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Adam Bjorndahl

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11 papers

2 author rows

TARK Conference 2023 Conference Paper

Sequential Language-based Decisions

Adam Bjorndahl
Joseph Y. Halpern

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 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.

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TARK Conference 2021 Conference Paper

Language-based Decisions

Adam Bjorndahl
Joseph Y. Halpern

In Savage's classic decision-theoretic framework, actions are formally defined as functions from states to outcomes. But where do the state space and outcome space come from? Expanding on recent work by Blume, Easley, and Halpern (BEH), we consider a language-based framework in which actions are identified with (conditional) descriptions in a simple underlying language, while states and outcomes (along with probabilities and utilities) are constructed as part of a representation theorem. Our work expands the role of language from that of BEH by using it not only for the conditions that determine which actions are taken, but also the effects. More precisely, we take the set of actions to be built from those of the form "do(phi)", for formulas phi in the underlying language. This presents a problem: how do we interpret the result of do(phi) when phi is underspecified (i. e. , compatible with multiple states)? We answer this using tools familiar from the semantics of counterfactuals: roughly speaking, do(phi) maps each state to the "closest" phi-state. This notion of "closest" is also something we construct as part of the representation theorem; in effect, then, we prove that (under appropriate assumptions) the agent is acting as if each underspecified action is first made definite and then evaluated (i. e. , by maximizing expected utility). Of course, actions in the real world are often not presented in a fully precise manner, yet agents reason about and form preferences among them all the same. Our work brings the abstract tools of decision theory into closer contact with such real-world scenarios.

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TARK Conference 2019 Conference Paper

Uncertainty About Evidence

Adam Bjorndahl
Aybüke Özgün

We develop a logical framework for reasoning about knowledge and evidence in which the agent may be uncertain about how to interpret their evidence. Rather than representing an evidential state as a fixed subset of the state space, our models allow the set of possible worlds that a piece of evidence corresponds to to vary from one possible world to another, and therefore itself be the subject of uncertainty. Such structures can be viewed as (epistemically motivated) generalizations of topological spaces. In this context, there arises a natural distinction between what is actually entailed by the evidence and what the agent knows is entailed by the evidence -- with the latter, in general, being much weaker. We provide a sound and complete axiomatization of the corresponding bi-modal logic of knowledge and evidence entailment, and investigate some natural extensions of this core system, including the addition of a belief modality and its interaction with evidence interpretation and entailment, and the addition of a "knowability" modality interpreted via a (generalized) interior operator.

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TARK Conference 2017 Conference Paper

Endogenizing Epistemic Actions

Will Nalls
Adam Bjorndahl

Through a series of examples, we illustrate some important drawbacks that the action logic framework suffers from in its ability to represent the dynamics of information updates. We argue that these problems stem from the fact that the action model, a central construct designed to encode agents' uncertainty about actions, is itself effectively common knowledge amongst the agents. In response to these difficulties, we motivate and propose an alternative semantics that avoids them by (roughly speaking) endogenizing the action model. We discuss the relationship to action logic, and provide a sound and complete axiomatization.

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TARK Conference 2017 Conference Paper

From Type Spaces to Probability Frames and Back, via Language

Adam Bjorndahl
Joseph Y. Halpern

We investigate the connection between the two major mathematical frameworks for modeling interactive beliefs: Harsanyi type spaces and possible-worlds style probability frames. While translating the former into the latter is straightforward, we demonstrate that the reverse translation relies implicitly on a background logical language. Once this "language parameter" is made explicit, it reveals a close relationship between universal type spaces and canonical models: namely, that they are essentially the same construct. As the nature of a canonical model depends heavily on the background logic used to generate it, this work suggests a new view into a corresponding landscape of universal type spaces.

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TARK Conference 2017 Conference Paper

Logic and Topology for Knowledge, Knowability, and Belief - Extended Abstract

Adam Bjorndahl
Aybüke Özgün

In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge. Building on Stalnaker's core insights, and using frameworks developed by Bjorndahl and Baltag et al. , we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and (roughly speaking) what is knowable; we argue that the foundational axioms of Stalnaker's system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an "evidence-in-hand" conception of knowledge and a weaker "evidence-out-there" notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for this logic as well as its uni-modal belief fragment. We then consider weaker logics that preserve suitable translations of Stalnaker's postulates, yet do not allow for any reduction of belief. We propose novel topological semantics for these irreducible notions of belief, generalizing our previous semantics, and provide sound and complete axiomatizations for the corresponding logics.

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TARK Conference 2015 Conference Paper

Bayesian Games with Intentions

Adam Bjorndahl
Joseph Y. Halpern
Rafael Pass

We show that standard Bayesian games cannot represent the full spectrum of belief-dependent preferences. However, by introducing a fundamental distinction between intended and actual strategies, we remove this limitation. We define Bayesian games with intentions, generalizing both Bayesian games and psychological games, and prove that Nash equilibria in psychological games correspond to a special class of equilibria as defined in our setting.

