# "Bayesian anchoring" and the fourfold pattern of risk attitudes

Francesco FumarolaΓfootnoteΓfootnoteΓfootnoteEqual contributors  RIKEN Center
for Brain Science, Wako, Saitama 351-0198, Japan  Lukasz
Kusmierz${}^{\ref{footnote 1}}$ Allen Institute, Seattle, WA 98109, United
States  Ronald B. Dekker  Department of Experimental Psychology, University of
Oxford, OX2 6BW Oxford, United Kingdom

(August 30, 2024)

###### Abstract

Experiments on decision making under uncertainty are known to display a
classical pattern of risk aversion and risk seeking referred to as "fourfold
pattern" (or "reflection effect") , but recent experiments varying the speed
and order of mental processing have brought to light a more nuanced
phenomenology. We model experiments though a Bayesian formalization of the
anchor-and-adjust heuristic observed in empirical studies on cognitive bias.
Using only elementary assumptions on constrained information processing, we
are able to infer three separate effects found in recent observations: (1) the
reported enhancement of the fourfold pattern for quicker decision processes;
(2) the observed decrease of fluctuations for slower decision-making trials;
(3) the reported dependence of the outcome on the order in which options are
processed. The application of Bayesian modeling offers a solution to recent
empirical riddles by bridging two heretofore separate domains of experimental
inquiry on bounded rationality.

##  I Introduction

One of the foundational results on decision making under uncertainty is the
fourfold pattern of risk attitudes often referred to as reflection effect [1,
2]. Given a choice between a risky and a risk-free choice with identical
average outcomes, subjects are risk-seeking when the possible gain is large
(and unlikely) and when the loss is likely (but relatively small); subjects
are risk-averse when the possible gain is likely but relatively small, and
when the loss is unlikely and large (this is summarized in Fig. 1). The neural
mechanisms underlying the effect have long been an object of debate among
neuroscientists [3, 4]. The effect is customarily described through
phenomenological models, the most relevant being Prospect Theory [5], which
posits that subjects compute expected utilities by weighing probabilities
through a nonlinear weight function. The function is unknown a priori and must
be specified as a parameter of the theory, though only its convexity/concavity
properties matter. Since the appearance of the theory, the literature has seen
the proposal of both variations [6] and alternatives [7].

Figure 1:  Schematic depiction of the fourfold pattern of risk attitudes.
Dashed lines outline risky choices with equal expected values; along such a
line, the green ellipse highlights the preferred region and the red ellipse
the disfavored region, for gains and losses separately. Although a cake and a
bomb will be used to picturize positive and negative outcome, the common
choice in experiments is for loss and gain to be monetary, easing the
comparison of expected values among options. In gain experiments, subjects
favor very unlikely but very large gain (risk seeking) or sure outcome; in
loss experiments, subjects favor likely but small losses (risk aversion).

One of the main recent developments in research on human decision making is
the emergence of a body of empirical evidence that Prospect Theory alone is
not able to account for. Such evidence was obtained by psychologists by
modifying details of the classic experimental procedure – most notably the
time allowed for decision-making and the order in which the options are
presented to subjects.

The former direction of inquiry concerns the influence of time constraints on
individual preferences and choices (for an overview, [8]). By randomly
allocating subjects into time pressure conditions (i.e. varying randomly the
completion deadline for tasks), the literature has identified the effects of
time constraints on risky, social, and strategic behavior in such contexts as
bargaining [9], beauty contests [10], decisions under ambiguity [11],
imitation in strategic games [12], and bidding in auctions [13].

What may be called "slow" risk tasks – risky decisions made with no hindrance
to tackling the full computation workload – have been carefully isolated in
experiments such as those of [14], which involve substantial monetary rewards
for completing the tasks correctly and allow the full necessary time for
decision making. These studies have confirmed the risk behavior predicted by
Prospect Theory, showing that such behavior arises as a result of underlying
prioritization, not short-term cognitive-load issues. It has long remained
inconclusive, however, whether constraints on information processing combine
synergically with the fourfold pattern or affect the decision outcome in more
complex ways [15].

Recently, "fast" risk-taking decisions have become the object of a dedicated
set of experimental tests by Kirchler et al. [15], who brought the random time
approach to the analysis of risk behavior. The major findings reported are
that (1) time pressure increases risk aversion for gains and risk taking for
losses compared to time delay; (2) fluctuations in the outcome are suppressed
by granting subjects a longer time to decide.

One framework that should be mentioned here as being applied to multiple
problems in psychology is the broad dichotomy between two alternative thinking
processes: "System 1" (fast, intuitive, and emotional) and "System 2" (slower,
deliberative, more logical) [1]. The System 1 vs System 2 duality offers a
convenient language to broadly discuss a variety of different experiments. For
instance, one may say that the role of time pressure [15], increased cognitive
load [16], or acute stress [17] during decision making is to increase the role
of System 1, as System 2 would be computationally and energetically more
expensive.

One would assume that fast decisions rely on a different psychological
attitude than slower decisions, and are computed therefore on the basis of a
different set of underlying risk perceptions. However, even in the absence of
a dual process faster decisions would be affected by computational limitations
and by the reduced ability to cope with cognitive load, leading to distinct
outcomes. It is unclear at present how much, among the features of fast
decision making, follows simply from information-processing constraints. The
fact that performance on the Cognitive Reflection Test (CRT) and variable
decision time correlate identically with the observed extent of cognitive bias
is intriguing but does not allow to draw definite conclusions [18, 15].

The finding that time pressure increases risk aversion for gains and risk
taking for losses can be phrased as stating that time constraints increase the
fourfold pattern predicted by Prospect Theory, but the reasons for this trend
remain open to multiple interpretations. Kirchler et al. interpret the finding
as confirming that System 1 gives rise to the fourfold pattern, but the
specific features of System 1 that would do so are currently unknown.

Asides from experiments on time pressure, another direction of recent
experimental inquiry concerns the effect of variations in the processing order
of the options. Various features of visual processing have long been known to
affect choice behavior [19, 20, 21, 22] but one crucial factor appears to be
the sequence in which information is presented; this has been shown to have an
impact in real-life setting such as personality judgment [23, 24], clinical
decision making [25, 26], and judicial decision making [27]).

Most recently, [28] replicated the classical decision-making experiment we
have described and additionally assessed the processing order of the options.
They did so in two different ways: through eye-tracking, and by presenting the
options sequentially. Both approaches led to the finding that the magnitude of
loss/gain asymmetry effects is larger when the risk-free option is processed
first. A mechanistic explanation for this asymmetry is lacking, but a
description in terms of System 1 vs System 2 dichotomy may be attempted. In
particular, [28] invokes "changes in arousal and/or emotion induced by the
initial perception of the frame information, i.e. to framing effects capable
of modulating between avoidance behaviors and approach behaviors.

In this paper, we point out a candidate for a common mechanism behind all of
the above findings. The mechanism we propose is not novel – it is the anchor-
and-adjust heuristic [29] widely documented by a separate lineage of
experimental literature (reviewed further below). Formalizing that heuristic
into what we term Bayesian Anchoring, we calculate the outcome of time-
constrained information processing in risky decision making and from that
premise alone we derive the findings of [15].

The outline of the paper is as follows:

  * •

We sketch what is arguably the most basic Bayesian framework for time-
pressured cognition (Sec. II), with a view to pursuing it to its furthest
logical conclusions. We feed into that framework (Sec. IV) a cognitive prior
taken from empirical observations (reviewed in Sec. III), thus setting the
structure of Bayesian Anchoring.

  * •

We reproduce the fourfold pattern of risk attitudes from Bayesian Anchoring
(Sec. V), providing a coherent explanation to experimental findings on risky
decision making under time pressure [15]. We show that multi-step iterations
gradually diminish the fourfold pattern, which explains the recent,
unexplained findings of Kirchler et al. (Sec. VI).

  * •

We then move to discuss what Ba ayesian anchoring theory would look like with
priors different from the one we selected initially (Sec. VII), and we
additionally present a number of theorems concerning the set of Bayesian
Anchoring priors that would yield predictions compatible with experiments.

  * •

We finally summarize our contribution, compare Bayesian Anchoring to other
modeling efforts in the literature, and discuss the new research directions it
opens.

##  II Modeling fast decisions

The problem we focus on is simply stated. The subject needs to make an
informed guess on some not-immediately-known variable $S$ ("decision
variable") that quantifies the stochastic gain or loss (utility) associated to
a decision problem. This guess will informed by some set of relevant knowledge
$K$, which may be thought of as the result of reviewing past experience and
simulating random outcomes accordingly.

This conceptual setup resembles for example that of Lieder et al. [29], who
proceed to develop a model based on the familiar theory of Markov Chain Monte
Carlo (MCMC). However, here our focus is on choices constrained by limits on
computation time, which makes a MCMC model inappropriate. Indeed, we cannot
assume that adjustments are implemented through steps of the Metropolis-
Hasting algorithm, which would require access to (possibly unnormalized)
conditional density $\mathbf{P}(S|K)$ and hence access to all relevant
knowledge at once. It would take substantial time and computational effort to
access, analyze, and synthesize all relevant knowledge. Instead, we assume
that in time-constrained decisions, adjustments are driven by an iterative
process wherein the belief over $S$ is updated by integrating at every step a
single piece of information from the set of relevant knowledge. We start by
focusing on a tightly time-constrained regime, analyzing therefore a single
such adjustment step. This may be thought of as a single round of simulations
of the random outcome, used to update the utility based on the posterior from
the realizations [30].

Utility is most often represented in economics as taking values on the real
line. We thus assume the random variable $S$ to be realized by real values
$s$. In the very first approximation, every piece of information $K$ can be
distilled into a pair of numbers $(x,\sigma)$ describing the mean and standard
deviation of possible values of $s$ that this piece information supports
($\sigma$ being the measurement error). The corresponding generative model, or
likelihood function, can be written as

$\rho((x,\sigma)|s)=g_{\sigma}(x-s)$ (1)

where $g_{\sigma}(z)=[\sqrt{2\pi}\sigma]^{-1}\exp(-z^{2}/(2\sigma^{2}))$ is a
normal distribution with standard deviation $\sigma$. The assumption that the
noise in the estimate of the value of interest is Gaussian is non-essential to
results and is used here for mathematical convenience.

The corresponding posterior probability density of $s$ can be expressed as

$\rho(s|(x,\sigma))=\frac{\rho((x,\sigma)|s)\rho_{0}(s)}{\rho(x,\sigma)}=\frac{g_{\sigma}(x-s)\rho_{0}(s)}{\int
dsg_{\sigma}(x-s)\rho_{0}(s)}$ (2)

where we are abusing notation by denoting both likelihood and posterior
densities as $\rho$, with arguments and conditionalities to distinguish the
distributions, while we call $\rho_{0}$ the distribution of prior beliefs. The
next step consists in determining an ansatz on the function $\rho_{0}(s)$
enforced by existing experimental knowledge.

To this aim, we will briefly review the relevant phenomenology.

##  III Review of anchoring and adjustment

The anchor-and-adjust heuristic is a simple heuristic widely observed in human
cognition [29]. It is usually associated with the purported irrationality of
human behavior as it leads to cognitive biases that do not seem to follow
rules of logic and probability as well as give rise to sub-optimal decision-
making. While the concept has wide applicability, the case most relevant to
this work is the one where subjects, faced with uncertainty, start their
estimation process from a very rough guess termed anchor, possibly suggested
by the experimenter, and then integrate additional information to adjust the
estimate. Since often these adjustments are not sufficient, this leads to an
observable bias that can be manipulated by the experimenter.

