11institutetext: Free University Berlin, Institute for Mathematics, Berlin,
Germany 22institutetext: Technical University Berlin, Efficient Algorithms
Research Group, Berlin, Germany 33institutetext: Potsdam Institute for Climate
Impact Research, Potsdam, Germany

# An Axiomatic Approach to Formalized Responsibility Ascription

Sarah Hiller111Corresponding author. Contact<EMAIL_ADDRESS>1133
0000-0002-4836-4036 Jonas Israel  22 0000-0002-3992-3203 Jobst Heitzig  33
0000-0002-0442-8077

###### Abstract

A formalized and quantifiable responsibility score is a crucial component in
many aspects of the development and application of multi-agent systems and
autonomous agents. We can employ it to inform decision making processes based
on ethical considerations, as a measure to ensure redundancy that helps us in
avoiding system failure, as well as for verifying that autonomous systems
remain trustworthy by testing for unwanted responsibility voids in advance. We
follow recent proposals to use probabilities as the basis for responsibility
ascription in uncertain environments rather than the deterministic causal
views employed in much of the previous formal philosophical literature. Using
an axiomatic approach we formally evaluate the qualities of (classes of)
proposed responsibility functions. To this end, we decompose the computation
of the responsibility a group carries for an outcome into the computation of
values that we assign to its members for individual decisions leading to that
outcome, paired with an appropriate aggregation function. Next, we discuss a
number of intuitively desirable properties for each of these contributing
functions. We find an incompatibility between axioms determining upper and
lower bounds for the values assigned at the member level. Regarding the
aggregation from member-level values to group-level responsibility we are able
to axiomatically characterise one promising aggregation function. Finally, we
present two maximally axiom compliant group-level responsibility measures –
one respecting the lower bound axioms at the member level and one respecting
the corresponding upper bound axioms.

###### Keywords:

Responsibility under Uncertainty, Agency, Formal Ethics, Trustworthy
Autonomous Systems, Axiomatic Evaluation

##  1 Background and Introduction

One important challenge in the development and deployment of trustworthy
autonomous and multi-agent systems is a formalized and quantifiable
responsibility measure. Internally to a system we can use responsibility as a
value which informs decision making processes, while from an external point of
view we can use shared responsibility as a measure to ensure redundancy and
avoid system failures in the case of an agent failure. Finally, in the
evaluation of autonomous agents interacting with humans it is important to
verify the requirement of meaningful human control [21]. The study of
responsibility in multi-agent systems lies at the intersection of game theory,
formal ethics, logic, and computer science. For a recent overview of open
questions regarding responsibility formalisation in autonomous systems
applications see [28]. In the current paper we focus on backward looking
responsibility for groups of agents in complex scenarios including
uncertainty.

In order to arrive at a formalized responsibility representation we need to
take two steps: First, we need to represent the situation in which
responsibility is to be ascribed. The way this is implemented varies
considerably between approaches from a philosophical tradition versus ones
from a computer science background. Second, we need to look at the criteria
for responsibility ascription and how these can be modelled in the given
situation.

Braham and van Hees [4] model the decision situations in which responsibility
is assigned using normal-form games. That is, several players may interact,
but decisions are all made at one single point in time, after which the
outcome is clear. Contrastingly, in the field of _stit_ logics, decisions take
place at a moment in an infinitely branching history. Every choice is
represented by a new branching off of possible futures [3, 14]. Finally,
Yazdanpanah et al. [27] use concurrent game structures where agents’ actions
initiate a change between finitely many possible states of a system, with the
possibility of revisiting a state more than once. Beyond these, there are many
other ways to model the relevant decision situations for the study of
responsibility ascription (e.g., see [1, 10]) which are, however, less
relevant to our approach. Our representation is somewhere in between the
previously mentioned approaches of using normal-form games, infinite Kripke
frames, and concurrent games. Namely, we use extensions of extensive-form game
trees. These extend normal-form games with a temporal component, or, viewed
from a different angle they can be conceived of as finite fragments of
infinite stit frames [8]. They are perhaps most similar to the concurrent
epistemic game structures used by [27], with several interacting agents having
the choice to influence the system. However, we include the option of
consequences being probabilistic or uncertain, as agents may not always know
about the consequences of their choices in interactive settings.

The criteria for responsibility ascription are mostly consistent throughout
recent philosophical literature. Free choice and the capability for moral
reasoning [20, 24] are generally assumed to hold for all agents and actions
(possibly after an appropriate restriction). Most attention tends to be
focused on the analysis of a causal relation between the action or event in
question and an undesirable outcome. Implementations of causality include NESS
causation [4, 26], or modelling causality as _seeing to it that_ a certain
outcome obtains in stit logics [3]. Very similar to this strict view, [27]
represent responsibility as the possibility to avoid a certain outcome, that
is, seeing to it that it’s opposite holds. Further elements such as (missing)
knowledge [6], intention [11] or other excuses [9] may be added or left out,
depending on the desired complexity of the representation.

In contrast to the often unquestioned assumption of actual causation as a
precondition for responsibility there are scenarios in which there is no
actual causality but we would still want to speak of responsibility. For
example, in legal contexts ‘attempts’ are well-known cases of non-actualized
causal relations [18]. [2] try to solve this by retroactively equipping stit
logics with a notion of _action failure_ or _attempt_. More comprehensively,
in some of the occasions in which we want to assign responsibility but cannot
fall back on actual causation it has been observed that the action in question
_increased the probability_ of the outcome [7]. Note, however, that we can
have both causation without probability increase – due to an initial lowering
of the probability of unlikely causes – and probability increase without
causation – due to fizzling or probabilistic preemption.222Fizzling here
describes an event in which the probability of the outcome was increased but
stays below 1 and the unwanted outcome does not occur due to chance.
Probabilistic preemption is simply preemption in a probabilistic context, that
is, the event which increased the probability of the outcome does not turn out
to be a cause as a different event interrupts the causal chain by causing the
outcome first. Thus, attempts to reduce causation to probability raising, as
described by [13, 16, 24], do not seem very promising.

We argue that causation is not probability increase, but that both are
separate grounds for responsibility, as was already suggested by [19]: “These
metaphysical conclusions are enough to make possible the moral thesis I now
want to advance: that chance-raising is an independent desert basis [for
responsibility], along with causation and counterfactual dependence.”333The
well worked out reduction by Halpern and Pearl of causation to counterfactual
dependence _in an appropriately modified context_ may serve as an argument to
disregard the third disjunct [12].

As responsibility due to actual causation has been the focus in the literature
so far, we now analyse the aspect of responsibility due to probability
increase. In a previous work (that is currently under review) a subset of the
authors suggested some functions for computing responsibility based on
probability increase and evaluated these according to a number of paradigmatic
example scenarios. The current work provides a more formalised and in-depth
examination of the properties of responsibility functions. We decompose the
responsibility computation for a group of agents into two parts: First we
compute values for individual decisions by group members leading to the
outcome, before aggregating these values into the final group responsibility.

In order to evaluate proposed responsibility functions we employ an axiomatic
method [23]. That is, we formulate desirable properties which our
responsibility functions are to have in the form of axioms, and assess the
quality of (sets of) proposed functions according to these axioms.
Additionally, we will present certain incompatibility and implication results
between the axioms and a characterisation of one specific subset of functions.
Two responsibility functions will be singled out to be maximally axiom-
compliant with respect to our impossibility result, one respecting the lower-
bound axioms and one respecting the upper-bound ones.

##  2 Framework and Responsibility Functions

We are interested in modelling decision situations of several agents
interacting over time. Additionally, we want to allow for the presence of
various forms of uncertainty, resulting from the interactive nature of the
scenario or from an inherent uncertainty (both probabilistic or not) regarding
the outcomes of actions. As noted earlier, the framework presented here also
occurs in a different work currently under review which is authored by a
subset of the authors.

###  2.1 Framework

In order to represent decision scenarios we use extensive-form game trees
equipped with specific additions for the features mentioned above. Formally,
the framework is the following.

######  Definition 1 (Decision tree)

A morally evaluated multi-agent decision tree with uncertainty (without
possible confusion in the remainder abbreviated as tree) is a tuple

$\mathcal{T}=\langle I,V,E,\sim,(A_{v}),(c_{v}),(p_{v}),\epsilon\rangle$

where:

  * •

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

  * •

$\langle V,E\rangle$ is a directed rooted tree with nodes $V$ and edges $E$.
$V=\bigcup_{i\in I}V_{i}\cup V_{a}\cup V_{p}\cup V_{o}$ is partitioned into a
set of decision nodes $V_{i}$ for each agent $i\in I$, a set of ambiguity
nodes $V_{a}$, a set of probabilistic uncertainty nodes $V_{p}$, and a set of
outcome nodes $V_{o}$, that are exactly the leaves of the tree. We denote the
set of all decision nodes by $V_{d}:=\bigcup_{i\in I}V_{i}$.

  * •

$\sim$ is an equivalence relation on $V_{d}$ so that $v\sim w$ implies $v,w\in
V_{i}$ for the same agent $i\in I$. We call $\sim$ information equivalence and
the equivalence classes of $\sim$ in $V_{i}$ the information sets of $i$.

  * •

$A_{v}$ is a nonempty, finite set of $i$’s possible actions in $v$, for each
agent $i\in I$ and decision node $v\in V_{i}$. Whenever $v\sim w$,
$A_{v}=A_{w}$. Let $\bigcup\limits_{v\in V_{d}}A_{v}=:\mathcal{A}$.

  * •

$c_{v}:A_{v}\to S_{v}$, where $S_{v}:=\\{w\in V:(v,w)\in E\\}$, is a bijective
consequence function mapping actions to successor nodes for each node $v\in
V\setminus V_{o}$.

  * •

$p_{v}\in\Delta(S_{v})$ for $v\in V_{p}$, with $\Delta(A)$ being the set of
all probability distributions on a given set $A$, is a probability
distribution on the set of possible successor nodes for each probability node.

  * •

$\epsilon\subseteq V_{o}$ is a set of ethically undesirable outcomes.

The set of agents and the directed tree encode a multi-agent decision
situation with the direction of the tree showing the temporal progression.
Decision, ambiguity and probability nodes receive their intuitive
interpretation. Information equivalence $\sim$ encodes which nodes are
indistinguishable to an agent at a given time. Note that only decision nodes
of one agent may be related – that is, i) agents always know whether or not it
is their turn, and ii) we do not need to index the equivalence relation with
the agent under consideration. We use the term uncertainty as an umbrella term
for information uncertainty, ambiguity and probabilistic uncertainty.444Note
that according to this representation we do not assign probabilities to any
player’s actions. While it can very well be argued that people do indeed
reason using subjective probability distributions over other players’ actions,
we do not include this here for several reasons. The first one is that it is
extremely difficult, if not impossible, to assess these probabilities
objectively. The other reason is that including reasoning via other agents’
expected actions leads to possible excuses in responsibility ascription that
we do not want to permit.

##### Graphical representation.

We represent these features graphically as follows. An example of two simple
scenarios is depicted in Figure 1.

Directed edges are shown as arrows labelled with actions, descriptions of the
situation, or probabilities, depending on the preceding node. Decision nodes
are depicted as diamonds labelled with the respective agent and ambiguity
nodes are empty diamonds, in line with the intuition that these can be
regarded as a decision node of a special agent that represents the decision
scenario’s environment. Probabilistic uncertainty nodes are depicted as empty
squares. Outcome nodes are represented as circles, with undesirable ones
shaded in gray. Information sets are connected via a dotted line.

(a) $i$$v_{1}$$v_{2}$$v_{3}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{evade\ \
}$$\mathtt{steady\ \ }$$p$$1-p$$\mathtt{deer\ moves\ \ }$$\mathtt{deer\ stays\
\ \ }$ (b)
$i$$v_{1}$$j$$v_{2}$$j$$v_{3}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{ignore}$$\mathtt{test}$$\mathtt{ignore}$$\mathtt{test}$$\mathtt{test}$$\mathtt{ignore}$

Figure 1:  Graphical representation of interactive decision scenarios with
uncertainty in the described framework. (a) An autonomous vehicle (agent $i$)
approaches a deer on a street. It can either try to evade the deer with a
computed success rate of $p$ or keep steady, resulting in a crash if the deer
does not move. (b) Agents $i$ and $j$ are two instances in a distributed
system which may perform a costly test for the failure of a component, with
$i$ coming before $j$. Both may either perform the test or ignore it, but $j$
has no way of knowing whether $i$ performed the test. If they both ignore the
test a failure is overlooked.

##### Further notions.

Additionally, we will need formal representations of the following notions
defined on this framework: that of _past_ and _future_ of a certain node, that
of _strategies_ , which tell us for a given set of agents $G$ all future
decisions of $G$, and that of _scenarios_ , which tell us about others’ future
decisions and resolve all non-probabilistic uncertainty. Formally, these are
defined as follows.

######  Definition 2 (History, Branch, Information Branch)

Let a tree $\mathcal{T}$ and a node $v\in V$ be given. Then the history of $v$
is

$H(v):=\\{v^{\prime}\in V\mid\exists v_{1},\ldots,v_{n}\in V\text{ s.t.
}(v^{\prime},v_{1}),(v_{1},v_{2}),\ldots,(v_{n},v)\in E\\},$

and the branch of $v$ is

$B(v):=\\{v^{\prime}\in V\mid v\in H(v^{\prime})\\},$

that is, all possible futures after $v$. The information branch of $v$ is
$B^{\sim}(v):=\bigcup_{v^{\prime}\sim v}B(v^{\prime})$, that is, all futures
considered possible in the information set of $v$.

Given a group $G$, we call their decision nodes $V_{G}:=\bigcup_{i\in
G}V_{i}$, and the decision nodes of all agents not in the group $I\setminus G$
together with all ambiguity nodes $V_{-G}:=\bigcup_{i\in I\setminus
G}V_{i}\cup V_{a}$. With these two notations we can define the terms
_strategy_ and _scenario_.

