Online research in philosophy

Games such as the St. Petersburg game present serious problems for decision theory.1 The St. Petersburg game invokes an unbounded utility function to produce an infinite expectation for playing the game. The problem is usually presented as a clash between decision theory and intuition: most people are not prepared to pay a large finite sum to buy into this game, yet this is precisely what decision theory suggests we ought to do. But there is another problem associated with the St. Petersburg game. The problem is that standard decision theory counsels us to be indifferent between any two actions that have infinite expected utility. So, for example, consider the decision problem of whether to play the St. Petersburg game or a game where every payoff is $1 higher. Let’s call this second game the Petrograd game (it’s the same as St. Petersburg but with a bit of twentieth century inflation). Standard decision theory is indifferent between these two options. Indeed, it might be argued that any intuition that the Petrograd game is better than the St. Petersburg game is a result of misguided and na¨ıve intuitions about infinity.2 But this argument against the intuition in question is misguided. The Petrograd game is clearly better than the St. Petersburg game. And what is more, there is no confusion about infinity involved in thinking this. When the series of coin tosses comes to an end (and it comes to an end with probability 1), no matter how many tails precede the first head, the payoff for the Petrograd game is one dollar higher than the St. Petersburg game. Whatever the outcome, you are better off playing the Petrograd game. Infinity has nothing to do with it. Indeed, a straightforward application of dominance reasoning backs up this line of reasoning.3 Standard decision theory.

In this paper, we explore Peirce's work for insights into a theory of learning and cognition for education. Our focus for this exploration is Peirce's paper The Fixation of Belief (FOB), originally published in 1877 in Popular Science Monthly. We begin by examining Peirce's assertion that the study of logic is essential for understanding thought and reasoning. We explicate Peirce's view of the nature of reasoning itself—the characteristic guiding principles or ‘habits of mind’ that underlie acts of inference, the dimensions of and interaction between doubt and belief, and his four methods of resolving or ‘fixing’ belief (i.e., tenacity, authority, a priori, and experimentation). The four methods are then juxtaposed against current models of teaching and learning such as constructivism, schema theory, situated cognition, and inquiry learning. Finally, we discuss Peirce's modes of inference as they relate educationally to the resolution of doubt and beliefs and offer an example of belief resolution from an experienced teacher in a professional development environment.

In his last dialogue, the Laws, Plato views citizens in the polis as players in a game. Just as contemporary game theory, Plato considers games to be states of strategic interaction. Yet the game of the Laws differs from those of game theory in one important respect. Where game theory assumes that players are rational--that they choose strategies, or rules for taking action at each instant of a game, in order to maximize payoffs--Plato explores the conditions under which rationality, as game theory defines it, is possible. Plato thus agrees with game theory that rational, maximizing behavior is a necessary constituent of civic order, but not that it may simply be assumed. He concurs that rationality can describe the behavior of citizens, but not under any circumstances. It is the task of politics in Plato's city to prepare the conditions for rationality. Only once politics has done its work is maximizing, utilitarian behavior possible. Yet political preparation of the game of the city is exactly what contemporary game theory assumes away. The Athenian, who leads the discussion, describes the political preparation for rationality as "this moderate old man's game concerning laws." Political preparation of the game of the city is itself a game, because it is never possible to escape strategic interaction. It is, however, a second-order game whose play paves the way for the first-order game of lawful strategic interaction. The second-order, political game at once completes the analysis of rationality and lays the groundwork for rationality to operate. The politics of the Laws is a game, but one whose actions and players differ from the actions and players of the first-order game. The action of the political game is lawgiving. The players are the gods and god-like men who serve as lawgivers. The second-order, political game also has its own distinct rationality, linked to a payoff that is qualitatively different from the utilitarian payoffs of the first-order game. The Athenian calls the rationality of the second-order game "intelligence." He calls the payoff associated with it "joy," as distinct from the "benefits" associated with utilitarian rationality. The players in the second-order game seek to maximize joy, not benefit. Joy is the experience of play. The payoff of players in the second-order game is the game itself, not a benefit collateral to playing it. The second-order game is a "true" game, one that the players enter in order to play, not to get utilitarian payoffs.

Game theory is a branch of economics that uses powerful mathematical models to predict what agents ought to do when interacting with other agents strategically. Bounded rationality is a sub-field of game theory that sets out to explain why, in some interesting cases, people don't act according their utility maximizing strategies, as described by game theory. Interactive Epistemology is formal tool used by Game Theorists and computer scientists to model interactive cases of knowledge. This interesting and useful tool has been previously ignored by philosophers. I'd like to introduce philosophers to interactive epistemology. After doing so, I'll go on to describe the way I've used this powerful formal tool in my own research, by giving some arguments about Bounded Epistemology, which is an analogue of Bounded Rationality, and, if I'm right, is explainable according to many, but not all, of the same models. Doing so, however, requires first setting out and explaining many of these concepts more fully.

