A trichotomy of attitudes for decision-making under complete ignorance

Abstract

We study decision criteria under complete ignorance, that is, when there is no available information regarding plausible probability distributions over the possible outcomes. We characterize the set of criteria satisfying quasi-transitivity, Savage’s independence, duplication, a strong version of dominance and scale invariance. Only three criteria satisfy these requirements. These criteria are the well-known protective criterion, its dual criterion which we call hazardous, and a neutral criterion which is the composition of both (a decision is strictly preferred according to this criterion to another one if both the protective and the hazardous criteria strictly prefer the former).

Introduction

A situation of uncertainty is said to be of complete ignorance when no information regarding the probabilities of the possible states is available. Understanding rational decision-making under uncertainty, and therefore, under complete ignorance as well, is an old objective in economic theory.

In the branch of the literature initiated by Savage (1954), rationality is assumed to imply that the decision criterion is transitive and satisfies Independence (his sure-thing principle). In another branch of the literature, the very essence of rationality under complete ignorance is captured by a Duplication property (see Luce and Raiffa, 1957 for an introduction to this classical literature). A decision criterion satisfies Duplication if the preference between two possible decisions is not affected by the replication of a state of nature, that is, by the appearance of a new state of nature which associates with each decision exactly the same utility level as an already known state. If, for instance, an expert provides a new scenario for the future, but the outcomes associated to each decision under this scenario are indistinguishable from the outcomes associated with the decisions under the scenario previously provided by another expert, then the new scenario should be considered uninformative and the preferences towards decisions should not change.

Early contributions in this line have shown that the criteria which end up being justified by Duplication have the following features: first, they are limited to using the information about the worst and the best outcomes associated with the decisions, and, second, they exhibit intransitivities (in the indifference relation) as soon as they also satisfy the indisputable requirement of Dominance (one decision dominates another if it always leads to a weakly larger and sometimes to a strictly larger utility level, and a decision criterion satisfies Dominance if it strictly prefers the dominating to the dominated decision) (Arrow and Hurwicz, 1972, Cohen and Jaffray, 1980).

Therefore, a crucial question of this classical literature has become to try to identify a good compromise between focussing only on the worst outcomes, which is often considered extremely pessimistic, and focussing only on the best outcomes, which is considered being extremely optimistic. A well-known early suggestion made by Hurwicz is to combine linearly the worst and the best outcomes (see Milnor, 1954 for an axiomatization of the so-called Hurwicz criterion).

Two subsequent contributions, however, have challenged the view that a compromise between the two extreme attitudes is possible. Cohen and Jaffray (1983) and Maskin (1979), indeed, prove that rationality requirements (together with the (strong) invariance requirement of ordinalism in Maskin) lead to variants of either the maximin or the maximax criteria, when rationality includes the requirement that the preference relation is transitive or can be approximated by a transitive relation. A similar conclusion is reached by Bossert (1997), even if comparisons between decisions are based only on the sets of outcomes, without reference to underlying states of nature.

Those results, however, are not fully satisfactory as, by disregarding the fundamental intransitivities resulting from the Duplication requirement, they fail to consider the entire spectrum of rational attitudes that may arise under complete ignorance. Following Cohen and Jaffray (1983), it has been remarked by Toulet (1986) that non-transitive but continuous approximations can also exhibit zones of local ‘indecisiveness’ but she does not characterize indecisiveness in terms of decision criteria.

In this paper, we prove that, indeed, no compromise between maximin and maximax types of criteria is possible, even if intransitivities are allowed. More precisely, we show that the combination of Quasi-transitivity (requiring transitivity of the strict preference relation only), the most demanding rationality requirements under complete ignorance (that is, Independence, Duplication and a strong version of Dominance) and the invariance requirement of Linearity (or Scale Invariance, requiring that a linear transformation of the utilities does not affect the preference) also used by Milnor (1954) characterizes three and only three criteria.

A common feature of the three criteria we obtain is to restrict comparisons between two decisions to states in which they lead to different outcomes. The first one is the protective criterion of Barberà and Jackson (1988). It consists of applying the maximin criterion under the above mentioned restriction. The second one, which we call the hazardous criterion, corresponds to the opposite attitude and resorts to the maximax criterion. The third one is the intersection of the two previous ones, as it only declares a decision strictly better than another one when both the protective and the hazardous criteria coincide with that recommendation.

The lesson to be drawn from our main result is that three and only three attitudes towards ignorance are rational, when rationality is interpreted as the combination of all the most demanding properties of the literature under the requirement that the strict preference relation be transitive. Unsurprisingly, two of these attitudes are extreme pessimism and extreme optimism. The new possibility, therefore, comes from the third criterion, which has never been studied before. We argue, however, that this criterion cannot be viewed as a compromise between the two extreme attitudes towards ignorance, as, instead of solving conflicting recommendations between them, it remains indecisive as soon as pessimism and optimism conflict with each other.

Another lesson is therefore that weakening the transitivity requirement is not the road to take if one wishes to avoid the dilemma between pessimism and optimism. A similar lesson is drawn by Barrett and Pattanaik (1994), using a different set of axioms, in the alternative framework where decisions are described in terms of sets of outcomes without reference to states of nature.

The paper is organized as follows. In Section 2, we introduce the model. In Section 3, we define the rationality and invariance axioms as well as our three decision criteria, and we prove our main result. In Section 4, we introduce three axioms of attitude towards ignorance and we show how to state our trichotomy in terms of these axioms. In Section 5, we provide the proofs. In Section 6, we conclude.