Degrees of Causation

Abstract

The primary aim of this paper is to analyze the concept of degrees of causal contribution for actual events and examine the way in which it can be formally defined. This should go some way to filling out a gap in the legal and philosophical literature on causation. By adopting the conception of a cause as a necessary element of a sufficient set (the so-called NESS test) we show that the concept of degrees of causation can be given clear and even empirical meaning. We then apply a game theoretical framework to derive a measure of causal contribution. Our favoured measure turns out to be a generalised version of the normalized Penrose–Banzhaf index of voting power.

1 Introduction

There is an important and vexing issue in legal and moral theory that has yet to be clarified in a satisfactory way, that of assigning degrees of causation to persons whose actions played a role in bringing about some outcome. That is, given all the information relevant to a particular circumstance and outcome, what is the meaning of the statement ‘Smith made a larger causal contribution to the realization of some state of affairs than Jones’? This paper is aimed at providing such a clarification.

One can of course immediately ask what, if any, meaning can be attached to the notion of degrees of causation for particular or singular events (or ‘cause in fact’ in legal terminology). ¹

For, if some event is an antecedent condition for a particular state of affairs, then that event is simply a cause or in complex situations a causal factor, and not more or less of a cause than some other antecedent condition. Like being pregnant, a causal ascription is categorical and thus has no magnitude.

Yet, the fact remains that not only do we make statements about the degrees of causation in ordinary speech but in many legal and moral contexts the concept is required for the practical purpose of distributing responsibility. This is especially the case if we take into account justice considerations in which punishment and reward are supposed to be at least in line with, if not proportional to, a person’s contribution to an outcome ²; and it is also the case for institutional design where we have to assign duties and obligations based upon institutionally determined capacities for action. ³

The contexts in which we need to make intelligible statements about degrees of causation are not uncommon and are particularly prominent in cases of collective action. These include those circumstances in which different parties (acting either alone or in concert) perform actions which together contribute to the emergence of a state of affairs but in which that outcome is not divisible (‘non-severable’ in legal terminology) in terms of those actions (i.e. aspects of the resulting state of affairs cannot be uniquely traced back to particular actions) ⁴; and they occur in cases of causal overdetermination or those circumstances in which the causes or causal factors of a state of affairs are cumulatively more than sufficient to generate the state of affairs and which may be asymmetric in character.

An example of a non-divisible (non-severable) outcome is the case discussed by Wright (1985) in which different firms cause a harm by simultaneously emitting amounts of effluent into a river and in which no single firm could cause the damage alone. Another version of this pollution problem is if the different firms emit the same amounts of effluent but of different kinds. One firm emits two different toxins into the river and a second firm only one, but the emission of three types form the post factum necessary and sufficient conditions for the harm. An example of overdetermination in which we may want to impute different degrees of causal impact to the agents involved is when two firms simultaneously pour toxins into a river with one firm dumping twice as much as the other, but in which the actions of both firms are in itself sufficient to cause a certain harm.

The common characteristic which all these examples exhibit is that in each there is an asymmetry in the actions available to, and performed by, the agents. This suggests that causal efficacy is not an all-or-nothing affair but a difference in degree. How are we to capture this asymmetry in our causal judgements? Even in relatively simple cases where there is no overdetermination, such as in the first two pollution examples above, it is not obvious whether the causal contributions to the harm should be taken to be equal or not.

Given the practical demand, can any factual content be given to the notion of degrees of causation? One prominent answer, which is the one that Hart and Honoré (1959, p. 233ff) provided in their seminal monograph Causation in the Law is to say that the concept, while valid, is inescapably vague with its substance being provided by attributive terms of ordinary language. That is, ‘degrees of causation’ is captured by locutions such as the ‘chief’ or ‘main’ or ‘principal’ cause, or of ‘more important’, ‘effective’ or ‘potent’ causes. Lifting from Hart and Honoré, we say, for instance, ‘His failure in the examination was due more to his not working than the difficulty of the papers’ or ‘The main [chief, principal] cause of [factor in] his success as a miler was his assiduous training.’

Critics of the concept of degrees of causation would be justified in retorting that what Hart and Honoré have to say is vacuous because it says nothing about why one causal factor is more important, effective, potent etc. than another. Ordinary language only teaches us how the concept of degrees of causation is commonly used but does not determine the truth-value of such quantitative attributions.

To determine the truth-value of quantitative attributions of causal impact we need to settle two issues, neither of which has been tackled in the literature. One is the units of measurement that are assignable to the respective actions that brought about an outcome; the other is the method of aggregating such units. With few exceptions, ⁵ philosophers have focussed exclusively on the meaning of the claim that an agent’s action was ‘causal’ or a ‘causal factor’ for an outcome and not on the meaning of the claim that one person’s action had, in some sense, a greater impact than another’s. This has left a significant void in our understanding of causality in general and responsibility in particular. The current situation is like saying that Smith is taller than Jones yet at the same time confessing that we have no clue how we arrive at such a conclusion. ⁶

Can this gap be filled? Can we render meaningful quantitative attributions of causal impact that distinguish between the various gradations of ‘more than’ in a non-arbitrary way? We believe the answer is a qualified ‘yes’ and will develop the argument in six steps. Our first task (Sect. 2) is to outline the concept of causal contribution that we will be working with. Here we will simply take on board what is increasingly becoming accepted—at least in the legal literature—as the most plausible and comprehensive account of actual causation, the so-called ‘NESS test’. We present two versions of it—a weak and a strong one. Next (in Sect. 3), and this is where our contribution gets off the ground, we will defend the idea that causal contribution has extensive magnitude by drawing a distinction between specific and overall causal impact of an action. While the former does not come in degrees, the latter does because, as we will demonstrate, overall causal impact is the aggregation of specific impacts as determined by the NESS test. In the third step (Sect. 4) we recast the two versions of the NESS test in the language of game theory and use this in our fourth step to construct and examine two corresponding and very simple functions for measuring overall causal impact (Sects. 5, 6). Finally we show how our analysis is related to existing measurements of individual power in voting games (Sect. 7). We conclude (Sect. 8) with some remarks about responsibility. ⁷

2 Unravelling Causal Ascriptions

We need to start with some preliminaries about the conception of causality that we will adopt. First, and very broadly, we take a ‘cause’ to be a relation between distinct events (in our case, actions) in the same time series, in which one ancestral event, c, has the efficacy to produce, or be part of the production, of another, the effect, e, i.e. c is a condition for e. Second, we will assume, as is generally the case in legal theory, and particularly in tort law, that causation is a relation of dependency to be understood in terms of necessary or sufficient conditions (Honoré 1995). This means we will set aside the definition of a cause or causal condition as being a ‘probability raising event’, i.e. c is a cause of e if the occurrence of c raised the probability that e would occur. ⁸