For example, in one well-known experiment [31] participants (high-school
students) were divided into two groups and were asked to multiply either

$1\times 2\times 3\times 4\times 5\times 6\times 7\times 8$ (3)

or

$8\times 7\times 6\times 5\times 4\times 3\times 2\times 1$ (4)

The trick was that the participants were given only 5 seconds to report the
answer—way below the time required for most people to perform such mental
calculations. Thus, they had to resort to heuristics. One may expect that
people would perform a few steps of computation, starting from numbers on the
left, and the extrapolate to estimate the full product. Since the adjustments
are often insufficient, we should expect students to underestimate these
products. Moreover, the first group should get lower values of the anchor and
thus underestimate more. Indeed, the median estimate for the first group was
$512$, compared to $2250$ for the other group. This is obviously much lower
than the correct result ($40320$).

##  IV Anchoring as Bayesian inference with a spike-and-slab prior

In this section we formalize the Bayesian Anchoring model, describe its
single-step realization, and discuss possible multi-step extensions.

###  IV.1 General considerations

The phenomenology we just summarized can be formalized into a choice for the
cognitive prior figuring in Eq. (2). Given an initial guess (anchor) $s_{0}$,
we posit that the belief over $X$ is stored as a (prior) density

$\rho_{0}(s)=p_{0}\delta(s-s_{0})+(1-p_{0})g_{\gamma_{0}}(s-s_{0})$ (5)

We will refer to this as spike-and-slab prior [32, 33], the "slab" component
being here not uniform but modulated (for simplicity, as a Gaussian with
variance $\gamma_{0}^{2}$).

This prior can be interpreted as a mixture of two hypotheses about $S$. Under
the first hypothesis ($\mathcal{H}_{0}$, with prior probability
$P(\mathcal{H}_{0})=p_{0}$) the initial guess is correct, i.e.,
$\rho(s|\mathcal{H}_{0})=\delta(s-s_{0})$. As a consequence, observing any
information from the set of accessible knowledge cannot influence the initial
guess, since we have

$\rho(s|(x,\sigma),\mathcal{H}_{0})=\frac{g_{\sigma}(x-s)\rho(s|\mathcal{H}_{0})}{\int
dsg_{\sigma}(x-s)\rho(s|\mathcal{H}_{0})}=\delta(s-s_{0})=\rho(s|\mathcal{H}_{0})$
(6)

In contrast to $\mathcal{H}_{0}$, the other hypothesis presumes non-zero
variance of the initial guess
$\rho(s|\mathcal{H}_{1})=g_{\gamma_{0}}(s-s_{0})$. Accordingly,
$\mathcal{H}_{1}$ acknowledges the possibility that relevant knowledge can
change the belief over $S$, since in general we have

$\rho(s|(x,\sigma),\mathcal{H}_{1})\neq\rho(s|\mathcal{H}_{1})$ (7)

The posterior distribution of $s$ has a form similar to the prior:

$\rho(s|(x,\sigma))=p_{1}\delta(s-s_{0})+(1-p_{1})g_{\gamma_{1}}(s-s_{1})$ (8)

where $p_{1}$, $\gamma_{1}$, and $s_{1}$ are updated values of parameters that
depend on the parameters of the prior as well as on the observation. In
particular, $p_{1}=P(\mathcal{H}_{0}|(x,\sigma))$ is the posterior probability
of the hypothesis $\mathcal{H}_{0}$:

$p_{1}=\frac{1}{1+K^{-1}}$ (9)

where

$K=\frac{p_{0}}{1-p_{0}}\frac{g_{\sigma}(x-s_{0})}{g_{\tilde{\sigma}}(x-s_{0})}$
(10)

and

$\tilde{\sigma}^{2}=\sigma^{2}+\gamma_{0}^{2}$ (11)

The width of the slab decreases according to the formula

$\gamma_{1}^{-2}=\gamma_{0}^{-2}+\sigma^{-2}$ (12)

The location of the slab shifts towards the observation as

$s_{1}=\frac{\gamma_{0}^{-2}}{\gamma_{0}^{-2}+\sigma^{-2}}s_{0}+\frac{\sigma^{-2}}{\gamma_{0}^{-2}+\sigma^{-2}}x$
(13)

Since the location of the spike is constant, in general the location of the
spike and Gaussian slab in the posterior do not coincide ($s_{1}\neq s_{0}$).

###  IV.2 Single adjustment step

Let us first assume that just a single adjustment step has been made–a likely
scenario when faced with very limited time. After the adjustment step the
posterior has been calculated, but what value should be used to compare
options and take a decision? Let us denote the point estimator used for
decision-making to compare options as $\hat{s}$. If the observation confirms
the anchor ($x=s_{0}$), the optimal estimator should remain the anchor, so we
must have $\hat{s}(0)=s_{0}$. On the other hand, any difference between
$\hat{s}$ and $x$ must clearly come from a bias in the prior distribution
$\rho_{0}(s)$, and in the limit where the prior is very flat
($p_{0}\rightarrow 0$), we must have the asymptotic behavior $\hat{s}\approx
x$.

These requirements are satisfied by a wide range of estimators and many
results presented in this work should extend to other such estimators. For
concreteness, in the following we choose the minimum mean squared error
estimator, which amounts to calculating the expectation $\hat{s}=\langle
s|x\rangle$ over the posterior distribution. Measuring the argument of the
estimator by its displacement from the anchor, we define
$\hat{s}(x-s_{0};s_{0}):=\langle s|x\rangle$, and will omit explicit mention
of the anchor as the second argument of $\hat{s}$ where obvious. The explicit
expression for $\hat{s}$ reads

$\hat{s}=s_{0}+\frac{1}{1+K}\frac{\gamma_{0}^{2}}{\sigma^{2}+\gamma_{0}^{2}}(x-s_{0}).$
(14)

Note that, due to the dependence on the Bayes factor $K$, $\hat{s}$ is a
nonlinear function of $x-s_{0}$. This nonlinearity will be essential when we
discuss the role of anchoring-with-adjustment in the fourfold pattern of risk
attitudes (Section V).

###  IV.3 Multiple adjustment steps

There are multiple ways in which one could extend our model to incorporate
multiple adjustments steps. We need to specify an update rule and a stopping
rule. The update rule describes how the new prior for the next step is formed
from the previous prior given the observation. The most natural update rule
seems to be to perform the exact Bayesian inference and to use the posterior
as the new prior. However, to perform Bayesian inference with multiple steps
of adjustments the agent should track values of $p_{n}$, $\gamma_{n}$,
$s_{n}$, and to remember the initial value of the anchor $s_{0}$.
Additionally, in general the Gaussian prior of $\mathcal{H}_{1}$ is not a
conjugate prior if the likelihood takes a different form. Thus, the inference
may become more complex and may require approximations or sampling.

Another possibility is to perform a simple average over $\\{x_{i}\\}_{i=1..T}$
(where $T$ denotes the number of pseudo-observations made in order to perform
adjustments) and then use this sample mean in the single adjustment step,
utilizing anchoring slab-and-spike priors as described above.

In the following we explore another hypothesis that humans rely on a
simplified recursive inference strategy. First note that the process of multi-
step anchoring-and-adjustment must at some point be terminated. The following
stochastic rule could utilize the probabilistic form of the posterior in order
to make a decision whether to continue or stop the process. A single sample is
drawn, taken from the posterior distribution over hypotheses (i.e.
$\mathcal{H}_{0}$ or $\mathcal{H}_{1}$). If $\mathcal{H}_{0}$ is chosen the
adjustment process is terminated and the value $\hat{s}$ is reported.
Otherwise, another adjustment step is taken with a new prior that takes the
form

$\rho_{1}(s)=p_{1}\delta(s-s_{1})+(1-p_{1})g_{\gamma_{1}}(s-s_{1}).$ (15)

This formula introduces an important modification compared to the optimal
Bayesian update. Namely, the position of the spike under $\mathcal{H}_{0}$ is
updated to reflect the new most likely position of $s$. Such reanchoring is
not essential for the consistency of Bayesian Anchoring theory but it
simplifies the calculations and may help account for the fact that the initial
anchor is rarely reported, even if it biases the final report. We hypothesize
that the reanchoring position corresponds to the mean of the posterior slab
($s_{1}$), but other reanchoring methods can be formulated. We performed
numerical simulations to check how expected bias and error change as functions
of the number of sampling and adjustment steps, depending on the reanchoring
method (Fig. 4). Our simulation results indicate that reanchoring to the
posterior slab mean quickly removes the bias. In contrast, reanchoring to the
overall posterior mean ($\hat{s}$) leads to an extremely persistent bias.

Additionally, it is possible that instead of updating $p_{0}$ to $p_{1}$
according to the rules of Bayesian inference, $p_{n}$ is modulated by the
internally or externally generated signal indicating a sense of urgency.
Intuitively, the more urgent it is to finish the inference, the higher should
be the value of $p_{n}$. This in turn makes it more likely to stop the
adjustment process in the following step.

Figure 2: Structure of cognitively biased decision making based on a
subjective update of utilities. The update may be thought of as the result of
reviewing past experience and simulating random outcomes accordingly. The
perceived utilities resulting from the process drive subjective choices.
Sections II-VI of this article follow the flow of the model along the brown
arrows in the drawing, from assumptions about the internal variables all the
way to inferred behavior. In Sec. VII, we show for completeness the results of
proceeding the opposite way, starting from the observed behavior and working
backwards to infer prior believes (gray dashed arrow).

##  V Emergence of the fourfold pattern of risk attitudes

We now apply the framework we developed to the experimental setting routinely
used to verify fourfold pattern of risk attitudes [5, 1, 2], where risky
choices are associated to a binary outcome as shown in Table 1. The utility
acquired though a risky choice stochastically turns out to be zero or to have
a positive/negative value. We will call $p$ the probability of the latter
outcome.

|  |  Gain/loss value $x$ |  Probability |  $\langle x\rangle_{\text{outcomes}}$
---|---|---|---|---
Option A (risky) $\longrightarrow$ |  Outcome $1$ | $r/p$ | $p$ | $r$
Outcome 2 | $0$ | $1-p$
Option B (risk-free) |  | $r$ | $1$ | $r$
Table 1: Structure of a choice options in traditional experiments on economic
decision [5]. Subjects are given two options to choose from, a risky and a
risk-free one. The last columns shows the mean gain/loss, i.e. the average of
the gain/loss value over objective outcomes, which is the same for both
options.

As shown in Table 1 subjects who choose the risk-free outcome receive or lose
a value $x=r$ with probability $1$; subjects who choose the risky outcome
receive or lose value $x=r/p$ with probability $p$ and $x=0$ with probability
$1-p$.

To yield the perceived utility associated to a choice option, the point
estimate $\hat{s}$ must be averaged over all outcomes of that option. The
perceived utility of a given option is therefore

$U^{\text{(option)}}=\langle\hat{s}(x-s_{0},s_{0})\rangle_{\text{outcomes}}$
(16)

Although the anchor is usually not reported, it is natural then to expect that
the gain/loss value stated as the sure outcome of the riskless choice would be
the anchor $s_{0}=r$. Replacing this into Eq. 16 for the two options $A$ and
$B$ in Table 1,

$\displaystyle U^{(A)}(p,r)$ $\displaystyle=$
$\displaystyle\langle\hat{s}(x-s_{0},s_{0})\rangle_{A}=\langle\hat{s}(x-\langle
x\rangle,\langle x\rangle)\rangle_{A}=p\ \hat{s}(r/p-r,r)+(1-p)\hat{s}(-r,r)$
(17) $\displaystyle U^{(B)}(r)$ $\displaystyle=$
$\displaystyle\langle\hat{s}(x-s_{0},s_{0})\rangle_{B}=\langle\hat{s}(x-\langle
x\rangle,\langle x\rangle)\rangle_{B}=\hat{s}(0,r)=r$ (18)

where we note that $U^{(B)}(r)=U^{(A)}(1,r)$ because setting the gain/loss
probability to unity removes all risk from the risky option.

A decision is made by comparing the perceived utility of Eqs. 17 over all
options available (Fig. 2), that is, the experimentally observed decision
depends on the sign of $U=U^{(A)}-U^{(B)}$, which is just $U^{A}$ measured
with its $p=1$ value for a a baseline.

