The observed sequences of statements on which climate was expected show a great deal of heterogeneity among subjects. In order to explain and classify patterns of expectation for-mation that may underlie participants' responses, these response paths are confronted with sequences of statements predicted by a number of models that incorporate rational patterns of expectation formation as well as different biases and heuristics.
Models of Expectation formation Rational Expectations
As mentioned before the rationality hypothesis suggests that subjects would use an objec-tively correct model that incorporates all available information to form their belief about the distribution to be in use. In the given case this means to assume that participants accept equal prior probabilities for favorable and adverse climate before having observed the first draw, and subsequently take into account all weather outcomes at any time to calculate posterior probabilities for each climate, applying Bayes' theorem. Due to different risk pref-erences, individuals may differ with regard to the posterior probability they take as suffi-cient to decide for a statement in favor of either one of the two climates. This circumstance can be reflected by defining a threshold posterior probability b, at which the predicted state-ment of expectation switches between undecided, adverse and favorable climate, allowing a number of different values for b. That means that the rational expectations model predicts a subject to expect adverse climate if 𝑃(𝑎𝑑𝑣|𝑘; 𝑛) > 𝑏, to expect favorable climate if 1 − 𝑃(𝑎𝑑𝑣|𝑘; 𝑛) > 𝑏 and to be undecided if 1 − 𝑏 < 𝑃(𝑎𝑑𝑣|𝑘; 𝑛) < 𝑏. Because variable risk preference with regard to expectation formation is not among the standard assumptions of rational expectation formation, evidence for this mechanism is reported separately from 'standard rational' behavior and referred to as rational expectation formation with elevated risk aversion, or in short expectation formation in line with an elevated threshold model.
One may object that participants were not necessarily able to fully compute posterior prob-abilities, and as such it would still be rational were they to make use of rule-of-thumb ap-proximations. The simplest approximation to calculating posterior probabilities would be to expect adverse climate in case that more bad than good weather realizations were ob-served and vice versa. Applying this simple rule-of-thumb in the experiment results in the same sequence of expectations as the rational expectation model with a threshold probabil-ity of 55% predicts (similar approximations can be thought of for higher thresholds). Even if participants were not able to exactly calculate posterior probabilities their behavior could still correspond to the predictions of the rational expectations model with different thresh-old probabilities, if they based their expectation formation on such an unbiased approxima-tion algorithm for the posterior probability.
Biases and heuristics
Despite the obvious random determination of the distribution of balls that represented cli-mate from two options at equal probability, some individuals seem to have assigned a higher probability for one distribution from the outset, as they stated an expectation of ei-ther climate before having observed the first draw. Maintaining the assumption of rational probability updating, this can be formally expressed using a biased prior probability in the application of Bayes' theorem. Four model specifications with biased prior probabilities featuring a prior probability for bad climate to occur 𝑃(𝑎𝑑𝑣) of 60%, 75%, 40% and 25%
are tested in the following, with the last two implying bias in favor of good climate, to reflect pre-set expectation bias of different direction and strength.35
Subjects who base their assessment on representativeness and/or availability heuristics would consider only the most recent draws. Formally, such a bias can be expressed using a model that applies Bayes' theorem for making predictions while only taking a small num-ber of recent observations previous to each decision into account. Two small sample mod-els (in the following also referred to as 'short-sighted modmod-els') that consider either the last three or five observations and feature threshold probabilities of b = 55% and 65%, respec-tively, are applied to reflect this behavior. The models assume indecision before the first draw and expectations based on the threshold probability when less than the stipulated number of observations are available. Depending on the switching thresholds used, this formal model can also be represented by a simple rule: For example, a formal model based on five years and a threshold probability of 55% corresponds to forming expectations based on the majority of weather observations within the last five years. A 65% probability thresh-old in the five years specification results in predicting the respective climate expectation statement if four or more weather realizations point toward it, otherwise in predicting a statement of ‘unknown’.
Such a recency bias might also be combined with an anchoring heuristic. The correspond-ing model assumes that individuals start with an initial expectation of either climate and maintain this expectation until two consecutive weather outcomes contradict this expecta-tion. In this case the predicted belief switches to the contrary. Because one weather outcome that contradicts the expectation held is tolerated, this model is referred to as tolerance model. This model features no statement of indecisiveness according to the setup described.
Finally, a naïve model is applied that assumes expectations to be based solely on the pre-vious year's observation. It is implemented as a model that predicts indecisiveness initially and subsequently adjusts predictions according to the weather outcome of the previous year. Table 11 summarizes the models developed and applied to the data.
35 As the possibility to combine different biases is restricted by the structure and amount of data gathered, all of these model specifications assume a threshold probability for switching statements of b = 55% as reflecting a variety of b-values in addition to the different prior probability values would blur the character of the models and result in misleading predictions.
Table 11: Overview of model specifications applied to analyze the experimental data for
over all observations 0.5 0.55
Rational with ele-vated threshold probability (b-value)
Bayesian probability updating
over all observations 0.5 0.65, 0.75, 0.85
Biased prior in fa-vor of adverse cli-mate
Bayesian probability updating
over all observations 0.6, 0.75 0.55 Biased prior in
fa-vor of fafa-vorable climate
Bayesian probability updating
over all observations 0.4, 0.25 0.55
Small sample Bayesian probability updating
over last 3 or 5 years 0.5 0.55, 0.65
Naїve Statement equal to weather outcome of previous year
0.5 -