######  Definition 3 (Strategy)

Let a tree $\mathcal{T}$, a group $G\subseteq I$, and a node $v\in V$ be
given. Then a strategy for $G$ at $v$ is a function

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

that chooses an action for every one of $G$’s future decision nodes considered
possible according to the information set in $v$, i.e., $\sigma(v)\in A_{v}$.
Additionally, we require that the actions selected are not just logically but
also practically possible, that is, a strategy must select the same action for
all nodes in one information set: $\sigma(v_{d})=\sigma(v^{\prime}_{d})$ for
all $v_{d},v^{\prime}_{d}\in V_{G}\cap B^{\sim}(v)\text{ s.t. }v_{d}\sim
v^{\prime}_{d}$.555This is sometimes called a _uniform strategy_ [27],
however, as this is the only kind of strategy we use here we leave out the
qualification. Also, our restriction is only concerned with individual
knowledge at the agent level and does not concern any kind of group knowledge
or coordination. Therefore, we can employ this restriction without going into
the debate about what actions a group can collectively select. We denote the
set of all these functions by $\Sigma(\mathcal{T},G,v)$.

######  Definition 4 (Scenario)

Again, let a tree $\mathcal{T}$, a group $G\subseteq I$, and a node $v\in
V_{G}$ be given. Then a scenario for $G$ at $v$ is a node $v_{\zeta}\sim v$
together with a function

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

that chooses successor nodes $\zeta(v^{\prime})$ for all future decision nodes
by non-members of G or ambiguity nodes, i.e., $\zeta(v)\in S_{v}$. The node
$v_{\zeta}$ is considered the actual node in scenario $\zeta$, thus resolving
information uncertainty. The function $\zeta$ resolves non-probabilistic
uncertainty regarding what will happen in the future (disregarding $G$’s own
actions).

We denote the set of all these functions by $Z^{\sim}(\mathcal{T},G,v)$. Note
that the specification of $v_{\zeta}$ is implicit in the definition of
$\zeta$.

Given a node $v\in V_{G}$, a strategy $\sigma\in\Sigma(\mathcal{T},G,v)$, and
a scenario $\zeta\in Z^{\sim}(\mathcal{T},G,v)$, one can easily compute in a
recursive way the conditional probability to end in an outcome node $w$ given
that the current node is actually $v_{\zeta}$ and the choices in non-
probabilistic nodes are made according to what $\sigma$ and $\zeta$ specify.
Let’s call this conditional probability the likelihood
$\ell(w|v,\sigma,\zeta)$. We are often interested in the likelihood of ending
in any undesirable outcome node $w\in\epsilon$ for which we slightly abuse
notation and write
$\ell(\epsilon|v,\sigma,\zeta)\coloneqq\sum_{w\in\epsilon}\ell(w|v,\sigma,\zeta)$.

In the remainder of the paper, unless specified otherwise, we will use
$v,v_{1},v_{2},\ldots$ to indicate decision nodes and $w,w_{1},w_{2},\ldots$
to indicate outcome nodes.

###  2.2 Responsibility Functions

We are interested in assigning backward looking moral responsibility to a
(possibly singleton) group for a certain outcome. In order to compute this, we
utilize the path from the root node of the decision tree to the given outcome
node. We will assign an intermediate value, which we call “member
contribution”, for single decisions by individual group members in a given
node. Consequently, we use an aggregation function to combine these values
into the final “outcome responsibility”.666By outcome responsibility we mean
moral responsibility assigned at an outcome node as opposed to at an
intermediate node, and not non-moral causal responsibility (as it is sometimes
understood in the literature, e.g. [25]).

Member contribution is not computed irrespective of an agent’s group
membership but precisely with this information in mind. Thus, the point of
view we take here does not stand in contrast to non-reducibility results known
from the literature [5, 8, 22].

It is important to note that we place no restrictions on which agents can form
a group. This is a decision the modeler takes at the time of the computation
of responsibility. Therefore, we also do not place any _a priori_ restrictions
on the level of coordination within an arbitrary group. Finally, we always
assume common knowledge of the full decision situation.

Member contribution is a function of the individual agent and the group that
they are a member of, the node in which the current decision is taken as well
as the action which the agent selects. All of this is considered within a
given decision situation. We measure contribution as an increase in
probability. Therefore the range of this function will be the interval
$[0,1]$.777Note that a member contribution function $\mathbf{r}$ only receives
information internal to the scenario modeled in the decision tree. The
ascribed contribution is thus invariant under any outside information. This
includes invariance with respect to the specific name an agent is given as
well as any form of duty or other historic information.

######  Definition 5 (Member Contribution)

Let a tree $\mathcal{T}$ be given. Member contribution is a function mapping a
set (‘group’) of agents $G\subseteq I$, an agent $i\in G$, a node $v\in V_{i}$
and an action $\mathtt{a}\in A_{v}$ to values in the interval $[0,1]$:

$\mathbf{r}\colon\mathcal{P}(I)\times I\times V_{d}\times\mathcal{A}\to[0,1].$

Whenever $i\notin G$, $v\notin V_{i}$, or $\mathtt{a}\notin A_{v}$ the
function is not defined.

Next we define the general form of the functions we use to aggregate member
contribution into outcome responsibility (see [17] for a general introduction
to aggregation functions).

######  Definition 6 (Aggregation function)

For $n\geq 1$ an n-ary aggregation function is a function
$\mathbf{agg}^{(n)}:[0,1]^{n}\to[0,1]$ that is non decreasing in each entry
and fulfills the following boundary conditions

$\inf_{\mathbf{x}\in[0,1]^{n}}\mathbf{agg}^{(n)}(\mathbf{x})=\inf[0,1]=0\quad\text{
and
}\quad\sup_{\mathbf{x}\in[0,1]^{n}}\mathbf{agg}^{(n)}(\mathbf{x})=\sup[0,1]=1.$

An extended aggregation function is a function
$\mathbf{agg}:\bigcup\limits_{n\in\mathbb{N}_{\geq 1}}[0,1]^{n}\to[0,1]$ such
that for all $n>1$, $\mathbf{agg}^{(n)}=\mathbf{agg}\restriction_{[0,1]^{n}}$
is an n-ary aggregation function and $\mathbf{agg}^{(1)}$ is the identity on
$[0,1]$.

Finally, outcome responsibility will be a function of a group of agents and an
outcome node, again in a specified decision situation.

######  Definition 7 (Outcome responsibility)

Let a tree $\mathcal{T}$ be given. Outcome responsibility is a function of a
group of agents $G\subseteq I$, and an outcome node $w\in V_{o}$ to the reals:

$\mathcal{R}\colon\mathcal{P}(I)\times V_{o}\to\mathbb{R}.$

We consider outcome responsibility for group $G\subseteq I$ at node $w$ to be
a function of the member contribution values of the individual decisions of
agents $i\in G$ which lead to the outcome $w$:

$\mathcal{R}(G,w):=\mathbf{agg}\left((\mathbf{r}(G,i,v,\mathtt{a}_{v\to
w}))_{i\in G,v\in V_{i}\cap H(w)}\right),$

for an appropriate aggregation function $\mathbf{agg}$. $\mathtt{a}_{v\to w}$
is that action available in $v$ for which the successor node lies on the path
to $w$.

In the following three sections we provide an axiomatic study of member
contribution, aggregation functions and outcome responsibility.

##  3 Member Contribution Functions

We start our analysis by considering member contribution functions in more
detail. After establishing a list of interesting axioms that reflect certain
moral desiderata we present three functions capturing different notions of
contribution. We conclude by qualitatively evaluating the presented functions
with respect to the axioms.

###  3.1 Axioms for Member Contribution Functions

The axioms described here demand certain contribution values in clearly
defined situations. This reflects some moral desiderata that might (or might
not) be important in different situations. The first axiom reflects the idea
that in situations the agent can not discern between they should not be
treated differently.

(KSym)

Knowledge Symmetry. The same action taken in the same information set produces
the same contribution value. Let $v\sim v^{\prime}$ and $\mathtt{a}\in
A_{v}=A_{v^{\prime}}$. Then
$\mathbf{r}(G,i,v,\mathtt{a})=\mathbf{r}(G,i,v^{\prime},\mathtt{a})$.

The following axioms bound the assigned contribution value either to 0 or to 1
for taking specific actions. For both pairs of axioms it is easy to see that
the version of the axiom considering all information branches (marked by the
∼) is logically implied by the original one.

(AMC)

Avoidance of member contribution. Consider a tree $\mathcal{T}$, a group
$G\subseteq I$ and a node $v\in V_{i}$ for some $i\in G$ with
$A_{v}=\\{\mathtt{a},\mathtt{b}_{1},\ldots,\mathtt{b}_{m}\\},$ where $m\geq
0,$ such that choosing $\mathtt{a}$ certainly leads to a desirable outcome and
choosing any other option certainly leads to an undesirable outcome. Then
$\mathbf{r}(G,i,v,\mathtt{a})=0$.

(AMC∼)

Avoidance of member contribution in information branches. Consider a tree
$\mathcal{T}$, a group $G\subseteq I$ and a node $v\in V_{i}$ for some $i\in
G$ with $A_{v}=\\{\mathtt{a},\mathtt{b}_{1},\ldots,\mathtt{b}_{m}\\},$ where
$m\geq 0,$ such that choosing $\mathtt{a}$ in any $v^{\prime}\sim v$ certainly
leads to a desirable outcome and choosing any other option certainly leads to
an undesirable outcome. Then $\mathbf{r}(G,i,v,\mathtt{a})=0$.

(FMC)

Full member contribution. Consider a tree $\mathcal{T}$, a group $G\subseteq
I$ and a node $v\in V_{i}$ for some $i\in G$ with
$A_{v}=\\{\mathtt{a},\mathtt{b}_{1},\ldots,\mathtt{b}_{m}\\},$ where
$m>0,$888Note that it is important here to enforce that there is actually a
choice for the agent, i.e., that $m\neq 0$. such that choosing $\mathtt{a}$
certainly leads to an undesirable outcome and choosing any other option
certainly leads to a desirable outcome. Then $\mathbf{r}(G,i,v,\mathtt{a})=1$.

(FMC∼)

Full member contribution in information branches. Consider a tree
$\mathcal{T}$, a group $G\subseteq I$ and a node $v\in V_{i}$ for some $i\in
G$ with $A_{v}=\\{\mathtt{a},\mathtt{b}_{1},\ldots,\mathtt{b}_{m}\\},$ where
$m>0$, such that choosing $\mathtt{a}$ in any $v^{\prime}\sim v$ certainly
leads to an undesirable outcome and choosing any other option certainly leads
to a desirable outcome. Then $\mathbf{r}(G,i,v,\mathtt{a})=1$.

The axioms (AMC) and (FMC) represent notions of seeing to it that [3]. They
request that if an agent ensures a desirable or undesirable outcome through a
single action in one decision node, i.e. she sees to it that this outcome
holds, she is to be assigned no or full contribution value, respectively.

###  3.2 Proposed Functions

We consider three different member contribution functions. The first follows
an ad-hoc way of measuring responsibility through increased likelihood of an
undesirable outcome. The other two use more elaborate notions of
responsibility through risk taking and through negligence. Let a tree
$\mathcal{T}$, group $G\subseteq I$, agent $i\in G$ and node $v\in V_{i}$ be
given.

##### Contribution through increase in guaranteed likelihood.

Here, agent $i$ is assigned contribution value for the undesirable outcome if,
and to the degree that, their action increased the _guaranteed likelihood_ of
$\epsilon$. This is often the first intuition when adopting a probabilistic
view of causation in responsibility ascription, see e.g. [15], or [24]: “I
shall assume that the relevant causal connection is that the choice increases
the objective chance that the outcome will occur”. We define the _known
guaranteed likelihood_ of $\epsilon$ at node $v\in V$ as

$\gamma(v)\coloneqq\min_{\sigma\in\Sigma(\mathcal{T},G,v)}\min_{\zeta\in
Z^{\sim}(\mathcal{T},G,v)}\ell(\epsilon|v,\sigma,\zeta).$

The member contribution we assign to agent $i\in G$ for performing action
$\mathtt{a}\in A_{v}$ in node $v\in V_{i}$ is measured as the increase in
guaranteed likelihood as follows

$\mathbf{r}^{\mathrm{like}}(G,i,v,\mathtt{a})\coloneqq\Delta\gamma(v,\mathtt{a})\coloneqq\gamma(c_{v}(\mathtt{a}))-\gamma(v).$

##### Contribution through risk taking.

This variant assigns a contribution value to an agent if their action can be
seen as some kind of risk taking. We define taking a risk here as not avoiding
a possible bad outcome. We define the _optimal avoidance_ of $\epsilon$ by
group $G$ given a scenario $\zeta\in Z^{\sim}(v)$ as

$\omega(v,\zeta)\coloneqq\min_{\sigma\in\Sigma(\mathcal{T},G,v)}\ell(\epsilon|v,\sigma,\zeta).$

We now measure member contribution as the shortfall in avoiding $\epsilon$ due
to action $\mathtt{a}$ at $v$

$\mathbf{r}^{\mathrm{risk}}(G,i,v,\mathtt{a})\coloneqq\max_{\zeta\in
Z^{\sim}(\mathcal{T},G,v)}\Delta\omega(v,\zeta,\mathtt{a})\coloneqq\max_{\zeta\in
Z^{\sim}(\mathcal{T},G,v)}[\omega(c_{v}(\mathtt{a}),\zeta)-\omega(v,\zeta)].$

That is, rather than assuming a single scenario we compare $G$’s strategies
over all scenarios.