Hence, the fourfold pattern of risk attitudes corresponds to the finding that
the relative perceived utility $U$ flips sign as a function of the gain/loss
probability $p$, becoming positive only for sufficiently large $p$ if $r$ is
negative, and becoming negative only for sufficiently large $p$ if $r$ is
positive. Subjects will reject the risk-free option in favor of an uncertain
gain only if that gain is sufficiently large, and will reject it in favor of a
purely potential loss (which may ultimately not take place) only if that loss
is sufficiently small (see Sec. S-III for details).

Simply plugging (5) into (2) and the result into (17) immediately yields the
comparative perceived utility plotted in Fig. 3, left-hand panel (see Sec.
S-IV.1 for details). The utility indeed flips sign as expected from
experiments, yielding risky choices in the gain domain only for high gains,
and risky choices in the loss domain only for small losses. Thus, Bayesian
Anchoring theory qualitatively reproduces the fourfold pattern of risk
attitudes with an arguably simpler set of assumptions compared to Prospect
Theory.

The main assumption has been the choice of the risk-free reward as an anchor,
which is motivated by the certainty of the reward within the risk-free option.
In other words, the reward associated to a risk-free option is assumed to be
more "salient". Within that assumption, the experimental results of [28] are
also arguably explained as a corollary.

Indeed, suppose that the risky option is presented first. Only after the risk-
free option is also given, its saliency is recognized and the anchor set (or
reset). This takes time and makes the process more liable to randomization. By
contrast, if the first option is the risk-free one, the anchor is set before
the second option is given, and the decision making is performed immediately
upon hearing the second option. An alternative way to think about this is that
two different aspects contribute to the determination of the anchor, saliency
and primacy. The interference between these two effects is constructive if the
risk-free option is presented first, destructive otherwise. This agrees with
the result of experiments where the processing order is monitored as in [28].

Experiments that vary the processing speed have a more elaborate output and
will be discussed next.

Figure 3: Perceived utility of a risky option with loss/gain probability $p$,
measured relative to the riskless option with equivalent expected outcome.
Left: Predictions of the anchoring theory. The main features of experiments,
such as the transition between risk-aversion and risk-seeking and the
associated reflection principle, are reproduced. Right: Predictions of the
phenomenological model of [7] shown for comparison. One difference between the
two predictions is that the Bayesian model does not lead to $U\to\pm\infty$
with $p\to 0$. In its basic version, the Bayesian model can only reproduce the
symmetric version with $k=1$ and the bias towards risk aversion must be
introduced ad hoc, similarly to the model of [7]. The parameter values used
here are $p_{0}=0.2$ and $\gamma_{0}=10$.

##  VI Time-dependent fluctuations: experiments and theory

Kirchler et al. [15] conducted a comprehensive study of the fourfold pattern
of risk attitudes over a large sample, with real financial rewards and across
multiple geographical areas.

One of their findings is that fluctuations in the response are enhanced by
time pressure. In section 5 of [15], the different components of such
fluctuations are teased out by breaking apart the effect of time pressure on
risk preferences and the effect of time pressure on noisiness.

Experiment includes a risky option and a non-risky option. The amount
suggested for the non-risky option is given and requires no calculation, so in
the Bayesian Anchoring framework it would correspond to the anchor $s_{0}$.
Thus, $s_{0}$ is induced externally and as such will not cause much
fluctuations.

However, whatever fluctuations there are will be dependent on the number of
steps you go through before issuing a verdict. The more time you have, the
more refined an estimation you will do. In the Bayesian Anchoring framework,
the number of adjustments you take in your process will affect how much
variance there is in the verdict. Indeed, our simulations of multistep
adjustment process (Fig. 4) confirm this intuition and show that the variance
of the estimate decreases with the number of adjustment steps. This result
shows that Bayesian Anchoring theory can qualitatively reproduce human risk
behavior under time pressure and suggests that experiments similar to the ones
performed in [15] could be designed to further validate the theory and
pinpoint its exact details including the form of the reanchoring.

Figure 4:  Bias and root-mean-square error as functions of the number of
adjustment steps. If reanchoring sets $s_{1}$ (and $s_{0}$ in the spike) to
the posterior mean $\hat{s}$, the anchor-induced bias persist for many steps
(top panels). In contrast, if $s_{1}$ is used for reanchoring, the bias is
reduced rapidly (bottom panels). Completely rational behavior corresponds to
zero bias as the two options to chose from have the same expected value. The
right-hand panels show that the root-mean-square error decreases as a function
of the number of adjustment steps. This provides a principled explanation to
the experimental finding, reported by [15], that fluctuations shrins as a
function of the number of steps. Simulation parameters: bias $\sigma=1$,
$p_{0}=0.9$, $s_{0}=0$, # of samples: $10^{4}$. Figure 5:  Key variables
emerging from the theory, plotted in upper row for a purely Gaussian prior
distribution and in the lower row for a leptokurtic prior. The different
columns show (a) the prior distribution itself, mesokurtic in the upper row
and leptokurtic in the lower row 2; (b) the internal measurement model, i.e.
the relevant likelihood function, normal in both cases; (c) the resulting
posterior distribution, which becomes bimodal in the leptokurtic case; (d) the
optimal estimator used to evaluate utility, i.e. the mean of the posterior
distribution; (e) the perceived utility as a function of the probability of a
non-null outcome in a decision-making test such as those of [15]; (6) the
perceived utility, as a function of the odds $q=\frac{1-p}{p}$. All results
come from the gaussian mixture model defined in Sec. S-IV.2), the parameter
values being $\alpha=0,\gamma=2\sigma=1$ in the upper row and
$\alpha=0.5,\gamma=2,\delta=0.3,\sigma=1$ in the lower row. We fixed $x=3$ in
panels b and c, $r=1$ in panels e and f of either row.

##  VII Robustness and generalizations

Most of the assumptions that went into the building of the theory are arguably
natural – such as the normality assumption on the model of inner measurement
error (which leads to the Gaussian likelihood in Eq. 2). However, to model the
prior distribution of the outcome we condensed experimental knowledge from the
anchoring-and-adjustment literature into a specific functional form (Eq. 5)
that has been rarely encountered in psychology so far, albeit being common in
studies on numerical methods [32, 33]. To test the robustness of the theory,
we must check whether it would still be able to predict experimental
observations if the prior distribution had a different form.

Therefore, while so far we have deduced behavior from assumptions on the
cognitive prior, we are now interested in the opposite logical path of
inferring the underlying cognitive prior from what the experimental literature
tells us on behavior (Fig. 2). The detailed investigation we carry out in the
Technical Supplement proves that indeed a more general class of priors can
account for the same fourfold pattern of risk attitudes, providing a more
nuanced model of resource-rational economic decision-making. However, all
cognitive priors compatible with the fourfold pattern of risk attitudes shares
some essential features of the spike-and-slab prior. We report three rigorous
results that clarify the connection between prior beliefs and behavior:

1\. Normality-rationality theorem. All options in the experiment described by
Table 1 correspond to the same expected gain/loss. In the language of bounded
rationality, rational behavior would be to rank all options equally. A
striking result of our framework is that normality equals rationality. As we
prove in Sec. S-V.2, rationality is equivalent to normality of the cognitive
prior. A subject’s preference over options is rational only if the cognitive
prior is a Gaussian distribution (Fig. 5, panels 1e,1f).

2\. Open mind theorem. We show in Sec. S-V.7 that a compact-support prior is
incompatible with experimental evidence. Since a Bayesian update cannot
restore probability where it is zero at the outset, we would interpret a
compact support as signifying that some potential values of the outcome have
been ruled out a priori. Similarly, a non-compact support encodes an open-
minded prior belief about possible outcome. Calculating the perceived
utilities curves for our theory shows that, if the prior has compact support
as in panel D of Fig. 6, the comparative utility function $U$ behaves as in
the dashed red curves of panels A and B of Fig. 6. The experimentally observed
pattern of risk attitudes implies, therefore, that the prior beliefs are
"open-minded".

2\. Leptokurticity of the cognitive prior. As an additional result, the
emergence of the fourfold pattern reveals that the cognitive prior is
leptokurtic. Leptokurtic distributions can be hand-wavingly described as
possessing thin tall peaks, and thick tails clearly separated from them [34].
Thus, a spike-and-slab prior is a caricature of what a leptokurtic
distribution looks like in the unimodal case; such distributions are less
restrictive than the spike-and-slab prior of Eq. 5 but share its essential
features (see e.g. Fig. 5, panel 2a). We can thus simply regard a leptokurtic
prior as a "softer" way to encode anchoring. Any other type of prior
distribution would not be able to yield the fourfold pattern of risk
attitudes, as proven in Sec. S-V.9 of the Technical Supplement.

In conclusion, the prior we used to encode the anchoring effect is not a
necessary choice, but any cognitive prior compatible with experiments shares
its essential features, which points back to anchoring-and-adjustment as a
mechanism supporting or enhancing the fourfold pattern in fast decision
making.

Figure 6:  Sketch depicting the open mind theorem. If the prior has compact
support as in panel d, the comparative utility function $U$ behaves as in the
dashed red curves of panels a and b. An "open-minded" prior excluding no
outcome a prior (panel c) is needed to give the perceived utility a slope
compatible with the fourfold pattern (blue curve in panels a and b).

##  VIII Discussion

We introduced Bayesian Anchoring, a formalization of the anchoring-and-
adjustment heuristic according to the Bayesian inference paradigm. In Bayesian
Anchoring, the anchoring effect is implemented at the level of a prior belief
utilized in the subsequent Bayesian inference that is assumed to take place
during adjustment steps.

In order to account for the experimentally observed features of the anchoring-
and-adjustment process, the prior should strongly bias the inference towards
the anchor. Arguably, the simplest prior satisfying this condition takes the
form of a "spike and slab", which we adopted in our base analysis. We showed
that this rather simple model reproduces the fourfold pattern of risk
attitudes mechanistically and without the somewhat arbitrary probability
weighting introduced in Prospect Theory. We also showed that less stringent
constraints on the initial prior lead to equivalent results. The latest
generation of decision-making experiments has provided new evidence not
explained by Prospect Theory, such as obtained by modulating processing speed
and processing order. We showed that Bayesian Anchoring provides a mechanistic
explanation for those results.

One of the central tenets of Prospect Theory is the idea that humans evaluate
their options in terms of gains and losses with respect to a reference point,
which depends on the context and can be manipulated from outside, for instance
by rephrasing the description of a decision problem. Since losses and gains
are not treated equally, the results of the evaluation depend on the reference
point, leading to framing effects. Our results suggest that the existence of
the reference point in Prospect Theory should be interpreted as an example of
anchoring effect.

The anchoring effect and other cognitive biases have been traditionally
explained in terms of bounded rationality [35, 36] which posits that humans,
given their imperfect knowledge of the world and limited cognitive resources,
attempt to make good-enough ("satisficing") rather than optimal decisions.
More recently this view has been enriched by quantitative models of cognitive
biases in the framework of resource-rationality [37], wherein each unit
computation has an associated cost and the optimal behavior has to balance
costs of computation and sub-optimal decisions. In this framework, anchoring-
and-adjustment has been interpreted as a costly MCMC sampling process [38,
29]. MCMC samplers start from an arbitrary initial guess and follow stochastic
dynamics to generate samples from a desired distribution. Since the values
generated in the initial phase are usually correlated with the initial guess
and may not be representative of the desired distribution, practical
implementations often incorporate a so-called burn-in period during which
values generated by the chain are discarded from the sampler. It has been
shown [38, 29] that "premature sampling", i.e. utilizing samples during the
initial phase of the MCMC sampler, leads to biases that are similar to those
generated due to anchoring-and-adjustment. Moreover, within this framework
such a strategy of impatient sampling is often actually the optimal strategy.
In this context, let us reflect upon the role of a spike-and-slab prior in our
theory as compared to its traditional applications in statistics, where it is
usually introduced as an element of sparsity-inducing variable-selection
techniques. We hypothesize that its normative role as used in the model of
anchoring-adjustment is to provide a simple greedy technique of reducing the
complexity of inference, as required by the rational utilization of limited
computational resources. This view should be explicated by studying multi-step
versions of our model in the context of resource-rationality, which we believe
is an exciting avenue for future research.

