07-23-2020, 08:43 PM

I have to admit, I rather love paradoxes (quite recently, I talked about the Unexpected Hanging Paradox). Now, it's time for me to introduce another one - this time, from game theory. It's called the "Allais Paradox" - and it concerns two choices between two life-saving pills.

Here's a video by BrainCraft, to explain the paradox:

Or, if you'd rather have it explained in text form... I'll do that too. You go to the doctor, and you're told that you're going to die - unless you seek treatment immediately. You're given a choice between two treatments:

In this case, I would choose the blue pill - and I think most of us would. That 1% chance of immediate death is scary - and, to me, it's not worth the 10% chance of another six years.

However, let's look at a second scenario. This time, your regular doctor is unavailable, and you have to go to a much worse one instead. They offer you the following two treatments:

In this case, I'd choose the yellow pill, and I think most of us would. If we live, then we get an extra six years - and that seems well worth the 1% increase in chance of death.

However, let's look at this mathematically. In the first scenario, both options have the same outcome 89% of the time (that is, living 12 years). Thus, we need only consider the 11% of cases where they differ: the blue pill has us living 12 years 11% of the time, while the red pill has us living 18 years 10% of the time, and dying 1% of the time. As for the second scenario... once again, both options have the same outcome 89% of the time (this time, dying immediately). Thus, we need only consider the 11% of cases where they differ: the green pill has us living 12 years 11% of the time, while the yellow pill has us living 18 years 10% of the time, and dying 1% of the time. But, that's the exact same decision as the first scenario... so, anybody who chooses the blue pill should also choose the green pill, and anybody who chooses the yellow pill should also choose the red pill. However, even after being exposed to this mathematical explanation, I would still choose the blue pill in the first scenario, and the yellow pill in the second scenario. So, what gives ?

I think the best way in which I can rationalise my choices is: I have a strong aversion to risk of death, and thus, I have a strong preference for options that don't contain a risk of death, over options that do. However, in cases where both options involve a risk of death anyway... the exact level of risk is less of a factor. Furthermore, when we compare the magnitude of two numbers, I think our minds tend to do it logarithmically. Thus, the difference between 89 and 90 doesn't feel anywhere near as significant as the difference between 0 and 1 - even though it's a difference of 1 in both cases.

So, I have three questions for you:

I'd be interested to hear your thoughts!

Here's a video by BrainCraft, to explain the paradox:

Or, if you'd rather have it explained in text form... I'll do that too. You go to the doctor, and you're told that you're going to die - unless you seek treatment immediately. You're given a choice between two treatments:

- Blue pill - 100% chance of giving you 12 years to live

- Red pill - 89% chance of giving you 12 years; 10% chance of giving you 18 years; 1% chance of immediate death

In this case, I would choose the blue pill - and I think most of us would. That 1% chance of immediate death is scary - and, to me, it's not worth the 10% chance of another six years.

However, let's look at a second scenario. This time, your regular doctor is unavailable, and you have to go to a much worse one instead. They offer you the following two treatments:

- Green pill - 89% chance of immediate death; 11% chance of living 12 years

- Yellow pill - 90% chance of immediate death; 10% chance of living 18 years

In this case, I'd choose the yellow pill, and I think most of us would. If we live, then we get an extra six years - and that seems well worth the 1% increase in chance of death.

However, let's look at this mathematically. In the first scenario, both options have the same outcome 89% of the time (that is, living 12 years). Thus, we need only consider the 11% of cases where they differ: the blue pill has us living 12 years 11% of the time, while the red pill has us living 18 years 10% of the time, and dying 1% of the time. As for the second scenario... once again, both options have the same outcome 89% of the time (this time, dying immediately). Thus, we need only consider the 11% of cases where they differ: the green pill has us living 12 years 11% of the time, while the yellow pill has us living 18 years 10% of the time, and dying 1% of the time. But, that's the exact same decision as the first scenario... so, anybody who chooses the blue pill should also choose the green pill, and anybody who chooses the yellow pill should also choose the red pill. However, even after being exposed to this mathematical explanation, I would still choose the blue pill in the first scenario, and the yellow pill in the second scenario. So, what gives ?

I think the best way in which I can rationalise my choices is: I have a strong aversion to risk of death, and thus, I have a strong preference for options that don't contain a risk of death, over options that do. However, in cases where both options involve a risk of death anyway... the exact level of risk is less of a factor. Furthermore, when we compare the magnitude of two numbers, I think our minds tend to do it logarithmically. Thus, the difference between 89 and 90 doesn't feel anywhere near as significant as the difference between 0 and 1 - even though it's a difference of 1 in both cases.

So, I have three questions for you:

- When you saw those scenarios for the first time, did you choose the blue pill and the yellow pill?

- After reading the mathematical breakdown, did you change your mind to either blue/green or red/yellow? If so, what did you change to?

- If you stuck with blue/yellow even after reading the explanation... why was this?

I'd be interested to hear your thoughts!

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Lurker101 Wrote:I wouldn't be surprised if there was a Mega Blok movie planned but the pieces wouldn't fit together.

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