A likelihood order is defined over linear subspaces of a finite dimensional Hilbert space. The question arises as to when such an order can be represented by a quantum probability. We introduce a few behaviorally plausible axioms that provide the answer in two cases: pure state and uniform measure. The general problem is answered by using duality-like conditions. The general problem of characterizing the partial orders that admit a quantum representation by behaviorally justified axioms remains open.

Two players are engaged in a zero-sum game with lack of information on one side, in which player 1 (the informed player) receives some stochastic signal about the state of nature. I consider the value of the game as a function of player 1’s information structure, and study the properties of this function. It turns out that these properties reflect the fact that in zero sum situation the value of information for each player is positive.

A theorem of Blackwell about comparison between information structures in classical statistics is given as an analogue in the quantum probabilistic set-up. The theorem provides an operational interpretation for trace-preserving completely positive maps, which are the natural quantum analogue of classical stochastic maps. The proof of the theorem relies on the separation theorem for convex sets and on quantum teleportation.

Two observers, who share a pair of particles in an entangled mixed state, can use it to perform some nonbilocal measurements over another bipartite system. In particular, one can construct a specific game played by the observers against a coordinator, in which they can score better than a pair of observers who only share a classical communication channel. The existence of such a game is an operational implication of an entanglement witness.

We prove that every two-player nonzero–sum stopping game in discrete time admits an ɛ-equilibrium in randomized strategies for every ɛ>0. We use a stochastic variation of Ramsey’s theorem, which enables us to reduce the problem to that of studying properties of ɛ-equilibria in a simple class of stochastic games with finite state space.

An infinite two-player zero-sum game with a Borel winning set, in which the opponent's actions are monitored eventually but not necessarily immediately after they are played, admits a value. The proof relies on a representation of the game as a stochastic game with perfect information, in which Nature operates as a delegate for the players and performs the randomizations for them.

The analysis of lab data entails a joint test of the underlying theory and of subjects' conjectures regarding the experimental design itself, how subjects frame the experiment. We provide a theoretical framework for analyzing the impacts of such conjectures or frames. We use experiments of decision making under uncertainty as a case study. Absent restrictions on subjects' framing of the experiment, we show that any behavior is consistent with standard updating ('anything goes'), including that suggestive of anomalies such as over-confidence, excess belief stickiness, etc. When the experimental protocol restricts subjects' conjectures (plausibly, by generating information during the experiment), standard updating has non-trivial testable implications. Such ``transparent'' protocols restrict action reversals that Bayesian subjects exhibit when they are provided with additional information. In the extreme case in which the amount of information revealed is conjectured to be independent of the underlying realized uncertainty, Bayesian updating is tantamount to dynamic consistency.

A decision maker faces a new theory of how certain events unfold over time. The theory matters for choices she needs to make, but possibly the theory is a fabrication. We show that there is a test which is guaranteed to pass a true theory, and which is also conservative: A false theory will only pass when adopting it over the decision maker’s initial theory would not cause substantial harm; if the agent is fooled she will not be harmed.

We also study a society of conservative decision makers with different initial theories. We uncover pathological instances of our test: a society collectively rejects most theories, be they true or false. But we also find well-behaved instances of our test, collectively accepting true theories and rejecting false. Our tests builds on tests studied in the literature in the context of non-strategic inspectors.