Our model was inspired by the anticipated surprise introduced by Chan and
Toyoizumi [7]. Their phenomenological model of economic decision making is
based on the assumption that a decision maker maximizes the anticipated
surprise defined as an expectation over a nonlinear function of the difference
between the value of each outcome and the expected value across all outcomes.
The form of the nonlinear function was chosen to qualitatively match
experimental results. Our proposal can be interpreted as the normative
underpinning of the anticipated surprise model, although this mapping comes
with a caveat: the original anticipated surprise model assumes a convex
nonlinear function for positive arguments, which the optimal estimator
$\hat{s}$ is not (see Sec. S-IV.1 of the technical supplement, or Fig. 5,
panel B4).

In deriving the results of section VII, the finding about the posterior (such
as leptokurticity) were translated into constraints on the prior by assuming a
perfectly Bayesian framework. Such deductions would have been completely
different if we had assumed a mechanism to go from prior to posterior that
includes costs for the process itself of hypothesis update, as in Ref. [39].
Such theories of "Costly Bayesian inference" may have the potential to yield
the observed behavior with different priors. Identifying the most plausible
way to explain observations with theories of this kind remains a problem open
to exploration.

Another future direction concerns the exploration of different anchoring
mechanisms. The model assumed that the initial guess (the anchor) coincides
with the risk-free option if available. However, there are settings where such
an anchor may not not available, and the gathering of experimental data on
such settings is a necessary next step to calculate the prior distributions
that lead to best reproducing empirical results.

Finally, the vanilla version of Bayesian Anchoring presented here does not
account for the overall risk aversion for losses observed in human decision
making. This will also constitute a motivation for additional research on
Bayesian Anchoring.

##  IX Acknowledgments

We would like to thank Taro Toyoizumi for introducing us to the problem, Ho Ka
Chan for discussions on prospect theory and its alternatives, and Manuel
Baltieri for inspiring conversations on the range of applicability of Bayesian
frameworks.

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TECHNICAL SUPPLEMENT

##  S-I Notation

This supplement contains mathematical derivation of various aspects of the
theory helpful for a fuller comprehension of its structure. To ease the
writing and reading of such derivations, in the supplement we will use a
leaner notation than in the main text.

  * •

The function $g_{\sigma}(x)$ is, as in the main text, a standard Gaussian
probability density with zero mean and standard deviation $\sigma$. To avoid
lengthy pedices, where the expressions for variances get long we will use the
notation $G(x;\sigma)=g_{\sigma}(x)$ and $G_{\mu\sigma}(s)=g_{\sigma}(s-\mu)$.

  * •

The prior distribution will be called simply $\rho(s)$ rather than
$\rho_{0}(s)$ as in the main text.

  * •

Random utilities are measured as shifts from the mean value of their prior
distribution (equal to the anchor $s_{0}$ for the spike-and-slab case), so
that the prior itself has zero mean. The variable $s$ (now meaning $s-s_{0}$)
will be referred to as the "deviation", and "zero-deviation" will mean "for
$s=0$. We will call $x$ the displacement of the outcome from that mean, which
in the main text would have been called $x-s_{0}$.

  * •

The posterior distribution conditioned to a given value $x$ of the measurement
will be called $\phi(s|x)$ (with its dependence on the measurement error
$\sigma$ being implicit) rather than $\rho(s|(x,\sigma))$.

  * •

We use overlines to indicate averages over $\rho$:

$\overline{h(s)}=\int_{-\infty}^{\infty}ds\ h(s)\rho(s)$ (S1)

  * •

We use the notation $\langle\ldots\rangle_{x}$ (or simply
$\langle\ldots\rangle$, leaving the value of the outcome implicit) to denote
the average over the posterior given outcome $x$.

  * •

Finally, assuming the measurement error to be Gaussian with variance
$\sigma^{2}$, we write the likelihood $\rho((x,\sigma)|s)$ as
$g_{\sigma}(x-s)$.

The one-step Bayesian update is thus described by:

$\phi(s|x)=\frac{\rho(s)g_{\sigma}(x-s)}{\int ds\ \rho(s)g_{\sigma}(x-s)}$
(S2)

##  S-II General properties

###  S-II.1 Symmetry of the prior

With the extra assumption that $\rho(s)$ is symmetric around 0, it follows
that the optimal estimator given by the posterior mean $\hat{s}(x)=\langle
s\rangle_{x}$ is antisymmetric.

Indeed write

$\hat{s}(x)=\frac{\int ds\ s\rho(s)g_{\sigma}(x-s)}{\int
ds\rho(s)g_{\sigma}(x-s)}$ (S3)

and notice that if we replace x with -x, but then flip the sign of the
integration variable and use the symmetry of $g_{\sigma}$ , we end up having
on the RHS the same object as before but with a minus in front.

The converse is also true. Indeed using the identity

$\displaystyle\partial_{x}g_{\sigma}(x-s)$ $\displaystyle=$
$\displaystyle\sigma^{-2}(s-x)g_{\sigma}(x-s)$ (S4)

we can write

$\hat{s}(x)=x+\sigma^{2}\partial_{x}\log\int ds\
\rho(s)g_{\sigma}(x-s)=x+\sigma^{2}\partial_{x}\log\left[\exp\left(\frac{\sigma^{2}\partial^{2}_{x}}{2}\right)\rho(x)\right]$
(S5)

where the second term represents the mean-value shift from measurement due to
the prior’s traction, expressed compactly via the differential formulation of
a Weierstrass transform [40].

As seen from (S5), antisymmetry of $\hat{s}(x)$ implies antisymmetry of the
second term on the RHS, hence symmetry of the function subjected to
$\partial_{x}$. Log being bijective on $\mathbb{R}^{+}$, that entails in turn
symmetry of the argument of the logarithm, that is, of the Weierstrass
transform of the prior. But the Weierstrass transforms commute with the parity
operator, from which we infer symmetry of the prior itself.

###  S-II.2 Gaussian identities

For some of the following manipulations it will be useful to recall, as a
preliminary, that the product of two Gaussian densities can be written as

$G_{\mu_{1},\sigma_{1}}(s)G_{\mu_{2},\sigma_{2}}(s)=G_{0,\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}}(\mu_{1}-\mu_{2})\
G_{\mu_{12},\sigma_{12}}(s)$ (S6)

where

$\mu_{12}=\frac{\mu_{1}\sigma_{2}^{2}+\mu_{2}\sigma_{1}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}\hskip
30.0pt\sigma_{12}^{2}=\frac{\sigma_{1}^{2}\sigma_{2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}$
(S7)

So if the prior is Gaussian with zero-mean and variance $\gamma^{2}$, applying
(S6) with $\sigma_{1}=\sigma$ and $\sigma_{2}=\gamma$ we see that the
resulting normalization prefactor is $g_{\Sigma}(0)$, where we are
establishing the notation $\Sigma^{2}=\gamma^{2}+\sigma^{2}$, and the
normalized posterior distribution is $\phi(s|x)=G_{\hat{s}(x),\Gamma}(s)$ with
the posterior variance

$\Gamma^{2}=\frac{\sigma^{2}\gamma^{2}}{\sigma^{2}+\gamma^{2}}$ (S8)

(note that $\sigma\gamma=\Sigma\Gamma$) and the posterior mean

$\hat{s}(x)\equiv\langle s|x\rangle=\langle
s\rangle_{x}=\frac{\gamma^{2}x}{\sigma^{2}+\gamma^{2}}$ (S9)

which is linear in $x$. In particular, it will be useful to remember that the
posterior mean is $\mathcal{O}(x)$ for large $x$ whereas the variance is
$\mathcal{O}(1)$.

###  S-II.3 Connection between perceived utility and posterior cumulants

We are going to prove the statement that

$\partial^{n}_{x}\hat{s}(x)=\sigma^{-2n}\mathcal{C}_{x}^{(n+1)}$ (S10)

where we defined the cumulants of the posterior distribution as

$\mathcal{C}_{x}^{(n)}=n!\left[\frac{\partial^{n}}{\partial t^{n}}\log\langle
e^{ts}\rangle_{s|x}\right]_{t=0}$ (S11)

We will prove relationship (S10) by induction. For $n=0$, (S10) reduces to
$\hat{s}(x)=\langle s\rangle_{s|x}$ and therefore is true. Assume that Eq.
(S10) is verified for $n-1$. We write then its LHS for $n$ as

$\partial^{n}_{x}\hat{s}(x)=\partial_{x}\left(\partial^{n-1}_{x}\hat{s}(x)\right)=\partial_{x}\left(\sigma^{-2n+2}\mathcal{C}_{x}^{(n)}\right)=\sigma^{-2n+2}\partial_{x}\mathcal{C}_{x}^{(n)}$
(S12)

To calculate this derivative of a cumulant, we begin by applying identity (S4)
to the derivative of the cumulant generator

$\partial_{x}\log\langle
e^{ts}\rangle=\sigma^{-2}\frac{\langle(s-x)e^{ts}\rangle_{s|x}}{\langle
e^{ts}\rangle_{s|x}}=-\sigma^{-2}x+\sigma^{-2}\partial_{t}\log\langle
e^{ts}\rangle=\sigma^{-2}\partial_{t}\sum_{n\geq
0}\frac{\mathcal{C}^{(n)}}{n!}t^{n}+h(x)=\sigma^{-2}\sum_{n\geq
0}\frac{\mathcal{C}^{(n+1)}}{n!}t^{n}+h(x)$ (S13)

where $h(x)$ is not a function of $t$ and therefore has no effect on
derivatives with respect to $t$.

The derivative of the cumulants is calculated by applying Eq. (S11) and then
Eq. (S13). We find that, for all $n\geq 1$,

$\partial_{x}\mathcal{C}_{x}^{(n)}=n!\left[\partial_{t}^{n}\partial_{x}\log\langle
e^{ts}\rangle\right]_{t=0}=\sigma^{-2}\mathcal{C}^{(n+1)}$ (S14)

while explicit calculation shows that equation (S14) is also valid for $n=0$.

Substituting relationship (S14) into Eq. (S12) leads straight to formula
(S10), thus proven by induction.

###  S-II.4 Range of the posterior mean

We show here that $\langle s\rangle_{x}$ lies between $\bar{s}$ and $x$ if the
prior distribution is symmetric and unimodal.