The purpose of this book is to develop a framework for analyzing strategic rationality, a notion central to contemporary game theory, which is the formal study of the interaction of rational agents, and which has proved extremely fruitful in economics, political theory, and business management. The author argues that a logical paradox (known since antiquity as "the Liar paradox") lies at the root of a number of persistent puzzles in game theory, in particular those concerning rational agents who seek to establish some kind of reputation. Building on the work of Parsons, Burge, Gaifman, and Barwise and Etchemendy, Robert Koons constructs a context-sensitive solution to the whole family of Liar-like paradoxes, including, for the first time, a detailed account of how the interpretation of paradoxial statements is fixed by context. This analysis provides a new understanding of how the rational agent model can account for the emergence of rules, practices, and institutions.

Contents. Introduction. 1. Preliminaries. 2. Normal Form Games. 3. Extensive Games. 4. Applications of Game Theory. 5. The Methodology of Game Theory. Conclusion. Appendix. Bibliography. Index. Does game theory—the mathematical theory of strategic interaction—provide genuine explanations of human behaviour? Can game theory be used in economic consultancy or other normative contexts? Explaining Games: The Epistemic Programme in Game Theory—the first monograph on the philosophy of game theory—is an attempt to combine insights from epistemic logic and the philosophy of science to investigate the applicability of game theory in such fields as economics, philosophy and strategic consultancy. I prove new mathematical theorems about the beliefs, desires and rationality principles of individual human beings, and explore in detail the logical form of game theory as it is used in explanatory and normative contexts. I argue that game theory reduces to rational choice theory if used as an explanatory device, and that game theory is nonsensical if used as a normative device. A provocative account of the history of game theory reveals that this is not bad news for all of game theory, though. Two central research programmes in game theory tried to find the ultimate characterisation of strategic interaction between rational agents. Yet, while the Nash Equilibrium Refinement Programme has done badly thanks to such research habits as overmathematisation, model-tinkering and introversion, the Epistemic Programme, I argue, has been rather successful in achieving this aim. "The 'epistemic' approach to game theory has emerged over the past twenty-five years. What is this approach? How does it differ from the conventional equilibrium-based approach to game theory? What have been its strengths and weaknesses to date? To find out, read this comprehensive and excellently written account". Adam Brandenburger, J. P. Valles Professor of Business Economics and Strategy, Stern School of Business, New York University "Reading Boudewijn de Bruin's book should be rewarding both for game theorists interested in the conceptual foundations of their discipline and for philosophers who want to learn more about formal analysis of strategic interaction. It provides an in-depth logical study of the currently dominant epistemic approaches to non-cooperative games, with an eye both to the attractions and to the serious challenges facing the Epistemic Programme". Wlodek Rabinowicz, Professor of Practical Philosophy, Department of Philosophy, Lund University .

The standard backward-induction reasoning in a game like the centipede assumes that the players maintain a common belief in rationality throughout the game. But that is a dubious assumption. Suppose the first player X didn't terminate the game in the first round; what would the second player Y think then? Since the backwards-induction argument says X should terminate the game, and it is supposed to be a sound argument, Y might be entitled to doubt X's rationality. Alternatively, Y might doubt that X believes Y is rational, or that X believes Y believes X is rational, or Y might have some higher-order doubt. X’s deviant first move might cause a breakdown in common belief in rationality, therefore. Once that goes, the entire argument fails. The argument also assumes that the players act rationally at each stage of the game, even if this stage could not be reached by rational play. But it is also dubious to assume that past irrationality never exerts a corrupting influence on present play. However, the backwards-induction argument can be reconstructed for the centipede game on a more secure basis.1 It may be implausible to assume a common belief in rationality throughout the game, however the game might go, but the argument requires less than this. The standard idealisations in game theory certainly allow us to assume a common belief in rationality at the beginning of the game. They also allow us to assume this common belief persists so long as no one makes an irrational move. That is enough for the argument to go through.

Game theory poses problems for modeling rational belief, but it does not need a new theory of rationality. Experimental results that suggest otherwise often reveal difficulties in testing game theory, rather than mistakes or paradoxes. Even though the puzzles Colman discusses show no inadequacy in the standard theory of rationality, they show that improved models of belief are needed.

Edited by three leading figures in the field, this exciting volume presents cutting-edge work in decision theory by a distinguished international roster of contributors. These mostly unpublished papers address a host of crucial areas in the contemporary philosophical study of rationality and knowledge. Topics include causal versus evidential decision theory, game theory, backwards induction, bounded rationality, counterfactual reasoning in games and in general, analyses of the famous common knowledge assumptions in game theory, and evaluations of the normal versus extensive form formulations of complex decision problems.

Formal epistemology is the study of crucial concepts in general or mainstream epistemology including knowledge, belief (-change), certainty, rationality, reasoning, decision, justi…cation, learning, agent interaction and information processing using a spread of di¤erent formal tools. The formal tools may be drawn from logic, probability theory, game theory, decision theory, formal learning theory, distributed computing and is thus not simply a purely philosophical province. Its practitioners include philosophers, computer scientists, social scientists, cognitive psychologists, theoretical economists, mathematicians, and theoretical linguists. Formal epistemology is a thoroughly interdisciplinary …eld with many agendas, actors and issues. What follows is a breezy overview of formal epistemology as organized around notions of agency and interaction.