# Smith’s Bayesian Decision Analysis

Jim Q. Smith Bayesian Decision Analysis: Principles and Practice 2000

Jim is a thoughtful and well-regarded academic advisor to Decision-Makers (DMs) faced with uncertainty. My own interest in this book is in how it helps us to discern when his approach is appropriate.

## Axioms

Jim’s ‘Probabilistic Determinism Axiom’ (pg 56) is that two gambles with the same associated reward distributions are necessarily considered as equivalent. He notes:

“Happily in my experience the times when an aided DM is unhappy to follow the rule of Probabilistic Determinism have been rare.”

This leads to the ‘weak substitution axiom’ (pg 61):

“For example suppose the DM states she is indifferent to having an orange or an apple. Then … the DM can conclude that if she will be indifferent between the option of receiving an apple with probability β and otherwise receiving a banana and the option of receiving an orange with probability β and otherwise a banana. My own view is that this is the most compelling of rules of rational choice. Sadly it is the rule most commonly violated by unaided DMs, even ones who are otherwise reasonable and well-trained, especially when combined with Probabilistic Determinism.”

Jim specifically discusses the Allais paradox, in which ‘most people’ make choices that conflict with Jim’s axioms, and hence which Jim regards as ‘irrational’.

## Justification

Jim justifies his axioms based on a notion of a DM with totally ordered preferences over decision rules to be enacted by an agent (pg 55):

“We ask the DM to imagine that she is to give instructions to an agent to enact her preferences in a hypothetical market place. Whenever she states that she strictly prefers d[2] to d[1] she is stating that she is instructing this agent to exchange the option of enacting d[1] for the option on d[2] if the trade becomes available. If two decisions are equally preferable then she is equally content for her agent to substitute one for the other or to retain the current one.”

“The DM is responsible for explaining the rationale behind her choice of decision in a compelling way to an auditor … an external regulator, a line manager or strategy team, a stakeholder, the DM herself or some combination of these characters. …”

That is, the DM must be sure of her preferences over all possibilities in advance, can only act via an impoverished specification of preferences and must be able to justify her actions to others. Jim gives some insightful remarks on this issue.

## Scope

The book assumes (pg 5) that:

“[T]he decision model is likely to have a limited shelf life and is intrinsically provisional. The DM simply strives to present an honest representation of her problem as she sees it at the current time. All accept that in the future her [judgments] may change [later].”

In this sense, the model is aimed at solving the smaller problem ‘as presented’, not some larger, unfolding, one.

Jim’s book is written for those, like him, who find his axioms reasonable and hence Bayesian Decision Analysis appropriate. But, being well-reasoned, the book can also be used by those – like me – who find (strong) Bayesianism problematic, to help identify those seductive ‘common-sense’ assumptions that may need to be challenged if we are to make progress.

### Axioms

The weak substitution axiom makes many ‘unaided’ DMs seem irrational. It is surely worrying that these DMs have intuitions that conflict with fundamental assumptions that their advisers find compelling. But suppose that:

1. We are considering what holiday to take next January.
2. We are indifferent between Madeira (sun) and the Alps (skiing) but need to know which it is to be so that we can prepare (e.g. by trimming flab or strengthening knees).
3. We now decide on the Canaries, even though there is a 10% probability that a volcano will prevent us.
4. The organizers offer us the option of diverting to Madeira or the Alps if we cannot get to the Canaries, but we have to choose which it would be now.

My reading of the weak substitution axiom is that we should be indifferent between Madeira and the Alps as a fall-back. And yet if we were ready for the Canaries there would be no problem switching to Madeira, whereas a visit to the Alps might be frustrating if we weren’t prepared for it. Hence I would have a definite preference for Madeira, even if it violates the axiom. If (as Jim does) we consider the rewards to be just numbers then this example also contradicts the Probabilistic Determinism Axiom. But if it seems more natural to take account of the conditionality of rewards, in which case the axiom is satisfied. Although we ‘value’ the Canaries and the Alps the same, they have different conditionalities. The weak substitution axiom could also be saved if we only take two reward distributions to be indifferent if they are indifferent under all conditions. But if the rewards are conditional, so are the ‘probabilities’ (as they are for Keynes).

Thus, while the weak substitution axiom is appealing, upon reflection it only seems to rule out some complexities and uncertainties. Given some of the critical decisions that Jim advises on one would hope that it is the case for his DMs, but it is perhaps not something to take for granted. This may explain the views of some of the DMs who otherwise appear irrational.

We can adapt Jim’s version of Allais as follows. I prefer the Canaries to Madeira, but there is a 20% chance that a holiday to the Canaries will be cancelled due to a volcano (not real figures). This just makes a certain holiday in Madeira preferable. But, suppose that we now learn that there is a 75% chance of an air-traffic controllers strike. As in Jim’s example it seems reasonable to me to now prefer the Canaries: it is up to me. But according to Jim this change is irrational. The solution to the apparent paradox may be that the axiom is too strong for one-off decisions.

### Justification

Jim supposes that the DM:

• must be sure of her preferences over all possibilities in advance
• can only act via an impoverished specification of preferences, granting no discretion to her agents
• must be able to justify her actions to others, and has no meaningful discretion herself.

This would seem reasonable if the DM had a small world problem, but Jim’s problems do not seem entirely small.

The restriction to enacting decision rules is possibly reasonable for Jim’s applications, but it leaves no room for discretion on the part of the agent, other than where it doesn’t matter. When deciding on a holiday I do not expect to have to consider all possible factors and give clear preferences: I expect my agent to show some initiative and judgment: Some of my best experiences on holiday have been things that I never would have thought were possible.

Where, as in the holiday examples above, I do have explicit preferences, they tend to be conditional and so I need a richer specification language, supporting conditionality. One could use a conditional version of Jim’s theory, perhaps yielding conditional or ‘imprecise’ decisions, allowing for some discretion or engagement. I rarely view holiday decisions as one-offs, and never seek the optimal holiday. Instead, I seek a near-optimal holiday strategy. In an organizational setting, suppose that the DM trades off short and long-run, and expected and reported. She:

• wants to maximize gain over a long term, subject to taking only reasonable risks.
• wants to report progress to a boss as soon as possible
• does not want to have to report negative progress.

It would seem natural for the DM to express these preferences and for the agent to develop a corresponding strategy, using their judgment. But instead the incentive is for the DM to express preferences that lead to probabilistically determined rules. This would seem to leave a role for some more trusted agent interfacing between the DMs actual preferences and the preferences required by the theory. It seems to me that this is an important part of what Jim actually does.

### Scope

The DM is focused on “her problem as she sees it at the current time”. But is she incentivized to focus on the whole, large problem, as those who may be affected by her decision would wish her to be, or just the smaller problem ‘as stated’? Should my travel agent be thinking of just my enjoyment of my next holiday, or (as she does) something more strategic?

To what extent were Chernobyl’s managers incentivized to think of Welsh sheep-farmers?

### Conclusion

The book lifts the carpet on uncertainty issues in decision analysis and gives us some good insights into how Jim deals with them. But in doing so what is important to me is not the Bayesian part of the analysis, but the discussion around the subject.

Jim takes the view that most DMs take a pragmatic ‘small world’ view, so that his methods are appropriate. I have found this to be true. But we can turn this around: in areas where his methods are not appropriate, Jim implicitly shows that pragmatic small world view are not adequate: one needs something ‘larger’, perhaps more ‘holistic’, in which agents are not so constrained by explicit targets and have more discretion: leadership, not management.