Under these conditions, we can write the $\rho(s)$ in terms of a zero-centered
distribution nonincreasing for positive argument, as
$\rho(s)=\hat{\rho}(s-\bar{s})$. We first notice that, defining the
normalization factor $\mathcal{N}=\int dsg_{\sigma}(x-s)\rho(s)$, we have

$\hat{s}(x)-x=\mathcal{N}^{-1}\int
ds(s-x)g_{\sigma}(x-s)\rho(s)=\mathcal{N}^{-1}\int dy\ y\ g_{\sigma}(y)\
\hat{\rho}(y+\hat{x})=\mathcal{N}^{-1}\int_{0}^{\infty}dy\ y\ g_{\sigma}(y)\
\Big{(}\ \hat{\rho}(\hat{x}+y)-\hat{\rho}(\hat{x}-y)\Big{)}$ (S15)

where $\hat{x}=x-\bar{s}$. It’s clear by symmetry that this integral vanishes
for $\hat{x}=0$. Without lack of generality, let’s check its sign for
$\hat{x}>0$. We have then

$\displaystyle\hat{s}(x)-x$ $\displaystyle=$
$\displaystyle\mathcal{N}^{-1}\Bigg{[}\int_{0}^{\hat{x}}\Big{(}\hat{\rho}(\hat{x}+y)-\hat{\rho}(\hat{x}-y)\Big{)}+\int_{\hat{x}}^{\infty}\Big{(}\hat{\rho}(\hat{x}+y)-\rho(\hat{x}-y)\Big{)}\Bigg{]}yg_{\sigma}(y)dy$
(S16) $\displaystyle=$
$\displaystyle\mathcal{N}^{-1}\int_{0}^{\hat{x}}\Big{[}\hat{\rho}(\hat{x}+y)-\hat{\rho}(\hat{x}-y)\Big{]}yg_{\sigma}(y)dy+\mathcal{N}^{-1}\int_{0}^{\infty}\Big{[}\hat{\rho}(2\hat{x}+v)-\hat{\rho}(-v)\Big{]}(v+\hat{x})g_{\sigma}(v+\hat{x})dv$
(S17)

The sign of this expression follows from the assumption that $\rho(s)$ is
symmetric and monotonous on either side of its mean. Indeed, in the first term
we notice $\hat{x}+y>\hat{x}-y>0$ and thus
$\hat{\rho}(\hat{x}-y)>\hat{\rho}(\hat{x}+y)$ and in the second term we notice
$\rho(-v)=\rho(v)>\rho(2\bar{x}+v)$. Thus, in both terms, the square brackets
have nonpositive content and multiply positive factors, leading to the
conclusion that expression (S17) is nonpositive, hence $\hat{s}(x)\leq x$.
Analogously, $\bar{x}<0$ leads to the result $\hat{s}(x)\geq x$, i.e.
$\hat{s}(x)$ lies on the same side of $x$ as $\bar{s}$ does.

To prove that, conversely, $\hat{s}(x)$ lies on the same side of $\bar{s}$ as
$x$, one proceeds as above with the roles of likelihood and prior swapped,
exploiting the monotonicity of the Gaussian density on either side of its
mean.

###  S-II.5 Bounds on the derivative of the posterior mean

The posterior mean can be written as
$\hat{s}=\overline{s\mathcal{L}}/\overline{\mathcal{L}}$ in terms of prior
averages, where $\mathcal{L}$ is an arbitrary likelihood
$\mathcal{L}(s):=\rho((x,\sigma)|s)$. Its derivative is therefore

$\hat{s}^{\prime}=\frac{\overline{\mathcal{L}}\
\overline{s\mathcal{L}^{\prime}}-\overline{s\mathcal{\mathcal{L}}}\
\overline{\mathcal{L}^{\prime}}}{\overline{\mathcal{\mathcal{L}}}^{2}}$ (S18)

The assumption of a Gaussian measurement model turns Eq. (S18) into the ratio
between the variance of the posterior and the variance of the likelihood

$\hat{s}^{\prime}(x)=\frac{\textrm{variance of posterior for outcome
{\hbox{x}}}}{\textrm{variance of likelihood}}$ (S19)

as also follows from Eq. (S10) evaluated with $n=1$.

Because this derivative is in principle measurable from experiments that probe
the perceived utility curve, its range of values may offer a further
experimental test for the theory. Obviously it can never be equal to zero,
because the numerator of Eq. (S19) vanishes only if the observations are
completely uninformative or the prior itself has zero variance. This marks a
difference between the theory presented here and the power-law ansatz seen in
[7], where this derivative vanishes in the origin.

Under the assumptions of Sec. S-II.4, it can be seen that the ratio (S19) is
always subunity at $x=0$. To prove that, it is sufficient to restate the main
result of Sec. S-II.4 as $\frac{\hat{s}(x)}{x}<1$. Since under the same
conditions $\hat{s}(0)=0$, we find

$\lim_{x\rightarrow 0}\hat{s}^{\prime}(x)=\lim_{x\rightarrow
0}\frac{\hat{s}(x)}{x}<1$ (S20)

###  S-II.6 Asymptotics of the posterior mean

The asymptotics of the posterior mean has asymptotic scaling contained between
$O(1)$ and $O(x)$ in the limit of large outcomes ($x\rightarrow\infty$).

To see this, begin by noticing that the posterior mean is a nondecreasing
function of the outcome $x$. From this, it follows that no scaling below
$O(1)$ is possible.

On the other hand, suppose the scaling was larger than $O(x)$. It would then
be possible to find a sufficiently large value $x*$, such that for $x>x*$ the
inequality $\langle s\rangle\geq x$ is violated. Since this inequality was
proven in Sec. S-II.4, it follows that ( under the same conditions given in
that section) the large-$x$ scaling of $\hat{s}(x)$ is linear at most.

###  S-II.7 Compact-support priors and posterior mean

A preliminary step to prove the open-mind theorem is to study the asymptotic
behavior of the posterior mean for compact-support priors.

In the following we consider the limit $x\rightarrow\infty$ for convenience,
although the same considerations apply to the limit $x\rightarrow-\infty$. We
prove that the two statements of $\hat{s}(x)$ having a finite limit for large
$x$ and the prior having compact support are equivalent, as long as $\rho\in
C^{0}$ and $\sigma<\infty$.

We start from the assumption that the prior has compact support. The set of
points $\\{s:\rho(s)>0\\}$ has therefore a supremum $S<\infty$, and from the
definition of $\hat{s}(x)$ it follows $\hat{s}(x)<S\ \forall x$, entailing
that the large-x limit of $\hat{s}(x)$ is finite.

Vice versa, to prove that a finite limit entails a compact support, we suppose
ad absurdum that the support is not compact. It follows that for every
$L\in\mathbb{R}$ we can find an increasing sequence $s_{1}<s_{2}<\ldots$ such
that $L<s_{1}$ and $\rho(s_{i})>0\ \forall i\geq 1$. Since $\rho\in C^{0}$, we
also have that $\rho(s)$ is positive in a neighborhood of each $s_{i}$.

The statement being true for any $L$, it holds in particular for
$L=\lim_{x\rightarrow\infty}\hat{s}(x)$. We write therefore

$\hat{s}(x)=\langle s\rangle_{s|x}^{[-\infty,s_{1}]}w(-\infty,s_{1}|x)+\langle
s\rangle_{s|x}^{[s_{1},\infty]}w(s_{1},\infty|x)$ (S21)

where

$\langle s\rangle_{s|x}^{[a,b]}=\frac{\int_{a}^{b}ds\ s\
\rho(s)g_{\sigma}(s-x)}{\int_{a}^{b}ds\ \rho(s)g_{\sigma}(s-x)}$ (S22)
$w(a,b|x)=\frac{\int_{a}^{b}ds\
\rho(s)g_{\sigma}(s-x)}{\int_{-\infty}^{\infty}ds\ \rho(s)g_{\sigma}(s-x)}$
(S23)

where $\rho(s)$ being positive in a right neighborhood of $s=s_{1}$ results in
$w(M,\infty|x)>0$ .

Given that formula (S22) is an average over the interval $[a,b]$ with the
distribution

$\psi(s;a,b)=\frac{\rho(s)g_{\sigma}(s-x)\theta(s-a)\theta(b-s)}{\int_{a}^{b}ds\
\rho(s)g_{\sigma}(s-x)}$ (S24)

it follows that

$\langle s\rangle_{s|x}^{[s_{1},\infty]}\geq\langle
s\rangle_{s|x}^{[-\infty,s_{1}]}$ (S25)

Let us now find bounds for the weights $w$’s. We note that

$w(-\infty,s_{1}|x)\leq\Big{(}\int_{-\infty}^{M}\rho(s)ds\Big{)}\
\mathrm{sup}_{s\in(-\infty,s_{1})}g_{\sigma}(s-x)=g_{\sigma}(s_{1}-x)\int_{-\infty}^{M}\rho(s)ds$
(S26)

where the last equality holds for sufficiently large $x$ ($x\geq s_{1}$). We
also note that there is a right neighborhood of $s_{2}$ with size $\delta$
where $\rho(s)>0$, hence

$w(s_{1},\infty|x)\geq\frac{\int_{s_{2}}^{\infty}ds\
\rho(s)g_{\sigma}(s-x)}{\int_{-\infty}^{\infty}ds\
\rho(s)g_{\sigma}(s-x)}\geq\frac{g_{\sigma}(s_{2}-x)\int_{s_{2}}^{\infty}\ ds\
\rho(s)}{\int_{-\infty}^{\infty}ds\ \rho(s)g_{\sigma}(s-x)}$ (S27)

where the second inequality holds for sufficiently large $x$ ($x\geq s_{2}$).

From Eqs. (S26) and (S27), it follows that

$\lim_{x\rightarrow\infty}\frac{w(-\infty,s_{1}|x)}{w(s_{1},\infty|x)}\leq\frac{\int_{-\infty}^{s_{1}}\rho(s)ds}{\int_{s_{2}}^{\infty}\rho(s)ds}\lim_{x\rightarrow\infty}\frac{g_{\sigma}(s_{1}-x)}{g_{\sigma}(s_{2}-x)}=0$
(S28)

On the other hand, for any $x$ we have
$w(-\infty,s_{1}|x)+w(s_{1},\infty|x)=1$, which together with Eq. (S28) yields

$\lim_{x\rightarrow\infty}w(-\infty,s_{1}|x)=0\hskip
40.0pt\lim_{x\rightarrow\infty}w(s_{1},\infty|x)=1$ (S29)

Applying (S28) and Eq. (S25) into Eq. (S21), we obtain

$\lim_{x\rightarrow\infty}\hat{s}(x)=\lim_{x\rightarrow\infty}\langle
s\rangle_{s|x}^{[s_{1},\infty]}w(s_{1},\infty|x)=\lim_{x\rightarrow\infty}\langle
s\rangle_{s|x}^{[s_{1},\infty]}$ (S30)

On the other hand, from Eq. (S22) we have $\langle
s\rangle_{s|x}^{[s_{1},\infty]}\geq s_{1}>L$, so we reach the contradictory
conclusion that $\lim_{x\rightarrow\infty}\hat{s}(x)>L$, which shows that a
finite limit $L$ cannot exist.

##  S-III Definition of the perceived utility function

The experimental setting used to verify the Reflection Effect [5] is described
by Table 1 of the main text. The function that determines the perceived
utility given a certain outcome is $\hat{s}(x)=\langle s\rangle_{x}$. This
function must be averaged over all outcomes to yield the actual value of the
perceived utility $U$ associated to any choice.

Therefore, the perceived utility becomes

$U(p,r)=U_{A}(p,r)-U_{B}(p,r)=p\hat{s}(r/p-r)+(1-p)\hat{s}(-r)$ (S31)

and is written most concisely in terms of the odds $q=\frac{1-p}{p}$ (ranging
between $0$ and $\infty$),

$U(q,r)=\frac{\langle s\rangle_{qr}+q\ \langle s\rangle_{-r}}{1+q}$ (S32)

The case of a risk-free outcome is recovered for $q=0$, when indeed we have
$U_{A}=U_{B}$ and thus $U(q,r)=0$.

In the main text we started out by deriving a theory based on a specific
cognitive prior (the spike-and-slab prior), and calculated from it a specific
form of the function S32, showing that it leads to experimentally verifiable
results. In this supplement, we would like to proceed the other way around.
Namely, we will now make some experimentally motivated demands on the
perceived utility function given in Eq. S32 (which constitutes of course the
final output of our theory) and will asking what cognitive priors would output
a perceived utility fulfilling those requirements.

The reflection effect amounts to the requirement that perceived utility $U$
flips sign as a function of the gain/loss probability $p$, becoming positive
for sufficiently large $p$ if $r$ is negative, and becoming negative for
sufficiently large $p$ if $r$ is positive.

The goal therefore is to characterize the set of prior densities $\rho(s)$
such that:

(A) For any $r>0$, the equation $U(q,r)=0$ is satisfied for a single
$q\in(0,\infty)$, let’s call it $q_{+}$. At $q_{+}$ the lowest-order
nonvanishing derivative of $U$ with respect to $q$ is positive.

(B) For any $r<0$, the equation $U(q,r)=0$ is satisfied for a single
$q\in(0,\infty)$, let’s call it $q_{-}$ . In $q_{-}$, the lowest-order
nonvanishing derivative of $U$ with respect to $q$ is negative.