While this is certainly a route to follow, it has a fundamental weakness that Lewis (2004, p. 79) has noted: not all probability raising events should count as a cause or causal condition because it is possible to raise the probability of an outcome via some unactualized event. ⁹

There are different ways of saying that c produced or contributed to the production of e. One way is to say that c is necessary for e: if e occurs then c must have preceded it. Another way to attribute causal status to c is to say that c is sufficient for e: if c occurs then e follows. Another way is to say that c was necessary and sufficient for e. Yet another way is to say a cause is conditional dependency that satisfies what is known as the ‘but for test’: c is a cause of e if and only if but for the occurrence of c, e would not (have) occur(red). ¹⁰

As is well known, necessity, sufficiency, and but-for dependence turn out to be unworkable. Sufficiency and the but-for test can fail if an outcome required a complex of events, such as collective action; and necessity as well as sufficiency and but-for dependency will fail whenever there is causal overdetermination. To elaborate, suppose three individuals are walking in the woods and they come across an injured jogger trapped under a fallen tree trunk. It takes at least two to lift the trunk and rescue the jogger but as it happens all three do the lifting. No individual’s action of lifting is sufficient, none is necessary, and none satisfies the ‘but-for’ condition. Hence nobody can be ascribed causal status for the rescue. A clearly unsatisfactory judgement. There are many other germane examples in the literature but we need not rehearse them here. ¹¹

The increasingly accepted solution to this conundrum is to use a form of dependence that subordinates the necessity criterion to the one of sufficiency and replaces the idea of identifying c as ‘the cause’ (‘sufficiency’ or ‘necessity and sufficiency’) or ‘a cause’ (‘necessity’) of e with that of identifying c as a ‘causally relevant factor’ of e. This conception ascribes c causal status for e if it satisfies the following criterion known as the NESS test (Wright 1988, p. 1020):

Definition 2.1

(Weak NESS test) There is a set of events that is sufficient for e such that: (i) c is a member of the set; (ii) all elements of the set obtain; (iii) c is necessary for the sufficiency of the set.

In words: c is a causal condition for e if c is a necessary element of a sufficient set of conditions for e. ¹² Here the ‘but for’ test is nested: ‘but for the presence of c the particular set of conditions containing c would not be sufficient for e’. In Pearl and Halpern’s (2005) language, a NESS test is a test of ‘contingent dependency’ which, roughly speaking, says that c is a cause of e if e depends on c under some contingency that was present on the occasion. ¹³

The NESS test as formulated is not the test as it is commonly used. A restriction that is generally made and accepted is that the sufficient sets must be minimal in the sense that no proper subset of the events is itself sufficient for the outcome in question. ¹⁴

This yields what we call the ‘strong’ NESS test because it has a strengthened third clause:

Definition 2.2

(Strong NESS test) There is a set of events that is sufficient for e such that: (i) c is a member of the set; (ii) all elements of the set obtain; (iii′) all elements of this set are necessary for its sufficiency.

There are a number of important and general remarks to be made with respect to both versions of the NESS test. The first is that because we are dealing with singular causation we assume that it is clear what the relevant events are that are to be taken into the picture of the particular case at hand. This means, for instance, that we know which part of the causal chain that led to some action is relevant for the assessment of a person’s causal contribution and which part is not. To establish, say, the causal impact of Smith’s driving on the accident in which he was involved, we assume that we can ignore the bad temper of Smith’s partner at breakfast even though that temper caused Smith to drive with less attention than he otherwise does. For our purposes we can make the additional simplifying assumption that the relevant facts about the world and the mechanisms are known to us. So, to return to our earlier example, when we say that one of the walkers makes a causal contribution to the rescue of the injured jogger we assume that we know which factors determine who can lift up the fallen tree. It is much like saying that we know the outcome of a vote (how everyone voted) but also the procedure and decision rule that was used (which alternative won).

Second, the generality of the NESS test means that it can collapse into the canonical dependencies of necessity and sufficiency: if c is a member of every sufficient set for e, then c is a necessary condition for e; and if c alone constitutes a sufficient set for e, then c is a sufficient condition for e.

Third, inclusiveness of the NESS test means that it easily accounts for cases of overdetermination because an event is attributed causal status even if, due to the presence of other actually or hypothetically sufficient sets, it was not necessary in the circumstances for the result. To see how this works, consider the injured jogger again. What the NESS test does is to resolve the excess sufficient set (the set of actions of ‘lifting the tree trunk’) into its component sufficient conditions and to check if an action is a necessary part thereof. Assuming, for the sake of simplicity, that each of the rescuers lifts with equal force, there are three possible sets of actions that are minimally sufficient for the rescue and each rescuer belongs to at least one (in fact two) of these. Consequently each of the rescuers’ actions can be attributed causal status even though none of those actions was itself a necessary, sufficient, or but-for condition for the rescue.

Regarding the difference between the weak and strong NESS test, the strengthening of the original NESS test appears to be eminently sensible because it prunes the extraneous events in the sufficient condition (i.e. those events that are devoid of any causal efficacy for the outcome such as a child picking up twigs in the injured jogger example). ¹⁵ Marc-Wogau (1962), for instance, gives an ordinary language justification which makes use of this intuition. He argues that irrelevant events are not intended when we speak of the elements of a ‘sufficient condition’. However, the intuitive sensibility seems to be the sole justification for the use of the strong version. The only other consideration that we have encountered is the one that Honoré (1995, p. 365) attributes to Mackie (1965, 1974). According to Honoré, for Mackie it is ideally the case that the sufficient condition is minimal because this is the way that we discover causal regularities. Mackie was, however, quite aware that statements of singular causation need not imply any such generalization. The lack of a proper justification of the use of the strong rather than the weak NESS test is all the more interesting because it turns out to be anything but innocuous when it comes to deriving an index of causal contribution. In fact, as we shall argue in Sect. 5, the measurement of degrees of causal contribution should be formulated in terms of the weak rather than in terms of the strong and standard version of the NESS test.

