## “A Bayesian Model of Risk and Uncertainty,” N. al-Najjar & J. Weinstein (2012)

In a Bayesian world with expected utility maximizers, you have a prior belief on the chance that certain events will occur, and you maximize utility subject to those beliefs. But what if you are “uncertain” about what your prior even is? Perhaps you think with 60 percent probability, peace negotiations will commence and there will be a .5 chance of war and a .5 chance of peace, but with 40 percent probability, war is guaranteed to occur. It turns out these types of compound lotteries don’t affect your decision if you’re just making a single choice: simply combine the compound lottery and use that as your prior. In this case, you think war will occur with .6*.5+.4*1=.7 probability. That is, the Bayesian world is great for discussing risk – decisionmaking with concave utility and known distributions – not that useful for talking about one-shot Knightian uncertainty, or decisionmaking when the distributions are not well-known.

al-Najjar and Weinstein show, however, that this logic does not hold when you take multiple decisions that depend on a parameter that is common (or at least correlated) across those decisions. Imagine that a stock has a daily return determined by some IID process which is bought by a risk-averse agent, and imagine that the agent doesn’t have a single prior about that parameter, but rather a prior over the set of possible priors. For instance, as above, with probability .6 you have a .5 chance of a 1 percent increase and a .5 chance of a 1 percent decrease, but with probability .4, a 1 percent increase is assured. Every period, I can update my “prior over priors”. Does the logic about the compound lottery collapsing still hold, or does this uncertainty matter for decisionmaking?

If utility is linear or separable over time, then uncertainty doesn’t matter, but otherwise it does. Why? Call the prior over your priors “uncertainty.” Mathematically, the expected utility is a double integral: the outer integral is over possible priors with respect to your uncertainty, and the inner integral is just standard expected utility over N time periods with respect to each prior currently being summed in the outer integral. In the linear or separable utility case, I can swap the position of the integrals with the summation of utility over time, making the problem equivalent to adding up N one-period decision problems; as before, having priors over your prior when only one decision is being made cannot affect the decision you make, since you can just collapse the compound lottery.

If utility is not linear or separable over time, uncertainty will affect your decision. In particular, with concave utility, you will be uncertainty averse in addition to being risk averse. Al-Najjar and Weinstein use a modified Dirichlet distribution to talk about this more concretely. In particular, assuming a uniform prior-over-priors is actually equivalent to assuming very little uncertainty: the uniform prior-over-priors will respond very slowly to information learned during the first few periods. Alternatively, if you have a lot of uncertainty (a low Dirichlet parameter), your prior-over-priors, and hence your decisions, will change rapidly in the first few periods.

So what’s the use of this model? First, it allows you to talk about dynamic uncertainty without invoking any of the standard explanations for ambiguity – the problems with the ambiguity models are discussed in a well-known 2009 article by the authors of the present paper. If you’re, say, an observer of people’s behavior on the stock market, and see actions in some sectors that suggests purchase variability that far exceeds the known ex-post underlying variability of the asset, you might want to infer that the prior-over-priors exhibited a lot of uncertainty during the time examined; the buyers were not necessarily irrational. In particular, during regime shifts or periods with new financial product introduction, even if the ex-post level of risk does not change, assets may move with much more variance than expected due to the underlying uncertainty. Alternatively, if new assets whose underlying parameters are likely to be subject to much Knightian uncertainty, this model gives you a perfectly Bayesian explanation for why returns on that asset are higher than seems justified given known levels of risk aversion.

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