The point that makes $U$ vanish will be called in either case the choice-
switching point. Experiments suggest that $q_{+}\approx q_{-}$ (reflection
principle) and that this is around $\approx 0.2$.

Assuming $\rho$ symmetric makes $\hat{s}(x)$ antisymmetric, and a consequence
of that is that conditions (B) are automatically satisfied if conditions (A)
are.

##  S-IV Particular cases

###  S-IV.1 Spike-and-slab prior

Although in this supplement we do not restrict ourselves to the spike-and-slab
form of the prior, that specific prior deserves special attention because it
encodes relevant experimental knowledge from anchor-and-adjust heuristic. We
review it briefly in this section.

The spike-and-slab prior already stated in the main text (Eq. 5) is

$\rho(s)=p_{0}\delta(s)+(1-p_{0})g_{\gamma}(s)$ (S33)

and leads to a posterior distribution of the form

$\phi(s)\propto
p_{0}g_{\sigma}(x)\delta(s)+(1-p_{0})g_{\sqrt{\gamma^{2}+\sigma^{2}}}(x)g_{\frac{\gamma\sigma}{\sqrt{\gamma^{2}+\sigma^{2}}}}\left(s-\frac{\gamma^{2}}{\gamma^{2}+\sigma^{2}}x\right)$
(S34)

The resulting optimal estimator (posterior mean) is then

$\hat{s}(x)=\frac{ax}{1+b\exp\left(-\frac{x^{2}}{2\tilde{\sigma}^{2}}\right)},$
(S35)

with positive constants $a,b,\tilde{\sigma}$ given by

$a=\frac{\gamma^{2}}{\gamma^{2}+\sigma^{2}},\hskip
10.0ptb=\frac{p_{0}}{1-p_{0}}\frac{\sqrt{\gamma^{2}+\sigma^{2}}}{\sigma},\hskip
10.0pt\tilde{\sigma}=\frac{\sigma}{\sqrt{a}}$ (S36)

For $x\ll\tilde{\sigma}$ this reduces to

$\hat{s}(x)\approx\frac{a}{1+b}x+\frac{ab}{(1+b)^{2}}\frac{x^{3}}{2\tilde{\sigma}^{2}}$
(S37)

which demonstrates that, around $x=0$, this estimator features a shape that is
convex for positive $x$’s and concave for negative $x$’s, suppressing small
fluctuations. However the convex-concave shape is lost for larger values of
$|x|$, and for large $x$ the estimator approaches a linear asymptote
$\hat{s}=ax$ from below.

Plugging into Eq. (S32), we obtain

$U(q,r)=\frac{aqr}{1+q}\left(\frac{1}{1+b\exp\left(-\frac{q^{2}r^{2}}{2\tilde{\sigma}^{2}}\right)}-\frac{1}{1+b\exp\left(-\frac{r^{2}}{2\tilde{\sigma}^{2}}\right)}\right)$
(S38)

which is the function shown in Fig. 3 of the main text.

###  S-IV.2 Gaussian-mixture prior

By a gaussian-mixture prior we mean a linear combinations of two gaussian
density functions.As seen in the main text, a gaussian-mixture prior with one
infinitely narrow gaussian ("spike") fullfills the phenomonological
requirement for any value of $\sigma$. On the other hand, assuming Gaussian
priors are an adequate model, we don’t expect that the value of $\sigma$ in
the actual experiment would ever vanish, and we may assume that in all
experiments conducted so far $\sigma$ was larger than some baseline value
$\delta$. Such a gaussian-mixture prior has been used to generate figure 5,
and for this reason we see it worthwhile to outline it here.

We write

$\rho(s)=\alpha g_{\delta}(s)+(1-\alpha)g_{\gamma}(s)$ (S39)

for a constant $\alpha\in]0,1[$, with $\gamma>\delta$.

Since the prior S39 is symmetric, all the odd cumulants vanish. Let us compute
the central moments

$\displaystyle\overline{s^{2}}$ $\displaystyle=$
$\displaystyle(1-\alpha)\gamma^{2}+\alpha\delta^{2}<\gamma^{2}$ (S40)
$\displaystyle\overline{s^{4}}$ $\displaystyle=$ $\displaystyle
3(1-\alpha)\gamma^{4}+3\alpha\delta^{4}<3\gamma^{4}$ (S41)

and the kurtosis

$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\overline{s^{4}}\big{/}(\overline{s^{2}})^{2}=\frac{3}{\alpha}\frac{1+\frac{1-\alpha}{\alpha}\eta^{4}}{1+2\frac{1-\alpha}{\alpha}\eta^{2}+\frac{(1-\alpha)^{2}}{\alpha^{2}}\eta^{4}}$
(S43)

where $\eta\equiv\gamma/\delta$. Since $\kappa>3$ for any $\eta\geq 0$, this
is a leptokurtic prior.

Applying Eq. S6 leads to the posterior distribution

$\phi(s)=(1-\beta(x))G\left(s-\frac{\gamma^{2}x}{\sigma^{2}+\gamma^{2}};\frac{\gamma\sigma}{\sqrt{\sigma^{2}+\gamma^{2}}}\right)+\beta(x)\
G\left(s-\frac{\delta^{2}x}{\sigma^{2}+\delta^{2}};\frac{\delta\sigma}{\sqrt{\sigma^{2}+\delta^{2}}}\right)$
(S44)

where

$\beta(x)=\frac{\alpha
G(x;\sqrt{\sigma^{2}+\delta^{2}})}{(1-\alpha)G\left(x;\sqrt{\sigma^{2}+\gamma^{2}}\right)+\alpha
G\left(x;\sqrt{\sigma^{2}+\delta^{2}}\right)}$ (S45)

Applying Eq. S6 leads to the optimal estimator

$\hat{s}(x)=\frac{\frac{(1-\alpha)\gamma^{2}}{(\sigma^{2}+\gamma^{2})^{3/2}}e^{\frac{x^{2}}{2\lambda^{2}}}+\frac{\alpha\delta^{2}}{(\sigma^{2}+\delta^{2})^{3/2}}}{\frac{1-\alpha}{(\sigma^{2}+\gamma^{2})^{1/2}}e^{\frac{x^{2}}{2\lambda^{2}}}+\frac{\alpha}{(\sigma^{2}+\delta^{2})^{1/2}}}\
x$ (S46)

where

$\lambda^{2}=\frac{(\delta^{2}+\sigma^{2})(\gamma^{2}+\sigma^{2})}{\gamma^{2}-\delta^{2}}.$
(S47)

From this, applying Eq. S32, we see that the formula for perceived utility has
the form

$U(q,r)=Kr\
\frac{e^{-\frac{q^{2}r^{2}}{2\lambda^{2}}}-e^{-\frac{r^{2}}{2\lambda^{2}}}}{Ae^{-\frac{(q^{2}+1)r^{2}}{2\lambda^{2}}}+Be^{-\frac{q^{2}r^{2}}{2\lambda^{2}}}+Be^{-\frac{r^{2}}{2\lambda^{2}}}+C}$
(S48)

where

$\displaystyle K$ $\displaystyle=$
$\displaystyle\alpha(1+\alpha)\frac{q}{1+q}\frac{\sigma^{2}(\gamma^{2}-\delta^{2})}{\sqrt{\sigma^{2}+\gamma^{2}}\sqrt{\sigma^{2}+\delta^{2}}}$
(S49) $\displaystyle A$ $\displaystyle=$
$\displaystyle(1-\alpha)^{2}(\sigma^{2}+\delta^{2})$ (S50) $\displaystyle B$
$\displaystyle=$
$\displaystyle\alpha(1-\alpha)\sqrt{\sigma^{2}+\delta^{2}}\sqrt{\sigma^{2}+\gamma^{2}}$
(S51) $\displaystyle C$ $\displaystyle=$
$\displaystyle\alpha^{2}(\sigma^{2}+\gamma^{2})$ (S52)

This indeed has a single zero at the half-point ($q=1$). As seen from the only
part of the numerator that changes sign, at that point the functions turns
from negative to positive if $r>0$, so it is indeed an increasing function of
q there (and a decreasing one if $r<0$) as desired.

All of the above formula also hold true in the limit $\delta\rightarrow 0$,
where one of the two gaussians becomes a delta function and the spike-and-slab
results are recovered with $p_{0}=\alpha$.

##  S-V General results

###  S-V.1 Posterior mean vanishes for zero deviations

The reflection effect is, in our framework, the fact that the perceived
utility $U(p,r)$ gets positive for sufficiently large $p$ if $r$ is negative,
and gets negative for sufficiently large $p$ if $r$ is positive. From that
fact, we infer in particular the inequalities

$U(p=1,r>0)\geq 0$ (S53) $U(p=1,r<0)\leq 0$ (S54)

On the other hand, from the definition of $U(p,r)$ we see that $U(p=1,r)$ is
always equal to $\langle s\rangle_{0}$ regardless of the sign of $r$. So the
only way to satisfy both inequalities is if $\langle s\rangle_{0}=0$.

That’s of course automatically satisfied if the prior is symmetric around
zero. But as seen here, it has to be true also for asymmetric priors.

###  S-V.2 Gaussian prior generates rational preferences

Eq. (S9) shows that the posterior mean is linear in the measurement.

Plugging that linear law into Eq. (S32), we find

$U(q,r)\equiv 0$ (S55)

for any value of $q$ and for any value of $r$, as shown in Fig. 5.

The finding is therefore that $U$ is a measure of nongaussianity – or, in
other words,the Kahneman-Tversky experiment itself measures the nongaussianity
of the prior. If the prior was Gaussian, experimental subjects would choose
indifferently among options with any risk level. Thus, the "linearity in
probability" that prospect theory problematizes proves to be here something
more familiar – normality of prior distributions.

In fact, a normal distribution is the only choice of a prior that would
annihilate the perceived utility – as we proceed to prove.

Imposing Eq. (S55) for all $(q,r)$ enforces $\langle s\rangle_{ax}=a\langle
s\rangle_{x}$ for all $a\leq 0$. This in turn entails the same equality
$\forall a>0$ as well (just apply it twice with $a_{1}=a_{2}=-\sqrt{a}$). For
a one-dimensional argument $x$, this scaling law implies linearity. If the
posterior mean has no $x$-derivatives beyond the first, Eq. (S10) implies that
the posterior distribution has no cumulants beyond the second one, i.e. it is
normal.

Gaussianity of the prior follows immediately through (2).

###  S-V.3 First tangent inequality

The derivative of the perceived utility is

$\partial_{p}U=\hat{s}(r/p-r)-\hat{s}(-r)-r\hat{s}^{\prime}(r/p-r)/p$ (S56)

where $\hat{s}(x)=\langle s\rangle_{x}$; or in terms of the odds

$\partial_{q}U=\frac{r\hat{s}^{\prime}(qr)+\hat{s}(-r)}{1+q}+\frac{-q\hat{s}(-r)-\hat{s}(qr)}{(1+q)^{2}}$
(S57)

Because $U$ must have the opposite sign to $r$ in a left neighborhood of $p=1$
(or right neighborhood of $q=0$), where it touches zero, it follows that the
derivative at that point is of particular importance. From (S57), we see that

$[\partial_{q}U]_{q=0}=r\hat{s}^{\prime}(0)+\hat{s}(-r)$ (S58)

Now, to comply with the reflection effect this must be negative for gains and
positive for losses which finally leads to

$\displaystyle\hat{s}(r)$ $\displaystyle>$
$\displaystyle\hat{s}^{\prime}(0)r\hskip 15.0pt\mathrm{if}\ r>0$ (S59)
$\displaystyle\hat{s}(r)$ $\displaystyle<$
$\displaystyle\hat{s}^{\prime}(0)r\hskip 15.0pt\mathrm{if}\ r<0$ (S60)

I will refer to these inequalities as "tangent inequalities" because they
state that the curve must lie above its zero-deviation tangent.