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KR Conference 2014 Conference Paper

Axiomatizing Rationality

Adam Bjorndahl
Joseph Halpern
Rafael Pass

complexities of higher-order beliefs in a game-theoretic context. Formal logic furnishes a powerful and versatile class of such models; namely, modal logics of belief appropriately specialized for reasoning about games. Rationality is no less important in this setting; however, while it has been incorporated into these models both syntactically and semantically, no axiomatization of the resulting logical systems has been provided. This paper fills this gap. We take as our point of departure axioms for rationality in the sense of expected utility maximization given in (Bjorndahl, Halpern, & Pass 2011). We extend these axioms to arbitrary decision rules, under the assumption that the players’ uncertainty is represented by a probability measure. This allows us to deal with not just expected utility maximization, but other standard rules such as maximin and minimax regret (see (Halpern 2003) for a discussion of all the decision rules mentioned in this paper). We then go on to consider what happens when the players’ uncertainty is represented by a set of probabilities, which allows us to capture well-known decision rules such as maxmin expected utility and minimax expected regret. Finally, we consider situations where a player might be uncertain about which decision rules his opponents are using. The rest of this paper is organized as follows. In Section 2, we define the core concepts formally: games, modal logics of belief appropriate for reasoning about games, and the incorporation of rationality into these logics. Section 3 gives the axiomatization, and proves that it is sound and complete. In Section 4, we provide sound and complete axiomatizations for the cases where the players’ uncertainty is represented by sets of probabilities, and where players may be uncertain about the decision rules used by other players. In Section 5 we discuss the role of language. Section 6 concludes with a discussion of future work. More detailed proofs and further discussion can be found in the full paper, which is available at http: //www. math. cornell. edu/∼abjorndahl/Site/CV files/Axiomatizing%20Rationality. pdf. We provide a sound and complete axiomatization for a class of logics appropriate for reasoning about the rationality of players in games. Essentially the same axiomatization applies to a wide class of decision rules.

TARK Conference 2013 Conference Paper

Language-based Games

Adam Bjorndahl
Joseph Y. Halpern
Rafael Pass

theory, beginning with [6] and expanded in [3], is an enrichment of the classical setting meant to capture these kinds of preferences and motivations. In a similar vein, work on reference-dependent preferences, as developed in [7], formalizes phenomena such as loss-aversion by augmenting players’ preferences with an additional sense of gain or loss derived by comparing the actual outcome to what was expected. In both of these theories, the method of generalization takes the same basic form: the domain of the utility functions is enlarged to include not only the outcomes of the game, but also the beliefs of the players. The resulting structure may be fairly complex; for instance, in psychological game theory, since the goal is to model preferences that depend not only on beliefs about outcomes, but also beliefs about beliefs, beliefs about beliefs about beliefs, and so on, the domain of the utility functions is extended to include infinite hierarchies of beliefs. The model we present in this paper, though motivated in part by a desire to capture belief-dependent preferences, is geared towards a much more general goal. Besides being expressive enough to subsume existing systems such as those described above, it establishes a general framework for modeling players with richer preferences. Moreover, it is equally capable of representing impoverished preferences, a canonical example of which are so-called “coarse beliefs” or “categorical thinking” [9]. More specifically, our formalism provides good practical and theoretical tools for handling beliefs as discrete rather than continuous objects, an advantage that is particularly relevant in the context of psychological effects in games. Despite this expressive power, the system is easy to use: player preferences are represented in a simple and natural manner, narrowing the divide between intuition and formalism. As a preliminary illustration of some of these points, consider the following simple example. We introduce language-based games, a generalization of psychological games [6] that can also capture referencedependent preferences [7]. The idea is to extend the domain of the utility function to situations, maximal consistent sets in some language. The role of the underlying language in this framework is thus particularly critical. Of special interest are languages that can express only coarse beliefs [9]. Despite the expressive power of the approach, we show that it can describe games in a simple, natural way. Nash equilibrium and rationalizability are generalized to this setting; Nash equilibrium is shown not to exist in general, while the existence of rationalizable strategies is proved under mild conditions.

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IJCAI Conference 2013 Conference Paper

Language-Based Games (Extended Abstract)

Adam Bjorndahl
Joseph Y. Halpern
Rafael Pass

We introduce language-based games, a generalization of psychological games [Geanakoplos et al. , 1989] that can also capture reference-dependent preferences [Kőszegi and Rabin, 2006], which extend the domain of the utility function to situations, maximal consistent sets in some language. The role of the underlying language in this framework is thus particularly critical. Of special interest are languages that can express only coarse beliefs [Mullainathan, 2002]. Despite the expressive power of the approach, we show that it can describe games in a simple, natural way. Nash equilibrium and rationalizability are generalized to this setting; Nash equilibrium is shown not to exist in general, while the existence of rationalizable strategies is proved under mild conditions.

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TARK Conference 2011 Conference Paper

Reasoning about justified belief

Adam Bjorndahl
Joseph Y. Halpern
Rafael Pass

Halpern and Pass [8] introduce a logic of justified belief and go on to prove that strong rationalizability is characterized in this logic in terms of common justified belief of rationality (CJBR). Their paper provides semantics for this logic but no axiomatization. We correct this deficiency by reformulating the definition of justified belief and providing a complete axiomatization of this new system. We then prove a result analogous to the characterization of strong rationalizability in terms of CJBR, and analyze the additional assumptions needed to do so.

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