##### Contribution through negligence.

The third and final variant we study in this paper assigns the contribution
value in a similar way to the previous variant but deducts a baseline of
unavoidable risk. This ensures that in situations where all available actions
produce some risk of leading to an undesirable outcome the agent is only
assigned a contribution value when choosing negligently. We define the minimal
risk formally through

$\underline{\rho}(G,i,v)\coloneqq\min_{\mathtt{a}\in
A_{v}}\rho(G,i,v,\mathtt{a})=\min_{\mathtt{a}\in A_{v}}\max_{\zeta\in
Z^{\sim}(\mathcal{T},G,v)}\Delta\omega(v,\zeta,\mathtt{a})=\min_{\mathtt{a}\in
A_{v}}\mathbf{r}^{\mathrm{risk}}(G,i,v,\mathtt{a}).$

With that we define this variant of member contribution as

$\mathbf{r}^{\mathrm{negl}}(G,i,v,\mathtt{a})\coloneqq\Delta\rho(G,i,v,\mathtt{a})\coloneqq\max_{\zeta\in
Z^{\sim}(\mathcal{T},G,v)}\Delta\omega(v,\zeta,\mathtt{a})-\underline{\rho}(G,i,v).$

###  3.3 Application to Example Scenarios

The suggested functions give the following results for the examples from
Figure 1 and Figure 2. The computations can be found in Appendix 0.A. We can
see that in many situations $\mathbf{r}^{\mathrm{like}}$ is not very fine-
grained. Contribution values for members of the group $\\{i,j\\}$ are often
zero as we can assume that the other group members will follow a strategy that
avoids the undesirable outcome. Also, we see that in accordance with our
intentions $\mathbf{r}^{\mathrm{risk}}$ may assign contribution values where
$\mathbf{r}^{\mathrm{negl}}$ fails to do so.

|  Figure 1(a) |  Figure 1(b) | Figure 2
---|---|---|---
| $G=\\{i\\}$ |  $G=\\{i\\}$ or $G=\\{j\\}$ | $G=\\{i,j\\}$ |  $G=\\{i\\}$ or $G=\\{j\\}$ | $G=\\{i,j\\}$
|  $\mathtt{evade}$ |  $\mathtt{steady}$ |  $\mathtt{ignore}$ |  $\mathtt{test}$ |  $\mathtt{ignore}$ |  $\mathtt{test}$ |  $\mathtt{left}$ |  $\mathtt{right}$ |  $\mathtt{left}$ |  $\mathtt{right}$
$\mathbf{r}^{\mathrm{like}}$ |  1-p |  0 |  0 |  0 |  0 |  0 |  0 |  0 |  0 |  0
$\mathbf{r}^{\mathrm{risk}}$ |  1-p |  p |  1 |  0 |  0 |  0 |  1 |  1 |  0 |  0
$\mathbf{r}^{\mathrm{negl}}$ |  1-2p |  0 |  1 |  0 |  0 |  0 |  0 |  0 |  0 |  0
Table 1: Evaluation of member contribution values with $\mathcal{T}$ as in the
examples from Figure 1 and Figure 2. $G$ and $\mathtt{a}$ are as specified,
$v$ is clear from the context. In cases where $G=\\{i,j\\}$ the values for
agents $i$ and $j$ are the same. We assume $p<1-p$.

###  3.4 Evaluation of Proposed Functions

In this section we check compliance of the member contribution functions
described earlier with our axioms. The proof of the following proposition can
be found in Appendix 0.B. Note that the coordination game as depicted in
Figure 2 works as a counterexample for all negative results.

| (KSym) | (AMC) |  (AMC∼) | (FMC) |  (FMC∼)
---|---|---|---|---|---
$\mathbf{r}^{\mathrm{like}}$ | $\times$ | ✓ | ✓ | ✓ | ✓
$\mathbf{r}^{\mathrm{risk}}$ | ✓ | $\times$ | ✓ | ✓ | ✓
$\mathbf{r}^{\mathrm{negl}}$ | ✓ | ✓ | ✓ | $\times$ | ✓
Table 2:  Summary of the compliance by the three proposed functions with the
considered axioms. “✓” denotes that the function satisfies the axiom whereas
“$\times$” denotes that there are instances where the axiom is not satisfied.

######  Proposition 1 ()

The three functions $\mathbf{r}^{\mathrm{like}},\mathbf{r}^{\mathrm{risk}}$
and $\mathbf{r}^{\mathrm{negl}}$ satisfy the given axioms as specified in
Table 2.

We now show that one has to choose between assigning unavoidable contribution
values in some situations – and thus violating an axiom like (AMC) – or having
situations where we cannot assign full contribution values – and in turn
violating an axiom like (FMC). We will come back to this point later for
similar axioms regarding aggregated outcome responsibility.

######  Proposition 2 ()

No function can simultaneously satisfy (KSym), (AMC), and (FMC).

###### Proof

Consider the coordination game depicted in Figure 2 and assume that the member
contribution function $\mathbf{r}$ satisfies (KSym), (AMC) and (FMC). By
(AMC),
$\mathbf{r}(j,v_{1},\mathtt{cinema})=\mathbf{r}(j,v_{2},\mathtt{theater})=0$
and by (FMC) we have
$\mathbf{r}(j,v_{1},\mathtt{theater})=\mathbf{r}(j,v_{2},\mathtt{cinema})=1$.
But this contradicts (KSym) as
$\mathbf{r}(j,v_{1},\mathtt{cinema})\neq\mathbf{r}(j,v_{2},\mathtt{cinema})$.∎

$i$$v_{1}$$j$$v_{2}$$j$$v_{3}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{left}$$\mathtt{right}$$\mathtt{left}$$\mathtt{right}$$\mathtt{right}$$\mathtt{left}$
Figure 2:  A coordination game where two autonomous vehicles (agents $i$ and
$j$) approach each other on a street with two lanes, destined to crash into
each other if they don’t evade. The crash is avoided if and only if both
vehicles move to their respective right or both move to their respective left.

On the other hand, we can see in Table 2 that each combination of two out of
the three axioms (AMC), (FMC), and (KSym) is possible. We would argue that
sacrificing (KSym) might not be satisfactory in most instances, as
responsibility assignment should not depend on factors outside the agents’
knowledge. Instead, one could require (KSym) together with the corresponding
weakened variants (AMC∼) and (FMC∼).

##  4 Aggregation Functions

After studying member contribution functions and their axiomatic compliance,
we will now turn to aggregation functions. Again, we start by defining a
number of axioms applicable to these, before presenting a set of possible
functions that could be used as aggregation functions and qualitatively
compare them using the axioms.

###  4.1 Axioms for Aggregation Functions

As a first axiom we consider monotonicity. If we input more entries of non-
zero member contribution into the computation, we will want the outcome
responsibility to increase as well. Similarly, if we input the same number of
member contributions but some of them have larger values, we also expect the
outcome responsibility to be larger (instead of just larger or equal, as is
demanded in the definition of aggregation functions).

(BSM+)

Bounded strict monotonicity under adding non-zero entries. For
$[x_{1},\ldots,x_{n}]\in[0,1]^{n}$ and $x_{n+1}\in(0,1]$ it holds that

$\text{ either }\quad\mathbf{agg}([x_{1},\ldots,x_{n}])=1\quad\text{ or
}\quad\mathbf{agg}([x_{1},\ldots,x_{n}])<\mathbf{agg}([x_{1},\ldots,x_{n},x_{n+1}]).$

(BSM>)

Bounded strict monotonicity under increasing entries. For
$[x_{1},\ldots,x_{n}],[y_{1},\ldots,y_{n}]\in[0,1]^{n}$ where there is a
$j\in\\{1,\dots,n\\}$ s.t. $x_{j}<y_{j}$ and $x_{i}=y_{i}$ for all $i\neq j$
it holds that

$\text{ either }\quad\mathbf{agg}([x_{1},\ldots,x_{n}])=1\quad\text{ or
}\quad\mathbf{agg}([x_{1},\ldots,x_{n}])<\mathbf{agg}([y_{1},\ldots,y_{n}]).$

(BSM)

Bounded strict monotonicity. Both of the above hold.

Strengthening this idea, we can ask for the aggregation function to not only
be monotone, but linear in each component. That is, if we keep the other input
values fixed, the influence of one member contribution on the outcome should
be linear. In order to define this we use the following notational convention.
For an n-ary aggregation function $\mathbf{agg}^{(n)}$, an n-dimensional
vector $\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})$, and an
index $i\leq n$, let
$\mathbf{agg}^{(n)}_{\overline{\mathbf{r}},-i}(x):=\mathbf{agg}^{(n)}(\mathbf{r}_{1},\ldots,\mathbf{r}_{i-1},x,\mathbf{r}_{i+1},\ldots,\mathbf{r}_{n})$
be the function obtained from $\mathbf{agg}^{(n)}$ by keeping all input values
fixed as given by $\overline{\mathbf{r}}$, except for the i-th entry.

(LIN)

Linearity in each component. For each $n\in\mathbb{N}_{\geq 1}$, for each
input vector of length $n$, $\overline{\mathbf{r}}\in[0,1]^{n}$ and for each
index $i\leq n$, $\mathbf{agg}^{(n)}_{\overline{\mathbf{r}},{-i}}$ is a linear
function. That is,

$\displaystyle\mathbf{agg}^{(n)}_{\overline{\mathbf{r}},{-i}}(x)$
$\displaystyle=\mathbf{agg}^{(n)}(\mathbf{r}_{1},\ldots,\mathbf{r}_{i-1},x,\mathbf{r}_{i+1},\ldots,\mathbf{r}_{n})$
$\displaystyle=ax+b\qquad\text{ for some }a,b\in\mathbb{R}.$

The third set of axioms concerns the behaviour of the aggregation function
under certain special input values and can be interpreted as follows. Firstly,
we can require that having been assigned full member contribution can not be
relativized by other actions. On the other hand, we can demand that adding a
decision which had a member contribution of 0 does not change an agents’, or
their groups’, assigned outcome responsibility. Lastly, one could require that
doing something “equally bad” multiple times does not change the ascribed
outcome responsibility.

(AN1)

_Annihilator 1._ For each $n\in\mathbb{N}_{\geq 1}$ and all
$[x_{1},\ldots,x_{n}]\in[0,1]^{n}$ it holds that if $x_{k}=1$ for some
$k\in\\{1,\ldots,n\\}$ then

$\mathbf{agg}([x_{1},\ldots,x_{n}])=1.$

(NE0)

_Neutral element 0._ For each $n\in\mathbb{N}_{\geq 1}$, all
$[x_{1},\ldots,x_{n}]\in[0,1]^{n}$ such that $x_{i}=0$ for some
$i\in\\{1,\ldots,n\\}$ it holds that

$\mathbf{agg}([x_{1},\ldots,x_{n}])=\mathbf{agg}([x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{n}]).$

(SIP)

_Strong idempotency._ For all $k\in\mathbb{N}$ and
$\mathbf{x}\in\bigcup_{n\in\mathbb{N}_{\geq 1}}[0,1]^{n}$ it holds that

$\mathbf{agg}(\underbrace{\mathbf{x}\oplus\ldots\oplus\mathbf{x}}_{\text{$k$
times}})=\mathbf{agg}(\mathbf{x}),$

where $\mathbf{x}\oplus\ldots\oplus\mathbf{x}\in[0,1]^{kn}$ is the vector
obtained by concatenating $\mathbf{x}$ with itself $k$ times.

The next axiom, _anonymity_ , requires that it does not matter who exactly
makes a certain decision, only the ‘value’ of their decision matters for the
final evaluation. Additionally, one can require that the time of the decision
making has no effect.

(AAT)

Anonymity (with respect to agents and time). For any $\mathbf{x}\in[0,1]^{n}$
and any permutation $\pi$ of the entries in $\mathbf{x}$ it holds that
$\mathbf{agg}(\mathbf{x})=\mathbf{agg}(\pi(\mathbf{x})).$

Finally, we consider an axiom which seems very natural if we hold that outcome
responsibility and member contribution are intended to capture aspects of the
same concept: that outcome responsibility is to be zero exactly in those cases
where all member contributions which we provide as an input are also zero.

(RED)

Reducibility. For any $\mathbf{x}\in[0,1]^{n}$ it holds that
$\mathbf{agg}(\mathbf{x})=0\text{ iff }\mathbf{x}=(0,\ldots,0).$

###  4.2 Proposed Aggregation Functions

Having developed a set of possibly desirable properties for aggregation
functions we now continue by suggesting specific functions. We will discuss
the full matrix of axiom compliance in Section 4.3. Let a tree $\mathcal{T}$,
group of agents $G\subseteq I$ and outcome node $w\in V_{o}$ be given. We drop
$\mathcal{T}$ and $G$ from the notation. Note that the first proposed
aggregator, the sum, is technically not an aggregation function in the sense
of our definition in Section 2, as it might exceed the interval $[0,1]$. We
still include it here as it is a natural way to aggregate member
contributions.

##### Variant 1, sum.

$\mathbf{sum}[(\mathbf{r}(i,v,\mathtt{a}_{v\to w}))_{i\in G,v\in V_{i}\cap
H(w)}]:=\sum_{i\in G,v\in V_{i}\cap H(w)}\mathbf{r}(i,v,\mathtt{a}_{v\to w})$

##### Variant 2, average.

$\mathbf{avg}[(\mathbf{r}(i,v,\mathtt{a}_{v\to w}))_{i\in G,v\in V_{i}\cap
H(w)}]:=\frac{1}{|\Set{v\in V_{i}\cap H(w)\mid i\in G}|}\sum_{i\in G,v\in
V_{i}\cap H(w)}\mathbf{r}(i,v,\mathtt{a}_{v\to w})$

##### Variant 3, maximum.

$\mathbf{max}[(\mathbf{r}(i,v,\mathtt{a}_{v\to w}))_{i\in G,v\in V_{i}\cap
H(w)}]:=\max_{i\in G,v\in V_{i}\cap H(w)}\mathbf{r}(i,v,\mathtt{a}_{v\to w})$

##### Variant 4, modified product.

$\mathbf{mprod}[(\mathbf{r}(i,v,\mathtt{a}_{v\to w}))_{i\in G,v\in V_{i}\cap
H(w)}]:=1-\prod_{i\in G,v\in V_{i}\cap H(w)}(1-\mathbf{r}(i,v,\mathtt{a}_{v\to
w})).$

###  4.3 Evaluation of Proposed Aggregation Functions

In addition to the axioms introduced in Section 4, we include one more
property in our axiomatic analysis. As we noted above, the sum technically is
not an aggregation function. To make this explicit we add the following
property that is satisfied by all aggregation functions in the sense of the
definition in Section 2 but is violated by the sum.

(01B)

0,1-boundedness. $\mathbf{agg}(\bigcup\limits_{n\in\mathbb{N}_{\geq
1}}[0,1])\subseteq[0,1]$.

###### Proposition 3

Compliance of the candidate aggregation functions with the axioms from Section
4.1 and (01B) is as depicted in Table 3.

