1 Introduction
The current climate crisis and its associated effects constitute one of the essential challenges for humanity and collective decision making in the upcoming years. An increase of greenhouse gas (GHG)^{1}^{1}1Prominently CO, but also methane, nitrous oxide and others. concentrations in the atmosphere attributable to human activity leads to a warming of Earth’s surface temperature by reducing the fraction of incoming solar radiation that is diffused back into space. An elevated mean earth surface temperature is however not a priori something reprehensible. Rather, it is the resultant effects that carry enormous dangers. Among these are the increased risk of extreme weather events such as storms and flooding, the rise of sealevels or the immense losses of biodiversity, which have repercussions not only for the physical integrity of the planet but which pose direct threats to human life.^{2}^{2}2See for example [19] for a concise overview of the relevant climate science explained for nonclimate scientists, or the IPCC and World Bank reports for more detail [23, 35].
Naturally, the public debate around this issue frequently invokes the question of responsibility: Who carries how much backwardlooking responsibility for the changes already inevitable, who is to blame; and who carries how much forwardlooking responsibility to realise changes, who has to act?^{3}^{3}3What we call “forwardlooking” or exante responsibility is closely linked to the idea of obligation or duty, whereas what we call “backwardlooking” or expost responsibility has also been called accountability, and relates to blame [39, 9] As the following citation from Mike Huckabee, twice candidate in the US Republican presidential primaries, shows, the concepts of both backward and forward responsibility is used throughout the political spectrum: “Whether humans are responsible for the bulk of climate change is going to be left to the scientists, but it’s all of our responsibility to leave this planet in better shape for the future generations than we found it.” [22]
Existing work.
The existing body of work regarding this question can roughly be divided into two categories, via the perspective from which this question is addressed. On the one side there are considerations focusing on applicability in the climate change context, computing tangible responsibility scores for countries or federations, with the aim of shaping the actions being taken and a lesser focus on conceptual elegance and consistency [6, 30]. On the other side there is considerable work in formal ethics, aiming at understanding and formally representing the concept of responsibility in general with a special focus on rigour and wellfoundedness, making it harder to account for messy real world scenarios (in realistic computation time) [5, 8, 13, 21].
It will be useful to highlight certain aspects of these works now. In the former set of works, and particularly also in public discourse, the degree of backward responsibility of a person, firm, or country for climate change is simply equated to cumulative past GHG emissions, or a slight variation of this measure [14]. Certainly, this approach has one clear benefit, namely that it is easy to compute on any scale, and also extremely easy to communicate to a nonscientific audience. Similarly, certain authors assume a country’s degree of forward responsibility to be proportional to population share, gross domestic product or some similar indicator, specifically in the debate about “fair” emissions allowances or caps [36, 34]. However, unfortunately, such ad hoc measures violate certain properties that one would ask of a generalised responsibility account.^{4}^{4}4For example, using cumulative past emissions, population shares or GDP ratios all result in a strictly additive responsibility measure. If agent has a responsibility score of and agent one of the group consisting of agents and has a score of . However, consider an example of two agents simultaneously shooting a third person. According to some intuitions, e.g., the legal theory of complicity [24], they would then both be responsible to a degree larger than just half of the responsibility of a lone shooter. So we would need to either allow for group responsibility measures above 100% (above total cumulative emissions/population share/GDP), or we would need to abandon additivity. Another issue of the cumulative emissions account is that many climate impacts are not directly proportional to emissions, a topic that will be discussed later on in this section.
In the latter body of work, a principled approach is taken. Starting from considerations regarding the general nature of the concept of responsibility, formalisms are set up to represent these. These comprise causal models [13], gametheoretical representations [46, 8] or logics [12, 39]. A vast number of different aspects have been included in certain formalisations, such as degrees of causation or responsibility, relations between individuals and groups, or epistemic states of the agents to name but a few. Generally, these are discussed using reduced, welldefined example scenarios and thought experiments capturing certain complicating aspects of responsibility ascription.
Additionally, there are investigations into the everyday understanding of the various meanings of the term ‘responsibility’ [43] as well as empirical studies regarding agents’ responsibility judgements in certain scenarios, showing a number of asymmetry results [32]. However, we are here not concerned with mirroring agent’s actual judgements, but rather with a normative account, so we will not go into detail about these.
Research question.
The present paper places itself in the category of a principled and formal approach, but aims at keeping in mind the practical applicability in complex scenarios. Also, we want to relocate the space of discussion in the formal community by proposing a set of responsibility functions that, rather than cautiously distributing responsibility and tolerating underdetermination (or voids), distribute responsibility somewhat more generously, evading certain forms of underdetermination, but sometimes resulting in what might be seen as overdetermination. The “correct” function is probably somewhere in between, and we think it is helpful to examine the space of possible solutions from several ends. It might be useful to add that our work is normative, not descriptive. We aim at representing ways in which responsibility should be ascribed, not the ways in which people in standard discussion generally do ascribe it or are psychologically inclined to perceive.
We introduce a suitable framework that is able to represent all relevant aspects of a decision scenario. In some core aspects this is an extension of existing frameworks, in others we deviate from the previous work. Subsequently, we will suggest candidate functions for assigning real numbers as degrees of responsibility (forward as well as backward looking) that have certain desirable properties.
Deliberation regarding which climate abatement goal is to be reached but also who will contribute how much in the joint effort to mitigate climate change is often carried out in the political sphere, with various voting mechanisms in place. It is therefore particularly interesting to determine measures of responsibility when the deliberation procedure is given by a specific voting rule. We will address this question for a set of voting scenarios and our proposed responsibility functions.
Method.
We will follow an axiomatic method as it is used in social choice theory in order to enable a wellstructured comparison between different candidates for responsibility functions [40]. That is, after determining a framework for the representation of multiagent decision situations with ambiguity and corresponding responsibility functions as well as their properties, we begin by determining a set of simple, intuitive and basic properties that one might want a prospective candidate for an appropriate responsibility function to fulfil. Our framework is based on the known concept of extensiveform games, with added features to represent the additional information, or rather lack thereof, that we want to include here.
Specific aspects to be considered.
