Requirements for a proof that the best chance you can get is 1/4:

We know that Evil has two things to keep hidden: their Evil status, and their favorite color.
If they ever reveal that they are Evil, we know we can trust the other two. If they ever say their favorite color, we can later determine the claimed favorite colors of the other two; if there are ever 3 distinct claimed favorite colors, we know the answer.

Strategies seem to consist of "trading" information that no one knows with information that Evil knows. In my strategy, Evil is first forced to give up 1 bit of information about which one they are in exchange for one bit of information that allows them to determine everyone's favorite colors.
At this point, we only need 1 more bit of information, and all the information we have left to "trade" is the true/false identities of the other two people.

It should be noted that at the start we have >2 bits of unknown information in each category (6 configurations) while Evil has only 1 bit of unknown information per category of color and status. Of course, we only have to discover one full category while Evil has to discover both.

Aside: I now actually have only 1 bit unknown in the color category, but unless I attempt to reveal statuses, Evil can always hide it from me by answering the same as their color partner. Therefore, I have to determine the final status bit before deducing the final color bit.

After "trading" >1 color bit for one status bit, we both need only one more bit of information to win. We have narrowed Evil to one of two friends, and we have to make sure to not give up anyone's status for free. The only way to try to obtain Evil's final status bit is to ask a question of them before asking a question of the other friend who might be Evil. This imposes a 1/2 chance of success. Given that we succeed in asking them first, Evil's goal is now to assume the identity of their color partner. They have no clue whether their partner is Truth or Lie, so they have to make a guess and stick with it. They have a 50% chance of guessing right, and if they do, this situation is the same as @AhronMaline's 1/2 chance upper bound and I can never obtain any more status information. If they guess wrong and are forced to trade their final status bit for our final status bit, we can now deduce everyone's status and then favorite colors.

I have to take a 50% chance, and so does Evil. That comes out to 1/4.


The above proof is based on the following assumptions:
1. You can only uncover information by giving Evil information and can therefore only uncover information twice (once for each bit they don't know.)
2. You cannot uncover two full bits of information by trading only one bit away.
3. There is not a less risky way to force Evil to trade information with you a second time. (The first time is a special case because you can threaten to directly uncover colors without caring about statuses.)
4. There is not a better way to leverage the reduced number of people who might be Evil in the second half of the strategy.

If you find more assumptions, please tell me.
If we can prove every assumption correct, then we can conclude that the maximum chance the optimal strategy can give you is 1/4.

If we just guess one person is Evil and then ask other two people the following 3 questions in any order: is your favourite colour red? Is your favourite colour Blue? Is your favourite colour green?

If we are right about A being evil, then the truthteller will say no twice and yes to one question and the odd answer is their favourite colour.
The liar will say yes twice and no to one question and the odd answer is their favourite colour.

We guess this is correct and we succeed in one third of cases where we have picked the right person as evil.

I expect we can do better than this but it is complicated.

@ChristopherRandles "guessing that it is correct" doesn't count. You need to get enough info to deduce, i.e. prove with certainty, who has which color. The probabilities are about whether you will succeed in extracting that info

@ChristopherRandles I figured out a way to explain what this deduce thing means. Imagine you are provided an opportunity to win 100$ if you can guess the friends' favorite colors, but if you guess wrong you die. You are free to walk away at any time without making a final guess. What is the highest chance you can give yourself of winning the money, without risking death?

@ChristopherRandles and even if we change the rules to allow guessing, you should still use @MaxE 's strategy! It gives you 1/4 chance of finding out the colors for sure, and in the other 3/4 of cases, it narrows you down to two possibilities. If you guess uniformly between those two, your total probability is 1/4+3/8=5/8.

But in fact you can do even better! In the cases where you don't get certainty, because your two suspects for being Evil gave the same answer to 1+1, you know that there are two possible reasons for that: either you asked Evil 1+1 first but he guessed right (1/4 of all cases) or you asked him second and he copied the other suspect (1/2 of all cases). So if you just guess that the second suspect is Evil, your guess will be right 2/3 of the time, half of all cases. So your total chances are 3/4.

But again, all this is with the alternative rules that allow guessing, not the actual rules of this market.

@ChristopherRandles ' strategy is my inspiration for this.

Start off by asking each player these questions:
"If I were to ask you 'Is your favorite color Red?' how would you answer?"
"If I were to ask you 'Is your favorite color Green?' how would you answer?"
"If I were to ask you 'Is your favorite color Blue?' how would you answer?"

