# Single-dimensional parametrization of risk / reward (short-run / long-run) trade-off



## BrahmsWasAGreatMelodist (Jan 13, 2019)

This seems to me to be as philosophical in nature as it is mathematical. I came across an economics paper a little while ago that suggested rationality as a measure of not how well a course of action is expected to perform, but of how your overall strategy (say, in terms of fractional resource allocation) might play out in the long run. I found it ridiculous at first - if something only happens once, then it only happens once - but upon further reflection I must admit I find it quite a thought provoking notion. The following example might help to illustrate this point.

Say you are faced with a wager that with 2/3 probability doubles your input and with 1/3 probability returns nothing. Suppose you can put fraction x in [0, 1] of your total portfolio (which initially has amount P) into the wager. If you're just trying to maximize expected return, obviously you choose x = 1. But suppose now that you play this game repeatedly. Above a certain threshold of x, you will with probability 1 tend towards going broke. There's some optimal x* well below that threshold which maximizes your long run growth rate. If we assume a general distribution of returns from the wager, this x* factors in both bias (expected return) and variance.

I find this kind of deep in a sense: just because something is theoretically good, that doesn't mean you should devote everything you can to it.

On a more general note, there's so much uncertainty in life. I mean, that's what makes it fascinating; how boring would life be if you always knew what was to come next? Yet we all try to strive for the "best" and hedge against the "worst". How do you define rationality? Does the idea of deriving rationality from extending single events _ad infinitum_ hold any value to you? How does the concept of rationality inform the decisions you make and the rivers you tread in life?


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