The above outline already shows several features of anthropogenic climate change that complicate responsibility assignments and occur in similar forms in other realworld multiagent decision problems in which uncertainty and timing play a significant role. We will now highlight and discuss several features that our framework will need to include, as well as certain aspects that we treat differently from existing work. One important idea is to avoid allowing agents to refuse taking on responsibility by recurring to a certain calculation, even though according to some intuitions they do carry (higher) responsibility. We will suggestively call such an argumentation scheme dodging, and the corresponding modelling aspect dodgingevasion.
First of all, the effects of climate change are the result of an interaction of many different actors: corporations, politicians, consumers, organisations, groups of these, etc. all play a role. Next, there is considerable uncertainty
regarding the impacts to be expected from a given amount of emissions or a given degree of global warming. While for some results we can assign probabilities and confidence intervals, for others this cannot be done in a wellfounded way and beyond specifying the set of possible alternatives one cannot resolve the
ambiguity with the given state of scientific knowledge.When several models give similar but slightly diverging predictions, for example, as is very often the case, we cannot assign probabilities to either of the models being ‘more right’ than the others. What we can say however, is that each of the predictions is within the set of possible outcomes (given the premises, such as a certain future behaviour). The same goes for varying parameters within one and the same model.
Contrastingly, in a large body of work concerning effects of pollution, or warming, predictions are associated with a specified probability. Take for example the IPCC reports, such as the well known statement about the remaining carbon budget if warming is to be limited to 1.5 degrees: “[…] gives an estimate of the remaining carbon budget of […] 420 GtCO
for a 66% probability [of limiting warming to 1.5°C above preindustrial levels]” [23]. In many cases, both aspects of uncertainty — ambiguity and probabilistic uncertainty — are combined by speaking about intervals of probabilities, which in particular the IPCC does pervasively [29].We argue that it is equally important to take note of the additional information in the probabilistic uncertainty case (often called ‘risk’ in economics^{5}^{5}5Since we use the term ‘risky’ in this article for a different concept, we stick to the term ‘probabilistic uncertainty’ here.) as of the lack thereof in the ambiguity case. It is known that the distinction between probabilistic and nonprobabilistic uncertainty is important in decision making processes, and we want this to be reflected in our attribution of responsibility [17].
As a further particularity, the effects of global warming do not scale in a linear way with respect to emissions. With rising temperatures, so called ‘tipping elements’ such as the Greenland or West Antarctic ice sheets risk being tipped [27]: once a particular (but imprecisely known) temperature threshold is crossed, positive feedback leads to an irreversible procession of higher local temperatures and accelerated degradation of the element.^{6}^{6}6Ice reflects more sunlight than water. Thus, if a body of ice melts and turns into water, this will retain more heat than the ice did, leading to higher temperatures and faster melting of the remaining ice. As is stated in [37]: “The keywords in this context are nonlinearity and irreversibility”. Note that not all tipping elements are bodies of ice — coral reefs or the Amazon rain forest also rank among them. The examples with corresponding explanation were chosen for their simplicity. This initially local effect then aggravates global warming and may contribute to the tipping of further elements [25, 45] adding up to the already immense direct impacts such as in case of these examples a sea level rise of several meters over the next centuries [37].
We think that this nonlinearity should be reflected at least to some extent in the resulting responsibility attribution. This constitutes another argument for deviating from the — linear — cumulative past emissions accounts mentioned above [6, 30].
In contrast to existing formalisations of moral responsibility in gametheoretic terminology [9], we include a temporal dimension in our representation of multiagent decisions by making use of extensiveform game trees rather than normalform games. This temporal component is also featured in formalisations using the branchingtime frames of stitlogics.^{7}^{7}7Note that normalform gametheoretical models correspond to a subclass of stit models [16]. Similarly, extensiveform games can also be represented as a stitlogic [11], but we don’t pursue this further here as the additional features that we will include would complicate a logical representation and this is not currently necessary to express what we want to.
However, we do not take into account the temporal distance of an outcome to the individual decisions that led to it. Unlike in the ongoing debate in the environmental economics community regarding the discounting factors to be employed when considering future damages, with the prominent opposition between William Nordhaus and Nicholas Stern [33, 38] and its “nondecision” by a large expert panel led by Ken Arrow [2]
, our account is not directly affected by any form of discounting. This is because while quantitative measures of welfare depend on notions of preferences, degrees of responsibility depend on notions of causation instead. Still, if the effects of an action disappear over time because of the underlying system dynamics (e.g., because pollutants eventually decay) and if this reduces the probability of causing harm much later, this fact can be reflected in the decision tree via probability nodes.
As another difference to existing formalisations we do not generally allow for assumptions regarding the likelihood of another agent’s actions. We consider every agent to have free will, which we interpret to imply that while agents might have beliefs about others’ behaviour, such beliefs cannot be seen as “reasonable” beliefs that provide justification in the sense of [3]. In other words, while beliefs about others’ actions may influence the psychologically perceived degrees of responsibility of the agents, it should not influence a normative assessment of their responsibility by an ideal ethical observer or “judge”.
Note that even [8] state that the important feature with respect to judging other’s actions in a tragedy of the commons application is that “[b]efore the game was played, each agent assigned at least some positive probability to the strategy combination the others actually did play”, i.e. an unstructured set of possible outcomes suffices. If we want to express probabilistic uncertainty, this can be done, but without specifying an actor.
Unlike those accounts of responsibility focusing on the socalled ‘necessary element of a sufficient set’ (NESS) test to represent causation, such as [8], we employ here the probability raising account as stated by [42]: “I shall assume that the relevant causal connection is that the choice increases the objective chance that the outcome will occur — where objective chances are understood as objective probabilities”. This account lends itself to our approach as we specifically want to discuss situations in which the outcome occurs with a given probability, and it enables a straightforward representation of degrees of responsibility. Note however, that unlike [42] we do not refer to agents’ beliefs regarding the probabilities.
Paradigmatic examples and their evaluation.
In order to better understand the proposed frameworks as well as the responsibility functions, we will refer to a number of paradigmatic examples, mostly known from the literature or moral theory folklore, for illustration purposes. Like thought experiments in other branches of philosophy, such as the famous trolley problem, these examples have been selected because they each represent an interesting aspect of responsibility attribution in interactive scenarios with uncertainty that will come up later in the delineation of the proposed responsibility functions.