The tangent inequalities are related to the concave-convex shape described in
[7]. Since in that theory $\hat{s}(0)=0$, a second derivative
$\hat{s}^{\prime\prime}(x)$ that has everywhere the same sign as $x$ is a
sufficient condition for the forbidden regions to stay untouched. However, it
is in general not a necessary condition. The graph of $\hat{s}(x)$ can
fluctuate without harm, changing the sign of all its derivatives, as long as
it never enters a forbidden wedge of the Cartesian plane – located beneath the
curve in the upper right quadrant, and above the curve in the lower left
quadrant.

One consequence of this has to do with the behavior at infinity. Namely, if
the asymptotic behavior of $\hat{s}(x)$ is linear $\hat{s}(x)=\beta x+o(x)$,
we must have necessarily $\beta\geq\hat{s}^{\prime}(0)$ and, if
$\beta=\hat{s}^{\prime}(0)$, a strictly positive subleading contribution in
the remainder.

###  S-V.4 Second tangent inequality

The above inequalities are a lower bound on $\hat{s}(r)/r$. Similarly, we also
have an upper bound on $\hat{s}(r)/r$. We will write down the derivation
briefly. For large q, we can write

$U\sim\hat{s}(-r)+\frac{1}{q}\hat{s}(qr)$ (S61)

With a symmetric prior, $\hat{s}(x)$ is antisymmetric, and $U>0$ can be
written as $\hat{s}(qr)>-q\hat{s}(-r)=q\hat{s}(r)$ which in turn can be
rearranged (setting $r^{\prime}=qr$) into

$\hat{s}(r^{\prime})>\hat{s}(r)+\frac{\hat{s}(r)}{r}(r^{\prime}-r)$ (S62)

or

$\frac{\hat{s}(r^{\prime})-\hat{s}(r)}{r^{\prime}-r}>\frac{\hat{s}(r)-\hat{s}(0)}{r-0}$
(S63)

which doesn’t need to be true for any pair of values $r$ and $r^{\prime}$, and
is only required to hold true in the limit $r^{\prime}\gg r$.

Now, because we already know that $f$ has to be asymptotically linear, calling
its asymptote $\hat{s}(r)\sim\hat{s}^{\prime}(\infty)x$, we can write this as

$\hat{s}^{\prime}(\infty)>\hat{s}(r)/r\ \ \ \ \forall r$ (S64)

or, given the continuity of the curve

$\lim_{r\rightarrow
0}\hat{s}^{\prime}(r)=\sup_{\mathbb{R}^{+}}\frac{\hat{s}(r)}{r}$ (S65)

###  S-V.5 Restating the two tangent inequalities

The tangent inequalities, shown graphically in Fig. S1, can be collected in
the form

$\boxed{\hat{s}^{\prime}(0)<\frac{\hat{s}(r)}{r}<\hat{s}^{\prime}(\infty)}$
(S66)

Obviously, requiring $\hat{s}^{\prime}(r)$ to be monotonically increasing
would be sufficient to ensure this. But it would also be overdemanding. The
ratio $\hat{s}(r)/r$ can range widely over the real axis as long as it never
becomes smaller than its zero-deviation value, or larger than the asymptotic
slope.

Notice that, applying (S10) with $n=1$, we can rewrite the tangent
inequalities as

$\frac{\text{Var}(s|x=0)}{\sigma^{2}}<\frac{\langle
s\rangle_{x}}{x}<\frac{\text{Var}(s|x\rightarrow\infty)}{\sigma^{2}}$ (S67)

which are direct constraints on the posterior distribution. The next question
to tackle is therefore what constraints on the prior are posed by such
constraints on the posterior.

###  S-V.6 Second derivative vanishes for zero-deviation outcome

The forbidden-region rules derived above teach us something about the
derivatives of $\hat{s}(x)$ in the origin. To see this, notice that the
inequalities (S59) must apply also for infinitesimally small $|r|$. In other
words, we must have

$\hat{s}(r)=\hat{s}(0)+\hat{s}^{\prime}(0)r+\frac{1}{2}\hat{s}^{\prime\prime}(0)r^{2}+o(r^{2})\begin{cases}>\hat{s}^{\prime}(0)r\hskip
10.0pt\mathrm{if}\ r>0\\\ <\hat{s}^{\prime}(0)r\hskip 10.0pt\mathrm{if}\
r<0\end{cases}$ (S68)

Now, we showed in Sec. S-V.1 that $\hat{s}(0)=0$. Ad absurdum suppose
$\hat{s}^{\prime\prime}(0)$ is different from zero. Then it becomes the
dominant term in these inequalities near the origin, because everything else
is zero or cancels out. The tangent inequalities require then the second
derivative to be at the same time nonnegative and nonpositive, so it must
vanish.

The leading term is therefore the one of order $r^{3}$ or higher. Applying Eq.
(S10) with $n=2$, notice that the vanishing of the second derivative is
equivalent to the vanishing of the skewness of the posterior in the particular
case of outcome $r=0$.

###  S-V.7 Open mind theorem (Incompatibility of compact priors)

Let us now apply the inequalities (S59) to the regime of large values of
$|x|$.

Because the variance of likelihood and prior is never zero,
$\hat{s}^{\prime}(0)$ is always positive. In the light of this and of the
tangent inequalities (S59), considered for large $|r|$, we reach the following
conclusion. A prior that leads to a sublinear asymptote for the posterior mean
is not admissible because it contradicts experiments.

Recall now that (as seen in Sec S-II.7) a compact-support prior leads to
sublinear asymptotic behavior for the posterior mean. We reach therefore the
conclusion ( "open-mind theorem") that a compact prior must be ruled out.

The open mind theorem is depicted in Fig. 6. If the prior has compact support
as in panel D, the comparative utility function $U$ behaves as in the dashed
red curves of panels A and B.

###  S-V.8 The zero-deviation posterior is leptokurtic

Assuming that the fourth central moment of the posterior at $x=0$ doesn’t
vanish, it must be positive. This is because if $\hat{s}(0)$ and
$\hat{s}^{\prime\prime}(0)$ both vanish, we can expand as
$\hat{s}(r)\sim\hat{s}^{\prime}(0)r+\frac{1}{6}\hat{s}^{\prime\prime\prime}(0)r^{3}$
and and sticking this into the tangent inequalities (S59) yields
$\mathrm{sign}(\hat{s}^{\prime\prime\prime}(0)r^{3})=\mathrm{sign}(r)$.

Applying now Eq. (S10) with $n=3$, we can write this result as

$\mathcal{K}-3=\frac{\big{\langle}\big{(}s-\langle
s\rangle\big{)}^{4}\big{\rangle}}{\big{\langle}\big{(}s-\langle
s\rangle\big{)}^{2}\big{\rangle}^{2}}-3\ >\ 0$ (S69)

or more concisely, $\mathrm{EK}>0$ if EK is the excess kurtosis, i.e. the
kurtosis measured from its Gaussian reference value.

In conclusion, the posterior distribution for zero outcome is leptokurtic i.e.
has positive excess kurtosis.

###  S-V.9 Kurtosis of the prior

As explained in S-III, we are pursuing a path inverse to that of the main
text, starting from empirically justified requirements on the perceived
utility and tracking back what constraints that entails on the prior
distribution. In Sec. S-V.8, we derived some simple constraints on the moments
of the posterior distribution. How do they transfer into constraints upon the
prior?

Above, when deducing that the excess kurtosis of the posterior had to be
positive, we did not mention for what value of $\sigma$ we are requiring that
positivity. Just like the value of the anchor, so the internally modeled
measurement error $\sigma$ is not communicated to experimenters, and we may
only reasonably assume that it fluctuates widely across subjects and contexts.

Therefore, to take seriously the empirical observations, a constraint derived
from them for the zero-deviation posterior should be taken as a statement
about all the zero-deviation posteriors corresponding to all values of
$\sigma$.

Now call $\vec{R}=(R_{1},R_{2},\ldots)$ the set of all the cumulants of the
prior distribution of $\rho(s)$. Knowing this infinite vector is largely
equivalent to knowing the distribution itself. If we call $F_{4}$ the excess
kurtosis of the posterior, now regarded as a function of both $\sigma$ and
$\vec{R}$, the resulting statement about $\vec{R}$ is then

$F_{4}(\sigma,\vec{R})>0\ \ \ \forall\sigma>0$ (S70)

We thus have to (1) minimize $F_{4}$ over $\sigma$, (2) require the minimum
value to be positive and (3) infer properties of $\vec{R}$ from that.

If we do that, we can easily say that the prior itself must be leptokurtic
($R_{4}>0$). The reason is that in the limit of broad likelihoods
($\sigma\rightarrow\infty$) the posterior is equal to the prior. If the
results of experiments have to be valid for large $\sigma$ as well (i.e. the
constraint of the posterior being leptokurtic must survive) then the prior
(which does not depend on $\sigma$) is leptokurtic.

In conclusion, the prior may not be of the spike-and-slab type but experiments
do reveal that, just like the spike-and-slab distribution, it has a positive
excess kurtosis (for a discussion see main text, Sec. VII).

Figure S1:  Depiction of the tangent inequalities.


living at the same time in the coreflexive equalizer $S\graytensoreq{B} T$.

Note that the two moves $n_1$ played by $\tau$
and $m_2$ played by $\sigma$ are equal moves
when seen from the point of view of the game $B$.

Similarly, the two moves $n_2$ played by $\sigma$
and $m_3$ played by $\tau$ are equal moves
when seen from the point of view of the game $B$.

This sequence of six moves may be thus described
in $S\graytensoreq{B} T$ as a sequence of four moves:

m_1\cdot m\cdot n\cdot n_3

as depicted below

The position of the four moves on the brown (grey) area
describing $S\graytensoreq{B} T$ indicate the polarity

\begin{array}{ccccccc}
m_1& \mapsto&\polarityminustarget
& \quad &
m &\mapsto&\polarityminusmid
\\
n &\mapsto&\polarityplusmid
& \quad &
\end{array}

assigned in the template of polarities

\anchorofasynch[2] = \twocatanchorof{\polarityminussource,\polarityplussource,
\polarityminusmid,\polarityplusmid,\polarityminustarget,\polarityplustarget}

itself defined as the coreflexive equalizer

\anchorofasynch[2] = {\moo[1]\graytensoreq{\moo[0]}\moo[1]}.

Note that postcomposing with multiplication

\begin{tikzcd}[column sep = .8em]
\mathsf{mult} \,\, : \,\, {\anchorofasynch[2]} \arrow[rr] && {\anchorofasynch[1]}
\end{tikzcd}

gives rise to the polarity function

\begin{tikzcd}[column sep = 1em]
\lambda_{\tau\circ\sigma} \,\, : \,\, {S\graytensoreq{B} T}
\arrow[rr] && {\anchorofasynch[2]} \arrow[rr] && {\anchorofasynch[1]}
\end{tikzcd}

on the four moves of the trajectory:

\begin{array}{ccccccc}
m_1& \mapsto&\polarityminustarget
& \quad &
m &\mapsto&\id{\lrangle{\ast}}
\\
n &\mapsto&\id{\lrangle{\ast}}
& \quad &
\end{array}

in the template of polarities $\anchorofasynch[1]=\anchorofstrat$.

This polarity on the trajectory of the composite strategy $\tau\circ\sigma$
reflects the fact that the two moves $m$ and $n$
are considered as internal moves in the interaction between $\sigma$ and $\tau$;
while the moves $m_1$ and $n_3$ of respective polarities $\polarityminustarget$
and $\polarityplustarget$ are the output of the interaction,
played by $\tau\circ\sigma$ on the target game $C$.

