Lotteries and Menus

A Comment on Unbounded Utilities

by Lloyd S. Shapley

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A sequel to P-4940, the author argues that the issue of whether utilities can be unbounded is independent of the considerations surrounding the St. Petersburg paradox. On the one hand, the Bernoulli lottery can be modified so that the utilities are bounded without losing whatever power the classical paradox may have to make a risk-linear utility for money look absurd. On the other hand, the logical challenge that the Bernoulli game raises against unbounded utilities (as distinct from unbelievably large, but mathematically bounded utilities) can be raised just as well in a more basic, non-lottery setting where ordinally ranked outcomes are selected from "menus." An illustrative game called "Blank Check" plays a central role in the discussion.

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