Degrees of individual and groupwise backward and forward responsibility in extensive-form games with ambiguity, and their application to social choice problems
DDegrees of individual and groupwise backward and forwardresponsibility in extensive-form games with ambiguity, and theirapplication to social choice problems
Jobst Heitzig and Sarah Hiller Potsdam Institute for Climate Impact Research, PO Box 60 12 03, 14412 Potsdam,Germany, [email protected] Free University Berlin, Institute for Mathematics, Arnimallee 3, 14195 Berlin,Germany, [email protected] version July 16, 2020
Abstract
Many real-world situations of ethical relevance, in particular those of large-scale social choice suchas mitigating climate change, involve not only many agents whose decisions interact in complicatedways, but also various forms of uncertainty, including quantifiable risk and unquantifiable ambiguity.In such problems, an assessment of individual and groupwise moral responsibility for ethically un-desired outcomes or their responsibility to avoid such is challenging and prone to the risk of under-or overdetermination of responsibility. In contrast to existing approaches based on strict causationor certain deontic logics that focus on a binary classification of ‘responsible’ vs ‘not responsible’, wehere present several different quantitative responsibility metrics that assess responsibility degrees inunits of probability. For this, we use a framework based on an adapted version of extensive-formgame trees and an axiomatic approach that specifies a number of potentially desirable propertiesof such metrics, and then test the developed candidate metrics by their application to a number ofparadigmatic social choice situations. We find that while most properties one might desire of suchresponsibility metrics can be fulfilled by some variant, an optimal metric that clearly outperformsothers has yet to be found.
The current climate crisis and its associated effects constitute one of the essential challenges for hu-manity and collective decision making in the upcoming years. An increase of greenhouse gas (GHG) concentrations in the atmosphere attributable to human activity leads to a warming of Earth’s surfacetemperature by reducing the fraction of incoming solar radiation that is diffused back into space. Anelevated mean earth surface temperature is however not a priori something reprehensible. Rather, itis the resultant effects that carry enormous dangers. Among these are the increased risk of extremeweather events such as storms and flooding, the rise of sea-levels or the immense losses of biodiversity,which have repercussions not only for the physical integrity of the planet but which pose direct threatsto human life. Prominently CO , but also methane, nitrous oxide and others. See for example [19] for a concise overview of the relevant climate science explained for non-climate scientists, or theIPCC and World Bank reports for more detail [28, 35]. a r X i v : . [ ec on . T H ] J u l aturally, the public debate around this issue frequently invokes the question of responsibility : Whocarries how much backward-looking responsibility for the changes already inevitable, who is to blame;and who carries how much forward-looking responsibility to realise changes, who has to act? As thefollowing 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 spec-trum: “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 wefound it.” [22]
Existing work.
The existing body of work regarding this question can roughly be divided into twocategories, via the perspective from which this question is addressed. On the one side there are con-siderations focusing on applicability in the climate change context, computing tangible responsibilityscores for countries or federations, with the aim of shaping the actions being taken and a lesser focuson conceptual elegance and consistency [6, 30]. On the other side there is considerable work in formalethics, aiming at understanding and formally representing the concept of responsibility in general with aspecial focus on rigour and well-foundedness, 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, andparticularly also in public discourse, the degree of backward responsibility of a person, firm, or countryfor climate change is simply equated to cumulative past GHG emissions, or a slight variation of thismeasure [14]. Certainly, this approach has one clear benefit, namely that it is easy to compute on anyscale, and also extremely easy to communicate to a non-scientific audience. Similarly, certain authorsassume a country’s degree of forward responsibility to be proportional to population share, gross domesticproduct 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 ofa generalised responsibility account. In the latter body of work, a principled approach is taken. Starting from considerations regarding thegeneral nature of the concept of responsibility, formalisms are set up to represent these. These comprisecausal models [13], game-theoretical representations [46, 8] or logics [12, 39]. A vast number of differentaspects have been included in certain formalisations, such as degrees of causation or responsibility,relations between individuals and groups, or epistemic states of the agents to name but a few. Generally,these are discussed using reduced, well-defined example scenarios and thought experiments capturingcertain complicating aspects of responsibility ascription.Additionally, there are investigations into the everyday understanding of the various meanings ofthe term ‘responsibility’ [43] as well as empirical studies regarding agents’ responsibility judgements incertain scenarios, showing a number of asymmetry results [32]. However, we are here not concerned withmirroring agent’s actual judgements, but rather with a normative account, so we will not go into detailabout these. What we call “forward-looking” or ex-ante responsibility is closely linked to the idea of obligation or duty, whereaswhat we call “backward-looking” or ex-post responsibility has also been called accountability, and relates to blame [39, 9] For example, using cumulative past emissions, population shares or GDP ratios all result in a strictly additive respon-sibility measure. If agent i has a responsibility score of R i and agent j one of R j the group consisting of agents i and j has a score of R i + R j . However, consider an example of two agents simultaneously shooting a third person. According tosome intuitions, e.g., the legal theory of complicity [23], they would then both be responsible to a degree larger than justhalf 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 thecumulative emissions account is that many climate impacts are not directly proportional to emissions, a topic that will bediscussed later on in this section. esearch question. The present paper places itself in the category of a principled and formal ap-proach, but aims at keeping in mind the practical applicability in complex scenarios. Also, we want torelocate the space of discussion in the formal community by proposing a set of responsibility functionsthat, rather than cautiously distributing responsibility and tolerating under-determination (or voids ),distribute responsibility somewhat more generously, evading certain forms of under-determination, butsometimes resulting in what might be seen as over-determination. The “correct” function is probablysomewhere in between, and we think it is helpful to examine the space of possible solutions from severalends. It might be useful to add that our work is normative, not descriptive. We aim at representing waysin 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 sce-nario. In some core aspects this is an extension of existing frameworks, in others we deviate from theprevious work. Subsequently, we will suggest candidate functions for assigning real numbers as degreesof 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 contributehow much in the joint effort to mitigate climate change is often carried out in the political sphere, withvarious voting mechanisms in place. It is therefore particularly interesting to determine measures ofresponsibility when the deliberation procedure is given by a specific voting rule. We will address thisquestion 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 enablea well-structured comparison between different candidates for responsibility functions [40]. That is,after determining a framework for the representation of multi-agent decision situations with ambiguityand corresponding responsibility functions as well as their properties, we begin by determining a set ofsimple, intuitive and basic properties that one might want a prospective candidate for an appropriateresponsibility function to fulfil. Our framework is based on the known concept of extensive-form games ,with added features to represent the additional information, or rather lack thereof, that we want toinclude here.
Specific aspects to be considered.
The above outline already shows several features of anthro-pogenic climate change that complicate responsibility assignments and occur in similar forms in otherreal-world multi-agent decision problems in which uncertainty and timing play a significant role. We willnow highlight and discuss several features that our framework will need to include, as well as certainaspects that we treat differently from existing work. One important idea is to avoid allowing agentsto refuse taking on responsibility by recurring to a certain calculation, even though according to someintuitions they do carry (higher) responsibility. We will suggestively call such an argumentation scheme dodging, and the corresponding modelling aspect dodging-evasion .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, thereis considerable uncertainty regarding the impacts to be expected from a given amount of emissions ora given degree of global warming. While for some results we can assign probabilities and confidenceintervals, for others this cannot be done in a well-founded way and beyond specifying the set of possiblealternatives 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 thecase, we cannot assign probabilities to either of the models being ‘more right’ than the others. Whatwe can say however, is that each of the predictions is within the set of possible outcomes (given the3remises, such as a certain future behaviour). The same goes for varying parameters within one and thesame model.Contrastingly, in a large body of work concerning effects of pollution, or warming, predictions areassociated with a specified probability. Take for example the IPCC reports, such as the well knownstatement about the remaining carbon budget if warming is to be limited to 1.5 degrees: “[. . . ] gives anestimate of the remaining carbon budget of [. . . ] 420 GtCO for a 66% probability [of limiting warmingto 1.5 ◦ C above pre-industrial levels]” [28]. In many cases, both aspects of uncertainty — ambiguity andprobabilistic uncertainty — are combined by speaking about intervals of probabilities, which in particularthe IPCC does pervasively [29].We argue that it is equally important to take note of the additional information in the probabilisticuncertainty case (often called ‘risk’ in economics ) as of the lack thereof in the ambiguity case. Itis known that the distinction between probabilistic and non-probabilistic uncertainty is important indecision 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 respectto emissions. With rising temperatures, so called ‘tipping elements’ such as the Greenland or WestAntarctic ice sheets risk being tipped [26]: once a particular (but imprecisely known) temperaturethreshold is crossed, positive feedback leads to an irreversible procession of higher local temperaturesand accelerated degradation of the element. This initially local effect then aggravates global warmingand may contribute to the tipping of further elements [24, 45] adding up to the already immense directimpacts 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 respon-sibility attribution. This constitutes another argument for deviating from the — linear — cumulativepast emissions accounts mentioned above [6, 30].In contrast to existing formalisations of moral responsibility in game-theoretic terminology [9], we in-clude a temporal dimension in our representation of multi-agent decisions by making use of extensive-formgame trees rather than normal-form games. This temporal component is also featured in formalisationsusing the branching-time frames of stit -logics. However, we do not take into account the temporal distance of an outcome to the individual decisionsthat led to it. Unlike in the ongoing debate in the environmental economics community regarding thediscounting factors to be employed when considering future damages, with the prominent oppositionbetween William Nordhaus and Nicholas Stern [33, 38] and its “non-decision” by a large expert panel ledby Ken Arrow [2], our account is not directly affected by any form of discounting. This is because whilequantitative measures of welfare depend on notions of preferences, degrees of responsibility depend onnotions of causation instead. Still, if the effects of an action disappear over time because of the underlyingsystem dynamics (e.g., because pollutants eventually decay) and if this reduces the probability of causingharm 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 regardingthe likelihood of another agent’s actions. We consider every agent to have free will, which we interpretto 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 aboutothers’ actions may influence the psychologically perceived degrees of responsibility of the agents, it Since we use the term ‘risky’ in this article for a different concept, we stick to the term ‘probabilistic uncertainty’ here. Ice reflects more sunlight than water. Thus, if a body of ice melts and turns into water, this will retain more heat thanthe ice did, leading to higher temperatures and faster melting of the remaining ice. As is stated in [37]: “The keywords inthis context are non-linearity and irreversibility”. Note that not all tipping elements are bodies of ice — coral reefs or theAmazon rain forest also rank among them. The examples with corresponding explanation were chosen for their simplicity. Note that normal-form game-theoretical models correspond to a subclass of stit models [16]. Similarly, extensive-formgames can also be represented as a stit -logic [11], but we don’t pursue this further here as the additional features that wewill include would complicate a logical representation and this is not currently necessary to express what we want to. degrees of responsibility. Note however, that unlike [42] we do not refer to agents’ beliefs regarding theprobabilities.
Paradigmatic examples and their evaluation.
In order to better understand the proposed frame-works as well as the responsibility functions, we will refer to a number of paradigmatic examples, mostlyknown from the literature or moral theory folklore, for illustration purposes. Like thought experimentsin other branches of philosophy, such as the famous trolley problem, these examples have been selectedbecause they each represent an interesting aspect of responsibility attribution in interactive scenarioswith 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 knowingwhether 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 knowingwhether 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 influ-ence the probability of a given event. That is, they have the choice between an option where theundesirable outcome has probability p and an option where it has probability q . Represented inFig. 1(c). • Hesitation I.
The agent has the choice to either rescue an innocent stranger immediately, orhesitate, in which case they might get another chance at rescuing the innocent stranger at a laterstage, but it might also already be too late. Represented in Fig. 1(d).If the agent does get a second chance and then decides to rescue the stranger, certain accounts willnot 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 strangerwho is seemingly having a heart attack. They have the choice to either help immediately by callingan 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 thiscase 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 While this example clearly seems somewhat odd in direct interaction contexts — imagine a scenario where someonehas the choice to save a person from drowning immediately or first finish off their ice-cream knowing that with probabilityp the other person will hold up long enough so they can still be rescued — it represents a common issue in climate changemitigation efforts.
5n uncertainty determining either before or after their second decision to help after all whetherthis decision is an option, or whether it is successful. While one might think two consecutivedecisions can be considered equivalent to one single combined decision it can be argued in this casethat if the agent does not end up helping they failed twice and should thus possibly carry higherresponsibility. • Climate Change.
Humanity (agent i ) has the choice to either heat up the earth or not, notknowing whether they are in a state of impending heating due to the greenhouse effect or a stateof impending cooling due to an onsetting ice age. • Knowledge gain.
