# Quantifying Responsibility with Probabilistic Causation — The Case of
Climate Action

Sarah Hiller  Free University Berlin, Institute for Mathematics, Arnimallee 3,
14195 Berlin, Germany  Potsdam Institute for Climate Impact Research, PO Box
60 12 03, 14412 Potsdam, Germany  Corresponding author. Email:
<EMAIL_ADDRESS>Jobst Heitzig  Potsdam Institute for Climate Impact
Research, PO Box 60 12 03, 14412 Potsdam, Germany

###### Abstract

Many real-world situations of ethical and economic relevance, such as
collective (in)action with respect to the climate crisis, involve not only
diverse agents whose decisions interact in complicated ways, but also various
forms of uncertainty, including both quantifiable risk and unquantifiable
ambiguity. In such cases, an assessment of moral responsibility for ethically
undesired outcomes or of the responsibility to avoid these is challenging and
prone to the risk of under- or overdetermination. In contrast to existing
approaches that employ notions of causation based on combinations of necessity
and sufficiency, or certain logics that focus on a binary classification of
‘responsible’ vs ‘not responsible’, we present a set of quantitative metrics
that assess responsibility degrees in units of probability. To this end, we
adapt extensive-form game trees as the framework for representing decision
scenarios and evaluate the proposed responsibility functions based on the
correct representation of a set of analytically assessed paradigmatic example
scenarios. We test the best performing metrics on a reduced representation of
a real-world decision scenario and are able to compute meaningful
responsibility scores.

##  1 Introduction

The current climate crisis and its associated effects constitute one of the
essential challenges for humanity and collective decision making in the
upcoming years. An increase of greenhouse gas (GHG)111Prominently CO2, but
also methane, nitrous oxide and others. concentrations in the atmosphere
attributable to human activity leads to a warming of Earth’s surface
temperature by reducing the fraction of incoming solar radiation that is
diffused back into space. An elevated mean Earth surface temperature is
however not a priori something reprehensible. Rather, it is the resultant
effects that carry enormous dangers. Among these are the increased risk of
extreme weather events such as storms and flooding, the rise of sea-levels or
the immense losses of biodiversity, which have repercussions not only for the
physical integrity of the planet but which pose direct threats to human
life.222See for example Gardiner (2004) for a concise overview of the relevant
climate science explained for non-climate scientists, or the IPCC and World
Bank reports for more detail (Masson-Delmotte et al., 2018; World Bank Group,
2014).

Naturally, the public debate around this issue frequently invokes the question
of _responsibility_ : Who carries how much backward-looking responsibility for
the changes already inevitable, who is to blame; and who carries how much
forward-looking responsibility to realise changes, who has to act?333What we
call “forward-looking” or ex-ante responsibility is closely linked to the idea
of obligation or duty, whereas what we call “backward-looking” or ex-post
responsibility has also been called accountability, and relates to blame
(Braham and van Hees, 2018; Tamminga and Hindriks, 2019). And to what degree?
In order to answer these questions, we need to clarify what exactly is meant
by the term _responsibility_ , to consequently work on devising a proper
quantification. Taxonomies of responsibility distinguish between six, nine, or
even fifteen different concepts all denominated by the same term (Fischer and
Tognazzini, 2011; van de Poel, 2011; Vincent, 2011). We need to specify which
specific meaning is being referred to before proceeding to elaborate on a
quantification, both of which we aim to achieve through an appropriate
formalisation.

###  1.1 Existing work

The existing body of work regarding the question of how to correctly model
responsibility in the context of climate change can roughly be divided into
two categories, via the perspective from which this question is addressed. On
the one side there are considerations focusing on applicability in the climate
change context, computing tangible responsibility scores for countries or
federations, with the aim of shaping the actions being taken and a lesser
focus on conceptual elegance and generalisability (Botzen et al., 2008; Müller
et al., 2009). On the other side there is considerable work in formal ethics,
aiming at understanding and formally representing the concept of
responsibility in general with a special focus on rigour and well-foundedness,
making it harder to account for messy real world scenarios (in realistic
computation time) (Belnap et al., 2001; Braham and van Hees, 2012; Chockler
and Halpern, 2004; Horty, 2001).

It will be useful to highlight certain aspects of these works now. In the
former set of works, and particularly also in public discourse, the degree of
backward responsibility for climate change of a person, firm, or country is
simply equated to cumulative past GHG emissions, or a slight variation of this
measure (Clark, 2011). Certainly, this approach has one clear benefit, namely
that it is easy to compute on any scale, and also extremely easy to
communicate to a non-scientific audience. Similarly, certain authors assume a
country’s degree of forward responsibility to be proportional to population
share, gross domestic product or some similar indicator, specifically in the
debate about “fair” emissions allowances or caps (Poetter, 2019; Ringius et
al., 2002). However, unfortunately, such ad hoc measures clearly fail to
generalise to other fields of application.

In the latter body of work, a principled approach is taken. Starting from
considerations regarding the general nature of the concept of responsibility,
formalisms are set up to represent these. These comprise causal models
(Chockler and Halpern, 2004), game-theoretical representations (Braham and van
Hees, 2012; Yazdanpanah and Dastani, 2016) or logics (Broersen, 2011; Tamminga
and Hindriks, 2019). A vast number of different aspects have been included in
certain formalisations, such as degrees of causation or responsibility,
relations between individuals and groups, or epistemic states of the agents to
name but a few. Generally, these are discussed using reduced, well-defined
example scenarios and thought experiments capturing certain complicating
aspects of responsibility ascription.

Additionally, there are investigations into the everyday understanding of the
various meanings of the term ‘responsibility’ (Vincent, 2011, e.g.) as well as
empirical studies regarding agents’ responsibility judgements in certain
scenarios, showing a number of asymmetry results (Nelkin, 2007, e.g.).
However, we are here not concerned with mirroring agent’s actual judgements,
but rather with a normative account, so we will not go into detail about
these.

###  1.2 Research question

As was stated above, we are interested in a formal representation and
quantification of the concept of both forward- and backward-looking
responsibility, with a specific focus on the application in the climate change
scenario. That is, we aim to bridge between the two lines of approaches stated
above, by staying within in the category of a principled and formal approach,
but keeping in mind the practical applicability in complex scenarios. Also, we
want to relocate the space of discussion in the formal community by proposing
a set of responsibility functions that differs from existing accounts in two
important ways. One criterion which is generally agreed to be a precondition
for responsibility ascription is some sort of relevant causal connection
between the agent and the outcome, with the specific interpretations of the
terms ‘relevant’ and ‘causal’ being up for discussion. In contrast to the
existing formal accounts of responsibility, where causality is treated via a
combination of necessity and sufficiency of sets of events for the occurrence
of the outcome, we instead take a probabilistic approach. As a consequence, we
are able to directly compute comparable numbers rather than having to manually
extend a binary measure. It might be useful to add that our work is normative,
not descriptive. We aim at representing ways in which responsibility should be
ascribed, not the ways in which people in standard discussion generally do
ascribe it or are psychologically inclined to perceive it.

We introduce a suitable framework that is able to represent all relevant
aspects of a decision scenario using an extension of the game-theoretic
framework of an extensive-form game tree. Subsequently, we will suggest
candidate functions for assigning real numbers as degrees of responsibility
(forward- as well as backward-looking) that are capable of correctly capturing
the effects of previously discussed emblematic example scenarios and that will
consequently be applied to a reduced representation of a climate change
decision situation.

###  1.3 Specific aspects to be considered

The above outline already shows several features of anthropogenic climate
change that complicate responsibility assignments and occur in similar forms
in other real-world decision problems. We will now highlight and discuss
several features that our framework will need to include, as well as certain
aspects that we treat differently from existing work.

#### Interaction, uncertainty.

First of all, the effects of climate change are the result of an _interaction_
of many different actors: corporations, politicians, consumers, organisations,
groups of these, etc. all play a role. Next, there is considerable
_uncertainty_ regarding the impacts to be expected from a given amount of
emissions or a given degree of global warming. While for some results we can
assign probabilities and confidence intervals, for others this cannot be done
in a well-founded way and beyond specifying the set of possible alternatives
one cannot resolve the _ambiguity_ with the given state of scientific
knowledge.

When several models give similar but slightly diverging predictions, for
example, as is very often the case, we cannot assign probabilities to either
of the models being ‘more right’ than the others. What we can say however, is
that each of the predictions is within the set of possible outcomes (given the
premises, such as a certain future behaviour). The same goes for varying
parameters within one and the same model.

Contrastingly, in a large body of work concerning effects of pollution, or
warming, predictions are associated with a specified probability. Take for
example the IPCC reports, such as the well known statement about the remaining
carbon budget if warming is to be limited to 1.5 degrees: “[…] gives an
estimate of the remaining carbon budget of […] 420 GtCO2 for a 66% probability
[of limiting warming to 1.5°C]” (Masson-Delmotte et al., 2018). In many cases,
both aspects of uncertainty — ambiguity and probabilistic uncertainty — are
combined by speaking about intervals of probabilities, which in particular the
IPCC does pervasively (Mastrandrea et al., 2011).

We argue that it is equally important to take note of the additional
information in the probabilistic uncertainty case444This is often called
‘risk’ in economics, however, since we use the term ‘risky’ in this article
for a different concept, we stick to the term ‘probabilistic uncertainty’ in
order to ease disambiguation. as of the lack thereof in the ambiguity case. It
is known that the distinction between probabilistic and non-probabilistic
uncertainty is important in decision making processes (Ellsberg, 1961), and we
want this to be reflected in our attribution of responsibility.

#### Nonlinearity.

As a further particularity, the effects of global warming do not scale in a
linear way with respect to emissions. With rising temperatures, so called
‘tipping elements’ such as the Greenland or West Antarctic ice sheets risk
being tipped (Lenton et al., 2019): once a particular (but imprecisely known)
temperature threshold is crossed, positive feedback leads to an irreversible
progress of higher local temperatures and accelerated degradation of the
element.555Ice reflects more sunlight than water. Thus, if a body of ice melts
and turns into water, this will retain more heat than the ice did, leading to
higher temperatures and faster melting of the remaining ice. As is stated in
(Schellnhuber et al., 2016): “The keywords in this context are non-linearity
and irreversibility”. Note that not all tipping elements are bodies of ice —
coral reefs or the Amazon rain forest also rank among them. The examples with
corresponding explanation were chosen for their simplicity. This initially
local effect then aggravates global warming and may contribute to the tipping
of further elements (Kriegler et al., 2009; Wunderling et al., 2019) adding up
to the already immense direct impacts such as in case of these examples a sea
level rise of several meters over the next centuries (Schellnhuber et al.,
2016).

We think that this nonlinearity should be reflected at least to some extent in
the resulting responsibility attribution. This constitutes another argument
for deviating from the — linear — cumulative past emissions accounts mentioned
above (Botzen et al., 2008; Müller et al., 2009).

#### Temporal progression.

In contrast to existing formalisations of moral responsibility in game-
theoretic terminology (Braham and van Hees, 2018), we include a temporal
dimension in our representation of multi-agent decisions by making use of
extensive-form game trees rather than normal-form games. This temporal
component is also featured in formalisations using the branching-time frames
of stit-logics.666Note that normal-form game-theoretical models correspond to
a subclass of _stit_ models (Duijf, 2018). Similarly, extensive-form games can
also be represented as a _stit_ -logic (Broersen, 2009), but we don’t pursue
this further here as the additional features that we will include would
complicate a logical representation and this is not currently necessary to
express what we want to.

#### Discounting of the future.

However, we do not take into account the temporal distance of an outcome to
the individual decisions that led to it. Unlike in the ongoing debate in the
environmental economics community regarding the discounting factors to be
employed when considering future damages, with the prominent opposition
between William Nordhaus and Nicholas Stern (Nordhaus, 2007; Stern, 2010) and
its “non-decision” by a large expert panel led by Ken Arrow (Arrow et al.,
2013), our account is not directly affected by any form of discounting. This
is due to the fact that while quantitative measures of welfare depend on
notions of preference (which is arguably skewed towards the present), degrees
of responsibility depend on causation (which does not decrease simply due to
time777The reason that events that are very far away from an outcome, such as
for example someone’s being born versus their committing a crime later on in
their life, tend to not be seen as causes is not due to the temporal distance,
but to the interfering effects. As a parallel example, consider someone
planting a bomb with a 50 year timer on it. This shows that it is not the
temporal distance in itself that diminishes the causal connection, such as it
is the case with preferences.). Still, if the effects of an action do
disappear over time because of the underlying system dynamics (e.g., because
pollutants eventually decay) and if this reduces the probability of causing
harm much later, this fact can be reflected in the decision tree via
probability nodes and the choice of the tree used to represent the situation
in question.

#### Beliefs about other agents’ actions.

As another difference to existing formalisations we do not generally allow for
assumptions regarding the likelihood of another agent’s actions. The reason
for this is that arguing with these probabilities enables evading a justified
responsibility ascription. This can be seen for example in the shooting squad
scenario, where an innocent prisoner is ordered to be shot by a group of
soldiers, and where it is arguably very likely that at least one of the other
soldiers shoots, but each individual shooter’s responsibility should not be
negated. Thus, while beliefs about others’ actions may influence the
psychologically perceived degrees of responsibility of the agents, it should
not influence a normative assessment of their responsibility by an ideal
ethical observer.

