Stochastic Dominance and Opaque Sweetening

ABSTRACT

This paper addresses the problem of opaque sweetening and argues that one should use stochastic dominance in comparing lotteries even when dealing with incomplete orderings that allow for non-comparable outcomes.

KEYWORDS:

• decision theory
• incompleteness
• stochastic dominance
• state-by-state dominance
• parity

Notes

¹ When comparability is understood in terms of determinate betterness, non-comparability can also be due to indeterminate betterness of the type discussed by Broome [1997].

² Local forms of non-comparability (that are best thought of in terms of intersection quasi-orderings that integrate different dimensions of evaluation) only allow for insensitivity to mild sweetening. By contrast, global forms of non-comparability (according to which comparability is an equivalence relation) allow for stronger forms of insensitivity. The arguments in this paper apply equally to local and to global forms of non-comparability.

³ Examples of supposedly non-comparable goods that are often cited include different careers as well as works of art. So one might, for instance, consider the contents of the boxes as being two paintings: A = Monet's ‘La Promenade’ and B = Manet's ‘Le Printemps’.

⁴ In order to avoid a non-comparability verdict in those cases, one will have to invoke different resources. For instance, one can apply dominance reasoning to the probabilistically discounted value vector.

as long as the difference in value between and A is sufficiently large to compensate for the slightly lower probability of ending up with the former rather than the latter. This is the case if . will then be at least as good as with respect to , and will be strictly better with respect to . Non-comparability then only arises in cases in which the difference in probability is too great relative to the evaluative difference between and A.

⁵ Whereas Jeffrey-Bolker theory is an evidential decision theory, the accounts considered below are non-standard versions of causal decision theory.

⁶ As they point out, this is the analogue of applying the distinction between optimising and maximising to cases of risk, where an optimising choice set presupposes completeness whilst a maximising choice set is compatible with incompleteness (cf. Sen [1997]).

⁷ Bales, Cohen, and Handfield [2014: 460] have advocated a very similar constraint: competitiveness. Hare [2010: 244] identifies, though ultimately does not endorse, a closely related principle: recognition.

⁸ As Gibbard and Harper [1978] have pointed out, the probabilistic independence condition can be interpreted in two ways, depending on whether one is using causal or evidential probabilities. Whilst the causal and evidential independence conditions can come apart in Newcomb-style cases, they coincide in the kinds of cases that are at issue in this paper.

⁹ Once it is recognized that state-by-state reasoning is not appropriate when probabilistic independence is not satisfied, we can see that sameness of state of nature by itself is not of significance. The question then arises as to what makes it the case that such reasoning is appropriate when probabilistic independence is satisfied. Is it the fact that we are dealing with the same state of nature together with probabilistic independence (which, as we have just seen, amounts to an equiprobability condition), or is it rather equiprobability by itself that is doing all of the work? The considerations about independent lotteries and about the evaluative irrelevance of the identity of states of nature adduced above suggest that it is equiprobability by itself.

¹⁰ For helpful discussions, I would like to thank Bassel Tarbush. I am particularly grateful to two anonymous referees for their detailed and helpful comments, which have led to substantial improvements.