3 Specific and Overall Causal Contribution

Now that we have a reasonably clear idea of what it means for an event to be ascribed causal status for some outcome, we can move on to the question of giving factual content to ‘degrees of causation’. To avoid any linguistic and conceptual confusions we need to make two clarifying statements. First, in line with the NESS test we take this locution to mean ‘degrees of causal contribution’ or ‘impact’ or such like. Second, when we speak of events we mean both the actions that have been performed by the agents and the combination of the actions which form a sufficient condition for the outcome. In the injured jogger case, the individual acts of lifting as well as the combination of these acts are events. Next, and this is the burden of this section, in order to defend the idea that causal contribution can come in degrees we need to identify the primitive events that are to be counted. To achieve this we will draw a distinction between specific and overall causal contributions of an outcome. The former is the primitive notion while the latter is derivative and obtained by aggregating the specific contributions. This aggregation gives us ‘degrees of causal contribution’.

When we use the NESS test to ascribe causal status to an action for some outcome we are making a very specific statement about that action. The test only informs us of the fact that the action was necessary for a condition to be sufficient for that outcome on a specific occasion. Hence the category of a specific causal contribution. Since the NESS test only checks whether or not a relation exists between a specific action and a specific outcome, being a causal factor does not have a magnitude if one assumes that ‘existence’ does not come in degrees.

The categorical nature of NESS conditions comes to the fore if we give it some formal structure. Anticipating the game-theoretic framework that we present later, we do so in set-theoretic terms. Define a set of actions S that have been performed as an event that is a critically sufficient condition for e if (i) it is sufficient for e and (ii) there is at least one s ∈ S such that S − {s} is not sufficient for e. S is a minimally sufficient condition if (i) it is sufficient for e and (ii) for all s ∈ S, S − {s} is not sufficient for e. For instance, assume $\{{\underline a},{\underline b}\}$ is the only critically (in fact minimally) sufficient condition for e (the criticality of an action is indicated by an underscore). Both of the actions a and b form a NESS for e by dint of them being elements of {a,b} which occurred. However neither of the two actions can be considered as being ‘more’ or ‘less’ of a cause than the other because regardless of how ‘big’ or ‘small’ a is, a could contribute to the production of e on the specific occasion only if {a,b} materialized. To see this, consider, for instance, a committee that uses a weighted voting rule (such as in the EU Council of Ministers) in which each member of the committee has a different number of votes and a proposal is approved if a particular quota of votes is reached. Suppose we have five voters with voting weights 35, 20, 15, 15, and 15 and a quota of 51. Suppose that the first two voters vote in favour of a given proposal and the remaining voters vote against it. The proposal is approved. Despite the fact that in terms of voting weights the first voter can be said to be ‘bigger’ (in fact ‘heavier’) than the second, it does not follow that in this contingency he is ‘more’ of a cause than the second voter. Both versions of the NESS test (in fact the but-for test as well) designate both voters as having made a causal contribution, irrespective of their ‘size’.

Accepting this straightforward point does not, however, rule out that the causal impact of an action can have magnitude if understood as a derivative concept. Consider the weak NESS test and assume, by way of example, that e and {a,b,c} have occurred. Assume that the critically sufficient conditions are $\{{\underline{a}, {b}, {c}}\},$ $\{{\underline{a}}, {\underline{b}}\}$ and $\{{\underline{a}},{\underline{c}}\}.$ It seems correct to say that a makes ‘more’ of a contribution to e on the occasion than either b or c. One way of explaining this is to say that this is true in virtue of the fact that a is necessary for e while b and c can be said to be merely ‘NESS causes’ (on the occasion). The ‘more’ in this scenario refers to the fact that a satisfies a stricter criterion. But this argument does not work in general. Consider the following: e and {a,b,c,d} occurred with the critically sufficient sets being $\{{\underline{a}},{\underline{b}},{\underline{c}}\}, \{{\underline{a}},{\underline{b}}\}, \{{\underline{a}},{\underline{c}}\}, \{{\underline{a}},{\underline{d}}\}, \{{\underline{a}},{\underline{b}},{\underline{d}}\}, \{{\underline{a}},{\underline{c}},{\underline{d}}\},$ and $\{{\underline{b}},{\underline{c}},{\underline{d}}\}.$ Here, we only have what can be termed ‘NESS causes’ and no ‘necessary’, ‘sufficient’ or ‘but-for’ causes, so the strictness condition will fail to generate an answer although it seems reasonable to postulate that a has more causal impact than b, c, or d. Why?

The answer lies with the implicit operation of quantifying over the number of critically sufficient sets for e that each action belongs to. Stated somewhat differently, a appears to make more of a contribution to e because there are more token events that are sufficient causes of e that require the presence of a. In the first example of the previous paragraph, a belongs to three such events while each of the other two actions belong to one. So a statement of the form ‘a makes more of a causal contribution than b to the occurrence of e’ can be given explicit and numeric sense in this framework because it takes as its primitive units the specific contributions that are picked out by the NESS test and those units can be counted. This gives us the concept of overall causal contribution. Note that this quantity can also be said to be ‘empirical’ in the sense that the contributions are based on individual actions which have occurred and can be observed to have occurred.

This complex quantitative attribute is what allows us to speak of degrees of a causation.¹⁶

To avoid confusion, and to clarify our language in this matter, we do not say that ‘a is more of a cause of e than b’ but rather ‘a has made a larger causal contribution than b to the production of e’. The reason for such a judgement is that there are a greater number of instances in which a forms part of an critically sufficient condition—a token event—for e. Clearly, the same applies if we take the strong instead of the weak NESS test; the only difference here is that we would focus on the number of instances in which an action forms a necessary part of a minimal rather than of a critical condition.

4 A Game-Theoretic Formulation of the NESS-Tests

The preceding sections have been concerned with supporting a single claim: because we have identified certain empirical objects that can be counted it is possible to speak in a qualified way about ‘degrees of causation’. We now have to consider how to aggregate these objects. More formally, we have to define a causality function on the space of actions which we take as the primitive events of a sufficient condition (a possibly composite event) that will yield a measure that expresses degrees of causal contribution. To do so, we present the outlines of the game-theoretic framework in terms of which we shall define our measure and give the game-theoretic renditions of the weak and strong forms of the NESS test (Definitions 2.1 and 2.2).

Let X be a set of outcomes with at least two members (i.e. #X ≥  2, where #(·) denotes a set’s cardinality) and N = {1,…,n} a set of individuals. G = (S ₁,…,S _n ,f) is a game form (on X and N): each S _i is a finite set of strategies for each individual i, and f is a function from the set of all strategy combinations, or plays, onto X. Since the function is onto X, each element of X is an outcome in at least one play.

