# The Perils of Overreaction

Konstantin von Beringe  Mark Whitmeyer Arizona State University. We thank
Ludvig Sinander for alerting us to the Dutch Book justification for
Bayesianism. We are also grateful to Geoffroy de Clippel and Collin Raymond
for suggestions that led to this paper and to Joseph Whitmeyer for his
feedback. Emails<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

In order to study updating rules, we consider the problem of a malevolent
principal screening an imperfectly Bayesian agent. We uncover a fundamental
dichotomy between underreaction and overreaction to information. If an agent’s
posterior is farther away from the prior than it should be under Bayes’ law,
she can always be exploited by the principal to an unfettered degree: the
agent’s ex ante expected loss can be made arbitrarily large. In stark
contrast, an agent who underreacts (whose posterior is closer to the prior
than the Bayesian posterior) cannot be exploited at all.

##  1 Exploiting Non-Bayesians

A classical justification for Bayesianism is a Dutch Book argument: a
malevolent (or at least selfish) bookie can offer a non-Bayesian a sequence of
bets that guarantee the bookie a strictly positive profit (Teller (1973);
Lewis (1999); Skyrms (1987)). Crucially, this bookie has a significant amount
of power: he offers bets both before and after the arrival of information,
observes the information himself, and knows the agent’s mistaken belief.

In this paper, in order to scrutinize and compare updating rules, we explore a
similarly adversarial scenario.111In the same vein, Echenique, Lee, and Shum
(2011) and Lanier, Polisson, and Quah (2024) illustrate how money-pumps are
useful in characterizing the magnitude of revealed preference violations. Our
environment contains a principal and an agent. They start with a common prior
about an unknown state of the world, and the agent alone privately observes
the realization of a signal. The agent updates her prior through a possibly
non-Bayesian updating rule. Then, at this interim stage, the principal offers
the agent a decision problem consisting of a collection of actions and a
utility function that delivers a state- and action-dependent payoff to the
agent.

The principal’s goal is to exploit the agent and make the agent’s ex ante
expected payoff as low as possible. Moreover, because the principal is unaware
of the agent’s private information, he faces a screening problem: generating a
decision problem is just the offering to the agent of a menu of actions.

What makes the problem interesting is that the agent’s choices are shaped by
her possibly non-Bayesian posteriors. That is, upon observing a piece of
information, she chooses an action according to her current belief, which is
potentially produced by means other than Bayes’ law. The principal knows the
updating rule, but does not observe the signal realization.

We distinguish between two varieties of reaction to new information;
overreaction and underreaction.

Overreaction: Given a prior and a signal, an agent overreacts to a piece of
information (a signal realization) if the Bayesian posterior lies on a line
segment between the prior and the agent’s posterior. That is, the agent
overreacts to a signal realization if her posterior is further away from the
prior than the Bayesian posterior.

Underreaction: An agent underreacts to a piece of information if her posterior
lies on a line segment between the prior and the Bayesian posterior.
Underreaction to a signal realization is when the agent’s posterior lies
closer to the prior than the Bayesian posterior.

We discover that there is a massive difference between the agent’s welfare in
these two classes. If there exists a signal realization that produces an
overly extreme posterior–formally, that produces a posterior that lies outside
of the closed convex hull of the support of the Bayesian posteriors–the
principal can exploit the agent to an arbitrarily large degree. Because the
posterior is so extreme, the principal can offer the agent a single action
that no Bayesian would take, but that the agent with the extreme (and
incorrect) posterior would. By scaling up the penalties to taking this action
in “bad states” the agent’s expected loss is limitless. A corollary of this
result is that if an agent overreacts to a signal realization, she can be
exploited.

On the other hand, if an agent underreacts to information, if all of her
posteriors are (weakly) closer to the prior than the Bayesian posteriors, she
cannot be exploited. The agent’s ex ante expected payoff cannot be worse than
that from her outside option (which we normalize to zero). The explanation for
this impossibility is more subtle than that for the overreaction result. If
the agent can be exploited, there must be some signal realization that leads
her to take the “wrong” action from the perspective of the Bayesian agent who
sees that signal realization. But for an underreacting agent, this means that
this “wrong action” is attractive at the prior and therefore at least one
other Bayesian posterior. To put differently, any action that hurts the agent
following some signal realization benefits her at another, and the latter gain
necessarily outweighs the former loss. This is a bit of an over-simplification
but the essence is that the “direction” of a mistake for an underreacting
agent precludes exploitation.

Our results suggest a comparative benefit to underreaction versus
overreaction. Interestingly, Epstein, Noor, and Sandroni (2010) also discover
an advantage to underreaction, though in a completely different sense. They
examine the long-run beliefs of an agent who repeatedly obtains information.
An underreacting agent eventually arrives at the truth, whereas an overreacter
may not. In contrast, we show that an underreacter’s mistakes are much less
costly than an overreacter’s.

One of the justifications for studying updating rules other than Bayes’ law is
that people, by and large, are not perfect adherents to Bayes’ law.
Importantly, as stressed by Benjamin (2019) and emphasized by Ba, Bohren, and
Imas (2022), in scenarios in which the environment is relatively
transparent–namely, the prior and signal are explicitly specified–agents tend
to underreact to information.222This stands in sharp contrast to the survey
responses of professional forecasters, households, and investors, who
overreact to news (Bordalo, Gennaioli, and Shleifer (2022)). Ba, Bohren, and
Imas (2022)’s thesis that this trend is driven by complexity is compelling. As
Benjamin (2019) states bluntly,

“Stylized Fact 1 Underinference is by far the dominant direction of bias.”

A contribution of our paper, therefore, is to provide a justification for why
underreaction is so dominant–it is safer.

##  2 Screening By a Malevolent Principal

There is an unknown state of the world $\theta$, drawn according to a full-
support prior $\mu$ from some finite set of states $\Theta$. The cardinality
of $\Theta$ is $n\geq 2$ and
$\Delta\left(\Theta\right)\subset\mathbb{R}^{n-1}$ denotes the $n$-state
probability simplex. An agent obtains information about the state by observing
the realization of a signal $\pi\colon\Theta\to\Delta(S)$, where $S$ is a
finite set of signal realizations. At the interim stage–viz., after the agent
observes the signal realization but before the state is known–a principal
offers the agent a menu of state dependent payoffs, i.e., a decision problem.

