We examine how causal beliefs affect an agent's choices and how feedback on those choices leads to updated causal beliefs. Building on the structural-equations framework for modeling causality, we first examine the general problem of updating causal beliefs in the face of novel (and possibly inexplicable) data. We model an agent who is uncertain of the true causal model, and therefore entertains a probabilistic belief over the set of possible models. We then consider how causal beliefs influence choices by building a model of agency and utility on top of the usual structural-equations framework. Using these two components, we propose a notion of steady state, where the feedback received from an agent's optimal action, given her current beliefs about the true causal model, can be rationalized by those beliefs.

Structural-equations models (SEMs) are perhaps the most commonly used framework for modeling causality, but they do not capture all domains of interest. For example, dynamical systems that evolve in continuous time are an important class of domains that are not (naturally) captured by SEMs. A wide variety of approaches have been proposed to fill the gap, including dynamical structural causal models (Bongers, Blom and Mooij 2018), causal constraints models (Blom, Bongers and Mooij 2019), and counterfactual resimulation (Laurent, Yang, and Fontana 2018). These models complement common-sense causal interpretations of specific dynamical systems, such as systems of ODEs. All these approaches look quite different from each other and from SEMs. They are hard to compare, and concepts developed for one approach may not make sense for another. But they are capturing the same notion of causality as SEMs do, in the sense that interventions map to outcomes. We propose a class of models that are, in a certain natural sense, the most expressive generalization of SEMs. Our generalized SEMs (GSEMs) can be viewed as a unifying framework that recovers structural dynamical causal models, causal constraints models, counterfactual resimulation, and common-sense causal interpretations of systems of ODEs and hybrid automata (Alur et al. 1992) as special cases. The input-output behavior, or "interface", of GSEMs is exactly that of SEMs, which means that definitions of concepts like actual cause, responsibility, blame, and explanation, can be immediately lifted from SEMs to GSEMs. The generality of GSEMs also makes them ideally suited to studying causality in the abstract; for example, they have been used to establish independence relationships among Halpern's axioms for SEMs (Peters and Halpern 2022).

Perhaps the most prominent current definition of (actual) causality is due to Halpern and Pearl. It is defined using causal models (also known as structural equations models). We abstract the definition, extracting its key features, so that it can be applied to any other model where counterfactuals are defined. By abstracting the definition, we gain a number of benefits. Not only can we apply the definition in a wider range of models, including ones that allow, for example, backtracking, but we can apply the definition to determine if A is a cause of B even if A and B are formulas involving disjunctions, negations, beliefs, and nested counterfactuals (none of which can be handled by the Halpern-Pearl definition). Moreover, we can extend the ideas to getting an abstract definition of explanation that can be applied beyond causal models. Finally, we gain a deeper understanding of features of the definition even in causal models.

The dynamics of random transitive delegations on a graph are of particular interest when viewed through the lens of an emerging voting paradigm: liquid democracy. This paradigm allows voters to choose between directly voting and transitively delegating their votes to other voters so that those selected cast a vote weighted by the number of delegations that they received. In the epistemic setting, where voters decide on a binary issue for which there is a ground truth, previous work showed that a few voters may amass such a large amount of influence that liquid democracy is less likely to identify the ground truth than direct voting. We quantify the amount of permissible concentration of power and examine more realistic delegation models, showing that they behave well by ensuring that (with high probability) there is a permissible limit on the maximum number of delegations received. Our theoretical results demonstrate that the delegation process is similar to well-known processes on random graphs that are sufficiently bounded for our purposes. Along the way, we prove new bounds on the size of the largest component in an infinite Pólya urn process, which may be of independent interest. In addition, we empirically validate the theoretical results, running six experiments (for a total of N = 168 participants, 62 delegation graphs, and over 11,000 votes collected). We find that empirical delegation behaviors meet the conditions for our positive theoretical guarantees. Overall, our work alleviates concerns raised about liquid democracy and bolsters the case for the applicability of this emerging paradigm. This paper was accepted by Martin Bichler, market design, platform, and demand analytics. Funding: This work was supported by the Michael Hammer Fellowship, the Office of Naval Research [2016 Vannevar Bush Faculty Fellowship, 2020 ONR Vannevar Bush Faculty Fellowship, and Grant N00014-20-1-2488], the Office of Secretary of Defense [Grants ARO MURI W911NF-19-0217 and ARO W911NF-17-1-0592], Simons [Investigator Award 622132], the Open Philanthropy Foundation, and the National Science Foundation [Grants IIS-178108, CCF-1733556, CCF-1918421, CCF-2007080, IIS-1703846, and IIS-2024287]. J. Y. Halpern was supported by a grant from the Cooperative AI Foundation [ARO Grant W911NF-22-1-0061]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.02470 .