Here Humanity (agent i ) is again posed before the same issue as in the previousexample. 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 situationsin which the agent might not actually be able to influence the outcome (because either the gun is notloaded so it does not matter whether they shoot or not, or because the other agent already threw a stonethat will shatter the window), but they do not know whether they are in this situation or in the one wheretheir action does have an impact. In both cases we argue that the responsibility ascription must take intoaccount the viable option that the agent’s action will have/would have had an impact. Therefore, theagent cannot dodge responsibility by referring to this uncertainty. They should be assigned full forwardand backward (if they select the possibly harmful action) responsibility. This relates to the discussionabout moral luck, and the case for disregarding factors that lie outside of the agent’s control is arguedin [31]: “Where a significant aspect of what someone does depends on factors beyond his control, yet wecontinue to treat him in that respect as an object of moral judgment, it can be called moral luck. Suchluck can be good or bad. [. . . ] If the condition of control is consistently applied, it threatens to erodemost of the moral assessments we find it natural to make.”This also relates to a prominent criticism of the probability raising account for causation, namelythat an agent may raise the probability of an event without this event actually occurring as a result, asthe probability stayed below 1. Similarly to situations in which the event does not end up occurring dueto the actions of others that the agent had no knowledge or influence over, we argue that this should notreduce 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 presentationof the proposed framework in which the responsibility functions as well as their desired properties will beformulated. Additionally, we explicate a number of desirable properties that will be important in drawinga difference between the various responsibility functions. In Sect. 3 we introduce four different candidateresponsibility functions (all differentiated between backward- and forward-looking formulations) anddetermine which of the axioms they fulfil. Subsequently, in Sect. 4 we present a number of votingscenarios known from social choice theory and determine agent’s responsibility ascription within thesescenarios. In Sect. 5 we discuss selected aspects of our results and finally conclude in Sect. 6.
We start this section by proposing a specific formal framework for the study of responsibility in multi-agent settings with stochasticity and ambiguities. It is based on the game-theoretical data structureof a game in extensive form, which is a multi-agent version of a decision tree, but with the additionalpossibility of encoding ambiguity via a special type of node. Also, in contrast to games, we do not specify6 a) i 1not loaded i 2loaded 3pass 4shoot 5pass 6shoot (b) j i 1don't throw i 2throw 3don't throw 4throw 5don't throw 6throw (c) i 1 21 – p 3p 41 – q 5q (d) i 1 3rescuepass i 2 4rescue 5passp > 0 61 – p < 1 Figure 1: Multi-agent decision situations that are paradigmatic for the assessment of responsibility,modelled by a suitable type of decision tree. Diamonds represent decisions and ambiguities, squaresstochastic uncertainty, circles outcomes, which are colored grey if ethically undesired. Dashed linesconnect nodes that an agent cannot distinguish when choosing. (a) Agent i may shoot a prisoner, notknowing whether the gun was loaded (node v ) or not ( v ), leading to the prisoner dead (node v ) oralive ( v , v , v ). (b) Agents i, j may each throw a stone into a window, not seeing the other’s action. (c)Agent i can choose between two probabilities of an undesired outcome. (d) Agent i may rescue someonenow or, with some probably, later. (a) i 4risk of warming i 5risk of cooling 9don't heat up 10heat up 11don't heat up 12heat up (b) i 1risk of warming i 2risk of cooling i 3learn i 4pass i 5pass i 6learn 7don't heat up 8heat up 9don't heat up 10heat up 11don't heat up 12heat up 13don't heat up 14heat up Figure 2: Stylized version of a decision problem related to climate change, used to study the effect ofoptions to reduce ambiguity on responsibility. Humanity (agent i ) must choose between heating up Earthor not, initially not knowing whether there is a risk of global warming or cooling (a), but potentiallybeing able to acquire this knowledge by learning (b). While at present, humanity is in node 3, in the1970’s they might rather have been in nodes 1. 7ndividual payoffs for all outcomes but only a set of ethically undesired outcomes. This is sufficient, as wewill not apply any game-theoretic analyses referring to rational courses of actions or utility maximisationbut rather use this data structure to talk about responsibility assignments. We use ∆( A ) to denote the set of all probability distributions on a set A , and use the abbreviations A + B := A ∪ B , A + a := A ∪ { a } , A − B := A \ B , A − a := A \ { a } . Trees.
We define a multi-agent decision-tree with ambiguity (or shortly, a tree ) to be a structure T = (cid:104) I, ( V i ) , V a , V p , V o , E, ∼ , ( A v ) , ( c v ) , ( p v ) (cid:105) consisting of: • A nonempty finite set I of agents (or players). • For each i ∈ I , a finite set V i of i ’s decision nodes , all disjoint. We denote the set of all decisionnodes by V d := (cid:83) i ∈ I V i . • Further disjoint finite sets of nodes: a set V a of ambiguity nodes , a set V p of probability nodes , anda nonempty set V o of outcome nodes . We denote the set of all nodes by V := V d + V a + V p + V o . • A set of directed edges E ⊂ V × V so that ( V, E ) is a directed tree whose leaves are exactly theoutcome nodes: V o = { v ∈ V : (cid:54) ∃ v (cid:48) ∈ V (( v, v (cid:48) ) ∈ E ) } . For all v ∈ V − V o , let S v := { v (cid:48) ∈ V : ( v, v (cid:48) ) ∈ E } denote the set of possible successor nodes of v . • An information equivalence relation ∼ on V d so that v (cid:48) ∼ v ∈ V i implies v (cid:48) ∈ V i . We call theequivalence classes of ∼ in V i the information sets of i . • For each agent i ∈ I and decision node v ∈ V i , a nonempty finite set A v of i ’s possible actions in v , so that A v = A v (cid:48) whenever v ∼ v (cid:48) , and a bijective consequence function mapping actions tosuccessor nodes, c v : A v → S v . • For each probability node v ∈ V p , a probability distribution p v ∈ ∆( S v ) on the set of possiblesuccessor nodes.Our interpretation of these ingredients is the following: • A tree encodes a multi-agent decision situation where certain agents can make certain choices in acertain order, and outcome node v ∈ V o represents a possible ethically relevant state of affairs thatmay result from these choices. • Each decision node v ∈ V i represents a point in time where agent i has the agency to make adecision at free will. The elements of A v are the mutually exclusive choices i can make, includingany form of “doing nothing”, and c v ( a ) encodes the immediate consequences of choosing a in v .Often, c v ( a ) will be an ambiguity or probability node to encode uncertain consequences of actions. • Probability and ambiguity nodes and information-equivalence are used to represent various types ofuncertainty 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, andagents’ future knowledge at later nodes. The agents are assumed to always commonly know thetree, and at every point in time to know in which information set they currently are. In particular,they know that at any probability node v ∈ V p , the possible successor nodes are given by S v and8ave probabilities p v ( v (cid:48) ), v (cid:48) ∈ S v . In contrast, about an ambiguity node v ∈ V a they only know thatthe possible successor nodes are given by S v , without being able to rightfully attach probabilitiesto them. Ambiguity nodes can also be thought of as decision nodes associated to a special agentone might term ‘nature’.In contrast to the universal uncertainty at the tree-level encoded by probability and ambiguitynodes, information-equivalence is used to encode uncertainty at the agent level. While in a certaininformation set of information-equivalent decision nodes, an agent i cannot distinguish betweennodes v ∼ v (cid:48) and has the same set of possible actions A v = A (cid:48) v . • When setting up a tree model to assess some agent i ’s responsibility, the modeler must carefullydecide which actions and ambiguities to include. If the modeler follows the basic idea that whatmatters is what i “reasonably believes” in any decision node v d (as in [3]), then A v d should consist ofthose options that i reasonably believes to have, S ( v ) for v ∈ V a ∪ V p should reflect what possibilities i reasonably beliefs exist at v , the choice whether v is an ambiguity or probability node shoulddepend on whether i can reasonably believe in certain probabilities of these possibilities, and ifso, then p v should reflect those subjective but reasonable probabilities. Likewise, if the modelerfollows the view that certain forms of ignorance may be a moral excuse (as in [47]), the informationequivalence 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 probabilitieslike those reported by the IPCC [29] or those corresponding to the assumption that “nature” uses anEllsberg strategy [15].Note that in contrast to some other frameworks, e.g., those using normal -form (instead of extensive-form) game forms such as [9], our trees do not directly allow for two agents to act at the exact sametime point. Indeed, in a real world in which time is continuous, one action will almost certainly precedeanother, if only by a minimal time interval. Still, as in the theory of extensive-form games, two actionsmay be considered “simultaneous” for the purpose of the analysis if they occur so close in time that thelater acting player cannot know what the earlier action was, and this ignorance can easily be encodedby means of information equivalence in a way similar to Fig. 6. Events, groups, responsibility functions (RFs).
As in probability theory, we call each subset ε ⊆ V o of outcomes a possible event. In the remainder of this paper, we will use ε to represent anethically undesirable event, such as the death of an innocent person, the occurrence of strong climatechange, or the election of an extremist candidate, whose probability might be influenced by the agents.Any nonempty subset G ⊆ I of agents is called a group in this article. Our main objects of interest are quantitative metrics of degrees of responsibility that we formalise as backward-responsibility functions (BRF) R b and forward-responsibility functions (FRF) R f .A BRF maps every combination of tree T , group G , event ε , and outcome node v ∈ V o to a real number R b ( T , v, G, ε ) meant to represent some form of degree of backward-looking (aka ex-post or retrospective)responsibility of G regarding ε in the multi-agent decision situation encoded by T when outcome v hasoccurred.An FRF maps every combination of tree T , group G , event ε , and decision node v ∈ V d to areal number R f ( T , v, G, ε ) meant to represent some form of degree of forward-looking (aka ex-ante)responsibility of G regarding ε in the multi-agent decision situation encoded by T when in decision node v . Note that we deliberately do not require that a set of agents shares any identity or possesses ways of communicationor coordination for an ethical observer to meaningfully attribute responsibility to this “group”. G = { i } , we also write R b/f ( T , v, i, ε ). Whenever any of the arguments T , v , G , ε are kept fixedand are thus obvious from the context, we omit to explicate them when writing R b/f or any of theauxiliary functions defined below. Graphical representation.
As exemplified in Fig. 1, we can represent a tree T and event ε graphicallyas follows. Edges are arrows, decision nodes are diamonds labelled by agents, with arrows labelled byactions, ambiguity nodes are unlabelled diamonds, probability nodes are squares with arrows labelledby probabilities, and outcome nodes are circles, filled in grey if the outcome belongs to ε . Finally,information equivalence is indicated by dashed lines connecting or surrounding the equivalent nodes. Auxiliary notation.
The set of decision nodes of a group G ⊆ I is V G := (cid:83) i ∈ G V i . To ease thedefinition of “scenario” below we denote the set of non-probabilistic uncertainty nodes other than V G (i.e., non- G decision and ambiguity nodes) by V − G := V d − V G + V a . If v (cid:48) ∈ S v , we call P ( v (cid:48) ) := v the predecessor of v (cid:48) . Let v ∈ V be the root node of ( V, E ), i.e., theonly node without predecessor. The history of v ∈ V is then H ( v ) := { v, P ( v ) , P ( P (( v )) , . . . , r } , where r is the root node of ( V, E ). In the other direction, we call B ( v ) := { v (cid:48) ∈ V : v ∈ H ( v (cid:48) ) } the (forward)branch of v . Taking into account information equivalence, we also define the information branch of v as B ∼ ( v ) := (cid:83) v (cid:48) ∼ v B ( v (cid:48) ).If v ∈ V , v d ∈ H ( v ) ∩ V d , and c v d ( a ) ∈ H ( v ), we call C v d ( v ) := a the choice at v d that ultimately ledto node v .A node v ∈ V d with { v (cid:48) : v (cid:48) ∼ v } = { v } is called a complete information node. Strategies, scenarios, likelihoods.