Load and shoot. An agent has the choice to shoot at an innocent prisoner or not, not knowing whether the gun was loaded. Represented in Fig. 1(a).

Rock throwing. An agent has the choice to throw a stone into a window or not, not knowing whether another agent already threw a stone before them. Represented in Fig. 1(b).

Choosing probabilities. An agent cannot select an outcome with certainty, but they can influence the probability of a given event. That is, they have the choice between an option where the undesirable outcome has probability and an option where it has probability . Represented in Fig. 1(c).

Hesitation I. The agent has the choice to either rescue an innocent stranger immediately, or hesitate, in which case they might get another chance at rescuing the innocent stranger at a later stage, but it might also already be too late.^{8}^{8}8
While this example clearly seems somewhat odd in direct interaction contexts — imagine a scenario where someone has the choice to save a person from drowning immediately or first finish off their icecream knowing that with probability p the other person will hold up long enough so they can still be rescued — it represents a common issue in climate change mitigation efforts.
Represented in Fig. 1(d).
If the agent does get a second chance and then decides to rescue the stranger, certain accounts will not assign backwards responsibility to them. However, they did in fact risk the stranger’s death, so it can also be argued that they should be held responsible to some degree. 
Hesitation II. An agent, who is a former lifeguard and thus trained in first aid, passes a stranger who is seemingly having a heart attack. They have the choice to either help immediately by calling an ambulance and keeping up CPR until the ambulance arrives, in which case the stranger survives. Alternatively they can hesitate, but decide again at a later stage whether to help after all. In this case it is not certain whether the stranger will survive. Represented in Fig. 3.
This example is parallel to the one before in the sense that the agent can in a first step hesitate, with an uncertainty determining either before or after their second decision to help after all whether this decision is an option, or whether it is successful. While one might think two consecutive decisions can be considered equivalent to one single combined decision it can be argued in this case that if the agent does not end up helping they failed twice and should thus possibly carry higher responsibility. 
Climate Change. Humanity (agent ) has the choice to either heat up the earth or not, not knowing whether they are in a state of impending heating due to the greenhouse effect or a state of impending cooling due to an onsetting ice age.

Knowledge gain. Here Humanity (agent ) is again posed before the same issue as in the previous example. But this time they have the added opportunity to learn about which state they are in (impending ice age or not) before deciding on an action.
The examples Load and shoot and Rock throwing are parallel to one another, both including situations in which the agent might not actually be able to influence the outcome (because either the gun is not loaded so it does not matter whether they shoot or not, or because the other agent already threw a stone that will shatter the window), but they do not know whether they are in this situation or in the one where their action does have an impact. In both cases we argue that the responsibility ascription must take into account the viable option that the agent’s action will have/would have had an impact. Therefore, the agent cannot dodge responsibility by referring to this uncertainty. They should be assigned full forward and backward (if they select the possibly harmful action) responsibility. This relates to the discussion about moral luck, and the case for disregarding factors that lie outside of the agent’s control is argued in [31]: “Where a significant aspect of what someone does depends on factors beyond his control, yet we continue to treat him in that respect as an object of moral judgment, it can be called moral luck. Such luck can be good or bad. […] If the condition of control is consistently applied, it threatens to erode most of the moral assessments we find it natural to make.”
This also relates to a prominent criticism of the probability raising account for causation, namely that an agent may raise the probability of an event without this event actually occurring as a result, as the probability stayed below 1. Similarly to situations in which the event does not end up occurring due to the actions of others that the agent had no knowledge or influence over, we argue that this should not reduce responsibility ascription but rather be interpreted as a form of ‘counterfactual’ responsibility.
Structure.
The rest of the paper is structured as follows. We will begin in Sect. 2 with a presentation of the proposed framework in which the responsibility functions as well as their desired properties will be formulated. Additionally, we explicate a number of desirable properties that will be important in drawing a difference between the various responsibility functions. In Sect. 3 we introduce four different candidate responsibility functions (all differentiated between backward and forwardlooking formulations) and determine which of the axioms they fulfil. Subsequently, in Sect. 4 we present a number of voting scenarios known from social choice theory and determine agent’s responsibility ascription within these scenarios. In Sect. 5 we discuss selected aspects of our results and finally conclude in Sect. 6.
2 Formal model
We start this section by proposing a specific formal framework for the study of responsibility in multiagent settings with stochasticity and ambiguities. It is based on the gametheoretical data structure of a game in extensive form, which is a multiagent version of a decision tree, but with the additional possibility of encoding ambiguity via a special type of node. Also, in contrast to games, we do not specify individual payoffs for all outcomes but only a set of ethically undesired outcomes. This is sufficient, as we will not apply any gametheoretic analyses referring to rational courses of actions or utility maximisation but rather use this data structure to talk about responsibility assignments.
2.1 Framework
We use
to denote the set of all probability distributions on a set
, and use the abbreviations , , , .Trees.
We define a multiagent decisiontree with ambiguity (or shortly, a tree) to be a structure
consisting of:

A nonempty finite set of agents (or players).