We study the strategic advantages of coarsening one's utility by clustering nearby payoffs together (i.e., classifying them the same way). Our solution concept, coarse-utility equilibrium (CUE) requires that (1) each player maximizes her coarse utility, given the opponent's strategy, and (2) the classifications form best replies to one another. We characterize CUEs in various games. In particular, we show that there is a qualitative difference between CUEs in which only one of the players clusters payoffs, and those in which all players cluster their payoffs, and that the latter type induce players to treat co-players better than in Nash equilibria in the large class of games with monotone externalities.

The evolution of cognition is frequently discussed as the evolution of cognitive abilities or the evolution of some neuronal structures in the brain. However, since such traits or abilities are often highly complex, understanding their evolution requires explaining how they could have gradually evolved through selection acting on heritable variations in simpler cognitive mechanisms. With this in mind, making use of a previously proposed theory, here, we show how the evolution of cognitive abilities can be captured by the fine-tuning of basic learning mechanisms and, in particular, chunking mechanisms. We use the term chunking broadly for all types of non-elemental learning, claiming that the process by which elements are combined into chunks and associated with other chunks, or elements, is critical for what the brain can do, and that it must be fine-tuned to ecological conditions. We discuss the relevance of this approach to studies in animal cognition, using examples from animal foraging and decision-making, problem-solving and cognitive flexibility. Finally, we explain how even the apparent human-animal gap in sequence learning ability can be explained in terms of different fine-tunings of a similar chunking process.This article is part of the Theo Murphy meeting issue 'Selection shapes diverse animal minds'.

Causal models are crucial for understanding complex systems and identifying causal relationships among variables. Even though causal models are extremely popular, conditional probability calculation of formulas involving interventions pose significant challenges. In case of Causal Bayesian Networks (CBNs), Pearl assumes autonomy of mechanisms that determine interventions to calculate a range of probabilities. We show that by making simple yet often realistic independence assumptions, it is possible to uniquely estimate the probability of an interventional formula (including the well-studied notions of probability of sufficiency and necessity). We discuss when these assumptions are appropriate. Importantly, in many cases of interest, when the assumptions are appropriate, these probability estimates can be evaluated using observational data, which carries immense significance in scenarios where conducting experiments is impractical or unfeasible.

We show that it is possible to understand and identify a decision maker’s subjective causal judgements by observing her preferences over interventions. Following Pearl [2000, DOI: doi.org/10.1017/S0266466603004109 ], we represent causality using causal models (also called structural equations models), where the world is described by a collection of variables, related by equations. We show that if a preference relation over interventions satisfies certain axioms (related to standard axioms regarding counterfactuals), then we can define (i) a causal model, (ii) a probability capturing the decision-maker’s uncertainty regarding the external factors in the world and (iii) a utility on outcomes such that each intervention is associated with an expected utility and such that intervention A is preferred to B iff the expected utility of A is greater than that of B. In addition, we characterize when the causal model is unique. Thus, our results allow a modeler to test the hypothesis that a decision maker’s preferences are consistent with some causal model and to identify causal judgements from observed behavior.

We show that it is possible to understand and identify a decision maker's subjective causal judgements by observing her preferences over interventions. Following Pearl [2000], we represent causality using causal models (also called structural equations models), where the world is described by a collection of variables, related by equations. We show that if a preference relation over interventions satisfies certain axioms (related to standard axioms regarding counterfactuals), then we can define (i) a causal model, (ii) a probability capturing the decision-maker's uncertainty regarding the external factors in the world and (iii) a utility on outcomes such that each intervention is associated with an expected utility and such that intervention $A$ is preferred to $B$ iff the expected utility of $A$ is greater than that of $B$. In addition, we characterize when the causal model is unique. Thus, our results allow a modeler to test the hypothesis that a decision maker's preferences are consistent with some causal model and to identify causal judgements from observed behavior.