We abstain from discussing the proof here and instead refer to Appendix 0.C.

| (01B) |  (BSM+) |  (BSM>) | (LIN) | (AN1) | (NE0) | (SIP) | (AAT) | (RED)
---|---|---|---|---|---|---|---|---|---
$\mathbf{sum}$ | ✓ | ✓ | ✓ | ✓ | $\times$ | ✓ | $\times$ | ✓ | ✓
$\mathbf{avg}$ | ✓ | $\times$ | ✓ | ✓ | $\times$ | $\times$ | ✓ | ✓ | ✓
$\mathbf{max}$ | ✓ | $\times$ | $\times$ | $\times$ | ✓ | $\times$ | ✓ | ✓ | ✓
$\mathbf{mprod}$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | $\times$ | ✓ | ✓
Table 3:  Summary of axiom compliance by the proposed aggregation functions.

The sum fulfills monotonicity, anonymity and reducibility. However, it fails
0,1-boundedness and thus is not an aggregation function. Averaging the sum
remains 0,1-bounded but fails monotonicity under adding entries. As the same
reasoning applies to other normalisation methods we move away from considering
normalisations of the sum. We mention the maximum for reasons of completeness,
despite it clearly not fulfilling any monotonicity requirements. The same
would hold for the product, which is why it is not not mentioned in detail and
does not appear in the table. However, a modification of the product seems
promising: The issue with the first three proposed functions seems to be that
they do not correctly include additional contributions when considering
several actions, thus either failing 0,1-boundedness or monotonicity. The
rationale behind the modified product becomes clearer when we look at an
example of a repeated action. Consider the scenario depicted in Figure 3, and
consider the path where agent $i$ selects continue twice and then repair. If
we assume that continuing once with a 90% chance of disaster produces a member
contribution of $0.9$, then in the given example continuing the first time
adds $0.9$ to the outcome responsibility, continuing the second times adds
$0.9$ of the remaining interval (i.e. $0.09$), and the final decision to
repair does not reduce any of the previously gained contributions. This
example also shows that the modified product does not fulfill (SIP) and hints
at non-compliance with (SIP) being in fact desirable in certain cases.

From the proposed aggregation functions, $\mathbf{mprod}$ is the only one that
is a proper aggregation function for any input member contribution values and
which fulfills all the described axioms except for (SIP).

Moreover, we can even show the stronger statement that $\mathbf{mprod}$ is
characterised by a combination of the discussed axioms and properties.999We
require (LIN) for this characterisation. As one might argue against this axiom
due to it being very strong and rather technical it is important to point out
that even without the characterisation result, $\mathbf{mprod}$ is the most
highly axiom compliant function out of the ones we suggested.

######  Theorem 4.1 ()

The axioms (LIN), (AAT), (NE0) and (AN1) together uniquely characterise
$\mathbf{mprod}$ among all aggregation functions.

We will only present the main idea of the proof here. For the full details we
refer to Appendix 0.C.

######  Proof (Sketch)

Linearity in each component together with a subset of the other axioms
constrains the function to

$\displaystyle\mathbf{agg}([$
$\displaystyle\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])=$ $\displaystyle
b+\sum\limits_{i=1}^{n}a_{i}\mathbf{r}_{i}+\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n}a_{i,i^{\prime}}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}+\ldots+\sum\limits_{\begin{subarray}{c}i^{(1)},\ldots,i^{(n-1)}=1\\\
i^{(1)}<\ldots<i^{(n-1)}\end{subarray}}^{n}a_{i^{(1)},\ldots,i^{(n)}}\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n-1)}}+a_{1,\ldots,n}\mathbf{r}_{1}\ldots\mathbf{r}_{n}.$

Further axioms restrict the values of the coefficients as follows. (AAT)
implies that $a_{1}=\ldots=a_{n}=:a^{(1)}$,
$a_{1,2}=\ldots=a_{n-1,n}=:a^{(2)}$ etc. Since an aggregation function is the
identity on $[0,1]$, we know that $a^{(1)}=1$, (NE0) additionally implies that
$b=0$ and with $(AN1)$ we can inductively prove that
$a^{(k)}=(-1)^{k-1}\quad\forall k\geq 1$, which results precisely in the
function $\mathbf{mprod}$.

$i$$v_{1}$$w_{1}$$v_{2}$$i$$v_{3}$$w_{2}$$w_{3}$$v_{4}$$i$$v_{5}$$w_{4}$$w_{5}$$w_{6}$$\mathtt{repair\qquad}$$\mathtt{continue\,\qquad}$$.1$$.9$$\mathtt{repair\qquad}$$\mathtt{continue\,\qquad}$$.1$$.9$$\mathtt{repair\qquad}$$\mathtt{continue\,\qquad}$
Figure 3:  Representation of an example scenario of a repeated action. The
operator of a large machine can either ignore a warning of the machine and
continue using it, knowing that this has a high chance of irreversible damage
or perform a repair. At some point in time the last chance to repair the
machine is reached.

##  5 Outcome Responsibility Functions

After exploring member contribution functions and aggregation functions on
their own we will now turn our attention to outcome responsibility as a
combination of the two previously mentioned classes of functions. Again, we
first propose a set of axioms that capture interesting properties before
checking axiom compliance for a set of proposed functions.

###  5.1 Axioms for Outcome Responsibility Functions

The following four axioms give upper and lower bounds on the assigned
responsibility and in that sense follow a similar rationale as (AMC) and (FMC)
for member contribution functions. The first two axioms present upper bounds
by determining that there are cases in which no responsibility is to be
ascribed. The first one captures the idea that if an agent has complete
control over the situation and steers it towards a desired outcome, that agent
should not be assigned responsibility. The second axiom requires that agents
have initial strategies which avoid responsibility. The fact that there is a
qualitatively different action available is seen by some authors as one of the
fundamental criteria for assigning responsibility. For example Braham and van
Hees [5] introduce an _avoidance opportunity condition_ , which requires that
a person has an opportunity to avoid contributing to an outcome in order for
responsibility ascription to be appropriate. A similar – but not equivalent –
idea is captured by the second axiom.

(CC)

Complete control. Given a tree $\mathcal{T}$, a singleton group $G=\\{i\\}$
and a desirable outcome node $w\in V_{o}\setminus\epsilon$ such that the path
to $w$ involves only decisions of agent $i$ and no decision nodes of other
agents nor ambiguity or probability nodes, $H(w)\subseteq V_{i}$. Then
$\mathcal{R}(w,i)=0$.

(NUR)

No unavoidable responsibility. Given a tree $\mathcal{T}$, each group
$G\subseteq I$ must have an original strategy that is guaranteed to avoid any
outcome responsibility. That is, denoting the root node of $\mathcal{T}$ by
$v_{r}$, for every group of agents $G\subseteq I$ there must be a strategy
$\sigma\in\Sigma(\mathcal{T},G,v_{r})$, such that $\mathcal{R}(G,w)=0$ for all
outcomes $w\in V_{o}\cap\sigma(V_{G}\cap B^{\sim}(v_{r}))$ reachable with that
strategy.

The next two axioms are lower bounds in the sense that they require a certain
amount of responsibility to be attributed. Specifically, they request the
absence of (individual or group-wise) responsibility voids.

(NRV)

No responsibility voids. Given a tree $\mathcal{T}$ with no uncertainty nodes,
$V_{a}=V_{p}=\emptyset$, and a set of undesirable outcomes $\epsilon\neq
V_{o}$. Then for each undesirable outcome $w\in\epsilon$, some group is at
least partially responsible, i.e., there exists a $G\subseteq I$ with
$\mathcal{R}(G,w)>0$.

(NIRV)

No individual responsibility voids. Given a tree $\mathcal{T}$ with no
uncertainty nodes, $V_{a}=V_{p}=\emptyset$, and a set of undesirable outcomes
$\epsilon\neq V_{o}$. Then for each undesirable outcome $w\in\epsilon$, there
is an individual agent $i\in I$ with $\mathcal{R}(\\{i\\},w)>0$.

Note that as we allow group composition to inform member contribution
functions it is not necessarily the case that (NRV) together with (RED)
implies (NIRV).

The absence of individual responsibility in the presence of group
responsibility is discussed repeatedly in the literature. That is, cases where
(NRV) holds, but (NIRV) for members of the same group does not. For example,
in the presentation by Duijf [8] the premises we provide in our axioms place
us in a cooperative decision context with no external uncertainty but with
coordination uncertainty. This combination is shown to allow for the absence
of individual responsibility in the presence of group responsibility.
Similarly, Braham and van Hees [5] show that this combination is possible if a
mechanism does not allow for the attribution of different aspects of an
outcome to varying individuals but only for an overall attribution of
responsibility at large, which is the case in our representation.

In addition to the above axioms that are specifically designed for outcome
responsibility functions, we also say that an outcome responsibility function
satisfies (KSym), (AMC) or (FMC) – as well as their weaker versions (AMC∼) and
(FMC∼) – if the underlying member contribution function satisfies the
respective axiom. Using this we prove the following impossibilities. All
proofs omitted in this section can be found in Appendix 0.D.

######  Proposition 4 ()

For any outcome responsibility function $\mathcal{R}$ the following
impossibilities hold.

  * (i)

$\mathcal{R}$ can not simultaneously satisfy (KSym), (FMC), and (NUR).

  * (ii)

$\mathcal{R}$ can not simultaneously satisfy (KSym), (NRV), and (NUR).

###  5.2 Evaluation

In the following we show the compliance of certain classes of responsibility
functions with these new axioms, with the full overview being visible in Table
4.

| (NRV) | (NIRV) | (NUR) | (CC) | (KSym) | (AMC) | (FMC)
---|---|---|---|---|---|---|---
$\mathbf{agg}\circ\mathbf{r}^{\mathrm{like}}$ | ✓ | ✓ | ✓ | ✓ | $\times$ | ✓ | ✓
$\mathbf{agg}\circ\mathbf{r}^{\mathrm{risk}}$ | ✓ | ✓ | $\times$ | ✓ | ✓ | $\times$ | ✓
$\mathbf{agg}\circ\mathbf{r}^{\mathrm{negl}}$ | $\times$ | $\times$ | ✓ | ✓ | ✓ | ✓ | $\times$
Table 4:  Summary of axiom compliance by the three variants of member
contribution from Section 3.1 combined with an aggregation functions
$\mathbf{agg}$ that satisfies (BSM) and (RED). For the axioms initially
defined for member contribution functions on the right hand side of the table
we say that the combined function satisfies the axiom if its member
contribution function satisfies it.

######  Proposition 5 ()

Let $\mathbf{agg}$ be an aggregation function fulfilling (BSM) and (RED) and
let $\mathbf{r}^{\triangle}$ for
$\triangle\in\Set{\mathrm{like},\mathrm{risk},\mathrm{negl}}$ be as defined
earlier. Then the compliance of the responsibility function
$\mathcal{R}^{\triangle}:=\mathbf{agg}\circ\mathbf{r}^{\triangle}$ with the
axioms (NRV), (NIRV), (NUR) and (CC) is as presented in Table 4.

This result immediately leads us to the following conclusion.

###### Corollary 1

Given a tree $\mathcal{T}$, a group of agents $G\subseteq I$ and an outcome
node $w\in V_{o}$. Then the following two statements hold.

  * (i)

$\mathbf{mprod}\circ\mathbf{r}^{\mathrm{risk}}$ fulfills all the axioms we
discussed in Section 4 (except for (SIP)) as well as (KSym), (FMC), (NRV),
(NIRV), and (CC).

  * (ii)

$\mathbf{mprod}\circ\mathbf{r}^{\mathrm{negl}}$ fulfills all the axioms we
discussed in Section 4 (except for (SIP)) as well as (KSym), (AMC), (NUR), and
(CC).

The results presented here show that if we constrain ourselves to functions
that satisfy (KSym), we have to choose between fulfilling (NUR) and thus
ensure some upper bound on the ascribed responsibility or fulfilling (NIRV)
and (FMC) and ensure some lower bound. The two combinations discussed in the
above lemma provide solutions to both choices, while being maximally compliant
with the other presented axioms.

###  5.3 Application to Example Scenario

In the example presented in Figure 3 we obtain the following outcome
responsibility values using the two functions determined above:
$\mathbf{mprod}\circ\mathbf{r}^{\mathrm{risk}}$ and
$\mathbf{mprod}\circ\mathbf{r}^{\mathrm{negl}}$. The results are summarised in
Table 5. Note that as selecting $\mathtt{repair}$ always results in a member
contribution of zero, $\mathbf{r}^{\mathrm{risk}}$ and
$\mathbf{r}^{\mathrm{negl}}$ actually return the same results.

|  $w_{1}$ |  $w_{2}$ |  $w_{3}$ |  $w_{4}$ |  $w_{5}$ |  $w_{6}$
---|---|---|---|---|---|---
$\mathbf{mprod}\circ\mathbf{r}^{\mathrm{like}}$ |  0 |  0.9 |  0.9 |  0.99 |  0.99 |  1
$\mathbf{mprod}\circ\mathbf{r}^{\mathrm{risk}}$ |  0 |  0.9 |  0.9 |  0.99 |  0.99 |  1
Table 5:  Summary of the outcome responsibility values for all outcomes in the
example in Figure 3.

##  6 Conclusion

We employed an axiomatic analysis of responsibility quantification in complex
decision situations containing interactions and uncertainty. The model is
based on representing appropriate decision scenarios via an extension of
extensive form game trees and assigning (moral) responsibility for a certain
outcome to a group of agents as a value between 0 and 1, based on the change
in the probability of an undesirable outcome.

We split responsibility functions for outcomes into member contribution
functions and aggregation functions. The former can be viewed as functions for
member contribution assigned to individuals as part of a group, while the
latter determine the aggregation of these individual values into a group
evaluation. This separation enables a more fine-grained analysis of the
functions’ properties. We translated different kinds of desiderata from the
literature on (moral) responsibility into this setting and formulated
respective axioms for the contributing functions.

We presented an impossibility result for three axioms regarding member
contribution. This elucidates the intuition that a principled and sensible
assignment of responsibility in coordination games is impossible. We were then
able to single out two promising candidates that satisfy the two opposing
points of view in situations such as the coordination game, and that satisfy
all of the other considered axioms for member contribution functions.