For each , a finite set of ’s decision nodes, all disjoint. We denote the set of all decision nodes by .

Further disjoint finite sets of nodes: a set of ambiguity nodes, a set of probability nodes, and a nonempty set of outcome nodes. We denote the set of all nodes by .

A set of directed edges so that is a directed tree whose leaves are exactly the outcome nodes:
For all , let denote the set of possible successor nodes of .

An information equivalence relation on so that implies . We call the equivalence classes of in the information sets of .

For each agent and decision node , a nonempty finite set of ’s possible actions in , so that whenever , and a bijective consequence function mapping actions to successor nodes, .

For each probability node , a probability distribution on the set of possible successor nodes.
Our interpretation of these ingredients is the following:

A tree encodes a multiagent decision situation where certain agents can make certain choices in a certain order, and outcome node represents a possible ethically relevant state of affairs that may result from these choices.

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

Probability and ambiguity nodes and informationequivalence are used to represent various types of uncertainty and agents’ knowledge at different points in time regarding the current state of affairs, immediate consequences of possible actions, future options and their possible consequences, and agents’ future knowledge at later nodes. The agents are assumed to always commonly know the tree, and at every point in time to know in which information set they currently are. In particular, they know that at any probability node , the possible successor nodes are given by and have probabilities , . In contrast, about an ambiguity node they only know that the possible successor nodes are given by , without being able to rightfully attach probabilities to them. Ambiguity nodes can also be thought of as decision nodes associated to a special agent one might term ‘nature’.
In contrast to the universal uncertainty at the treelevel encoded by probability and ambiguity nodes, informationequivalence is used to encode uncertainty at the agent level. While in a certain information set of informationequivalent decision nodes, an agent cannot distinguish between nodes and has the same set of possible actions .

When setting up a tree model to assess some agent ’s responsibility, the modeler must carefully decide which actions and ambiguities to include. If the modeler follows the basic idea that what matters is what “reasonably believes” in any decision node (as in [3]), then should consist of those options that reasonably believes to have, for should reflect what possibilities reasonably beliefs exist at , the choice whether is an ambiguity or probability node should depend on whether can reasonably believe in certain probabilities of these possibilities, and if so, then should reflect those subjective but reasonable probabilities. Likewise, if the modeler follows the view that certain forms of ignorance may be a moral excuse (as in [47]), the information equivalence relation should reflect what ignorance of this type the agents have.
An ambiguity node whose successors are probability nodes can be used to encode uncertain probabilities like those reported by the IPCC [29] or those corresponding to the assumption that “nature” uses an Ellsberg strategy [15].
Note that in contrast to some other frameworks, e.g., those using normalform (instead of extensiveform) game forms such as [9], our trees do not directly allow for two agents to act at the exact same time point. Indeed, in a real world in which time is continuous, one action will almost certainly precede another, if only by a minimal time interval. Still, as in the theory of extensiveform games, two actions may be considered “simultaneous” for the purpose of the analysis if they occur so close in time that the later acting player cannot know what the earlier action was, and this ignorance can easily be encoded by means of information equivalence in a way similar to Fig. 6.
Events, groups, responsibility functions (RFs).
As in probability theory, we call each subset
of outcomes a possible event. In the remainder of this paper, we will use to represent an ethically undesirable event, such as the death of an innocent person, the occurrence of strong climate change, or the election of an extremist candidate, whose probability might be influenced by the agents.Any nonempty subset of agents is called a group in this article.^{9}^{9}9Note that we deliberately do not require that a set of agents shares any identity or possesses ways of communication or coordination for an ethical observer to meaningfully attribute responsibility to this “group”.
Our main objects of interest are quantitative metrics of degrees of responsibility that we formalise as backwardresponsibility functions (BRF) and forwardresponsibility functions (FRF) .
A BRF maps every combination of tree , group , event , and outcome node to a real number meant to represent some form of degree of backwardlooking (aka expost or retrospective) responsibility of regarding in the multiagent decision situation encoded by when outcome has occurred.
An FRF maps every combination of tree , group , event , and decision node to a real number meant to represent some form of degree of forwardlooking (aka exante) responsibility of regarding in the multiagent decision situation encoded by when in decision node .
If , we also write . Whenever any of the arguments , , , are kept fixed and are thus obvious from the context, we omit to explicate them when writing or any of the auxiliary functions defined below.
Graphical representation.
As exemplified in Fig. 1, we can represent a tree and event graphically as follows. Edges are arrows, decision nodes are diamonds labelled by agents, with arrows labelled by actions, ambiguity nodes are unlabelled diamonds, probability nodes are squares with arrows labelled by probabilities, and outcome nodes are circles, filled in grey if the outcome belongs to . Finally, information equivalence is indicated by dashed lines connecting or surrounding the equivalent nodes.
Auxiliary notation.
The set of decision nodes of a group is . To ease the definition of “scenario” below we denote the set of nonprobabilistic uncertainty nodes other than (i.e., non decision and ambiguity nodes) by .^{10}^{10}10This can be thought of as nodes where someone who is not part of group G — another agent or Nature — takes a decision.
If , we call the predecessor of . Let be the root node of , i.e., the only node without predecessor. The history of is then , where is the root node of . In the other direction, we call the (forward) branch of . Taking into account information equivalence, we also define the information branch of as .
If , , and , we call the choice at that ultimately led to node .
A node with is called a complete information node.
Strategies, scenarios, likelihoods.
We call a function that chooses actions for some set of ’s decision nodes a partial strategy for at iff , , whenever , and for all and . The latter condition says that does not specify actions for decision nodes that become unreachable by earlier choices made by . A strategy for at is a partial strategy with a maximal domain . This means that a strategy specifies actions for all decision nodes that can be reached from the information set containing given the strategy.
Let (or shortly if are fixed) be the set of all those strategies. For , let
i.e., the set of possible outcomes when follows from on.
Complementary, consider a function that chooses successor nodes for a set of ambiguity or others’ decision nodes, and some node . Then we call a partial scenario for at iff , or , , and for some whenever , and for all and . The latter condition says that does not specify successors for nodes becoming unreachable under . A scenario for at is a partial scenario with a maximal domain . This means that a scenario specifies successors for all ambiguity and others’ decision nodes that can be reached from or the information set containing given the scenario.
Let (or shortly ) be the set of all scenarios at and (or shortly ) that of all scenarios at with .
Each strategyscenario pair induces a Markov process on leading to a prospect, i.e., a probability distribution on the potential future outcome nodes, that can be computed recursively in the following straightforward way:
(1)  
(2)  
(3)  
(4)  
(5) 
Let us denote the resulting likelihood of by
2.2 Axioms
Following an axiomatic approach similar to what social choice theory does for group decision methods and welfare functions, we study RFs by means of a number of potentially desirable properties formalized as axioms.
In the main text, we focus on a selection of axioms which turn out to motivate or distinguish between certain variants of RFs that we will develop in the next section and then apply to social choice mechanisms. In the Appendix, a larger list of plausible axioms is assembled and discussed.
All studied RFs fulfill a number of basic symmetry axioms such as anonymity (treating all agents the same way), and a number of independence axioms such as the independence of branches with zero probability, and, more notably, also the following two axioms:
 (IOA)