Each player will answer "Yes" to exactly one question. If Evil doesn't do that, you know their identity and can easily win. This will ensure that you know Truth and Lie's favorite colors..

If everyone answers a different color, then Evil has answered their own color, and you know the answer, so Evil has to copy someone else's favorite color. (This can be done if they are asked first by saying a color that isn't their own.) Now that we have a pair of people with the same "favorite color", we will ask one at random to answer " 1+1=?" to determine their liar status, and then ask the same of the other. Note that no one yet knows anyone else's status. Each person will say a truth or a lie, and if they say different things, you win.

If you asked Evil first, they have to guess whether the other is Truth or Lie.
If the two you ask say different things, then you can ask "1+1" to the person with a unique stated fav color (they definitely are not Evil). Whatever they say, you know the one who matched them is Evil. (Ex. first person says 2, second says 3, unique says 2 => first is Evil. OTHER EX. first says 2, second says 3, unique says 3 => second is Evil) If Evil is the second person you ask, they will always copy the first and you will never determine who is who. If Evil was asked first and guessed the truth value of the second one, you also lose.

At this point, if they answered the same, I believe you lose. The only way I have found to get good information from Evil is by forcing them to reveal information without knowing everyone's truth value or color. Now, those are all known to Evil and I have no way to get anything else out of them, who will act exactly like the person they mimicked at the start forever.

Therefore the chance of knowing all colors with 100% confidence with this strategy is 1/2 (Evil is asked for their truth value first) * 1/2 (they guess incorrectly) = 1/4

I was at first very skeptical that there would be any non-deterministic strategies that could have a chance to produce perfect knowledge, but I was wrong. I ended up really enjoying this market.

@MaxE I think that this works. I will give others time to object.

@MaxE I am wondering about asking questions to people in different orders to have a better chance of asking evil something first when he doesn't know.

e.g. Ask A B and C in that order Is C the truthteller?
Ask B C and A in that order is A the liar?
Ask C A and B in that order is B evil?

Unfortunately I think evil is usually quickest to work out who is truth and liar so this is probably not as good as pick one person and hope it is evil.

So taking another line
If we start with questions like is you favourite colour red? If Evil is asked first he is going to have to decide whether to emulate which different colour to his own and who is the truthteller or liar, a 1 in 4 chance. If he is asked second he has gained some knowledge so he now has a 50%? chance of guessing correctly? Can this result in an answer that is higher than the 25% that I think we have arrived at.

@ChristopherRandles if evil ever tells their true favorite color we instantly win. We can obtain the colors of Truth and Lie from them and then we know the answer. Evil has to pick someone else's color, which they can do 100% of the time because they know their own color.

I dont think I believe in your strategy of asking other people what their opinion is, as we either have access to all the information they have or they are Evil.

I dont have an easy counter to your idea about asking in that order so I'll just walk through it.

A will say no if they are Truth, decline if they are Lie, and be evil if they are Evil.

B will do the same (evil could have done something weird so they dont know anything new)

C will say yes if they are Truth or Lie, so Evil would do the same.

B will answer yes if they are Lie, decline if they are Truth, and do something evil if they are Evil

C will answer the same

A will answer no

C will decline (given that this pattern doesn't force evil to reveal, but I dont think it does)

A will decline

B will decline

Unless I made a mistake I dont think that pattern works because evil has so many chances to learn information and we dont gain very much. If I missed something, you should make a comment where you go over it more explicitly.

@MaxE very nice!

@MaxE The strategy makes sense but I'm not sure that it's probability to succeed is 1/4 under the rules of this question because of this clarification: https://manifold.markets/robert/with-what-probability-does-the-opti#n9ye8jhey8

Since we cannot assume a distribution of colors, we also cannot assume what Evil's probability of guessing correctly is. Maybe Evil happens to decide to always guess Red, and that happens to always be the correct guess that foils us. It would make sense to say that the colors are distributed uniformly so Evil's chance to guess correctly is 50%, but the clarification seems to explicitly disavow that.

Even if the colors are distributed non-uniformly, Evil could force the probability to be 50% by choosing to guess randomly, but that is Evil's choice not ours so we cannot guarantee it. To force it, we need to either prove that by guessing non-randomly Evil would improve our chances (which seems impossible to prove without knowing the color distribution) or we need to make our own random choice part of the calculus somehow. For example, our chance of asking evil first is a true probability because regardless of the distribution of players we can forcibly randomize who we pick for questioning, but this seems to only work for the first part not the second part.

