Decision Theory III - Optimality Criteria

We briefly introduce and discuss some criteria for optimal decisions. For the majority of this module, we consider a person’s utility for different options (or ‘rewards’ or ‘actions’) and, with uncertainty about the reward which will follow a specific decision, we aim at maximisation of the expected utility, which is actually the utility of the optimal decision (note that a decision is a gamble if the outcome is uncertain due to uncertainty about the state). However, the next topic will be Nash’s theory for two-person zero-sum games, where a minimax (or ‘maximin’) criterion is used, which is quite pessimistic: it assumes a very clever opponent who can enforce the worst pay-off to you once you have made your decision, so not the expected pay-off but the minimum pay-off corresponding to each decision will be maximised. In the literature, a range of optimality criteria for decision making have been proposed, where maximisation of expected utility and the minimax criterion are probably the most commonly used. There is a very substantial literature on such criteria, e.g. Simon French in his excellent 1988 book Decision Theory: An Introduction to the Mathematics of Rationality discusses a range of criteria in detail with an investigation into whether or not they satisfy a range of axioms. Martin Peterson also discusses criteria in a nice manner in his 2009 book An Introduction to Decision Theory (this is nicely priced and from next year a recommended text for this module). No criterion satisfies all axioms listed by French, which just indicates that there is not a single criterion that is widely accepted as ‘best’. For our module, we just briefly mention some criteria to raise awareness of this important topic, and we illustrate them with an example, mainly to show that they can lead to quite different decisions. Of course, they also can lead to the same decision, e.g. if one alternative dominates all other ones then the criteria discussed here (and probably any sensible criterion..) would all select the dominating strategy. To illustrate the different criteria, we will use the decision problem with pay-off table given below. You have the choice between four actions (a1, . . . , a4) and there are four unknown states (θ1, . . . , θ4). The pay-offs given can be considered to be utilities, with ‘more’ being better for you. Assume further that uncertainty of the real state is reflected by each being assigned probability 1/4, this is only used in the first criterion below.