MichaelStJules comments on Impossibility results for unbounded utilities

MichaelStJules 2 Feb 2022 20:14 UTC
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From section 3 of Tarsney’s paper:
Option O first-order stochastically dominates option P if and only if
1. For any payoff x, the probability that O yields a payoff at least as good as x is equal to or greater than the probability that P yields a payoff at least as good as x, and
2. For some payoff x, the probability that O yields a payoff at least as good as x is strictly greater than the probability that P yields a payoff at least as good as x.
(...)
Stochastic dominance is a generalization of the familiar statewise dominance relation that holds between O and P whenever O yields at least as good a payoff as P in every possible state, and a strictly better payoff in some state. To illustrate: Suppose that I am going to flip a fair coin, and I offer you a choice of two tickets. The Heads ticket will pay $1 for heads and nothing for tails, while the Tails ticket will pay $2 for tails and nothing for heads. The Tails ticket does not statewise dominate the Heads ticket because, if the coin lands Heads, the Heads ticket yields a better payoff. But the Tails ticket does stochastically dominate the Heads ticket. There are three possible payoffs: winning $0, winning $1, and winning $2. The two tickets offer the same probability of a payoff at least as good as $0, namely 1. And they offer the same probability of an payoff at least as good as $1, namely 0.5. But the Tails ticket offers a greater probability of a payoff at least as good as $2, namely 0.5 rather than 0. Stochastic dominance is generally seen as giving a necessary condition for rational choice:
Stochastic Dominance Requirement (SDR) An option O is rationally permissible in situation S only if it is not stochastically dominated by any other option in S.
This principle is on a strong a priori footing. Various formal arguments can be made in its favor. For instance, if O stochastically dominates P, then O can be made to statewise dominate P by an appropriate permutation of equiprobable states in a sufficiently finegrained partition of the state space (Easwaran, 2014; Bader, 2018).