Electronic Proceedings in Theoretical Computer Science
In Savage's classic decision-theoretic framework [12], actions are formally defined as functions ... more In Savage's classic decision-theoretic framework [12], 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 [3], 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 in [3] 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(ϕ), for formulas ϕ in the underlying language. This presents a problem: how do we interpret the result of do(ϕ) when ϕ is underspecified (i.e., compatible with multiple states)? We answer this using tools familiar from the semantics of counterfactuals [13]: roughly speaking, do(ϕ) maps each state to the "closest" ϕ-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.
We introduce language-based games, a generalization of psychological games [6] that can also capt... more 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.
This paper proposes new semantics for nondeterministic program execution, replacing the standard ... more This paper proposes new semantics for nondeterministic program execution, replacing the standard relational semantics for propositional dynamic logic (PDL). Under these new semantics, program execution is represented as fundamentally deterministic (i.e., functional), while nondeterminism emerges as an epistemic relationship between the agent and the system: intuitively, the nondeterministic outcomes of a given process are precisely those that cannot be ruled out in advance. We formalize these notions using topology and the framework of dynamic topological logic (DTL) [1]. We show that DTL can be used to interpret the language of PDL in a manner that captures the intuition above, and moreover that continuous functions in this setting correspond exactly to deterministic processes. We also prove that certain axiomatizations of PDL remain sound and complete with respect to the corresponding classes of dynamic topological models. Finally, we extend the framework to incorporate knowledge using the machinery of subset space logic [2], and show that the topological interpretation of public announcements as given in [3] coincides exactly with a natural interpretation of test programs. 1 See [4] for an overview of this branch of modal logic.
International Joint Conference on Artificial Intelligence, 2013
We introduce language-based games, a generalization of psychological games [6] that can also capt... more 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.
theoretical aspects of rationality and knowledge, 2017
We investigate the connection between the two major mathematical frameworks for modeling interact... more 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.
We introduce language-based games, a generalization of psychological games [Geanakoplos et al., 1... more 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. 1
The idea that there is a fundamental difference in value between persons and things and that resp... more The idea that there is a fundamental difference in value between persons and things and that respecting this difference is an important moral requirement has strong intuitive appeal. Kantian ethics is unique in placing this requirement at the center of a moral system and for explicating the conditions for complying with it. Unlike challenges to Kantian ethics that focus on tragic cases that pit respect for one person against respect for another, this paper focuses on the question of how we can respect the value distinction between persons and things under conditions of uncertainty. After exploring why decision making under uncertainty is a neglected topic among Kantians and demonstrating how uncertainty challenges our ability to comply with this norm, we propose a notion of morally insignificant risk within a framework that allows agents to navigate real world decisions involving material benefit and some risk to dignity without violating the Kantian’s core commitments. We conclude ...
We provide a sound and complete axiomatization for a class of logics appropriate for reasoning ab... more 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.
Through a series of examples, we illustrate some important drawbacks that the action logic framew... more 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 relationshop to action logic, and provide a sound and complete axiomatization.
In recent work, Robert Stalnaker proposes a logical framework in which belief is realized as a we... more In recent work, Robert Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge [28]. Building on Stalnaker’s core insights, 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...
We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public a... more We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ϕ to be the "local truth" of ϕ, semantically rendered via an interior operator. This is closely related to the notion of ϕ being "knowable". We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness.
This paper proposes new semantics for nondeterministic program execution, replacing the standard ... more This paper proposes new semantics for nondeterministic program execution, replacing the standard relational semantics for propositional dynamic logic (PDL). Under these new semantics, program execution is represented as fundamentally deterministic (i.e., functional), while nondeterminism emerges as an epistemic relationship between the agent and the system: intuitively, the nondeterministic outcomes of a given process are precisely those that cannot be ruled out in advance. We formalize these notions using topology and the framework of dynamic topological logic (DTL) [1]. We show that DTL can be used to interpret the language of PDL in a manner that captures the intuition above, and moreover that continuous functions in this setting correspond exactly to deterministic processes. We also prove that certain axiomatizations of PDL remain sound and complete with respect to the corresponding classes of dynamic topological models. Finally, we extend the framework to incorporate knowledge using the machinery of subset space logic [2], and show that the topological interpretation of public announcements as given in [3] coincides exactly with a natural interpretation of test programs. 1 See [4] for an overview of this branch of modal logic.
Electronic Proceedings in Theoretical Computer Science, 2019
We develop a logical framework for reasoning about knowledge and evidence in which the agent may ... more 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.
Jaakko Hintikka on Knowledge and Game-Theoretical Semantics, 2018
We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public a... more We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ϕ to be the "local truth" of ϕ, semantically rendered via an interior operator. This is closely related to the notion of ϕ being "knowable". We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness.
Electronic Proceedings in Theoretical Computer Science, 2017
Through a series of examples, we illustrate some important drawbacks that the action logic framew... more 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 relationshop to action logic, and provide a sound and complete axiomatization.
We provide a sound and complete axiomatization for a class of logics appropriate for reasoning ab... more We provide a sound and complete axiomatization for a class of logics appropriate for reasoning about the rationality of players in games, and show that essentially the same axiomatization applies to a very wide class of decision rules. We also consider games in which players may be uncertain as to what decision rules their opponents are using, and define in this context a new solution concept, D-rationalizability.
Electronic Proceedings in Theoretical Computer Science, 2016
We show that standard Bayesian games cannot represent the full spectrum of belief-dependent prefe... more 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 [5], and prove that Nash equilibria in psychological games correspond to a special class of equilibria as defined in our setting.
We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public a... more We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ϕ to be the "local truth" of ϕ, semantically rendered via an interior operator. This is closely related to the notion of ϕ being "knowable". We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness.