Here is a strategy that works with probability 1/6: randomize an order between the three friends, and ask the first one, "is your favorite color red? Is it green? Is it blue?" And then the same for each of the other two.

Truth will say yes to one color and no to the other two, Lie will say yes to two of them and no to his actual color, and Evil will need to choose a strategy.

If Evil is second or third, then you lose. Evil knows from the first friend's answers whether they are Truth or Lie and what their color is, so he can make himself indistinguishable from them. But if Evil is first, then he needs to decide what to pretend his color is, and he has a half chance of guessing wrong, in which case you win.

For example, if the true colors are RGB for True, Lie, and Evil respectively, then Evil doesn't know who has red or green. If he pretends to be Truth and red, or to be Lie and green, then you lose. But if he pretends to be Lie and red (by saying yes to green and blue, and no to red), then you get two friends who are either Lie or Evil, so the other one must be Truth. Then you know Truth is actually red, and so Evil told the truth when saying no to red, so you know he isn't Lie. Then you can trust Lie's answer to be false and you know all the colors. Similarly, if Evil pretends to be Truth and green, you will catch him by seeing that Lie is actually green.

So either way, your probability is 1/6: 1/3 chance that you will ask Evil first, and then 1/2 chance he will pick the wrong color to predend.

[If Evil tries pretending his actual color (blue), playing either as Truth or as Lie, then it's even easier: you know that some two of the three pretended colors are correct (those of Truth and Lie, whoever they are) and since all three are different they must all be correct. You don't even need to find Evil. So this is a bad strategy for him.]

Is that a 1 in 3 chance of having a 50% success rate and a 2 in 3 chance of having 0% success rate.
On the logic of
"As an example, a strategy that only provides a 100% certain answer for one specific assignment of colors (out of six possibilities) has a success rate of 0%, not 1/6."

That seems like a 0% answer for the question though you do have a 1 in 6 chance of getting the right answer.

Or am I misinterpreting this?

@ChristopherRandles the strategy includes randomizing the order of the three friends to ask - that's what the RNG is for. So the 1/3 probability that you will ask Evil first goes into you success rate, giving a rate of 1/6, no matter which of A,B,C is actually Evil.

@AhronMaline I think this works, but I would rather not resolve anything that is at all controversial.

@AhronMaline
Update 2025-08-02 (PST) (AI summary of creator comment): The creator has confirmed that if it is proven that no strategy can exist that results in 100% confidence, the market will resolve to 0%.

Your strategy results in 16.7% confidence so if that is the best that can be done the question resolves at 0%

But I am struggling to interpret this.
@robert Does this mean if we found a solution where if we ask evil first we have 100% chance of having 100% confidence in our answer and a 2 in 3 chance of not having full confidence does this mean the question resolves to >=1/3 or to 0%?

If we have a strategy where asking evil first gives us 75% chance of getting answer we are 100% certain of and failing to be sure in the other 2 in 3 cases does the question resolve at 0%?

@ChristopherRandles It would resolve to 1/3. The market is how good of a chance can you make it that your strategy produces 100% confidence. If you can insta win by asking Evil first, you then have a 1/3 strategy.

@ChristopherRandles I think that the AI is misinterpreting what I said. The actual question I responded "yes" was:

If I were to prove that there is no strategy that has any chance of creating 100% confidence in a correct answer, would this resolve to 0%?

@robert Does this mean if we found a solution where if we ask evil first we have 100% chance of having 100% confidence in our answer and a 2 in 3 chance of not having full confidence does this mean the question resolves to >=1/3 or to 0%?

If we have a strategy where asking evil first gives us 75% chance of getting answer we are 100% certain of and failing to be sure in the other 2 in 3 cases does the question resolve at 0%?

The probability of success would be 1/3 and 1/3 * 3/4 = 1/4 in those cases.

@MaxE Then I think I can get to a 3 in 12 ie 25% chance.: If I ask Evil questions first evit has to guess truthfulness and colours of other two each a 50% chance. However there could be better yet.

First ask each person what is their favourite colour. then ask their best guess at the truthfulness and favourite colours of each of the other two. I am hoping to find this useful later.