Note that even Braham and van Hees (2012), who do consider beliefs about
other’s actions in their model, state that the important feature in a tragedy
of the commons application is that “[b]efore the game was played, each agent
assigned at least some positive probability to the strategy combination the
others actually did play”. That is, an unstructured set of possible outcomes
suffices.

#### Causation.

As mentioned previously, our account of causation differs from those used in
many previous formal representations. Namely, while in most previous accounts
the clear insufficiency of a but-for condition is avoided by employing NESS
(Wright, 1988) or INUS (Mackie, 1965) conditions, these remain binary,
deterministic and based on the concepts of necessity and sufficiency. We
instead use a probabilistic account, where an event is seen as a cause of an
outcome to the degree that it raised its probability, as suggested for example
by Vallentyne (Vallentyne, 2008).

The but-for condition claims that an event is a cause of an effect if the
effect had not occurred, were it not for the event in question. This is at the
same time too strict, failing to capture overdetermination, and too inclusive,
including for example very indirect but necessary events such as someone’s
being born related to an action they perform in their 50’s. NESS is an acronym
for ‘necessary element of a sufficient subset’, that is, an event is seen as a
cause for an outcome if and only if it is a necessary element of a subset of
all the events leading to the outcome, and the subset together is sufficient
for the outcome to occur. For example, consider five people pushing a car down
a cliff, but it would only take three people to move it. Then one single
person pushing is a necessary element of the sufficient subset of any group of
three people (that they are part of) pushing the car. Similarly, INUS is an
acronym for ‘insufficient but necessary element of an unnecessary but
sufficient set of events’. This view captures the intuition that an outcome
might have several sets of events that can explain it, of which either is
sufficient (but neither is necessary), and a cause is any necessary element of
each of these sets888Reducing this notion to one possible set of explanatory
events by removing the ‘insufficient’ and ‘unnecessary’ conditions results in
the NESS condition.

Despite discussions and suggestions for using probabilistic approaches to
represent causation (Fenton-Glynn, 2016; Hitchcock, 2021; Kvart, 2002), these
views have to our knowledge not yet been employed in generalised
formalisations of responsibility. We employ here the account as stated by
Vallentyne (2008) in relation to responsibility: “I shall assume that the
relevant causal connection is that the choice increases the objective chance
that the outcome will occur — where objective chances are understood as
objective probabilities”. This account lends itself to our approach as we
specifically want to discuss situations in which the outcome occurs with a
given probability, and it enables a straightforward representation of degrees
of responsibility. Note however, that unlike Vallentyne (2008) we do not refer
to agents’ beliefs regarding the probabilities.

###  1.4 Structure of the paper

The remainder of the paper is structured as follows. We will begin in Sect. 2
by discussing a set of paradigmatic example scenarios and the results that a
responsibility function should assign in them. In Sect. 3 we continue with a
presentation of the framework we use to represent relevant decision scenarios
in order to formulate responsibility functions. In Sect. 4 we introduce four
different candidate responsibility functions (all differentiated between
backward- and forward-looking formulations) and evaluate them by determining
to what extent they are able to correctly capture the aforementioned
paradigmatic example scenarios. In Sect. 5 we present a reduced real-world
application scenario in which we apply the best-performing responsibility
functions. Finally, we conclude in Sect. 6.

##  2 Paradigmatic examples

In this section we will have a look at a few paradigmatic example scenarios,
mostly known from the literature, together with their proposed evaluation. We
will then use these examples to assess our proposed formal representation by
testing whether our model outputs the same results as the ‘analytic’
evaluation we present here.

###  2.1 Common scenarios and their evaluation

The ‘training set’ examples have been selected because they each represent an
interesting aspect of responsibility attribution that we will want to capture
correctly with the proposed responsibility functions. Additionally, some
scenarios have been adjusted in order to showcase every possible kind of
uncertainty node that we make available in our framework. Let us now describe
the scenarios. Graphical representations can be seen in Fig. 1.

  * •

Load and shoot. Agent $i$ has the choice to either load a gun that they pass
to a second agent, $j$, or to not load it. Agent $j$ then has the choice to
shoot an innocent victim with said gun or not, not knowing whether the gun is
loaded. Represented in Fig. 1(a).

  * •

Rock throwing. Agent $i$ has the choice to either throw a rock into a window,
thus shattering the window, or not. A second agent $j$ then has the same
choice, not knowing whether agent $i$ already threw a stone before them.
Represented in Fig. 1(b).

  * •

Cooling vs. Heating. It is unclear whether humanity (agent $i$) is faced with
an impending ice age or not. They nevertheless have to select whether or not
to heat up the earth through green house gas emissions, which, in case of an
absent ice age, will lead to catastrophic heating. Represented in Fig. 1(c).

  * •

Hesitation. Agent $i$ has the choice to either rescue an innocent stranger
immediately, or hesitate, in which case they might get another chance at
rescuing the innocent stranger at a later stage, but it might also already be
too late.999While this example might seem somewhat odd in direct interaction
contexts — imagine the scenario described above where someone has the choice
to save a person from drowning immediately or first finish off their ice-cream
knowing that with probability $p$ the other person will drown in the meantime
— it represents a common issue in climate change mitigation efforts.
Represented in Fig. 1(d).

We argue that these examples should be analysed as follows. These analyses are
closely related to questions of moral luck (Andre, 1983; Nagel, 1979; Tong,
2004), reasonable beliefs (Baron, 2016), and ignorance as an excuse
(Zimmerman, 2016).

#### Load and shoot.

The examples Load and shoot and Rock throwing are parallel to one another,
both including situations in which agent $j$ might not actually be able to
influence the outcome (because either the gun is not loaded so it does not
matter whether they shoot or not, or because the other agent already threw a
stone that will shatter the window), but they do not know whether they are in
this situation or in the one where their action does have an impact. In both
cases we argue that the responsibility ascription must take into account the
viable option that the agent’s action will have/would have had an impact.
Therefore, the agent cannot dodge responsibility by referring to this
uncertainty. They should be assigned full forward and backward (if they select
the possibly harmful action) responsibility. This relates to the discussion
about moral luck, and the case for disregarding factors that lie outside of
the agent’s control is argued in Nagel (1979): “Where a significant aspect of
what someone does depends on factors beyond [their] control, yet we continue
to treat [them] in that respect as an object of moral judgment, it can be
called moral luck. Such luck can be good or bad. […] If the condition of
control is consistently applied, it threatens to erode most of the moral
assessments we find it natural to make.”

This also relates to a prominent criticism of the probability raising account
for causation, namely that an agent may raise the probability of an event
without this event actually occurring as a result, as the probability stayed
below 1. Similarly to situations in which the event does not end up occurring
due to the actions of others that the agent had no knowledge of or influence
over, we argue that this should not reduce responsibility ascription but
rather lead to what can be interpreted as a form of ‘counterfactual’
responsibility.

#### Rock throwing.

As stated previously, this example is symmetric to the ‘load and shoot’ one.
Again, agent $j$ does not know whether they are in a situation in which their
action can have an effect on the outcome or not. We argue that due to this
lack of knowledge, agent $j$ must assume the possibility that their actions
will have negative consequences. Thus, again, they should be assigned full
forward and backward (if they choose to throw the rock) responsibility. This
analysis also relates to the issue of assigning probabilities to other’s
actions. If $j$ were able to plausibly assume that $i$ did/did not throw the
rock with probability 50/50 or 90/10, then this could be taken into account
for a computation of partial responsibility instead of staying within the
absolute case as we do. However, recall that we exclude such a reasoning, as
argued in Sect. 1.

#### Cooling vs. Heating.

This situation is somewhat non-trivial with respect to responsibility
ascription. In the two cases, if there is an impending ice age and if there is
none, the agent has the choice between the same actions, but the actions have
directly opposing outcomes. Additionally, the agent does not know which
situation they are in. Therefore, no matter what they do, they must always
assume that their action may lead to the undesired outcome, but they also must
always assume that it might be the only option to achieve the desired outcome.
Whether or not we should assign responsibility in all nodes, or in none of
them, depends on whether we want to prevent responsibility voids, or whether
we want to ensure agents always have the option to avoid responsibility.

#### Hesitation.

This scenario helps determine the effect of repeated actions and shows whether
it is possible to redeem a negatively evaluated action. Let us first compare
the backwards-looking responsibility in the two morally desirable outcomes:
$w_{1}$, where the agent decided to immediately save the stranger, and
$w_{2}$, where the agent first hesitated but was lucky to get a second chance
and ended up rescuing the stranger. Clearly, if the agent decides to
immediately rescue the stranger, they should be assigned zero backwards-
looking responsibility. However, as we discussed previously, events which the
agent has no influence over, such as the outcome of a probabilistic event,
should not modify responsibility ascription. Therefore, even if the agent does
in the end save the stranger after an initial hesitation, we argue that they
should be assigned at least some backwards-looking responsibility for risking
the stranger’s death. This differs from other accounts that put more emphasis
on the actualised outcome, such as Braham and Van Hees (2009).101010Note that
according some accounts, such as the one applied in the parable of the lost
son in the New Testament, one might even consider that the actions leading to
$w_{2}$ are morally superior to those leading to $w_{1}$. That is, this
account claims it is more praiseworthy to select a morally reproachable action
and subsequently select that action which still results in the good outcome,
rather than to have immediately selected the morally praiseworthy action.
Similarly, if we look at the undesirable outcome $w_{4}$ where the agent also
initially hesitated but was not lucky and the stranger died, we argue that
they should not carry full responsibility for the stranger’s death, as there
was the possibility they might get another chance at rescuing them. Lastly, in
the situation where the agent repeatedly decided not to help, they should be
assigned full responsibility. This is again quite clear. With respect to
forward-looking responsibility, the situation again requires some thought. The
second decision node does not differ, in a forward-looking way, from any other
node which offers the decision between an action that immediately leads to a
desired outcome and one that immediately leads to an undesired outcome.
Therefore, in this node the agent has full forward-looking responsibility.
Contrastingly, in the first node the agent has the choice between an action
that immediately leads to a good outcome and a probabilistically uncertain
outcome. While it might seem like the agent does not have full influence over
the situation, we argue that indeed they do, as the second decision node
belongs to the same agent, and agent $i$ can therefore ensure both that the
stranger survives, by immediately rescuing them, and that they die, by passing
twice. Thus, we argue that agent $i$ carries full forward-looking
responsibility in this case.

(a)
$i$$j$$v_{1}$$j$$v_{2}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{load}$$\mathtt{pass}$$\mathtt{shoot}$$\mathtt{pass}$$\mathtt{pass}$$\mathtt{shoot}$

(b)
$i$$j$$v_{1}$$j$$v_{2}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{throw}$$\mathtt{pass}$$\mathtt{throw}$$\mathtt{pass}$$\mathtt{pass}$$\mathtt{throw}$

(c) $i$$v_{1}$$i$$v_{2}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{no\
risk}$$\mathtt{cooling}$$\mathtt{pass}$$\mathtt{heat\ up}$$\mathtt{heat\
up}$$\mathtt{pass}$

(d)
$i$$v_{1}$$w_{1}$$i$$v_{2}$$w_{4}$$w_{2}$$w_{3}$$\mathtt{rescue}$$\mathtt{pass}$$1-p$$p\
\ $$\mathtt{rescue}$$\mathtt{pass}$

Figure 1:  Paradigmatic example scenarios. Diamond-shaped nodes labelled with
an agent represent a decision node of this agent, with outgoing arrows
labelled with the available actions. Empty diamonds represent ambiguity nodes.
Empty squares represent probabilistic uncertainty nodes, with the outgoing
edges labelled with the corresponding probability. Circles represent outcomes,
with those shaded in being undesirable. Dotted lines between nodes represent
indistinguishability for the deciding agent. (a) Load and shoot. One agent $i$
may load a gun or not before passing it to another agent $j$ who can shoot an
innocent victim or not. (b) Rock throwing. Two agents have the choice to throw
a rock into a window, thus shattering it. Agent $j$ does not know whether the
other agent, $i$, already threw their stone. (c) Cooling vs. Heating. Agent
$i$ can select whether to induce global warming, not knowing whether there is
a threat of an impending ice age that this would alleviate or not, in which
case the warming leads to a climate crisis. (d) Hesitation. Agent $i$ can
choose to save someone from drowning immediately, or to hesitate. If they
hesitate they might get another chance to save the person from drowning with a
certain probability, but it might also be too late.

##  3 Formal model

In this section we will propose a formal framework for the study of
responsibility in multi-agent settings with various forms of uncertainty. For
reasons of completeness, we spell out the full formal representation. However,
readers unfamiliar with this notation may wish to skip those parts and instead
focus on the interpretation and the graphical representation.

In order to represent decisions of an interacting set of agents in a temporal
progression, the game-theoretical data structure of a game in extensive form
has proven effective. We therefore base our representation on this structure.
However, we need to extend it in order to be able to represent all the
features described above. Specifically, we add special types of nodes for
encoding uncertainty. Also, in contrast to games, our structures do not
specify individual payoffs for all outcomes but only a set of ethically
undesired outcomes. This is sufficient, as we will not apply any game-
theoretic analyses referring to rational courses of actions or utility
maximisation but rather use this data structure to talk about responsibility
assignments.

