistics for risk aversion, PDI scores and APM scores
PDI, 21-Item Peters et al. Delusions Inventory; APM, 12-item version of Raven's Advanced Progressive Matrices; s.d., standard deviation.
Table 3. Optimal decision points and mean ( s.d. ) draws to decision in each sequence, for both the full sample (n = 112) and the two outer quartiles reflecting low-delusion-prone (n = 26) and high-delusion-prone participants (n = 29)
s.d., Standard deviation.
Lakes and fish task
Relative effects
PDI scores were a significant predictor of draws to decision F 1,110 = 5.520, p = 0.021, R adjusted 2 = 0.039; see Table 4, model 1). The higher a participant's PDI score, the earlier decisions were made.
Table 4. B values, standard errors ( s.e. ), β values and p values for each of the predictors in the steps of the regression models for draws to decision
PDI, 21-Item Peters et al. Delusions Inventory; APM, 12-item version of Raven's Advanced Progressive Matrices.
A sequential regression was subsequently run to investigate whether adding PDI scores could predict additional variance not already accounted for by intelligence and risk aversion. Intelligence was a significant predictor of draws to decision in both models, but the addition of PDI scores significantly improved the model (ΔF 1,108 = 6.128, Δp = 0.015, ΔR 2 = 0.042; F 3,108 = 12.076, p < 0.001, R adjusted 2 = 0.230; see Table 4, model 2; and Fig. 2)Footnote 5 . Risk aversion was not a significant predictor of draws to decision in either model and did not correlate with draws to decision when considered alone (r 112 = 0.032, p = 0.741). We thus replicate the standard, relative, JTC finding in a fully incentivized paradigm: higher delusion proneness was associated with earlier decisions on the task, even accounting for risk aversion and intelligenceFootnote 6 .
Fig. 2. Average number of fish seen before making a decision plotted against standardized scores of intelligence (12-item version of Raven's Advanced Progressive Matrices, 12-APM), risk aversion and delusion proneness (square-root transformed scores on the 21-Item Peters et al. Delusions Inventory, PDI). The vast majority of participants decided before the optimal number of fish (indicated by the dotted line).
Absolute effects
Considered as a whole, participants in our sample gathered significantly fewer pieces of information than would have been optimal, that is they jumped to conclusions (mean deviation from optimal number of fish = −2.566, s.d. = 1.75, t 111 = 15.506, p < 0.001)Footnote 7 , Footnote 8 . To further investigate whether this effect was limited to participants high in delusion proneness, we divided participants into low-delusion-prone and high-delusion-prone groups based on a quartile split of their continuous PDI scores (as in Colbert & Peters, Reference Colbert and Peters2002). We then conducted separate one-sample t tests to investigate whether each group drew significantly fewer fish than would have been optimal. These tests revealed that both groups of participants jumped to conclusions [mean deviation from optimal for low-delusion-prone participants (PDI < 46) = −1.946, s.d. = 1.679, t 25 = 5.912, p < 0.001; mean deviation for high-delusion-prone participants (PDI > 95) = −2.993, s.d. = 2.018, t 28 = 7.987, p < 0.001]Footnote 9 , Footnote 10 .
These deviations from optimal decision points could have had considerable financial consequences. For Bayesian individuals, deciding at the optimal point would have led to a gain in expected value of 10.6 points compared to deciding after seeing just one fish (averaged across the non-random trials). On average, however, our participants would have forgone more than 80% of this potential gain if these trials had been used for payout.
Discussion
That delusional and delusion-prone individuals ‘jump to conclusions’ on probabilistic reasoning tasks is perhaps the most important and influential claim in the entire literature on cognitive theories of delusions. Surprisingly, however, although the notion of JTC implies that delusion-prone individuals gather insufficient evidence and reach premature decisions, no previous study has investigated whether the evidence gathering of such individuals is in fact suboptimal. The standard JTC effect is a relative effect but using relative comparisons to substantiate absolute claims is problematic (Shah, Reference Shah2012). Bankers may earn less money than film stars but this does not mean they have ‘insufficient money’; likewise, the finding that delusion-prone individuals gather fewer data than controls does not mean they gather insufficient data; this is an empirical question, hitherto unaddressed.
