This chapter explores how state evolution impacts uncertainty in classical systems using majorization and related principles.

This chapter introduces entropy as a fundamental measure of information, linking it to uncertainty and informational tasks.

This chapter is the introduction to Part I, “The early history of game theory from Borel”. Starting from a 1921 paper by Émile Borel, it surveys the historical development of game theory during its first few decades.

Game theory in biology emerged as a theoretical and modeling approach to the evolutionary study of behavior and other phenotypes. It was inspired by the prominence of game theory in economics. This chapter gives a perspective on this influence of economic thinking on biology. Which ideas were taken from economics and the social sciences to biology, how were they modified, which biological phenomena were they applied to, and what are the successes, challenges, and the possible future of the field? It focuses on two of the earliest applications of game theory in biology, namely sex-ratio theory and animal contests.

This chapter introduces relative entropy measures, including Shannon, Tsallis, and Rényi, and generalizations using the f-divergence framework.

We end this book with a chapter on the connection between Vaught’s conjecture and computable structure theory. This chapter does not contain fully detailed proofs, and it is mostly an expository chapter on what is know and not know.

This chapter examines the distinctions between quantum and classical information, highlighting quantum states and measurement properties.

This chapter reviews literature on game-theoretic analysis of voting. Both cooperative and noncooperative concepts are used to answer questions, such as, How do candidates or parties propose alternatives to voters in strategic interactions? Why do voters vote? What are the implications of asymmetric information for candidates’ and voters’ incentives? Do prevoting deliberations improve information sharing? If so, through what type of rules? Sophisticated voters may act strategically, and therefore it matters whether one’s choices are pivotal. In the presence of private information, the mechanism design approach is highly appropriate, as voters’ incentives can be heavily influenced by the institutional settings that determine how votes are transformed to election outcomes. The analysis of information aggregation in large-scale elections brings important insights to our understanding of representative democracy. Due to the nonexistence of a core and the cyclical structure of pairwise comparison, there may be a fundamental difficulty in the preference aggregation by majoritarian democracy in large-scale elections. The chapter concludes with questions for future research: How does the limitation of preference/information aggregation in large-scale elections affect the stability of representative democracy? What determines the robustness of democratic norms? What is the role of the media in the presence of information asymmetry, particularly in ideological battles where information filtering can play an exacerbating role?