Essays on Indices and Matching

In many decision problems, agents base their actions on a simple objective index, a single number that summarizes the available information about objects of choice independently of their particular preferences. The first chapter proposes an axiomatic approach for deriving an index which is objective and, nevertheless, can serve as a guide for decision making for decision makers with different preferences. Unique indices are derived for five decision making settings: the Aumann and Serrano (2008) index of riskiness (additive gambles), a novel generalized Sharpe ratio (for a standard portfolio allocation problem), Schreiber’s (2013) index of relative riskiness (multiplicative gambles), a novel index of delay embedded in investment cashflows (for a standard capital budgeting problem), and the index of appeal of information transactions (Cabrales et al., 2014). All indices share several attractive properties in addition to satisfying the axioms. The approach may be applicable in other settings in which indices are needed. The second chapter uses conditions from previous literature on complete orders to generate partial orders in two settings: information acquisition and segregation. In the setting of information acquisition, I show that the partialorder prior independent investment dominance (Cabrales et al., 2013) refines Blackwell’s partial order in the strict sense. In the segregation setting, I show that without the requirement of completeness, all of the axioms suggested in Frankel and Volij (2011) are satisfied simultaneously by a partial order which refines the standard partial order (Lasso de la Vega and Volij, 2014). In the third and fourth chapters, I turn to examine matching markets. Although no stable matching mechanism can induce truth-telling as a dominant strategy for all participants (Roth, 1982), recent studies have presented conditions under which truthful reporting by all agents is close to optimal (Immorlica and Mahdian, 2005; Kojima and Pathak, 2009; Lee, 2011). The third chapter demonstrates that in large, balanced, uniform markets using the Men-Proposing Deferred Acceptance Algorithm, each woman’s best response to truthful behavior by all other agents is to truncate her list substantially. In fact, the optimal degree of truncation for such a woman goes to 100% of her list as the market size grows large. Comparative statics for optimal truncation strategies in general one-to-one markets are also provide: reduction in risk aversion and reduced correlation across preferences each lead agents to truncate more. So while several recent papers focused on the limits of strategic manipulation, the results serve as a reminder that without preconditions ensuring truthful reporting, there exists a potential for significant manipulation even in settings where agents have little information. Recent findings of Ashlagi et al. (2013) demonstrate that in unbalanced random markets, the change in expected payoffs is small when one reverses which side of the market “proposes,” suggesting there is little potential gain from manipulation. Inspired by these findings, the fourth chapter studies the implications of imbalance on strategic behavior in the incomplete information setting. I show that the “long” side has significantly reduced incentives for manipulation in this setting, but that the same doesn’t always apply to the “short” side. I also show that risk aversion and correlation in preferences affect the extent of optimal manipulation as in the balanced case.Business Economic