###  3.1 Framework

We use $\Delta(A)$ to denote the set of all probability distributions on a set
$A$.

#### Trees.

We define a multi-agent decision-tree with ambiguity (or shortly, a tree) to
be a structure
$\mathcal{T}=\langle
I,(V_{i}),V_{a},V_{p},V_{o},E,\sim,(A_{v}),(c_{v}),(p_{v})\rangle$ consisting
of:

  * •

A nonempty finite set $I$ of agents (or players).

  * •

For each agent $i\in I$, a finite set $V_{i}$ of $i$’s decision nodes. At each
decision node, only one agent acts, so all these sets $V_{i}$ are disjoint. We
denote the set of all decision nodes by $V_{d}:=\bigcup_{i\in I}V_{i}$.

  * •

Further disjoint finite sets of nodes: a set $V_{a}$ of ambiguity nodes, a set
$V_{p}$ of probability nodes, and a nonempty set $V_{o}$ of outcome nodes. We
denote the set of all nodes by $V:=V_{d}\cup V_{a}\cup V_{p}\cup V_{o}$.

  * •

A set of directed edges $E\subset V\times V$ so that $(V,E)$ is a directed
tree whose leaves are exactly the outcome nodes:

$V_{o}=\\{v\in V:\not\exists v^{\prime}\in V((v,v^{\prime})\in E)\\}.$

For all $v\in V\setminus V_{o}$, let $S_{v}:=\\{v^{\prime}\in
V:(v,v^{\prime})\in E\\}$ denote the set of possible successor nodes of $v$.

  * •

An information equivalence relation $\sim$ on $V_{d}$ so that $v\in V_{i}$ and
$v^{\prime}\sim v$ implies $v^{\prime}\in V_{i}$. We call the equivalence
classes of $\sim$ in $V_{i}$ the information sets of $i$.

  * •

For each agent $i\in I$ and decision node $v\in V_{i}$, a nonempty finite set
$A_{v}$ of $i$’s possible actions in $v$, so that $A_{v}=A_{v^{\prime}}$
whenever $v\sim v^{\prime}$, and a bijective consequence function mapping
actions to successor nodes, $c_{v}:A_{v}\to S_{v}$.

  * •

For each probability node $v\in V_{p}$, a probability distribution
$p_{v}\in\Delta(S_{v})$ on the set of possible successor nodes.

Our interpretation of these ingredients is the following:

  * •

A tree encodes a multi-agent decision situation where agents can make choices
in a given order. An outcome node $v\in V_{o}$ represents an ethically
relevant state of affairs that may result from these choices.

  * •

Each decision node $v\in V_{i}$ represents a point in time where agent $i$ has
the agency to make a decision at free will. The elements of $A_{v}$ are the
mutually exclusive choices $i$ can make, including any form of “doing
nothing”, and $c_{v}(a)$ encodes the immediate consequences of choosing $a$ in
$v$. Often, $c_{v}(a)$ will be an ambiguity or probability node to encode
uncertain consequences of actions.

  * •

Probability and ambiguity nodes as well as information equivalence are used to
represent various types of uncertainty. The agents are assumed to always
commonly know the full tree. For any probability node $v\in V_{p}$, they know
that the possible successor nodes are given by $S_{v}$ and have probabilities
$p_{v}(v^{\prime})$ for $v^{\prime}\in S_{v}$. Note that these probabilities
are universal, rather than individual. About an ambiguity node $v\in V_{a}$
they only know that the possible successor nodes are given by $S_{v}$, without
being able to rightfully attach probabilities to them. Ambiguity nodes can
also be thought of as decision nodes associated to a special agent called
‘nature’.

In contrast to the universal uncertainty at the tree-level encoded by
probability and ambiguity nodes, information-equivalence is used to encode
uncertainty at the agent level. An agent $i$ cannot distinguish between their
information-equivalent decision nodes $v\sim v^{\prime}$, with
$v,v^{\prime}\in V_{i}$. Information-equivalent nodes $v\sim v^{\prime}$ have
the same set of possible actions $A_{v}=A_{v}^{\prime}$, as otherwise the
agent would be able to distinguish between them via the available actions.

An ambiguity node whose successors are probability nodes can be used to encode
uncertain probabilities like those reported by the IPCC (Mastrandrea et al.,
2011) or those corresponding to the assumption that “nature” uses an Ellsberg
strategy (Decerf and Riedel, 2019).

Note that in contrast to some other frameworks, e.g., those using normal-form
(instead of extensive-form) game forms such as (Braham and van Hees, 2018),
our trees do not directly allow for two agents to act at the exact same time
point. Indeed, in a real world in which time is continuous, one action will
almost certainly precede another, if only by a minimal time interval. Still,
two actions may be considered “simultaneous” if they occur so close in time
that the later acting player cannot know what the earlier action was. This
ignorance can easily be encoded by means of information equivalence in a way
similar to Fig. 1 (a) – (c).

As with any modelling exercise, the result of the representation depends
strongly on the point of view of the modeller and the corresponding selection
of what to include in the representation. If the modeler follows the basic
idea that what matters for the assessment of an agent $i$’s responsibility is
what $i$ “reasonably believes” in any decision node $v_{d}$ (as in Baron
(2016)), then the available actions, consequences thereof and probability
distributions should be represented accordingly. If, on the other hand, the
modeler follows the idea that some “objective reality” is the basis for
evaluation, even if it might have been unknown to the agent in question (as
for the strict liability described by Broersen (2011) or the causal
responsibility concept mentioned in Vallentyne (2011)) then this, too, will be
reflected in the available actions and consequences included in the model.

#### Events, groups, responsibility functions.

As in probability theory, we call each subset of outcomes a possible event. In
the remainder of this paper, we will use $\varepsilon$, with
$\varepsilon\subseteq V_{o}$ to represent an ethically undesirable event,
those events not in $\varepsilon$ are considered ethically desirable. Any
nonempty set $G\subseteq I$ of agents is called a group in this
article.111111Note that we deliberately do not require that a set of agents
shares any identity or possesses ways of communication or coordination for an
ethical observer to meaningfully attribute responsibility to this “group”.

Our main objects of interest are quantitative metrics of degrees of
responsibility that we formalise as backward-looking responsibility functions
$\mathcal{R}_{b}$ and forward-looking responsibility functions
$\mathcal{R}_{f}$.

A backward-looking responsibility function maps a group $G$, an outcome node
$v\in V_{o}$, the decision situation $\mathcal{T}$ and the undesirable outcome
$\varepsilon$ to a real number $\mathcal{R}_{b}(\mathcal{T},v,G,\varepsilon)$.
It is meant to represent some form of degree of backward-looking
responsibility of $G$ regarding $\varepsilon$ in the multi-agent decision
situation encoded by $\mathcal{T}$ when outcome $v$ has occurred.

A forward-looking responsibility function maps every combination of tree
$\mathcal{T}$, group $G$, event $\varepsilon$, and decision node $v\in V_{d}$
to a real number $\mathcal{R}_{f}(\mathcal{T},v,G,\varepsilon)$ meant to
represent some form of degree of forward-looking responsibility of $G$
regarding $\varepsilon$ in the multi-agent decision situation encoded by
$\mathcal{T}$ when in decision node $v$.

When we are concerned with only one agent, i.e. $G=\\{i\\}$, we also write
$\mathcal{R}_{b/f}(\mathcal{T},v,i,\varepsilon)$. Whenever any of the
arguments $\mathcal{T}$, $v$, $G$, $\varepsilon$ are kept fixed and are thus
obvious from the context, we omit to explicate them when writing
$\mathcal{R}_{b/f}$ or any of the auxiliary functions defined below.

#### Graphical representation.

As exemplified in Fig. 1, we can represent a tree $\mathcal{T}$ and event
$\varepsilon$ graphically as follows. Edges are shown as arrows, decision
nodes are diamonds labelled by agents, with outgoing arrows labelled by
actions, ambiguity nodes are unlabelled diamonds, probability nodes are
squares with outgoing arrows labelled by probabilities, and outcome nodes are
circles, filled in grey if the outcome belongs to $\varepsilon$. Finally,
information equivalence is indicated by dashed lines connecting or surrounding
the equivalent nodes.

#### Auxiliary notation.

The set of decision nodes of a group $G\subseteq I$ is the union of the
decision nodes of its members, $V_{G}:=\bigcup_{i\in G}V_{i}$. To ease the
definition of “scenario” below we denote the set of nodes that are not in
$V_{G}$ and are not probabilistic uncertainty nodes (i.e., all non-$G$
decision nodes and all ambiguity nodes) by $V_{-G}:=V_{d}\setminus(V_{G}\cup
V_{a})$. This can be thought of as nodes where someone who is not part of
group G – another agent or Nature – makes a decision.

The history of a node $v\in V$ are all those nodes that lie on the path from
the start node to this node. If $v^{\prime}\in S_{v}$, we call
$P(v^{\prime}):=v$ the predecessor of $v^{\prime}$. Let $v_{0}\in V$ be the
root node of $(V,E)$, i.e., the only node without predecessor. The history of
$v\in V$ is then $H(v):=\\{v,P(v),P(P((v)),\dots,v_{0}\\}$. In the other
direction, we call the nodes accessible after $v$ $B(v):=\\{v^{\prime}\in
V:v\in H(v^{\prime})\\}$ the (forward) branch of $v$. Note that there is only
one past but possibly a branching future. Taking into account information
equivalence, we also define the information branch of $v$
$B^{\sim}(v):=\bigcup_{v^{\prime}\sim v}B(v^{\prime})$, as all those nodes
that, for all the player deciding in $v$ knows, may be reachable from their
current position.

For a node $v\in V$, a decision node in its history $v^{\prime}\in H(v)\cap
V_{d}$ and that action $a$ available in $v^{\prime}$ which takes us on the
path to $v$, i.e. $c_{v^{\prime}}(a)\in H(v)$, we introduce the notation
$a_{v^{\prime}\to v}$.

A decision node $v\in V_{d}$ where the agent has no information uncertainty
$\\{v^{\prime}:v^{\prime}\sim v\\}=\\{v\\}$ is called a complete information
node.

From now on, unless stated otherwise, we will use $w$ to refer to outcome
nodes $w\in V_{o}$ and $v$ to refer to non-outcome nodes $v\in V\setminus
V_{o}$. Mostly, $v$ will be used to designate decision nodes $v\in V_{d}$.

#### Strategies, scenarios, likelihoods.

In order to be able to quantify responsibility ascription, we will need to be
able to range over all the ways that the uncertain evolution of the future
plays out. To this end, we will use the notions of _strategy_ and _scenario_ ,
with a strategy telling us what agents within a given group will be doing, and
a scenario resolving all other unquantified uncertainty by telling us what
agents outside of the group (including ‘nature’) will be doing as well as
revealing the current node within an information set. We do not need to
resolve quantified probabilistic uncertainty, as we can use _likelihoods_ in
the computation of responsibility.

Formally, given a tree $\mathcal{T}$, a group $G\subseteq I$, and a node $v\in
V$, a _strategy_ for $G$ at $v$ is a function

$\sigma\colon V_{G}\cap B^{\sim}(v)\to\bigcup_{v^{\prime}\in V_{G}\cap
B^{\sim}(v)}A_{v^{\prime}}$

that chooses an action for every one of $G$’s future decision nodes considered
possible according to the information set in $v$. Let
$\Sigma(\mathcal{T},G,v)$ (or shortly $\Sigma(v)$ if $\mathcal{T},G$ are
fixed) be the set of all those strategies. For
$\sigma\in\Sigma(\mathcal{T},G,v)$, let

$V_{o}^{\sigma}:=\\{w\in B^{\sim}(v)\cap V_{o}:a_{v^{\prime}\to
w}=\sigma(v^{\prime})\text{~{}for all~{}}v^{\prime}\in B^{\sim}(v)\cap
H(w)\cap V_{G}\\},$

i.e., the set of possible outcomes when group $G$ follows strategy $\sigma$
from node $v$ on.

Complementarily, a _scenario_ for $G$ at $v$ is a node $v_{\zeta}\sim v$
together with a function

$\zeta\colon V_{-G}\cap B(v_{\zeta})\to\bigcup_{v^{\prime}\in V_{-G}\cap
B(v_{\zeta})}S_{v^{\prime}}$

such that $\zeta(v^{\prime})=c_{v^{\prime}}(a)$ and
$\zeta(v^{\prime\prime})=c_{v^{\prime\prime}}(a)$ for some $a\in
A_{v^{\prime}}$ whenever $v^{\prime}\sim v^{\prime\prime}\in V_{-G}\cap
B(v_{\zeta})$.

