5,185 research outputs found
On the strength of the finite intersection principle
We study the logical content of several maximality principles related to the
finite intersection principle (F\IP) in set theory. Classically, these are
all equivalent to the axiom of choice, but in the context of reverse
mathematics their strengths vary: some are equivalent to \ACA over \RCA,
while others are strictly weaker, and incomparable with \WKL. We show that
there is a computable instance of F\IP all of whose solutions have
hyperimmune degree, and that every computable instance has a solution in every
nonzero c.e.\ degree. In terms of other weak principles previously studied in
the literature, the former result translates to F\IP implying the omitting
partial types principle (). We also show that, modulo
induction, F\IP lies strictly below the atomic model theorem
().Comment: This paper corresponds to section 3 of arXiv:1009.3242, "Reverse
mathematics and equivalents of the axiom of choice", which has been
abbreviated and divided into two pieces for publicatio
Reverse mathematics and equivalents of the axiom of choice
We study the reverse mathematics of countable analogues of several maximality
principles that are equivalent to the axiom of choice in set theory. Among
these are the principle asserting that every family of sets has a
-maximal subfamily with the finite intersection property and the
principle asserting that if is a property of finite character then every
set has a -maximal subset of which holds. We show that these
principles and their variations have a wide range of strengths in the context
of second-order arithmetic, from being equivalent to to being
weaker than and incomparable with . In
particular, we identify a choice principle that, modulo induction,
lies strictly below the atomic model theorem principle and
implies the omitting partial types principle
A Logic for Reasoning about Group Norms
We present a number of modal logics to reason about group norms. As a preliminary
step, we discuss the ontological status of the group to which the norms are
applied, by adapting the classification made by Christian List of collective attitudes
into aggregated, common, and corporate attitudes. Accordingly, we shall introduce
modality to capture aggregated, common, and corporate group norms. We investigate
then the principles for reasoning about those types of modalities. Finally, we discuss
the relationship between group norms and types of collective responsibility
Stronger Utility
Empirical research often requires a method how to convert a deterministic economic theory into an econometric model. A popular method is to add a random error term on the utility scale. This method, however, violates stochastic dominance. A modification of this method is proposed to avoid violations of dominance. The modified model compares favorably to other existing models in terms of goodness of fit to experimental data. The modified model can rationalize the preference reversal phenomenon. An intuitive axiomatic characterization of the modified model is provided. Important microeconomic concept of risk aversion is well-defined in the modified model.Decision Theory, Probabilistic Choice, Stochastic Dominance, Strong Utility, Risk Aversion
History-Dependent Risk Attitude
We propose a model of history-dependent risk attitude (HDRA), allowing the attitude of a decision-maker (DM) towards risk at each stage of a T-stage lottery to evolve as a function of his history of disappointments and elations in prior stages. We establish an equivalence between the existence of an HDRA representation and two documented cognitive biases. First, the DM’s risk attitudes are reinforced by prior experiences: he becomes more risk averse after suffering a disappointment and less risk averse after being elated. Second, the DM displays a primacy effect: early outcomes have the strongest effect on risk attitude. Furthermore, the DM lowers his threshold for elation after a disappointing outcome and raises it after an elating outcome; this makes disappointment more likely after elation and vice-versa, leading to statistically reversing risk attitudes. “Gray areas” in the elation-disappointment assignment are connected to optimism and pessimism in determining endogenous reference points.history-dependent risk attitude, statistically reversing risk attitudes, reinforcement effect, primacy effect, endogenous reference dependence, betweenness, optimism, pessimism
Generalized Disappointment Aversion and Asset Prices
We provide an axiomatic model of preferences over atemporal risks that generalizes Gul (1991) A Theory of Disappointment Aversion' by allowing risk aversion to be first order' at locations in the state space that do not correspond to certainty. Since the lotteries being valued by an agent in an asset-pricing context are not typically local to certainty, our generalization, when embedded in a dynamic recursive utility model, has important quantitative implications for financial markets. We show that the state-price process, or asset-pricing kernel, in a Lucas-tree economy in which the representative agent has generalized disappointment aversion preferences is consistent with the pricing kernel that resolves the equity-premium puzzle. We also demonstrate that a small amount of conditional heteroskedasticity in the endowment-growth process is necessary to generate these favorable results. In addition, we show that risk aversion in our model can be both state-dependent and counter-cyclical, which empirical research has demonstrated is necessary for explaining observed asset-pricing behavior.
A subjective spin on roulette wheels.
We provide a behavioral foundation to the notion of ‘mixture’ of acts, which is used to great advantage in he decision setting introduced by Anscombe and Aumann. Our construction allows one to formulate mixture-space axioms even in a fully sub-jective setting, without assuming the existence of randomizing devices. This simplifies the task of developing axiomatic models which only use behavioral data. Moreover, it is immune from the difficulty that agents may ‘distort’ the probabilities associated with randomizing devices. For illustration, we present simple subjective axiomatizations of some models of choice under uncertainty, including the maxmin expected utility model of Gilboa and Schmeidler, and Bewley’s model of choice with incomplete preferences.
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