Formally, the principal offers the agent $\left(A,u\right)$, where $A$ is a
compact set of actions and $u\colon A\times\Theta\to\mathbb{R}$ is a
collection of state- and action-dependent payoffs (in utils). The agent
selects at most one of the actions $a\in A$, with her outside option’s
payoff–that that she obtains from taking none of the proffered
actions–normalized to $0$. We denote the outside option action by
$\varnothing$ and extend any $u$ offered by the principal to
$A\cup\left\\{\varnothing\right\\}$ by defining $u(\varnothing,\theta)=0$ for
all $\theta\in\Theta$.

The principal shares the agent’s prior, and given this, wishes to minimize the
agent’s ex ante expected payoff. What makes the problem nontrivial, and what
potentially gives the principal scope for exploiting the agent, is that the
agent’s updating rule may not be Bayes’ law. That is, given any signal
realization $s$, the agent’s posterior $\hat{x}(s)$ may not equal $x(s)$,
where

$x(s)(\theta)\coloneqq\frac{\mu(\theta)\pi\left(\left.s\right|\theta\right)}{\sum_{\theta\in\Theta}\mu(\theta)\pi\left(\left.s\right|\theta\right)}$

is the Bayesian posterior. We specify that the signal $\pi$ is known to the
principal, as is how the agent interprets signal realizations,333That is, we
specify that if the principal could observe $s$ she would know the agent’s
posterior belief. but that the signal realization, $s$, itself is observed
only by the agent.

Consequently, the principal faces a potentially non-trivial screening problem.
We specify further that $\pi$ is such that the Bayesian posteriors are
affinely independent,444A justification for this is that any Bayes-plausible
(Kamenica and Gentzkow (2011)) distribution over (Bayesian) posteriors can be
obtained by randomizing over Bayes-plausible distributions with affinely
independent support. and that each $s\in S$ realizes with strictly positive
probability. That is,
$\sum_{\theta\in\Theta}\mu(\theta)\pi\left(\left.s\right|\theta\right)>0$ for
all $s\in S$.

All in all, here is the timing of the scenario:

  1. 1.

$\theta$ is drawn according to $\mu\in\operatorname{int}\Delta(\Theta)$ and is
observed by neither principal nor agent.

  2. 2.

$s$ is drawn from $S$ according to $\pi\left(\left.\cdot\right|\theta\right)$
and is observed privately by the agent.

  3. 3.

The principal offers the agent a decision problem $\left(A,u\right)$.

  4. 4.

The agent chooses a single element from $A\cup\left\\{\varnothing\right\\}$.

  5. 5.

Payoffs realize.

The agent’s payoff is her ex ante expected payoff

$\sum_{\theta\in\Theta}\sum_{s\in
S}\mu(\theta)\pi\left(\left.s\right|\theta\right)u(a^{*}(s),\theta)\text{,}$

where $a^{*}(s)$ is (without loss of generality)555Please refer to the
discussion at the beginning of $\S$3.2. the principal-preferred selection from

$\operatorname*{arg\,max}_{a\in
A\cup\left\\{\varnothing\right\\}}\mathbb{E}_{\hat{x}(s)}u(a,\theta)\text{.}$

The principal wishes to infimize the agent’s payoff.

######  Definition 2.1.

Given $\pi$ and $\mu$, an agent can be Exploited if for any $K>0$, the
principal can offer the agent a decision problem yielding her an ex ante
expected payoff of $-K$.

Our imposition that the support of the distribution of Bayesian posteriors is
affinely independent implies there are at most $n$ posteriors. We let the
agent’s type denote her Bayesian posterior. Henceforth, in our notation, we
also suppress the dependence of the posteriors on the signal realization,
i.e., $x_{i}=x(s_{i})$ and $\hat{x}_{i}=\hat{x}(s_{i})$, where beliefs without
hats are those produced by Bayes’ law.

Let us make two easy observations. The first is an immediate consequence of
the fact that the agent’s payoff can be scaled without affecting incentives.

######  Remark 2.2.

An agent can be exploited if and only if the principal can offer the agent a
decision problem yielding her an ex ante expected payoff of $-K$ for some
$K>0$.

The second remark is the revelation principle, more-or-less. Viz., we can
always prune excess actions and reduce the agent’s posterior-dependent choice
to one that is deterministic.

######  Remark 2.3.

If an agent can be exploited, she can be exploited by a decision problem with
at most $n$ actions in which the agent’s choice of action following any signal
realization is deterministic.

##  3 Over- Versus Underreaction

Given a collection of posteriors $\left\\{x_{1},\dots,x_{t}\right\\}$ let
$\operatorname*{conv}\left\\{x_{1},\dots,x_{t}\right\\}$ denote the closed
convex hull of this set.

######  Definition 3.1.

Fix $\mu$ and $\pi$. An agent Underreacts to realization $s$ if there exists a
$\lambda\in\left[0,1\right]$ such that
$\hat{x}_{s}=\lambda\mu+(1-\lambda)x_{s}$. An agent Underreacts to Information
if she underreacts to each realization $s\in S$.

Our definition of overreaction to information is the easily-anticipated
mirror:

######  Definition 3.2.

Fix $\mu$ and $\pi$. An agent Overreacts to realization $s$ if there exists a
$\lambda\in\left(0,1\right)$ such that
$x_{s}=\lambda\mu+(1-\lambda)\hat{x}_{s}$. An agent Overreacts to Information
if she overreacts to each realization $s\in S$.

###  3.1 Some Non-Bayesian Updating Rules

Before exploring the main results in $\S$3.2, let us briefly discuss some
well-known updating rules. As highlighted by de Clippel and Zhang (2022), an
oft-observed non-Bayesian rule observed in the literature is Grether (1980)’s
$\alpha-\beta$ Model of Updating (see, e.g., Benjamin (2019), Benjamin, Bodoh-
Creed, and Rabin (2019), Augenblick and Rabin (2021), and Benjamin, Rabin, and
Raymond (2016)). As revealed by de Clippel and Zhang (2022), an $\alpha-\beta$
distorted posterior $\hat{x}_{s}$ is a function of the Bayesian posterior
$x_{s}$, which simplifies to

$\hat{x}_{s}(x_{s})=\frac{x_{s}^{\beta}}{x_{s}^{\beta}+(1-x_{s})^{\beta}}$

when there are just two states and the prior is $1/2$. Evidently, if
$\beta>1$, the agent overreacts ($x_{s}<\hat{x}_{s}$ if $x_{s}>1/2$ and
$x_{s}>\hat{x}_{s}$ if $x_{s}<1/2$) and underreacts if $\beta<1$.

