The subjective probabilities implied by Savage's (1954, 1972) Postulates are finitely but not countably additive. The failure of countable additivity leads to two known classes of dominance paradoxes, money pumps and indifference between an act and one that pointwise dominates it. There is a common resolution to these classes of paradoxes and to any others that might arise from failures of countably additivity. It consists of reinterpreting finitely additive probabilities as the "traces" of countably additive probabilities on larger state spaces. The new and larger state spaces preserve the essential decision-theoretic structures of the original spaces.

We consider the question, “Under what conditions is the extremum of a random function over a random set itself a random object?” The answer is relevant to problems in both game theory and econometrics, as we illustrate with examples. Our purpose here is to bring the powerful tools of the theory of analytic sets as developed by Dellacherie and Meyer (1978) to the wider attention of the economics profession and to distill Dellacherie and Meyer's work in such a way as to provide some readily accessible theoretical results that will permit relatively easy treatment of economically or econometrically relevant applications.

The authors present three distinct approaches to perfect and proper equilibria for infinite normal form games. In the first two approaches, players 'tremble' in the infinite game playing full support approximate best responses to others' strategies. In the strong approach, a tremble assigns high probability to the set of pure best responses; in the weak approach, it assigns high probability to a neighborhood of this set. The third, limit-of-finite approach applies traditional refinements to sequences of successively larger finite games. Overall, the strong approach to equilibrium refinement most fully respects the structure of infinite games.

A new framework for games in continuous time is proposed. The continuous-time model conforms as closely as possible to the conventional discrete-time framework. Indeed, continuous time is viewed as "discrete time, but with a grid that is infinitely fine." The paper presents several examples illustrating the difficulties that arise in continuous-time game theory. Theorems relate the equilibria of continuous time games to the equilibria of approximating discrete time games. A variety of industrial organization applications are studied, yielding sharp predictions. Applications include continuously repeated games, preemption models, and patent races.

This paper is organized as follows. In Section 2, we motivate and discuss the sequential testing approach. Section 3 discusses invariance principles of the past and present, and the CUSUM and fluctuation instability detectors. Section 4 contains some illustrative Monte Carlo experiments. A summary and concluding remarks are given in Section 5. Proofs are gathered into the Mathematical Appendix

In many economic, political, and social situations, circumstances change at random points in time, reacting is costly, and reactions appropriate to present circumstances may become inappropriate upon future changes, requiring further costly reaction. Waiting is informative if arrival of the next change has non-constant hazard rate. We identify two classes of situations: in the first, delayed reaction is optimal only when the hazard rate of further changes is decreasing; in the second, it is optimal only when the hazard rate of further changes is increasing. These results in semi-Markovian decision theory provide motivations for building delay into decision systems.