For all $T \subseteq N,$ we call an element s _T of Π _i∈T S _i a T-event: it describes the event of the members of T performing the actions described by s _T (if $T=\emptyset$ we may call s _T a non-event). Given an event s _T , s _i denotes the strategy of i ∈ T, for event s _T , s′ _i is the element played by i ∈ T in s′ _T , etc. Furthermore, we write (s _T ,s _N-T) to denote the play of G which consists of the combination of the (mutually exclusive) events s _T and s _N-T.

We let π(s _T ) denote the set of outcomes that can result from the event s _T : $\pi(s_T)= \{f(s_T,s_{N-T}) \mid s_{N-T} \in \Pi_{i \not \in T} S_i\}$ .

Definition 4.1

A T-event s _T is a sufficient condition for $A \subseteq X$ if and only if $\pi(s_T) \subseteq A.$

If A is a singleton set, say A = {x}, we drop the curly brackets and simply say that the event is a sufficient condition for x. Similarly, we shall often write in such cases π(s _T ) = x rather than π(s _T ) = {x} or f(s _T ) = x for the case in which T = N.

For any s _U and s _T , call s _U a subevent of s _T if $U \subseteq T$ and if each member of U adopts the same strategy in s _U as in s _T . Abusing notation, we shall write $s_U \subseteq s_T$ to indicate that s _U is a subevent of s _T . Similarly, we say that s _U is a proper subevent, and write $s_U \subsetneq s_T,$ if U is a proper subset of T. The critical and minimally sufficient conditions can now be formulated as:

Definition 4.2

An event s _T is a critically sufficient condition for A if and only if (i) s _T is a sufficient condition for A and (ii) there is at least one i ∈ N such that the proper subevent s _T-{i} is not sufficient for A (i is called A-critical for s _T ).

Definition 4.3

An event s _T is a minimally sufficient condition for A if and only if (i) s _T is a sufficient condition for A and (ii) for all i ∈ N the proper subevent s _T-{i} is not sufficient for A.

We now define the game-theoretic versions of the weak and strong NESS conditions as follows:

Definition 4.4

(Weak NESS test) Given a play s _N , an individual strategy s _i is a weak NESS condition for A if, and only if, there is an event $s_T \subseteq s_N$ such that (i) s _T is a critically sufficient condition for A, (ii) i is A-critical for s _T .

Definition 4.5

(Strong NESS test) Given a play s _N , an individual strategy s _i is a strong NESS condition for A if, and only if, there is an event $s_T \subseteq s_N$ such that (i) s _T is a minimally sufficient condition for A, (ii) i is a member of T.

5 Causation Indices for Simple Strategies

There are two basic approaches to fashion a measure describing degrees of causation. One is to try to derive an ordinal comparison of degrees of causation attributed to agents and their actions. A more demanding approach is to define a measure that also allows for cardinal comparisons. Given that our ultimate target (although not in this paper) is to be able to say something about legal and moral responsibility and therefore be able to say something about the distribution of punishment, rewards, or burdens, defining a cardinal measure is the approach we will follow. The problem with an ordering is that it generates insufficient information to be an argument in a responsibility function because, it will only inform us which agents made a larger causal impact than others but not tell us by how much. That is, the deficiency of an ordinal measure is its insensitivity to the extent to which a person participated in bringing about an outcome.

Given a game form G and a play s _N = (s ₁,…,s _n ) that results in some outcome π(s _N ), how might such a cardinal value function be defined? Firstly, we want such a function to assign a value between 0 and 1 for each action such that if the action was necessary and sufficient on the occasion it takes a value of 1 and, at the other extreme, if it was not a NESS condition then it takes a value of 0.

Secondly, in order that we can make interpersonal comparisons, we want a function that assigns to each player i in a play of G a value that describes the share that i’s strategy had of the total causal condition, i.e. the sum of values is equal to 1.¹⁷

Before deriving our measure we need to make one further distinction between two types of games, because this will have a significant implication for the measures. The distinction is the one between situations in which each strategy describes one specific action and situations in which a strategy may stand for a combination of actions that can be carried out simultaneously. In the first case we speak about the individuals having simple strategies whereas in the second case they are said to have complex strategies. As complex strategies pose extra technical difficulties, we will first derive our metric for simple strategies and treat the complex case separately in the next section.

Very generally, we want to obtain a function that expresses the share of the total set of NESS conditions on a specific instance of an outcome.¹⁸ Because of the general acceptance of the strong version of the NESS test, it is natural to focus on the frequency by which an action satisfies the strong NESS test.

Given s _N , let ${\mathcal{M}}_i$ be the set of all sub-events s _U that form a minimal sufficient condition for π(s _N ) and in which i performs an action:

$${\mathcal{M}}_i= \{s_U \mid {i \in U \; \hbox{and}\; s_U \subseteq s_N \; \hbox{is\;a\;minimally\;sufficient\;condition\;for}\;\pi(s_N)}\}.$$

The (normalized) frequency by which an action is a strong NESS condition is given by:

$$ \alpha_i(G,s_N):=\frac{\# {\mathcal{M}}_i} {\mathop{\sum}\limits_{{j \in N}} \# {\mathcal{M}}_j}. $$

(5.1)

Despite its simplicity and intuitive appeal, the measure is problematic. It can generate highly unreasonable rankings as the following example shows. The basic problem with the focus on minimally sufficient conditions is that it does not discriminate between the causal contribution made by an action that is sufficient and one that is ‘merely’ necessary.

