The utility of being hanged on the gallows

Abstract

1. I thank the editors for permitting me to reply to criticisms of my views (Blatt 1979-80 and 1980). Ulph (1981-82) shows, in his Figure 1, that a von Neumann individual may accept the gamble for small probability p of disaster and reject it for large p; yet this man has an unbounded utility of money. However, Ulph's man exhibits some highly peculiar preferences. Suppose the money gain M becomes very large. In his Figure 1 the point labeled V(Wo + M, 0) moves ever higher on the vertical axis, hence the critical probability p* (below which the gamble is accepted) moves ever closer to unity on the horizontal axis. Thus, by merely promising him enough money M upon success, Ulph's man can be induced to accept the gamble, no matter how poor his chances of escaping the gallows! While some criminals may be so utterly foolhardy, not all rational people act that way; this is all that is needed to refute the objection. 2. Let me make this argument more formal. Define a "greedy but cautious criminal," abbreviated GCC henceforth, by the three properties: (1) His utility of money u(M) is unbounded (Blatt, 1980); (2) There exists a maximum probability Pma Pmax, no matter how big the money sum M; (3) He accepts the gamble for small enough, but nonzero, p. THEOREM: The preference scale of a GCC is inconsistent with expected utility theory. PROOF: Expected utility is E(U) = (1 p) u(M) + pu(G). Let