Thinking Outside the Euclidean Box: Riemannian Geometry and Inter-Temporal Decision-Making

PLOS ONE, Forthcoming

52 Pages Posted: 7 Dec 2015

See all articles by Himanshu Mishra

Himanshu Mishra

University of Utah - David Eccles School of Business

Arul Mishra

University of Utah - David Eccles School of Business

Date Written: December 6, 2015

Abstract

Inter-temporal decisions involves assigning values to various payoffs occurring at different temporal distances. Past research has used different approaches to study these decisions made by humans and animals. For instance, considering that people discount future payoffs at a constant rate (e.g., exponential discounting) or at variable rate (e.g., hyperbolic discounting). In this research, we question the widely assumed, but seldom questioned, notion across many of the existing approaches that the decision space, where the decision-maker perceives time and payoffs, is a Euclidean space. By relaxing the rigid assumption of Euclidean space, we propose that the decision space is a more flexible Riemannian space of Constant Negative Curvature. We test our proposal by deriving a discount function, which uses the distance in the Negative Curvature space instead of Euclidean temporal distance. The distance function includes both perceived values of time as well as money, unlike past work which has considered just time. By doing so we are able to explain many of the empirical findings in inter-temporal decision-making literature. We provide converging evidence for our proposal by estimating the curvature of the decision space utilizing manifold learning algorithm and showing that the characteristics (i.e., metric properties) of the decision space resembles those of the Negative Curvature space rather than the Euclidean space. We conclude by presenting new theoretical predictions derived from our proposal and implications for how non-normative behavior is defined.

Keywords: Riemannian Geometry; Inter-temporal Decisions; Discounted Utility; Gaussian Curvature; Manifold learning; Non-Euclidean Geometries; Exponential Discounting; Hyperbolic Discounting

JEL Classification: D90, D91, M31

Suggested Citation

Mishra, Himanshu and Mishra, Arul, Thinking Outside the Euclidean Box: Riemannian Geometry and Inter-Temporal Decision-Making (December 6, 2015). PLOS ONE, Forthcoming, Available at SSRN: https://ssrn.com/abstract=2699851

Himanshu Mishra (Contact Author)

University of Utah - David Eccles School of Business ( email )

1645 E. Campus Center Drive
Salt Lake City, UT 84112-9304
United States

HOME PAGE: http://himanshumishra.com

Arul Mishra

University of Utah - David Eccles School of Business ( email )

1645 E. Campus Center
Salt Lake City, UT 84112
United States

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