We call a function σ : V σG → (cid:83) v d ∈ V σG A v d that chooses actions σ ( v d ) ∈ A v d for some set V σG of G ’s decision nodes a partial strategy for G at v iff v ∈ V , V σG ⊆ V G ∩ B ∼ ( v ), σ ( v d ) = σ ( v (cid:48) d ) whenever v d ∼ v (cid:48) d , and V σG ∩ B ∼ ( c v d ( a )) = ∅ for all v d ∈ V σG and a ∈ A v d − σ ( v d ). Thelatter condition says that σ does not specify actions for decision nodes that become unreachable byearlier choices made by σ . A strategy for G at v is a partial strategy with a maximal domain V σG . Thismeans that a strategy specifies actions for all decision nodes that can be reached from the informationset containing v given the strategy.Let Σ( T , v, G ) (or shortly Σ( v ) if T , G are fixed) be the set of all those strategies. For σ ∈ Σ( T , v, G ),let V σo := { v o ∈ B ∼ ( v ) ∩ V o : C v d ( v o ) = σ ( v d ) for all v d ∈ B ∼ ( v ) ∩ H ( v o ) ∩ V G } , i.e., the set of possible outcomes when G follows σ from v on.Complementary, consider a function ζ : V ζ → (cid:83) v (cid:48) ∈ V ζ S v (cid:48) that chooses successor nodes ζ ( v (cid:48) ) ∈ S v (cid:48) for a set V ζ of ambiguity or others’ decision nodes, and some node v ζ ∈ V d . Then we call ζ a partialscenario for G at v iff v ∈ V , v ζ = v or v ζ ∼ v , V ζ ⊆ V − G ∩ B ( v ζ ), ζ ( v (cid:48) ) = c v (cid:48) ( a ) and ζ ( v (cid:48)(cid:48) ) = c v (cid:48)(cid:48) ( a )for some a ∈ A v (cid:48) whenever v (cid:48) ∼ v (cid:48)(cid:48) ∈ V ζ , and V ζ ∩ B ∼ ( v (cid:48)(cid:48) ) = ∅ for all v (cid:48) ∈ V ζ and v (cid:48)(cid:48) ∈ S ( v (cid:48) ) − ζ ( v (cid:48) ).The latter condition says that ζ does not specify successors for nodes becoming unreachable under ζ . A scenario for G at v is a partial scenario with a maximal domain V ζ . This means that a scenario specifiessuccessors for all ambiguity and others’ decision nodes that can be reached from v or the information setcontaining v given the scenario.Let Z ∼ ( T , v, G ) (or shortly Z ∼ ( v )) be the set of all scenarios at v and Z ( T , v, G ) ⊆ Z ∼ ( T , v, G ) (orshortly Z ( v )) that of all scenarios at v with v ζ = v .Each strategy-scenario pair ( σ, ζ ) ∈ Σ( v ) × Z ∼ ( v ) induces a Markov process on B ∼ ( v ) leading to a prospect , i.e., a probability distribution π v,σ,ζ ∈ ∆( V o ∩ B ∼ ( v )) on the potential future outcome nodes, This can be thought of as nodes where someone who is not part of group G — another agent or Nature — takes adecision. ψ ( v ζ ) = 1 , (1) ψ ( v (cid:48)(cid:48) ) = ψ ( v d ) if [ v d ∈ V G ∧ v (cid:48)(cid:48) = c v d ( σ ( v d ))] ∨ [ v (cid:48) ∈ V − G ∧ v (cid:48)(cid:48) = ζ ( v (cid:48) )] , (2) ψ ( v (cid:48)(cid:48) ) = ψ ( v (cid:48) ) p v (cid:48) ( v (cid:48)(cid:48) ) for v (cid:48) ∈ V p , v (cid:48)(cid:48) ∈ S v (cid:48) , (3) ψ ( v (cid:48)(cid:48) ) = 0 for all other v (cid:48)(cid:48) ∈ B ∼ ( v ) , (4) π v,σ,ζ ( v o ) = ψ ( v o ) for all v o ∈ V o ∩ B ∼ ( v ) . (5)Let us denote the resulting likelihood of ε by (cid:96) ( ε | v, σ, ζ ) := (cid:88) v o ∈ ε π v,σ,ζ ( v o ) . Following an axiomatic approach similar to what social choice theory does for group decision methodsand welfare functions, we study RFs by means of a number of potentially desirable properties formalizedas axioms.
In the main text, we focus on a selection of axioms which turn out to motivate or distinguish be-tween certain variants of RFs that we will develop in the next section and then apply to social choicemechanisms. 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 agentsthe same way), and a number of independence axioms such as the independence of branches with zeroprobability, and, more notably, also the following two axioms: (IOA)
Independence of Others’ Agency. If i ∈ I − G , and some of i ’s decision nodes v d ∈ V i is turnedinto an ambiguity node v a with S v a = S v d , then R ( G ) remains unchanged (i.e., it is irrelevantwhether uncertain consequences are due to choices of other agents or some non-agent mechanismwith ambiguous consequences). (IGC) Independence of Group Composition. If i, i (cid:48) ∈ G and all occurrences of i (cid:48) are replaced in T by i , R ( G ) remains unchanged.Note that these two conditions preclude dividing a group’s responsibility equally between its membersor following other agent- or group-counting 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 complete-information decision node v d ∈ V i is succeededvia some action a ∈ A v d by another complete-information decision node v (cid:48) d = c v d ( a ) ∈ V i of thesame agent, then the two decisions may be treated as part of a single decision, i.e., v (cid:48) d may bepulled back into v d : v (cid:48) d may be eliminated, S v (cid:48) d added to S v d , { a } × A v (cid:48) d added to A v d , and c v d extended by c v d ( a, a (cid:48) ) = c v (cid:48) d ( a (cid:48) ) for all a (cid:48) ∈ A v (cid:48) d . (IAT) Independence of Ambiguity Timing.
Assume some probability node v ∈ V p or complete-information decision node v ∈ V d is succeeded by an ambiguity node v a ∈ V a ∩ S v . Let B ( v ), B ( v a ) , B ( v (cid:48) ) be the original branches of the tree ( V, E ) starting at v , v a and any v (cid:48) ∈ S v a . For each v (cid:48) ∈ S v a , let B (cid:48) ( v (cid:48) ) be a new copy of the original B ( v ) in which the subbranch B ( v a ) is replacedby a copy of B ( v (cid:48) ); let f ( v (cid:48) ) be that copy of v that serves as the root of this new branch B (cid:48) ( v (cid:48) ).If v ∈ V d , put f ( v (cid:48) ) ∼ f ( v (cid:48)(cid:48) ) for all v (cid:48) , v (cid:48)(cid:48) ∈ S v a Let B (cid:48) ( v a ) be a new branch starting with v a andthen splitting into all these new branches B (cid:48) ( v (cid:48) ). Then v a may be “pulled before” v by replacingthe original B ( v ) by the new B (cid:48) ( v a ), as exemplified in Fig. 4.11 Figure 3: Situation related to the Independence of Nested Decisions (IND) axiom. The agent seessomeone having a heart attack and may either try to rescue them without hesitation, applying CPUuntil the ambulance arrives, or hesitate and then reconsider and try rescuing them after all, in whichcase it is ambiguous whether the attempt can still succeed. v v a p wq v’’v’ v a v’’v’v p wqv p wqv a a wb v’’v’ v a v’’v’a wba wbi v i vi v Figure 4: Explanation of the “pulling back” transformation described in the (IAT) axiom. Top: pullingback an ambiguity node before a probability node; bottom: pulling back an ambiguity node before adecision node, leading to information equivalence.(IND) may seem plausible if one imagines, say, a decision to turn either left or right directly followedby a decision to stop at 45 or 90 degrees rotation, since these two may more naturally be considered asingle decision between four possible actions, turning 90 or 45 degrees left or right. But in the situationof Fig. 3, it may rather seem that when hesitating and then passing, i has failed twice in a row, whichshould 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, allother actions being represented as simple ambiguity). If one accepts a number of similar further axiomslisted in the Appendix, one can also assume the reduced tree has only properly branching non-outcomenodes, has at most one ambiguity node and only as its root node, and has no two consecutive probabilitynodes and no zero probability edges.The next pair of axiom state that responsibility must react in the right direction under certainmodifications: (GSM) Group Size Monotonicity. If G ⊆ G (cid:48) then R ( G ) (cid:54) R ( G (cid:48) ) (i.e., larger groups have no lessresponsibility). (AMF) Ambiguity Monotonicity of Forward Responsibility.
If, from an ambiguity node v a ∈ V a , we12emove a possibility v (cid:48) ∈ S v a and its branch B ( v (cid:48) ), then forward responsibility R f ( v ) in anyremaining node v / ∈ B ( v (cid:48) ) does not increase.In other words, (AMF) requires that increasing ambiguity should not lower forward responsibility (be-cause 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 conditionfrom [9]: (NRV) No Responsibility Voids.
If there is no uncertainty, V a = V p = ∅ , and if ε (cid:54) = V o , then for eachundesired outcome v o ∈ ε , some group G ⊆ I is at least partially responsible, R b ( v o , G ) > (NUR) No Unavoidable Backward Responsibility.
Each group G ⊆ I must have an original strategy σ ∈ Σ( v , G ) that is guaranteed to avoid any backward responsibility, i.e., so that R b ( v o , G ) = 0for all v o ∈ V σo . (MBF) Maximal Backward Responsibility Bounds Forward Responsibility.
For all v, G , there must be σ ∈ Σ( v, G ) and v o ∈ V σo so that R b ( v o , G ) (cid:62) R f ( v, G ) (i.e., forward responsibility is bounded bypotential backward responsibility).Finally, we consider four axioms that require certain assessments in the paradigmatic situations fromFig. 1 which are closely related to questions of moral luck [31, 1, 41], reasonable beliefs [3], and ignoranceas an excuse [47]: (NFT) No Fearful Thinking.
With T and ε as depicted in Fig. 1(b), R f ( v , { i } ) = 1 since i ’s actionmakes a difference even though she thinks acting might not help, since it would be unreasonableto believe this must be the case. (NUD) No Unfounded Distrust.
With T and ε as depicted in Fig. 1(b), R f ( v , { i } ) = 1 since i cannotknow that acting cannot help. (MFR) Multicausal Factual Responsibility.
With T and ε as depicted in Fig. 1(b), R b ( v , { i } ) = 1 since i ’s action was necessary even though not sufficient. (CFR) Counterfactual Responsibility.
With T and ε as depicted in Fig. 1(a), R b ( v , { i } ) = 1 since i could not know that her action would not cause ε so she must reasonably have taken into accountthat it might.Before turning to the definition of candidate RFs and study their axiom compliance, we brieflymention 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. Here we will introduce four pairs of responsibility functions ( R f , R b ) that fulfill most of the above axiomsbut each also violate a few, and a reference function R b related to strict causation.These candidate responsibility functions will measure degrees of responsibility in terms of differencesin likelihoods between available strategies in all possible scenarios.To define them, we need some additional auxiliary notation and terminology. For now, let us keep T , G , and ε fixed and drop them from notation.Since the below definitions typically involve several nodes, we denote the decision node at which R f is evaluated by v d ∈ V d , the outcome node at which R b is evaluated by v o ∈ V o , and other nodes by v, v (cid:48) ∈ V so that v comes before v (cid:48) (i.e., v ∈ H ( v (cid:48) ), v (cid:48) ∈ B ( v )).13 enchmark variant: strict causation. The most straightforward definition of a backward respon-sibility function in our framework that resembles the strict causation view, as employed for example inthe most basic way of ‘seeing to it that’, is to set R b ( v o ) := 1 iff there is a past node v ∈ H ( v o ) at which ε was certain, B ( v ) ∩ V o ⊆ ε , directly following a decision node v d = P ( v ) ∈ V G at which ε was notcertain, B ( v d ) ∩ V o (cid:54)⊆ ε , and to put R b ( v o ) := 0 otherwise.It is easy to see that given v o , there is at most one such v d regardless of G , and exactly those G aredeemed responsible which contain the agent choosing at v d , i.e., for which v d ∈ V G . The rationale for this variant, which tries to translate the basic idea of the stit approach into a proba-bilistic context, is that backward responsibility can be seen as arising from having caused an increase inthe guaranteed likelihood of an undesired outcome.
Guaranteed likelihood, caused increase, backwards responsibility.
We measure the guaranteedlikelihood of ε at some node v ∈ V by the quantity γ ( v ) := min σ ∈ Σ( v ) min ζ ∈ Z ( v ) (cid:96) ( ε | v, σ, ζ ) . (6)We measure the caused increase in guaranteed likelihood in choosing a ∈ A v d at decision node v d ∈ V d by the difference ∆ γ ( v d , a ) := γ ( c v d ( a )) − γ ( v d ) . (7)Note that since v d ∈ V d rather than v d ∈ V p , we have ∆ γ ( v d ) (cid:62) G ’s backward responsibility regarding ε in outcome node v o ∈ V o , in this variant we taketheir aggregate caused increases over all choices C v d ( v o ) taken by G that led to v o , R b ( v o ) := (cid:88) v d ∈ H ( v o ) ∩ V G ∆ γ ( v d , C v d ( v o )) . (8) Maximum caused increase, forward responsibility.