§.§.§ Template games continued

Template games continued:
In order to compose strategies, one needs to consider the pullback

Two strategies

\begin{array}{c}
\begin{tikzcd}[column sep=1em]
\sigma=(S,s_S,t_S,\lambda_{\sigma}) \,\,:\,\, A\arrow[rr,spanmap]&&B
\end{tikzcd}
\\
\begin{tikzcd}[column sep=1em]
\tau=(T,s_T,t_T,\lambda_{\tau}) \,\,:\,\, B\arrow[rr,spanmap]&&C
\end{tikzcd}
\end{array}

of $\SpanGames{\anchor}$ are composed in the following way:

\begin{tikzcd}[column sep=1.1em, row sep=1.4em]
\Sgame\times_{\Bgame}\Tgame
\arrow[lldd,"{\pi_1}"{swap}]\arrow[rrdd,"{\pi_2}"]
\arrow[dd,"{\interaction{\sigma}\,\|\,\interaction{\tau}}"{description}]
\arrow[rrdd,"{\pi_2}"]
\\
\\
\Sgame
\arrow[lldd,"{s_S}"{swap}]
\arrow[dd,lightgray,"\interaction{\sigma}"{swap}]
\arrow[lldd,lightgray,"{\pi_1}"{swap}]
\arrow[rrdd,lightgray,"{\pi_2}"]
\arrow[dddd,"{m}"]
\Tgame
\arrow[dd,lightgray,"\interaction{\tau}"]
\arrow[rrdd,"{t_T}"]
\\
\\
\Agame \arrow[dd,"\lambda_{\Agame}"{swap}]
\color{lightgray}
\arrow[lldd,lightgray,"s"{swap}]
\color{lightgray}
\arrow[rrdd,lightgray,"t"]
\Cgame \arrow[dd,"\lambda_{\Cgame}"]
\\
\\
{\anchor[0]} &&&&
\arrow[rrrr,"{t}"]
\arrow[llll,"{s}"{swap}]
&&&& {\anchor[0]}
\end{tikzcd}

Here, $\lambda_{\sigma}\|\lambda_{\tau}$ denotes the map of $\Scategorysurround$
induced by the pullback diagram (<ref>)
and uniquely determined by the equations

\lambda_{\sigma} \circ \pi_1 = \pi_1\circ (\lambda_{\sigma}\|\lambda_{\tau})
\quad\quad\quad\quad
\lambda_{\tau} \circ \pi_2 = \pi_2\circ (\lambda_{\sigma}\|\lambda_{\tau})

Accordingly, the identity map called the copycat strategy

$$\begin{tikzcd}[column sep=1em]
\identityspan{A}=(A,id_A,id_A,e\circ\lambda_{A}) \,\,:\,\, A\arrow[rr,spanmap]&&A
\end{tikzcd}

is constructed in the following way:

\begin{tikzcd}[column sep=3em, row sep=1em]
\Agame
\arrow[llddd,"id"{swap}]
\arrow[dd,"\lambda_{\Agame}"]
\arrow[rrddd,"id"]
\\
\\
\arrow[ddd,"e"]
\arrow[llddd,lightgray,"id"{swap}]
\arrow[rrddd,lightgray,"id"]
\\
\Agame \arrow[dd,"\lambda_{\Agame}"{swap}]
\Agame \arrow[dd,"\lambda_{\Agame}"]
\\
\\
\arrow[ll,"s"{swap}]
\arrow[rr,"t"]
\end{tikzcd}

More generally, in every category $\Scategorysurround$ with finite limits,

Every internal category $\moo$ defines a weak 2-category $\Games{\moo}$.

§.§ Internal weak categories

The 2-category $\Delta$ of simplices is the strict monoidal 2-category
freely generated by

m_k : 1\tensor \cdot \tensor 1 \to 1

has non-empty finite ordinals $[p]=\{0,\dots,p\}$
as objects and monotone functions $f:[p]\to[q]$ between them as morphisms.

The category $\Delta$ is monoidal with tensor product noted $\oplus$
defined as addition $[p]\oplus [q]=[p+q]$ and tensorial unit
defined as the singleton set $[0]$.

A distinctive property of the category $\Delta$ is that the diagram

\begin{equation}\label{equation/segal-pushout}
\begin{tikzcd}[column sep = .8em, row sep=.8em]
{[0]} \arrow[lldd,"{t}"{swap}]\arrow[rrdd,"{s}"]
\\
\\
\arrow[rrdd,"{inl}"{swap}]
\arrow[lldd,"{inr}"]
\\
\\
&& {[p]\oplus[q]}
\end{tikzcd}
\end{equation}

is a pushout diagram in the category $\Delta$,
for every pair of natural numbers $p,q\in\mathbb{N}$,

t:0\mapsto p
\quad\quad
s:0\mapsto 0
\quad\quad
inl:k\mapsto k
\quad\quad
inr:k\mapsto p+k

The pushout diagram describes very neatly how one obtains
an interval of length $p+q$ by gluing together two intervals
of length $p$ and $q$ respectively.

Then, a simplicial object $\anchor$
in a category $\Scategorysurround$ is a functor

\begin{equation}\label{equation/simplicial-object}
\begin{tikzcd}[column sep = .8em, row sep=.8em]
{\anchor[-]} \quad : \quad \Delta^{op} \arrow[rr]
\Scategorysurround
\end{tikzcd}
\end{equation}

which transports every natural number $p\in\mathbb{N}$
to an object noted $\anchor[p]$ of the category $\Scategorysurround$.

In that context, we often find convenient to call space
any object of the category $\Scategorysurround$,
and in particular the objects of the form $\anchor[p]$.

Note that every simplicial object $\anchor$ comes equipped
with a multiplication map and a unit map:

\begin{tikzcd}[column sep = .8em, row sep=.8em]
m \quad : \quad {\anchor[2]} \arrow[rr] && {\anchor[1]}
\end{tikzcd}
\quad\quad\quad\quad\quad\quad
\begin{tikzcd}[column sep = .8em, row sep=.8em]
e \quad : \quad {\anchor[0]} \arrow[rr] && {\anchor[1]}
\end{tikzcd}

living in the category $\Scategorysurround$.

An internal category $\anchor$ in a category $\Scategorysurround$
with finite limits is a simplicial object (<ref>)
such that every pushout diagram of the form (<ref>)
in the category $\Delta$ of simplices is transported to a pullback diagram

\begin{tikzcd}[column sep = .8em, row sep=.8em]
{\anchor[p+q]} \arrow[lldd,"{\pi_1}",swap]\arrow[rrdd,"{\pi_2}"]
\\
\\
\arrow[rrdd,"{t}",swap]
\arrow[lldd,"{s}"]
\\
\\
&& {\anchor[0]}
\end{tikzcd}

in the category $\Scategorysurround$.

We write $\anchor[0]$ and $\anchor[1]$ for the two objects
of the category $\Scategorysurround$
defining respectively the space of objects $\anchor[0]$
and the space of morphisms $\anchor[1]$
of the internal category $\anchor$.

§ THE CANONICAL EMBEDDING OF A WEAK DOUBLE CATEGORY

For every weak double category, there is a homomorphism in the weak double category $\Span{\Cat}$.
Every object $A$ is mapped to $Vert/A$.
Every vertical map $a:A\to A'$ is mapped to $Vert/a:Vert/A\to Vert/A'$.
Every horizontal map $S:A\spanmap B$ is mapped to ...
A monad is mapped into a monad in $\Span{\Cat}$.
A monad in $\Span{\Cat}$ is a double category.

§ THE FRAMED BICATEGORY OF GRAY COMONOIDS AND GRAY BICOMODULES

In this section, we establish the fact that the double category is framed.

The framing plays a fundamental role in the construction of the tensor product
of asynchronous template games.

§.§ Pseudo double categories

A pseudo double category $\Vbicategory$ is defined as

* a vertical category $\Vbicategory_0$ of objects and vertical maps,

* a vertical category $\Vbicategory_1$ of horizontal maps and double cells,

* source and target functors $s,t:\Vbicategory_1\to\Vbicategory_0$,

together with horizontal unit and composition functors,
associative up to special natural isomorphisms,
these data satisfying moreover a number of expected coherence diagrams.

We refer the reader to Garner's PhD thesis <cit.>
for a detailed definition of the notion of pseudo double category.

§.§ Framed bicategories

A framed bicategory is a pseudo double category where the source and target functors

are Grothendieck bifibrations.
This means that...

§.§ The framed bicategory of comonoids and bicomodules

We suppose that we work in a monoidal category with equalizers,
preserved by the tensor product.

A construction by Marcelo Aguiar.

The category of comonoids has pullbacks (to check!).

§.§ Internal categories

Every internal category $\moo[0]$ can be pulled along a vertical map $\moo'[0]\to\moo[0]$.

The definition is based on the framed structure.

§.§ Regular maps between internal categories

§.§ ...

In any monoidal category with equalizers, one has a framed double category.

In this case:
a specific property of pulls along monos!!!

§ ASYNCHRONOUS TEMPLATE GAMES

We use the 2-category $\moo[\epsilon-,\epsilon+]$ as template of asynchronous games.

An asynchronous template game is defined as a Gray comonoid $A$
equipped with a 2-functor

A\to \moo[\epsilon-,\epsilon+]

preserving the comonoid structure, in the sense that the diagrams below commute:

§.§ Gray comonoids and polarity structures

We are interested in comonoids
in the symmetric monoidal category $\TwoCat$
of 2-categories, equipped with the Gray tensor product.

One typical example is $\anchorminus$ and $\anchorplus$

A polarity structure on an asynchronous graph $(G,\diamond)$
is a function $\lambda:E_G\to\{+1,-1\}$ which transports
every edge $u\in E$ of the graph $G=(V,E)$ to a polarity $\lambda(u)=+1$
or $\lambda(u)=-1$.

A polarity structure is called asynchronous when
$\lambda(u_1)=\lambda(v_2)$ and $\lambda(u_2)=\lambda(v_1)$
for all permutation tiles $u_1\cdot u_2\diamond v_1\cdot v_2$
of the asynchronous graph $G=(G,\diamond)$.

A\to A\graytensor A

suppose that $A$ comes from an asynchronous graph.
Then each edge should be polarized left-to-right or right-to-left.
Conversely, every signed set induces a Gray comonoid.

Moreover, in a symmetric monoidal category,
every comonoid induces an opposite comonoid
obtained by permuting the two outputs using the symmetry.

This induces the remarkable equation:

\twocatanchorof{\Label}^{op} =\twocatanchorof{\Label^{op}}.

Given an asynchronous graph $(G,\diamond)$,
there is a one-to-one relationship between:

* a polarity structure $\lambda:E_G\to\{+1,-1\}$
on the asynchronous graph $(G,\diamond)$,

* a Gray comonoid structure on the associated
2-category $[G,\diamond]$ of positions, trajectories and reshufflings.

Note that the polarity structure does not have to be asynchronous.

§.§ Asynchronous games and strategies

An asynchronous game $(A,\lambda_A)$
is a pair consisting of a Gray comonoid equipped
with a comonoid map into the template 2-category $\moo[+-]$.

An asynchronous strategy $\sigma=(S,\lambda_{\sigma})$
between asynchronous games $A$ and $B$ is a pair
consisting of a Gray $A,B$-bicomodule

S\to A\Box S\Box B

together with a bicomodule morphism
to the $\moo[+-],\moo[+-]$-bicomodule $\moo[+-+-]$

with a comonoid map into the template 2-category $\moo[+-]$.

\begin{tikzcd}[column sep = 1em]
\arrow[dd,"{\lambda_{\sigma}}"{swap}]
\arrow[rr]
\moo[+-]\Box S \Box\moo[+-]
\arrow[dd,"{\moo[+-]\Box\lambda_{\sigma}\Box\moo[+-]}"]
\\
\\
\moo[+-+-]
\arrow[rr]
\moo[+-]\Box \moo[+-+-]\Box\moo[+-]
\end{tikzcd}

Composition as follows ...

§.§ The weak double category

For any internal category = horizontal monad in the double category,
there is a double category. Similar to Hirschowitz.

§ ASYNCHRONOUS TEMPLATE GAMES

§.§ The template of asynchronous games

§.§ The template of tensor and cotensor games

§.§ The star-autonomous structure