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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. 

For over 25 years, common belief has been widely viewed as necessary for joint behavior. But this is not quite correct. We show by example that what can naturally be thought of as joint behavior can occur without common belief. We then present two variants of common belief that can lead to joint behavior, even without standard common belief ever being achieved, and show that one of them, action-stamped}common belief, is in a sense necessary and sufficient for joint behavior. These observations are significant because, as is well known, common belief is quite difficult to achieve in practice, whereas these variants are more easily achievable. 

In earlier work, we introduced the framework of language-based decisions, the core idea of which was to modify Savage's classical decision-theoretic framework by taking actions to be descriptions in some language, rather than functions from states to outcomes, as they are defined classically. Actions had the form ``if psi then do phi''', where psi and phi$ were formulas in some underlying language, specifying what effects would be brought about under what circumstances. The earlier work allowed only one-step actions. But, in practice, plans are typically composed of a sequence of steps. Here, we extend the earlier framework to \emph{sequential} actions, making it much more broadly applicable. Our technical contribution is a representation theorem in the classical spirit: agents whose preferences over actions satisfy certain constraints can be modeled as if they are expected utility maximizers. As in the earlier work, due to the language-based specification of the actions, the representation theorem requires a construction not only of the probability and utility functions representing the agent's beliefs and preferences, but also the state and outcomes spaces over which these are defined, as well as a ``selection function'' which intuitively captures how agents disambiguate coarse descriptions. The (unbounded) depth of action sequencing adds substantial interest (and complexity!) to the proof. 

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. 

Work on optimal protocols for \emph{Eventual Byzantine Agreement} (EBA)---protocols that, in a precise sense, decide as soon as possible in every run and guarantee that all nonfaulty agents decide on the same value---has focused on full-information protocols} (FIPs), where agents repeatedly send messages that completely describe their past observations to every other agent. While it can be shown that, without loss of generality, we can take an optimal protocol to be an FIP, full information exchange is impractical to implement for many applications due to the required message size. We separate protocols into two parts, the information-exchange protocol and the action protocol, so as to be able to examine the effects of more limited information exchange. We then define a notion of optimality with respect to an information-exchange protocol. Roughly speaking, an action protocol P is optimal with respect to an information-exchange protocol E if, with P, agents decide as soon as possible among action protocols that exchange information according to E. We present a knowledge-based EBA program for omission failures all of whose implementations are guaranteed to be correct and are optimal if the information exchange satisfies a certain safety condition. We then construct concrete programs that implement this knowledge-based program in two settings of interest that are shown to satisfy the safety condition. Finally, we show that a small modification of our program results in an FIP that s both optimal and efficiently implementable, settling an open problem posed by Halpern, Moses, and Waarts (SIAM J. Comput., 2001). 

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=\OT. 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. 

As autonomous systems rapidly become ubiquitous, there is a growing need for a legal and regulatory framework that addresses when and how such a system harms someone. There have been several attempts within the philosophy literature to define harm, but none of them has proven capable of dealing with the many examples that have been presented, leading some to suggest that the notion of harm should be abandoned and ``replaced by more well-behaved notions''. As harm is generally something that is caused, most of these definitions have involved causality at some level. Yet surprisingly, none of them makes use of causal models and the definitions of actual causality that they can express. In this paper we formally define a qualitative notion of harm that uses causal models and is based on a well-known definition of actual causality due to Halpern and Pearl. The key features of our definition are that it is based on contrastive causation and uses a default utility to which the utility of actual outcomes is compared. We show that our definition is able to handle the examples from the literature, and illustrate its importance for reasoning about situations involving autonomous systems. 

Generalized structural equations models (GSEMs) are, as the name suggests, a generalization of structural equations models (SEMs). They can deal with (among other things) infinitely many variables with infinite ranges, which is critical for capturing dynamical systems. We provide a sound and complete axiomatization of causal reasoning in GSEMs that is an extension of the sound and complete axiomatization provided by Halpern for SEMs. Considering GSEMs helps clarify what properties Halpern's axioms capture. 

Consider a bank that uses an AI system to decide which loan applications to approve. We want to ensure that the system is fair, that is, it does not discriminate against applicants based on a predefined list of sensitive attributes, such as gender and ethnicity. We expect there to be a regulator whose job it is to certify the bank's system as fair or unfair. We consider issues that the regulator will have to confront when making such a decision, including the precise definition of fairness, dealing with proxy variables, and dealing with what we call allowed variables, that is, variables such as salary on which the decision is allowed to depend, despite being correlated with sensitive variables. We show (among other things) that the problem of deciding fairness as we have defined it is co-NP-complete, but then argue that, despite that, in practice the problem should be manageable.