In the present study we investigated evidence gathering using an incentivized task modelled after the classic beads task, the paradigm most commonly used to study the JTC bias. After observing the colour of a fish caught from one of two lakes, participants could choose to see further fish or to decide on one of the lakes as the source of the sequence of fish. We rewarded correct decisions but imposed costs for gathering extra information (seeing additional fish). This combination of rewards and costs generated an optimal number of ‘draws to decision’ for each sequence of fish, that is a number of draws that would maximize the expected payoff. We found that more delusion-prone participants made earlier decisions, even when accounting for risk aversion and intelligence. This result replicates the standard relative JTC finding and indicates that this standard finding is not attributable to differences in intelligence or risk preferences; although, like Falcone et al. (Reference Falcone, Murray, Wiffen, O'Connor, Russo, Kolliakou, Stilo, Taylor, Gardner-Sood, Paparelli, Jichi, Di Forti, David, Freeman and Jolley2014), we did find that intelligence significantly affected JTC. Importantly, high-delusion-prone participants also decided in advance of an objective rational optimum, gathering fewer data than they should have to maximize their expected payoff. This reflects bona fide JTC. Surprisingly, we found that even low-delusion-prone participants jumped to conclusions in this objective, absolute sense, gathering significantly fewer pieces of information than would have been rationally optimal. By deciding too quickly, participants forwent more than 80% of the gain in expected value they could have obtained if they had decided at the optimal decision points.
Our results count against one alternate explanation of the standard relative JTC finding using the unincentivized beads task. The finding that deluded and delusion-prone participants take fewer draws to decision in unincentivized variants could simply be due to these participants being less motivated than controls to persevere with a seemingly worthless task and in more of a ‘rush’ to end the study (White & Mansell, Reference White and Mansell2009). Although our replication of the standard relative finding in a fully incentivized variant of the task casts doubt on this explanation, it is possible that exogenous incentives or disincentives (e.g. intrinsic motivation to perform well, fatigue) were nevertheless still confounded with delusion proneness in this study. In other words, the subjective cost of sampling further information could have been greater for delusion-prone participants, equal explicit incentives notwithstanding, and might have led them to decide earlier. In an important study, however, Moutoussis et al. (Reference Moutoussis, Bentall, El-Deredy and Dayan2011) analysed data from an unincentivized beads task using a costed-Bayesian model and found no support for this ‘high-sampling-cost hypothesis’.
Although our findings align with the notion that low-delusion-prone participants are more ‘conservative’ than high-delusion-prone participants (Dudley et al. Reference Dudley, John, Young and Over1997b ), the fact that even low-delusion-prone participants jumped to conclusions in our incentivized draws-to-decision paradigm is surprising given previous evidence of generally ‘conservative’ belief revision in healthy participants (e.g. Phillips & Edwards, Reference Phillips and Edwards1966). However, van der Leer & McKay (Reference van der Leer and McKay2014) have recently shown that incentivizing a probability estimate variant of the task, in which participants were shown a fish from the fisherman's catch and then supplied probability estimates for each lake, led low-delusion-prone participants to provide probability estimates not significantly different from Bayesian probabilities, whereas high-delusion-prone participants still deviated from this objectively rational norm despite the included incentives. These findings resonate with a ‘liberal acceptance account’ (Moritz et al. Reference Moritz, Woodward and Hausmann2006, Reference Moritz, Woodward and Lambert2007; White & Mansell, Reference White and Mansell2009; Averbeck et al. Reference Averbeck, Evans, Chouhan, Bristow and Shergill2011), implying that participants, particularly those prone to delusions, might behave less conservatively than their estimates of relevant probabilities suggest they should. Future investigations with clinical populations may shed further light on this possibility. Given that delusional patients gather even less evidence than delusion-prone participants (Warman et al. Reference Warman, Lysaker, Martin, Davis and Haudenschield2007), we might expect deviations from optimal decisions to be even more pronounced for delusional patients than for our delusion-prone participants.