An agent exhibits Extreme Belief Aversion (DuCharme and Peterson (1968),
DuCharme (1970), and Benjamin, Rabin, and Raymond (2016)) if she is averse to
holding beliefs close to certainty. This yields, from the perspective of our
framework, underreaction to information, particularly when the agent’s signal
is extremely informative. Our main results, therefore, provide a rationale for
such behavior:666Whitmeyer (2023) shows that when there are two states, an
agent who is averse to extreme beliefs nevertheless prefers more information
(Blackwell (1951), Blackwell (1953)) to less. Notably, extreme belief aversion
is the only class of non-Bayesian updating rules compatible with the Blackwell
order. extreme belief aversion is relatively innocuous, and better than the
opposite variety.

In Rabin and Schrag (1999)’s Model of Confirmatory Bias (with two states and
two signal realizations) an agent correctly processes the signal realization
that confirms her prior but only obtains the correct posterior following the
signal realization that contradicts her prior with some probability. With its
complement, her posterior following this conflicting signal realization is the
confirming-realization posterior instead. Formally, given $\mu>1/2$ and
Bayesian posteriors $x_{H}$ and $x_{L}$ ($0\leq x_{L}<1/2<x_{H}\leq 1$),
$\hat{x}_{H}=x_{H}$ and $\hat{x}_{L}=x_{H}$ with some probability
$q\in\left(0,1\right)$ and $\hat{x}_{L}=x_{L}$ otherwise. Though this type of
random error does not fit within our main specification, we study a
generalized version of this environment in $\S$3.4.1.

An agent’s malformed posterior could also come from having a Misspecified
Prior, from the perspective of the principal, at least. Let us again
specialize to two states and binary experiments. In this case, there is always
one signal realization $s$ with respect to which the agent overreacts. If she
is not too misspecified, she underreacts to the other signal realization, but
otherwise does not.

###  3.2 Main Results

Throughout this section, it is important to keep in mind that the agent
chooses her action following a signal realization based on her posterior
$\hat{x}_{s}$ and not the Bayesian posterior $x_{s}$. Furthermore, in
principle, how we break ties when the agent is indifferent between actions
could affect our results. Here is why how we break such ties does not matter.
First, in our sufficiency results for when the agent can be exploited, in a
decision problem that exploits the agent, incentives can be made strict for
the posterior that matters, eliminated the importance of our tie-breaking
rule. Second, our necessity results for the agent to be exploited, we break
ties in a principal-optimal way as a different rule would only make it more
difficult for him to exploit the agent.

Our first result identifies a sufficient condition for exploitation. Fixing
$\mu$ and $\pi$, and letting
$X\coloneqq\left\\{x_{1},x_{2},\dots,x_{n}\right\\}$ denote the set of
Bayesian posteriors,

######  Lemma 3.3.

If there exists an $s\in S$ for which
$\hat{x}_{s}\notin\operatorname*{conv}X$, the agent can be exploited.

###### Proof.

Suppose there exists an $s\in S$ for which
$\hat{x}_{s}\notin\operatorname*{conv}X$. Without loss of generality
$\hat{x}_{s^{\prime}}=x_{s^{\prime}}$ for all $s^{\prime}\in
S\setminus\left\\{s\right\\}$. As $\hat{x}_{s}\notin\operatorname*{conv}X$,
$\hat{x}_{s}$ and $\operatorname*{conv}X$ can be strictly separated by a
hyperplane
$H_{\alpha,\beta}\coloneqq\left\\{x\in\left.\mathbb{R}^{n-1}\right|\alpha\cdot
x=\beta\right\\}$. Suppose the principal offers a single-action decision
problem $\left(\left\\{a\right\\},u\right)$ where $u$ is such that
$\mathbb{E}_{x}u(a,\theta)=\lambda(\alpha\cdot x-\beta)$ ($\lambda>0$) and
$\alpha\cdot\hat{x}_{s}-\beta>0$. Observe that this implies $\alpha\cdot
x_{s^{\prime}}-\beta<0$ for all $s^{\prime}\in S$. The agent’s ex ante
expected payoff is

$\lambda\sum_{\theta\in\Theta}\mu(\theta)\pi(\left.s\right|\theta)\left(\alpha\cdot
x_{s}-\beta\right)\text{.}$

Then for any $K>0$ we just set

$\lambda=\frac{-K}{\sum_{\theta\in\Theta}\mu(\theta)\pi(\left.s\right|\theta)\left(\alpha\cdot
x_{s}-\beta\right)}\text{.}$

Exploitation achieved.∎

Indeed, observe that a single-action decision problem will do. The key part of
the construction is that this action is strictly suboptimal at every one of
the agent’s Bayesian posteriors, yet her extreme posterior makes her strictly
prefer to take it following some signal realization. Big mistake. We also wish
to highlight that the affine independence of $X$ is not needed for this
result: so long as there is some $\hat{x}_{s}$ outside of
$\operatorname*{conv}X$, the result goes through.

This lemma implies a peril of overreaction. Note that this corollary does rely
on the affine independence of $X$. If the set is not affinely independent, the
agent can overreact to some $s$ yet $\hat{x}_{s}\in\operatorname*{conv}X$.

######  Corollary 3.4.

Fix $\mu$ and $\pi$. If an agent overreacts to some realization $s$, she can
be exploited.

On the other hand, it is impossible to exploit an agent who underreacts to
information. Letting $\alpha^{a}\cdot x-\beta^{a}$ denote the expected payoff
in belief $x$ to action $a$, here is the lemma central to establishing
Proposition 3.7.

######  Lemma 3.5.

If an underreacting agent can be exploited, there exists a contract that
exploits the agent that does not contain any action $a$ for which
$\alpha^{a}\cdot x_{i}-\beta^{a}>0\geq\alpha^{a}\cdot\hat{x}_{i}-\beta^{a}$
for some type $x_{i}$.

