Generalized Arrow-Pratt Theory

Abstract

Given a utility defined on a Hilbert outcome space, we define at each outcome a generalized Arrow-Pratt (GAP) coefficient belonging to the Hilbert space. Comparing the risk aversion of such utilities using their GAP coefficients is equivalent to doing so in terms of other standard, decision-theoretically persuasive, criteria. The resulting GAP theory of risk aversion significantly expands the scope of the classical Arrow-Pratt theory by admitting a large class of risks with vector outcomes. This allows the theory to address risks that are embodied in a significant class of random processes. The GAP theory's implications are studied in five contexts with Hilbert outcome spaces, namely, the theories of portfolio choice, insurance, asset valuation, auctions and moral hazard in teams. We also show a duality between utility functions on Euclidean spaces and GAP coefficients. Finally, we provide a theoretically well-founded and computationally tractable method for estimating the expected GAP coefficient from empirical data.