If you ask what is 1+1 to each person, the truthteller tells truth the liar lies and evil is left with a choice to pretend to be either truthteller or the liar to frustrate your efforts. Whichever he chooses you will either have 2 people saying 2 and one says something else in which case you have identified the liar or only 1 says 2 and you have identified the truthteller.

If you have identified the truthteller you can ask the truthteller what is your favourite colour?
If you have identified the truthteller you can ask Is your favourite colour red? and if yes ask Is your favourite colour blue? and this identifies the liar's favourite colour.

Having identified one colour of one person who you know to be either liar or truthteller, there are only two possible choices so all answers >= a number of half or less must be true.

But can you get any further given that evil is trying to frustrate your efforts and can answer to do so?

It seems difficult, but what if the Evil person was the one we asked their favourite colour first? At that stage he won't know what liar will reply nor the colour of the truthteller. Does this mean he is more likely to try to emulate liar's answers having a 2 in 3 chance of guessing ok rather than emulating truthteller where he only has a 1 in 3 chance of guessing correctly?

So I think the answer is >= a number higher than 0.5. In addition to the 50% chance there is a 1 in 3 chance we asked the evil person first making his job of avoiding being caught out telling a truth when he is pretending to be the liar harder and we may have asked enough questions that we have a good chance of catching him out. If we don't catch him out in this way it may be because he ensured his answers were lies by saying his favourite colour was what it actually was. If the liar has given the truthtellers favourite colour as his own this gives us another way of figuring it out.

@ChristopherRandles Your reasoning up to the 3rd last paragraph is the same as mine. Given the first part, the chance is still 0, given the markets resolution criteria. I will respond to the 2nd part in a second comment.

@MaxE If we are asking Evil first to say what the truthfulness of other players is as well as their favourite colours as well as ask A for A's best guess at what B would say if B is the truthteller/liar/Evil and so on.

Evil needs to guess the truthfulness of the other players as well as the favourite colour of each person. so only a 1 in 4 chance of evil guessing both correctly.

So if we are 1 in 3 lucky and ask Evil questions first, there is a good chance of figuring it out but a good chance of figuring it out is not enough and >= 1/3 and in fact all answers including >=1/12 resolve as no?

@ChristopherRandles You need a strategy that has a chance of you being 100% confident in an answer. As far as I can tell, you need to find out Evil. I dont really understand your logic, but im cooking

@ChristopherRandles This is a very smart idea.

I have written too much for a reply. I'm sorry but im going to make my own comment.

I can prove that there is no strategy giving probability more than half: Evil can, at the beginning, decide to pretend to be Truth. For this he needs to guess what Truth's favorite color is, and pretend that it is his own. Evil knows his own true color, so he can guess one of the two others. With 50% probability he will guess Truth's actual color, and then his answers and those of Truth will be identical for every possible question. Since the third friend, Lie, doesn't know which of Truth and Evil is which, nor which actually has Truth's favorite color, you will never be able to break the symmetry between those two friends.

Of course Evil may have other strategies, that doom you with greater probability. But you certainly cannot bring your chances above 1/2.

@AhronMaline I reasoned this too, but I didn't share it because I wanted to confirm how that information would resolve the market.

Your second paragraph doesn't influence the market because as far as I understand these resolution criteria, Evil only has to always create any amount of doubt in order for this to resolve 0%.

The question is, "With what success rate can the optimal strategy tell you the correct answer with 100% certainty?"

@AhronMaline Resolving 1, 2/3, 3/4 NO

@MaxE FWIW I did understand that to be the intention. The goal is to reach certain knowledge of the colors, and strategies to acheive that will necessarily have a limited success rate.

@AhronMaline So you believe this means it should resolve 0%?

@MaxE I'll send you 100 mana if you find a deterministic optimal strategy (or a strategy that has the highest success rate of any strategy posted before this market resolves).

I never specified how roles/favorite colors are assigned. You don't know how, and nor does Evil. Both of you are trying to do your best in the worst case for you. This means that a strategy that gets it right (with 100% certainty and logically deduces it) on A-Red B-Blue C-Green resolves, but in all other cases, does not logically deduce the favorite colors has a success rate of 0, not 1/6.

@MaxE no, if Evil tries the strategy I suggested but picks the wrong color (the one that actually belongs to Lie) then I think you should be able to catch him with certainty.

I misinterpreted your solution. My solution (lower in the comment thread, never responded to by Robert) had evil match with Truth with everything except for fav color.