We focus on explaining image classifiers, taking the work of Mothilal et al. [2021] (MMTS) as our point of departure. We observe that, although MMTS claim to be using the definition of explanation proposed by Halpern [2016], they do not quite do so. Roughly speaking, Halpern's definition has a necessity clause and a sufficiency clause. MMTS replace the necessity clause by a requirement that, as we show, implies it. Halpern's definition also allows agents to restrict the set of options considered. While these difference may seem minor, as we show, they can have a nontrivial impact on explanations. We also show that, essentially without change, Halpern's definition can handle two issues that have proved difficult for other approaches: explanations of absence (when, for example, an image classifier for tumors outputs "no tumor") and explanations of rare events (such as tumors).

Protocols for tossing a common coin play a key role in the vast majority of implementations of consensus. Even though the common coins in the literature are usually fair (they have equal chance of landing heads or tails), we focus on the problem of implementing a biased common coin such that the probability of landing heads is p ∈ [0, 1]. Even though biased common coins can be implemented using fair common coins, we show that this can require significant inter-party communication. In fact, we show that there is no bound on the number of messages needed to generate a common coin of bias p in a way that tolerates even one malicious agent, even if we restrict p to an arbitrary infinite subset of [0, 1] (e.g., {1/2n: n ∈ {0, 1, 2, …}}) and assume that the system is synchronous. By way of contrast, if we do not require the protocol to tolerate a faulty agent, we can do this. Thus, the cause of the message complexity is the requirement of fault tolerance.

We review the literature on models that try to explain human behavior in social interactions described by normal-form games with monetary payoffs. We start by covering social and moral preferences. We then focus on the growing body of research showing that people react to the language in which actions are described, especially when it activates moral concerns. We conclude by arguing that behavioral economics is in the midst of a paradigm shift towards language-based preferences, which will require an exploration of new models and experimental setups.

The increasing wireless communication capabilities of vehicles creates opportunities for more efficient intersection management strategies. One promising approach is the replacement of traffic lights with a system wherein vehicles run protocols among themselves to determine right of way. In this paper, we define the intersection problem to model this scenario abstractly, without any assumptions on the specific structure of the intersection or a bound on the number of vehicles. Protocols solving the intersection problem must guarantee safety (no collisions) and liveness (every vehicle eventually goes through). In addition, we would like these protocols to satisfy various optimality criteria, some of which turn out to be achievable only in a subset of the contexts. In particular, we show a partial equivalence between eliminating unnecessary waiting, a criterion of interest in the distributed mutual-exclusion literature, and a notion of optimality that we define called lexicographical optimality. We then introduce a framework to design protocols for the intersection problem by converting an intersection policy, which is based on a global view of the intersection, to a protocol that can be run by the vehicles through the use of knowledge-based programs. Our protocols are shown to guarantee safety and liveness while also being optimal under sufficient conditions on the context. Finally, we investigate protocols in the presence of faulty vehicles that experience communication failures and older vehicles with limited communication capabilities. We show that intersection protocols can be made safe, live and optimal even in the presence of faulty behavior.

We consider $k$-resilient sequential equilibria, strategy profiles where no player in a coalition of at most $k$ players believes that it can increase its utility by deviating, regardless of its local state. We prove that all $k$-resilient sequential equilibria that can be implemented with a trusted mediator can also be implemented without the mediator in a synchronous system of $n$ players if $n >3k$. In asynchronous systems, where there is no global notion of time and messages may take arbitrarily long to get to their recipient, we prove that a $k$-resilient sequential equilibrium with a mediator can be implemented without the mediator if $n > 4k$. These results match the lower bounds given by Abraham, Dolev, and Halpern (2008) and Geffner and Halpern (2023) for implementing a Nash equilibrium without a mediator (which are easily seen to apply to implementing a sequential equilibrium) and improve the results of Gerardi, who showed that, in the case that $k=1$, a sequential equilibrium can be implemented in synchronous systems if $n \ge 5$.

