Decision trees are a very simple tool for framing decision problems with a discrete set of alternatives and a discrete representation of uncertainty. We may start moving to more complicated cases by expressing a decision problem under risk in a more general way:
It is important to understand what problem (13.1) represents:
- First, we pick up a feasible solution x in the feasible set S.
- Then, a random event ω occurs, random variables are realized, and we have a cost f(x, ω) depending on both our decision (variables under our control) and random risk factors (variables not under our control). If the function f(x, ω) represent a profit, we should change the problem to a maximization.
This is a here-and-now decision, as we must make a decision before observing the realization of random risk factors; all we can do is look for a solution that is the best one “on average,” which is what is obtained by minimizing the expected value of the cost; the notation Eω[⋅] points out that expectation depends on random event ω. The optimal solution yields an expected value f*.
It would be very, very nice to postpone decisions and make them after we observe the realization of risk factors. This wait-and-see solution would, no doubt, be better than the here-and-now decision, as we could adapt our choice to the specific realized contingency. Unfortunately, in most real-life situations, we cannot wait and see and decide under perfect information. Nevertheless, we can estimate the theoretical value of perfect information. This is obtained by swapping minimization and expectation in (13.1):
The subscript in 
tells us that this is the expected value of cost if we could optimize with perfect information, after observing event ω. It stands to reason, and it can be shown formally, that for a minimization problem
The difference in cost is the expected value of perfect information (EVPI):
When dealing with a maximization problem, the terms in the difference represent profits, rather than costs, and should be swapped. The EVPI tells us something about the impact of uncertainty and is best illustrated by a toy example.
Fig. 13.5 Schematic illustration of EVPI – the value of (waiting for) perfect information: (a) here-and-now and (b) wait-and-see decisions.
Example 13.3 Consider a stylized investment problem. We must select an investment strategy between two possibilities, aggressive and conservative. These two choices are associated with decisions x1 and x2, respectively; so, the feasible set is S = {x1, x2}. Uncertainty is represented by three possible states of the economy, which we interpret as follows:
- ω1, with probability 0.1, is a “very bullish” economy, in which the aggressive strategy yields 30% and the conservative one yields a much less exciting +6%.
- ω2, with probability 0.7, is a “moderately bullish” economy, in which the aggressive strategy yields 8% and the conservative one yields +4%.
- ω3, with probability 0.2, is a “bearish” economy, in which the aggressive strategy loses an epic 40% and the conservative one limits the loss to a moderate 2%.
The here-and-now decision consists of selecting a strategy before observing returns. The corresponding decision tree is depicted in Fig. 13.5(a). If we take expected return as our objective function, to be maximized, we should take the maximum between
and
So, we would choose the aggressive strategy, and f* = 6%. Now, how much would be the value of clairvoyance? We should restructure the decision tree as in Fig. 13.5(b). If we could invest after observing return, the expected return before the realization of the state of the economy would be
The EVPI, in terms of percentage returns, is
Of course, we swap terms with respect to Definition (13.3), as we seek to maximize return. The return could be translated into monetary terms by assuming an invested wealth.
The careful reader, of course, will notice that there is a missing piece in the above example: If an investor really had clairvoyance, in the case of the very bullish scenario, she would actually borrow money to pursue an aggressive strategy; even a small difference in return can yield a huge amount of money, if a correspondingly huge wealth is invested. This strategy is based on leveraging, and has been pursued in recent times of low interest rates. As the 2008 financial crisis has clearly shown, such leveraged strategies are quite risky. In fact, another fundamental missing piece in the example is the role of risk: Wise investors are risk-averse. In the next section we take a look at modeling risk aversion and measuring risk.
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