Smooth sets for borel equivalence relations and the covering property for σ-ideals of compact sets

This thesis is divided into three chapters. In the first chapter we study the smooth sets with respect to a Borel equivalence realtion E on a Polish space X. The collection of smooth sets forms σ-ideal. We think of smooth sets as analogs of countable sets and we show that an analog of the perfect set theorem for Σ¹₁ sets holds in the context of smooth sets. We also show that the collection of Σ¹₁ smooth sets is ∏¹₁ on the codes. The analogs of thin sets are called sparse sets. We prove that there is a largest ∏¹₁ sparse set and we give a characterization of it. We show that in L there is a ∏¹₁ sparse set which is not smooth. These results are analogs of the results known for the ideal of countable sets, but it remains open to determine if large cardinal axioms imply that ∏¹₁ sparse sets are smooth. Some more specific results are proved for the case of a countable Borel equivalence relation. We also study I(E), the σ-ideal of closed E-smooth sets. Among other things we prove that E is smooth iff I(E) is Borel.

In chapter 2 we study σ-ideals of compact sets. We are interested in the relationship between some descriptive set theoretic properties like thinness, strong calibration and the covering property. We also study products of σ-ideals from the same point of view. In chapter 3 we show that if a σ-ideal I has the covering property (which is an abstract version of the perfect set theorem for Σ¹₁ sets), then there is a largest ∏¹₁ set in I^int (i.e., every closed subset of it is in I). For σ-ideals on 2^ω we present a characterization of this set in a similar way as for C₁, the largest thin ∏¹₁ set. As a corollary we get that if there are only countable many reals in L, then the covering property holds for Σ¹₂ sets.

Essays in revealed preference theory and behavioral economics

Time, risk, and attention are all integral to economic decision making. The aim of this work is to understand those key components of decision making using a variety of approaches: providing axiomatic characterizations to investigate time discounting, generating measures of visual attention to infer consumers' intentions, and examining data from unique field settings.

Chapter 2, co-authored with Federico Echenique and Kota Saito, presents the first revealed-preference characterizations of exponentially-discounted utility model and its generalizations. My characterizations provide non-parametric revealed-preference tests. I apply the tests to data from a recent experiment, and find that the axiomatization delivers new insights on a dataset that had been analyzed by traditional parametric methods.

Chapter 3, co-authored with Min Jeong Kang and Colin Camerer, investigates whether "pre-choice" measures of visual attention improve in prediction of consumers' purchase intentions. We measure participants' visual attention using eyetracking or mousetracking while they make hypothetical as well as real purchase decisions. I find that different patterns of visual attention are associated with hypothetical and real decisions. I then demonstrate that including information on visual attention improves prediction of purchase decisions when attention is measured with mousetracking.

Chapter 4 investigates individuals' attitudes towards risk in a high-stakes environment using data from a TV game show, Jeopardy!. I first quantify players' subjective beliefs about answering questions correctly. Using those beliefs in estimation, I find that the representative player is risk averse. I then find that trailing players tend to wager more than "folk" strategies that are known among the community of contestants and fans, and this tendency is related to their confidence. I also find gender differences: male players take more risk than female players, and even more so when they are competing against two other male players.

Chapter 5, co-authored with Colin Camerer, investigates the dynamics of the favorite-longshot bias (FLB) using data on horse race betting from an online exchange that allows bettors to trade "in-play." I find that probabilistic forecasts implied by market prices before start of the races are well-calibrated, but the degree of FLB increases significantly as the events approach toward the end.