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This dissertation consists of three essays that investigate decisions under limited attention. Chapter $1$ provides an introduction for this dissertation. In Chapter 2, I study additive representations under idempotent attention. Additive representations suggest that the utility of a menu equals the sum of the utility of alternatives in it, which implicitly assumes that a decision maker (DM) has full attention. However, many theoretical and empirical studies have argued that the DM does not always pay full attention. The purpose of this paper is to extend our understanding of additive representations to the context of limited attention. Specifically, we focus on additive representations when attention rules are idempotent, which are generalizations of attention filters. We propose the characterization of additive representations under idempotent attention as well as special cases of it: additive representations under attention filter. This model can also be applied to simple cardinality-based ordering and Borda scores when the DM has limited attention.In Chapter 3, I focus on satisficing under idempotent attention. Satisficing choice pattern has been accused of a lack of cognitive ability in perceiving and analyzing alternatives. In an effort to disentangle the limitations of noticing alternatives from other factors, this paper proposes a model of satisficing under limited attention. Our focus centers on the idempotent attention rules, leading to Satisficing under Idempotent Attention (SIA). Our study provides a characterization of SIA as well as a discussion of revealed attention and preference practices. Notably, these outcomes stem from choices made on a subset of menus. The revealed attention and preference, however, remain inherently non-unique. Additionally, considering the idempotent nature of attention filters and competition filters, we also present distinctive characterizations of satisficing under these two attention rules.Chapter 4 investigates choices under limited attention where attention is formed by interactions between alternatives. Specifically, given a choice problem, the presence of an alternative can trigger or block the attention of other options. An alternative captures the DM's attention when the effects of triggers outweigh the effects of blockers. This paper also provides valuable insights into the concept of signed orders within the decision-making process.

Standard decision theory has some striking consequences. First, if you have any irrational preferences, it does not make sense to ascribe credences to you---decision theory treats you as having no opinions at all. Second, you should always prefer to look at more information before making a decision if the information is free, even if you think the information is likely biased.These consequences seem wrong. Your preferences are sometimes irrational, but you clearly still have opinions about things. If we could only ascribe credences to agents with perfectly rational preferences, decision theory couldn't give any useful advice to imperfect agents like us. Furthermore, people sometimes prefer not to look at more information before making a decision. For example, when grading a paper, I prefer not to know the student's name. This behavior seems perfectly rational, even laudable.Does this mean that decision theory is broken? No. I argue that we can fix its problems. The key to ascribing credences to non-ideal agents is to start by looking at comparative probability judgments, like thinking it's equally likely to be sunny and rainy tomorrow. Comparative probability is tied to preferences in a straightforward way---you think that sunshine and rain are equally likely if you are indifferent between betting on them. If sunshine and rain are the only two possibilities for the weather tomorrow and you think they're equally likely, you assign probability .5 to both. I show that if your preferences satisfy some minimal constraints, we can extend this procedure to fix your entire credence function while allowing many of your preferences to be irrational (Chapter 2).The key to explaining why it can be rational to reject free information is to recognize that we can be uncertain about how we will react to evidence. If I were certain that I always respond to evidence in a perfectly rational way (by 'conditionalization'), then perhaps it would be rational to look at my students' names before grading papers. Indeed, certainty that one will react to evidence in a perfectly rational way is an assumption built into Good's well-known theorem about the value of information. But this assumption might fail. I know that I'm not always rational---for example, I might give too much weight to the fact that George got an `A' on the first paper and treat this as better evidence than it is that his current paper deserves a high mark. Once I take this possibility into account, I might be better off ignorant (Chapter 3).Uncertainty about how we will react to evidence has other consequences as well. For example, it can lead us to make sequences of choices which, taken together, yield sure loss. Many decision theorists have claimed that such choices always indicate irrationality. I argue that this bit of conventional wisdom needs revision (Chapter 4).One upshot is that by widening the scope of decision theory to include non-ideal agents, we enable the decision theorist to give vindicating explanations of common phenomena, like having opinions without being fully rational and avoiding information before making a decision. Another upshot is that a more sophisticated decision theory is relevant for designing beneficial AI systems, since existing ideas for doing so assume an implausibly strong conception of rationality (Chapter 5).

Among many things, Frank Ramsey is famous for his invention of a logical device called the Ramsey sentence. The Ramsey sentence of a scientific theory is a conjunction of the theory’s propositions where the theoretical vocabulary is existentially quantified out. The sentence appears in an obscure paper titled “Theories”. This paper is extremely difficult because Ramsey likely did not finish it before his death and it mostly consists of a mathematical model of scientific theories and a labyrinthine toy example. Consequently, philosophers have proposed many radically different and incoherent interpretations of the Ramsey sentence and Ramsey’s philosophy of science. An important fact interpreters have ignored is that Ramsey had intended “Theories” to be a chapter in a book centered on a revised version of his formal theory for decision-making under uncertainty. The principal accomplishment of my dissertation is to leverage this fact to provide the first complete and philosophically satisfying interpretation of “Theories”, including Ramsey’s toy example and the Ramsey sentence.Ramsey’s theory for decision-making is incompatible with his philosophical commitments in “Theories” and contemporary papers. In my dissertation, I revise that theory to make it consistent with those commitments. I combine this with the formal model he describes in “Theories” to provide a unified reconstruction of Ramsey’s intended model for scientific theories. I use the reconstructed model to explain Ramsey’s toy example, and I furnish a complete interpretation of Ramsey’s arguments in “Theories”. With these parts, I develop a detailed account of the laws and chances derived from scientific theories and a novel interpretation of the Ramsey sentence. This has several important philosophical consequences.First, Ramsey did not use the Ramsey sentence in the ways philosophers have previously proposed. Philosophers view the Ramsey sentence either as a tool for eliminating theoretical terms or for asserting the existence of theoretical properties. Neither of these claims is correct. Instead, I argue that the Ramsey sentence is a communication and deliberation device aimed at representing the propositions an individual believes with her behavioral dispositions. Informally, it expresses as new propositions the mixture of rules a person uses to guide her behavior so that she may deliberate and communicate those rules with herself and other people. This suggests the Ramsey sentence can be used to read off the content of an agent’s belief from their behavior.Second, Ramsey formulated a novel and interesting type of non-reductionistic scientific anti-realism. That is, for Ramsey theoretical propositions are fictitious and yet not eliminable for observational propositions. Theoretical propositions are fictitious because their meaning depends on theoretical laws, which, being universally quantified, are also fictitious. But theoretical propositions cannot be eliminated because they entail conceptual possibilities that are richer than those given by the observation language.Third, Ramsey held laws and chances to be rules of deference for degrees of belief. His account of laws and chances foreshadows pragmatic, subjectivist accounts of laws and chances from later in the twentieth century. Degrees of belief are representations of preferences over gambles involving observable propositions. A chance then is a rule for preferring some gambles over others, and a law is a chance that assigns unity to its degrees of beliefs. This prefigures the views that chances and subjunctive conditionals are reducible to properties of degrees of belief.