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Abraham, Dolev, Geffner, and Halpern [ 1 ] proved that, in asynchronous systems, a (k, t)-robust equilibrium for n players and a trusted mediator can be implemented without the mediator as long as n > 4( k+t ), where an equilibrium is ( k, t )-robust if, roughly speaking, no coalition of t players can decrease the payoff of any of the other players, and no coalition of k players can increase their payoff by deviating. We prove that this bound is tight, in the sense that if n ≤ 4( k+t ) there exist ( k, t )-robust equilibria with a mediator that cannot be implemented by the players alone. Even though implementing ( k, t )-robust mediators seems closely related to implementing asynchronous multiparty ( k+t )-secure computation [ 6 ], to the best of our knowledge there is no known straightforward reduction from one problem to another. Nevertheless, we show that there is a non-trivial reduction from a slightly weaker notion of ( k+t )-secure computation, which we call ( k+t )-strict secure computation , to implementing ( k, t )-robust mediators. We prove the desired lower bound by showing that there are functions on n variables that cannot be ( k+t )-strictly securely computed if n ≤ 4( k+t ). This also provides a simple alternative proof for the well-known lower bound of 4 t +1 on asynchronous secure computation in the presence of up to t malicious agents [ 4 , 8 , 10 ]. 

We introduce a theoretical model of information acquisition under resource limitations in a noisy environment. An agent must guess the truth value of a given Boolean formula \( \varphi \) after performing a bounded number of noisy tests of the truth values of variables in the formula. We observe that, in general, the problem of finding an optimal testing strategy for \( \varphi \) is hard, but we suggest a useful heuristic. The techniques we use also give insight into two apparently unrelated but well-studied problems: (1) rational inattention , that is, when it is rational to ignore pertinent information (the optimal strategy may involve hardly ever testing variables that are clearly relevant to \( \varphi \) ), and (2) what makes a formula hard to learn/remember. 

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. 

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. 

Although the definition of sequential equilibrium can be applied without change to games of imperfect recall, doing so leads to arguably inappropriate results. We redefine sequential equilibrium so that the definition agrees with the standard definition in games of perfect recall while still giving reasonable results in games of imperfect recall. The definition can be viewed as trying to capture a notion of ex ante sequential equilibrium. The picture here is that players choose their strategies before the game starts and are committed to it, but they choose it in such a way that it remains optimal even off the equilibrium path. A notion of interim sequential equilibrium is also considered. 

Secure function computation has been thoroughly studied and optimized in the past decades. We extend techniques used for secure computation to simulate arbitrary protocols involving a mediator. The key feature of our notion of simulation is that it is bidirectional: not only does the simulation produce only outputs that could happen in the original protocol, but the simulation produces all such outputs. In asynchronous systems there are also new subtleties that arise because the scheduler can influence the output. Thus, these requirements cannot be achieved by the standard notion of secure computation. We provide a construction that is secure if n > 4t, where t is the number of malicious agents, which is provably the best possible. We also show that our construction is secure in the universal composability model and that it satisfies additional security properties even if 3t < n \le 4t. 

We investigate how to model the beliefs of an agent who becomes more aware. We use the framework of Halpern and Rego (2013) by adding probability, and define a notion of a model transition that describes constraints on how, if an agent becomes aware of a new formula φ in state s of a model M, she transitions to state s* in a model M*. We then discuss how such a model can be applied to information disclosure.