###### Proof.

As we could just prune any actions that are not chosen by any type, we start
with a decision problem that exploit the agent in which each action is chosen
by at least one type. The set of actions available to the agent is $A$. The
following observation will be useful.

######  Claim 3.6.

If $a\in A$ is such that $\alpha^{a}\cdot
x_{i}-\beta^{a}>0\geq\alpha^{a}\cdot\hat{x}_{i}-\beta^{a}$ for some type
$x_{i}$, we cannot have $\alpha^{a}\cdot\hat{x}_{j}-\beta^{a}\geq
0>\alpha^{a}\cdot x_{j}-\beta^{a}$ for any $x_{j}\in X$.

###### Proof.

Because the agent is underreacting, $\alpha^{a}\cdot
x_{i}-\beta^{a}>0\geq\alpha^{a}\cdot\hat{x}_{i}-\beta^{a}$ for some type
$x_{i}$ implies that

$0\geq\alpha^{a}\cdot\hat{x}_{i}-\beta^{a}\geq\alpha^{a}\cdot\mu-\beta^{a}\text{.}$

Suppose for the sake of contradiction that
$\alpha^{a}\cdot\hat{x}_{j}-\beta^{a}\geq 0>\alpha^{a}\cdot x_{j}-\beta^{a}$
for some $x_{j}\in X$. Again, by underreaction, this implies

$\alpha^{a}\cdot\mu-\beta^{a}\geq\alpha^{a}\cdot\hat{x}_{j}-\beta^{a}\geq
0\text{,}$

a contradiction.∎

Let $A^{\dagger}\subseteq A$ denote the subset of actions in the agent’s
decision problem for which $\alpha^{a}\cdot
x_{i}-\beta^{a}>0\geq\alpha^{a}\cdot\hat{x}_{i}-\beta^{a}$ for some type
$x_{i}$, i.e., formally,

$A^{\dagger}\coloneqq\left\\{a\in\left.A\right|\exists x_{i}\in
X\colon\alpha^{a}\cdot
x_{i}-\beta^{a}>0\geq\alpha^{a}\cdot\hat{x}_{i}-\beta^{a}\right\\}\text{.}$

Without loss of generality we assume that $A^{\dagger}$ is non-empty, or else
we are done.

Claim 3.6 implies that for any $a^{\dagger}\in A^{\dagger}$ and each $x_{i}\in
X$, either

$\begin{split}&1.\quad\alpha^{a^{\dagger}}\cdot
x_{i}-\beta^{a^{\dagger}}>0\geq\alpha^{a^{\dagger}}\cdot\hat{x}_{i}-\beta^{a^{\dagger}}\
\text{; or}\\\ &2.\quad 0>\alpha^{a^{\dagger}}\cdot
x_{i}-\beta^{a^{\dagger}}\geq\alpha^{a^{\dagger}}\cdot\hat{x}_{i}-\beta^{a^{\dagger}}\
\text{; or}\\\ &3.\quad\alpha^{a^{\dagger}}\cdot
x_{i}-\beta^{a^{\dagger}}\geq\alpha^{a^{\dagger}}\cdot\hat{x}_{i}-\beta^{a^{\dagger}}>0\text{.}\end{split}$

Immediately, we see that we cannot have $A^{\dagger}=A$, or else the agent’s
ex ante expected payoff would be weakly positive, contradicting that she is
being exploited. Accordingly $A^{\dagger}\subset A$ (but remember
$A^{\dagger}$ is non-empty). Moreover for any $a\in A\setminus A^{\dagger}$
and any $x_{i}\in X$, either

$\begin{split}&1.\quad\alpha^{a}\cdot\hat{x}_{i}-\beta^{a}\geq
0>\alpha^{a}\cdot x_{i}-\beta^{a}\ \text{; or}\\\ &2.\quad
0>\alpha^{a}\cdot\hat{x}_{i}-\beta^{a}\geq\alpha^{a}\cdot x_{i}-\beta^{a}\
\text{; or}\\\ &3.\quad\alpha^{a}\cdot\hat{x}_{i}-\beta^{a}\geq\alpha^{a}\cdot
x_{i}-\beta^{a}>0\text{.}\end{split}$

Because the agent is being exploited, we must have

$0>\sum_{i=1}^{k}p_{i}\left(\alpha_{i}\cdot
x_{i}-\beta_{i}\right)+{\color[rgb]{0.31,1,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.31,1,0.5}\pgfsys@color@cmyk@stroke{0.69}{0}{0.50}{0}\pgfsys@color@cmyk@fill{0.69}{0}{0.50}{0}\sum_{i=k+1}^{n}p_{i}\left(\alpha_{i}\cdot
x_{i}-\beta_{i}\right)}\text{,}$

where $\alpha_{i}\cdot x-\beta_{i}$ is the expected payoff to type $x_{i}$
from choosing optimally at belief $\hat{x}_{i}$. Keep in mind that it is
possible that a type $x_{i}$ is choosing the outside option, in which case
$\alpha_{i}=\underline{0}$ and $\beta_{i}=0$. Moreover, the set of types
$\left\\{x_{1},\dots,x_{k}\right\\}$ ($1\leq k<n$) are those whose choice of
action (assuming any indifferences are broken in a principal-optimal way) is
different when the set of available actions is $A\setminus A^{\dagger}$ rather
than $A$.

Suppose that the principal prunes away the set of actions $A^{\dagger}$,
offering the agent $A\setminus A^{\dagger}$ instead. Observe that agent’s
payoff (breaking ties in a principal-optimal way) is now

$\sum_{i=1}^{k}p_{i}\left(\tilde{\alpha}_{i}\cdot
x_{i}-\tilde{\beta}_{i}\right)+{\color[rgb]{0.31,1,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.31,1,0.5}\pgfsys@color@cmyk@stroke{0.69}{0}{0.50}{0}\pgfsys@color@cmyk@fill{0.69}{0}{0.50}{0}\sum_{i=k+1}^{n}p_{i}\left(\alpha_{i}\cdot
x_{i}-\beta_{i}\right)}\text{,}$

where $\tilde{\alpha}_{i}\cdot x-\tilde{\beta}_{i}$ is the expected payoff to
type $x_{i}$ ($i\leq k$) from choosing optimally at belief $\hat{x}_{i}$ when
the set of available actions is $A\setminus A^{\dagger}$.