The node $v_{\zeta}$ resolves uncertainty regarding the current information
set, while the function $\zeta$ resolves non-probabilistic uncertainty
regarding what will happen in the future (disregarding $G$’s own actions), by
choosing successor nodes $\zeta(v^{\prime})$ for all future decision nodes by
non-members of G or ambiguity nodes. The condition on information-equivalent
nodes states that a scenario cannot differentiate between them and select two
different actions in each node but must select the same one.
Let $Z^{\sim}(\mathcal{T},v,G)$ (or shortly $Z^{\sim}(v)$) be the set of all
scenarios at $v$ and $Z(\mathcal{T},v,G)\subseteq Z^{\sim}(\mathcal{T},v,G)$
(or shortly $Z(v)$) that of all scenarios at $v$ with $v_{\zeta}=v$.

Given a starting node $v$, we can define a probability distribution
$\pi_{v}\in\Delta(V_{o}\cap B^{\sim}(v))$ on the set of possible outcome nodes
in the following straightforward way:

$\displaystyle\psi(v)$ $\displaystyle=1,$ (1)
$\displaystyle\psi(v^{\prime\prime})$
$\displaystyle=\psi(v^{\prime})\quad\text{for~{}}v^{\prime}\in V_{d}\text{ or
}v^{\prime}\in V_{a},\ v^{\prime\prime}\in S_{v^{\prime}},$ (2)
$\displaystyle\psi(v^{\prime\prime})$
$\displaystyle=\psi(v^{\prime})p_{v^{\prime}}(v^{\prime\prime})\quad\text{for~{}}v^{\prime}\in
V_{p},v^{\prime\prime}\in S_{v^{\prime}},$ (3) $\displaystyle\pi_{v}(w)$
$\displaystyle=\psi(w)\quad\text{for all~{}}w\in V_{o}\cap B^{\sim}(v).$ (4)

Let us denote the resulting likelihood of $\varepsilon$ given the starting
node $v$ by

$\ell(\varepsilon|v):=\sum_{w\in\varepsilon}\pi_{v}(w).$

Consequently, to define the likelihood of $\varepsilon$ given a starting node
$v$ together with scenario $\zeta$ and strategy $\sigma$, we will in each step
consider only those successor nodes that can be reached when following
$\sigma$ or $\zeta$, and assign zero likelihood to all other nodes:

$\displaystyle\chi(v_{\zeta})$ $\displaystyle=1,$ (5)
$\displaystyle\chi(v^{\prime\prime})$
$\displaystyle=\chi(v^{\prime})\quad\text{for~{}}[v^{\prime}\in V_{G}\text{
and }v^{\prime\prime}=c_{v^{\prime}}(\sigma(v^{\prime}))]\text{ or
}[v^{\prime}\in V_{-G}\text{ and }v^{\prime\prime}=\zeta(v^{\prime})],$ (6)
$\displaystyle\chi(v^{\prime\prime})$
$\displaystyle=\chi(v^{\prime})p_{v^{\prime}}(v^{\prime\prime})\quad\text{for~{}}v^{\prime}\in
V_{p},\ v^{\prime\prime}\in S_{v^{\prime}},$ (7)
$\displaystyle\chi(v^{\prime\prime})$ $\displaystyle=0\quad\text{for all
other~{}}v^{\prime\prime}\in B^{\sim}(v),$ (8)
$\displaystyle\pi_{v,\sigma,\zeta}(v_{o})$
$\displaystyle=\chi(w)\quad\text{for all~{}}w\in V_{o}\cap B^{\sim}(v).$ (9)

Recall that $v_{\zeta}$ is that node in $v$’s information set which according
to scenario $\zeta$ is the actual one. Only this node, and not $v$ itself, is
relevant in the computation of $\chi$.

Again, we denote the resulting likelihood of $\varepsilon$ given starting node
$v$, strategy $\sigma$ and scenario $\zeta$ by

$\ell(\varepsilon|v,\sigma,\zeta):=\sum_{w\in\varepsilon}\pi_{v,\sigma,\zeta}(w).$

##  4 Candidate responsibility functions

In this section we will introduce four pairs of responsibility functions
$(\mathcal{R}_{f},\mathcal{R}_{b})$, that measure degrees of responsibility in
terms of differences in likelihoods of the undesired outcome, together with a
reference function $\mathcal{R}_{b}^{0}$ related to strict causation. We will
examine whether the proposed functions output the right results for the
paradigmatic example scenarios described and analysed in section 2, and see
that we need to progressively adjust them in order to achieve the desired
outcomes.

In general, we will mostly view backwards responsibility as some measure
related to ‘having influenced the outcome towards the undesired one’, for
example by increasing likelihood or risk-taking. Correspondingly, we view
forward-looking responsibility as a measure related to influence. Thus, the
forward responsibility in a certain node will often be the maximal possible
value of backwards responsibility arising from actions taken in that node.

To simplify the notation we will use some conventions in the remainder of this
chapter. Let $\mathcal{T}$, $G$, and $\varepsilon$ be given; we will drop them
from notation. With respect to the various different nodes appearing in the
definitions, we stick with the previously mentioned convention and denote the
decision node at which $\mathcal{R}_{f}$ is evaluated by $v\in V_{d}$, the
outcome node at which $\mathcal{R}_{b}$ is evaluated by $w\in V_{o}$, and
other nodes also by $v\in V$ or $v^{\prime},v^{\prime\prime}\in V$ so that $v$
comes before $v^{\prime}$ comes before $v^{\prime\prime}$ (i.e., $v\in
H(v^{\prime})$, $v^{\prime}\in B(v)$ etc.).

#### Benchmark variant: strict causation.

The most straightforward definition of a forward responsibility function in
our framework that resembles the strict causation view, as employed for
example in the most basic way of (being able to) ‘see to it that’ is to set

$\mathcal{R}^{0}_{f}(v):=\begin{cases}1\text{ iff }B(v)\cap
V_{o}\not\subseteq\varepsilon\text{ and }B(c_{v}(a))\cap
V_{o}\subseteq\varepsilon\text{ for some }a\in A_{v},\\\ 0\text{
else}\end{cases}$

that is, if at decision node $v_{d}$, the undesired outcome is not certain,
but there is an action the agent can select which makes it certain. The
corresponding backward responsibility function would be to assign full
responsibility to G if at any point they toggled the undesired outcome from
uncertain to happening.

$\mathcal{R}^{0}_{b}(w):=\begin{cases}1\text{ iff }B(v)\cap
V_{o}\not\subseteq\varepsilon\text{ and }B(v^{\prime})\cap
V_{o}\subseteq\varepsilon\text{ for some }v\in H(v_{o})\cap
V_{G},v^{\prime}=c_{v}(a_{v\to w}),\\\ 0\text{ else}\end{cases}$

It is easy to see that given $w$, there is at most one such $v$ regardless of
$G$, and exactly those $G$ are deemed responsible which contain the agent
choosing at $v$, i.e., for which $v\in V_{G}$.

Clearly, this account does not take any form of uncertainty into account, and
therefore gets neither the ‘load and shoot’ nor the ‘rock throwing’ examples
correctly. In those nodes where agent $j$ does not actually have an influence,
but they do not know this for sure we argue that they should assume
responsibility while the strict causation account assigns zero. This is one of
the motivations why we argue that a probabilistic account of causation taking
into account uncertainty is more valuable.

###  4.1 Variant 1: increasing guaranteed likelihood.

In this variant we begin by taking uncertainty into account, thus eliminating
the very first criticism we had for the benchmark account. This reasoning
serves as the very motivation for why we chose to include uncertainty in our
model and why we employ a probabilistic account of causation. The rationale
for this variant is to translate the basic idea of the stit approach into a
probabilistic context, by sticking to a ‘certain direct influence’ view of
responsibility. That is, backward responsibility can be seen as arising from
having caused an increase in the guaranteed likelihood of an undesired
outcome. This is also in line with the conception of Vallentyne (2008).

#### Guaranteed likelihood, caused increase, backward responsibility.

We measure the known guaranteed likelihood of $\varepsilon$ at some node $v\in
V$ by the quantity

$\displaystyle\gamma(v)$
$\displaystyle:=\min_{\sigma\in\Sigma(v)}\min_{\zeta\in
Z^{\sim}(v)}\ell(\varepsilon|v,\sigma,\zeta).$ (10)

We measure the caused increase in known guaranteed likelihood in choosing
$a\in A_{v_{d}}$ at decision node $v\in V_{d}$ by the difference

$\displaystyle\Delta\gamma(v,a)$ $\displaystyle:=\gamma(c_{v}(a))-\gamma(v).$
(11)

Note that by definition of $\gamma$, $\Delta\gamma(v,a)\geqslant 0$ for all
$a\in A_{v}$.

To measure $G$’s backward responsibility regarding $\varepsilon$ in outcome
node $w\in V_{o}$, in this variant we take their aggregate caused increases
over all decisions $a_{v\to w}$ for $v\in V_{G}$.

$\displaystyle\mathcal{R}_{b}^{1}(w)$ $\displaystyle:=\sum_{v\in H(w)\cap
V_{G}}\Delta\gamma(v,a_{v\to w}).$ (12)

#### Maximum caused increase, forward responsibility.

Finally, to measure $G$’s forward responsibility regarding $\varepsilon$ in
decision node $v\in V_{G}$, we take the maximal possible caused increase,

$\displaystyle\mathcal{R}_{f}^{1}(v)$ $\displaystyle:=\max_{a\in
A_{v}}\Delta\gamma(v,a).$ (13)

#### Application to paradigmatic examples.

This variant of the responsibility functions fails for the ‘load and shoot’
example. The example is repeated in Fig. 2. In node $v_{2}$, where the gun was
not loaded by agent $i$, agent $j$ cannot actually worsen the worst-case
likelihood of the undesired outcome $\varepsilon$, as it always stays zero.
However, as we argued previously, agent $j$ should nevertheless be assigned
responsibility here, as they cannot be sure that their action will not have a
negative influence.

This shows the two main drawbacks of this account: it does not sufficiently
take information equivalence into account and it is in a sense too
“optimistic” by allowing agents to assume the best case. The next variant
tries to resolve these issues.

(a)
$i$$j$$v_{1}$$j$$v_{2}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{load}$$\mathtt{pass}$$\mathtt{shoot}$$\mathtt{pass}$$\mathtt{pass}$$\boldsymbol{\mathtt{shoot}}$

(b)
$i$$j$$v_{1}$$j$$v_{2}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{throw}$$\mathtt{pass}$$\mathtt{pass}$$\mathtt{pass}$$\mathtt{throw}$$\boldsymbol{\mathtt{throw}}$

Figure 2:  Repetition of the paradigmatic example scenarios. Highlighted
choices are those for which the influence on responsibility ascription is
difficult to capture correctly. (a) Load and shoot.
$\mathcal{R}^{1}_{b}(w_{3})=0$, while we argue that it should be non-zero, as
agent $j$ must consider the possibility that their action has negative
consequences. (b) Rock throwing. Similarly, $\mathcal{R}^{2}_{b}(w_{1})=0$,
while we argue that it should be non-zero.

###  4.2 Variant 2: increasing worst-case likelihood

The previous variant, as we discussed, was in a sense ‘too optimistic’ by
assigning responsibility only to the degree that the agent did not minimize
_guaranteed_ likelihood of the undesired outcome. As any ambiguity or
knowledge uncertainty with a possible good outcome immediately sets guaranteed
likelihood to zero, this can be seen as a focus on the ‘best-case’ view. Now
we want to take the opposite view by looking at the worst case instead. That
is, in this variant, we assign responsibility to the degree that the agent did
not minimize worst-case likelihood of the undesired outcome. In defining the
worst case, however, we assume a group $G$ can plan and commit to optimal
future behaviour, so that we will look at strategies $\sigma$ rather than
actions $a$.

#### Worst case and minimax likelihoods.

We define $G$’s worst-case likelihood of $\varepsilon$ at any node $v\in V$
given some strategy $\sigma\in\Sigma(v)$ by

$\displaystyle\lambda(v,\sigma)$ $\displaystyle:=\max_{\zeta\in
Z^{\sim}(v)}\ell(\varepsilon|v,\sigma,\zeta).$ (14)

$G$’s minimax likelihood regarding $\varepsilon$ at $v$ is taken to be the
smallest achievable worst-case likelihood,

$\displaystyle\mu(v)$
$\displaystyle:=\min_{\sigma\in\Sigma(v)}\lambda(v,\sigma)=\min_{\sigma\in\Sigma(v)}\max_{\zeta\in
Z^{\sim}(v)}\ell(\varepsilon|v,\sigma,\zeta).$ (15)

#### Worst-case increase, backward responsibility.

We measure $G$’s worst-case increase in minimax likelihood in choosing $a\in
A_{v}$ at node $v\in V_{G}$ by taking the difference in minimax likelihood
before and after $a$. Again we assume the worst case, by taking the highest
value over all nodes in $v$’s information set.121212Note that while we did
already range over all nodes in the information set in the definition of
worst-case likelihood, we need to include them again here. We are now
comparing the states after an action with the states before, which can
evidently differ depending on the decision node in which the action was
effected.

$\displaystyle\Delta\mu(v,a)$ $\displaystyle:=\max_{v^{\prime}\sim
v}[\mu(c_{v^{\prime}}(a))-\mu(v^{\prime})]\geqslant 0$ (16)

Similarly to before, to measure $G$’s backward responsibility for
$\varepsilon$ in node $w$, we take the aggregate of the worst-case increases
in minimax likelihood.