Finally, to measure G ’s forward responsibility regarding ε in decision node v d ∈ V G , we take the maximal possible caused increase, R f ( v d ) := max a ∈ A vd ∆ γ ( v d , a ) . (9)At this point, we notice to potential drawbacks of this variant. For one thing, it fails (IAT), mainlybecause it does not take into account any information equivalence and thus depends too much on subtletiming issues that the agents information does not depend on and that hence any responsibility assess-ments should maybe also not depend on. On the other hand, it is in a sense too “optimistic” by allowingagents to ignore the possibility that their action might make a negative difference if this is not guaranteedto be the case. The next variant tries to resolve these two issues. This variant is in a sense the opposite of variant 1 with respect to its ambiguity attitude. To understandtheir relationship, consider the tree in Fig. 5 which shows that variant 1 can be interpreted as suggesting14
Figure 5: Situation related to ambiguity aversion in which the complementarity of variants 1 and 2 ofour responsibility functions can be seen. The agent must choose between an ambiguous course and arisky course. The ambiguous course seems the right choice in variant 1 since it does not increase theguaranteed likelihood of a bad outcome, which remains zero, while the risky course seems right in variant2 since it reduces the minimax likelihood of a bad outcome from 1 to p .an ambiguity-affine strategy while variant 2 suggests an ambiguity-averse strategy.In this variant, the rationale is that backward responsibility can be seen as arising from havingdeviated from behaviour that would have seemed optimal in minimizing the worst-case (rather than theguaranteed) likelihood of an undesired outcome in view of the information available at the time of thedecision. In defining the worst-case, however, we assume a group G can plan and commit to optimalfuture behaviour, so some of the involved quantities are in now terms of strategies σ rather than actions a . Worst case and minimax likelihoods. G ’s worst-case likelihood of ε at any node v ∈ V given somestrategy σ ∈ Σ( v ) is given by λ ( v, σ ) := max ζ ∈ Z ∼ ( v ) (cid:96) ( ε | v, σ, ζ ) . (10) G ’s minimax likelihood regarding ε at v is the smallest achievable worst-case likelihood, µ ( v ) := min σ ∈ Σ( v ) λ ( v, σ ) = min σ ∈ Σ( v ) max ζ ∈ Z ∼ ( v ) (cid:96) ( ε | v, σ, ζ ) . (11)Note that (11) differs from (6) not only in using a maximum but also in taking into account possibleignorance about the true node by using Z ∼ instead of Z . Caused increase, backward responsibility.
We measure G ’s caused increase in minimax likelihood in choosing a ∈ A v d at node v d ∈ V G by taking the difference∆ µ ( v d , a ) := max v (cid:48) d ∼ v d µ ( c v (cid:48) d ( a )) − µ ( v d ) (cid:62) , (12)again now taking information equivalence into account. Similar to before, to measure G ’s backwardresponsibility regarding ε in node v o , we here take their aggregate caused increases in minimax likelihood, R b ( v o ) := (cid:88) v d ∈ H ( v o ) ∩ V G ∆ µ ( v d , C v d ( v o )) . (13) Maximum caused increase, forward responsibility.
In analogy to variant 1, to measure G ’s forward responsibility regarding ε in node v d ∈ V G , we take the maximal possible caused increase in15inimax likelihood, R f ( v d ) := max a ∈ A vd ∆ µ ( v d , a ) . (14)While this variant seems well related to the maximin-type 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 agentsto ignore the possibility that their action might make a positive difference. While variants 1 and 2 can be interpreted as measuring the deviation from a single optimal strategy thatminimizes either the guaranteed (best-case) or the worst-case likelihood of a bad outcome taking intoaccount all ambiguities, our next variant is based on families of scenario-dependent 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 arisesfrom taking risks to not avoid an undesirable outcome.
Optimum, shortfall, risk, backward responsibility.
Given a scenario ζ ∈ Z ∼ ( v ) at any node v ∈ V , the optimum G could achieve for avoiding ε at that node in that scenario is the minimumlikelihood over G ’s strategies at v , ω ( v, ζ ) := min σ ∈ Σ( v ) (cid:96) ( ε | v, σ, ζ ) . (15)So let us measure G ’s hypothetical shortfall in avoiding ε in scenario ζ due to their choice a ∈ A v d atnode v d ∈ V G by the difference in optima∆ ω ( v d , ζ, a ) := ω ( c v ζ ( a ) , ζ ) − ω ( v d , ζ ) (cid:62) . (16)Then then risk taken by G in choosing a is the maximum shortfall over all scenarios at v d , (cid:37) ( v d , a ) := max ζ ∈ Z ∼ ( v d ) ∆ ω ( v d , ζ, a ) . (17)To measure G ’s backward responsibility regarding ε in node v o ∈ V o , we now take their aggregate risktaken over all choices they made, R b ( v o ) := (cid:88) v d ∈ H ( v o ) ∩ V G (cid:37) ( v d , C v d ( v o )) . (18) Influence, forward responsibility.
Regarding forward responsibility, we test a different approachthan before, which is simpler but less strongly linked to backward responsibility. The rationale is thatsince G does not know which scenario applies, they must take into account that their actual influenceon the likelihood of ε might be as large as the maximum of this over all possible scenarios, so the largerthis value is the more careful G need to make their choices.Let us measure G ’s influence regarding ε in scenario ζ at any node v ∈ V by the range of likelihoodsspanned by G ’s strategies at v , ∆ (cid:96) ( v, ζ ) := max( L ) − min( L ) , (19) L := { (cid:96) ( ε | v, σ, ζ ) : σ ∈ Σ( v ) } . (20)16o measure G ’s forward responsibility regarding ε at node v d ∈ V G , we this time simply take theirmaximum influence over all scenarios at v d , R f ( v d ) := max ζ ∈ Z ∼ ( v d ) ∆ (cid:96) ( v d , ζ ) . (21)A main problem with R b is that it fails (NUR), so that in situations like Fig. 2(a), it will assign fullbackward responsibility no matter what i did. This “tragic” assessment arises because in such situations,there is no weakly dominant strategy that is optimal in all scenarios, hence risk-taking cannot be avoided. In our final variant, we turn the “tragic” assessments of variant 3 into “realistic” ones, making it fulfil(NUR), by using risk-minimizing 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 responsibilityapplied in variants 1 and 2 to fulfil (MBF), but at the cost of losing compliance with (AMF).
Risk-minimizing action, negligence, backward responsibility.
The minimal risk and set of risk-minimizing actions of G in decision node v d ∈ V G is (cid:37) ( v d ) := min a ∈ A vd (cid:37) ( v d , a ) , (22) α ( v d ) := arg min a ∈ A vd (cid:37) ( v d , a ) , (23)where the latter is nonempty but might contain several elements.We now suggest to measure G ’s degree of negligence in choosing a ∈ A v d at v d by the excess riskw.r.t. the minimum possible risk, ∆ (cid:37) ( v d , a ) := (cid:37) ( v d , a ) − (cid:37) ( v d ) (24)Comparing (24) with (7) and (12), we see that this variant is still sensitive to all scenarios (like variant3) rather than just the best-case (as in (7)) or the worst-case (as in (12)). In particular, if a strategy σ is weakly dominated by some undominated strategy σ (cid:48) , then using σ is considered negligent even if thedifference between σ and σ (cid:48) only matters in cases other than the best or worst.Now, to measure G ’s backward responsibility regarding ε in node v o ∈ V o , we suggest to take theiraggregate negligence over all choices taken, R b ( v o ) := (cid:88) v d ∈ H ( v ) ∩ V G ∆ (cid:37) ( v d , C v d ( v o )) . (25)This now fulfils (NUR) again since by using a risk-minimizing strategy σ for which σ ( v d ) ∈ α ( v d ) for all v d ∈ V G , G can avoid all backward responsibility. Maximum degree of negligence, forward responsibility.
In analogy to variants 1 and 2, tomeasure G ’s forward responsibility regarding ε in node v d ∈ V G , we suggest to take the maximal possibledegree of negligence, R f ( v d ) := max a ∈ A vd ∆ (cid:37) ( v d , a ) = max a ∈ A vd (cid:37) ( v d , a ) − (cid:37) ( v d ) . (26)17 ariant (IND) (IAT) (GSM) (AMF) (NRV) (NUR) (MBF) (NFT) (NUD) (MFR) (CFR)0 (cid:88) — (cid:88) n/a (cid:88) (cid:88) n/a n/a n/a (cid:88) —1 (cid:88) — (cid:88) — (cid:88) (cid:88) (cid:88) (cid:88) — (cid:88) —2 (cid:88) (cid:88) — — — (cid:88) (cid:88) — — — (cid:88) (cid:88) — (cid:88) (cid:88) — — (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) — — — (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) Table 1: Summary of selected axiom compliance by the suggested variants of ( R f , R b )We can now summarize some first results before turning to applying the above RFs in the socialchoice 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.
In this section, we apply the above-defined responsibility functions for measuring degrees of forward andbackward responsibility to a number of social choice problems in which an electorate of N voters usessome election or decision method or social choice rule to choose exactly one out of a number of candidatesor options, one of which, U , is ethically undesired. We are interested in the forward responsibility of agroup G of m voters to avoid the election of U at each stage of the decision process, and the backwardresponsibility of G for U being elected.We first consider deterministic single-round decision methods in which all voters vote simultaneouslyand probability plays a marginal role only to resolve ties, and significantly probabilistic single-rounddecision methods. Afterwards, we study a selection of two-round methods in which voters act twice withsome sharing of information between the two rounds. Finally, we turn to an example of a specific stylizedsocial choice problem related to climate policy making.We exploit all symmetry and independence properties heavily when modeling the otherwise ratherlarge decision trees. In particular, in each round, we implicitly treat a group G of m (cid:62) G ’s ballots. We then model the simultaneous decision of all voters in a certain round by a single decisionnode for G , followed by ambiguity nodes representing the choices of the m (cid:48) := N − m many other voters,one for each possible way or class of ways in which the members of V − G might vote, as exemplified inFig. 6.For simplicity, we do not discuss bordering cases in which ties may occur, in particular by assumingthe number of voters N is odd. Our results are summarized in Table 2. Two-option majority voting.
This is the simplest classical case. Besides the ethically undesiredoption U , there is only one other, ethically acceptable option A , and the event ε to be avoided is theelection of U . Each voter votes for either U or A , with no abstentions allowed, and the option with morevotes is elected.We find that R f ( G ) = 1 no matter how small m is, since in the scenario where about half of theother voters vote for U , G ’s voting determines whether U is elected ( (cid:96) = 1) or A ( (cid:96) = 0). By contrast, R f ( G ) = 1 only if m > N/
2, otherwise R f ( G ) = 0 since then G ’s worst-case likelihood is always 1.Similarly, R f ( G ) = 1 only if m > N/ (others)other (others)U 4< (N–1)/2 vote U 6(N–1)/2 vote U 8> (N–1)/2 vote U 5< (N–1)/2 vote U 7(N–1)/2 vote U 9> (N–1)/2 vote U Figure 6: Two-option majority voting from the perspective of a single voter i who can either voter forthe ethically undesired option U or another, acceptable option, not knowing how the other N − u voters from G (and an arbitrary number of the other voters) have voted for U .Obviously, R b = R b ( G ) = 1 iff u > N/
2, since only that guarantees a likelihood of 1.To determine R b ( G ), we notice that for m < N/ G ’s worst-case likelihood is always 1, so G has zerodegree of deviation and R b ( G ) = 0; for m > N/
2, the (unconditional) minimax likelihood is µ = 0; theconditional minimax likelihood given u is 1 if m (cid:48) > N/ − u , otherwise 0. This implies that R b ( G ) = 1only if u > m − N/ >
0, otherwise 0.To determine R b ( G ), we first consider a scenario ζ in which v of the others have voted for U ; then G ’s optimum is ω ( ζ ) = 1 iff v > N/
2, otherwise ω ( ζ ) = 0, and G ’s optimum after choosing u is 1 iff v > N/ − u , otherwise 0; hence G ’s shortfall in ζ is ∆ ω ( ζ ) = 1 iff N/ > v > N/ − u , otherwise∆ ω ( ζ ) = 0. For m = 1, the relevant distinction w.r.t. v is depicted in Fig. 6. So G ’s risk taken bychoosing u was (cid:37) = 1 iff such a scenario exists, i.e., iff u > m (cid:48) > N/ − u , otherwise (cid:37) = 0. Thisimplies that R b ( G ) = 1 if u > max( m − N/ , R b ( G ) = 0. Since putting u = 0 is a weaklydominant strategy, R b = R b here.By comparison, we see that in variants 3 and 4 of our responsibility functions, minorities can havenonzero forward and backward responsibility for the election outcome, while in variant 2 only majori-ties can. In particular, under variants 3 and 4 every single voter who voted for U has full backwardresponsibility 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 be-haviour of the others is irrelevant for the assessment of backward responsibility.Note that this is in contrast to the ad-hoc idea that backward responsibility of G should be amore smoothly increasing function of the number u of voters from G that voted for U or maybe evenproportional to u . Random dictator.
A major contrast is given by a method that is rarely used in practise but oftenused as a theoretical benchmark in social choice theory, the “random dictator” method. In addition tooption U , 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 G controls exactly a share m/N of the winning probability, their influence on U ’s likelihood is m/N in all scenarios, hence R f = m/N . Also R f = R f = m/N since their actions span a range ofguaranteed likelihoods or worst-case likelihoods of width | m/N | . When u in G have voted for U , their19hortfall is u/N in all scenarios, hence R b = u/N . Also R b = R b = u/N since their action increased theguaranteed or worst-case likelihood by u/N .But R b = 0 unless u = m = N since for u < N a positive probability for ¬ U remains. This showsthat in situations with considerable stochasticity, assessments based on deterministic causation such as R b differentiate too little to be of any practical use. Multi-option simple majority.