We were able to derive a characterisation of an appealing aggregation
function, the modified product, using five intuitive axioms. For outcome
responsibility functions we prove compliance with respective axioms for
classes of combinations of member contribution functions and aggregation
functions. The previously mentioned impossibility result transfers to the case
of outcome responsibility. When we require that agents are assigned the same
member contribution in all situations they cannot discern between, then a
responsibility function has to either allow for voids or for unavoidable
responsibility. The same pair of member contribution functions which we
determined earlier, paired with the selected aggregation function, is able to
capture each side of this division.

The work presented in this paper can be continued in various directions. First
of all, we can use the functions which we determined here as fulfilling most
of the desired properties in real-world application scenarios. Next, a
characterisation of the space of all three functions using appropriate axioms
would be ideal. Another promising direction for future research is the
introduction of weighted aggregation functions that are not indifferent to the
time at which a decision was taken. This enables the study of responsibility
in settings where more recent actions or decisions should carry more weight
than more distant ones.

#### Acknowledgements.

We thank Alexandru Baltag, Markus Brill, Rupert Klein, the rest of the
Efficient Algorithm research group at Technical University Berlin, and the
copan collaboration at the Potsdam Institute for Climate Impact Research for
fruitful discussions and comments.

The research leading to these results received funding from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – The Berlin Mathematics Research Center MATH+
(EXC-2046/1, project ID: 390685689) and under grant BR 4744/2-1.

#### Statements and Declarations.

The authors have no financial or proprietary interests in any material
discussed in this article.

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##  Appendix 0.A Example Evaluations of Member Contribution Functions

Recall the examples.

(a) $i$$v_{1}$$v_{2}$$v_{3}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{evade\ \
}$$\mathtt{steady\ \ }$$p$$1-p$$\mathtt{deer\ moves\ \ }$$\mathtt{deer\ stays\
\ \ }$ (b)
$i$$v_{1}$$j$$v_{2}$$j$$v_{3}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{ignore}$$\mathtt{test}$$\mathtt{ignore}$$\mathtt{test}$$\mathtt{test}$$\mathtt{ignore}$

Figure 1: (a) An autonomous vehicle (agent $i$) approaches a deer on a street.
It can either try to evade the deer with a computed success rate of $p$ or
keep steady, resulting in a crash if the deer does not move. (b) $i$ and $j$
are two instances in a distributed system which may perform a costly test for
the failure of a component, with $i$ coming before $j$. Both may either
perform the test or pass, but $j$ has no way of knowing whether $i$ performed
the test. If they both pass a failure is overlooked.
$i$$v_{1}$$j$$v_{2}$$j$$v_{3}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{left}$$\mathtt{right}$$\mathtt{left}$$\mathtt{right}$$\mathtt{right}$$\mathtt{left}$
Figure 2: A coordination game where two autonomous vehicles (agents $i$ and
$j$) approach each other on a street with two lanes, destined to crash into
each other if they don’t evade. The crash is avoided if and only if both
vehicles move to their respective right or both move to their respective left
side.

We will now compute the member contribution values for these examples given by
the three different functions proposed in Section 3.2.

#####  $\mathcal{T}$ as in Figure 1 (a):

$\displaystyle\mathbf{r}^{\mathrm{like}}(\\{i\\},i,v_{1},\mathtt{evade})$
$\displaystyle=\Delta\gamma(v_{1},\mathtt{evade})=\gamma(v_{2})-\gamma(v_{1})$
$\displaystyle=\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{2})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)$
$\displaystyle=(1-p)-0=1-p$

$\displaystyle\mathbf{r}^{\mathrm{like}}(\\{i\\},i,v_{1},\mathtt{steady})$
$\displaystyle=\Delta\gamma(v_{1},\mathtt{evade})=\gamma(v_{3})-\gamma(v_{1})$
$\displaystyle=\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{3})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{evade})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{evade})=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\omega(v_{2},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max_{\zeta\in\\{\mathtt{deer\ moves,deer\
stays}\\}}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max\\{[(1-p)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{deer\
moves})],$ $\displaystyle\
[(1-p)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{deer\
stays})]\\}$ $\displaystyle=\max\\{(1-p)-0,(1-p)-(1-p)\\}=1-p$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{steady})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{steady})=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\omega(v_{3},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max_{\zeta\in\\{\mathtt{deer\ moves,deer\
stays}\\}}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max\\{[0-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{deer\
moves})],$ $\displaystyle\
[1-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{deer\
stays})]\\}$ $\displaystyle=\max\\{0-0,1-(1-p)\\}=p$

Assume $p<1-p$, thus
$\underline{\rho}(\\{i\\},i,v_{1})=\min\limits_{\mathtt{a}\in
A_{v_{1}}}\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{a})=p$.

$\displaystyle\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,v_{1},\mathtt{evade})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{evade})-\underline{\rho}(v_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{evade})-\underline{\rho}(v_{1})=(1-p)-p=1-2p$

This is larger than $0$, as we assumed that $p<1-p$.

$\displaystyle\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,v_{1},\mathtt{steady})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{steady})-\underline{\rho}(v_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{steady})-\underline{\rho}(v_{1})=p-p=0$

#####  $\mathcal{T}$ as in Figure 1 (b):

$\displaystyle\mathbf{r}^{\mathrm{like}}(\\{i\\},i,v_{1},\mathtt{ignore})$
$\displaystyle=\Delta\gamma(v_{1},\mathtt{ignore})=\gamma(v_{2})-\gamma(v_{1})$
$\displaystyle=\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{2})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{like}}(\\{i\\},i,v_{1},\mathtt{test})$
$\displaystyle=\Delta\gamma(v_{1},\mathtt{evade})=\gamma(v_{3})-\gamma(v_{1})$
$\displaystyle=\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{3})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{like}}(\\{i,j\\},i,v_{1},\mathtt{ignore})$
$\displaystyle=\Delta\gamma(v_{1},\mathtt{ignore})$
$\displaystyle=\gamma(v_{2})-\gamma(v_{1})$
$\displaystyle=\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{2})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)$
$\displaystyle\phantom{\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{2})}}-\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{1})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{like}}(\\{i,j\\},i,v_{1},\mathtt{test})$
$\displaystyle=\Delta\gamma(v_{1},\mathtt{evade})$
$\displaystyle=\gamma(v_{3})-\gamma(v_{1})$
$\displaystyle=\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{3})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)$
$\displaystyle\phantom{\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{2})}}-\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{1})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{ignore})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{ignore})=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\omega(v_{2},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max_{\zeta\in\\{\mathtt{pass,test}\\}}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max\\{[1-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{ignore})],$
$\displaystyle\
[0-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{test})]\\}$
$\displaystyle=\max\\{1-0,0-0\\}=1$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{test})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{test})=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\omega(v_{3},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max_{\zeta\in\\{\mathtt{pass,test}\\}}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max\\{[0-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{ignore})],[0-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{test})]\\}$
$\displaystyle=\max\\{0-0,0-0\\}=0$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i,j\\},i,v_{1},\mathtt{ignore})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{ignore})=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}[\omega(v_{2},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i,j\\},i,v_{1},\mathtt{test})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{test})=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}[\omega(v_{3},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=0-0=0$

$\underline{\rho}(\\{i\\},i,v_{1})=\min\limits_{\mathtt{a}\in
A_{v_{1}}}\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{a})=0$, hence

$\displaystyle\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,v_{1},\mathtt{ignore})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{ignore})-\underline{\rho}(v_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{ignore})-\underline{\rho}(v_{1})=1-0=1$

$\displaystyle\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,v_{1},\mathtt{test})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{test})-\underline{\rho}(v_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{test})-\underline{\rho}(v_{1})=0-0=0$

And $\underline{\rho}(\\{i,j\\},i,v_{1})=\min\limits_{\mathtt{a}\in
A_{v_{1}}}\mathbf{r}^{\mathrm{risk}}(\\{i,j\\},i,v_{1},\mathtt{a})=0$, hence

$\displaystyle\mathbf{r}^{\mathrm{negl}}(\\{i,j\\},i,v_{1},\mathtt{ignore})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{ignore})-\underline{\rho}(v_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i,j\\},i,v_{1},\mathtt{ignore})-\underline{\rho}(\\{i,j\\},i,v_{1})=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{negl}}(\\{i,j\\},i,v_{1},\mathtt{test})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{test})-\underline{\rho}(\\{i,j\\},i,v_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i,j\\},i,v_{1},\mathtt{test})-\underline{\rho}(\\{i,j\\},i,v_{1})=0-0=0$

#####  $\mathcal{T}$ as in Figure 2:

$\displaystyle\mathbf{r}^{\mathrm{like}}(\\{i\\},i,v_{1},\mathtt{left})$
$\displaystyle=\Delta\gamma(v_{1},\mathtt{left})=\gamma(v_{2})-\gamma(v_{1})$
$\displaystyle=\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{2})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{like}}(\\{i\\},i,v_{1},\mathtt{right})$
$\displaystyle=\Delta\gamma(v_{1},\mathtt{right})=\gamma(v_{3})-\gamma(v_{1})$
$\displaystyle=\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{3})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{like}}(\\{i,j\\},i,v_{1},\mathtt{left})$
$\displaystyle=\Delta\gamma(v_{1},\mathtt{left})$
$\displaystyle=\gamma(v_{2})-\gamma(v_{1})$
$\displaystyle=\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{2})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)$
$\displaystyle\phantom{\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{2})}}-\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{1})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{like}}(\\{i,j\\},i,v_{1},\mathtt{right})$
$\displaystyle=\Delta\gamma(v_{1},\mathtt{right})$
$\displaystyle=\gamma(v_{3})-\gamma(v_{1})$
$\displaystyle=\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{3})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)$
$\displaystyle\phantom{\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{2})}}-\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{1})}\min_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{left})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{left})=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\omega(v_{2},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max_{\zeta\in\\{\mathtt{left,right}\\}}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max\\{[0-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{left})],$
$\displaystyle\
[1-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{right})]\\}$
$\displaystyle=\max\\{0-0,1-0\\}=1$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{right})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{right})=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\omega(v_{3},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max_{\zeta\in\\{\mathtt{left,right}\\}}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=\max\\{[1-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{left})],[0-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\mathtt{right})]\\}$
$\displaystyle=\max\\{1-0,0-0\\}=1$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i,j\\},i,v_{1},\mathtt{left})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{left})=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}[\omega(v_{2},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{2})}\ell(\epsilon|v_{2},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i,j\\},i,v_{1},\mathtt{right})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{right})=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}[\omega(v_{3},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{3})}\ell(\epsilon|v_{3},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i,j\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=0-0=0$

$\underline{\rho}(\\{i\\},i,v_{1})=\min\limits_{\mathtt{a}\in
A_{v_{1}}}\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{a})=1$, hence

$\displaystyle\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,v_{1},\mathtt{left})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{left})-\underline{\rho}(\\{i\\},i,v_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{left})-\underline{\rho}(\\{i\\},i,v_{1})=1-1=0$

$\displaystyle\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,v_{1},\mathtt{right})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{right})-\underline{\rho}(\\{i\\},i,v_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{right})-\underline{\rho}(\\{i\\},i,v_{1})=1-1=0$

And $\underline{\rho}(\\{i,j\\},i,v_{1})=\min\limits_{\mathtt{a}\in
A_{v_{1}}}\mathbf{r}^{\mathrm{risk}}(\\{i,j\\},i,v_{1},\mathtt{a})=0$, hence

$\displaystyle\mathbf{r}^{\mathrm{negl}}(\\{i,j\\},i,v_{1},\mathtt{left})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{left})-\underline{\rho}(\\{i,j\\},i,v_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i,j\\},i,v_{1},\mathtt{left})-\underline{\rho}(\\{i,j\\},i,v_{1})=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{negl}}(\\{i,j\\},i,v_{1},\mathtt{right})$
$\displaystyle=\max_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i,j\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{right})-\underline{\rho}(\\{i,j\\},i,v_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i,j\\},i,v_{1},\mathtt{right})-\underline{\rho}(\\{i,j\\},i,v_{1})=0-0=0$

##  Appendix 0.B Axiom Compliance for Member Contribution Functions

Here we present the proof omitted in Section 3.

See 1

###### Proof

We consider the axioms one by one. For (KSym) first note that
$\mathbf{r}^{\mathrm{risk}}$ and $\mathbf{r}^{\mathrm{negl}}$ fulfill this
axiom by definition. The avoidance $\omega$ as well as the minimal risk
$\underline{\rho}$ are defined over the whole information set of a given node
and both member contribution functions then maximise over $Z^{\sim}(v)$, which
also takes into account all vertices in the same information set as $v$. For
$\mathbf{r}^{\mathrm{like}}$ this is different. Even though the known
guaranteed likelihood $\gamma(v)$ is defined over $\Sigma(v)$ and
$Z^{\sim}(v)$, the member contribution function then measures the difference
in $\gamma$ between the node $c_{v}(\mathtt{a})$, the agent ends up in after
taking the action $\mathtt{a}$, and $v$, the node the agent was before taking
action $\mathtt{a}$. Since $c_{v}(\mathtt{a})$ depends on the actual node $v$
– and not its information set – we cannot guarantee (KSym). For an actual
example where (KSym) is violated consider the coordination game depicted in
Figure 2. Here we have

$\mathbf{r}^{\mathrm{like}}(j,v_{1},\mathtt{cinema})=0\neq
1=\mathbf{r}^{\mathrm{like}}(j,v_{2},\mathtt{cinema}).$

It is clear from definition that $\mathbf{r}^{\mathrm{like}}$ satisfy (AMC) as
it measures the actual increase in guaranteed known likelihood which is 0 in
the setting of (AMC). To prove compliance of $\mathbf{r}^{\mathrm{negl}}$ with
(AMC) assume we are in a situation as described in the definition of the
axiom, i.e., a node $v\in V_{i}$ is given for an agent $i$ and
$A_{v}=\\{\mathtt{a},\mathtt{b}_{1},\ldots,\mathtt{b}_{m}\\}$ such that
choosing $\mathtt{a}$ certainly leads to a desirable outcome and choosing any
other action certainly leads to an undesirable outcome. We first note that

$\min_{\mathtt{b}\in
A_{v}\setminus\\{\mathtt{a}\\}}\rho(v,\mathtt{b})=\min_{b\in
A_{v}\setminus\\{\mathtt{a}\\}}\max_{\zeta\in
Z^{\sim}}\omega(c_{v}(\mathtt{b})\zeta)-\omega(v,\zeta)\geq\omega(c_{v}(\mathtt{b}),\zeta^{\prime})-\omega(v,\zeta^{\prime})=1-0=1,$

where $\zeta^{\prime}$ is a scenario where agent $i$ is actually in node $v$.
Thus w.l.o.g. we can assume that for computation of $\underline{\rho}$ action
$\mathtt{a}$ is used. Then we have
$\mathbf{r}^{\mathrm{negl}}(i,v,\mathtt{a})=\rho(v,\mathtt{a})-\rho(v,\mathtt{a})=0$,
which proves compliance. $\mathbf{r}^{\mathrm{risk}}$ however does not satisfy
(AMC). To see this consider again the coordination game depicted in Figure 2.
Here,

$\mathbf{r}^{\mathrm{risk}}(j,v_{1},\mathtt{cinema})=\max_{\zeta\in
Z^{\sim}(v_{1})}\omega(c_{v_{1}}(\mathtt{cinema}),\zeta)-\omega(v_{1},\zeta)=1.$

(AMC∼) is by definition satisfied by all functions that satisfy (AMC).
Additionally, also $\mathbf{r}^{\mathrm{risk}}$ satisfies it since choosing
action $\mathtt{a}$ is the best action in all nodes $v^{\prime}\sim v$.