Example 5.1

Let N = {1,2,3,4,5} be a five person committee having to make a choice about X = {x,y}. The voting rule specifies that x is chosen if, and only if, (i) 1 votes for x, or (ii) at least three of the players 2–5 vote for x. Assume all individuals vote for x: s _N  = (x,x,x,x,x). Hence, we have a case of overdetermination: 1’s vote is a sufficient condition for the realization of x, and so is any combination of the votes of at least three of the other individuals. The action of voter 1 is a member of only one minimally sufficient condition, i.e. the event consisting only of his action of voting for x. Since each of the actions of the other individuals is a member of three minimally sufficient conditions (each such action is a member of three ‘minimal’ majorities not containing player 1) the measure defined in Eq. 1 yields:

$$ \alpha_1=\frac{1}{13},\;\alpha_2 = \alpha_3 = \alpha_4 = \alpha_5 = \frac{3}{13}. $$

This is a questionable allocation. By focusing on minimally sufficient conditions, the measure ignores the fact that anything that players 2–5 can do to achieve x, player 1 can do, and in fact more—he can do it alone. So, where does this leave us? It has to be borne in mind that the justification for α _i lies first and foremost with the NESS condition and then with the minimality restriction. That is, to measure causal contribution we are not a priori wedded to the minimality condition. The obvious solution here is to notch up the causal contributions by taking the weak NESS test as the appropriate rendition of causal factors. That is, for any game form G and a play of it s _N  = (s ₁,…,s _n ) that results in some outcome π(s _N ), we count the instances in which an action is critical (Definition 4.2) irrespective of the causal contribution of other actions. This gives us a measure analogous to α _i but with the summation taking place over $s_U \in {\mathcal{C}}_i$ instead of $s_U\in {\mathcal{M}}_i,$ where ${\mathcal{C}}_i$ is the set of all sub-events s _U that form a critically sufficient condition for π(s _N ) and which contain i’s strategy:

$$ {\mathcal{C}}_i= \{s_U \mid {s_U \subseteq s_N\;\hbox{is\;a\;CSC\;for}\;\pi(s_N)\;\hbox{and}\;i\;\hbox{is}\;\pi(s_N)\hbox{-critical\;for}\;s_U}\}.$$

We put,

$$ \beta_i(G,s_N) := \frac{\# {\mathcal{C}}_i}{\sum_{j \in N} \#{\mathcal{C}}_j}. $$

(5.2)

Consider Example 5.1 once again. The new measure now takes into account the full range of sufficient events in which at least one action is necessary: all events in which 1 votes for x and at most two others as well, and all events in which exactly three individuals other than 1 vote for x. As can be checked, 1’s vote for x is a critical member of 11 (not necessarily minimal) sufficient conditions, and a vote for x of any of the other players is so of exactly 3 such conditions. Thus we get:

$$ \beta_1 = \frac{11}{23},\;\beta_2 = \beta_3 = \beta_4 = \beta_5 = \frac{3}{23}. $$

It is interesting to note that there is a close relation between the two indices that we presented and the indices that have been presented as measures of power in voting bodies. The metric α _i corresponds with the so-called ‘Public Good Index’ introduced by Holler (1982) and Holler and Packel (1983), whereas β _i is closely related to the so-called normalized Penrose–Banzhaf index (Felsenthal and Machover 1998). In Sect. 7 we shall make these relationships more precise and discuss the relation between causation and power. We now turn to the analysis of complex strategies.

6 Causation and Complex Strategies

Irrespective of the exact way one derives a value describing the degree of causal contribution, any such value will have to take into account that individual strategies can be complex in the sense described in the previous section. To see why this is relevant, we examine a number of examples. We start first with the overdetermined case and then show that the same problems emerge even for the more elementary in situations in which all actions are necessary. Consider first:

Example 6.1

Players (say firms) 1, 2 and 3 dumped different toxins in a river. Each of the firms’ strategies consisted of a single action: dumping a fixed quantity of the toxin denoted by T ₁, T ₂, and T ₃ respectively. Any of the three actions was in itself sufficient to kill all the fish in the river. Applying the causation index β _i , each firm is allocated a causal contribution of 1/3 to the outcome of the fish being dead (the result is the same if we use the strong NESS index, α _i ).

The allocation is to be expected: it is a case of overdetermination in which each of the companies has contributed ‘equally’ to the resulting outcome. It is the same as the classic over-determination case in which two assassins simultaneously shoot and kill their victim. Each act of shooting is sufficient for the death of the victim, and each act thus is a NESS condition. However, to tease out the effect of complex strategies, consider the following variation of the above example:

Example 6.2

Firm 3 transfers its activities to Firm 1 (e.g. Firm 1 buys out the business of Firm 3) so that Firm 3 ceases to dump T ₃ in the river. Firm 1 now dumps toxins T ₁ and T ₃ into the river and Firm 2 dumps T ₂. All else remains as in Example 6.1. Both the strategy of player 1 (dumping T ₁ as well as T ₃) and the strategy of player 2 (dumping T ₂) form NESS conditions for the resulting outcome. The causation indices α _i and β _i allocate equal shares of the cause for the dead fish—each will have a value of 1/2 (on either index).

Given the allocation of causal contributions in Example 6.1, this allocation is counterintuitive. Firstly, there has been no change in the type and quantity of toxins that have been dumped in the river, and secondly there has been no change in the number of events that have brought about the pollution of the river and the death of the fish. The only feature that has changed is who performed the actions of dumping. Firm 1 could only cause the death of the fish with a single action, it can now do it with two because Firm 3’s action has shifted to Firm 1. We thus would expect that the causal contribution of Firm 1 now is 2/3: the sum of Firm 1’s and Firm 2’s causal contribution in Example 6.1. The reason this allocation does not emerge is that the measures we presented only focus on strategies as such and thereby ignore the components of those strategies. By not discriminating between the differences in causal contributions made by those components we lose important information.

If we want to distinguish the various actions that different strategies comprise, we need to have a richer informational framework. For this purpose we introduce the notion of an inner game form of a play s _N = (s ₁,…,s _n ) of a game form G. The inner game form will be used by us to distill information about the causal impact of the various actions constituting a person’s complex strategy, information that cannot be obtained from the original game form. Once we have that information we use it to obtain a more adequate value of the causal contributions made by the various individuals in the original game form.

To account for the complexity of strategies, we now assume that each strategy s _i in s _N is a set, to wit a set of actions a. Without loss of generality, we thereby assume that the actions played by different individuals are different from each other, i.e. for all i, j ∈ N, $s_i \cap s_j=\emptyset.$ Let m be the total number of actions being performed in s _N and let ${\mathcal{A}} = \cup_{i \in N} s_i = \{a_1,\ldots,a_m\}$ denote the set of all those actions. Finally, let t denote some action which is not played in s _N ; t can be interpreted as ‘staying passive (with respect to π(s _N ))’ but this is not necessary.