Independence of Others’ Agency. If , and some of ’s decision nodes is turned into an ambiguity node with , then remains unchanged (i.e., it is irrelevant whether uncertain consequences are due to choices of other agents or some nonagent mechanism with ambiguous consequences).
 (IGC)

Independence of Group Composition. If and all occurrences of are replaced in by , remains unchanged.
Note that these two conditions preclude dividing a group’s responsibility equally between its members or following other agent or groupcounting approaches similar to Banzhaf’s or other power indices.
The first two axioms that only some of our candidate RFs will fulfill are the following:
 (IND)

Independence of Nested Decisions. If a completeinformation decision node is succeeded via some action by another completeinformation decision node of the same agent, then the two decisions may be treated as part of a single decision, i.e., may be pulled back into : may be eliminated, added to , added to , and extended by for all .
 (IAT)

Independence of Ambiguity Timing. Assume some probability node or completeinformation decision node is succeeded by an ambiguity node . Let , be the original branches of the tree starting at , and any . For each , let be a new copy of the original in which the subbranch is replaced by a copy of ; let be that copy of that serves as the root of this new branch . If , put for all Let be a new branch starting with and then splitting into all these new branches . Then may be “pulled before” by replacing the original by the new , as exemplified in Fig. 4.
(IND) may seem plausible if one imagines, say, a decision to turn either left or right directly followed by a decision to stop at 45 or 90 degrees rotation, since these two may more naturally be considered a single decision between four possible actions, turning 90 or 45 degrees left or right. But in the situation of Fig. 3, it may rather seem that when hesitating and then passing, has failed twice in a row, which should perhaps be assessed differently from having failed only once.
When using an RF with all the above properties to assess responsibility of a particular group of agents, one can “reduce” the original tree to one that has only a single agent (representing the whole group, all other actions being represented as simple ambiguity). If one accepts a number of similar further axioms listed in the Appendix, one can also assume the reduced tree has only properly branching nonoutcome nodes, has at most one ambiguity node and only as its root node, and has no two consecutive probability nodes and no zero probability edges.
The next pair of axiom state that responsibility must react in the right direction under certain modifications:
 (GSM)

Group Size Monotonicity. If then (i.e., larger groups have no less responsibility).
 (AMF)

Ambiguity Monotonicity of Forward Responsibility. If, from an ambiguity node , we remove a possibility and its branch , then forward responsibility in any remaining node does not increase.
In other words, (AMF) requires that increasing ambiguity should not lower forward responsibility (because that might create an incentive to not reduce ambiguity).
The next three axioms set lower and upper bounds for responsibility, the first taking up a condition from [9]:
 (NRV)

No Responsibility Voids. If there is no uncertainty, , and if , then for each undesired outcome , some group is at least partially responsible, .
 (NUR)

No Unavoidable Backward Responsibility. Each group must have an original strategy that is guaranteed to avoid any backward responsibility, i.e., so that for all .
 (MBF)

Maximal Backward Responsibility Bounds Forward Responsibility. For all , there must be and so that (i.e., forward responsibility is bounded by potential backward responsibility).
Finally, we consider four axioms that require certain assessments in the paradigmatic situations from Fig. 1 which are closely related to questions of moral luck [31, 1, 41], reasonable beliefs [3], and ignorance as an excuse [47]:
 (NFT)

No Fearful Thinking. With and as depicted in Fig. 1(b), since ’s action makes a difference even though she thinks acting might not help, since it would be unreasonable to believe this must be the case.
 (NUD)

No Unfounded Distrust. With and as depicted in Fig. 1(b), since cannot know that acting cannot help.
 (MFR)

Multicausal Factual Responsibility. With and as depicted in Fig. 1(b), since ’s action was necessary even though not sufficient.
 (CFR)