(FMC) is again clearly satisfied by $\mathbf{r}^{\mathrm{like}}$.
$\mathbf{r}^{\mathrm{negl}}$ does not satisfy it, however. To see this
consider the same coordination game as above where
$\mathbf{r}^{\mathrm{negl}}(j,v_{1},\mathtt{theater})=0$.
$\mathbf{r}^{\mathrm{risk}}$ on the other hand satisfies the axiom. To see
this assume we are in a situation as described in the definition of the axiom,
i.e., a node $v\in V_{i}$ is given for an agent $i$ and
$A_{v}=\\{\mathtt{a},\mathtt{b}_{1},\ldots,\mathtt{b}_{m}\\}$ such that
choosing $\mathtt{a}$ certainly leads to an undesirable outcome and choosing
any other action certainly leads to a desirable outcome. Let $\zeta^{\prime}$
be the scenario where agent $i$ is actually in node $v$ again. Then

$\mathbf{r}^{\mathrm{risk}}(i,v,\mathtt{a})=\max_{\zeta\in
Z^{\sim}(v)}\omega(c_{v}(\mathtt{a}),\zeta)-\omega(v,\zeta)\leq\omega(c_{v}(\mathtt{a}),\zeta^{\prime})-\omega(v,\zeta)=1-0=1.$

As was the case with (AMC∼), for (FMC∼) we only have to check the function
that does not satisfy the stronger version of the axiom. Here, this is
$\mathbf{r}^{\mathrm{negl}}$. For this member contribution function we have
$\mathbf{r}^{\mathrm{negl}}(i,v,\mathtt{a})=\mathbf{r}^{\mathrm{risk}}(i,v,\mathtt{a})-\min_{\mathtt{b}\in
A}\mathbf{r}^{\mathrm{risk}}(v,\mathtt{b})=1-0=1$.

##  Appendix 0.C Axiom Compliance for Aggregation Functions

Here we present the proofs omitted in Section 4. For each aggregation function
we prove the stated axiom compliance resulting in Proposition 3.

###### Lemma 1

$\mathbf{sum}$ fulfills all of the given axioms except (AN1) and (SIP).
Further, it fails (01B) and is thus not a proper aggregation function.

###### Proof

We provide arguments for each axiom individually.

(01B)

It is clear by definition that this is not satisfied..

(BSM+)

Let $\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
and
$\overline{\mathbf{r}}^{\prime}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n},\mathbf{r}_{n+1})\in[0,1]^{n+1}$
be point-wise responsibilities, s.t. $\mathbf{sum}(\overline{\mathbf{r}})<1$
and $\mathbf{r}_{n+1}>0$. Then

$\mathbf{sum}(\overline{\mathbf{r}}^{\prime})=\sum_{k=1}^{n+1}\mathbf{r}_{k}=\sum_{k=1}^{n}\mathbf{r}_{k}+\mathbf{r}_{n+1}=\mathbf{sum}(\overline{\mathbf{r}})+\mathbf{r}_{n+1}>\mathbf{sum}(\overline{\mathbf{r}}).$

(BSM>)

Let $\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
and
$\overline{\mathbf{r}}^{\prime}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{m-1},\mathbf{r}^{\prime}_{m},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
with $m\leq n$ be point-wise responsibilities such that
$\mathbf{sum}(\overline{\mathbf{r}})<1$ and
$\mathbf{r}^{\prime}_{m}>\mathbf{r}_{m}$. Then

$\mathbf{sum}(\overline{r}^{\prime})=\sum_{k=1}^{m-1}\mathbf{r}_{k}+\mathbf{r}^{\prime}_{m}+\sum_{k=m+1}^{n}\mathbf{r}_{k}=\mathbf{sum}(\overline{\mathbf{r}})+(\mathbf{r}^{\prime}_{m}-\mathbf{r}_{m})>\mathbf{sum}(\overline{\mathbf{r}}).$

(LIN)

Let $n\in\mathbb{N}_{\geq 1}$,
$\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
point-wise responsibilities and $i\leq n$ an index. Then

$\mathbf{sum}^{(n)}_{\overline{\mathbf{r}},-i}(x)=\mathbf{sum}((\mathbf{r}_{1},\ldots,\mathbf{r}_{i-1},x,\mathbf{r}_{i},\ldots,r_{n}))=\underbrace{\sum\limits_{\begin{subarray}{c}k=1\\\
k\neq i\end{subarray}}^{n}r_{k}}_{=:b}+\underbrace{1}_{=:a}x=b+ax.$

(AN1)

This is not satisfied since $\mathbf{sum}((1,1))=2\neq 1$.

(NE0)

Clear by definition.

(SIP)

This is not satisfied since $\mathbf{sum}((1,1))=2\neq 1$.

(AAT)

Holds due to commutativity of the sum.

(RED)

For the first direction observe that $\sum_{k=1}^{n}0=0$. For the other
direction let
$\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$ be
point-wise responsibilities such that $\sum_{k=1}^{n}\mathbf{r}_{k}=0$. Then

$\displaystyle\sum_{k=1}^{n}\mathbf{r}_{k}$ $\displaystyle=0\text{ and
}\mathbf{r}_{k}\in[0,1]\text{ for all }k=1,\ldots,n$
$\displaystyle\Rightarrow\ \mathbf{r}_{k}$ $\displaystyle=0\text{ for all
}k=1,\ldots,n$ $\displaystyle\Rightarrow\ \overline{\mathbf{r}}$
$\displaystyle=(0,\ldots,0).$

###### Lemma 2

$\mathbf{avg}$ fulfills (01B), (BSM>), (AAT) and (RED), but does not fulfill
(BSM+), (LIN) and (NE0).

###### Proof

We provide arguments for each axiom individually.

(01B)

Let
$\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$,
i.e. $r_{k}\in[0,1]$ for all $k=1,\ldots,n$. Then
$\frac{1}{n}\sum_{k=1}^{n}\mathbf{r}_{k}\in[0,1]$.

(BSM+)

To see that this is violated let $\overline{\mathbf{r}}=(0.9)\in[0,1]^{1}$ and
$\overline{\mathbf{r}}^{\prime}=(0.9,0.1)\in[0,1]^{2}$ be point-wise
responsibilities. $0.1>0$. Then

$\mathbf{avg}(\overline{\mathbf{r}}^{\prime})=\frac{0.9+0.1}{2}=0.5<0.9=\mathbf{avg}(\overline{\mathbf{r}}).$

(BSM>)

Let $\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
and
$\overline{\mathbf{r}}^{\prime}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{m-1},\mathbf{r}^{\prime}_{m},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
with $m\leq n$ be point-wise responsibilities such that
$\mathbf{avg}(\overline{\mathbf{r}})<1$ and
$\mathbf{r}^{\prime}_{m}>\mathbf{r}_{m}$. Then

$\mathbf{avg}(\overline{r}^{\prime})=\frac{1}{n}[\sum_{k=1}^{m-1}\mathbf{r}_{k}+\mathbf{r}^{\prime}_{m}+\sum_{k=m+1}^{n}\mathbf{r}_{k}]=\mathbf{avg}(\overline{\mathbf{r}})+\frac{1}{n}(\mathbf{r}^{\prime}_{m}-\mathbf{r}_{m})>\mathbf{avg}(\overline{\mathbf{r}}).$

(LIN)

To see why this is violated let $n\in\mathbb{N}_{\geq 1}$,
$\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
point-wise responsibilities and $i\leq n$ an index. Then

$\mathbf{avg}^{(n)}_{\overline{\mathbf{r}},-i}(x)=\mathbf{avg}((\mathbf{r}_{1},\ldots,\mathbf{r}_{i-1},x,\mathbf{r}_{i},\ldots,r_{n}))=\underbrace{\frac{1}{n}\sum\limits_{\begin{subarray}{c}k=1\\\
k\neq i\end{subarray}}^{n}r_{k}}_{=:b}+\underbrace{\frac{1}{n}}_{=:a}x=b+ax.$

(AN1)

It holds that $\mathbf{avg}((1,0.2))=\frac{1}{2}\times(1+0.2)=0.6\neq 1$.

(NE0)

Clear by definition that this is not satisfied.

(SIP)

Clear by definition.

(AAT)

Holds due to commutativity of the sum.

(RED)

For one direction observe that $\frac{\sum_{k=1}^{n}0}{n}=0$. For the other
direction let
$\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$ be
point-wise responsibilities such that
$\frac{1}{n}\sum_{k=1}^{n}\mathbf{r}_{k}=0$. Then

$\displaystyle\frac{1}{n}\sum_{k=1}^{n}\mathbf{r}_{k}$ $\displaystyle=0\text{
and }\mathbf{r}_{k}\in[0,1]\text{ for all }k=1,\ldots,n$
$\displaystyle\Rightarrow$ $\displaystyle\sum_{k=1}^{n}\mathbf{r}_{k}$
$\displaystyle=0\text{ and }\mathbf{r}_{k}\in[0,1]\text{ for all
}k=1,\ldots,n$ $\displaystyle\Rightarrow$ $\displaystyle\mathbf{r}_{k}$
$\displaystyle=0\text{ for all }k=1,\ldots,n$ $\displaystyle\Rightarrow$
$\displaystyle\overline{\mathbf{r}}$ $\displaystyle=(0,\ldots,0).$

###### Lemma 3

$\mathbf{max}$ fulfills (01B), (1ID), (AAT) and (RED), but does not fulfill
(BSM+) nor (BSM>), (LIN), and (NE0).

###### Proof

We provide arguments for each axiom individually.

(01B)

Clear by definition.

(BSM+)

To see why this is violated let $\overline{\mathbf{r}}=(0.5)\in[0,1]^{1}$ and
$\overline{\mathbf{r}}^{\prime}=(0.5,0.4)\in[0,1]^{2}$ be point-wise
responsibilities. $0.4>0$. Then

$\mathbf{max}(\overline{\mathbf{r}}^{\prime})=0.5=\mathbf{max}(\overline{\mathbf{r}}).$

(BSM>)

To see why this is violated let $\overline{\mathbf{r}}=(0.5,0.8)\in[0,1]^{2}$
and $\overline{\mathbf{r}}^{\prime}=(0.6,0.8)\in[0,1]^{2}$. $0.6>0.5$. Then

$\mathbf{max}(\overline{r}^{\prime})=0.8=\mathbf{max}(\overline{r}).$

(LIN)

To see why this is violated let $n=2$,
$\overline{\mathbf{r}}=(0.5,0.5)\in[0,1]^{2}$ and $i=1$. Then

$\mathbf{max}^{(n)}_{\overline{\mathbf{r}},-i}(x)=\mathbf{max}((x,0.5))=\max\\{x,0.5\\}=\begin{cases}x\qquad\text{
if }x\geq 0.5\\\ 0.5\qquad\text{ else.}\end{cases}$

This is not linear.

(AN1)

Clear by definition.

(NE0)

Clear by definition.

(SIP)

Clear by definition.

(AAT)

Holds due to commutativity of the maximum.

(RED)

For one direction observe that $\max(0,\ldots,0)=0$. For the other direction
let $\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
be point-wise responsibilities such that $\max(\overline{\mathbf{r}})=0$.
Assume $\mathbf{r}_{k}>0$ for some $k\in\\{1,\ldots,n\\}$. Then
$\max(\overline{\mathbf{r}})=r_{k}>0$. Contradiction.

###### Lemma 4

$\mathbf{mprod}$ fulfills all of the mentioned axioms except for (SIP).

###### Proof

We provide arguments for each axiom individually.

(01B)

Let $\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
be point-wise responsibilities. $\mathbf{r}_{k}\in[0,1]$ for all
$k=1,\ldots,n$, hence $1-\mathbf{r}_{k}\in[0,1]$ for all $k=1,\ldots,n$,
$\prod_{k=1}^{n}(1-\mathbf{r}_{k})\in[0,1]$, and finally
$\mathbf{mprod}(\overline{\mathbf{r}})=1-[\prod_{k=1}^{n}(1-\mathbf{r}_{k})]\in[0,1]$.

