EF5 – Concepts of Change in Historical Processes



A Mathematical Theory of Responsibility in Complex Multi-Agent Decision Problems with Uncertainties

Project Heads

Rupert Klein, Jobst Heitzig, Markus Brill

Project Members

Sarah Hiller (FU) 

Project Duration

15.09.2019 – 14.09.2022

Located at

FU Berlin


Motivated by the needs of climate change research, this project aims to formalize the concept of moral responsibility, both backward- and forward-looking, in interactive multi-agent decision scenarios with various levels of uncertainty.




The current climate crisis and its associated effects constitute one of the essential challenges for humanity and collective decision making in the upcoming years. Naturally, the public debate around this issue frequently invokes the question of responsibility: Who carries how much backward-looking responsibility for the changes already inevitable, who is to blame; and who carries how much forward-looking responsibility to realize changes, who has to act?

In the current project we aim for a formal representation of the concept of moral responsibility in particular, in order to make precise what is being talked about and to allow for a quantified measure for responsibility ascription. As collective action in face of the climate crisis is the initial motivation for this research question, it will also be the application scenario that we ultimately test our achievements on. However, several features inherent to this scenario complicate responsibility assignments:

  • Anthropogenic climate change as well as its possible mitigation is innately the result of a collective and interactive action.
  • There is considerable uncertainty in several areas such as the precise results of specific actions or the actions of other agents.

  • The decision making is not a one-off event but rather embedded into a temporal progression.

Previous work regarding formalizations of moral responsibility in the context of climate change can roughly be divided into two categories, via the perspective from which this question is addressed. On the one side there are considerations focusing on applicability in the climate change context, using naïve ad-hoc measures for this specific area with the advantage of being easy to compute but disregarding generalisability. On the other side there is considerable work in formal ethics, aiming at understanding and formally representing the concept of responsibility in general with a special focus on rigor and well-foundedness, making it harder to account for messy real world scenarios.


Project goals and methods

The project’s goals are to

  • contribute to a mathematical theory of responsibility by developing a framework for the representation of relevant decision scenarios
  • define representations of responsibility ascription within this framework
  • identify from the literature as well as from additional considerations paradigmatic example scenarios and other desirable properties in order to evaluate proposed responsibility functions
  • evaluate and ideally characterise the set of responsibility functions according to these features
  • apply the derived measure to real-world problems such as climate change.

As a baseline, such a formalization should account for ethical evaluations of the uncertain consequences of possible and actual actions of agents.

We employ a variety of methods from different fields throughout the analysis. First, the representation of decision situations in which we want to assign responsibility is achieved through an extension of extensive-form games, giving us the necessary tools to talk about responsibility in a formal way. Proposed responsibility functions are then evaluated according to their performance in selected paradigmatic example scenarios or via an axiomatic method. Finally, the selected functions will be applied in a real-world scenario, which will be computed using climate modelling. 



As a first step, we identified taxonomies of the various meanings of the term ‘responsibility’. The kind that we are interested in concerns the moral evaluation of actions of intentionally acting agents and can be referred to as ‘moral responsibility’.

To begin, we translated relevant decision scenarios for which we want to compute responsibility scores into a data structure adapted from extensive-form games. 

Example of ex post responsibility assessment

Assessment of the degree of backwards-looking responsibility in an example situation discussed in the moral philosophy literature. A robber (1) has stabbed a victim, knowing that a doctor (2) would administer either a negligent or a regular treatment, leading to different probabilities of survival (★). Actually, the doctor chooses the regular treatment and the victim lives. In one variant of the responsibility measure, the robber is assigned 50% “counterfactual” responsibility for the victim’s possible death. This is the largest difference in the probability of death arising from the robber’s action, compared in all possible scenarios. We argue that the robber was not allowed to count on the doctor’s treatment, as this constitutes a case of ‘moral luck’. Other variants come to different assessments since they treat the unquantifiable uncertainty of what the doctor would do in a different way.


Subsequently, we examined criteria for responsibility ascription to be used in our formalisation. In contrast to previous work in the formal ethical literature, we do not assume that responsibility requires actual causality. Several situations, such as attempts, preempted actions, or the case of moral luck described above, show that there are indeed situations where actual causality between the action and the effect is not given while we would still like to be able to assign responsibility. Therefore, we base responsibility ascription directly on probability raising. This also allows us to immediately infer a graded measure, rather than having to manually extend a two-valued score.

We investigated what baseline to use for the comparison of a probability increase using paradigmatic example scenarios. These were manually evaluated and employed for testing proposed responsibility functions. Results from this investigation showed that neither a comparison reduced to best-case nor worst-case scenarios were sufficiently effective, but that instead possible actions need to be evaluated regarding their effects in all possible scenarios. The two best performing functions were sucessfully applied to a simplified emissions reduction decision making situation as a proof of concept.

In a next step the comparison of proposed functions was carried out more formally using an axiomatic method. That is, we determined desirable properties of the functions as ‘axioms’ and evaluated implications, incompatibilities and characterizations of these axioms. Computation of outcome responsibility scores for a group of actors was divided into the computation of member contributions for individual actions within the decision making chain, paired with an aggregation function. For both of these functions, sets of desirable properties were devised and analyzed formally. Axioms for the member contribution function revealed an incompatibility between a lower-bound requirement and an upper-bound requirement in a restricted search space. Specifically, for every member contribution function which assigns equal scores for decisions that are taken with the same prior knowledge (a property called ”Knowledge Symmetry”) we need to choose between either allowing for responsibility voids, i.e. no member contribution despite an undersirable outcome, or unavoidable responsibility, i.e. an agent being faced with a choice where every possible action results in some contribution value strictly bigger than 0. We were able to characterize one specific aggregation function using a set of intuitive axioms such as anonymity and explicit results for limit cases such as an input of all zeros or an additional input of value one. One of the axioms required for the characterization – linearity in each component – is rather strong, but the singled out aggregation function is also maximally axiom compliant in comparison to all other proposed functions when considering weaker axioms such as monotonicity.

A computational implementation in order to allow for more complex and intricate interactive application scenarios is currently ongoing.




Project Webpages

Selected Publications

1. Hiller S, Heitzig J (2021) Quantifying Responsibility with Probabilistic Causation — The Case of Climate Action. arXiv preprint https://arxiv.org/abs/2111.02304

2. Hiller S, Israel J, Heitzig J (2022) An Axiomatic Approach to Formalized Responsibility Ascription. In PRIMA 2022: Principles and Practice of Multi-Agent Systems, volume 13753 of Lecture Notes in Computer Science, pages 435 – 457. Springer, 2022. URL: https://link.springer. com/book/10.1007/978-3-031-21203-1.

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