Conclusions
That delusional ideation is associated with JTC on probabilistic reasoning tasks is almost a received view in psychiatry (Huq et al. Reference Huq, Garety and Hemsley1988; Colbert & Peters, Reference Colbert and Peters2002; Fine et al. Reference Fine, Gardner, Craigie and Gold2007; Garety & Freeman, Reference Garety and Freeman2013). We adopted stringent experimental economics methods to minimize effects of potential confounds and, crucially, to generate rationally optimal decision points. Our procedure provides a generalized method of instantiating and computing optimal decision points, given any permutation of ratios, costs and rewards (see the Appendix). We found that nearly all of our participants jumped to conclusions, deciding before it was optimal to do so (from a risk-neutral perspective). Nevertheless, this tendency was greater the more delusion prone the participants were. No previous beads task investigation has included both rewards for correct decisions and costs for gathering information; the most that could justifiably be claimed from draws-to-decision studies using standard unincentivized paradigms is that delusional and delusion-prone individuals ‘reach conclusions more quickly’ than controls. Our findings, however, support and clarify the claim that delusional ideation is associated with a tendency to ‘jump to conclusions’.
Appendix
Calculation of optimal decision pointsFootnote 11
First we introduce some notation:
- w
-
: the event that the next fish caught is white
- b
-
: the event that the next fish caught is black
- W
-
: the event that the true lake is ‘White’, the lake with predominately white fish
- B
-
: the event that the true lake is ‘Black’, the lake with predominately black fish
- n w
-
: the number of white fish caught so far
- n b
-
: the number of black fish caught so far
- Δ
-
= n w −n b
- Δ′
-
= the value of Δ after catching one more fish
- l
-
: the event that the next fish is of the currently ‘leading’ fish colour (l = w if n w > n b and l = b if n w < n b )
- L
-
: the event that the true lake is the currently ‘leading’ lake (L = W if n w > n b and L = B if n w < n b )
- p
-
= Pr(w|W) = Pr(b|B) > 0.5
- ρ
-
= p/(1−p) > 1
- π
-
: the probability of making a correct guess if guessing now
- c
-
: the cost of seeing one more fish
- R
-
: the reward for a correct guess
Suppose that n fish have been caught so far, of which n w are white and n b are black. We are interested in the probability that the true lake is White, conditional on having caught n w white fish: Pr(W|(n w , nb )) [note that Pr(B|(n w , nb )) = 1−Pr(W|(n w , nb ))]. We find this using Bayes’ rule:
$$\eqalign{&{\rm Pr}(W \vert ({n_w},{n_b})) \cr &\quad= \displaystyle{{\rm Pr}(({n_w},{n_b}) \vert W){\rm Pr}(W) \over {\rm Pr}(({n_w},{n_b}) \vert W){\rm Pr}(W) + {\rm Pr}(({n_w},{n_b}) \vert B){\rm Pr}(B)}}$$
Because we have assumed a diffuse prior [i.e. participants are informed that both lakes are a priori equally likely, Pr(W) = Pr(B) = 0.5], the formula simplifies to:
$${\rm Pr}(W \vert ({n_w},{n_b})) = \displaystyle{{{\rm Pr}(({n_w},{n_b}) \vert W)} \over {{\rm Pr}(({n_w},{n_b}) \vert W) + ({\rm Pr}({n_w},{n_b}) \vert B)}}$$
The conditional probabilities that n w of the n fish are white given the type of lake are:
$${\rm Pr}(({n_w},{n_b}) \vert W) = {\,p^{{n_w}}}{(1 - p)^{{n_b}}}\left( {\matrix{ {{n_w}} \cr {{n_w} + {n_b}} \cr}} \right)$$
$${\rm Pr}(({n_w},{n_b}) \vert B) = (1 - p{)^{{n_w}}}{(p)^{{n_b}}}\left( {\matrix{ {{n_w}} \cr {{n_w} + {n_b}} \cr}} \right)$$
Inserting these expressions into the formula for Pr(W|(n
w
, nb
)) and dividing through by
${p^{{n_w}}}{(1 - p)^{{n_b}}}$
, we finally obtain:
$${{\rm Pr}(W \vert ({n_w},{n_b})) = \displaystyle{1 \over {1 + {\rho ^{{n_b} - {n_w}}}}},\quad {\rm where}\quad \rho = \displaystyle{\,p \over {1 - p}} \gt 1}$$
Note that
${\rm Pr}(B \vert ({n_w},{n_b})) = 1 - {\rm Pr}(W \vert ({n_w},{n_b})) =1/(1 + $
$ {\rho ^{{n_w} - {n_b}}})$
. Because p > 0.5, ρ > 1, so that Pr(W|(n
w
, nb
)) > Pr(B|(n
w
, nb
)) if and only if n
w
> n
b
. Therefore, if the decision maker decides to make a guess, they should always guess the lake corresponding to the most fish caught so far, and the probability of a correct guess is:
$$\pi = \left( {\matrix{ {{\rm Pr}(W \vert ({n_w},{n_b})) = \displaystyle{1 \over {1 + {\rho ^{{n_b} - {n_w}}}}}} & {{\rm if} \,{n_w}{\rm \gt} {n_b}} \cr {{\rm Pr}(B \vert ({n_w},{n_b})) = \displaystyle{1 \over {1 + {\rho ^{{n_w} - {n_b}}}}}} & {{\rm if} \,{n_b}{\rm \gt} {n_w}} \cr {{\rm Pr}(W \vert ({n_w},{n_b})) = Pr(B \vert ({n_w},{n_b})) = \displaystyle{1 \over 2}} & {{\rm if} \,{n_w}\;{\rm =} \;{n_b}} \cr}} \right.$$
Note that this simplifies to the following extremely simple rule, where the probability of a correct guess depends only on the absolute value of the difference of the numbers of white and black fish caught so far:
$${\pi (\Delta ) = \displaystyle{1 \over {1 + {\rho ^{ - \Delta}}}}, \quad {\rm where}\quad \Delta = {n_w} - {n_b}}$$
It follows that the only relevant state variable for our problem is Δ, and we can write the value function as V(Δ). Here V(Δ) is the expected value to the decision maker of having observed Δ more fish of one colour than of the other. As in any stopping-time problem, this value is the maximum of (a) the expected value of guessing immediately and (b) the expected option value of seeing one more fish.
The only missing element remaining in the formulation of this problem is the state transition matrix. This, however, is easy to find. Clearly, if one more fish is caught, Δ will change to either Δ + 1 or Δ − 1. Furthermore, if Δ = 0, the change will be to Δ′ = Δ + 1 with probability one. If Δ > 1, the probability of it changing to Δ + 1 is simply the probability of getting one more fish of the currently leading colour:
$${\rm Pr}\left( {\Delta ' = \Delta + 1} \right) = {\rm Pr}(l \vert L) + {\rm Pr}(l \vert L) = p\pi (\Delta ) + (1 - p)(1 - \pi (\Delta ))$$
To summarize:
$${\rm Pr}(\Delta {\rm ^{\prime}} = \Delta + 1) = \left\{ {\matrix{ 1 & {{\rm if}\,\Delta = 0} \cr {\,p\pi (\Delta ) + (1 - p)(1 - \pi (\Delta ))} & {{\rm otherwise}} \cr}} \right.$$
Now, we are ready to formulate the Bellman equation for the optimal stopping time problem. The expected value of guessing now is π(Δ)·R. The expected value of drawing one more fish is Pr(Δ′ = Δ + 1)V(Δ + 1) + (1−Pr(Δ′ = Δ + 1)V(Δ − 1) − c). The value function is therefore defined recursively by:
$$\eqalign{V(\Delta) &= \max \{ \pi (\Delta )R;{\rm Pr} (\Delta^{\prime} = \Delta + 1) V(\Delta + 1) \cr &\quad + (1 - {\rm Pr} (\Delta^{\prime} = \Delta + 1))V(\Delta - 1) - c\} \cr}$$
This equation is easy to solve by value function iteration. It can also be proven analytically that the stopping rule will always take the form ‘Stop if and only if Δ > Δ0’ for some Δ0.
Acknowledgements
This research received no specific grant from any funding agency, commercial or not-for-profit sectors. We thank Sir Robin M. Murray and three anonymous reviewers for valuable feedback on a draft of this manuscript. We also thank K. Levine and R. J. Banis for their generous advice, and V. Bourne and R. Ross for helpful discussions. Finally, we are grateful to M. Naef for the opportunity to use the experimental economics laboratory (EconLab) at Royal Holloway, University of London, and to T. S. Woodward for providing the images that formed the basis of our stimuli.
Declaration of Interest
None.