(BSM+)

Let $\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
and
$\overline{\mathbf{r}}^{\prime}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n},\mathbf{r}_{n+1})\in[0,1]^{n+1}$
be point-wise responsibilities, s.t. $\mathbf{mprod}(\overline{\mathbf{r}})<1$
and $\mathbf{r}_{n+1}>0$. Then

$\mathbf{mprod}(\overline{\mathbf{r}}^{\prime})=1-[\prod_{k=1}^{n+1}(1-\mathbf{r}_{k})]=1-[\prod_{k=1}^{n}(1-\mathbf{r}_{k})\times(1-\mathbf{r}_{n+1})].$

It holds that $\mathbf{r}_{n+1}\in(0,1]$, so $(1-\mathbf{r}_{n+1})\in[0,1)$,
and
$\prod_{k=1}^{n}(1-\mathbf{r}_{k})\times(1-\mathbf{r}_{n+1})<\prod_{k=1}^{n}(1-\mathbf{r}_{k})$.
Therefore
$\mathbf{mprod}(\overline{\mathbf{r}}^{\prime})=1-[\prod_{k=1}^{n}(1-\mathbf{r}_{k})\times(1-\mathbf{r}_{n+1})]>1-[\prod_{k=1}^{n}(1-\mathbf{r}_{k})]=\mathbf{mprod}(\overline{\mathbf{r}})$.

(BSM>)

Let $\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
and
$\overline{\mathbf{r}}^{\prime}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{m-1},\mathbf{r}^{\prime}_{m},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$
with $m\leq n$ be point-wise responsibilities such that
$\mathbf{mprod}(\overline{\mathbf{r}})<1$ and
$\mathbf{r}^{\prime}_{m}>\mathbf{r}_{m}$. Then

$\mathbf{mprod}(\overline{r}^{\prime})=1-[\prod_{k=1}^{m-1}(1-\mathbf{r}_{k})\times(1-r^{\prime}_{m})\times\prod_{k=m+1}^{n}\mathbf{r}_{k}]=1-[\prod_{k=1}^{m-1}(1-\mathbf{r}_{k})\times\prod_{k=m+1}^{n}\mathbf{r}_{k}\times(1-r^{\prime}_{m})].$

It holds that $\mathbf{r}^{\prime}_{m}>\mathbf{r}_{m}$ and
$\mathbf{r}_{m},\mathbf{r}^{\prime}_{m}\in[0,1]$, hence
$1-\mathbf{r}^{\prime}_{m}<1-\mathbf{r}_{m}$, and
$\prod_{k=1}^{m-1}(1-\mathbf{r}_{k})\times\prod_{k=m+1}^{n}\mathbf{r}_{k}\times(1-r^{\prime}_{m})<\prod_{k=1}^{m-1}(1-\mathbf{r}_{k})\times\prod_{k=m+1}^{n}\mathbf{r}_{k}\times(1-r_{m})$.
Thus
$\mathbf{mprod}(\overline{r}^{\prime})=1-[\prod_{k=1}^{m-1}(1-\mathbf{r}_{k})\times\prod_{k=m+1}^{n}\mathbf{r}_{k}\times(1-r^{\prime}_{m})]>1-[\prod_{k=1}^{m-1}(1-\mathbf{r}_{k})\times\prod_{k=m+1}^{n}\mathbf{r}_{k}\times(1-r_{m})]=\mathbf{mprod}(\overline{r})$.

(LIN)

Let $n\in\mathbb{N}_{\geq 1}$,
$\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$ be
given point-wise responsibilities and $i\leq n$ be an index. Then

$\displaystyle\mathbf{mprod}^{(n)}_{\overline{\mathbf{r}},-i}(x)$
$\displaystyle=\mathbf{mprod}((\mathbf{r}_{1},\ldots,\mathbf{r}_{i-1},x,\mathbf{r}_{i},\ldots,r_{n}))$
$\displaystyle=1-\prod\limits_{\begin{subarray}{c}k=1\\\ k\neq
i\end{subarray}}^{n}(1-r_{k})\times(1-x)$
$\displaystyle=\underbrace{1-\prod\limits_{\begin{subarray}{c}k=1\\\ k\neq
i\end{subarray}}^{n}(1-r_{k})}_{=:b}+\underbrace{\prod\limits_{\begin{subarray}{c}k=1\\\
k\neq i\end{subarray}}^{n}(1-r_{k})}_{=:a}\times x$ $\displaystyle=b+ax.$

(AN1)

Let $\overline{r}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$ be
point-wise responsibilities such that $\mathbf{r}_{i}=1$ for some $i\leq n$.
Then $1-\mathbf{r}_{i}=0$, hence $\prod_{k=1}^{n}(1-\mathbf{r}_{k})=0$ and
$\mathbf{mprod}(\overline{\mathbf{r}})=1-\prod_{k=1}^{n}(1-\mathbf{r}_{k})=1$.

(NE0)

Let $\overline{r}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$ be
point-wise responsibilities such that $\mathbf{r}_{i}=0$ for some $i\leq n$.
Then we have $1-\mathbf{r}_{i}=1$ and thus
$\prod\limits_{k=1}^{n}(1-\mathbf{r}_{k})=\prod\limits_{\begin{subarray}{c}k=1\\\
k\neq i\end{subarray}}^{n}(1-\mathbf{r}_{k})$, giving us
$\mathbf{mprod}((\mathbf{r}_{1},\ldots,\mathbf{r}_{n}))=\mathbf{mprod}((\mathbf{r}_{1},\ldots,\mathbf{r}_{i-1},\mathbf{r}_{i+1},\ldots,\mathbf{r}_{n}))$.

(SIP)

To see why this is violated observe that $1-(1-0.9)\times(1-0.9)=1-0.1\times
0.1=1-0.01=0.99\neq 0.9$.

(AAT)

Holds due to commutativity of the product.

(RED)

For one direction observe that $1-[\prod_{k=1}^{n}(1-0)]=1-[1]=0$. For the
other direction let
$\overline{\mathbf{r}}=(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})\in[0,1]^{n}$ be
point-wise responsibilities such that
$1-[\prod_{k=1}^{n}(1-\mathbf{r}_{k})]=0$.

$\displaystyle 1-[\prod_{k=1}^{n}(1-\mathbf{r}_{k})]$ $\displaystyle=0$
$\displaystyle\Leftrightarrow$
$\displaystyle\qquad\prod_{k=1}^{n}(1-\mathbf{r}_{k})$ $\displaystyle=1$
$\displaystyle\xRightarrow{1-\mathbf{r}_{k}\in[0,1]\forall k}$
$\displaystyle\qquad 1-\mathbf{r}_{k}$ $\displaystyle=1\quad\text{ for all
}k=1,\ldots,n$ $\displaystyle\Leftrightarrow$
$\displaystyle\qquad\qquad\mathbf{r}_{k}$ $\displaystyle=0\quad\text{ for all
}k=1,\ldots,n.$

See 4.1

For ease of readability we break up the proof into two lemmata. We start out
by showing that linearity in each component together with the other axioms
prescribes a specific linear combination of the input values as aggregation
function, before showing that this equates exactly to $\mathbf{mprod}$.

###### Lemma 5

The axioms (LIN), (AAT), (NE0), and (AN1) uniquely characterise the
aggregation function

$\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])=\sum\limits_{i=1}^{n}\mathbf{r}_{i}-\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}+\ldots\pm\sum\limits_{\begin{subarray}{c}i^{(1)},\ldots,i^{(n-1)}=1\\\
i^{(1)}<\ldots<i^{(n-1)}\end{subarray}}^{n}\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n-1)}}\mp\mathbf{r}_{1}\ldots\mathbf{r}_{n}.$

###### Proof

The proof proceeds via induction on n. The property (1ID) of the definition of
aggregation functions will play a crucial role and we will refer to it
specifically. We explicitly name the step $n=2$ as it uses a similar reasoning
to the general case of $n\to n+1$ and thus makes the latter step easier to
follow.

n=1:

$\mathbf{agg}([\mathbf{r}_{1}])\stackrel{{\scriptstyle\text{(1ID)}}}{{=}}\mathbf{r}_{1}.$
✓

n=2:

$\displaystyle\mathbf{agg}([\mathbf{r}_{1},\mathbf{r}_{2}])\stackrel{{\scriptstyle\text{(LIN)}}}{{=}}$
$\displaystyle
f(\mathbf{r}_{2})+g(\mathbf{r}_{2})\mathbf{r}_{1}=h(\mathbf{r}_{1})+k(\mathbf{r}_{1})\mathbf{r}_{2}$
$\displaystyle\stackrel{{\scriptstyle\text{(AAT)}}}{{\Rightarrow}}\qquad\mathbf{agg}([\mathbf{r}_{1},\mathbf{r}_{2}])\
=\ $ $\displaystyle
f(\mathbf{r}_{2})+g(\mathbf{r}_{2})\mathbf{r}_{1}=f(\mathbf{r}_{1})+g(\mathbf{r}_{1})\mathbf{r}_{2}.$

For some functions $f,g,h$ and $k$.
Now we can use the axioms (NE0) and (AN1) to solve these equations for
specific values.

$\displaystyle\mathbf{agg}([0,\mathbf{r}_{2}])=f(\mathbf{r}_{2})+g(\mathbf{r}_{2})\cdot
0=f(\mathbf{r}_{2})\stackrel{{\scriptstyle\text{(NE0)}}}{{=}}\mathbf{agg}([\mathbf{r}_{2}])\stackrel{{\scriptstyle\text{(1ID)}}}{{=}}\
$ $\displaystyle\mathbf{r}_{2}$
$\displaystyle\text{and}\qquad\mathbf{agg}([1,\mathbf{r}_{2}])=f(\mathbf{r}_{2})+g(\mathbf{r}_{2})\cdot
1=\mathbf{r}_{2}+g(\mathbf{r}_{2})\stackrel{{\scriptstyle\text{(AN1)}}}{{=}}$
$\displaystyle 1$ $\displaystyle\Leftrightarrow\qquad g(\mathbf{r}_{2})=\ $
$\displaystyle 1-\mathbf{r}_{2}.$

Thus

$\mathbf{agg}([\mathbf{r}_{1},\mathbf{r}_{2}])=f(\mathbf{r}_{2})+g(\mathbf{r}_{2})\mathbf{r}_{1}=\mathbf{r}_{2}+(1-\mathbf{r}_{2})\mathbf{r}_{1}=\mathbf{r}_{2}+\mathbf{r}_{1}-\mathbf{r}_{1}\mathbf{r}_{2}.\quad\text{\checkmark}$

n$\to$ n+1:

Assume
$\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])=\sum\limits_{i=1}^{n}\mathbf{r}_{i}-\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}+\ldots\pm\mathbf{r}_{1}\ldots\mathbf{r}_{n}.$
As before

$\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n},\mathbf{r}_{n+1}])\stackrel{{\scriptstyle\text{(LIN)}}}{{=}}f([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])+g([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])\mathbf{r}_{n+1}$

for some functions $f$ and $g$. Thus

$\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])\stackrel{{\scriptstyle\text{(NE0)}}}{{=}}\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n},0])=f([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])+0,$

but also

$\displaystyle\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n},1])$
$\displaystyle=f([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])+g([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])$
$\displaystyle\stackrel{{\scriptstyle\text{(AN1)}}}{{=}}$ $\displaystyle 1$
$\displaystyle\Leftrightarrow\qquad$
$\displaystyle\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])+g([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])$
$\displaystyle\ \ =$ $\displaystyle 1$ $\displaystyle\Leftrightarrow\qquad
g([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])$ $\displaystyle\ \ =$ $\displaystyle
1-\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}]).$

Hence

$\displaystyle\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n+1}])$
$\displaystyle=\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])+\mathbf{r}_{n+1}[1-\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])]$
$\displaystyle=\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])+\mathbf{r}_{n+1}-\mathbf{r}_{n+1}\mathbf{agg}([\mathbf{r}_{1},\ldots,\mathbf{r}_{n}])]$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\text{ind.h.}}}}{{=}}\,\,\,{\color[rgb]{0.1171875,0.2421875,0.8125}\definecolor[named]{pgfstrokecolor}{rgb}{0.1171875,0.2421875,0.8125}\sum\limits_{i=1}^{n}\mathbf{r}_{i}}{\color[rgb]{0.25390625,0.9296875,0.51953125}\definecolor[named]{pgfstrokecolor}{rgb}{0.25390625,0.9296875,0.51953125}-\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}}{\color[rgb]{0.84765625,0.03515625,0.6328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.03515625,0.6328125}+\sum\limits_{\begin{subarray}{c}i,i^{\prime},i^{\prime\prime}=1\\\
i<i^{\prime}<i^{\prime\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}\mathbf{r}_{i^{\prime\prime}}}-\ldots{\color[rgb]{0.9609375,0.546875,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9609375,0.546875,0}\pm\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}}{\color[rgb]{0.1171875,0.2421875,0.8125}\definecolor[named]{pgfstrokecolor}{rgb}{0.1171875,0.2421875,0.8125}+\mathbf{r}_{n+1}}$
$\displaystyle\phantom{=M}{\color[rgb]{0.25390625,0.9296875,0.51953125}\definecolor[named]{pgfstrokecolor}{rgb}{0.25390625,0.9296875,0.51953125}-\mathbf{r}_{n+1}\sum\limits_{i=1}^{n}\mathbf{r}_{i}}{\color[rgb]{0.84765625,0.03515625,0.6328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.03515625,0.6328125}+\mathbf{r}_{n+1}\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}}-\ldots{\color[rgb]{0.9609375,0.546875,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9609375,0.546875,0}\pm\mathbf{r}_{n+1}\sum\limits_{\begin{subarray}{c}i^{(1)},\ldots,i^{(n-1)}=1\\\
i^{(1)}<\ldots<i^{(n-1)}\end{subarray}}^{n}\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n-1)}}}$
$\displaystyle\phantom{=M}\mp\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}\mathbf{r}_{n+1}$
$\displaystyle={\color[rgb]{0.1171875,0.2421875,0.8125}\definecolor[named]{pgfstrokecolor}{rgb}{0.1171875,0.2421875,0.8125}\sum\limits_{i=1}^{n+1}\mathbf{r}_{i}}{\color[rgb]{0.25390625,0.9296875,0.51953125}\definecolor[named]{pgfstrokecolor}{rgb}{0.25390625,0.9296875,0.51953125}-\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n+1}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}}{\color[rgb]{0.84765625,0.03515625,0.6328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.03515625,0.6328125}+\sum\limits_{\begin{subarray}{c}i,i^{\prime},i^{\prime\prime}=1\\\
i<i^{\prime}<i^{\prime\prime}\end{subarray}}^{n+1}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}\mathbf{r}_{i^{\prime\prime}}}-\ldots{\color[rgb]{0.9609375,0.546875,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9609375,0.546875,0}\pm\sum\limits_{\begin{subarray}{c}i^{(1)},\ldots,i^{(n)}=1\\\
i^{(1)}<\ldots<i^{(n)}\end{subarray}}^{n+1}\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}}\mp\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}\mathbf{r}_{n+1}.$

This completes the proof.