$\displaystyle\mathcal{R}_{b}^{2}(w)$ $\displaystyle:=\sum_{v\in H(w)\cap
V_{G}}\Delta\mu(v,a_{v\to w})$ (17)

#### Maximum caused increase, forward responsibility.

In analogy to variant 1, to measure $G$’s forward responsibility regarding
$\varepsilon$ in node $v_{d}\in V_{G}$, we take the maximal possible worst-
case increase in minimax likelihood,

$\displaystyle\mathcal{R}_{f}^{2}(v)$ $\displaystyle:=\max_{a\in
A_{v}}\Delta\mu(v,a).$ (18)

#### Application to paradigmatic examples, discussion.

This variant correctly captures the intuition that we should assign
responsibility to agent $j$ in the ‘load and shoot’ scenario if they do shoot,
even if the gun was not loaded. However, it fails for the ‘rock throwing’
example. The worst-case likelihood of the undesirable outcome when the second
agent $j$ has to decide is always one, as there is a scenario in which agent
$i$ threw the rock and the undesired outcome will certainly happen. Thus, so
is the minimax likelihood and no further increase is possible.

This variant is in a sense the opposite of variant 1 with respect to its
ambiguity attitude. To understand their relationship, consider the situation
depicted in Fig. 3. Variant 1 is ambiguity-affine by favoring ambiguity over
risk, while variant 2 is ambiguity-averse, by favoring risk over ambiguity.

In a way, both the previous variant and this one do not sufficiently embrace
the complexity arising from uncertainty, but try to reduce the situation to a
simple comparison with a best- or worst-case scenario. In the next iteration
we aim at improving this.

$i$$v_{1}$$v_{2}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{a}$$\mathtt{b}$$p$$1-p$
Figure 3:  Situation related to ambiguity aversion in which the
complementarity of variants 1 and 2 of our responsibility functions can be
seen. The agent must choose between an ambiguous course $\mathtt{a}$ and a
risky course $\mathtt{b}$. The ambiguous course leads to a lower
responsibility ascription in variant 1 since it does not increase the
guaranteed likelihood of a bad outcome. Contrastingly, in variant 2 it is the
risky course that leads to a lower responsibility ascription, since this
action reduces the minimax likelihood of a bad outcome from 1 to $p$.

###  4.3 Variant 3: measuring responsibility in terms of influence and risk-
taking

While variants 1 and 2 can be interpreted as measuring the deviation from a
single optimal strategy that minimizes either the guaranteed (best-case) or
the worst-case likelihood of the undesired outcome, our next variant is based
on families of scenario-dependent optimal strategies. That is, rather than
first reducing to a single scenario and evaluation the choice within this, we
evaluate possible strategies in all available scenarios. In this way, this
variant partially manages to avoid being too optimistic or too pessimistic and
thereby captures both the ‘load and shoot’ as well as the ‘rock throwing’
examples correctly. The main idea is that backward responsibility arises from
taking risks to not avoid an undesirable outcome.

In addition to this new approach to backward responsibility ascription, we
also change the way that we treat forward responsibility. In the previous
variants, we computed forward responsibility in a certain node as the maximal
contribution to backward responsibility that the agent may effect through
actions in this node. However, as van de Poel (2011) pointed out, the relation
between these two measures of responsibility need not be so straightforward.
Therefore, we now employ a separate, independent measure.

#### Optimum, shortfall, risk, backward responsibility.

Given a node $v\in V$ and a scenario at $v$, $\zeta\in Z^{\sim}(v)$, we call
the optimum $G$ could achieve for avoiding $\varepsilon$ at that node in that
scenario the minimum likelihood for $\varepsilon$ over $G$’s strategies at
$v$,

$\displaystyle\omega(v,\zeta)$
$\displaystyle:=\min_{\sigma\in\Sigma(v)}\ell(\varepsilon|v,\sigma,\zeta).$
(19)

So let us measure $G$’s hypothetical shortfall in avoiding $\varepsilon$ in
scenario $\zeta$ due to their choice $a\in A_{v}$ at node $v\in V_{G}$ by the
difference in optima

$\displaystyle\Delta\omega(v,\zeta,a)$
$\displaystyle:=\omega(c_{{v_{\zeta}}}(a),\zeta)-\omega(v,\zeta)\geqslant 0.$
(20)

Then let the risk taken by $G$ in choosing $a$ be the maximum shortfall over
all scenarios at $v$,

$\displaystyle\varrho(v,a)$ $\displaystyle:=\max_{\zeta\in
Z^{\sim}(v)}\Delta\omega(v,\zeta,a).$ (21)

That is, rather than first reducing the situation to a simple best or worst-
case scenario and then comparing strategies within this reduced setting, in
this definition we look at strategies for all scenarios.

To measure $G$’s backward responsibility regarding $\varepsilon$ in node $w\in
V_{o}$, we again take their aggregate risk taken over all choices they made,

$\displaystyle\mathcal{R}_{b}^{3}(w)$ $\displaystyle:=\sum_{v\in H(w)\cap
V_{G}}\varrho(v,a_{v\to w}).$ (22)

#### Influence, forward responsibility.

Regarding forward responsibility, we test a different approach than before,
which is simpler but less strongly linked to backward responsibility. We now
consider forward responsibility to be strongly related to the concept of
influence. The rationale is that since $G$ does not know which scenario
applies, they must take into account that their actual influence on the
likelihood of $\varepsilon$ might be as large as the maximum of this over all
possible scenarios, so the larger this value is, the more careful $G$ need to
make their choices.

Let us measure $G$’s influence regarding $\varepsilon$ in scenario $\zeta$ at
any node $v\in V$ by the range of likelihoods spanned by $G$’s strategies at
$v$,

$\displaystyle\Delta\ell(v,\zeta)$
$\displaystyle:=\max\\{\ell(\varepsilon|v,\sigma_{1},\zeta)-\ell(\varepsilon|v,\sigma_{2},\zeta):\sigma_{1},\sigma_{2}\in\Sigma(v)\\}$
(23)

To measure $G$’s forward responsibility regarding $\varepsilon$ at node $v\in
V_{G}$, we this time simply take their maximum influence over all scenarios at
$v$,

$\displaystyle\mathcal{R}_{f}^{3}(v)$ $\displaystyle:=\max_{\zeta\in
Z^{\sim}(v)}\Delta\ell(v,\zeta).$ (24)

#### Application to paradigmatic examples, discussion.

With this variant of responsibility functions we manage to capture both the
‘load and shoot’ as well as the ‘rock throwing’ examples correctly.

Let us go through the analysis for the ‘load and shoot’ example in detail.
Recall that we are specifically interested in that path in which agent $i$ did
not load the gun and agent $j$ decided to shoot. When agent $j$ has to decide,
there are two possible scenarios: the one where agent $i$ did load the gun,
let us call this $\zeta_{\mathtt{load}}$, and the one where they did not,
$\zeta_{\mathtt{pass}}$. There are also two strategies, shooting
$\sigma_{\mathtt{shoot}}$ and not shooting $\sigma_{\mathtt{pass}}$. Then

$\omega(v_{2},\zeta_{\mathtt{load}})=\min_{\sigma\in\Sigma(v_{2})}\ell(\varepsilon|v_{2},\sigma,\zeta_{\mathtt{load}})=\ell(\varepsilon|v_{2},\sigma_{\mathtt{pass}},\zeta_{\mathtt{load}})=0$
$\Delta\omega(v_{2},\zeta_{\mathtt{load}},\mathtt{shoot})=\omega(c_{v_{1}}(\mathtt{shoot}),\zeta_{\mathtt{load}})-\omega(v_{2},\zeta_{\mathtt{load}})=\omega(w_{1},\zeta_{\mathtt{load}})-0=1$

Thus the risk $\varrho(v_{2},\mathtt{shoot})=1$, and the same holds for the
backward responsibility

$\mathcal{R}^{3}_{b}(w_{3},\\{j\\})=\sum_{v\in H(w_{3})\cap
V_{j}}\varrho(v,a_{v\to w_{3}})=\varrho(v_{2},\mathtt{shoot})=1$

Note that for the computation of the shortfall the scenario
$\zeta_{\mathtt{load}}$ is more important than the actual node $v_{2}$. This
is due to the fact that the agent does not know which node they are actually
in, so placing emphasis on this node in the reasoning would assume a knowledge
the agent does not have.

The analysis for the ‘rock throwing’ example is parallel to this one and left
as an easy exercise to the reader.

From the paradigmatic examples discussed in section 2, we have thus far left
out the ‘cooling vs. heating’ and ‘hesitation’ examples. These are shown again
in Fig. 4. Consider now the ‘cooling vs. heating’ example. The current variant
of responsibility functions will assign full backwards- and forwards
responsibility in all situations. Backwards responsibility arises from the
fact that is impossible for the agents to avoid taking risks. Forward
responsibility arises from the realisation that nevertheless, there are
scenario-strategy combinations that do lead to the desired outcome. With the
next variant we aim at preventing situations like these, where it is
impossible for agents to avoid responsibility ascription.

Just like the two variants before, the current variant correctly represents
that in the ‘hesitation’ example we do want to assign some backwards
responsibility to agent $i$ if they initially hesitated, even if they do in
the end rescue the stranger. All three accounts assign responsibility of $p$
in this case. The same holds for outcome node $w_{4}$ where the agent
hesitated and was unlucky. However, unlike the previous two variants, the
current variant also correctly captures that agent $i$ should be assigned full
forward responsibility in the initial node $v_{1}$, as they can ensure any
outcome if we take into account the full tree.

(a) $i$$v_{1}$$i$$v_{2}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{no\
risk}$$\mathtt{cooling}$$\boldsymbol{\mathtt{heat\ up\
}}$$\boldsymbol{\mathtt{pass}}$$\boldsymbol{\mathtt{pass}}$$\boldsymbol{\mathtt{heat\
up\ }}$

(b) $i$$v_{1}$$w_{1}$$i$$v_{2}$$w_{4}$$w_{2}$$w_{3}$$\mathtt{rescue}$$1-p$$p\
\ $$\mathtt{pass}$$\boldsymbol{\mathtt{pass\ }}$$\boldsymbol{\mathtt{rescue}}$

Figure 4:  Repetition of the paradigmatic example scenarios. Highlighted
choices are those for which the influence on responsibility ascription is
difficult to capture correctly. (a) Cooling vs. Heating. With variant 3 of our
proposed responsibility functions, agent $j$ will always be assigned
responsibility, even if they do select the action that finally leads to the
desired outcome (marked by thick dashed arrows). Reversely, with variant 4
agent $j$ will never be assigned responsibility, even if they selec the action
that finally leads to the undesired outcome (marked by thick solid arrows).
(b) Hesitation. Using variants 1 and 2, the agents will be assigned partial
forward responsibility in node $v_{1}$ and no backward responsibility in node
$w_{2}$. Variants 3 and 4 capture this correctly by assigning full forward
responsibility, and partial backwards responsibility.

###  4.4 Variant 4: measuring responsibility in terms of negligence and
action-prescribing influence

In our final variant, we turn the strict assessments of variant 3 in the
‘cooling vs. heating’ coordination example into a more lenient one. We achieve
this by using the actions with the minimum risk as a baseline reference and
only ‘punish’ risk-taking exceeding this baseline. The downside of this is
that we are consequently faced with responsibility voids. We do not see how
any reasonable responsibility ascribing function could be designed that leads
neither to voids nor to unavoidable responsibility in coordination examples,
so we believe that one has to make a choice between the two.

We also stick with the idea presented in the previous variant of decoupling
forwards from backwards responsibility. The issue that we saw previously with
the forward responsibility ascription is that agent $i$ was assigned full
forward responsibility in the ‘cooling vs. heating’ example, due to the fact
that in every scenario their actions can fully influence the outcome. However,
this does not give the agent any information on which strategy to select in
order to achieve a certain outcome. They cannot willfully ensure a certain
outcome. We aim at reflecting this intuition in our definition.

#### Risk-minimizing action, negligence, backward responsibility.

We define the minimal risk for $G$ in decision node $v\in V_{G}$ as

$\displaystyle\underline{\varrho}(v)$ $\displaystyle:=\min_{a\in
A_{v}}\varrho(v,a).$ (25)

We now suggest to measure $G$’s degree of negligence in choosing $a\in A_{v}$
at $v$ by the excess risk with respect to the minimum possible risk,

$\displaystyle\Delta\varrho(v,a)$
$\displaystyle:=\varrho(v,a)-\underline{\varrho}(v).$ (26)

We see that this variant is still sensitive to all scenarios (like variant 3)
rather than just the best-case or the worst-case.

Now, to measure $G$’s backward responsibility regarding $\varepsilon$ in node
$w\in V_{o}$, we suggest again to take their aggregate negligence over all
choices taken,

$\displaystyle\mathcal{R}_{b}^{4}(w)$ $\displaystyle:=\sum_{v\in H(w)\cap
V_{G}}\Delta\varrho(v,a_{v\to w}).$ (27)

#### Consistent influence, forward responsibility.

Recall from variant 3 that we measured $G$’s influence regarding $\varepsilon$
in scenario $\zeta$ at any node $v\in V$ by

$\displaystyle\Delta\ell(v,\zeta)$
$\displaystyle:=\max\\{\ell(\varepsilon|v,\sigma_{1},\zeta)-\ell(\varepsilon|v,\sigma_{2},\zeta):\sigma_{1},\sigma_{2}\in\Sigma(v)\\},$
(28)

and took the maximum of this value over all scenarios to arrive at the forward
responsibility.

