Abstract
We introduce a monotone class theory of Prospect Theory's value functions, which shows that they can be replaced almost surely by a topological lifting comprised of a class of compact isomorphic maps that embed weakly co-monotonic probability measures, attached to state space, in outcome space. Thus, agents solve a signal extraction problem to obtain estimates of empirical probability weights for prospects under risk and uncertainty. By virtue of the topological lifting, we prove an almost sure isomorphism theorem between compact stochastic choice operators, and well defined outcomes which, under Brouwer-Schauder theory, guarantees fixed point convergence in convex choice sets. Along the way we introduce a risk operator in the Hoffman-Jorgensen class of lifting operators, and value function [averaging] operators with respect to Radon measure. In that set up, suitable binary operations on gain-loss space show that our risk operator is isometric for gains and skewed for losses. The point spectrum from this operator constitutes the range of admissible observations for loss aversion index in a well designed experiment.
Highlights
At issue is the following excerpt from (Tversky and Khaneman, 1992, pg. 300): Let S be a finite set of states of nature; subsets of S are called events
Why do prospect theory’s agents map from state space to outcome space, and map a value function from outcome space to the reals? Why can’t they map directly from state space to the reals? In other words, are value functions irrelevant? (Luce and Narens, 2008, pg. 1) characterized problems of this type thusly: Most mathematical sciences rest upon quantitative models, and the theory of measurement is devoted to making explicit the qualitative assumptions that underlie them
We make slight modifications to a monotone class theorem in (Blumenthal and Geetor, 1968, pg. 7, Prop. 2.7) which, in the context of our model, essentially states that our direct map is measureable with respect to KT92 choice functions iff there is a monotone sequence of functions on outcome space X, i.e. strictly increasing value functions
Summary
At issue is the following excerpt from (Tversky and Khaneman, 1992, pg. 300): Let S be a finite set of states of nature; subsets of S are called events. This paper comprises topological analysis of prospect theory’s function space under risk and uncertainty It does not address axiomatic foundations of stochastic choice. Our procedure exploits the equivalence class sistencies and the best known theories for decision making, for example, those of von Neumann and Morgenstern [12] or Savage [15], base the existence of a measurable utility upon a pattern of invariant two-place relations, sometimes called ’preference’ and ’indifference’ 2.7) which, in the context of our model, essentially states that our direct map is measureable with respect to KT92 choice functions iff there is a monotone sequence of functions on outcome space X, i.e. strictly increasing value functions.
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