The green sums in Expressions 3.2 and 3.2 are the same. Moreover, by the
Consequences of Claim 3.6 enumerated in Lists 3.2 and 3.2, we must have

$\tilde{\alpha}_{i}\cdot
x_{i}-\tilde{\beta}_{i}\leq\tilde{\alpha}_{i}\cdot\hat{x}_{i}-\tilde{\beta}_{i}\leq\alpha_{i}\cdot\hat{x}_{i}-\beta_{i}\leq\alpha_{i}\cdot
x_{i}-\beta_{i}\text{,}$

for each $i\leq k$ and so

$\sum_{i=1}^{k}p_{i}\left(\tilde{\alpha}_{i}\cdot
x_{i}-\tilde{\beta}_{i}\right)+{\color[rgb]{0.31,1,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.31,1,0.5}\pgfsys@color@cmyk@stroke{0.69}{0}{0.50}{0}\pgfsys@color@cmyk@fill{0.69}{0}{0.50}{0}\sum_{i=k+1}^{n}p_{i}\left(\alpha_{i}\cdot
x_{i}-\beta_{i}\right)}\leq\sum_{i=1}^{k}p_{i}\left(\alpha_{i}\cdot
x_{i}-\beta_{i}\right)+{\color[rgb]{0.31,1,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.31,1,0.5}\pgfsys@color@cmyk@stroke{0.69}{0}{0.50}{0}\pgfsys@color@cmyk@fill{0.69}{0}{0.50}{0}\sum_{i=k+1}^{n}p_{i}\left(\alpha_{i}\cdot
x_{i}-\beta_{i}\right)}<0\text{,}$

completing the proof.∎

######  Theorem 3.7.

Fix $\mu$ and $\pi$. If an agent underreacts to information, she cannot be
exploited.

###### Proof.

Suppose for all $s\in S$ $\hat{x}_{s}$ lies on the line segment between $\mu$
and $x_{s}$.

Our proof approach is via induction. We shall first show that a single-action
decision problem is incapable of exploiting the underreacting agent. We will
then argue that if a $t$-action ($1\leq t$) decision problem cannot exploit
the agent, neither can a $(t+1)$-action decision problem.

Step 1. Base case: A single-action decision problem cannot exploit the agent.
As each $\hat{x}_{s}$ lies on the line segment between $\mu$ and $x_{s}$, for
any hyperplane
$H_{\alpha,\beta}\coloneqq\left\\{x\in\left.\mathbb{R}^{n-1}\right|\alpha\cdot
x=\beta\right\\}$ that strictly separates some $x_{s^{\prime}}$ from
$\hat{x}_{s^{\prime}}$, for which $\alpha\cdot\hat{x}_{s^{\prime}}-\beta>0$,
we must have $\alpha\cdot\mu-\beta>0$. Accordingly, as the average posterior
is the prior $\mu$, the agent’s payoff is bounded (weakly) below by
$\alpha\cdot\mu-\beta>0$.

Step 2. Induction hypothesis and step: if a $t$-action ($1\leq t$) decision
problem cannot exploit the agent, neither can a $(t+1)$-action decision
problem. Suppose for the sake of contradiction that there exists a
$(t+1)$-action decision problem that exploits the agent. By Lemma 3.5, we
assume without loss of generality that there does not exist an action $a$ for
which $\alpha^{a}\cdot
x_{i}-\beta^{a}>0\geq\alpha^{a}\cdot\hat{x}_{i}-\beta^{a}$ for some type
$x_{i}$.

Denote the actions in this decision problem by $\tilde{a},a^{1},\dots,a^{t}$,
keeping in mind that the set of actions available to the agent is
$\left\\{\tilde{a},a^{1},\dots,a^{t},\varnothing\right\\}$. In the specified
$(t+1)$-action decision problem, let $\left\\{x_{1},\dots,x_{k}\right\\}$
($1\leq k$) be the types who select action $\tilde{a}$ and
$\left\\{x_{s},\dots,x_{n}\right\\}$, those who select $\varnothing$. By
assumption,

$0>\sum_{i=1}^{k}p_{i}(\tilde{\alpha}\cdot
x_{i}-\tilde{\beta})+{\color[rgb]{0.31,1,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.31,1,0.5}\pgfsys@color@cmyk@stroke{0.69}{0}{0.50}{0}\pgfsys@color@cmyk@fill{0.69}{0}{0.50}{0}\sum_{i=k+1}^{s-1}p_{i}(\alpha_{i}\cdot
x_{i}-\beta_{i})}\text{,}$

where $\tilde{\alpha}\cdot x-\tilde{\beta}$ is the expected payoff from taking
action $\tilde{a}$ and $\alpha_{i}\cdot x-\beta_{i}$ is the expected payoff
from taking the optimal action for type $x_{i}$ given her posterior
$\hat{x}_{i}$.

As no $t$ action decision problem exploits the agent, we must have (when
$\tilde{a}$ is removed)

$\sum_{i=1}^{k}p_{i}(\alpha_{i}\cdot
x_{i}-\beta_{i})+{\color[rgb]{0.31,1,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.31,1,0.5}\pgfsys@color@cmyk@stroke{0.69}{0}{0.50}{0}\pgfsys@color@cmyk@fill{0.69}{0}{0.50}{0}\sum_{i=k+1}^{s-1}p_{i}(\alpha_{i}\cdot
x_{i}-\beta_{i})}\geq 0\text{.}$

where $\alpha_{i}\cdot x-\beta_{i}$ is the expected payoff, in the posterior
$x$, for type $x_{i}$’s optimal action in the set
$\left\\{a^{1},\dots,a^{t},\varnothing\right\\}$ given her belief
$\hat{x}_{i}$. Note that green expressions in Inequalities 3.2 and 3.2 are the
same.