###### Lemma 6

Let $[\mathbf{r}_{1},\ldots,\mathbf{r}_{n}]\in[0,1]^{n}$. Then

$1-\prod\limits_{i=1}^{n}(1-\mathbf{r}_{i})=\sum\limits_{i=1}^{n}\mathbf{r}_{i}-\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}+\ldots\pm\sum\limits_{\begin{subarray}{c}i^{(1)},\ldots,i^{(n-1)}=1\\\
i^{(1)}<\ldots<i^{(n-1)}\end{subarray}}^{n}\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n-1)}}\mp\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}.$

###### Proof

The proof proceeds via induction on n and uses the same observation as the
previous proof. Again, we explicitly name the step $n=2$ as it uses a similar
reasoning to the general case of $n\to n+1$ and thus makes the latter step
easier to follow.

n=1:

Clear.

n=2:

$1-\prod_{i=1}^{2}(1-\mathbf{r}_{i})=1-(1-\mathbf{r}_{1})(1-\mathbf{r}_{2})=1-(1-\mathbf{r}_{1}-\mathbf{r}_{2}+\mathbf{r}_{1}\mathbf{r}_{2})=\mathbf{r}_{1}+\mathbf{r}_{2}-\mathbf{r}_{1}\mathbf{r}_{2}.$

n$\to$ n+1:

Let
$1-\prod\limits_{i=1}^{n}(1-\mathbf{r}_{i})=\sum\limits_{i=1}^{n}\mathbf{r}_{i}-\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}+\ldots\pm\sum\limits_{\begin{subarray}{c}i^{(1)},\ldots,i^{(n-1)}=1\\\
i^{(1)}<\ldots<i^{(n-1)}\end{subarray}}^{n}\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n-1)}}\mp\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}$
Which is equivalent to
$\prod\limits_{i=1}^{n}(1-\mathbf{r}_{i})=1-\underbrace{\left(\sum\limits_{i=1}^{n}\mathbf{r}_{i}-\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}+\ldots\pm\sum\limits_{\begin{subarray}{c}i^{(1)},\ldots,i^{(n-1)}=1\\\
i^{(1)}<\ldots<i^{(n-1)}\end{subarray}}^{n}\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n-1)}}\mp\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}\right)}_{=:y}$.
With this we obtain

$\displaystyle 1-\prod\limits_{i=1}^{n+1}(1-\mathbf{r}_{i})$ $\displaystyle=$
$\displaystyle
1-\prod\limits_{i=1}^{n}(1-\mathbf{r}_{i})\times(1-\mathbf{r}_{n+1})=1-[(1-y)(1-\mathbf{r}_{n+1})]$
$\displaystyle=$ $\displaystyle
1-[1-y-\mathbf{r}_{n+1}+y\mathbf{r}_{n+1}]=y+\mathbf{r}_{n+1}-y\mathbf{r}_{n+1}$
$\displaystyle=$
$\displaystyle{\color[rgb]{0.1171875,0.2421875,0.8125}\definecolor[named]{pgfstrokecolor}{rgb}{0.1171875,0.2421875,0.8125}\sum\limits_{i=1}^{n}\mathbf{r}_{i}}{\color[rgb]{0.25390625,0.9296875,0.51953125}\definecolor[named]{pgfstrokecolor}{rgb}{0.25390625,0.9296875,0.51953125}-\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}}{\color[rgb]{0.84765625,0.03515625,0.6328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.03515625,0.6328125}+\sum\limits_{\begin{subarray}{c}i,i^{\prime},i^{\prime\prime}=1\\\
i<i^{\prime}<i^{\prime\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}\mathbf{r}_{i^{\prime\prime}}}-\ldots{\color[rgb]{0.9609375,0.546875,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9609375,0.546875,0}\pm\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}}{\color[rgb]{0.1171875,0.2421875,0.8125}\definecolor[named]{pgfstrokecolor}{rgb}{0.1171875,0.2421875,0.8125}+\mathbf{r}_{n+1}}$
$\displaystyle{\color[rgb]{0.25390625,0.9296875,0.51953125}\definecolor[named]{pgfstrokecolor}{rgb}{0.25390625,0.9296875,0.51953125}-\mathbf{r}_{n+1}\sum\limits_{i=1}^{n}\mathbf{r}_{i}}{\color[rgb]{0.84765625,0.03515625,0.6328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.03515625,0.6328125}+\mathbf{r}_{n+1}\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}}-\ldots{\color[rgb]{0.9609375,0.546875,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9609375,0.546875,0}\pm\mathbf{r}_{n+1}\sum\limits_{\begin{subarray}{c}i^{(1)},\ldots,i^{(n-1)}=1\\\
i^{(1)}<\ldots<i^{(n-1)}\end{subarray}}^{n}\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n-1)}}}\mp\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}\mathbf{r}_{n+1}$
$\displaystyle=$
$\displaystyle{\color[rgb]{0.1171875,0.2421875,0.8125}\definecolor[named]{pgfstrokecolor}{rgb}{0.1171875,0.2421875,0.8125}\sum\limits_{i=1}^{n+1}\mathbf{r}_{i}}{\color[rgb]{0.25390625,0.9296875,0.51953125}\definecolor[named]{pgfstrokecolor}{rgb}{0.25390625,0.9296875,0.51953125}-\sum\limits_{\begin{subarray}{c}i,i^{\prime}=1\\\
i<i^{\prime}\end{subarray}}^{n+1}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}}{\color[rgb]{0.84765625,0.03515625,0.6328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.03515625,0.6328125}+\sum\limits_{\begin{subarray}{c}i,i^{\prime},i^{\prime\prime}=1\\\
i<i^{\prime}<i^{\prime\prime}\end{subarray}}^{n+1}\mathbf{r}_{i}\mathbf{r}_{i^{\prime}}\mathbf{r}_{i^{\prime\prime}}}-\ldots{\color[rgb]{0.9609375,0.546875,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9609375,0.546875,0}\pm\sum\limits_{\begin{subarray}{c}i^{(1)},\ldots,i^{(n)}=1\\\
i^{(1)}<\ldots<i^{(n)}\end{subarray}}^{n+1}\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}}\mp\mathbf{r}_{i^{(1)}}\ldots\mathbf{r}_{i^{(n)}}\mathbf{r}_{n+1}.$

This completes the proof. ∎

The theorem now follows immediately from Lemma 5 and Lemma 6.

##  Appendix 0.D Axiom Compliance for Outcome Responsibility Functions

Here we present the proofs omitted in Section 5.

See 4

###### Proof

To prove result (i), consider the coordination game depicted in Figure 2 and
assume that the responsibility function $\mathcal{R}$ satisfies (KSym).
Towards a contradiction assume that $\mathcal{R}$ satisfies (FMC) and (NUR) in
addition to (KSym). Again, by (FMC) we have
$\mathcal{R}(w_{2},j)=\mathcal{R}(w_{3},j)=1$ and by (KSym) it additionaly has
to hold that $\mathcal{R}(w_{1},j)=\mathcal{R}(w_{4},j)=1$. But this leaves
agent $j$ without a strategy to avoid responsibility and thus violates (NUR).
Concerning (ii), consider the adapted decision tree in Figure 4. By symmetry
of the tree, (NUR) and (KSym) agent $i$ gets assigned responsibility of 0 for
all four outcome nodes. This violates (NRV) as no agent or group of agents is
assigned any responsibility.

$i$$i$$v_{1}$$i$$v_{2}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$\mathtt{no\,risk}$$\mathtt{cooling}$$\mathtt{ignore}$$\mathtt{heat\,up}$$\mathtt{heat\,up}$$\mathtt{ignore}$
Figure 4:  An adaption of the coordination game from Figure 2. 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.

See 5

###### Proof

Note that (NRV) is implied by (NIRV). The coordination game depicted in Figure
2 shows that $\mathcal{R}^{\mathrm{negl}}$ does not satisfy (NRV) or (NIRV)
since both agents get assigned responsibility 0 in all their decision nodes no
matter what action they take. To prove (NIRV) for the other responsibility
functions we single out the decision node $v\in V_{d}$ that is closest to some
$w\in\epsilon$ while still achieving $B(v)\nsubseteq\epsilon$. Each of the
remaining two responsibility functions assigns positive responsibility to the
agent that corresponds to that decision node. Regarding (NUR) it is easy to
check for $\mathcal{R}^{\mathrm{like}}$ and $\mathcal{R}^{\mathrm{negl}}$ that
an agent always has an action available with zero responsibility. The
exception is $\mathcal{R}^{\mathrm{risk}}$, which satisfies (FMC) and (KSym)
and thus by Proposition 4 can not satisfy (NUR). Concerning (CC), it can be
shown for all three responsibility functions that they do not assign
responsibility along the path of the strategy that leads to $w\in
V_{o}\setminus\epsilon$.

##  Appendix 0.E Computation of Outcome Responsibility Values

Recall the example.

$i$$v_{1}$$w_{1}$$v_{2}$$i$$v_{3}$$w_{2}$$w_{3}$$v_{4}$$i$$v_{5}$$w_{4}$$w_{5}$$w_{6}$$\mathtt{repair\qquad}$$\mathtt{continue\,\qquad}$$.1$$.9$$\mathtt{repair\qquad}$$\mathtt{continue\,\qquad}$$.1$$.9$$\mathtt{repair\qquad}$$\mathtt{continue\,\qquad}$
Figure 3: Scenario of a repeated action. The operator of a large machine can
either ignore a warning of the machine and continue using it, knowing that
this has a high chance of irreversible damage, or perform a repair. At some
point in time the last chance to repair the machine is reached.

Let $\mathcal{T}$ as in the example and $G=\\{i\\}$. We first need to compute
the values for $\mathbf{r}^{\mathrm{risk}}$ and $\mathbf{r}^{\mathrm{negl}}$
for every action in each decision node.

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{repair})$
$\displaystyle=\max\limits_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{repair})=\max\limits_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\omega(w_{1},\zeta)-\omega(v_{1},\zeta)]$
$\displaystyle=\max\limits_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}[\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},w_{1})}\ell(\epsilon|w_{1},\sigma,\zeta)-\min_{\sigma\in\Sigma(\mathcal{T},\\{i\\},v_{1})}\ell(\epsilon|v_{1},\sigma,\zeta)]$
$\displaystyle=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{continue})$
$\displaystyle=\max\limits_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{1})}\Delta\omega(v_{1},\zeta,\mathtt{continue})=0.9-0=0.9$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{3},\mathtt{repair})$
$\displaystyle=\max\limits_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{3})}\Delta\omega(v_{3},\zeta,\mathtt{repair})=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{3},\mathtt{continue})$
$\displaystyle=\max\limits_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{3})}\Delta\omega(v_{3},\zeta,\mathtt{continue})=0.9-0=0.9$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{5},\mathtt{repair})$
$\displaystyle=\max\limits_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{5})}\Delta\omega(v_{5},\zeta,\mathtt{repair})=0-0=0$

$\displaystyle\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{5},\mathtt{continue})$
$\displaystyle=\max\limits_{\zeta\in
Z^{\sim}(\mathcal{T},\\{i\\},v_{5})}\Delta\omega(v_{5},\zeta,\mathtt{continue})=1-0=1$

As the member contributions for selecting $\mathtt{repair}$ are always 0, i.e.
$\underline{\rho}(G,i,v)=0$, it holds that
$\mathbf{r}^{\mathrm{risk}}(G,i,v,\mathtt{a})=\mathbf{r}^{\mathrm{negl}}(G,i,v,\mathtt{a})$
for all $G\subseteq I$, $i\in G$, $v\in V_{i}$ and $\mathtt{a}\in A_{v}$ in
this example.

Now we can use the member contribution values together with the aggregation
function $\mathbf{mprod}$ to compute the outcome responsibility.

$\displaystyle\mathbf{mprod}\circ\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,w_{1})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{repair})=0=\mathbf{mprod}\circ\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,w_{1})$

$\displaystyle\mathbf{mprod}\circ\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,w_{2})$
$\displaystyle=\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{continue})=0.9=\mathbf{mprod}\circ\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,w_{2})$

$\displaystyle\mathbf{mprod}\circ\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,w_{3})$
$\displaystyle=1-[(1-\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{continue}))(1-\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{3},\mathtt{repair})]$
$\displaystyle=1-[(1-0.9)(1-0)]=1-[0.1]=0.9$
$\displaystyle=\mathbf{mprod}\circ\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,w_{3})$

$\displaystyle\mathbf{mprod}\circ\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,w_{4})$
$\displaystyle=1-[(1-\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{continue}))(1-\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{3},\mathtt{continue})]$
$\displaystyle=1-[(1-0.9)(1-0.9)]=1-[0.01]=0.99$
$\displaystyle=\mathbf{mprod}\circ\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,w_{4})$

$\displaystyle\mathbf{mprod}\circ\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,w_{5})$
$\displaystyle=1-[(1-\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{continue}))$
$\displaystyle\phantom{=1-[(1-}\cdot(1-\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{3},\mathtt{continue})(1-\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{5},\mathtt{repair}))]$
$\displaystyle=1-[(1-0.9)(1-0.9)(1-0)]=1-[0.01]=0.99$
$\displaystyle=\mathbf{mprod}\circ\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,w_{5})$

$\displaystyle\mathbf{mprod}\circ\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,w_{6})$
$\displaystyle=1-[(1-\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{1},\mathtt{continue}))$
$\displaystyle\phantom{=1-[(1-}\cdot(1-\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{3},\mathtt{continue})(1-\mathbf{r}^{\mathrm{risk}}(\\{i\\},i,v_{5},\mathtt{continue}))]$
$\displaystyle=1-[(1-0.9)(1-0.9)(1-1)]=1-[0]=1$
$\displaystyle=\mathbf{mprod}\circ\mathbf{r}^{\mathrm{negl}}(\\{i\\},i,w_{6})$