Counterfactual Responsibility. With and as depicted in Fig. 1(a), since could not know that her action would not cause so she must reasonably have taken into account that it might.
Before turning to the definition of candidate RFs and study their axiom compliance, we briefly mention that while there obviously exist certain logical relationships between subsets of the above axioms (and the further axioms listed in the Appendix), they are not the scope of this article.
3 Candidate responsibility functions
Here we will introduce four pairs of responsibility functions that fulfill most of the above axioms but each also violate a few, and a reference function related to strict causation.
These candidate responsibility functions will measure degrees of responsibility in terms of differences in likelihoods between available strategies in all possible scenarios.
To define them, we need some additional auxiliary notation and terminology. For now, let us keep , , and fixed and drop them from notation.
Since the below definitions typically involve several nodes, we denote the decision node at which is evaluated by , the outcome node at which is evaluated by , and other nodes by so that comes before (i.e., , ).
Benchmark variant: strict causation.
The most straightforward definition of a backward responsibility function in our framework that resembles the strict causation view, as employed for example in the most basic way of ‘seeing to it that’, is to set iff there is a past node at which was certain, , directly following a decision node at which was not certain, , and to put otherwise.
It is easy to see that given , there is at most one such regardless of , and exactly those are deemed responsible which contain the agent choosing at , i.e., for which .
3.1 Variant 1: measuring responsibility in terms of causation of increased likelihood
The rationale for this variant, which tries to translate the basic idea of the stit approach into a probabilistic context, is that backward responsibility can be seen as arising from having caused an increase in the guaranteed likelihood of an undesired outcome.
Guaranteed likelihood, caused increase, backwards responsibility.
We measure the guaranteed likelihood of at some node by the quantity
(6) 
We measure the caused increase in guaranteed likelihood in choosing at decision node by the difference
(7) 
Note that since rather than , we have .
To measure ’s backward responsibility regarding in outcome node , in this variant we take their aggregate caused increases over all choices taken by that led to ,
(8) 
Maximum caused increase, forward responsibility.
Finally, to measure ’s forward responsibility regarding in decision node , we take the maximal possible caused increase,
(9) 
At this point, we notice to potential drawbacks of this variant. For one thing, it fails (IAT), mainly because it does not take into account any information equivalence and thus depends too much on subtle timing issues that the agents information does not depend on and that hence any responsibility assessments should maybe also not depend on. On the other hand, it is in a sense too “optimistic” by allowing agents to ignore the possibility that their action might make a negative difference if this is not guaranteed to be the case. The next variant tries to resolve these two issues.
3.2 Variant 2: measuring responsibility in terms of increases in minimax likelihood
This variant is in a sense the opposite of variant 1 with respect to its ambiguity attitude. To understand their relationship, consider the tree in Fig. 5 which shows that variant 1 can be interpreted as suggesting an ambiguityaffine strategy while variant 2 suggests an ambiguityaverse strategy.
In this variant, the rationale is that backward responsibility can be seen as arising from having deviated from behaviour that would have seemed optimal in minimizing the worstcase (rather than the guaranteed) likelihood of an undesired outcome in view of the information available at the time of the decision. In defining the worstcase, however, we assume a group can plan and commit to optimal future behaviour, so some of the involved quantities are in now terms of strategies rather than actions .
Worst case and minimax likelihoods.
’s worstcase likelihood of at any node given some strategy is given by
(10) 
’s minimax likelihood regarding at is the smallest achievable worstcase likelihood,
(11) 
Note that (11) differs from (6) not only in using a maximum but also in taking into account possible ignorance about the true node by using instead of .
Caused increase, backward responsibility.
We measure ’s caused increase in minimax likelihood in choosing at node by taking the difference
(12) 
again now taking information equivalence into account. Similar to before, to measure ’s backward responsibility regarding in node , we here take their aggregate caused increases in minimax likelihood,
(13) 
Maximum caused increase, forward responsibility.
In analogy to variant 1, to measure ’s forward responsibility regarding in node , we take the maximal possible caused increase in minimax likelihood,
(14) 
While this variant seems well related to the maximintype of analysis known from early game theory, it still fails (NUD) and (MFR), both because it is now in a sense too “pessimistic” by allowing agents to ignore the possibility that their action might make a positive difference.
3.3 Variant 3: measuring responsibility in terms of influence and risktaking
While variants 1 and 2 can be interpreted as measuring the deviation from a single optimal strategy that minimizes either the guaranteed (bestcase) or the worstcase likelihood of a bad outcome taking into account all ambiguities, our next variant is based on families of scenariodependent optimal strategies. In this way, it partially manages to avoid being too optimistic or too pessimistic and thereby fulfil both (MFR) like variant 1 and (CFR) like variant 2. The main idea is that backward responsibility arises from taking risks to not avoid an undesirable outcome.
Optimum, shortfall, risk, backward responsibility.
Given a scenario at any node , the optimum could achieve for avoiding at that node in that scenario is the minimum likelihood over ’s strategies at ,
(15) 
So let us measure ’s hypothetical shortfall in avoiding in scenario due to their choice at node by the difference in optima
(16) 
Then then risk taken by in choosing is the maximum shortfall over all scenarios at ,
(17) 
To measure ’s backward responsibility regarding in node , we now take their aggregate risk taken over all choices they made,
(18) 
Influence, forward responsibility.
Regarding forward responsibility, we test a different approach than before, which is simpler but less strongly linked to backward responsibility. The rationale is that since does not know which scenario applies, they must take into account that their actual influence on the likelihood of might be as large as the maximum of this over all possible scenarios, so the larger this value is the more careful need to make their choices.
Let us measure ’s influence regarding in scenario at any node by the range of likelihoods spanned by ’s strategies at ,
(19)  
(20) 
To measure ’s forward responsibility regarding at node , we this time simply take their maximum influence over all scenarios at ,
(21) 
A main problem with is that it fails (NUR), so that in situations like Fig. 2(a), it will assign full backward responsibility no matter what did. This “tragic” assessment arises because in such situations, there is no weakly dominant strategy that is optimal in all scenarios, hence risktaking cannot be avoided.
3.4 Variant 4: measuring responsibility in terms of negligence
In our final variant, we turn the “tragic” assessments of variant 3 into “realistic” ones, making it fulfil (NUR), by using riskminimizing actions as a reference, but at the cost of losing compliance with (NRV). We also return to the original idea of basing forward responsibility on potential backward responsibility applied in variants 1 and 2 to fulfil (MBF), but at the cost of losing compliance with (AMF).
Riskminimizing action, negligence, backward responsibility.
The minimal risk and set of riskminimizing actions of in decision node is
(22)  
(23) 
where the latter is nonempty but might contain several elements.
We now suggest to measure ’s degree of negligence in choosing at by the excess risk w.r.t. the minimum possible risk,
(24) 
Comparing (24) with (7) and (12), we see that this variant is still sensitive to all scenarios (like variant 3) rather than just the bestcase (as in (7)) or the worstcase (as in (12)). In particular, if a strategy is weakly dominated by some undominated strategy , then using is considered negligent even if the difference between and only matters in cases other than the best or worst.
Now, to measure ’s backward responsibility regarding in node , we suggest to take their aggregate negligence over all choices taken,
(25) 
This now fulfils (NUR) again since by using a riskminimizing strategy for which for all , can avoid all backward responsibility.
Maximum degree of negligence, forward responsibility.
In analogy to variants 1 and 2, to measure ’s forward responsibility regarding in node , we suggest to take the maximal possible degree of negligence,
(26) 
We can now summarize some first results before turning to applying the above RFs in the social choice context.
Proposition 1
Compliance of variants 0–4 with axioms (IND), (IAT), (GSM), (AMF), (NRV), (NUR), (MBF), (NFT), (NUD), (MFR), and (CFR) is as stated in Table 1.
Variant  (IND)  (IAT)  (GSM)  (AMF)  (NRV)  (NUR)  (MBF)  (NFT)  (NUD)  (MFR)  (CFR) 