Definition 6.1

(Inner Game Form) Given some game form G, a play s _N of G (with outcome denoted by x), and the associated ${\mathcal{A}},$ G’s inner game form is a game form $G^{\ast}=(S^{\ast}_1,\ldots,S^{\ast}_m,f^{\ast})$ on $M = \{1,\ldots,m = \# {\mathcal{A}}\}$ and $A \subseteq X$ such that:

• 1. For all i ∈ M, $S^{\ast}_i =\{a_i,t\};$

• 2. For all $i, j \in M (i \not= j),$ $a_i \not = a_j;$

• 3. $f^{\ast}(s^{\ast}_1,\ldots,s^{\ast}_m) =x$ if $s^{\ast}_i =a_i$ for all i ∈ M;

• 4. For all plays $s^{\ast}_M, \bar{s}^{\ast}_M$ of the inner game form: if $\{ i \in M \mid s^{\ast}_i = t\} \subseteq \{ i \in M \mid \bar{s}^{\ast}_i = t\},$ then $f^{\ast}(s^{\ast}_M) \not= x$ implies $f^{\ast}(\bar{s}^{\ast}_M) \not= x.$

Each individual in the inner game form has only two strategies, one that corresponds with an action played in the play s _N of the original game form and one consisting of an action, t, that was not played in s _N (Clause 1). Furthermore, each individual has been assigned a distinct element of ${\mathcal{A}}$ (Clause 2) and each action being played is thus part of some individual strategy set of the inner game form. In case each action performed in s _N is played in the inner game form as well, the outcome of the play of the inner game form is the same as that of s _N (Clause 3). If some play $s^{\ast}_M$ leads to an outcome different from x, then any other play that only differs because some members who played the same action in s _N no longer do so, will not lead to x either (Clause 4). The rationale for the last condition is to exclude ‘perverse’ strategies such as Firm 1 in Examples 6.1 and 6.2 simultaneously dumping a toxin and an antidote into the river. The clause thus precludes the existence of ‘preventive’ actions. Note that since any game form has at least two distinct outcomes, the clause also entails that the play in which all individuals play t will have an outcome different from x.

Though different inner game forms can be associated with a particular play s _N of a standard game form G, we assume that there is exactly one ‘appropriate’ one. To illustrate the definition, consider again the game form described by Example 6.2.

Example 6.2

(continued) If we let T ₀ denote the action of not dumping any toxin, the relevant inner game form is like a three-person game form in which one player can play T ₀ or T ₁, one can play T ₀ or T ₂, and one can play T ₀ or T ₃, i.e. in the inner game form each action of each strategy that was played in the original game form now forms a separate strategy of a ‘separate player’. By assumption, a play of the inner game form will lead to the outcome in which the fish are killed if, and only if, at least one of the players adopts a strategy other than T ₀.

Given some G and s _N and associated inner game form G*, we can now define the refined causation indices $\alpha^{\ast}_i$ and $\beta^{\ast}_i.$ Let a _M denote the play of G* consisting of all actions that are played in s _N , i.e. a _M  = (a ₁,…,a _m ). Furthermore, for each i ∈ N, let $\theta(i) = \{j \in M \mid a_j \in s_i\}.$ Hence, θ assigns to each i the set of individuals of the inner game form who are able to play one of the actions comprised by s _i . We define:

$$ \alpha^{\ast}_i(G,s_N) := \sum_{j \in \theta(i)} \alpha_j(G^{\ast},a_M)\;\hbox{and}\;\beta^{\ast}_i(G,s_N) := \sum_{j \in \theta(i)} \beta_j (G^{\ast},a_M)$$

(6.1)

The refined causation indices are thus obtained by simply adding up in the inner game form the (non-refined) causation values of the players corresponding to the various actions in the original strategies. Obviously, in case each of the strategies in the play s _N is a simple rather than a complex strategy (consists of only one action), the refined causation values coincide with the non-refined ones. The refined indices can therefore be seen as generalizations of the non-refined ones.

Example 6.2

(continued) We now first calculate the non-refined causation values for the three individuals in the play (T ₁, T ₂, T ₃) of the inner game form. On both the α _i and the β _i versions, the value is 1/3 for each player. Summing up for each player in N the causation values of the actions performed by him yields the refined values:

$$ \alpha^{\ast}_1=\beta^{\ast}_1 = \frac{2}{3}\;\hbox{and}\;\alpha^{\ast}_2=\beta^{\ast}_2=\frac{1}{3}.$$

Now consider a more complicated instance of overdetermination.

Example 6.3

Firm 1 dumped toxin T ₁ and T ₃ into the river while Firm 2 dumped toxin T ₂. The minimal conditions for the death of the fish now are (i) T ₁ and (ii) T ₂ and T ₃ together. Assume all toxins are dumped. Since there are two minimal sufficient conditions in the corresponding inner game form (the events (T ₁) and (T ₂,T ₃)), the shares in the cause for the death of the fish on the strong NESS test are:

$$ \alpha^{\ast}_1 = \frac{1} {3} + \frac{1} {3} = \frac{2}{3}\;\hbox{and}\;\alpha^{\ast}_2=\frac{1}{3}$$

Taking the weak NESS test yields a different allocation since there are three critical sufficient conditions (the minimal one plus (T ₁,T ₂) and (T ₁,T ₃)) in which T ₁ is the critical strategy. As there is only one critical sufficient condition containing both T ₂ and T ₃ (viz. (T ₂,T ₃)) we get:

$$ \beta^{\ast}_1 = \frac{3}{5} + \frac{1} {5} = \frac{4}{5}\;\hbox{and}\;\beta^{\ast}_2=\frac{1} {5}.$$

Thus far we have considered only inner game forms in cases of overdetermination. They are also important if there is no overdetermination:

Example 6.4

Firm 1 dumped toxin T ₁ and T ₃ into the river while Firm 2 dumped toxin T ₂. The three actions form together the necessary and sufficient condition for the outcome. Using the non-refined values yields a causal impact of 1/2 for each player. If we take the inner game form however, we derive the same conclusion as with the refined values in Example 6.2: ${\alpha^{\ast}_1}= {\beta^{\ast}_1}=2/3,$ and ${\alpha^{\ast}_2}={\beta^{\ast}_2}=1/3.$

The various examples that we used thus far concern the combination of qualitatively different types of actions. We assumed that it is indeed possible to make such qualitative distinctions, that is, we assumed that we can associate a unique inner game form to the original game form. Sometimes, however, the actions performed by individuals differ only quantitatively, say when the firms in our various examples submit different quantities of the same toxin. By assuming that we can treat the emission of certain quantities of toxin (or the exertion of a certain degree of strength when one lifts a tree trunk, or the intake of a certain quantity of wine, etc.) as if they constitute different actions, inner game forms are helpful here as well. To see how this works, consider the following example:

Example 6.5

Firm 1 dumps 2 liters of toxin T in the river, whereas Firm 2 dumps 0.5 liter of the same toxin. The threshold value for killing all fish is the emission of 1 liter of T into the river (an emission of 1 liter is the necessary and sufficient condition for the fish being killed). To yield the inner game form we assume that each emission of k liters of toxin T forms a separate action, where k is the greatest common divisor of the various quantities of T dumped by the agents and of the threshold value. Thus k will equal 0.5.

