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What is the name of the theory which states that even if the chances of something happening are so low (like 1%) but said event, if it were to happen, it is so disproportionate in character that even a 1% chance is still to be considered?

Like taking a gamble to which the chances are: 99% winning 100k $ 1% an atomic bomb goes off in the city.

Where the outcome of the least favored scenario is so significant that it should be considered as much (or more?) as the other scenario.

I know it may seem trivial, but it makes decision making based solely on probability irrelevant.

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This is related to the concept of utility, which is the subjective measure of the "value" of a probabilistic outcome. For small sums of money, for example, utility is well-correlated with nominal value, meaning that \$2 has twice the "value" of \$1. For large sums of money, that's not so true anymore, as most people would not consider \$20M to have double the "value" of \$10M - going from \$0 to \$10M will have a far greater subjective impact on your life than going from \$10M to \$20M.

By assigning utility to non-numeric outcomes, we can get an idea of whether a gamble is "worth it". An atomic bomb going off has a very high negative utility, which is orders of magnitude higher than the positive utility of \$100k. So, even with only a 1% chance of the bomb going off, the wager has a negative utility, meaning it's better off to not take the bet.

This does not "make our rules of what is called rational decision making based on probability irrelevant" as you suggest - rather, it recognizes that not all outcomes have the same value. Decisions should not be made solely on the likelihood of an outcome, but also how much that outcome is worth. On a roulette table, betting on black will win 18/38 times, while betting a single number will win only 1/38 times. But since the payout for a single number bet is much larger, the two bets are roughly equally rational - the single number bet isn't more irrational just because it's less likely to hit (both are equally irrational, since they both have negative expected utility).

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Nuclear Wang's answser is good, I am not sure if you just mean the theory of utility.

Decision theory starts with some rules about how people evaluate gambles $\{p(x)\}_{x \in X}$ over outcomes $x \in X$, then shows that there is some numerical representation $$ U(p) = \int_{x \in X} u(x) p(x) dx $$ where $u$ is the utility of event $x$ occurring. That's the von Neumann-Morgenstern theorem that Nuclear Wang probably means.

Then there's the question of where beliefs come from. Leonard Savage uses your behavior to deduce what your beliefs must have been to rationalize those choices. The idea is that we all have beliefs that are subjective, and he would like to understand where those beliefs come from, or at least give a meaningful account of how people behave as if they had subjective beliefs like a vNM decision-maker.

Then there's a behavioral literature that considers whether the axioms of the previous theories are consistent with observed behavior, and tries to adjust the axioms to capture the weird quirks we observe. This includes people like Kahneman, Tversky, Thaler, and Rabin. So your gambles above are super extreme: get 20 bucks or the world ends. What people noticed is that humans tend to under-weight likely events (win a coin flip) and over-weight unlikely ones (the world ends in nuclear fire). This leads to a representation $$ U(p) = \int_{x\in X} u(x) \phi(p(x))dx $$ where $\phi:[0,1]\rightarrow 1$, $\phi$ is continuous and increasing, $\phi(1/2) = 1/2$, but $\phi(0)>0$ and $\phi(1) < 1$. There are many theories like this that try to explain paradoxes in human decision-making that seem to contradict reasonable axioms.

The vNM and Savage-type papers are typically called expected utility theory or decisionmaking under uncertainty, and the Kahneman-Tversky-type stuff is often called behavioral economics.

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