Analogously, as our choice of an outside option of $0$ was merely a
normalization,

$\sum_{i=1}^{k}p_{i}(\tilde{\alpha}x_{i}-\tilde{\beta})+\sum_{i=k+1}^{s-1}p_{i}(\alpha_{i}\cdot
x_{i}-\beta_{i})+\sum_{i=s}^{n}p_{i}(\alpha_{i}\cdot
x_{i}-\beta_{i})\geq\tilde{\alpha}\cdot\mu-\tilde{\beta}\geq 0\text{.}$

Crucially, for each $i\geq s$, $\alpha_{i}\cdot\hat{x}_{i}-\beta_{i}\leq 0$.
Thus, for each $i\geq s$, $\alpha_{i}\cdot x_{i}-\beta_{i}\leq 0$, which
implies

$\sum_{i=1}^{k}p_{i}(\tilde{\alpha}x_{i}-\tilde{\beta})+\sum_{i=k+1}^{s-1}p_{i}(\alpha_{i}\cdot
x_{i}-\beta_{i})\geq 0\text{,}$

a contradiction.∎

It turns out that if we specialize to experiments with just two signal
realizations, we can completely characterize updating behavior that leaves an
agent vulnerable (or not) to exploitation.

######  Proposition 3.8.

Fix $\mu$ and binary $\pi$. An agent cannot be exploited if and only if she
underreacts to information.

###### Proof.

Sufficiency follows from Theorem 3.7. By Lemma 3.3, we need only check the
situation in which one of the two posteriors $\hat{x}_{0}$ can be strictly
separated from $\operatorname*{conv}\left\\{x_{0},\mu\right\\}$ by some
hyperplane $\alpha\cdot x=\beta$. However, in this instance, there exists a
single-action decision problem, such that the agent’s payoff is bounded above
by $\alpha\cdot\mu-\beta<0$. ∎

This proposition levers the fact that if the agent’s experiment is binary, and
if an agent’s posterior lies within the convex hull of the Bayesian
posteriors, it is either an under-reaction, or it “skips over the prior.” As
the proof reveals, such a skip renders the agent vulnerable. In the next
section, we use this proposition to shed light on a broad class of updating
rules.

###  3.3 Systematic Distortions

A special class of updating rules are those for which the posterior produced
by the updating rule is a function of the Bayesian posterior, irrespective of
the prior or the signal.

######  Definition 3.9.

An updating rule systematically distorts beliefs if for any full-support prior
$\mu$, signal $\pi$, and signal realization $s$, the posterior produced by the
agent’s updating rule $\hat{x}_{s}$ equals $\varphi(x_{s})$ for some function
$\varphi\colon\Delta(\Theta)\to\Delta(\Theta)$.

de Clippel and Zhang (2022) introduce this class of updating rules, which has
subsequently been studied by, e.g., Chambers, Masatlioglu, and Raymond (2023)
and Whitmeyer (2023). Jakobsen (2022) explores a particular class of such
systematic distortions.

######  Definition 3.10.

An agent is Exploitable if for any $K>0$, there exists a full-support prior
$\mu$, a signal $\pi$, and a decision problem yielding the agent an ex ante
expected payoff of $-K$.

We term an agent who is not exploitable Unexploitable. Repeating our earlier
remark almost verbatim, and via the exact same logic,

######  Remark 3.11.

An agent is exploitable if and only if the principal can offer the agent a
decision problem yielding her an ex ante expected payoff of $-K$ for some
$K>0$.

Then,

######  Proposition 3.12.

An agent whose updating rule systematically distorts beliefs is unexploitable
if and only if for any full-support prior $\mu$ and signal $\pi$ she
underreacts to information.

###### Proof.

$\left(\Leftarrow\right)$ Suppose for the sake of contraposition there exists
a full-support prior $\mu$ and signal $\pi$ for which the agent does not
underreact to information. Because the agent’s updating rule systematically
distorts beliefs, this implies there exists a binary experiment and a signal
realization $s$ for which $\hat{x}_{s}$ does not lie within the closed convex
hull of the set of Bayesian posteriors. Accordingly, Lemma 3.3 implies that
the agent can be exploited and so she is exploitable.

$\left(\Rightarrow\right)$ This is an immediate consequence of Theorem 3.7. ∎

###  3.4 Uncertain Rules and Random Mistakes

It is possible that the principal does not know the updating rule used by the
agent, or that agent’s updating rule maps at least one signal realization at
random to posteriors. These two cases are formally equivalent: for each signal
realization $s\in S$, the agent’s posterior is a random vector
$\hat{\mathcal{X}}_{s}$ distributed according to some $\gamma_{s}$ on
$\Delta(\Theta)$. Recalling that $X$ denotes the set of Bayesian posteriors
and $\operatorname*{conv}X$ its closed convex hull, we have

######  Lemma 3.13.

Fix $\mu$ and $\pi$. If there exists a closed convex set
$B\subset\Delta(\Theta)$ disjoint from $X$ for which
$\mathbb{P}(\hat{\mathcal{X}}_{s})\in B>0$, the agent can be exploited.

###### Proof.

By the strict separating hyperplane theorem, there exists a hyperplane that
strictly separates $B$ from $X$. Using this, the remainder of the proof
mirrors that for Lemma 3.3 and so we omit it. ∎

It is also easy to derive an analog of Theorem 3.7.

######  Corollary 3.14.

Fix $\mu$ and $\pi$. If for each $s$ $\hat{\mathcal{X}}_{s}$ is supported on a
finite subset of $\operatorname*{conv}\left\\{x_{s},\mu\right\\}$, the agent
cannot be exploited.

###### Proof.

This result is exactly Theorem 3.7 once we redefine an agent’s type to be her
realized posterior. Given this, there will obviously be more than $n$ types
(unless the updating is deterministic), but there are still only finitely many
types. ∎

####  3.4.1 Rabin and Schrag’s Model of Confirmatory Bias

One type of updating rule that does not fit our main specification, nor the
extension of this subsection is Rabin and Schrag (1999)’s model of
confirmatory bias. A more general version of this is as follows: there are $n$
states, a full-support prior $\mu$ and a signal $\pi$ that produces a
collection of affinely-independent Bayesian posteriors. Then, for any signal
realization $s$, the agent’s posterior is a random variable
$\hat{\mathcal{X}}_{s}$ supported on the Bayesian posteriors
$X=\left\\{x_{1},\dots,x_{n}\right\\}$. For each $s$, we let
$q_{s}\coloneqq\mathbb{P}(\hat{\mathcal{X}}_{s}=x_{s})\in\left[0,1\right]$,
i.e., $q_{s}$ is the probability $s$ is interpreted correctly. We call this
class of updating rules Generalized Confirmatory Bias.