0  ✓  —  ✓  n/a  ✓  ✓  n/a  n/a  n/a  ✓  — 
1  ✓  —  ✓  —  ✓  ✓  ✓  ✓  —  ✓  — 
2  ✓  ✓  —  —  —  ✓  ✓  —  —  —  ✓ 
3  —  ✓  —  ✓  ✓  —  —  ✓  ✓  ✓  ✓ 
4  —  ✓  —  —  —  ✓  ✓  ✓  ✓  ✓  ✓ 
4 Application to social choice problems
In this section, we apply the abovedefined responsibility functions for measuring degrees of forward and backward responsibility to a number of social choice problems in which an electorate of voters uses some election or decision method or social choice rule to choose exactly one out of a number of candidates or options, one of which, , is ethically undesired. We are interested in the forward responsibility of a group of voters to avoid the election of at each stage of the decision process, and the backward responsibility of for being elected.
We first consider deterministic singleround decision methods in which all voters vote simultaneously and probability plays a marginal role only to resolve ties, and significantly probabilistic singleround decision methods. Afterwards, we study a selection of tworound methods in which voters act twice with some sharing of information between the two rounds. Finally, we turn to an example of a specific stylized social choice problem related to climate policy making.
We exploit all symmetry and independence properties heavily when modeling the otherwise rather large decision trees. In particular, in each round, we implicitly treat a group of many voters as equivalent to a single agent whose action set in a certain round consists of all possible combinations of ’s ballots. We then model the simultaneous decision of all voters in a certain round by a single decision node for , followed by ambiguity nodes representing the choices of the many other voters, one for each possible way or class of ways in which the members of might vote, as exemplified in Fig. 6.
For simplicity, we do not discuss bordering cases in which ties may occur, in particular by assuming the number of voters is odd. Our results are summarized in Table 2.
4.1 Singleround methods
Twooption majority voting.
This is the simplest classical case. Besides the ethically undesired option , there is only one other, ethically acceptable option , and the event to be avoided is the election of . Each voter votes for either or , with no abstentions allowed, and the option with more votes is elected.
We find that no matter how small is, since in the scenario where about half of the other voters vote for , ’s voting determines whether is elected () or (). By contrast, only if , otherwise since then ’s worstcase likelihood is always 1. Similarly, only if since only then they can guarantee a likelihood of 1.
Now assume voters from (and an arbitrary number of the other voters) have voted for . Obviously, iff , since only that guarantees a likelihood of 1.
To determine , we notice that for , ’s worstcase likelihood is always 1, so has zero degree of deviation and ; for , the (unconditional) minimax likelihood is ; the conditional minimax likelihood given is if , otherwise . This implies that only if , otherwise .
To determine , we first consider a scenario in which of the others have voted for ; then ’s optimum is iff , otherwise , and ’s optimum after choosing is iff , otherwise ; hence ’s shortfall in is iff , otherwise . For , the relevant distinction w.r.t. is depicted in Fig. 6. So ’s risk taken by choosing was iff such a scenario exists, i.e., iff and , otherwise . This implies that if , otherwise . Since putting is a weakly dominant strategy, here.
By comparison, we see that in variants 3 and 4 of our responsibility functions, minorities can have nonzero forward and backward responsibility for the election outcome, while in variant 2 only majorities can. In particular, under variants 3 and 4 every single voter who voted for has full backward responsibility since they took the risk that theirs would be the deciding vote.
Also, in all variants all degrees of responsibilities are either zero or one, and the actual voting behaviour of the others is irrelevant for the assessment of backward responsibility.
Note that this is in contrast to the adhoc idea that backward responsibility of should be a more smoothly increasing function of the number of voters from that voted for or maybe even proportional to .
Random dictator.
A major contrast is given by a method that is rarely used in practise but often used as a theoretical benchmark in social choice theory, the “random dictator” method. In addition to option , there are any number of other, ethically acceptable options. Each voter votes for one option, then a voter is drawn at random and their vote decides the election.
As controls exactly a share of the winning probability, their influence on ’s likelihood is in all scenarios, hence . Also since their actions span a range of guaranteed likelihoods or worstcase likelihoods of width . When in have voted for , their shortfall is in all scenarios, hence . Also since their action increased the guaranteed or worstcase likelihood by .
But unless since for a positive probability for remains. This shows that in situations with considerable stochasticity, assessments based on deterministic causation such as differentiate too little to be of any practical use.
Multioption simple majority.
Coming back to majority voting, we next study the case of more than two options, and will see that this leads to much more complicated analysis. With an undesirable option , acceptable options , and the possibility to abstain, we assume the winner is elected by lot from those that got the largest number of votes.
Suppose of the voters from vote for and for , with . Then the guaranteed likelihood of and the value of are 0 (since the others can avoid for sure), iff , they are in iff , and they are 1 iff . So can increase the guaranteed likelihood by 1 (i.e., ) iff (since only then they can make both and ), and otherwise (iff ), .
