I have a d3 (three-sided die) and I’m trying to get the highest possible outcome. Should I choose to reroll a 2?

Question

As the title says, if I roll a 2 on a d3 and am allowed to reroll it, is there an advantage to re-rolling?

It seems like the choice doesn’t matter, or is purely preferential but what is wrong with this way of framing the problem:

On a roll of 2, if I reroll, I have a 66% chance of doing the same or better. ‘Same’ is an outcome I would be happy with. ‘Better’ is an outcome I would be happy with. I only have a 33% chance of a bad outcome. This makes it sound like I should reroll.

Is it wrong to consider ‘same’ a good outcome because of the cost of risk?

What’s the math?

Edit: in addition, how does this change if I am alowed to re-roll it 2 times or 10 times?

Answer 1

If your goal is merely to maximize the number received on the die, your current value is a $2$, and if you were to reroll it the Expected Value is also a $2$.

When deciding whether or not you should reroll, you should in general reroll if the expected value is strictly greater than your current roll and you should in general not reroll if your current value is strictly greater than the expected value.

In the event that you are in, your current roll is equal to the expected value which means that in the interest of maximizing your roll it is neither an advantage nor a disadvantage to roll again and you are free to choose whichever you wish at the moment.

Now... in certain situations, you may have additional things to consider when deciding to reroll or not... In many games it might be to your benefit to reduce the variance of your return and so should choose not to reroll. In others, perhaps getting such a strong early lead will help you tremendously towards winning the game and so rerolling is preferred. Neither of these "secondary objectives" however are mentioned in your problem and so can be ignored for now.

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TLDR: It is neither better nor worse to reroll in this situation.

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As for the expanded question of if you are allowed multiple rerolls... as you should guess yes you should reroll. We are no longer in this case comparing the outcome of having rolled a $2$ as compared to the expected value of another single roll, but now we are now comparing the outcome of having rolled a $2$ as compared to the expected value of the maximum of ten future rolls. If you were to run the calculations, you would find that the maximum of those ten rolls is a $1$ with probability merely $\frac{1}{3^{10}}$, the maximum of those ten rolls is a $2$ with probability $\frac{2^{10}-1}{3^{10}}$ and the maximum of those ten rolls is a $3$ with probability $\frac{3^{10}-2^{10}}{3^{10}}$. Calculating the expected value with these numbers I leave to you but it will most certainly be above $2$ and so it is to your benefit to try and try again., only possibly not choosing to continue rerolling in the event that you have used all but one of your chances and have currently ended with a $2$, but in that situation we are back to the original problem and so it is your choice whether you reroll or not.

Answer 2

The 'math' of this is down to what we call the utility of the different outcomes. It is possible to associate a value (the utility) with each outcome. Having done that, the usual procedure is to identify the action that has the maximum expected utility.

In this case a simple measure of utility would be 1 for a result of 1, 2 for a result of 2 and 3 for a result of 3. In that case the expected utility if you roll the dice is $1 \times \frac 13 + 2 \times \frac 13 + 3 \times \frac 13=2$. The expected utility if you don't roll is also 2, so there is no difference between rolling and not rolling.

In your question you mention the 'cost of risk'. To include a consideration of risk we can tweak the utility of the different outcomes. Let's keep the utility of a result of 2 at 2, but let's suppose that getting a 1 is a really bad outcome. We can then set the utility of getting a result of 1 as 0. If we still keep the utility of a result of 3 as 3, then the calculation becomes:

Expected utility if you roll the dice is $0 \times \frac 13 + 2 \times \frac 13 + 3 \times \frac 13=\frac 53$. The expected utility if you don't roll is 2, so the maximum expected utilty is achieved by not rolling.