Any error in this sense allows the principal to exploit the agent.

######  Proposition 3.15.

Fix $\mu$ and $\pi$. If the agent’s updating rule is generalized confirmatory
bias and there exists an $s$ for which $q_{s}<1$, the agent can be exploited.

###### Proof.

Suppose there exists an $s$ for which $q_{s}<1$. Then there exists an
$s^{\prime}\neq s$ for which
$\mathbb{P}(\hat{\mathcal{X}}_{s}=x_{s^{\prime}})>0$. As $x_{s^{\prime}}$ is
an exposed point of $X$, there exists a supporting hyperplane
$H_{\alpha,\beta}\coloneqq\left\\{x\in\left.\mathbb{R}^{n-1}\right|\alpha\cdot
x=\beta\right\\}$ for which
$H_{\alpha,\beta}\cap\operatorname*{conv}X=\left\\{x_{s^{\prime}}\right\\}$.
Accordingly, there exists an action yielding payoff $\alpha\cdot x-\beta$,
where $\alpha\cdot x_{s}-\beta<0$ for all $s\neq s^{\prime}$ and $\alpha\cdot
x_{s^{\prime}}-\beta=0$. As the agent’s payoff is the average of a strictly
negative number with $0$, her expected payoff is strictly negative; that is,
she can be exploited.∎

## References

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DSS

    Decision Support System
HDS

    Human Digital Shadow
IoP

    Internet of Production
CNN

    Convolutional Neural Network
DS

    Digital Shadow
DT

    Digital Twin
ML

    Machine Learning
BT

    Behaviour Tree
BTs

    Behaviour Trees
HTNs

    Hierarchical Task Networks
HTN

    Hierarchical Task Network
FSM

    Finite State Machine
FSMs

    Finite State Machines
PLC

    Product Life Cycle
DCC

    Digital Capability Centre
BOL

    Beginning of Life
AI

    Artificial Intelligence
XAI

    eXplainable Artificial Intelligence
HRI

    Human-Robot Interaction
HRC

    Human-Robot Collaboration
IoT

    Internet of Things
IIoT

    Industrial Internet of Things
WWL

    World Wide Lab
FRP

    Fibre Reinforced Plastics
SUS

    System Usability Scale
TFD

    Temporal Fast Downward
PDDL

    Planning Domain Definition Language
LMP

    Laser Material Processing
IR

    Industrial Robot
TCP

    Tool Centre Point
EL

    Euler Lagrange
RNN

    Recurrent Neural Network
RL

    Reinforcement Learning
AM

    Additive Manufacturing
LPBF

    Laser Powder Bed Fusion
AE

    Auto Encoder
DL

    Deep Learning
CPPS

    Cyber-Physical Production System
MTL

    Metric Temporal Logic
SME

    Small and Medium sized Enterprise
OCELs

    Object-Centric Event Log
GAN

    Generative Adversarial Network
ANN

    Artificial Neural Network
OT

    Optical Tomography
KPI

    Key Performance Indicator
OCPC

    Object-Centric Process Cube
LfD

    Learning from Demonstration
MPC

    Model Predictive Control
MIP

    Mixed Initiative Planning
NPC

    Non-Player Character
FEM

    Finite Element Method
KB

    Knowledge Base
ROS

    Robot Operating System
MAS

    Multi-Agent System
MRS

    Multi-Robot System
LiDAR

    Light Detection And Ranging
SLAM

    Simultaneous Localisation And Mapping
IQA

    Information Quality Assessment
HDS

    Human Digital Shadow
AGV

    Automated Guided Vehicle
API

    Abstract Programming Interface
W3C

    World Wide Web Consortium
CBF

    Control Barrier Function
CBF-BT

    Control Barrier Function Behavior Tree
CARE

    Constraint-Based Activity REpresentation
w.r.t.

    With Respect To

# Digital Shadows of Safety for Human Robot Collaboration in the World-Wide
Lab

Mohamed Behery, Gerhard Lakemeyer

###### Abstract

The World Wide Lab (WWL) connects the Digital Shadows of processes, products,
companies, and other entities allowing the exchange of information across
company boundaries. Since DSs are context- and purpose-specific
representations of a process, as opposed to Digital Twins which offer a full
simulation, the integration of a process into the WWL requires the creation of
DSs representing different aspects of the process. Human-Robot Collaboration
(HRC) for assembly processes was recently studied in the context of the WWL
where Behaviour Trees were proposed as a standard task-level representation of
these processes. We extend previous work by proposing to standardise safety
functions that can be directly integrated into these BTs. This addition uses
the WWL as a communication and information exchange platform allowing
industrial and academic practitioners to exchange, reuse, and experiment with
different safety requirements and solutions in the WWL.

## Introduction

Industrial robotics processes, particularly those involving Human-Robot
Interaction (HRI), have recently become widespread in many manufacturing
domains and interaction paradigms (Baier et al. 2022). Despite being dynamic
and safety critical applications of robotics, these processes usually involve
ad hoc programming which is often verified empirically on a case-by-case
basis, especially when the process relies on Deep Learning (DL) applications.
In this paper, we propose standardizing safety functions for HRI processes in
the WWL. Having a standard representation offers the ability to reuse these
functions as well as producing reusable code for representing the tasks
involved in a process.

Previous work has introduced the Digital Shadow (DS) of production along with
requirements and challenges facing a standard DS (Bauernhansl, Hartleif, and
Felix 2018). Further research with a focus on integrating HRI assembly into
the WWL suggested using Behaviour Trees (BTs) (Colledanchise and Ögren 2018)
as a standard task-level representation for these processes (Behery et al.
under review 2023).

BTs model a robot’s behavior in a tree structure where the leaves represent
atomic actions and conditions while internal nodes are used for control flow.
Execution begins by ticking the root which propagates this tick down to the
leaves. When a node is ticked, it returns its execution status to its parent
allowing real-time execution monitoring as well as plan synthesis and repair.

This paper proposes using standard parameterisable safety functions
encapsulated in BT nodes. We propose using Control Barrier Functions (Ames et
al. 2019) which have been combined with traditional BTs (Özkahraman and Ögren
2020) producing Control Barrier Function Behavior Trees that guarantee
convergence and safety.