Electronic Proceedings in Theoretical Computer Science
In Savage's classic decision-theoretic framework [12], actions are formally defined as functions ... more In Savage's classic decision-theoretic framework [12], 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 [3], 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 in [3] 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(ϕ), for formulas ϕ in the underlying language. This presents a problem: how do we interpret the result of do(ϕ) when ϕ is underspecified (i.e., compatible with multiple states)? We answer this using tools familiar from the semantics of counterfactuals [13]: roughly speaking, do(ϕ) maps each state to the "closest" ϕ-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.
We introduce language-based games, a generalization of psychological games [6] that can also capt... more 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.
This paper proposes new semantics for nondeterministic program execution, replacing the standard ... more This paper proposes new semantics for nondeterministic program execution, replacing the standard relational semantics for propositional dynamic logic (PDL). Under these new semantics, program execution is represented as fundamentally deterministic (i.e., functional), while nondeterminism emerges as an epistemic relationship between the agent and the system: intuitively, the nondeterministic outcomes of a given process are precisely those that cannot be ruled out in advance. We formalize these notions using topology and the framework of dynamic topological logic (DTL) [1]. We show that DTL can be used to interpret the language of PDL in a manner that captures the intuition above, and moreover that continuous functions in this setting correspond exactly to deterministic processes. We also prove that certain axiomatizations of PDL remain sound and complete with respect to the corresponding classes of dynamic topological models. Finally, we extend the framework to incorporate knowledge using the machinery of subset space logic [2], and show that the topological interpretation of public announcements as given in [3] coincides exactly with a natural interpretation of test programs. 1 See [4] for an overview of this branch of modal logic.
International Joint Conference on Artificial Intelligence, 2013
We introduce language-based games, a generalization of psychological games [6] that can also capt... more 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.
theoretical aspects of rationality and knowledge, 2017
We investigate the connection between the two major mathematical frameworks for modeling interact... more 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.
We introduce language-based games, a generalization of psychological games [Geanakoplos et al., 1... more 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. 1
The idea that there is a fundamental difference in value between persons and things and that resp... more The idea that there is a fundamental difference in value between persons and things and that respecting this difference is an important moral requirement has strong intuitive appeal. Kantian ethics is unique in placing this requirement at the center of a moral system and for explicating the conditions for complying with it. Unlike challenges to Kantian ethics that focus on tragic cases that pit respect for one person against respect for another, this paper focuses on the question of how we can respect the value distinction between persons and things under conditions of uncertainty. After exploring why decision making under uncertainty is a neglected topic among Kantians and demonstrating how uncertainty challenges our ability to comply with this norm, we propose a notion of morally insignificant risk within a framework that allows agents to navigate real world decisions involving material benefit and some risk to dignity without violating the Kantian’s core commitments. We conclude ...
We provide a sound and complete axiomatization for a class of logics appropriate for reasoning ab... more 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.
Through a series of examples, we illustrate some important drawbacks that the action logic framew... more 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 relationshop to action logic, and provide a sound and complete axiomatization.
In recent work, Robert Stalnaker proposes a logical framework in which belief is realized as a we... more In recent work, Robert Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge [28]. Building on Stalnaker’s core insights, 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...
We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public a... more We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ϕ to be the "local truth" of ϕ, semantically rendered via an interior operator. This is closely related to the notion of ϕ being "knowable". We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness.
This paper proposes new semantics for nondeterministic program execution, replacing the standard ... more This paper proposes new semantics for nondeterministic program execution, replacing the standard relational semantics for propositional dynamic logic (PDL). Under these new semantics, program execution is represented as fundamentally deterministic (i.e., functional), while nondeterminism emerges as an epistemic relationship between the agent and the system: intuitively, the nondeterministic outcomes of a given process are precisely those that cannot be ruled out in advance. We formalize these notions using topology and the framework of dynamic topological logic (DTL) [1]. We show that DTL can be used to interpret the language of PDL in a manner that captures the intuition above, and moreover that continuous functions in this setting correspond exactly to deterministic processes. We also prove that certain axiomatizations of PDL remain sound and complete with respect to the corresponding classes of dynamic topological models. Finally, we extend the framework to incorporate knowledge using the machinery of subset space logic [2], and show that the topological interpretation of public announcements as given in [3] coincides exactly with a natural interpretation of test programs. 1 See [4] for an overview of this branch of modal logic.
Electronic Proceedings in Theoretical Computer Science, 2019
We develop a logical framework for reasoning about knowledge and evidence in which the agent may ... more 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.
Jaakko Hintikka on Knowledge and Game-Theoretical Semantics, 2018
We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public a... more We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ϕ to be the "local truth" of ϕ, semantically rendered via an interior operator. This is closely related to the notion of ϕ being "knowable". We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness.
Electronic Proceedings in Theoretical Computer Science, 2017
Through a series of examples, we illustrate some important drawbacks that the action logic framew... more 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 relationshop to action logic, and provide a sound and complete axiomatization.
We provide a sound and complete axiomatization for a class of logics appropriate for reasoning ab... more We provide a sound and complete axiomatization for a class of logics appropriate for reasoning about the rationality of players in games, and show that essentially the same axiomatization applies to a very wide class of decision rules. We also consider games in which players may be uncertain as to what decision rules their opponents are using, and define in this context a new solution concept, D-rationalizability.
Electronic Proceedings in Theoretical Computer Science, 2016
We show that standard Bayesian games cannot represent the full spectrum of belief-dependent prefe... more 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 [5], and prove that Nash equilibria in psychological games correspond to a special class of equilibria as defined in our setting.
We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public a... more We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ϕ to be the "local truth" of ϕ, semantically rendered via an interior operator. This is closely related to the notion of ϕ being "knowable". We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness.
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Papers by Adam Bjorndahl