The solution is as follows. In the inner game form each player dumps either 0.5 liter or nothing (0.5 being the greatest common divisor). Thus there are 4+1 = 5 ‘players’. Any combination of two players forms a sufficient condition for the threshold. Take any player. He is a member of a minimal and critically sufficient condition in exactly four cases (there are four ways in which exactly one other action of dumping 0.5 liter is performed by one of the others). Hence, we obtain the values:

$$ \alpha^{\ast}_1 = \beta^{\ast}_1 = \frac{16}{20}= \frac{4}{5}\;\hbox{and}\;\alpha^{\ast}_2 = \beta^{\ast}_2 = \frac{4}{20} = \frac{1}{5}. $$

Note that k being a common divisor entails that the α* and β* values will always coincide if there is a single type of ‘quantifiable’ action involved.¹⁹

The surprising feature of the last example as well as of Example 6.3 is that despite Firm 2’s powerlessness to either unilaterally pollute the river or to prevent the river from being polluted, Firm 2 still has a share of the causal contribution. The examples reveal that a person may be causally efficacious for an outcome without the possibility to force or prevent that outcome against the will of others.²⁰ Any sense of paradox can, however, be quickly dismissed because we can easily find an explanation in a standard Lewis type counterfactual analysis. In the case of Example 6.3, when we determine the efficacy of the actions in the subevent (T ₂,T ₃) we do so by comparing it to a near-by possible world in which the only difference to the actual world is one in which Firm 1 only dumped Toxin 3. In this case, Firm 2’s dumping of Toxin 2 is a necessary condition for the death of the fish in that world, despite the fact that Firm 1 could unilaterally pollute the river and kill the fish.

7 Causation and Power

Examples 6.3 and 6.5 raised the issue of power and its relationship to causation. The examples show that making a causal contribution does not necessarily imply preventive power. This does not contradict the intuitively close relationship between ‘power’ and ‘causation’.²¹ In fact in this section we shall show that the relationship holds for an important class of games, so-called ‘simple games’, which are used to model collective decision making in committees.

A simple game is a game in coalitional form which means that the set of players is divided into subsets of players $S \subseteq N$ known as ‘coalitions’. It is ‘simple’ because these coalitions are either ‘winning’ or ‘losing’. Although the game forms that we have used are non-cooperative, we show how these can be seen to ‘induce’ a simple game. We then show that the causal contribution index based on the strong NESS test, α _i , corresponds to Holler’s power-index. The weak NESS test index, β _i , is shown to correspond to the (normalized) Penrose–Banzhaf power-index.²²

Definition 7.1

A simple game (on N) is an ordered pair $(N,{\mathcal{W}})$ with ${\mathcal{W}}$ being a set of subsets of N (the set of ‘winning’ coalitions) satisfying: (i) $\emptyset \not \in {\mathcal{W}},$ and (ii) for all $S \in {\mathcal{W}}$ and all $T \supset S,$ $T \in {\mathcal{W}}$ (monotonicity).

Let ${\mathcal{G}}$ be the set of all simple games. A power index is a function that assigns for each game in ${\mathcal{G}}$ a vector of real numbers each component of which represents the power of a player in the game $(N,{\mathcal{W}}).$ Given a simple game $(N,{\mathcal{W}}),$ let ${\mathcal{M}}_i({\mathcal{W}})$ be the set of all minimal winning coalitions in ${\mathcal{W}}$ which contains i, i.e. ${\mathcal{M}}_i({\mathcal{W}}) = \{S \in {\mathcal{W}} \mid$ i ∈ S and there is no $T \in {\mathcal{W}}$ for which $T \subsetneq S\}$ .

Definition 7.2

The Holler power index is the function h _i defined as:

$$ h_i(N,{\mathcal{W}}):= \frac{\# {\mathcal{M}}_i({\mathcal{W}})}{\sum\limits_{j \in N} \# {\mathcal{M}}_j({\mathcal{W}})}. $$

(7.1)

The (normalized) Penrose–Banzhaf index is not defined in terms of the number of minimal winning coalitions an individual is a member of, but in terms of the number of times the individual is ‘critical’: that is, how often it can turn a winning coalition into a losing one. Let ${\mathcal{C}}_i(W)$ be the set of all winning coalitions in ${\mathcal{W}}$ in which i is critical, i.e. ${\mathcal{C}}_i({\mathcal{W}}) = \{ S \in {\mathcal{W}} \mid S - \{i\} \not\in {\mathcal{W}}\}.$

Definition 7.3

The Penrose–Banzhaf power index is the function b _i defined as:

$$ b_i(N,{\mathcal{W}}) := \frac{\# {\mathcal{C}}_i({\mathcal{W}})}{\sum\limits_{j \in N} \#C_j(W)}. $$

(7.2)

There is an obvious formal similarity between h _i and the α indices (given by 5.1 and 6.1) and between b _i and the β indices (given by 5.2 and 6.1). We now establish more precisely the relation between a power index and causation index by transforming a game form and a play of it into a simple game. We do so by (i) viewing each action that has been performed in the game form as an individual player and (ii) stipulating that a subevent of the play s _N is a sufficient condition for π(s _N ) if, and only if, the actions form a winning coalition in the corresponding simple game.

Let G be a game form (on N and X), s _N a play of G and ${\mathcal{A}}$ the set of actions played in s _N . We define $({\mathcal{A}},{\mathcal{W}})$ as the simple game in which $T \in {\mathcal{W}}$ if and only if (i) T is a non-empty subset of ${\mathcal{A}}$ and (ii) the event in which each action in T is played is a sufficient condition for π(s _N ) in the inner game form G*. We then have:

Proposition 7.1

For any G and s _N :

$$ \alpha^{\ast}_i(G,s_N) = \sum_{a \in s_i}\; h_{a}({\mathcal{A}},{\mathcal{W}})\;\hbox{and}\;\beta^{\ast}_i(G,s_N) = \sum_{a \in s_i} b_{a}({\mathcal{A}},{\mathcal{W}}).$$

Proof

The proof is straightforward.□

For any game form and any play s _N of it in which an individual i plays a simple strategy, the relation between causation and power is even more obvious. Letting a denote the unique element of i’s strategy s _i , we then obtain the following corollary of Proposition 7.1:

Corollary 7.1

For anyG and s _N in whichs _i = {a}:

$$ \alpha^{\ast}_i(G,s_N) = h_{a}({\mathcal{A}},{\mathcal{W}})\;\hbox{and}\;\beta^{\ast}_i(G,s_N) = b_{a}({\mathcal{A}},{\mathcal{W}}). $$

Given these results, we may summarize the relation between causal contribution and power as: the overall causal contribution to some outcome made by an individual equals the sum of the power of the actions performed by that individual. In the words of Hobbes: ‘Power and cause are the same thing. Correspondent to cause and effect are POWER and ACT; nay, these and these are the same things’ (English Works, 1, X).