The rest of the paper is structured as follows: The next section Background
and Related Work describes the background knowledge and related work on DSs in
the WWL, BTs, and CBFs. After that we discuss our proposal for standard CBFs
to represent different safety aspects in HRI processes. Lastly, we conclude
the paper with a summary and recommendations for future work.

Figure 1: The architecture proposed by Behery et al. (under review 2023)
integrated into the WWL (Liebenberg and Jarke 2020). Human operators add,
introspect, and query DSs of task representations of different agents. They
can also share and use condition nodes encapsulating a CBF to an existing BT.
Additionally, DSs can communicate with each other and self-optimize as
proposed by (Bauernhansl, Hartleif, and Felix 2018).

## Background and Related Work

The WWL provides an infrastructure for communication between different
entities within a company as well as beyond its boarders (Liebenberg and Jarke
2020). Entities in the WWL are represented using their DSs which are task- and
purpose-specific projections of entities such as processes, machines, and
products to name a few. Compared to the widely used DTs, which represent a
full simulation of an entity (Liebenberg and Jarke 2020), DSs can be used in
real-time applications such as quality prediction and plan repair (Liebenberg
2021). Previous work (Bauernhansl, Hartleif, and Felix 2018) defined
requirements for DSs split into different levels of complexity and development
milestones.

Since robotics and particularly HRC-based processes span different
disciplines: motor control, perception, planning, and human safety among
others, they require multiple DSs to be fully integrated into the WWL. Task
level representations of different robotic activities have been in development
over the past several decades. They usually involve modeling the domain, e.g.,
Planning Domain Definition Language (PDDL) (McDermott 2003) where a developer
has to model the robot’s capabilities as well as the initial and goal states
of the world. After that the agent can create a plan to get from the former to
the latter. Recent approaches have shown combinations and transformations
between these systems and rule based systems (Niemueller, Hofmann, and
Lakemeyer 2019) that combines goal reasoning and teleoreactive programs.
However, these paradigms are not modular enough to allow code reuse and
exchange due to their dependence on problem, domain, and robot descriptions.

BTs (Colledanchise and Ögren 2018) were recently adapted by the robotics
community to overcome these problems. Their structure allows entire tree
branches to be changed or removed without affecting the rest of tree.
Modifying the tree requires less effort and time compared to other graph-based
models such as Finite State Machines or Hierarchical Task Networks, since the
transitions are encoded in the tree structure and do not have to be explicitly
modeled and maintained with every change.

They were recently combined with CBFs to guarantee their concurrent goal
satisfaction allowing a tree to work toward multiple given goals without
breaking safety rules (Özkahraman and Ögren 2020). A CBF is a function
$h:R^{n}\rightarrow R$ is used to evaluate the system state $x$ w.r.t. safety
where $h(x)\geq 0$ implies $x$ is a safe state. They were encapsulated in
condition nodes of the BT which can then select appropriate actions to keep
the robot within a safe set of states without contradicting previously
satisfied conditions.

##  CBFs with Predefined APIs as Safety DSs

Özkahraman and Ögren (2020) combined CBFs with BTs to achieve goals with
different priorities. Goal conflicts are resolved using the tree’s structure
to allow the robots to favor safety over task execution. We propose
maintaining a pool of condition nodes encapsuating CBFs through the WWL
allowing practitioners from academia and industry to share and exchange
standard and explicit safety requirements across company boundaries.
Exchanging process DSs through the WWL has been discussed and demonstrated in
previous work (Liebenberg 2021) where process DS were used for real-time plan
repair in several use-cases.

Behery et al. (under review 2023) proposed standardizing BTs as a task-level
DS for assembly processes involving HRI. The modularity of BTs allows nodes,
branches, and entire sub-trees to be added, removed, and exchanged thus adding
new levels of safety (e.g., guarding actions with preconditions) or
alternative behaviors in case of failure.

Our extension of the WWL architecture proposed in (Liebenberg 2021; Behery et
al. under review 2023) is shown in Figure 1. In addition to storing, sharing,
and querying task level description, we allow human operators to exchange
safety functions such as checking battery, joint velocities, or whether a
human is in the cell of the robot. Each of these functions can have it’s own
interface to the middle-ware (e.g., ROS) of a robot allowing it to read
battery or joint state or other perception information such as 3D point cloud
data. As a parameter, they can take topic names of these types of inputs and
connect to them directly, since they are condition nodes, their return status
encoding whether the system is in a safe state. Since CBFs are composable,
they can form safety branches and be added to a BT for added safety without
affecting the core tasks of the tree.

## Conclusion and Future Work

We propose using standard DSs for safety in HRC-based assembly processes.
Storing CBFs encapsulated in BT condition nodes as proposed by (Özkahraman and
Ögren 2020) allows exchanging and reusing them within the WWL. These functions
can be used in benchmarks for evaluating the safety of newly proposed
approaches in different paradigms of HRC. Some topics still remain open such
as the creation of a general Abstract Programming Interface (API) between the
CBF and the perception modules or the underlying knowledge-bases. Another
direction is the interface for defining these functions, their parameters,
particularly involving training requiring generalization guarantees. This
aligns with the road map presented by Behery et al. (under review 2023)
allowing automatic registration, modification, and optimization of DSs in the
WWL.

## Acknowledgements

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC-2023 Internet of
Production – 390621612. This work is supported by the EU ICT-48 2020 project
TAILOR (No. 952215).

## References

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  * Behery et al. (under review 2023) Behery, M.; Brauner, P.; Kluge-Wilkes, A.; Baier, R.; Mertens, C., Alexander Brecher; Ziefle, M.; and Lakemeyer, G. under review 2023.  The Digital Shadow for Robotic Assembly in the World Wide Lab.  _Procedia CIRP_.
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  *[WWL]: World Wide Lab
  *[DSs]: Digital Shadow
  *[HRC]: Human-Robot Collaboration
  *[BTs]: Behaviour Tree
  *[HRI]: Human-Robot Interaction
  *[DL]: Deep Learning
  *[DS]: Digital Shadow
  *[BT]: Behaviour Tree
  *[CBFs]: Control Barrier Function
  *[CBF]: Control Barrier Function
  *[DTs]: Digital Twin
  *[PDDL]: Planning Domain Definition Language
  *[w.r.t.]: With Respect To
  *[APIs]: Abstract Programming Interface
  *[API]: Abstract Programming Interface