However, this does not tell the agent which action actually leads to the
desired or undesired outcomes, as the difference might go in diverging
directions for the same strategy pair depending on the scenario.

Therefore, we are interested in whether a certain strategy consistently leads
to high or low likelihoods of $\varepsilon$ starting from a given node $v$,
meaning in its peak conclusion that even its minimum is high, or its maximum
is low.131313Evidently, intermediate approaches would also be an option.
However, as we will see below, for the current application scenario
$\mathcal{R}^{3}$ and $\mathcal{R}^{4}$ are exactly the same. Therefore, a
differentiation between these marginal and other intermediate measures is not
currently crucial. To this account, we define

$\displaystyle\underline{\Delta}\mathcal{L}(v)$
$\displaystyle:=\max\\{\min_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{1},\zeta^{\prime})-\max_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{2},\zeta^{\prime}):\sigma_{1},\sigma_{2}\in\Sigma(v)\\},$
(29)

which can be seen to be at most as large as $\mathcal{R}_{f}^{3}(v)$ because

$\displaystyle\underline{\Delta}\mathcal{L}(v)$
$\displaystyle=\max\\{\min_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{1},\zeta^{\prime})+\min_{\zeta^{\prime\prime}\in
Z^{\sim}}[-\ell(\varepsilon|v,\sigma_{2},\zeta^{\prime\prime})]:\sigma_{1},\sigma_{2}\in\Sigma(v)\\}$
(30) $\displaystyle\leqslant\max\\{\min_{\zeta^{\prime}\in
Z^{\sim}}[\ell(\varepsilon|v,\sigma_{1},\zeta^{\prime})-\ell(\varepsilon|v,\sigma_{2},\zeta^{\prime})]:\sigma_{1},\sigma_{2}\in\Sigma(v)\\}$
(31)
$\displaystyle\leqslant\max\\{\ell(\varepsilon|v,\sigma_{1},\zeta^{\prime})-\ell(\varepsilon|v,\sigma_{2},\zeta^{\prime}):\sigma_{1},\sigma_{2}\in\Sigma(v),\zeta^{\prime}\in
Z^{\sim}\\}=\mathcal{R}_{f}^{3}(v).$ (32)

In case there is no pair of strategies such that one consistently leads to
high likelihoods of $\varepsilon$ while the other consistently leads to low
likelihoods of $\varepsilon$, then the value of
$\underline{\Delta}\mathcal{L}(v)$ will be negative. Hence we take

$\displaystyle\mathcal{R}^{4}_{f}(v)$
$\displaystyle:=\max\\{0,\underline{\Delta}\mathcal{L}(v)\\}\leqslant\mathcal{R}_{f}^{3}(v).$
(33)

#### Application to paradigmatic examples, discussion.

As we intended, this variant assigns zero responsibility, both forward- and
backward-looking, in the ‘cooling vs. heating’ scenario. It still correctly
represents both the ‘load and shoot’ and the ‘rock throwing’ examples as well.
Finally, it also captures the intuition that in the ‘hesitation’ example agent
$i$ should be assigned full forward responsibility in the initial node $v_{1}$
and partial backward-looking responsibility in outcome node $w_{2}$ where they
initially hesitated and thus risked the stranger’s death.

###  4.5 Summary of the section

In this section we presented four different variants of forward- and backward-
looking responsibility functions, together with one benchmark variant related
to strict causation. We were able to show that initial simple approaches do
not correctly capture the previously discussed paradigmatic examples and thus
moved on to more refined representations. We saw that in coordination
scenarios such as the ‘cooling vs. heating’ example it is inevitable to be
faced with either responsibility voids or unavoidable responsibility. We
presented functions for each of these options. In the following, we will apply
these two variants to the computation of responsibility in our reduced climate
change application scenario.

##  5 Application

In this section we will apply the well-performing proposed responsibility
functions to a stylized version of a real-world scenario and compute
quantified responsibility scores. To this end, we start out by defining the
specific example. This climate decision scenario will not be evaluated
‘analytically’; rather, we use this as a first example to apply the developed
representation and determine responsibility results based on the proposed
functions. For this, we will use those responsibility functions that were able
to correctly represent the paradigmatic examples, variants 3 and 4. This
example utilizes actual emission values and likelihood ranges resulting from
climate modelling. However, it is extremely reduced and simplified and only
meant as a proof of concept.

###  5.1 Application scenario

The application scenario within which we analyse the results of our
representation is depicted graphically in Fig. 6. We use an abbreviation node
for outcomes in a known probability range for a more succinct representation.
See Fig. 5 for the details of this abbreviation. This example showcases a
reduced climate change decision making scenario: Two agents,
$i=\mathtt{EU+USA}$ and $j=\mathtt{China}$, consecutively get to decide on
high or low emissions, depending on their respective emission standards. Agent
$i$ obviously does not know yet what the other will decide. However, also
agent $j$ cannot be sure what $i$ will do, as even if countries sign, say, the
Paris agreement, it remains unclear whether they will actually follow through
with the corresponding actions. Therefore, we assume that the second agent $j$
does not know about agent $i$’s decision. Afterwards a third agent
$k=\mathtt{RoW}$, the rest of the world, faces the same decision, but already
knowing what the previous agents selected. These groups of countries were
selected to represent the ‘early-onset industrialised countries’ in contrast
to a ‘growing economy’. China, Europe and the United States combined emitted
more than half of country specific CO2 emissions in 2017 (Muntean et al.,
2018). Thus, the rest of the world was arbitrarily grouped into one for this
reduced example. No responsibility ascription will be computed for this third
agent.

The example is set from the beginning of the year 2018 until the end of the
year 2040. This allows us to use the statements made in the 2018 IPCC report
regarding carbon budgets, that were computed starting from 2018 (Masson-
Delmotte et al., 2018). Setting the end-point at 2040 also allows us to use
the given likelihoods of warming, that are more precise in a multi-decadal
rather than a longer time scale, while also ensuring that with a ‘high’
emissions strategy the carbon budget would be depleted. Setting the starting
time to 2018 evidently sets aside any kind of historical responsibility. For a
realistic evaluation of responsibilities in the climate crisis, however, such
a historical perspective should not be neglected.

The undesired outcome is a warming of the global mean surface air temperature
bigger or equal to 1.5°C above pre-industrial levels. Fig. 1 shows the
relations between the remaining carbon emissions starting in 2018 and the
likelihood of the undesired outcome as they were assessed in 2017/18.

Carbon emissions after 2017 | Likelihood of warming $<$ 1.5°C | Likelihood of warming $\geqslant$ 1.5°C
---|---|---
0 – 420 Gt CO2 | 66% – 100% | 0% – 34%
420 – 580 Gt CO2 | 50% – 65% | 35% – 50%
$>$ 580 Gt CO2 | 0% – 49% | 51% – 100%
Table 1: Relations between remaining carbon emissions and likelihoods of
warming above 1.5°C, evaluated in 2017/2018.

‘High’ emissions will be conceptualised as continuing to emit CO2 at 2017
levels for the 22 years covered by our example. We call this ‘business as
usual’ or $\mathtt{bau}$. These levels are taken from the “2018 report on
fossil CO2 emissions of all world countries” issued by the publications office
of the European Union (Muntean et al., 2018). This means for $\mathtt{EU+USA}$
emissions of $\sim 9$ GtCO2 per year, for $\mathtt{China}$ $\sim 11$ GtCO2 per
year and for the rest of the world, $\mathtt{RoW}$, $\sim 17$ GtCO2 per year.

‘Low’ emissions will be conceptualised as a reduction such that the total
contribution to the global emissions do not exceed the countries’ per capita
share of the remaining carbon budget for a 66% chance of staying within 1.5°C
warming. Again we make things simple by assuming no population growth. We call
this strategy $\mathtt{reduce}$. With the population numbers also given in
Muntean et al. (2018) this amounts to $46.4$ GtCO2 for $\mathtt{EU+USA}$,
$83.5$ GtCO2 for $\mathtt{China}$ and $289.9$ GtCO2 for the rest of the world
$\mathtt{RoW}$. This results in the total emissions given in Fig. 2 for each
strategy.141414The large discrepancy for the three groups between the
difference in amounts given for the business as usual strategies and the
reduction strategies shows the unequal distribution in current emissions.

|  $\mathtt{EU+USA}$ |  $\mathtt{China}$ |  $\mathtt{RoW}$
---|---|---|---
$\sigma=\mathtt{reduce}$ | 46.4 Gt CO2 | 83.5 Gt CO2 | 289.9 Gt CO2
$\sigma=\mathtt{bau}$ | 207 Gt CO2 | 253 Gt CO2 | 391 Gt CO2
Table 2: Emissions per agent for both strategies, over the whole period of
time covered by the example.

$p$ to $q$_abbreviates_

$\mathtt{low}$$\mathtt{high}$$p$$1-p$$1-q$$q$

Figure 5:  Abbreviation node used in the depiction of the climate decision
scenario. A triangular node with a range of likelihoods noted at is right end
is an abbreviation for an ambiguity node, for the range of likelihoods,
followed by two probability nodes with the corresponding chances for the
undesired outcome.

$\mathtt{RoW}$$v_{4}$$\mathtt{RoW}$$v_{5}$$\mathtt{RoW}$$v_{6}$$\mathtt{RoW}$$v_{7}$
$419.8$ Gt CO${}_{2}\qquad 0-34\%$$w_{1}$ $520.9$ Gt CO${}_{2}\qquad
35-50\%$$w_{2}$ $589.3$ Gt CO${}_{2}\qquad 51-100\%$$w_{3}$ $690.4$ Gt
CO${}_{2}\qquad 51-100\%$$w_{4}$ $580.4$ Gt CO${}_{2}\qquad 35-50\%$$w_{5}$
$681.5$ Gt CO${}_{2}\qquad 51-100\%$$w_{6}$ $749.9$ Gt CO${}_{2}\qquad
51-100\%$$w_{7}$ $851$ Gt CO${}_{2}\qquad 51-100\%$$w_{8}$Total emissions
Likelihood of
$>1.5$°warming$\mathtt{CHINA}$$v_{2}$$\mathtt{CHINA}$$v_{3}$$\mathtt{EU+USA}$$v_{1}$$\mathtt{reduce}\
\ \ $$\mathtt{bau}$$\mathtt{reduce}\ \ \ $$\mathtt{bau}$$\mathtt{reduce}\ \ \
$$\mathtt{bau}$$\mathtt{reduce}\ \ \ $$\mathtt{bau}$$\mathtt{reduce}\ \ \
$$\mathtt{bau}$$\mathtt{reduce}\ \ \ $$\mathtt{bau}$$\mathtt{reduce}\ \ \
$$\mathtt{bau}$ Figure 6:  Reduced version of a decision problem related to
the climate crisis. First the EU+USA, agent $i$, get to choose between high or
low emissions, then CHINA, agent $j$, gets to make the same choice (adjusted
to their emission level), without previously knowing about the other agent’s
selection. Finally the rest of the world, RoW or agent $k$, makes the same
choice but knowing about the previous’ agents choices. Consequently, undesired
climate tipping occurs within a known range of probabilities.

###  5.2 Evaluation

Now we compute the forward- and backward-looking responsibilities for
$\mathtt{CHINA}$ and $\mathtt{EU+USA}$ in the described example using variants
$\mathcal{R}^{3}$ and $\mathcal{R}^{4}$ of the proposed responsibility
functions, those variants that managed to get the analytically evaluated
paradigmatic examples right. It turns out that in this example, variants 3 and
4 are completely unanimous in their ascription.

As we used abbreviation nodes to make the representation of the tree more
succinct, we did not name every single outcome node. However, as we consider
that actions not under the influence of the agent do not affect responsibility
ascription, the backward-looking responsibility will actually be the same for
all four outcome nodes contained in one abbreviation node. The name we gave
the abbreviation node in Fig. 6 will be used to talk about those outcome
nodes. Thus $w_{1}$ actually stands for four outcome nodes
$\\{w_{1,1},w_{1,2},w_{1,3},w_{1,4}\\}$, and a statement such as
$\mathcal{R}^{4}_{b}(w_{1})=x$ is short for $R^{4}_{b}(w)=x\ \forall w\in
w_{1}=\\{w_{1,1},w_{1,2},w_{1,3},w_{1,4}\\}$.
The resulting responsibility scores are summarised in table 3. The detailed
computation can be found in the Appendix.

It was visible that only in case every agent selects ‘low’ emissions is it
possible to keep the likelihood of undesired heating below 34%, while a
combination of China and one other agent selecting ‘low’ emissions with the
third agent selecting ‘high’ emissions leads to a likelihood of 35–50% of
undesired warming. All other combinations, that is, all situations in which
more agents selected high emissions, lead to a chance of the undesired warming
which is greater than 50%. In particular, as soon as $\mathtt{China}$ selects
‘high’ emissions, the likelihood for undesired warming is above 50%.