8 Conclusion

Earlier in this essay (Sect. 5) we stated that our aim in studying degrees of causation is to be able to say something about the attribution of (retrospective) responsibility. We want to wind up with a number of remarks on the implication of our analysis for our understanding of responsibility.

First, whichever causation index one would favour—and we favour the β indices for the reasons given—the values themselves say nothing in general about the degree of responsibility. More often than not, responsibility is a matter of culpability, which is commonly seen as a qualitative rather than quantitative attribute. The common law, for instance, divides guilty felons into four categories: ‘perpetrators’, ‘abettors’, ‘inciters’ (all of these are ‘accomplices’) and ‘criminal protectors’ (Feinberg 1968, p. 684). Thus the problem of assessing degrees of responsibility for an outcome is not only a matter of assessing the extent of each individual’s causal contribution to that outcome but also involves an assessment and integration of various other dimensions such as degrees of initiative, degrees of authority, the gains from the activities involved, and perhaps most difficult of all, the degree of voluntariness.

Consider, for example, the case in which an order is issued to a group of soldiers. The military code stipulates that in the event of a conflict between orders issued by their superiors the soldiers must (and it is assumed that they will) obey the highest ranked officer. As it happens there is no conflict: the senior officer who is present on the occasion remains silent; it is only the subordinate officer who issued the command. Given that both the issuing of the command by the subordinate officer and the silence on the part of the senior officer are NESS-conditions, and in fact the only ones, our measures will allocate each officer equal shares of the causal impact. If the outcome is the wrongful execution of a prisoner of war, then despite the equal causal shares, the senior officer is clearly more responsible given the authority that he had to prevent the execution. Consequently, a causation index will only be useful for distinguishing between individuals in cases where the culpability of the actions are alike or where responsibility and causal contribution are taken to be synonymous. ²³ Example 6.4 is the illustrative case. If emitting toxins into a river is a culpable act, then, ceteris paribus, Firm 1 is more responsible than Firm 2 because it performed two culpable acts each of which was a weak NESS condition for the death of the fish compared to Firm 1’s single act of performing a weak NESS condition. Note that this result required the innovation of the inner game form. Otherwise both firms would be attributed equal responsibility.

Second, our framework indicates that holding a person morally responsible in the sense of blameworthiness appears to require something weaker than actual causal contribution to some state of affairs. A person may be blameworthy if, inter alia, the action they performed is at least a potential causal factor. Here is an intuitive case. Consider again the example of the weighted voting game that we discussed in Sect. 3 in which there are five autonomous and rational voters A, B, C, D, E with voting weights of 35, 20, 15, 15, 15 respectively and a quota of 51. Suppose that A, B, and C vote in favour of a proposal that will result in a foreseeable harm to some third party. Players D and E vote against it. Now in this configeration of votes, C makes no causal contribution to the outcome because it does not form a NESS condition under this contingency. But one could argue that C is blameworthy given his vote in favour of the proposal on the grounds that the harm was foreseeable and the only reason that he did not make a causal contribution is because of the particular configuration of votes that occurred—it was a matter of luck. The fact is, had another configeration of votes occurred, such as A, B, C, and D voting in favour, C’s vote is a NESS condition. Under this view, what is required for moral responsibility is only that there exists a possible configeration of votes in which a player’s strategy choice makes a causal contribution to the realization of the outcome.

Third, another feature of responsibility that the derivation of the causation indices brings to light is that it tests a certain aspect of the power–responsibility relationship. Although it is common practice to associate responsibility with power, this is mistaken. Indeed, as Peter Morriss (1987: 39) argues, ‘[t]he connection between power and responsibility is, then, essentially negative: you can deny all responsibility by demonstrating lack of power.’ That is, you can prove your innocence for the outcome by being able to demonstrate that you could not have prevented it. ‘[P]ower’, says Morriss, ‘is a necessary (but not sufficient) condition for blame: if you didn’t have the power, you are blameless.’ ²⁴ Obviously, if an agent had the power to force the particular outcome in question, such as Firm 1 in Example 6.3, that agent will be responsible if the action is culpable by some standard. However, the fact that power may imply responsibility does not mean that powerlessness to prevent can be used to exonerate agents. If preventive power is a necessary condition for responsibility, then Firm 2 in Example 6.3 is not accountable for the pollution of the river. This appears to be wrong, given that there is—even if it is relatively weak—a causal connection between Firm 2’s action and the pollution of the river and the death of the fish. ²⁵

Fourth, there is a significant methodological by-product that has been thrown up. The fact that certain voting power indices can be shown to be a special case of causation indices suggests an additional use and interpretation for these measures. Note that in the case of simple games there is a overlap between causal contribution and power. This permits us to apply the normalized Penrose–Banzhaf index as the appropriate means for allocating responsibility for the outcome of committee votes. Although Holler (2007) has recently made a similar proposal and suggests the use of his index for this purpose, our justification fundamentally differs from his. Holler argues that we should use power to attribute responsibility. Given Proposition 7.1, this means Holler takes the strong NESS-test as the basis for responsibility allocations. As explained in Sect. 5, we reject the strong NESS-test. Furthermore, although it is not the topic of this paper and we thus will not argue for it here, we also reject the claim that power is in general either a necessary or sufficient condition for attributing responsibility. Nonetheless, what Holler and ourselves agree upon, and for which we believe that we have delivered the correct theoretical foundations, is the idea that power indices can be used for more than designing fair voting systems.

Last, we have thus far only discussed issues involved in relating singular causal contribution to retrospective responsibility. As pointed out in the introduction, responsibility is, however, not just about what has happened, but also about prospective events and outcomes. What remains open is the derivation of a measure that captures what we can do, not just what we have done. This is vital to our understanding of our positive duties to others.