Likewise, the worstcase likelihood of is 1 (since the others can make win for sure) iff , it is in iff , and it is 0 iff . So can increase the minimax likelihood by 1 (i.e., ) iff (since only then they can make both and ), and otherwise (iff ), . Hence iff and , iff and , and otherwise.
If all others abstain, ’s influence is 1, hence no matter how small . Assume . Then of the others, many could vote for and all others could abstain, so that gets elected for sure. If , could then have avoided this outcome by increasing by 2. Hence if , takes risk 1 and gets . But also if , , and , takes risk 1, since of the others, many could voter for , two for a third option that did not vote for, and all others could abstain, so that again gets elected for sure and could have avoided it by voting for the same as the others. This shows that for , no matter what does, hence no matter what does, and hence .
So, in contrast to the twooption case, in the multicanditate case only variant 3 assigns full backward responsibility to a single voter who votes for , while variant 4 acknowledges the possible excuse that, because there is not a unique contender to , and hence no weakly dominating strategy, also any other way of voting of the single voter could have helped win.
But variant 3 has the major problem that it assigns full backward responsibility to every minority regardless of their behaviour.
Approval voting.
Here everything is just as in multioption simple majority, except that a voter can now vote for any number of options at the same time [10]. Also the analysis is the same as before, except that now also a minority has a weakly dominating strategy that reduces their risk to zero, namely voting for all options but . As a consequence, now again, and a single voter voting for but no other option has full backward responsibility in variants 3 and 4.
Full consensus or random dictator.
As another probabilistic voting method, let us look at a method studied in [20] that was designed to give each group of voters an effective decision power proportional to their size (in contrast to majoritarian methods which give each majority full effective power and no minority any effective power).
In this method, each voter marks one option as “favourite” and one as “consensus”. If all mark the same option as consensus, is elected, otherwise the option marked as favourite on a random ballot is elected. Let be ’s “favourite” votes for and ’s “consensus” votes for option .
If no member of distinguished between her two votes, the analysis is the same as for the random dictator method. If all in put some as consensus (), the guaranteed likelihood of stays at zero since might still get 100% winning probability. In that case, even if wins.
All worstcase likelihoods come from scenarios where the others specify as consensus, so the assessment in variant 2 is the same as in random dictator.
Regarding risk, however, we find that always . This is because there is a scenario where everyone not in elects as favourite and the same option as consensus. In this case ’s optimal strategies are those where everyone also selects as consensus, leading to a zero probability of winning. If some of ’s members select another option as consensus, the resulting likelihood of being elected is , which amounts to the risk taken by .
Hence and . Since the least possible risk is , and .
Note that as cannot know which option the others select they cannot in the scenario described above know which option to select as consensus.
Other majoritarian methods.
In any singleround method in which any group of many members have a way of voting that enforces any option they might choose, we will have and .
Other proportional power allocating methods.
In any singleround method in which any group of many members have a way of voting that guarantees any option they might choose a probability of at least , we will have , .
4.2 Tworound methods
Realworld social choice situations often turn out to consist of several stages upon closer examination even when the “main” voting activity consists of all voters acting “simulteneously”. There are many ways in which decisions taken before or after the main voting stage may be relevant, including the preselection of options or candidates put on the menu e.g. via “primaries”, taking and publishing any preelection polls, seeing an election in the context of previous and future elections, using one or several runoffs rounds to narrow down the final choice, challenging a decision afterwards in courts, etc.
We select here three paradigmatic examples that we believe cover most of the essential aspects: (i) a round of preelection polling as a very common example of “cheap talk” before the actual decision that has no formal influence on the result; (ii) the possibility of amending an option as an example of influencing the menu of a decision that is very common in committee and parliamentary proceedings; (iii) a simple runoff round after taking the main vote, as an example of an iterative procedure commonly used in public elections in order to ascertain a majority.
Simple majority with a preelection poll.
Before the actual voting by simple majority, a poll is performed and the options’ total vote shares are published, so voters might form beliefs about each others’ eventual voting. But since our responsibility measures are independent on any beliefs the agents might form about the probabilities of other agents’ unknown choices at free will, which are rather treated like any other ambiguity, the polling has no influence on our assessment. Backward responsibility depends only on actual voting behaviour, and forward responsibility is zero when answering the poll. The same is true of any other form of prevoting “cheap talk”.
In particular, in all our variants, even the prediction of a landslide victory of does not reduce the responsibility to help avoiding .
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