Within this scenario, we computed the backward- and forward-looking
responsibilities of $\mathtt{EU+USA}$ as well as $\mathtt{China}$, and were
able to show that if they select ‘low’ emissions, neither agent will carry
backward-looking responsibility. However, if they select ‘high’ emissions both
agents carry backward-looking responsibility $\geqslant 0.5$, with the score
being slightly higher for China, due to its higher impact. Again, recall that
we artificially set our starting point to 2018 and disregarded any historical
considerations. Forward-looking responsibility in this case coincided
completely with maximal backward-looking one.

This shows that, even though China is the only agent who can ‘see to it that’
the highest chance for the undesired outcome obtains by selecting ‘high’
emissions, it is not the only agent who is assigned responsibility.
Additionally, despite no one being able to single-handedly ‘see to it that’
the lowest chance for the undesirable outcome obtains, we are still able to
assign responsibility.

As a more interesting comparison, using a NESS account of causation would also
result in every agent who selects ‘high’ emissions being assigned backward-
looking responsibility, albeit without discrimination regarding the likelihood
ranges that their action entails. Consider for example the situation in which
$\mathtt{EU+USA}$ and $\mathtt{China}$ select ‘high’ emissions, while the rest
of the world selects ‘low’ emissions. In this case, $\mathtt{EU+USA}$ is not a
NESS-cause for the likelihood of warming being above 50%. If they selected
‘low’ emissions the same likelihood range would obtain, while if China
selected ‘low’ emissions, a lower likelihood range would obtain. However,
going back to a NESS account of causation for undesired warming, this
distinction would have to be disregarded. This is because the undesired
outcome still lies along the same path of further events, despite being less
likely.

|  | $\mathtt{EU+USA}$ | $\mathtt{CHINA}$
---|---|---|---
$\mathcal{R}_{b}$ | $\sigma=\mathtt{reduce}$ | $v\in\\{w_{1},\ldots,w_{4}\\}$ |  0 % | $v\in\\{w_{1},w_{2},w_{5},w_{6}\\}$ |  0 %
$\sigma=\mathtt{bau}$ | $v\in\\{w_{5},\ldots,w_{8}\\}$ |  50 % | $v\in\\{w_{3},w_{4},w_{7},w_{8}\\}$ |  66 %
$\mathcal{R}_{f}$ |  | 50 % | $v=v_{2}$ |  66 %
|  |  | $v=v_{3}$ |  66 %
Table 3: Responsibility scores for both agents in the described example when
evaluated at node $v$. There is no need to differentiate between
$\mathcal{R}^{3}$ and $\mathcal{R}^{4}$, as in the current example the two
give exactly the same results.

##  6 Conclusion and Outlook

We aimed at providing a representation of degrees of responsibility in complex
situations, with specifically the current climate crisis as one application
scenario in mind. Complicating features in the relevant decision scenarios are
multi-agent interactions, uncertainty, both probabilistic as well as
ambiguous, and a temporal progression between decisions.

#### Framework.

We provided a framework to represent all these features of the relevant
decision scenarios. The framework we used is based on extensive-form games, as
these provide a tool for representing multi-agent interactive decision
scenarios over time. We added specific nodes to represent probabilistic and
ambiguous uncertainty about the future, and an equivalence relation to
represent agents’ information uncertainty. A universal ‘ethical desirability’
assessment designates a subset of outcome nodes as undesirable.

#### Paradigmatic examples.

Afterwards, we represented a set of example scenarios in this framework. Three
of the scenarios represent two-stage interactions of two agents, with the
second unaware of the first’s decision. In the first example (‘load and
shoot’) the second agent does not know whether each of their actions will lead
to a _good_ outcome, or if their actions select between good and bad outcomes.
Due to this uncertainty, we argue that the agent must take into account that
their action has an effect and should carry responsibility. The second example
(‘rock throwing’) is parallel, but this time the second agent does not know
whether all of their actions lead to a bad outcome or if they can
differentiate between good and bad outcomes. Our argument regarding
responsibility ascription is the same as before. The third example (‘cooling
vs. heating’) is a coordination game, where the second agent knows for sure
that their actions can differentiate between good and bad outcomes, however,
they do not know which actions leads where. This scenario is hard to analyse,
as it is impossible to eschew both responsibility voids and unavoidable
responsibility. Finally the last example (‘hesitation’) is one of a repeated
action by the same agent with a probabilistic uncertainty node determining
whether the agent ‘is allowed’ a second choice. We argue that repeated actions
towards a bad outcome, or away from a good one, should lead to higher
responsibility scores.

#### Candidate functions.

Subsequently, we defined candidate responsibility functions within the
previously presented framework. In contrast to existing representations in the
area of formal ethics we took a probabilistic approach to causation.

As a benchmark variant we employed a representation of responsibility related
to the most basic view of strict causation. As this variant takes no
uncertainty into account, it gets neither of the paradigmatic examples right.
Next we continued with variants using the aforementioned probabilistic account
of causation and taking uncertainty into account. In a first variant
($\mathcal{R}_{f/b}^{1}$) we assumed the best case scenario, and evaluated
from there. This was insufficient for the ‘load and shoot’ example, where in
the best case scenario the second agent has no influence but in the other
scenario they do. We managed to correctly represent the ‘load and shoot’
example by taking the opposite approach and assuming the worst case scenario
in a second variant ($\mathcal{R}_{f/b}^{2}$). The prescribed action rationale
behind this function can be seen as ‘avoiding the worst’ (in order to avoid
carrying responsibility), rather than optimising for the best. However, this
approach fails for the ‘rock throwing’ example, with parallel reasoning as
before.

Both of the above functions reduce the conceptual complexity of responsibility
ascription by relying on comparisons between the action taken at a specific
node and an assumed ’baseline case’ (specifying both a scenario and the
desirable action within this scenario). In order to surpass this limitation,
we subsequently compare agents’ _strategies_ , that is, full plans of action
for all future decisions to be taken, within _families of scenarios_. In the
next variant ($\mathcal{R}_{f/b}^{3}$) we assigned responsibility whenever a
hypothetical minimum was not reached. This correctly captured both the ‘load
and shoot’ as well as the ‘rock throwing’ examples, and prevented
responsibility voids in the coordination game ‘cooling vs. heating’. However,
in this last example it also resulted in $i$ being assigned responsibility no
matter what their action was. In the final variant we therefore set a group’s
best option as the baseline to be compared to ($\mathcal{R}_{f/b}^{4}$). This
still correctly captured the first two examples and avoided $i$ being always
to some extent responsible in the coordination example. However, it re-
introduced voids that were absent with the preceding function. Both of these
functions also capture the last example correctly, and together manage to
represent either approach to the coordination example that a modeller might
want to take.

#### Application

Finally, the last two responsibility variants, those that captured the
paradigmatic examples correctly, were applied to a reduced scenario inspired
by real-life decision making. The scenario we examined was one where three
agents, EU+USA, China and the rest of the world, successively decided on high
or low emissions in the beginning of the year 2018. Outcome nodes were set in
the year 2040. High emissions were implemented as continued emissions at 2017
levels for the 22 years covered by the scenario, while low emissions were
implemented as a reduction in line with limiting total emissions to the cap
given by the 2018 IPCC report for a 66% chance of staying below 1.5°C warming
(Masson-Delmotte et al., 2018). The undesired outcome was a warming of earth’s
surface temperature higher than 1.5°C above pre-industrial levels. The
corresponding likelihood ranges were again taken from the 2018 IPCC report. We
computed backward- and forward-looking responsibility scores for EU+USA and
China. It turned out that the two variants of the responsibility function that
we used gave the same results in this example. We also showed that while the
agent who has a higher total influence carries a slightly higher forward- as
well as backward-looking responsibility, this does not suffice to relieve the
other agent from their responsibility, as is sometimes argued.

Overall, we were able to provide a proof of concept that the suggested
responsibility functions are applicable to real-world decision scenarios and
provide sufficiently differentiated responsibility scores. Specifically for an
application to climate action, likelihoods of outcomes are a significant
informant for decision making and this should be reflected in responsibility
ascription.

#### Conclusion.

In conclusion, we were able to provide a framework for representing decision
scenarios which is able to capture all relevant features for responsibility
ascription in complex interactive situations such as collective action with
respect to climate change. Within this framework we defined a set of possible
responsibility functions, giving a quantitative measure based on a
probabilistic view of causation. We discussed a set of paradigmatic example
scenarios to evaluate the proposed functions and saw that there are situations
in which it is impossible to avoid both responsibility voids as well as
unavoidable responsibility for the agents. In those cases, we need to select
either always assigning responsibility, or to never do. Aside from this
selection we showed that two of our proposed responsibility functions, one for
each of the mentioned choices, correctly captures all the discussed examples.
We applied those functions to a reduced real-world application scenario based
on international climate action. We were able to show that our functions
provide differentiating responsibility scores in this case, both forward- and
backward-looking, and that our approach provides a more fine-grained analysis
than an account using, for example, NESS-causation would be able to.

#### Outlook.

Two central continuations of the work presented here immediately come to mind.
First of all, we might wish to get a better understanding of which
responsibility functions are better suited in which situations. In the current
paper we did this by analytically analysing example scenarios and comparing
proposed functions by asking whether their evaluation of these scenarios
coincides with the previously established one. This can be generalised by
abstracting away from concrete application scenarios to universal desirable
properties. One example could be an equivalence requirement for decisions
taken in the same information set, which would cover both the ‘load and shoot’
as well as the ‘rock throwing’ example. Using these desirable properties in an
axiomatic method, ideally the space of possible responsibility functions could
be characterised.

Second, clearly the reduced application example presented in section 5 leads
one to consider a large-scale representation of climate relevant decisions of
various agents and a subsequent computation of applicable responsibility
scores. In line with this, possibly an evaluation and reduction of the
complexity of our proposed functions will become necessary. As our
calculations often employ minimal or maximal likelihoods of the undesired
outcome the calculation will quickly require many steps if we take into
account every action by every agent. However, applying heuristics concerning
what the worst and best scenarios will be or means of reducing the underlying
tree, should allow for the application of these functions to actual real world
decision problems currently at hand.

#### Acknowledgements.

We thank Markus Brill, Jonas Israel, Rupert Klein, Ulrike Kornek, Martin
Schönfeld, two anonymous referees of an earlier version of the manuscript, the
rest of the ALGO group at Technical University Berlin, the copan collaboration
at the Potsdam Institute for Climate Impact Research and the participants of
the Formal Ethics workshop in Ghent 2019 for fruitful discussions and
comments.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – The Berlin
Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).

#### Declarations of interest.

None.

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## Appendix

### Computations of the Responsibility Scores

Now we will give the details of the computations of the responsibility
assignments, both forward- and backward-looking for both agents considered in
the example. Recall the notice on abbreviations of outcome nodes. As we used
abbreviation nodes to make the representation of the tree more succinct, we
did not name every single outcome node. However, as we consider that actions
not under the influence of the agent do not affect responsibility ascription,
the backwards-looking responsibility will actually be the same for all four
outcome nodes contained in one abbreviation node. The name we gave the
abbreviation node will be used to talk about those outcome nodes. Thus $w_{1}$
actually stands for four outcome nodes
$\\{w_{1,1},w_{1,2},w_{1,3},w_{1,4}\\}$, and a statement such as
$\mathcal{R}^{4}_{b}(w_{1})=x$ is short for $R^{4}_{b}(w)=x\ \forall w\in
w_{1}=\\{w_{1,1},w_{1,2},w_{1,3},w_{1,4}\\}$. Additionally we will use the
notations $\sigma_{red}$ or $\sigma_{bau}$ for those strategies where the
agent selects reduced or business as usual emissions, respectively. Similarly,
$\zeta_{red,bau,low}$ stands for that scenario where the other agents select
reduced emissions, business as usual emissions and where the ambiguity node is
followed by the lower probability branch, in that order.151515Note that,
depending on the agent under consideration $\zeta{red,bau,low}$ could mean
that $\mathtt{EU+USA}$ selects $\mathtt{reduce}$d emissions or that
$\mathtt{CHINA}$ selects $\mathtt{reduce}$d emissions, then $\mathtt{RoW}$
selects $\mathtt{bau}$ emissions and the ambiguity node is followed by the
lower probability branch.

Before we start, recall the definitions of the responsibility functions. Let a
tree $\mathcal{T}$, event $\varepsilon$ and group $G$ be given.