Coming back to majority voting, we next study the case of morethan two options, and will see that this leads to much more complicated analysis. With an undesirableoption U , k (cid:62) A j , and the possibility to abstain, we assume the winner is electedby lot from those that got the largest number of votes.Suppose u of the m voters from G vote for U and a j for A j , with max j a j = a . Then the guaranteedlikelihood of U and the value of R b are 0 (since the others can avoid U for sure), iff u − a < m (cid:48) , theyare in [1 / ( k + 1) , /
2] iff u − a = m (cid:48) , and they are 1 iff u − a > m (cid:48) . So G can increase the guaranteedlikelihood by 1 (i.e., R f = 1) iff m > N (since only then they can make both u − a < m (cid:48) and u − a > m (cid:48) ),and otherwise (iff m < N ), R f = 0.Likewise, the worst-case likelihood of U is 1 (since the others can make U win for sure) iff a − u < m (cid:48) ,it is in [1 / ( k + 1) , /
2] iff a − u = m (cid:48) , and it is 0 iff a − u > m (cid:48) . So G can increase the minimax likelihoodby 1 (i.e., R f = 1) iff m > N (since only then they can make both a − u < m (cid:48) and a − u > m (cid:48) ), andotherwise (iff m < N ), R f = 0. Hence R b = 1 iff m > N and a − u < m (cid:48) , R b ∈ [1 / ( k + 1) , /
2] iff m > N and a − u = m (cid:48) , and R b = 0 otherwise.If all others abstain, G ’s influence is 1, hence R f = 1 no matter how small G . Assume a − u < m (cid:48) .Then of the others, v = a − u + 1 many could vote for U and all others could abstain, so that U getselected for sure. If a − u < m − G could then have avoided this outcome by increasing a − u by2. Hence if a − u < min( m (cid:48) , m − G takes risk 1 and gets R b = 1. But also if u = 0, a = m , and m (cid:48) (cid:62) m + 3, G takes risk 1, since of the others, v = m + 1 many could voter for u , two for a third option A i that G did not vote for, and all others could abstain, so that again U gets elected for sure and G could have avoided it by voting for the same A i as the others. This shows that for m < N − R b = 1no matter what G does, hence R b = 0 no matter what G does, and hence R f = 0.So, in contrast to the two-option case, in the multi-canditate case only variant 3 assigns full backwardresponsibility to a single voter who votes for U , while variant 4 acknowledges the possible excuse that,because there is not a unique contender to U , and hence no weakly dominating strategy, also any otherway of voting of the single voter could have helped U win.But variant 3 has the major problem that it assigns full backward responsibility to every minorityregardless of their behaviour. Approval voting.
Here everything is just as in multi-option simple majority, except that a voter cannow vote for any number of options at the same time [10]. Also the analysis is the same as before, exceptthat now also a minority G has a weakly dominating strategy that reduces their risk to zero, namelyvoting for all options but U . As a consequence, now R = R again, and a single voter voting for U butno 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 methodstudied in [20] that was designed to give each group of voters an effective decision power proportionalto their size (in contrast to majoritarian methods which give each majority full effective power and nominority any effective power).In this method, each voter marks one option as “favourite” and one as “consensus”. If all mark thesame option X as consensus, X is elected, otherwise the option marked as favourite on a random ballot20s elected. Let u be G ’s “favourite” votes for U and a j G ’s “consensus” votes for option A j (cid:54) = U .If no member of G distinguished between her two votes, the analysis is the same as for the randomdictator method. If all in G put some A (cid:54) = U as consensus ( a = m ), the guaranteed likelihood of U staysat zero since A might still get 100% winning probability. In that case, R b = 0 even if U wins.All worst-case likelihoods come from scenarios where the others specify U as consensus, so the as-sessment in variant 2 is the same as in random dictator.Regarding risk, however, we find that always (cid:37) = ( u + m (cid:48) ) /N . This is because there is a scenariowhere everyone not in G elects U as favourite and the same option A as consensus. In this case G ’soptimal strategies are those where everyone also selects A as consensus, leading to a zero probability of U winning. If some of G ’s members select another option as consensus, the resulting likelihood of U being elected is ( u + m (cid:48) ) /N , which amounts to the risk taken by G .Hence R b = ( u + m (cid:48) ) /N and R f = 1. Since the least possible risk is (cid:37) = ( m (cid:48) ) /N , R b = R b − (cid:37) = u/N and R f = m/N .Note that as G cannot know which option the others select they cannot in the scenario describedabove know which option to select as consensus. Other majoritarian methods.
In any single-round method in which any group of m > N manymembers have a way of voting that enforces any option they might choose, we will have R f = R f = 1 m> N and R f = 1. Other proportional power allocating methods.
In any single-round method in which any groupof m many members have a way of voting that guarantees any option they might choose a probabilityof at least m/N , we will have R f = R f = m/N , R b (cid:54) m = N . Real-world social choice situations often turn out to consist of several stages upon closer examinationeven when the “main” voting activity consists of all voters acting “simulteneously”. There are manyways in which decisions taken before or after the main voting stage may be relevant, including thepre-selection of options or candidates put on the menu e.g. via “primaries”, taking and publishing anypre-election polls, seeing an election in the context of previous and future elections, using one or severalrun-offs 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 pre-election polling as a very common example of “cheap talk” before the actual decisionthat has no formal influence on the result; (ii) the possibility of amending an option as an example ofinfluencing 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 commonlyused in public elections in order to ascertain a majority.
Simple majority with a pre-election poll.
Before the actual voting by simple majority, a poll isperformed 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 mightform about the probabilities of other agents’ unknown choices at free will, which are rather treated likeany other ambiguity, the polling has no influence on our assessment. Backward responsibility dependsonly on actual voting behaviour, and forward responsibility is zero when answering the poll. The sameis true of any other form of pre-voting “cheap talk”.21n particular, in all our variants, even the prediction of a landslide victory of U does not reduce theresponsibility to help avoiding U . Two-option majority with an amendment round.
Now we turn to a case where the repeatedchoices can really lead to changed responsibilities which might even exceed 1 in case of repeated failureto avoid U .In round 1, U is compared to an amended, ethically acceptable version A . If U wins, it is comparedto another acceptable option B in round 2. Abstentions are not allowed. Assume a of the m voted A inround 1 and b of the m voted B in round 2 (if round 2 is not reached since A won in round 1, then weput b = m ). U can only be caused for sure or its guaranteed or worst-case likelihood increased in round 2 if m > N ,by putting b < N ; hence in round 1, R f = R f = 0; in round 2, R f = R f = 1 m> N ; and eventually R b = R b = R b = 1 b< N . Also, G ’s maximal influence is R f = 1 in both rounds since their votes mightmake a difference.From the simple majority analysis, we know already that R f = R f = 1 in round 2 and R b , R b havea summand 1 b< min( N ,m ) from their action in round 2. Assume of the m (cid:48) others, a (cid:48) will vote A in round1 and b (cid:48) would vote B in round 2. In round 1, G ’s optimum likelihood is then 0 iff they can prevent U , i.e., iff max( a (cid:48) , b (cid:48) ) > N − m , otherwise it is 1. After voting in round 1, the optimum changes from 0to 1 (so that G has a shortfall of 1) iff a + a (cid:48) < N and max( a (cid:48) , b (cid:48) ) > N − m but b (cid:48) < N − m , i.e., iff b (cid:48) < N − m < a (cid:48) < N − a ; otherwise G ’s shortfall is 0. G ’s risk taken by choosing a is then 1 iff theothers can choose a (cid:48) , b (cid:48) so that b (cid:48) < N − m < a (cid:48) < N − a , i.e., iff a < m < N . Hence R b has a summand1 a
After a simple vote on the three options
U, A, B without abstentions (round 1), eitherthe one with an absolute majority (more than N votes) wins or U has the fewest votes, or a round 2 istaken where U is compared to the other front-runner.Let a, b be G ’s votes for A, B in round 1 and u their votes for U in round 2 (or 0 if there is no round2). For simplicity, we ignore ties here. U can be caused if m > N , by putting a + b < N in round 1 or u > N in round 2, giving the valuesfor variants 0–1 (see Table 2), and again maximal influence is 1 in both rounds. By choosing a, b inround 1, G increases minimax likelihood from 0 to 1 iff a + b < N < m .Again, R f = R f = 1 in round 2 and R b , R b have a summand 1 u> max( m − N , from round 2. Assumethe scenario in which the corresponding vote counts of the m (cid:48) others are a (cid:48) , b (cid:48) , u (cid:48) . In round 1, G ’s optimumlikelihood is then 0 if they can either exclude U in round 1 by putting min( a + a (cid:48) , b + b (cid:48) ) > N − a − b − a (cid:48) − b (cid:48) ,which is possible iff a (cid:48) + b (cid:48) > N/ − m , or if they can make U lose in round 2, which is possible iff u (cid:48) < N .By choosing a, b , they increase the optimum likelihood to 1 if they either make U win in round 1 byputting a + b < N − a (cid:48) − b (cid:48) or if they let U get to round 2 by putting min( a + a (cid:48) , b + b (cid:48) ) < N − a − b − a (cid:48) − b (cid:48) in asituation where they cannot avoid that U will win in round 2 since u (cid:48) > N . In all, their shortfall in round1 will be 1 iff ( a (cid:48) + b (cid:48) > N/ − m or u (cid:48) < N ) and ( a + b < N − a (cid:48) − b (cid:48) or (min( a + a (cid:48) , b + b (cid:48) ) < N − a − b − a (cid:48) − b (cid:48) and u (cid:48) > N )). This is equivalent to 2 N/ − m < a (cid:48) + b (cid:48) < N/ − a − b or ( a (cid:48) + b (cid:48) > N/ − m andmin( a + a (cid:48) , b + b (cid:48) ) < N − a − b − a (cid:48) − b (cid:48) and u (cid:48) > N ) or ( u (cid:48) < N and a + b < N − a (cid:48) − b (cid:48) ). Such ascenario ( a (cid:48) , b (cid:48) , u (cid:48) ) exists iff a + b < max( m − N/ , N ), so this is the condition for having taken a risk of1 in round 1, contributing to R b . Since it can be avoided by putting a + b = m , R b = R b .22 .3 Median voting for an emissions cap We finally analyze a stylized probabilistic example from global climate policy making inspired by Weitz-man’s discussion of a fictitious World Climate Assembly [44]. Assume countries vote on a global green-house gas emissions cap (or, alternatively, a carbon price) by median voting. Each country i specifies anamount a i (cid:62) a , . . . , a N ) is realized as a global cap. Thiscan be seen as a shortcut to taking a series of binary majority decisions in each of which the cap maybe lowered or raised by some amount.As a consequence of the resulting emissions-induced temperature increase, a certain climatic tippingelement [25] may tip ( ε ), leading to undesired economic damages and loss of life. Let f ( a ) be the bestestimate of the probability of tipping given cap a , based on the current state of scientific knowledge, andassume f ( a ) is weakly increasing in a , f ( c ) = 0 and f ( c ) = 1 for some values c , c .Tipping can only be caused only if m > N , e.g. by putting a i (cid:62) c for all i ∈ G . Assuming the m members of G voted a (cid:54) · · · (cid:54) a m , we hence get R b = 1 if m > N and a m − ( N − / (cid:62) c , else 0.Similarly, R b = f ( a m − ( N − / ) if m > N , else 0; hence R f = 1 m> N .The worst-case likelihood before voting is 1 m< N . After voting, it is f ( a ( N +1) / ) if m > N , else 1.Hence R b = f ( a ( N +1) / ) if m > N , else 0, and also R f = 1 m> N .In the scenario where the others vote a (cid:48) (cid:54) · · · (cid:54) a (cid:48) m (cid:48) , G ’s optimum likelihood before voting is ω = f ( a (cid:48) m (cid:48) − ( N − / ) if m < N , else 0. Their shortfall is ∆ ω = med( a , . . . , a N ) − ω . Given G ’s votes, if m > N , the scenario that maximizes this shortfall is when all others vote c so that ∆ ω = f ( a ( N +1) / ).If m < N , it is when ( N + 1) / − m many of the others vote c and the rest c so that ∆ ω = f ( a m ). Inall, G ’s risk by voting a G is R b = f ( a min( m, ( N +1) / ). This is minimized to 0 by the weakly dominantstrategy of putting a G ≡ c , hence R b = R b , R f = R f = 1. In this section we will present a discussion of a selection of responsibility ascriptions resulting from theapplication of our proposed functions to the paradigmatic examples presented in the beginning. This willinclude reference to certain of the desired axioms as well as to properties of existing formalisations, andthe question of whether or not there are fulfilled by the corresponding functions. We will also discuss aselection of results of the application of the responsibility functions in the social choice scenarios. (MFR), (CFR) and luck.
