# The Family of Alpha,[a,b] Stochastic Orders: Risk vs. Expected Value

## TL;DR

This paper introduces a new family of stochastic orders, called $oldsymbol{ ext{α,[a,b]}}$-concave stochastic orders, which generalize second order stochastic dominance and enable novel economic comparative statics analysis.

## Contribution

The paper develops a new class of stochastic orders based on a novel set of very concave functions, allowing for advanced economic comparisons involving risk and expected value.

## Key findings

- Derived new comparative statics results for economics applications.
- Provided tools to analyze situations where riskiness and expected value have opposite effects.
- Applied the stochastic orders to problems in consumption-savings, self-protection, and Bayesian games.

## Abstract

In this paper we provide a novel family of stochastic orders that generalizes second order stochastic dominance, which we call the $\alpha,[a,b]$-concave stochastic orders. These stochastic orders are generated by a novel set of "very" concave functions where $\alpha$ parameterizes the degree of concavity. The $\alpha,[a,b]$-concave stochastic orders allow us to derive novel comparative statics results for important applications in economics that cannot be derived using previous stochastic orders. In particular, our comparative statics results are useful when an increase in a lottery's riskiness changes the agent's optimal action in the opposite direction to an increase in the lottery's expected value. For this kind of situation, we provide a tool to determine which of these two forces dominates -- riskiness or expected value. We apply our results in consumption-savings problems, self-protection problems, and in a Bayesian game.