Many decisions in life involve the tradeoff between risk and reward. Perhaps one has to choose between a "low risk, low reward" course of action, that plays it safe, but does not achieve any big wins; or a "high risk, high reward" choice which could potentially give greater benefits, but also greater downsides.

Risk management is a complex task, but one can explore it initially with some very simplified models, always keeping in mind George Box's dictum that "all models are wrong, but some are useful". For this simple model, one can assume that each course of action comes with two numbers: the "reward", which is the average positive net gain (benefits minus costs) one expects to get from pursuing the action, and the "risk", which in this model can be thought of as a standard deviation of the fluctuation from the mean. For instance, a "safe" action might have a reward of 5 and a risk of 3, which I would represent as 5±3 in this model, the return would typically range anywhere from 2 to 8 units of net benefit. A "bold" action might have a reward of 9 and a risk of 10, which I would represent as 9±10; in this model, the return would typically range from -1 to 19 units of net benefit. Here we imagine that there is some sort of bell curve in effect; while a 5±3 action will *usually* give a net benefit between 2 and 8, there could be exceptional events that lead the return to be less than 2 or greater than 8. But I will gloss over these tail events for sake of this discussion.

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Given the general tendency to prefer risk aversion, one possible strategy is to try to minimize "value at risk" - the value that one might lose in a bad (though not absolutely worst case) scenario. Here, a simple such metric would be "risk minus minus reward" - the higher end of the range of typical losses. For instance, in the above model, the "safe" action has a value at risk of 3-5 = -2: even in a somewhat bad-luck scenario, one would still expect to make a small gain. In contrast, the value at risk for the "bold" would have a value at risk of 10-9=1, so with bad luck one would actually have a small loss. The advice would then be to take the safe action instead of the bold one.

One interesting thing about this model is that it is not affected by predictable exogenous shocks. Suppose for instance there is an event that causes all projected outcomes to be revised downward by the same constant factor, let's say -5 to sake of argument. The projected return from the safe action now revises downward from 5±3 to 0±3, with value at risk going up from -2 to +3; and the projected return from the bold action similarly revises downward from 9±10 to 4±10, with the value at risk going down up +1 to +6. This is obviously not good news for the person having to make the choice, but the shock does not actually change one's decision making; the safe action continues to have less value at risk than the bold one, and so this decision framework would still recommend taking the safe action. (2/4)

It has been noted in the behavioral economics literature that actual human decisions do not, actually, behave like this: external shocks do in fact have psychological impacts that can distort the decision making process. But there is actually a rational basis for having such shocks alter one's decisions, *if* the shocks also themselves carry risk. Suppose for instance that the external shock does not simply add -5 to all outcomes, but rather −5±10; the added impact of the external shock in fact ranges in most scenarios from a very negative addition of -15 to even a slight positive outcome of +5. How does this change the calculus of risk and reward?

Thanks to the law of linearity of expectation in probability, the reward side of the equation is easy to analyze: just add together the two component rewards. For instance, the safe action initially had a reward of 5, and the shock imposes an additional reward of -5, so the combined reward is 0. Similarly, the combined reward of the bold action is now 9-5=4. But what about the risk? Risk behaves differently from reward in that it is not additive; it is possible for one risk to sometimes cancel another. A more realistic rule is given by the law of additivity of variance, that says that if two risks behave in an uncorrelated fashion, then it is the *square* of the combined risk which is the sum of the *squares* of the individual risks. This should remind one of the Pythagorean law for computing the length of the hypotenuse of a right-angled triangle, and indeed in some sense the additivity of variance is an instance of the Pythagorean law (but now in an infinite-dimensional Hilbert space of random variables); but I digress. (3/4)