As was discussed above when introducing the paradigmatic examplescenarios, we believe that in a situation where an agent did not know whether their action was going tohave an effect, such as those represented in Figure 1(a), node 4, and Figure 1(b), node 6, the responsibilityascription should be made on the basis of their having to assume that their action was going to havean effect. The rationale behind this is precisely to disable dodging by referring to certain assumptionsabout the state of the world, since such assumptions would form an “unreasonable belief” in certaintyin an actually uncertain situation. This relates to the discussion on moral luck , and the statementby [42] that we mentioned in the introduction, arguing for disregarding effects outside of the agents’control. these considerations are reflected in the axioms (MFR) and (CFR). R b and R b diverge on thisquestion, with R b assigning full backwards responsibility in the case of throwing a rock even though thewindow would have shattered anyways, but not assigning any responsibility for shooting when the gunwas not loaded, and R b giving inverse results. Ideally, both axioms would be fulfilled by a responsibilityfunction. In variant 3, we managed this by basing it on a maximum over likelihood differences (whichwe call ‘risk’ in this context) rather than on a difference of minimax likelihoods. Since this introducedsome overdetermination, we further modified the formula in variant 4 to give agents again a way to avoid23lame. Voids.
One important topic when talking about responsibility ascription in voting scenarios and otherinteractive settings is the potential of responsibility voids, as discussed in [9]. While such situations,in which one does not assign responsibility to any single agent due to the interactions of several agentsand/or nature, cannot occur in our variants 0, 1, and 3, they do exist in our variants 2 and 4. It is,however, our intuition that this is not a serious problem as long as one assigns responsiblity to at leastsome nonempty group of agents, which variants 2 and 4 do. Still, it makes these variants fail groupsubadditivity (see Appendix).
Effect of reducing ambiguity by learning.
Consider Fig. 2, which is a stylized version of thedecision situation humanity faced around the 1970’s regarding climate change, when it was already clearthat humanity can influence global mean temperature via greenhouse gas emissions, but when it was stillunclear that there was a risk of undesired global warming rather than one of undesired global coolingdue to the onset of glaciation.Let us at first assume that the option to learn about which of the two scenario was correct wasunavailable, Fig. 2(a). Then, in nodes 4 and 5, where humanity chooses to either heat-up the Earth(via high GHG emissions) or not (via low emissions), we have R f = 1 since their choice would definitelymake a difference, but R f = 0 since they don’t know what the right choice is. Likewise, in nodes 10 and11 we have R b = 1 and R b = 1 , since then they made the wrong choice in node 4 or 5, but R b = 0since they didn’t know it was the wrong choice. Even in nodes 9 and 12, we then have R b = 1 since eventhough their choice was right, it could have been wrong, while R b = 0 since their choice was not wrongfrom a worst-case avoidance perspective, and R b = 0 since their choice was right in the true scenario.The fact that variant 2 does not assign responsibility in nodes 10 and 11 since G did not know theywere causing the undesired event of large-scale climate change might seem problematic since it mightseem that such a method of assessing responsibility gives perverse incentives to remain ignorant to avoidresponsibility.However, we will see that when we take into account that an agent has chosen to remain ignorant,the responsibility assessment will reflect this. Let us therefore now take into account the learning optionin nodes 1 and 2 in Fig. 2(b). While this does not change R f and R f in nodes 4 and 5, it changes theremaining values. In nodes 1 and 2, we get R f = 1 since their future choices will make a difference,and also R f = 1 since the worst-case likelihood is 0 for the learning option but 1 for the passing option.So learning is a minimax strategy here and consequently R b counts the choice to pass as a deviation ofdegree 1, now leading to R b = 1 in nodes 10 and 11 (as well as nodes 8, 9, 12, and 13). R b = 0 only innodes 7 and 14 where both choices made were correct. R b behaves the same as R b here. R b however,caring only about causation, is unaffected by knowledge and so still has value 1 exactly in those nodesbelonging to ε : 8, 10, 11, 13.Needless to say, today humanity is in node 3, where both variants 1 and 2 assign full forward respon-sibility, even though climate science “sceptics” claim we’re in information set { , } or still in { , } , oreven deny the whole model. Effect of reducing ambiguity coordination.
When we replace the initial ambiguity node of Fig.2(a) and (b) by another agent j ’s decision node, (a) becomes formally equivalent to a pure coordinationgame such as choosing one of two possible places to meet, and (b) can represent the possibility that i can call j to ask where j will go, in order to coordinate. This means that also coordination can reduceresponsibility. Likewise, in a social choice situation, voters who could coordinate their votes but fail to24o so will be more responsible. The same will be true when voters could inform themselves about thelikely consequences of the given options but fail to do so. Relationship between individual and group responsibility.
In [7], the example of three walkerstogether freeing a jogger from below a fallen tree is discussed, under the assumption that all three helplifting the trunk while two would have sufficed to lift it. In that example, which is formally equivalentto the two-option majority voting with three voters, if we assume neither walker sees whether the othersare really lifting the trunk or just pretending to, our variants 0–2 judge no single walker but each pair ofwalkers (as well as all three together) as responsible for freeing the jogger. Variants 3 and 4 also judgeeach single walker as responsible.The two-option majority example is also enlightening regarding the monotonicity demanded by(GSM). As one can see from the dependence on m shown in the top row of Table 2, variants 2–4 fail(GSM) since they may assign a group less responsibility than its members in cases where one member’sactions makes harmless the possibly bad consequences of another member’s action who could howevernot trust that he will be this lucky. In such cases, these variants judge the latter member responsible,thereby fulfilling (CFR), but not the group. Indeed it seems difficult to fulfill both (GSM) and (CFR)without also assigning a group responsibility in cases where they were not simply lucky.Another type of relationship between a group and a group member appears in [27]. In this example,a group orders one member to do something who can then decide to follow the order or not, but whowill not act without the order. Since the order does not guarantee any positive probability of action, ourvariants 0 and 1 will only hold the member responsible. Since it increases the worst-case probability andis risky, variants 2–4 will also hold the group responsible, which seems to conform better to the generalintuition. Influence of timing.
As we discussed when arguing for the use of extensive-form games rather thanthe more common normal-form games, actions in real life hardly ever happen at the exact same time.Considering for example a situation with two agents, their actions can be regarded as practically co-incidental if they do not know about the other’s choice before deciding on their own action. This isrepresented in our model using information sets. However, according to the argument given here, itshould not matter whether we represent the situation as one agent acting first and the other followingup, or the other way around. However, in variants R b and R b , this change of representation wouldcounter-intuitively shift responsibility assignments between the agents. Variants R and R manage toavoid this fallacy by taking into account the agent’s information set. We have established that determining a representation of degrees of responsibility within interactivescenarios playing out over time with probabilistic uncertainty as well as ambiguity is an importantendeavour. Specifically in light of the current climate crisis it becomes of a special interest to providecalculations applicable to a set of election scenarios that take into account all of these complexities.Certain existing calculations are of an ad hoc nature, reducing the issue of responsibility for climatechange to cumulative past emissions or population shares. Others are of a foundational nature, providingrepresentations for the concept of responsibility in general, but not considering all of the complicationsat the same time or distributing responsibility rather cautiously, leading to voids.In the present paper we followed the second route, by providing an account that applies to questionsof responsibility in general, but specifically aims at accounting for complexities in real-world applications.We suggested a number of responsibility functions extending those previously presented in the literature,25o open the discussion on available representations. We used an axiomatic method, as it is known fromsocial choice theory, to evaluate the proposed functions in a rigorous way.
Framework.
The framework used in this paper is an extension of extensive-form games. This game-form includes a temporal aspect, allowing for agents to make choices successively. We used specific nodesto represent ambiguity and probabilistic uncertainty with respect to the state of the world after this node,and an equivalence relation to express agents’ perception (equivalent nodes cannot be distinguished). Aswe do not apply game theoretic analyses, such as evaluations of strategies of rational agents, we didnot require individual utility functions. Instead, a universal ‘ethical desirability’ assessment was made,selecting a subset of the outcome nodes as undesirable.
Axioms.
Having introduced the framework we presented some potentially desirable properties forprospective responsibility functions. Clearly, a large number of such properties come to mind. Theones that we decided to present in detail were mainly ones that differentiate between the proposedresponsibility functions. One important feature that we kept in mind was that we wanted to avoiddodging of responsibility. That is, we wished to reduce the number of situations where an undesirableoutcome occurs or can occur but an agent potentially involved in its bringing about can claim to haveno responsibility.The first desirable properties that we presented were a set of independence axioms regarding thespecific representation of the decision scenario. That is, if one and the same situation can be representedusing slightly different game trees, this should not influence the resultant responsibility assignments.While these properties are certainly desirable, they are not trivial, as it is well known that in a formali-sation the specific choice of representation can have repercussions on the outcome (consider the ‘Queenof England’ example from [4] repeated in [8]).The next set of desirable properties were monotonicity requirements: first with respect to increasinggroup size, and second with respect to increasing knowledge.Subsequently we included another set of very intuitive considerations that have already been discussedin the literature. These conditions are the avoidance of responsibility voids (in the absence of uncertainty someone must be responsible for an undesirable outcome) and the possibility to avoid responsibility byfollowing some strategy that is optimal in this respect.Next, an axiom to relate forward and backward responsibility ascription was introduced: the degreeof forward responsibility of a group is bounded by their maximal degree of backward responsibility.Lastly, we presented a set of axioms relating to situations where the agent is unsure about the actualstate of the world and they do not know whether they are in a position to have any effect at all onthe outcome. One can argue that they need to take into account the possibility of their action beingsignificant, and accept the corresponding responsibility.
Candidate responsibility functions.
This set of axioms allowed for a fruitful comparison of severalcandidate responsibility functions. As the benchmark variant ( R f/b ) we studied a representation of‘strict causation’ in our framework: ascription of full backwards-looking responsibility to a group if andonly if there was a specific node at which an agent from the group took a decision determining theundesired outcome. Clearly for this variant does no axiom relating to forward-looking responsibilityapplies, nor does it include a sensible idea of degrees of responsibility. More importantly, however, itassigns zero responsibility if the agent’s action, unbeknownst to the agent, was not actually going toaffect the outcome. This, as we stated above, was something we wanted to avoid.The next suggestion ( R f/b ) was a function that extended the idea of ‘strict causation’ to a prob-abilistic context: we assigned responsibility to the degree that an agent has caused an increase in the26uaranteed likelihood of the undesired event. Group responsibility arises as an aggregate of its members’responsibility. While this function does include a notion of degree, it may still assign no responsibility ifthere was uncertainty regarding the effects of an action.The previous function preferred ambiguity over probabilistic uncertainty when trying to avoid respon-sibility, as the goal then has to be to avoid increasing guaranteed likelihood of the undesired outcome.As no probabilities are available for ambiguity nodes, the guaranteed likelihood of each of its successorsremains zero. On the contrary, one can consider a responsibility function which acts in the oppositedirection: preferring probabilistic uncertainty over ambiguity (as is the case for most human decisionmakers). We achieved this by considering increases in the minimal worst-case likelihood of the undesiredevent to lead to responsibility ascription ( R f/b ). The prescribed action rationale behind this functioncan be seen as ‘avoiding the worst’ (in order to avoid carrying responsibility), rather than optimisingfor the best, similar to the game-theoretic notion of maximin strategies. In situations where an agentdoes not know whether their action leads to an undesirable outcome or not this seems to be a reasonableconsideration. Surprisingly, however, this function may lead to responsibility voids.Both of the above functions reduce the conceptual complexity of responsibility ascription by relyingon comparisons between the action taken at a specific node and a ‘baseline case’. In order to exploitthe full weight of the extensive-form game structure that we have at hand, we decided to refer to theinformation sets as well as to strategies , i.e., full plans of action including reactions to future outcomes ofother agent’s choices or uncertainty resolutions. By employing these two features we managed to escapethe responsibility voids due to uncertainty that we experienced with the previous function.In the next variant ( R f/b ) we assigned responsibility whenever a hypothetical minimum was notreached. This avoided responsibility voids, however, it also resulted in groups sometimes being assignedresponsibility no matter what their action was. In the final variant we therefore set their best option asthe baseline to be compared to ( R f/b ). This avoided certain situations in which a group is always tosome extent responsible, but it re-introduced voids that were absent with the preceding function.As a bottom-line, even though we managed to fulfil all of the desired properties by one or anothervariant of responsibility functions, none of the functions studied so far complies to all of them. Social choice.