Many studies have shown that humans are "predictably irrational": they do not act in a fully rational way, but their deviations from rational behavior are quite systematic. Our goal is to see the extent to which we can explain and justify these deviations as the outcome of rational but resource-bounded agents doing as well as they can, given their limitations. We focus on the well-studied ranger-poacher game, where rangers are trying to protect a number of sites from poaching. We capture the computational limitations by modeling the poacher and the ranger as probabilistic finite automata (PFAs). We show that, with sufficiently large memory, PFAs learn to play the Nash equilibrium (NE) strategies of the game and achieve the NE utility. However, if we restrict the memory, we get more "human-like" behaviors, such as probability matching (i.e., visiting sites in proportion to the probability of a rhino being there), and avoiding sites where there was a bad outcome (e.g., the poacher was caught by the ranger), that we also observed in experiments conducted on Amazon Mechanical Turk. Interestingly, we find that adding human-like behaviors such as probability matching and overweighting significant events (like getting caught) actually improves performance, showing that this seemingly irrational behavior can be quite rational.

Causal models have proven extremely useful in offering formal representations of causal relationships between a set of variables. Yet in many situations, there are non-causal relationships among variables. For example, we may want variables LDL, HDL, and TOT that represent the level of low-density lipoprotein cholesterol, the level of lipoprotein high-density lipoprotein cholesterol, and total cholesterol level, with the relation LDL+HDL=TOT. This cannot be done in standard causal models, because we can intervene simultaneously on all three variables. The goal of this paper is to extend standard causal models to allow for constraints on settings of variables. Although the extension is relatively straightforward, to make it useful we have to define a new intervention operation that disconnects a variable from a causal equation. We give examples showing the usefulness of this extension, and provide a sound and complete axiomatization for causal models with constraints.

Fluid democracy is a voting paradigm that allows voters to choose between directly voting and transitively delegating their votes to other voters. While fluid democracy has been viewed as a system that can combine the best aspects of direct and representative democracy, it can also result in situations where few voters amass a large amount of influence. To analyze the impact of this shortcoming, we consider what has been called an epistemic setting, where voters decide on a binary issue for which there is a ground truth. Previous work has shown that under certain assumptions on the delegation mechanism, the concentration of power is so severe that fluid democracy is less likely to identify the ground truth than direct voting. We examine different, arguably more realistic, classes of mechanisms, and prove they behave well by ensuring that (with high probability) there is a limit on concentration of power. Our proofs demonstrate that delegations can be treated as stochastic processes and that they can be compared to well-known processes from the literature -- such as preferential attachment and multi-types branching process -- that are sufficiently bounded for our purposes. Our results suggest that the concerns raised about fluid democracy can be overcome, thereby bolstering the case for this emerging paradigm.

In earlier work we defined a qualitative notion of harm: either harm is caused, or it is not. For practical applications, we often need to quantify harm; for example, we may want to choose the least harmful of a set of possible interventions. We first present a quantitative definition of harm in a deterministic context involving a single individual, then we consider the issues involved in dealing with uncertainty regarding the context and going from a notion of harm for a single individual to a notion of "societal harm", which involves aggregating the harm to individuals. We show that the "obvious" way of doing this (just taking the expected harm for an individual and then summing the expected harm over all individuals) can lead to counterintuitive or inappropriate answers, and discuss alternatives, drawing on work from the decision-theory literature.

We present Colordag, a blockchain protocol where following the prescribed strategy is, with high probability, a best response as long as all miners have less than 1/2 of the mining power. We prove the correctness of Colordag even if there is an extremely powerful adversary who knows future actions of the scheduler: specifically, when agents will generate blocks and when messages will arrive. The state-of-the-art protocol, Fruitchain, is an epsilon-Nash equilibrium as long as all miners have less than 1/2 of the mining power. However, there is a simple deviation that guarantees that deviators are never worse off than they would be by following Fruitchain, and can sometimes do better. Thus, agents are motivated to deviate. Colordag implements a solution concept that we call epsilon-sure Nash equilibrium and does not suffer from this problem. Because it is an epsilon-sure Nash equilibrium, Colordag is an epsilon Nash equilibrium and with probability (1 - epsilon) is a best response.