#### Variant 3, backward responsibility.

Given a node $v\in V$ and a scenario at $v$, $\zeta\in Z^{\sim}(v)$, we call
the optimum $G$ could achieve for avoiding $\varepsilon$ at that node in that
scenario the minimum likelihood for $\varepsilon$ over $G$’s strategies at
$v$,

$\displaystyle\omega(v,\zeta)$
$\displaystyle:=\min_{\sigma\in\Sigma(v)}\ell(\varepsilon|v,\sigma,\zeta).$

So we measure $G$’s hypothetical shortfall in avoiding $\varepsilon$ in
scenario $\zeta$ due to their choice $a\in A_{v_{d}}$ at node $v_{d}\in V_{G}$
by the difference in optima

$\displaystyle\Delta\omega(v_{d},\zeta,a)$
$\displaystyle:=\omega(c_{{v_{\zeta}}}(a),\zeta)-\omega(v_{d},\zeta)\geqslant
0.$

Then we defined the risk taken by $G$ in choosing $a$ to be the maximum
shortfall over all scenarios at $v_{d}$,

$\displaystyle\varrho(v_{d},a)$ $\displaystyle:=\max_{\zeta\in
Z^{\sim}(v_{d})}\Delta\omega(v_{d},\zeta,a).$

To measure $G$’s backward responsibility regarding $\varepsilon$ in node
$v_{o}\in V_{o}$, we took their aggregate risk taken over all choices they
made,

$\displaystyle\mathcal{R}_{b}^{3}(v_{o})$ $\displaystyle:=\sum_{v_{d}\in
H(v_{o})\cap V_{G}}\varrho(v_{d},a_{v_{d}\to v_{o}}).$

#### Variant 3, forward responsibility.

We measure $G$’s influence regarding $\varepsilon$ in scenario $\zeta$ at any
node $v\in V$ by the range of likelihoods spanned by $G$’s strategies at $v$,

$\displaystyle\Delta\ell(v,\zeta)$
$\displaystyle:=\max\\{\ell(\varepsilon|v,\sigma_{1},\zeta)-\ell(\varepsilon|v,\sigma_{2},\zeta):\sigma_{1},\sigma_{2}\in\Sigma(v)\\}$

To measure $G$’s forward responsibility regarding $\varepsilon$ at node
$v_{d}\in V_{G}$, we then took their maximum influence over all scenarios at
$v_{d}$,

$\displaystyle\mathcal{R}_{f}^{3}(v_{d})$ $\displaystyle:=\max_{\zeta\in
Z^{\sim}(v_{d})}\Delta\ell(v_{d},\zeta).$

#### Variant 4, backward responsibility.

We define the minimal risk and set of risk-minimizing actions of $G$ in
decision node $v_{d}\in V_{G}$ as

$\displaystyle\underline{\varrho}(v_{d})$ $\displaystyle:=\min_{a\in
A_{v_{d}}}\varrho(v_{d},a).$

We now measure $G$’s degree of negligence in choosing $a\in A_{v_{d}}$ at
$v_{d}$ by the excess risk with respect to the minimum possible risk,

$\displaystyle\Delta\varrho(v_{d},a)$
$\displaystyle:=\varrho(v_{d},a)-\underline{\varrho}(v_{d}).$

Now, to measure $G$’s backward responsibility regarding $\varepsilon$ in node
$v_{o}\in V_{o}$, we took their aggregate negligence over all choices taken,

$\displaystyle\mathcal{R}_{b}^{4}(v_{o})$ $\displaystyle:=\sum_{v_{d}\in
H(v)\cap V_{G}}\Delta\varrho(v_{d},a_{v_{d}\to v_{o}}).$ (34)

#### Variant 4, forward responsibility.

Here, we were interested in whether a certain strategy consistently leads to
high or low likelihoods of $\varepsilon$ starting from a given node $v$,
meaning even it’s minimum is high, or it’s maximum is low.

$\displaystyle\underline{\Delta}\mathcal{L}(v)$
$\displaystyle:=\max\\{\min_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{1},\zeta^{\prime})-\max_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{2},\zeta^{\prime}):\sigma_{1},\sigma_{2}\in\Sigma(v)\\}$

To avoid a negative value we take

$\displaystyle\mathcal{R}^{4}_{f}(v)$
$\displaystyle:=\max\\{0,\underline{\Delta}\mathcal{L}(v)\\}\leqslant\mathcal{R}_{f}^{3}(v).$
(35)

#### Backward-looking responsibility, EU+USA.

Now we begin with computing the actual scores in the described example.
Let $G=\\{\mathtt{EU+USA}\\}$, $\sigma^{*}=\mathtt{reduce}$, $v=v_{1}$. Then
$\min\limits_{\sigma\in\Sigma(v)}\ell(\varepsilon|v,\sigma,\zeta)=\ell(\varepsilon|v,\sigma_{red},\zeta)\quad\forall\zeta\in
Z$, thus

$\omega(v,\zeta)=\min_{\sigma\in\Sigma(v)}\ell(\varepsilon|v,\sigma,\zeta)=\begin{cases}0\%,\
\ \qquad\qquad\qquad\zeta_{red,red,low}\\\
34\%,\qquad\qquad\qquad\zeta_{red,red,high}\\\ 35\%,\quad\text{ for
}\zeta=\quad\zeta_{red,bau,low}\\\
50\%,\qquad\qquad\qquad\zeta_{red,bau,high}\\\
51\%,\qquad\qquad\qquad\zeta_{bau,red,low}\zeta_{bau,bau,low}\\\
100\%\qquad\qquad\qquad\zeta_{bau,red,high}\zeta_{bau,bau,high}\end{cases}$

And equivalently

$\omega(c_{v_{\zeta}}(\mathtt{reduce}),\zeta)=\min_{\sigma\in\Sigma(v_{2})}\ell(\varepsilon|v_{2},\sigma,\zeta)=\begin{cases}0\%,\
\ \qquad\qquad\qquad\zeta_{red,red,low}\\\
34\%,\qquad\qquad\qquad\zeta_{red,red,high}\\\ 35\%,\quad\text{ for
}\zeta=\quad\zeta_{red,bau,low}\\\
50\%,\qquad\qquad\qquad\zeta_{red,bau,high}\\\
51\%,\qquad\qquad\qquad\zeta_{bau,red,low}\zeta_{bau,bau,low}\\\
100\%\qquad\qquad\qquad\zeta_{bau,red,high}\zeta_{bau,bau,high}\end{cases}$

$\displaystyle\text{Thus }\ \omega(c_{v_{\zeta}}(\mathtt{reduce}),\zeta)$
$\displaystyle=\omega(v,\zeta)\qquad\forall\zeta\in Z$
$\displaystyle\Rightarrow\varrho(\mathtt{EU+USA},v,\mathtt{reduce})$
$\displaystyle=0$
$\displaystyle\Rightarrow\underline{\varrho}(\mathtt{EU+USA},v_{1})$
$\displaystyle=0$ $\displaystyle\Rightarrow\mathcal{R}^{3}_{b}(w)$
$\displaystyle=R^{4}_{b}(w)\qquad\forall w\in V_{o}$ $\displaystyle\text{and
}\mathcal{R}^{3}_{b}(\mathtt{EU+USA},w)$
$\displaystyle=\mathcal{R}^{4}_{b}(\mathtt{EU+USA},w)=0\qquad\text{ for
}w\in\\{w_{1},\ldots,w_{4}\\}$

Now let $\sigma^{*}=\mathtt{bau}$. $\omega(v,\zeta)$ remains unchanged.
However,

$\omega(c_{v_{\zeta}}(\mathtt{bau}),\zeta)=\min_{\sigma\in\Sigma(v_{3})}\ell(\varepsilon|v_{3},\sigma,\zeta)=\begin{cases}0\%,\
\ \qquad\qquad\qquad-\\\ 34\%,\qquad\qquad\qquad-\\\ 35\%,\quad\text{ for
}\zeta=\quad\zeta_{red,red,low}\\\
50\%,\qquad\qquad\qquad\zeta_{red,red,high}\\\
51\%,\qquad\qquad\qquad\zeta_{red,bau,low},\zeta_{bau,red,low}\zeta_{bau,bau,low}\\\
100\%\qquad\qquad\qquad\zeta_{red,bau,high},\zeta_{bau,red,high}\zeta_{bau,bau,high}\end{cases}$

Thus

$\Delta\omega(v_{1},\zeta,\mathtt{bau})=\omega(v_{3},\zeta)-\omega(v_{1},\zeta)=\begin{cases}0\%,\
\
\qquad\qquad\quad\zeta_{bau,red,low},\zeta_{bau,red,high},\zeta_{bau,bau,low},\zeta_{bau,bau,high}\\\
16\%,\quad\text{ for }\zeta=\quad\zeta_{red,red,high},\zeta_{red,bau,low}\\\
35\%,\qquad\qquad\qquad\zeta_{red,red,low}\\\
50\%,\qquad\qquad\qquad\zeta_{red,bau,high}\end{cases}$

$\displaystyle\text{Hence }\ \varrho(v_{1},\mathtt{bau})$
$\displaystyle=\max\limits_{\zeta\in
Z}\Delta\omega(v_{1},\zeta,\mathtt{bau})=50\%$ $\displaystyle\text{and
}\mathcal{R}^{3}_{b}(\mathtt{EU+USA},w)$
$\displaystyle=\mathcal{R}^{4}_{b}(\mathtt{EU+USA},w)=0.5\quad\text{ for
}w\in\\{w_{5},\ldots,w_{8}\\}.$

#### Backward-looking responsibility, China.

The computation for $\mathtt{CHINA}$ is equivalent to that of
$\mathtt{EU+USA}$ with the following results

$\displaystyle\mathcal{R}^{3}_{b}(\mathtt{CHINA},w)$
$\displaystyle=\mathcal{R}^{4}_{b}(\mathtt{CHINA},w)=0\quad\text{ for
}w\in\\{w_{1},w_{2},w_{5},w_{6}\\}$ $\displaystyle\text{and
}\mathcal{R}^{3}_{b}(\mathtt{CHINA},w)$
$\displaystyle=\mathcal{R}^{4}_{b}(\mathtt{CHINA},w)=0.66\quad\text{ for
}w\in\\{w_{3},w_{4},w_{7},w_{8}\\}.$

#### Forward-looking responsibility, EU+USA.

Let $G=\mathtt{EU+USA}$ and $v=v_{1}$. We start out by computing
$\mathcal{R}^{3}_{f}(v)$.

$\Delta\ell(v,\zeta)=\max\\{\ell(\varepsilon|v,\sigma_{1},\zeta)-\ell(\varepsilon|v,\sigma_{2},\zeta)|\sigma_{1},\sigma_{2}\in\Sigma(v)\\}=$
$=\begin{cases}0\%,\
\qquad\qquad\qquad\zeta_{bau,red,low},\zeta_{bau,red,low},\zeta_{bau,bau,low},\zeta_{bau,bau,high}\\\
16\%,\qquad\qquad\qquad\zeta_{bau,red,high},\zeta_{red,baulow}\\\
35\%,\quad\text{ for }\zeta=\quad\zeta_{red,red,low}\\\
50\%,\qquad\qquad\qquad\zeta_{red,bau,high}\\\
66\%,\qquad\qquad\qquad-\end{cases}$

$\displaystyle\text{Hence }\quad\mathcal{R}^{3}_{f}(v)=\max_{\zeta\in
Z(v)}\Delta\ell(v_{d},\zeta)=0.5.$

So this value is actually equivalent to maximum backward-looking
responsibility in this specific case. Now let us continue to computing
$\mathcal{R}^{4}_{f}(v)$.

$\ell(\varepsilon\mid
v,\sigma_{red},\zeta)-\ell(\varepsilon|v_{1},\sigma_{bau},\zeta)=\begin{cases}0\%,\
\qquad\qquad\qquad\zeta_{bau,bau,low},\zeta_{bau,bau,high},\zeta_{bau,red,low},\zeta_{bau,red,high}\\\
-16\%,\qquad\qquad\qquad\zeta_{red,red,high},\zeta_{red,baulow}\\\
-35\%,\quad\text{ for }\zeta=\quad\zeta_{red,red,low}\\\
-50\%,\qquad\qquad\qquad\zeta_{red,bau,high}\\\
-66\%,\qquad\qquad\qquad-\end{cases}$ And $\displaystyle\ell(\varepsilon\mid
v,\sigma_{bau},\zeta)-\ell(\varepsilon|v_{1},\sigma_{red},\zeta)=-[\ell(\varepsilon\mid
v,\sigma_{red},\zeta)-\ell(\varepsilon|v_{1},\sigma_{bau},\zeta)].$ Thus
$\displaystyle\min_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{red},\zeta^{\prime})-\max_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{bau},\zeta^{\prime})=-0.5,$ while
$\displaystyle\min_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{bau},\zeta^{\prime})-\max_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{red},\zeta^{\prime})=0.5.$ Hence
$\displaystyle\max\\{\min_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{1},\zeta^{\prime})-\max_{\zeta^{\prime}\in
Z^{\sim}}\ell(\varepsilon|v,\sigma_{2},\zeta^{\prime})|\sigma_{1},\sigma_{2}\in\Sigma(v)\\}=0.5=\mathcal{R}^{4}_{f}(\mathtt{EU+USA},v1)$

#### Forward-looking responsibility, China.

Again, the computation for agent $\mathtt{CHINA}$ runs equivalent to that of
agent $\mathtt{EU+USA}$, with the following result

$\displaystyle\mathcal{R}^{3}_{f}(\mathtt{CHINA},v_{2})$
$\displaystyle=\mathcal{R}^{3}_{f}(\mathtt{CHINA},v_{3})=\mathcal{R}^{4}_{f}(\mathtt{CHINA},v_{2})=\mathcal{R}^{4}_{f}(\mathtt{CHINA},v_{3})=0.66.$