In a next step we determined the responsibility ascription the proposed functions offer inspecific social choice settings. The first set of methods we examined were single-round methods, startingwith simple two-option majority voting. In line with our considerations when suggesting the differentresponsibility functions, it does not actually matter for responsibility ascription (in either function) whatthe others voted. Also, considering that majority voting contains no probabilistic component out of theproposed functions only R f and R f can assign non-zero forward responsibility to non-majority groups.This consideration was one of the reasons we introduced these measures, as they represent a very commonintuition: even if a minority group (say one voter) cannot influence the outcome with certainty, they stillcarry a responsibility to avoid the undesirable candidate.As was to be expected, the benchmark responsibility function based on deterministic causation doesnot represent intuitions very well when we look at voting mechanisms that make use of probabilisticprocedures.Multi-option majority voting is somewhat more complicated than the single-option case, and, notably,differentiates between responsibility functions R and R . In the case of several alternatives with one ofthem being undesirable, it is not clear which other option to vote for in order to avoid the election of theundesired candidate, as any of the other candidates might turn out to be the strongest opponent. Our‘strictest’ function R b does not allow for this excuse, but assigns full responsibility to every minoritygroup, regardless of their behaviour. R b does allow for the described excuse and omits the unavoidableresponsibility of minorities. 27s a second set of voting methods we examined two-round methods, such as voting with a pre-electionpoll or simple runoff (between the two preferred candidates from the first round). As we explicitly didnot consider assumptions about other’s behaviour to have an influence on responsibility ascription, pollsor any other means for forming beliefs about other’s voting behaviour have no influence, in neither of ourfunctions. If the ethically undesirable candidate is elected in a runoff scenario, a group’s responsibilitycan rise above 1.As a last voting scenario, and to get back to our initial application of climate policy, we examinedresponsibility ascriptions in a hypothetical situation of median voting for an emissions cap (or carbonprice). The unwanted tipping of a certain element is induced with a certain probability, depending onthe elected cap. This example neatly represents the direct reflection of the probability measures in theresponsibility ascription. R b , R b and R b assign zero probability to minority groups, as their votes canguarantee neither a positive nor a below-one probability of tipping. In contrast, variants R b and R b assign a minority group a responsibility that equals that tipping probability which corresponds to thelargest cap that any of the group members suggested. In particular, a single voter suggesting some valueof the cap is responsible to the exact degree that this cap would make tipping likely. Outlook.
Due to the integration of methods from different disciplines several paths to continue thework presented here offer themselves.First of all, to give full credit to the axiomatic method employed here, it would be natural to determinelogical implications or exclusions between the axioms, as well as a characterisation of certain groups ofresponsibility functions with respect to sets of axioms.An alternative account of causation using a variant of the NESS test, which has a direct representationin our formalism, could be compared to the resulting functions developed here.The differences between variants 0 and 1 on one hand and 2–4 on the other suggest looking forcompromise variants, either by taking a closer look at the literature on choice under ambiguity [18], orby combining several functions into one, e.g. R (cid:48) = ( R + R ) / References [1] Judith Andre. Nagel, Williams, and moral luck.
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Appendix
Longer list of axioms
In this section, we compile a longer list of axioms which we believe may be relevant for the design ofplausible RFs. Similar to axioms in other branches of social choice theory, most of the axioms state thatthe value of an RF should not change or should change in a certain direction when some of its arguments T , v, G, ε are changed in certain simple ways. We group the axioms roughly into categories, beginningwith basic symmetry and independence axioms, then listing certain possible monotonicity properties,and finally some that suggest certain values in specific situations. We do not mean to suggest that allthese axioms should be fulfilled, only that there may be reasonable arguments why one might think itplausible to desire them. Symmetry axioms, independence, and simplification axioms.
Our first four axioms are similarto social choice theory’s anonymity and neutrality axioms. (Anon)
Anonymity.
If every occurrence of a certain individual i ∈ I is replaced in both T and ( V, E ) bya new individual i (cid:48) / ∈ I , then R ( G ) remains unchanged. (I.e., individuals’ identities are irrelevantbeyond their influence on the outcome.) (ACon) Action-Related Consequentialism.
If for some v ∈ V d , a certain action a ∈ A v is replaced by anew action a (cid:48) / ∈ A v in both A v (cid:48) and c v (cid:48) for all v (cid:48) ∼ v , then R ( G ) remains unchanged. (I.e., actionsare only relevant via their (potential or actual) consequences.) (OCon) Outcome-Related Consequentialism.
If a certain outcome v o ∈ V o − v is replaced in both T and ε by a new outcome v (cid:48) o / ∈ V , then R ( v ) remains unchanged; and if v ∈ V o and it is replaced inboth T and ε by a new outcome v (cid:48) o / ∈ V , then the new R ( v (cid:48) o ) equals the old R ( v ). (I.e., outcomesare only relevant via their belonging to ε .) (FCS) Forward Complementation Symmetry. If ε is replaced by its complement ε (cid:48) = V o − ε , forwardresponsibility does not change: R f ( ε (cid:48) ) = R f ( ε ). (I.e., forward responsibility is about G ’s influence31n which of the two mutually exclusive events ε, ε (cid:48) obtains, not about which of the two is ethicallydesirable.)Note that (ACon) in particular implies that there is no inherent difference between “doing something”and “doing nothing” despite their consequences.The next five allow us to trim and coarse-grain a tree in certain ways, and are ruling out RFs thatare based on some form of merely “counting possibilities”: (IST) Independence of Sure Thing Nodes.
If a node v ∈ V − V d or complete-information node v ∈ V d has only one successor, S v = { v (cid:48) } , it may be eliminated and replaced by v (cid:48) in S P ( v ) and c P ( v ) .(I.e., nodes with only one possible successor are irrelevant for responsibility assessments.) (IZP) Independence of Zero Probabilities.
If a successor v (cid:48) ∈ S v p of a probability node v p ∈ V p has zeroprobability, p v p ( v (cid:48) ) = 0, then v (cid:48) and its branch may be ignored in assessing R ( v ) for any v thatis not contained in the branch B ( v (cid:48) ). (I.e., possibilities that are “almost surely” not occurring areirrelevant for responsibility assessments.) (ICP) Independence of Cloned Possibilities.
Let v a ∈ V a be an ambiguity node and v (cid:48) ∈ S v a one ofits successors. Assume we add to S v a another node v (cid:48)(cid:48) which is an exact copy of v (cid:48) , followed bya branch B ( v (cid:48)(cid:48) ) that is an exact copy of B ( v (cid:48) ). Then R ( v ) must not change. (I.e., two identicalpossibilities are equivalent to just one copy of this possibility.) (INA) Independence of Nested Ambiguities.
If an ambiguity node v a ∈ V a is succeeded by anotherambiguity node v (cid:48) a ∈ V a ∩ S v a , v (cid:48) a may be “pulled back” into v a , i.e., v (cid:48) a may be eliminated and S v (cid:48) a added to S v a . (INP) Independence of Nested Probabilities.
If a probability node v p ∈ V p is succeeded by anotherprobability node v (cid:48) p ∈ V p ∩ S v p , v (cid:48) p may be pulled back into v p , i.e., v (cid:48) p may be eliminated, S v (cid:48) p added to S v p , and p v p extended to S v (cid:48) p via p v p ( v (cid:48)(cid:48) ) = p v p ( v (cid:48) p ) p v (cid:48) p ( v (cid:48)(cid:48) ) for all v (cid:48)(cid:48) ∈ S v (cid:48) p .The next one is in a similar spirit but likely more debatable (see main text): (IND) Independence of Nested Decisions.
If a complete-information decision node v d ∈ V i is succeededvia some action a ∈ A v d by another complete-information decision node v (cid:48) d = c v d ( a ) ∈ V i of thesame agent i , then v (cid:48) d may be pulled back into v d , i.e., v (cid:48) d may be eliminated, S v (cid:48) d added to S v d , { a } × A v (cid:48) d added to A v d , and c v d extended by c v d ( a, a (cid:48) ) = c v (cid:48) d ( a (cid:48) ) for all a (cid:48) ∈ A v (cid:48) d .Also more debatable are the following axioms which can be seen as treating the relationship of RFs tocertain game-theoretic concepts. The first one basically states that like in most equilibrium concepts forextensive-form games, ambiguity and information-equivalence play a kind of complementary role: (IAT) Independence of Ambiguity Timing.
Assume some probability node v ∈ V p or complete-information decision node v ∈ V d is succeeded by an ambiguity node v a ∈ V a ∩ S v . Let B ( v ), B ( v a ) , B ( v (cid:48) ) be the original branches of the tree ( V, E ) starting at v , v a and any v (cid:48) ∈ S v a . For each v (cid:48) ∈ S v a , let B (cid:48) ( v (cid:48) ) be a new copy of the original B ( v ) in which the subbranch B ( v a ) is replacedby a copy of B ( v (cid:48) ); let f ( v (cid:48) ) be that copy of v that serves as the root of this new branch B (cid:48) ( v (cid:48) ).If v ∈ V d , put f ( v (cid:48) ) ∼ f ( v (cid:48)(cid:48) ) for all v (cid:48) , v (cid:48)(cid:48) ∈ S v a Let B (cid:48) ( v a ) be a new branch starting with v a andthen splitting into all these new branches B (cid:48) ( v (cid:48) ). Then v a may be “pulled before” v by replacingthe original B ( v ) by the new B (cid:48) ( v a ).The second one states that, in contrast to the most common game theoretic approach where players’“optimal” behaviour depends on the subjective probabilities they attach to others’ behaviours, for nor-mative responsibility assessments only others’ possible actions should play a role, not the agent’s beliefs32bout their likelihoods; as a consequence, other agents’ actions could be seen as just another source ofambiguity: (IOA) Independence of Others’ Agency. If i ∈ I − G , and some of i ’s decision nodes v d ∈ V i is replacedin T by a new ambiguity node v a / ∈ V with S v a = S v d , then R ( G ) remains unchanged (i.e., it isirrelevant whether uncertain consequences are due to choices of other agents or some non-agentmechanism with ambiguous consequences).The third one is related to the branch of game-theory that studies group strategies and group deviationsin that it allows us to treat a group of agents like a single agent when it comes to assessing that group’sresponsibility: (IGC) Independence of Group Composition. If i, i (cid:48) ∈ G and all occurrences of i (cid:48) are replaced in T by i , R ( G ) remains unchanged.The final two in this category of axioms basically state that certain forms of luck should not influenceresponsibility assessments: (FIU) Forward Independence of Unknowns. If v d ∼ v (cid:48) d ∈ V i then R f ( v d , { i } ) = R f ( v (cid:48) d , { i } ) (i.e., forwardresponsibility is the same in decision nodes the agent cannot distinguish). (BIL) Backward Independence of Luck. If v o , v (cid:48) o ∈ V o , H ( v o ) ∩ V d = H ( v (cid:48) o ) ∩ V d = W , and C v (cid:48)(cid:48) ( v o ) = C v (cid:48)(cid:48) ( v (cid:48) o ) for all v (cid:48)(cid:48) ∈ W , then R b ( v o ) = R b ( v (cid:48) o ) (i.e., backward responsibility is the same in outcomenodes that have the same choice history).The combination of all the above axioms would allow us to restrict our interest to single-agent situationsthat have I = G = { i } , have only properly branching non-outcome nodes, at most one ambiguity nodeand only as their root node, have no two consecutive probability nodes, no zero probabilities, and noconsecutive decision nodes. Continuity, monotonicity, and other inequality axioms.
The axioms in this category state howan RF may change when certain features of the situation change. The first one disallows slight changesin probabilities to have large impacts on assessments: (PCont)
Probability Continuity. If v ∈ V and for any v p ∈ V p , the probability distribution p v p is variedcontinuously, then R ( v ) does not change discontinuously in dependence on p v p .The next four basically state that reduced agency or certain forms of ambiguity should not increaseresponsibility: (CAM) Current Agency Monotonicity.
If we remove some action a ∈ A v d and its branch B ( c v d ( a ))from the current decision node v d ∈ V G and the latter is complete-information, then R f ( v d ) doesnot increase. (PAM) Past Agency Monotonicity.
If we remove some non-taken action a ∈ A v (cid:48) d , C v (cid:48) ( v o ) (cid:54) = a , and itsbranch from a past decision node v (cid:48) d ∈ V G ∩ H ( v o ) and the latter is complete-information, then R b ( v o ) does not increase. (AMF) Ambiguity Monotonicity of Forward Responsibility.
If we remove the branch B ( v (cid:48) ) of a possiblesuccessor v (cid:48) ∈ S v a of an ambiguity node v a ∈ V a that does not contain v , v / ∈ B ( v a ), then R f ( v )does not increase.The last four relate responsibilities of groups to their subgroups, and backward to forward responsi-bility: 33 GSM)
Group Size Monotonicity. If G ⊆ G (cid:48) then R ( G ) (cid:54) R ( G (cid:48) ) (i.e., larger groups have no lessresponsibility). (GSA) Group Subadditivity. R ( G + G (cid:48) ) (cid:54) R ( G ) + R ( G (cid:48) ) for all G, G (cid:48) ⊆ I . (GPA) Group Superadditivity. R ( G + G (cid:48) ) (cid:62) R ( G ) + R ( G (cid:48) ) for all G, G (cid:48) ⊆ I . (GA) Group Additivity. R ( G + G (cid:48) ) = R ( G ) + R ( G (cid:48) ) for all disjoint G, G (cid:48) ⊆ I . (MBF) Maximal Backward Responsibility Bounds Forward Responsibility.
