# Rational Choice, Decision Theory and Game Theory

In the past century, philosophers and social scientists have given theories of individual and interactive decision making a rigorous foundation. Indeed, contemporary decision and game theory have revolutionized our understanding of rational choice in ways that parallel the concurrent revolution in philosophical logic. Carnegie Mellon's philosophy department is recognized as one of the foremost departments in the world in decision and game theory. Primary research at Carnegie Mellon in decision and game theory focuses on the foundations of Bayesian decision theory, interactive knowledge concepts and their applications in game theory and equilibrium selection in games.

Decision theory is motivated to a large extent by the consequentialist, and especially the utilitarian, traditions in moral philosophy. To bring about the best consequences, one must know what they are. From the beginning, both critics and defenders of moral consequentialism raised skeptical doubts about the possibility of ever deriving a satisfactory procedure for rank ordering alternatives so as to identify the best choice. In the special case of utilitarianism, the great 19th century utilitarians John Stuart Mill and Henry Sidgwick themselves thought that an exact calculus of utility that would enable society's members to know precisely how to produce the greatest overall welfare might not be possible. Any proposal for a utilitarian calculus raises two fundamental questions: (1) How are quantities of utility to be ascribed to alternatives in a nonarbitrary way?, and (2) How are likelihoods to be ascribed to alternatives in a nonarbitrary way? A decision theory based upon utility is intimately related to theories of probability, which are needed for the calculation of expected consequences. In 1926, Frank Ramsey presented a monumental essay Truth and Probability, which laid the cornerstones of contemporary decision theory. Ramsey proved a representation theorem that enables one to derive both quantitative utilities and probabilities over alternatives that uniquely cohere with one's qualitative preferences over these alternatives. The work of Ramsey and his successors, most notably Leonard Savage, has resulted in modern Bayesian decision theory, which provides a precise account of how to choose so as to maximize expected utility. This work has also sparked a great flowering of alternative decision theories, some of which generalize Ramsey's and Savage's treatments and some of which constitute alternatives to standard Bayesian decision theory.

Game theory considers cases in which decision problems interact. The mathematician John von Neumann and the economist Oskar Morgenstern established game theory as an important branch of social science in 1944 with the publication of their treatise Theory of Games and Economic Behavior. Von Neumann and Morgenstern presented a precise mathematical account of situations in which agents make interdependent decisions. To be sure, von Neumann and Morgenstern had intellectual precursors. In the 1910s and 1920s, the mathematicians Ernst Zermelo and Emile Borel as well as von Neumann himself give mathematical analyses of some game-theoretic problems. Prior to the 20th century certain philosophers, including Thomas Hobbes, David Hume, Jean Jacques Rousseau and Adam Smith, presented arguments that employ informal game-theoretic insights that help to explain forms of social coordination. Indeed, game theory gets its name from an insight expressed by the 17th century philosopher Gottfried Leibniz. Leibniz suggested that philosophers should attempt to better understand the reasoning of the players engaged in games, since people appear to devote special energy to their deliberations when they must choose strategies in the games they play. Contemporary game theorists use games as motivating examples, although the theory models all varieties of social interaction. Von Neumann and Morgenstern conjectured that game theory would ultimately develop into a general theory of social interaction as rigorous and predictively powerful as the mathematical physics of their time. While we are still very far from the scientific theory of social interaction von Neumann and Morgenstern envisioned, game theory has developed far beyond the classical theory of von Neumann and Morgenstern, and now spans many disciplines.

At Carnegie Mellon, Teddy Seidenfeld works on the analytical foundations of Bayesian decision theory. Seidenfeld has developed a number of extensions of Savage's theory that apply to decision-making agents who do not satisfy the requirements of the Savage theory.

Adam Bjorndahl works on the epistemic analysis of games, with particular focus on generalizations of the classical setting that capture such phenomena as belief-dependent preferences, the effects of coarseness, and models of intention. The “language-based games” project he initiated (jointly with Joseph Y. Halpern and Rafael Pass) integrates several such generalizations under a single framework in which a logical language parametrizes what is relevant to players’ preferences.

Peter Vanderschraaf has worked in branches of game theory. Vanderschraaf has worked on dynamical adjustment models of the beliefs of agents engaged in games and has investigated how agents who update their beliefs dynamically can learn to follow equilibrium points of games.

Horacio Arlo-Costa has worked on the epistemic foundations of the theory of games. He has also investigated the extent to which belief and desire are separable and inter-reducible in models of rational choice as well as the functional role of valence in those models. Recent work focused on applying and generalizing the theory of conjoint measurement.