For all v, G , there must be σ ∈ Σ( v, G ) and v o ∈ V σo so that R b ( v o , G ) (cid:62) R f ( v, G ) (i.e., forward responsibility is bounded bypotential backward responsibility). Existence and special situation axioms.
The first two axioms in this category require the existenceof responsible groups and responsibility-avoiding strategies. (NRV)
No Responsibility Voids. If V a = V p = ∅ and ε (cid:54) = V o , then for each v o ∈ ε , there is G ⊆ I with R b ( v o , G ) > (NUR) No Unavoidable Backward Responsibility.
For each G ⊆ I , there exists a strategy σ ∈ Σ( v , G )so that R b ( v o , G ) = 0 for all v o ∈ V σo (i.e., G must have a way of avoiding backward responsibility).Finally, we consider a number of axioms which require certain values of R b or R f for the paradigmaticexample situations discussed informally in the Introduction. (Norm) Responsibility Degree Normalization.
With T and ε as depicted in Fig. 1(c) with p = 0 and q = 1, R f ( v , { i } ) = R b ( v , { i } ) = 1 and R b ( v , { i } ) = 0. (NWT) No Wishful Thinking.
With T and ε as depicted in Fig. 1(a), R f ( v , { i } ) = 1. (NUT) No Unfounded Trust.
With T and ε as depicted in Fig. 1(a), R f ( v , { i } ) = 1. (NFT) No Fearful Thinking.
With T and ε as depicted in Fig. 1(b), R f ( v , { i } ) = 1. (NUD) No Unfounded Distrust.
With T and ε as depicted in Fig. 1(b), R f ( v , { i } ) = 1. (UFR) Undivided Factual Responsibility.
With T and ε as depicted in Fig. 1(a), R b ( v , { i } ) = 1. (MFR) Multicausal Factual Responsibility.
With T and ε as depicted in Fig. 1(b), R b ( v , { i } ) = 1. (CFR) Counterfactual Responsibility.
With T and ε as depicted in Fig. 1(a), R b ( v , { i } ) = 1. (OPR) Ordered Probability Responsiveness. If T and ε are as depicted in Fig. 1(c), p < q , and we eitherincrease q or decrease p , then R f ( v , { i } ) and R b ( v , { i } ) strictly increase. (MAR) Multiple Aberration Responsiveness. If T and ε are as depicted in Fig. 1(d) and p > R b ( v , { i } ) > R b ( v , { i } ) (i.e., an uncertain additional chance to avoid ε strictly increases backwardresponsibility if missed).Note that (AMF) and (Norm) together imply (NWT) and (NFT) but not (NUT) or (NUD), while(NWT) and (FIU) imply (NUT), and (NFT) and (FIU) imply (NUD). Also, given (FCS), (NWT) and(NFT) become equivalent, and (NUT) and (NUD) become equivalent. Finally, (BIL), (IAT), and (UFR)together imply (CFR). 34 roofs and further propositions Proof of Proposition 1. Whenever the consequences of some change in T are discussed, quantities afterthe change are marked by ˆ. Let B ( v ) = B ( v ) ∩ V o . (IND) Fig. 3 is a counterexample for R and R . Compliance of R b is straightforward. Because of thecanonical bijection between the strategy sets Σ before and after the change, at no node v (cid:54) = v (cid:48) d , γ ( v )or µ ( v ) change, hence at no node v (cid:54) = v d , R f or R f change. Hence, if C v d ( v o ) (cid:54) = a , no summand∆ γ or ∆ µ occurring in R b ( v o ) or R b ( v o ) changes. If C v d ( v o ) = a , C v (cid:48) d ( v o ) = a (cid:48) and c v (cid:48) d ( a (cid:48) ) = v (cid:48)(cid:48) ,the only change is that the old ∆ γ ( v d , a ) + ∆ γ ( v (cid:48) d , a (cid:48) ) = γ ( c v d ( a )) − γ ( v d ) + γ ( c v (cid:48) d ( a (cid:48) )) − γ ( v (cid:48) d ) = γ ( v (cid:48)(cid:48) ) − γ ( v d ) is replaced by the new ∆ γ ( v d , ( a, a (cid:48) )) = γ ( c v d ( a, a (cid:48) )) − γ ( v d ) = γ ( v (cid:48)(cid:48) ) − γ ( v d ), whichis the same value. The same holds for µ instead of γ . (IAT) Fig. 4 becomes a counterexample for R and R if one puts ε = { v (cid:48) } and v o = v (cid:48) . Because ofthe canonical bijection between the scenario sets Z ∼ before and after the change and since R , R , and R use Z ∼ rather than just Z , the values of µ and ω do not change, hence these variantycomply with (IAT). (Note that R would also comply if we had used Z ∼ rather than Z in thedefinition of γ , and R would comply if we had used B ∼ rather than B in its definition.) (GSM) Counterexamples for R , R , and R can be constructed easily. (AMF) If v a / ∈ B ( v d ), none of the variants of R f ( v d ) are affected by the change. So assume v a ∈ B ( v d ).Since the change reduces the scenario set Z ∼ ( v d ) but does not alter any ∆ (cid:96) ( v, ζ ) for any remainingscenario ζ , the value of the maximum R f ( v d ) cannot increase.A counterexample for R , R and R f is the situation where the two available choices at somedecision node v d lead to two ambiguity nodes v d , v (cid:48) d , each of which has two successors, one ofwhich is in ε and the other not in ε . Then both choices have guaranteed likelihood of 0, a minimaxlikelihood 1, and are equally risky, hence R f ( v d ) = R f ( v d ) = R f ( v d ) = 0. But after removing thesuccessor of v d that is not in ε , the first choice has guaranteed likelihood 1, hence R f ( v d ) = 1 afterthe removal. Likewise, after removing the successor of v d that is in ε , the first choice has minimaxlikelihood 0 and ceases to be risky, hence R f ( v d ) = R f ( v d ) = 1 after the removal. (NRV) Fig. 2(a) is a counterexample for R and R . (NUR) Fig. 2(a) is a counterexample for R .All other properties should be obvious from the definitions. Q.E.D.
Proposition 2
All variants R , R , R , R , R fulfill (Anon), (ACon), (OCon), (IST), (ICP), (INA),(INP), (IOA), (IGC), (BIL), and (UFR).The variants R , R , R , R also fulfill (IZP), (PCont), (CAM), (Norm), (OPR), and (MAR),while R fulfills neither.The variants R f , R f , R f , R f also fulfill (NWT).(FIU) and (NUT) are fulfilled by R f , R f , R f but not R f .Proof. (IST) R : If S v = { v (cid:48) } , B o ( v ) = B o ( v (cid:48) ), so the change does not affect R . R : There are obvious canonical bijections F between the scenario sets Z ∼ ( v ) and ˆ Z ∼ ( v ) andbetween the strategy sets Σ( v ), ˆΣ( v ) before and after the change. Since ∆ γ ( v d , a ) is unaffected for35 d (cid:54) = v , while ∆ γ ( v d , a ) = 0 and ∆ˆ γ ( v d , a ) = ∆ γ ( v (cid:48) ) if v d = v , its sum along H ( v o ) ∩ V G (giving R b ( v o )) is unaffected, and R f ( v d ) is unaffected for v d (cid:54) = v , while R f ( v d ) = 0 and ˆ R f ( v d ) = R f ( v (cid:48) )if v d = v . R : As for R , with ∆ µ ( v d , a ) instead of ∆ γ ( v d , a ). R , R : As for R , with (cid:37) ( v d , a ) instead of ∆ γ ( v d , a ). (INA) R : Note that B o ( v ) for any v (cid:54) = v (cid:48) a remains unchanged when pulling v (cid:48) a back into v a . Let v o ∈ V o , v ∈ H ( v o ), and v d = P ( v ) ∈ V d . Since v d ∈ V d , v d (cid:54) = v a , v (cid:48) a and thus v (cid:54) = v (cid:48) a . Hence B o ( v ) ⊆ ε (cid:54)⊇ B o ( v d ) after the change if and only if this is so before the change. R , R : Since there is an obvious canonical bijection between the scenario sets before and afterthe change, γ ( v ) and µ ( v ) do not change for any v (cid:54) = v (cid:48) a , ∆ γ ( v d , a ) and ∆ µ ( v d , a ) not for any v d ,and H ( v o ) ∩ V G not for any v o , hence R and R are unaffected. R , R : Let F be the above-mentioned bijection between scenarios. Since ˆ ω ( v, F ( ζ )) = ω ( v, ζ )and ∆ˆ (cid:96) ( v, F ( ζ )) = ∆ (cid:96) ( v, ζ ) for all v (cid:54) = v (cid:48) a , also ˆ (cid:37) ( v d , a ) = (cid:37) ( v d , a ), so ∆ (cid:37) ( v d ) and thus both R and R are unaffected. (INP) This is completely analogous to (INA). (IZP), (PCont)
The change discussed in (IZP) does not affect any (cid:96) ( ε | v (cid:48)(cid:48) , σ, ζ ) for v (cid:48)(cid:48) / ∈ B ( v (cid:48) ), andthat in (PCont) affects (cid:96) continuosly. Since R – R are based on (cid:96) alone and are continuous in (cid:96) ,the claim follows. Minimal counterexamples for R are trivial to find. (CAM) The change removes some strategies, hence γ ( v d ), µ ( v d ), and ω ( v d , ζ ) can not decrease, while γ ( c v d ( a (cid:48) )), µ ( c v d ( a (cid:48) )), and ω ( c v d ( a (cid:48) )) , ζ ) are unaffected for all a (cid:48) (cid:54) = a , so that ∆ γ ( v d , a (cid:48) ), ∆ µ ( v d , a (cid:48) ),and ∆ ω ( v d , ζ, a (cid:48) ) cannot increase for any a (cid:48) (cid:54) = a . The variants of R f are weakly monontonicfunctions of the latter. (FIU) R , R , R are based on Z ∼ , while R uses Z and thereby ignores information equivalence.All other properties should be obvious from the definitions. Q.E.D.
Regarding (PAM), we abstain from proving the following conjecture.
Conjecture 1
All variants R , R , R , R , R fulfill (PAM). Non-graded variant based on the NESS condition
Here we finally sketch some BRF variant based on the idea of the NESS criterion [7], interpreting theirnotion of ‘event’ in our context as a single decision taken by some agent, acknowledging the informationsets of agents.An information set for i is a ∼ -equivalence-class y ⊆ V i . Let Y i be the set of all information sets for i and Y G = (cid:83) i ∈ G Y i . Then Y = (cid:83) i Y i is the partition of V d into ∼ -equivalence classes. For v d ∈ V d , let y ( v d ) be that y ∈ Y with v d ∈ y . A decision is a pair d = ( y, a ) with y ∈ Y and a ∈ A v d for all v d ∈ y .For an outcome v o ∈ V o , let D ( v o ) = { ( y ( v d ) , C v d ( v o )) : v d ∈ H ( v o ) ∩ V d } be the set of all taken decisions that led to v o . For a set D of decisions, let V Do = { v o ∈ V o : D ⊆ D ( v o ) }
36e the set of all outcomes that may occur if all the decisions in D are actually taken. Note that V Do is aweakly decreasing set function of D .Let some v o ∈ ε be fixed.A subset D ⊆ D ( v o ) of the decisions that led to v o is called sufficient iff V Do ⊆ ε . Note that if D issufficient, so is every larger D (cid:48) ⊇ D , but there need not be any sufficient set since the whole D ( v o ) mightnot be sufficient if luck plays a role (e.g., in Fig. 1(c)).A decision d is necessary for the sufficiency of D iff d ∈ D , D is sufficient, but D − d is not sufficient.A decision d ∈ D ( v o ) was a NESS-cause for ε if there is some D ⊆ D ( v o ) such that d is necessary forthe sufficiency of D .A group G is NESS-responsible for ε , denoted R Nb ( i, v o ) = 1, iff they took some decision ( y, a ) ∈ D ( v o ), y ∈ Y G , that was a NESS-cause for ε .It appears that the resulting BRF R Nb probably fulfills the axioms (IGC), (IND), (IAT), (GSM),(NRV), (MFR) but probably violates (CFR) and (IOA) (if the converted node belonged to a non-singleton information set). Which of the other axioms it fulfills is beyond the scope of this paper.37 ethod R f R f R f R f R b R b R b R b R b two-option majority voting 1 m> N m> N u> N u> N u>m − N > u> max( m − N , u> max( m − N , multi-option simple majority 1 m> N m> N m (cid:62) N − u> N u − a>m (cid:48) a − u